Mechanisms of Suppression and Enhancement of Photocurrent ...
Suppression and enhancement of van der Waals...
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Suppression and enhancement of van der Waals interactions Minhaeng Cho and Robert J. Silbey Citation: J. Chem. Phys. 104, 8730 (1996); doi: 10.1063/1.471562 View online: http://dx.doi.org/10.1063/1.471562 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v104/i21 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 27 Oct 2012 to 18.189.115.229. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions
Transcript of Suppression and enhancement of van der Waals...
Suppression and enhancement of van der Waals interactionsMinhaeng Cho and Robert J. Silbey Citation: J. Chem. Phys. 104, 8730 (1996); doi: 10.1063/1.471562 View online: http://dx.doi.org/10.1063/1.471562 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v104/i21 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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Suppression and enhancement of van der Waals interactionsMinhaeng Choa) and Robert J. SilbeyDepartment of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
~Received 18 January 1996; accepted 12 February 1996!
The van der Waals interaction of a pair of anisotropic molecules near planar dielectric surfaces isstudied by using the linear response formalism. The spatial correlation function~Green function! ofthe vacuum electric field in the presence of dielectric surfaces is obtained by using suitable Fresnelmode functions of the quantized electric field. In the short-distance limit, it is observed that thelong-range interaction potential is significantly modified by the dielectric surfaces and stronglydepends on the geometry of the two molecules near dielectric surfaces. When the two molecules areanisotropic, depending on the molecular alignments with respect to the surfaces, the van der Waalsinteraction is enhanced or suppressed by the existence of the surfaces. When the two molecules arein between two dielectric surfaces, the overall magnitude of the van der Waals interaction issuppressed in comparison to that in the free space because the vacuum electromagnetic fieldintensity is reduced by the transmissivities that are generally less than unity. ©1996 AmericanInstitute of Physics.@S0021-9606~96!00219-8#
I. INTRODUCTION
Boundary effects on various quantum phenomena, suchas level shifts of an atom in various types of cavities, en-hancement or suppression of spontaneous emission rate, ex-citation transfer etc., are of current interests both theoreti-cally and experimentally.1 The modifications by theboundary conditions are the manifestation of the changes inmode structures of electromagnetic fields near boundaries.For example, the spontaneous emission rate of an excitedatom near a dielectric surface can be enhanced or suppresseddepending on the distance from the surface.1–3 Based on theFermi Golden rule, the spontaneous emission rate is propor-tional to the mean square fluctuation amplitude of the quan-tum vacuum field.4–6 In the vicinity of a dielectric surface,7
the spatial variation of the field fluctuation amplitude differsfrom that of the free space, which in turn cause modulationsof spontaneous emission rate as a function of distance fromthe surface. In this paper, we shall show how these changesin modal structures near planar surfaces modify the van derWaals interaction between two polarizable molecules.
It was perhaps MacLachlan8 who first showed that thevan der Waals interaction between two neutral polarizablemolecules can be significantly altered by the existence of thedielectric surface compared to that of the free space. Heshowed that the van der Waals interaction potential of a pairof molecules in the vicinity of a dielectric surface dependson the detailed geometry. If the two molecules are locatedside by side parallel to the dielectric surface, the intermo-lecular interaction decreases in comparison to that in the freespace. In contrast, when one is on top of the other, the inter-action potential increases. In case when the anisotropy of themolecular polarizability is large, this boundary effect, in-duced by a single dielectric surface, on the van der Waals
interaction of a pair of molecules can be very large so thatthe van der Waals interaction could be suppressed or en-hanced dramatically. Although this case was theoreticallystudied by Imura and Okano,9 the clear physical interpreta-tions and implications of these results were not fully ex-plored in their paper.
In this paper, we focus on the more general situationwhen two molecules are located in a planar microcavity~seeFig. 1!. Particularly, the distance between the two surfaces ismuch shorter than the most important wavelength of the sys-tem, that is, the wavelength of the optical transition fre-quency of the molecules or dielectric media. Noting that usu-ally the optical transition wavelength is order of hundreds ofnanometer, the short distance limit we shall consider is thecase when both the intermolecular distance and intersurfacedistance are smaller than this length scale. If a pair of mol-ecules are located in a planar microcavity~Fabry–Perot cav-ity!, one can expect that the van der Waals interaction issimilarly but more strongly modified by the two planar di-electric surfaces, though there are distinctive differences inmodal structures in comparison to the case of a single sur-face. A related problem, the properties of liquid confined bytwo dielectric surfaces, has been extensively studied by mea-suring normal and shear forces between the two surfaces as afunction of distance between the two surfaces.10–12This caseis found to be very important in the surface sciences of liquidlubricants, adhesion and wear properties,13 wetting and dew-etting of surfaces,14 liquid films,15 etc. Despite of the exten-sive theoretical studies using Lifshitz’s theory16 and molecu-lar dynamics simulations as well as experimental studies,fundamental aspect of the intermolecular interaction betweenmolecules in a plane-confined geometry has not been stud-ied.
