Effective boundary conditions in optics of thin surface overlayersEmmanuel I. Rashba Citation: Journal of Applied Physics 79, 4306 (1996); doi: 10.1063/1.361799 View online: http://dx.doi.org/10.1063/1.361799 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/79/8?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of adlayer dimer orientation on the optical anisotropy of single domain Si(001) Appl. Phys. Lett. 69, 176 (1996); 10.1063/1.117363 Optical response of Cu3Ge thin films J. Appl. Phys. 79, 8115 (1996); 10.1063/1.362370 Structural, optical, and electronic properties of magnetronsputtered platinum oxide films J. Appl. Phys. 79, 7672 (1996); 10.1063/1.362341 Surface photoabsorption study of the effect of substrate misorientation on ordering in GaInP Appl. Phys. Lett. 68, 2237 (1996); 10.1063/1.115870 Interband optical absorption in amorphous semiconductors AIP Conf. Proc. 120, 371 (1984); 10.1063/1.34766

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Effective boundary conditions in optics of thin surface overlayersEmmanuel I. Rashbaa)Department of Physics, University of Utah, Salt Lake City, Utah 84112 and L. D. Landau Institute forTheoretical Physics, Moscow 117940, Russia

~Received 30 October 1995; accepted for publication 15 January 1996!

Narrow surface overlayers contribute significantly to optical spectra. Effective boundary conditionsare derived which generalize the usual boundary conditions of crystal optics and include correctionsto them up to the order (kd)2, wherek is the wave vector, andd is the layer width. Electromagneticresponse functions of a surface layer as well as a bulk dielectric function enter as parameters intothese boundary conditions. Equations are presented which relate parameters of the reflected wavesand their angular dependences to the surface response functions. These equations indicate that thespectral shapes of those features in reflection spectra which originate from the surface absorptiondepend critically on the bulk dielectric function. The proposed formalism can be used forreconstructing surface response functions from experimental data and applied for investigatingsurface layers, surface-controlled ordering effects, etc. ©1996 American Institute of Physics.@S0021-8979~96!09008-4#

I. INTRODUCTION

Investigation of surface reconstruction of III-V semicon-ductor compounds is of great physical interestper se, and itacquired special importance after the relation between sur-face reconstruction and the growth of ordered ternary com-pounds of the Ga0.5In0.5P type has been established.1 Com-bining reflection high-energy electron-diffraction~RHEED!and optical measurements proved to be highly successful.2–4

The latter permit one to observein situ not only themolecular-beam epitaxy~MBE! growth, but also the surfacetransformations under the conditions of organometallic vaporphase epitaxy~OMVPE!. Application of optical methods isbased on the fact that the symmetry of surface overlayers,both reconstructed and nonreconstructed ones, is usuallylower than the bulk symmetry of a substrate, e.g., directions@110# and @ 110# are nonequivalent at the (001) face of aGaAs-type crystal. Different dimers typical of reconstructedsurfaces, III or V stabilized, are oriented in these directions.The width of a reconstructed overlayer~including a strainedstratum under the reconstructed one! where the symmetry islowered as compared to the bulk, is typically about 10atomic layers. A layer of such a width makes an observablecontribution, typically about 1%, into a reflection coefficient.Two optical techniques being used currently for investigationof the surface overlayers are based on reflectivity measure-ments. Reflectance-difference~RD! spectroscopy2 is basedon measuring the reflectance difference for two beams polar-ized in the@110# and@ 110# planes under the conditions of anormal incidence upon the face (001). The surface photo-absorption~SPA! technique3 measures the optical-reflectiondifference between III and V stabilized surfaces by irradiat-ing a surface by polarized light at a shallow angle of inci-dence. The latter technique also permits one to detect thosetransitions in the surface layer which are polarized along anormal, z, to the face. Surface reconstruction of InP5 andGa0.5In0.5P,

6 and near surface ordering in disorderedGa0.5In0.5P

7 were recently observed by optical techniques.

