Interface ellipsometry: An overview

Surface Science @ North-Holland

INTERFACE D.E.

ELLIPSOMETRY;

AN OVERVIEW

ASPNES

Bell Laboratories,

Received

101 (1980) 84-98 Publishing Company

Murray

15 October

Hill, New Jersey 07974. USA

1979; accepted

for publication

7 December

197Y

The principles of ellipsometric measurements, their relationship to reflectance and Raman scattering. the reasons for their sensitivity to monolayer films, and the meaning and modeling of the optical response of thin films are discussed in qualitative terms. Methods for reducing ellipsometric data. including recently developed multilayer analysis techniques and combined ellipsometry/reflectance measurements, are also discussed. Several examples of the analysis of solid+Aectrolyte interfaces, taken from the Fourth International Ellipsometry Conference. illustrate the capabilities of these techniques.

1. Introduction Owing to the limited transparency range of the ambient, in situ spectroscopic studies of the solid-electrolyte interface are presently limited to photons-in-photons-out techniques in approximately the 1-6 eV range. The available techniques are Raman scattering and polarimetry. Raman scattering determines vibrational spectra and provides information about local bonding, whereas polarimetry determines dielectric properties and provides information about the more extended electronic wavefunctions. Thus the two approaches are roughly complementary. Polarimetry is a general category which includes all Stokes-parameter measuring techniques. The currently significant ones are reflectometry and ellipsometry. The former measures power (Stokes parameter so) while the latter measures impedance (Stokes parameter ratios s,/s,,, sZ/sCI,sJ/so), or, alternatively, the size and shape, respectively, of the polarization ellipse. When reflectometry is performed with a linear polarizer in the reflected beam, the two are related in that reflectometry then measures a projection of the ellipse if the incident light is polarized, or of a distribution of ellipses if the incident light is unpolarized or partially polarized. If three different polarization azimuths in the reflected beam are used with polarized incident light, non-normal incidence reflectometry can also map the polarization ellipse and then becomes equivalent to ellipsometry. However, reflectance measurements are typically limited to polarizations parallel (p) and per84

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pendicular (s) to the plane of incidence in which case the mapping is incomplete. The relative capabilities of reflectometry and ellipsometry can be visualized with reference to the black-box problem of electrical engineering. Normal-incidence reflectometry, s- and p-wave reflectometry, and ellipsometry are analogous to deducing the contents of the box using a wattmeter, a wattmeter with a filter, and an impedance bridge, respectively. Clearly, the advantage lies with impedance measurements. The point can be emphasized by noting that no reflectance studies are done at a single wavelength, while the literature is full of results deduced from fixedwavelength ellipsometric measurements. Also, ellipsometry is relatively insensitive to macroscopic roughness which scatters light out of the measurement system. The greater contributions of reflectometry to the understanding of interfaces derives largely from the fact that reflectometry provides spectroscopic information while ellipsometric measurements generally have been performed at single wavelengths. The sensitivities of Raman scattering, reflectometry, and ellipsometry to surface and interface films also differ substantially. A 50 A film represents the present detectivity limit for conventional Raman scattering owing to the relatively inefficient inelastic scattering processes involved [l]. In contrast, changes in optical response resulting from changes in film thickness of the order of fractions of a monolayer can be detected routinely in rellectometry and ellipsometry. Despite the apparent contradiction of being able to detect a fraction of an A using a probe of characteristic length A = 0.1 to 1.0 pm, there is nothing mysterious about it as the following argument shows. If a thin transparent film of index it and thickness d is present on a reflecting surface and if the light is incident at Brewster’s angle for the film, the p-wave component does not “see” the film while, for simplicity, we suppose that the s-wave component is completely reflected at the air-film interface. Thus the p-wave component travels -2d farther, resulting in a relative s-p phase shift A8 = 4n-rid/A.. In ellipsometry this phase shift is measured directly. A good null instrument is capable of resolving changes of O.Ol”, and a good photometric instrument about an order of magnitude smaller. Taking A0 = 0.01” with A = 5000 8, and n = 1.5, we calculate immediately that d = 0.05 A, which is well under atomic dimensions. The role of the phase shift shows that the relevant length is not A but rather the s-p relative coherence length, which must be adequate to define relative phase shifts of millidegrees. Further, the argument illustrates that the change is totally surface-related, so the presence of like material in the ambient, which affects s- and p-polarized components equally, does not influence the measurement. Similar sensitivities to surface films arise in reflectometry through the action of thin films to change the impedance

