THE INFLUENCE OF SAMPLE STRUCTURE ON THE OPTICAL PROPERTIES OF SOLIDS

(1)

HAL Id: jpa-00217157

https://hal.archives-ouvertes.fr/jpa-00217157

Submitted on 1 Jan 1977

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THE INFLUENCE OF SAMPLE STRUCTURE ON

THE OPTICAL PROPERTIES OF SOLIDS

O. Hunderi

To cite this version:

(2)

JOURNAL DE PHYSIQUE Colloque C5, supplkment au no 1 1 , Tome 38, Nocembre 1977, page C5-89

THE INFLUENCE OF SAMPLE STRUCTURE

ON THE OPTICAL PROPERTIES OF SOLIDS

0. HUNDERI

Department of Physical Metallurgy, Norwegian Institute of Technology, 7034 Trondheim-NTH, Norway

Rkumb. - On discute l'influence de la structure en surface et en volume sur les propriktks optiques des solides. On considbre essentiellement les khantillons m&talliques, et plus particulibre- ment les mkanismes d'absorption additionnelle introduits par I'excitation des plasmons de surface puisque leur effet sur les spectres optiques est le plus prononc6.

Abstract.

-

We discuss the influence of surface and internal structure on the optical properties of solids. The emphasis is on metal samples and particularly on additional absorption mechanisms caused by the excitation of surface plasmons since these have the most pronounced effect on the optical spectra.

1. Introduction.

-

When an electromagnetic wave is incident upon a plane interface between two homogeneous media, it is reflected according to well known laws : The reflection coefficients are given by Fresnels equations and depend upon the angle of incidence of light upon the sample and upon the dielectric properties of the two adjoining media. In fact, the laws of reflection at a plane boundary are so well understood that the dielectric properties (optical constants) of materials are often determined by measuring its reflection coefficients. In this paper we will discuss what happens if the boundary between the two media is not plane, but rough, or i the sample is strongly inhomogenous.

The most striking difference between a smooth homogenous surface and a rough surface

-

or a flat surface of an inhomogenous material, is that the former only reflects an incident wave in the specular direction, while the latter will scatter light into various directions. This property is generally taken as a definition of a rough surface : It is a surface which will scatter energy from an incident plane wave into various directions, while a sur-ace which reflects only in the specular direction will be called smooth and homogeneous. The problem of rough surfaces has been discussed by physisists for at least 75 years and has become of particular interest in recent years for a number of reasons : When the dielectric properties of substances are determined by measuring reflection coefficients one always assumes that the surface under investigation is smooth and homogeneous. This, however, is not always so, and it is therefore of interest to be able to estimate the effect of sur.ace roughness and bulk inhomogeneities on the measured parameters.

Surface film formation during corrosion is often

followed by ellipsometry. When the surface under investigation is polycrystalline surface film formation

is

generally accompanied by an increase in surface roughness since high and low index faces do not show the same corrosion rate. This surface roughening can contribute significantly to the change in ellipsometer readings and, if not accounted for properly, it will introduce large errors in the corrosion film thickness determination.

Many optical experiments require high quality optical components. The quality of these components may be severly degraded by surface roughness, not only will they scatter light into various directions; but they will also affect the degree of coherence of the specular beam. It is therefore important to develop methods whereby the quality of optical surfaces can be tested routinely and also to understand how the remaining roughness may affect the experiment.

Under the definition of surface roughness given above, the same surface may be rough for some wavelengths and smooth for others. For this reason the selectivity of materials, when used as solar energy convertors, is improved when the absorbing surface is made rough and the roughness is properly chosen. The solar spectrum peaks at about 0.5 ym while the thermal reradiation has its maximum in the 8-10 ym range for practical solar energy convectors. It is possible to choose the roughness so that the surface looks relatively rough at 0.5 pm while it looks rather smooth at 10 ym. Surface roughness generally increa- ses the optical absorption, hence the solar energy absorption by such surfaces is increased while the thermal radiation is essentially unaffected.

Ultrafine metal particles are also of interest as solar energy absorbers, we will therefore briefly mention the optical properties of compositionally inhomogeneous

(3)

C5-90 0. HUNDERI

materials. Here one should also point out that many corrosion products are deposited as particles or as inhomogeneous films (e. g. rust).

