Photoemission studies of quantum well states in thin...

Photoemission studies ofquantum well states in thin ®lms

T.-C. Chianga,b

aDepartment of Physics, University of Illinois, 1110 West Green Street,Urbana, IL 61801-3080, USA

bFrederick Seitz Materials Research Laboratory, University of Illinois,104 South Goodwin Avenue, Urbana, IL 61801-2902, USA

Amsterdam±Lausanne±New York±Oxford±Shannon±Tokyo

Contents

1. Introduction 184

2. Basic properties of thin ®lm quantum wells: Ag on Au(1 1 1) 187

2.1. Con®nement by a relative gap 187

2.2. Model calculations 191

2.3. Wave functions and quantum numbers 193

3. Resonances, coupled quantum wells, and superlattices 195

3.1. Incomplete con®nement and resonance states: Ag(1 1 1) + Au + Ag 195

3.2. Coupling between quantum wells: Au(1 1 1) + Ag + Au + Ag 198

3.3. Superlattices 201

4. Mismatched interfaces 203

4.1. Ag on Cu(1 1 1) 205

4.2. Ratcheting quantum well peaks Ð Ag on Ni(1 1 1) 206

5. Atomically uniform ®lms as quantum wells and electron interferometers 208

5.1. Discrete layer thicknesses 208

5.2. Preparing atomically uniform ®lms 210

5.3. Intensity modulations 214

5.4. The Bohr±Sommerfeld quantization rule and band structure determination 214

5.5. Quantum wells as electron interferometers 219

5.6. Temperature dependence of the band structure 222

5.7. Quasiparticle lifetime and scattering by defects, electrons, and phonons 224

5.8. Interfacial re¯ectivity and phase shift 226

6. Magnetic effects and spin polarization 227

7. Summary and conclusions 231

Acknowledgements 232

References 232

182 T.-C. Chiang / Surface Science Reports 39 (2000) 181±235

Photoemission studies of quantum well states in thin ®lms

T.-C. Chianga,b,*

aDepartment of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801-3080, USAbFrederick Seitz Materials Research Laboratory, University of Illinois, 104 South Goodwin Avenue,

Urbana, IL 61801-2902, USA

Manuscript received in final form 9 May 2000

Abstract

Con®nement of electrons in small structures such as a thin ®lm results in discrete quantum well states. Such

states can be probed by angle-resolved photoemission, as ®rst predicted by theory in 1983 and later observed

experimentally in 1986. Since then, numerous advances have been made in this ®eld, and the purpose of this

report is to review the basic physics and applications of quantum well spectroscopy. The energies and lifetime

widths of quantum well states in a ®lm depend on the ®lm thickness, the dynamics of electron motion in the ®lm,

and the con®nement potential. A detailed study allows a determination of the bulk band structure of the ®lm

material, the lifetime broadening of the quasiparticle, and the interfacial re¯ectivity and phase shift, as will be

demonstrated with simple examples. Quanti®cation of the photoemission results can be achieved via a simple

phase analysis based on the Bohr±Sommerfeld quantization rule. Explicit forms of wave functions can also be

constructed for additional information regarding the spatial distribution of the electronic states. From such

studies, a detailed understanding of the behavior of simple quantum wells including the effects of lattice

mismatch can be developed, which provides a useful basis for investigating the properties of multilayers.

Examples of multilayer systems including coupled quantum wells and superlattices will be presented and

discussed. An important recent development in this ®eld is the preparation of atomically uniform ®lms, as

demonstrated for Ag(1 0 0) grown on Fe(1 0 0) whisker substrates. The elimination of atomic layer ¯uctuation

allows a precise measurement of the intrinsic line widths of quantum well states, which are related to the

quasiparticle lifetime and the interfacial re¯ectivity. The underlying physics can be described in terms of electron

interferometry, and a Fabry±PeÂrot analysis yields the quasiparticle lifetime. It consists of three major

contributions including electron±electron scattering, electron±phonon scattering, and defect scattering. The

experimental results are compared with theoretical predications based on the Fermi-liquid theory and a

perturbative treatment of the electron±phonon coupling. Since the absolute ®lm thickness in terms of monolayers

is known from layer counting, the same interferometric analysis yields an electronic band structure of Ag with

unprecedented accuracy (< 30 meV). The resulting Fermi wave vector challenges the de Haas±van Alphen value.

Film structures containing magnetic materials can exhibit spin-split quantum well states, and their magnetic

Surface Science Reports 39 (2000) 181±235

0167-5729/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 7 - 5 7 2 9 ( 0 0 ) 0 0 0 0 6 - 6

* Corresponding address: Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801-3080,

USA. Tel.: �1-217-333-2593; fax: �1-217-244-2278.

E-mail address: [email protected] (T.-C. Chiang).

sensitivity makes them suitable candidates for device applications. Quantum well spectroscopy is useful for

clarifying the relationship between quantum con®nement and translayer magnetic coupling effects. # 2000 Else-

vier Science B.V. All rights reserved.

1. Introduction

Discrete states form when electrons are con®ned in space by a potential well. An elementary exampleis the hydrogen atom problem described in standard quantum mechanics textbooks. The electron in ahydrogen atom is bounded by a three-dimensional Coulomb well, and the resulting states form a set ofenergy levels that are well described by the Bohr theory, which is a major cornerstone for modernquantum mechanics. Likewise, electronic states are quantized in molecules, which represent morecomplex con®nement potentials.

Quantum well states, the subject of this report, are usually associated with discrete quantizedelectronic states in small arti®cial structures with adjustable physical dimensions. Although the basicphysics is similar, energy levels in atoms or molecules are generally not referred to as quantum wellstates. Arti®cial structures allow tailoring of properties, and metastable and nonequilibrium structuresthat are not found in nature due to thermodynamic requirements may offer unique opportunities forapplications. Many electronic devices are made of thin ®lms. When ®lms become thin enough, quantummechanical effects become important. Because of a widespread interest in electronic deviceapplications and a long history of research and development in this area, thin ®lms have been amajor playground and test ground for quantum well effects. However, in recent years, interest inquantum structures of other forms has been growing. These include quantum dots, corrals, wires,stripes, etc. Research in such low-dimensional systems has been greatly aided by advances innanofabrication techniques including self-assembly, template growth, and direct atomic manipulation.As the system dimensions reduce, there is an increased localization and overlap of electronic wavefunctions, and consequently, an increase in electron correlation. Thus, low-dimensional systems offer aconvenient platform for experimenting with many-body effects.

This paper will focus on quantum well effects in thin ®lms. A simple beginner's model for electronicmotion perpendicular to the ®lm surface is that of a free electron con®ned in a one-dimensional box.Although this is a very crude model, it serves to illustrate the basic ideas and provides a good startingpoint. The allowed wave vectors k for stationary states, or quantum well states, are determined by therequirement that standing wave patterns ®t into the geometry

k � npd; (1)

where n is an integer quantum number and d is the ®lm thickness or box dimension. The energy levelsare given by

E � �h2k2

2m� �h2

2m

npd

� �2

; (2)

where m is the free electron mass, and the wave functions are given by

c�z� / sinnpz

d

� �: (3)


Fig. 1 shows the probability density for the ®rst three quantum well states n � 1, 2, and 3. Thequantum number n is just the number of antinodes (or maxima). These results can be found in standardtextbooks of quantum mechanics. Although this is effectively a one-dimensional problem, theelectronic wave functions are extended within the ®lm along the x and y directions. The con®nementdoes not generally lead to enhanced electron correlation effects compared to the bulk case (exceptpossibly when the ®lm thickness is reduced to atomic dimensions).

For a solid ®lm, the E(k) band dispersion relation is generally different from the free electrondispersion given in Eq. (2). This has a direct bearing on the measured energies of the quantum wellstates, En. Thus, a measurement of En can provide useful information about E(k). Determination of E(k)has been a major issue in solid state physics [1±6]. It will be shown in the ensuing discussion thatquantum well spectroscopy is a powerful method for bulk band structure determination. Note that thequantization condition given in Eq. (1) is valid only for an abrupt in®nite barrier. For solid±solid andsolid±vacuum interfaces, the con®nement potential is generally ®nite and rounded. Thus, thequantization condition must be modi®ed by a phase shift. This phase shift information can be deducedfrom quantum well spectroscopy. It is an important quantity characterizing the boundary potential.

Quantum well lasers and quantum devices are familiar to all, and many would associate quantumwell effects with semiconductor systems. The fundamental gap in semiconductors plays a critical rolein many device applications. Electrons with energies in a gap cannot propagate, and therefore, it is quitecommon to employ the fundamental gap as a means for carrier con®nement. The situation with a metalis less obvious. There are no absolute gaps at the Fermi level in metals, and electrical conductionthrough a metal-to-metal junction is an everyday experience. There are no such things as thresholdvoltages or rectifying junctions. For these reasons, one might have the impression that con®nement in ametallic system would be impossible or rare. It is not. For epitaxial ®lms, all that is required is a`̀ relative gap'' in the substrate. Namely, if there is a gap for a particular direction (usuallyperpendicular to the ®lm), electrons propagating along that particular direction can be con®ned. If oneperforms direction-speci®c spectroscopic measurements, such as angle-resolved photoemission, such

Fig. 1. Probability densities for the n � 1, 2, and 3 quantum well states of a particle con®ned in a one-dimensional box of

size d.

T.-C. Chiang / Surface Science Reports 39 (2000) 181±235 185

con®nement effects can be easily observed. Relative gaps are actually fairly common in metals [7,8].Electrical conduction through a metal-to-metal junction is not a direction-speci®c probe of the junctionproperties. It involves a wide range in k space, and therefore the gap effect is generally not apparent.

In addition to the relative gap, con®nement can be accomplished by the so-called symmetry gap orhybridization gap in a metallic system. An example to be presented below is Ag on Fe. The d bands inFe are near the Fermi level, where the sp band exhibits an effective gap due to hybridization with the dbands. This is not a true gap, but the re¯ectivity for an sp electron incident on a Fe substrate can behigh, and spectroscopic studies can yield results very similar to the truly con®ned case. These partiallycon®ned states are often referred to as quantum well resonances.

A gap is effective for electron con®nement if the overlayer and substrate are lattice matched or have acommensurate epitaxial relationship. However, more often than not, the interface is incommensurate orhas a rather complex atomic arrangement. This can cause non-specular or umklapp re¯ections, resultingin damping of the wave function. How this affects the photoemission spectra is an interesting issue andwill be addressed.

Most of the examples below are drawn from metallic systems. In many of them, a noble metal such asAg is chosen as the overlayer material. The reason for this choice is simplicity. The band structure of anoble metal near the Fermi level is nearly free-electron-like, and there is only one band crossing theFermi level. One does not need to worry about issues such as different band masses for the heavy andlight holes as in typical semiconductor structures [9]. The spectra are thus much simpler, andinterpretation of experimental data is straightforward. Likewise, the alkali metals are excellentcandidates for exploring the basic behaviors of quantum well states.

Here, we will make some historical remarks in regard to the development of quantum wellspectroscopy for ®lm studies. In 1975, Jaklevic and Lambe reported the observation of oscillatory I±Vcurves in a transport measurement of a ®lm structure [10]. They recognized this as a manifestation ofquantum size effects, and pointed out the connection among the oscillations, ®lm thickness, and bandstructure. Later, a group led by Park observed oscillatory I±V transmission curves for a ®lm using a lowenergy electron diffraction setup [11±14]. This was attributed to quantum interference effects. Theseearly measurements clearly established the importance of quantum size effects in ®lms. At about thesame time, semiconductor devices based on quantum con®nement, such as the quantum well laser, werebeing developed [15]. An important signature was the blue shift of lasing frequencies as the activeregion became thin.

Angle-resolved photoemission is an extremely powerful technique for measuring the electronicproperties of solids [1±6]. Unlike photoluminescence, which is widely employed for studies ofsemiconductor structures and mostly yields the energy difference between the highest occupied stateand the lowest unoccupied state, photoemission yields information about the occupied states directly. Inthe early 1980s, a large number of solid ®lms were examined with this technique. The ®rst paperrecognizing the importance of quantum size effects was a theoretical paper by Loly and Pendry [16].They pointed out that photoemission from thin ®lms should reveal a set of sharp peaks, and ameasurement of the peak positions and widths should permit a precise determination of the bandstructure and photohole lifetime. However, their paper was largely ignored by the community at thattime because nobody had reported such observations despite the many photoemission experimentsperformed on ®lms. In fact, it was often argued, even demonstrated experimentally [17], thatphotoemission was such a surface sensitive probe that a ®lm as thin as a few monolayers should exhibitspectral lineshapes indistinguishable from the bulk. In retrospect, many of the failures were likely


caused by ®lm roughness. The ®lms were probably so rough that signatures of quantum size effectswere smeared out beyond recognition.

The ®rst photoemission observation of quantum size effects was reported in 1986 [18]. The evidencewas clear but the quantum well peaks were very broad, again due to ®lm roughness. Later work,however, clearly established the importance of quantum size effects in ®lms [19±25]. The argument thatphotoemission senses only the top few atomic layers and is therefore insensitive to the ®lm thickness is¯awed. This will become clear from an interferometric formulation of the problem to be presentedbelow. Other related methods for experimental observation of quantum size effects in ®lms includere¯ection high energy electron diffraction [26], inverse photoemission [27±29], low energy electronmicroscopy [30], scanning tunneling microscopy [31,32], low energy electron re¯ection [33], etc. Anyexperiment involving either an electron beam incident on a ®lm or electron emission from a ®lm islikely to reveal quantum interference effects under appropriate conditions.

