LOCAL FIELDS IN THE ELECTRODYNAMICS

LOCAL FIELDS IN THE ELECTRODYNAMICS OF MESOSCOPIC MEDIA

OleKELLER Institute of Physics, Aalborg University, Pontoppidanstrazde 103, DK-9220 Aalborg 0st, Denmark

ELSEWIER

AMSTERDAM - LAUSANNE - NEW YORK - OXFORD ~ SHANNON - TOKYO

PHYSICS REPORTS

ELSEVIER Physics Reports 268 (1996) 85-262

Local fields in the electrodynamics of mesoscopic media

Ole Keller

Institute of Physics, Aalborg Unioersit_v, Pontoppidanstnede 103, DK-9220 Aalborg 0st, Denmark

Received June 1995: editor: D.L. Mills

Contents

1. Introduction

PART A. FUNDAMENTAL THEORY 2. Electromagnetic propagator formalism

2.1. Integral relation between the local field and the prevailing current density

2.2. Microscopic Ewald-Oseen extinction

theorem 2.3. Transverse and longitudinal vacuum

propagators 2.4. Pseudo-vacuum propagators

3. Linear and nonlocal response theory 3.1. Pauli Hamiltonian in the presence of

a prescribed external field 3.2. Many-body density matrix operator

approach 3.3. Many-body conductivity tensor. The

response to the transverse local field plus the longitudinal external field

3.4. Gauge invariance 3.5. One-electron theories

4. Local-field calculations and underlying physics 4.1. Integral equations for the local field 4.2. External conductivity tensors 4.3. Coupled-antenna theory 4.4. Local-field resonance conditions

88

93

94

98

100 102

106

106

108

111 117 122

124 125 132 135 144

PART B. LINEAR MESO-ELECTRODYNAMICS 5. Local-field phenomena in quantum wells 147

5.1. Diamagnetic electrodynamics of a metallic quantum well 148

5.2. Paramagnetic electrodynamics of metallic and semiconducting quantum wells 154

5.3. Local-field eigenmodes 5.4. Sheet-model electrodynamics and saltus

conditions 5.5. Nonretarded response, radiation reaction

and Lamb shift 6. On the role of local fields in small particles

and quantum dots 6.1. Optical polarizability of small particles 6.2. Optical response of single-electron

spherical quantum dots 6.3. On the spontaneous decay and Lamb

shift 7. Near-field electrodynamics and surface

dressing of particles in motion 7.1. Local fields in the context of near-field

optics 7.2. Local-field electrodynamics of moving

electron wave packets subjected to surface dressing

PART C. NONLINEAR MESO-ELECTRODYNAMICS

8. Optical second-harmonic generation 8.1. Longitudinal and transverse source fields 8.2. 2o-generation in quantum wells

9. Photon drag 9.1. Nonlinear DC current density 9.2. Photon drag in a single-level metallic

quantum well 9.3. Photon drag in the vicinity of an

intersubband transition 9.4. Photon drag in a mesoscopic ring

10. Optical phase conjugation of the field radiated by a mesoscopic particle

159

168

174

178 178

186

190

193

194

203

212 212 215 222 223

225

229 231

236

0370-1573/96/$32.00 $2 1996 Elsevier Science B.V. All rights reserved c

0. Keller 1 Physics Reports 268 (1996) 85-262 87

10.1.

10.2.

10.3.

Electromagnetic propagator for a phase-conjugated field RPA integral equation for the local field. and its solution in the paramagnetic limit On phase conjugation of evanescent waves. Quantum dots of light

Appendix A. Integral relation between the transverse local field and the

microscopic current density A.l. Derivation of Eq. (2.13) A.2. Derivation of Eq. (2.14)

A.3. From the scalar Green’s function g,, to the dyadic propagator D,

Appendix B. Eigenvector expansion of the

transverse vacuum propagator

239

241

243

247 247

248

249

249

Appendix C. Single- and double-commutator

relations between the particle- density and free-Hamiltonian operators 251

Appendix D. Zero and first-order moments of the current density of a moving electron wave packet 252

D.l. Zero-order moment 252 D.2. First-order moment 252

Appendix E. Introduction of the mean values

(IIr) and (rlZ) 253 Appendix F. Derivation of expressions for

d’(Q)(t)/dt’ and dU x (L)(t)/dt 254 References 255

Abstract

To understand the electrodynamics of mesoscopic media it is in general necessary to take into account local-field effects. This article presents a review of the role played by local fields in the high-frequency electrodynamics of systems exhibiting essential quantum confinement of the electron motion, In Part A, the fundamental local-field theory is described. By combining an electromagnetic propagator formalism with a microscopic linear and nonlocal response theory the basic loop equation for the local field is established and some of its implications studied. Various kinds of local-field calculations are presented and the underlying physical interpretations discussed. In Part B, the basic theory is

used to study the linear local-field electrodynamics of a few, but representative and varied, mesoscopic systems. Special emphasis is devoted to investigations of the local-field phenomena in quantum wells and small particles (quantum dots),

and to studies of optical near-field electrodynamics and surface dressing of charged wave packets in motion. In Part C, important features of the nonlinear local-field electrodynamics of mesoscopic media are described on the basis of selected examples. Thus, a description of optical second-harmonic generation in quantum wells is followed by a discussion of the photon-drag effect in one- and two-level quantum wells, and in mesoscopic metallic and semiconducting rings. Finally,

a local-field study of the optical phase conjugation of the field radiated by a mesoscopic particle is undertaken, and a new route leading to confinement of electromagnetic fields into the so-called quantum dots of light is presented.

88 0. Kellrr / Physics Reports 26X (1996) X5-262

1. Introduction

In the macroscopic theory of the electrodynamics of condensed media, the electromagnetic state of a medium is described in terms of the macroscopic electric and magnetic field strengths,

E,X,, and &,,,, , and the electric and magnetic displacement fields, D,,,,, and H,,,,,. The macroscopic Muxwell equations relate specific time and space derivatives of these fields to each other, and to the macroscopic charge and current densities, P,,,_, and J,,,,,. From a knowledge of

E macr” > B,,,,,, D,,,,, and H,,,,,, the macroscopic polarization, P,,,,,, and magnetization Mmacro, can be obtained. In the macroscopic description it is imagined that the system under consideration is divided into elementary volume elements (building blocks). On the one hand, the diameters of the volume elements are small compared with distances over which the macroscopic quantities mentioned above vary appreciably, and on the other hand, the building blocks contain many atoms or molecules.

The range of validity of the macroscopic description is to be determined from underlying microscopic considerations as already emphasized by Lorentz, who originally initiated a program to “separate matter and ether” [l, 21. The investigations of Lorentz were based on the hypothesis that the seat of the electromagnetic field is empty space. In the Lorentz formulation there is only one electric, El,,, and one magnetic, Bloc, field vector, and the electromagnetic state of the medium is characterized by these so-called focaf-jetd quantities. The electromagnetic field (&,, B,,,) is created by atomistic electric charges in motion. In the Lorentz program, Newtons second law with a Lorentz force 4 (&,, + 2) x Bloc) was used to describe the self-consistent motion of the individual atomic particles. The microscopic (local) fields created by the atomistic particles satisfy the so-called microscopic Maxwell-Lorentz equations (see Section 2.1, Eqs. (2.1)-(2.4)). In these equations only the local fields Eloc and B,,, enter together with the microscopic charge and current- density distributions, p and J. The microscopic fields and the atomic charge- and current-density distributions are rapidly varying functions of space and time, and an exact determination of these in general is an impossible task. To go beyond the macroscopic approach in a rigorous manner one thus has to limit oneselves to particularly simple (model) systems from the outset. It is remarkable how close the Lorentz program for determining the local fields of condensed matter systems is to the modern approach. In a sense, the most rigorous modern calculations only deviate from those of Lorentz due to the use of the (many-body) SchrSdinger equation instead of Newtons law for a point

particle. For a detailed account of the foundation of the macroscopic electromagnetic theory and the Lorentz program the reader is referred ,to the book by De Groot [3] and the review article by Van Kranendonk and Sipe [4].

A survey of the textbook literature on physical optics (linear as well as nonlinear) and solid-state optics reveals that most texts contain a mandatory and brief description of local-field effects following in most cases the approach indicated below. Despite of the obvious shortcomings of the heuristic approach to follow even in much of the research literature on the linear and nonlinear bulk and surface electrodynamics (optics) of condensed media the heuristic description is used to account for local-field effects. A legitimate reason for doing so in many cases originates in the complexity of the local-field problem rather than in a qualified believe in the quantitative correctness of the approach. As an introduction to a rigorous description of local-field effects in mesoscopic media (as well as in macroscopic systems) the heuristic textbook formulation is of interest since to some extent it displays important aspects of the entire complex of problems.

0. Keller / Ph_vsics Reports 268 (1996) 85-262 89

In the heuristic description the local electric field, Eloc, acting on a reference atom (molecule) inside a dielectric medium is related to the macroscopic field, EmacrO, via [S]

(1.1)

and the argumentation is as follows. The selected atom is regarded as being surrounded by a sphere (reference atom at the centre) with a radius large in comparison to the distances between the atoms but small compared with the macroscopic dimension of the sample. Insofar as the electromagnetic interaction with the reference atom is concerned the dielectrics outside the sphere is treated as a continuum giving rise to a macroscopic field, E,,,,,, on the position of the atom. The atoms inside the sphere are so close to the reference atoms that their electromagnetic interaction with this must be considered on an individual microscopic basis. Thus, by imagining that all atomic charges inside the sphere are removed, before being put back particle by particle, we obtain two additional contributions to the local field. The first one, named the Lorentz cavity field, E,,,, originates in the polarization charges on the surface of our fictitious cavity. Neglecting retardation effects, the cavity field acting on the reference atoms can be calculated using Coulomb’s law. In terms of the macroscopic polarization one obtains in SI units for a spherical cavity [S]

E,,, = Pmacro/3ao > (1.2)

where 8. is the vacuum permittivity. The individual atoms within the cavity contribute a field Edip to the local field acting on the reference atom. In the heuristic approach the various atoms inside the cavity are considered as electric point-dipoles. Atom number i located at Ri is assumed to have an electric-dipole moment pi, and the reference atom is placed at R. The field E,,,(R) thus becomes

E,,,(R) = C F(R - Ri) *pi 2 (1.3)

where F(R - Ri) is the dipole field tensor. It is often sufficient to approximate F(R - Ri) by its nonretarded near-field form. In such cases one has in dyadic notation

3(R - Ri)(R -Ri)- IR - Ril’U

IR -Ril' 1 *pi(Ri) 2 (1.4)

where U is a unit tensor. It may happen that the field E,,,(R) vanishes. As an example, this is the case for an atom site with cubic environment. In such a situation the local field is given by [6]

E,oc = E,,,,, + Rmacrol3~o .

If one introduces the macroscopic relative dielectric tensor amacro as follows:

(1.5)

P macrcl = ~o(~macro - U)~Emacr, > (1.6)

the relation between the local and macroscopic fields in Eq. (1.5) takes the standard form often used in phenomenological descriptions, i.e. [7-g],

&c = 3 (c,,,,, + 2U) ~&mm . (1.7)

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Though widely used in linear and nonlinear optics, it is certainly obvious that one should not rely too much on the result obtained in Eq. (1.7) for the local-field correction to the macroscopic field. Basically, it is clear that an expression for the local field containing only macroscopic

quantities (emacro, G,,,,) cannot represent any “deep” understanding of the problem. The introduction of the Lorentz cavity concept as such is problematic. For instance, how do we decide on the radius of the sphere? Also, a rigorous introduction of the macroscopic dielectric constant concept runs into troubles. In most situations it is preferable to relate the local field not to the macroscopic field but to the external field, E,,,, impressed on the system, and driving it. Such a relation is adequate because E,,, is a controllable parameter, at least within the framework of response theory. Written in the form

(1.8)

T’ is the so-called field-field response tensor. Since Eloc must be related to E,,, in a causal manner, r satisfies the Kramers-Kroenig relations. Also the neglect of the dipole field Edir usually is a too severe restriction.

In molecular optics [4] one aims at a microsopic description of local-field effects on the basis of the dipolar interaction model already presented in Eqs. (1.3) and (1.4). In the presence of an external field, the molecular point-dipole moments are assumed to satisfy the so-called molecular equations

c41

pi(Ri) = cI [ Eext(Ri) + 1 F(Ri - Rj) *Pj(Rj) .

j( i 0 1 (1.9)

As the index i runs over the various molecular positions Ri a set of coupled linear algebraic equations among the unknown pi's is generated. In Eq. (1.9), u is the (linear) polarizability of the individual molecules. For simplicity, all molecules are assumed to be identical here. Despite of its immediate attractiveness, the molecular approach has some obvious shortcomings. First of all, the assumption that the atom (molecule) is a point-particle object is in general inadequate. After all, even in a dielectrics, where the molecular model may be most accurate (gas and plasma systems not in consideration), the “size” of the molecule usually is of the same order of magnitude as the intermolecular distance. Albeit the fact that the optical response is not directly related to the spatial extension of the quantum mechanical charge-density distribution but to the extension of the relevant transition current densities, it is still correct to doubt whether the point-particle model is a good starting point for local-field calculations in electrodynamics. Secondly, in molecular optics also the calculation of the polarizability runs into serious troubles. The question of how to determine a in fact is directly related to the local-field problem itself. Once we go beyond the point-dipole model it is fair to ask whether the local-field calculation should incorporate the electrodynamic interaction of the reference molecule with its own field? As we shall perceive, such a question brings the Lamb shift mechanism and the radiation reaction process in contact with the local-field problem.

In metallic and semiconducting media where the highly delocalized conduction and valence band electrons usually are responsible for the optical properties it is from the outset doubtful whether the Lorentz model is useful at all. Bearing in mind that (condensed) matter in a quantum mechanical picture in a statistical sense forms a continuum it might be a better starting point for

0. Keller /Physics Reports 268 (1996) 85-262 91

a local-field calculation to set out from a continuum assumption. To examplify this let us briefly consider the heuristic determination of the local field inside a small spherical particle characterized by the (complex) isotropic, homogeneous and local dielectric constant E. A simple electrostatic calculation shows that the relation between the local and the spatially constant external fields is

given by [6]

J%oc = [3/(s + 2)l &ct (1.10)

in this case. The textbook model also leads to the famous, and often used Rayleigh expression [lo] for the polarizability, i.e.,

a = 47n+z3(8 - l)/(C + 2) ) (1.11)

n being the radius of the sphere. As we shall see, the use of the Rayleigh expression for the polarizability can in some cases be justified on the basis of a rigorous microscopic local-field calculation.

In the basic local-field theory described in Part A of this monograph we abandon from the outset any discretization of the matter field. Starting from a continuum distribution of the induced current density in the quantum mechanical sense, we derive an integral relation between the local field and the prevailing current density of the (mesoscopic) medium under study. By dividing the local field into its transverse (divergence-free) and longitudinal (rotational-free) parts we are able to introduce in a rigorous manner an electromagnetic vacuum propagator containing the transverse propagation characteristics of the field as well as the longitudinal and transverse self-field responses. By means of a microscopic version of the Ewald-Oseen extinction theorem [ 11, 121 it is possible in a rigorous manner to identify the interaction volume for the field-particle coupling. It turns out that this volume does not coincide with the volume occupied by the induced current density itself, but rather with the transverse part (or equivalently the longitudinal part) of this current density. A clear distinguishing between these two volumes is necessary in order to make a rigorous division between external and internal fields. In cases where the mesoscopic system is in contact with a macroscopic object such a distinction is of utmost importance. Starting from the Pauli Hamil- tonian in the presence of a prescribed external field, a many-body density matrix approach is used to obtain the nonlocal but linear relation between the induced current density and the electromagnetic field. This so-called constitutive relation relates the induced current density at a given point in space to the sum of the transverse localfield and the external lonyitudinaljield in the surroundings. At the core of this relation we find the microscopic many-body conductivity tensor, including space as well as spin parts. In the one-electron limit the new formalism presented in this article contains the density-functional approach and the random-phase-approximation model as special cases.

By combining the integral relation between the local field and the induced current density with the constitutive relation it is possible to set up an integral equation for the local field. Since, for a macroscopic medium it is usually an impossible task to solve this integral equation rigorously the local-field problem of macroscopic systems does not seem to be a good starting point for understanding the basic physics attached to the electrodynamics on a small (atomic) length scale. The reason that the integral equation problem is so difficult to solve in a macrosystem originates in the fact that the induced current density normally is composed of contributions from extremely many electronic transitions, each having their own transition current density. From self-energy

92 0. Keller __/ Physics Reports 268 (1996) 85-262

quantum electrodynamics we know that the transition current densities of the SchrGdinger equation do play the role of real current-density sources for the electromagnetic field. The fact that a mesoscopic object from an electromagnetic point of view often can be characterized by a few (one, two, . . . ) electronic levels, and thus also by few transition current densities only, makes such an object particularly useful for rigorous microscopic calculations of the local field. In a number of cases the kernel of the integral equation can be separated and a new so-called coupled-antenna theory can be used to perform a rigorous microscopic calculation of the local field. The coupled- antenna theory in a beautiful manner also allows one to study the local-field resonances in mesoscopic systems.

In Part B, the basic theory is applied to study the linear electrodynamics of various kinds of mesoscopic systems. In a quantum-well system the local-field problem is essentially reduced to a one-dimensional problem, and this fact makes it possible to investigate a number of local-field problems in depth. As examples, I discuss the diamagnetic local-field electrodynamics of metallic quantum wells, and the paramagnetic electrodynamics of semiconducting wells. A quantum well deposited on a dielectric substrate is a good candidate for obtaining a better understanding of the boundary (‘jump, saltus) conditions in surface (interface) electrodynamics. At the core of the boundary condition analysis is an integral equation for the local field. The radiation reaction and Lamb shift effects so often discussed in atomic physics on the basis of self-energy quantum electrodynamics are linked to the local-field problem, and these phenomena may be studied within the framework of a particularly simple (one-dimensional) model in quantum-well systems. The role of local fields in small particles and single-electron quantum dots are investigated also. Part B is concluded by an examination of the relation between the growing research field called near-field optics and local-field electrodynamics, and a conceptual analysis of the local-field problem attached to mesoscopic particles which are in motion and at the same time subjected to electromagnetic surface dressing.

In Part C, the noniinear local-field electrodynamics of mesoscopic systems is studied. Due to the fact that it is possible in the nonlinear domain to excite the mesoscopic object internally with longitudinal as well as transverse source fields, the combination of nonlinearity and nonlocality turns out to be particularly useful for deepening our understanding of local fields in condensed matter systems. Via a number of carefully selected examples I seek to demonstrate this aspect in Part C. Following a general discussion of optical second-harmonic generation particular emphasis is devoted to studies of the 2to-generation in quantum wells possibly subjected to the influence of external DC electric fields. Upon a heuristic discussion of the 2cti-process in a two-level quantum- well system, the photon-drag phenomenon in a single-level quantum well is analysed. The photon-drag effect par excellence is a local-field phenomena because it requires a momentum transfer between particle and field, and hence necessitates from the outset a nonlocal treatment. If one forms a closed mesoscopic ring it is sufficient to transfer angular momentum from the electromagnetic field to the mobile electrons of the ring in order to create a photon-drag current. A heuristic description of the photon drag in mesoscopic rings terminates my discussion of local-field phenomena associated with the second-order nonlinearities. The monograph finally deals with a bright new subject in local-field electrodynamics, viz., optical phase conjugation of the field radiation from mesoscopic particles. In the wake of a general discussion, the problem of phase conjugation of evanescent waves is touched upon, and it is argued that a rigorous analysis of this theme necessitates a local-field approach. The phase conjugation of evanescent waves has recently

0. Keller J Physics Reports 268 (1996) 85-262 93

been observed experimentally using the technique of near-field optics [13]. Letting the fibre tip of the optical near-field microscope play the role of a mesoscopic object it is possible to phase- conjugate a broad part of the outgoing angular spectrum from the tip, and in this manner produce long-living small light spots. Since these light spots, due to the phase conjugation of also evanescent components of the field from the mesoscopic tip, may be of subwavelength size, and thus smaller than the smallest ones allowed by classical diffraction theory, I have suggested that these small light spots are referred to as quantum dots of light. To adress the electromagnetic confinement problem of quantum dots of light in a rigorous way a local-field formulation is certainly needed.

PART A. FUNDAMENTAL THEORY

2. Electromagnetic propagator formalism

In order to study the electrodynamic properties of a mesoscopic system it is adequate to take as a starting point an investigation of the relation between the local electric field and the field-induced microscopic current density prevailing inside the mesoscopic domain. The relation is necessarily a nonlocal one, coupling the local field at a given point in space not only to the current density at this point but also to the current distribution in the surroundings. The coupling is provided by an electromagnetic Green’s function (propagator) of one sort or another. Basically, it is always the vacuum propagator which is responsible for the coupling, but in may practical studies it is often convenient to replace this propagator by a so-called pseudo-vacuum propagator which includes in its propagation characteristic, e.g., the electrodynamic influence from macroscopic objects in contact with the mesoscopic system in consideration. Integral relations between the local field and the induced current density are often established in a non rigorous manner. If so, one is lead not only to limitations in the usefulness, in particular, for studies of the electrodynamics of mesoscopic objects, but also often to erroneous results. As we shall realize, there is a tendency that the errors occurring from the use of heuristic (nonrigorous) propagators become more pronounced when the medium under study is reduced to a mesoscopic size. In this section I shall present the main steps in a rigorous establishment of the fundamental integral relation between the prevailing local field and the induced current density. The relation derived provides us with the correct electromagnetic vacuum propagator. This propagator possesses a number of interesting features, and we shall realize that an understanding of the interplay between the longitudinal and transverse self-field parts of the vacuum propagator is a core issue for local-field studies of mesoscopic objects. The integral relation between the field and current density is of the inhomogeneous type, and the inhomogeneous term is obtained via an integration over the surface of the mesoscopic medium. The integrand of the surface integral can be constructed from a pre-knowledge of(i) the transverse part of the local field on the surface and its first-order derivative in the direction perpendicular to the surface plane, and (ii) the electromagnetic vacuum propagator with the self-field parts omitted. In an equivalent description the transversality can be transferred from the field to the propagator. By generalizing the famous Ewald-Oseen extinction theorem [ll, 12, 141 in macroscopic electrodynamics to the microscopic domain the inhomogeneous term mentioned above is identified with the incident field acting on the mesoscopic system. The extinction theorem in an appealing manner allows one to introduce the so-called pseudo-vacuum propagators to be used extensively in

94 0. Keller/Physics Reports 268 (1996) KS-262

Parts B and C of this article. Section 2 is finished by a discussion of electromagnetic pseudo-

vacuum propagators.

2.1. Integral relation between the local field and the prevailing current density

In local-field electrodynamics it is necessary eo ipso to focus the attention on the spatial (Y) behaviour of the microscopic electric field and therefore take as a starting point the microscopic Maxwell-Lorentz equations. After an adequate Fourier transformation to the frequency (0) domain these equations take the form

I7 xE(r; U) = icoB(r; w) , (2.1)

F xB(r; co) = poJ(r; w) - (io/cj+)E(r; U) , (2.2)

v .E(r; co) = (l/E,)/+ 0) ) (2.3)

V .B(r; w) = 0 ) (2.4)

where E(r; o) and B(v; o) are the microsopic electric and magnetic field vectors, respectively, and J(r; co) and p(r; 03) are the microscopic (many-body) current and charge densities. By combining Eqs. (2.1) and (2.2) one obtains the following wave equation for the microscopic electric field:

V x (V x E(r; co)) - (cc)/~)~ E(r; w) = ipooJ(r; co) . (2.5)

Already at this point I divide the field and current density into their divergence-free (T) and rotational-free (L) parts, i.e.

E(r; 0) = E,(r; co) + &(r; co) , (2.6)

J(r; co) = &(r; co) + J,(r; co) , (2.7)

where V .Er = V - JT = 0, and V xEL = V x JL = 0. In the space Fourier domain the T- and L-parts of a vector field are transverse and longitudinal, respectively; hence the subscripts. Before proceeding let me emphasize that the harmless looking (unique) division of the vector fields into their transverse and longitudinal parts is of utmost importance for a correct understanding of the electrodynamics of mesoscopic media, the reason being that even in classical (i.e. field unquantized descriptions) the field-matter interaction at small distances is qualitatively different for transverse and longitudinal vector fields. If this fact is not appreciated essential errors may (and have) appear(ed) in the local-field formalism, as I shall demonstrate later on in this article. In standard optics one may normally neglect the longitudinal vector fields, and in low-frequency (quasi-static) studies of for instance condensed matter systems of macroscopic dimensions transverse effects (propagating with the (vacuum) velocity of light) are omitted. In mesoscopic local-field studies a fascinating and important interplay occurs between the transverse and longitudinal electrodynamics. By inserting Eqs. (2.6) and (2.7) into Eq. (2.5) this splits into two, viz.,

(2.8)

(2.9)

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where q. = o/c,, is the vacuum wave number of light. The differential equation for ET can be written in an equivalent form

(V2 + &)&(Y; co) = - ipooJT(y; Q) , (2.10)

which is particularly adequate for the following calculation. To transform Eq. (2.10) into an integral relation between ET and JT one introduces the well-known scalar Green’s function go(R; co) [14], R = IRI = IY - J 1, which for R # 0 is given by

go(R; w) = eiqoR/4nR , R # 0

and which outside the singular point R = 0 satisfies the differential equation

(2.11)

(V’ + &)g,,(R; w) = 0, R # 0 . (2.12)

Let us consider now the calculation of the transverse part of the electric field, ET(r; o), inside the domain V, in which the transverse current density, J,(r; w), is different from zero. The closed surface bounding V, is denoted by CT, and the domain outside V, by P,. I stress at this point that due to the fact that the transverse and total current densities are related nonlocally (see e.g. Eqs. (2.20) and (2.48)), the domains V, and I/ in which the two current densities are different from zero are not identical. Especially for mesoscopic and microscopic objects the domains may differ significantly; compare e.g. (7.26), (7.28), and (7.30). As a first step in the calculation ofET(r; CD), Green’s theorem is applied to the two functions go(R; co) and ET@; w). To apply this theorem in a rigorous manner it is necessary to avoid the singularity in go(R) at R = 0. Thus, we cut out from the domain V, a (small) spherical exclusion volume F(r) centred on the point r inside V, at which we want to calculate ET(r; cc)), see Fig. 2.1. The surface of the sphere is denoted by (T, and the outwards directed unit

Fig. 2.1. Schematic diagram showing the domain V, in which the transverse current density is different from zero. The V, - domain is bounded by the closed surface C T, and the domain outside is denoted by VT. From the domain V, a spherical exclusion volume, centered on the space point where the transverse microscopic electric field is sought for, is cut out. The sphere has a radius s, and its surface is denoted by cr. The outward directed unit normals from the shaded volume are denoted by n.

96 0. Keller / Physics Reports 268 (1996) 85-262

normal from the shaded volume (i.e. V, minus the spherical exclusion volume) is denoted by II = PZ(~) [or n’ = n(i)]. As demonstrated in Appendix Al, Green’s theorem when applied to go and ET over the shaded volume of Fig. 2.1 gives

J

V, ipL0u go(lv - i/)Jr(J) d3r’

E(T)

=i

Z,+U [ET(r')n' . V’yo( Ir - ~‘1) - go( Ir - r’l)n’ . l”ET(r’)] dS’ (2.13)

in the notation of Appendix Al. For brevity, I have omitted here and shall do so in the following, the reference to w from the notation. To determine ET(r) we calculate the surface integral over g in the limit where the spherical exclusion volume shrinks to zero, while keeping its form. Denoting the radius of the sphere by s one finds as shown in Appendix A2

lim s

V [ET(r’)n’.V’gO(Ir - iI) - go(Ir - r’I)n’.17’ET(r’)] dS’ = ET(r) , (2.14) s-0

and thus

ET(r) = i,noco go( Jr - r’I)&(r’) d3r’

1,

+ i’[ go(lr - /I) y - ET(i) ““I’([ny ‘I)] dS’ , (2.15)

with the notations lim p+Ojp ( . ..) = l:Ao( ...), an d n’s V’ = d/an.‘. It is appealing at this point to

introduce the well-known dyadic electromagnetic vacuum propagator

Do(r - r’) = - (U + (l/q:) VF)go(lr - r’l), r - r’ # 0 (2.16)

instead of go in Eq. (2.15). In Eq. (2.16), U denotes the unit tensor of dimension 3 x 3, and the minus sign is a matter of pure convention. I describe how Do can be introduced in the formalism in Appendix A3, and the final result is as follows:

ET(r) = Z,(r) + & A(r) - i/coo i

V,

Do(r - r’) .&(r’) d3r’ ,

0 Ed0

where

ET(r) = aD,(r - r’)

ET@‘) - an, - D,(r - r’) .%$ dS’. 1

(2.17)

(2.18)

At this point in our analysis the connection to the phenomenological theory of molecular optics described in Section 1 starts to emerge. Hence, in the last term of Eq. (2.17) the (harmonically oscillating) transverse current density &(r’) in the infinitesimal volume element d3r’ located at r’ gives rise to a contribution to the transverse electric field ET(r) at the observation point r which is

0. Keller / Physics Reports 268 (I 996) X-262 91

signalled by the usual vacuum propagator DO(r - i) which is closely related to the dipole field tensor in Eq. (1.3). Although the integrated contribution in the last term from the neighbourhood of r cancels, this does not mean that the transverse current density close to r does not contribute to the field at Y. On the contrary, this contribution is expressed explicitly via the second term, J,(r)/(3ic:,,w), on the right-hand side of Eq. (2.17). This term represents in a rigorous calculation the so-called transverse self-field contribution to the field. The physical meaning of the term Z,(v) in Eq. (2.17), which in explicit form is given by Eq. (2.18) will appear once we have established the microscopic Ewald-Oseen extinction theorem (see Section 2.2). In the context of macroscopic electrodynamics an equation analogous to (2.17) has been derived previously [15]. The macroscopic and microscopic equations, however, deviate on three essential points. Thus, first of all in the macroscopic theories the electric and magnetic fields and their first derivatives in space will exhibit jumps at the (macroscopically) sharp boundary separating the medium under study from the surroundings (e.g. vacuum). This in turn implies that Z,(v) can depend on whether the surface (C,) is approached from the inside or the outside [ 161. Secondly, the need for considering the effects of induced polarizations, free current densities, and magnetizations in macroscopic studies is absent in the microscopic theory. Only the induced microscopic current density is needed (and in fact is the only meaningful physical quantity). Thirdly, in macroscopic studies it is assumed a priori that the fields and current densities are strictly transverse (i.e. ET = E and JT = J), and so, the macroscopic relation equivalent to Eq. (2.17) is among the total vector fields E and J. Since ipso facto the interplay between the transverse and longitudinal electrodynamics is of the utmost importance in mesoscopic electrodynamics, it is crucial that one appreciates that Eq. (2.17) is an integral relation

among the transverse vector fields. The longitudinal part of the local electric field is related to the L-part of the induced current

density as follows:

EL(r) = (l/kOo) JL(r) , (2.19)

cf. Eq. (2.9). In the terminology used to discuss the relation between ET and JT one may say that the longitudinal field consists solely of a longitudinal self-field contribution JL(r)/(i(:Ow). The fact that the longitudinal and transverse self-fields deviate by a factor of three leads to important new consequences in near-field electrodynamics as I shall demonstrate in Sections 6 and 7.

In linear electrodynamics the (many-body) Schrodinger equation predicts the existence of a nonlocal relation between the total current density J = JT + J,_ and the sum of the prevailing transverse local field ET and the longitudinal part of the external field, Ey’, cf. the comprehensive discussion presented in Section 3. In turn this means that Eqs. (2.17) and (2.19) constitute a set of coupled integral equations among the transverse and longitudinal parts of the local electric field.

For compactness it is useful to rewrite Eqs. (2.17) and (2.19) in the form of a single integral relation. To do so we start by introducing the transverse (&(r - Y’)) and longitudinal @,(r - i)) delta functions (tensors) by means of the formal equations

s

XC

Jdr) = &(r - r’) -J(i) d3r’ , (2.20) -2

JL(r) = &(r - r') . J(r’) d3r’ . (2.21)

98 0. Keller / Physics Reports 268 (I 996) 85-262

By addition of Eqs. (2.20) and (2.21) it is realized that

&r(Y - i”‘) + &(Y - i) = U6(r - J) ) (2.22)

where 6(r - i) is the usual Dirac delta function. In Section 2.3 a few of the properties of hT and ~5~ will be discussed. Secondly, we use the relation

i

V, Do@ - i) . J,(J) d3r’ =

Ii-0 i

V

D;(r - r’) . J(r’) d3r’ (2.23) C+O

to replace Jr(i) by J(r’) in the last term on the right-hand side of Eq. (2.17). To do so we must also replace the vacuum propagator Do(r - r’) by its so-called transverse part D~(Y - Y’), and Vr by P’. The prove that Eq. (2.23) is correct, as well as an analysis of DE will be given in Section 2.3. Now, by adding Eqs. (2.17) and (2.19) and making use of Eqs. (2.6) (2.20), (2.21), and (2.23) one obtains

V

E(r) = Z,(r) - ipom s

Go@ - r’) .J(r’) d3+, r E V, , (2.24) E-0

where the dyadic electromagnetic propagator Go(r - Y’) is given by

Go@ - r’) = D;f(r - i) + g,(r - i) + g,(r - i)

with

(2.25)

gT@" -r') = (l/3& 6T+" - f"') > (2.26)

g,(r - i) = (l/q;) i$(r - r’) . (2.27)

In the following we shall refer to gT and gL as the transverse and longitudinal self-field propagator, respectively. The integral relation in Eq. (2.24) between the local field and the prevailing current density is the one we have sought for in this section.

2.2. Microscopic Ewald-Oseen extinction theorem

Up to now, I have considered the current density, J(r), induced in the (mesoscopic) medium under study without discussing its origin. In the following I shall assume that this current stems from the radiation produced by a distribution of external (ext) charges oscillating at the frequency CCL The transverse current-density distribution of the external source field, which is responsible for the radiation, will be denoted by J?‘(r; o) = J+“‘( r ), and it will be assumed that the two transverse current-density distributions &(Y) and J?‘(r) do not overlap in space. The domains occupied by JT and JFt will be denoted by VT and VFt, respectively, and the vacuum domain filling the remaining part of space is named FT. The closed surfaces bounding VT and I”+” are denoted by ,& (as till now) and CF’, and the outwards directed unit normals from the two transverse current- density domains by II. In Section 2.1 Green’s theorem was applied to the domain vr and it was found that

ET(r) = z,(r) + &.'T(') - ivOw I

V, Do@ - i) *JT(Y’) d3r’, r E VT (2.28)

E+O


if, as indicated, the observation point is located inside Vr(r E Vr). If instead the point of observation lies outside Vr, i.e. in pr or I’+Xt, Green’s theorem when applied to Vr once again obviously gives since the surface integral over cr is absent

s

V, 0 = Z,(r) - ipOo Do@ - i) .&(r’) d3r’, r E pr or V+“. (2.29)

Since Do has no singular points in this case there is no need for the spherical exclusion volume, and the transverse self-field contribution must be absent also. If Green’s theorem is applied to the vacuum domain VT, two equations analogous to (2.28) and (2.29) are obtained, viz.,

ET(Y) = - L,(r) - Z?“‘(r), YE l& ( (2.30)

0 = - Z,(v) - Z+“‘(Y), Y E Vr or Vyt . (2.3 1)

In Eq. (2.30) and (2.31) Z+“(Y) has the form given in Eq. (2.18) with the integration extending over the surface ,Z+“. Finally, by application of Green’s theorem to the source domain V;,’ it follows immediately that

0 = I?'(Y) - ipoO Do@ - i) . J+“‘(i) d3r’ ,

- r’) .J;,‘(,)d3r’, Y E V+” , (2.32)

r~ Vr or VT. (2.33)

The physical meaning of L,(r) given in Eq. (2.18) becomes clear if Eqs. (2.3 1) and (2.33) are added. Hence,

s

VY &(r) + iln0u Do@ - r’).J+“(r’) d3r’ = 0, YE V, . (2.34)

It is obvious that by adding the two equations one must require that r E Vr, as indicated above, In Eq. (2.34) the term

E;"'(r) = - ipocL) i

Vs"

Do(r - r’).JF’(r’) d3r’, YE P, or Vr (2.35)

can be identified as the (necessarily transverse) source field (E?'(r)), i.e. the (transverse) field outside

the source region originating in the external sources. With this identification, it follows that

Z,(Y) = E;"'(r), VE I', . (2.36)

The term X,(r) in Eq. (2.32) thus equals the external field incident on the mesoscopic system. Eq. (2.34) is an example of a microscopic Ewald-Oseen extinction theorem. In the classical formulation of the Ewald-Oseen extinction theorem in molecular optics it is asserted that Eq. (2.34) implies that the incident field is extinguished entirely by those molecular dipoles that are situated

100 0. Keller /Physics Reports 268 (1996) 85-262

on the boundary of the medium (dielectric) in consideration [ll, 121. By addition of Eqs. (2.29) and (2.31), and of Eqs. (2.29) and (2.33) extinction theorems equivalent to the one discussed above for V, can be established for the domains I’+” and P,, respectively.

The original Ewald-Oseen extinction theorem has been extended and applied to several problems of recent interest in the research literature [4, 15, 161. Among these, the electrodynamics of spatially dispersive (nonlocal) media [17-201 is of primary interest in the present context. The above-mentioned studies [ 177201 deal with sharp-boundary problems. Generalized extinction theorems established by the present author [21] enable one to study the selvedge response and the coupling between the selvedge and the bulk in nonlocal metal optics. A time-dependent version of the extinction theorem has been presented also [22]. In the context of optical multistability also the possibilities of generalizing the extinction theorem to the nonlinear regime have been investigated

~231. By inserting Eq. (2.36) into Eq. (2.24), we have finally by a rigorous calculation obtained the

following integral relation between the local field and the induced current density:

j

V

E(r) = E'"'(r) - ipow G,,(v - Y’) .J(r’) d3r’, r E I’, , (2.37) Cd0

where we have omitted the subscript T on the external field due to the fact that the field radiated from the external source distribution is necessarily transverse outside the source region, i.e. E+“‘(r) = Eext(y) for Y E V, or 8,. For field points in the vacuum domain the field-current density integral relation can be obtained via Eq. (2.30). Thus, since - Z?‘(v) = E?‘(r), cf. Eqs. (2.33) and (2.35) and because C,(r) is related to JT by means of Eq. (2.29), one obtains after use of Eq. (2.23) the result

1

V

E(r) = EeXt(v) - ipoO D;(Y - Y’). J(f) d3r’, YE VT) (2.38)

where the subscripts T on the local and external fields have been omitted due to the fact that the fields in the vacuum domain are transverse.

2.3. Transz;evse and longitudinal uacuum propagators

To put the Green’s function formalism described in the previous two sections in perspective, and to gain further insight in the characteristics of the vacuum propagators introduced, it is fruitful to consider the propagator formalism from the point of view of distribution theory. Thus, as a starting point we take, upon comparison to Eq. (2.5), the equation

- V x (V x D,,(r, ro)) + q;Do(r, ro) = U6(r - ro) . (2.39)

By division of the propagator Do into its transverse (D;f) and longitudinal (Dk) parts, i.e.

DoP, ~0) = I%, c,) + D:(y, ~0) 3 (2.40)

where P - Di = V x Dfj = 0, Eq. (2.39) can be split into two equations

0” + q;)D&, ro) = MY - ro) , (2.41)

&?D:(r, ~0) = %(y - ~0) (2.42)


By comparison to Eq. (2.27) it is realized that

Db, ~0) = g,(r - yo) , (2.43)

which means that the longitudinal part of Do is just the longitudinal self-field propagator. To investigate the properties of the transverse part of Do it is adequate to expand Dlf in terms of

the transverse eigenvectors ({;f) and eigenvalues (qn) of the equation

(V2 + 435: = 0. (2.44)

Being interested only in the transverse spectrum, the eigenvectors must satisfy the requirement Fe 9: = 0. Over the domain (Q) in consideration, the eigenvectors obey the orthonormality condition

s R

(C:(r))* - 5%) d3r = bm > (2.45)

and are assumed to form a complete set. The quantity S,,, is the Kronecker delta. The completeness is expressed through the completeness theorem for the transverse spectrum, viz.

~5%“)m”o))* = w - ro) (2.46) n

This theorem can also be considered as an eigenfunction expansion for the transverse delta function. As proven in Appendix B the eigenvector expansion of the transverse propagator of Eq. (2.4 1) becomes

D&,, yo) = c

n

(2.47)

A particularly useful expression for Di is obtained if a plane-wave eigenvector expansion over the infinite domain is employed (continuous “orthonormal” basis). Thus, associated with each wave vector q one has two transversely polarized and linearly independent eigenvectors {i,(r) = (2~)-~!~e,~ exp(iq. r), and <zI(r) = (27~) 3’2eq, exp(iq - r), with e,,, * eq, = 6ij. Since

%%I + e&71 = U - e4e,,, where eq is a unit vector in the q-direction, one immediately arrives via Eq. (2.46) at the well-known plane-wave expansion of the transverse delta function, i.e.

IX &(r - ro) = (2~)~~ (U - e,e,)eiY.(rPG dzq ) rfb, (2.48) -ZC

and at the following expansion for the transverse vacuum propagator, cf. Eq. (2.47):

x Dz(r - ro) = (2~)~~ u - eqeq $q.rr-r,j dsq ,

~ jc’ q; - q2 r#ro. (2.49)

102 0. Keller, Pltysics Reports 268 (I 996) KS-262

By means of Eq. (4.22) and the plane-wave expansion of the Dirac delta function, it is realized that the longitudinal delta function has a plane-wave expansion of the form

W hL(y - Ye) = (27r) 3 i

eqepei‘7’(’ - ml d3q , y # yg (2.50) -ZX

which, in turn, implies that the longitudinal vacuum propagator takes the following form in our

continous “orthonormal” basis:

D:(r - yo) = (2~))~ y _# y. . (2.51)

In the presence of only outgoing waves (Sommerfeld radiation condition at infinity) it follows upon an explicit integration over q-space (compare Section 6.1.2, Eq. (6.12)) that the resulting expression for D,(r - ro) = D~(Y - Y,,) + D~(Y - Y,-,) is identical to that given in Eq. (2.16). Since the integrals in Eqs. (2.49) and (2.51) are singular at I’ = yo, special precautions must be taken at this point. I shall return to this problem in connection with the discussion of local-field effects in mesoscopic particles and quantum dots in Section 6.

By considering the expansions in Eqs. (2.49) and (2.51) as Fourier integrals, the Fourier ampli- tudes of the transverse and longitudinal vacuum propagators are D:(q) = (U - e,e,)/(qi - q2) and D;(q) = eqeq/qir respectively. If one denotes the Fourier amplitude of the induced current density J(r) by J(q), the transverse part &(Y) has a Fourier amplitude J,(q) = (U - eqeq) -J(q), cf. Eq. (2.20). Introducing also the Fourier amplitude D,(q) = D;(q) + D:(q), the above-mentioned expressions show that

Do(q)-J,(q) = D&&J(q) > (2.52)

a relation which together with the folding theorem leads to the result postulated in Eq. (2.23).

2.4. Pseudo-vacuum propagators

The mesoscopic object under study often is in contact with a macroscopic system. As examples of this one could mention (a) a (few) monolayer thick film (quantum well) on top of a substrate, (b) a quantum dot (or ring) embedded in a solid matrix, and (c) small particles on a surface. To cope with such a situation it is useful to make use of what I call electromagnetic pseudo-vacuum propagators. Conceptually, such propagators can be introduced in the manner described below provided the transverse parts of the field-induced current densities of the mesoscopic object (JT), the macrosystem (JT macro), and the external sources (JF’) do not overlap. The domains occupied by the associated transverse current densities are denoted by I$, I’TmaCr’-‘, and Vqxt, respectively. The essential idea is to eliminate the explicit appearence of the induced current density of the macrosystem from the formalism. The price paid for this elimination is a replacement of the electromagnetic vacuum propagator by a new one, which in its propagating characteristics incorporates all effects of the macrosystem. Having achieved once the elimination of Jmacro, the dynamics of the mesoscopic system is effectively driven only by the external source distribution, but with the field radiation described by a new, dressed propagator named the pseudo-vacuum


propagator. Since a given pseudo-vacuum propagator electronically depends only on the electronic properties of the macrosystem, the same pseudo-vacuum propagator can be used for different mesoscopic arrangements provided that the macrosystem has the same electronic properties in all cases.

To eliminate the current-density distribution of the macrosystem let us write down an expression for the local electric field inside VraCro. In the wake of the analysis presented in Sections 2.2 and 2.3 it is realized that this field is given by

1 E(r) = & J+=‘O(r) + 7 JL”““‘“(r)

0 l&Oco

[s V”““” s V

- ipow D;s(r - r’) .Jmacro(r’) d3r’ + D;f(r - r’) .J(r’) d3r’ E-0

s V’“’

+ D;(r - i). P(r’) d3r’ 1 , Y E V;- (2.53)

To proceed from here one needs a relation between the induced current density in the macrosystem

(J”““‘” ) and the prevaling field given above. I shall assume here that this relation is given by linear response theory in the random-phase approximation (RPA) approach, i.e.

