Current State of the Theory of Optical Properties of Semitransparent Metal Coatings

G. V. Rozenberg

1. Introduction

Until comparatively recently, one of the principal tasks of experimental metal optics was seen as the determination of the spectral dependences of the optical constants of a massive metal, i.e., of its refractive index $n$ and absorption coefficient $\chi$, related to the complex refractive index $\mathfrak{n}$ by

\[ \mathfrak{n}=n-i\chi . \tag{1} \]

In the case of bodies that do not possess such strong absorption as metals, the solution of this problem does not encounter insurmountable difficulties. By using specimens of sufficiently large thickness and studying the conditions under which light waves penetrate through their thickness, it is always possible to reveal the properties of the bulk material itself and confidently to eliminate the distortions introduced by its surface layers. The situation is quite different in the case of metals. Here one must be content with studying only their reflectivity, i.e., ultimately, the phenomena occurring in a very thin surface layer. Let us recall that the thickness of the skin layer for most metals in the visible region of the spectrum amounts to only tens of angstroms. Thus the possibility of an optical investigation of the internal regions of the specimen is obviously excluded, and the distortions characteristic of its surface layers, rather than the properties of the bulk substance itself, already come to the fore. As a result, the results obtained prove to be extremely sensitive to the method of formation and treatment of the surface and are difficult to subject to theoretical interpretation. The situation is further complicated by the fact that the anomalous character of the skin effect, characteristic of most metals in the visible region of the spectrum, generally

deprives the very formulation of the problem of content, since in the surface layer of a metal the field equations cease to be linear and, at the same time, the possibility is lost of characterizing the metal by its optical constants (for details see ^1).

It is quite natural that, in seeking ways to overcome these difficulties, many experimenters began to try to apply to metals the same methods as for the investigation of transparent substances, namely, they turned to the study of metals “in transmission.” At present it is perfectly clear that such attempts were doomed to failure from the outset. Indeed, in order for a specimen to possess appreciable transparency, it must be a film whose thickness is, in order of magnitude, comparable with the thickness of the skin layer and, in particular, much smaller than the mean free path of the electron. Consequently, here too the investigation is limited to surface layers and can yield no information about the properties of the bulk substance.

Although this circumstance was not duly appreciated by most experimenters, which often led to an unfounded interpretation of the data they obtained, it did not deprive the study of the optical properties of thin metallic films of theoretical interest. In addition to clarifying a number of questions connected with practical applications of thin-layer coatings (for example, in interferometry, in bolometers, and as photocathodes), important consequences may be expected from it also from the standpoint of the new problems posed before metal optics by the development of the electron theory of metals. What is involved is the experimental verification of theoretical conclusions and the determination of the constants with which the theory operates.

Here, however, too, a certain disappointment set in. The very first experiments showed that in thin semitransparent films a metal possesses entirely different characteristics than in bulk specimens and even in their surface layers. It further turned out that these characteristics, extremely sensitive to the conditions of formation of the layer, generally cannot be reconciled with the conception of the film as a continuous formation. Meanwhile, irrefutable evidence was obtained that semitransparent metallic layers do indeed have a distinctly expressed granular structure. Thus a new problem arose—to connect the observed optical properties with the structure of the film and to find out what information, concerning the substance forming the granules (the metal), can be drawn from their analysis. The present article is devoted to a brief survey of the current state of this range of problems. In doing so, we shall confine ourselves to considering only those questions that have a direct bearing on the theory of the optical properties of granular layers. The reader will find a survey of experimental data on the structure of layers and their optical properties, for example, in reviews ^2,^3 and monographs ^4,^5.

The need for a special review of the basic concepts of the theory of the optical properties of thin metallic films we see not only in the fact that the study of these properties is now at the center of attention of a wide circle of investigators, but also, and chiefly, in the fact that the majority of investigators, in interpreting experimental data, continue to use obsolete theoretical concepts, which, of course, ultimately leads to a devaluation of the data themselves. In addition, we wish to draw attention to a number of new problems arising in connection with the study of metallic films both in the field of experiment and before theoretical metal optics. In particular, the latter is confronted with the problem of studying the optical properties of colloidal particles of metal consisting of a few hundreds or thousands of atoms. In this direction, which is of fundamental importance for the optics of colloids, very little has yet been done. Let us also point out that most of the theoretical considerations discussed below apply not only to metals, but equally to any granular coatings of strongly absorbing substances, for example to adsorbed layers of dyes.

2. STRUCTURE OF SEMITRANSPARENT METALLIC LAYERS AND THE QUESTION OF THE EFFECTIVE PARAMETERS OF THE LAYER

At the present time we already possess sufficiently complete data to outline in general terms the process of formation of a metallic coating on a dielectric or on another metal under conditions of cathodic sputtering or thermal evaporation in vacuum. In accordance with the theoretical predictions of Ya. I. Frenkel,^6 confirmed by numerous experiments, deposition of the layer begins only upon attainment of a certain critical density of the atomic beam irradiating the surface to be coated. At the same time deposition of the precipitate begins suddenly with the formation of numerous granules, isolated from one another and of comparatively large size (several tens of ångströms in diameter). The number, shape, and initial dimensions of the granules depend on the nature of the metal being evaporated and on the conditions of its deposition (temperature, cleanliness of the surface, density and angle of inclination of the atomic beam, etc.). The granules of some metals, such as silver, gold, and tin, have an approximately spherical shape, whereas in other metals (for example, zinc, cadmium) they immediately acquire a distinctly expressed crystalline faceting.

Thus, in the initial stage the metal is deposited on the surface in the form of a two-dimensional colloid. Subsequently, as the amount of deposited material increases, the sizes grow, but not the number of granules (no new centers of crystallization arise). As a result, at a certain “critical” thickness of the layer the granules

come into contact with one another, and the process of their coalescence begins. In this process the layer acquires a loose porous structure corresponding to a two-dimensional gel. Almost simultaneously with the transformation of the two-dimensional colloid into a two-dimensional gel, deposition of the subsequent tiers of particles begins, i.e., the transformation of the two-dimensional gel into a three-dimensional one. This process is accompanied by compaction of the coating, which leads (with further deposition of metal) to the formation of a continuous layer, which gradually, as its thickness increases, acquires the properties of a massive metal. As a rule, the transparency of the layer is lost much earlier than this compaction process is completed. It should be noted that the transformation of a granular structure into a continuous one is accompanied by the displacement of various impurities contaminating the metal, inevitably present (and, apparently, in large quantities) in the granular stage of layer deposition. At the same time, a restructuring of the crystalline structure of the metal also often takes place.

