The effect of gross roughness on ellipsometry

J. Electroanal. Chem., 150 (1983) 277-290 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

277

THE EFFECT OF GROSS R O U G H N E S S ON ELLIPSOMETRY

TENNYSON SMITH Rockwell International Science Center, Thousand Oaks, CA 91360 (U.S.A.) (Received 27th July 1982)

ABSTRACT Ellipsometry and eUipsometric spectroscopy are powerful tools for investigating the molecular structure of electrode-electrolyte interfaces. However, a serious drawback is that exact relationships between measured ellipsometric parameters and the optical properties of a surface have only been developed for ideal perfectly smooth surfaces. Although accurate ellipsometric data can be obtained with grossly rough surfaces, theoretical interpretation has not yet been developed. This paper considers the effect of shadowing on ellipsometry, due to roughness of pits, ridges and bumps. These three types represent roughness for most real surfaces. For example, etched surfaces are represented by the pit model, rolled, abraded surfaces are represented by the ridge model and nucleation and growth on surfaces are represented by the bump model. Experimental results for surfaces prepared in various ways, to simulate these models, justify the theoretical predictions of shadowing. Four theoretical approaches, the Kirchoff scalar diffraction (KSD) theory, the vector-perturbation (VP) theory, the amplitude scatter factor (ASF) theory and the effective film (EF) theory were tested with ellipsometric results from real surfaces. It was discovered that no theory (e.g. KSD, VP) that multiplies or adds to the total reflection coefficient for a smooth surface, can correct for gross roughness, whereas a phenomenological theory, e.g. ASF, with shadowing, can.

INTRODUCTION

In the past, ellipsometric data have often been misinterpreted because the available computer programs consider only film thickness and optical properties, and neglect surface roughness. This has been a natural result because exact equations relate A and ~p to film properties only if the substrate is perfectly smooth. The tendency has been to interpret transparent films on rough surfaces as absorbing films on smooth surfaces. No exact relationships have been developed to account for gross roughness. However, no substrate is perfectly smooth, and most of the ellipsometric data reported so far have been for substrates of sufficient roughness to affect A and ~b significantly (particularly qJ). Although theoretical corrections have been derived [1-4] for very slight roughness, this paper is primarily concerned with gross roughness. The need for empirical relationships to account for roughness is great because precise ellipsometric data can be obtained for rough as well as smooth surfaces. Additionally, much useful physical information can be obtained, other than that concerning roughness itself. 0022-0728/83/$03.00

© 1983 Elsevier Sequoia S.A.

278 THEORIES FOR R O U G H N E S S CORRECTION

A correction factor, a, has been derived by various authors (see Table 1) to correct ellipsometric calculations for the effect of surface roughness. These correction factors range all the way from very complex fundamental derivations (e.g. O - L and C - Z ) to more phenomenological derivations (e.g. B - J and F - M ) . The T-S theory, outlined here, has been developed for macroscopic roughness and requires one of the other correction factors for microscopic roughness. However, if macroscopic roughness exists, it is necessary to account for its effect prior to correction for microscopic roughness. It is not necessary to know the theoretical expressions for a and their associated parameters in order to discover whether or not the correction will be useful. It is only necessary to know the relationship between a, the total reflection coefficients and their ratio O, as indicated in Table 1. The approach for testing the theories is twofold: first, calculations of A and ~p are made as a function of h s, h f, d, O and a to see the trends to expect from a; second, experimental data from rough surfaces are used to calculate d, ~ f and a by the McCrackin [8] computer program that has been modified by the introduction of a. Here, t~s and /lf a r e the complex index of refraction for the substrate and film respectively, d is the film thickness, and 0 the angle of incidence. Since a enters the equation for p (--- tan ~p exp(iA)) in a different way for each theory, it is possible to check the utility of one theory with respect to another. For example, in the O - L theory, a p and a s are added to R p and R s values for a smooth surface; in the C - Z theory, R P / R s is multiplied by a; and for the B - J and F - M theories, a is within the expressions for R. For details, refer to ref. 7.

