Optical properties of spatially dispersive surfaces and thin films

Thm Sohd Fdrns. 89 (1982)271-275 ELECTRONICSAND OPTICS

271

OPTICAL PROPERTIES OF SPATIALLY DISPERSIVE SURFACES AND THIN FILMS* P. H A L E V I , G.

HERN]kNDEZ-COCOLETZIAND J. A. GASPAR-ARMENTA

Instituto de Ctenctas de la UmverstdadAut6noma de Puebla, Apdo Postal J-48, Puebla, Puebla (Mexico) (Received August 3, 1981, accepted September 21, 1981)

The generalized additional boundary conditions are applied to the calculation of reflectivlty spectra R(co) m the wcinlty of an excttontc transition Results are presented for three systems: a ZnSe surface, a CdS thin film and the arrangement prism/(atr gap)/(ZnO surface). In the last case comparison is made with the attenuated total reflectivity measurements of Lagois and Fischer.

]. INTRODUCTION We are concerned with reflectivtty spectra of undoped I I - V I semiconductors in the vicinity of an isolated excitonlc transition. The interpretations of these spectra require the mtroduction of a dielectric function ~ which depends on the wavevector q as well as on the frequency o9. We shall adopt the Hopfield-Thomas model 1 for e(co, q). In additmn, we shall follow the approach of Halevi and Fuchs z and assume a linear response

D,(z) = jo~ e,(z, z')E,(z') dz'

(1)

where ~(z, z') = ~ ( z - z') + U, ~(z + z')

t = x, z

Here e(zTz') are Fourier transforms of e(co,q) with respect to the coordinate z normal to the surface; e ( z - z ' ) represents the bulk response, while e(z+z') is an approximation to the surface response. The parameters Ux and U~ account for the intensity of scattermg and the change in phase of an exciton which approaches the surface region of the crystal. If the numerical values of Ux and U~ are specified then the parallel component Dx and the normal component D. of the displacement vector are completely determined. In general Ux and Us are complex numbers whose absolute values do not exceed unity 2. Other workers 3 have adopted simdar models, although they have assumed that Ux = Uz. * Paper presented at the Fifth International Thin Films Congress, Herzha-on-Sea, Israel, September 21-25, 1981

0040-6090/82/0000-0000/$02 75

~ Elsevier Sequoia/Printed in The Netherlands

p HALEVIet al

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A n o n - l o c a l dielectric function implies that, for a given frequency, several waves m a y p r o p a g a t e simultaneously. F o r T E or s-polarized light there are two m o d e s w h o s e d i s p e r s i o n r e l a t m n s are given by e(~,),q) = cZqZ/~o z. F o r T M or p - p o l a r i z e d light there is an a d d i t i o n a l m o d e given by ~(o),q) = 0 T h e d e t e r m i n a t i o n of the a m p l i t u d e s of these waves necessitates one "'additional b o u n d a r y c o n d i t i o n " (ABC) for s p o l a r i z a t i o n a n d two A B C s for p p o l a r i z a t i o n . These A B C s are u n a m b i g u o u s l y d e t e r m i n e d by the dielectric response, n a m e l y eqn. (I). If we also assume the existence of an exclton-free surface layer ("dead layer"} of thickness l then the excltonlc p o l a r i z a t i o n P(z) m u s t satisfy the following c o n d m o n s at z = l

(I + u , ) d ~ ( I ) + t F ( 1 - U , ) P , ( I ) = O

t = ,c,z

(2)

CI.2

where ['2 =

(0 2 + IVO) - - ¢OT2 -- D q : , 2

D E q u a t m n s (21 are o u r g e n e r a h z e d ABCs. T h e y reduce to the P e k a r A B C ~'4 for U , = U_ = - 1, to the A g a r w a l et al. A B C ("dielectric a p p r o x i m a t i o n ' ) ~ for U , = U: = 0, to the F u c h s - K h e w e r A B C 6 for U , = - U _ = 1, a n d to the R i m b e y - M a h a n A B C 7 for U~ = - U. = - 1 T h e p a r a m e t e r s U, can be d e t e r m i n e d from best fits of theoretical a n d e x p e r i m e n t a l spectra, this a m o u n t s to the d e t e r m i n a t i o n of the A B C s for a given crystalline surface W e shall n o w a p p l y eqns (2) to the calculation of the reflectlvltles R(co) for three different systems. 2 a Z n S e SURFACE WITH ALI,OWANCE FOR AN EXCITON-FREE LAYER 148/~ THICK W e use the exclton p a r a m e t e r s m e a s u r e d by Hlte et al 8 (although they c a n n o t be c o n s i d e r e d to be reliable) a n d v/(ox = 10 ~ T h e p - p o l a r i z e d light is incident on the surface at 45 T h e results for three different ABCs, as well as for the local case, are s h o w n in F i g 1 It s h o u l d be n o t e d t h a t all the curves have been c a l c u l a t e d from a single f o r m u l a for Rp(o), Ux, U,_) The n u m b e r s in parentheses m the c a p t i o n are the c h o s e n values of U_, a n d U: respectively. It Is evident that the four spectra are q u a l i t a t i v e l y different, this is caused by the existence of the d e a d layer, m spite of the fact t h a t this layer is c h a r a c t e r i z e d by the dielectric c o n s t a n t ~,o which is i n d e p e n d e n t of(,) a n d q. 3. A PRISM/(AIR G A P ) / ( Z n O SURFACE)