The formation of the layer, as a rule, is not completed with the cessation of the sputtering process, but often extends over a long time, during which the structure of the layer undergoes further irreversible changes (aging), apparently akin to creep and increasing the nonuniformity of the distribution of material over the surface. Significant and fairly rapid changes, evidently of a corrosion type, are undergone by metallic layers when they are taken out of the vacuum into air.

Thus, sufficiently thin (semitransparent) metallic films are by no means a homogeneous layer of metal, and their optical properties must be determined entirely by the nature of light scattering at the boundaries. However, the sizes of the particles and their mutual distances are much smaller than the wavelength of light (usually they are measured in tens, more rarely hundreds of angstroms), i.e., from the optical point of view the layer appears as a quasi-homogeneous formation (but only in two dimensions!). This is manifested, for example, in the fact that the scattering of light by the layer is extremely small. At the same time this means that the effective field in which the particles are located must differ substantially from the field of the wave irradiating the layer.

In studying the optical properties of such a granulated layer, the first question that arises is: to what extent is it possible to model it by means of a homogeneous layer with certain effective parameters—the refractive index \((n_{\mathrm{eff}})\) and absorption \((\chi_{\mathrm{eff}})\) and thickness \((t_{\mathrm{eff}})\)? In most works devoted to the optics of metallic films, the possibility of such modeling, i.e., of selecting values of \(n_{\mathrm{eff}}\), \(\chi_{\mathrm{eff}}\), and \(t_{\mathrm{eff}}\) for which a homogeneous layer would be optically completely equivalent to the real metallic film, is taken for granted and as requiring no discussion. However, such a point of view is completely unfounded. In the theory of multilayer coatings (see, for example, \(^{7}\)) it is proved;

that an arbitrary layered medium can be modeled not by a single-layer, but by a two-layer coating, and moreover for different angles of incidence the model parameters may be different. On the other hand, there are numerous experimental facts indicating that it is inadmissible to model a real metallic film by means of a single homogeneous layer. This is evidenced first of all by the impossibility, well known to many experimentalists, of reconciling the effective optical constants of one and the same film, determined on the basis of its different optical properties (transmission and reflection coefficients, phase shifts and their angular dependences). Thus, for example, according to the data of Harris and Loeb⁸, the values of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) for a gold film, calculated using data on its reflecting powers from different sides of the layer, differ sharply from one another. Similarly, calculations of \(t_{\mathrm{eff}}\) carried out by Abelès⁹ for one and the same film, but using data on its optical properties in different regions of the spectrum, led him to values differing by several times. The number of such examples known from the literature is very large; however, a special test of the applicability of the single-layer model has not yet been made, and it continues to be used almost everywhere. It is therefore advisable to recall briefly the main conclusions obtained with its aid, especially since in what follows we shall repeatedly have to use the notions associated with it.

It has long been established that the optical properties of a thin metallic film depend very strongly, and in a rather peculiar way, on its thickness \(t\). However, the very concept of thickness as applied to a granulated layer loses its direct definiteness and requires additional qualifications. More precisely, it becomes necessary to establish an unambiguous correspondence between the values of \(t\) measured by different methods. As for the choice of one or another of these values as the basic one, it must ultimately be determined by the requirement that the method of measuring \(t\) correspond to the way in which this quantity is used.

For purposes of theoretical analysis, the greatest definiteness can apparently be achieved if by the thickness of the layer one understands the thickness \(t_v\) that the layer would have if the same amount of metal were uniformly distributed over the surface with a density equal to that of the massive metal \((\rho_M)\). This value \(t_v\) may be measured by directly or indirectly weighing the layer and is equal to the volume of massive metal expended on coating a unit area of the underlying surface (but not to the total volume of all granules on this area, since the density of the metal in the granules \(\rho\) may differ from \(\rho_M\)).

Of course, \(t_v\) will coincide with the effective thickness \(t_{\mathrm{eff}}\) of the modeling layer only in the case when the layer actually

forms a continuous structure with \(\rho = \rho_m\), i.e., only for sufficiently thick (opaque) layers—usually at \(t \gtrsim 400 \div 500\) Å. For thin granulated layers one should expect a substantial difference between \(t_v\) and \(t_{\mathrm{eff}}\). The degree of this difference is conveniently characterized by the quantity \(q=\dfrac{t_v}{t_{\mathrm{eff}}}\), which we shall henceforth call the “filling factor.” The actually observed character of the dependence of \(q\) on \(t_{\mathrm{eff}}\) for layers of gold\(^{10,11}\) and silver\(^{12,13}\) is shown in Fig. 1.

a)

b)

Fig. 1. Dependence of the filling factor \(q\) on the effective layer thickness \(t_{\mathrm{eff}}\); \(a\)—gold (crosses—Schopper’s data\(^{10}\), circles—Male’s data\(^{11}\)); \(b\)—silver (crosses—recalculation of Philina’s data\(^{12}\), triangles—recalculation of Isiguro and Kuwabara’s data\(^{13}\)).

Let us note that the data of different authors differ substantially, which is quite natural if one takes into account the structural differences of the films they investigated, caused by the nonidentity of the technology of their preparation (see, for example, \(^{14}\)), as well as the uncertainty of the quantities \(t_{\mathrm{eff}}\) themselves, calculated on the basis of completely different initial ...

THEORY OF THE OPTICAL PROPERTIES OF SEMITRANSPARENT METALLIC COATINGS

data (transparency and reflectance in some works and phase shifts in others). Therefore Fig. 1 should be regarded only as indicative, making it possible to judge the character of the process of layer compaction, but not describing it quantitatively. This must be borne in mind when comparing theoretical expectations with experiment.