J”‘-+-) = jv- cmacro(r, f) . )$J) d3r’ , (2.54)

where cmacro (r, r’) is the linear and nonlocal conductivity response tensor of the macrosystem in the RPA limit. Since Section 3 is devoted to an extensive analysis of the linear response theory for mesoscopic media exhibiting nonlocal electrodynamics I shall not dwell on the physical basis for the relation in Eq. (2.54) nor shall I comment on the structure of ~~~~~~~~~ Y’). By inserting Eq. (2.53) into Eq. (2.54) we obtain an implicit relation among the current densities J, Jmacro, and Jext, viz.,

V”““”

(r’)d3r’ = ‘P(r,r’)-J(Yi)d3r’ + s s

V’“’

a(r, i) -J”““‘” P(Y, i) - Jext(r’) d3r’,

r E V;aCro )

where

a(r, r’) = 6(r - r’)U + ipoco d macro(r, y”) . Go (y” _ r’) d3r”

s V”““”

p(Y, d) = - i,uocL, CJ~~‘~“(~, r”). Di(r” - r’) d3r” .

(2.55)

(2.56)

(2.57)

104 0. Keller / Ph?;sics Reports 268 (1996) KS-262

As indicated, the relation in Eq. (2.55) is valid for observation points (r) inside the domain of the macrosystem. For compactness, I have in the expression for OI(Y, r’) made use of the propagator Go of Eq. (2.25). By defining the inverse (a- ‘(Y, r’)) tensor of a(r, r’) via the integral equation

i

V”‘““” u- l(r, Y”) - a@“, r’) d3r” = 6(r - r’)U , (2.58)

a formal but explicit expression can be obtained for the induced current density inside the macrosystem. Hence,

” “‘XC

J macryr) = R(r, r’). J(r’) d3r’ + R(r, r’). Jext(r’) d3r’ (2.59)

with

v ““‘Lro

R(r, r’) = 01~ ‘(r, r”) * @(r”, r’) d3r”. (2.60)

Eq. (2.59) relates J”“c’o to J and Jext in a linear and nonlocal fashion, as one would expect. The relation between the current densities is given via the tensorial function R(r, r’) which in principle is known if the electronic response properties of the macrosystem are given (through tsmacro(r, r’)), and of course also the electromagnetic vacuum propagator including its self-field terms. The relation in Eq. (2.59) is the key which allows us to eliminate the current density of the macrosystem. To realize this, let us consider the integral relation between the local field inside the mesoscopic system and the prevailing current densities of the three domains i.e.

E(r) = &J&9 + 0

&J&) TO

- ipoo r IV DE(r - r’). J(r’) d3r’ + [‘-lir” Dz(r - r’) - Jmacro(r’) d3r’ L JE-o

i

1/“’

+ Dz(r - r’) - Jext(r’) d3r’ 1

,

By inserting in this equation the expression in

J

Eq. (2.59) for J”““‘” one obtains

E(r) = & Jdr) + 7 ’ JL(r) ? lE()O

r fv

(2.61)

- i~occ~ 1J

DT(r, r’) - J(r’) d3r’ + J

DT(r, r’) - Jext(r’) d3r’ , r E VT , (2.62) &+O

where

DT(r, r’) = Dz(r - r’) + ) D;f(r - r”) .R(r”, r’) d3r” . (2.63)

0. Keller / Physics Reports 26X (1996) 85-262 105

Fig. 2.2. Schematic diagrams showing the transverse current-density domains of the mesoscopic system (VT), the external sources (V,‘“‘), and the macrosystem ( VTmacra), possibly in contact with the mesoscopic medium. As described in the main

text, and indicated by the drawn lines, the pseudo-vacuum propagator consists of two parts, viz. a direct part, describing the unperturbed field propagation between a source point at r’ and an observation point at r, and an indirect part involving an interaction with the macrosystem at r”. In the upper diagram the source point is located inside the external current-density domain, and in the lower diagram the source point lies inside the mesoscopic medium.

The screened electromagnetic propagator DT(r, r’) given in Eq. (2.63) is the pseudo-vacuum propagator we have searched for. It consists of two parts, namely a direct part Di(r - r’) (identical to the transverse part of the vacuum propagator discussed in Section 4.3.) describing the unperturbed field propagation between Y’ and r, and an indirect part ~vm“r”D~. R d3r” accounting for all field-propagation channels (from Y’ to Y) involving interactions with the macrosystem, cf. Fig. 2.2.

For compactness, it is useful to add to the pseudo-vacuum propagator DT also its self-field parts, so as to obtain a new pseudo-vacuum propagator

G(r, r’) = DT(r, r’) + gT(r - r’) + g,(r - r’) ,

cf. Eq. (2.25). Introducing also the so-called background

“‘X’ EB(r) = - &0 DT(r, i) . Jext(r’) d3r’ ,

(B) field

(2.64)

(2.65)

106 0. Keller i Phyics Reports 268 (I 996) 85-262

the integral relation between the local field and the prevailing current density inside the mesoscopic medium finally can be written as follows:

i

V

E(r) = E”(r) - ipOo G(r, r’).J(v’) d3r’ , Y E VT . (2.66) C+O

By a comparison to Eq. (2.37) it is seen that the formal elimination of the induced current density of the macrosystem in electromagnetic contact with the mesoscopic object under study implies that the external driving field Eex’(r) must be replaced by the background field, EB(r), and that the pseudo-vacuum propagator G(r, v’) must be substituted for the electromagnetic vacuum propagator Go@ - i).

3. Linear and nonlocal response theory

3.1. Pauli Hamiltonian in the presence of a prescribed external field

Let us consider the electrodynamics of a mesoscopic system of particles subjected to an externally applied field, and let us assume that the sources of the external field are not part of the dynamical system. We use the word “external” here in a wider sense than in the previous Section, cf. for instance the use of this word in the studies described in Part C. The current and charge densities of these sources, Jext(r, t) and peXt(r, t), are thus considered as prescribed functions of time, fixed independently of the field of the mesoscopic system, that is to say the reaction of this field back on the external sources is negligible (or compensated by the experimental set-up). The particles of the mesoscopic system hence evolve under the simultaneous action of the external field, the field created by the particle dynamics induced by the external field, and the free field.

As a starting point for our analysis of the electrodynamics of the mesoscopic system we thus adequately take the following Pauli Hamiltonian:

cr, .B(Y,, t) + 2 s

(E+(r, t) + c$I12(r, t)) d3r , (3.1)

where m,, q,,p,, yX, and O, are the mass, the charge, the conjugate momentum, the g-factor, and the Pauli spin operator, respectively, of particle number a. The vector (A) and scalar (cp) potentials of the total electromagnetic field are given by

A = Aext + Aind + free , (3.2)

cp = qtxt + (pind + cpfree , (3.3)

0. Keller/Physics Reports 268 (1996) X5-262 107

where, as indicated (Aext, (pext), (Aind, qnind) and (Afree, qfree) are the external, induced, and free potentials, respectively. In the present case where the external field is prescribed, the dynamical variables of the electromagnetic field are {Aind + Afree, qind + cpfree, a(A’“d + Fe)/&, 8((pind + qfree)/at}. The Hamiltonian in Eq. (3.1) is written in the Coulomb gauge as fur as the dynurnicaljeld vuriubles are concerned. The Coulomb gauge is economic for the present purpose because the redundant dynamical part of the scalar potential, qind + q ‘Ice, is eliminated in favour of the particle variables {Y,.. The interaction of the particles with the dynamical scalar potential thus enters via the Coulomb-energy part of the Hamiltonian. The Coulomb energy consists of two parts, viz., the Coulomb self energy of the particles, CXc&,,, and the Coulomb interaction energy between pairs (SI, /?) of particles, (%cE~)) ‘C 3: + II q,qll/lr, - ~~1. In the Coulomb gauge the vector potentials of the induced and free fields are hence transverse, i.e. Aind = AFd, and Afree = -4;““. Having eliminated the potentials qind and (pfree we only need to account for the potential energy of the particles in the external scalar potential, hence the presence of the term Cs qa(pext(r,, t) in the Pauli Hamiltonian. Note that to describe the jield of the sources, via (Aext, vex’) there is no need to choose the Coulomb gauge. In the context of the optical studies of mesoscopic systems I want to address in this monograph the transverse free-field effects will be negligible. Thus, it suffices to take

A = A”“’ + Ayd (3.4)

in the Hamiltonian, remembering that AFd = 0 in the Coulomb gauge. Unless we choose the Coulomb gauge also for the external field, the self-consistent vector potential A in Eq. (3.4) will not be transverse. In fact, it is often useful to choose an external-field gauge in which qext = 0 so that the potential energy term C, ~,,J@~~(Y~, t) is eliminated. In such a gauge A”“’ has a longitudinal as well as a transverse part, i.e. A?’ + Ay’. The interaction energy of the electromagnetic field with the particle spins is given by the terms in Eq. (3.1) which contain the g,-factors. Neglecting again the free-field part of the magnetic field, this takes the form

B = V x (A?’ + Avd) (3.5)

since V x Art = 0. The last term on the right-hand side of Eq. (3.1) gives the energy of the transverse field. In terms of the potentials and particle variables {Y?}, the transverse and longitudinal parts of the electric field are (leaving out the transverse part of the free field).

ET = - (a/at) (A?’ + AFd) .

8AY’ -- EL = - at vp + & ‘o ; qa rrr:;;3 .

a

(3.6)

(3.7)

Let us now divide the Hamiltonian into a free part (HF), consisting of the Hamiltonian of the particles evolving in the Coulomb field plus the Hamiltonian of the transverse field, and a part (Hi) describing the external and internal coupling between the particles and the external field plus the induced radiative field, i.e.

H = HF + H, ) (3.8)

10X 0. Keller j Physics Reports 268 (1996) X5-262

(3.91

(3.10)

Having neglected the transverse part of the free field the retardation in the free-field electron-electron interaction has been neglected in HF. In the following where the electromagnetic field is treated as a classical nonquantized quantity there is no need to keep the Hamiltonian of the transverse field in HF.

3.2. Many-body density matrix operator approach

In the electrodynamics of mesoscopic systems, e.g. metallic objects, time-dependent electron-electron interaction effects often play a significant role, and for this reason it is important to study the current density induced by the local electric field from the point of view of many-body theory. In the following we shall denote the various many-body states of the free Hamiltonian by capital letters i.e. II), IJ), etc., and operators written in second-quantized form by calligraphic letters. In the absence of the external field, the energy eigenstates thus satisfy the equation

PFll) = EIIJ), (3.11)

where EI is the energy eigenvalue belonging to state 11). Under the assumption that only the electrons are mobile in the external field, the index CI can be omitted from the particle charge, mass, and g-factor i.e. qz * - e(e > 0), m, * m, and go * g. The interaction operator in Eq. (3.10) thus is a one-body Fermion operator. A general one-body Fermion operator

O(rl, r2, . . . ,rN, t) = fj Otri, t,

i=l

(3.12)

of an N-particle system can in second-quantized form, be written

C(t) = 1 (Ic’, s’ IO I ii, s) ads,s ak,s , k,k s, s’

(3.13)

where lk, s)( Ik’, s’)) is a time-independent single-particle state vector characterized by the triple set of spatial quantum numbers k(k’) and the spin-state label s(s’), and a,_(&, ,Y’) and a~,,{a~,,sJ) are the Fermion annihilation and creation operators belonging to state {k, s} ({k’, s’)), respectively. The single-particle state vectors belonging to a given observable form a complete set.

0. Keller / Physics Reports 268 (1996) 85-262 109

Let us now focus the attention on the interaction Hamiltonian in Eq. (3.10), and let us choose for the external electromagnetic field, a gauge in which the scalar potential qext is zero. In this gauge the vector potential takes the form

&‘&‘+A~‘+/&?. (3.14)

Although the induced vector potential is still transverse, the vector potential of the external field has both transverse and longitudinal parts. Since in this section we are interested only in the linear response of the electron system we can neglect the A *A-term in Ht. In second-quantized form the interaction Hamiltonian thus becomes

(3.15)

where FB = eh/(2m) is the Bohr magneton. By choosing as single-particle state vectors those belonging to a spin-independent free Hamiltonian, the eigenkets lk, s) can be taken as tensor products (0) of kets belonging to the wave function and spin-state spaces, i.e.

Ik,s)=lk)Ols). (3.16)

Below we shall use as basis states for the spin the eigenstates 11)) 1 - 1) for the z-component of the spin operator, i.e. crZ 1 s) = s 1 s), where s = + 1 and - 1 for, respectively, the spin-up and spin-down state. By utilizing the orthonormality of the spin eigenstates ((s’ ) s) = 6,,,) and the result

1 <S'lblS)d',s'ak.s = c [cex + isey) d’, -sak,s + ezsd',s~k.sl , (3.17) s,s’ s= + 1 _

where ei(i = x, y, z) denote the unit vectors along the axes of the Cartesian xyz-coordinate system used, the interaction Hamiltonian can be written as follows:

X,(t) = 21(t) + X!(t) 9 (3.18)

where

(3.19)

(3.20)

are the space and spin parts of Xi, respectively. To calculate the linear response of the electron system via a density matrix approach one needs

the relevant current-density operator. In first-quantized form this is conveniently written as

follows:

j =jF +jG +jl , (3.21)


where

jF = - (e/2m) Cp6(r - r,) + 6(r - rJp] , (3.22)

jg = - (ie/2m) [ (5 (r - r,)p x d - p x ab(r - r,)] , (3.23)

are the contributions from the spatial motion of the electron (jr) and the electron spin (j;) to the free (F), i.e. the field-independent, part of the current-density operator, and

j, = - (e’/m) A6(r - r,) (3.24)

is the field-dependent part ofj. This part is necessary in order to preserve the gauge invariance of the current-density operator. In second-quantized form the one-body operators in Eqs. (3.21)-(3.24) are given by

#‘F = - & ,$, (k’ [P&r - r,) + 6(r - r,)p I k)akT,,sak,s , > 1

(3.25)

x [(e, + iSey)d-,ak., + ezwI~,s~k,sl > ,

$1 = -~~~~(k’lA5(r-r,)lk)a:,,a,,. > 3

(3.26)

(3.27)

To derive the expression for $g use has been made of Eq. (3.17). Knowing the Hamiltonian Z = Xr + .X1, the many-body density matrix operator y in prin-

ciple can be obtained from the Liouville equation [24]

iti ap/i3t = [.X?, p] , (3.28)

where [ ...I stands for commutator. If the mesoscopic system under study is driven only slightly away from (thermal) equilibrium by the externally impressed electromagnetic field, the Liouville equation is adequately solved by expanding the density matrix operator in a power series of the interaction Hamiltonian. Within the framework of linear response theory only zero- and first-order terms in Xi are needed. Hence, we take

P=PF+Pl +P?> (3.29)

where PF is the free density matrix of the field-unperturbed system, and pi and p; are the parts which are linear in X1 and 2:. The free part of p satisfies the eigenvalue equation

PFiI) = pIII) > (3.30)

where PI = P(E,) is the probability for the energy eigenstate 11 > being occupied. In the frequency domain the first-order perturbations of the density matrix fulfil the equations

(3.3 1)

0. Keller / Physim Reports 268 (1996) 85-262 111

By making use of Eqs. (3.11) and%f3.30) it readily follows that the many-body matrix elements of pr and p: are given by

(3.32)

The current density J(r, t) prevailing in the mesoscopic system now can be calculated from the general expression [24]

J(r, t) = Trip!} , (3.33)

where Tr{ ... > means the trace of the operator inside { ... ). By the assumption that the electron system in the absence of the impressed electromagnetic field is in thermal equilibrium one has

Tr{prj) = 0. (3.34)

The formal expression for the linear current-density response J(r; co) (adequately calculated in the frequency domain) in turn becomes

J(r; m) = JSPACE + JSPACE-SPIN + JSPIN , (3.35)

where

JSPACE(r; ~0) = Tr{PrJ’r > + Tr{pr jr} ,

JSPACE-SPIN(r; co) = Tr{pIj$} + Tr{pTjF} ,

JSPIN(r; cc,) = Tr{pyj;) .

(3.36)

(3.37)

(3.38)

In the subsequent subsection we shall calculate the induced current density J(r, a), and then via its relation to the vector potential in Eq. (3.14) the relevant linear conductivity tensor.

3.3. Many-body conductivity tensor. The response to the transverse local field plus the IongitudinaI external jield

3.3. I. Space conductivity Since the conductivity tensor associated with the part JSPACE(r; co) of the induced current

density given in Eq. (3.36) and related to the spatial parts of the density matrix and current- density operator, has been discussed often in the literature (see e.g. Refs. [25-28-J) I shall limit myself here to a brief summary necessary for the subsequent studies of mesoscopic objects. The contribution

(3.39)

112 0. Keller i Physics Reports 268 (I 996) 85-262

is calculated by inserting the expression for $I (Eq. (3.27) into Eq. (3.39)). Since one must require that k’ = k in order that the matrix element (I 1 aJ,,sak.s 1 I) be different from zero, one obtains

WPFA > = -; ,4, cl$kP)12M~ I Jl./k,s I r>lA(f-; 0) > , ,

where ATk,, = aJ,sak,s is the number operator for particles in state (k, s), and

$/C(y) = (rlk)

(3.40)

(3.41)

is the Schrodinger wave function in the (Iv) l-representation. The number operator satisfies the eigenvalue equation

,+-k. s I I> = N:, s I I> (3.42)

with N:,, = 1 or 0 depending on whether the single-particle state (k, s) in 11) is occupied or empty, respectively. Since in the @“’ = O-gauge

A(r; 0) = AY’ + Art + AFd = (io)- ’ (&(Y; 0) + Ey’(r; co)) , (3.43)

where Er’ is the longitudinal part of the external field, and ET is the transverse part of the local field, it is realized that

Tr{pF&l > = s

&i(r, r’; co) + (ET@'; CO) + EF'(r'; w)) d3Y’ , (3.44)

where

dia 6MB ( 6 r ‘; co) = (ie*/Mzo) NO(r)G(v - i)U (3.45)

is the so-called diamagnetic (dia) many-body (MB) conductivity tensor. The diamagnetic response is isotropic, local (in space) and structure-independent in the sense that it depends only on the electron density

N,(r) = c Nk.s I U9l” k.s

(3.46)

at space point r in the field-unperturbed system. In the equation above

Nk,s = c N:,J’I I

(3.47)

gives the number of particles in the single-particle state (k, s) when the system is in the mixed many-body state characterized by the Pl’s. The term

(3.48)


is calculated by inserting Eqs. (3.25) and (3.32) into Eq. (3.48), and making use of the fact that in order to have a nonzero value for the quantity (I 1 ait,sak,s 1 J)(J 1 a~...,sIak.,,ss 1 I) one must demand that k”’ = k, k” = k’, and s’ = s. By introducing also the usual one-electron transition space-current density from state lar) to state Ifi), i.e.

j:!!?(r) = - (e/2m) (p lp6(r - r,) + 6(r - r,)p I a)

it is realized that

(3.49)

(3.50)

where

ccia(r, r’; co) = C LZ!~,~(W) j~_AkCE(r)j~5~~E(r') (3.51) k.k’

is the so-called paramagnetic (para) many-body conductivity tensor. In contrast to the diamagnetic response, the paramagnetic response is manifestly nonlocal in space, and the nonlocal character is expressed via a superposition of tensor products of the relevant one-electron transition current densities. In the individual tensor products the first vector is a function of r, the second of r’. It appears that there is a tensor product related to each pair (k%.k’) of one-electron levels, and the weighting factors in the tensor-product superposition are given by the many-body expression

(3.52)

For a macroscopic system it is normally necessary to take into account a huge number of transitions when studying in, e.g., spectroscopy the optical paramagnetic response. For a mesoscopic system one can often do with a relatively small number of levels and in these cases advantage can be taken of the basic tensor-product structure so as to obtain a rigorous solution for the prevailing local field (and hence for the various quantities related directly to optical experiments). In Section 4, I shall present in some detail the physical picture originating from the form of the parametric response. The sum

f$kCE(r, r’; 0) = ozi(r, r’; (0) + ogia(r, r’; u) (3.53)

constitutes the total many-body space conductivity.

3.3.2. Spin conductivi@ In the interaction Hamiltonian of Eq. (3.15) a term involving the dynamic spin-field interaction

was included. While spin effects associated with externally impressed DC (or low-frequency) magnetic fields in many cases are important in linear (and nonlinear) optics, the dynamic spin effects, i.e. those originating in the interaction of the electron spin with the high-frequency magnetic

114 0. Keller / Physics Reports 268 (I 996) 85.-262

field accompanying the prevailing optical field, usually are negligible in macroscopic systems. That this is so can be estimated from the fact that the ratio between the optical spin and space conductivities is of the order [hy2/(mo)12 M 10-i’ in the optical region, 4 being the magnitude of the (complex) wave number of light. In a mesoscopic system it seems that the situation might change dramatically, however. Let us consider for instance a metallic quantum well of thickness k 10 A. If such a well is excited by p-polarized light the electric field across the quantum well can

change significantly. Since the spin effect is proportional to V x E z q x E (cf. Eq. (3.15)), one might expect that the spin effect increases w (102)4 = 10’ times because the effective q’s will be of the order of the reciprocal well thickness, i.e. 4 zz lo9 rn- ‘. A strong electron confinement thus raises the spin-to-space conductivity ratio to [tiq2/(mco)]' z 10-2. Is it likely that the relative importance of the dynamic spin interaction could be further increased? I believe so provided nonlinear experiments of even order are investigated. Let us consider, e.g., the optical second-harmonic generation in a medium exhibiting centrosymmetric bulk properties. It is known that in such a medium a competition exists between the local electric-dipole contribution from the regions of broken cenrosymmetry and the nonlocal higher-order multipole (electro-quadrupole, magnetic dipole, etc.) contributions. For a review of 2o-generation in centrosymmetric media the reader is referred to Ref. [29]. The dynamic spin interaction is inherently nonlocal (cf. the presence of the factor V x A, and the discussion to follow below), and so, by depressing via the centrosymmetry the electric-dipole interaction always present in the linear regime, the relative importance of the spin-field coupling should increase. The qualitative increase is expected to be 5 l/(fine structure constant) z 137. Since the linear spin conductivity response thus is expected to be of importance in mesoscopic systems, and since this response has not attracted general attention in the literature I shall briefly present the basic analysis of the spin conductivity in this and the subsequent subsection.

In analogy with the treatment presented in the previous subsection let us now establish the relation between the induced spin-current density

(3.54)

and the field ET + E?.“‘. It appears from Eq. (3.26) that it is adequate to introduce the one-electron transition spin-current density

j E:(r) = - (ie/2m) (p [6(r - r,)p -p6(r - Y,) 1 a)

= - (eWJm) C$&)~$B(r) + tis*(rY$&)l = - (efiPW W&)tip*W) (3.55)

in the formalism. In passing it is worth noticing that the expressions for the quantities ijzyi(r) andjzcAFE deviate only through the signs between the two terms in the respective parentheses, cf. Eqs. (3.49) and (3.55). A straightforward calculation shows that the one-body matrix element of V XA entering the expression for 3’; can be expressed in terms of jf!$. Thus,

(3.56)


It is convenient also to introduce the tensor

3k.k’(u) F & c pJ - ‘I I,.,.s ho + EJ - EI

(e,e,(~l~X~.,~k.,I~>~JI~X~..~~k~.,I~)

+ (e, + ise,)(e, - ise,)(Z Iukt,,,~~.-~IJ)(JIukt.-~u~,~I Z)) . (3.57)

Utilizing Eq. (3.56) and the abbreviation in Eq. (3.57) a tedious calculation shows that the induced spin-current density becomes

JSp’N(r; co) = c j;Y~(r) x &,.(u). j”y,(r’) x (E&J; co) k.k’ s

+ Er’(r’;co))d3r’ . (3.58)

Simple algebraic manipulations in turn allow us to rewrite the constitutive relation in the standard form, i.e.

Jsp’N(r, 0) = o~~N(y,y’;~).(ET(~‘;(l)) + E~“(r’;w))d3r’ , (3.59)

where

k.k’

is the many-body spin conductivity tensor. In the present context, where we have limited ourselves to spin-independent free Hamiltonians (spin-orbit effects, ets. being neglected), the .ik,kJ (o)-tensor necessarily must be proportional to the unit tensor, i.e. ak,k! ((0) = .%k,k’((l))U. In turn, it then suffices to consider the e,e,-tensor element of Eq. (3.57) when writing down the simplified expression for ,%?k. k’ (w). Hence, upon a comparison to the formula for P&k.k,((o) (Eq. (3.52)), it is realized that

&&,(w) = - (y/2)&k,k’(u)U . (3.61)

By inserting Eq. (3.61) into Eq. (3.60), and making use of the tensor relation Q x U x fi = Pa - U(a+ /I), it is seen that the spin-conductivity tensor can be written in the alternative form

o$iN(r,r’;u) = t c .~k.k’(u)[u(j,Sp~~(r).j,S~~,(r’)) k.k’

-j”!t~,(r’)j,SY!(r)] . (3.62)

The similarity between the structures of the paramagnetic part of the space conductivity, cgia (Eq. (3.51)) and the spin conductivity cr 5:” (Eq. (3.62)) should be noted.


The absence of spin-dependent terms in the free Hamiltonian also has the consequence that

JSPAC” SP’N(y; (0) = 0 ) (3.63)

as can be demonstrated by an explicit calculation. In conclusion, we thus have realized that within the framework of the Pauli Hamiltonian including the presence of a prescribed externally control- led field, the linear and nonlocal constitutive relation takes the natural form

J(r; 01) = s

cMB(r, r’; co) - (ET(r’, w) + Ey’(r’; (u)) d3r’ , (3.64)

where

(3.65)

is the relevant many-body conductivity tensor.

3.3.3. TT, LL and TL-response.functions The constitutive relation in Eq. (3.64) relates the induced current density to the relevant

transverse (E,) and longitudinal (Er”‘) parts of the electric field. It is of interest for our discussion of the gauge invariance of the linear and nonlocal response theory described in this section, and in order to establish the connection between Eq. (3.64) and simpler versions of the constitutive relation often used in the literature, to divide the induced current density into its transverse and longitudinal parts and obtain contributions to JT and J,_ from, respectively, ET and ET”‘. Hence by inserting the delta function relation in Eq. (2.22) into the identity

cMB(r,v’) = s

6(r - r”)U~csMB(r”,~“‘).U~(r”’ - ,‘)d3r”‘d3r” , (3.66)

the many-body conductivity tensor is split into four relevant pieces, viz.

(3.67)

i

ii/‘@ - Y”) . aMB(J’, r”‘). a&” - r’) d3r”‘d3r” , A, B = T or L . (3.68)

The division in Eq. (3.67) formally allows one to obtain the individual contributions to the transverse part of the induced current density from ET and ET’

JT(r) = s

o~;(r,r’).ET(i.‘)d3r’ + s

(T~LB(r,y’).E~‘(y’)d3).’ :

and also the longitudinal part

JL(r) = s

o~‘(Y,+ET(+)d3r’ + s

a”;“L”(V,r’).E~‘(r’)d3r’ .

(3.69)

(3.70)

0. Keller J Physics Reports 268 (1996) KS-262 117

The division in Eq. (3.67) is particularly illustrative when applied to the spin conductivity tensor (Eq. (3.60)). Hence, by writing this (in the absence of spin-orbit coupling effects) in the form

ag$N(r,r’; 0) = i=x.y.z k,k’

(3.71)

and noting that j;!‘:!(r) x ei and ei ~j~pf~~(r') are transverse vectors [because V . (j”Y; x ei) =

ei .(v Xjff?!!) = - [di/(h)]f?i*(v X v(Gkl t,@)) = 0, etc.] it iS realized that

og”(r, r’; 0) = rJy;3sp’N(r, r’; to) . (3.72)

The spin system thus only contributes to the purely transverse electrodynamics. Due to the fact that the spin part of the interaction Hamiltonian is proportional to V XA = (io)-‘(V xET + V xEr”‘) = (io)-‘V xET’t 1 was obvious of course from the outset that the longitudinal part of the external electric field cannot contribute to the spin-current density.

3.4. Gauge invariance

To determine the induced current density in a prescribed external field E’“’ = ,!?yt + IV??’ we took as a starting point the Pauli Hamiltonian given in Eqs. (3.8)-(3.10). In a Hamiltonian formalism the external field enters via the vector and scalar potentials (Aext, CJ?). Since physically measurable quantities do not change under a gauge transformation, the response of the mesoscopic system must be independent of the gauge used to describe the external electromagnetic field. Although electrodynamics as such is gauge invariant it is not obvious that the linearization of the current-density response invoked in the preceeding sections does not lead to a breaking of the gauge-invariance principle. Fortunately, the gauge-invariance principle survives within the framework of the linear response theory established in this section, as I shall demonstrate below. The proof given here holds for a general many-body system and thus generalizes that established by Bagchi [26], previously. When changing from an old (OLD) to a new (NEW) gauge, the transverse part of the vector potential is conserved, i.e. [A?’ + AF’(OLD), @“‘(OLD)] = [A?;” +

Ay”‘(NEW), @“‘(NEW)]. It is thus necessary only to establish the gauge invariance for a longitudinal field stimulus. Within the linear response formalism, which I am considering in this work, it is sufficient for a guage-invariance proof to compare the responses to a pure longitudinal vector potential (Art) and a pure scalar potential ((pext), cf. also Eq. (3.7). In the many-body calculation carried out in Sections 3.2 ad 3.3, in fact the pext = O-gauge was used (see Eqs. (3.14) and (3.15)). This means that we need to calculate the response of the system to a pure scalar potential (q,““‘) and then prove that this response is in agreement with that already obtained in Section 3.3.1. Only the space conductivity needs to be considered since the spin system only reacts to the transverse part of the vector potential.

Let us begin with the constitutive equation relating the induced current density to a purely longitudinal external field, i.e.

J@;(o) = (a) + of;lLB(r,r’; co)).Ef.Y(r’; co)d3r’ , (3.73)

118 0. Keller / Ph_vsics Reports 268 (1996) 85-262

where the relevant expressions for the conductivities are to be derived from Eq. (3.53) with Eqs. (3.45) and (3.51) inserted. In deriving ogkCE, the reader should remember that the 9”“’ = O-gauge

(Ey”’ = icAr”‘) was employed. Using the equation of continuity for the electric charge, i.e.

V -J(r; a) + ieoN(r; 0) = 0 , (3.74)

where N(r;co) is the induced particle density in the frequency domain, and the relation Ey”‘(r’; w) = - V’4ext(r’; w), Eq. (3.73) implies that

(3.75)

since V . &‘T = 0. By means of an integration by parts, Eq. (3.75) can be rewritten in the form

(3.76)

extending the integration over a volume slightly larger than (and enclosing) the volume occupied by the mesoscopic medium. In Eq. (3.76) the response function [i/(eco)] 8’ - V + opF(r, r’; co) relating N(r; w) and cpext(r’; o) is obtained in the @“’ = O-gauge. Below, we shall establish an alternative relation between the induced particle density and the external scalar potential, viz.

N(r; co) = xMB(r, r’; ~)qF(r’; co) d3r’ , (3.77)

starting from theA?“’ = O-gauge. In order for the formalism to be gauge invariant it is required that the many-body density-density ( xMB(r,r’; co)) and current-current (ay’(r, r’; 0)) response functions satisfy the equality

X&l”, r’; co) = & V’ . v . [cJy[,dia(r, r’; co) + CTyyr, r’; co)] . (3.78)

Due to the presence of the operator V’ - V . , the tensor sum cy’,dia + ~~~~~~~~ could of course have been replaced by aft;‘; + OK”;;“.

To determine the many-body density-density response function entering Eq: (3.77) it is needed to calculate the induced particle density, N(r; ~0). Since the particle-density operator ~,6(r - Y,) is independent of the electromagnetic field, N(r; co) may be obtained, for instance, from the expression

(3.79)

0. Keller / Phy,s?cs Reports 268 (1996) 85-262 119

where

.J”(r) = c (k’,s’ld(r - r,) 1 k,s)a:S,,Sa,,,

(3.80)

is the number operator in second quantization, and X1 is the space part of the interaction Hamiltonian. In the A@’ = O-gauge, X1 is given by (cf. Eq. (3.10))

~Tifl = -e C(k’,s’I~ex’Ik,s)Ka:,,,,ak,s k,k’ s.s’

=- e c d’,sak,s

s

~2(r)~k(r)~ext(r;W)d3y’ . (3.81) k,k’,s

By inserting the expressions for N(r) (Eq. (3.80)) and %I (Eq. (3.81)) into Eq. (3.79) it is a straightforward matter to obtain a relation of the form given in Eq. (3.77). By means of the abbreviation in Eq. (3.52), one finally gets the standard expression

(3.82)

It is left to demonstrate by now that a direct calculation of the right-hand side of Eq. (3.78) leads precisely to the expression given in Eq. (3.82). As a first step, I consider the paramagnetic contribution. Since, in second quantization

ye, = ; s

,$F(r’). (ET(f) + Ey’(r’)) d3r’ ,

the paramagnetic part of the conductivity adequately can be written as follows:

ogG”(r, r’; w) = _!_ 1 pI - pJ

o 1.J tier, + EI - EJ (zI~F(r)lJ)<JI yF(r')II) .

(3.83)

(3.84)

Denoting the longitudinal part of the free current-density operator by [$r]L, the paramagnetic LL-response tensor becomes

l$py#yJ; a) = J_ c pI - pJ

w I,J ZIO + EI - EJ (II [bFk)lLIJ>

x <JIC&Fb”)lLl~) * (3.85)

In the present context, Eq. (3.85) is not of importance, since, in order to calculate the paramagnetic contribution in Eq. (3.78), one just needs V - [fFIL = V - { [BFIL + [$F]T} = P -&F because the

120 0. Keller- ! Physics Reports 268 (I!?&) KS-262

divergence of the transverse part of &r is zero. Hence, we obtain

(3.86)

To relate the contribution in Eq. (3.86) to xMB(r,r’; o) it is natural to attempt to introduce the number operator (Eq. (3.80)) in the formalism instead of %r, cf. the equation of continuity (Eq. (3.74)). To do so, I make use of the commutator relation

[.%?r, .,a;-(r)] = (h/ie)V ‘RF(r) . (3.87)

The proof that this relation is correct is given in Appendix C. By replacing V - 2F(r) and V’ - $F(r’)

in Eq. (3.86) by the two commutators, a simple calculation shows that

i v ’ . v . bpF.*ara(r, r’; w) eco

e c pI - pJ

(ht~)~ l,J tie + El - EJ (E, - EJ)2(Z 1 csd”‘(r) 1 J)(J 1 ./f’.(r’) 1 I) . (3.88)

Before making a comparison to the expression for xMB(r, r’; to) let me as the second step consider the diamagnetic contribution. To rewrite this contribution in an adequate form, I take as a starting point the double-commutator relation (for a proof, see Appendix C).

[~:~F7_i(~~(r)],rl~“(r’)] = - (h’/m)~‘.~(Al”(r)G(r - r’)) . (3.89)

The mean values of the right- and left-hand sides of Eq. (3.89) in the thermal equilibrium state, i.e.

Tr(PF[[~~?.‘(‘(r)],~,l ‘(r’)]) = - (h2/nz)TrIp,~‘.~(.,1 ‘(r)S(r - r’))} (3.90)

can easily be calculated, using the completeness relation for the many-body states, i.e. x1 I I) (I I = 1 (1 being the identity operator), twice. Hence, one obtains

1 (P, - PJ)(E, - E,T)(I I +1/'(r) 1 J)(J I l’-(r’) I I) I,.1

= -G V’.V(N,(r)G(r - r’)) , (3.9 1)

whereN,(r)=~,(zI..l”(r)lz) =L,sNk.sI$k(r)I 2 is the electron density in the field-unperturbed

state, cf. the analysis in Section 3.3.1. By means of the result in Eq. (3.91), the diamagnetic contribution to the right-hand side of Eq. (3.78), i.e.

(i/eo)V’.V .#F,dia (r,r’;o) = (i/ew)V’.V .n$i(r,r’;cI))

= - (e/mo2)V’~V(No(r)G(r - r’)) , (3.92)

0. Keller/Physics Reports 26X (1996) 85-262

can now be put in a form adequate for an addition to the paramagnetic term. Hence,

121

$jT c (PI - I. .l

and then, in turn after

PI - PJ =- 4z 1.3 flu + EI - EJ

(I 1 -/V(r) 1 J)(J I .J’(r’)IZ)

+& C(PI - P,)(~I~~~(r)IJ)(JI~,~‘(r’)I~) I,J

(3.93)

(3.94)

The last double sum in this equation is zero as one realizes by rewriting it as follows:

~(P,-P,)(II.J-(r)(~)(Jl.‘(r’)II)

= C PI(I I [-V(r), ,V(r’)] I I) . (3.95)

To obtain this result, in the term containing PJ we have made the interchange I $ J, and thereafter in both terms used the completeness relation on the J-summation. Since the particle number operators commute, i.e.

[X(r), A’(r’)] = 0 ,

the right-hand side of Eq. (3.95) vanishes. This, then implies that

(3.96)

pI - PJ =- c I,J ho + EI - EJ

(I I,+/(r) I J)(J I J’(r’) ) I) (3.97)

If finally the expression (see Eqs. (3.80) and (3.81))

XI = - e s

AT(r’)4ext(r’)d3r’ (3.98)

is inserted into Eq. (3.79) it is realized immediately that the right-hand side of Eq. (3.97) equals XMe(r,r;co). This brings to an end my proof of Eq. (3.78), an equation which ensures the gauge invariance of the linear, nonlocal response formalism in the many-body case.

3.5. One-electron theories

3.5. I. Density functional approach Virtually all of the approximations to the many-body Hamiltonian in Eq. (3.1) have aimed at

constructing an accurate Hamiltonian for a single electron (see e.g. [30]). In the context of mesoscopic electrodynamics the longitudinal part of the local field often plays a significant (and even dominating) role as we shall see from some of the examples treated in Parts B and C of this monograph. Keeping only the longitudinal part of the response it is adequate to switch to the scalar response formalism, and hence take the constitutive relation between the induced particle density and the external scalar potential (Eq. (3.77)) as the starting point.

In the famous density-functional theory [31-351 one makes the assumption that the correct density N(r; (0) can be reproduced in an appropriate single-particle potential $(Y; 0). The constitutive relation in Eq. (3.77), one hence replaces by

N(r; co) = xKS(r, r’; CL))&Y’; a) d3r’ , (3.99)

where x~~(Y,Y’; o) is the density-density response function of the noninteracting (Kohn-Sham (KS)) ground state. If we denote the KS (single-particle) eigenfunctions and eigenvalues by IC/K and &(K = k or k’), XKs is giVen by

(3.100)

where .fK = (exp[(& - ,~~u)/(kT)l f l> ~ 1 is the Fermi-Dirac distribution factor. The factor of 2 in Eq. (3.100) originates in the spin-summation. In the KS approach, the exchange-correlation (xc) part of $(Y; w), named qX,(r; co) is dejned by the equation

s A’@‘; (0) d3 r’

Ir - Y’I + cpX,(G~) . (3.101)

In passing it should be noted that omission of cp&;o) in the equation above yields the Hartree relation between @ and q”“‘. A formal nonlocal relation

cpx,(r;co) = s

fxc(Y,y’;o)N(r’;Lr))d3r’ (3.102)

between the exchange-correlation potential and the induced particle density can easily be established provided the inverse response functions xi,’ (r, r’; co) and xii (r, r’; co) given via

(3.103)

(3.104)

and

@(Y;w) = s

X~~(r,r’;co)N(r’;w)d”r’

0. Keller /Physics Reports 268 (1996) X-262 123

exist. Thus, by inserting the expressions in Eqs. (3.102)-(3.104) into Eq. (3.101) one obtains

fx.(r, r’; cd) = xii (r, r'; co) - & (r, r’; co) + e/(47tco ( r - r’ I) _ (3.105)

For a system with an assumed homogeneous (No(r) = constant) ground state fXC -S$‘” is explicitly known because xMB(r, r’; o) = x&r - r’; tu) and xKs(r, r’; w) = lKs(r - r’; co) are just the density-density response function of the homogeneous electron gas [36], and the Lindhard function [37], respectively. In the Fourier domain we thus have

3.5.2. Random-phase-approximation approach

For many purposes it is sufficient to calculate the conductivity response tensor under the assumption that the eigenstates of the many-electron Hamiltonian X$, consist of direct products of single-particle-like states. Within this so-called random-phase-approximation (RPA) [38-401, the ground state IG) is given by

(3.107)

where IO> denotes the particle-vacuum state. The notation k < kF is meant to indicate that only the lowest lying energy eigenstates are filled (in a condensed-matter system kF is the Fermi wave number, and at T = 0 K only states at and below k, are occupied). In the RPA approach the excited eigenstates are of the particle-hole type, i.e.

Ik’, s’; k, s> = a~j,s.uk.rl G). (3.108)

The energy differences EJ - EI now mUSt be equal to EJ - El = &k - &‘, where &k and Ek’ are the energies of the single-particle-like states (k, s) and (k’, s’) satisfying a single-electron SchrSdinger equation

(3.109)

the particles being subjected to an effective potential I/eff, which one usually calculates from the Hartree-Fock orbitals [41] (so that the resulting Schrodinger equation becomes a multidimen- sional nonlinear integrodifferential equation). Simpler schemes based on, e.g., (i) the Hartree approximation [42], in which the effective potential energy for the electron is determined by the average motion of the other electrons, or (ii) a model, where the V& is taken as the ionic potential itself (possibly in a jellium approximation) are also often used. Since IJ) CC a&k’.s) I>, one has PJ - P, = Py”(l - P:‘%‘) - I’:‘,“(1 - I’!*“) = Pi,’ - I’:‘,“, where P:,” is the probability that there is an electron in the one-electron level (k,s) of the many-body state 1 I). Utilizing next that

Cr V:*” - P:‘.“) = fk -fk’, ‘t 1 is realized that _&$‘(Qj given in Eq. (3.52) takes the form

dk,k’(d = (2i/0>(fk -.fdh~ + Ek - Ek') (3.110)

in the RPA approach (the factor of 2 again originating in the summation over the spin states).

124 0. Keller / Ph_vsics Reports 268 (1996) 85-262

In the RPA model the diamagnetic contribution to the conductivity tensor still has the form given in Eq. (3.45) but with an electron density N,(r) = N,,,,,(r) calculated from

The paramagnetic part of the conductivity takes the well-known form

and, finally, the spin conductivity becomes

hw + Ek - ck’

(3.111)

(3.112)

(3.113)

In the RPA approach the density-density response function is identical in form to that of Eq. (3.100), except from the fact that one uses HartreeeFock (or Hartree, etc.) orbitals instead of Kohn-Sham eigenfunctions. It is possible to demonstrate that the RPA calculation leads to gauge-invariant results also [26].

4. Local-field calculations and underlying physics

In Section 3 an integral relation between the local field and the current density prevailing inside a mesoscopic system was established and discussed. Due to the fact that the transverse and longitudinal field-matter interaction are qualitatively different at small distances it was shown that as far as electrodynamics is concerned the spatial domain occupied by the mesoscopic medium has to be identified with the transverse (or equivalently longitudinal) current-density domain and not with the domain occupied by the total (transverse plus longitudinal) particle (here electron) current density. In particular for objects of mesoscopic (or microscopic) sizes it is important to distinguish between the domains of the current density itself and its transverse (longitudinal) part. The basic integral relation is given in Eq. (2.37). In establishing this relation it was assumed that the transverse current densities of the external source field and the mesoscopic system do not overlap in space. When this is so, the field radiated from the external sources is necessarily transverse inside the transverse current-density domain of the mesoscopic medium. If the mesoscopic object under study is in electromagnetic contact with a macroscopic system the prevailing local field and current density inside the mesoscopic system are related via the integral equation in (2.66). The two integral relations in Eqs. (2.37) and (2.66) essentially are identical in form. To switch from Eq. (2.37) to Eq. (2.66) one just has to replace the external field (E”“‘(r) = EFt(r)) and the vacuum propagator (G,(r - r’)) by the background field (EB(r) = E;(r)) and the pseudo-vacuum propagator (G(r, r’)), respectively.

0. Keller / Physics Reports 268 (1996) KS-262 125

In order to determine the local field in the mesoscopic medium via the integral relation in Eq. (2.66) (or Eq. (2.37)) it is necessary to relate the induced current density to the prevailing field. Taking as a starting point the Pauli Hamiltonian in the presence of a prescribed external field, we obtained within the framework of linear many-body response theory a general nonlocal constitutive relation (see Eq. (3.64). By combining Eq. (2.66) (or possibly Eq. (2.37)) and Eq. (3.64) a fundamental integral equation for the local field inside the transverse current-density domain can be established. In passing, I stress that the word “external” was used in a wider sense in Section 3 than in Section 2. Thus, in the context of the constitutive relation, by an “external” field we mean a prescribed field for which the dynamic evolution is unaffected by the electrodynamics of the mesoscopic medium under study. As we shall discuss, part of the prescribed current density might be located inside the transverse current-density domain of the mesoscopic system. Since it will be obvious from the given context we shall in the following, for brevity, omit writing as superscript on the various integral signs the domain over which the integration is to be performed. For definite- ness we shall relate the local field and the current density of the mesoscopic object by means of Eq. (2.37). If necessary, the pseudo-vacuum integral relation in Eq. (2.66) can of course be used instead.

4. I. Integral equations for the local field

In the present section the basic coupled integral equations for the transverse and longitudinal parts of the local field will be established and discussed. I devote particular attention to the setting up of these equations in various important limits and under different approximations.

4.1.1. Driving current sources located outside the mesoscopic medium If the transverse current-density distributions of the driving source (e.g. a laser source) and the

mesoscopic medium do not overlap in space, the longitudinal part of the source current density (FL”) is zero inside the mesoscopic system. Since the Maxwell equations for the source system imply that E?“(r) = Jy.“‘(r)/( &u), cf. Eq. (2.19), the external field prevailing inside the mesoscopic system must necessarily be transverse, i.e. Ey”’ = 0. In turn, this means that the constitutive relation in Eq. (3.64) takes the form

J(r) = s aMB(r,r')-ET(r')d3r' . (4.1)

By inserting the equation above into the integral relation in Eq. (2.37) one obtains after a division of the resulting equation into its transverse and longitudinal parts

ET(r) = E?‘(r) - ip,,w s

(D;f(r - i)

+ g,(r - Y’)) . o~;(Y’, r”) . ET(r”) d3 r” d3 r’ , (4.2)

(4.3) J&_(r) = & s oFf(r, r') -ET(r') d3 r’ .