This report is organized as follows. In Section 2 the results from a photoemission study of the Ag±Ausystem will be reviewed. This is a very simple model system, and provides a nice illustration of thebasic features including the thickness dependence and the effect of the con®nement gap. The analysiswill be based on a two-band model, which yields realistic wave functions that can be used for analyzingmore complex structures. The topics for discussion in Section 3 include partial con®nement, quantumwell resonances, coupled quantum wells, and superlattices. Section 4 focuses on issues related to latticemismatch. Section 5 presents results obtained from Ag on Fe(1 0 0), which can be prepared in the formof atomically uniform ®lms. An interferometric formulation will be developed and used to analyze thelineshape of the quantum well peaks, from which the quasiparticle lifetime widths can be deduced anddecomposed into various contributions in terms of the interactions of elementary excitations. Section 6contains a discussion of the spin polarization of quantum well states in systems containing magneticmaterials. The relationship between quantum well effects and magnetic behavior will be addressed.

2. Basic properties of thin ®lm quantum wells: Ag on Au(1 1 1)

The lattices of Ag and Au are almost perfectly matched. Ag grows epitaxially on Au(1 1 1) to form a(1 1 1) ®lm. The interface is abrupt for growth at room temperature or below. For such a commensurateinterface, the momentum component parallel to the surface and the interface, kk, is a conservedquantity. Most of the experimental work to be discussed below employs a normal-emission geometry.The photoemitted electron has kk � 0, and so does the initial state. The only momentum component ofinterest would be that perpendicular to the surface, k?, along the [1 1 1] direction. For simplicity, wewill use k to denote this perpendicular component, as in the discussion above for the one-dimensional box.

2.1. Con®nement by a relative gap

Before proceeding, it is useful to review the band structures of Ag and Au. Fig. 2 shows the bandstructure of Ag along major symmetry directions [7,8,34]. The complicated manifold at energies 4±8 eV below the Fermi level consists of the Ag 4d bands. Between these bands and the Fermi level, thereis only one band, the free-electron-like sp band, in each direction. It crosses the Fermi level along[1 0 0] and [1 1 0], but does not quite reach the Fermi level along the [1 1 1] direction, the directionprobed by the normal-emission geometry. Thus, the Fermi level lies in a relative gap along the [1 1 1]


direction. The dashed circle in Fig. 2 highlights the region of interest. The band structure of Au issimilar, with the d bands being closer to the Fermi level.

Fig. 3 shows a magni®ed view within the region of interest for Ag and Au. Here, we have changedthe vertical axis from `̀ energy'' as used in Fig. 2 to `̀ binding energy''. The two scales are related by asign reversal. For photoemission, it is customary to work with the binding energy scale, since the maininterest is in the occupied states. The top of the Au sp band is at 1.1 eV binding energy (marked by ahorizontal dashed line), compared to 0.3 eV for Ag. Along this particular direction, each crystal lookslike a semiconductor with a gap. This relative gap supports an occupied Shockley surface state [35±38].

Fig. 2. Band structure of Ag. The dashed circle indicates the region of interest for normal-emission from Ag(1 1 1).

Fig. 3. Band structure of Ag and Au near the Fermi level along the [1 1 1] direction. The dashed line indicates the valence

band maximum of Au, which is also the threshold of con®nement for the quantum well states in Ag.


Imagine sending an electron with energy within the gap from vacuum towards the crystal surface. Theelectron cannot propagate inside the crystal and must be re¯ected backwards. If the energy is alsobelow the vacuum level (work function), the electron cannot escape and is therefore con®ned. Thisresults in a surface quantum well, which can support a set of quantized states, or surface states. ForAg(1 1 1) and Au(1 1 1), there is one such surface state lying between the top of the sp band and theFermi level [39±41].

Fig. 4 shows normal-emission spectra taken from Au(1 1 1), Ag(1 1 1) � 20 ML Au, Ag(1 1 1), andAu(1 1 1) � 20 ML Ag [20], where ML denotes a monolayer. Here, the Au(1 1 1) substrate is really athick Au ®lm grown on a bulk crystal of Ag(1 1 1). The Ag(1 1 1) spectrum shows a peak just below theFermi level followed by a smooth background-like continuum emission. The peak represents emissionfrom the Shockley surface state just mentioned. The continuum emission is mostly derived from surfacephotoemission from the sp band, and the contribution from inelastic scattering events is relativelyminor [42±46]. The rise at high binding energy is caused by a tail from a direct-transition peak. Surfacephotoemission is nonselective in k?, and thus the entire sp band contributes, giving rise to a continuumemission. The spectrum from Au(1 1 1) is very similar. The surface state is at a larger binding energy,and the rise at the high binding energy end is due to d band emission.

The spectrum for Ag(1 1 1) � 20 ML Au looks very similar to that from bulk Au(1 1 1). The samesurface state is observed. The wave function of the surface state does not penetrate very far into the®lm, and is therefore insensitive to the ®lm±substrate interface. For this reason, the binding energyshould be essentially the same, as the experiment shows. Likewise, the surface state peak forAu(1 1 1) � 20 ML Ag looks the same as that for Ag(1 1 1). However, the part of the spectrum belowthe surface state looks different, with the ®lm sample showing three peaks. These are quantum wellpeaks corresponding to electrons in the Ag ®lm con®ned by the relative gap in Au. These peaks arelabeled n � 1, 2, and 3, where the state label n denotes the nth peak below the surface state. The n used

Fig. 4. Normal-emission spectra taken with hn � 10 eV for Au(1 1 1), Ag(1 1 1) � 20 ML Au, Ag(1 1 1), and

Au(1 1 1) � 20 ML Ag.


here is related to, but not the same as, the quantum number n used in Eqs. (1)±(3), and its physicalmeaning will be clari®ed below.

A simple test of the quantum size effect is to change the ®lm thickness. Eq. (1) shows that as dincreases, the difference between neighboring allowed values of k decreases, and so does the energydifference. In the limit of a very large d, the allowed states should form a continuum, and one shouldrecover the band structure of the bulk solid. Fig. 5 shows a set of normal-emission spectra for Ag ®lmsof various thicknesses. Clearly, the peaks become more crowded as d increases, in qualitativeagreement with the expectation. As the ®lm thickness increases, the observed quantum well peaksevolve and converge toward the top of the sp valence band. In the limit of an in®nitely thick ®lm, onecan imagine that the peaks merge to form a continuum.

The vertical dashed line in Fig. 5 indicates the threshold for con®nement. Intense quantum well peaksare observed only to the right of this line. This line, at a binding energy of 1.1 eV, corresponds to the topof the Au sp band (dashed horizontal line in Fig. 3). Above this energy, the electrons in the Ag ®lm arewithin the relative gap of Au and therefore con®ned by the Au potential, forming quantum well states.Below this, the electrons can couple to the bulk states in the Au substrate and are therefore uncon®ned.Some weak and broad features are present in the spectra, which can be attributed to quantum wellresonances due to partial re¯ection by the boundary potential. The same reasoning explains why thespectrum for Ag(1 1 1) � 20 ML Au in Fig. 4 shows no quantum well peaks. The Au sp band does notoverlap the gap in Ag. None of the electrons in the Au ®lm is con®ned, and therefore there should notbe any quantum well states. These results illustrate the concept of con®nement by a relative gap. It isimportant to note that con®nement by a relative gap is generally dependent on the probing direction.Here, if the detection direction is moved off normal, there might not be a gap for a suf®ciently largeemission angle [47]. Again, if one measures the electrical resistance across a Ag±Au junction, onewould not detect a gap, because such a measurement is not k selective.

Fig. 5. Normal-emission spectra taken with hn � 10 eV for Ag overlayers with indicated thicknesses on Au(1 1 1). The

dashed curves indicate the evolution of the n � 1 ± 3 quantum well peaks.


2.2. Model calculations

The sp band structures of Ag and Au are nearly free-electron-like, and can be well described by theusual two-band model [35,38,48] with a minimal number of parameters. Despite the simplicity, thismodel is extremely powerful for simulating experimental results, and can be easily extended tocalculations involving multilayer ®lm structures. In the two-band model, two plane wave componentsand a single pseudopotential form factor are employed. The wave vector k as a function of bindingenergy E (referred to the Fermi level) is given by

ki;f � pÿ 2mi;f

�h2

� �1=2 �h2p2

mi;fÿ V � Evbm ÿ E ÿ V2 � 4

�h2p2

2mi;fÿ V � Evbm ÿ E

� ��h2p2

2mi;f

� �1=2( )1=2

;

(4)

where p B p/t, t is the monolayer thickness and p is the wave vector at the Brillouin zone boundary (theL point for the [1 1 1] direction). The subscripts `̀ i'' and `̀ f'' refer to the lower and upper sp bands,respectively, separated by the relative gap. For photoemission from a bulk single crystal using lowphoton energies for excitations across the gap, bands i and f are just the initial and ®nal bands [42,43].The quantities mi,f are the effective masses associated with the two bands. The quantity V is the absolutevalue of the pseudopotential form factor; it equals one-half of the gap at the zone boundary. Thequantity Evbm is the binding energy of the valence band maximum (the top of the lower sp band). It ispositive for the [1 1 1] direction, and negative for the [1 0 0] direction (see Fig. 2).

For each crystallographic direction, this model contains just four adjustable parameters mi,f, V, andEvbm. The parameters V and Evbm determine the positions of the band edges, and they are the onlyparameters used in the standard nearly free-electron model. The free electron mass is replaced by theparameters mi,f here in order to set the curvatures of the lower and upper bands correctly. Theyrepresent higher order corrections from multi-band effects. Note that mi,f do not equal the inversecurvatures of the bands. There are other ways to parameterize the band structure [38], but fourparameters represent the minimum requirement. Eq. (4) can be inverted to express the binding energy E(increasing downward) in terms of k. The formula is

E � Evbm ÿ V ÿ �h2�pÿ k�22mi;f

� 4�h2p2

2mi;f

�h2�pÿ k�22mi;f

� V2

!1=2

; (5)

where the � sign and mi should be used for the lower band and the ÿ sign and mf should be used for theupper band.

The wave functions are given by

ci;f�z� / exp�ikz� ÿ ��h2p2=2mi;f� ÿ V � Evbm ÿ E ÿ ��h2k2

i;f=2mi;f�V

exp�i�k ÿ 2p�z�; (6)

where the two plane wave components are explicitly shown. This wave function can be either apropagating Bloch state or an exponentially damped (or exponentially growing) oscillatory statedepending on the energy relative to the gap. If the energy is within the gap, k is complex, giving rise toan exponential factor. Only the exponentially damped term should be included in the solution for asemi-in®nite substrate (corresponding to total re¯ection). For a ®lm, however, both the exponentially


damped and growing terms must be included, and the coef®cients are determined by the boundaryconditions.

Since the two-band model is based on the nearly free-electron approximation (or the lowest orderpseudopotential model), it provides a good representation of the wave function only in the interstitialregion. Orthogonalization to the core wave functions (including the d wave functions) is not taken intoaccount explicitly, and the real wave functions could look much more complicated near the cores. Thesize of the core region is a fraction of the inter-atomic distance. It has little effect on the matching ofwave functions across a boundary, which is in the interstitial region. Photoemission intensity isgoverned by the square of the transition matrix element, and the contribution from the core region,proportional to the core volume, is generally negligible.

Another simple model that is often used for quantum well calculations is the tight-binding model[49,50]. A set of overlap integrals is used as parameters in a model Hamiltonian in the form of a matrix.These parameters are adjusted to reproduce the bulk band structure. Diagonalizing the Hamiltonian yieldsthe wave functions and eigenvalues of the system. Additional parameters may be used for coupling termsacross an interface. This model is better suited for bands with small dispersions, such as the d states.

2.3. Wave functions and quantum numbers

The following discussion will be based on the two-band model. To construct the wave functions forquantum wells such as Ag(1 1 1) on Au(1 1 1), one begins with appropriate choices of the fourparameters each for Ag(1 1 1) and Au(1 1 1) to reproduce the bulk band structure. First, let us considerthe symmetric case where a Ag ®lm is sandwiched between two semi-in®nite Au(1 1 1) crystals. Thewave function in the Ag ®lm is a linear combination of ci(z) and ci(ÿ z). This wave function must bejoined to an exponentially damped wave function in the Au crystal to the right. The usual boundaryconditions apply; namely, the wave function and its ®rst derivative must be continuous (or thelogarithmic derivative must be continuous if the normalization of the wave function is of no concern).This ®xes the mixing ratio between ci(z) and ci(ÿ z) in Ag. The same boundary condition must alsoapply to the interface at the left. In general, this cannot be satis®ed except at certain energies. Thesespecial energies de®ne the allowed energies of quantum well states.

The probability densities for several quantum well states obtained from such a calculation for a24 ML Ag ®lm sandwiched between two semi-in®nite Au crystals are shown in the left-hand side panelof Fig. 6. Here the state label n is the same as that used in Fig. 5. In other words, the quantum number nis used to label the nth state below the valence band maximum, and is not the same as that used inEqs. (1)±(3). Going back to the particle-in-a-box model, the quantum number n is the number ofantinodes in the probability density. Counting the number of antinodes between the two vertical linesrepresenting the two interface boundaries, there are 23, 22, and 21 of them for the bottom, middle, andtop states depicted in the left-hand side panel of Fig. 6. Thus, the quantum numbers would be 23, 22,and 21, respectively, based on that scheme. If we call the quantum number used in Fig. 6, n, and thequantum number based on the number of antinodes in the probability density, n0, the relationship isn � N ÿ n0, where N � 24 is the number of monolayers in the ®lm. The quantum number n0 is large inthis case, because we are probing states near the zone boundary, where the wave vector k � p/t is large.Substituting k � p/t in Eq. (1) yields n0 � N.