Many works are devoted to the determination of the effective optical constants \(n_{\mathrm{eff}}\) and \(\varkappa_{\mathrm{eff}}\) for films made of various metals under various conditions (see \(^{2-5}\)). The main conclusion that can be drawn from their analysis is that these quantities, which vary substantially depending on the technology of preparing the film and on the method of determining the constants themselves, undergo strong and, in general, regular changes as a function of the layer thickness, and that their spectral dependences, again different for layers of different thickness, have little in common with the spectral dependences of the optical constants of the original material.

If one disregards the scatter of values associated with differences in the individuality of the films themselves and in the methods of measuring them, then a clear tendency is observed toward a decrease of \(\varkappa_{\mathrm{eff}}\) and an increase of \(n_{\mathrm{eff}}\) as \(t\) decreases. This is illustrated by Fig. 2, which convincingly shows that for \(t < 100\ \text{\AA}\) the substance of the film loses metallic properties and acquires semiconductor and even dielectric properties.

Fig. 2. Dependence of the effective refractive and absorption indices of a silver film on its thickness \(t_B\), according to the data of Ishiguro and Kuwabara\(^{13}\).

It is curious to note that measurements of the electrical conductivity of thin films\(^{3}\) lead to an analogous result. However, in the latter case the decisive factor is apparently not a change in the properties of the substance itself, but an increase in the role of contacts between granules.

It is very important that significant (by a factor of \(1.5\)—\(2\)) changes in the effective optical constants occur with comparatively insignificant changes of \(t_{\mathrm{eff}}\). There is no reason to suppose that in small granules with diameters of tens of angstroms, i.e. consisting of many thousands of atoms, the metal has the same optical characteristics as in massive specimens. A whole series of factors, such as the dependence of the electron mean free path on the dimensions of the specimen, the hypertrophied role of surface energy levels, the presence of inevitable impurities disrupting the structure of the metal, the presence of powerful surface forces, etc., must lead

to their radical change. At the same time, however, all these changes must, in the main, occur during the disintegration of massive material into granules, whereas, for comparatively small changes in the sizes of the granules themselves, they can be expected only to a very insignificant degree. Thus we inevitably arrive at the conclusion that the dependence of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) on \(t_{\mathrm{eff}}\) cannot be attributed to changes in the properties of the substance forming the granules themselves, and that the effective values \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) can in no way be identified with the optical constants of the metal itself.

3. THE MAXWELL–GARNETT THEORY OF THE “ARTIFICIAL DIELECTRIC”

Long before the granularity of covering metallic layers was reliably discovered with the aid of the electron microscope, the peculiar coloration of metallic films led a number of investigators \(^{15-18}\) to the idea of their colloidal structure. The first theory of the optical properties of colloidal solutions was developed and applied to the use of the optical properties of semitransparent metallic films by Maxwell Garnett \(^{19}\).

The basic idea of Maxwell Garnett was that sufficiently small colloidal metal particles can, like molecules of a dielectric, be regarded as dipoles. Therefore a medium with metallic inclusions randomly embedded in it should behave like an ordinary dielectric, but with a changed refractive index. The validity of the idea itself is not in doubt, and at the present time it has received a sufficiently detailed development in the theory of so-called “artificial dielectrics,” widely used in radio engineering. At the same time, the development of the theory, taking into account a number of effects (for example, the Hall effect) that are substantial for the radio range, has not affected the conclusions relating to the visible region of light. Therefore the original Maxwell–Garnett theory fully retains its significance, at any rate to the extent that one may neglect electric and magnetic moments of higher multipolarity induced in the metallic particles.

As is known, allowance for higher multipoles may be made on the basis of the theory of diffraction by a sphere developed by Mie. This has given rise to the erroneous opposition, widespread in the literature, between the theories of Mie and Maxwell–Garnett. In reality the named theories refer to completely different phenomena. Mie’s theory is confined to considering the scattering of light by a single particle, whereas Maxwell–Garnett’s theory, proceeding from the notion of Rayleigh scattering, takes account of the mutual irradiation of particles, i.e. it is a theory of the propagation of light in a medium with dipole-scattering particles and, consequently, belongs among theories of dispersion. In a more general form, with allowance for multi-

THEORY OF THE OPTICAL PROPERTIES OF SEMITRANSPARENT METALLIC COATINGS

complete scattering, the theory of the propagation of light in scattering media was developed by G. V. Rozenberg\(^{20,21}\). However, as applied to thin metallic films, taking account of the Mi effects, as direct calculation shows\(^{22}\), cannot lead to corrections exceeding a few percent, i.e., in the present state of the theory it is superfluous.

If the volume occupied by the colloid has, in all three dimensions, sizes much exceeding the distances between the colloidal particles, and if the latter are incomparably smaller than the wavelength of light, then, following Maxwell-Garnett, one may use the Lorentz–Lorenz formula, i.e., assume that the effective complex dielectric constant of the colloidal medium is equal to

\[ \varepsilon_{\mathrm{eff}} = n_{\mathrm{eff}}^{2} = 1 + \frac{4\pi N' \alpha}{1 - \frac{4\pi}{3} N' \alpha}, \tag{2} \]

where

\[ \alpha = \frac{n'^2 - 1}{n'^2 + 2}\, a^3 \tag{3} \]

is the polarizability of metallic spheres having radius \(a\), \(n'\) is the complex refractive index of the metal in the granules, \(N'\) is the number of particles per unit volume of the colloid, and it is assumed that the metal particles are suspended in air. Allowing for the dielectric constant of the medium in which the particles are suspended presents no difficulty and was also carried out by Maxwell-Garnett. If the density of the metal in the granules is \(\rho\), then the total volume of massive metal sprayed into a unit volume of the colloid, i.e., the filling factor \(q\), is obviously equal to

\[ q = \frac{4\pi}{3}\, N'a^3\, \frac{\rho_{\mathrm{M}}}{\rho}, \tag{4} \]

whence

\[ n_{\mathrm{eff}}^{2} = (n_{\mathrm{eff}} - i\chi_{\mathrm{eff}})^2 = 1 + \frac{3\,\frac{\rho}{\rho_{\mathrm{M}}}\, q\, \frac{n'^2 - 1}{n'^2 + 2}} {1 - \frac{\rho}{\rho_{\mathrm{M}}}\, q\, \frac{n'^2 - 1}{n'^2 + 2}} . \tag{5} \]

In other words, the effective optical constants of the colloid depend exclusively on the filling factor \(q\) and on the characteristics of the substance forming the particles \((\rho, n')\), but do not depend either on the concentration or on the sizes of the particles separately. This, in particular, eliminates the need to take into account the distribution of particles by size, which, of course, is connected with neglecting multipole effects. Further, it follows from (5) that the resonant character of the dependence of \(n_{\mathrm{eff}}\) on the film thickness is due not to dipole resonance, but to the resonant character of the mutual influence of the particles.