In the following section we will present a number of experimental results related to what has been discussed above. In section 3 we will present a number of theore- tical models, discuss their merits and their disadvan- tages, and finally fit some of these models to experimental data.

2. Experimental resuIts. - As indicated above, we may divide sample imperfections into two general categories : volume effects or internal roughness and surface effects. Volume effects may arise either from impurities in the sample, or from structural inhomogeneities or imperfections. The former will not be discussed here. Surface effects arise primarily from surface irregularities, either in the sample itself or in a surface film.

2.1 VOLUME EFFECTS - INTERNAL ROUGHNESS.

-

When studying the optical properties of materials by means of thin evaporated films it is often necessary to cool the substrate, if not very rough sample surfaces are obtained. This is the case for the alkali metals, for lead and for indium amongst others. But cooling the substrate is in principle the way of making an amorphous film. The optical properties of the amorphous state is, as is well known, quite different from that of the corresponding crystalline material, but as the true amorphous state is a homogeneous phase, it falls outside the scope of this paper. Most pure metals, however, do not form amorphous films when evaporated onto cooled substrates, even at liquid helium temperatures. Instead, a microcrystalline sample results, with grain diameters in the 50-100

A

range at 4

K.

This may seriously affect the optical properties as is shown in figure 1, where we present the optical conductivity in aluminium for films evaporated onto substrated at 25 K and for well annealed films [I].

The conductivity of bulk aluminium is characterized by a sharp parallel band peak at 1.55 eV. In the quenched films, this peak is completeley absent.

The microcrystalline state will also promote surface plasmon absorption. The rise in conductivity towards high energies in quenched aluminium in figure 1 is a manifestation of this, if the measurements were exten- ded to higher energies we would probably find a peak in a(o) in the 6-8 eV range. Such a peak, arising from the excitation of surface plasmons propagating along the grain boundaries, is shown in figure 2, which shows the optical conductivity in silver films quenched onto substrates held at liquid nitrogen temperature 121.

When discussing volume effects, one should also mention the void network found in some amorphous germanium film [3]. This void network will influence the optical properties, particularly near the absorption egde. The void free film egde is nearly as sharp as that of the single crystal material, and the absorption falls

FIG. 1.

-

Triangles, optical absorption at 25 K of an alumi- nium film deposited at RT. Circles, optical absorption at 25&

of an aluminium film deposited at 25 K ; the slight kink at 1.6 eV is due to detector changeover. Squares, optical absorption of liquid aluminium at 900 OC (after Miller, Ref. [28]). Dashed line, Drude absorption of crystalline aluminium, m* = 1.5 and

z = 1.2 x 10-14 S.

FIG. 2. -The real part of the optical conductivity of an opaque silver film, o, measured immediately after evaporation onto a sapphire substrate held at 140 K ; measured after the

(4)

INFLUENCE OF SAMPLE STRUCTI

off sharply below the egde, whereas for the films with voids, the egde is shifted to longer wavelengths and an absorption tail appears.

The optical absorption spectrum of gas evaporated ultrafine metal particles differ considerably from the corresponding bulk absorption spectrum [4], but has features in common with quenched samples of the same material. The enormous amount of work on the optical properties of small metallic particles is too extensive to be included here, the interested reader is referred to reference [4] and the many references therein. The same also applies to the optical properties of cosputtered metal-insulator samples [5] and to coeva- porated metal-solid noble gas samples [6], both of which shown segregation into two phases. All these types of systems show additional absorption due to the excitation of surface plasmons. In addition to this, the interband absorption may also be seriously affected as we have seen in the case of aluminium.

2.2 SURFACE EFFECTS - SPECULAR REFLECTANCE.

-

Surface roughness has an important effect on the optical properties of materials. Samples prepared by mechanical and electrochemical polishing of bulk materials will unavoidably have a considerable amount of scratches and orange peel in the surface. Evaporated films of many metals contours the substrate surface quite well, the surface roughness of the evaporated film is roughly the same as the substrate roughness. Others, as we have seen above, form very rough surfaces when evaporated onto substrates held at room temperature. It is thus clear that all samples used in optical studies have some degree of microrough- ness, and it is important to be able to predict the effect of surface roughness on the optical measurements, wheter these be reflectance, transmittance or ellipsometric measurements. Generally, one can classify surface roughness in three regimes :