The rapid oscillations of the wave functions in Fig. 6 are associated with the zone boundary wavevector. These oscillations are modulated by a slowly varying envelope function [9] due to beating


between the plane wave components. Focusing on the envelope function, one sees that the three states,from bottom to top, have 1±3 antinodes, respectively. Thus, the quantum number n shown in Fig. 6 canbe identi®ed as the number of antinodes in the envelope function. Clearly, as the quantum number nincreases, the envelope function will vary more rapidly, and eventually will lose its meaning. For thisreason, envelope functions are useful only for quantum well states near the zone boundary.

Referring back to Fig. 3, the band structure near the valence band maximum is roughly an invertedparabola (hole-like). One can imagine reformulating the problem by making a local expansion aboutthe valence band maximum relative to k measured from the zone boundary. This is basically a Wanniertransform to remove the rapid oscillatory part of the wave function, and the resulting wave function willresemble the envelope function. The quantum number n is then again compatible with the particle-in-a-box model provided that the energy scale is inverted with the zero at the valence band maximum. Thisapproach is often adopted in semiconductor physics.

There is a good reason that we will want to use the quantum number n as shown in Figs. 5 and 6rather than the other. In Fig. 5, the state n � 1 is always the one closest to the valence band maximum.As the ®lm thickness increases, the ®rst peak evolves smoothly, and always has the same quantumnumber. If we use the other scheme, the quantum number would be changing from 9, to 11, to 14, etc.Changing the state label for a seemingly continuously evolving peak is a little awkward. It is also morenatural to start counting the quantum numbers from unity. Of course, this is largely a matter of personaltaste.

The calculation for a Ag ®lm on a Au substrate is similar. The boundary condition at the Ag±vacuuminterface can be modeled in many ways. The simplest approximation is an abrupt interface [38]. Thewave function in vacuum is an exponential function damped towards the vacuum, and the damping isdetermined by the difference between the energy of the electron and the vacuum level. There is,however, a question about the location of the boundary. Physically, the electrons in the Ag spill outa little into vacuum, and the effective boundary should be slightly outside the boundary de®ned by theso-called positive background. The positive background refers to the region in space covered bymodeling each atomic layer as a uniform slab of thickness t centered about the atomic plane. Thus, the

Fig. 6. Left-hand side panel: theoretical probability densities for the n � 1 ± 3 quantum well states in a 24 ML Ag(1 1 1) ®lm

sandwiched in-between two semi-in®nite Au(1 1 1) crystals. Right-hand side panel: the same except that the Au(1 1 1) crystal

on the right is replaced by vacuum. The vertical lines indicate the Ag±Au and Ag±vacuum boundaries.


edge of the positive background is just 12

t beyond the last atomic plane. For the two-band calculation,this vacuum boundary position is adjusted to reproduce the measured binding energy of the Shockleysurface state.

The calculation for the Ag surface state is straightforward. An exponentially damped gap state in Agis matched to an exponentially damped plane wave state in vacuum. Solving the resulting eigenvalueequation gives the surface state energy. Matching the surface state energy to the experimental valueguarantees that the phase shift is exactly correct at the energy of the surface state. Since we areconcerned mostly with energies near the Fermi level (also near the Shockley surface state), the errorintroduced by using this model should be minimal. Other boundary potentials such as the imagepotential, a linear potential, etc., can be employed, but the differences are minor and do not warrant theextra effort at this level of approximation [51±53].

The wave functions for the n � 1 ± 3 quantum well states for a 24 ML Ag ®lm on a Au substrate areshown in the right-hand side panel of Fig. 6. The wave functions look similar to the ones in the left-hand side panel, but are no longer symmetric. They are damped very rapidly on the vacuum side,because the energy involved is far below the vacuum level. It is interesting to note that the n � 3 statehas a long tail going into the Au substrate. Referring to Fig. 5, this state at N � 24 is just slightly abovethe threshold of con®nement. In other words, this state is just barely con®ned, and there should be along exponential tail going into the substrate. At a slightly smaller ®lm thickness, this state becomes

Fig. 7. (a) Left-hand side panel: sample con®guration involving a Ni monolayer sandwiched in-between two Cu wedges.

Right-hand side panel: the Ni layer position within the Cu quantum well across the diagonal BD. (b) Photoemission intensity

map at the Fermi level across the sample surface (®gure taken from [57]).


uncon®ned, and the wave function extends to z � ÿ1. For the same reason, the wave function forn � 2 penetrates a little deeper into the Au substrate than n � 1. Similar calculations for this system canbe found in [54,55]. The evolution of the quantum well states from a ®lm exposed to vacuum to a ®lmcapped by the substrate material has been examined experimentally in a related system [56].

The surface state in Fig. 5 is labeled as n � 0. This state can be regarded as one of the quantum wellstates. The main difference from the other states is that its binding energy and wave function becomeindependent of the ®lm thickness for ®lms thicker than just a few monolayers.

An interesting way to probe the envelope function experimentally is illustrated in Fig. 7(a) [57]. Thesample con®guration consists of two mutually orthogonal Cu(1 0 0) wedges separated by a 1 ML Nilayer grown on a Co(1 0 0) substrate. The Ni layer here is employed as a perturbing layer in a Cuquantum well made of the two wedges. Along the diagonal AC, the Ni layer is always at the center ofthe Cu quantum well as depicted, and the well itself has a continuously varying thickness. Along theother diagonal BD, the total Cu thickness is constant, while the Ni layer is swept continuously from oneside of the Cu quantum well to the other. The grey scale image in Fig. 7(b) is a measure of the normalphotoemission intensity at the Fermi level across the sample surface. Two types of oscillations aredetected. The oscillation along AC is due to different quantum well peaks moving through the Fermilevel for increasing quantum well width (thin at A, thicker at C). The oscillation along BD can berelated to the envelope function of the quantum well state. The interpretation is that, as the Ni layerposition in the quantum well varies continuously, it perturbs the wave function in different ways. Whenits position coincides with a node in the envelope function, the perturbation is a minimum, and thephotoemission intensity remains a maximum at the Fermi level crossing. When it coincides with anantinode, the perturbation is a maximum, causing the emission intensity to decrease signi®cantly. Thisexample demonstrates that wedge samples can greatly facilitate experiments designed to probe thequantum well behavior as a function of layer con®guration.

3. Resonances, coupled quantum wells, and superlattices

The above example illustrates the basic physics of thin ®lm quantum wells. One can go beyond andexplore the physics of multilayers and superlattices. Layering of different materials offers opportunitiesfor tailoring properties and for exploring interesting physical phenomena. A few examples based on theAg±Au system will be given to cover some major topics of interest, including leaky quantum wells,coupling between quantum wells, and superlattice effects [58±64]. Again, some of the advantages ofAg±Au are a close lattice match, a simple nearly free-electron-like band structure, and the availabilityof high quality ®lms. For Ag±Au multilayer systems, the two-band model described above can beextended to simulate the results and to provide a framework for making predictions.

3.1. Incomplete con®nement and resonance states: Ag(1 1 1) � Au � Ag

The system under consideration is Ag(1 1 1) � x ML Au � y ML Ag [58]. When x is large, thesituation reduces to the previous case of a simple quantum well as far as the electrons in the Agoverlayer are concerned. What happens when x is small? The quantum well states in the Ag overlayerdo penetrate somewhat into the Au. If the Au layer thickness becomes comparable to the penetrationdepth, the wave function in the Ag overlayer can couple to the continuum states in the Ag substrate.


Effectively, one will have a leaky quantum well, and the quantum well states will become resonancestates as the con®nement becomes incomplete. The Au layer is employed here as a barrier, and varyingits thickness allows a continuous change in the degree of con®nement. The decay of the Ag quantumwell states is exponential in the Au, and therefore even a very thin Au layer can be an effective barrier.The degree of con®nement also is a function of binding energy. A state near the Au band edge (1.1 eV)would be much less con®ned than a state near the Fermi level (see Fig. 6).

Fig. 8 shows representative spectra. The bottom spectrum was taken with high resolution from a bulksingle crystal Ag(1 1 1). In addition to the surface state, the spectrum shows a direct-transition peak,which corresponds to a vertical optical transition from band i to f as discussed above in connection withthe two-band model. The continuum part of the spectrum, between the surface state peak and the direct-transition peak, is dominated by surface photoemission as discussed above [42±46]. A threshold for thisemission is seen just below the surface state peak, and corresponds to the valence band maximum. Thiswas not seen in Fig. 4, because those spectra were taken with a lower resolution. The other two spectrain Fig. 8 are for Ag(1 1 1) � x ML Ag � 15 ML Ag, with x � 2 and 3, respectively. Other than thequantum well peaks labeled n � 1 ± 3, these spectra resemble the single crystal spectrum. The surfacestate is minimally affected by the Au barrier because the overlayer thickness of 15 ML is much largerthan the penetration depth of the surface state. The direct-transition peak involves the sp wave functionover many atomic planes (limited by the ®nal state mean free path), which should be fairly insensitiveto the 2 or 3 ML of Au inserted into the lattice.

Fig. 8. Normal-emission spectra taken with hn � 10 eV for the leaky quantum well system Ag(1 1 1) � x ML Au � 15 ML

Ag. The three spectra are for x � 0, 3, and 2 as indicated. Also shown are theoretical spectra for x � 3 and 2.


Note that the Au band edge is at 1.1 eV (Fig. 3). Only the n � 1 peak has an energy within the Auband gap. Nevertheless, the Au layer is so thin that none of the three peaks n � 1 ± 3 are truly con®ned.These are all resonance states. Measurements carried out at other photon energies indicate that theseresonance peaks have ®xed binding energies. Also shown in the ®gure are calculated spectra based onthe two-band model. The ®nal state for the photoemission process in the calculation is taken to be thetime-reversed low-energy-electron-diffraction state in accordance with the one-step model [3,6,65]. Thedipole matrix element for optical transition is evaluated using the two-band wave functions, and thespectral lineshape is calculated using Fermi's golden rule. The quantum well peak positions are wellreproduced by the calculation. However, the observed quantum well peak widths are much broader thanpredicted. This broadening is most severe for larger quantum numbers at higher binding energies. Then � 3 peak is broadened almost beyond recognition, an effect that can be attributed to sampleimperfection. Despite this extrinsic broadening, it is clear that the peaks for x � 3 ML are sharper andmore intense than x � 2 ML, both experimentally and theoretically. This is because a larger x providesa higher degree of con®nement; a less con®ned state should have a larger width.

The n � 1 and 2 peaks are at binding energies of 0.60 and 1.23 eV, respectively. Fig. 9 shows somecalculated initial state wave functions at these and other energies for x � 2 ML. None of the states withenergies below the Ag valence band maximum are truly con®ned, and the allowed energies form acontinuous band. In this ®gure, the vertical dashed lines indicate the boundaries between Ag, Au, andvacuum. The wave functions again appear as short-period oscillations superimposed on an envelopefunction because the wave vector is near the zone boundary. Focusing on the envelope function, it isclear that the n � 1 resonance state is characterized by a good ®t of the ®rst antinode of the envelopefunction into the Ag slab. When this happens, the electron becomes partially trapped in the slab, as

Fig. 9. Probability densities at binding energies 0.40, 0.60, 0.90, and 1.23 eV for the leaky quantum well system

Ag(1 1 1) � 2 ML Au � 15 ML Ag. The n � 1 and 2 resonances are indicated.


indicated by a higher probability density. Likewise, the n � 2 resonance state is characterized by a good®t of the ®rst two antinodes into the Ag slab. Between the two resonances, the wave function has areduced probability density within the Ag slab, as evidenced by the curve for 0.9 eV binding energy.This example illustrates that quantum well resonances are associated with enhanced probabilitydensities within the Ag slab when the wave functions can ®t into the slab geometry forming a patternresembling a standing wave. Because of the ®nite probing depth governed by the mean free path in the®nal state, an enhanced probability density within the ®lm should lead to a higher photoemissionintensity and thus a peak in the spectrum. In a sense, quantum well resonances are just broadenedquantum well states superimposed on a background of continuum.

When n becomes large, more antinodes in the envelope function are cramped into the Ag slab. TheAg slab is likely to have imperfections such as atomic steps or layer thickness ¯uctuations. Thescattering probability caused by the imperfections is proportional to the thickness scale of theimperfections divided by the wave length of the envelope function. Thus, imperfections in the Ag slabstructure at either the surface or the interface will cause a stronger damping for quantum wellresonances with larger n's. In the limiting case where the wavelength of the envelope function becomesthe same as the layer thickness ¯uctuation, the n and (n � 1) resonances will become indistinguishable,and the peaks will simply merge. Although a quantitative measure of the ®lm roughness is not availablein this case, it is conceivable that atomic steps and incomplete layering during ®lm growth can easilylead to an effective layer thickness ¯uctuation of � 1 ML or more. For the n � 3 state, this would giverise to a perturbation on the order of 2

15n � 40%, and the binding energy of the resonance peak can

become broadened by the same order, or � 1 eV. A 1 eV broadening would make the n � 3 peak nearlyunrecognizable, as seen in our data.

3.2. Coupling between quantum wells: Au(1 1 1) � Ag � Au � Ag

Here, we will consider the coupling between two Ag quantum wells [59±61]. The sample is made bystarting with a Au(1 1 1) substrate and sequentially depositing 8 ML of Ag, 3 ML of Au, and x ML ofAg. The two Ag slabs form quantum wells, and the 3 ML Au is a barrier layer. The Au substrate and thevacuum provide con®nement for all occupied states up to the Au band edged at 1.1 eV binding energy.The quantum well states in the two Ag wells are decoupled in the limit of an in®nitely thick Au barrier.As the thickness of the outer well is varied, the binding energies of the outer well states evolve andsweep across the energies of the inner well states. Such level crossings must be avoided if there is anycoupling between the two wells. By using a thin Au barrier to facilitate the coupling, the anticrossingbehavior can be probed directly with photoemission, and the dispersion is a direct measure of thecoupling strength. This coupling strength is an essential parameter for modeling more complex layerstructures.