Maxwell-Garnett assumes that the properties of the metal do not change when it is crushed into granules, i.e., \(\rho=\rho_m\) and \(n'=n\), and considers a homogeneous covering layer of thickness \(t_{\mathrm{eff}}\), made of a substance with complex refractive index \(n_{\mathrm{eff}}\), determined by expression (5). In other words, here again a real metallic layer is modeled by a homogeneous single-layer coating, whose effective parameters this time are known functions of the filling factor \(q\). Thus the Maxwell-Garnett theory, already in its initial assumptions, suffers from all the shortcomings of the single-layer model.

Fig. 3. Dependences of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) on the filling factor \(q\) for \(\chi=4\), \(n=0, 1, 2, 4\) (after Malé\(^{23}\)).

The character of the dependence of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) on \(q\) was calculated by Malé\(^{23}\) for \(\chi=4\) and \(n=0, 1, 2, 4\). The curves he obtained (Fig. 3) show that the effective constants can differ substantially from the constants of the substance forming the particles. In particular, metallic particles at \(q \lesssim 0.8\) form a colloid with typical dielectric properties.

Using (5), it is not difficult to show\(^{14}\) that

\[ \left. \begin{aligned} 2n_{\mathrm{eff}}\chi_{\mathrm{eff}} &= \frac{6qB}{(1-qA)^2+4q^2B^2},\\ \chi_{\mathrm{eff}}^2-n_{\mathrm{eff}}^2 &= 2-\frac{3(1-qA)}{(1-qA)^2+4q^2B^2}, \end{aligned} \right\} \tag{6} \]

where

\[ \left. \begin{aligned} A&=\frac{(\chi^{2}-n^{2}-1)(\chi^{2}-n^{2}-2)-4n^{2}\chi^{2}} {(\chi^{2}-n^{2}-2)^{2}+4n^{2}\chi^{2}},\\[4pt] B&=\frac{3n\chi} {(\chi^{2}-n^{2}-2)^{2}+4n^{2}\chi^{2}} . \end{aligned} \right\} \tag{7} \]

Direct differentiation of expressions (6) leads to the conclusion that the dependence of \(n_{\mathrm{eff}}\chi_{\mathrm{eff}}\) (and consequently also of the absorptive power of the layer) on \(q\), i.e. ultimately on \(t\), has a maximum under the condition

\[ A^{2}+4B^{2}>1, \tag{8} \]

and, if \(B\ll A\), this maximum is situated at \(q=\dfrac{1}{A}\) and is proportional to \(\dfrac{1}{qB}\). Comparison of these conclusions with data on the dependence of the absorptive power of films of various metals on their thickness and wavelength has shown\(^{14,24-26}\) that there is qualitative agreement. Maxima on the curves of the dependence of the absorptive power of a film on its thickness are observed only in those cases when condition (8) is satisfied, and their position and height depend on the wavelength approximately as required by the theory.

Fig. 4. Comparison of spectral dependences of the quantities \(2n_{\mathrm{eff}}\chi_{\mathrm{eff}}\) (solid curves) and \(n_{\mathrm{eff}}^{2}-\chi_{\mathrm{eff}}^{2}\) (dashed curves) for the oscillator and for gold layers of various thicknesses according to the data of Goos\(^{24}\).

Thus, the Maxwell–Garnett theory makes it possible to form a qualitatively correct picture of the optical properties of thin-layer coatings. This convincingly indicates that the principal factor determining these properties is granularity, with resonance effects clearly coming to the fore.

The latter is illustrated, for example, by Fig. 4, borrowed from the work of David\(^{26}\), and representing a comparison of the spectral

dependencies of the quantities \(2n_{\mathrm{eff}}\chi_{\mathrm{eff}}\) and \(n_{\mathrm{eff}}^{2}-\chi_{\mathrm{eff}}^{2}\), calculated from Goos’s data\({}^{24}\) for gold layers of different thicknesses, with the corresponding curves for an oscillator. It should be noted that attention was drawn to the existence, and also to the decisive role, of resonance phenomena already in the early works of Kosonogov\({}^{16}\) and Wood\({}^{17}\).

However, while correctly revealing the nature of the optical “anomalies” characteristic of thin metal films, the Maxwell–Garnett theory does not provide a quantitative explanation of them.

In Fig. 5 the dependences \(n_{\mathrm{eff}}\chi_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}^{2}-n_{\mathrm{eff}}^{2}\), calculated according to the Maxwell–Garnett theory and determined from experiment, are compared as functions of \(q\) for gold films\({}^{10}\).

Fig. 5. Comparison of the dependences \(n_{\mathrm{eff}}\chi_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}^{2}-n_{\mathrm{eff}}^{2}\) on \(q\), calculated from experimental data and according to the Maxwell–Garnett theory for gold, according to Shopper’s data\({}^{10}\).

\(\chi_{\mathrm{eff}}^{2}-n_{\mathrm{eff}}^{2}\) on \(q\) for gold films\({}^{10}\). Comparison of these curves indicates a complete absence of quantitative agreement. Indeed, such agreement was hardly to be expected. We have already said above that the optical properties of a metal in a granulated state must differ from those in massive samples, and also that the model of a homogeneous layer is inapplicable to real metallic coatings. However, the most essential circumstance, in our opinion, is the inadequacy of the basic premise of the Maxwell–Garnett theory

on the applicability of the Lorentz–Lorenz formula. The condition for its applicability in the present case reduces to the requirement of the three-dimensionality of the colloid, i.e., that the thickness of the film be much greater than the distances

Fig. 6. Electron microphotographs of silver layers of various thicknesses[^14].

between the particles. In reality, however, for \(t \lesssim 100\,\text{Å}\) we are dealing with a two-dimensional colloid, as is clearly evidenced, for example, by Fig. 6, which is a series of electron microphotographs

microphotographs of silver coatings of various thicknesses^14. However, before turning to the clarification of ways of overcoming this limitation of the Maxwell–Garnett theory, we must briefly dwell on attempts to take into account the nonsphericity of the shape of colloidal particles.