In this paper, we shall focus on the van der Waals inter-action between two nonpolar molecules when they are inclose proximity to the surfaces, treating the surfaces in amacroscopic manner. There exist however numerous effects
a!Present address: Department of Chemistry, Korea University, Seoul,Korea.
8730 J. Chem. Phys. 104 (21), 1 June 1996 0021-9606/96/104(21)/8730/12/$10.00 © 1996 American Institute of Physics
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that cannot be taken into account by using the macroscopicpicture of a solid surface with the phenomenological Max-well’s equations. For example, the nonlocality of the dielec-tric function near a surface17–22may play a crucial role whenthe distance between the molecules and a surface is order ofthat between atoms of the solid. Second, the surface caninduce a permanent dipole moment by distorting the groundstate wave function of the molecule.17 The magnitude of theinduced permanent dipole moment increases as the moleculeapproaches the surface. Besides of the two complicationsmentioned above, there exist various mechanisms causingdeviations from the van der Waals interaction. In order toremedy the effect of the nonlocality of the dielectric func-tion, Zaremba and Kohn23 used the jellium model as an ap-proximate microscopic model for a metal surface. Theyshowed, when studying the interaction of a single atom witha metal surface, that the reference~dynamical image! planeposition has to be calculated by taking into account the mi-croscopic nature of the metal surface. Thus the effective dis-tance between the molecule and the surface differs from thedistance between the molecule and the jellium surface. How-ever, the basic theories developed by Lifshitz are still appro-priate in describing key aspects of physisorption and the role
of the dielectric surface. Thus, although the results given inthis paper will suffer from the lack of a realistic quantumdescription of the solid surface, they should be useful inunderstanding fundamental aspects of the boundary effectsand in developing proper intermolecular potential functionsused in molecular dynamics simulation studies.
This paper is organized as follows. In Sec. II, the effec-tive Hamiltonian of nonpolar polarizable molecules is dis-cussed. The van der Waals interaction between general an-isotropic molecules in a planar cavity is considered in Sec.III. Two limiting cases of a single dielectric surface and twodielectric surfaces are discussed in Secs. IV and V, respec-tively. Finally we summarize the results in Sec. VI.
II. EFFECTIVE INTERACTION HAMILTONIAN
The usual minimal coupling Hamiltonian24 contains twointeraction terms that are linearly and quadratically propor-tional to the vector potential of the quantized field. By usingthe Power–Zienau–Wooley transformation,24–26one can ob-tain the multipolar Hamiltonian,
H5Hmol1HF1H I . ~1!
The molecular Hamiltonian is, in the second quantized form,
Hmol5(m
(j~m!
\VmjBmj1 Bmj1(
m2pE dr uPm
'~r !u2.
~2!
The subscriptsm and z denote themth molecule andzthstate of the mth molecule, respectively.Vmz is the transitionfrequency between the ground stateug(m)& and thezth stateuj(m)&. Bmj
1 and Bmj are transition operators defined byBmj
1 5uj(m)&^g(m)u and Bmj5ug(m)&^j(m)u, respectively.The transition operators obey the usual commutation rela-tions,
@Bmj ,Bm8j81
#5~122Bmj1 Bmj!dmm8djj8 . ~3!
We assumed that the overlap between molecules is ignored.One of the notable difference of the multipolar Hamiltonianfrom the minimal coupling form is that the intermolecularinteraction in the multipolar form is purely determined by theretarded exchange of the virtual transverse photons, whereasthe minimal coupling Hamiltonian contains additional in-stantaneous intermolecular interaction terms.25,26
The quantized field Hamiltonian is as usual,4
HF5(kl
\vk~ akl1 akl11/2!, ~4!
where the photon creation and annihilation operators are de-noted byakl
1 andakl . The interaction HamiltonianH I in Eq.~1! is
H I52E dr P~r !•D~r !. ~5!
Here the polarization operator, that is associated with thetransitions between states of a given molecule, is defined as
FIG. 1. ~a! Two types of modes are shown. Type I modes are associatedwith the incident modes propagating downward from the dielectric 1 to theinterface between the dielectric 1 and the free space, whereas type II modesare associated with the incident modes propagating upward from the dielec-tric 3 to the interface between the dielectric 3 and the free space.~b! De-tailed geometry of the two molecules with respect to the two surfaces aredrawn. Two close circles represent the two real molecules in the free spaceregion, whereas the two circles in the dielectric media are pictorial repre-sentations of the image molecules of the B-molecule.