The same two-band model can be employed to make predictions. The left-hand side panel of Fig. 10summarizes the behavior of single quantum wells. The dashed curves labeled On, for n � 1 ± 4, indicatethe binding energies of Ag single quantum well states for various thicknesses x. They represent theouter well states of a double quantum well in the limit of an in®nitely thick Au barrier. They allconverge to the Ag valence band maximum as x becomes large. For an 8 ML Ag quantum well boundedon both sides by semi-in®nite Au, only one inner well state is allowed (labeled I). In this limit, it isdecoupled from the outer well, and its binding energy is independent of x, as indicated by the horizontaldashed line at E � 0.67 eV in the ®gure. Note that state I intersects O1 and O2 at x � 14 and 28 ML,


respectively, so the states involved are degenerated for decoupled quantum wells. In these places weexpect to observe avoided crossings if the Au barrier thickness is ®nite (3 ML in the experiment).

In the two-band calculation, the coupling is mediated by the wave functions in the Au barrier layer,which are oscillatory wave functions modulated by exponential functions. Solving the boundaryconditions at all four boundaries in the system yields an eigenvalue equation. The solutions are plottedin the left-hand side panel of Fig. 10 using circles at integer values of x. These represent the energies ofthe coupled quantum well states, and are labeled A±E. The avoided crossings are apparent. Forexample, state A starts out as state I for small x, becomes a mixture of I and O1 at x � 14, and ®nallybecomes O1 for large x. Likewise, state B evolves from O1, to a mixture of I and O1 at x � 14, to mostlyI at x � 20, to a mixture of I and O2 at x � 28, and ®nally to O2. The two gaps of the avoided crossingsat x � 14 and 28 ML are 0.17 and 0.12 eV, respectively.

The measured dispersions are shown as circles in the right-hand side panel of Fig. 10. The solidcurves connect the data points, and there is a good overall agreement with the two-band predictionas indicated by the dashed curves. While the experimental curves are about 40 meV higher in energythan the predictions, the sizes of the gap and the rates of dispersion are all well reproduced. The reasonfor the slight discrepancy is likely that the modeling of the 3 ML Au barrier layer is not entirelyaccurate. In particular, the Ag±Au boundary potential must be somewhat rounded due to a ®nitemetallic screening length, but in the model, this is assumed to be abrupt. The sizes of the circles in the®gure are a rough indication of the predicted (left-hand side panel) and measured (right-hand side

Fig. 10. Left-hand side panel: theoretical binding energies (circles) for the Au(1 1 1) � 8 ML Ag � 3 ML Au � x ML Ag

double quantum well system as a function of the outer well thickness x. The area of each circle is proportional to the

photoemission intensity. The dashed curves show binding energies for isolated inner (I) and outer (On, n � 1 ± 4) well states.

The energy positions of the Au and Ag valence band maxima (VBM) and the Shockley surface state (S) are also indicated.

Right-hand side panel: the experimental results compared to the theoretical dispersions (dashed curves).


panel) photoemission intensities. Again, the trends of the intensity variation are well reproduced by themodel.

Additional insight can be gained from an examination of the wave functions. The calculatedprobability densities for a few representative cases are shown in Fig. 11. For x � 10 ML, state A

roughly corresponds to state I, and state B to state O1 (see Fig. 10). Thus, the wave functions are mostlyconcentrated in the inner and outer wells, respectively. At x � 14 ML, states I and O1 are fully mixed.The higher energy state A has an antinode in its envelope function at the barrier, while the lower energystate B has a node. They resemble the n � 1 and 2 states in the combined well with the Au replaced byAg. At x � 20 ML and beyond, state A becomes essentially the n � 1 state in the outer well, O1. StateB, at x � 20 ML, becomes essentially the inner well state I, and upon further increasing of x, evolvesinto O2.

The above illustrates how coupling between quantum wells can lead to signi®cant modi®cations ofthe bound state energies. In the two-band model, the coupling is mediated by the wave functions in thebarrier layer, and the coupling strength is determined by wave function matching at the boundaries.There are no additional parameters speci®c to the interface. It is remarkable that the anticrossing gapsare well reproduced by this model where the only inputs are the bulk band structures of the constituentmaterials. This is unlike the tight-binding method where the Ag±Au overlap integrals must beintroduced as additional parameters for the interface.

Fig. 11. Probability densities of states A and B (see Fig. 10) for a few representative outer well thicknesses x in the double

quantum well system Au(1 1 1) � 8 ML Ag � 3 ML Au � x ML Ag. All states are normalized. The vertical dashed lines

indicate the various Ag±Au and Ag±vacuum boundaries in the system.


The coupling between the two quantum wells decays rather rapidly as the Au barrier thickness isincreased, because the interaction is via evanescent (exponentially decaying) waves within the Aubarrier. Longer-range interaction with oscillatory structures can be expected if the coupling is viapropagating waves, i.e., quantum well states or resonances. Later in Section 6, this point will beaddressed again in connection with the magnetic behavior of multilayer systems.

3.3. Superlattices

Superlattices are often employed in device structures. The main effects of superlattice modulationinclude band folding and mini-gap formation. There are two relevant parameters for a binarysuperlattice, namely, the thicknesses of the two repeating constituent layers. Varying these parametersallows detailed tuning of the band structure, providing opportunities for property enhancement andavenues for device engineering. Here, we consider a superlattice made of repeating layers of Ag and Au[62±64]. It is expected that the sp band dispersion should lie somewhere between those of Ag and Au,with superlattice gaps at the mini-zone boundaries.

The same two-band model can be easily applied to the superlattice. Assume that each superlatticeperiod is made of N1 monolayers of Ag and N2 monolayers of Au. The wave functions in the Ag and Auare linear combinations of ci(z) and ci(ÿz), and there are four coef®cients in each period. Theboundary condition at the Ag±Au interface requires that the wave function and its derivative becontinuous. Further, the Bloch theorem requires that the wave function and its derivative advance by aphase factor of exp[ik(N1 � N2)t] for a displacement by a period. Thus, the four coef®cients mustsatisfy a set of four homogeneous algebraic equations. For a solution, the determinant must vanish, andthe resulting eigenvalue equation generally has only one solution at each k in the extended zone. Thiseigenvalue equation can be cast into a simple transcendental form, and the reader is referred to theoriginal publication for details [64].

Fig. 12 shows a set of normal-emission spectra for a superlattice with N1 � 8 ML (Ag) andN2 � 4 ML (Au). These spectra were taken with various photon energies in the usual range for `̀ bandmapping''. This particular superlattice is terminated by a Ag slab, and the surface state looks like thatof pure Ag(1 1 1). The dispersive peak labeled N (for normal) is very similar to the direct-transitionpeak seen in pure Ag (see the bottom spectrum in Fig. 8). This superlattice is made of more Ag thanAu, and therefore it is not surprising that the peak position is closer to that of Ag than Au. As thephoton energy is varied, the direct transition moves to different positions in k (see Fig. 2), and the peakposition moves in accordance with the dispersions of both the initial and ®nal bands. The weakerdispersive peaks labeled U (for umklapp) are derived from the folded bands of the superlattice, andhave no counterparts in pure Ag or Au. The two U branches are separated by a mini-gap, in which thereis a non-dispersive peak (S). This peak, observed over a wide photon energy range, is a surface statepeak speci®c to the superlattice geometry.

Fig. 13 displays the measured positions of the N and U peaks as a function of k in the extended zone,assuming that the ®nal band is the same as that for pure Ag. The N and U peak positions agree wherethey overlap. The vertical dashed lines indicate the expected superlattice zone boundaries, where mini-gaps should form. The data indeed show gaps at these positions. The Ag and Au band dispersions areincluded for comparison. The superlattice band dispersion falls in-between, and is closer to the Agdispersion as expected. Also shown in Fig. 13 are solid curves that represent the calculated superlatticeband structure based on the two-band model as discussed above. The overall behavior of the data is well


reproduced by the calculation. Most encouraging is that the measured gaps are in excellent agreementwith the calculation, which predicts that the upper gap is 0.30 eV and the lower gap is 0.12 eV. Thelower gap can be detected if the data points are viewed at a glancing angle relative to the page; it can bequanti®ed by curve ®tting of the two adjacent branches. The measured dispersion curves are slightlyhigher than predicted. Discrepancy of this magnitude should not be surprising, considering the extremesimplicity of the model (abrupt, ideal interfaces).

Extensive data show that the energy position of peak S (see Fig. 12) is independent of photon energyand falls within the superlattice gap. This establishes that peak S is derived from a surface state ratherthan a bulk state. Fig. 14 focuses on this state and shows a set of spectra taken at hn � 11.6 eV so thatthe N and U peaks are well separated from peak S. The bottom spectrum is taken from the Ag-terminated sample, as before. The other spectra, from bottom to top, are taken from the same sampleafter adding one atomic layer at a time in going through one superlattice period. While the position ofpeak S depends on the surface termination, it always remains within the superlattice gap, as indicatedby the vertical dashed lines. As the surface termination is changed, peak S moves across the superlatticeband gap over one half of the superlattice period, disappears for the other half of the period, andreappears as the surface termination is restored after one complete period (the top spectrum in the®gure). The triangles indicated the predicted positions of the surface state based on a two-band

Fig. 12. Normal-emission spectra for an 8 ML Ag � 4 ML Au superlattice taken with various photon energies as indicated.

The dashed curves are guides to the eye for the various peaks in the spectra.


calculation and a phase shift analysis. Again, there are no adjustable parameters beyond the bulk bandstructures of Ag and Au, and yet the agreement is excellent. The success of such a simple model shouldnot be surprising, because the behavior follows some very general principles and should be independentof the numerical details [63].

4. Mismatched interfaces

The overwhelming majority of elemental pairs in the periodic table are lattice mismatched, and thismismatch affects overlayer growth. Some exhibit strained growth to force a match. Defects form inothers to accommodate the mismatch. Interfacial reactions, intermixing, and segregation representcomplicating factors. For metal epitaxial systems, it is common to have unstrained overlayer growth.For example, Ag grows unstrained along [1 1 1], the low energy direction, on Cu(1 1 1), Ni(1 1 1), andCo(1 1 1). The mismatch is too large for a strained growth. Unlike semiconductors where the bonding iscovalent and highly directional, metals are held together by metallic bonding involving a sea ofelectrons. These electrons are fairly diffuse and can easily accommodate the bonding requirements atan interface, even for the highly directional dangling bonds on a semiconductor substrate. In all of thecases mentioned above, there is an orientational relationship between the overlayer and the substrate.The major crystallographic axes are parallel, and in some cases, there is signi®cant twin formationsuggesting that second nearest neighbor interaction is a weak effect.

What are the rami®cations of a mismatched interface for quantum well states? The mismatch maygive rise to an incommensurate interface or a reconstructed interface involving a large unit cell. The

Fig. 13. Experimental band dispersion (diamonds) derived from the normal and umklapp peaks in Fig. 12. The dashed curves

are the bands for pure Ag and Au. The solid curves are derived from a two-band calculation.


interface potential is characterized by a linear combination of the two sets of interface reciprocal latticevectors from the two materials, and can cause an electron incident normally on the interface to scatteroff to many different directions. A quantum well state is essentially a standing wave involving multiplere¯ections of the electron between the two ®lm boundaries. Interface scattering to other directionscauses the standing wave to decay over time, and the result is similar to partial con®nement. Anotherissue of importance is the probability for electron transmission through an interface. In the Ag±Au casediscussed above, the gap in the substrate allows no transmission. For a lattice mismatched case,however, the matching of the wave functions at the interface becomes more complicated. The kk � 0wave functions on the two sides of the interface are no longer phase matched at every lattice site. A fullcalculation would consider coupling of wave functions involving many different kk. An enhancedre¯ection probability can result if the wave functions do not match well. Specular re¯ectivity at theinterface is unity reduced by the transmissivity and the non-specular re¯ectivity. If this reduction is nottoo large, the system should support well-de®ned quantum well resonances.

Fig. 14. Normal-emission spectra for various surface terminations of the Ag±Au superlattice taken with a photon energy of

11.6 eV. Starting from the bottom spectrum, which is for the Ag-terminated con®guration, each successive spectrum

corresponds to the addition of one atomic layer in going through one superlattice period. The top and bottom spectra

correspond to the same sample con®guration. The superlattice gap is indicated by two vertical dashed lines. The triangles are

the energy positions of the surface state from a calculation.


4.1. Ag on Cu(1 1 1)

Ag on Cu(1 1 1) represents a case that supports well-de®ned quantum well resonances [24]. Fig. 15shows a set of normal-emission spectra for Ag ®lms of various thicknesses on Cu(1 1 1) taken with aphoton energy of 10 eV. The main features include a Shockley surface state S and quantum well peaksthat evolve in an expected manner. Note that the valence band maximum of Cu(1 1 1) is at about0.85 eV, above which there is a relative gap. Quantum well peaks below this can still be seen, althoughbroader. Data taken at different photon energies indicate that the peak positions are essentiallyindependent of the photon energy. This is consistent with earlier ®ndings for the leaky quantum wellsystem of Ag(1 1 1) � x ML Au � 15 ML Ag. In both cases, the resonance peak positions are governedby the standing wave patterns in the initial state, and are independent of the photon energy.