TABLE 1 Ellipsometry roughness correction-theories Model

Reference

Relationship of correction factor a to R (p ~- t a n ~ exp(iA)--= f(R, a)

K.irchhoff scalar diffraction Vector-perturbation

Ohlidal and Luk~s O - L theory [4] Church and Zavada, C - Z theory [1,3] Bradshaw and Jerrard, B - J theory [5] Fenstermaker and McCrackin, F - M theory [6] Tennyson Smith T-S theory [7]

( R p 4- Otp)//( R s + a 2)

Amplitude-scattering Film-model

Shadow-model

(RP/RS)a R = f(a) R = f(~)

Corrects for heterogeneous surface

279

T-S theory Shadowing All of the theories for roughness have concerned the effect of light scattering by very limited roughness (rms deviation from a mean plane o < h, where h is the light wavelength), i.e. shadowing has been neglected. The shadowing effect (T-S model) can generally be divided into three regimes--ridges, pits and bumps. Ridges are generally formed by machine or rolling grooves, pits are formed by etching and bumps are formed by nucleation and growth of deposits. One might expect that in many cases the tops of the ridges, pits or bumps would have different optical properties (i.e. N1, dl, nfl and eq of type 1) than would the bottoms or valleys (i.e. N 2, d 2, n f2 and a 2 of type 2). The property N is the number of facets in area 1 or 2 that are in the mean plane of the surface and have proper orientation (i.e. yield the proper angle of incidence), d is the film thickness and n f the film index of refraction (maybe complex). As a first approximation, we assume that N~ = N 2 , n f l = nf2 , but a 1 ~ ot2 and d 1 * d 2. Consequently, the average properties are expected to change with angle of incidence as the valleys are shadowed out. Figure la illustrates roughness on two scales. On the large scale, o and T are larger than the light wavelength; on the small scale, o and T are smaller than the light wavelength. The small-scale roughness factor, a I, can be different on the top surface than the roughness factor a 2 in the valleys. This is expected for real surfaces because the tops and valleys would not exist if the chemistry were not different in these two regions. Figure,1 a has facets, some of which provide the proper angle of incidence in all regions. Any light that does not strike one of these facets will be reflected away from the ellipsometer detector. Consequently, contrary to reflectivity measurements, ellipsometric measurements selectively monitor only that set of facets that provides the proper angle of incidence. The light received by the detector can be divided into the fraction fl that comes from the top with scatter factor otI and the fraction f2 that comes from the valleys with a2, i.e. an effective scatter factor is

a =fla, +fzaz

(1)

where fl + fz = 1

(2)

Therefore, a = a 1+f2(a2-al)

(3)

and, similarly, the effective film thickness is d = d, + f z ( d 2 - d,)

(4)

For 0 < 01, no shadowing occurs and f2 =f2,1 where f2,1 is the fraction of the surface that has ridges, pits or bumps. For 0 > 01, shadowing does occur, and f2 is a function

280

al 02 " ~ ' ° 1

a2

I

'

I

--I

7

al

,

---

c

2T

(a)

,/ *dy

÷

I

x-~

~f

tx

/ n

/

2 T + 2C

(b)

Fig. 1. Schematic representation of a ridge on an etch pit: (a) cross-section; (b) plan view for circular pits.

of the angle of incidence and the roughness geometry. For O > 02, shadowing is complete and only reflections from the top surface are detected.

Ridges Considering the tops in Fig. 1a to be the tops of ridges that run perpendicular to the page, with scattering factor a l, and the valleys to have scattering factor a 2, the top area unit width A 1 (A 1 = 2C) is independent of/9, whereas the valley area per unit width, A 2, that is not shadowed, is dependent on 8 by the equation A 2 = ( 2 T - 2t) = ( 2 T - 2o tan 0)

(5)

Here, t is doubled to account for shadowing on one side of the ridge and backscattering on the other. That is, as the light beam approaches the right-hand ridge at t, it will be scattered back as for beam 3 in Fig. la. The fraction f 2 = A 2 / ( A l + A2), i.e. dividing .4 2 and (A 1 + Az) by 2T, f2 = (1 - ( o / T ) tan 0 ) / [ 1 + ( C / T ) - ( o / T ) tan 0]

(6)

281

Pits Consider the surface to be an array of areas 4 ( T + C) 2 c o m p o s e d of one circular pit of depth, 0, and diameter 2 T (see Fig. lb). The flat top region A 1 with al is independent of 0,

(7)

A 1 = 4 ( T + C ) 2 - ~rT 2 T h e unshadowed area d a y of width d y is

(8)

dAy= 2 ( B - t ) d y

where t is a function of o and 0, i.e. t = o tan 0, B is a function of y and the pit radius T ( y = v @ 5 - B 2 ) , and d y = - B d B / f T 2 - B 2. The total u n s h a d o w e d area A 2 = 2Ay is