8731M. Cho and R. J. Silbey: Enhancement of van der Waals interactions
J. Chem. Phys., Vol. 104, No. 21, 1 June 1996
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P~r !5(m
mmd~r2Rm!, ~6!
where the dipole operatormm is
mm5 (j~m!
mmj~Bmj1Bmj1 !. ~7!
mmj is the electric dipole transition matrix element.D~r ! inthe interaction Hamiltonian is thetransversedisplacementfield operator, that is the conjugate momentum of the vectorpotential in the multipolar Hamiltonian,
D~r !5 i(kl
S 2p\vk
V D 1/2$fkl~r !akl2fkl* ~r !akl1 %, ~8!
and the vector potential is
A~r !5(kl
S 2p\c2
VvkD 1/2$fkl~r !akl1fkl* ~r !akl
1 %. ~9!
fkl~r ! are the normalized mode functions satisfying the cor-responding Helmholtz equations with the proper boundaryconditions.5
The interactions between nonpolar polarizable moleculescan be described by the second-order or fourth-order pertur-bation theories depending on the situation. For example, theattractive interaction between a molecule and a dielectricwall is a second-order problem, i.e., quadratic Stark shift,27,28
that is, the interaction potential is linearly proportional to themolecular polarizability. The interaction of a pair of mol-ecules can be, as shown by London,29 obtained by using thefourth-order perturbation theory, where the energy shift isproportional to both polarizabilities of the two molecules.Therefore we find that it is useful to derive an effectiveHamiltonian by eliminating a one-photon vertex in the Feyn-man diagrams.26 By using the Heisenberg equations of mo-tion of the transition operators and introducing theWeisskopf–Wigner approximation ~Adiabaticapproximation!,30 the effective interaction Hamiltonian canbe obtained as
H Ieff52(
mF(kl
am~vk!:Dkl~Rm!GD~Rm!, ~10!
where the term in the square bracket in Eq.~10! is the effec-tive induced dipole operator and the molecular polarizabilityoperators are defined as
am~vk!51
\ (j~m!
@12Wmj~0!#mmjmmjS 2Vmj
Vmj2 2vk
2D ,~11!
Wmj is the population operator,Wmj52Bmj1 Bmj . The spon-
taneous emission contribution to the effective dipole operatorwas neglected, since we focus on the molecular interactionbetween two molecules in the ground states. As expected, theeffective interaction Hamiltonian given above involves twophoton processes. A vacuum field with wave vectork andpolarization directionl creates an induced dipole at the cen-ter of themth molecule, and in turn this induced dipole in-teracts with vacuum field at the same position. We shall use
a second-order perturbation theory with respect to this effec-tive interaction Hamiltonian to obtain the interaction poten-tial of a pair of nonpolar polarizable molecules.
III. VAN DER WAALS INTERACTION
The van der Waals interaction of a pair of molecules hasbeen a subject of extensive studies since the seminal paper ofLondon.29 Recently Power and Thirunamachandran31
showed that the van der Waals–Casimir interaction can becalculated by considering two induced dipoles at the centersof the two molecules and then the two induced dipoles inter-act through the well-known retarded dipole–dipole interac-tion tensor obtained by McLone and Power.32 For a pair ofmolecules A and B, the interaction energy is thus given bythe vacuum expectation value,
DE5(kl
^mAeff~k,l!•V~k!•mB
eff~k,l!&vac, ~12!
where theeffectiveinduced dipole operators are defined as
mmeff~k,l!5am~vk!•Dkl~Rm!. ~13!
This result, Eq.~12!, can also be obtained by using thesecond-order perturbation theory with respect to the effectiveinteraction Hamiltonian given in Eq.~10!. Inserting the dis-placement field operator given in terms of mode functions inEq. ~8! into Eqs.~12! and ~13! gives
DE52\c
p E0
`
dk Tr@aA~vk!V~k!aB~vk!F1~k!#,
~14!
where the dipole–dipole interaction tensorV(k) and its Hil-bert transform,F(k), can be written in terms of classicalmode functions
F~k
trized time correlation function of the electric field operator.A detailed derivation ofF(k) by using Eq.~15! with propermode functions of the quantized electromagnetic fields isgiven in Appendix C. ThenV(k) can be calculated from Eq.~15!.