Fig. 16 shows the dependence of these quantum well peaks on the polar emission angle. As theanalyzer is moved off normal to either side, the peaks move toward the Fermi surface following aroughly parabolic behavior, and so is the surface state S. One by one, these peaks cross the Fermi leveland become invisible. The measured peak dispersions can be used to deduce the Ag sp band dispersionsalong the direction of kk. This is suf®cient to provide a complete speci®cation for the surface statebecause it has no k? dependence. Likewise, the quantum well states are completely speci®ed by n andkk. In other words, the quantum number k? for a bulk single crystal is replaced by n in the case of a®lm.

It is clear from Fig. 16 that the quantum well peaks become weaker as the polar emission anglebecomes larger. This can be explained as follows. Each quantum well peak corresponds to a standingwave pattern involving multiple re¯ections of the electron between the two ®lm boundaries. With anoff-normal geometry, each round trip of the electron within the ®lm is associated with a sideways

Fig. 15. Normal-emission spectra taken with hn � 10 eV for various Ag coverages on Cu(1 1 1).


displacement. This displacement accumulates until it becomes larger than the lateral coherence lengthof the wave function. At that point, multiple beam interference terminates. The effective number ofbeams contributing to the interference determines the intensity and sharpness of the quantum well peak.For a large polar angle, this number is small, and therefore the peak becomes weaker and broader. Theargument here is similar to that for a leaky quantum well.

4.2. Ratcheting quantum well peaks Ð Ag on Ni(1 1 1)

One would expect Ag on Ni(1 1 1) to be very similar to Ag on Cu(1 1 1) because both systems have alarge lattice mismatch. A detailed examination of the spectra shows interesting differences [66]. Theleft-hand side panel of Fig. 17 shows photoemission spectra from a 14 ML Ag ®lm on Ni(1 1 1) takenwith photon energies ranging from 5.5 to 13.75 eV in 0.25 eV steps. In each spectrum, the sharp peakjust below the Fermi level is emission from the Ag(1 1 1) surface state. In addition, four quantum wellpeaks, n � 1 ± 4, are observed. There is also an intense, broad feature that moves toward higher bindingenergies as the photon energy is increased. This is the counterpart of the sp direct-transition peak seenin Ag(1 1 1) (see Fig. 12 for a similar peak for the Ag±Au superlattice discussed above). At a highphoton energy, where the direct-transition feature is away from the region of interest, one can clearlysee that the quantum well peaks evolve toward the Ag valence band maximum for increasing ®lmthicknesses, as in Ag on Cu(1 1 1).

One important difference for Ag on Ni is the cyclic or `̀ ratcheting'' behavior of the quantum wellpeaks as a function of photon energy. This is indicated by the dashed curves in Fig. 17. As the photon

Fig. 16. Photoemission spectra for 30 ML of Ag on Cu(1 1 1) taken with hn � 11 eV. The polar emission angle y and the

direction of azimuth are indicated. The dashed curve is a guide to the eye, and shows the approximately parabolic dispersion of

a quantum well peak.


energy is varied, each peak emerges from one end of the range of movement, moves to the other end,pops back, and repeats its motion. The range of peak movement is a fraction of an electron volt for eachpeak, and is different for different peaks. Its intensity diminishes near the two ends of the movement,and reaches a maximum at the midpoint. Such a ratcheting behavior is not evident in the case of Ag onCu(1 1 1).

The right-hand side panel of Fig. 17 shows results from a model calculation based on themethodology discussed above. The initial state is again a linear combination of ci(z) and ci(ÿz), withthe coef®cients determined by the vacuum boundary condition and by appropriate normalization. The®nal state is the time-reversed low-energy-electron-diffraction state made of a two-band Bloch state inthe crystal and plane wave states in vacuum normalized to a unity outgoing wave. The Ag±Ni interface,being incommensurate, is assumed to be a perfectly diffuse boundary, so that an electron incident on theboundary is completely lost by incoherent scattering. Thus, the matrix element integral for opticaltransition between the initial and ®nal states is truncated at the Ag±Ni interface. This is a physicallyreasonable model and conveniently bypasses the problem of setting up the wave functions in the Nisubstrate. The calculated matrix element, together with the density of states, yields the photoemissionspectra shown at the right of Fig. 17. The results reproduce all of the essential features of the data,including the ratcheting behavior of the peak positions, the cyclic evolution of the intensities, and themovement of the direct-transition-like peak. Even the amplitudes of the ratcheting for different peaksand the relative widths of the various peaks are fairly well reproduced. The good overall agreement isgratifying, considering that the only parameters entering the calculation are the bulk band structure of

Fig. 17. Left-hand side panel: normal-emission spectra for 14 ML Ag on Ni(1 1 1). Scans are shown for each 14

eV photon

energy from 5.5 to 13.75 eV. The dashed lines are guides to the eye showing the peak motions. Right-hand side panel:

calculated spectra presented in the same fashion as the left-hand side panel. The surface state is not included in the calculation.


Ag and the work function. This provides strong support for the assumed diffuse scattering boundarycondition.

The difference between Ag on Ni and Ag on Cu is interesting. Ag on Cu behaves like a leakyquantum well, where the quantum well peaks are characterized by standing wave patterns that arebroadened by coupling effects. The Ag on Ni case does not support such standing wave patterns.Instead, the quantum-well-like features are a consequence of the truncation of the matrix element forthe optical transition, as would be the case for a negligible specular re¯ectivity at the Ag±Ni interface.This truncation creates an interference effect, and yields the unusual ratcheting peak positions andintensities. While both the Ag±Cu and Ag±Ni interfaces are mismatched, the results suggest that thespecular re¯ectivity for Ag±Cu is signi®cantly higher than that for Ag±Ni. This difference may berelated to the presence of a relative gap in Cu, but not in Ni. The gap in Cu suppresses the interfacialtransmissivity, and thus the specular re¯ectivity is higher.

5. Atomically uniform ®lms as quantum wells and electron interferometers

The quantum well peaks seen so far are broad, with widths that are generally much larger thanexpected based on the photohole lifetime. This broadening can be attributed to ®lm roughness. Thesame broadening explains why the quantum well peaks appear to evolve continuously as a function ofcoverage. The ®lm thickness should be discrete, and the quantum well peak positions should be discreteas well. Variations in the ®lm thickness on a real structure smear out such discreteness, and thepositions of the broad quantum well peaks simply re¯ect the average ®lm thickness. Until recently, ithas been generally thought that roughness on the atomic scale would be inevitable over a macroscopiclateral distance. If this roughness can be eliminated somehow, highly accurate determination of theelectronic properties will become feasible. These include the band structure, photohole (orquasiparticle) lifetime, interfacial re¯ectivity, and phase shift. These properties are of basic importanceto solid state physics and interface science. This section will focus on the making of atomically uniform®lms and using these ®lms for precision spectroscopic studies [67±72]. An analysis in terms of electroninterferometry will be presented, which provides a useful framework for understanding the quantumwell peak positions and lineshapes.

5.1. Discrete layer thicknesses

Countless investigations by scanning tunneling microscopy have shown that surfaces contain stepsand defects. Film growth involves a stochastic deposition process and the resulting ®lms are alsoaffected by steps, defects, and impurities. Thermodynamic ¯uctuations and growth kinetics tend tocreate roughness, and the roughness tends to increase as a function of thickness, often according tocertain scaling laws [73±75]. A ®lm with a nominal thickness of N atomic layers typically consists ofmultiple thicknesses including N, N � 1, N � 2, . . . over different domains. If the domain sizes arelarge, photoemission spectra should exhibit a linear combination of quantum well peaks derived fromthese different ®lm thicknesses (as well as effects unique to the steps).

An example is shown in Fig. 18 [76]. The spectrum in the lower panel is taken from a Ag ®lm of8 ML nominal thickness grown on a cleaved graphite substrate. The peak just below the Fermi level isthe Shockley surface state. The broad hump centered about 1.3 eV is the usual n � 1 quantum well peak.


The little bumps superimposed on the broad peak correspond to individual quantum well peaks derivedfrom N � 6; 7; . . . ; 10. The lower part of the panel shows the Ag band structure and the predicted peakpositions for different layer thicknesses. These individual peaks are barely resolved in this case. As the®lm gets thicker, the peak spacing becomes smaller, and eventually the individual peaks can no longerbe resolved. This is illustrated by the results shown in the upper panel of the ®gure for a ®lm of 28 MLnominal thickness. A survey of the photoemission literature shows that such atomic layer resolvedquantum well peaks are the exception rather than the norm. It is likely that most ®lms have such a highstep and/or defect density that random lateral con®nement and scattering cause signi®cant smearing ofthe quantum well peaks, rendering peaks from different thicknesses unresolvable.

Another example of such atomic layer resolved quantum well peaks is shown in Fig. 19. The spectraare for a Ag(1 0 0) ®lm of 12 ML nominal thickness deposited on a Fe(1 0 0) whisker substrate at room

Fig. 18. Measured and calculated photoemission spectra for Ag on graphite. The quantum number n and the layer thickness

N are indicated. In the lower panel the dispersion is shown to illustrate that the different sub-peaks are derived from different

discrete layer thicknesses. In the upper panel, the ®lm is too thick for the sub-peaks to be resolved (®gure taken from [76]).


temperature. Within the binding energy ranging 0±2 eV, there should be just two quantum well peaksfor a thickness of 12 ML, and yet many more peaks are observed due to the presence of severalthicknesses around 12 ML. Note that in this case, the ®lm orientation is (1 0 0). Referring back to theband structure of Ag shown in Fig. 2, the Ag sp band crosses the Fermi level along the [1 0 0] direction.The gap is above the Fermi level, and there is no surface state peak in the spectrum as in the case of a(1 1 1) ®lm.

The reason for using a Fe whisker for Ag growth is that such whisker crystals represent perhaps thebest possible substrates available. Each whisker contains just one bulk defect by nature. Theobservation of atomic layer resolved quantum well peaks suggests that the domain sizes in the Ag ®lmare unusually large, and the system looks promising. The growth of Ag on Fe(1 0 0) has beeninvestigated extensively. The interest stems from a good lattice match (within 0.8%) and a nearly idealepitaxial relationship. There is no intermixing or reaction between Ag and Fe even at elevatedtemperatures.

5.2. Preparing atomically uniform ®lms

The technique of molecular beam epitaxy has been widely employed for growth of semiconductordevice con®gurations. The basic strategy is to grow the ®lm at an elevated temperature to ensure

Fig. 19. Normal-emission spectra taken from 12 ML Ag deposited on Fe(1 0 0) at room temperature using various photon

energies as indicated. The vertical dashed lines indicate the positions of quantum well peaks. Several thicknesses are present in

this ®lm.


adequate adatom mobility, thus promoting layer-by-layer growth and minimizing the chance of defectformation caused by statistical ¯uctuation of the deposition process. The growth temperature is usuallyset as high as possible to optimize adatom diffusion, but within the limits determined by re-evaporationand intermixing. This strategy, however, does not work for Ag on Fe(1 0 0). Growth at roomtemperature or higher always results in ®lms with multiple thicknesses, as illustrated by the spectrashown in Fig. 19. It was noticed in several recent experiments that a highly nonequilibrium pathwayoften led to smoother ®lms in related systems [77±79]. This involves the deposition of the overlayer atvery low temperatures followed by annealing. The deposition temperature is so low that the resulting®lm has a ®ne-grained texture, and epitaxial crystalline order is restored by post-deposition annealing.This technique has been demonstrated in the growth of Ag on GaAs and Si. In those cases, the ®lms aremuch better, but not atomically uniform.

Employing this low temperature growth technique for Ag on Fe(1 0 0) resulted in dramaticimprovement in ®lm structure relative to growth at room temperature. In the experiment, the basetemperature of the substrate was 100 K, and the annealing temperature employed was � 600 K. Fig. 20shows normal-emission spectra taken at various photon energies for a 6.6 ML Ag ®lm deposited onFe(1 0 0) [67]. There are two main peaks, and the peak intensities vary signi®cantly as a function ofphoton energy due to cross section variations. The two peaks arise from quantum well statescorresponding to N � 6 and 7, as indicated in the ®gure. Hence, the ®lm exhibits only two thicknesses.This strongly suggests that if the amount of deposition is just right (an integer of monolayers), theresulting ®lm will become atomically uniform, as has been veri®ed by experiment. However, it is nevereasy to lay down the exact amount of coverage in one step, and a correction is often required.

Fig. 20. Normal-emission spectra taken from 6.6 ML Ag on Fe(1 0 0) prepared by low temperature deposition followed by

annealing. The two peaks can be assigned to ®lm thicknesses of 6 and 7 ML, respectively. The peak heights oscillate as a

function of photon energy.


Fortunately, the same procedure of low temperature deposition followed by annealing can be repeatedto build up a ®lm gradually without any adverse effect on ®lm roughness. Atomic layer ¯uctuationremains suppressed even after many cycles of deposition and annealing, and the ®lm thickness isalways either an integer, uniform across the surface, or a linear combination of two neighboring integerthicknesses.

Fig. 21 demonstrates the atomic uniformity of ®lms of integer monolayer coverages [68]. The bottomspectrum is for a ®lm made by adding incremental amounts to a thinner ®lm to reach a coverage of38 ML. Five very sharp quantum well peaks are observed. Adding 0.5 ML to this ®lm yields the middlespectrum, which exhibits two sets of quantum well peaks. One set is at the same positions as the 38 MLcase, and the other corresponds to a thickness of 39 ML. This is veri®ed by adding another 0.5 MLbecause the peaks corresponding to 38 ML are suppressed, and only the 39 ML peaks remain. The samediscrete layer behavior has been seen for many different starting thicknesses. Since the area probed inthe photoemission experiment is about � 1 mm in size, the above result establishes that the ®lm isuniform on an atomic scale over a macroscopic distance (� 1 mm). If the overlayer or substrate is notoptimally prepared (contamination of the substrate surface, too high a substrate temperature, etc.), therewill be a multiple set of thicknesses. Once this happens, no amount of annealing or deposition canrestore the atomic-level uniformity, and the sample must be stripped clean for a new attempt. For moreseverely contaminated substrates (by deliberate gas dosing), the quantum well peaks can become verybroad.