0

126 0. Keller /Physics Reports 268 (1996) KS-262

Introducing the dyadic kernel KTT(r,r’) via the definition

K&r, r’) = - i,uOcu s

(D:(r - Y”) + g,(r - r”)). d$(r”,d) d3r” , (4.4)

it is seen that the transverse part of the local field satisfies the integral equation

ET(r) = Ey’(r) + s

K&r,r’) &(r’)d3r , r E VT (4.5)

The formal solution of this equation is given by

ET(r) = r,,(r,r’)dyt(r’)d3r , (4.6)

where, by inserting this solution into Eq. (4.5), it is realized that the so-called transverse-transverse (TT) field response tensor FrT(r,y’) satisfies the dyadic integral equation

r&r,r’) = ijT(r - r’) + s

KTT(r, r”). rTT(r”, r’) d3 y” . (4.7)

Once the solution for the transverse part of the local field has been obtained (in practice usually an approximate one, of course) the longitudinal part of the local field can be found by inserting Eq. (4.6) into Eq. (4.3). Hence,

EL(r) = r,,(r,r’)-E~t(r)d3r’ , s

where the longitudinal-transverse (LT-) field-field response tensor is given by

FL&, r’) = 1 IFOW s

c$f(r, r”). rTT(d’, r’) d3 Y” .

(4.8)

(4‘9)

4.1.2. Prescribed current sources located inside the mesoscopic medium The current-density sources driving the mesoscopic medium are often located inside the trans-

verse current-density domain (Vr). As an example, one might mention the case where a (fast) moving charged particle is penetrating the mesoscopic system (see Fig. 4.1). Provided that one neglects the modification induced in the dynamics of this particle by the mesoscopic medium, the current density of the particle is prescribed and can be considered as an “external” current-density source giving rise to the presence of a prescribed external field in the sence of the word used in Section 3. As another example, let me mention the case where the optical second-harmonic generation from a mesoscopic medium is studied. Provided that the second-harmonic generation processes are considered within the framework of a parametric approach, i.e. an approach in which the dynamics of the fundamental field evolves independently of the second-harmonic dynamics, the driving current-density sources at the second-harmonic frequency are (i) prescribed (generated


Fig. 4.1. Schematic diagrams showing two examples in which the current-density sources driving the electrodynamics of the mesoscopic medium are located inside the transverse current-density domain (VT) of the mesoscopic system itself. In the upper part of the figure the “external” current-density source is that of a moving charged particle (black dot). In the lower part of the figure the monochromatic field from an external current-density distribution, Y’(W), gives rise to a prescribed “external” current density Jext(20j) at the second-harmonic frequency inside the nonlinear mesoscopic medium. In this case the current-density sources driving the dynamics of the mesoscopic system at 201 are located inside

VT.

parametrically by the fundamental field) and (ii) located inside the transverse current-density domain of the mesoscopic medium (see Fig. 4.1). In both of the above-mentioned examples the external (in the wide sense of the word) source field has a longitudinal component.

To establish the integral equations for the transverse and longitudinal parts of the local field in this case, let us take as a starting point Eq. (2.37). Since Ey1 = 0 in this equation under the assumption that there are no sources outside V,, this equation takes the form

E(r; w) = - ipOo s

GO(r - r’; w) - JTOTAL(r’; co) d3r’ . (4.10)

To emphasize the fact that the current density (at co) inside the mesoscopic system, named J TOTAL(r; a~), in the pr esent case has two contributions, viz. one (Jprescr(r; co) F Jext(r; co)) stemming from the prescribed (in the wide sense external) source dynamics and one (Jind(r;Co) E J(r; CL))) originating in the induced dynamics of the electrons of the mesoscopic system, I have used a slightly different notation from that of Eq. (2.37). By means of the division

JTOTAL(r; co) = Jext(r; co) + J(r; co) , (4.11)

the integral relation in Eq. (4.10) now can be written, leaving out again the reference to the frequency from the notation

E(r) = Eext(r) - ipOo s

Go(r - r’). J(r’)d3r’ , r E VT , (4.12)


where the driving field (in this context the wide-sense external field) is given by

.,X,(r) = - ipOo-) GO(r - r’).Jext(r’)d3r’ . s

(4.13)

The induced current density in Eq. (4.12) is the one which within the framework of linear response theory is given by Eq. (3.64). By dividing the induced current density into its transverse (Eq. (3.69)) and longitudinal (Eq. (3.70)) components, one readily obtains the following coupled integral equations for the transverse and longitudinal parts of the local field:

ET(r) = Eyt(r) - ip0w s

(Di(r - r’) + g-& - r’)).&‘!(r’,r”)

-Er”‘(r”) d3r” d3r’ - ipOco I

(D;f(v - r’) + g,(r - i))

- oyff (r', r”) - ET(r”) d3 r” d3 r’ ,

EL(r) = Er”‘(r) + & s orf(r,r’) - Ey’(r’)d3r’ 0

1 +V

l&g(“) s

oyf(r, r') . ET(r’) d3 r’ .

(4.14)

(4.15)

The division used above for the external field is achieved in the usual manner. Thus

Eyt(r) = - ipocu s

(Dz(r - r’) + g,(r - r’)) .P(r’) d3r’ (4.16)

E?‘(r) = - i,uoo s

g,(r - r’) ..F(r’) d3r’ = : (4.17)

By a comparison to Eqs. (4.2), and (4.4)-(4.6), it readily follows that the transverse part of the local field is given by

ET(r) = s

lYTT(r, r’) .EyLff(r’) d3 r’ , (4.18)

where

E i%(r) = E?;“‘(r)

- iyoo (Di(r - r’) + g,(r - r’)).o~~(r’,r”).E~‘(r”)d3r”d3r (4.19)


is the effective (eff) transverse driving field. Once the solution for ET(r) has been obtained, the longitudinal local field can be calculated from

EL(Y) = Ey&(r) + s rL&“,r’) *E+y&“‘) d3r’ ,

where

E T&(Y) = E?"'(r) + 1 1EoW s o~~(r,r’).E~t(y’)d3y’ .

(4.20)

(4.21)

In both the present case and in the one analysed in Section 4.1.1 the coupling between ET and EL in the relevant integral equations is a one-way coupling (from the transverse to the longitudinal field). In the fundamental theory this must necessarily be so, because the interaction of the electrons with the dynamical part of the scalar potential enters via the Coulomb-energy part of the Pauli Hamiltonian. In the Coulomb gauge the induced (plus free) part of the scalar potential is eliminated in favour of the particle position variables, so that only E yp’ enters. In contrast, the screened field ET = Ey’ + Eyd always enters the transverse electrodynamics. As a consequence of this asym- metry, the transverse local field appears in the integral equation for the longitudinal local field, whereas EL does not enter the integral equation for ET. This one-way coupling then forces us first to solve the integral equation for ET, which basically is the only one of the two integral equations which involves the self-consistency principle between the induced current density and the local field. Once the transverse problem has been solved in a self-consistent manner, the longitudinal part of the local field can be obtained directly, without any need of self-consistency. Hence, in a sense it is correct to say that the longitudinal local field is driven by the longitudinal part of the external (or possibly effective external) field plus the self-consistently determined transverse field. Before proceeding I would like to emphasize that the above-mentioned conclusion is based on the fact that it is the correct many-body wave functions that enter the constitutive relation. As we shall see below, the replacement of these wave functions by one-electron ones (a replacement almost always necessary to do in order to perform a numerical calculation) leads to a self-consistency problem for also the longitudinal part of the local field.

4.1.3. Longitudinal electrodynamics in the density-functional approach In a number of situations one expects that the electrodynamic properties of the mesoscopic

system under investigation are dominated by the longitudinal electrodynamics. In such cases the longitudinal local field, in principle, can be obtained from the expression

EL(r) = ET”‘(r) + & s o”;‘,“(r, Y’) - E;f;‘(r’) d3 r’ ,

0

(4.22)

cf. Eq. (4.15). However, when trying to obtain EL(r) from this equation we are confronted with the problem that the many-body wave functions needed in order to determine oF,“(~,r’) are seldom known, nor can be calculated numerically.

To establish an integral equation for the longitudinal local field within the framework of the density-functional theory, the expression for the appropriate single-particle potential Cp(r) in

130 0. Keller il Ph,vsics Reports 268 (IYY6) KS--262

Eq. (3.104) is inserted into Eq. (3.101). Hence, one has

cp(r) = - (P&) + s

&(r,+V~‘)d3r’ , (4.23)

where

s N(r’)d3r’

Ir - r’) (4.24)

is the prevailing local scalar potential. By replacing the exchange-correlation potential by the induced electron density by means of Eq. (3.102), and by making use of the Maxwell equation

V *(EL(r) - &x”‘(r)) = - (e/c,)N(r) ) (4.25)

Eq. (4.23) can be written in the form

q(r) = -: (x~~(Y,Y’) -fxc(y,r’))~‘.(EL(y’) - Ert(r’))d3r’ . (4.26)

An integration by parts of Eq. (4.26) followed by a gradient operation to obtain EL(r) = - V 4(r), finally leads to the following integral equation for the longitudinal local field:

EL(Y) = K?f,&)

+: s

[VV’(.&(r,r’) - ~~~(r,r’))] .EL(r’)d3r’ , (4.27)

where the effective driving field is given by

n

(4.28)

If one compares Eqs. (4.22) and (4.27) it appears that in going from a many-body to a one-electron approach in an attempt to calculateE,(r), the direct evaluation ofEL(r) from the right-hand side of Eq. (4.22) has been replaced by a self-consistent loop calculation of the local field (Eq. (4.27)). In other words, in the density-functional theory approach EL(r) is to be obtained from a loop calculation, just as ET(r) has to be calculated from a loop in the many-body case.

4.1.4. Random-phase-approximation loop In the RPA approach one assumes that the induced current-density response of the electrons

effectively is to the local field ET(r) + EL(r) instead of to the field ET(r) + Ep”‘(r), and that the corresponding conductivity response tensor is the one-particle aRpA(r, r’) discussed in Section 3.5.2, and not the many-body conductivity t~~~(r,r’) of Sections 3.3 and 3.4. In the RPA theory the constitutive equation in (3.64) is thus replaced by

J(r; co) = s

cKPA(r,r’; u) -E(r’; co) d3r’ . (4.29)

0. Keller / Physics Repom 268 (1996j 85-262 131

By inserting this relation into Eq. (2.37) one obtains the following integral equation for the total local field:

E(v) = E‘=‘(v) + I

KRpA(r,+E(&)d3r’ , (4.30)

with an RPA-kernel

KRPA(r,r’) = - ipOcr-, s

G& - Y”). crRpA(r”, 8) d3r” . (4.3 1)

The formal solution of Eq. (4.30) now can be established in an equivalent manner to that discussed in Section 4.1.1, if necessary. Within the framework of the RPA theory the starting point for a calculation of the local field in a mesoscopic medium thus is a self-consistent (integral equation) loop for the total field. In certain applications of the RPA model it is useful to divide the integral equation in (4.30) into two coupled ones, viz. one for ET(r) and one for EL(r), as we shall see in Parts B and C.

4.1.5. Extended densityTfunctional integral equation The density-functional approach discussed in Section 4.1.3 has turned out to be extremely useful

in low-frequency studies where the longitudinal fields dominate the electrodynamic properties. However, in the context of optics, and here the optics of mesoscopic systems, one cannot in general justify a neglect of the transverse dynamics. Furthermore, it is also known that the RPA approach described in Section 4.1.4 can lead to quantitatively wrong results in both linear and nonlinear optics of mesoscopic systems in cases where exchange-correlation effects in the electron-electron interaction play a role. In these situations it seems natural to replace the RPA constitutive equation in Eq. (4.29) by the relation

J(r; co) = s

o&r, r’; w) - (E(r’; co) + &Jr’; co)) d” r’ , (4.32)

where

E,,(r) = -l’&.(r) = - VfX,(Y,r’)N(r’)d3r’ (4.33)

is the exchange-correlation field, and <rKS(r,r’) is the Kohn-Sham conductivity tensor, to be obtained by inserting the KS (single-particle) eigenfunctions and eigenvalues in the one-body expressions for the dia and paramagnetic conductivities, and the spin conductivity, cf. Eqs. (3.45) (with (3.111)), (3.112), and (3.113). To determine E,,(r) one needs a calculation of the induced particle density N(r). At this stage, I assume that the contribution to the induced density from the “cross-coupling” term j(~yf(r,r’) -&jr’) d”r’ is negligible. If so assumed, N(r) can be obtained following the procedure outlined in Sections 3.5.1 and 4.1.3.

By inserting the constitutive relation of Eq. (4.32) into Eq. (2.37) one obtains what I shall call the extended density-functional (EDF) integral equation for the local field, i.e.

E(r) = Eg&(r) + s

K&r, r') - E(r') d3 1.’ , (4.34)

132 0. Keller j Physics Reports 268 (I 996) KS-262

where

K&r, r’) = - ipOcr, i

GO(r - r”) .cks(r”, r’) d3r” (4.35)

is the KohnSham kernel, and

E&-(r) = E’“‘(r)

- i/cOcfl i

G,,(r - r’) - cr&r’,r”) .&,(r”)d3r”d3r.’ (4.36)

is the effective driving field in the EDF approach. If the exchange-correlation potential, taken as a functional of the prevailing electron density, is Taylor expanded to first order around the equilibrium density, i.e.

cpXC(Nll + N) = 4”XC(N&)) + (a(pxclan)In=No(r)N(r;w) 3

one obtains the commonly used approximation

.fxc(r,r’;~) = (a(PxclaYt)ln=N,,(r~fi(r - r’) >

and hence the following formula for the xc-field

&&o) = - N(r;(~)B(a(pxclaY1)In=N,,(r) - (a(pxclaY1)I,,=No(r)~N(r;Q) .

Once the EDF integral equation has been established one can proceed

(4.37)

(4.38)

(4.39)

along the previously sketched lines to obtain the formal solution for E(r), to set-up the coupled integral equations for ET(r) and EL(r), etc.

4.2. External conductivity tensors

On the basis of the Pauli Hamiltonian in the presence of a prescribed electromagnetic field, called the external field in the wide sense understanding, we have established and discussed in Section 3 the general linear and nonlocal constitutive relation (see Eq. (3.64)). It appears from the many-body approach that the induced current density J(r; co) is obtained as the response to the sum of the longitudinal part of the external electric field and the prevailing (local) transverse field, Ey’(r; co) + E&;co). Since E,(r; co) is not known a priori a pre-knowledge of the external field distribution, Ey’(r; a) + Er”‘(r; co) in itself does not allow one to calculate J(r; co) from Eq. (3.64) alone. To determine ET(r;co), one has to combine the constitutive equation in (3.64) and the microscopic Maxwell equations as described in Section 4.1. Once the relevant (nonlocal) relation between ET(r; co) and Eext(r; co) has been established, the induced current density in Eq (3.64) can be obtained by a direct integration.

4.2.1. External many-body conductivity In the general case where the external field has both a transverse and a longitudinal part the

relation between I?&; co) and EF’(r; w) and Ey”(r; co) is given in Eq. (4.18), with Eq. (4.19) inserted.

0. Keller 1 Physics Reports 268 (1996) KS-262 133

Although this relation has been obtained under the assumption that the prescribed current sources were located inside the transverse current-density domain of the mesoscopic medium under study, it is obvious that the relation is correct also in the case where there is an additional prescribed source distribution outside Vr, provided that one extends the domain of integration in Eq. (4.13) to cover all regions where P’(r; w) # 0. To simplify the notation let us introduce the quantity

K&r, r’) = - i,uoo s

(DT(r - r”) + g,(r - r”)) - c$F(r”, r’) d3r” , (4.40)

cf. the analogous abbreviation in Eq. (4.4), and then rewrite Eq. (4.19) in the form

E ‘;-?kff(r) = E?-“‘(r) + s

K&,r’) .Er”‘(r’) d3r’ . (4.41)

By insertion of the above expression for the effective transverse driving field into Eq. (4.18). and in turn the resulting equation for ET(r;m) into the constitutive relation in Eq. (3.64) this becomes

J(r) = IS~&,~“) ’ r&r”,r’) Ey-“‘(r’) d3r”d3r’

+ s

IS~~(Y,~“). [rTT(r”,r”‘).KTL(r”‘,r’) + U&r” - r”)6(r” - r’)]

.E~X’(r’)d3r”‘d3r’rd3r’ . (4.42)

By making use of the division of (T MB given in Eq. (3.67) it is realized that the constitutive relation can be written in the compact form

J(r; w) = s

(r~~(r,r’;O).Eext(r’;w)d3r’ , (4.43)

where (omitting o from the notation)

oR;:(r, r’) = $f(r, r’) + oyF(r, r’)

+ s

cMB(r, r”) - r&r”, r”‘) - [&(r” - r’) + KTL(r”‘, r’)] d3 r”’ d3 1”’ , (4.44)

is the so-called external conductivity tensor. Albeit the fact that cgi(r,r’;o) has an extremely complicated structure, in principle it can be obtained from a knowledge of the many-body conductivity cMB (r, r’; co) and the transverse part, Di(r - r’; co) + g,(r - r’; w), of the electromagnetic vacuum propagator. However, from a conceptual point of view 0;; takes up an important position, and in cases where it can be calculated by approximate methods to a sufficient degree of accuracy it often simplifies the overall analysis.

134 0. Keller J Ph,vsics Reports 268 (1996) 85-262

To appreciate the conceptual importance of G gi, we transform Eq. (4.43) into the time domain, i.e.

J(r, t) = s

aE;(r, r’, t - t’) .lF(r’, t’) dt’ d3r’. (4.45)

Since the stimulus lF(r’, t’) necessarily preceeds the response J(v, t) one must have o~;(Y, Y’, r - t’) = 0 for t’ > t, or in other words, it follows that o~~(Y,Y’, t - t’) plays the role of a causal response function. Causality and linearity imply that o$;(Y, r’; (II) satisfies the Kramers-Kroenig relations.

4.2.2. Transverse stimulus and moment expansion of the response In the important case where the driving sources are located outside the transverse current-

density domain one has EeXt(v;~) = E~‘(~;LL)) so that

J(r) = s a~~((r,r’).E~‘(r’)d3r’ (4.46)

with

oggr, r’) = s cMB(r, r”). r&r”, r’) d3r” , (4.47)

cf. Eqs (4.42) and (4.44). In the optical regime, where the relevant wavelength of the external field, E;“‘(~;u), often but not always is large in comparison to the (relevant) extension of the system under consideration, it is natural to make a Taylor series expansion of EFt(r’) around the point r where the current density is sought for. Hence, by inserting

E.yyr’) = f ; [(i - r). lq”Eyyr) n=O .

(4.48)

into Eq. (4.46) one obtains

J(v) = do'(r)-ET'(r) + O(')(Y): VI??-"'(r) + ... ,

where

(4.49)

b(O)(r) = cMB(r, r”) - r-&r”, r’) d3r” d3rr (4.50)

is the local (zeroth-order) part of the external conductivity, and

&)(r) = s cMB(r, r”) - rTT(r”, r’)(r’ - r) d3 r” d3 r’ (4.5 1)

is the so-called near-local (first-order) part of 0;;. Though the long-wavelength expansion in Eq. (4.49) apparently leads to a simplification of the constitutive equation in (4.46) it should be


remembered that the nth r’-moment of the external conductivity, O’“‘(Y), is a tensor of rank y1 + 2. The moment expansion thus results in the introduction of response tensors of increasing rank, whereas the fully nonlocal conductivity tensor GE; is of the lowest (second) rank. The various moments of the external conductivity are of course interrelated, cf, Eqs. (4.50) and (4.51), but when it comes to comparisons with experimental data one often inserts fitting numbers for the relevant tensor elements. The increasing number of fitting parameters in practice makes it difficult to handle a moment expansion beyond first order.

From time to time physicists have been tempted to make the long-wavelength expansion directly in the constitutive relation in Eq. (3.64) (or in the RPA-version given in Eq. (4.29)) before inserting this into the Maxwell equations to close the loop for the local-field calculation. Such an approach can give rise to significant errors in the final result for several reasons. First of all, the presence of longitudinal external fields, originating in prescribed current-density sources inside the mesoscopic system, inevitably implies that the field components rapidly varying in space across Vr take part in the electrodynamics, and thus prevent us from making the moment expansion on <TMB (or oRPA). Secondly, even in the absence of Ey’, the transverse part of the local field, ET, might vary rapidly inside the mesoscopic domain. This occurs, e.g., in the context of optical (parametric) second- harmonic generation where the transverse part of the external driving field at the second-harmonic frequency, EF-“‘(r; 20), necessarily must vary significantly (at least in certain regions) inside V, since it is zero outside.

4.2.3. External RPA conductivity

In view of the importance of the simple RPA approach for a number of practical studies, c.f., e.g., a number of those discussed in Parts B and C of this article, let me finish this section by writing down the RPA expression for the external conductivity. In the RPA description ogi(v,r’) of Eq. (4.43) is replaced by

og;(r,r’) = s Q RPA(~, r”) - rRPA(r”, r’) d3 r” , (4.52)

where rRPA(r”,r’) is the field-field response tensor of the RPA model. Formally, FRPA can be obtained by solving the tensorial integral equation

rRPA(r, Y’) = Ud(r - r’) + I KRpA(r, r”) a rRPA(r”, r’) d3 r” , (4.53)

where the RPA-kernel is given in Eq. (4.31). Particularly in Part B, I shall make use of Eq. (4.52).

4.3. Coupled-antenna theory

To determine the prevailing electromagnetic field inside a medium one has to solve an appropriate set of coupled integral equations for the transverse and longitudinal parts of the local electric field as we have seen in Section 4.1. For a macroscopic medium the extremely large number of electronic levels participating in the dynamics usually makes it impossible to carry out the local-field calculation from first-principles. In a mesoscopic system, however, normally one can

136 0. Keller : Physics Reports 26X f 1996) 85-262

justify to keep only a limited number of levels in the analysis at hand. In such cases the dynamical calculation of the local field can often be carried out in a rigorous manner as I shall demonstrate below, and in Parts B and C.

4.3. I. Dominating paramagnetic response In many cases of interest it is possible to neglect the diamagnetic and spin contributions to the

many-body conductivity tensor. Generally speaking, the diamagnetic contribution is most important in media where the number of mobile electrons is large (e.g. in metals highly doped semiconductors, and superconductors), and/or at low frequencies (in the infrared region in the metallic case, e.g.). Even in cases where the overall contribution from the diamagnetic effect is considerable, it can often be neglected in comparison to the paramagnetic contribution if the system is excited at a frequency close to one of the participating inter-level transitions. Unless special arrangements are made (see Section 3.3.2) the contribution from the spinfield coupling is expected to be negligible.

To prepare for the calculation of the local field of a mesoscopic medium in the case where the paramagnetic coupling dominates, let us rewrite Eq. (3.51) (with Eq. (3.52)) as follows:

where

(4.55)

is the many-body transition current density from state IA) to state lB> (Yu, and Yflg being the associated many-electron wave functions), and the factor of two originates in the spin summation. For the sake of notational simplicity let us next write Eq. (4.54) in the form

cpara(r, r’) = 2 A,,(~)J,~,(v)J,,,(r’) (4.56) m,n

with

A,,(~) = (2i/(u) (P, - P,)/(tio + E, - E,) . (4.57)

In the many-body formalism the loop for the field is always for the transverse part of the local field, and independent of whether the source distribution is located inside or outside V, (or both), the integral-equation loop has the form (cf. Eqs. (4.2) and (4.14))

ET(r) = W)

- ipOo s

(DO(r - Y”) + g,(r - r”)).crF;a(r”,r’).ET(r’)d3r”d3r’ , (4.58)

where the effective driving field has been denoted by Et(r), and

+“(r, r’) = 2 &..(~)J;f,(W~,(r’) , m.n

(4.59)

0. Keller J Physim Reports 268 (I 996) 85-262 137

JTm(J’,,) denoting the transverse part of J,,(Jmn). By inserting Eq. (4.59) into Eq. (4.58) one obtains

ET(Y) = G(r) + ~Gl(~)Bfnn > (4.60) m,n

where

Fzm(r) = - i,uowA,,(w) s

(D;f(r - r’) + gT(r - r’)).Jzm(r’)d3r’ , (4.61)

Pmn = s

Ji,(r)‘ET(r)d3r. (4.62)

It appears from Eq. (4.60) that the deviation between the transverse parts of the local field and driving field, ET(r) - E;(r), is given by a linear superposition of (transverse) functions F;r,(r). For a given free Hamiltonian, H r, the transverse transition current densities J:,,,(r) belonging to the various pairs (n,m) of levels are in principle known (in practice they can be calculated only in simplified situations, of course). According to Eq. (4.61) this means that the spatial forms of the F;f,(r)‘s are known. So, to determine ET(r) - Ei( r one needs only to calculate the various fimn’s. ) The quantity Pm,, gives what one might call the strength of the FT,Jr)-mode. Since the individual pm,, depends on the so far unknown local field ET(r) through the integral in Eq. (4.62) the fimn’s need still to be determined. Before proceeding let me stress the fact that the transverse conductivity oyGa(r, r’) consists of a sum of tensor products, where in the individual tensor product, A,,*(o)J;r,(r)J~,(r’), the two vectors JTm(r) and J:,,(J) entering are functions only of r and r’, respectively. The remarkable circumstance that the tensor products of the paramagnetic conductivity are separable with respect to the two variables r and r’ has made possible a substantial simplification of the integral-equation problem for the transverse part of the local field.

Returning to Eqs. (4.60)-(4.62) it transpires that the prevailing trasverse field at r, ET(r) consists of the driving field, E!(r), plus the field stemming from the radiation accompanying the various electronic transitions. The effective current-density distribution giving rise to the radiation from the many-body transition y1 -+ m is A,,fi,,,nJ;fm(r’). The form, JL(r’),of this distribution is already known so what is left is to determine its strength A,,/&,,, or essentially fim,,. The problem of determining the Pm,,‘s from the integral equation of Eqs. (4.60) and (4.62) is readily converted to a matrix equation problem by inserting Eq. (4.60) into Eq. (4.62). Hence, one obtains

Li - 1 JC$:pnlBop = H,, > 0.P

where

(4.63)

No”,” = s

J’,,(r).FTo(r)d3r , (4.64)

H,, = s

J:,(r) SE?(r) d3 r . (4.65)

By letting the indices m and n run through the relevant possibilities, Eq. (4.63) gives a set of inhomogeneous, linear algebraic equations among the unknown Pmn’s. Conceptually, the set

138 0. Keller- :I Physics Reports 268 (I 996) 85-262

consists of infinitely many equations since the complete set of many-body energy eigenstates contains an infinite number of states (discrete plus continuous basis). For a mesoscopic system, however, it is often sufficient to take into account only a few energy eigenstates, and in consequence, the dimension of the matrix problem is reduced to a size where it might be handled numerically (or possibly analytically). Since flrnln = 0 for all 112 if the wave functions are assumed to be real, the set of equations in (4.63) is among k(k - l), in general complex, unknown Pln,,‘s if the number of relevant levels is k. Once the Pm,,‘s have been determined from Eq. (4.60), and the longitudinal part of the local field can be obtained from

o:f(r, r’) - E;(r’) d3r,’

(4.66)

cf. Eqs. (4.15) and (4.60). It is instructive at this point to apply the formalism developed in this subsection to the case

where the mesoscopic medium electronically can be considered as a two-level system. By denoting the lower and upper many-body quantum states by 11) and 12), respectively, Eq. (4.60) becomes

ET(Y) = G(r) + J%Qi* + JU%Li . (4.67)

The transverse field prevailing at an arbitrary space point r inside the mesoscopic medium thus equals the sum of the driving field (E:(r)) and the fields generated by the downwards (Fz,(r)P,J and upwards (FT2(r)B2i) transitions of the electron system. In the rotating-wave approximation (RWA) [24] the last term on the right-hand side of Eq. (4.67) is neglected. The RWA model tends to be accurate when the optical excitation frequency approaches to the electronic transition frequency, i.e w =(E2 - E,)/h, since the denominator of Ai2 (which occurs in FT,) is zero at resonance (damping effects neglected). However, one should bear in mind that close to resonance, other things being equal, it is easier to drive the two-level system into the nonlinear regime, where the linear response formalism fails to describe the dynamics. The field (propagating plus self-field parts) emitted in the downwards (12) + II)) and upwards (I 1) + 12)) transitions are accompanied by light absorbed in the opposite transitions, i.e. in 11) -+ 12) and 12) + 11 ), respectively. The amplitude strengths of these absorption processes are A12fi12 and AZ1bZ1, where

Pi2 = (l/D)IH,,(l - M) + &Cl > (4.68)

P21 = (lID)CH,,(l - N;) + fh2~::l > (4.69)

with

D = (1 - Ni<)(l - N;:) - N::N:: . (4.70)

It appears from Eq. (4.68) that the electronic upward-transition is stimulated by the coherent sum of the driving field (strength: Ai2Hlz/D), the field emitted in the 12) -+ 11) transition (strength: - A12H12Ns:/D), and the field emitted in the (1) -+ 12) transition (strength: A,,H,,N::/D), as shown schematically in Fig. 4.2. In this figure a graphical exposure is given of also the light

0. Keller/Physics Reports 268 (1996) 85--262 139

Fig. 4.2. Schematic diagrams illustrating the coupled-antenna theory for a mesoscopic two-level system. The lower and upper many-body energy eigenstates are denoted by 1 I) and I2), respectively. A black arrow indicates that the transition in question acts as an emitter (source) of the electromagnetic field, and a white arrow indicates that the transition in concern acts as an absorber of the field. The thin lines with arrows show the various emissionabsorption channels. It appears from the left part of the upper diagram that the 11) + 12) transition is stimulated by the sum of the external driving field (channel marked by a zero), the field emitted in the downward ( 12) + I 1)) transition, and the field emitted in the upward ( 11) + / 2)) transition. The right part of the upper diagram shows that the 12) + I 1) transition is driven by the sum of the external field, and the fields emitted in the down and upward transitions. In the lower diagram the rotating-wave approximation (RWA) is displayed. In this approximation the upward transition is driven by coherent

contributions from the external field and the field emitted in the downward transition.

absorption processes stimulating the electronic downward-transition (as described via Eq. (4.69)). In the RWA approach, the transverse field is given by

ET(Y) = G(r) + r_H,,l(l - Nmm9 3 (4.7 1)

and as shown in Fig. 4.2, the field absorption driving the upward transition consists of coherent contributions from the driving field (strength: Ai2Hr2/D) and the field emitted in the 12)+ 11) transition (strength: - AlzHlz Ni:/D). The sum of these contributions has a strength

A,,H,,I(l - N$). It is physically appealing at this stage to consider the pair (I 1 ), 12)) of states as equivalent to

a single antenna. Initially the driving field E!(r) induces current-density flows (Jlz and Jzl) in the antenna, and these flows in turn give rise to emission of electromagnetic fields. The emitted fields act back on the antenna and changes its current-density distribution. In a self-consistent description the prevailing local field is given by Eq. (4.67). The entire process thus can be considered as a classical radiation-reaction problem for a two-level system (dominated by paramagnetic interaction), and as such the process resembles the classical radiation reaction on an accelerated (point) particle, say electron [43], cf. the analysis in Section 7.2. The antenna point of view also is fruitful if more than two levels participate in the electrodynamics. Thus, in this case each pair (mn) of levels constitute One antenna. The current in a given antenna is now driven by the sum of the effective external field, the fields radiated by all other antennas, and the field radiated by the antenna itself (radiative reaction). The requirement of self-consistency in the entire process leads to Eq. (4.60) (with Eq. (4.62) inserted). A schematical illustration of the self-consistent antenna problem for a three-level mesoscopic medium is shown in Fig. 4.3.


-L_

l3> a ll>

Fig. 4.3. Schematic illustration of the coupled-antenna theory for a mesoscopic three-level system. Each pair ( ( 1) s 12). 11) F? /3), 12) P 13)) of levels is equivalent to one antenna, and the spatial form of the current-density distribution of the individual antennas is given by the upward and downward electronic transition current densities (indicated by the white arrows). As indicated by the black (double) arrows a given antenna is driven by the field emitted by the other two antennas, the field radiated by the antenna itself (radiative reaction), and the external field (not shown).

Our ability to perform a rigorous calculation of the local field inside a mesoscopic medium was, as described above, based on the special tensor product structure of the paramagnetic conductivity, and on the criterion that only a limited number of energy eigenstates needs to be incorporated in the dynamic analysis. Instead of using a superposition of the many-body transition-current densities to obtain ogi”(r, r’) (Eq. (4.54)) it is in principle possible to use a tensor-product structure based on the one-electron transtion-current densities (Eq. (3.51)). Which of these two alternatives should one use in practice? The answer to this question depends on the properties of the actual system under study. Thus, use of the many-body transition-current densities often would allow one to study the electrodynamics incorporating fewer levels than would be needed in the one-body approach. On the other hand, unless the number of movable electrons is low (not more than a few) it is not possible to calculate the many-body wave functions of the field-unperturbed system accurately. In cases where the longitudinal field dynamics dominates the interaction a good compromise is obtained using the transition-current densities of the density-functional theory. As we shall realize in the subsequent subsection this also enables one to incorporate the diamagnetic response in the coupled-antenna theory.

The separable-kernel technique studied in this section in the many-body case can readily be applied to an RPA calculation. Basically, the main difference lies in the fact that the coupled- antenna approach is related to the transverse fields in the many-body case and to the total fields in the RPA formulation.

4.3.2. Combined paru and diamagnetic response in the density-functional approach It is often necessary in mesoscopic electrodynamics to take into account also the diamagnetic

response. To a certain extent the coupled-antenna theory described in the previous subsection can be used to treat the combined para and diamagnetic response. Below, we shall see how this is done within the framework of the density-functional theory. In this theory only the longitudinal parts of the involved fields are considered, and hence, retardation effects are neglected in the inter and intra antenna-coupling approach.


As a starting point, let us consider the integral relation between the effective single-particle potential G(r) and the induced electron density N(r), i.e.

@(f-l = qf”‘(4 + SC Lc(r,r’) - 4nc

0 ,“, _ r,, ww3r . > (4.72)

To obtain Eq. (4.72) I inserted Eq. (3.102) into Eq. (3.101). Another relation between @ and N is

obtained inserting the Kohn-Sham response function xks(~,r’) of Eq. (3.100) into the constitutive relation in Eq. (3.99). Thus, with the notational changes k + m and k’ -+ ~1, and the abbreviation

%W((fi) = - 2e(f, -,L)/(~~ + snl - c,) ,

one has

(4.73)

Using the KS response function, x&r, v’), both the para and diamagnetic effect has been included, cf. Eq. (3.78). By combining Eqs. (4.72) and (4.74) one readily obtains an integral equation for 6 which basically has the same structure as that for ET (Eq. (4.60)), viz.,

G(r) = cp’“‘(~) + C &&)a,, , m,n

with the abbreviations

(4.75)

(4.76)

!?I mn = J h,dr)ti,*(r)@(r)d3r. (4.77)

As usual, the unknown constants u,, have to be determined from a set of linear algebraic equations, namely

a ,,,,, - c n,mPnaop = hmn , (4.78) 0.P

where

(4.79)

h,, = $drM,*(r)@Yr)d3r . s (4.80)

Once the cxmn’s have been found, the local potential, related to the induced particle density via Eq. (4.24) is given by

dr) = FV) - & c umn(~)u,, j ‘iy!?i” d3r' . m,n

(4.81)


The coupled-antenna theory may sometimes be extended to treat also the diamagnetic response. This extension is needed in cases where both the longitudinal and transverse part of the local field are important for understanding the electrodynamics of the mesoscopic system. Thus, let us briefly consider the one-electron KohnSham expression for the diamagnetic conductivity, viz.

cg(r,r’) = f-g 6(r - r’)U cfn 1 $n(r)12 . n

(4.82)

By means of the completeness theorem for the single-particle wave functions, i.e.

c ~~(r)~mV) = W - r’) > M

(4.83)

the diamagnetic conductivity can be put in separable form. If the diamagnetic coupling dominates, the local field in the extended density-functional approach is given by

E(r) = G&(~) + c F,(r) - Pm , (4.84) m

where

F,(r) = $ s

t,b;(r’)No(r’)Go(~ - r’)d3r’ , (4.85)

Pm = s

$,(r)E(r) d3 p . (4.86)

In the present case the yet unknown p,-vectors are obtained by solving the super-matrix-equation set

z[ U&, -

s i,MVX9d3r

1 s .p,, = tim(r)EF&(r)d3r . (4.87)

n

By a super-matrix equation we mean a matrix equation in which each “vector’‘-component in itself is a vector and each “matrix” element itself a matrix. In sofar it is a reasonable approximation to apply the completeness theorem to a truncated (few-level) system the local-field problem might be handled as described above.

4.3.3. Combined paramagnetic and spin response in the random-phase-approximation approach It is possible to extend the coupled-antenna theory to take into account spin effects on top of the

paramagnetic response. To illustrate this in a qualitative manner I shall indicate below how the calculation of the local field is performed within the framework of the RPA model.

In order to solve the RPA integral equation for the local field (Eq., (4.30)) it is adequate to rewrite the constitutive equation as follows:

ji\*““(r’) .E(r’) d3 r’

[ sj-N(r’) xE(r’)d’f] . (4.88)


Using this relation between J and E, the RPA-loop leads to a local field of the form (driving field

E”)

E(r) = E’(r) + c F tLAcE(~)Pmn + 1 CZN(r) x amn , (4.89) m,n m,n


Go(r - r’).jzLAcE(r’)d3r’ ,

plnn = s

ji\AcE(r).E(r)d3r, (4.92)

a mn = j::,‘,‘“(r) x E(r) d3 r . (4.93)

By making use of the vectorial relation A x (I? x C) = [BA - U(A .B)] . C, and with the introduction of the abbreviations

HSPACE _ mn - ji\ACE(r)SEo(r)d3r,

s

HS,“N _ - mn s

j:\“(r) x E’(r) d3r ,

Sy; = s

ji:AcE(r) - FEACE (r) d3 r ,

Vy; = i‘

jz:AcE(r) x FEN(r) d3r ,

WY; = s

j”,‘,‘“(r) xFpCE(y)d3r ,

T,“,” = [F~‘N(r)j~~*N(r) - U(j~~N(r).F~(~))] d3r ,

(4.94)

(4.95)

(4.96)

(4.97)

(4.98)

(4.99)

one obtains the following set of linear and inhomogeneous algebraic equations among the unknown a,,,,,‘s and b,,,,,‘s:

(4.100)

(4.101)

144 0. Keller i Ph_vsics Reports 268 (1996) 85-262

After a moment of contemplation. one realizes that the number of unknowns is 4N(N - 1) if the mesoscopic medium is considered as an N-level system. The dimension of the matrix in question hence is 16N2(N - 1)2. Thus, since a 16-level system, for instance, requires a solution of a 960 x 960-matrix problem, it is apparent that the coupled-antenna method suggested above to tackle the local-field problem in the presence of dynamic spin effects in practice is usable only for few-level systems. In the absence of the spin dynamics a calculation of the local field for an N-level system would require the solution of a matrix problem of dimension N2(N - 1)2. For comparison, 16 levels thus give a 240 x 240-matrix problem. It is important to notice that although the space and spin spaces give additive (noninterfering) contributions to the electronic conductivity, the radiations accompanying the space and spin dynamics interfere so that the local field finally obtained is not just the sum of the local fields accompanying the space and spin dynamics. In the context of Eq. (4.89) the amplitude strengths pm,, and amn of, respectively, the space

(I,,, CACE Bmn) and spin (c~.,,~%‘” x a,,) terms are determined from the coupled space and spin equations in (4.100) and (4.101).

4.4. Local-jeld resonance conditions

The electromagnetic frequency response of a (mesoscopic) medium often shows a resonance behaviour. In situations where the local-field corrections to the externally impressed field are negligible the resonance positions are direct fingerprints of the level structure of the system under study. Thus, in the many-body formalism, the electronic resonance condition reads, cf. Eq. (4.54)

fio+E,-E1=O, (4.102)

provided irreversible damping mechanisms are unimportant. If the irreversible damping cannot be neglected the individual denominators of Eq. (4.54) can no longer be exactly zero. However, for small dampings the various denominators can still exhibit pronounced minima, which depending on the magnitude of the damping will be more or less displaced from the minima described by Eq. (4.102). In cases where the local-field corrections are significant, the electronic resonance peaks are displaced and new peaks may appear in the experimental spectra. Some of the local-field resonances are commonly known, e.g. the polariton (T-mode) and plasmon (L-mode) resonances in bulk and in surface solid-state optics, and the plasma resonance in small spherical metal and semiconductor particles. In the present subsection I shall present a brief and general discussion of the resonance condition(s) for the local field, and in subsequent sections I shall demonstrate that the general resonance condition in a physically appealing manner leads to a rich variety of well-known resonances in appropriate limits, as well as to a number of new ones.

We have realized in Section 4.1 that in order to determine the local field in a many-body formulation one has to solve the integral equation

J!&+;(I)) = @(r;o) + s

KTT(~,y’;f~~).ET(~‘;~)d3r’ (4.103)

for the transverse part of the local field. Once ET(r) has been obtained, the longitudinal part of E(r) can be calculated by a direct integration. The condition for having a local-field resonance in the


mesoscopic system is tantamount to the requirement that a local field can exist in the absence of the effective driving field E:(v). Consequently, the spatial distribution of the transverse part of the local field @““(v), has to satisfy the homogeneous integral equation

[KTT(r, r’; co) - &Jr - r’)] - E$ES(r’; Q) d3 r’ = 0 (4.104)

at resonance (RES), with KTT(r, v’; (0) given by Eq. (4.4). It is important to realize at this point that the resonance condition in Eq. (4.104) can never be completely met experimentally. This is so, because in order to study the electrodynamical behaviour of our system it is necessary, after having decided how to define the system in respect to the surroundings, to excite the system, and afterwards probe its response. The excitation is done by means of @(r’) and the response is described via E,(r). Formally, the relation between the two fields is given by

ET(r) = [&(Y - r’) - KTT(r, r’)] - l &(r’) d3 r’ , (4.105)

where the so-called reciprocal kernel, [i&(r - i) - K-&,r’)]- ‘, of &(r - r’) - Krr(~,r’) satisfies the integral relation

[s,(r - r”) - K&r, r”)] - [h,(r” - r’) - K&“, i)] - 1 d3r” = &(Y - Y’) . (4.106)

Roughly speaking, the resonance condition, related to 6r - K rr, operationally appears as a pole condition on (6, - KTT)- ‘. To illustrate the resonance principle, let us consider the case where the paramagnetic response is the dominating one. At resonance the local field is given by

(4.107)

where the resonant values of the /$,n’~ are to be determined from

cf. Eqs. (4.60) and (4.63). It readily appears that to obtain a local-field resonance in this case the determinant (Det) of the system of homogeneous equations among the fiiE”s in (4.108) must be zero, i.e.

Det {&n.op - NY;‘,“4 ) = 0 . (4.109)

Once the possible values of the flkfs’ s have been determined, the accompanying resonance distributions for the transverse part of the local field can be obtained, to within a multiplicative factor, from Eq. (4.107). For a two-level mesoscopic system the resonance condition is

(1 - N:;)(l - N;:) - N::N:: = 0 ) (4.110)

146 0. Keller ! Physics Reports 268 (1996) 85-262

Fig. 4.4. Schematic illustration of the self-sustaining driving loop of a two-level system. As indicated in the top diagram, the upward transition (and also the downward transition) is driven by the sum of the fields emitted in the upward and downward transitions. In the RWA approach the upward transition is driven only by the downward transition and vice

versa, as indicated in the bottom diagram.

cf. Eq. (4.70). Although the Det( ... } = 0- con i ion was established for the case where the d t. paramagnetic interaction dominates it is obvious that in all cases where the coupled-antenna theory can be applied the resonance condition can be expressed as a null-condition for an appropriate determinant. From a physical point of view resonance is obtained if the electromagnetic field emitted from the prevailing current-density distribution of the mesoscopic medium upon absorption can create precisely the above-mentioned current-density distribution, cf. Eq. (4.103) with Et(r) = 0. This self-sustaining driving loop is illustrated schematically for a two-level system in Fig. 4.4.

Within the framework of the density-functional theory the resonance condition for the longitudinal local field takes the form

r,r’;co) - &(r,r’;co)) - tiL(r - r’) 3

.E:ES(r’;u)d3r’ = 0 : (4.111)

cf. Eq. (4.27). On the basis of the analysis in Section 4.3.2 it is realized (see Eq. (4.78)) that the resonance condition in Eq. (4.111) for the combined para and diamagnetic response is given by

Det (6,,, 0P - PI&‘:,“) = 0 . (4.112)

Once the resonance values of the CI,,‘s have been determined (to within a multiplicative factor) the local scalar potential can be obtained from

(4.113)

In the many-body and density-functional theories the resonance loops are for the transverse and longitudinal parts of the local field, respectively. In the extended density-functional approach the self-sustaining loop is for the total field, and the kernel involved is the KS-kernel, i.e.

s [KKS(r,r’;co) - 6(r - r’)U] .ERES(y’;w)d3r’ = 0 , (4.114)

0. Keller /Physics Reports 268 (I 996) KS-262 147

and in the RPA approach the resonance condition is as in Eq. (4.114) provided one makes the replacement KKS =s- KRPA.