In order to rewrite the van der Waals interaction energyin Eq. ~14!, we define
G~k
where, form, n50,I,II,
Lm,m~ ik!5Tr$aAaBTmTm12aAbBTmtBTm
1
2bAaBtATmTm11bAbBtATmtBTm
1%
Lm,n~ ik!5Tr$aAaB~ TmTn11TnTm
1!2aAbB~ TmtBTn1
1TntBTm12bAaB~ tATmTn
11 tATnTm1!
1bAbB~ tATmtBTn11 tATntBTm
1! ~24!
with
tA23uAuA ,
tB5 I23uBuB ,
T05T23r r , ~25!
TI5 I 823RIRI8 ,
TII5 I 823RIIRII8 .
Equations~23! with ~24! and~25! are the main results in thissection, and show complex dependence on the distances be-tween real molecules as well as between real and image mol-ecules. Only the first term is inversely proportional to thesixth power of the direct intermolecular distance. Therefore,the first term is the largest one since bothRI and RII arealways larger than the direct distance through the free space.The physical meaning of each term can be understood byconsidering the exchange process of a couple of virtual pho-tons. As shown in Fig. 2~a!, the first term on the right-handside of Eq.~23! corresponds to the case when the two virtualphotons are exchanged via free space. On the other hand, thesecond and third terms@Figs. 2~b! and 2~c!, respectively#involve a single reflection by the lower and upper surfaces,respectively. The fourth and fifth terms@Figs. 2~d! and 2~e!#involve two reflections by the same surface, whereas thesixth term @Fig. 2~f!# does two reflections—once by eachsurface.
The general expression given in Eq.~23! is very compli-cated because of the geometrical factors, though it is quite
straightforward to rewrite them in terms of angles and dis-tances defined in Fig. 1~b!. Instead of discussing Eq.~23!more in detail, we shall consider several limiting cases thatshould be found useful in practical calculations. Since thevan der Waals interaction between either isotropic or aniso-tropic molecules in vacuum has been studied extensively, weshall not consider those cases in the following sections.
IV. SINGLE DIELECTRIC SURFACE
We shall consider, in this section, the case when the twomolecules are in the vicinity of a single dielectric surface.Let us assume that the upper dielectric medium is replacedwith an infinite vacuum. In this case, the dielectric functione1( ik) equals to 1, that is the transmission coefficientt12becomes unity whereasr 2150. Then the third, fifth, and sixthterms on the right-hand side of Eq.~23! vanish.
A. Isotropic molecules
If the two molecular polarizabilities are isotropic, fromEq. ~23! we find
DE523\
pr 6I 0
aa1\~223 cos 2c23 cos 2x!
2pr 3RI3 I 1
aa
23\
pRI6 I 2
aa , ~26!
where
I 0aa5E
0
`
dkaA~ ik!aB~ ik!,
I 1aa5E
0
`
dkFe3~ ik!21
e3~ ik!11GaA~ ik!aB~ ik!, ~27!
I 2aa5E
0
`
dkFe3~ ik!21
e3~ ik!11G2aA~ ik!aB~ ik!.
Equations ~26! with ~27! are exactly identical toMacLachlan’s results8,34,37$note that the angles defining thegeometry @see Fig. 1~b!# are defined differently fromMacLachlan’s8%. In the definitions of three integrals, the su-perscript, ‘‘aa,’’ denotes that the integrals involve the iso-tropic polarizabilities in the integrand. The first term is justthe van der Waals interaction in the free space—London’sresult can be obtained by assuming that the molecular polar-izability is determined by a single frequency, so that theintegration over the imaginary frequency can be performedby using residue theorem. The second term represents thevdW interaction when one of the two virtual photons is re-flected by the surface. Because of the interference effect be-tween the direct dipolar field and the reflected dipolar field,the magnitude of the second term becomes strongly depen-dent on the detailed geometry. For example when the twomolecules are located parallel to the surface and separatedfar apart, the second term becomes positive so that the over-all vdW interaction decreases compared to that in the freespace. The third term corresponds to the contribution fromthe interaction between the induced dipole in the free space
FIG. 2. Six different contributions to the van der Waals interaction areshown@see Eq.~23! and discussion below Eq.~25!#. Exchange of two vir-tual photons are represented by arrows.~a!–~f! correspond to the first to thesixth terms in Eq.~23!, respectively.
8734 M. Cho and R. J. Silbey: Enhancement of van der Waals interactions
J. Chem. Phys., Vol. 104, No. 21, 1 June 1996
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and the inducedimagedipole created in the dielectric me-dium. They are separated by the distanceRI . Usually thethird term is much smaller than the second one, except whenthe two molecules are far apart as well as when they both areclose to the surface.