Despite the fact that RD and SPA spectroscopies arewithout any doubt highly effective diagnostic tools, no sys-tematic efforts have been made until now to find a consistentmacroscopic description of the overlayer contribution intothe reflection spectra and to reconstruct dielectric responsefunctions of the overlayers from experimental data. Withoutsuch a reconstruction even the position of surface-absorptionmaxima cannot be found reliably since the shape of the re-flection spectrum depends critically on both~i! the z depen-dence of the dielectric polarizability of the overlayer and~ii !the dielectric function of the substrate, and also on the ge-ometry of the reflection experiment. From the macroscopicpoint of view the underlying problem is a stratified mediumproblem8–11 in which the overlayer is usually absorbing, an-isotropic in thexy plane and highly inhomogeneous inzdirection, while the substrate is usually absorbing and can beboth isotropic ~binary III-V crystals! and anisotropic~or-dered ternary compounds!. Under these conditions the gen-eral equations of crystal optics are rather cumbersome. It isalso worth mentioning that the above problem differs signifi-cantly from the traditional problems of the optics of multi-layer media.9,10 The latter usually deals either with interfer-ence phenomena typical of layers having a width comparableto the wavelength~and uses a model of a pile of homoge-neous films! or with the eikonal limit. The overlayer problemcorresponds to the opposite limit when the polarizabilityshows a steep change within an overlayer, while the over-layer widthd is small as compared to the wavelength. Underthese conditions it is possible to take advantage of the exist-ence of a small parameterukud!1, whereuku is a complexwave vector, and to derive effective boundary conditions~EBCs! which reduce the three-media problem~ambient, in-homogeneous overlayer, and substrate! to a two-media one~ambient and substrate!. The derivation takes into accountthe z dependence of the dielectric response inside the over-layer and clarifies the explicit meaning and the number ofindependent parameters entering the EBCs. In general, thisnumber is larger than in the model three-media problem, i.e.,in the model with a homogeneous overlayer. Such a rigorousanalysis is necessary since for inhomogeneous thin layers ofa!Electronic mail: rashba@physics.utah.edu

4306 J. Appl. Phys. 79 (8), 15 April 1996 0021-8979/96/79(8)/4306/6/$10.00 © 1996 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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a typical width of about 10 atomic layers the notions of thelayer width, positions of the layer edges, etc., cannot be welldefined. Therefore, the equations for physical quantities~re-flectivities, phase shifts, etc.! should be expressed only interms of the quantities having an exact meaning.

We also derive a number of simple explicit expressionsfor the amplitudes and phases of the reflected waves in termsof the surface response functions and the bulk dielectricfunction. These equations can hopefully be directly appliedfor a treatment of experimental data.

II. EFFECTIVE BOUNDARY CONDITIONS

Two homogeneous half-spaces,z<j andz>h, are sepa-rated by a layer,j<z<h, where the dielectric susceptibilitytensor,k(z), depends on the coordinatez; more general non-local z dependence ofk(z) can also be considered in thesame way. Near the boundaries the tensork(z) merges con-tinuously with the dielectric functions of both subspaces. Wesuppose that the main-axes directions ofk(z) are determinedby symmetry requirements and therefore do not depend onz and that thez axis is one of the main axes, e.g., for the(001) face of a GaAs crystal the main axes are@110#,@ 110#, and@001#. A generalization of the theory for arbitrarydirections of main axes is straightforward but equations arerather cumbersome. It is the main assumption of the theorythat the layer width,d5h2j, is small as compared to thewavelength, i.e.,ukud!1, wherek is a complex wave vectorof the light wave. Light is incident from the lower half-space,z<j, having a real dielectric susceptibility tensork1 . Actually it will be taken to be vacuum, (k1)ab5dab .Here and below Greek indices take valuesx, y, andz, whileLatin indices only the valuesx andy. The dielectric tensork2 of the upper subspace~substrate!, z>h, can be complex.The matching conditions for bulk and surface polarizabilitiesare k(h)5k2 and k(j)5k1 .

Maxwellian equations of crystal optics for the fieldsEandH and the displacementD in the regionj<z<h, whenwritten in the main axes ofk(z), have the form:

ik0H5curl E, ik0D52curl H, Da~r !

5ka~r !Ea~r !, ~1!

where k05v/c. The weak effect of the gyrotropy of theoverlayer is neglected. If the coordinate dependence ofE,H, andD in x and y directions is chosen in the form of aplane wave, exp(ikir ) with ki5(kx ,ky), thez dependence offields is determined by in-plane components of equation~1!:

]zEj52 ik0~ z3H! j1 ik jEz ,

]zH j5 ik0~ z3D! j1 ik jHz .~2!