4. THE INFLUENCE OF PARTICLE SHAPE ^26,10

The question of the influence of particle shape on the optical properties of the layer arose precisely as an attempt to correct the Maxwell–Garnett theory in order to achieve better agreement with experiment. Having no sufficiently reliable data on the actual shape of the particles, David^26 assumed that the latter are not spheres, but ellipsoids of revolution with an axis of symmetry located normal to the surface of the layer. The polarizability of such an ellipsoid under conditions in which the electric vector of the light wave is perpendicular or parallel to the axis of symmetry of the ellipsoid may be written in the form

\[ \alpha=\frac{1}{4\pi}\, \frac{n'^2-n_0^2}{\dfrac{n'^2-n_0^2}{n_0^2}\,f+1}\,v, \tag{9} \]

where \(n'\) is the complex refractive index of the substance forming the particle, \(n_0\) is the refractive index of the surrounding medium, \(v\) is the volume of the particle, and \(f\) is a parameter that depends uniquely on the ratio \(s\) of the length of the axis of revolution of the ellipsoid to the length of the axis perpendicular to it. The dependences of \(f\) on \(s\) for a wave whose electric vector is directed along and across the axis of revolution of the ellipsoid are shown in Fig. 7.

Fig. 7. Dependences of the quantity \(f\) on the ratio \(s\) of the axes of an ellipsoid: \(f_1\)—the electric vector of the wave is perpendicular to the axis of revolution of the ellipsoid; \(f_2\)—the electric vector of the wave is parallel to the axis of revolution of the ellipsoid.

David clearly recognized the two-dimensional character of the problem of a covering film and the necessity of abandoning the Lorentz–Lorenz formula. However, he went too far along this path, completely neglecting the mutual influence of the particles, i.e. the difference between the effective field and the field of the wave irradiating the layer. With a comparatively dense packing of particles in the layer such neglect, as we shall see below, is inadmissible. At the same time, David did not understand that the two-dimensionality of the structure is incompatible with the representation

of a continuous covering layer possessing certain effective optical constants. Therefore, the problem posed and solved by him does not in essence relate to a two-dimensional colloid, but reduces to the determination of \(n_{\mathrm{eff}}\) and \(\chi_{\mathrm{eff}}\) of a three-dimensional medium consisting of a suspension of oriented, but mutually noninteracting, ellipsoidal particles.

Under these assumptions,

\[ n_{\mathrm{eff}}^{2}-n_{0}^{2}=4\pi\alpha N', \tag{10} \]

where \(N'\) is the number of particles per unit volume, or, using (9) and recalling that the filling factor \(q\) in the present case is equal to (if \(\rho=\rho_M\))

\[ q=N'v, \tag{11} \]

we have:

\[ n_{\mathrm{eff}}^{2}-n_{0}^{2} = \frac{n'^{2}-n_{0}^{2}} {\dfrac{n'^{2}-n_{0}^{2}}{n_{0}^{2}}\,f+1} \,q. \tag{12} \]

Thus, besides \(q\), one more parameter, \(f\), has entered the formula; it characterizes the shape of the particles and, since the latter is unknown, plays the role of an adjustable parameter.

Schopper\(^{10}\) experimentally determined the effective parameters and the filling factor for several layers of gold and, taking \(n_{0}=1\) (air), calculated for each of the layers the value of the parameter \(f\), using the real part of equality (12) (i.e. the value

\[ \frac{1}{q}\left(n_{\mathrm{eff}}^{2}-\chi_{\mathrm{eff}}^{2}\right), \]

and the values of \(n\) and \(\chi\) for the bulk metal. However, substitution of these values into the imaginary part of equality (12) showed that the quantities \(n_{\mathrm{eff}}\chi_{\mathrm{eff}}\) calculated in this way do not correspond to what is obtained from the experimental data. In order to improve the agreement with experiment, Schopper took into account the dependence of \(f\) on the orientation of the particles, introducing (without proper justification) a distribution function of the particles over angles of the form

\[ g(f)\,df=\mathrm{const}\cdot f\exp\left(-\frac{f^{2}}{\bar f^{\,2}}\right)\,df, \]

where \(\bar f\) is a parameter characterizing the position of the maximum of the distribution, while the constant is determined from the normalization condition. Then, calculating the statistical mean of the right-hand side of expression (12) and comparing its real part with the values obtained from experiment \(\left(n_{\mathrm{eff}}^{2}-\chi_{\mathrm{eff}}^{2}-1\right)\), Schopper found \(\bar f\) for each of the layers, and then, using these values, calculated the quantity \(\dfrac{1}{q}\,n_{\mathrm{eff}}\chi_{\mathrm{eff}}\).

A comparison of the values of this quantity obtained in this way with its values calculated directly from the experimental data is shown in Fig. 8. At first glance the agreement seems more than satisfactory, but upon closer examination

it is found that the discrepancies are very large and cannot be eliminated without completely destroying the agreement in the real part of equality (12). In this connection it should also be taken into account that the product \(n_{\mathrm{eff}}x_{\mathrm{eff}}\) is much less sensitive to changes in the film thickness than each of the factors separately. Thus the David–Schopper theory, while revealing the great importance and the character of the influence of particle shape on the optical properties of the layer, can in no way claim to give a quantitative explanation of the latter. Let us note that the resonance character of the dependence of \(\eta_{\mathrm{eff}}\) on \(t\), according to David–Schopper, is determined not by the interaction of particles but by the dipole resonance on isolated particles (the resonance character of the dependence of the polarizability of particles on their shape).