These results has been used in understanding lateral in-teractions between adatoms,35–37where the two-dimensionalflat film is normally assumed. For example, Rauberet al.38
showed that the reduction in the potential well depth of15%–20% from the free space values for the inert gas atomsnear graphite surface can be assigned to the additional~sec-ond! term in Eq.~26!.
In order to obtain simpler results with retaining salientfeatures of a dielectric~metal! surface, we further use a free-electron-like~Thomas–Fermi model! dielectric function39,40
e~ ik!511vp2
k2 , ~28!
wherevp is the plasma frequency, which is the characteristicresonance frequency of absorption of the electrons in themetal. For the sake of simplicity, if the two molecules areidentical and the molecular polarizability is determined by asingle transition frequency,v0, i.e.,
a~ ik!52umu2v0
\~v021k2!
, ~29!
the three integrals defined in Eq.~37! can be calculated,within these approximations to the dielectric function and themolecular polarizabilities,
I 0aa5
pv0a2~0!
4,
I 1aa5
pv0a2~0!
4 F g212g
~g11!2G , ~30!
I 2aa5
pv0a2~0!
4 Fg~g213g11!
~g11!3 G .Hereg is the ratio of the surface-plasmon frequency to themolecular transition frequency,
g5vp /&
v0. ~31!
If the ratio g becomes large, that is, the surface acts like agood mirror, all three integrals given in Eqs.~30! becomeidentical. In this limit, the surface boundary effects will bemaximized. On the other hand, as the ratiog approacheszero, the last two integrals,I 1
aa andI 2aa , decreases to zero, so
that the boundary effects become negligible, as expected.In the limiting case both when the two molecules are far
apart and located side by side~‘‘lateral’’ ! on the dielectricsurface ~c5x>p/2 and r>RI! and when the ratiog ap-proaches a large number, the total vdW interaction energyreduces to approximately two-thirds of that in the free space.This is demonstrated by a numerical evaluation of Eq.~26!with Eq. ~30! in Fig. 3~a!, where the ratio of the vdW inter-action between two isotropic molecules near a surface to that
in the free space is calculated. Here the dielectric surface isfully characterized by the ratiog that is assumed to be 2.Thedistance between the molecule A and the surface is assumedto be unity, and the distance between molecules is scaled bythis unit distance in the following calculations. As the twomolecules approaches to each other the surface boundary ef-fect decreases since the additional terms@the second andthird terms in Eq.~26!# are negligibly small in this limit. Ifone of the two molecules is on top of the other~‘‘vertical’’ !,the intermolecular interaction becomes larger than that in thefree space as shown in Fig. 3~a!.
B. Anisotropic molecules
In case when the two molecules near a surface are an-isotropic, one needs to consider the complicated geometricalfactors given in Eqs.~23!. Since it is very tedious to rewritethe geometrical factors in terms of angles of the two mol-ecules with respect to the surfaces, we instead present nu-
FIG. 3. The ratios of the van der Waals interaction between two isotropicmolecules~a! or two anisotropic molecules~b! near asingledielectric sur-face to that in thefree space are plotted. The ratio of the surface-plasmonfrequency to the optical transition frequency,g, is assumed to be 2@see Eq.~31!#. ai~v!/a'~v!53. The distance between the molecule A and the surfaceis assumed to be unity, and the distance between molecules is scaled by thisunit distance.
8735M. Cho and R. J. Silbey: Enhancement of van der Waals interactions
J. Chem. Phys., Vol. 104, No. 21, 1 June 1996
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merical calculations to survey the general trends. To this end,we assume that the surface and molecular polarizabilities aregiven by the forms in Eqs.~28! and~29!, respectively, exceptthat the ratio of the parallel polarizabilityai~v! to the per-pendicular componenta'~v! is assumed to be 3. The ratiogdefined in Eq.~31! equals 2.
In Fig. 3~b!, the ratios of the vdW interactions near asurface to those in the free space are shown, where we con-sider three different cases when the molecular principal axeslie along three different axes in the laboratory coordinatespace. Again it should be noted that the two molecules are inthey–z plane~see Fig. 1!. When the two molecular axes areparallel to either thex- or y-axis, the vdW interaction de-creases compared to that in the free space. In contrast, thez–z interaction is enhanced by the existence of the surface.This can be understood by using the image picture. Supposethe two molecular axes lie along thez-axis—that is, the prin-cipal axes are perpendicular to the surface plane. Then theinstantaneous induced dipole at the center of a molecule ismore likely to be aligned to be perpendicular to the surfaceplane, because the parallel component of the polarizability islarger than the perpendicular one. The image dipole thusinduced in the dielectric medium is oriented with the samedirection with the induced dipole of the molecule A. There-fore, the magnitude of the combined dipole moment is largerthan that in the free space, which will be shown up as anincreased vdW interaction energy when the two moleculesare well separated. However when they are in close proxim-ity, the overall interaction is not monotonic as can be seen inFig. 3~b!. Similar discussions with the image picture are ap-plicable to the other two cases,x–x and y–y interactions,though in these cases the cancellation between the induceddipole of a real molecule and the inducedimagedipole willmake the overall vdW interactions decrease in comparison tothose in the free space.