The z components of the Eq.~1! determineEz andHz :

k0Hz5~k3E!z , k0Dz52~k3H!z . ~3!

These equations permit one to eliminateEz andHz from Eq.~2!.

If one takes advantage of the existence of a small param-eter ukud!1, one can solve Eqs.~2!–~3! by iterations. Thisprocedure is more general than solving the crystal-optics

equations for the model three-media problem since it takesinto account the inhomogeneity of the overlayer. It is alsotechnically simpler than the solution of that model problem.Integration of Eq.~2! results in:

Ej~z!5Ej~h!1 ik0Ez

h

~ z3H~z8!! jdz8

1 ikjk0Ez

h dz8

kz~z8!~k3H~z8!!z , ~4!

Hj~z!5Hj~h!1 ik0(l

«z j lEz

hk l~z8!El~z8!dz8

2 ikjk0Ez

h

~k3E~z8!!zdz8, ~5!

where«abg is the Levi-Civitatensor andz is a unit vector inthe z direction. These equations are exact. Iterating them,one obtains approximate equations in powers ofk0d:

Ej~j!5Ej~h!1 ik0d~ z3H~h!! j1 ikjk0

~kH ~h!!z

3Ej

h dz

kz~z!1Ej

~2! , ~6!

Hj~j!5Hj~h!1 ik0(l

«z j lEl~h!Ej

hk l~z!dz2 i

kjd

k0

3~k3E~h!!z1Hj~2! . ~7!

Here terms linear ind are written down explicitly, whileterms quadratic ind are included inEj

(2) andHj(2)

Ej~2!52k0

2Ej~h!Ej

hdz~z2j!k j~z!1

d2

2~ z3k! j

3~k3E~h!!z1kj(lklEl~h!

3Ej

h dz

kz~z!Ez

hdz8k l~z8!, ~8!

Hj~2!52k0

2Hj~h!(l

~12d j l !Ej

hdz~h2z!k l~z!

1d2

2kjklHl~h!2~k3H~h!!z(

l«z j lkl

3Ej

hdz k l~z!E

z

h dz8

kz~z8!. ~9!

Equations~6!–~9! determine the transfer matrix relating in-plane components of the fields,$Ej ,Hj%, in the z5j andz5h planes, or, what is the same, the characteristic matrix ofthe stratified medium.9 These equations serve as EBCs whichreduce the three-media problem to a two-media one. De-pending on the polarization of the incident light, it is conve-nient to express the right-hand sides of these equations as thefunctions of eitherE(h) or H(h). Such a form of EBCs willbe used in the next sections. It is seen from equations~6!–~9!

4307J. Appl. Phys., Vol. 79, No. 8, 15 April 1996 Emmanuel I. Rashba [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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that only a single integral from each of the functionsk l(z)andkz(z) enters into thed linear terms of the EBCs, and thatintegrals fromk l(z) and kz(z) are of quite different form.One additional onefold integral from each of the functionsk l(z) appears ind2 terms. These terms also include twoiterated integrals which are not independent since there ex-ists one connection between them.

If either k(z) or k2 is complex, the overlayer contributesterms linear ind in the reflection coefficientR. On the con-trary, if both k(z) and k2 are real, the corrections toR onlyappear in the order (k0d)

2. Therefore, we consider these twocases separately in what follows.

III. ABSORBING MEDIA

If at least one of the dielectric functions,k(z,v) ork2(v), has a non-zero imaginary part in the spectral regionof interest, there exist corrections linear ind to the intensityand to the phase of the reflected light. Therefore, one canneglect in Eqs.~6! and ~7! quadratic ind terms when con-sidering absorbing media. For simplicity purposes we sup-pose that the substrate is optically isotropic,(k2(v))ab5k2(v)dab . As usual,8 we consider separatelythe components of the incident light polarized in the inci-dence plane and along the normal to it.

A. s -polarization

Electric vectorE0 of the incident wave is normal to theincidence plane which we choose as thexz-plane, hence,E05E0y. Since H5(k3E)/k0 , the fields at the frontboundary of the overlayer are equal to

Ey~j!5E01E1 , Hx~j!52~k0z /k0!~E02E1!. ~10!