The density-functional theory leads to the resonance condition in Eq. (4.111) for the longitudinal part of the local field. Since EL RES(r; co) is a nonretarded field it is of interest to set up also a resonance condition for the nonretarded (self-field) part of the transverse local field. Hence, by omitting the retarded part, D~(Y - r”), of the propagator from the expression for KTT(r,r’) (Eq. (4.4)) the resonance condition in Eq. (4.104) is reduced to the form

(4.115)

in the transverse self-field limit.

PART B. LINEAR MESO-ELECTRODYNAMICS

5. Local-field phenomena in quantum wells

In the present section the fundamental theory developed in Part A shall be used to study the linear electrodynamics of quantum-well systems. Instead of embarking on a comprehensive analysis of local-field phenomena in these systems, an analysis which would lead us far beyond the scope of this monograph, we shall concentrate our attention on a few selected examples which in an illustrative manner show the fingerprint of the role of local-field effects in quantum-well structures. Due to the fact that the electron motion in a quantum-well system is confined in the direction perpendicular to the plane of the quantum well the integral-equation problem for the local field, which in general involves all three spatial coordinates (r), is effectively reduced to a one-coordinate (say z) problem. The confinement gives rise to a pronounced discretization of the energy-level structure relevant for the particle motion across the quantum well. In turn this implies that the electrodynamics in the z-direction often may be treated within the framework of a few-level model. The field-induced motion of the electrons parallel to the quantum-well plane naturally resembles that of two-dimensional “bulk” systems. In a rigorous description of the electrodynamics of quantum wells it is usually necessary also to take into account the coupling between the atomic-like behaviour perpendicular to the quantum well and the two-dimensional bulk behaviour along the well plane.

In recent years the linear optical reflection properties of ultrathin metallic films deposited on metallic [44] and dielectric [45,46] substrates have been investigated experimentally by means of reflection spectroscopy [44,45] and ellipsometry [46]. In the metal on metal case, Cs overlayers on Ag with a half-, a full- and two-monolayer coverage were studied by measuring the changes in the s- and p-polarized reflectivity caused by the absorption of Cs [44]. In the paper by Alieva et al. [45], the linear optical properties of ultrathin Nb films deposited on crystalline quartz were investigated in the infrared frequency region. In situ ellipsometry measurements at 0.6328 urn were performed on a series of metal films (Au, Ag, Ni, Ru, Rh, Pa, Re, Nb, MO, W), all deposited on glass (BK7) substrates by Yamamoto and Namioka [46].

Also the optical properties of semiconductor quantum-well structures have recently attracted attention. It has been shown that quantum confinement of carriers leads to the existence of pronounced resonances in the optical excitation spectra. These resonances can be attributed to both conduction-to-valence band [47] and intersubband [48-50-J transitions.

Along with the above-mentioned studies experimental investigations of semiconductor multiple quantum wells and superlattice structures have been carried out [48,50-551.

To obtain a fundamental understanding of the elementary electronic resonance excitations in the semiconducting quantum-well structures the density-density correlation function was calculated within the framework of the RPA approach, and the dispersion relation of the collective excitations in turn was obtained [56-581. As we have seen, retardation effects associated with electromagnetic interaction phenomena are neglected in the above-mentioned approach. The experimental results demonstrating the strong optical absorption arising from an electronic transition from the ground state to the first excited state of GaAs/AlGaAs quantum wells [48,50-531 were analysed on the basis of calculations of the electronic states for a square-well potential both with [53,59] and without [48,50] electron-electron interactions taken into account. The optical absorption spectra of metallic quantum wells have been investigated theoretically by Silberberg and Sands [60]. In their calculation, the nonretarded electron-electron interaction is partially included by solving simultaneously the relevant SchrGdinger and Poisson equations, whereas retardation effects associated with transverse electromagnetic interactions are neglected. In a series of papers, Liu and the present author used the local-field formalism described in Part A of this monograph to study metallic [61-641 as well as semiconducting [65,66] quantum-well structures.

5.1. Diamagnetic electrodynamics qf a metallic quantum well

Let us consider a quantum-well system consisting of an ultrathin metallic film of thickness d placed on top of a dielectric substrate, and let us assume that the system exhibits translational invariance against infinitesimal displacements parallel to the film plane. Described in a Cartesian xllz-coordinate system the sharp surface of the dielectrics coincides with xq’-plane, and the substrate occupies the half-space z > 0. When the quantum-weil system is excited by a monochromatic plane wave of cyclic frequency LC) the local electric field inside the quantum well takes the form

E(r; w) = E(z; qll, cO)e’yll ” , (5.1)

where qll is the wave-vector component of the field parallel to the film plane, and E(z; qll, co) is the amplitude, which satisfies within the framework of the RPA approach (which we shall adopt here) the vectorial integral equation (see Eqs. (4.30) and (4.31))

- i/lloa GB(z,z”; q,l,co)-cr(z”,z’;q,l,u$E(z’;q,l,co)dz”dz’ , (5.2)

where ER(z;ql,,~o) = EB(z) is the background field, consisting of the incident field plus the field reflected by the vaccum/substrate surface in the absence of the quantum well, and

0. Keller ! Ph.vsics Reports 268 (I 996) KS-262 149

GB(z, z”; q ;, o) = GB(z, 2”) is the electromagnetic pseudo-vacuum propagator of the vacuum/substrate system. This propagator consists of three pieces, i.e.

GB(z, z’) = D;f(z - z’) + I(z + z’) + 0 z ’ 6(z - z’)eze; , (5.3)

where Di(z - z’) is the direct part, describing the free propagation of the field from z’ to z, I(z + z’) is the indirect part, describing the propagation from z’ to z via the surface, and (c,,/u)~ (5 (z - z’)eZeZ is the self-field part. For brevity, we shall denote the sum of the first two terms on the right-hand side of Eq. (5.3) by PB(z,z’). The explicit expressions for EB(z), Dz(z - z’), and I(z + z’) can be found in, e.g., [67]. If the optical diamagnetic response is the dominating one, as will often be the case for metallic quantum wells subjected to radiation having frequencies in the mid- and far-infrared parts of the electromagnetic spectrum, the s-polarized (here y-polarized) part of the local field fulfils the integral equation

E,(y) = E;(z) + a G;Jz, z’)n(z’)E,(z’) dz’ ,

and the p-polarized part the coupled set

E,(z) = E;(z) + a s

G;,(z,z’)n(z’)E,(z’) dz’

+ a s

GzJz, z’)n(z’)E=(z’) dz’ ,

E,(z) = E;(z) + a ?’

G&(z,z’)n(z’)E,(z’)dz

+ a G:Z(z,z’)n(z’)E,(z’) dz’ , s

(5.4)

(5.5)

(5.6)

where a = ~,,~oe%/[m(i + CM)], z being a (phenomenological) momentum relaxation time of the electrons. In the low-temperature limit (T + 0 K), the conduction electron density is given by

n(z)=+3(, F - h)(~F - En) 1 sl/&) 1 2 , n

(5.7)

where $,(z) and c,* are, respectively, the energy eigenfunction and eigenvalue of state number 11, i;F is the Fermi energy, and 0 is the Heaviside unit step function.

Limiting ourselves to studies of thin (todjc, 4 1) quantum wells, the background field and the propagator G,B will vary slowly across the well. This implies that the s-polarized part of the local field in lowest order is constant across the quantum well, and thus is given by

E,(O) = E;(O)l(I - K,,) , (5.8)

150 0. Keller i PI~wics Reports 268 (1996) 85-262

with

K,.,. = ~~oe2tor(l + rs)N+d/[2mqy(ioz - l)] . (5.9)

In Eq. (5.9), rs and N, denote the s-polarized vacuum/substrate amplitude reflection coefficient, and the concentration of the ions of the quantum well. The quantity & = [(co/c~)~ - &J1” is the wave-vector component of the incident field perpendicular to the film plane. The local-field correction to the background field hence is given by the factor (1 - KJ ‘. Usually, 1 KY,/ $ 1 so that the local-field correction is small for s-polarized excitations.

To obtain the p-polarized part of the local field from the coupled integral equations in (5.5) and (5.6) one normally has to resort to numerical methods. However, in some cases the solution can be obtained using an approximation I have named the slave approximation. In this approximation the term a j Gz,(z, _ 7’)n(z’)E,(z’)dz’ in Eq. (5.5) is neglected. Physically this means that the z- component of the local field, and thus the z-component of the driving background field does not influence the electron dynamics along the film plane. In the diamagnetic case, where the conductivity tensor is diagonal, the cross coupling between the x- and z-dynamics solely originates in a radiative cross-coupling (via G:! and GzX). Neglecting the radiative coupling from x to z, the x-component of the local field inside the quantum well is to be obtained from the integral equation

E,(Z) = E:(Z) + u i

G~.~(z,z’)n(z’)E,(z’)dz’ . (5.10)

Within the framework of the slave approximation we have thus obtained an integral equation for E,(z) which in form is identical to that for E,(z) (compare Eqs. (5.4) and (5.10)). Utilizing the slow variation of Ez and Gf, across the quantum well, E, becomes in lowest order independent of z and hence given by

L(O) = fiY(O)l(I - KXX) 3 (5.11)

with

K,, = e2z&l - r,)N+d/[2mcoo(icti~ - l)] , (5.12)

I’~ being the p-polarized amplitude reflection coefficient of the vaccum/substrate surface. The local-field correction factor for the x-component of the field thus is (1 - KJ ‘. As we shall see in Section 5.3, it is possible in the optical region to achieve a resonant enhancement of the local field in the x-direction, but usually not in the lj-direction.

By inserting Eq. (5.11) into Eq. (5.6), and taking slowly varying quantities outside the integrals, it is realized that the z-component of the local field in the quantum well satisfies the integral equation

[l - (c&~)~Iz(z)]&(z) = EEff + aP:JO,O) s

n(z’)E,(z’)dz’ , (5.13)

where

E;‘f(z) = E;(o) + l”“fp s Gf&, z’)n(z’) dz’ X.X

(5.14)

0. Keller i Physics Reports 268 (1996) 85-262 151

c., considered as an effective (eff) background field driving the z-component of the local field. Since G&(z, z’) contains step functions of the type O(z - z’), EEff(z) usually cannot be considered as a slowly varying function of z inside the well. The slave approximation is so named because the dynamics in the z-direction is essentially enslaved by the dynamics in the quantum-well plane when the term as Gf,(z, z’)n(z’)E,(z’) d z’ is omitted. Situations where the dynamics perpendicular to the well-plane enslaves the x-direction dynamics can also occur. The solution of Eq. (5.13) has the form

E,(z) = [E:ff(z) + NaP::(O,o)]/[l - a(c&)‘n(z)] )

where

(5.15)

1 N=

~(z)E,‘~~(z) dz

1 - K,z 1 - a(c&)2n(z) .

As we shall realize in Section 5.3, where local-field resonances are discussed, the quantity

K,, = UPfJO, 0) s n(z) dz

1 - a(c&)%z(z)

(5.16)

(5.17)

plays a role for the local-field dynamics perpendicular to the quantum-well plane, which is equivalent to the roles played by K,, and K,, for the in-plane dynamics.

In the so-called self-field approximation, only the self-field part of the electromagnetic propagator is kept. In this approximation Eqs. (5.5) and (5.6) decouple, and one immediately finds

E,(z) = E:(O) > (5.18)

and

E,(z) = E:(O)/[ 1 - Use] , (5.19)

The self-field approximation used previously (see [27], and references therein) in studies of the optical reflection from metal surfaces having a smooth electron-density profile, is less general than the slave approximation, and is normally too restrictive to be able to account for the experimental infrared-reflection data, cf. [SS]. One can compare the slave and self-field models in a physically appealing way by means of the electric displacement field inside the quantum well. Thus, since the dielectric tensor, E(Z, z’; q,,, o) = E(Z, z’), of the quantum well is isotropic, the normal (z) component of the electric displacement field, D,(z), is related to the normal component of the electric field by the constitutive relation

DJZ) = Eo c,,(z, z’)E,(z’) dz’ . (5.20)

Utilizing the fact that c,,(z) is related to n(z) via

E==(z,z’) = 6(z - z’) [l + ie’rn(z)/c,mco(l - icon)] , (5.21)

the normal component of the electric displacement field is given by (cf. Eqs. (5.19))(5.21))

D=(z) = coE:(0) (5.22)

152 0. Keller I Physics Reports 268 (1996) KS-262

in the self-field (SF) approximation. In this approximation, the z-component of the electric displacement field thus is constant across the quantum well. In the slave approximation, it follows from Eq. (5.13) upon neglect of the usually small term proportional to P!=(O, 0) that

D,(z) -E&ff(Z) . (5.23)

In an obvious notation, the variation of D,(z) across the well thus is given by

Dz(z)(s,ave - D,(z)IsF = coa 1 YE). jG:X(z,z’)~(~‘) dz’ xx

(5.24)

To illustrate in a qualitative manner the local-field effect in the case of dominating diamagnetic coupling, I shall present a few results for metallic quantum wells of niobium deposited on crystalline quartz substrates. In the spectral range around 10 pm the diamagnetic effect dominates and the quartz substrate exhibits a pronounced resonance, a favourable feature for optical reflection studies as we shall see. The standard Lorentz oscillator expression

(5.25)

is used to represent the dielectric constant of quartz, inserting the appropriate values [45,68]. For simplicity it is assumed that the conduction electrons are confined in a square-well potential of finite height. In Fig. 5.1 is shown, for different film thicknesses, the z-component of the normalized p-polarized local field as a function of the position across the Nb quantum well. Since in the present case the relative deviation ((exact-slave)/exact) between the “exact” calculation and that obtained on the basis of the slave model is always less than 10P3, the slave model is extremely good. It

-0.5’ I I -1.00 -0.98 -0.96 -0.94

zjtl

-0.06 -0.04 -0.02 -000

Z/d

Fig. 5.1. The z-component of the normalized p-polarized local field, E,(z)/&, as a function of the position (given in terms of z/d) across the Nb quantum well for three different film thicknesses, viz. (1) 18, (2) 12, and (3) 6 A. The real and imaginary parts of the local field are represented by the solid and dashed lines, respectively. The frequency of light is 1.2 x 10” cm ‘, and the electron collision frequency is I .O x 10“ cm- I.

0. Keller j Physics Reports 268 (I 996) 85-262 153

appears from Fig. 5.1 that the z-component of the local field varies dramatically with z only in narrow regions near the edges of the potential well. In the main part of the quantum well the local field is predicted to be almost zero, in the RPA approach at least. A comparison of the slave and self-field approximations for E=(z) is presented in Fig. 5.2, and in Fig. 5.3, the variation of the z-component of the electric displacement field across the well is shown.

Im \

--

Re \ 7 \

Fig. 5.2. Real and imaginary parts of the z component of the normalized p-pojarized local field, E,(Z)/&, as a function of the normalized position, z/d, across a Nb quantum well of thickness rl = 18 A. The solid and dashed lines represent the results obtained on the basis of the slave and self-field approximations, respectively. The electron collision frequency and the wavelength of the light is the same as used in Fig. 5.1.

-0.55 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

Z/d

Fig. 5.3. The z-component of the normalized local electric displacement field, DZ(z)!(F,,EO),Oas a function of the position, z/d, across the Nb quantum well for three film thicknesses, namely, (1) 18, (2) 12, and (3) 6 A. The solid and dashed lines correspond to the real and imaginary parts of the local field, respectively. Remaining input data are as in Fig. 5.1.

Once the local field inside the quantum well has been obtained the field can be calculated everywhere in space by means of a direct integration using the appropriate electromagnetic Green’s function, cf. the description presented in Section 2. In turn, quantities, such as the s- and p-polarized reflection and transmission coefficients, which are directly accessible to experimental tests can be calculated. As an example is shown in Fig. 5.4 experimentally and theoretically data for the energy reflection coefficient for s-polarized light as a function of frequency for the Nb/quartz system in the vicinity of the quartz resonance.

5.2. Paramagnetic electrodynamics of metallic and semiconducting quantum wells

Let us turn the attention now towards a situation where the electrodynamic interaction is dominated by paramagnetic effects. As a starting point for the analysis we take the RPA integral equation in Eq. (5.2) inserting the appropriate expression for the paramagnetic conductivity tensor. Limiting ourselves to optical studies, it is sufficient to calculate the paramagnetic conductivity tensor in the long-wavelength (y,, + 0) limit. In this limit only the diagonal elements of the conductivity tensor are different from zero, and given by [63]

2ir2(kBT)2 Xti2to c @(&l - &n)

(&l - &l)CF(&J - ~(&I)1 = m.n [h(o + i/r,n,)]2 - (c, - c,)’ ‘mn(Z)9mn(Z’)’

(5.26)

= gg c O(c, - c;,,) (cl - G?JC~hl) - ~(~,)I

m.n [ti(to + i/rMn)]” - (c,, - z,,)~ @M(~)@~‘(~‘) ’ (5.27)

assuming free-electron dynamics along the quantum well. For brevity we have introduced the quantities

F(c) = xd.u

1 + exp[(c - /l)/(k,T)]e” ’ (5.28)

H(C) = In {l + exp[(/l - c)/(knT)]} , (5.29)

&n(~) = $M(z)$,,(z) and Qm,,(z) = $,(z)d$,(z)/dz - $n(z)d$m(z)/dz. The quantity T,,,~ is the lifetime associated with intersubband transition between the stationary states m and ~1. The chemical potential, p, of the electron system is to be derived from the global charge neutrality condition

$ c H(c,) = N+d. II

(5.30)


0.8

i

A-‘(cm’) Fig. 5.4. The energy reflection coefficient for s-polarized light, I?,, as a function of the reciprocal light waveltngth, i,- I, for a bare quartz substrate (0) and for a quartz substrate covered with Nb films of the following thicknesses (in A): (1) 3, (2)

12, (3) 25, and (4) 50. The points (with various symbols) represent the experimental data of Alieva et al. 1451, and the solid lines shows the results of the theoretical calculations of Liu and the present author. [64].

Now, let us focus our attention on the case where the incident field is p-polarized. By inserting the explicit expressions for the conductivity tensor elements given in Eqs. (5.26) and (5.27) into Eq. (5.2) the coupled-antenna formalism procedure described in Section 4.3 shows that the local field inside the quantum well takes the form

E(z) = EB(4 + c Pm,,(z) *rm, > (5.3 1) m,n

where the r,,-vectors have to be determined from the following set of algebraic equations:

rmn - c Knn.,~,~ - rmw = Km . (5.32) m’,n’

The quantities S,, and Km,,,8,f are given by

K mn.m’n’ = J &Jz)P$;!(z) dz s &,n(~)P$;!(~) dz

> J @,,(z)P:;!(z) dz 1 @,,(z)Pz:(z) dz ’

(5.33)

(5.34)

156 0. Keller / Ph,vsics Reports 26X (1996) KS-262

where

Pm&) = i

P::‘(z) PrJIxnZ)(z) P:;‘(z) P::‘(z)

= arm l G,B,(z, z’) b&‘W b,,, i G,B,(z, z’) @mn(z’)dz’ an 1 G,B,(z, z’) 4,dz’)dz b,,,, J GZB,(z, 2’) @mn(z’)dz’ > ’

(5.35)

The rather lengthy expressions for the (space-independent) quantities umn and h,, appearing in Eqs. (5.35) can be found in [63]. Once the local field in Eq. (5.31) has been obtained it is a straightforward matter to calculate the p-polarized amplitude reflection and transmission coefficients, see Ref. [63]. To illustrate in a qualitative manner the local-field theory for the paramagnetic coupling 1 shown in Fig. 5.5 the energy reflection (R,) and transmission (T,)

0.08

0.06

CT 0.04

0.02

0.80 -

0.60 -

! 2.7 3.2 3.7 4.2

Photon energy (eV)

Fig. 5.5. Energy reflection (R,) and transmission (T,) coefficients as well as the absorptance (A,) of a Nb quantum well as a function of the photon energy for 3, 4, and 5 monolayer (ML) thick wells. After Ref. [63].


coefficients, and the absorptance (A,) of a superthin niobium film embedded in fused quartz. The electromagnetic propagator relevant to this case can be obtained from Eq. (5.3) by leaving out the indirect term, and by replacing (at the appropriate places) the vacuum speed of light (co) by the speed of light in the dielectrics (c). Details of the numerical calculation as well as the values of the relevant parameters can be found in [63]. In Fig. 5.5 results are presented for three film thicknesses, viz. 3,4, and 5 monolayers, and the angle of incidence of the light is 70’. In the view of the result of Fig. 5.5, it is of interest to notice that the reflection, transmission, and absorption spectra (for each thickness) exhibit only one resonance peak in the frequency range 2.2-4.2 eV, although the electromagnetic field certainly excites more than one electronic transition in this range. The resonance appearing in the optical spectrum results from the collective excitation of the entire quantum-well system, and the dynamic screening effect of the electrons plays an important role for the character of the resonance, which one should refer to as a local-field resonance. By a comparison of the spectra belonging to different thicknesses, it is seen that the resonance excitation energy decreases when the film thickness increases. The strong thickness dependence of the resonant optical transition frequency indicates that quantum size effects in the niobium/fused quartz system are pronounced. In passing, it is worth mentioning that although many-electron effects are ignored in the calculation shown in Fig. 5.5 an inclusion of these effects does not change the theoretical predictions from a qualitatived point of view. The electron-electron interaction only modifies the unscreened energy eigenvalues and wave functions of the system in the absence of the optical field. In consequence, this modification leads to an extra shift of the resonance peak location in the spectra. To describe quantitatively the optical paramagnetic response of metallic quantum wells, it is necessary to take into account both electrostatic and dynamic screening effects.

Let me finish this section by a brief and heuristic discussion of the local-field effects associated with the optical paramagnetic response in a two-level quantum well, and let us focus our attention on the p-polarized case. In the s-polarized case the local-field effects are much smaller and thus we omit discussing this case here. The relevant coupled integral equations for the x- and z-components of the local field are in this case (in a slightly different (and compact) notation than used above)

E,(z) = E:(z) + F,,(z) s

$(z”)E,(z”) dz” + F,,(z) s

@(z”)E,(z”)dz” (5.36)

E=(Z) = E:(Z) + FJZ) s

$(z”)E,(z”) dz” + F,,(z) s

@(z”)E,(z”)dz” > (5.37)

where

al G,,(z,z’)4(z’)dz’ hl G,,(z,z’)@(z’)dz’

> aj G,,(z,z’)&z’)dz’ bJ G,,(z,z’)@(z’)dz’ ’ (5.38)

with 4(u) = $1(u)$2(u), and Q(u) = $l(~)d$Z(u)/du - $z(LL)dtjl(u)/du. Above $I and $2 denote the (real) wave functions of the lower and upper state, respectively, and a and b are space- independent constants, the explicit expressions of which can be found in [65]. Using for heuristic purposes infinite-barrier wave functions and energies it is a straightforward matter to obtain analytical expressions for the local-field components in Eqs. (5.36) and (5.37), see Ref. [65]. In turn,

15x 0. Keller / Ph_vsics Reports 268 (1996) K5--262

these “exact” expressions can then be compared to those obtained via simplified schemes and commonly used [65]. Firstly, I consider the self-field (SF) approximation. In this approximation only the self-field part, (c/u)2CS(z - z’)eZeZ, of the propagator is kept. From Eq. (5.36) one immediately finds that the x-component of the local field equals the background field, i.e.

E;F(z) = E_!(z) . (5.39)

By means of the coupled-antenna procedure one readily realizes that the z-component of the field is

E;“(z) = E!(z) + b(c/oJ)2 Q(z) s @(z)E:(z)dz

1 - (c,‘co)~ s Q2(z) dz . (5.40)

Mainly due to the fact that the term proportional to F,,(z) in Eq. (5.37) is neglected, the self-field approximation does not in general predict the correct form of the local-field variation across the quantum well. The strength of the local-field correction to the background field normally also will be quite different from that obtained in the exact theory. Secondly, let us consider the first Born (1B) approximation. In this approximation one replaces E,(z”) and EZ(z”) on the right-hand side of Eqs. (5.36) and (5.37) by E:(z”) and Ef(z”). The exact theory and the lB-approximation give the same spatial form of the local-field variation in the quantum well, but the strengths of the corrections usually are quite different. Thirdly, let us consider the so-called self-field first Born (SF-1B) approximation. In this approximation one takes advantage of the facts that (i) use of the first Born approximation implies that the correct,fir.m of the local-field correction is obtained, and (ii) the self-field approximation takes into account the main contribution to the often dominating z-component of the local field. Using the SF-1B approximation one would expect to obtain results close to those obtained from the “exact” calculation. In the self-field first Born approximation one thus replaces E,(z”) and E,(z”) under the integral signs in Eqs. (5.36) and (5.37) by Ef(z”) and EiF(z”). The only difference between the 1 B and SF-l B approximations hence is the strength of the local-field correction perpendicular to the plane of the quantum well. The numerical results presented below demonstrates that this difference can lead to quite different predictions for the local field.

To illustrate the heuristic findings described above let me present a few numerical results for the important semiconducting AlGaAs/GaAs/AlGaAs quantum-well structure. The parameters used in the calculations described below (as well as a number of details) can be found in Ref. [65]. In Fig. 5.6 is shown the real and imaginary parts of the local-field correction to the background field as obtained in the exact two-level theory as a function of the position across the quantum well. The results are presented for three different optical frequencies, viz. those given indirectly as tiw/(~~ - cl) = 1.1, 1.2, and 1.3. It appears from these figures that the x-component of the local-field correction is small, whereas the correction of the z-component is large. The local-field correction clearly depends on the frequency of light, and in Fig. 5.7 this dependence is shown in a frequency range above the electronic transition frequency. It is seen from Fig. 5.7 that there exists a pronounced resonance in the frequency spectrum of the local field, and it is of interest to note that the resonance excitation energy does not equal the energy separation between the upper and lower electronic level, but lies somewhat above the energy spacing of the two levels. Following up on the discussion of the three important approximations, namely the SF, lB, and SF-1B approximations,

0. Keller / PhIsirs Reports 268 (I 996) 85-262 159

3

I

2

1

40 GO IL 4, -1 N I -2

g -3

2 -4

0.0 0.2 0.4 0.6 0.6 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Z/d - z/d -

Fig. 5.6. Real (a) and imaginary (b) parts of the Y (figure to the left) and z (figure to the right) components of the normalized p-polarized local-field correction to the background field as a function of the normalized position, z;‘d, across a two-level GaAs quantum well for three (normalized) photon energies, namely ho)l(~~ - I:~) = (I) 1.1, (2) 1.2, and (3) 1.3.

After Ref. [65].

the local-field differences EZ - EzF and E, - EfFP lB are shown on normalized form as a function of the z-coordinate for three different photon energies (the same as in Fig. 5.6) in Fig. 5.X. The difference E, - EiB has not been plotted because the first Born approximation leads to extremely inaccurate results when applied to the z-dynamics (see Ref. [65]). It appears from Fig. 5.8 that the SF-1B approximation only improves the pure SF results slightly. As far as the x-component of the local-field correction is concerned it appears from Fig. 5.9 that the first Born approximation gives a fairly correct result for the local field across the quantum well, and if needed extremely accurate results can be obtained using the SF-1B approximation.

5.3. Local:field eigenmodes

The general criteria for obtaining self-sustaining frequency oscillations in the local field were discussed in Section 4.4, and special emphasis was devoted to an analysis of the local-field resonance condition in the case where the paramagnetic coupling dominates. In this section the

160 0. Keller / Ph_vsics Reports 268 (I 996) KS-262

t

0.03

0.02

TJ

> 0.01 c_ cr;

' 0.00 14

-0.01

-O.O?

----------

-. I

*\ ’

\ I .

1.0 1.1 1.2 1.3 1.4 1.5

FLW/(f~ - F,) -

Fig. 5.7. The x- and z-components of the normalized local-field correction to the background field as a function of the normalized photon energy for the GaAs quantum well of Fig. 5.6. The field has been calculated at the position z = 0.2d, and the real and imaginary parts of the local-field correction are represented by the solid and dashed lines, respectively. After Ref. 1651.

considerations of Section 4.4 are complemented, and applied to a study of the resonance condition in metallic and semiconducting quantum wells dominated by the diamagnetic response.

For thin (wd/cO 4 1) quantum wells, where advantage can be taken of the slow variation across the well of the background field and certain components of the electromagnetic propagator, the resonance condition for the s-polarized part of the field may readily be obtained. Thus, by taking the limit E:(O) + 0, Eq. (5.8) shows that the resonance condition is

&&,,,4 = 1 3 (5.41)


t

0.07

0.05

=:o h

F 0.03 6

1

y" 0.01 z E + -0.01 ' I

0.0 0.2 0.4 0.6 0.6 1.0 I/d -

3

A (a)

c -o.ol: 1

t 0.07 I

h 0.05 2.7 i 9

7 z,

0.03

h

-0.01 ’ I 0.0 0.2 0.4 0.6 0.8 1.0

z/d -

Fig, 5.8. Real (Re) and imaginary (Im) parts of the normalized local-field differences E; - Es” and E, - E:Fm lB plotted

as functions of the position across a two-level GaAs quantum well for three different photon energies (normalized to the transition energy), i.e. hcr@(~;~ - el) = (1) 1.1, (2) 1.2, and (3) 1.3. After Ref. [65].

a result which of course displays the criterion for being able to obtain a nonzero solution to the homogeneous part of the integral equation in (5.4). Usually, the resonance condition in Eq. (5.41) can be fulfilled only for modes having q,,-values substantially larger than the vacuum wave number of light. In fact, the relevant wave numbers are often comparable to the electron Fermi wave number, and thus of less interest in the present context. If one adopts the slave approximation, the resonance condition for the p-polarized case splits into two, namely one for the x-component of the field (see Eq. (5.11))

K&9 Q) = 1 1

and one for the z-component (see Eqs. (5.15)-(5.17))

(5.42)

Al = 1 . (5.43)

It is important to emphasize here that the local-field resonance in the z-component of the field, as well as in the two other components, is a distributed resonance occurring simultaneously all over space. Another type of resonance, i.e. a spatially localized resonance, may occur at a certain depth

162 0. Keller i’ Physics Reports 268 (I 996) 85-262

-0.014 ’ I 0.0 0.2 0.4 0.6 0.8 1.0

z/d -

Fig. 5.9. Real (Re) and imaginary (Im) parts of the normalized local-field differences E, - E_JB and E, - EzFmlB as functions of the position across a two-level GaAs quantum well for the same three photon energies as used in Fig. 5.8.

After Ref. [65].

z = z. inside the quantum well if the condition

a(c&~)2n(zo) = 1 (5.44)

is fulfilled, cf. Eq. (5.15). This type of resonance usually does not cause a pronounced resonance in the p-polarized diamagnetic reflection coefficient because E,(z) only appears in rP via some weighted integral over the quantum well, cf. the discussion in Ref. [64]. By inserting the explicit expression for a, Eq. (5.44) can be rewritten in the form

1 - w,‘(z~)/[o((L) + i/z)] = 0 , (5.45)

where We = [~(z~)e’/(m~~)] 1/2 is the so-called plasma frequency. The relation in Eq. (5.45) can be recognized as the resonance condition for excitation of localized plasmons. Before proceeding I emphasize that the localized plasmon resonance and the local-field resonance are two different concepts, a conclusion which is apparent from the discussion above. In the literature these concepts are often mixed up.

Before taking a closer look on the resonance condition for p-polarized light in the slave approximation, let us generalize our model a bit so that it can comprehend an extra (local and

0. Keller / Phjxics Reports 268 (1996) 85-262 163

isotropic) contribution to the electrodynamic response. This contribution, which may originate in ionic excitations, vertical interband transitions, etc., is described by means of a loal dielectric constant C,,(U). As the relevant dielectric tensor we thus take

E(z,z’) = I:, U1s(z - z’) + (i/cO(o)bdia(z,z’) , (5.46)

or equivalently

E(Z, z’) = U6(z - z’) + (i/coto)a$~(z, z’) , (5.47)

where

d$F(z,z’) = (e2z/[m*(1 - itnz)]j neff(z)6(z - z’)U (5.48)

is the effective diamagnetic conductivity tensor. The inclusion of the extra contribution to E(z,z’) implies that the density of the mobile electrons is reduced to the effective (screened) value

n,rr(z) = M(Z) + (&))2(1 - Em)/U (5.49)

with

a = /10e2c0z/[m*(i + or)] . (5.50)

Also we have replaced the free-electron mass by an effective one, denoted by m*, above. By replacing n(z) by neff(z) in Eq. (5.10) one an easily derive the following resonance condition for the x--component of the local field:

(5.51)

where

N:if (CD) = id [c, - 1 - (c.~/u)~,N+] . (5.52)

In Eq. (5.51), Ed is the appropriate relative dielectric constant of the substrate, and Kl = [(O/C())2CI - @l/2 is the wave-vector component perpendicular to the plane of the quantum well inside the substrate. It is adequate to write the resonance condition for the local field in the form given in Eq. (5.51). Thus, in the limit d + 0, Eq. (5.51) is reduced to qyc, + tiL = 0, or equivalently

q,, = $J(l + cl)l':' . (5.53)

The expression in Eq. (5.53) is just the well-known dispersion relation for electromagnetic surface waves on the bare vacuum/substrate surface. In the presence of the quantum well, Eq. (5.51) hence is the dispersion relation for guided electromagnetic waves in the vacuum/quantum well/substrate system. Ifc, = 1, Eq. (5.51) can readily be shown to be identical to Eq. (5.42). In conclusion we have thus realized that the local-field resonance condition for the dynamics parallel to the quantum well is identical to the eigenmode condition (guided wave dispersion relation) for guided waves on the structure. As a practical example is shown in Fig. 5.10 the real and imaginary parts of the

I64 0. Keller: Pigsics Reports .?hX (1996) KS262

1180 r

1160

:

1140/-

g .. 1120- _- <

llOO-

1080-

1060 I I I , 1 I 0.8 10 12 14 1.6 1.8 2.0

XP((‘I,(I,l/~/

Fig. 5.10. Real (Re) and imaginary (Im) parts of the normalized wave number, cOq, /w, of an electromagnetic surface wave as a function of the reciprocal wavelength, iL- ‘, on a bare quartz substrate (O), and on quartz substrates covered with Nb films of the following thicknesses (in A): (1) 3, (2) 6, (3) 12, (4) 25, and (5) 50. After Ref. [64].

dispersion relation given in Eq. (5.51) for a quartz substrate covered with Nb quantum wells of different thicknesses. Experimental data obtained for the Nb/quartz system are in qualitative agreement with the theoretical predictions of Eq. (5.51), [45, 641. The resonance condition for the

0. Keller / Ph>csics Reports 268 (I 996) 85-262 165

dynamics perpendicular to the quantum well, given in Eq. (5.43) with K,, taken from Eq. (5.17) inserted, is readily generalized by replacing n(z) with n,rf in Eq. (5.17). Hence, one obtains the following local-field resonance condition

aP,H, (0, 0) 1’ n(z) + cc1 - ~:dm~l~~o)2 dz = 1

- u(L.&!l)2n(z) 3

c 7;

or after a few algebraic calculations

where

dz

_d 1 - &,~(z)/[o(w + i/z)] 1 ’

(5.54)

(5.55)

(5.56)

In Eq. (5.56) we have introduced the so-called screened local plasma frequency

(Z,(Z) = [n(z)e2/(m*coc,)]‘12. In the limit d + 0, the electron-density approaches

n(z;d+O) = N+d6(z). (5.57)

In turn, this implies that the integral in Eq. (5.56) asymptotically equals d/~,~ so that N$‘(to;d -+ 0) = 0. In this limit one thus regains the bare vacuum/substrate dispersion relation of Eq. (5.53) cf. (5.55). Usually, the dispersion relation exhibits a pronounced resonance above the bulk plasma frequency, i.e. in the ultraviolet part of the electromagnetic spectrum for metallic quantum wells. In semiconductors, where the number of free carriers is much less, the bulk plasma frequency is located in the infrared region, and hence one would expect a resonance behaviour of Eq. (5.55) in the infrared part of the spectrum. An even more dramatic dispersion is expected to occur if quartz is used as the substrate material, and the semiconductor is doped so as to place the bulk plasma frequency in the resonance region for quartz. As an example is shown in Fig. 5.11 the real and imaginary parts of the dispersion relation of Eq. (5.55) in a vacuum/GaAs/quartz quantum-well system for different film thicknesses. It appears from this figure that as the quantum-well thickness is increased the dispersion-relation resonance moves downwards in frequency towards the bulk plasma frequency. If one compares the dispersion relations for the z-dynamics with that of the x-dynamics (shown in the inserts of Fig. 5.11) it is realized that only the electrodynamics perpendicular to the well plane exhibits this strong plasma-resonance behaviour. In a sense this is what one would expect because induced motions of the electrons perpendicular to the quantum well due to confinement leads to build-up of a strongly inhomogeneous charge distribution across the well plane. In Fig. 5.12 the real and imaginary parts of the dispersion relation belonging to the z-dynamics are shown for a 800 A wide GaAs quantum well with the doping rate as a parameter. As the doping rate is increased the plasma-resonance frequency also increases as expected. The material parameters used in Figs. 5.11 and 5.12, as well as a detailed discussion of the local-field resonance condition in Eq. (5.55) can be found in Ref. [69].

166 0. Keller / Ph,vsics Reports 26X (1996) X5-262

147

145

143

S’ u 141 E

Y

2

139

137

135 r 1.60 i.60 i.ia 124 1 .i2 I io

147 ,

I 0.2

-I 1. 2

Fig. 5.11. Real (Re) and imaginary (Im) parts of the zz-dispersion relation in Eq. (5.55) for a vacuum/GaAs/quartz quantum-well system with different film thicknesses (in A), viz., (1) 500, (2) 600, (3) 700, (4) 800 and (5) 900. The plasma frequency is ho, = 138.9 meV and the relaxation energy is h/t = 2 meV. In the insets are shown the xx-dispersion relation in Eq. (5.51). for comparison. The dashed lines represent the pure vacuum/quartz dispersion relation.

0. Keller / Phvsics Reports 268 (1996) 85-262 167

.I

I 1.25

[coq,,b 1 133, ! I I 1 1 I

0.9 1.1 1.3 1.5 1.7 1.9 2.1

Re [ c,4,,/ (JJ 1

1 (b)

I I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 114

Fig. 5.12. Real (Re) and imaginary (Im) parts of the zz-dispersion relation in Eq. (5.55) for the GaAs quantum-well system of Fig. 5.11 with the doping rate as a parameter. The doping rate is expressed implicitly via the plasma energy and the curves correspond!0 the following hw, values (in meV): (1) 148.9, (2) 141.3, (3) 138.9, and (4) 136.4. The thickness of the GaAs well is 800 A, and h/~ = 2 meV. For comparison are shown in the insets the real and imaginary parts of the xx-dispersion relation (Eq. (5.51)) for the aforementioned doping rates.

16X 0. Keller : Physics Reports 2hK (1996) 85-262

5.4. Sheet-model electrodynamics and saltus conditions

To describe the linear response (reflection and transmission) of an ultrathin film possibly deposited on a substrate it is tempting from the outset to assume tha the film from an electrodynamic point of view can be considered as an infinitely thin sheet carrying field-induced currents and charges. This so-called sheet-model approach is widely used inphenomenological studies of the optical response of quantum wells, superlattices and (a few) monolayer thick films as such (see e.g. [70,71]). From an intuitive point of view it seems reasonable to adopt the sheet-model description when the film thickness is much smaller than the wavelength of the external field. However, it is certainly not easy to choose adequate induced currents and charges for the sheet. The choice made for these quantities in turn enters (and affects) the calculation of the amplitude reflection and transmission coefficients via a set of electromagnetic boundary (jump/saltus) conditions for the various components of the electromagnetic field. The current and charges induced in the sheet have to be determined in a self-consistent manner from the electrodynamics inside the sheet. In the phenomenological description there is essentially no ways to approach this problem. By taking as a starting point the basic integral equation for the local field inside a mesoscopic film it is demonstrated in this section that the electromagnetic response of the film always can be represented in terms of a sheet-model description. Furthermore, we shall briefly consider the general form of the sheet conductivity tensor and also derive a consistent set of jump conditions for the field across the film. Finally, I shall identify the inconsistencies attached to the use of standard (textbook) boundary conditions. For simplicity, we shall base our studies on the RPA approach throughout this section and thus leave out the reference to RPA from the notation.

In the present context it is useful to replace the local field appearing under the integral sign in Eq. (5.2) by the background field. This is done by means of the RPA field-field response tensor, see Section 4.2.3, and leads us to the following expression for the local field inside the quantum well (leaving out the reference to q,, and CL) from the notation):

E(z) = E”(z) - i/row s

GB(~,~‘)~r?t(~‘,~“)~EB(~“)dz”dz’ , (5.58)

where

P(z, z’) = s

(T(z, z”) - I-(z”, z’)dz” (5.59)

is the external conductivity tensor of the quantum well. For observation points outside the quantum well in the vacuum domain (z 5 - d), the prevailing field is given by

E(z) = EB(z) - i,uOcg i

PB(z,z’)Vt(z’,z”)~EB(z”)dz”dz’, zs -d . (5.60)

Introducing a reference plane located at an arbitrary position z = z0 inside the quantum well, Eq. (5.60) can be rewritten in the physically appealing form

E(z) = EB(z) - i~uowPB(z,zO)*JS(zO) , (5.61)

where

J%o) = S(zo)-EB(zo) >

0. Keller / Ph,vsics Reports 26X (1996) X-262 169

(5.62)

with

S(Q) = N(z’,zO)Y-Xt(z’,z”).N(z”,zo)dz”dz’ . (5.63)

The internal electrodynamics of the quantum-well sheet has recently been discussed in Ref. [72]. The central problem for the internal electrodynamics is the determination of the field-field response tensor r(z,z’). Once this has been calculated from the basic integral equation (see Ref. [72]) the external conductivity and the sheet conductivity may be obtained by direct integration, see Eqs (5.59) and (5.63). The physical interpretation of the result in Eq. (5.61) is as follows. The prevailing field outside the quantum well at z( 5 - d) consists of a sum of the background field and the field radiated from a current sheet carrying a sheet (S) current density Js(zo) and placed at an arbitrary chosen position inside the quantum-well domain. For field observation points outsille (and only outside) the quantum well (in the vacuum domain, here) the radiation from the well is completely equivalent to that from a current sheet placed in the vacuum domain just outside the substrate surface. As illustrated in Fig. 5.13, the radiation from the sheet (detected at z) is composed of a direct part and an indirect part reaching z upon reflection from the vacuum/substrate surface. The current density of the sheet is driven in a linear fashion by the background field at z. . The sheet conductivity tensor, S(zo), is given by the integral expression in Eq. (5.63) and contains besides the external conductivity tensor, two displacement factors (tensors) denoted by N. These are related to the background field and the propagator as follows: EB(z”) = N(z”,zo)~EB(zO) and P’(z,z’) = P”(z, zo) * N (z’, zo). The explicit expression for N can be found in Ref. [73], but is not need here. Although the sheet conductivity tensor depends on the arbitrary chosen portion zO, the field radiated from the sheet into the vacuum, i.e. - ipow PB(z, zo) - Js(zo), is independent of z,, [73], as it must be. In terms of the components of the sheet-conductivity tensor the p- and s-polarized amplitude reflection coefficients are given by

Qo-1 = r,(W + (~04wx(q~)2(1 - r,(o-))2&x(o-)

- 4131 + ~,(o-))2~zzK-) + 4,,431 - ~;(o-MLm) - sxm)) 3 (5.64)

Fs(o-) = rs(O_) - (#U()CU/2q:) (1 + rs(0-))2Sy)>(OP) . (5.65)

Let me now relate the findings in Eqs. (5.64) and (5.65) to the experimental studies on thin films. For film thicknesses comparable to (or larger than) the optical wavelength the description of the linear reflection usually can be based on macroscopic electrodynamics, and a parametrization in terms of the (local) dielectric constant (tensor) of the film, sfiim(Q), and the film thickness, d, provides a natural and adequate link between theory and experiment. For the mesoscopic films considered here it is in general impossible to use the refractive index, ~lr~,,,,(~), concept (or equivalent the ~film(CO) concept), and one can then ask whether or not it will be possible to parametrize the microscopic description in such a manner that the introduced parameters are directly accessible to an

170 0. Keller ii Physics Reports 268 (1996) 85-262

Fig. 5.13. Schematic illustration showing the sheet-model radiation channels The quantum well occupies the region _ tl 2 3 < 0 and the electrodynamically equivalent current-density sheet (gray area) is placed at z = zO. Radiation from

a source point located at r’ can reach the observation point at r along various paths. Thus the contribution from the background field (given by the dashed lines) consists of a direct term plus an indirect one, reaching r via a reflection from the substrate. As indicated by the dotted lines, secondary sources are created at the various points rg in the sheet by the primary radiation from r’, reaching r. by the direct and indirect paths. The radiation emitted from r,] reaches the observation point at r by direct or indirect propagation, as illustrated by the solid lines.

experimental determination. If not, one faces the problem that each experiment has to be linked directly to the full microscopic theory, a situation which often is uncomfortable for experimentalists (and theorists). If one tries to introduce effective (eff) dielectric constants, $!;‘,(w), and thicknesses, den, it turns out that both of these may depend on the light frequency, the polarization and the angle of incidence, a fact which makes this so-called effective medium approach physically erroneous, and useless. By means of linear, phase sensitive reflection experiments, e.g. ellipsometry, one can measure Fl, and I~,~. From a macroscopic point of view, these quantities basically are related to the optical properties of the quantum well through S(O-), cf. Eqs. (5.64) and (5.65). For the p-polarized case the conductivity parameters S,,, SZZ, and SZ, - S,, enter, and for the s-polarized case only one parameter S,,. is needed. In the s-polarized case the situation thus is clear. Measurements of the real and imaginary parts of FS, directly allow one to determine the complex number S,.,. In the p-polarized case one faces a complicated situation, i.e. three unknown complex numbers appear, and only one complex number, FP, is obtained experimentally. To tackle this problem one might try to use the so-called Feibelman d-function approach, which up to now has been extensively used to study the optical response of metal surfaces in the visible and ultraviolet frequency range [27,74]. Recently, this approach has also been applied to analyse the electromagnetic response of semiconductor heterostructures in the infrared spectral region [75]. In the d-function approach two frequency-dependent parameters, named d,, and dl, are introduced. These parameters characterize the electromagnetic responses parallel (d,,) and perpendicular (d,) to the

0. Keller ! Physics Reports 268 (I 996) 85~-262 171

film (surface) plane. The Feibelman d-parameters can be obtained from the sheet conductivity tensor provided the following assumptions are made: (i) SXZ = S,, = 0, (ii) S,, = S,,, and (ii) S,., and SZZ are independent of the angle of incidence. By adopting the above-mentioned simplifications, and by assuming that S,, = d,, has been obtained by a fit to the experimental data for the s-polarized case, SZZ = dL can be determined from the p-polarized data. In situations where the

off-diagonal elements of the sheet conductivity tensor, S,, and SZ,, are negligible one expects the Feibelman d-function approach to work. Since the off-diagonal elements of S originates in the electronic and electromagnetic cross-couplings between the dynamics parallel and perpendicular to the film plane, it is of interest to test the limits for application of the Feibelman d-function approach. This was done in a recent study of the linear optical properties of a Si/SiOZ multiple quantum well structure [SS]. In Fig. 5.14 experimentally obtained reflection spectra for s- and p-polarized light are shown along with the best-fitting curves, and in Fig. 5.15 the obtained frequency dependence of S,, is presented.