mismatch between ambient and substrate and thus change the efficiency of dissipating the incident radiation in the substrate. In fairness, it should be noted that polarimetric sensitivity to thin films is somewhat misleading because it applies to changes only. Absolute measurements cannot be made yet to these accuracies. Moreover, because all layers to which light penetrates contribute to the overall dielectric response, the interpretation of polarimetric data becomes a sorting problem. Recent efforts have been directed toward systematic separation, and also to the microscopic description of the effect of films on measured optical properties. The rest of the paper concerns ellipsometry, with emphasis on current status and application to interfaces. A brief description of instrumentation is followed by current approaches to modeling and a discussion of the theory of the optical response of adsorbed layers. A summary of representative results, drawn from the just-completed Fourth International Conference on Ellipsometry, concludes the paper. We shall not attempt to cover fundamentals and mathematical equations, as these are treated thoroughiy by Azzam and Bashara [2]. Worth mentioning are the recent reviews by Neal [3] on applications of ellipsometry to a wide range of bulk and interface systems.

2. Instrumentation This is basically a solved problem, even though spectroscopic instruments are not yet available commercially. The first fully automatic instrument. reported by Cahan and Spanier [4], was a photometric design that employed anatog, rather than digital, detection and analysis techniques. Since then a wide range of config~ations has been realized [5], the most versatile and precise being based on mini- or microcomputers for digital data acquisition, reduction, and analysis. Computer-based systems are also well adapted for modification such as to configurations based on achromatic compensators which may be available in the future. Photometric instruments rely on the modulation-spectroscopic principle of impressing a time dependence on the transmitted optical flux, for example by rotating a polarizer prism, that can be decoded later for the parameters of interest. They are fast and operate at high optical efficiency, and therefore can be used with weak continuum sources for spectroscopy. If the source is stabihzed, it is possible to measure simultaneously the smali change in reflectance induced by the formation of a film at an interface and thus to obtain a third independent constraint through the Stokes parameter SO[6]. Alternatively, a third independent constraint can be obtained by varying the angle of incidence [7]. In contrast, the older null instruments extinguish the

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signal to obtain the data and are most effective with high-intensity laser or atomic line sources. However, null-ellipsometer data are independent of detector nonlinearities and thus are more accurate within their wavelength limitations than photometric instruments. A typical photometric instrument is diagramed in fig. 1. This is a rotating-analyzer ellipsometer, which is somewhat limited as it is capable of measuring directly only the Stokes parameter ratios s,/so and sZ/so. However, it offers a significant advantage for ellipsometric spectroscopy because it is totally achromatic - the only wavelength-dependent element is the sample itself. In one design [8], the source is a 75 W short-arc Xe lamp for high intensity with small spot size, the monochromator is a Cary 14 for excellent stray-light rejection, the polarizing prisms are quartz Rochons for wide bandwidth and outbound polarization scrambling, and the detector is an EM1 9558 QB end window photomultiplier giving a useful spectral range of 1.5-6.0eV. With an attached shutter for stray light and baseline determination, linearized electronics, and Fourier-transform detection, the accuracy approaches that of a good null ellipsometer. Aside from the development of rotating-compensator photopolarimeters capable of measuring all Stokes parameters, future advances will include rapid-scan polarimetry, where the rotating element(s) will be synchronized with a rapid-scan spectrometer and the multiplexed data reduced by Fourier

MONOCHROMATOR

POLARIZER

(COMPENSATOR)

AhALYZER---_?