HereE1 is the amplitude of the reflected wave. The fieldsEy(h) andHx(h) at the back side of the overlayer are

Ey~h!5E2 , Hx~h!52~k2z /k0!E2 , ~11!

whereE2 is the transmitted wave amplitude. Vectorsk0 andk2 are wave vectors of the incident and transmitted waves,andk0z andk2z are theirz projections. Feeding Eqs.~10! and~11! into Eqs.~6! and~7! and expressing right-hand sides interms ofE2 , one obtains

E01E15 f 1E2 , k0z~E02E1!5k2zf 2E2 , ~12!

where

f 1512 ik2zd, f 2512 i ~k02^ky&2kx

2!d/k2z

^ky&d5Ej

hky~z!dz.

~13!

It follows from Eq. ~12! that in the leading order ink0d theratio of amplitudes equals

~E1 /E0!s5~E1 /E0!0s~11 iFs!, ~14!

where

SE1

E0D0s

5k0z2k2zk0z1k2z

, Fs52k0zdk22^ky&

k221. ~15!

The relationk225k2k0

2 was used when deriving Eq.~15!.Therefore, the linear ind contribution,dRs , to the reflectioncoefficient is equal to

dRs522RsIm$Fs%, Rs5u~k0z2k2z!/~k0z1k2z!u2,~16!

whereRs is Fresnel’s reflection coefficient for ans - polar-ized wave. The real part ofFs determines the phase shift aswill be discussed in Sec. III C below.

If ky(z)5const within the layerj<z<h, Eq. ~15! fordRs reduces to the expression which can be obtained for themodel three-media problem. For normal incidence,k0z5k0 , it coincides with the expression given in Ref. 2, buthas an opposite sign.

Dielectric functionky(z) changes continuously withinthe overlayer from the valuek151 for z,j to k2 forz.h. The overlayer widthd enters into Eq.~15! explicitly.Nevertheless, neither this quantity which lacks an exact defi-nition, nor a specific choice of the positions of the layerboundaries,j and h, influence the magnitude ofdRs . In-deed,Fs can be rewritten as

Fs52k0zEj

h

~k22ky~z!!dz/~k221!

52k0zH 2j1F E2`

`

dz z dky~z!/dzG /~k221!J .~17!

In Eq. ~17! the integration by parts was performed, and inte-gration limits were extended over the regions where the in-tegrand is equal to zero. The first term in Eq.~17! makescontributions only to the real part ofFs , while the secondterm does not depend ofj andh. Therefore, Im$Fs% is welldefined: it is a functional ofky(z), but it is independent of aspecific choice of the layer boundaries,j and h, which issomewhat arbitrary. If the absorption band of the overlayerfalls into the absorption area of the substrate, Im$Fs% deter-mines completely the imaginary part of^ky(v)&, i. e., of theresponse function of the overlayer; they differ only by thereal factor (k221). To find the real part ofky(v)& whichcannot be determined fromdRs , one can apply Kramers-Kronig dispersion relations toky(v)& and find the contribu-tion of the overlayer absorption band into Re$^ky(v)&%. Ifabsorption bands of the overlayer and substrate overlap, dis-persion relations also can be applied for reconstructing^ky(v)& but the procedure becomes more involved.

The frequency dependence ofdRs /Rs is strongly influ-enced by the properties ofk2(v). If k2(v) is real, thendRs /Rs54k0zd Im$^ky(v)&%/(k2(v)21). When k2(v), 0 ~metallic reflection from the substrate!, dRs is negativeinside the peak of the function Im$^ky(v)&% describing theabsorption by the overlayer. The decrease in the reflectioncaused by presence of the overlayer results from the dissipa-tive losses in it. When (k2(v)21)&0, dRs(v)/Rs(v) re-peats approximately the shape of the absorption curve,Im$^ky(v)&%. If imaginary part ofk2(v) dominates overRe$k2(v)21%, the reflection curvedRs(v)/Rs(v) re-sembles the inverted dispersion curve of the layer,2Re$^ky(v)&%.