If the quantum well is sufficiently thin the phase shift of the background field across the quantum well can be neglected and the displacement tensors in Eq. (5.63) hence replaced by unit tensors. The resulting sheet conductivity

S ED-ED = next (z, z’) dz’ dz , (5.66)

which is given by the zero-order moments (in z and z’) of cext(z, z’) over the quantum well, can be characterized as an electric-dipoleeelectric-dipole (ED-ED) conductivity since SEDPED describes the current sheet as an electric-dipole (ED) receiver and an electric-dipole (ED) radiator. Within the framework of the ED-ED approximation to the sheet conductivity it is possibly to derive in a rigorous manner the correct macroscopic boundary conditions for the electromagnetic field across the sheet [73]. Although these conditions are generally referred to as bounllary conditions, they might more correctly by calledjump conditions, or as in the older literature saltus conditions. The rigorous result obtained for the six saltus conditions is as follows:

E,(O+) - E,(O -) = 0 , (5.67)

B,(O+) - II,(O-) = p&(0-) ) (5.68)

B,(Of) - BZ(OK) = 0 (5.69)

for s-polarized fields, and

&(O+ 1 - LK-) = (l/co) (&I J”(O-) > (5.70)

&(O+ 1 - DzV-) = (@I J,s(O-) 3 (5.71)

II, - By(O-) = - pfJJx”(o-) (5.72)

for p-polarized fields. The jump conditions above are different from those derived in the macroscopic standard (textbook) description. The reason for this difference can be traced back to the fact

172 0. Keller- ! Physics Reports 268 (I 996) 85-262

0.6

i

/,‘__., \ _ \\

‘\ ‘, , \ ‘.___- _*-------I /km I’ \ \ \

\ ‘C ,*

__-----__ _’ _____---

‘.__’ >-

0.5

! 0.4

Q?

12 1.6 2.0 2.4 2.8 3.2

hd (PIT) -

0.3

0.2

0.1

0.0 1

1

.2 1.6 2.0 2.4 2.8 3.2

h&L (CV) __L

Fig. 5.14. Energy reflection spectra of an amorphous Si/Si02 multiple quantum-well structure for s (upper figure) and p (lower figure) - polarized light at different angles of incidence: (1) 15’, (2) 30’, (3) 50’, and (4) 70”. The experimental and calculated spectra are displayed by the solid and dashed lines, respectively. For the s-polarized data the complex fitting parameter S,, was used, and to fit the p-polarized spectra the Feibelman d-function approach was employed. For angles of incidence around 45’, the cross-coupling between the x- and z-components of the local field is of importance. The neglect of S,, and S’?, in the Feibelman approach hence is responsible for the discrepancy between theory and experiment

for curves (2) and (3) in the bottom figure.

that oscillating surface currents flowing perpendicular to the surface plane are neglected in the traditional approach. A neglect of Jf leads to inconsistencies among the jump conditions. Not only is Eq. (5.70) different from the textbook relation, also Eq. (5.71) has been modified so as to replace the sheet charge density ,oS by (Q/u)J~ on the right-hand side of the equality sign. To match the fields in the vacuum and substrate across the well it is necessary to use two of the three jump conditions in both the s- and p-polarized case. How does one pick the two equations from the three


-\ ’ *._I

-1.0 I I, I I I

1.2 1.6 2.0 2.4 2.8 3.2

TLW (e\J -

Fig. 5.15. Real (solid line) and imaginary (dashed line) parts of the sheet conductivity S,, as a function of the photon energy. The arrows indicate the valence-to-conduction subband transition energies of the amorphous Si/Si02 quantum well as calculated on the basis of the square potential-well model.

possible combinations? To answer this question let us first consider the s-polarized case. Since J;(O-) enters only in Eq. (5.68) this equation must be used as the one equation. Since the Maxwell equation V x E = iuB leads to qll E, = COB,, the jumps in E, and (u/~,,)B, must be identical. In turn, this implies that the jump conditions in Eqs. (5.67) and (5.69) are equivalent. Thus in the field matching procedure for s-polarized fields one has to use Eq. (5.68) together with either Eq. (5.67) or Eq. (5.69). The two choices give consistent results. For p-polarized fields, Eq. (5.70) has to be used as one of the two equations since the z-component of the sheet current density only enters this equation. Furthermore, it appears from the Maxwell equation V x B = ,uoJ - &wD that V x B = - ipOuD in the vacuum and in the dielectric substrate, so that QB, = - ,u~uD, outside the well. The jump conditions for B, and - ,uo(~/q,,)D, hence are the same. Next, this implies that Eqs. (5.71) and (5.72) are equivalent. In the matching procedure for p-polarized fields Eq. (5.70) thus shall be used together with either Eq. (5.71) or (5.72). If wrongly one had used the charge density ps in Eq. (5.71) instead of (~/o)J,s, the relations in Eqs. (5.71) and (5.72) would no longer have been consistent with the Maxwell equation demand q,, B, = - ,uo c&, . The use of ps then would have led to two mutually conflicting results in problems where boundary matching of p-polarized fields are needed. Also, one could start to wonder why Eqs. (5.71) and (5.72) could not be used as a set (giving a third result !). The jump conditions in Eqs. (5.67)-(5.72) are not in final form before the current density of the sheet has been eliminated in favour of the local field acting on the sheet, i.e. E(O-). In compact form the final results for the four saltus conditions involving the sheet conductivity are

e;[E(O’) -E(O-)] =+?z.so.E(O-)) 0

(5.73)

174 0. Keller : Plyics Reports 268 (1996) KS-262

e_*[D(O+) - D(O-)] = -J-J-e,-So-D(OF) , (5.74)

ez x [B(O+) - B(OV)] = /L~(U - eZeZ)*S,*E(Op) , (5.75)

where

so = SE-ED. [U - i/~OcOPB(O-,O~)*SED-ED]-l , (5.76)

In recent analyses of the s- and p-polarized reflectivity from Nb/quartz single quantum wells [61,62,64], and Si/SiOZ multiple quantum wells [55] the rigorous sheet jump conditions dicussed above were used.

5.5. Nonvetarded response, radiation reaction and Lamb shijt

Since the thickness of a quantum well is many times less than the wavelength of the incoming optical field one might expect that the local field inside the quantum well is independent of electromagnetic retardation effects across the well. This expectation broadly speaking is correct. However, if one excites the quantum-well system at a frequency close to a local-field resonance and by some means suppresses the electronic loss mechanisms (phonon scattering, impurity scattering, boundary scattering, etc.) retardation phenomena may be of importance. Below, I shall illustrate the role played by electromagnetic retardation effects in a particular simple case, viz. that of a two-level quantum well in which the paramagnetic coupling dominates. Because the propagation properties of the electromagnetic field across the quantum well are hidden in the Green’s function of Eq. (5.3) let us briefly consider this, and let us for simplicity assume that the quantum well is embedded in a homogeneous, isotropic, nonabsorbing and local (background) medium so that the indirect term is absent. The speed of light in the substrate medium is denoted by c. By comparison with Eq. (2.49) it appears that the transverse direct propagator is given by

(5.77)

where q. = w/c and e4 = qll e, + q1 e, in the plane-wave expansion. Since the integrand of Eq. (5.77) has first-order poles on the real q,-axis at q1 = i q(j = 5 (qg - q,f)1/2, one obtains via contour integration

Di(z - z’; qll, w) = (ei4”1’-z’1/2iq~) [(qy)“exex + (w/c)2eye,, + qifezez

- qYqllw(z - z’)ke, + e,e,) 1 (5.78)

Formally, the nonretarded (NR) part of the propagator Dt” can be obtained from Eq. (5.78) letting c --+ x . Since in this limit qy + iq,,, Eq. (5.78) becomes

Dt”(z - z’;qll,o) = (C/0)2(q,,/2)e-q11’2-““[exex - ezez

+ isgn(z - z’)(e,e, + e,e,)] . (5.79)

0. Keller / Physics Reports 268 (1996) H5-~262 175

In the nonretarded limit only the longitudinal part of the local field survives. This means that one should be able to study the local-field electrodynamics of the quantum level starting from the longitudinal propagator in Eq. (2.51) in the limit c’ --+ x.

The relevant part of Eq. (2.51) is given by

Db(z - z’;q,,,4 = (5.80)

The integrand has first-order poles at q I = ~fr iq,\, and upon a simple contour-integration calculation one realizes that

D;(z - z';q,+)lc+m = D;"(z - z';qll,cu) = D,“(z - z’;ql, cd . (5.8 1)

In the nonretarded limit the prevailing local field thus must necessarily be both divergence-free and rotational-free. In consequence of this fact DtR(r - r’; 10) = DtR(z - z’; qll, (0) exp [iq,, (X - x’)] has to satisfy the criteria

V l Dr” (r - r’; q,,w) = V x DtR (r - r’; yIl w) = 0 , (5.82)

and also V’.DtR = V’ x DtR = 0, of course. A direct calculation readily proves that the equations in (5.82) in fact are correct. In passing we note that only p-polarized fields can survive in the nonretarded regime, cf. Eq. (5.79). Altogether, we thus have seen that the background Green’s function of relevance for the present quantum-well problem is given by

G~R(z - z’;Q,w) = D~“(z - z’;cJ~,o) + (c/(o)~~(z - z’)eZe, (5.83)

in the nonretarded limit. Let us return now to the two-level AlGaAs/GaAs/AlGaAs quantum-well system discussed in the

last part of Section 5.2. Numerical calculations show that local-field calculations performed on the basis of the exact (retarded) propagator (Eq. (5.78) plus the self-field term) and by means of the nonretarded propagator (Eq. (5.81)) give identical results except near the local-field resonance. To emphasis this I have plotted in Figs. 5.16 and 5.17 the real and imaginary part of the (normalized) z-component of the local field in the middle of the well (z = 0.5 d) as a function of the (normalized) photon energy in a frequency range around the local-field resonance. The angle of incidence was chosen to be 0 = 7c/4. It appears from these figures that the (upward) shift of the resonance peak away from the electronic transition frequency is essentially the same in the full theory and in the

NR-theory. In a consistent nonretarded description the plane-wave background field, here identical to the

incident (inc) field

EinC(x,z)= E;;TS(eZ -(qY/q,,)e,)e i(q~z+qllx) 2 (5.84)

which is transverse (V *E inc = 0) when the sources are located outside the transverse current- density domain of the quantum well, should be approximated by its nonretarded form

E;-;RC(x,z) = Er;f”,(e, - ie,)e-qlI”ei411” , (5.85)

176 0. Keller : Physics Reports 268 (1996) 85-262

150

‘00

- 50

4

3 O

e -50

0

-50

ra 6 -100

\

5 -150

E 4 -200

-250

1.08 1.09 1.10 1.11 42

Fig. 5.16. Real (Re) and imaginary (Im) parts of the (normalized) z-component of the local field in the middle (z = 0.5~f) of a two-level AlGaAs/GaAs/AlGaAs quantum-well system (the same as described in Sections 5.2) as a function of the (normalized) photon energy in the vicinity of the local-field resonance. The angle of incidence is 0 = rr.i4. and the calculations have been performed for five different (intersubband) relaxatioOn energies (h/r) namely (0) 0, (1) 0.05, (2) 0.1, (3) 0.5, and (4) 1.0 meV. The thickness of the quantum well is d = 125A, and the two-dimensional density (related to the donor concentration) was taken to be 1.39 x 1Or2 cm- ‘. The electronic transition energy used is

c2 r = Ed - or = 108.6 meV. The numerical calculations shown were performed using the exact (retarded) electromagnetic propagator.

which is both divergence-free (V *E &‘i = 0) and rotational-free (V x EKi = 0). In contrast to the linearly p-polarized and propagating field given in Eq. (5.84) the nonretarded part of the incident field is circularly polarized and evanescent. At resonance an electromagnetic surface wave (SEW) is also circularly polarized, and in fact the field in Eq. (5.85) represents one of the possible solutions to


-1501 3 I 1” 1.08 1.09 1.10 1.11 1.12

-300' I I J

1 .08 1.09 1.10 1.11 i.12

Fig. 5.17. Real (Re) and imaginary (Im) parts of the (normalized) z-component of the local field of a two-level

AlGaAs/GaAs/AlGaAs quantum-well system for five different relaxation energies (the same as those in Fig. 5.16) as a function of the normalized photon energy. The calculations were performed using the nonretarded electromagnetic propagator. Besides this the remaining data are identical to those used in Fig. 5.16.

the local SEW dispersion relation at resonance, viz. the solution belonging to the SEW branch for which the field is decaying towards the surface on the vacuum side [76].

To stress the importance of retardation effects in the vicinity of the local-field resonance, the curves in Figs. 5.16 and 5.17 have been plotted for different relaxation times. In particular, one should notice that even in the absence of all electronic relaxation mechanisms (impurity scattering, phonon scattering, boundary scattering, etc.), i.e. for t + cc, the peak height at resonance is finite

178 0. Keller- ! Ph_vsics Reports 26X 11996) KS-262

in the presence of retardation effects. The half-width of the r + 8~ -peak, which in the present case is approximately 0.1 meV, thus might be called the natural linewidth of the two-level quantum-well system. This linewidth, which determines the spontaneous lifetime of an electron in the upper state and hence the rate of spontaneous emission from the well, originates in the backaction of the field created by the quantum well upon the well. This effect thus is analogues to the famous radiation reaction describing within a semiclassical framework the spontaneous emission of an excited atom. The frequency shift of the local-field resonance from the electronic transition which occurs in the z + x limit might be called the Lamb shift of the quantum well. The Lamb shift stems essentially from the nonretarded part of the field created by the quantum well itself. In the present example this shift is - 10.7 meV. 1 shall return to the Lamb shift and the spontaneous emission when discussing the electrodynamics of quantum dots (see Sections 6.2 and 6.3).

6. On the role of local fields in small particles and quantum dots

In this section I shall analyse in a qualitative manner the importance of local-field effects in the optics of mesoscopic particles ranging from quantum dots containing only a single electron to small metal particles having so many conduction electrons that the classical Lorenz-Mie theory [77,78] in the Rayleigh approximation [79] is approached. Following a brief but general discussion of the local-field correction to the so-called electric-dipole-electric-dipole polarizability the diamagnetic polarizability of small metal particles is studied. Thereupon, we turn our attention towards the paramagnetic polarizability of small semiconducting particles, for which the discreteness of the energy level structure plays a decisive role for the local-field effects. Next, I discuss the optical response of single-electron (spherical) quantum dots dominated by the paramagnetic coupling. Finally, the vector potential accompanying the interaction of a single-particle quantum dot with its own field is discussed within the framework of the semiclassical approach well known from atomic physics and often named the self-field approach. The terms in the self-field vector potential which give rise to the Lamb shift and spontaneous emission are identified and discussed.

6.1. Optical polarizabilit,v qf small particles

The purpose of the present section is to give a qualitative description of the role of local-field effects in the electrodynamics of particles so small that (if the inhomogeneity of the electron density inside the particle and/or (ii) the discreteness of the energy-level structure plays a role. In what follows, I shall from time to time use the name quantum particles for these particles. In particular, we shall seek the fingerprints of local-field electrodynamics in the particles optical polarizability, a quantity accessible to experimental investigations.

Optical effects associated with small particles have been of interest to researchers for many years, and to catch a glimpse of this huge field the reader is urged to consult the milestone books by van de Hulst [80] and Kerker [Sl], or the recent book by Bohren and Huffman [82]. Comprehensive reviews describing fundamental approaches as well as the latest developments can be found in the book edited by Barber and Chang [83]. As a good introduction to optical phenomena of particular interest in the present context the reader might consult the review of Huffman [84] on the applicability of bulk optical constants in studies of small particles. The evidence for non-bulk-like

0. Keller / Ph_wics Reports 268 (1996) 85~-262

behaviour in semiconductor clusters has been surveyed by Brus [SS]. As emphasized originally by Kawabata and Kubo [SS], as well as by Gor’kov and Eliashberg [87] the discreteness of the energy-level structure should show up in the electrodynamics of small particles. A comprehensive discussion of quantum size effects as such in small particles is given in the review article by Halperin

WI. In the context of the local-field approach presented below the papers mentioned below are of

special interest. A major group of authors has focused their studies on self-consistent, e.g., density-functional, calculations of the static polarizability [89-961. Although these density- response calculations have also been used to predict the dynamic polarizability, it is not obvious that scalar-potential formalisms can give a fair account of the optical polarizability induced by transversely polarized electromagnetic waves. In extended versions of the Lorenz-Mie theory [77,78] researchers have aimed at obtaining a complete solution for the propagation of the eletromagnetic field inside the particle. To achieve such a goal simple dielectric response functions, e.g. homogeneous [77,78] or inhomogeneous [97] local ones, or nonlocal transverse and longitudinal bulk response functions of the hydrodynamic type or of the Lindhard type [90,988101], were used. In a recent article by the present author [ 1023, a local-field study of the optical polarizability of small quantum particles has been carried out. emphasizing mainly a new self-field approach. In connection to this work, numerical studies of the local-field corrections to (i) the diamagnetic polarizability of small metallic spheres [ 1031, and (ii) the paramagnetic polarizability of semiconducting particles of (assumed) cubic form [104] have been performed. The potential power of the local-field formalism was illustrated by comparing the diamagnetic calculation with available data on small Au particles [105,106]. In recent years also various of the nonlinear optical properties of droplets and small particles have been investigated, cf. e.g. the review paper by Hill and Chang [107], and the article by Hache et al. on third-order nonlinearities [lOS].

6.1.1. Electric-dipoleeelectric-dipole polarizahiliq As a starting point for our analysis we take the RPA expressions for the external conductivity

given in Eq. (4.52). Furthermore, we assume that the particle under consideration is of mesoscopic size so that the external field is essentially constant across the particle domain. If the centre of mass of the particle is located at the origin of our coordinate system we thus put E’“‘(r) = E”“‘(O) for all Y inside the particle. The induced current density of the mesoscopic particle hence is given by

J(r) = [I crRPA (v, i). rRpA (i, y”)d%“d ‘1.’ 1 -E”‘(O) . (6.1)

In the terms of J(r) the polarization induced in the particle is P(v) = (i/cjj)J(v), and from P(r) one can calculate the electric-dipole moment

J(r)d3T (6.2)

of the induced polarization. By combining Eqs. (6.1) and (6.2) one obtains

(6.3)

180 0. Keller : Phjisicx Reports 268 (I 996) 85-262

where

is the dynamic RPA polarizability in the so-called electric-dipole-electric-dipole (ED-ED) ap- proximatin. In the ED-ED approximation the quantum particle acts as an electric-dipole receiver clrzd radiator, cf. the discussion of the ED-ED sheet conductivity of a quantum well (Eq. (5.66)) presented in Section 5.4. To facilitate the notation we shall omit the references to ED-ED and RPA on the various quantities in the following. The local-field correction, A(W), to the polarizability, a(~), can be obtained directly from Eq. (6.4) by inserting the implicit expression for F given in Eq. (4.53). Thus,

a(cti) = OIL + A(O)) , (6.5)

where

a(r,r’)d3r’d3r

is the polarizability in the absence of local-field corrections, and

A(o) =; ‘i’

o(r,r’).K( r’,,“).y(r”)d3r”d3~‘d3r

is the local-field correction. As indicated, A(W) only depends on the integral

(6.7)

y(r) = F(r,r’)d3r.’ (6.8)

of the field-field response tensor when the quantum particle behaves like an electric-dipole receiver. The tensor y(v) satisfies the integral equation

y(r) = U + s

K(r,r’).y(v’)d3r’ (6.9)

6.1.2. Diamagnetic polavizahility qf small metal particles For a mesoscopic metallic particle the optical polarizability often will be determined by the

diamagnetic effect at least at infrared frequencies. By retaining only the diamagnetic part of the conductivity tensor (given in Eq. (3.45) with the light-unperturbed electron density taken from Eq. (3.111) in RPA) the so-called diamagnetic polarizability, udia(cs), becomes

adia((o) = - 2 jy(r)NO(r)d3r (6.10)

Instead of embarking directly on a comprehensive numerical analysis of Eq. (6.10), let us in two steps simplify the calculational scheme in a manner which allows us to preserve the main features of

0. Keller / Physics Repwts 268 (1996) 85-262 181

the system and, in a physically appealing way, shows the relationship to the classical Lorenz-Mie theory [77,78] for the polarizability of a small and homogeneous spherical metal particle. In the Lorenz-Mie scattering theory [77,78] the excitation of the spherical particle by a transversely polarized electromagnetic waves gives rise to creation of a divergence-free local field, and in the limit where the particle radius is much less than the optical wavelength the theory reproduces the quasi-static Rayleigh result [79]. When the particle size becomes much smaller than the wavelength of the impressed field it is expected that the overall influence of retardation effects tends to vanish, with the reservations indicated in Section 5.5 for quantum-well systems and discussed for quantum dots (atoms) in Section 6.3. So, as a first step let us consider the retardation as it appears in the electromagnetic propagator of Eq. (2.25). The response originating in the transverse and longitudinal self-field propagators is nonretarded, as one readily realizes from the fact that both of these are proportional to ~0’ (see Eqs. (2.26) and (2.27)), and thus upon multiplication by /lo (see Eq. (2.24)) the light velocity disappears in favour of the electrostatic r:; ‘-factor. Thus, to estimate the role of retardation effects one has to consider the transverse vacuum propagator D~(Y - r’; CO). As a starting point let us take the plane-wave expansion of Di(r - r’; CO), given in Eq. (2.49). and let us carry out the integrations in spherical coordinates. Performing the angular integrations one gets

D;(R) = (27~) - * ~;/{(“-eRe~)[&-&+&]

2 iqK

e ~ : 2 dq 3 R f 0 3 40 - 4

(6.11)

where R = r - r' and eH = R/R. The integral over 4 is performed by contour integration, noting

that the integrand has poles on the real q-axis at q = + q. and q = 0. Hence, we finally have

DT(R) = 40 0 47ci &(“-eneR)-[&-& (U-?e,eRj)

0 1 x eiqOR - L (u - 3eReR)

(iqo RI3 , RfO, (6.12)

where the term (U - 3eReR)/(iyoR)3 stems from the pole at q = 0. To elucidate to the role of the field retardation a power expansion in orders of ~0’ is made, leading to

D;(R) = (U + eReR)/8nR - itaU/67rco + 0(ci2) (6.13)

By a comparison to the self-field terms which are of the order ~02, it is realized that the in-phase retarded contribution is a second-order correction (of the order ~0”) and the out-of-phase retarded contribution is a third-order correction (of the order ~‘0 ’ ). In first order there thus is no corrections to the self-field dynamics from retardation for a mesoscopic particle (with the proviso discussed in Section 6.3). Neglecting retardation effects, the local field will be given by

E(r) = E?‘(r) + (1/3&c!))&(r) + (l/ieotu)J&) , (6.14)

182 0. Keller ! Physics Reports 268 (1996) X-262

where we have stressed that the external field in mind is assumed to be transverse. When the size of the metal particle increases one approaches the classical regime (Lorenz-Mie, Rayleigh) where the local field tends to be transverse, and when the particle size approaches the atomic limit, the longitudinal field again play a less direct role for the self-consistent field electrodynamics, because it is eliminated as a dynamical variable in favour of the electron coordinates, cf. the discussions in Sections 3.1 and 4.1. So, let us as a second step ignore the longitudinal part of the field and thus assume that the transverse dynamics is the dynamics of the full system. Mathematically, we hence neglect a small term, [2/(3ie0w)]JL, in Eq. (6.14) (but keep the small term [1/(3&cti)]J,J. Doing this, Eq. (6.14) is replaced by the relation

E(r) = Ey’(r) + (1/3&o) J(r) . (6.15)

Albeit certainly not rigorous, let us see what kind of result the heuristic formula in (6.15) leads to. By utilizing the diamagnetic relation between J(r) and E(r) to eliminate J(r) in Eq. (6.15) it readily follows that the field-field response tensor (Eq. (4.53)) is given by

r(V,r’) = (1 - eVV,(v)/3c()mo2)- l 6(r - r’)U (6.16)

in the present case. By combining Eqs. (6.Q (6.10), and (6.16) one obtains the following heuristic result for the isotropic, diamagnetic polarizability:

adia(Q) = _ u /!f s N&)d3r

mu2 1 - e2N&/(3comcu2)

In terms of the local dielectric function .sdia(y) = 1 - an illustrative manner, i.e.,

_ e2No(r)/(eom~~2), Eq. (6.17) can be written in

(6.17)

(6.18)

For a spherical particle with an assumed homogeneous electron density, Eq. (6.18) leads immediately to the Rayleigh expression [79] for the polarizabilities, i.e.

I+$,_)) = 4rQ43 {[Ed’“@) - l]/[c:d’“(W) + 21) u ) (6.19)

where u is the classic radius of the particle. In the Rayleigh expression a seemingly too strong localized resonance occurs for cdia((ti) = - 2 if the damping rate is small. In the more general expression of Eq. (6.18) this resonance is softened because of the involved integration. Using the

Drude expression cdia(~) = 1 - o~/[o(o + i/r)] the resonance occurs at o 2: wF/$. The softening of the localized resonance in our small metallic particle should be compared to the resonance softening discussed for a metallic quantum well in Section 5.3, cf. Eqs. (5.43)-(5.45), and (5.54). In [ 1031, a quantitative calculation of the polarizability of a spherical particle, based on the so-called transverse self-field approximation (the model above), has been carried out within the framework of the RPA approach. It was assumed in this calculation that the ionic potential is flat in the region r < R (jellium model) and infinite outside (infinite-barrier model). The calculation shows a distinct

0. Keller/Physics Reports 268 (I 996) 85-262 183

peak in the frequency spectrum of the imaginary part of the polarizability. The peak is blue-shifted with respect to the classical Rayleigh peak, and when the particle radius is decreased the shift of the resonance peak becomes larger. In Fig. 6.1 is shown the height of the resonance peak (and part of its fine structure) normalized to the Rayleigh peaks as a function of the particle diameter. The data, in which the electron density of gold has been used, show that the overall tendency is a smooth

0 2 4 6 8 10 12

diameter of gold sphere (nm)

b) 00 I 1 I I I I 1 I

2.0 2.1 22 2.3 2 4 2.5 diameter of gold sphere (nm)

Fig. 6.1. (a) The height of the resonance peak in the imaginary part of the polarizability of a gold sphere as a function of the sphere diameter for three different damping rates, i.e. y = O.lw,(O), 0.0304,(O), and O.Olw,(~). For comparison to the classical result the peak height has been normalized to the value of the Lorenz-Mie peak. (b) A detailed view of the normalized height of the resonance peak in the diameter range 2.OG2.5 nm.

increase of the peak height towards the Rayleigh value as the sphere diameter increased;. However, one should notice that the particle diameter has to be considerably larger than 100 A before the classical limit is reached, especially for the smallest damping rates. By comparison to the experimental data of Kreibig [ 105,106] the overall agreement is good if one chooses a damping rate - 0.0301,. A careful look of the data of Kreibig reveals a fine structure on the steep part of the

curve. This fine structure is found also in the data of Fig. 6.1. The fine structure stems from the variation in the electron density and might be referred to as a quantum-size effect. Starting from the left in Fig. 6.1 (b), each new point corresponds to an addition of two new electrons (two because of the spin degeneracy). Each time one starts to fill electrons into a new (higher) energy level Ekl,

a discontinuity is obtained in the slope of the fine structure peak height. The overall increase of the peak height with increasing sphere diameter originates mainly in the change of the surface density profile. It seems that the quantum-size effect discussed above has recently been observed also in optical second-harmonic generation from metallic nanocrystals [109].

Before closing this subsection, let me make a comment concerning the part of the transverse propagator in Eq. (6.12) which does not contain the exponential factor, exp(iy,R). Thus, if one in the plane-wave expansion for the longitudinal propagator given in Eq. (2.51) carries out the integrations in spherical coordinates in the same manner as was done for D:(R) (in the D:(R) case the q-integrand has a first-order pole only at q = 0), one obtains

D,$(R) = (4,-,/4ni) [l/(iq0R)3] (U - 3e,e,), R # 0. (6.20)

By a comparison to Eq. (6.12), it is realized that the term without the exp(iq,R) factor just equals Dk(R). By addition of Eqs. (6.12) and (6.20) one readily obtains an explicit expression for Do(R) = D:(R) + D;(R) valid for R # 0, viz. that of Eq. (2.16).

6.1.3. Pcrramagnetic polarizability of small semiconductor particles Let us turn our attention now towards the case where the ED-ED polarizability is dominated by

the paramagnetic response, and let us for heuristic purposes perform the analysis within the framework of the RPA formalism. To obtain the local-field correction to the optical polarizability the coupled-antenna approach described in Section 4.3 is used in the conventional manner. Details of the calculation are not given here but can be found in Ref. [104]. Before giving the explicit expression for A(W), let me present the well-known dyadic expression for the bare polarizability, i.e.

c911

(6.21)

where,/; and i-:i (i = yz or MI) denote, respectively, the Fermi-Dirac distribution function and the single-particle energy of energy eigenstate number i. The quantity

Px,j = - e s

~@(v)r$,(r)d~r (6.22)

can be recognized as the electric-dipole transition matrix element between the energy eigenstates r and p. Using the notation above, the local-field correction takes the explicit form [102] (in [lo23

0. Keller !I Physics Reports 268 (I 996) X5-262

a factor of i is missing in A due to a typographical error)

A(u) =& c (“6% -ML -.m(~~1 - Ekbc! 0 k.l.m,n [h(u + i/z) + cli - Ed] [h(to + i/z) + E,, - c,~] ‘lkPmn ’

1x5

(6.23)

with the abbreviation

ML; = J,&) [+.I,T&) + J,L,(y)]d3r . (6.24)

In Eq. (6.24), J,‘, and J,“, denote the divergence-free (T) and irrotational (L) parts of the one- electron transition current density J,,,n given in Eq. (3.49). In deriving Eq. (6.23) it has been assumed that the local-field interaction is dominated by self-field effects (compare Eqs. (6.14) and (6.24)). The fit,,,-vectors appearing in Eq. (6.23) essentially play the role of self-consistently determined vectorial strengths of the integrated field-field response tensor Y(Y) and have to be following set of equations (letting (MZ,Y~) run through all possible (relevant) (nz = n excluded)):

determined from the combinations of M, II

(6.25)

By combining Eqs. (6.21), (6.23), and (6.25) the expression for A(U) can be rewritten in a form

A(w) = -& 1 (h -fi)h - Ek)

0 k.l [h(u + i/r) + &k - El] ‘lkPk’ - ao(o) ’ (6.26)

particularly adequate for numerical calculations. On the basis of the bare polarizability, Wood and Ashcroft [91] have studied the paramagnetic

response of small particles assuming for computational simplicity that the electrons are confined in a mesoscopic cubic box (flat potential bounded by infinitely high barriers at the faces of the box). Due to the fact that it recently has become possible to produce semiconductor quantum dots of almost cubic form it might be possible to confront this extremely simple model with experimental data in the near future. Although it has been shown [91] that the quantitative differences between the results obtained on the basis of the cubic box model and other geometrical shapes of confinement are small in the local limit, one would expect the local-field corrections to depend in a crucial manner on the shape of the mesoscopic particle. Within the framework of the cubic box model it seems possible also to carry out a rather complete calculation of the local-field correction. As a first step, such a calculation was recently performed in the transverse self-field approximation [104], described via Eq. (6.15). To determine the &,‘s from the set of equations in (6.25) one has to solve the related super-matrix problem. Since the dimension of the matrix increases rapidly with the number of involved energy levels in practice one is limited to examining systems having relatively few relevant levels. Ten levels for instance result in a super-matrix problem of dimension 90 x 90, and hence a standard matrix problem of dimension 270 x 270.

In my discussion of the coupled-antenna theory (Section 4.3) the radiative-reaction coupling (self-coupling) was described for a two-level system. It might happen that the self-coupling

186 0. Kellrr- Plq,sics Reports 268 (1996) X5-262

dominates the local-field interaction in the present case. If only the self-coupling mechanism is retained all M& elements apart from M,yi,l and M,!$A are zero so that the explicit solution for

P ,,L,l may readily be obtained. In turn, the polarizability becomes

(6.27)

where c,,, = c,, - c,, and SC stands for self-coupling approximation. The notation MZ< and 11’ attached to the summation sign(s) mean that the m- and n-states must be filled and empty, respectively. The self-coupling expression for the polarizability displays an important feature in the sense that associated with each electronic transition frequency c?-),,~ = c,,/ii one has a self-field resonance located in the small damping limit at a frequency

(6.28)

always fulfilling the inequality &,;s”, > (L),~,~. If the local-field coupling among all levels are taken into account the self-coupling resonance might be displaced, and sometimes split into two.

To illustrate the findings discussed above let me present a few numerical results for the polarizability of a GaAs box. The parameters used to obtain these results as well as a detailed discussion can be found in [104]. In Fig. 6.2, the box has a side length 200A and contains eight conduction electrons, occupying the four lowest lying energy eigenstates. The 28 lowest lying unoccupied energy eigenstates were taken into account, and there are six electric-dipole allowed transitions. Due to the local-field correction the resonance positions are displaced from the electronic transition resonances as shown in Fig. 6.2(b). As shown in Fig. 6.2, the frequency dependences of the polarizability calculated within the self-coupling approximation and with all transitions taken into account are considerably different in the present case. It thus appears that the local-field resonances obtained in the self-coupling approximation are frequency shifted, changed in form, and one of the peaks even split into two when the couplings between the various transitions are incorporated. When the box size, and hence the number of transitions, increases a distinct envelope of the resonance peaks appears. When the box size becomes sufficiently large the individual resonances are effectively smeared. The smearing leaves only a smooth envelope, and in this way the transition to the macroscopic domain is attained for the paramagnetic polarizability. When the energy level distances becomes sufficiently small the summation used above can be replaced by an integration. In the absence of local-field corrections a discussion of the summation- to-integration conversion is given in the Wood-Ashcroft paper [91].

6.2. Optical response qf single-electron spherical quantum dots

In the preceding Section I have discussed the optical polarizability of mesoscopic particles. Special emphasis was devoted to studies of particles containing a relatively large number of mobile electrons. For these particles electron-electron coupling effects such as Coulomb interaction and exchange-correlation phenomena usually play a significant role. Once the electron-electron dynamics is of importance it is normally necessary to resort to the one-electron theory when calculating in practice the conductivity tensor. In turn this implies that the integral equation loop for the local field involves also the longitudinal part of the field, cf. the analyses presented in Section 4.

0. Keller / Physics Reports 268 (1996) K5S2h2

0.06 ---_ $- f 1124) - - 006

187

0.05 0.10

Tlw (eky 015

Fig. 6.2. Numerical magnitude of the normalized polarizability of a cubic GaAs particle, with side length u = 200 A, as

a function of the photon energy calculated with (solid lines in (b) and (c)) and without the local-field correction (dashed line in (b)), as well as within the self-coupling approximation (dashed line in (c)). The appropriate transitions and the various energy eigenstates are shown in (a). The classification of the levels is given to the right in (a), and the curly bracket is meant to indicate the set (with 1,3, or 6 members) of equivalent quantum numbers for a given energy. With i < j < kin

iijk] the six levels have from left to right the quantum numbers (ijk), (ilcj), (jik), (jki), (kij), and (!xji). If i = j < k the left to right order is (iik), (iki), and (kii) [and if i <j = k, (ijj), (jij), and (jji)].

Recent progress in technology, however, has made it possible to produce particles so small that they only contain a single or a few mobile electrons [110-l 151. These particles have been named quantum dots, and because the dots are embedded in a condensed matter medium they can be considered as a kind of artificial solid-state atoms. A typical example is a GaAs quantum dot embedded in a Gal _,AI,As host medium.

In recent years, experimental absorption [ 113,114,116] and transmission [ 110,111,115] spectra of quantum dots clearly showing electron transitions between discrete states have been presented, and also photoluminescence studies [ 112,116,117] have been carried out. In several papers the problem of calculating the energy spectra and wave functions for electrons, holes, and excitons in quantum dots has been addressed. Once the light-unperturbed properties of the quantum dot in

188 0. Keller / Ph_vsics Reports 268 (I 9%) KS-262

consideration have been determined [ 116,118-l 221 the optical properties may be calculated using one or another kind of optical response theory [116,119,1233130].

For a single-electron quantum dot excited by an external field located outside the transverse current-density domain, only the transverse part of the local field enters the integral-equation loop since the electron does not “see” its own longitudinal field, cf. the discussion of Section 3.1. For a single-electron quantum dot, as well as for a few-electron dot, the diamagnetic part of the conductivity is usually negligible, and therefore we leave it out in the following. Omitting also the small contribution from the spin conductivity we are left with the one-electron expression for the paramagnetic conductivity. In the low-temperature limit where only the lowest lying energy eigenstate (n = 0) is occupied, the paramagnetic conductivity tensor can be written in the form

(6.29)

taking the wave functions entering the one-electron transition current density from the ground state (0) to the nth excited state (n), jon, as real. All electronic relaxation mechanisms are treated phenomenologically in Eq. (6.29) by means of the collision frequency V.

To determine the local field in a quantum dot dominated by the paramagnetic response we follow the coupled-antenna-theory method of solution described in detail in Section 4.3.1. Thus, in a slightly different notation, the basic integral equation for the transverse field becomes (compare to Eq. (4.60))

where

&l(o) = 2ti(C, - 60)

t12(cc, + iv)2 - (6, - &0)2 ’

Y =

and

(6.30)

(6.3 1)

(6.32)

(6.33)

In Eq. (6.33),jo,(q) and G:(q) are the Fourier transforms of the transition current density and the transverse part of the electromagnetic propagator of Eq. (2.25) respectively. The matrix equation problem for the present case takes a particularly simple form in comparison to that of Eq. (4.63), viz.

(6.34)

0. Keller I Physics Reports 268 (1996) X-262

where

and

(6.35)

(6.36)

Recently, the formalism described above has been used to study the electromagnetic response of a single-electron quantum dot with an isotropic parabolic confinement potential [130]. The dot is

embedded in a homogeneous medium described by a space-independent dielectric constant e(u). Without loss of generality we let the (plane-wave) incident field be polarized along the x-direction of our Cartesian coordinate system. Isotropy then implies that the only relevant excited states are those with excited orbitals along e,. The composite quantum numbers 0 and y1 in Eq. (6.29) thus stands for “000” and “~~00”. Retaining only the leading power in P/Q, in the real and imaginary parts of N,” it is found that [130]

nrn( - l)WW

X3’*!? [(n + m)’ - 1](2”+m+ln!m!)i’2 (6.37)

where Q,+, = -(-1).1.3.5...(n+ m - 3), if y1 + m is an even number. If y1 + m is odd N,” vanishes. In Eq. (6.37), the quantity p is given by /J = (mwo/h)1i2, ha, being the energy separation between the neighbouring eigenstates of the oscillator, and q. = (o/c,)E”*(u). Intro- ducing the matrix

rI(c0) =

1 + a,(w)N; 0 us(o) N; ...

0 1 + az(o)N; 0 . . .

ai MN: 0 1 + a&)N; ...

-1

(6.38)

a straightforward calculation now gives

Yn = Klb4YY ?

where

1’: = (eRB/i$m)exp C - (&/4P’)] E” ,

(6.39)

(6.40)

E O being the amplitude of the incident plane wave. Having determined the various y,,‘s, the transverse part of the local field can be found from Eq. (6.30). The local-field resonances are

190 0. Keller / Ph.vsics Reports 268 (1996) KS-262

obtained from the poles of n((ti). In [ 1301 the local-field calculation presented above has been used to study the absorption cross-section of a GaAs quantum dot embedded in a Ga, _,Al,As medium. The nonlocal response of an assembly of two-level semiconducting spheres placed on a D- dimensional lattice (D = 0 (single sphere), 1,2,3) has recently been investigated by Ohfuti and Cho [131] on the basis of the transverse local-field loop [ 1321.

6.3. On the spontaneous decay and Lamb shif2

In atomic physics the radiative transition process of an electron between two atomic electronic levels has been studied for many years using various theories. The approaches fall in two main families, viz. “the perturbative QED family” [133-1361 where the field is quantized, and “the semiclassical family” [137-1391 where the field is not quantized. In the semiclassical approach (also called the self-field approach) one views radiative corrections as arising from radiation reaction effects due to the interaction of the atom with is own self-field [140-1461. The self-field approach (and also QED) predicts that an atom will decay “spontaneously” from an excited state with a characteristic time constant equal to the reciprocal of the Einstein A coefficient Cl473 for the transition. The self-field model also predicts that the light radiated during the transition will have a frequency slightly different from the electronic transition frequency (Lamb shift [148-1501). In the present section it is suggested that the spontaneous emission and the Lamb shift might be of importance in the context of the electrodynamics of quantum dots and small particles. In Section 5.5, the role of the above-mentioned processes was discussed briefly for a two-level quantum well

system. Let us now briefly review in a heuristic manner the self-field description leading to a determina-

tion of the Lamb shift and the spontaneous decay of a single-electron quantum dot (or atom) (see also the paper by Crisp and Jaynes [151]). By the assumption that the continuum states are not significantly excited by the electromagnetic field any state of the one-electron dot may be expressed as

y(r~ t) = C aj(t)$j(r) , (6.41)

where $j(r) are stationary-state eigenfunctions of the field unperturbed Hamiltonian. In the state Y (r, t) the quantum dot carries a current density given by

J(r, t) = - (e/2m) { Y *(v, t) [(h/i) V + eA (v, t)] Y (Y, t)

+ Y(r,t)[ - (h/i)V + eA(r,t)] Y*(r,t)} . (6.42)

By inserting Eq. (6.41) into Eq. (6.42) and assuming that A (v, t) is so small that the corresponding terms in J(r, t) can be neglected, one obtains

(6.43)

wherejpa(r) is the spatial part of the transition current density from state I/j’) to state la), and

pp&) = a&MW (6.44)


is the (pa)-matrix element of the density matrix. The transverse part of the current density, JT(r, t), creates a transverse electromagnetic self-field which when added to the transverse external field constitutes the prevailing field. The associated transverse vector potential, AT(r, t), in the Coulomb gauge is given by

PO A, (r, t) = A ;“’ (r, t) + & s JT(r’,t - Ir - r’lIc0) d3r,

Jr - r’) (6.45)

For source (Y’) and observation (r) points located inside (or in the vicinity of) the quantum dot the retardation Ir - Y’ I/co is so small compared with the characteristic time variation in JT(y, t) that Eq. (6.45) can be approximated by

J#, t)d3r’ . (6.46)

assuming the (monochromatic) external field to be constant across the quantum dot, also. At this stage it is adequate to rewrite the time dependence of the two terms in Eq. (6.46)

describing the interaction of the quantum dot with its own self-field in terms of the elements of the density matrix. In order to rewrite the last term on the right-hand side of Eq. (6.46) we use the result

s hT(r - r’)d3r’ = +U (6.47)

to replace JT by J under the integral sign, i.e.

J&‘,t)d3r = ki,(~‘-+J(~,t)d~rd~~‘=+

Next, by inserting Eq. (6.43) into Eq. (6.48), and utilizing the relation

c jdr) d 3 I = iw,,, P,, ,

where Q,~ = (c, - c,)/h, and Pox is given by Eq. (6.22), it is realized that

s JT(r', t)d3r’ = $ c pBa(t)%gPol . 3L.P

(6.48)

(6.49)

(6.50)

Since dpBa(t)/dt 2: - iugaPBJ (neglecting the small perturbation stemming from AT (see below)), the last term in Eq. (6.46) can be written in terms of the ppil(t)‘s as follows:

PO d I 3 I --_

47x, dt JT (r’, t) d ?” = & Ix Pax(t)Q$&, *

1.P (6.5 1)

192 0. Keller / Ph_vsics Reports 268 (I 996) X-262

The middle term on the right-hand side of Eq. (6.46) can be rewritten starting from the identity

where the Qq-integration is over the entire 47r solid angle in q-space. Hence,

JT(q, t)eiq”dSZq dy ,

(6.52)

(6.53)

where JT(q, t) is the Fourier transform of JT(r’, t). Since J(q, t) = C,,pppz(t)jpl(q), it follows that

JT (r’, 0 ~ d3r’ = $ C psi(t) IY - Y’I l,lr

jL(q)eiq.‘dQqdq , (6.54)

wherej;f,(q) is the Fourier transform of the transverse part,j&(r), of the transition current density, j,,,(r). Introducing finally the Fourier transform ofjz(q)/q’, i.e.