2A =4 f [-Bt Z;-

(9)

and the unshadowed fraction f2 = A2/(A 1 + A2). Let f3={sin

l(1)-sin-'[(o/T)tanO]-(o/T)tanO~l-(o/T)

2 tan20}

(10)

then, dividing A 1 and A 2 by 2 T 2, f2 becomes f2 = f 3 / ( [ 2 - (¢r/2) + (4C/T)] +f3)

(11)

Bumps

Considd: surface roughness on two levels, gross roughness as protrusions or b u m p s with al on a flat with ~x2. Figure 2a shows the plan view, and Fig. 2b, the cross-section of the model. The b u m p s are p y r a m i d s of base T, height o, separated b y distance C. As for pits, for 0 < 01, the scatter factor remains constant, i.e. the sum of the fraction fl (with a l ) for the p y r a m i d s and the fraction f2 (with % ) for the flat area is unity. For 0 < 0 1 (where 01 = t a n - l ( T / 2 o ) ) , no shadowing occurs. For 0 > 01, shadowing will occur over the entire projected area of one side of the p y r a m i d (cross-hatched area in Fig. 2a) for area 1 (i.e. for fl and al), and shadowing will occur over region 2 (dotted area) for area 2. F r o m Fig. 2a, the area of type 1 that is not shadowed is A 1 = T 2 / 2 and that of type 2 is

A 2 = (C + T) z - T 2 - T(o tan 0 - T/2) since t = a tan 0 - T/2. The fraction f2 = Az/(A1 (4) for a and d is therefore

[ 2 ( C / T ) + ( C / T ) 2 + 1 / 2 - ( o / T ) tan 0]

f2

(12) + A2) to

be used in eqns. (3) and

(13)

[1 + 2 ( C / T ) + ( C / T ) 2 - ( o / T ) tan 0]

In each case, ridges, pits or bumps, the u n k n o w n coefficients are a 1, a 2, C / T and o / T . These coefficients are c o m p u t e d from a set of values of a vs. 0 by using an

282

(b)

/..,',

.1..\

~:.:'...

I..."

;,'.%

~.:.~ \.~

...../ :.

L__, (a) Fig. 2. (a) Plan view of pyramidal bumps; (b) side view of pyramidal bumps.

IMSL library routine that finds the minimum of a function of n variables with a quasi-Newton method. TEST OF ROUGHNESS THEORIES

The author has prepared surfaces [9] that represent the ridge, pit and bump models. Electron microscope scans of these surfaces can be seen in Figs. 3 and 4. Figure 3a shows a polished surface of copper, and Fig. 3b is for a copper surface that had been sanded with 600 grit carbide paper to represent ridges. Figure 4 shows low (left side) and high (right side) magnification SEM pictures of aluminium after various exposure to a sulfuric acid-dichromate etch. The low-magnification pictures represent pits with roughness that is large with respect to the light wavelength, and the high-magnification pictures represent roughness (within the pits) on a scale that is small with respect to the light wavelength. Figures 3c and 3d show a low and high magnification of vapor-deposited aluminium that represents bumps. In order to test the models, the McCrackin program [8] was modified such that iterations were made on a rather than n f to find solutions of d and n f that satisfy the criteria of convergence with the experimental values of A and ~p.

283 ~c8}:! ~49i

(b) Cu

~3

x

4000

(d) AI Fig. 3. (a) SEM pictures of polished copper; (b) sanded copper; (c, d) vapor-deposited aluminium.

Ridges (600 grit-sanded Cu) F o r p o l i s h e d copper, calculations of film thickness a n d c o m p l e x refractive i n d e x yield average values of d = 10.5 nm, n~ --- 2.83 (1 - i 0.29). These calculations were m a d e with no roughness corrections, a n d with i t e r a t i o n on the real p a r t of /~f (i.e.

284

X500

ELECTROPOL] SH

(E.P.)

X10,000

a

4

~4 ~ ,~

n~ .~.

X500

E.P.