In comparison to the vdW interaction between twoiso-tropic molecules, one of the most distinctive features in theinteraction betweenanisotropicmolecules is that the align-ment of the two molecules is very important factor in deter-mining the role of the dielectric surface. Depending on themolecular alignments, the surface enhances or suppresses thevdW interaction in the vicinity of a dielectric surface—notethat a similar observation was also made in studies on thespontaneous emission rate near a dielectric surface.1–3
V. TWO DIELECTRIC SURFACES
When the two molecules are located in between two sur-faces, one can expect that the boundary effects induced by asingle surface discussed above would perhaps be enhanced.Indeed as we show in Figs. 4~a! and 4~b!, there are strongmodifications of the vdW interaction strength. For example,when two isotropic molecules are located exactly in betweenthe two surfaces with their intermolecular axis parallel to thesurfaces~lying along thex-axis!, the ratio of the vdW inter-action to that in the free space is shown in Fig. 4~a!. Here weassume that the two dielectric media are identical. The trendobserved in Fig. 4~a! is identical to the case when two iso-
tropic molecules are near a single surface@see Fig. 3~a! ‘‘lat-eral’’#. Similarly, we calculated the vdW interaction betweentwo anisotropic molecules in a planar cavity in Fig. 4~b!.Since we assumed that the geometry is symmetric—the twodielectrics are identical and distances between each surfaceand the two molecules are also identical, the quantitativetrends of enhancement or suppression are found to be iden-tical to Fig. 3~b!.
Although the boundary effect induced by two surfaces isqualitatively identical to that by a single surface, the overallmagnitude of the vdW interaction energy in a planar cavity issuppressed in comparison to those in vacuum or in the pres-ence of a single dielectric surface, and also strongly dependson the dielectric properties of the two media. In order toshow this tendency, we numerically calculate Eq.~23! whenthe intermolecular distance between two isotropic moleculesis much smaller than the distances between molecules andsurfaces—this is the short-intermolecular-distance limit in
FIG. 4. The ratios of the van der Waals interaction between two isotropicmolecules~a! or two anisotropic molecules~b! in betweentwo dielectricsurfacesto that in the free space are plotted. The ratio of the surface-plasmon frequency to the optical transition frequency,g, is assumed to be 2.ai~v!/a'~v!53. The two molecules are located exactly in between the twosurfaces so that the lateral intermolecular interaction is only considered inthis figure.
8736 M. Cho and R. J. Silbey: Enhancement of van der Waals interactions
J. Chem. Phys., Vol. 104, No. 21, 1 June 1996
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Fig. 4~a!. Here the Thomas–Fermi model@see Eq.~28!# andEq. ~29! are used for the dielectric functions and polarizabil-ities, respectively. The two dielectrics are identical. In thislimit, the first term in Eq.~23! dominates. The ratio of thevdW interaction energy in between two surfaces to that inthe vacuum space as a function ofg~5vp/&v0! is plotted inFig. 5. As the transmissivity decreases, that is,g increases,the vdW interaction in between two dielectrics decreases.For instance, the vdW interaction energy of two isotropicmolecules is about one-fifth of that in the free space, whenwe assume that the ratio~g! of the surface-plasmon fre-quency to that of the molecular transition frequency@see Eq.~31!# equals to 2. This behavior can be understood as follow-ing. When there is asinglesurface, the amplitude of the fieldmode around the surface is identical to that in the free space,though the mean square fluctuation amplitude of the electro-magnetic field in the vicinity of the surface dramatically dif-fers from that in the free space. In order to illustrate thispoint, let us consider the temperature Green functions, Eqs.~21! and ~22!, in the cases of both the free space[ e1( ik)5e3( ik)51] and the single dielectric surface@e1( ik)51 ande3( ik).1#, which are, respectively,
G0~ ik!5I23r r
r 3
and
Gs.s.~ ik!5I23r r
r 32Fe3~ ik!21
e3~ ik!11G I 823RIRI8
RI3 .