1 G(r) = (2Tc)3 ~ qp2j,&(q)eiq”‘d3q ,

s

the second term on the right-hand side of Eq. (6.46) becomes

Altogether, we have thus found

A*(r,t) =Ay(O,t) + PO 1 P/3&) G(r) +zpu. . a.B [ 1

The elements of the density matrix evolve in time according to

ih dP,, dt = C CHljPjm - pljH' ] .P 7

j

(6.55)

(6.56)

(6.57)

(6.58)

and to see the influence of self-field interactions in the single-electron quantum dot on the time evolution, the expression for Ar(r, t) given in Eq. (6.57) IS inserted in the interaction Hamiltonian (e/2vn)(p *AT. + AT-p) (the small nonlinear term e2 A+/(2m) being neglected). Doing so, it turns out that the Lamb shift originates in the term proportional to I&(Y) in Eq. (6.57), and the spontaneous decay of the quantum dot in the term proportional to 0,&P~~/(67r~~) [151]. It is interesting to notice that the Lamb shift essentially can be calculated neglecting retardation effects, whereas the spontaneous decay can be obtained only if the field retardation inside the quantum dot is kept.


Let me terminate this section by a brief return to the analysis of Section 6.2. As shown in Ref. [130] the meaning of the N,“-terms in Eq. (6.35) is related to the Lamb shift and the rate of spontaneous emission. Thus, it is found that

IV: = An0 + (i/2)r,, . (6.59)

where An0 is the Lamb shift of the jOO0) + ln00) transition, and Tno is the decay rate of the associated spontaneous emission. Specializing ourselves to a two-level system (excited state n = 1) the resonance condition for the local field is given by

1 + ai(oRES)N: = 0 . (6.60)

For an electronic collision frequency small compared to o 2 c&s, the complex resonance frequency becomes

(6.61)

It appears from this equation that the spontaneous emission decay rate adds directly to the electronic collision frequency, and this, as emphasized previously in this section, again demonstrates that it is wrong to include the spontaneous decay rate in the collision rate V. If special precautions are taken to reduce the electronic relaxation mechanisms one might be able to measure directly the rate of spontaneous emission from the dot. Using typical data for a GaAs dot, the Lamb shift is small, i.e. Alo/calo rr lop3 for realistic dot sizes [130].

7. Near-field electrodynamics and surface dressing of particles in motion

In recent years the interest in near-field optics [1.52-1721 has grown so dramatically that the domain by now appears as a new branch of physical optics [ 173,174]. One of the basic goals in the near-field optics is to achieve a spatial resolution on the atomic level, and important progress in this direction has been reported, lately [168, 1691. Once the spatial resolution enters the atomic length scale, macroscopic electrodynamics can no longer be used to analyse and predict near-field optical phenomena. At the present state of the art it thus appears meaningful and necessary to employ local-field techniques in the studies. In the present section, I thus present a brief and qualitative discussion of the role of local-field effects in near-field optics. After having analysed the electrodynamics of the so-called quantum tips, I turn my attention towards the most popular microscopic model in near-field optics, viz. the point-dipole model [ 175-1811. I use this model to illustrate the presence of the so-called configurational resonances [181-1831. We finish the discussion with a few remarks on the question: Where is the theoretical limit for the spatial

resolution in near-field optics? In the second half of this section, I discuss aspects of the local-field electrodynamics of charged

particles in motion. In particular, the electromagnetic dressing problem for particles moving in the vicinity of a surface is addressed. Although the analysis is limited to studies of well-localized electron wave packets, its basic aspects might be useful also for more complicated wave packet

194 0. Keller i Ph.vsics Reports 268 (I 996) 85-262

systems, e.g. mesoscopic ones. The section is finished by a brief discussion of the possibility that the electromagnetic surface dressing might result in the excitation of the so-called Cherenkov-Landau surface shock waves, provided the phase velocity of the moving particle exceeds that of the relevant surface eigenmodes.

7. I. Local jields in the context of near-Jield optics

In order to obtain a better understanding of the electrodynamics governing the interactions in near-field optics a variety of theoretical studies has been carried out in recent years. Broadly speaking, the studies might be divided into two main categories depending on whether they are based on macroscopic electrodynamics Cl8441871 or semiclassical microscopic electrodynamics [175,177-179, 181-183,188, 1891. Up to now the macroscopic investigations have constituted by far the predominant part. This is understandably since the discipline of near-field optics has evolved from classical optics. However, as the desire for achieving higher spatial resolutions increases the need for attacking the problem from a microscopic point of view tends to be more urgent. Once a microscopic approach is needed the local-field concept and the techniques for calculating local fields become of central importance. Although, in principle, the microscopic theory comprises the macroscopic approach in the limit, the numerical calculations involved in the microscopic formulation become so comprehensive that they often prevent us from studying the microscopic-to-macroscopic transition. Focusing for simplicity the efforts on the self-consistent electrodynamics of the tip, I shall in a qualitative manner seek to establish the link between near-field optics and local-field electrodynamics. Afterwards, I shall demonstrate how the microscopic standard model in near-field optics, viz. the point-particle model may be obtained from the general model. Then, I shall briefly discuss the so-called configurational resonances, and finally, a few remarks concerning the limits for the spatial resolution will be given.

7. I. I. Quantum tip

It is of key importance in near-field optics to understand on the microscopic level the electrodynamic interaction between the probe tip and the particles in the surface structure under investigation. For a macroscopic probe, the geometrical form of the tip and its refractive index are the quantities characterizing the electrodynamic properties of the object. Once these quantities are given one seeks a rigorous solution of the macroscopic Maxwell equations for the tip-surface (plus possibly bulk) system. For the macroscopic surface-bulk system being probed by the light one needs as input parameters the topography and the (inhomogeneous) dielectric tensor. To be able to solve the macroscopic Maxwell equations in the above-mentioned situation it is usually necessary to assume that the tip has a particular simple form, e.g. spherical (also the surface topography and the inhomogeneity of the dielectric tensor of the bulk of course have to be simple). For spherical tips of macroscopic size one often uses the well-known long wavelength electrostatic expression for the polarizability of a homogeneous sphere. In the limit where the radius of the sphere is much less than the optical wavelength the polarizability thus is given by the Rayleigh expression in Eq. (1.11).

Once the Rayleigh formula is used in combination with the assumption that the tip radiates as an electric point dipole the internal dynamics of the probe is neglected. To account for the internal dynamics of the probe one often models the tip by an assembly of electromagnetically interacting point dipoles emitting electric-dipole and possibly magnetic-dipole plus electric-quadropole and

0. Keller I/ PhJassic.s Repwts 268 (1996) X-262 195

higher-order multipole fields [175, 17771791. While such an approach might be adequate for mesoscopic tips consisting of units with weakly overlapping electronic orbitals it certainly would be too simple for systems, e.g. metal or free-electron like semiconductor probes, with strong electronic coupling across the domain of the tip. Furthermore, the point-dipole model, and its extension to an assembly of point dipoles, generally is not able to account for quantum size phenomena, known to be important in a number of mesoscopic systems.

To go beyond the above-mentioned framework it appears natural to take as a starting point the local-field formalism described in Part A, and, in particular, its application to mesoscopic particle electrodynamics (Section 6) see also [190]. Thus let us consider the situation depicted in Fig. 7.1. A monochromatic electromagnetic field of angular frequency CO is incident upon a system composed of a flat surface, a number of point dipoles placed on top of the surface, and a so-called quantum tip (mesoscopic particle) of finite size. To determine in a self-consistent manner the local field inside the quantum tip one begins with the integral relation (see Eq. (2.66))

E(r) = P(r) - ilOw GB (Y, v’) .J(v’) d”l-’ , (7.1)

where EB(r) denotes the background field, here the sum of the incident field and the held reflected from the surface in the absence of the surface dipoles and the quantum tip, and GB(r,r’) is the associated pseudo-vacuum (background) propagator. In terms of the propagator GB(s, z’) given in Eq. (5.3) the present Green’s function is

GB(r, i) = (27r- 2 s

S-l.GB(z,Z’).Seiq~~‘~‘~~~ri’d2y, , (7.2)

where S is the rotation matrix characterizing the appropriate rotation around the z-axis (the rotational transformation brings the vector ql into qlle,). The light-induced current-density distribution inside the quantum tip, in the following often named the quantum particle (QP), and on the

a b

Fig. 7.1. Schematic diagrams illustrating the various channels of electromagnetic field propagation from a source point r’ to an observation point r, both points being located inside the quantum particle (tip) (QP). The channels shown in (a) are those which do not involve scattering on the point dipoles (PD) of the surface (S) (the point dipoles are indicated by the blackened circles). In (b) are shown the four channels (r’ +PD+r,r’+PD+S+r, r’+S+PD+r,

r’ + S + PD 4 S + r) which involve scattering on the point dipoles. The scattering strengths of the point dipoles and the source strengths at the various points inside the quantum tip are calculated in a self-consistent manner, which takes into account multiple scattering effects to infinite order, cf. the description in Section 7.1.1.

196 0. Keller : PIl?ssics Reports 26X (1996) 85-262

point dipoles at the top of the surface we assume is given by

>V J(V’) = 2 Sj(CL))sE(rj)G(r - Yj) +

.j= 1 s o(r’, Y”). E(r”) d”r” , (7.3)

QP

where S,(U) is the conductivity tensor of the jth point dipole, located at position ~j, and (T(Y’, v”) is the conductivity tensor of the quantum particle in the RPA approach. To proceed from here one needs an explicit expression for the RPA conductivity tensor. Let us hence assume that the paramagnetic contribution (given by Eq. (3.112)) is the dominating one. In this case, Eq. (7.1) with, Eq. (7.3) inserted takes the form

E(r) = EV) + 1 P,?I,iF,&) t i T(r,rj)*E(rj) ) (7.4) ,?I. ,t j= 1

where in the usual notation

GB(r, v’) +j,Jr’) d3y’ , (7.5)

and

T(v, rj) = - ipocOPB (y, Vj) ’ Sj , (7.6)

PB being equal to G” minus its self-field part. To obtain the local field from Eq. (7.4) we need to calculate the prevailing fields, E(rj), ( j = 1,2, . . . , N), on the sites of the N dipoles, and the unknown

constants

Pm,, = s

j,,(r) - E(v) d3r . (7.7) QP

If one compares with the coupled-antenna technique, used in, e.g., Section 4.3.1 to obtain the local field inside a mesoscopic medium, a new feature appears here since we have to determine the yet unknown fields on the surface dipoles. We achieve our goal in two steps, Firstly, we calculate the E(ri)‘s in terms of the unknown Bmn’s. This is done by letting r in Eq. (7.4) in turn be equal to

Tl,YZ, ... 3 rN. From the resulting super-matrix problem the E(r,j)‘s are found in terms of the /jmn’s. Secondly, by inserting the expressions obtained in this manner for the E(rj)‘s into Eq. (7.4) one is able to determine the Brnn’s in the standard way from a set of linear and inhomogeneous algebraic equations. The final result one obtains is as follows:

E(v) = F(Y) + c pm,* Fnm(Y) + z”(r). [i? - .F] ~ l * [a” + 1 ljMn’F,*)n] . (7.8) in. II m.n

where

(7.9)

0. Keller i Physics Reports 268 (199~3) KS-262 197

(7.10)

are supervectors and supertensors, respectively, and )? is a super unit tensor, The set of linear algebraic equations from which the Pmn’s can be calculated is

bmn - 1 N&“&, = H,,, , (7.1 1) 0.P


N$’ = .bAW’po(r) d3r + j,Jr) . &(r) d”r 1 .[L.F]p.Fp, (7.12)

Hmn = s QP j,,(r)@+) d3r. + 1

.[+-.?]-l.&B. (7.13)

The relations in Eqs. (7.8) and (7.11) form an adequate starting point for a quantitative numerical calculation of the optical near-field (and middle- and far-field) interaction between a quantum tip and an assembly of surface point-particles.

In the description given above it was assumed that the surface particles electrodynamically

behave like point dipoles. This assumption allowed us to focus the attention on the internal dynamics of the quantum tip. It is apparent, however, that it often will be necessary to go beyond the point-particle approach when treating the dynamics of the surface particles. For a dielectric surface, where the electronic overlap between the atomic (molecular) orbitals of different particles is small each surface particle might be treated as a quantum particle (quantum dot) and the scheme layed out above can be followed provided each term in the summation in Eq. (7.3) is replaced by an integral over the particle concerned. It might happen also of course that not only the top layer of surface particles has to be treated as composed of discrete units. In cases where the electron overlap between neighbouring particles is considerable it tends to be meaningless in general to start with a treatment in which a discretization is invoked from the outset. Abandoning the individual particle approach an adequate starting point might be one in which the surface (and possibly the bulk) is described via the underlying band structure. Provided that the transverse-current density domain of the quantum tip does not overlap the surface (or rigorously speaking the transverse current-density domain occupied by the surface structure), a division of the electron current density of Eq. (7.3) (into two parts) is still possible. The first part, replacing the summation over the N individual surface particles, is determined via a surface band structure calculation. The second part, which is due to the quantum tip, is unchanged. The resonance condition for the local field of the entire tippsurface particle-bulk system is given by Eq. (4.109) inserting for NY; the expression in Eq. (7.12). Inside this general resonance condition is hidden the resonances of the various subsystems, e.g. the configurational resonances of the system of surface dipoles (to be discussed in

198 0. Keller ! Physics Reports 268 (1996) 85-262

the subsequent subsection), the bulk and surface plasmon and polariton resonances of the vacuum-surface system, and the single-particle (possibly inter-level transition) and collective resonances (localized plasmons) of the quantum particle. Whether or not these subsystem resonances play a significant role for the resonances of the entire system among other things depend on how strongly coupled the subsystems are.

In some cases, e.g., for a dielectric quantum tip consisting of weakly bound molecules or atoms, the effective dimension of the matrix NT; appearing in Eq. (7.11) is drastically reduced, and thus the quantitative numerical calculation of the local field a priori becomes easier. To realize this let us compare the energy level diagram for the entire quantum tip with that of a system of subparticles without wave function overlap. If the quantum tip has N energy levels, one has N(N - 1) unknown

fintn’s since the diagonal elements, /3,,,, are zero. Now, if the tip consists of M electronically decoupled subparticles and subparticle number i has Ni energy levels, the number of unknown constants in Eq. (9.11) will be C;E 1 Ni(Ni - 1). With the constraint N = 1;: 1 IV, it is obvious that the effective dimension of the matrix NY; is reduced when the inter-subparticle electronic coupling tends to zero. As an example, if N = 4 and N1 = N2 = 2 the number of unknown constants is reduced from 4(4 - 1) = 12 to 2(2 - 1) + 2(2 - 1) = 4. The replacement of a four-level quantum particle by two two-level subparticles thus reduces the dimensionality of the matrix problem considerably. Also symmetry can lead to a reduction of the effective dimensionality of the problem

at hand.

7. I .2. Point-particle approach and configurational resonances

Among the microscopic models in near-field optics the most popular one with no doubt has been (and still is) the so-called point-particle (or electric point-dipole) model [175-181-J. Its popularity is associated with the fact that it usually leads to manageable numerical problems, and also I believe it is appreciated among experimentalists because of its conceptual simplicity. As the research in near-field optics moves towards higher spatial resolutions it seems to me that the point-dipole model becomes less useful.

Whether or not a finite-sized particle (quantum tip, subparticle of a tip, or surface particle) electrodynamically can be considered as a point-like entity in a given case is a basic but difficult question of interest in its own right. To indicate the challenge one is confronted with it is sufficient to consider the quantum tip problem discussed in Section 7.1.1, since an analogous discussion can be carried out for a surface particle or subparticle of the quantum tip. It appears from Eq. (7.4) that the external field acting on the quantum particle is given by

EeXt(r) = EB(r) + 2 T(r, Vj) ‘E(rj) . (7.14) .j= 1

The quantum particle behave as a point-particle if the external field can be considered constant across the particle domain, cf. the discussion in Section 61.1. Although the background field usually is the same everywhere inside the particle, the last term in Eq. (7.14) certainly may give rise to appreciable field variations across the particle region when the near-field zone of the neighbouring particle(s) tends to overlap the quantum particle. Provided the quantum particle can be considered as an electric dipole receiver and radiator the current density JO@‘) of the quantum


particle is

J,(f”‘) = &((0).E(ro)G(r’ - r()) )

where, cf. the expression for aiFiED(w) in Eq. (6.4)

(7.15)

So(w) = i‘

oRpA@, r’) . rRPA@‘, Y”) d3r” d3r’ d”r (7.16) QP

is the conductivity tensor of the quantum particle (particle number equal to 0), which is supposed to be located at ro. To line up with the Eq. (7.3) we have also used the notation EeX’(vO) E E(r,). If local-field effects inside the quantum particle are neglected one has

So(o)) = s

csRPA(r, r’) d3r’ d3r . (7.17)

By inserting the RPA expressions for the paramagnetic and diamagnetic conductivities into Eq. (7.17) one obtains

So(o) = ie2U

m(w + +)

No(r)d3r +&C h;t,f:"r;, ""E P',P,,,, , ‘rn n

(7.18)

i.e., the phenomonological result of the electric point-dipole approach. Having established the transition from the mesoscopic particle approach to the electric point-

dipole model, let us consider now the local-field problem for N point particles (including the probe tip, also if this is composed of a number of subparticles). In this case, Eq. (7.4) is replaced by

E(r) = EB(r) + : T(r,rj)*E(rj) . j= 1

(7.19)

By letting r coincide in turn with each of the dipole positions one obtains a set of N linear, algebraic equations among the fields E(ri) s Ei (i = 1,2, . . . , N) prevailing at the dipole positions, i.e.

Ei=Es + ; Tij*Ej) (7.20) j= I

where EB G EB(ri) and Tij G T(ri,rj). When using the point-particle approach it is necessary by brute force to leave out the direct part and the self-field part of the propagator from Tij because those parts diverge. As we know, this divergence is an artifact stemming from the assumed delta function character of the induced current density. Written in supertensor notation, Eq. (7.20) takes a particularly simple form, namely

&&B+&g,

where

(7.21)

(7.22)

If the system is far from resonance the coupling is weak and the local fields at the dipole sites thus can be obtained directly from a Born series expansion

(7.23)

The weak-coupling approximation, however, breaks down when the system is close to resonance. In such a case the exact solution

&‘=(&~)-r.&~ (7.24)

must be used. It appears from Eq. (7.24) that the condition for obtaining resonant interaction between the tip and the system of surface particles (plus the bulk) is given by

Det{;M’-;$] =O. (7.25)

It is a demanding task to analyse the resonance condition in detail. In the context of near-field optical microscopy there is however one type of resonances of particular interest namely the so-called configurational resonances [lSl, 1831. To introduce the configurational-resonance concept let us imagine that a specific frequency of light has been chosen. In general this frequency does not lead to a fulfillment of the resonance condition in Eq. (7.25). By changing the spatial configuration {rr, r2, . _ , Y,~ > of the particles one might be able to meet the condition in Eq. (7.25). If so, one has found a configurational resonance. In practice, the relative positions of the surface particles are fixed, so that configurational resonances can be sought for only by moving the tip particle(s) around. If the tip needs to be modelled by a set of subparticles the relative positions of the subparticles are kept fixed. In near-field microscopy information on the surface structure is usually obtained by scanning the probe tip over a certain surface area. In such scans it is in many cases necessary to have as strong a coupling as possible between the probe and the surface system. Studies of the configurational resonances are often helpful in this respect because they might allow one to find the optimal distance(s) of the tip from the surface. Afterwards, one can perform the experiments under resonance (or near-resonance) conditions. Even if one confines oneself to investigations of configurational resonances it is a tedious (or impossible) task to carry out a resonance analysis for a large assembly of dipoles, in particular if one wants to incorporate the middle- and far-field parts of the propagator and/or realistic expressions for the p- and s-polarized bulk reflection coefficients and for the polarizability of the dipoles. A quantitative study of configurational resonances can be carried out in cases where only a few dipoles and the bulk are present. A conceptually simple case is the one where the system consists of a tip dipole and single surface dipole placed above a flat surface. The most simple case is that of a single tip particle above a plane surface. Schematic diagrams showing the various interaction channels for the one and two dipole cases are shown in Fig. 7.2. A detailed account of the quantitative aspects appearing in these few-dipole systems can be found in 11821. Quite recently, also the configurational resonances appearing in near-field optical microscopy with a mesoscopic metallic probe have been studied [183].

Although from the name near-field optics one might believe that only the near-field part of the electromagnetic propagator, i.e. cg(U - 3eReR)/(4 mo2R3), is of importance this is certainly not necessarily the case. Also the middle-field and the interference between near- and middle field effects seem to be of significance in this new field of optics, at least from theoretical estimates.

0. Keller / Physics Reports 268 (1996) X5-262 201

Fig. 7.2. Schematic diagrams showing systems with one or two dipoles on top of a surface. The z-coordinates of the tip and surface dipole are denoted by ~ R, and - R,, respectively. The actual electromagnetic interaction channels are shown by the various lines. Thus: (a) one dipole interacting with the surface; (b) two dipoles on top of each other with the interaction between the upper dipole and the surface neglected; (c) two dipoles on top of each other with all interactions

taken into account, (d) two dipoles having different x coordinates with all interactions included.

7.1.3. On the spatial resolution in near-jield spectroscopy

A number of recent experiments have demonstrated that it is possible in near-field optics to overcome the spatial resolution limit set by classical diffraction theory. From a theoretical point of view it has also been clear for a long time that it certainly should be possible to by-pass the classical diffraction limit. Having surmounted this limit, it is natural to ponder over(i) where the limit for the spatial resolution in near-field optics then is and (ii) which kind of physical mechanisms determine this limit. Although different answers to these questions have been put forward in recent years, it seems to me that we still lack the analyses to provide us with a satisfactory answer to the above- mentioned questions. With emphasis on the qualitative physical aspects, let me finish this section on near-field optics with a brief discussion of the basic problems we are facing when addressing the question of the spatial resolution.

One might be tempted to believe that there is a close relation between the geometrical size of the probe tip and the spatial resolution. For a macroscopic sized probe the spatial resolution for a fixed wavelength is certainly improved if the tip is made smaller. Once the tip reaches a mesoscopic size the situation changes however. Let us assume for a moment that the tip is a point-dipole. In general, the solutions to the characteristic equation (Eq. (7.25)) are complex so that the self- consistent field at resonance has a finite value. Even if all electronic damping mechanisms in the system were removed the peak heights of the resonances and their half-widths would be finite. Such a conclusion cannot be reached if one at the outset approximate the electromagnetic propagator by its near-field part ( cc RP3), as is often done, because, as we have realized from the discussion of the Lamb shift and the spontaneous emission in Sections 5.5 and 6.3, peak heights will always be finite, due to the inevitable presence of retardation effects. So, even if one adopts the point-dipole model the peak heights are always finite, and a calculation of the “natural linewidth” requires that one goes beyond the near-field approximation for the electromagnetic propagator.

Approaching the point-dipole limit we run into other problems as well. A classical analysis already points to one of those problems, viz. the polarizability of the probe tip. From the Rayleigh

expression (Eq. (1.11)) for a spherical tip it thus appears that the polarizability goes to zero as (i3 (a being the particle radius). From a quantum mechanical point of view the situation may be even worse as we have seen in Section 6.1. It is known for instance that the diamagnetic polarizability of a metallic sphere goes to zero faster than a3, cf. Fig. 6.1.

Instead of applying a mesoscopic tip it seems promising to use a molecular (or atomic) tip in order to reach atomic resolutions in near-field optics. In such a case it would be tempting to guess that single molecules (or atoms) of the surface could be resolved. One must remember however that on the microscopic level the electrodynamics is governed not by the induced microscopic (molecular, atomic) current-density distribution itself but rather by its transverse and longitudinal parts. It occurs to me that this fact indicates that the resolution is lost before the induced current-density distributions of the tip molecule (atom) and surface molecule (atom) in question start to overlap. It might thus be correct to claim that the spatial resolution in this context is limited by the spatial extension of the induced transverse (or equivalently longitudinal) current density. Following up on this claim it is of interest to make quantitative comparisons between the spatial distributions of the induced current density and its transverse and longitudinal parts for the system under

consideration. In actual cases it is of course difficult, even by numerical methods to determine the transverse

and longitudinal parts of a given current-density distribution. In the present context it is sufficient to consider a heuristic case namely that of a hydrogen atom (or hydrogen-like quantum dot) excited from the 1s to 2p, energy level. Described in a spherical coordinate system (I, 8, q) with unit vectors e,.,~, e,, the 1s + 2p, transition current density is given by

J ls.zPZ(r) = A[e,(2 + ~/uco)cosO - 2e0sin0]rPh’, (7.26)

where A = &/(16$imoz) and h = 3/(2a0), a0 being the (effective) Bohr radius. Due to the

rotational symmetry of the involved states around the z-axis, the current density is independent of the azimuth angle 43, and is for an arbitrary position vector r always confined to the e,-eZ plane. A tedious calculation shows that the transverse and longitudinal parts of the current density in Eq. (7.26) are

where

JT= 0

-!!!sin() l+L+_ 2 __ 2

hr (hr)2 + (hr)3 1 I _~ 2

3 (br)3 ’

1 ___ 8 1 (br)3 ’

3+br 4 8 8 8 - ~ __

+ G + (br)2 + (br)3 1 1 (br)3

(7.28)

(7.29)

(7.30)

(7.3 1)

0. Keller / Ph,vsics Reports 268 (1996) 85-262 203

It appears from the expressions in Eqs. (7.28))(7.31) that the tail of the transverse (and longitudinal) part of the transition current density has the form

= (rA/3,1Zir,zLz~)(l/(hr)“)(essinH + 2e,cos0) (7.32)

The transverse current density thus decays with a rate ye3 which is much slower than the exponential decay rate of the transition current density itself. The electromagnetic field radiated during a 2p, + 1s transition of the electron spreads out from the atom as time goes on, but Eq. (6.14) tells us that the field can never be confined to a region smaller than that of the transverse current density. Since photons are generated by JT and not by J, one might argue that the best spatial photon confinement which can be achieved (at least in the present case) in a field-quantized description is given in a qualitative sense by the C3 -tail behaviour. Thus, when the overlap of the I -3-tails of the tip and surface atoms increases the spatial resolution is gradually lost. In a photon statistical sense tail overlap means that one cannot distinguish whether an emitted photon originates from the tip or surface atom.

7.2. Local-jield electrodynamics oj‘moving electron wave packets subjected to surface dressing

In Section 6, recent efforts to elucidate the role of local-field effects in small particles and single-electron quantum dots were described, and in Section 7.1 microscopic theories of near-field optics with mesoscopic tips were reviewed. In the above-mentioned studies it was implicitly assumed that the individual particles had no centre-of-mass motion. It is obvious, however, that local-field electrodynamics might be of central importance also in mesoscopic systems where it is indispensable to take into account centre-of-mass motions. Let me mention just two examples. Thus if an oscillating atomic dipole is incident on a surface at almost grazing incidence, and if the surface carries an electromagnetic surface wave (or the evanescent tail of an electromagnetic wave subjected to total reflection from the medium side) the path of the atomic particle can be bend in the spatially inhomogeneous local field prevailing outside the surface [191]. The second example is related to the research on light emission from tunnelling electrons [167, 169, 192-2071. Hence, during its motion in the tunneling process the electron is subjected to the local field in the gap, and as far as the motion parallel to the surface is concerned it seems that the electromagnetic surface dressing might play an important role for the character of the emitted light as we shall briefly discuss in Section 7.3. Below, elements of a new local-field theory describing in a self-consistent manner the external electromagnetic dressing of a moving and well-localized electron wave packet are presented. The new theory may enable one to study the dynamic radiative reaction from the walls on the motion of a charged particle in a microwave or optical cavity.

7.2.1. The time-dependent local-jield problem Let us consider an electron wave packet propagating in space initially under the influence of

a prescribed external electromagnetic field, and let us assume that in the vicinity of the wave packet we have a macroscopic condensed medium (see Fig. 7.3). In the absence of the electron wave packet, the local field equals the background field E”(r, t). This field induces a time and space varying

204 0. Keller ! Ph.vsics Reports 268 (I 996) 85-262

Fig. 7.3. Schematic illustration of an electron wave packet moving under the simultaneous influence of the so-called interna electromagnetic dressing, stemming from the direct interaction of the electron wave packet with its own field, and surface dressing, associated with the radiation reaction on the electron intermediated by a nearby macroscopic medium.

current-density distribution inside the wave packet, and in turn this distribution gives rise to electromagnetic radiation from the electron. Upon reflection from the macroscopic system the radiation acts back on the electron, modifies the internal current density and changes the propagation of the wave packet. The radiation reaction on the electron caused by interaction with the macroscopic medium we shall call the external electromagnetic dressing. In a Green’s function approach the electromagnetic interaction with the macroscopic medium is described in terms of the indirect propagator, here named I@, r’, t - t’). In letting the indirect propagator depend only on the time difference t - t’ it has been assumed that the macroscopic medium possesses time-independent electromagnetic properties. As a starting point for our analysis we thus take the following integral relation between the local field and the current density of the electron wave packet:

E(r, t) = EB(V, t) + /Lo sI^ I(r, r’, t - t’) * Wr’, t’) d3r’ dt’

at, 2 (7.33)

where the spatial integration extends over the domain of the electron wave packet, and the t’-integration over the interval - x < t’ d t in order to satisfy the principle of causality. In writing down Eq. (7.33) we have neglected the internal electromagnetic dressing stemming from the direct interaction of the electron wave packet with its own field. The internal dressing gives rise to a radiation reaction on the electron. For a strongly localized wave packet, this reaction can be described via the second term on the right-hand side of the expression for the direct propagator in Eq. (6.13). In the frequency domain the radiation reaction (RR) thus gives a contribution

ERR(W) = - &- ‘0 ?I ! J(r; w) d3r (7.34)

0. Keller ! Physics Reports 268 (I 996) 85-262 205

to the local field. In one inserts the electric-dipole momentp(o) (given in Eq. (6.2)) in Eq. (7.34) and afterwards transforms the resulting equation back to the time domain one obtains

ERR(t) = (1 /6ne0c;)d”p(t)/dt3 , (7.35)

i.e. the classical expression for the radiation reaction on a point-particle [43]. It is a straightforward matter to show that the local field associated with the radiation reaction on the electron wave packet is identical to that obtained in the atomic case (the last term on the right-hand side of Eq. (6.46)). In the atomic case the internal dressing gives rise to a Lamb shift also, as we have discussed in Section 6.3. The Lamb shift is related to the first term on the right-hand side of Eq. (6.13), or equivalently the second term on the right-hand side of Eq. (6.46). The corresponding term in the electron wave packet case causes a renormalization of the electron mass.

In the nonrelativistic domain the current density of the electron is given by Eq. (6.42). The wave function Y(r, t) of the electron satisfies the time-dependent Schrodinger equation

HY(r, t) = ih(aY(Y, t)/at) , (7.36)

and in the present case it is adequate to take the Hamilton operator in minimal-coupling form, i.e.

H = (1/2YM) (p + e‘4 (r, t))2 . (7.37)

Since the scaler potential is zero in the minimal-coupling guage, one has E(r, t) = - M(r, t)/at.

7.2.2. Electric-quadropole and magnetic-dipole current densities

In the following it is assumed that the electron wave packet is so well-localized in space that it is adequate to make an expansion of the current density in Eq. (6.42) around the centre, R(t), of the wave packet, the position of which is given by the mean value of the position operator, i.e.

R(t) = (r)(r) = s

Y*(r,t)rY(r,t) d3r (7.38)

The expansion is in terms of the Dirac delta function and its spatial derivatives of increasing order. Hence,

J(r,t) = [ jJ(r.t)d’+-R(A)

-is J(r, t)(r - R(t)) d3r 1 . V&r - R(t)) + ... (7.39)

In this equation appears the zero (1 Jd3 ) r and first-order (l Jr d3v) (tensorial) moments of the current density. As demonstrated below these have a clear physical interpretation. Before establishing this interpretation it is convenient, however, to introduce the standard abbreviation

(O)(r) = j

Y*(~,t)OY(r,t)d~r (7.40)

206 0. Keller 1 Physics Reports 268 (1996) 85-262

for the mean value of the operator 0 (cf. Eq. (7.3X)), and also the mechanical momentum operator

ll=p+eA. (7.41)

The mean value of n is related to R via (n)(t) = m dR(t)/dt. Returning to Eq. (7.39) one obtains (see Appendix Dl)

s dR(t) J(r,t)d3r= -;(“)(+ -edt, (7.42)

showing as expected that the zero-order moment of the current density equals the time-derivative of the electric-dipole moment of the wave packet around the origin of the coordinate system. A tedious calculation, of which a few of the crucial steps are presented in Appendix D2, results in the following expression for the first-order tensorial moment of the current density:

I Jk t)rd3r = ; (Q>(t) - & U x (L)(t) , (7.43)

where

Q = - (e/2)rr

is the electric-quadropole operator, and

(7.44)

L=rxll (7.45)

is the mechanical angular momentum operator, both operators taken with respect to the origion. The first-order moment of J thus consists of a sum of electric-quadropole and magnetic-dipole contributions. The first one equals the time-derivative of the mean value of the electric-quadropole operator, and the second one is proportional to the vectorial product of the unit tensor and the mean value of the mechanical angular momentum operator. As one would expect only the sum of the EQ and MD contributions occurs.

It is of interest for the discussion of the subsequent subsection to analyse Eq. (7.43) a bit further. Thus, from the equation of continuity

I7 - J(r, t) + (a/&)( --el Y(r, t)12) = 0 , (7.46)

follows the identity

-; nr;l'I'(r,t),'d"r= -; s s

rrV.J(r,t)d3r. (7.47)

The left-hand side of this equation is just the time-derivative of the mean value of the electric- quadropole moment operator. By performing a partial integration on the right-hand side of Eq. (7.47) one thus obtains

$(Q)(t) = l s

[rJ(r, t) + J(r, t)r] d3r , (7.48)


since J vanishes at the integration limits. The equation above shows that the time-derivative of the electric-quadropole moment is equal to a symmetrized first-order current-density moment. In turn, one finds immediately by inserting Eq. (7.48) into Eq. (7.43)

-$Jx(L)(t)=; [J(r,t)r-rJ(r,t)]d3r. i‘

(7.49)

As anticipated, the magnetic-dipole term is related to the antisymmetric part of the first-order moment of J.

It is instructive, and of importance for the analysis of the source field to be presented in the following subsection, to rewrite the two terms on the right-hand side of Eq. (7.43) in a different form. Hence, starting from the general equation for the time evolution of the mean value of the observable 0, i.e.

it is shown in Appendix E that

(7.51)

Inserting the expression for L given in Eq. (7.45) into the last term of Eq. (7.43) one obtains after some manipulations of the double cross product (cf. Appendix E)

-(e/2m)U x (L)(t) = - (e/2m)((Z7r)* - (rZZ)) (7.52)

Adding Eqs. (7.51) and (7.52) one then finds

s J(r, t)r d3v = - &( (LV) + (LJr)*) , (7.53)

7.2.3. Source jield of the electron wave packet

It appears from the analysis of the preceding subsection that the current density of the electron wave packet is given by

J(r, t) = JED(r, t) + JEQiMD(r, t) , (7.54)

if second- and higher-order spatial derivatives in (5 are neglected. In explicit form the ED and EQ/MD contributions are

JED(r, t) = - (e/m)(ZI)(t)S(r - R(t)) , (7.55)

JEQIMD(r, t) = - T(t)-C78(r - R(t)) , (7.56)

with

T(t) = (d/W (Q>(t) - WW-J x (L)(t) + WmW > (W(t) . (7.57)

208 0. Keller /Physics Reports 268 (1996) KS-262

To determine the local field in the various time-space points one needs to know the source field of the wave packet. According to Eq. (7.33) this is given by CV(v, t)/at, essentially. To calculate aJED/CIt, one makes use of the relations

(d/dt)(Z7)(t) = m(d2R(t)/dt2) = (F)(t) , (7.58)

where

F= -e[E+(1/2m)(zzxB-Bxn)] (7.59)

is the Lorentz force operator. The electric (E) and magnetic (B) field operators appearing in Eq. (7.59) are those of the local electromagnetic field. Since &?(r - R)/& = - dR/dt . P6(r - R) one obtains

t3JED(r, t)/at = - (e,‘m)[ (F)(t)(s(r - R(t))

- (llm)(n>(t)(n>(t).~~(r - R(t))1 . (7.60)

If, consistently, one neglects the second-order spatial derivatives in 6, the EQ/MD source field becomes

aJEQiMD(r, t)/at = - (dT(t)/dt).VcZ(r -R(t)) . (7.61)

To determine dT/dt we start with a calculation of d2(Q)(t)/dt2. Using Eq. (7.50), with 0 = [ - e/(2m)] (nv + vZI), one obtains (for details of the derivation, see Appendix F)

(d2/dt2)(Q)(t) = - (e/m) (1 /mZlll + i(Fr + rF)) (t) (7.62)

Next, by utilizing the equation of motion for the mean value of the mechanical angular momentum, viz.

(d/dt)(L)(t) = 4 (r x F - P x r)(t) , (7.63)

one obtains, as shown in Appendix F

(d/dt)(-(e/2m)U x (L)(t)) = - (e/2m)$(Fr - rF)(t) + c.c.) . (7.64)

Since

(7.65)

and (Fr + rF) is a real quantity, the expression for aJ EQiMD/i3t thus can be written in the form

aJEQiMD(r, tyat = (e/m)[(l/m)((nII)(t) - (ZI)(t)(I7>(t))

+ f((Fr)(t) + C.C.) - (r;)(t)(r)(t)] *Vd(r - (r)(t)) . (7.66)

0. Keller / Physics Reports 26X (1996) 85-262 209

Our final expression for the time-derivative of the current density of the electron wave packet is obtained by adding Eqs. (7.60) and (7.66). Hence,

aqr, t)jat = - (e/m)(F)(t)G(r - (r)(t))

+ (e/m)[(l/m)(ZIZI>(t) + t((Fr)(t) + cc.)

- vxt)(rXt)l *me” - (r>(t)) . (7.67)

7.2.4. Electrodynamics of a surjbce-dressed ED wave packet

The local-field electrodynamics of our strongly localized electron wave packet may be investigated starting from Eq. (7.1) with Eq. (7.67) inserted in the following manner. Using the Maxwell equation V x E = - aB/at one derives from Eq. (7.1) an expression for the magnetic field. With a knowledge of E and B, implicit expressions for the mean values (F)(t), (Fr)(t), and (Fr)*(t) can be established. These expressions in fact constitute a set of coupled integral equations among the above-mentioned mean values.

To achieve conceptual insight in the surface-dressing problem we now assume that the electron wave packet is so well localized that only the electric-dipole electrodynamics needs to be addressed [208]. Neglecting also the magnetic field part of the Lorentz force operator, one obtains from Eqs. (7.42) and (7.60)

aJED(r’, t’)/at’ = e(dR(t’)/dt’)(dR(t’)/dt’). V’d(r’ - R(t’))

+ (e”/m)(E)(f’);S(r’ - R(t’)) . (7.68)

Inserting this expression into Eq. (7.1) and performing the integration over r’, one obtains

E(r,t) = EB(r,t) + 5 s

I(r,R(t’), t - t’) - (E)(t’)dt’

- I”oe dR(t’) dR(t’) dt’ -.B’I(r,R(t’),t - t’)dt’, dt, (7.69)

where V’I(r,R(t’),t - t’) = V’I(r,r’,t - t’)l,.,=RCt,J. It appears from Eq. (7.69) that once (E)(t) and R(t) are known, the prevailing electric field can be calculated everywhere in space by direct integration. The central problem of calculating (E)(t) and R(t) in a self-consistent manner is addressed by multiplying Eq. (7.69) by 1 $(r, t)l*, and then integrating the resulting equation over the space domain. Since 1 $(r, t)12 = 6(r - R(t)) in the lowest-order approximation one obtains

(E)(t) = (EB)(t) + $ s

I(R(t),R(r’), t - t’) - (E)(t’)dt’

- he s dR(t’) dR(t’) dt’ -.V’I(R(t),R(t’), t - t’)dt’ . df (7.70)

Neglecting the magnetic field operator, the equation of motion for the centre of the wave packet is, cf. Eq. (7.58)

m(d’R(t)/dt’) = - e(E)(t) . (7.71)

To determine (E)(t) and R(t) in a self-consistent manner one thus has to solve the set of coupled equations in (7.70) and (7.71). Although it is a formidable task to solve this set of equations even for a flat surface, it seems to me that in order to understand in near-field electrodynamics for instance the light emission from tunnelling electrons or electrons moving close to a surface as such, a rigorous investigation of Eqs. (7.70) and (7.71) is needed. Also in the context of the cavity quantum electrodynamics, Eq. (7.70) (with Eq. (7.71)) appears to be important for addressing the question of how the reflection from the cavity walls in a dynamic manner affects the motion of an electron in the cavity. In the next subsection we shall use Eq. (7.70) to study the electromagnetic surface dressing of a charged particle moving with a constant velocity parallel to a flat surface.

7.2.5. Resonant dressing by Cherenkou-Landau surface shock waues

Let us consider a situation where an ED wave packet of charge Q moves with a constant velocity dR/dr = I/ = I/e, parallel to a flat jellium surface at a height z. above this, and let us assume that no external fields act on the particle. As usual, we let the metal occupy the half space z 3 0. At time t = 0 the particle is located at the position R(t = 0) = (O,O, -zo). When the particle velocity is constant, Eq. (7.70) is reduced to [neglecting the second (“diamagnetic”) term on the right hand

side]

(E) = -Poe s VV~V’I(R(r),R(t’),t - t’) dr’ . (7.72)

In the present case, the field originating in the surface dressing is time independent, and given by

~091

i

(&)2t’p(qI,Ix) COSCL - (V/C0)2Ylfrs(Yi~,~)Sin2aCOSa

X 0 dq,, dr > (7.73)

4’, 4Yi?,(% ‘4 J

where qy = + iq, [l - (V/co)* COS~CI]““~. To derive Eq. (7.73) the indirect propagator was expanded in a complete set of undamped plane-wave components over the surface plane (Weyl expansion). The double integral in Eq. (7.73) reflects a polar coordinate description of this expansion. In passing we note that qy is a purely imaginary number. In order that the field decays away from the particle (towards the surface) it is as indicated necessary that a plus sign is used when taking the square root of (c&~.

0. Keller / Ph,vsics Reports 268 (1996) 85-262 211

When the charged particle moves along the surface electron-hole and collective excitations are created in the jellium. In Eq. (7.73) the coupling between the particle and the jellium excitations is hidden in the reflection coefficients rs and rP. Of particular importance are the jellium excitations associated with the poles of the amplitude reflection coefficients, especially if the dampings of these modes are small. This is so because a small damping implies that the pole in consideration is located near the real axis in the complex q,,-plane. Since the pole location depends parametrically on the polar angle X, the resonance contributions to the field dressing stem from surface modes tracing out specific curves (narrow stripes) in the (q,,, &)-plane, one stripe for each pole. The collective eigenmodes are all hidden in rP, and resonant dressing is obtained when rg + rx). Within the framework of the semiclassical infinite barrier (SCIB) model the resonance condition for the collective modes is given by [67, 2101

inserting tr) = ql, I/ cos X. The quantities cT and cL are the transverse and longitudinal bulk dielectric functions in the collective mode approximation, and KT and ICY are the complex wave number of the bulk polariton and plasmon, respectively. The projection of the polariton and plasmon wave vectors perpendicular to the surface are denoted by KT and ~4. It appears from Eq. (7.74) that the collective surface excitations providing resonant dressing are neither transverse nor longitudinal. From studies of the dispersion relation qll = q,(o), given implicitly in Eq. (7.74), it appears that more than two branches exist in the frequency range where the surface polariton and plasmon are strongly coupled [211]. Using a hydrodynamic approach a qualitative description of the surface dressing problem is presented in [209].

In bulk matter strong light (polariton) emission can be observed when a charged particle moves with a speed exceeding the phase velocity of the transverse (polariton) modes of the medium. This is the famous Cherenkov effect [212]. A closely related phenomenon, sometimes named the Landau effect [213] can occur when the particle moves with a velocity slightly larger than the longitudinal (plasmon) phase velocity of the medium. The surface dressing briefly discussed in this subsection may lead to shock wave excitation when the particle velocity exceeds one of the characteristic phase velocities of the surface mode dispersion relation given implicitly in Eq. (7.74). Since the surface modes cannot be classified as purely transverse (divergence-free) or longitudinal (rotational-free) the resonant coupling is neither of the Cherenkov type nor of the Landau type. Only outside the region of strong coupling between the surface polaritons and plasmons the particle-surface interaction takes the character of a simple surface Cherenkov or Landau mechanism. The new shock waves might be named Cherenkov-Landau surface shock waves. The surface dressing investigated above also might be of importance when an electron is tunnelling between a metal tip and a metal surface. In k-space the electron is transferred between states located near the Fermi surfaces of the respective metals. The tunnelling electron thus is expected to have a velocity of the order of the Fermi velocity. Hence a substantial fraction of the electrons emitted from the tip should have projections of their phase velocities along the surface plane comparable in magnitude to the Fermi velocity. Due to the fact that the plasmon phase velocity is typically of the order of the square root of the diffusion coefficient (hydrodynamic estimate), and thus comparable to the Fermi velocity, one may expect that Cherenkov-Landau surface shock waves are excited in the tunnelling process [214-2161.