Fig. 21. Normal-emission spectra for a 38 ML (bottom), 38.5 ML (center), and 39 ML (top) Ag ®lm on Fe(1 0 0) taken

at a photon energy of 13 eV. The quantum well peak positions are indicated by vertical dashed lines. The spectrum for the

38.5 ML ®lm shows two sets of quantum well peaks indicating the simultaneous presence of areas covered by 38 and 39 ML

of Ag.


Fig. 22 shows the results from a least-squares ®t, using Voigt line shapes, to the quantum well peaksfor ®lms of thicknesses 12 and 38 ML [68]. The Gaussian width is the instrumental resolutiondetermined from a ®t to the Fermi edge line shape. The Lorentzian width is a ®tting parameter, and isgiven in the ®gure for each ®tted peak. These peaks are much sharper than the usual direct-transitionpeaks seen in bulk single crystals. The trend is that the peak width is smaller at lower binding energiesand for larger ®lm thicknesses. A quantitative discussion of the peak width in terms of the quasiparticlelifetime will be given below.

Clearly, growth at low temperatures is key to the making of such uniform ®lms. It is likely thatthermal atomic diffusion is suppressed at low temperatures, allowing the ®lm to build up `̀ uniformly''before annealing to restore the atomic order. In contrast, growth at higher temperatures involves a largediffusion length, and kinetic effects related to the interaction of diffusing atoms and atomic steps canlead to the formation of surface roughness [73±75]. For example, an adatom approaching an atomicstep can sense a potential barrier (Ehrlich±Schwoebel barrier) [80,81]. Such a barrier can cause islandformation on an incomplete layer, and continued growth can lead to the formation of multi-levelterraces.

Fig. 22. Normal-emission spectra for atomically uniform ®lms of 12 and 38 ML of Ag on Fe(1 0 0). The curves are results

from a ®t using Voigt lineshapes. The Lorentzian full width is indicated for each ®tted peak, and the Gaussian width is the

instrumental resolution obtained from a ®t to the Fermi edge.


5.3. Intensity modulations

Fig. 20 shows that the quantum well peak intensities can depend strongly on the photon energy andthe ®lm thickness. Fig. 23 presents the measured intensities for ®lm thicknesses 6, 7, 8, and 9 ML as afunction of the ®nal state energy [67]. The intensity modulation is almost 100%. In other words, atcertain photon energies, the emission intensity is essentially zero. For example, the 7 ML peak vanishesat hn � 16.5 eV, and the 6 ML peak vanishes at hn � 18 eV. An important conclusion is that theobserved photoemission intensities are not necessarily a good measure of the area covered by aparticular layer thickness. For example, based on the spectrum taken with hn � 16.5 eV alone (seeFig. 20), one might erroneously think that the 6.6 ML ®lm is a pure 6 ML ®lm.

The results in Fig. 23 demonstrate that the cross section depends on the ®nal state energy. Thisbehavior is somewhat similar to the case of Ag on Ni(1 1 1) discussed earlier, where the photoemissionintensity of each quantum well peak is cyclic. For Ag on Fe(1 0 0), the ®nal state of photoemissioninvolves an interference pattern related to the two boundaries of the ®lm, just as the initial state. Thisinterference pattern evolves as a function of energy, and the dipole matrix element between the initialand ®nal states oscillates depending on the relative phases. This gives rise to an intensity modulation.Similar intensity modulations have been reported for Ag on V [82], Cu on Co [83], and Na and Cs onCu [84,85].

Because the cross section modulations are different for different thicknesses, a rough ®lm consistingof multiple thicknesses will exhibit a reduced intensity modulation representing the average behavior.The modulation also depends on the photon energy and ®lm thickness. When the photon energy is high,the ®nal state mean free path becomes short. This effectively introduces a broadening in k, and theintensity modulations are reduced. Likewise, if the ®lm is thick, the interference pattern in the ®nalstate becomes less pronounced, leading to a reduced intensity modulation.

5.4. The Bohr±Sommerfeld quantization rule and band structure determination

An important application of angle-resolved photoemission is band structure determination [1±6].Photoemission involves an optical transition from an occupied (initial) state to an unoccupied (®nal)

Fig. 23. Quantum well peak intensities in normal-emission for ®lms of thicknesses 6, 7, 8, and 9 ML of Ag on Fe(1 0 0). The

horizontal axis is the ®nal state energy.


state, and the resulting spectra generally depend on both the initial and ®nal band properties. Theinformation is thus convoluted, and much of the historical development of the angle-resolvedphotoemission technique has focussed on methods to untangle this information in order to extract theinitial state properties. While this is straightforward for two-dimensional systems such as layercompounds and surface states, it is a major problem for three-dimensional systems. Because the surfaceof a crystal breaks the translational symmetry, momentum conservation does not hold along the surfacenormal direction. The momentum component perpendicular to the surface, k?, of the photoelectronoutside the crystal can be measured accurately, but this information is generally insuf®cient for adetermination of k? for the initial state inside the crystal. In contrast, the parallel component of themomentum, kk, is conserved, and this is the only component of interest for two-dimensional systems.This `̀ k? problem'' for three-dimensional systems has been the subject of much research, and manymethods have been devised to overcome this problem with varying degrees of success and utility.Approximations, interpolations, and/or theoretical calculations are often invoked in these methods,resulting in an uncertainty of Dk? typically on the order of one-tenth of the Brillouin zone size at anarbitrary point in k space. Another related problem is that the measured photoemission lineshape isoften quite broad because it is dominated by a very large ®nal state lifetime width [86±88].Furthermore, the lineshape can be distorted by interference from surface photoemission [42±46]. As aresult of these complications, an energy uncertainty of DE � 0.1±0.2 eV is typical. These energy andmomentum uncertainties are much too large for modern research, and essentially all recent high-resolution studies have been limited to two-dimensional systems.

Quantum well spectroscopy provides a solution to this k? problem. The basic idea is that k? isquantized in a ®lm as a result of electron con®nement. The allowed k? values are determined by the®lm thickness and boundary conditions. Measurements of quantum well peak positions for manydifferent ®lm thicknesses should permit a unique solution of E(k?). Although this method was proposedand implemented rather early on [16,23,24], the uncertainty Dk? remained quite large in those earlystudies due to ®lm thickness ¯uctuation and uncertainty. Recent success in preparing atomicallyuniform ®lms has ®nally made it possible to carry out accurate band structure determination usingquantum well spectroscopy. This procedure will be presented below for Ag on Fe(1 0 0). It is importantthat the absolute ®lm thickness N be known. In the actual experiment, a series of absolute ®lm thicknesscalibrations was made layer by layer at N � 1; 2; 3; . . . ; 20 as the ®lm was built up gradually bysubmonolayer depositions. This discrete layer counting provided a data set large enough to establisha unique functional relationship between the absolute ®lm thickness N and quantum well peakpositions, which could be used for higher thicknesses by extrapolation [69,70]. Additional layers atN > 20 were prepared and the peak positions were veri®ed to be consistent with this relationship. Fig. 24presents one subset of such data taken at normal-emission with a relatively high photon energy of55 eV.

While it is easy to carry out accurate calculations for Ag±Au as discussed above, Ag±Fe presents amuch more dif®cult case. The d wave functions in Fe near the Fermi level are much more complicated.Furthermore, several d bands can be simultaneously involved in the interfacial coupling. Fortunately,inasmuch as the peak positions and band structure are concerned, there is no need to construct the wavefunctions explicitly. It suf®ces to carry out a simple analysis based on the Bohr±Sommerfeldquantization rule

2k�E�Nt � F�E� � 2np; (7)


where F, a function of the binding energy E, is the total phase shift of re¯ection at the surface and theAg±Fe interface. This equation states that the total phase shift in a round trip perpendicular to thesurface is equal to an integer (quantum number n) times 2p. This is the same as requiring the de Brogliewave associated with the electron to form a standing wave. Eq. (7) is sometimes referred to as the phaseaccumulation rule in the literature [38], and its origin dates back to the beginning of quantummechanics. Eq. (1) is a special case of Eq. (7) with F � 0 (or 2p).

In Eq. (7), both k and F are dependent on E, and it is impossible to solve this equation uniquely basedon a single measurement. However, there is a multiplicative factor N associated with k only, and thusmeasurements at different values of N should lead to additional independent equations that can be usedto extract a unique solution. If quantum well state n for thickness N happens to be at the same energy E

as that of quantum well state n0 for thickness N0, we have the additional phase relation

2kN 0t � F � 2n0p; (8)

where k and F are the same as before in Eq. (7) because E is the same. Eqs. (7) and (8) can be solved toyield k and F at E in terms of the known quantities N, N0, n, n0, and t. Of course, it is rather seldom thatquantum well peaks from different thicknesses happen to have the same energy, but mathematicalinterpolation can be employed because k and F are continuous functions of E. In practice, a number ofthicknesses are used in the experiment to determine k(E) and F(E) via a least-squares ®tting procedure.The band dispersion E(k) is obtained by inverting k(E).

Fig. 24. Normal-emission spectra for atomically uniform ®lms of Ag on Fe(1 0 0) taken with a photon energy of hn � 55 eV.

The ®lm thicknesses are indicated.


Additional information is available from normal-emission data taken from bulk single crystalAg(1 0 0). The direct-transition peaks, shown in Fig. 25, are determined by

ÿEi�k� � hn � ÿEf�k�; (9)

where hv is the photon energy, and Ei,f are the initial and ®nal binding energies, respectively (Ei ispositive and Ef is negative). A unique solution of the band structure is impossible with Eq. (9) alone.One could rescale k in Eq. (9) by an arbitrary factor, and the resulting equation would still be consistentwith the normal-emission data. The lack of k constraints is the origin of the k? problem mentionedearlier. If the initial band structure is known, as would be the case with the quantum well data analyzedin the manner outlined above, the ®nal band structure is then uniquely determined by Eq. (9). Note thatthe peaks in Fig. 25 are broad and asymmetric. The large width is caused by a large ®nal state lifetimebroadening, and the asymmetry is due to surface photoemission. A careful analysis of the lineshape isneeded to yield an accurate measure of Ei [42±46].

In the actual data analysis [70], analytic expressions of Ei,f(k) from the two-band model are used. Thephase shift F(E) is modeled by a third order polynomial. Eqs. (4), (7) and (9) are used to ®t the quantumwell data and the normal-emission data from bulk Ag(1 0 0) simultaneously (with N � 1 ± 3 quantumwell data excluded; see below). In all, a total of 46 quantum well peak positions and nine normal-emission peak positions are used in this simultaneous ®t. The resulting four band structure parametersare presented in Table 1. As mentioned above, the quantum well data alone are suf®cient to determinethe initial band dispersion. The addition of the normal-emission data from the bulk crystal to the ®ttingprocedure allows the ®nal band dispersion to be determined.

The circles in Fig. 26 are the measured quantum well peak positions used as input to the ®t, plotted asa function of N, and the solid curves are results derived from the ®t. N, an integer by de®nition, is taken

Fig. 25. Normal-emission spectra for a bulk single crystal Ag(1 0 0). The photon energies are indicated.


to be a continuous variable in this calculation. These continuous curves illustrate the evolution of thepeak position for each quantum number n, which is given in the ®gure and corresponds to the schemethat k is measured from the zone boundary. The differences between the curves and the data points arevery small. The bottom panel in the same ®gure shows the differences using an ampli®ed vertical scale.There are no systematic deviations except for N � 1, 2, and possibly 3. Ignoring these data points, theaverage deviation is � 20 meV. The larger and systematic deviation for N � 1 and 2 can be attributed tooverlap of the surface and interface potentials. The screening length in a metal is short. Nevertheless,there can be a signi®cant overlap at very small ®lm thicknesses, and Eq. (7), based on the assumptionthat the phase shifts at the two boundaries are independent and additive, is no longer valid. For thisreason, the quantum well data for N � 1 ± 3 are excluded from the ®t as mentioned above. The samereason explains why this model fails at N � 0 (the quantum well states reduce to a surface state) [49].

Table 1

Parameter values for the Ag(1 0 0) band structure (me denotes the free electron mass)

Parameter Value

V 3.033 eV

Evbm ÿ1.721 eV

mi 0.759me

mf 0.890me

Fig. 26. The top panel is a structure plot showing the quantum well peak binding energies as a function of ®lm thickness of

Ag on Fe(1 0 0). The circles are data points and the curves are computed using parameters from a best ®t. The quantum

number n for each curve is shown. The bottom panel shows the difference between the best ®t and experiment using an

ampli®ed vertical scale.


The quantum well data can also be modeled by the tight-binding method [49]. While the generalbehavior is correctly reproduced, the agreement is not quite as good. A recent ab initio layer Korringa±Kohn±Rostoker calculation, on the other hand, has yielded quantum well peak positions in reasonablygood agreement with the experiment [89].

The initial and ®nal band dispersion relations of Ag(1 0 0) deduced from the ®t are plotted as solidcurves in Fig. 27. The shaded region indicates the k range spanned by the quantum well data. Thecurves outside this region represent an extrapolation. The circles indicate the ®nal states reached bydirect transitions from the initial band based on the normal-emission data from bulk Ag(1 0 0).Additional data analyses show that the band structure within the shaded region is accurate to within30 meV, which sets a new standard in bulk band structure determination. The dashed and dash-dottedcurves indicate the results of two band structure calculations [90,91], which are chosen asrepresentative for the large number of available results. The differences between these two calculationsare much larger than our experimental accuracy, and illustrate the level of accuracy of modern bandstructure calculations.