SAMPLE APERTURE APERTURE =

Fig. 1. Schematic

PHOTOMULTIPLIER

diagram

of a typical

rotating-analyzer

ellipsometer

(after ref. [S]).

88 decomposition. prove response

D.E. Aspnes

Fourier-transform in the infrared.

I Interface ellipsometry

spectrometers

will also be used

to im-

3. Optical models for interfaces Actual interfaces are complicated, consisting of several types of dielectrics such as substrate, ambient, and interface. The distinction may be blurred, with one material grading into the other. Moreover, interfaces, by their nature, are expected to be anisotropic, and the substrate, interface species, and ambient may not be pure but may consist of mixtures of separate phases. n-phase mixtures are relatively common, the simplest example being a polycrystalline metal film where the two principal constituents are the metal itself and voids. Reflection and transmission are treated for bulk-phase configurations by solving Maxwell’s equations for the reflected wave where all polarizable phases are represented by individual dielectric functions or conductivities. This is a purely macroscopic approach which is acceptable phenomenologically because one measures only the far-field radiation pattern of microscopic charges set in motion by the incident electromagnetic wave; details on the atomic scale are not observable directly but only through their effects on the radiated light. When dealing with bulk systems it is generally unnecessary to go beyond a macroscopic description unless heterogeneous dielectrics are involved. But because interfaces are of the scale of atomic dimensions, atomic polarizabilities are often more relevant parameters, in which case a microscopic description is essential. It is useful to discuss briefly the connection between macroscopic optics and microscopic models before treating interface applications themselves. For bulk systems, the polarization response problem connecting microstructure and atomic polarizabilities to the macroscopic dielectric function E consists of two distinct parts. First, one solves the “local-field” problem for the exact microscopic field e(r) and polarization p(r) in terms of the microscopic parameters of the system. Second, one makes the connection to the macroscopic quantity E by simply volume-averaging everything to obtain the macroscopic (observable) field E = (e(r)) and polarization P = (p(r)) and using the definition D = EE = E + 41rP. Exactly the same approach is used in the theory of the dielectric response of heterogeneous materials consisting of aggregates of dimension large enough to have their own dielectric identity, but still small compared to the wavelength of light. With the substitution of dielectric function for atomic polarizability the heterogeneous materials and local-field problems are isomorphic, which is not surprising, since everything is heterogeneous on an atomic scale. The difficulty with the solve-average approach is that the first part cannot

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be done exactly except in highly idealized representations, such as a simple cubic lattice of polarizable points - the Clausius-Mossotti local-field problem - or other systems consisting of weakly interacting subunits repeated in a specific long-range pattern, even in the usual uniform-field approximation that takes advantage of the gross difference in scale between atomic dimensions and the wavelength of light to assume that the latter is infinite. Macroscopic optics - by analogy to hydrodynamics and thermodynamicsis based entirely on the fact that the solve-average procedure can be reversed and the microscopic equations of electrodynamics modified so as to obtain corresponding expressions written in terms of macroscopic variables only. Thus follow Maxwell’s equations for macroscopic media and the suppression of microscopic properties into constitutive relations based on the dielectric function or conductivity and the magnetic susceptibility [9]. Microscopic solutions, if invoked at all, are used only to the extent necessary to relate a microscopic parameter to a macroscopic quantity. A familiar example of the average-solve approach is the sphericalcavity derivation of the Clausius-Mossotti expression relating the microscopic polarizability of a point particle to the macroscopic dielectric function of a lattice of such particles. While applicable (at the cost of some physics) to bulk systems the average-solve philosophy is virtually impossible to apply to films of atomic dimensions-as can be appreciated by trying to define the equivalent of a Clausius-Mossotti sphere for an atom adsorbed on a surface. Thus first-principles interface calculations are restricted to the solve-average approach. 3.1. Substrates The need to obtain accurate bulk values of E before attempting interface analysis is perhaps obvious, yet this aspect is often overlooked. For metal substrates differences in measured or apparent dielectric properties may reach factors of two. These differences are hardly ever due to the optical technique used to determine E, as is commonly supposed, but rather due to sample imperfections such as voids, rough surfaces, or the presence of unintended surface films. For example, representative E spectra for gold are shown in fig. 2 [lo]. Significant differences are seen despite the fact that all data are measured on nominally ideal samples. These differences can be related to the microstructure of the respective samples [lo]. Above the 2.5 eV interband transition threshold differences are determined by void fractions and surface films (including microscopic roughness). Voids simply reduce the polarizability per unit volume, the definition of E, and in effective-medium theories they scale E between the true bulk values and 1 + i0. The difference between the Theye and JohnsonChristy data in fig. 2 represents a 9% difference in void volume. This value