Fig. 8. Comparison of the experimentally observed dependence \(\frac{1}{q} n_{\mathrm{eff}} x_{\mathrm{eff}}\) on the shape factor \(\bar f\) (according to Schopper’s data) with the theoretically expected one.

Knowledge of \(f\) (or \(\bar f\)) enabled Schopper to determine, with the aid of the graph in Fig. 7, the ratio \(s\), i.e. the shape of the particles. He thereby found that the gold particles on the average have a strongly flattened shape \(\left(s \simeq \frac{1}{8}\right)\). This result is difficult to reconcile with the data of electron-microscopic and electron-diffraction investigations of gold layers. Although there are a number of indications of the flattening of gold particles forming the layer, it is apparently very small and arises chiefly in the process of their coalescence into a two-dimensional gel. As for the range of thicknesses in which the particles may be regarded as isolated from one another, i.e. where the David–Schopper theory may be applied, the gold particles in it apparently have a shape only slightly different from spherical. Thus this result of Schopper should be counted among the facts testifying rather to the inadequacy of the theory than to its successes.

5. REJECTION OF THE MODEL OF A HOMOGENEOUS COVERING LAYER AND THE THEORY OF THE OPTICAL PROPERTIES OF A TWO-DIMENSIONAL COLLOIDAL COATING

The necessity of abandoning the modeling of a two-dimensional covering layer by a homogeneous three-dimensional one, and the impossibility of assigning effective optical constants to this layer, was first clearly revealed in connection with the theory of the transition layer on the surface of a solid...

dies or liquids. The most complete theory of a surface transition layer was developed by D. V. Sivukhin^27, who showed that the optical properties of such a layer are completely characterized by its specific polarizability, i.e., by the electric moment per unit surface area of the layer referred to the unit field strength of the wave irradiating the layer (in D. V. Sivukhin’s theory the layer parameters are the ratios of the components of this quantity along the coordinate axes to the components of the polarization vector of the medium on which the layer rests). At the same time D. V. Sivukhin showed that the main problem in the theory of the optical properties of a cover layer is the calculation of the effective field in which the particles are situated, i.e., the establishment of the relation between the polarizability of the layer as a whole and the polarizability of the particles constituting it. The solution of this problem already requires specific model concepts concerning the structure of the layer, and it cannot be solved in general form.

Since D. V. Sivukhin’s theory applies only to transition layers having the same nature as the underlying substance, and since the method he used for calculating the effective field is inapplicable in the case of granulated metallic coatings, we shall not give the details here and shall refer the reader to the original papers.

The optical properties of a two-dimensional colloidal coating having a nature different from that of the underlying medium were considered by G. V. Rozenberg^22,20.

In accordance with the data of electron-microscopic investigations of metallic films (see, for example, Fig. 6), the particles were assumed to be so small that one may confine oneself to taking into account only their dipole moments and completely neglect Mie effects. At the same time it was assumed that the surface concentration of the particles \(N\) is sufficiently large and that the distances between them are incomparably smaller than the wavelength of light. This assumption is of fundamental importance, since it permits the layer to be regarded as a quasi-homogeneous structure in its plane, which radically simplifies the mathematical treatment. A direct calculation of the interference of waves scattered by all the particles of the layer under conditions in which the layer is irradiated by a plane wave (more precisely, by a wave emitted by an infinitely remote dipole) makes it possible to relate the induced radiation (scattering) of the layer forward and backward to the mean polarizability of the particles, their surface concentration, and the magnitude of the effective field in the layer \((E^{\mathrm{eff}})\). If the layer is located in the plane \(z=0\) and the plane of incidence coincides with the plane \(yz\), then the components of the electric-field strength of the wave scattered by the layer are equal to

\[ \left. \begin{aligned} E_x^{\mathrm{scat}}&=E_x^{\mathrm{eff}}\eta C_x\,\frac{1}{\cos\vartheta}\,e^{\mp ikz\cos\vartheta},\\ E_y^{\mathrm{scat}}&=\left(E_y^{\mathrm{eff}}\eta C_y\cos\vartheta \pm E_z^{\mathrm{eff}}\eta C_z\sin\vartheta\right)e^{\mp ikz\cos\vartheta},\\ E_z^{\mathrm{scat}}&=\left(E_z^{\mathrm{eff}}\eta C_z\,\frac{\sin^2\vartheta}{\cos\vartheta} \pm E_y^{\mathrm{eff}}\eta C_y\sin\vartheta\right)e^{\mp ikz\cos\vartheta}, \end{aligned} \right\} \tag{13} \]

where

\[ \eta=\frac{2\pi N}{k^2} \tag{14} \]

is the specific surface concentration of particles,

\[ C_{x,y,z}=-ik^3\alpha_{x,y,z}, \tag{15} \]

\(\alpha_x,\ \alpha_y,\ \alpha_z\) are the components of the mean polarizability of the particles along the coordinate axes, \(k=\dfrac{2\pi}{\lambda}\) is the wave number, \(\vartheta\) is the angle of incidence, positive for the wave scattered forward and negative for the wave reflected by the layer, while the sign in the exponent is chosen in accordance with the requirement that the wave propagate away from the layer.

The calculation of the effective field in the layer can be carried out by three essentially different methods. First, one may directly compute the total radiation field of all dipoles at the place where one of them is located, if it is assumed that the particles are distributed chaotically in the layer and that each of them, on the average, occupies in the layer an area of magnitude \(\dfrac{1}{N}\). Second, one may regard the three-dimensional colloid as a set of two-dimensional colloidal layers separated from one another by a distance

\[ 2\xi=\frac{N}{N'}=\frac{4}{3\sqrt{\pi N}}, \tag{16} \]

where \(N'\) is the volume concentration of particles. To such a three-dimensional colloid the Lorentz–Lorenz formula is applicable. Next, removing one of the layers and taking into account the interference in the gap thus formed, one can find the field inside the gap and the wave reflected from it in the medium. Finally, once again introducing the removed layer and imposing the condition that the waves scattered by it, after leaving the gap, extinguish the wave reflected from it, we obtain an expression relating the effective field in the layer to the field of the wave illuminating it. (The same result is obtained if the layer is placed on the surface of a continuous medium of the same nature and it is assumed that the influence of the layer reduces only to a displacement of the surface of the medium by a distance \(2r_1^{22}\).) Third, taking into account the interference of the illuminating and scattered waves, one can find the absorption in the layer, which, obviously, is equal to the total absorption of all particles located in the effective field; this again makes it possible to determine the latter, if the mean absorbing capacity of an individual particle is known. All these methods, according to