5.5. Quantum wells as electron interferometers

The above analysis is simple and yet powerful. However, it leaves out the lineshape, or the width ofthe quantum well peak, which is related to the quasiparticle lifetime. A more detailed analysis willrequire an examination of the wave function. The photoemission signal from a quantum well state istypically dominated by surface photoemission. Consider the time-reversed process. The photoelectronis sent back from the detector to the ®lm, and makes an optical transition to the initial state at the

Fig. 27. Band dispersions of Ag from a best ®t to the photoemission data (solid curves) and from calculations by Eckardt

et al. [90] (dash-dotted curves) and Fuster et al. [91] (dashed curves). The circles indicate the ®nal states based on the normal-

emission spectra from bulk Ag(1 0 0). The shaded region indicates data from both quantum wells and bulk single crystals are

available. Outside this range, the dispersion curves are simply an extrapolation based on the best-®t two-band model.


surface. The electron then travels to z � ÿ1 to complete the electrical circuit. In the case of aquantum well, the electron must undergo multiple re¯ections between the two boundaries, as illustratedby the diagram in Fig. 28. The initial state wave function is thus modulated by an interference factor

1

1ÿ R exp�i�2kNt � F�� exp�ÿNt=l� ; (10)

where R B r1r2 is the product of the re¯ectivities at the surface and the interface, and l is thequasiparticle mean free path. The mean free path gives rise to damping of the wave function, and isrelated to the quasiparticle inverse lifetime G via the group velocity v by G � v/l.

The photoemission intensity is modulated by the absolute square of the factor in Eq. (10), and thespectrum for a quantum well becomes

I / 1

1� �4f 2=p2� sin2�kNt � �F=2��A�E� � B�E�; (11)

where A is a smooth function of E, its prefactor comes from the absolute square of the interferencefactor, and B is a smooth background function due to inelastic scattering and incoherent emission. Thequantity f is the Fabry±PeÂrot ®nesse (ratio of peak separation to peak width) given by

f � p��Rp

exp�ÿNt=2l�1ÿ R exp� ÿ Nt=l� : (12)

A short mean free path corresponds to a low ®nesse, so does a low re¯ectivity. Eqs. (11) and (12) arethe same as the usual Fabry±PeÂrot formula for an optical interferometer ®lled with an absorptivemedium [92,93].

Eq. (11) yields a set of peaks at positions where the sine function in the denominator equals zero. Theresulting condition is just the Bohr±Sommerfeld quantization rule given in Eq. (7). The peak width dEdepends on N, R, and l

dE � GZ1ÿ R exp� ÿ 1=Z��

Rp

exp�ÿ1=2Z� ; (13)

where Z � l/Nt. Generally, dE > G. The only exception is dE � G, when R � 1 and l @ Nt. In otherwords, the measured width is just the quasiparticle inverse lifetime for an ideal quantum well (R � 1)with a long mean free path. The Ag±Fe quantum well has R � 0.8. The peaks in Fig. 22 are thus

Fig. 28. A schematic diagram illustrating the time-reversed photoemission process for a Ag ®lm on Fe. The time-reversed

photoelectron travels from the detector to the surface, where it makes an optical transition to the initial state. It then travels to

z � ÿ1, and during this process, it undergoes multiple re¯ections.


broadened by this effect. This broadening is relatively minor for the thicker ®lms, but can be quitesigni®cant for the thinner ®lms (due to loss caused by more frequent re¯ections). As mentioned earlier,the peaks become narrower near the Fermi level. This is mostly due to a smaller G at lower bindingenergies.

The quantities k and G (related to the real and imaginary parts of the band dispersion relations) and Rand F (related to the boundary con®nement potentials) are of basic interest, and completely specify theinterferometer properties. While k and F are related to the peak positions through Eq. (7), G and R arerelated to the peak width through Eq. (13). They all depend on E, but not on N (except possibly forN � 1 ± 3). Thus, a set of data spanning a wide range in N (> 3) allows a unique determination, andcrosschecks, of these four quantities at a given E. A ®tting procedure is adopted, and some of thespectra employed in this ®t together with the ®tting results (solid curves through the data points) areshown in Fig. 29 [71]. In this ®gure, all of the spectra with integer monolayer coverages correspond toatomically uniform ®lms. The 27.5 and 42.5 ML spectra show a mixture of peaks derived fromneighboring integer thicknesses and are not included in the ®t. The band structure k(E) is parameterizedusing the two-band model, G and R are modeled by quadratic functions of E, and F is modeled by acubic function of E. In addition, A in Eq. (11) is taken to be a polynomial of E (up to fourth order andno higher than the number of peaks in each spectrum), and B is also a polynomial (fourth order for the119-ML spectrum, and third order for the rest). The resulting energy dependence of k (band structure),

Fig. 29. Normal-emission spectra for Ag on Fe(1 0 0) at various coverages as indicated. The photon energy is hn � 16 eV,

and the spectra are normalized to the incident ¯ux. The data points are shown as dots, while the ®t to the data and the

background function are shown as curves.


G (quasiparticle lifetime), F (phase shift), and R (re¯ectivity) is shown in Fig. 30 [71]. The banddispersion is essentially indistinguishable from that shown in Fig. 27.

Fig. 31 illustrates the effects of surface roughness on the quantum well peak width [71]. The bottomspectrum is for a 29 ML atomically uniform ®lm, where the circles are data points and the solid curvesindicate the ®t and the background function. After adding 0.05 ML Ag to the sample at the basetemperature without annealing, the quantum well peaks become much weaker and broader, as shown bythe top spectrum. This is caused by a mere � 5% defect on the surface, and the large effectdemonstrates the high sensitivity of the quantum well peaks to sample imperfection. The increasedwidth can be modeled by a decrease in specular re¯ectivity at the surface [71].

5.6. Temperature dependence of the band structure

Fig. 32 shows the effect of raising the sample temperature [72]. The quantum well peaks for a 19 ML®lm, compared to the vertical reference lines, are seen to shift slightly to lower binding energies forincreasing temperatures. More noticeable are the reduction in peak height and increase in peak width.The shifts can be attributed to three possible causes: (1) thermal expansion of the Ag lattice resulting in

Fig. 30. Results deduced from Ag(1 0 0) quantum well data: (a) band dispersion relation for the Ag sp band; (b) quasiparticle

inverse lifetime as a function of binding energy; (c) re¯ectivity as a function of binding energy; (d) phase shift in units of p as

a function of binding energy (solid curve). Phase shift deduced from a semi-empirical formula, with an arbitrary vertical

offset, is shown as a dashed curve.


a change in the interferometer path length, (2) temperature dependence of the band structure, and (3) achange in phase shift at the Ag±Fe boundary. These factors could contribute to the shift according to theBohr±Sommerfeld quantization rule, Eq. (7). A detailed analysis shows that the thermal expansionaccounts only for part of the observed shifts. The remaining two contributions can be distinguished bythe fact that a change in k is ampli®ed by a factor of N in the phase shift, while there is no suchampli®cation for F. An analysis of data taken at different ®lm thicknesses shows that F does not changeas a function of T, and the remaining shift must be caused by a band structure change.

The same interferometric analysis, with the thermal expansion taken into account, yields thetemperature dependence of the band structure. The results are summarized in Fig. 33. The top panelshows the dispersion of Ag along [1 0 0] at two temperatures as a function of the reduced wave vector(k normalized to the Brillouin zone size kGX). The dispersion is slightly reduced at higher temperatures,because the interatomic overlap integral decreases as the lattice expands. Due to charge conservation,the low and high temperature bands must coincide near the Fermi level, leading to a very smalltemperature dependence of the reduced Fermi wave vector kF/kGX. This temperature dependence isshown in the lower panel. The extrapolated value at absolute zero temperature is kF/kGX �0.829 � 0.001. The error represents the systematic error deduced from an analysis of Ag layers ofvarious thicknesses. This result challenges the value kF/kGX � 0.819 deduced from a de Haas±vanAlphen measurement [94]. The de Haas±van Alphen method is the standard method for Fermi surfacemeasurements, and its development is widely regarded as a milestone in solid state physics. Thismethod, however, gives only the circumference of the Fermi surface, and the determination of theFermi wave vector depends on the precision of the parameterization of the Fermi surface. It has beenshown that in a study of the radio-frequency size effect, the Fermi surface of Au as determined by thede Haas±van Alphen method is in fact not entirely accurate. Namely, the Fermi wave vector in the[1 0 0] direction is too small by 1%. This is very similar to the 1% discrepancy, with the same sign,found for Ag in this study [95].

Fig. 31. Normal-emission spectra taken at a photon energy of 16 eV from a 29 ML atomically uniform ®lm of Ag on

Fe(1 0 0) (bottom) and the same ®lm after the addition of 0.05 ML deposited at 100 K (top). The solid curves represent a ®t to

the data and the background function for each case.


An issue of concern is the possibility of a lattice distortion in the ®lm. Ag(1 0 0) and Fe(1 0 0) have asmall lattice mismatch of 0.8%. For semiconductor epitaxial systems with such a small mismatch,strained growth to a critical thickness followed by unstrained growth and defect formation is typical.For metal epitaxial systems, the situation is less clear. Since metallic bonding is not directional as thecovalent bonds in semiconductors, lattice relaxation is likely to occur already at the early stages of ®lmgrowth. The post-deposition anneal to 3008C is likely to further suppress lattice strain. Assuming theworst case that the ®lms are fully strained by 0.8% in the interface plane, this translates into a 0.6%expansion along the surface normal direction with a Poisson ratio of 0.37. Incorporating this change inlattice constant in the analysis leaves the band edge parameters in the ®t unchanged. However, itchanges the effective masses by about 1.5%, and the effect is essentially the same as a rescaling in k.The band structure, when plotted against the normalized wave vector k/kGX, changes by less than 1 meVin the energy range of interest, and the change in the normalized Fermi wave vector is negligible. Thisanalysis suggests that lattice strain is very unlikely an explanation for the discrepancy between the deHaas±van Alphen and quantum well results.

5.7. Quasiparticle lifetime and scattering by defects, electrons, and phonons

Eq. (13) shows that the quantum well peak width is determined by the quasiparticle inverse lifetimeand the re¯ectivity at the interface. Since the re¯ectivity is tied to the band structure mismatch betweenAg and Fe, this is not expected to change signi®cantly as a function of temperature. The thermally

Fig. 32. Normal-emission spectra (dots) from a 19 ML thick Ag ®lm on Fe(1 0 0) taken at various temperatures. The curves

represent a ®t using a generalized Fabry±PeÂrot formula. The vertical reference lines highlight peak shifts.


induced peak broadening seen in Fig. 32 simply re¯ects an increase in lifetime broadening due tophonon scattering. As the temperature rises, the phonon population increases, leading to an enhancedscattering rate.

The quasiparticle inverse lifetime is approximately given by

G�E; T� � G0 � G1�E;T� � 2bE2; (14)

where E is the binding energy [96±99]. The ®rst term, being independent of T and E, represents thecontribution from defect scattering. The second term represents phonon scattering, and a perturbationcalculation yields

G1�E;T� � 2plZ ED

0

E0

ED

� �2

�1ÿ f �E ÿ E0� � 2b�E0� � f �E � E0�� dE0; (15)

where the electron±phonon mass enhancement parameter l enters as a proportionality constant, ED isthe Debye energy (0.0194 eV for Ag), and f and b are the Fermi±Dirac and Bose±Einstein distributionfunctions, respectively. For E > � 50 meV, G1(E, T) � G1(T); in other words, the phonon contributionbecomes asymptotically independent of E as long as E is much larger than ED. The third term in Eq. (4)

Fig. 33. (a) Ag sp band dispersion at 100 and 400 K in the GX-direction plotted as a function of the reduced wave vector

k/kGX. These curves are deduced from an interferometric analysis of the quantum well data. (b) The reduced Fermi wave vector

k/kGX as a function of T. The straight line represents a linear regression of the data.


represents the contribution of electron±electron scattering. The E2 dependence is often regarded as atest of the Fermi-liquid theory [96]. An experiment carried out at the base temperature of 100 K yields2b � 25.6 meV/eV2 (see Fig. 30(b)). Similar experiments carried out at elevated temperatures allow adeduction of G0 � G1(T), provided E > � 50 meV. This last condition is easily satis®ed if quantum wellpeaks very close to the Fermi level are ignored.

Fig. 34 shows such an analysis [72]. The circles represent G ÿ 2bE2, where G is determined from aFabry±PeÂrot ®t to the data. The solid curve is a ®t using G0 � G1(T), and the ®tting parameters arel � 0.29 � 0.05 and G0 � 8 meV. The latter corresponds to a quasiparticle coherence length of about1000 AÊ in the absence of phonon scattering and electron±electron scattering. The parameter l is ameasure of the strength of electron±phonon coupling, and is closely related to the transitiontemperatures of traditional BCS superconductors. Theoretical and experimental determination of l isdif®cult, and often only rough estimates are available. The above result appears to be the onlyexperimental determination of l involving a bulk-derived state for Ag. It is larger than the availabletheoretical estimates by about a factor two [97,98], and puts Ag very close to the borderline ofsuperconductivity. This study demonstrates the utility of quantum well spectroscopy for detailed studiesof the various scattering contributions to the quasiparticle lifetime. This information is critical forunderstanding the interactions among elementary excitations in solids.