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6

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4 -----l l l l l

2

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dielectric

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Fig. 2. Measured

Pells and Shigo Johnson and Christy This Work

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51

(eV)

functions

for Au (after

ref. [lo])

was obtained in the Bruggeman effective-medium approximation (EMA) [ll], which appears to be the best of the simple effective medium theories for general application [12]. The existence of voids implies a microstructure which from the preceding section implies microscopic field distributions quite different from the macroscopic average field. While not so important for linear optical processes, such variations are expected to be more critical for highly nonlinear processes such as surface-enhanced Raman scattering, for which the assumption that the observed response arises uniformly over the surface may not apply. Surface films such as oxides, contaminants, or microscopic roughness on metals and semiconductors invariably have dielectric functions smaller than substrate values and consequently act to impedance-match the substrate to the ambient. The effect on the measured E spectrum is not simple scaling as with voids, but the existence of surface films can also be deduced from spectroscopic analysis [lo]. At the peak of the .e2 spectrum either films or voids result in measured lZ values less than the true bulk value. This leads to a “biggest-is-best” rule-of-thumb which is extremely useful for monitoring surface cleaning processes in any ambient, given an automatic ellipsometer

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equipped with a real-time readout. The sample is simply processed until the highest l2 value is obtained. In the Drude region below 2.5 eV in fig. 2, the data provide information on grain size and surface scattering, as both increase scattering and therefore E2

WI.

3.2. Thick interfaces If interface layer thicknesses are of the order of 10 A or more, a uniform dielectric function representation for the interface response may be adequate. This is the three-phase (substrate, overlayer, ambient) model. We note two possibilities: either the dielectric properties of all constituentssubstrate, interface layer(s), and ambient -are known a priori from independent measurements and one must determine layer thicknesses, compositions, etc., or else nothing is known about the interface and all its properties must be determined. The former case is completely equivalent to the black-box problem where the frequency-independent lumped parameters corresponding to resistors, capacitors, and inductors are now layer thicknesses, compositions, and void fractions. The parameters can be determined by approximating the actual physical situation with an n-phase model or its anisotropic elaborations [2], then using this model to reproduce actual data with parameters varied to minimize the mean-square deviation. Sufficient data must be available to provide a meaningful comparison. This is most conveniently done spectroscopically, particularly in semiconductors where the optical response is dielectric-like in the visible and changes to qualitatively different metalliclike behavior in the ultraviolet. This approach has recently been used to show that the Si-Si02 interface consists of a 7 + 2 A wide layer of chemically mixed Si and 0 of approximate stoichiometry SiOo.4~0.?[13]. Because meaning is given to the model parameters, it is important to establish confidence limits for them by linear regression analysis [14] as well as simply determining their values by minimizing the mean-square deviation. Confidence limits provide the added function of limiting the number of parameters in a model, and showing which are being determined by the data and which are not [12]. While the mean-square deviation always decreases as parameters are added, the confidence limits first decrease and then increase (usually catastrophically) if the number of parameters exceeds the capability of the data to determine them, or if they become correlated, or if irrelevant parameters are added. The validity of a given model can be assessed in spectroscopic applications by the goodness of fit over an entire spectrum and not just at selected wavelengths. In the second case where nothing is known about the interface, one starts with the dielectric response of the substrate and ambient and inverts the