4308 J. Appl. Phys., Vol. 79, No. 8, 15 April 1996 Emmanuel I. Rashba [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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B. p -polarization

Magnetic vectorH0 of the incident wave is normal to theincidence planexz, hence,H05H0y; electric vectorE0 islying in the incidence plane. SinceE52(k3H)/k0 invacuum, the fields at the front boundary,z5j, are equal to

Hy~j!5H01H1 , Ex~j!5~k0z /k0!~H02H1!, ~18!

whereH1 is the amplitude of the reflected wave. In the sub-stratek lEl52(k3H) l /k0 , and the fields at the back side ofthe layer, z5h, can be expressed through the amplitudeH2 of the transmitted wave:

Hy~h!5H2 , Ex~h!5~k2z /k2k0!H2 . ~19!

Feeding of Eqs.~18! and ~19! into Eqs.~6! and ~7! and ex-pressing the right-hand sides in terms ofH2 results in:

H01H15g1H2 , k0z~H02H1!5k2zg2H2 /k2 , ~20!

where

g1512 ik2zd^kx&k2

, g2512ik2d

k2z~k0

22^kz21&kx

2!,

~21!

and

^kx&d5Ej

hkx~z!dz, ^kz

21&d5Ej

hdz/kz~z!. ~22!

It follows from Eq. ~20! that the ratio of amplitudes is equalto

~H1 /H0!p5~H1 /H0!0p~11 iFp! ~23!

in the leading order ink0d!1. Here

SH1

H0D0p

5k0zk22k2zk0zk21k2z

,

Fp52k0zd

3~k22^kx&!1~^kx&2k2

2^kz21&!kx

2/k22

~k221!@12~k211!kx2/k2

2#. ~24!

The correction linear ink0d, dRp , to the Fresnel’s reflectioncoefficientRp equals

dRp522RpIm$Fp%,

Rp5u~k0zk22k2z!/~k0zk21k2z!u2.~25!

For normal incidence,kx50, Eqs.~24!–~25! coincide withEqs.~15!–~16! ~with kx substituted byky). However, as dis-tinct fromFs the phaseFp shows a strongkx dependence. Ifk2 is real and positive,Rp turns into zero andFp divergesnear the Brewster angle; the latter is determined by the con-dition kx /k0z5Ak2. In the vicinity of this angle

dRp~v!

Rp'2kxd

k2

k221

Im$^kx~v!&1k2^kz21~v!&%

Ak22kx /k0z. ~26!

It is seen from Eq.~26! that the spectral shape of the reflec-tion spectrum in the vicinity of a surface resonance, a peak ora dip in the reflectivity, depends on the fact which of thecomponents of the tensork(z,v) shows a resonance~in-plane component or out-of-plane one!. If k2 is negative andlarge ~metallic reflection from the substrate!, the absorptionband of the layer manifests itself in a reflection spectrum as

dRp~v!/Rp}2Im$^kx~v!&~11kx2/uk2uk0

2!

2uk2u^kz21~v!&kx

2/k02%. ~27!

The shape of the feature depends on the ratiokx /k0 and onthe polarization of the transition~along x or z). WhenIm$k2(v)% has a considerable magnitude, the correctiondRp(v)/Rp acquires a contribution having a dispersion-curve shape ~cf. the previous section!. Even whenIm$k2(v)% is small, a considerable dispersion-curve shapecontribution todRp(v)/Rp should be typical of incidence inthe vicinity of the Brewster angle.

The equation for the phaseFp written in terms of de-rivatives ofkx(z) andkz

21(z) has a form

Fp52k0zH 2j1*2`

` dz z@dkx~z!/dz2~dkx~z!/dz2k22dkz

21~z!/dz!kx2/k2

2#

~k221!@12~k211!kx2/k2

2# J , ~28!

which is similar to Eq.~17!. Therefore,dRp } Im$Fp% is welldefined and does not depend of a specific choice of theboundaries,j andh.

C. Phase shift

Reflection coefficientsdRs anddRp permit one to mea-sure imaginary parts of the functionsFs andFp . Real partsof them determine surface contributions to the phase shiftsbetween reflected and incident waves. These quantities areproportional tod which is only about several atomic spac-ings and cannot be determined with an accuracy better than

one to two atomic spacings. Therefore, Re$Fs% andRe$Fp% are not well defined quantities. However, their dif-ference

Fps5Fp2Fs ~29!

is a well defined quantity since the uncertainty in the frontboundary position,j, is canceled out from it. This propertyseems obvious from physical arguments and can be verifiedby comparison of Eqs.~17! and ~28!.