212 0. Keller / Physics Reports 268 (1996) X5-262

PART C. NONLINEAR MESO-ELECTRODYNAMICS

To describe the electrodynamics of a mesoscopic medium it is of central importance to study the local field inside the system, as we have seen. The external field impressed on the medium gives rise to induced currents and these in turn cause the emission of electromagnetic waves. The emitted and external fields add to produce the local field which finally drives the motion of the electrons of the mesoscopic objects in a self-consistent manner. In Part A, the self-consistent loop for the local field was discussed within the framework of linear response theory, and in Part B the basic theory was used to study linear local-field phenomena in various kinds of mesoscopic systems.

It is of substantial interest also to understand the nonlinear electrodynamics of mesoscopic systems. The theoretical framework established in Part A does not allow us to study nonlinear local-field phenomena in a systematic manner, however. Furthermore, such a study would lead us far beyond the scope of this article. It is possible though to uncover a corner of the area of nonlinear local-field electrodynamics of mesoscopic media on the basis of the theory presented in Part A. Thus, if the nonlinearities are not too large it is meaningful to expand the induced current-density distribution in powers of the prevailing electric field. Such an expansion is often used in nonlinear optics, and it leads in lowest (nonlinear) order to optical second-harmonic generation and rectification (photon drag). The formalism described in Part A can be used to study the local-field loop at the rzth-harmonic frequency provided the fields causing the production of the nth-harmonic component can be treated parametrically. Even in the simple parametric approach where the dynamics of the field(s) generating the lzth harmonic evolves independently of the local-field-loop solution, important fingerprints of nonlinearities are present in the local-field electrodynamics as we shall realize from our studies of a few selected examples here in Part C.

8. Optical second-harmonic generation

8. I. Longitudinal and transcerse source ,fields

In linear optics the current-density sources driving the electrodynamics of the mesoscopic system usually are located outside the transverse current-density domain of the mesoscopic medium, as we have seen in the examples treated in Part B. If this is so, the driving field in most cases will vary slowly across the mesoscopic object under study. The slow spatial variation has advantages and disadvantages. From a theoretical point of view it is an advantage that a moment expansion of the external (many-body) conductivity tensor can be introduced since such an expansion simplifies the overall calculation significantly without loosing the fingerprint of the local field. In the context of local-field resonances the slow spatial variation of the source field may be a disadvantage because it prevents us from exciting resonances with a significant content of high spatial harmonics in an effective manner. In linear electrodynamics excitation of such resonances usually requires the presence of Coulomb-like (longitudinal) components in the source field. These components are conveniently obtained sending charged particles through or into the vicinity of the mesoscopic object. If only the light (transverse field) from external source distributions, like those in a laser or lamp, are present it is difficult to excite short wavelength local-field resonances to a sufficient degree in linear optics.

0. Keller / Physics Reports 268 (I 996) 85-262 213

In nonlinear optics of the situation can change completely, as we shall realize below. Starting from the transverse source field belonging to the fundamental light frequency, one might generate longitudinal as well as transverse components in the nonlinear driving field, and hence simultaneously achieve the advantages of light (transverse field) and particle (longitudinal field) excitation in linear electrodynamics.

To illustrate the above-mentioned premises it is sufficient to consider the case of optical second-harmonic generation. Starting from the Liouville equation for the second-harmonic part, p2, of the density matrix operator in the frequency domain i.e.

2fw2 = c-*F,P21 + 3C~~I,YIl + c-*,,PFl >

where in second-quantized form

(8.1)

.Fz = 2 c (k’IA(r; cd) .A@; f2l)lk) a~‘,sak,s ) k,k’s

W)

it is a straightforward matter to obtain the second-harmonic (2~) part of the forced nonlinear current density from the formula

.F(r; 2~) = Tr(pz2F) + fTr{p,$i) . (8.3)

Albeit of general interest we have neglected all spin effects in Eqs. (8.1) and (8.3). I have attached the superscript “ext” to the forced part of the nonlinear current density because Jext(r; 2~0) in the wide sense of the word external is the prescribed source current-density distribution at 2~, provided the generated second-harmonic field is so weak that the fundamental field can be considered as a parametric quantity (see below). In terms of the fundamental field one obtains

J’“‘(v; 20) = s

X MB ( r, I”‘, r”; u -+ 2CO) : (E#‘; 01)

+ Ey’(r; CL)))&(Y’; o) + Ey’(v’; 01)) d3r”d3r’ , (8.4)

where the nonlinear many-body conductivity tensor in triadic notation is given by

XMB (r, r’, r”; 0 + 2(B) = s(r’ - r”) C”,, (r, r’; co + 2w)U

+ X”,, (r, r’, r”; 0 + 2W) + 6(r - r’)U X”,, (r, iJ’; C!J + 2to) ) (8.5)

with

C”,, (r,r’,r”; (0 -+ 2~0) = 4 1 1 PK - PI

CO ,,,,,2tic0+EJ-E1 ko+EK-EI

PJ - PK - htrj + EJ - EK 1 J~YJcE(r)J~~E(r’)JSKP_A~E(r”) , (8.7)

214 0. Keller/Physics Reports 268 (1996) k-262

XT ;P_AJCE(r)Jy?yE(r”) ) (8.8)

with

T S5ryE(r) = c I$:@‘- . . . ,ra, . . . )6(r - Y,)l+bI(Y1, . . . ,r,, . . . )n, d%, . (8.9) 1

In the equations above I have used a notation analogous to that of Eqs. (4.54) and (4.55). Once the forced part of the induced current density has been calculated, the driving field at 2w can be determined from the equation

Eext(y; 20~) = - 2i,u0cr, G(r, v’; 20) - Jext(r’; 20) d 3r’ . (8.10)

The transverse and longitudinal parts of E”“‘(r; 2~0) are immediately obtained using the division of the pseudo-vacuum propagator given in Eq. (2.64). Hence,

Eext(y; 2C0) = EG”‘(r; 20) + E,““‘(r; 20) (8.11)

with

&“‘(r; 2co) = & JF’(r; 20) - 2ip0w DT(r, r’; 2~0). Jext(r’; 2m)d 3yr , 0 s

(8.12)

Et”‘@; 20) = (1/2ic,o) Jy’(r; 2~) . (8.13)

Two important aspects appear from Eq. (8.12) and (8.13). Firstly, it is seen that the driving field has a longitudinal component if the associated external current density has an L-part, cf. also Eq. (2.19). By inserting Eq. (8.5) with Eqs. (8.6)-(8.8) into Eq. (8.4) it is seen that in order for the first two terms on the right-hand side of Eq. (8.5) to contribute to Jlxt at least one of the transition current densities

Jyp,““(r) must have a longitudinal part. Utilizing the relation in Eq. (2.22) it is obvious that the last term in Eq. (8.5) always contributes to JF’. Secondly, it is seen that the driving field, not least because of its self-field contributions, will vary significantly across the mesoscopic object. This is so

since Jext vanishes outside the mesoscopic system, and because the individual transition current

densities J;yy are rapidly varying in space and only relatively few of these are present in a few-level system.

With a prescribed driving field, EF’(r; 2co), the loop equation at the second-harmonic frequency takes the general form

E(r; 20) = Eext(r; 2a) - 2ipoo c

G(r, r’; 20J) - cMB(r’, r”; 20) - (&(r”; 2co)

+ EtXf(r”; 2w))d3r”d3r’ . (8.14)

0. Keller/Physics Reports 268 (1996) 85-262 215

If the generated second-harmonic field is sufficiently strong it is necessary to take into account the nonlinear depletion of the fundamental field. In such cases the local-field analysis must start at least from a set of two “both-way coupled” integral equations for E(r; IX) and E(r; 20). Though of substantial interest in its own right I shall not go into a discussion of local-field phenomena in these nonparametric cases, since this would lead us far beyond the scope of this monograph.

Having established the loop equation for the second-harmonic part of the field, one may proceed in one of the ways discussed in the context of linear meso-electrodynamics. This is so since Eq. (8.14) is a linear equation in the local field at 20. In the many-body case one starts with a solution of the integral equation for the transverse field IT,+; 2(o), as described in Section 4.1.2. Once E-r has been obtained the longitudinal field follows upon a direct integration. In cases where the longitudinal electrodynamics dominates the second-harmonic process, it is reasonable to start from the density-functional approach of Section 4.1.3, or its extended version, discussed in Section 4.1.5. Finally, one may also take as a starting point the RPA loop, cf. Section 4.1.4. The external conductivity tensor concept is of importance in the second-harmonic studies, and the coupled-antenna theory and the local-field analysis may be used in the nonlinear problem also.

8.2. Zco-generation in quantum wells

It is obvious from the consideration of Section 8.1 that it is more difficult to perform a local-field analysis in the nonlinear than in the linear regime of electrodynamics, even if the calculation is carried out in the parametric approximation. Just as in the linear case a substantial reduction of the problem is achieved in quantum-well systems, where the basic integral equations for both the first- and second-harmonic components of the field involve only one of the Cartesian coordinates.

At the time of writing the nonlinear optical properties of quantum wells are of substantial interest to the scientific community, see e.g. [217-2261.

In the present section we study optical second-harmonic in quantum wells. For simplicity, we start from the random-phase-approximation description, and throughout the section we adopt as far as possible the notation of Section 5.1. The basic integral equation of the second-harmonic frequency hence reads [28,227,228]

E*(Z) = E;(z) - 2ip0co s

Gf(z, z’).az(z’, ~“).E~(z”)dz”dz’ , (8.15)

where instead of writing 2q,, and 2w in the arguments of the various quantities we have added a subscript “2” to remember that the loop in Eq. (8.15) is at the second-harmonic frequency. The integral equation for the fundamental field is given in Eq. (5.2).

8.2.1. Two-level metallic quantum well. A heuristic approach To illustrate some of the basic physics I shall here consider a simple two-level metallic quantum

well, and assume that the paramagnetic coupling dominates the linear responses at o and 20. Furthermore, I shall assume that the QW electrons exhibit free-particle-like behaviour along the well plane and consider only the long-wavelength (q,, --* 0) and low-temperature (T + 0 K) limits.

216 0. Keller/Physics Reports 268 (1996) 85-262

Finally, the electromagnetic field propagator is approximated by its self-field part. In the self-field approximation for Gf, Eq. (8.15) is reduced to

[i c2(z, z') .E2(z’)dz’ . 1 (8.16)

In this approximation the local field along the quantum well thus is equal to the background field in the plane of the well. In the long-wavelength limit the paramagnetic conductivity tensor is diagonal, see Eqs. (5.26) and (5.27) and for a two-level system one has for T --f 0 K and e, < & < 62

fJ;;ra(Z, z’; 2q,, -+ 0,2W) = b(2w)@(z)@(z’) ) (8.17)

where

b2 - Ed(EF - &I) (8.18)

Combining Eqs. (8.16) and (8.17), it is realized that the local-field component perpendicular to the quantum-well plane is given by

E2,JZ) = E;,=(Z) + fg C@(z) . 0

(8.19)

Due to the presence of Q(z), the local-field correction to the background field, E&z) - Et,=(z), varies rapidly across the quantum well. The strength of the correction is proportional to

C = @(z)E2.-(z)dz . s

(8.20)

Using the coupled-antenna technique, the explicit expression for C can be obtained. Thus,

C = [I - 2 j@2(z)dz]-1j@(z)E&(z)dz (8.21)

The local-field resonance condition at 20 is determined from the condition C --f ~1, and hence is

We) 2ir Q2(z)dz = 1 .

0 s

(8.22)

Provided the occupied electronic state is located near the Fermi surface (ci % &F), the local-field dynamics causes a slight blue shift of the electronic resonance. In the collisionless limit the shift approximately equals

2tio - (Ez - t.1) = e2ti2(&F - al)

16~~~45~ - E~)~ j Q2(z)dz . (8.23)

0. Keller / Ph_vsics Reports 268 (I 996) X5-262 217

In the self-field approximation the z-component of the background field is given by

Et,Z(~) = (1/2&co) J:,,(Z) , (8.24)

and the two other Cartesian components vanish. To calculate the z-component of the background current density at 201, Jt.Z(z), let us first consider the relation between the local field and background field at cc). This relation was obtained in Section 5.2 (Eq. (5.40)). Since at optical frequencies the background field at the fundamental frequency, E;(z), is essentially constant across the quantum well, Eq. (5.40) gives

E,,;(z) = (1 + M(WW)E::. > (8.25)

where

M(u) = [z - Jb’(z)dz]-’ $(c, - a2)P12,Z, (8.26)

P 12.2 being the z-component of the electric-dipole transition matrix element. The quantity b(o) is obtained from Eq. (8.18) making the substitution 2~ -+ o. The local-field resonance condition at the fundamental frequency is determined from the condition M + co. By comparing Eqs. (8.22) and (8.26) it is realized that the resonance condition at w can be obtained from Eq. (8.22) making just the replacement 201-+ w. Having determined the relation between E,,,(z) and Ey._, the background current density can be calculated from the one-coordinate RPA version of Eq. (8.4). Since CB(z, z’, z”) vanishes for a two-level system, and ZA(z, z’) = CA(z, z’)e, and Ic(z, z”) = Cc(z, z”)e= in the long-wavelength limit [229] it is possible to show that

J;,-(z) = W(Z)(E,B)~ ,

where

W(z) = ~(20)M(cfi)@(z) $(z’)@(z’)(2 + M(u)@(z’))dz’ $

+ 4 a(~r,)4(z)(l + M(o)@(z)) @(z’) (1 + M@)@(z’))dz’

with

U(Q) = ie3(cF - el) !L? + i/r

xmL?' [h(i2 + i/r)]’ - (az - &I)2 ’ ’ = O’ 2co .

(8.27)

(8.28)

(8.29)

Once the two wave functions and the associated energies have been determined the local field at the second-harmonic frequency inside the quantum well can be calculated from the expression

E2(Z) = & W”(z) + @(z) J @(z’)W(z’) dz’

0 2iEoco/b(2w) - J Q2(z’)dz’ (EZB)2 ’ 1 (8.30)

218 0. Keller / Ph,vsics Reports 268 (1996) 85-262

8.2.2. Thickness dependence of the second-harmonic generation from metallic quantum wells

The two-level calculation described above can readily be generalized to N( >2) levels. In the linear case the N-level paramagnetic problem was treated in Section 5.2. We realized there that the local field was given by Eq. (5.31) with the r,, -vectors determined from the set of algebraic vector equations in (5.32). In the self-field approximation, only the z-component of Eq. (5.31) contain a local-field correction to the background field, and the rmn’s are replaced by scalars. Since the structure of the local-field problem essentially is the same at Q and 20~ in the parametric limit, the local-field analysis presented in Section 5.2 can be applied directly to the integral equation at the second-harmonic frequency, provided one makes the appropriate replacements. The background field at 2~ only has a z-component, and to calculate this one just needs the z-component of the driving current density, cf. Eq. (8.24). In the RPA description the dominating part of the nonlinear response turns out to have the following nonzero elements in the q, + 0 limit [229]:

c czyy = e3(2ru + i/z)ksT zxx =

47cimw2

c @(c,, - GJ[IH(&?l) - ffbn)l m,n [ti(2~ + i/z)12 - (e, - E,)~

x @?&)4?&‘)w’ - 2’) 3 (8.31)

c cyyz = ie3(u + i/7)&T

xxz = 2xrnu2

1 @(&a - GJCH(&J - ffbz)l m.n CfG + i/W - Cc, - d2

x &&)@m&“)w - z) > (8.32)

c zzz = czxx + ~xxz . (8.33)

The formalism briefly described above has recently been used to study the thickness dependence of the optical second-harmonic generation from ultrathin niobium films sandwiched between amorphous A1,03 layers [230]. By having the same kind of dielectric medium on both side of the quantum well a symmetric quantum-well potential is obtained. The A1203/Nb/Al,0, quantum- well system was chosen because experimental second-harmonic data are available for this structure [230]. In the calculation of [230] simple infinite-barrier wave functions and associated energies were used. To solve the matrix problem of the coupled-antenna theory the number of levels must not be too high. In turn this implies that the calculations can be carried out only up to a certain well thickness. For a given thickness the maximum quantum number used was adjusted so that the second-harmonic energy reflection coefficient is insensitive to a further increase in the number of levels incorporated. In Fig. 8.1. is shown the calculation of the second-harmonic energy reflection coefficient q, defined as the ratio between the light intensity at 2w and the square of the incident intensity at o as a function of the quantum-well thickness in the case where the first- and second-harmonic fields are p-polarized (pm +pzw). The energies of the incident photons are tic0 = 2.34 eV and the angle of incidence is 60”. Results are shown for three different relaxation times. At small thicknesses, ye exhibits rapid and almost periodic oscillations as a function of the thickness, provided the collision frequency is not too high. This type of in-and out-of-resonance behaviour can be ascribed directly to the structure of the electronic level transitions in >he nonlinear conductivity tensor. The pronounced maximum appearing at a thickness of - 23 A, if the damping in the system is not too large, is due to the local-field resonance effect.


In Fig. 8.2 the experimentally obtained data for the thickness dependence of the second- harmonic intensity are compared to the theoretical predictions. Theoretical curves are plotted for four different relaxation times, and all calculated intensities have been normalized in such a way that the local-field peaks have the same height %s the experimentally observed peak. It appears ihat the observed resonance peak appears at - 15 A while the theory predicts it to occur at - 23 A. It is not clear at the time of writing how this discrepancy can be explained. It seems however that inclusion of the diamagnetic effect, partially screened by the Nb ion cores, would move the theoretical peak towards lower thicknesses and at the same time increase the peak half-width. If the ion screening is neglected the inclusion of the diamagnetic effect would make the peak disappear in the present case. It is also expected that use of a finite-barrier potential and inclusion of electron-electron interaction effects (not already present in the RPA formalism) will push the local-field peak towards lower thicknesses. A change of the relaxation time cannot explain the discrepancy as it appears from Fig. 8.1. Newly obtained experimental data for Au/Si quantum-well systems show the same tendency, i.e. the observed local-field peak appears at a smaller well thickness than predicted by the simple local-field theory; see Fig. 8.2.

The optical second-harmonic generation in ultrathin metallic films has also been studied in [231-2331. In the metal on metal case [231,233] the electronic states responsible for the optical response are usually (but not always) highly delocalized, i.e. spread over the entire film-substrate region. The delocalization makes it conceptually impossible to isolate the film response, and

5 10

0.4

2 0.3

9 L?

.s 5 0.2

C .d LC m 0.1

0.c I 0

QW thickness (a)

0

Fig. 8.1. Thickness dependence of the second-harmonic energy reflection coefficient of a Nb quantum well calculated from the microscopic local-field model for different relaxation times (in lo- l4 s), viz., r = 0.33 (curve l), 0.13 (curve 2), and 0.07 (curve 3). Both the fundamental and second-harmonic field are p-polarized (p, * pzrn configuration), and the angle of incidence is 0 = 60’.

Fig. 8.2. Plots of the experimental (circles) and theoretical (solid lines) second-harmonic intensities versus the Nb film thickness for the pw = p2,,, scattering configuration. Curves l-4 correspond to the following relaxation times (in lo- l4 s) z = 0.22, 0.17, 0.13, and 0.11, respectively. To compare the shape of the theoretical and experimental thickness dependencies the calculated second-harmonic intensities have for each r been normalized in such a way that the peak height of the resonance is the same as in the experiment.

220 0. Keller / Ph,vsics Reports 268 (1996) KS-262

therefore the microscopic analysis becomes extremely difficult to carry out; cf. the second-harmonic studies from free metal surfaces (see [234] and references therein). Provided that localized electronic states exist in the film region, and provided the optical excitationdeexcitation process takes place between these states, it is possible to speak of an isolated film response even in the metal-on-metal case. Also in the case where a metal film is deposited on a semiconducting substrate [232] one has to worry about the role of delocalized electronic states.

8.2.3. Second-harmonic generation jiom a symmetric semiconducting quantum well.

Injuence of DC electric ,$elds In the last decade there has been an increasing interest in studying the nonlinear optical

properties of semiconductor quantum-well structures [217-223,225,226,235]. The main reason for this is the many possible applications of these systems in areas such as long-wavelength infrared detection, integrated optics, and optical communications. The overwhelming majority of the optical second-harmonic generation studies has been carried out on asymmetric quantum-well structures and symmetric quantum wells subjected to applied DC electric fields. In the above- mentioned investigations the theoretical calculations were carried out using the electric-dipole interaction Hamiltonian (local approximation). Large second-order nonlinearities associated with various intersubband transitions were predicted, and it was often claimed that in order to obtain a large nonlinearity the electric-dipole interaction need to be present. In a symmetric quantum-well structure, due to the definite parity of the wave functions of the bound states, the 2w-polarizability vanishes in the local limit. Thus, in order to analyse the optical second harmonic generation from symmetric quantum-well structures, a nonlocal approach has to be used. Theoretical studies by now have predicted that the nonlocal 2to-generation in a symmetric quantum well can be comparable in magnitude with second-harmonic generation from naturally or artificially asymmetric quantum wells, the artificial ones being symmetric wells subjected to DC electric fields.

A theoretical treatment of the intersubband optical second-harmonic generation in a symmetric Al,Ga, _.As/GaAs/Al,Ga, _,As two-level quantum-well structure is presented in [225]. In the calculation carried out in this reference also retardation effects were included. The theory in [225] predicts a frequency dependence of the nonlinear reflection coefficient for the pcu + p2,,, configuration at 60’ angle of incidence as shown in Fig. 8.3. For convenience the optical frequency has been normalized by division with the transition frequency LL)~~ = (Ed - cI)/ti, and curves are plotted for different donor concentrations. For a given donor concentration the spectrum exhibits two pronounced resonance peaks, one located somewhat above ~0~ 1, the other a little above (11~ ,/2. The two resonances stem from the local-field resonances at the fundamental (the peak with the highest frequency) and the second-harmonic frequencies. As expected, the two local-field resonances are shifted to the blue side of the respective electronic resonances. One also notices from Fig. 8.3 that the second-harmonic energy reflection spectrum does not exhibit any resonant behaviour at the frequencies 0~~~ and m2J2, although the second-harmonic conductivity response function is resonantly enhanced at these frequencies. This fact is in agreement with our overall picture of the role of local-field corrections. It appears from Fig. 8.3 that the resonance frequencies decrease when the doping concentration is decreased. This is to be expected since the magnitude of the induced current-density oscillation in the quantum well decreases when fewer carriers are available to contribute. For a smaller current flow the field stemming from this current in turn becomes less and, hence, smaller relative to the external field. Altogether, the local-field effect becomes less and


0.4 0.6 0.8 1.0 1.2 14

+%I

Fig. 8.3. Second-harmonic pW to pZW energy reflection coefficient, q, as a function of the normalized fundamental frequency, oj/w21, in a symmetric Al, Gal -,As/GaAs/Al, Gal _.As, two-level quantum-well structure for five different donor concentrations (in 10” cmm3), namely, (1) 2.5, (2) 2.0, (3) 1.5, (4) 1.0, and (5) 0.5. After Ref. [225].

less important when the doping is reduced and displacements of the peak positions towards cc)21 and 0j2J2 emerge. With increasing donor concentration the conversion efficiency also increases as one would expect beforehand.

It is of interest to investigate the influence of an external DC electric field on the second- harmonic conversion efficiency. Such an investigation can easily be carried out within the framework of the local-field formalism presented in this section. The only new ingredient is to replace the old wave functions by new ones satisfying the time-independent Schrodinger equation

[ - (ti2/2m*) (d2/dz2) + V(z) + eF.z - e,] t/3,,(z) = 0 , (8.34)

where V(z) is the self-consistent one-electron potential of the quantum well (below, we use infinite square-well potential and thus we take V(z) = 0 inside the well), and F is the magnitude of the DC electric field. For V(z) = 0, the wave functions inside the quantum well can be taken as linear combinations of two independent Airy functions Ai[Z,(z)] and Bi[Z,(z)], i.e. [221]

tin(z) = cIn~~cuz)l + c2n~~KI(~)1 , (8.35)

where the independent variable Z,(z) is given by

Z,(z) = - [2m*/(ehF)2]‘13(c, - eFz) . (8.36)

The possible eigenenergies E, are determined from the boundary conditions for the wave function, and if also the normalization of the wave function is taken into account the unknown constants Ci, and CZn can be determined.

222 0. Keller / Physics Reports 268 (1996) KS-262

50 70 90 110 130 150 170 190 210 Photon energy (meV)

Fig. 8.4. Optical second-harmonic energy reflection coefficient, q, in a AlGaAs/GaAs/AlGaAs quantum well as a function of the photon energy at the fundamental frequency for the following four values of the applied ?C electric field (in keV/cm): (1) 0, (2) 50, (3) 70, and (4) 100. The parameters used in the calculation are: d = 200 A, m* = 0.0665 m, h/z = 5.0 meV, and Ed (dielectric constant of the background medium) = 11 .l. In the calculation only the intersubband transition between the two lowest lying bound states were taken into account. After Ref. [236].

In Ref. [236], the optical second-harmonic generation in a symmetric Al,Ga, _.As/GaAs/ Al,Gal_,As quantum-well structure subjected to a DC electric field across the well was investigated theoretically. A typical example taken from this reference is presented in Fig. 8.4, where the nonlinear energy reflection coefficient for the pw + pzw configuration at an angle of incidence equal to 45” has been plotted as a function of the photon energy for four different values of the applied DC field. It appears from this figure that the two local-field resonances are blue-shifted in the presence of the DC field. The blue shift which becomes larger with increasing DC field is due to the so-called quantum confined Stark effect, i.e. the presence of the DC field leads to an increase of the energy spacing between the discrete sublevels [237], and hence also to higher local-field resonance frequencies. For a more detailed account of the predictions offered by local-field theory for the optical second-harmonic generation in the Al,Gal _.As/GaAs/Al,Ga, -,As quantum-well systems subjected to DC electric fields the reader is referred to Ref. [236].

9. Photon drag

By mixing two monochromatic electromagnetic fields of cyclic frequencies w1 and co2 in a nonlinear medium one might generate fields at the sum (co1 + cc)J and difference (I co1 - o2 I) frequencies in the lowest order of nonlinearity. In the special case where the two frequencies coincide the above-mentioned processes are replaced by optical second-harmonic generation and the formation of a DC electric field in the medium, respectively. If no net momentum is transferred between the particle and field subsystems the DC-field formation process is named optical

0. Keller 1 Physics Reports 268 (I 996) 85-262 223

rectification. The DC-field generation process, however, normally will be accompanied by some momentum transfer between the subsystems. The part of the DC process which requires a net momentum exchange (or in special cases just an angular momentum exchange) is called photon drag. Since the wave vector of the electromagnetic field hence plays no role for the optical rectification process, this can be described within the framework of local response theory. To study the photon-drag phenomenon a nonlocal response theory is required a priori. Once a nonlocal description is needed local-field effects are of importance. In media without inversion symmetry the electric-dipole interaction between particles and fields is allowed and this usually implies that the optical rectification dominates the DC-field formation process. To investigate the photon-drag effect it might thus be advantageous to consider the local-field electrodynamics of centrosymmetric media. The photon-drag phenomenon is accompanied by the generation of a forced DC current in the medium because a net momentum transfer inevitable leads to induced charged-particle flows in medium initially in thermal equilibrium. For electrons in the superconducting state the photon drag may give rise to the generation of a DC current flow without an accompanying DC electric field. The photon-drag phenomenon in a superconductor can be considered as a kind of nonlinear Meissner effect. This is so because the self-consistently determined photon-drag current precisely cancels the forced nonlinear DC magnetic field generated in the nonlinear mixing process [238,239].

9.1. Nonlinear DC current density

To determine the forced part of the nonlinear DC current density, Jo(r), induced by the fundamental field inside the mesoscopic medium under consideration one starts from the iterative- ly obtained expression

J&) = Tr{p&G> + 4Tr {pl$:} + $Tr(p:$%} , (9.1)

where p. is the nonlinear DC part of the density matrix operator. The Liouville equation for p. is as follows:

o= CflF,POl +dC%,dl +tL-~:,PIl+ C=@O,PFI,

where

(9.2)

.x0 = g 1 (k’IA(r; a)-A*@; O)Ik)akf,sak.s k,k', s

(9.3)

is the DC part of the interaction Hamiltonian e2A .A/(2m). A calculation shows that the first term on the right-hand side of Eq. (9.1) gives a current-density contribution of the form

Trbo9,) = CW - r”)n&( s f”, f”‘; co + 0) u + rI&(Y, Y’, r”; cll + O)]

: [ET(r)‘; co) + EF’(r”; co)] [Eq(r’; co) + (E~“‘(r’; cc)))*] d3y” d 3r’ ,

where

(9.4)

(9.5)

224 0. Keller /Physics Reports 26X (1996) 85-262

Though easy to derive the explicit expression for LIiB(rY v’; CL) -+ 0) is rather complicated, and since it is not needed in what follows I shall sustain from presenting it here. Readers who want the formula for Hi, are referred to Section 5.1 of Ref. [28]. From the expression for Tr {pO$r} given in this reference, the explicit formula for Hi, can readily be obtained. The second term on the right-hand side of Eq. (9.1) gives a current-density contribution

+ E~*(Y”; u)] [E;(r’; o) + (Ef”‘(r’; u))*] d3r”d3r’ , (9.6)

with

n;,(r,r”:w+tl)~e2~ PJ - PI 2mco2 I,., ho + EJ - EI

7yyE(r)J,SP_A,CE(tq . (9.7)

If one compares Eqs. (8.8) and (9.7) it appears that

n&Jr, ,“; Q + 0) = - 2 Zi&, Y”; 0 + 20) ) (9.8)

a result which in fact is obvious remembering that the formulas for Hz, and I:, were derived from the expressions iTr{p,&:) and ~Tr{p,~, >, respectively. Since

Tr{&#%) = (Tr{pr&:})* , (9.9)

the explicit expression for the contribution from the last term on the right-hand side of Eq. (9.1) to x0(r) can readily be written down starting from Eqs. (9.6) and (9.7).

Above we have discussed theforced part of the DC current density. To this part one has to add an as yet unknownj+ee (F) part JF in order to obtain a self-consistent solution to the combined set of the Schrodinger and Maxwell equations including the appropriate boundary conditions. A particular beautiful example demonstrating the above-mentioned matter appears in studies of the photon-drag phenomenon in a BCS-paired superconductor. Thus, by combining the magnetos- tatic Maxwell equation V x B,(v) = pO(Jo(r) + J&)), where B,(v) = V x&(r) with the constitutive equation for the Meissner (free) current density, i.e.

JF(r) = s

So(r, r’) - A,(r’) d3r’ , (9.10)

where SO(r, r’) is the linear and nonlocal Meissner response tensor, one obtains the following inhomogeneous integro-differential equation for the DC vector potential

V x(F xA,(r)) -/co s

SO(Y,r’).Ao(r’)d3r’ = ,u,,Jo(r) . (9.11)

Starting from Eq. (9.11) the photon drag of BCS-paired electrons at a superconductor surface was studied in [238,239]. Instead of embarking on an elaborate investigation of the photon-drag effect in mesoscopic superconductors in the subsequent section I shall discuss a particular simple example, namely, the photon drag in a one-level metallic quantum well.

0. Keller / Physics Reports 268 (I 996) 85-262 225

9.2. Photon drag in a single-level metallic quantum well

In recent years a number of studies have appeared in which the photon-drag phenomenon in confined structures is analysed and measured. In two-dimensional electron gas systems the photon-drag effect was studied theoretically by Vas’ko [240], Luryi [241], and Grinberg and Luryi [242], and experimentally by Wieck et al. [243]. For semiconductor heterostructures, Stockman et al. [244] carried out a theoretical analysis. In the above-mentioned studies main emphasis was devoted to the effects that occur when the optical excitation frequency is close to an intersubband transition. The theory of the two-band light-induced free-electron drift in metals was studied by Shalaev et al. [245], who also presented experimental evidence for this effect in the form of spatially asymmetric photoemission from silver films. An experimental study of the photon-drag response of Al,Gal _,As/GaAs multiple quantum-well detectors in the picosecond region and in the 10 urn wavelength range was presented by Kesselring et al. [246]. On the basis of the local-field formalism the present author studied the photon drag in metallic quantum wells possessing one bound state, only [247]. For the moment the local-field studies of the photon-drag effects is being extended to two-level quantum-well systems, and to multiple quantum-well media [248].

Let us now consider the metallic quantum-well system introduced already in Section 5.1. We assume that there exists only one bound state of energy t: in the quantum well and that this level lies below the Fermi level, i.e. c < cr. Parallel to the plane of the film the electrons exhibit free-electron- like behaviour. Limiting ourselves to a one-electron RPA description, the particles in the quantum well have stationary-state wave functions of the form

$(r) = (27~)) ’ $(z)eikll.’ . (9.12)

If the frequency of the external electromagnetic field is so low that the electrons in the well cannot be excited into the continuum the nonlinear dynamics of the electrons in the external field is essentially two-dimensional. This is so because the z-dependent part of the wave function cannot be modified (it is frozen to the form $(z)). For a single-level quantum-well system the forced part of the photon-drag current density is given by

Jo = iTr{pi$:} + C.C. , (9.13)

since the term Tr(~)O,#F) vanishes in this case. The translational invariance along the film plane implies that the photon-drag current density in Eq. (9.13) is independent of x and y. In explicit form, the RPA expression for Jo(z) is [247]

Jo(z) =; n”(z,z’,z”):E1(z”)E;(z’)dz”dz’ + C.C. , c

with the photon-drag response tensor given by

(9.14)

X x fk + P2/2m) lkll - 411 I ‘) -f@ + h2kf/2m) (2k,,

-m h(tr, + i/r) + (ti2/2m) Ik,, - qil I* - ti2kf/2m

d2k, 4’1) (j$ ’ (9.15)


thef’s being the relevant Fermi-Dirac distribution functions. For the normal metallic state it is a good approximation to assume that the distribution function is a step function. Employing this T = 0 K approximation it is possible to carry out the double integral in Eq. (9.15). Doing this, one obtains a photon-drag response tensor

I~“(z,z’,z~) = (i?e3/4m2c02)$J(z)~2(z”)~(z - z’)(R+ - R_)Ue,, (9.16)

where


I<\\ = [(2@‘)(& - &)]1’2 .

(9.17)

(9.18)

(9.19)

The photon-drag response tensor for the one-level quantum-well system given in Eq. (9.16) has a particularly simple form. Thus, since intersubband transitions do not occur the z-dynamics is frozen down. The nonlinear dynamics along the plane of the well is determined by the quantity R+ - R_, cf. Eq. (9.15). Since the symmetry of I7 is given by the third-rank tensor Ue, it appears that an s-polarized incident field cannot give rise to a photon-drag current. By inserting Eq. (9.16) into Eq. (9.14) one obtains the following expression for the photon-drag current density:

Jo(z) = iCK(q,,, 4$2(4E:(z) $“(W,,.(z’)dz + c.cl , f

(9.20)

with the abbreviation

K(q,,, o) = (he3/4m2w2)(R+ - R_) . (9.21)

Since our two-dimensional jellium system possesses inversion symmetry, optical rectification cannot occur in the plane of the quantum well. This in turn means that the current density Jo(z) should vanish in the long-wavelength (local) limit, where there is no net momentum exchange between the electrons and light. Since

K(q,, -+ 0, w) = 0 (9.22)

the above-mentioned conclusion is verified. So far we have considered only the forced part of the photon-drag current density in this section.

To this part one has to add an as yet unknown free part Jr(z) in order to obtain a self-consistent solution to the combined set of the Schrodinger and Maxwell equations, including the appropriate

0. Keller/Physics Reports 268 (I 996) 85-262 22-l

boundary conditions. In the present case the total current density in the direction perpendicular to the quantum-well plane vanishes in every space point, i.e.

e, - (JF(z) + JO(Z)) = 0 . (9.23)

If the electrons are allowed to flow freely along the quantum well the associated prevailing current density is simply (U - e,e,) - Jo(z). In cases where the slave model can be used to calculate the local field at the fundamental frequency the photon-drag current density takes a particular simple form, viz.

(U - e,e,)+J&) = 4 K(qll, u)ti2(z) 1 F’i xx

EF,,e, * 1 - K,, > 1 + c.c . (9.24)

By integration of the result in Eq. (9.24) over z, one obtains the following expression for the integrated photon-drag current density, 1,:

B

“‘; xx

(9.25)

It is adequate to characterize the strength of the photon-drag effect in terms of ratio between the magnitude ofI,, and the magnitude of the Poynting vector of the incident field, Sine. For p-polarized light this ratio becomes [247]

10 211 -r,(q,,,w)12cos2tf -=

Sinc eoco I 1 - Kx(q,, > a) I 2 Re Wll, 4 (9.26)

within the framework of the slave approximation, 8 being the angle of incidence of the incoming field. In Fig. 9.1 is presented the result of a numerical calculation of the ratio ZO/‘Sinc as a function of the photon energy fo; the niobium/quartz system considered in Section 5.1. By choosing a film thickness of d = 3 A there exists only one bound state below the Fermi level, and & - + N - 3.31 eV. In addition, one has an unoccupied level (of energy f;2) above ‘!+ at &2 - &F ‘v 1.91 eV, but in th e photon-energy range of interest here, namely 0.15 ho LO.18 eV, the optical excitation of this level is negligible. The ratio ZO/S’inc is shown for three different angles of incidence, viz. 6 = 20”, 50”, and 70”. It appears that the drag current is particularly large in the frequency region where the p-polarized reflection coefficient of the quartz substrate has a resonance, as one would expect. If El,, is calculated on the basis of the “exact” equations in (5.5) and (5.6) it turns out [64] that the relative deviation [(“exact”-slave)/slave] between the ZO/Sinc calculations for all w and 8 is less than lo- 3. The slave approximation hence is an extremely good approximation in the present case. As illustrated in Fig. 9.1, even the self-field approximation, obtained by setting K&q,,, co) = 0 in Eq. (9.26), gives a quite accurate result.

Since the local-field resonance condition K,,(y, , co) = 1, as discussed in Section 5.3, essentially is the dispersion relation for guided electromagnetic waves in the vacuum/quantum well/substrate


system, one should anticipate the photon-drag effect to be resonantly enhanced if the excitation is done by a p-polarized surface electromagnetic wave (SEW). If the SEW is excited by means of the Otto prism configuration [249], the Poynting vector of the incident field is parallel to the plane of the quantum well in the gap, i.e. Sine(Z) = Si”,(Z)e,. In terms of the magnitude of the Poynting vector of the incident field just outside the quartz surface, Sinc(Z = O-), one can show [247] that the normalized integrated photon-drag current density is given by

IO 8 [qt - (o~/c~)“] [(w/co)*aa - ~$1 Re K(q, co)

Sinc(Z = O-) = crJtoq~ l&o + k’l - K&!%, I 2 ’ (9.27)

where N,,(U) = N_:_t’(cti; c, = 1) = &N+/(i$,). In Fig. 9.2 is shown the ratio ZO/‘Sinc(Z = O-) as a function of coq ,/o for five different photon energies, viz. tie = 0.134,O. 136,O. 139, 0.141, and 0.144 eV. The results presented were obtained on the basis of a slave-model calculation, which also in this case is extremely accurate. In the case of SEW excitation local-field effects play an important role for the photon-drag current. That this is so becomes obvious if one compares the slave-model

I F ; 0.10

4 ; 0 c;

.c e ,”

0.05

0.00. 0.10 0.12 0.14 0.16 0.18

fiwlevl- CoqlJ~ -

Fig. 9.1, Magnitude of the normalized integrated photon-drag current density, IO/&, in a single-level Nb quantum well as a function of the photon energy, ha), for three different angles of incidence, viz., (1) H = 20”, (2) 50 , and (3) 70 Fully drawn curves: slave model; dashed curves: self-field model.

Fig. 9.2. Magnitude of the normalized integrated photon-drag current density, 10/Si,, (Z = O-), as a function of the (normalized) real wave number, c’,,q /(II, parallel to the plane of a single-level Nb quantum well in the vicinity of the SEW resonance for five different photon energies, namely (1) hm = 0.134, (2) 0.136, and (3) 0.139, (4) 0.141. and (5) 0.144 eV. Solid lines: slave model; dashed lines: self-field model.


results with those obtained from a self-field calculation. It hence appears from Fig. 9.2 that the self-consistent local-field calculation leads to an integrated photon-drag current density which is approximately a factor of two lower near resonance than that obtained in the self-field approach. The resonance behaviour of ZO/Sinc(z = O-) displayed in the various curves of Fig. 9.2 is closely related to the excitation of guided modes on the structure. That this is so is obvious from Eq. (9.27) because the condition yycQ + xl = N,,&K, is precisely identical to the dispersion relation for SEWS on the vacuum/niobium/quartz system, as we have realized via the discussion in Section 5.3. By comparison to the bulk excitation method it appears that SEW excitation causes an overall increase in the photon-drag current density by two orders of magnitude for the same incident field at the surface.

So far, I have considered only the current density induced along the quantum-well plane. In order to balance the forced part of the photon-drag current in cases where the electrons are prevented from flowing in the z-direction a charge build-up takes place across the well. The induced charge distribution will attain such a value that the accompanying DC electric field,

E,(z) = &(z)e,, g ives rise to a free current density ezez ..JF(z) which precisely can cancel the forced part of the photon-drag current density, cf. Eq. (9.23). For the one-level quantum well the DC field induced across the quantum well is given by

Eo,JZ) = &* 2e2T(e - +)

Wq,,, ~)Et,z(z) ~~2(z’)E~.x(z’W + C.C. s 1 (9.28)

Upon integration of this expression, the DC voltage, V0 = - s Ea._(z) dz, induced across the well can be obtained. A new resonance not present in the photon-drag current may appear in the induced voltage, because the z-component of the fundamental field appears in Eq. (9.28). Using the slave model, E,,=(z) is given by Eq. (5.15), and the new resonance thus is determined from the condition K,,(qil, CO) = 1, already analysed in Section 5.3.

9.3. Photon drag in the vicinity of an intersubband transition

It is of conceptual importance to investigate whether or not the induced electron dynamics in the direction perpendicular to the quantum-well plane plays a role for the photon-drag phenomenon. To study this question one has to go beyond the one-level calculation described in the previous subsection. Once we introduce an extra bound-state level above the Fermi surface, electron motions in the z-direction are no longer prohibited. From a heuristic point of view the two-level quantum-well thus appears adequate for the above-mentioned purpose. Further, if one excites the quantum-well system near an electronic resonance the results of a two-level model calculation may even be quantitatively correct.

In a two-level system the contribution to the photon-drag current density from niB(z, z’, z”) is absent. The term l7& now contributes to Jo(z), however. The tensorial structure of the A and C contributions to the photon-drag response tensor is shown in Fig. 9.3. Though the nonlinear DC current density does depend on the A term, this term does not contribute to the integrated photon-drag current density along the well plane, however. One can easily prove this by noting that the integrated response is proportional to the integral of the x-component of the RPA

230 0. Keller J Physics Reports 268 (1996) 85-262

Fig. 9.3. Symmetry schemes for the so-called A and C contributions to the photon-drag response tensor of a two-level

quantum well.

transition current density, cf Eq. (9.5). Thus, since the orthogonality of the two energy eigenfunctions implies that e, - sjlz(z) dz = 0, it appears that the A contribution to the integrated x- component of the current density is indeed zero. Focusing our attention on the C term, an explicit RPA calculation shows that this gives the following contributions to the photon-drag current density of the two-level well:

llc = n,c,, = II,“,, XXX

= (tie3/4m2W2)6(z - z’) [$:(z)$:(z”)(RYi) - R!!i’)

+ &*(Z)@r2(Z”)(R(:i) - R?‘)] ) (9.29)

nc =nc =flc XXZ ,‘YZ ZZZ

= (tie3/4iun202)6(z - ~‘)~~~(z)@i~(z”)(H!:*) + H’2l’) (9.30)

in the notation of Section 5.2. The intrasubband two-dimensional dynamics of the (occupied) lower level is described by the functions R yl) = R+ and R!!” = R_ already introduced in Section 9.2. -

The presence of two levels gives rise also to intersubband effects. The intersubband dynamics is hidden in the quantities R(?” - R(!*) and H$! ‘) + H? l), the explicit expressions of which can be

found in [248]. To illustrate the photon-drag phenomenon in a two-level quantum well let us once again

consider the niobium/quartz system. Using the infinite-barrier model, a 15 A wide quantum well has an adequate occupied - to unoccupied transition at a2 - cl = 1.838 eV. Using p- polarized light incident at an angle 8 = 60”, the frequency dependence in the vicinity of the intersubband resonance of the normalized integrated photon-drag current density, IO/Sine, is shown in Fig. 9.4 for two different intersubband relaxation energies, viz. 0.4 and 0.6 meV. In the present case the contribution from the intrasubband transitions is two orders of magnitude less than that of the intersubband transitions, and hence negligible. It appears from Fig. 9.4 that the photon-rag current may be resonantly enhanced in the vicinity of the intersubband transition, and that the current changes direction when the frequency of light is tuned through the resonance.


Fig. 9.4. Frequency dependence of the normalized integrated photon-drag current density, 10/Si.,, of a two-level Nb/ quartz quantum-well system in the vicinity of the intersubband resonance sl - e1 = 1.838 eV for two different relaxation energies, viz. 0.4 meV (dashed curve) and 0.6 meV (fully drawn curve). After Ref. [248].