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ellipsometric data within the three-phase model for the effective dielectric function and thickness of the interface. It is clear that polarimeters measuring three parameters are better suited to this application than two-parameter instruments although a spectroscopic capability can be used to advantage if it is known that the film is transparent in some region in which case n and d can be obtained uniquely. The best that one can do, however, is a single effective dielectric function describing the average response of the entire interface region. 3.3. Thin interfaces The simplest approach is to use the three-phase model taken to arbitrarily small thicknesses [15], although the assumption of a uniform bulk dielectric constant, or equivalently a surface conductivity [16], is not so easy to justify for monolayer or submonolayer films where the polarizability is varying over the interface width and probably is also anisotropic. While anisotropy can always be included empirically, recent theoretical investigations have taken a more fundamental approach by considering the polarizability to be graded smoothly between substrate and ambient, and solving Maxwell’s equations under this assumption to determine the consequences (17,181. Ptieth and Naegele f18] have shown in this case that the three-phase model is applicable for s-polarization, where the thickness of the film is the width of the graded region and the dielectric function the spatial average d

However, this result is not valid for p-polarized light, basically because for this polarization H is the component that is slowly varying across the interface instead of E owing to the boundary conditions on tangential fields. Here, the expressions are more complicated and the relevant average is (E)-’ = f f [E(Z)]-’ dz 0

This automatically introduces an anisotropy but also new parameters because it is necessary now to specify the spatial variation itself. A model of this type has been applied to describe the dependence of tan 4 and h versus angle of incidence for dispersed Au particles on a glass substrate and significantly better agreement was found with respect to the isotropic three-phase model [19]. It is not clear, however, how much of the improvement was due to the model and how much was due to the two extra free parameters varied to fit the data.

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If boundaries are incorrectly assigned to be deeper into the substrate or ambient so that the “interface” actually includes regions of bulk or ambient, the exact solution [18] will correct for this error by yielding zero contribution over those regions where “interface” values do not differ from bulk or ambient. Such corrections do not occur in the three-phase model. However, the exact solution also shows that linear grading is sufficiently gradual to be approximated quite accurately by the three-phase model, which explains why the latter approximation works so well in practice. Purely microscopic formulations of mono- and submonolayer arrays have recently been reviewed rather comprehensively by Habraken et al. [20], and we shall not repeat their analysis here. These models usually consist of polarizable points arrayed some distance above a smooth dielectric substrate according to a two-dimensional lattice. The principal objective is to calculate the local-field enhancement of the polarization response of a single point due to the proximity of other points and the image dipoles. These represent an improvement over the dielectric models only to the extent that the intrinsic polarization response of the point is not modified by bonding to the substrate. While this may occur in weak physisorption it is clearly not valid for chemisorption. When applied to chemisorption, the polarization parameters determined by inversion of the results should be viewed as phenomenological. The theoretical description of the optical response of chemisorbed systems is a currently developing subject.

4. Examples 4.1. Passivation films on iron This work by Chen, Cahan, and Yeager [21] was initiated to determine the nature of passivating films on iron. It is significant not only for its spectral identification of the film species, but also because it is the first work to use successfully a full spectroscopic three-constraint process, where the change in reflectance was measured together with the standard ellipsometric data. The experimental configuration involved a 99.998% pure zone-melted iron ingot, mechanically polished ending with 0.05 pm alumina and measured in a cylindrical cell containing a mixture of 0.15M H3B03 and 0.075M Na2B407. The optical spectrum of the sample after cleaning was measured in situ under conditions expected to yield a film-free surface and the optical properties from 1.8-3.5 eV were found to compare favorably with those measured in ultrahigh vacuum. The dielectric function and thickness data for a film grown at 1.35 V and measured at 1.25 V are shown in fig. 3. These data were calculated independently at each energy by solving for n, k, and d from ellipsometric and