If to combine Eq.~24! with the Eq.~15! written for as -polarized wave withE0ix, one gets

4309J. Appl. Phys., Vol. 79, No. 8, 15 April 1996 Emmanuel I. Rashba [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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Fps}2k0zdki2

k02 3ReH ~k211!2~^kx&1k2^kz

21&!

~k221!@12~k211!ki2/k2k0

2# J ,~30!

whereki is the in-plane projection ofk0 . The numerator ofEq. ~30! is the integral from the functionfps(z)5(k211)2@kx(z)1k2 /kz(z)# over the layer(j,h). The functionfps(z) is equal to zero both forz<j@when kx(z)5kz(z)51# and for z>h @whenkx(z)5kz(z)5k2#, hence, the integration limits can be ex-tended up to (2`,1`). Therefore, both the real and imagi-nary parts of the phase differenceFps are well defined quan-tities and do not depend ofj andh. It is remarkable that anexplicitly invariant expression for the functionFps can bewritten in the form of Eq.~30! which includes components ofthe tensork(z,v). Therefore, the transformation ofFps to aform including derivatives ofk(z,v) is not needed.

IV. TRANSPARENT MEDIA

The imaginary part of the phaseF, determining a cor-rection dR to the reflection coefficient, is non-zero in thelinear ink0d approximation only for absorbing media as it isseen from Eqs.~15! and~24!. If both the substrate and over-layer are transparent, the correctiondR has the order of mag-nitude of (k0d)

2. Since the second order equations are morecumbersome than the first order ones, we restrict ourselves tothe normal incidence. Opposite todR, the phase shift whichis determined by Re$F% remains a first order effect and canbe found from Eq.~15!, e.g., the difference in the phaseshifts of the reflected waves polarized alongx andy is equalto

Fxy52k0d^ky&2^kx&

k221. ~31!

Equations~8! and~9! add (k0d)2 corrections to Eq.~13!

for f 1 and f 2 . For normal incidence,kx50, these generalizedequations are:

f 1512 ik2d2~k0d!2^~z2j!ky&,~32!

f 2512 i ~k0d/Ak2!^ky&2~k0d!2^~h2z!ky&,

where

^zky&d25E

j

hzky~z!dz. ~33!

There exists a connection between first and second ordercoefficients in Eq.~32! which follows from the energy con-servation law in transparent media. The continuity of thePoynting vector condition is:

k0~ uE0u22uE1u2!5k2uE2u2. ~34!

When written in terms of the coefficientsf 1 and f 2 this equa-tion takes the form

12 ~ f 1f 2*1 f 1* f 2!51. ~35!

It is easy to check thatf 1 and f 2 determined by Eq.~32! obeythis condition with the accuracy up to terms of the order of(k0d)

2.

The reflection coefficient can be found from Eq.~32!quite analogously to Sec. III A, but calculations are morelaborious. For a wave withE0iy the final result is:

dRy

R52

4k0k2d2

~k221!2$~^ky&21!~k22^ky&!1~k221!

3^~j1h22z!ky&%. ~36!

Variations of the right hand side of Eq.~36! overj andh areequal to zero whenj andh are chosen in the regions whereky(z) reaches its asymptotic values,ky(j)5k1 andky(h)5k2 . Therefore, this expression does not depend ofj andh, i.e., it is well defined. Equation~36! can be trans-formed to the form in which this property can be seen ex-plicitly. By performing all integrals in Eq.~36! by parts oneobtains:

dRy

R52

4k0k2~k221!2 H ~k221!E

j

hdz z2dky~z!/dz

2S Ej

hdz z dky~z!/dzD 2J

522k0k2

~k221!2E

j

hdzE

j

hdz8~z2z8!2

dky~z!

dz

dky~z8!

dz8.

~37!

Since both limits,j and h, belong to the regions wheredky(z)/dz50, the independence ofdRy of the limits is eas-ily seen. The second part of Eq.~37! makes it also obviousthat the integral does not depend on the choice of the originsincez andz8 enter into the integrand only in a combinationz2z8. If the overlayer is homogeneous in thez direction,Eq. ~36! takes the form:

dRy

R524k0k2d

2~ky21!~k22ky!