9.4. Photon drag in a mesoscopic ring

In the quantum-well case it is necessary to transfer linear momentum from the electromagnetic field to the mobile electrons in order to create a DC current along the well plane. The photon-drag phenomenon in this case is an essentially nonlocal effect. However, if the mesoscopic medium forms a closed ring a photon-drag current may be excited transferring only angular momentum from the field to the electrons of the ring. For charged particles moving in a closed loop the photon-drag phenomenon thus may exist even in the local limit (the net linear momentum transfer from the field to the centre of mass motion of the ring is of no concern here). To create a photon-drag current in a closed ring it is thus necessary that the incident electromagnetic field is elliptically (or circularly) polarized. In the last decade persistent currents induced by externally impressed magnetic fields in small metallic loops see e.g. Refs. [250-2631 as well as the Aharonov-Bohm effect [264,265], appearing in the flux-periodic (period It/e) oscillations of the magnetoresistance of mesoscopic metallic rings, have been extensively investigated. Only recently the possible occurrence of the photon drag in mesoscopic loops seems to have been predicted and studied [266].

Let me now present a heuristic calculation of the photon-drag effect in a circular mesoscopic ring of radius ro, and let me for simplicity assume that the cross-sectional area of the ring is infinitesimal small. When the cross-section tends to zero only transitions between quantum states along the ring become allowed. In a Cartesian xyz-coordinate system we place the ring in the z = 0 plane, with the centre in the origin (see Fig. 9.5). Under the assumption that the electron dynamics is

232 0. Keller / PhJaics Reports 26X (I 996) 85-262

Fig. 9.5. Schematic illustration of a mesoscopic ring subjected to a right-(R) or left-(L) hand circular polarized plane electromagnetic wave propagating in a direction perpendicular to the plane of the ring. The photon drag current in the ring, I,‘, induced by R (plus sign) or L (minus sign) circular polarized fields, may circulate in the same or opposite direction as the electromagnetic field vector, depending on the frequency of the field.

free-electron-like along the ring, the relevant eigenfunctions and energies thus are

I/I~(Y) = (27~r&‘;~~(r, z)einH , II = f 1, * 2, . . . .

with

(9.3 1)

I&, z)12 = a(r - r0)Q) , (9.32)

6, = h2n2/2mr~ , (9.33)

in cylindrical (r, 0, z)-coordinates. Without restriction one can assume that the wave function cp is real, i.e. cp = cp *. We excite the ring by a right (plus sign) or left (minus sign)-hand circular polarized external field in the form of a plane-wave propagating in the z-direction, i.e.

Eext(y; CO) = (I&/$) (e, * &)e + i’eiqz , (9.34)

where e, and e. are unit vectors belonging to the cylindrical coordinate system. Since it has been assumed that the cross-sectional area of the ring is zero, local-field effects at the fundamental frequency are absent a priori, cf. also the discussion of the point-particle model in near-field optics in Section 7.1.2.

The photon-drag current along the ring named 18 (the plus and minus sign referring to the two circular polarizations) is given by

I$ =eo- Jo(r) dr dz , (9.35)

0. Keller / Physics Reports 268 (1996) K-262 233

in the free-flow case. The radial photon-drag current density must necessary vanish (the forced and free parts balance each other). Since the A contribution to Z$ is proportional to

(.f,,, - fiJ s $Z ti,i I E ext I ’ d 3r, it appears that the orthogonality of the eigenstates makes the A term vanish in the limit where the external field is constant across the ring. It can be shown that the selection rules for the angular momentum implies that also the B contribution to ZO* is zero in the present case. Altogether, this means that the photon-drag current along the ring may be obtained from the expression

x ~C/~(V)~~(Y)~~.(E~~‘(~))* drdz s J,,(v)* Eext(r)d3r 1 + C.C. . Since the transition current density is given by

&(r) = - [efi(cl + /?)/47clfl~,‘] 6(r - rO)b(z)ei(“-b’oe, ,

it is easy to show that

(9.36)

(9.37)

(9.38)

where a,,, r 1 is the Kronecker delta. The relation in Eq. (9.38) expresses the angular momentum

conservation of the field-particle system. Utilizing also the fact that

$$(r)tj~~(r)e~ - (E”“‘(r))* dr dz = f 2 & ei(flma F *)’ , J

(9.39)

it is readily realized that

1; = - he31Eo12 1 CL -.L1)(2~ * 1)

32rcm2c02ri n h(co+i/z)+~,--~~~ + cc. . (9.40)

At this point we must consider the question concerning how many mobile electrons we hold in the ring. When the cross-sectional area is zero the logic answer is zero electrons. However, once we let the cross-sectional area be finite one can estimate the number of conduction electrons from the magnitude of the ring volume. Let us thus assume that the ring holds two electrons (of opposite spin) in each of the states e’” and eei”. To hold so few electrons the cross-section of the ring must be extremely small. This implies that the separation between the energy eigenstates across the ring is extremely large compared to the separation between energy eigenstates along ring. If the ring thus has a very small cross-sectional area the excitation amplitude to states across the ring is negligible, the number of electrons hold in the ring is small, and the approximation 1 cp 1 2 = 6(r - ro)6(z) is adequate. The various assumptions hence fit together. If only the states a = + 1 are occupied (each with both a spin-up and spin-down electron) in the absence of the electromagnetic field the


photon-drag current takes the value

I+ = 3e31-&12 A2-m2_2-2

0 8rcm20r; ((B’ + - r 2 + A 2)2 - 4u2,4 2 ’ (9.41)

for the right-hand circular polarization, and

1; = - Iof , (9.42)

for the left-hand polarization. In Eq. (9.41), the quantity A denotes the transition frequency between the states a = 1 (or - 1) and a = 2 (or - 2), i.e. A = (c2 - cI)/h = (E_~ - e_ I)/h. In the collision- free limit the photon-drag current takes a particular simple form, viz.

I$ = - 1, = (3e3 ) I&, 1 2/8~m2r~~) [l/(A2 - co”)] . (9.43)

In Fig. 9.6 is shown the ratio between the photon-drag current Z,’ and the magnitude of the Poynting vector of the incident field, I&,,/, as function of the photon energy for different ring diameters. As input data we have used the effective mass of GaAs, i.e. m* = 0.07 m, and a relaxation time r = 6.3 x lo- l2 s. It appears from the figure that the various IO+/‘Sinc - curves exhibit a resonance behaviour in the vicinity of the electronic transition frequency. The sign reversal in lO+/‘Sinc occurs slightly below hA, namely for ho = h(A2 - z-2)112, cf. Eq. (9.41). In Fig. 9.7 the normalized photon-drag current, IO+/Sinc, is plotted as a function of the GaAs ring radius with the photon energy as a parameter. As in the previous figure resonances occur in the vicinity of the matching condition Zzo = h(A2 - T-~)~“.

Let us turn the attention now towards the photon-drag effect in metallic rings, and to be specific let us consider a gold ring of radius r. = 0.41 um,Oand 1 e us assvme that the cross-section of the t

ring is rectangular with the dimensions a = 49 A and h = 38 A. To simplify the approach in a manner which keeps the qualitative features of the photon-drag phenomenon, we make use of the free-electron infinite-barrier model to calculate 1,’ for the ring. For the box model the energy eigenvalues are given by

E P,4,n = (rc2h2/2m) [p2/a2 + q2/h2 + n”/x”ri] , (9.44)

wherep,q,n = f 1, k2, . . . In a low-temperature (T = 0 K) calculation, which in the metallic case also is adequate at room temperatures, all electronic states below the Fermi energy (cr z 5.51 eV in Au) are occupied, and all states above are empty. This means that for a given (chosen) set of {p, q}-quantum numbers among the positive n-values only the one satisfying the inequalities

+,q,n < EF < f$,q,nt 1 (9.45)

can contribute to the photon-drag effect, cf. Eq. (9.40). Letting the quantum numbers p and q run through all relevant combinations, i.e. those satisfying the inequality rc2tt2( p2/a2 + q2/b2)/(2m) < +, all the relevant (positive) n-values can be determined. For the present Au ring the actual number of n-values is 199. For each positive-n transition n = n + 1, there is of course an associated transition -n= -(n+l) between negative-quantum-numbers states, i.e. 1 -+ 2( - 1 + - 2),

2 + 3( - 2 + - 3) etc. Altogether the summation in Eq. (9.40) is over a$nite number of cI’s (n’s), in


0.2

-0.2

-0.3 ( I , I / I 10 15 20 25 30 35 1

Fig. 9.6. Ratio between the photon-drag current, Ii, and the magnitude, Sincr of the Poynting vector of the incident electromagnetic field in a GaAs ring as a function of the photon energy for different ring radii.

0.3

0.2 -_

r 3 0.1 z

s : 0.0

2 k

-0.1

Fig. 9.7. Normalized photon-drag current, l~/S,,,C, as a function of the radius of a mesoscopic GaAs ring calculated for three different photon energies.

the present case 2 x 199. The number of (double degenerated) electronic transitions contributing to the photon-drag effect in the gold ring under consideration hence is 199. In Figs. 9.8-9.10 are shown the results of numerical calculations of the normalized photon-drag current ZO+/Sinc, as a function of the photon energy, tiw, for three different relaxation times. The small vertical lines inserted in the various figures mark the various electronic transition energies which contribute to the photon-drag current along the ring. It appears from the figures that 1,’ goes rapidly towards zero with increasing photon energy once tic0 is larger than the largest of the relevant transition energies, as one would expect. At low frequencies one has I$ > 0 due to the fact that all transitions give a positive contribution to I,‘, cf. Eq. (9.40) and the fact that tin + E, < E,+~ for all a. As the

236 0. Keller ! Physics Reports 26X (1996) 85-262

-0.25 0.20 i.i5 2.io 3.65 4.

tl”J ;m I-] )O

Fig. 9.8. Normalized photon-drag current, I J/C&, in a mesoscopic Au ring of radius r. = 0.41 urn as a function of the

photon energy, ho. The cross-sectional area of the ring is 49 x 38 A’, and the relaxation time used is t = 6.6 x IO- l2 s. The small straight lines at the bottom of the figure mark the relevant electronic transition energies.

frequency is increased more and more transitions give negative contributions to Zof (because hcc, + E, > e,, 1 for these transitions), and hence the current exhibits an overall but slow decrease with increasing photon energy. At a certain photon energy the current changes sign. Apart from the above-mentioned slow decrease, an overall o ~ ’ decrease is present. This decrease is due to the COP2 -factor in front of the summation sign in Eq. (9.40). As the relaxation time is increased, a more and more “noisy” photon-drag response appears in the frequency range below the highest electronic transition frequency. The “noisy” structure however is a nice fingerprint of the individual electronic transitions as one can see from the lower part of Fig. 9.10, for instance. For large electronic collision rates, only groups of narrow-lying transitions reflect themselves in I:, cf. the lower part of Fig. 9.9. Finally, if the collision rate is too high the photon-drag current just exhibits a smooth variation with the photon energy, cf. Fig. 9.8. Albeit the analysis described above, and numerically illustrated in Figs. 9S9.10, from a quantitative point of view may be too simple, it seems to me that measurements of the photon-drag current as a function of the photon energy in mesoscopic rings with low electronic damping might turn out to be a valuable tool for electromagnetic spectroscopy. Roughly speaking, mesoscopic metal rings should be adequate for microwave and far-infrared spectroscopy, whereas semiconducting rings seem particularly useful for the mid- and near-infrared parts of the spectrum, and possibly also for the visible region.

10. Optical phase conjugation of the field radiated by a mesoscopic particle

Theoretical and experimental studies of the linear optical response of atoms and molecules placed in front of a mirror have been carried out for many years as it appears for instance from the review paper of Drexhage [267]. Within the framework of a local approach Morowitz [268,269] analysed the distance-dependent modulation of the molecular fluorescence decay time in terms of


-0.25 iI- 0.20 1.15 2.10 3.05 4.00

FL,, [rnrl‘]

1.25

-0.25 .:' 0.20 0.40 0.60 0.80 1.00

h." [IWI‘]

Fig. 9.9. Normalized photon-drag current, I gf/S’inc, as a function of the photon energy, ho, for the same ring as used in Fig. 9.8. The electron relaxation time is chosen to be 5 = 3 x lo-” s. The bottom figure, showing resolved groups of transitions, is a detailed plot of part of the spectrum of the upper figure.

the image theory, and predicted a cooperative level shift analogous to the Lamb shift. The model by Morowitz was generalized by Kuhn [270] to take into account the finite reflectivity of the surface. In a paper by Morowitz and Philpott [271] the coupling of an excited molecule to surface plasmons was considered, and the theory including the surface plasmon coupling was compared to the pure image theory. Replacing the fixed-amplitude dipole radiator by a fixed-power dipole radiator the steady-state fluorescence emission of a fluorophore located outside a dielectric or metallic surface has been discussed by Hellen and Axelrod [272]. When the dipole is close to a metal surface it may be necessary to invoke the nonlocal part of the metal response in the calculation. Within the framework of the hydrodynamic model this was done by Agarwal and Wolmer [273] in their calculation of the radiative lifetime of an atom near a metal surface. With main emphasis on a determination of the surface-dressed electric-dipole polarizability of the atom


1.60

; ;;_ .$ 0.80 x1

0.20 l.i5 2.io 3.65 4

LJ (IIWI-]

1.60 T = 6.6 x lo-"SFC.

-0.40 ill1 0.20 0.40 0.60 0.80 1.

Fig. 9.10. Normalized photon-drag current, I:/.‘&, as a function of the photon energy, hw, for the Au ring of Figs. 9.8 and 9.9. The used electron relaxation time is r = 6.6 x 10-r’ s. The figure at the bottom, which shows part of the upper-figure spectrum in an enlarged horizontal scale, demonstrates that individual electronic transitions may be resolved in mesoscopic rings with low electron collision rates.

the semiclassical infinite-barrier model was used by Fuchs and Barrera [274] to describe the nonlocal response of the metal. The SCIB-model allowed them to treat the nonlocal surface dressing in a scheme which goes beyond that of the near-local (hydrodynamic) approach. On the basis of a screened nonlocal electromagnetic propagator formalism the present author [67] extended the work of Fuchs and Barrera [274] so as to take into account retardation effects, still treating the metal as a nonlocal reflector. Using the infinite-barrier model, which allows one to treat the inhomogeneity of the electron density at the metal surface, Korzeniewski et al. [275], and Maniv and Metiu [276] developed a theory for the surface screening of the dipole radiation. Using the integral equation formalism, Keller and Sondergaard [277,278] studied (i) the elastic scattering of light from a few atomic dipoles placed outside a flat metal surface, and (ii) the electromagnetic

0. Keller / Physics Reports 268 (I 996) 8.5-262 239

interaction between two dipoles. On the basis of the well-known Bloch formalism, and within the framework of the rotating-wave approximation (RWA), the surface dressing of an electric-dipole oscillator driven coherently by an external field having a frequency lying close to an atomic transition frequency was studied by Huang et al. [279]. The nonlocal electrodynamics of an atom (or a small particle) in front of a mirror was examined by the present author [280].

Despite of the fact that the amount of literature on the electrodynamics of atoms and molecules in front of a normal mirror is quite large, the electrodynamics of microscopic particles in front of a phase-conjugating mirror has only been treated by relatively few authors. It seems that the first article on the subject was that of Agarwal [281]. Following his suggestions Hendriks and Nienhuis [282], Milonni et al. [283], Arnoldus and George [284,28.5], and I [280] have predicted that an atom placed near a phase-conjugating mirror may behave electrodynamically quite different from an atom in empty space or in the vicinity of an ordinary mirror. Experimental studies of the electrodynamics of a single atom in front of a phase-conjugating mirror appears difficult to carry out and so far no one seems to have achieved this goal. Recently, however, the group of the present author succeeded in phase conjugating the spatially divergent field from the tip of a near-field microscope [286,287]. The phase conjugation of an optical near-field in some respects conceptually resembles that of atomic phase conjugation. This is so because the fibre tip acts much like a point radiator sending out multipole fields. Since the tip of the microscope is placed very near the surface of the phase conjugator it appears that a substantial fraction of the angular spectrum of the outgoing (almost) spherical field from the tip can be converted into a converging (spherical) wave.

In the case of an atom in front of a normal mirror it is often sufficient to consider the atom as a point-particle. However, in the case of the phase-conjugating mirror the internal dynamics of the atom a priori appears more important because the phase-conjugated radiation is focused on the atom. For a mesoscopic-particle of an optical near-field microscope local-field effects may be of substantial importance for several reasons. Firstly, the mere size of the tip object may necessitate the consideration of the internal dynamics. Secondly, the fact that the spot size of the phase- conjugated signal may be smaller than predicted by classical diffraction theory, since it appears that evanescent field components in the outgoing field from the tip can be phase conjugated, makes a nonlocal treatment more urgent. Thirdly, taking into account that things may be arranged in such a manner that the phase conjugator amplifies the radiation from the tip, it is hard to believe that a local treatment is sufficient for studying the electrodynamics of a mesoscopic particle in front of a phase conjugating mirror.

10.1. Electromagnetic propagator for a phase-conjugated field

The integral equation formalism described in Part A of this monograph may be applied to study the local-field problem associated with the phase conjugation of the field radiated by a mesoscopic particle provided we add to the pseudo-vacuum propagator in Eq. (2.64) an extra term accounting for the phase conjugation of the field from the particle. The dyadic electromagnetic propagator of the phase conjugator/vacuum system, G(v, r’; co), hence consists of five pieces, i.e.

(10.1)


The two last terms on the right-hand side of Eq. (10.1) originate in the presence of the nonlinear medium. The first one, I, is associated with the normal (Fresnels) reflection of the field from the surface, and the second one, GPC, accounts for the propagation of the phase conjugated field. For simplicity, we only treat the case where the phase conjugation stems from degenerate four-wave mixing. This implies that the incident and phase-conjugated fields have the same frequency. In the following it is also assumed that the phase conjugator occupies the half-space z > 0. For this geometry it is adequate to use the Weyl expansion for the transverse direct propagator, i.e. Eq. (7.2) with Dz(z - z’; q,,, co) given in Eq. (5.78) inserted for GB(z, z’; Q, u). Let us now establish a simple explicit expression for the propagator G pc, following a procedure originally suggested by Agarwal et al. [288]. We start by decomposing the propagator as follows:

D,‘(r - r’) = D!“)(r - r’)O(z - z’) + Dy’(r - r’)@(z’ - z) + D”‘@ _ r’) , (10.2)

where D !“’ and D!“’ represent the contribution to D;f from all homogeneous (H) plane waves propagating towards (D(F)) or away (D(F)) from the surface, and DC” gives the contribution from all inhomogeneous (I), i.e. evanescent plane waves. Since the homogeneous and inhomogeneous parts of the angular spectrum are characterized by the conditions ql < o/c0 and qll > co/co, respectively, the exprerssion for the various terms in Eq. (10.2) are

D?l<,(r - r’; co) = (2~)-~ i

S- ’ - Di!‘;<)(z - z’; qll, w) 4,, 6 w/co

x Se’? .b -r/j dzq,, , (10.3)

D”‘(r - r’; o) = (27~) - 2 S- ’ . D”‘(z - z’; qll, co)

(10.4)

with

+ iq’:(z-2’) (10.5)

and D(‘)(z - z’; ql , co) = Di(z - z’; qll, co) (Eq. (5.78)), remembering that qll > Q/C. In Eq. (10.5), the plus and minus signs in the exponential function are associated to D!“’ and D!“‘, respectively. At this stage Agarwal et al. [288] makes a drastic reduction of the problem assuming that only the homogeneous part of the incident field is phase conjugated. I shall return to this assumption in the subsequent subsection. Furthermore, Agarwal et al. [2SS] assume that the nonlinear reflection coefficient of the phase conjugator is independent of the angle of incidence and state of polarization of the incoming field. In a formal scheme the last assumption can be removed without difficulty. Thus, for the ideal situation sketched above an incident electric field &(r; o) is replaced by ,&?*(r; w) in the phase conjugation process (PC), i.e.

Q(r; <II) E$_ pF*(r; (0)) ) (10.6)

0. Keller 1 Ph?xics Reports 268 (1996) 85-262 241

where ,U is the complex reflection coefficient of the phase conjugator. The reflection coefficient in a phenomenological manner accounts for the losses (I p 1 < 1) or gains (1 p 1 > 1) that may arise in the phase conjugation process. Denoting as usual the prevailing current density of the mesoscopic medium by J(r; w), the homogeneous part of the field radiated from the mesoscopic particle towards the surface is given by

b(r; u) = - &to O(z - z’) D(c’(r - r’; co). J(u’; co) d3r’ . (10.7)

The phase conjugation process creates from this field a replica

,d*(r; u) = - &,~/~ [D’$‘(r - r’; a)]* .J*(r’; a)d33r (10.8)

in the vacuum half-space. The result in Eq. (10.8) shows that the electromagnetic Green’s function describing the propagation of the phase conjugated signal must be

GPC(r - v’; tn) = P(r - r’; u)c , (10.9)

where the tensor

P(r - r'; to) = - ,u[D!H)(r - r'; o)]” = ,uD!H)(r - r'; u) ,

and the operator c is defined by the property

(10.10)

tf(r; co) =,f*(r; 0) . (10.11)

A schematic illustration of the angular plane-wave spectra of the propagators D!H’(r - r’; co) O(z - z’), D!“‘(r - r’; u) O(z’ - z), and P(r - r’; 01) is shown in Fig. 10.1.

Having determined the explicit form of the electromagnetic propagator describing the phase- conjugation process, albeit under idealized circumstances, we just need to give the expression for the background field driving the dynamics. For the degenerate four-wave mixing process considered above this field is

EB(r; m) = E’(r; co) + ER(r; w) + EPC(r; to) , (10.12)

where E’ is the incident field, ER is the field linearly reflected from the surface, and Epc is the phase conjugated replica of the incident field.

10.2. RPA integral equation for the local ,field and its solution in the paramagnetic limit

In the presence of a phase conjugating mirror it appears from the discussion of the previous subsection that the basic integral equation for the local field inside the mesoscopic particle takes the form

E(r) = EB(r) + s

KN( r, r’).E(r’)d3r’ + KpC(r, r’).E*(r’)d3r’ s

(10.13)


Fig. 10.1. Schematic illustrations showing the plane-wave angular spectra of the D(y)O(z - z’) (b), and P (c).

propagators D’~‘@(z’ - 4 (4,

in the RPA approach. In the absence of the phase conjugation process, the Green’s function would be GN(r, r’) = G(r, r’) - G&r, r’), and the associated so-called normal (N) kernel is KN(r, r’). Essentially, the K,-kernel is obtained by integrating the product of GN(rr r”) and uRPA(r”, r’) over the r”-domain, cf. Eq. (4.31). The kernel KPC(r, r’) originates in the phase-conjugation process, and is given by

K&r, r’) = - iy,o s

P(r - r”) - tsiPA(r”, r’) d3r” . (10.14)

By assuming that the paramagnetic interaction dominates the coupling, the local field inside the mesoscopic particle takes the form

E(r) = EB(r) + C (PmnL(r) + BPmCX9) , m,n

where

fig: = p$,(r).E*(r)d3r

15) (10.

(10. 16)

and

F::(r) = - 2~~ _Ll -xl hw + E, - e, s

P(r - r’).Jnm(r’)d3r’ . (10.17)


The explicit expressions for F,,(r) and b,,, are given in Eqs. (7.5) and (7.7), respectively. Upon a comparison of Eqs. (7.7) and (10.16) it follows that

(10.18)

are to be obtained from the following set of algebraic In the present case, the unknown Pmn’s equations:

Pm - c OT.L”Pop + Po”,“P:p, = H,,

with

> (10.19)

(10.20)

and NY: and H,, given by the first integral on the right-hand side of Eqs. (7.12) and (7.13). Letting m and y1 run through the possible values, Eq. (10.19) gives a set of inhomogeneous, linear algebraic equations among the unknown Pmn’s. If the number of relevant levels is k, the number of algebraic equations among the complex coefficients will be k(k - 1). Since each of the coefficients has a real and imaginary part (even in the absence of irreversible losses) the number of equations among unknown real coefficients will be 2k(k - 1). If the mesoscopic particle interacts only with a normal mirror, the number of unknown coefficients will be k(k - 1) in the lossless limit.

10.3. On phase conjugation of evanescent waves. Quantum dots of light

Following the recipe of Agarwal et al. [288], I assumed in Section 10.1 that only the homogeneous part of the field radiated by the mesoscopic particle was phase conjugated. The homogeneous part of the field consists of the spectrum of propagating, undamped plane waves, since ql, < o/c. If one considers the monochromatic radiation from a point-particle, and assumes that all Weyl components, homogeneous as well as inhomogeneous, are phase conjugated with the same complex reflection coefficient p, it is obvious that the phase conjugated field is focused to a point, located on the site of the point-particle, cf. Fig. 10.1. If instead one phase conjugates (with a constant ,u) only the propagating (homogeneous) part of the point-particle radiation, it is clear that the focus of the phase conjugated spot will have a finite size. Since the largest wave number along the surface in the homogeneous group of plane waves equals u/co, qualitatively speaking, the spot size cannot be smaller than dictated by classical diffraction theory. Provided phase conjugation of inhomogeneous components of the point-dipole field is achievable one might expect that spot sizes smaller than the smallest ones allowed by classical diffraction theory can be obtained.

Is it possible thus to phase conjugate evanescent components of the radiated field, and if so, to which extent? To address these questions we consider the standard theory of four-wave mixing in a transparent medium having a linear refractive index n. In this theory the two pumps and the probe beam are undamped plane waves, and the phase conjugated wave emerging in the nonlinear process then also becomes an undamped plane wave. Let us consider now just one of the evanescent waves in the angular spectrum of the mesoscopic particle. If this wave upon linear


transmission through the vacuum/medium boundary is converted into a homogeneous wave in the nonlinear medium phase conjugation takes. The phase conjugated beam upon transmission through the boundary in turn is converted into an evanescent wave. Since freely propagating monochromatic plane waves inside the phase conjugator have wave numbers, (o/c)n, which are independent of their direction of propagation, a homogeneous wave inside the medium never can possess a wave number larger than (w/c)n along the surface (if the wave number component is

(co/c)n the wave propagates along the boundary). Using the law of refraction it in turn follows that at least all waves in the angular spectrum from the mesoscopic medium which satisfy the criterion qll < (o/c)n can be phase conjugated. The standard theory of phase conjugation hence predicts that evanescent waves from the mesoscopic object within the angular range u/c < ql, < (o/c)n may be phase conjugated. If an effective phase conjugation of modes in the range 4, < (o/c)n is achieved a subwavelength spot size may emerge. Recently, it has been demonstrated experimentally that evanescent waves can be phase conjugated [286]. In the above-mentioned experiment the light emitted from the tip of an external-reflection near-field optical microscope was phase conjugated via degenerate four-wave mixing in a Fe: LiNb03 crystal. It was observed that at a wavelength of 633 nm a spot size less than 180 nm could be produced. A few characteristic near-field optical images of subwavelength-sized spots are shown in Figs. 10.2 and 10.3. It is convenient for several reasons to use the radiation from the tip of an optical near-field microscope as the source field. First of all, it is possible to launch from a subwavelength-sized fibre tip a large angular spectrum of homogeneous and inhomogeneous plane waves. Secondly, by placing the tip in a near-field (or middle-field) distance from the phase conjugator an appreciable part of the evanescent spectrum emitted from the tip can reach the surface of the nonlinear medium. A direct experimental demonstration of this transpires from fact that the size of the phase conjugated spot becomes larger when the tip-phase conjugator distance is increased [286]. Thirdly, in order to be able to measure a subwavelength spot size one needs an optical detector with a spatial resolution exceeding the classical resolution limit. An optical near-field microscope can achieve this!

Various nonlinear crystals have so long memory times that these small light spots, typically located at distances less than a wavelength from the surface, can stay “alive” long time after the fibre tip has been removed from the surface [286,287]. In Section 7.1.3, I discussed in relation to near-field spectroscopy the confinement problem in electrodynamics. We know that in space regions occupied by matter (electron) waves, the electromagnetic field cannot be confined to a region of extension smaller than that of the transverse current-density distribution. We have seen above that the confined fields of mesoscopic objects via phase-conjugation processes with long memory times can be used to confine fields to spatial domains of subwavelength extension located in material (electron)-free regions of space. If light is confined to a region of subwavelength extension in a sense one may say that a mesoscopic light spot has been produced. Recently, I have suggested that the name a quantum dot of light is used for such a spot.

Let us finally for a short while turn our attention towards the possibility for phase conjugating the remaining part of the angular spectrum from the mesoscopic object, i.e. those evanescent wave- vector components which satisfy the inequality q,, > (co/+ Inside the phase conjugator these modes are also evanescent, so, the question arises whether or not evanescent probe-beam modes can be phase conjugated? The question is modified a bit if one takes into account the fact that the losses in the phase-conjugating medium are never completely absent. Because of this the probe field is really a more or less strongly decaying inhomogeneous wave. The real part of the

0. Keller /Physics Reports 268 (1996) k-262 245

( a ) (b)

Fig. 10.2. First experiment (performed on 5 May 1994) demonstrating the production of a quantum dot of light by phase conjugation of an optical near field from a scanning near-field microscope. The left figure (a) shows a surface area of dimension 2 x 2 urn2 before exposure to light. The right figure (b) shows the same area after exposure. A quantum dot of

size - 180 x 250 nm’ has been created. After Ref. [13].

wave vector has components both parallel (Q) and perpendicular (4:) to the (flat) boundary, whereas the imaginary part of the wave vector (4:) only has a component perpendicular to the surface. It is certainly not obvious that the standard recipe given in Eq. (10.6) can be employed in the present case, especially not if the exponential decay constant is large. In an attempt to circumvent this problem let us imagine that we have resolved the inhomogeneous wave field of the probe

and &(r; CL)) = 0 for z < 0 into an angular spectrum of undamped plane waves, i.e.

s cc

&(r; 0) = &f. eiqll “11

eiql= dql -CC KY + i(qL - ~7) 2~ .

(10.22)

For each plane-wave component in Eq. (10.22) which has a positive q1 (wave propagation into the phase conjugator away from the surface) we now use the standard theory for degenerate four-wave mixing, generalized to take into account that we expect the nonlinear reflection coefficient to depend not only on the frequency (0) hut also OYE the (real) wave vector q,l + eZqy. We also account

246 0. Keller i Ph_vsics Reports 268 (1996) 85-262

(b)

Fig. 10.3. Two quantum dots produced by degenerate four-wave mixing using the field from an optical near-field microscope as probe field. In the upper figure (a) a strong light exposure (over exposure) has been used, and in the lower figure (b) normal exposure was applied. The distance between the two dots is approximately 0.5 pm, and the areas shown in (a) and (b) are 2 x 2 pm2 and 1 x 1 pm2, respectively.

for the fact that the state of polarization of the field may change in the phase conjugation process by replacing the scalar p by a tensorial reflection matrix, ~(4 , yl, co). In the present context the tensorial character of p is not that important. The nonlinear reflection matrix of course also depends on the characteristics of the pump beams but we keep this dependence implicit in the notation. Based on the considerations above the phase-conjugated field takes the form

(10.23)

In the limit, where the damping of the probe beam is small (tc:z 4 I), one may assume as in the standard theory, that p only depends on the frequency of light. If this is so we retain the phenomenological theory for phase conjugation of weakly damped plane waves [289]. Thus, altogether it seems to me that phase conjugation of evanescent waves is possible.

The degree of complexity of a not yet developed theory of phase conjugation of evanescent waves necessarily depends on the magnitude of the decay constant K :. Basically, one should start from a nonlocal quantum mechanical response formalism. By analogy with the response theories for the linear conductivity, the second-harmonic generation, and the photon-drag effect, which lead to microscopic expressions for (T(z, z’), Z;(z, z’, z”), and KI(z, z’, z”), such a formalism would result in an explicit expression for the nonlinear response function Epc(z, z’, z”, z”‘) responsible for the phase conjugation process on the microscopic level. The entire scenario hidden in Epc(z, z’, z”, z”‘) might be needed if one aims at a description of the phase conjugation process on the atomic monolayer


level, a description which is necessary in order to investigate the possibilities for phase conjugation in quantum wells, and in the profile region of the electron density distribution at a metal or semiconductor surface, say. In many cases a refinement of the theory as described above is not needed. Hence, if the probe beam does not decay spatially too fast, a nonlocal bulk theory for an assumed translationally invariant medium may be sufficient. For the above mentioned first- and second-order processes the associated response functions are a(& C(q, q’), and II(q, q’), and in analogy with this the corresponding fourth rank response tensor for the phase-conjugation process is of the form Epc(q, q’, 4”). Using this form, a significant simplification of the expression for p(q,:, ql, co) will emerge. While phase matching appears to be needed not only parallel to the surface but also in the direction perpendicular to the boundary if one relies on the standard (macroscopic) approach [289], I do believe on the basis of microscopic considerations as the ones indicated above, that the phase matching in the normal direction to the surface can be relaxed without a total damage of the possibilities for phase conjugation. Only if this prediction holds we can hope for phase conjugation of evanescent waves with exponential decay lengths comparable to the characteristic wavelength of the four-wave mixing process in consideration.

Acknowledgements

I am indebted to the friends and colleagues from abroad and from Denmark who over the years by virtue of their work and ability of inspiration, and through numerous discussions have helped me to sharpen my understanding of the role of local-field phenomena in the electrodynamics of mesoscopic media. Without their fruitful contributions I would not have been able to write the present review.

Appendix A. Integral relation between the transverse local field and the microscopic current density

A.I. Derivation of Eq. (2.13)

Taking as a starting point the relation

one obtains upon integration over the shaded volume shown in Fig. 2.1, and use of Gauss theorem to the integral of the left-hand side of the resulting equation

(A.21

248 0. Keller /Physics Reports 26X (1996) KS-262

By making use of the ith component of Eq. (2.10) and Eq. (2.12) the volume integral in Eq. (A.2) can be written in the form

s “’ CET,~(Y’)(V’)~Y~(IY-~‘I) -g~(l~-~‘l)(v’)*ET,i(~‘)l d3r’

8 (4

Vl = ipOa

i yO( 1 r - ~‘1) JT.;(r’) d3r’ .

F(I) (A.3)

By combining Eqs. (A.2) and (A.3) one gets

i?’

z.1 + (r [ET@‘) n’ . V’go(jr -r’l) -go(lr- r’l)(n’. V’)E,(r’)]dS’

1 i

u

VT

= ipOcl-, go(lr - ~‘I)J~(~‘)d3r’ ,

E 6-j I i

i.e. precisely the ith component of Eq. (2.13).

(A.4)

A.2. Derimtion of Eq. (2.14)

An explicit calculation of the surface integral over CT in Eq. (2.13) can be performed in the limit E(Y) -+ 0 as follows. Starting from the equations

d

lim E#)n’. V’yO(lr - - r’j) dS’ s+O /

0

= ET(r) lim IZ’. V’gO(lr - r’l) dS’ ,

S+O 11 (A.9

c I-” lim s+O U

go((r - r’l) n’s VIE+) dS’ i

4

yo( Ir - r’l) n’ dS’ 11

- V&(r) , (A.61

and making use of the results

4 lim s+O U

n’. V’g,(lr - r’j)dS’ = 1 , 1

(A.71

0

lim S-r0 is

g,(Ir-r’I)n’dS’ =O, I

(A4

one readily can obtain Eq. (2.14). By performing the integrations in spherical coordinates centred on the point r, it is a straightforward matter to prove that Eqs. (A.7) and (A.@ are correct.

0. Keller/Physics Reports 26‘7 (1996) X5-262 249

A.3. From the scalar Green’s function go to the dyadic propagator Do

By making use of the relations [4]

?’

VI vv. yo( Ir - r’l) J,(r’) d3r’

F(r) + 0

=s

VT VV(LJ~(~Y-~‘~).J~(~‘))~~~‘-+J~(~),

c(r) + 0

VT VI V2

s go( 1 Y - ~‘1) &(r’) d3r’ =

E(r) + 0 i‘ V2(go(Ir - 4I.W)) d3r’ -A(r) ,

i.(r)-0

(A.9)

(A.lO)

one obtains upon performing the operation Vx ( Vx ( ... )) on both sides of Eq. (2.15)

Vx ( VxE,(r)) = ipocL) ?’

V1 (VV-UV2)(~0(I~-~‘~)Jr(~‘))d3r’

E(l)+0

+ $i~ocL)Jr(y) + (VV- U V’).&(Y), (A.1 1)

where

(A.12)

Now, by utilizing the wave equation Vx (VxE,) = qiET + ipoe&, and the relation V 2g0 = - q?jgo (Eq. (2.12)) Eq. (A.1 1) can be rewritten as follows:

+1 Zr

4; go(,~-F1/)~-ET(T’)Bgo(~~~r”)

> dS’. (A. 13)

In writing the last term on the right-hand side of Eq. (A.13) I have made use of the fact that the field point r is not located on the surface CT which means that the operator VV- U V2 can be moved inside the integral sign, directly. By making use of Eq. (2.16) it readily appears that Eq. (A.13) may be written in the form presented in Eq. (2.18).

Appendix B. Eigenvector expansion of the transverse vacuum propagator

To derive Eq. (2.47) let us write the dyadic expansion of the transverse vacuum propagator

D~(Y, ~0) = C ‘i,ij(Y> ~0) eiej i,j= x,y.z

(B.1)

250 0. Keller / Ph.vsics Reports 268 (1996) 85-262

in the form

where

Fj(r, r~) = C DE,ij(Y> ~0) ei . (B.3)

Next, we expand Fj(r, rO) in terms of the complete set of transverse eigenvectors { cz > of Eq. (2.44), i.e.

Fj(r, r0) = C Aj,(rO) 5fCr) . (B.4) M

By inserting Eq. (B.4) into (B.2) we thus obtain

Operating with V2 on both sides of Eq. (B.5), we get

V’Di(r, ~0) = - C C 4: Ai(r,) G(r) ej .i m

(B.6)

after having used V2 <f = - L$,<: (Eq. (2.44)). By inserting Eqs. (B.5) and (B.6) into Eq. (2.41) one then obtains

CC(qt - 415) Aj,(f”o) SI(r)ej = ~T(Y - ro) . (B.7)

In the next step we make a scalar multiplication of the left-and right-hand sides of Eq. (B.7) with (c,‘(r))* from the left followed by an integration over the domain Q. This gives

C x(4; - 4:) Ai R (<T(r))* * <L(r)d3y ej = * ({z(r))* * ST(r - ro) d3r . j m 1 s 03.8)

The orthonormality of the eigenvectors over the domain 52 (see Eq. (2.45)) simplifies Eq. (B.8) to

C (4; - Cl,‘) 4iPo) ej = (C(ro))* 2

or equivalently

(B.lO)

By inserting Eq. (B.lO) into Eq. (B.5) we finally obtain Eq. (2.47).


Appendix C. Single and double commutator relations between the particle-density and free-Hamiltonian operators

To derive the commutator relation postulated in Eq. (3.87) it is adequate to carry out the calculations with the operators given in first-quantized form. Hence, one immediately obtains

CHF, N(r)1 = &C Cd, &r - rdl . x

In turn, one finds via a few elementary manipulations

CL d(r - rJ1 = W) Wd(r - rdx +b.(VdW - rd)l

= - (k/i) V-(&r - ror)pa +paG(r - r,)) .

By inserting Eq. (C.2) into Eq. (C.l), and making use of the expression

h(r) = - & 1 l&Q - r,) + 6(r - r,)p,] , a

one readily obtains the first-quantized form of Eq. (3.87). Following the procedure outlined above the double commutator

[CHF? N(r)l, N(r’)l = &c [Cd, &r - ra)l, &r’ - rx)l . L1

is easily transformed. Thus, we start from

[ [pt, 6(r - r,)], 6(r’ - r,)] = ih V. [S(r - r&&r’ - r,)

+pb6(r - rJd(r’ - ra) - d(r’ - r,)d(r - r,)p, - 6(r’ - r&&r - r,)l

= 2ti2 Vu [6(r - r,) Va6(r’ - rJ]

= - 2ti2 V. [6(r - rJ V’6(r’ - ra)]

= - 2ti2 V. V’(J(r - r,)S(r’ - r,))

= - 2h2 V. V’((s(r - r’)h(r - ra)) .

Performing a summation of Eq. (C.5) over D! we obtain

[ [HF, N(r)], N(r’)] = - (h2/m) v- v’ 6(r - r’) c 6(r - r,) dl >

= - (h2/m) V. V’(6(r - r’) N(r)) ,

cc.11

(C.2)

(C.3)

CC.41

(C.5)

(C.6)

252 0. Keller 111_ Ph_vsics Reports 268 (I 996) 85-262

Appendix D. Zero- and first-order moments of the current density of a moving electron wave packet

D. I. Zero-order moment

To derive Eq. (7.42) one may start from the commutator relation

[r, H] = (iti/m) I7 .

By means of this one obtains

(IZ)=; Y*[r,H] Y’d3r. s

Using the Schrodinger equation on the right-hand side of Eq. (D.2), it is realized that

(Iz)=m a s dR rzIY-1*d3r=mdl.

CD.11

U3.2)

(D.3)

To relate (ZI) to the zero-order moment of the current density the equation of continuity (Eq. (7.46)) is used to transform the expression in the middle of Eq. (D.3). Hence, one obtains

(n)=p rV.Jd3y= -p Jd3r, e s e s (D.4)

as indicated in Eq. (7.42).

0.2. First-order moment

To verify the expression presented in Eq. (7.43) for the first-order moment of the current density, it is adequate to start by considering the commutator [H, rr]. Using the division H = HF + H,,

with HF = [ - h2/(2m)] V* and HI = [etz/(2mi)] (V-A + A - V) + [e2/(2m)] A .A, it is known (see e.g. [290]) that

[HF, rr]ij = - (h2/m) { (U)ij + 2(r V)ji - [U X (r X V)]ij} . (D.5)

A straightforward, but tedious, calculation gives

[H,, rr] = (eti/mi) [2Ar - U x (r x A)] P.6)

By combining the results of Eqs. (D.5) and (D.6) one obtains

[H, rr] Y’ = - (h”/m) U Y + (2ti/mi) [((A/i) V+ e_4) Y] r

+ (ih/m) U x [r x ((h/i) V + eA) Y] (D.7)

0. Keller J Physics Reports 268 (1996) 85-262

Multiplication of Eq. (D.7) by Y*, followed by an integration over space give

253

s Y*[H,vr]Yd3r= -;U+$ j

Y*(17Y)rd3r+~U~(L). (D.8)

By means of, e.g., the general equation in (7.50) for the time evolution of the mean value of the observable 0 = (- e/2)vr it follows that

combining Eqs. (D.8) and (D.9) we thus have

(d,dl)<@-(e,2m)Ux(L)=$/-; jY*(flY)rd”r.

Using the explicit expression for the current density (Eq. (6.42)), and the relation

- Y(VY*)rd3r = U + s s

Y*(VY)rd3r,

one can readily show that

s Jrd3r=$U-E Y*llYd3r. 1 m j

Together, Eqs. (D.lO) an (D.12) lead to Eq. (7.43).

(D.9)

(D. 10)

(D.ll)

(D.12)

Appendix E. Introduction of the mean values (l7r) and (m)

To rewrite the expression for d(Q)/dt given in Eq. (D.9), we use the commutator identity

[H, rr] = [H, Y] Y + r[H, r] . (E.1)

Inserting Eq. (D.l) into Eq. (E.l), we then obtain

[H, rr] = (h/im) (ZIr + m) . (J3-4

By combining Eqs. (D.9) and (E.2), Eq. (7.51) is readily verified.

254 0. Keller / Ph.vsics Reports 268 (I 996) 85-262

To prove the correctness of Eq. (7.52) let us make use of the dyadic expansion of the unit tensor, and thus write

UX(L)=C Y*eieiX(~xnY)d3r i s

Y*[eiei*(l7Y)r - r*eieiHY] d3r. (E.3)

Interchanging the integration and summation procedures, we obtain using once more the expansion U = Cif?ik?i

[Y*(ZIY)r- Y*rZIY]d3r. (E.4)

Finally, we transform the first term on the right-hand side of Eq. (E.4) as follows:

s (LfY)rY* d3r = (nY)*rY d3 r]*=[/Y*ZZrYd’r]l. (E.5)

To obtain the last result in Eq. (ES) the hermiticity of the operator n was used. By inserting Eq. (E.5) into Eq. (E.4), we arrive at the relation in Eq. (7.52).

Appendix F. Derivation of expressions for d2(Q)(t)ldt2 and dUx <L)(t)ldl

Let us consider the double commutator [[H, rr], H]. Starting from the result in Eq. (E.2) the double commutator can be written in the form

CM 4, HI = Wim) ( W, HI r + fl Cc HI

+ [r,H]Il +r[zI,H]} . F.1)

Since one can show that

[ZI, H] = iWb , F.2)

where

Fb= -(e/2m)(ZlxB--BxZI) (F.3)

is the part of the Lorentz-force operator which belongs to the magnetic field (B), the double commutator becomes

[[H, rr], H] = (2ti2/m) i nn + 4 (Fbr + rFb) 1 . CF.41


We also need the time derivative

(a/at) [H, rr] = @/im) ((an/at) Y + r(anjat)) .

Since II =p + eA implies that

an/at = e&4/& = - eE ,

one obtains

(a/at) [H, ~1 = (ti/im) (F,r + rF,) ,

(F.5)

F-6)

(F.7)

where

F, = -eE (F.8)

is the electric-field part of the Lorentz-force operator. Since, according to Eq. (D.9)

(d2/dt2) <Q>(t) = (e/W WW C CK 4 > , (F.9)

one finds by applying the time evolution equation in (7.50) to the operator 0 = [H, ur], and by using the results in Eqs. (F.4) and (F.7), precisely the formula in Eq. (7.62).

To derive the result in Eq. (7.64), Eq. (7.63) is multiplied from the left by U x . Employing next the dyadic expansion of the unit tensor, and performing then manipulations analogous to those used to proceed from Eq. (E.2) to (E.3), we obtain

(Ux(rxF)) = (Fr)* - (rF), (F.lO)

and

(Ux(Fxr)) = (rF)* - (Fr) . (F.ll)

A straightforward manipulation involving the equations above finally leads to the result given in Eq. (7.64).

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