D.E. Aspnes

I Interface

elfipsometry

- 40

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Fig. 3. E]. ~2, and d data and modulated reflectance

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20

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for a passive film on iron, by simultaneous spectra (after ref. [211).

solution

of eliipsometric

simultaneous modulated reflectance data with no assumptions other than the applicability of the three-phase model. This model is expected to be valid here because the film is relatively thick. Ideally d should be constant: the fluctuations in fact are less than 10% providing an independent assessment of the results. For comparison, it often is difficult to determine d for thin films to within a factor of two if all three data are not available. These data have everywhere smaller values than equivalent E data taken on sintered samples, and the peak is shifted to higher energy. The first observation allows the conclusion that the film is amorphous and of much lower density than the polycrystalline phase, i.e., that it is quite porous. The shift of the 2.85 eV peak in sintered material to higher energies was interpreted as evidence of incorporation of protons in the film, but a more likely explanation is the loss of long-range order causing a coalescence of band structure into a single peak. 4.2. Film growth under organic

coatings

This work, by Ritter and Kruger [22], is significant because it is the first to show that ellipsometric measurements can follow changes in the interface between a substrate and a passivating overlayer in electrolyte solution. The

D.E. Aspnes 1 Interface ellipsometry

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experimental configuration consisted of a Fe substrate polished with 0.05 pm alumina and coated with a collodion overlayer lo-30pm thick intended to simulate the protection offered by a standard paint film. Ellipsometric measurements were performed at a fixed wavelength, and the evolution of tan $I, A data were plotted as a function of time after immersion in 0.025N NaCl for a control sample and a test sample that had K2Cr04 embedded at selected points around the measurement area. The K2Cr04 leaches slowly and acts as a corrosion inhibitor. The results are shown in fig. 4. The interface pH and the potential of the Fe electrode were both monitored as well as the optical parameters. The essential changes in all sets of data occur near lo3 min and can be understood qualitatively by noting that a decrease in A and approximate constancy of I) for these films is a signal that an interface film is developing between the Fe substrate and the protective overlayer. Computer modeling showed

----

UNPROTECTED

-

CrOi-PROTECTED II-_

11

t (min)

Fig. 4. Variation with time of A, IJG,and interface pH for collodion (---) impregnated collodion (-) films on iron (after ref. [22]).

and chromate-

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/ Interface ellipsometry

that an increase of 18, in an assumed aqueous interface film resulted in a decrease of 0.28” in A. If chromate ion is present, the predictions of the model are relatively well reproduced. At l@min, the ambient electrolyte penetrates the overlayer and begins to accumulate at the inner interfaces. The accumulation reaches an average thickness of the order of 30 8, by 104min, although the film is probably not uniformly thick. If chromate is not present, the change is much more pronounced although in this case it is not clear whether the change is due to a physically thicker interface film or to a different spectral response (the corrosion reaction products are expected to absorb in this spectral region). Without CrO:-, the pH continues to increase into the basic regime where corrosion is expected to occur. 4.3. Anodic oxidation of silver Muller and Smith [23] have recently investigated the anodic oxidation of single-crystal Ag in KOH solutions under varying electrochemical conditions. The final state of the oxidized surface is extremely complex, consis85 -

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(1030) Aq SINGLE

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i 80

&kg)

of Ag under constant-voltage conditions (after ref. Data points correspond to times 0, 2, 4, 5. 10, 15, 20,