~k221!2. ~38!

This equation can also be derived directly for the modelthree-media problem. It is a remarkable property of the exactEq. ~37! that dRy can be expressed through a single param-eter including the properties of the overlayer. However, asdistinct from Eq.~38! this parameter cannot be expressedthrough the parameter^ky& which enters into the phase shift,Eqs. ~15! and ~31!. This fact emphasizes the importance ofthe general theory for establishing the number of indepen-dent parameters of the problem.

Since terms linear ink0d cancel fromdR, surface gyrot-ropy effects may become of importance. This effect needsspecial consideration.

V. DISCUSSION

Optical methods which recently have become an impor-tant tool in investigating narrow surface overlayers are usedpredominantly for diagnostic purposes. The equations estab-lished in this article are both sufficiently general and conve-nient. Therefore, they can be applied for solving the inverseproblem, i.e., for reconstructing surface dielectric-responsefunctions from experimental data.

Effective boundary conditions derived in Sec. II reduce athree-media problem of the propagation of electromagnetic

4310 J. Appl. Phys., Vol. 79, No. 8, 15 April 1996 Emmanuel I. Rashba [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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waves to a two-media one when the width of the overlayerseparating two bulk phases is small as compared to thewavelength,k0d!1. For absorbing media EBCs includethree response functions, k j (v)& with j5x,y and^kz

21(v)&, describing dielectric susceptibility of the over-layer, and also the dielectric function of the substrate,k2(v). Explicit dependencies of EBCs on the in-plane com-ponentskj of the wave vector and the polarization of thewave follow directly from the solution of Maxwellian equa-tions inside the overlayer. EBCs allow one to find the con-tribution of the surface overlayer into the reflection spectra,in particular, as applied to the basic experimental technics~RD and SPA spectroscopies!. The equations of Sec. III re-late the dielectric response functions of the overlayer to theRD and SPA signals and can be used for the reconstruction ofresponse functions from experimental data. This reconstruc-tion would always result in a combination of the surfaceresponse functions and the bulk susceptibility which are welldefined, as it was discussed in Secs. III and IV. Dispersionrelations can be applied for getting a more complete infor-mation, cf. Sec. III A.

It was shown above that the spectral shape of those fea-tures in reflection spectra which originate from the surfacepolarizability are strongly influenced by the bulk susceptibil-ity k2(v). Depending on the relative magnitude of the realand imaginary parts ofk2(v), the sign of its real part, etc., aresonance in the surface polarizability can manifest itself inreflection spectra as a peak or a dip, or a dispersion curve, orsome superposition of them. For this reason a systematicquantitative treatment of experimental data is needed for re-constructing surface response functions from them. It is de-sirable to investigate experimentally both the reflectivity andthe phase shift of the reflected wave since they give comple-mentary information on response functions, more specific, ontheir imaginary and real parts. Finding the surface responsefunctions can hopefully become an important step in the op-tical investigation of the microscopic properties of surfacestates, e.g., some data in Ref. 12 seem indicative of the pres-ence of the Fano profiles in the reflection spectra.

It was supposed in this article that the anisotropy of op-tical spectra originates completely from the surface polariz-ability described by EBCs. Therefore, the equations of Sec.III are directly applicable to cubic crystals like Si and GaAs.Ternary compounds of the In0.5Ga0.5P type may show theanisotropy of the reflection, transmission, andphotoluminescence-excitation spectra originating fromordering.13 The effect of strains due to lattice mismatch onthe optical properties of ordered systems was alsoinvestigated.14 Both these effects are usually small in theatomic scale, and the theory of them is based on thek•papproach.15 The EBCs of Sec. II are applicable to these sys-tems, but some generalization of the equations of Sec. III isneeded. One can expect that the effects of surface and bulkmechanisms are additive in the leading approximation in theparameter of a weak anisotropy. Different approaches can be

used for separating these contributions, e.g., one can takeadvantage of the different spectral dependencies of thesecontributions. Experimental techniques employing the isola-tion of the surface contribution by comparing spectra takenfrom III and V stabilized surfaces are an invaluable tool inthe spectroscopy of surface overlayers.