+ 13 min FPL e t c h

XIO.O00

X500

E.P. + 60 min FPL etch

XIO,O00

C

Fig. 4. SEM photographs of A12024-T3: (a) electropolished, EP; (b) EP + 13 min FPL etch; (c) E.P. + 60 rain FPL etch.

n f). The values of n f and Kf are close to those reported in the literature [10-12] for oxides of copper (i.e. nf ~ 2.8 and rf = 0.3). The gross roughening of the sanded copper dramatically increased ~. Calculations of d and #lf with no roughness corrections yields unrealistic values (i.e. d - 68.8 nm, n f - 1.7, E f - 0). Both the B - J and the C - Z theories yield reasonable results for sanded copper, if the film is considered to be absorbing, as experimentally observed for smooth

285

copper. The film thickness is estimated to be between 4 nm and 7 nm with n f 2.8 and xf - 0.3-0.4. With the values of a calculated with the B - J roughness correction, n r - 2.8 and xf nearest to 0.3 (i.e. 0.4), the best-fit values of a], %, o/T and C/T prove to be 1.03, 1.28, 0.38 and 0.32 respectively. Figure 5 shows a plot of a calculated from eqns. (3) and (6), and these values, along with the values of a, are used to obtain c%, %, o / T and C/T. Both a and d increase with O, and Fig. 6 shows that for copper, c~ increases approximately linearly with d. The equations for a and d, the ridge model, - -

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,~

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,~

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H

II

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H

0.72 <

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H

THEOR RIDGES

Q7

_ EXPTL RIDGES

•

0_64

• = THEORRIDGES H = EXPTL. RIDGES

= SMOOTH 0.6

40

oo 20 ' 3'0 ' - 4 b ' 5 0 FILM THICKNESS

50 60 70 80 90 ANGLE OF INCIDENCE

70 (A)

Fig. 5. Alpha vs. theta, Cu, 600 grit sand. Fig. 6. Alpha vs. film thickness, Cu, 600 grit sand. ,

160

,

,

-~- . . . . • = THEOR. RIDGES H = EXPTL. RIDGES

,

,

~

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o

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~

i

,

i

,

i

I

i

,

i

50 60 70 80 90 ANGLE OF INCIDENCE

Fig. 7. Delta vs. theta, Cu, 600 grit sand. Fig. 8. Psi vs. theta for Cu, 600 grit sand.

~

42

38 i

40

~

i

40

, ~ h ,

i ,

50 60 70 80 90 ANGLE OF INCIDENCE

80

286

are a = 1.03 + f 2 ( 1 . 2 8 - 1 . 0 3 )

(14)

d = 74 + f 2 ( 2 8 - 74)

(15)

where f2 is obtained from eqn. (6) with o / T ~ 0.30 and C / T - 0.32. Figures 7 and 8 are plots of the experimental values of A and ~b for the smooth and sanded copper respectively. The curves are theoretical with values of a and d calculated from eqns. (13) and (14). Pits (etched AI)

Experimental values of A and ~p vs. 8 for s m o o t h (electropolished) and acid-etched aluminium alloy (A17075-T6) correspond to SEM pictures representing progressive etch time in Fig. 4. The smooth aluminium yields values of d - 8.3 nm, n f--- 1.6, xf = 0.12, at 8 = 80 ° only. The literature values [13,14] are rtf = 1.6, xf - 0. With the B - J roughness correction, a proves to be near unity, consistent with an 8.5 n m transparent oxide film on a s m o o t h substrate. After 5 min acid etch, A and ~p have b o t h dropped. Ellipsometric calculations without a roughness correction yield d - 30 nm, n e - 1.6, tee-0.2, i.e. an absorbing film. With the B - J roughness correction, the film thickness d - 32 nm, h e - 1.6, xf - 0 and a - 0.8. After a 10 and 40 min acid etch, the film thickness is about the same, and the roughness factor a has decreased to about 0.76. With the C - Z roughness correction, the 10 rain etch yields d - 4 0 - 1 0 0 nm, n r - 1.2-1.6 and a - 0.85 for/£f = 0. Figure 9 is a plot of a vs. 9 for smooth and 10 min etched aluminium. The solid line through the experimental points for the etched aluminium was calculated with the best-fit values of a I = 0.74, a 2 = 0.84, a / T = 0.25 and C / T = 0.11. Figure 10 is

< T ~

1,1

•

= o'/T

31 .°c/T

1.0

,

A,---..@

U

.A

et ~ O.g

I

u./

11

zn - o.8 O

~0.7 H = ALPHA = ALPHA

O.E

;

0

'

6 'o ANGLE

ROUGH SMOOTH

. . 7 0. . .