1. Small roughness

2. Medium roughness o = @(Ai). 3. Large roughness a 9 Ai, o = O(1,).

Here

L

is the wavelength of light in vacuo, ;ti is the wavelength in the material, I, is the coherence

length of the incident light and a is a roughness para- meter. The only data available for common metals are for type 1) and type 2) roughness, although some data on relatively large rhenium whiskers excist [ I l l . However, the selective nature of rough metal surfaces, when used as solar energy absorbers will probably lead to and increased interest in the optical properties of all classes of rough surfaces. For metals, there are two striking differences between the optical properties of smooth and rough surfaces. Firstly, as already stated, the former only reflects an incident wave in the specular direction, while the latter will

scatter light into various directions. Secondly, on a rough surface, an incident wave may excite surface plasmons, this may alter the reflection coefficients of metals dramatically in the neighbourhood of the surface plasmon cut off frequency given by the relation :

Re(&(o)) = - 1. This is demonstrated in figures 3 and 4 which shows the change in the normal incidence

FIG. 3. - Experimentally observed reflectance drops in Ag as a function of surface roughness cr (After Stanford et al., ref. [7]).

FIG. 4. - Experimentally observed relative decrease in reflec- tance AR/Ro for a silver sample of medium roughness.

(5)

C5-92 0. HUNDERI too FEUERBACHER

.

STEINMANN 7 201 1 1 I I I I I I I

I

3.0 4.0 50 6.0 7.0 8D 30 tOD I I W 12.0

PHOTON ENERGY lev1 _+W,dfi

FIG. 5.

-

A1 reflectance for films of varying roughness. The rougher films are characterized by their measured rms roughness

o. Also shown is the smooth surface A1 reflectance obtained by

Feuerbacher and Steinman (After Endriz and Spicer, ref.

[83.

reflectance of aluminium in the 0.2-0.3 pm range drops from the smooth surface reflectance of over 90

%

to below a few percent, (Fig. 6 ) a dramatic decrease indeed.

FIG. 6.

-

Experimentally observed relative decrease in reflec- tance AR/Ro for two aluminium samples of medium roughness.

In regions of high interband absorption surface plasmons are strongly damped, however, the absorption in rough samples is still considerably increased compared to the smooth surface absorption. An example of this is given in figure 7 which shows the difference in absorptance between a rough and smooth copper film divided by the smooth surface reflectance. The rms roughness has not been explicitly determined, but the sample is certainly in the small roughness regime.

Most controlled surface roughness experiments have been performed to determine the effect of roughness on the normal incidence reflectance. Some controlled experiments on the effect of surface roughness on the ellipsometric parameters

+

and A have, however,

Fig. 7.

-

Experimentally observed relative decrease in reflec- tance AR/Ro for a copper sample with small roughness.

also been carried out. An example of such a study is a report recently published by S,xnith (I). His samples

were mechanically polished aluminium samples, subse- quently electropolished and exposed to etch solutions for various periods. His results are shown in figure 8.

0 I I I I I

I

0 10 20 30 40 50 60 FPL etch t l m e (mmn)

FIG. 8.

-

Experimentally observed change in ellipsometric parameters y and A and rms roughness o as a function of etch

time (After Smith).

There is approximately a linear increase in root mean square surface roughness o with time. The change in $

from that of a smooth surface is also approximately linear with etch time. The cause of the sharp increase

(6)

in

-

61,b between 0 and 2 min is believed to be related to the removal of organic contamination. The major change in I,b during etch is belived to be related to surface roughenning, while most of the change in A is related to the simultaneous growth of an oxide layer as indicated on the left hand site in figure 8. This study was undertaken on very rough surfaces. No fully satisfactory theory for this roughness region which include multiple reflections and shadowing effects has yet been developed, and that complicates the inter- pretation of these data. A further complication arises because the reflected light consists of two parts, a coherent part and an incoherent part-surface roughness will give an underlying component of unpolarized light. This component will in fact be dominating for large roughness. Conventional ellipsometry can not properly analyze a light flux in a general state of polarization. An ellipsometer for measuring all Stokes parameters of a general polarization state should therefore be employed. Such ellipsometers have recently been described in the literature [9].

2.3 SURFACE EFFECTS

-

LIGHT SCATTERING EXPE- RIMENTS.