After this manuscript was submitted to the Journal, twoimportant relevant papers came to my attention. Hingerlet al.16 developed a reflection theory for a three-media modelin the linear ind approximation and applied it to comparisonof the available experimental data on RD and SPA spectra ofGaAs. Aspneset al.17 succeeded in reconstructing surfacedielectric anisotropy spectra of several structures on the~001! face of GaAs using their experimental data on RDspectra.

ACKNOWLEDGMENTS

I am grateful to Professor P. C. Taylor for suggestingdiscussions, to Professor G. B. Stringfellow and Dr. J. M.Olson for making preprints of the papers—Refs. 6 and 7,respectively, available to me, and to Colin Inglefield for criti-cal reading the manuscript and bringing the Refs. 16 and 17to my attention. The support of the Office of Naval Researchunder Contract No. N000149410941 is acknowledged.

1A. Gomio, T. Suzuki, and S. Iijima, Phys. Rev. Lett.60, 2645~1988!; I. J.Murgatroyd, A. G. Norman, and G. R. Booker, J. Appl. Phys.67, 2310~1990!; G. S. Chen, D. H. Jaw, and G. B. Stringfellow,ibid. 69, 4263~1991!; A. Zunger and S. Mahajan, inHandbook of Semiconductors,2nded., edited by S. Mahajan~Elsevier, Amsterdam, 1994!, Vol. 3, p. 1399,and references therein.

2D. E. Aspnes, J. P. Harbinson, A. A. Studna, and L. T. Florez, Phys. Rev.Lett. 59, 1687~1987!.

3N. Kobayashi and Y. Horikoshi, Jpn. J. Appl. Phys.28, L1880 ~1989!.4A. Gomio, K. Makita, I. Hino, and T. Suzuki, Phys. Rev. Lett.72, 673~1994!.

5Y. Kobayashi and N. Kobayashi,Proceedings of the 7th InternationalConference on InP and Related Materials~The Institute of Electrical andElectronics Engineers, New York, 1995!, p. 225.

6H. Murata, I. H. Ho, T. H. Hsu, and G. B. Stringfellow, J. Vac. Sci.Technol. B13, 1755~1995!.

7J. S. Luo, J. M. Olson, and M.-C. Wu, Appl. Phys. Lett.67, 3747~1995!.8J. C. Stratton,Electromagnetic Theory~McGraw-Hill, New York, 1941!.9M. Born and E. Wolf,Principles of Optics~Pergamon, Oxford, 1970!.10Z. Knittl, Optics of Thin Films~Wiley, London, 1976!.11Thin Films in Optical Systems, edited by F. R. Flory~Dekker, New York,1995!.

12Y. Yamauchi, K. Uvai, and N. Kobayashi, Jpn. J. Appl. Phys.,32, 3363~1993!.

13M. C. Delong, D. J. Mowbray, R. A. Hogg, M. S. Skolnick, M. Hopkin-son, J. R. David, P. C. Taylor, S. R. Kurtz, and J. M. Olson, J. Appl. Phys.73, 5163 ~1993!; J. S. Luo, J. M. Olson, S. R. Kurtz, D. J. Arent, K. A.Bertness, M. E. Raikh, and E. V. Tsiper, Phys. Rev. B51, 7603 ~1995!;S.-H. Wei and A. Zunger,ibid. 51, 14 110~1995!; M. Schubert, B. Rhei-nlander, and V. Gottschalch, Solid State Commun.95, 723 ~1995!, andreferences therein.

14Y. Ueno, Appl. Phys. Lett.62, 553~1993!; E. P. O’Reilly and A. T. Meney,Phys. Rev. B51, 7566~1995!, and references therein.

15G. L. Bir and G. E. Pikus,Symmetry and Strain-Induced Effects in Semi-conductors~Wiley, New York, 1974!.

16K. Hingerl, D. E. Aspnes, and I. Kamiya, Surf. Sci.287/288, 686 ~1993!.17D. E. Aspnes, Y. C. Chang, A. A. Studna, L. T. Florez, H. H. Farrell, andJ. P. Harbinson, Phys. Rev. Lett.64, 192 ~1990!.

4311J. Appl. Phys., Vol. 79, No. 8, 15 April 1996 Emmanuel I. Rashba [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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