80

90

OF iNCIDENCE

,

1/

o

i

~

i

30

i

ETCH TIME ! rain

Fig. 9. Roughness factor vs. theta AI7075-T6, etched and polished. Fig. 10. Roughness parameters vs. etch time for A1.

i

2(3

i

40

287

M=

160''

48

12(

PSI

= PSI

ROUGH SMOOTH

44 e

40

40 3E

~: i

50

DELTA SMOOTH i

i

60

i

i

70

\ i

i

l

BO

~"

90

ANGLE OF INCIDENCE

' 70

50

'sb

90

ANGLE OF INCIDENCE

Fig. 11. Delta vs. angle of incidence for etched AI. Fig. 12. Psi vs. theta for A17075-T6 etched and polished, curves.

a plot of o/T, C / T and o / C as a function of etch time. The small change of o / T and much larger changes of C / T indicate that the depth of the pits does not increase much, but the width and number of pits increase with etch time after the first 5 min. This appears to be the trend in the SEM pictures of Fig. 4. Figures 11 and 12 plot the experimental A and ~kvalues vs. 0; the curves are theoretical with the B - J correction.

Bumps (vapor-deposited A1/ Si) Smooth and rough aluminium, vapor deposited on smooth silicon substrates, represent the smooth and b u m p model (see Fig. 3c, d). With no roughness corrections, the smooth aluminium yields d - 5 nm, n f - 1.6, /£f- 0.02. With the B - J correction, d - 5 nm and a - 1 as it should be. With the bumps, and no correction, the film d - 5 0 - 1 0 0 nm, n f - 1-1.9 and K f - 0-0.1. With the B - J correction, the oxide film varies from d - 3 4 . 2 to 89.4 nm, and a - 1.09 to 0.46. With the C - Z correction, d - 17.1-38.1 nm, nf - 4 and a - 0.4; i.e. the refractive index of the film is not reasonable. Figure 13 is a plot of experimental and theoretical a vs. 0 for aluminium bumps. The theoretical curve was calculated from best-fit values of a I = 0.44, ct2 = 1.59, o/T= 0.29, C/T= 0.22. The value of o/T-0.3 is reasonable, but cannot be checked with the plan-view SEM picture (Fig. 3c, d). However, an estimate (from Fig. 3d) of the fraction of the area covered with b u m p s (i.e. (2T)2/(2T+ 2 C ) 2) of about 0.6, yielding C / T - 0.5, is in fair agreement with 0.22 from the ellipsometric results. Figure 14 is a plot of a vs. film thickness, as calculated with the B - J corrections. The theoretical curve is calculated with the best-fit values of d] = 62.8 nm and

288 1.2

1.2

,

,

,

,

,

,

.

.

.

= T H E O R . , BUMPS NTAL

.

.

.

i

,=THEOR. H=EXPTL.

1.0

1.OF

<~ 0.8

0.8F

I (3_

0.6

0.61"

0.4

~

0.41-

i

,

40

i

,

t

i

i

i

i

~)

~

50 60 70 80 90 A N G L E OF INCIDENCE

i

i

i

i

i

,

i

i

,

,

i

i

40 50 60 70 IbO 9 0 FILM THICKNESS / m m

F i g . 13. A l p h a vs. theta, A1, bumps. F i g . 14. A l p h a vs. film thickness, A1, b u m p s .

d 2 = 14.4 nm, o / T = 0.29, C / T = 0.22, from eqns. (4) and (12). From this analysis, the oxide film between the bumps is about 62.8 nm and on the bumps about 14.4 nm. It should be noted that the relationship between a and d in Fig. 14 is completely different from that between a and d in Fig. 6, indicating their independence. This favors, but does not prove, the general form of the model. Finally, Figs. 15 and 16 are experimental and theoretical plots of A and ~p vs. 0. The theoretical curves are based on the values for al, a2, o/T, C/T, d~ and d 2 mentioned above. The theoretical curves do not predict the experimental points of q~ beyond 0 = 75 ° due to the inability of eqn. (4) to predict the film thickness beyond 75 o.

50

160

,

• = THEOR.. BUMPS RIMENTAL

~

•

, =

i

r

i

,

,

,

,

H

PI

,

T H E O R I . B U M P S

H = EXPERIMENTAL 48

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4O

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4C i

40

i

5;

,

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6C) 70 86 ANGLE OF INCIDENCE

i

90

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'

5 0'

'6"0 ' 70 ' 8"0' 9 0' ANGLE o F INCIDENCE

F i g . 15. D e l t a vs. theta, A1, n s = 1.43, x s = 5.17, n f = 1.6, w a v e l e n g t h = 6 3 2 8 . 2 nm. F i g . 16. Psi vs. theta, A1, n s = 1.43, r s = 5.17, n f = 1.6, w a v e l e n g t h = 6 3 2 8 . 2 nm.