-

The presence of surface roughness with

correlations in the order of the wavelength of light will, as we have seen above, change the specular reflection properties of surfaces. However, by far the simplest and most sensitive way to detect the presence of surface roughness is to measure the amount of diffusely scattered light. A perfectly flat, infinite surface does not give any diffusely scattered light, any amount of scattered light will therefore signal the presence of surface roughness, and measurements of the intensity of this scattered light allow you to evaluate the root mean square surface roughness. Such measurements of the surface roughness of aluminized ground glass have been performed in the infrared by Bennett and Porteus [lo]. In this region the metals can be regarded essentially as perfect conductors and the socalled scalar scattering theory seems to apply well. This is a theory based on an evaluation of Kirchhoff's scalar diffraction integral over a randomly rough, perfectly conducting, surface. For metals in the visible spectral region the diffuse scattering is strongly influenced by the excitation of surface plasmons and by the finite conductivity of the surface. However, theories have been developed which allows us to account for the effect of surface plasmons and to extract information about the nature of the rough surface even in this case. As a typical example of light scattering by rough metal surfaces in the visible we show the angular and spectral variation of the scattered intensity for a random rough silver surface. In figure 9 we show how the intensity of scattered light varies with emission angle for different polarizations of the incident and scattered light with respect to the emission plane. In first order scattering theory, the scattered intensity is proportional to the absolute square

I

g(k)

1'

of the Fourier component of the roughness for the spatial frequency k, where for a

FIG. 9.

-

Angular distribution of the fractional scattered intensity measured with polarizer and analyzer parallel ; left, for a rough silver surface ; right with crossed polarizer and analyzer. The curves are marked with the settings of the polarizer. The dotted curve shows a fit to the scalar theory (After Hunderi

and Beaglehole, ref. [22]).

normally incident photon emitted at an angle 8 with the mean surface normal

The angular variation of the scattered intensity therefore presents us with a way of directly measuring the frequency spectrum of the roughness. We will return to this later. We also observe the considerable degree of depolarization of the scattered light.

In figure 10 we have plotted the wavelength depen- dence of the total amount of scattered light against

FIG. 10. -The total amount of scattered light plotted as Rr/(Rr

+

Rso) versus 1-2 for a rough silver sample. Rr is the measured reflectivity of the rough silver sample and RSc is the fraction of diffusely scattered light (After Hunderi and Bea-

glehole, ref. [22]).

(7)

C5-94 0. HUNDERI

3. Theoretical models. - The theoretical approaches towards a description of the effect of surface roughness on the optical properties can be divided into three main categories

1) Approaches within the scalar and vector theory of diffraction.

2) Perturbation expansions around a mean flat surface.

3) Model surface treatments, effective medium theories.

The first theories of the optical properties of rough surfaces were scalar diffraction theories, a typical example of such a theory is the work by Davies [12] in 1953. He calculated the reflectivity

R

and the angular variation of the scattered intensity from a perfectly conducting random rough surface described by a Gaussian height distribution and a Gaussian autocorrelation function and found the now well known result

R = e - ( 4 n a ~ n ) 2

Generalization to a finite conductivity is simple in the basic integrals, but the ensuing integrals are difficult to evaluate. Since a description of such theories will be given by Lukes, we will only make a few comments on their applicability. First of all, the theories assume a local reflectivity, therefore at all points the local curvature of the surface must be small compared to the wavelength of the light. Secondly such theories do not allow for the possibility of surface plasmon generation. For metals this is very important in energy regions where surface plasmons can be generated. For a comprehensive list of references, the reader is referred to Beckmann and Spizzichino [13] and to a number of recent papers by Ohlidal and Lukes [14-161. A number of recent papers have treated the problem by the use of scattering theory or by simple classical perturbation expansions around the mean surface. Maradudin and Mills [17] treated the problem as a problem in classical electromagnetic theory, working directly with Maxwell's equations. They formally expanded the dielectric constant in a Taylor series in the amplitude [ ( x , y) of the surface roughness, conver- ted Maxwells equations to integral form, with the term proportional to [ (x, y) treated by the methods of scattering theory. Within the first Born approximation, one obtains from this approach the contributions to the roughness-induced scattering and absorption rate proportional to o2 where B is the rms surface roughness.

Kretschmann and Kroger [IS] and Juranek [19] have also examined the problem using classical electromagnetic theory. They start with the usual boundary conditions for the tangential components of the fields at the interface. They assume that the fields can be continued continuously from the mean surface z = 0 to the actual surface z = [(x, y) and the boundary conditions are now invoked at the actual surface.