In both cases, the leading term is the usual dipole–dipoleinteraction tensor in the free space. However, in the casewhen the vacuum space is confined by two dielectric media,the intensity of the mode in the vacuum region becomesproportional to the square of the transmission coefficients,which appears in Eqs.~22! as
t122 5F 2Ae1~ ik!
e1~ ik!11G2
and t322 5F 2Ae3~ ik!
e3~ ik!11G2
,
which are always less than or equal to unity. Thus,in theshort distance limit, the magnitude of the van der Waalsinteraction can be suppressed by the two dielectric surfaceswith a large dielectric function. Here we should emphasizeagain that, because Eq.~21! was obtained in the short-distance limit, contributions from multiple reflections wereignored. If the reflectivity of the two dielectric media be-comes close to one~that is the transmissivity from the dielec-tric to the free space region is equal to zero!, one has to takeinto account the multiple reflection effects. In particular,these effects will be important when the two molecules arefar apart and the distances between molecules and surfacesare comparatively small.
VI. SUMMARY
Using the mode functions of the quantized electromag-netic field, the Green function, in the imaginary frequency, ina planar cavity was calculated in the short-distance limit. Thevan der Waals interaction between two isotropic or aniso-tropic molecules are calculated by considering the dipole–dipole interaction between two vacuum-field-induced di-poles. When a pair of molecules are in the vicinity ofsurfaces, the interference effects between the dipolar fieldand the reflected dipolar field are found to be important. Inparticular, when the two molecules are anisotropic, the sur-face can enhance or suppress the van der Waals interactiondepending on the molecular alignments. A simple image pic-ture was used to describe these effects. It was shown that, incase of two planar surfaces, the general trends are found tobe identical to the case of a single surface. However, sincethe mode intensity between two dielectric media is decreasedrelative to that near a single surface, the absolute magnitudeof the van der Waals interaction in a planar cavity is smallerthan those in the free space or near a single surface. Conse-quently, the estimate of the van der Waals interaction be-tween atoms or molecules in the confined geometry by usingthe parameters obtained from the gas-phase experiment islikely to be strongly deviated from the actual value in thiscircumstance.
ACKNOWLEDGMENT
Financial support by the National Science Foundation isgratefully acknowledged.
APPENDIX A: CALCULATION OF THEDIPOLE–DIPOLE INTERACTION TENSOR
In this appendix we derive an expression for the generaldipole–dipole interaction tensor, which is valid for any arbi-trary boundary conditions. McLone and Power32 consideredthe resonance interaction of a pair of dipoles including theretardation effect in thefreespace. We generalize their resultfor arbitrary planar boundary conditions imposed by themacroscopic dielectric surfaces. In order to obtain thedipole–dipole potential tensor we need to use second-order
FIG. 5. The ratio of the van der Waals interaction energy between twoisotropic molecules in between two dielectric surfaces to that in the freespace is plotted as a function ofg in the short-intermolecular-distance limit.The two dielectrics are identical. The dielectric functions and polarizabilitiesare assumed to be described by Eqs.~28! and ~29!, respectively.
8737M. Cho and R. J. Silbey: Enhancement of van der Waals interactions
J. Chem. Phys., Vol. 104, No. 21, 1 June 1996
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perturbation theory with respect to the original interactionHamiltonian, H I52mA•D~RA!2mB•D~RB!. From thesecond-order perturbation theory, the dipole coupling inter-action is
M5(I
^ f uH IuI &^I uH Iu i &EI2Em
, ~A1!
where the initial and final states are product states given by
u i &5ueA,gB;$0%&,~A2!u f &5ugA,eB;$0%&.
Here, for example,gA andeB denote the ground state of theA molecule and the excited state of the B molecule, respec-tively. $0% denotes the vacuum state of the field. For initialand final states given above, there are two possible interme-diate states,
uI &5ueA,eB;1k& and uI &5ugA,gB;1k&. ~A3!
The former intermediate state involves two excited states ofboth A and B molecules and one excitation of thekth field.The latter one involves two ground states of A and B mol-ecules and one excitation of thekth field. Although we con-sidered vacuum states for the both initial and final fieldstates, we found that the dipole coupling interactionM de-fined in Eq.~A1! does not depend on the initial photon num-bers as long as the final field state is identical to that of theinitial state.
Inserting the electric displacement field operator given inEq. ~8! and electric dipole operators in Eq.~7!, we find thedipole coupling interaction is
M5mA•V~kp!•mB , ~A4!
wherem is the electric dipole matrix element, and the gen-eral dipole–dipole interaction tensor can be defined as
V~kp5vp /c
Dl[12r 21l r 23
l exp~2ikd cosu!. ~B4!