289 CONCLUSIONS

Although roughness greatly complicates the interpretation of ellipsometry, and no precise theoretical corrections have been derived for gross roughness, approximate theories and empirical approaches can be used to determine accurately the thickness of films on grossly rough surfaces. Ellipsometry, with the roughness corrections, is extremely useful (especially in conjunction with other surface tools) for elucidating surface properties as they change during chemical and physical processes of industrial importance. The best approximation for the interpretation of ellipsometric data for grossly rough surfaces has been developed by Bradshaw and Jerrard [5]. With their scattering factor and a shadowing factor developed in this report by the author, most film thicknesses can be determined; this has been verified by experimental data for the various types of roughness, ridges, pits and bumps. There are numerous other papers [15-22] concerning the effect of roughness on the interaction of light with solid substrates. Most of these papers are difficult to interpret with respect to ellipsometry. There is one paper that is of particular interest, Aspnes and Theeten's [22] treatment of the rough surface as an "equivalent film". Performing ellipsometric spectroscopy enables the authors to choose between various models and reveal details about the equivalent film that is not possible wikh monochromatic light. ACKNOWLEDGEMENT

Much of this work was supported by Rockwell International Science Center IR&D funds. G.W. Lindberg has been a great help with the experimental research. REFERENCES 1 E.L. Church and J.M. Zavada, Effects of surface microroughness in ellipsometry, Optical Society of America Annual Meeting, Oct. 1976. 2 G.R. Valenzuela, Proc. I.E.E.E., 58 (1970) 1279. 3 E.L. Church and J.M. Zavada, J. Opt. Soc. Am., 64 (1974) 547A;~Appl. Opt., 14 (1975) 1788. 4 I. Ohlidal and F. Luk~s, Opt. Acta, 19 (1975) 817; Opt. Commun., 7 (1973) 76. 5 D.J. Bradshaw and H.G. Jerrard, Surf. Sci., submitted for publication. 6 C.A. Fenstermaker and F.L. McCrackin, Surf. Sci., 16 (1969) 85. 7 T. Smith in W.E.J. Neal (Ed.), Ellipsometry: Theory and Applications, Plenum, New York, to be published. 8 F.L. McCrackin, Natl. Bur. Stand. Tech. Note 479 (April 1969). 9 T. Smith, Surf. Sci., 56 (1976) 252; Surf. Tech., 8 (1979) 1. 10 T. Smith, J. Opt. Soc. Am., 67 (1977) 48. 11 P.C.S. Hayfield and G.W.T. White in E. Passaglia, R.R. Stromberg and J. Kruger (Eds.), Ellipsometry in the Measurement of Surfaces and Thin Films, N.B.S., Washington, DC, 256, 1963, pp. 44 and 173. 12 R.H. Muller and C.G. Smith, Surf. Sci., 56 (1976) 440. 13 W.E.J. Neal, Surf. Sci., 6 (1977) 81. 14 I.H. Malitson, F.V. Murphy, Jr. and W.S. Rodney, J. Opt. Soc. Am., 48 (1958) 72; M.A. Barrett and A.B. Winterbottom, First International Congress on Metallic Corrosion, Butterworths, London, 1962, p. 657.

290 15 16 17 18 19 20 21 22

A.A. Maradudin and D.L. Mills, Phys. Rev. (B), 11 (1975) 1392. D. Beaglehole and O. Hunderi, Phys. Rev. (B), 2 (1970) 309. E. Kretschmann and E. Kroger, J. Opt. Soc. Am., 65 (1974) 150; also Z. Phys. 237 (1970) D.W. Berreman, J. Opt. Soc. Am., 60 (1970) 499. J. Bodesheim and A. Otto, Surf. Sci., 45 (1974) 441. A.F. Houchen and R.G. Hering, Prog. Astronaut. Aeronaut., 20 (1967) 65. R.C. Birkebak and E.R.G. Eckart, J. Heat. Transfer, (Feb. 1965)185. D.E. Aspnes and J.B. Theeten, Phys. Rev. (B), 20 (1979) 3292.