This leads to a perturbation expansion for the fields and Kretschmann and Kroger solved these classical perturbation equations to second order. Their results are similar to those of Maradudin and Mills. Some of the results of Kretschmann and Kroger is presented in figure 11 which shows the calculated relative change

FIG. 11. - Relative change of reflectance

-

AR/Ro = - ( R

-

Ro)/Ro

from a rough-surface vacuum, free-electron-like metal (y = 0.04) for various correlation lengths.

of reflectance for a free electron like metal for various correlation lengths. A Gaussian autocorrelation function was assumed in these calculations. Although no detailed comparison between theory and experiments was performed, we see that the general trend of the experimental curves in figures 3 and 4 is well reproduced.

A quantum mechanical calculation of the effect of surface roughness on the reflectance has been performed by Ritchie and Elson [20]. Their results disagree with those of the previous two papers and Maradudin and Mills have pointed out that some of the assump- tions in Ritchies calculation are invalid. The essential point of Ritchie and Elson's theory is a transformation to a curvilinear coordinate system within which the rough surface is mapped into a smooth plane. In a second paper Elson and Ritchie [21] use the same coordinate transformation in a classical calculation. In that case, they obtain results similar to those of Mara- dudin and Mills.

(8)

C5-95

gives the same result for the scattered intensity as found Mills. For normally incident light, this gives a scatte- by Kretschmann and Kroger and by Maradudin and ring pr. sterradian :

I

sin2 0

-

E

I

sin2 V )

₊

cos2 V )

(

e 90s 0

+

(8

-

sin2 p)lf2

I2

I

cos 6

+

( E

-

sin2 0)'"

/'

8 and q are the polar scattering angles. The term in figure 13 where we compare the calculated change in the sin2 9 is the p polarized fraction of the scattered light, reflectivity with experimental values for a rough silver and the term in cos2 cp is the s polarized fraction. The surface. The agreement is reasonably good, all relevant function g(k) is the Fourier transform of the norma- features of the experimental data are reproduced by lized autocorrelation function of surface height. A the calculation, although the surface plasmon absorp- comparison between this expression and experimental tion maximum is slightly shifted to lower energies. results on silver is shown in figure 12. A Gaussian auto- This is a manifestation of the neglect of multiple scat-

tering effects in these models.

FIG. 12.

-

Angular distribution of the relative scattered intensity measured with polarizer and analyzer parallel (-0-0-) ; and theoretical prediction with correlation length a = 1 100 A

(-

-

- -) (After Hunderi and Beaglehole ref. 1221).

correlation function was assumed. The general beha- viour of the observed scattering is fairly well reproduced in this theory. One must bear in mind, however, that these theories are perturbation expansions and thus only valid for small roughness.

A number of authors have represented the rough surface by a specific model such as hemispherical bosses [24], round protrutions [25] or spherical particles [22] on an otherwise smooth surface. The scattering and extinction cross-sections of these particles were calculated and expressions for the change in reflectance were formulated in terms of these cross- sections. An example of such a calculation is given in

FIG. 13. - Comparison of experiment and best theoretical fit. The relative decrease in reflectivity AR/Ro for a rough silver sample ; - experiment, -

-

- model surface theory

(After Hunderi and Beaglehole ref. 1221).

A similar approach is to represent the roughness by an equivalent surface layer of small particles, calculate the effective dielectric constant of this surface layer by a Maxwell Garnett type effective medium theory, and use the standard three layer equations to calcu- late the resulting reflection coefficients [26]. The same - -

model has a ~ s d been used by Brusic et al. 1271 to discuss the influence of surface roughness on the ellipsometric parameters. Work is presently in pro- gress to calculate the influence of surface roughness on the ellipsometric parameters within the framework of classical perturbation theory (2) as has been done for the reflectance.

Effective medium theories have also been used to account for the effect of internal roughness on the reflectivity of silver [2], to account for the effect of the void network in evaporated germanium films [3], to

(9)

C5-96 0. HUNDERI

describe the optical properties of gas evaporated ultrafine metal particles [4] and discontinous thin films. For further details of these works the readers are referred to the referenced publications and references therein.