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ting of a roughened substrate, at least two distinct layers, microcrystallite formation (under certain conditions), and a transport layer in the electrolyte with optical properties different from that of the ambient electrolyte. This work is significant because it represents the most ambitious modeling effort to date on any interface, and the authors show that the optical data allow at least seven parameters describing the interface to be determined. A polar curve measured for anodic oxidation under constant-voltage conditions is shown in fig. 5. It is worth noting that the oxidation process took only 21.5 s, indicating the necessity of a fast, automatic instrument. The oxidation begins with the growth of a “type 2” film at the silver-electrolyte interface, followed about 6 s later by the onset of secondary crystal growth. By determining the degree of hydration of the secondary crystals from their effect on the optical data, they were shown to correspond to approximately Ag,O. Under constant-current conditions the initial surge and resulting supersaturation of the boundary layer were absent and the secondary crystals were Ag,O-HzO. Acknowledgments I wish to thank B.D. Cahan, J.J. Ritter and J. Kruger, and R.H. and C.G. Smith for preprints of their work prior to publication.

Muller

References developing stimulated Raman gain techniques may improve this limit substantially. See, e.g., B.F. Levine and C.G. Bethea, Appl. Phys. Letters 37 (1980).

[l] Currently

[2] R.M.A. Azzam and N.M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977). [3] W.E.J. Neal, Surface Technol. 6 (1977) 81; Appl. Surface Sci. 2 (1979) 445. [4] B.D. Cahan and R.F. Spanier, Surface Sci. 16 (1969) 166. [S] P.S. Haughe, Surface Sci. 96 (1980) 108. [6] B.D. Cahan, Surface Sci. 56 (1976) 354. [7] 0. Hunderi, Surface Sci. 61 (1976) 515. [8] D.E. Aspnes and A.A. Studna, Appl. Opt. 14 (1975) 220; Rev. Sci. Instr. 49 (1978) 291. [9] A comprehensive discussion of the foundations of macroscopic optics is given by J. van Kranendonk and J.E. Sipe, in: Progress in Optics, Vol. 15, Ed. E. Wolf (North-Holland, Amsterdam, 1977) p. 245. [lo] D.E. Aspnes, E. Kinsbron and D.D. Bacon, Phys. Rev. B21 (1980) 3290. [ll] D.A.G. Bruggeman, Ann. Physik (Leipzig) 24 (1935) 636. (121 D.E. Aspnes, J.B. Theeten and F. Hottier, Phys. Rev. B20 (1979) 3292. [13] D.E. Aspnes and J.B. Theeten, Phys. Rev. Letters 43 (1979) 1046. [14] See, e.g., E.S. Keeping, Introduction to Statistical Inference (Van Nostrand, Princeton, NJ, 1962) ch. 12. [15] See, e.g., J.D.E. McIntyre and D.E. Aspnes, Surface Sci. 24 (1971) 417. [16] R. Kofman, R. Garrigos and P. Cheyssac. Surface Sci. 44 (1974) 170.

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[17] J.R. Bower, Surface Sci. 49 (1975) 253; P. Feibelman. Phys. Rev. B14 (1976) 762; S.P. Timashev and M.A. Krykin. Phys. Status Solidi (b) 76 (1976) 67. [18] W.J. Plieth and K. Naegele, Surface Sci. 64 (1977) 84. [19] A.C. Grivtsov, R.M. Ergunova. Z.M. Zorin. M.A. Drykin. Yu.N. Mikhailovskii. A.A. Nechaev, S.F. Timashev and A.E. Chalykh, Opt. Spektrosk. 45 (197X) 73X. [20] F.H.P.M. Habraken. O.L.J. Gijzeman and G.A. Bootsma. Surface Sci. 96 (19X0) 482. [21] C.T. Chew thesis, Case Western Reserve University, unpublished; C.T. Chew B.D. Cahan and E. Yeager. presented at 4th Intern. Conf. on Ellipsometry, Berkeley. CA, 1979. [22] J.J. Ritter and J. Kruger. Surface Sci. 96 (1980) 364. [23] R.H. Muller and C.G. Smith, Surface Sci. Y6 (lYX0) 375.