5.8. Interfacial re¯ectivity and phase shift

The phase shift function F(E) for Ag on Fe(1 0 0) is shown in Fig. 30(d) as a solid curve. Much ofthis variation is associated with a hybridization gap in Fe. This gap is de®ned by the effective width ofthe d bands in Fe. The phase shift variation at the interface is expected to change by p across thegap. The phase shift variation at the Ag surface over the same energy range is much smaller. A detailed

Fig. 34. The circles represent experimental results for the sum of phonon and defect scattering contributions to the

quasiparticle inverse lifetime. The solid curve is a ®t, and the dashed curves indicate the separate phonon and defect

contributions. Note that the phonon contribution does not vanish at T � 0 because phonon emission is still possible.


calculation of the phase shift will require matching wave functions in Ag and Fe, which is dif®cult.Smith et al. [49] has derived a semi-empirical formula for F(E) with the upper and lower edges of theFe hybridization gap, Eu and El, respectively, as the only two adjustable parameters. If Eu � 0 eV andEl � 2 eV, are assumed, the result of the semi-empirical formula, with the addition of an arbitraryvertical offset, yields the dashed curve in Fig. 30(d). It agrees very well with the experiment, thussuggesting that the hybridization gap in Fe covers the range from 0 to 2 eV below the Fermi level. Thelower edge position is con®rmed approximately by the observation that quantum well peaks becomesigni®cantly weaker and broader at binding energies greater than 2 eV, where the electrons in Agbecome uncon®ned.

The hybridization gap in Fe(1 0 0) has been previously identi®ed as the range between the G12

critical point and the highest point of the lowest lying D1 band [49,100]. Based on available bandstructure calculations, this estimate would yield Eu � E(G12) � ÿ 1.3 eV and El � E�D1

max� � 2:7 eV,or a gap about 4 eV. This is much larger than the value of 2 eV noted above. The factor of twodiscrepancy is too large to be accounted for by inaccuracies in band structure calculations. A likelyexplanation is that the hybridization between the sp and d states involves a gradual shift in orbitalcharacter, and estimating the gap boundary by visual inspection of the band dispersions is not accurate.

The re¯ectivity shown in Fig. 30(c) is another important interfacial property. It is close to, but lessthan, unity, suggesting that the interface potential at the Ag±Fe interface is not fully con®ning. This isnot surprising, since a hybridization gap, rather than an absolute gap, provides the con®nementpotential. Also, the lattice mismatch between Ag and Fe, though small, could lead to non-specularre¯ection, resulting in a reduced specular re¯ectivity. The ®nesse of the electron interferometer withR � 0.8 is about 20; this is almost comparable to that of a simple optical etalon.

6. Magnetic effects and spin polarization

The discussion so far has ignored the spin polarization, and yet spin effects can be very important forlayer systems containing magnetic materials. Much of the interest in this area is driven by applicationsin `̀ spintronics'' (electronics based on the spin rather than the charge of carriers) or `̀ magnetoelec-tronics'' [101±105]. A well-known example is the use of the giant magnetoresistance effect for deviceapplications. This effect relates to the phenomenon that two ferromagnetic layers separated by anonmagnetic layer can exhibit oscillatory magnetic coupling depending on the thickness of theinterlayer. If the magnetic alignment is ferromagnetic, carriers can travel across the interfaces without aspin ¯ip transition, and the electrical resistance is thus signi®cantly lower than the case of anantiferromagnetic alignment. A carefully engineered multilayer structure that exhibits a large resistancechange upon the application of a magnetic ®eld can be used as a reader for magnetic recording media[101±109].

As discussed earlier in connection with multilayer systems, translayer coupling can be mediated bystates that are either propagating or nonpropagating. The propagating states can be quantum-well-likeor continuum-like. In most magnetic systems of technological interest, it seems that translayer couplingvia quantum-well-like states is the most common. The coupling is generally believed to be of theRKKY type, and the oscillation periods are thus determined by the nesting Fermi wave vectors of theinterlayer material. An attempt to describe such magnetic coupling effects in terms of electronic states(quantum well states) has led to much interest in the spin properties of simple quantum wells made of


nonmagnetic ®lms on magnetic substrates. Translayer magnetic coupling is a rapidly growing area ofresearch. A detailed review of the applications and technologies is beyond the scope of this paper. Thefocus here will be on the basic physics of spin-polarized quantum wells.

As seen earlier, quantum well spectroscopy is an excellent tool for measuring k(E), including theFermi wave vector kF � k(0). At the Fermi level, the Bohr±Sommerfeld quantization rule becomes

2kFNt � F�0� � 2np: (16)

As N (the average thickness of a ®lm) increases, quantum well peaks move through the Fermi level one-by-one. Each passage is marked by a maximum in photoemission intensity at the Fermi level. Twoneighboring maxima (Dn � 1) are separated by

DN � ptkF

: (17)

A measurement of DN gives kF. The appearance of intensity maxima at the Fermi level is thus a fairlydirect measure of the oscillation periods [28,29]. This argument can be generalized to emissiondirections other than the surface normal.

Fig. 35. Spin-resolved normal-emission spectra from Cu ®lms of various thicknesses taken with hn � 13 eV. The minority

quantum well states are marked by shaded areas (®gure taken from [118]).


Cu on Co [28,29,100,110±127] and Ag on Fe [28,29,67±72,89,100,128,129] represent two magneticquantum well systems that have attracted much interest. In both cases, the lattices are nearly matched,and high quality ®lms can be prepared by standard growth techniques (although only Ag on Fe(1 0 0)has been made with atomic uniformity). Fe and Co are ferromagnetic, while Ag and Cu are not. Themagnetic substrate materials are characterized by an exchange splitting of the bands. As a result, boththe hybridization gap and the window of con®nement depend on the spin. The phase shifts can bedifferent for the two spin orientations as well. Thus, photoemission should reveal two sets of quantumwell states, one spin up and the other spin down, generally at different binding energies. Spin-polarizedphotoemission should yield unambiguous signatures for these splittings [100].

Fig. 35 shows spin-resolved photoemission spectra from Cu ®lms of different thicknesses onCo(1 0 0) [118]. The majority spin spectrum shows a broad hump, with no clear evidence for quantumwell peaks, while the minority spin spectrum shows well-de®ned quantum well peaks. This behaviorcan be explained in terms of the Co band structure. The Cu sp band near the Fermi level overlaps withthe minority hybridization gap in Co, and thus the minority electrons in Cu are well con®ned. Thecorresponding Co majority gap is lower and overlaps the Cu d band region, and thus no sp quantumwell peaks near the Fermi level are expected.

For the Ag on Fe(1 0 0) system, the situation is similar. A recent study has shown that the observedquantum well peaks are of the minority spin character at low coverages. Fig. 36 shows spin-resolvedphotoemission spectra taken from a 1 ML Ag ®lm on Fe(1 0 0) [100,129]. The quantum well peak isclearly of the minority spin character. As mentioned above, the Fe hybridization gap covers a range of2 eV below the Fermi level. This is actually the minority gap, and so it is not surprising that minorityquantum well states are observed in this energy range. Available band structure calculations show that

Fig. 36. Normal-emission spectra for one monolayer of Ag on Fe(1 0 0). The spin-integrated spectrum is indicated by the

curve, and the majority and minority spin components are indicated by full and open triangles, respectively (®gure taken from

[129]).


Fig. 37. (a) Sample con®guration. (b) Photoemission intensity at the belly of the Fermi surface oscillates with 5.6 ML

periodicity of the Cu thickness. (c) Photoemission intensity at the neck of the Fermi surface oscillates with 2.7 ML periodicity

of the Cu thickness. (d) Interlayer coupling from a magnetic X-ray linear dichroism measurement. The bright and dark regions

correspond to antiferromagnetic and ferromagnetic couplings, respectively. (e) Calculated interlayer coupling based on a

phased sum of the two oscillations (®gure taken from [127]).


G12 and D1max are at about 0.8 and 3.2 eV binding energies, respectively, for the majority spin states in

Fe. Assuming that the same factor of two correction is needed as discussed above for the minority gap,the majority gap should be about 1.2 eV wide. Applying the same linear mapping needed to go from thecalculated G12 and D1

max to Eu and El for the minority states, we can estimate that Eu � 1.6 eV andEl � 2.8 eV for the majority states. This range, though far below the Fermi level and relatively small,should support some majority quantum well states. However, experimental search in the form of extrapeaks not explainable by the minority spin states has not turned up anything that can be identi®ed asmajority spin states. Perhaps they are just too weak or broad (the Bohr±Sommerfeld quantization rulesays nothing about the peak intensity). However, evidence for majority unoccupied states has beendetected by spin-polarized inverse photoemission [89,128].

We close this section by presenting an example in which quantum well measurements provide anelegant explanation of the magnetic coupling effect in a Co±Cu±Co(1 0 0) sandwich structure.Translayer coupling of the RKKY type can involve multiple nesting Fermi wave vectors. For Cu(1 0 0),there are two relevant Fermi wave vectors, one associated with the `̀ belly'', and the other associatedwith the `̀ neck''. This can be probed by photoemission from quantum wells as discussed before. Datataken from a linear wedge sample with a con®guration depicted in Fig. 37(a) are shown in Fig. 37(b)and (c) for the belly and neck oscillations, respectively [127]. One can use a simple semi-empiricalformula to estimate the relative phases between the two oscillations. A phased linear superposition ofthe two oscillations, shown in Fig. 37(e), should reproduce the observed magnetic coupling as afunction of interlayer thickness. This is indeed the case, as revealed by a comparison with Fig. 37(d),which is obtained by a magnetic X-ray linear dichroism measurement.

7. Summary and conclusions

This paper reviews the physics of quantum wells made of thin ®lms. The basic manifestation ofquantum con®nement is the formation of discrete states as observed by angle-resolved photoemission.Concepts including con®nement by gaps, interfacial re¯ection, phase shift, barrier penetration, electronpropagation and scattering, layer±layer coupling are addressed in terms of the band structure and wavefunctions. For commensurate overlayers, all that is required is a relative gap for con®nement. Partialcon®nement by hybridization gaps and thin barriers give rise to quantum well resonances. A signi®cantlattice mismatch can cause a substantial reduction in specular re¯ectivity, and the resulting spectralresponse can be qualitatively different. The foundation laid by studies of simple quantum wellsprovides the knowledge base needed for understanding the properties of multilayer systems. Theconcepts of interlayer coupling, translayer coupling, and band folding are clari®ed and illustrated byexamples involving multilayer and superlattice con®gurations.

An important application of quantum well spectroscopy is to determine the band structure,quasiparticle lifetime, interface re¯ectivity, and phase shift, which are fundamental to solid statephysics and interface science. These quantities completely specify the electron dynamics within a ®lmand at the ®lm interfaces, but are dif®cult to determine using any other means. The accuracy ofquantum well measurements is greatly improved by the use of atomically uniform ®lms. Aninterferometric analysis of the photoemission data yields a band structure that is suf®ciently accurate tochallenge the de Haas±van Alphen method for Fermi surface determination. The fundamental reasonfor this improvement is that k?, being a continuous variable in a bulk crystal and highly uncertain in


photoemission, becomes quantized and precisely determined by the geometry of a ®lm. Temperature-dependent line width determination yields accurate measures of the quasiparticle lifetime, allowing adetailed analysis of the electron±electron, electron±phonon, and electron±defect interactions. The phaseshift and re¯ectivity deduced from quantum well spectroscopy can be used to extract useful informationabout the interface potential and the band structure of the substrate material.

Much of the current interest in quantum wells is driven by technological issues related to devicecon®gurations made of magnetic layer structures. In such systems, quantum well states can be spinpolarized, giving rise to a magnetically sensitive response. The giant magnetoresistance effect is aprominent case, and many technological issues can be addressed by photoemission measurements usingmodel structures. This provides an interesting example for a fairly direct connection between basicresearch and device design and industrial applications.

Atomically uniform ®lms could become a dominant theme in this area of research if such ®lms canbe routinely prepared for materials other than just Ag on Fe whiskers. Beyond issues of academicinterest, atomically uniform ®lms represent the ultimate limit for device con®gurations made of ®lms.The response function can be much sharper, and modeling can be much simpler. The current push fornanotechnology will undoubtedly continue, and soon it would be up against the atomic limit. Atomiclevel manipulation and control will remain a subject of intense research in the foreseeable future. Ofgreat interest would be the fabrication of other forms of quantum structures such as dots, wires, stripes,etc., with atomic uniformity. Controlled dimensionality allows ®ne tuning of properties includingelectron correlation. Studies of the interplay between quantum con®nement and magnetism in ®lmshave already resulted in useful applications. Similar studies involving other physical phenomena suchas superconductivity, charge ordering, and colossal magnetoresistance may be equally fruitful for bothbasic science and technology.

Acknowledgements

The author wishes to thank present and former group members T. Miller, J.J. Paggel, A.L. Wachs,W.E. McMahon, E.D. Hansen, M.A. Mueller, A. Samsavar, A.P. Shapiro, T.C. Hsieh, G.E. Franklin,and D.-A. Luh for their contributions to the work on quantum wells. He also wishes to thank J. Weaverfor reading the manuscript and providing feedback. Much of the material presented here is based uponwork supported by the US National Science Foundation, under grant Nos. DMR-99-75470 and DMR-99-75182. An acknowledgment is made to the Donors of the Petroleum Research Fund, administeredby the American Chemical Society, and to the US Department of Energy, Division of MaterialsSciences, (grant No. DEFG02-91ER45439) for partial support of the synchrotron beamline operationand for support of the central facilities of the Frederick Seitz Materials Research Laboratory. TheSynchrotron Radiation Center of the University of Wisconsin is supported by the National ScienceFoundation under grant No. DMR-95-31009.

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