A p - p o l a r i z e d b e a m of hght passes t h r o u g h the prism a n d is incident on the p r l s m - ( a l r gap) interface at an angle which exceeds the critical angle. The evanescent fields in the air g a p excite a surface p o l a r i t o n at the surface of the crystal. C o n s e q u e n t l y , a characteristic m i n i m u m a p p e a r s in the " a t t e n u a t e d total reflectivlty" (ATR) s p e c t r u m In F i g 2 we show A T R spectra for U: = 0 a n d 16 different values of U , The a b s o l u t e value of U:, is either 1 (elastic scattering; full curves) or 0.5 (inelastic scattering; b r o k e n curves) The angles i n d i c a t e d are the phases 4> of U,, t.e U., = ]U,] exp0qS). W e note that the p o s i t i o n of the m i n i m u m is quite insensitive to the value of U~ a n d varies between 1.0017~o x and 1 O019~,)x,

273

OPTICAL PROPERTIES OF SPATIALLY DISPERSIVE SURFACES AND FILMS

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corresponding to a difference of about 0.7 meV In contrast, the width and depth of the spectral hnes vary by as much as a factor of 4. The greatest hnew~dth ~s obtained for IUxl = 1 and ~b = r~, i.e. Ux = - 1 (this does not correspond to the Pekar ABC because U= = 0). This hnewidth is about 3 meV, We find similar results for U: = + 1 ; however, the hnewidth is, as a rule, somewhat smaller than the above value. In contrast, the llnewldth measured by Lagois and Fischer 9 ]s approximately 5 eV. (Our parameters for the C,_ 1 exc~ton of ZnO are taken from the same reference ) C o n t n b u h o n s to the hnewldth which we have not allowed for in our calculanon derive from an inhomogeneous air gap, divergence of the laser beam and a small dead layer. However, these contributions turn out to be too small to account for the discrepancy, 4

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THIN FILM WITH

ALLOWANCE

FOR EXCITON-FREE SURFACh LAYERS

The thickness of the film is 1200/k, the thickness of each dead layer is 100/~, and we have used the A,_ 1 exclton parameters measured by Yu and Evangehsti ~° by the technique of Brlllouin scattering. Normal incidence reflectiwhes for the local case (D = 0) and for five real values of U are shown in F~g. 3 These spectra are products of the interference between four plane waves which coemst m the spatially dispersive medium. They are quahtahvely dxfferent because the relanve amphtudes of these waves depend strongly on the ABCs employed Comparison with an experimental spectrum will be reported m a future pubhcanon. We also wish to note that, for other

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OPTICAL PROPERTIES OF SPATIALLY DISPERSIVE SURFACES AND FILMS

275

values of the parameters, our spectra agree with those calculated by Linton Johnson 11, although they strongly disagree with the calculated spectra of Bishop 12. ACKNOWLEDGMENTS

We wish to acknowledge very useful talks with and personal communications by R. Fuchs, B. Fischer and J. Lagols. REFERENCES

1 J J HopfieldandD G Thomas, Phy~ Rev,132(1963) 563 2 P Halevl and R Fuchs, Proc 14th Int Conf on the Physws' o f Semtconductor~, Edinburgh, 1978, in Inst Phys Conf Ser 43 (1979) 863, to be pubhshed 3 N N Akhmedlev and V V Yatslshen, Soy Phys SolldState, 18 (1976) 975 F Garc[a-Mohner and F Flores, J Phys (Paris), 38 (1977) 851 R Monreal, F Garcia-Mohner and F Flores, Solid State Commun., 32 (1979) 613

4

S I Pekar, Sov. Phys

JETP, 6(1957)785

G S Agarwal, D N Pattanayak and E Wolf, Phy~ Rev Lett, 27(1971)1022 J L B i r m a n a n d J J Sem, Phys Rev B, 6(1972)2482 A A Maradudm and D L Mdls, Phys Rev B, 7(1973)2787 6 K L K h e w e r a n d R Fuchs, Phys Rev ,172 (1968) 607 7 P R R l m b e y a n d G D Mahan, SohdStateCommun,15(1974) 35 8 G E Hlte, D T F Marple, M AvenandB. Segall, Phys Rev ,156 (1967) 850 9 J Lagols and B Fischer, SohdState Commun., 18 (1976) 1519 10 P Y Yu and F Evangehstl, Phy~ Rev Lett, 42 (1979) 1642 11 D LmtonJohnson, Phys Rev B, 18(1978) 1942 12 M F Bishop, SohdState Commun,, 20 (1976) 779 5