4. Concluding remarks. - Surface and internal roughness may, as we have seen, change the reflection coefficients of metals considerably, particularly in the

vicinity of the surface plasmon frequency. This is of particular importance in corrosion studies, where film formation is often accompanied by an increase in surface roughness. However, theoretical models exist today which allow us to account for the effect of surface roughness upon the reflection coefficients, as long as the roughness is not too large, particularly if reflection measurements are combined with measurements of the diffusely scattered light.

References [I] BERNLAND, L. G., HUNDERI, 0. and MYERS, H. P., Phys.

Rev. Lett. 31 (1973) 363 ;

HUNDERI, O., Thin Solid ~ i l m s 37 (1976) 275.

[2] HUNDERI. 0. and MYERS, H. P., J. Phys. F. : Metal Phys. 3 (1973) 683.

[3] GALEENER, F. L., Phys. Rev. Lett. 27 (1971) 1716.

[4] GRANQVIST, C. G. and HUNDERI, O., Phys. Rev. B 15

(1977).

[5] COHEN, R. W., CODY, G. D., COUTTS, M. S. and ABELES, B.,

Phys. Rev. B 8 (1973) 3689.

[6] RYBERG, R. and HUNDERI, O., J. Phys. C. In press. [7] STANFORD, J. L., BEN NET^, H. E., BENNETT, 3. M.,

ASHLEY, E. J. and ARAKAWA, E. T., Bull. Am. Phys.

SOC. 13 (1968) 989.

[8] ENDRIZ, J. G. and SPICER, W. E., Phys. Rev. B 4 (1971) 4144.

[9] ASPENES, D. E., J, Opt. SOC. Am. 65 (1975) 1274 ; ASPENES, D. E. and HATJGE, P. S., :bid. 66 (1976) 949. [lo] BEN NET^, H. E. and PORTEUS, J. O., J. Opt. SOC. Am. 51

(1961) 123.

I l l ] SERAPHIN, B. 0. in : ROC. Symp. on the Material, Science Aspects of Solar Energy Conversion, Tucson (Ed. B. 0. Seraphin) 1974.

[12] DAVIES, H., Proc. Insf. Elecfr. Eng. 101 (1954) 209. [I31 BECKMANN, P. and SPIZZICHINO, A., The Scattering of

Electromagnetic Waves from Rough Surfaces (Pergamon Press) 1963.

[14] OHLIDAL, I. and LUKES, F., Opt. Acta 19 (1972) 817. [15] OHLIDAL, I. and LUKES, F., Opt. Commun. 5 (1972) 323. 1161 OHL~DAL, I. and N A V R ~ I L , K., Thin Solid Films 31

(1976) 223.

[17] MARADUDIW, A. A. and MILLS, D. L., Phys. Rev. B 11

(1975) 1392.

[IS] KRETSCHMANN, E. and KROGER, E., J. Opt. SOC. Am. 65 (1975) 150.

[19J JURANEK, H. J., 2. Phys. 233 (1970) 324.

[20] ELSON, J. M. and RITCHIE, R. H., Phys. Rev. B 4 (1971) 41 29.

1211 ELSON, J. M. and RITCHIE, R. H., Phys. Status SoIidi

B 62 (1974) 461.

[22] HUNDERI, 0. and BEAGLEHOLE, D., Phys. Rev. B 2 (1970) 309, ibid. B 2 (1970) 321.

[23] VLIEGER, J. and BEDAUX, D., Physica 82A (1976) 221,

ibid. 85A (1976) 389.

[24] TWERSKY, V., Trans. I. R. E. AP 5 (1957) 81.

[25] BERREMANN, D. W., Phys. Rev. 163 (1967) 855, ibid. B 1

(1970) 381.

1261 CHAN, E. C. and MARTON, J. P., J. Appl. Phys. 45 (1974) 5004, ibid., 45 (1974) 5008.

1271 BRUSIC, V., GENSHAW, M. A. and BOCKRIS, J. O'M.,

Surf: Sci. 29 (1972) 653.

[281 MILLER, J. C., Phil. Mag. 20 (1969) 1115.

DISCUSSION

A. HUGOT LE GOFF.

-

Quelles sont les possibilit6s 0. HUNDERI. - The combined effect of surface d'6tendre le modBle (qui permet de traiter des cas de roughness and on overlayer have been published by rugositks plus r&alistes que le modele proposC par le Elson, J. Opt. Soc. Am. (1976), there, is also a forth- Dr. Lukes) au cas de surfaces rugueuses recouvertes coming paper by Bedaux and Vlieger.