By defining the Lifshitz variables as
p5cosu,
s15Ae1~v!211p2, ~B5!
s35Ae3~v!211p2,
the Fresnel reflection and transmission coefficients for thenormal and parallel transverse modes are written as,43
~i! reflection coefficients forn51 or 3,
r 2n' 52r n2
' 5p2snp1sn
,
~B6!
r 2ni
52r n2i
5enp2snenp1sn
;
~ii ! transmission coefficients forn51 or 3,
tn2' 5
2snp1sn
,
t2n' 5
2p
p1sn,
~B7!
tn2i
52Aensnenp1sn
,
t2ni
52Aenp
enp1sn.
It should be noted that the reflection coefficients are fre-quency dependent via the frequency dependent dielectricconstant,e~vk!. By expanding 1/Dl in a geometrical series, itis clear that the mode function between the two planar sur-faces includes multiple reflection contributions.
APPENDIX C: CALCULATIONS OF F(k ) AND D(ik ) INSHORT DISTANCE LIMIT
Using the mode functions, given in Appendix~B!, in theplanar microcavity, we calculateF(k) first. From the defini-tion of F(k) given in Eq. ~15! and mode function in Eq.~B1!, we find
F I~k!5F Ieven~k!1F I
odd~k!, ~C1!
where
F Ieven~k
1 to 0, represents the contribution from the propagatingmodes with real wave vector, whereas that over the pureimaginary axis corresponds to the evanescent mode contri-bution to the spatial correlation function, particularly, theimaginary part of the Green function. Detailed discussionson the integration contour has been given by Lifshiftz16 andTikochinsky and Spruch44 in the contexts of the vdW inter-actions between two macroscopic dielectric surfaces and be-tween an atom and a dielectric surface, respectively.
Although the general expressions given in Eqs.~C3! arelengthy and cumbersome, in the short distance limit, onemay find very simplified results. Here the short distance limitmeans that the distances between the two molecules as wellas between the two surfaces are much shorter than the mostimportant wavelength of the whole system. For instance thewavelength of the dominant electronic transition of mol-ecules or dielectric media is order of hundreds of nanometer.Thus if the intermolecular and intersurface distance is shorterthan this length scale, the short distance approximation canbe invoked. In the calculation of the van der Waals interac-tion defined in Eq.~14! or ~17!, the most important wavevector is approximately equal to the inverse of intermolecu-lar distance. In this high frequency region, the dielectricfunction is close to unity, so that one can make the followingapproximations:~i! sn>p for n51 and 3,~ii ! the Fresnelreflection and transmission coefficients defined in Eqs.~B6!and~B7! are independent onp~5cosu! as a consequence ofthe first approximation~i!, finally ~iii ! we ignore multiplereflection contributions, i.e., the denominatorDl>1 in Eqs.~C3!. Particularly, the last approximation can be understoodby expanding 1/Dl as a geometrical series,
1
Dl511 (
n51
`
~r 21l r 23
l !n exp~2inkd cosu!. ~C5!
Inserting this expanded form into Eq.~C3! and using theapproximations~i! and ~ii !, we may findF(k) and V(k) aregiven by sums overn. If one focuses on the terms inverselyproportional to the cubic of distance—here the distance rep-resents the shortest distance that a virtual photon may travelwith multiple reflections by the two surfaces—thenth termwith nonzeron can be shown to be approximately equal to1/(r1nd)3. Therefore, one can ignore terms withn>1. Thisapproximation means we ignore contributions from multiple~more than once! reflections in calculatingF(k), V(k), andG( ik). After some algebra, we finally find thatGI( ik),which is associated with the type I mode contributions, isgiven by
GI~ ik!5GIeven~ ik!1GI
odd~ ik!, ~C6!
where
GIeven~ ik!>
1
2 F 2Ae1~ ik!
e1~ ik!11G2H 11Fe3~ ik!21
e3~ ik!11G2J I23r r
r 3,
GIodd~ ik!>2F 2Ae1~ ik!
e1~ ik!11G2Fe3~ ik!21
e3~ ik!11G I 823RIRI8
RI3 . ~C7!
Also one can immediately find that the corresponding tensorsassociated with the type II modes are
GIIeven~ ik!>
1
2 F 2Ae3~ ik!
e3~ ik!11G2H 11Fe1~ ik!21
e1~ ik!11G2J I23r r
r 3,
GIIodd~ ik!>2F 2Ae3~ ik!
e3~ ik!11G2Fe1~ ik!21
e1~ ik!11G I 823RIIRII8
RII3 . ~C8!
Equations~C7! and~C8! are the main results, Eq.~21!, usedin Sec. III.
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