lead to the expressions

\[ \left. \begin{aligned} E_x^{\mathrm{eff}} &= \frac{E_{0s} e^{-iky\sin\vartheta}} {1-\left(\dfrac{i\eta^{1/2}}{2\sqrt{2}}+\dfrac{1}{\cos\vartheta}\right)\eta C_x}, \\[6pt] E_y^{\mathrm{eff}} &= \frac{E_{0p}\cos\vartheta\, e^{-iky\sin\vartheta}} {1-\left(\dfrac{i\eta^{1/2}}{2\sqrt{2}}+\cos\vartheta\right)\eta C_y}, \\[6pt] E_z^{\mathrm{eff}} &= \frac{E_{0p}\sin\vartheta\, e^{-iky\sin\vartheta}} {1-\left(-\dfrac{i\eta^{1/2}}{\sqrt{2}}+\dfrac{\sin^2\vartheta}{\cos\vartheta}\right)\eta C_z}. \end{aligned} \right\} \tag{17} \]

where \(E_{0s}\) and \(E_{0p}\) are the \(s\)- and \(p\)-components of the electric-field strength of the wave irradiating the layer at the origin of coordinates.

Substitution of (17) into (13) then makes it possible to find the \(s\)- and \(p\)-components of the wave scattered by the layer:

\[ \left. \begin{aligned} E_s^{\mathrm{scatt}} &= P_s E_{0s} e^{\mp ikz\cos\vartheta-iky\sin\vartheta},\\ E_p^{\mathrm{scatt}} &= (P_{p1}\pm P_{p2})E_{0p} e^{\mp ikz\cos\vartheta-iky\sin\vartheta}, \end{aligned} \right\} \tag{18} \]

where

\[ \left. \begin{aligned} P_s &= \frac{\eta C_x} {\cos\vartheta-\eta C_x\left(1+\dfrac{i\eta^{1/2}\cos\vartheta}{2\sqrt{2}}\right)}, \\[6pt] P_{p1} &= \frac{\eta C_z\sin^2\vartheta} {\cos\vartheta-\eta C_z\left(\sin^2\vartheta-\dfrac{i\eta^{1/2}\cos\vartheta}{\sqrt{2}}\right)}, \\[6pt] P_{p2} &= \frac{\eta C_y\cos^2\vartheta} {\cos\vartheta-\eta C_y\left(\cos^2\vartheta+\dfrac{i\eta^{1/2}\cos\vartheta}{2\sqrt{2}}\right)}. \end{aligned} \right\} \tag{19} \]

and the signs \(+\) and \(-\) refer respectively to waves scattered by the layer forward and backward.

So far we have considered a two-dimensional colloid in the absence of a substrate surface. The influence of the latter is not difficult to take into account if one recalls the quasi-homogeneity of the layer in its plane. Since the particles are simply scattered over the surface, it should be assumed that the center of the layer is separated from the surface by a distance \(\xi\), equal to the mean radius of a particle (or to the mean value of the length perpendicular

to the surface of the axis of the ellipsoid of revolution, if a nonspherical shape is assigned to the particles; owing to the symmetry of the conditions it should be assumed that, on the average, \(C_x=C_y\) and that the axis of symmetry of the ellipsoid is normal to the surface. Then, taking into account the boundary conditions at the surface of the underlying medium and the fact that the layer is irradiated not only by the incident wave, but also by the wave reflected from this surface, we obtain, for the amplitude coefficients of transparency \((\tau)\) and reflection \((r)\) of the coating (for the \(s\)- and \(p\)-components of the light wave), the expressions

\[ \left. \begin{aligned} r_s &= \frac{r_{s0}(1+2P_s)+P_s e^{2ik\xi\cos\vartheta}} {1-r_{s0}P_s e^{-2ik\xi\cos\vartheta}}, \\[6pt] r'_s &= \frac{-r_{s0}+P_s e^{-2ik\xi\cos\vartheta}} {1-r_{s0}P_s e^{-2ik\xi\cos\vartheta}}, \\[6pt] \tau_s &= \tau_{s0}\frac{1+P_s} {1-r_{s0}P_s e^{-2ik\xi\cos\vartheta}} = \frac{\tau_{s0}}{\tau'_{s0}}\,\tau'_s, \\[8pt] r_p &= \frac{(1+P_{p1}+P_{p2})^2} {1-r_{p0}(P_{p1}-P_{p2})e^{-2ik\xi\cos\vartheta}} +(P_{p1}-P_{p2})e^{2ik\xi\cos\vartheta}, \\[6pt] r'_p &= \frac{-r_{p0}+(P_{p1}-P_{p2})e^{-2ik\xi\cos\vartheta}} {1-r_{p0}(P_{p1}-P_{p2})e^{-2ik\xi\cos\vartheta}}, \\[6pt] \tau_p &= \tau_{p0} \frac{1+P_{p1}-P_{p2}} {1-r_{p0}(P_{p1}-P_{p2})e^{-2ik\xi\cos\vartheta}} = \frac{\tau_{p0}}{\tau'_{p0}}\,\tau'_p, \end{aligned} \right\} \tag{20} \]

where \(r_{s0}\), \(r_{p0}\), \(\tau_{s0}\), and \(\tau_{p0}\) are the Fresnel coefficients of reflection and transparency for the \(s\)- and \(p\)-components in the absence of the coating, and, as usual, quantities corresponding to irradiation of the layer from the side of the underlying surface are marked by a prime.

The transition from the amplitude coefficients of transparency and reflection to the energy coefficients may be made by the usual formulas (if the media bounding the layer are free of absorption):

\[ \left. \begin{aligned} &\text{reflection coefficients} \\[-2pt] &\qquad R=rr^{*}, \qquad R'=r'r'^{*}, \\[6pt] &\text{transparency coefficient (by flux)} \\[-2pt] &\qquad T=n_0\frac{\cos\vartheta'}{\cos\vartheta}\,\tau\tau^{*} = \frac{\cos\vartheta}{n_0\cos\vartheta'}\,\tau'\tau'^{*}, \end{aligned} \right\} \tag{21} \]

where \(n_0\) is the refractive index of the underlying medium and \(\vartheta'\) is the angle of refraction in this medium. The phase shifts can be calculated from (20) in the usual way.

It remains now to relate the parameters \(P_s\), \(P_{p1}\), and \(P_{p2}\) to the thickness of the layer and to the optical constants of the substance forming the particles. For this purpose we introduce the notation

\[ \begin{aligned} a_1+i b_1&=\frac{v}{4\pi\alpha_x}=\frac{v}{4\pi\alpha_y},\\ a_2+i b_2&=\frac{v}{4\pi\alpha_z}, \end{aligned} \tag{22} \]

where, as before, \(v\) is the mean volume of a particle. Then, taking into account (14) and (15), we find:

\[ \begin{aligned} \eta C_x=\eta C_y&=-i\,\frac{k}{2}\,\frac{t_{\mathrm{в}}}{a_1+i b_1},\\ \eta C_z&=-i\,\frac{k}{2}\,\frac{t_{\mathrm{в}}}{a_2+i b_2}, \end{aligned} \tag{23} \]

where, by definition, the weight thickness of the layer is

\[ t_{\mathrm{в}}=Nv \tag{24} \]

(it is assumed that \(\rho=\rho_m\)).

Substituting (23) into (13), we find:

\[ P_s=-\frac{i}{2}\, \frac{k t_{\mathrm{в}}} {\left(a_1-\frac{\eta^{1/2}}{4\sqrt{2}}k t_{\mathrm{в}}\right)\cos\vartheta +i\left(b_1\cos\vartheta+\frac{1}{2}k t_{\mathrm{в}}\right)}, \]

\[ P_{p1}=-\frac{i}{2}\, \frac{k t_{\mathrm{в}}\sin^2\vartheta} {\left(a_2+\frac{\eta^{1/2}}{2\sqrt{2}}k t_{\mathrm{в}}\right)\cos\vartheta +i\left(b_2\cos\vartheta+\frac{1}{2}k t_{\mathrm{в}}\sin^2\vartheta\right)}, \]

\[ P_{p2}=-\frac{i}{2}\, \frac{k t_{\mathrm{в}}\cos^2\vartheta} {\left(a_1-\frac{\eta^{1/2}}{4\sqrt{2}}k t_{\mathrm{в}}\right)\cos\vartheta +i\left(b_1\cos\vartheta+\frac{1}{2}k t_{\mathrm{в}}\cos^3\vartheta\right)}. \tag{25} \]

In accordance with (9), for particles having, on the average, the form of an ellipsoid of revolution about an axis normal to the surface,

\[ a_j+i b_j=f_j-\frac{1}{n^2-1}, \tag{26} \]

where the dependences of \(f_j\) \((j=1,2)\) on the ratio \(s\) of the axes of the ellipsoid are shown in Fig. 7.

Thus, formulas (20), (25), and (26) make it possible to relate the optical characteristics of a granulated coating to the thickness of the layer \(t_{\mathrm{в}}\), the specific surface concentration of particles \(\eta\), the shape

particles ($f$), by the optical constants ($n$ and $\chi$) of the substance forming them, and also by the refractive index $n_0$ of the underlying medium for any angles of incidence of the light beam. At the same time, it is seen from (25) that the role of the interaction of the particles, manifested in the difference between the effective field in the layer and the field of the incident wave, is very large and is even decisive with respect to a number of optical anomalies characteristic of thin layers.

Let us also note that formulas (17), in the case of a two-dimensional colloid, are the equivalent of the Lorentz–Lorenz formula pertaining to the three-dimensional case. The transition from one formula to the other can be carried out if one takes into account that, for a three-dimensional colloid formed by spherical particles embedded in a medium with refractive index $n_0$, the effective refractive index is determined by the relation[^22]

\[ n_{\mathrm{eff}}^2 - n_0^2 = \frac{iP}{k\xi}, \tag{27} \]

where $\xi$ is determined by formula (16), $P=P_s$ under the condition $\cos \vartheta = 1$, and $k$ is the wave number in the medium with refractive index $n_0$. In the case of molecules, formula (27) is identical with the Lorentz–Lorenz formula; in the case of colloidal particles, however, it contains an additional correction, which has a noticeable influence when the colloid concentration is decreased.

For the purpose of comparing the theoretical conclusions with experimental data, G. V. Rozenberg[^22],[^20] calculated the case of normal incidence of light on a silver film at $\lambda = 5400$ Å. Since, in accordance with the theory of Ya. I. Frenkel and the experiment of Canadian physicists,[^28] an increase in the thickness of the film is not accompanied by a change in the number of particles forming it, the calculations were based on the value $N = 5.5 \cdot 10^{11}\ \mathrm{cm}^{-2}$, obtained by direct counting of the particles in the upper photograph of Fig. 6, corresponding, according to the data,[^14] to $t \simeq 50$ Å. Further, since neither the shape of the particles nor the optical constants of the metal in the finely dispersed state are known, the constants $a_1$ and $b_1$ were regarded as fitting parameters (the only ones) and, in accordance with the considerations set out in Section 2, were considered independent of thickness. This meant that both the shape of the particles and the optical constants of the substance forming them were assumed to be independent of the film thickness.

The comparison was carried out with the dependences on $t$ of the transparency $T$, the absorptive capacity $A$, the reflecting powers $R$ and $R'$ and the corresponding phase shifts $\alpha$ and $\alpha'$ upon reflection, obtained for silver films by Fost[^29], as well as with the dependence of the phase shift upon the passage of light through a silver film ($\beta$) on $t$ according to Isiguro’s data[^30]—Figs. 9–14 (dotted lines). For the choice of the parameters $a_1$ and $b_1$, the condition of the minimum was used

Fig. 9. Theoretical and experimental dependences of \(T\) and \(A\) on \(t\) for silver at \(\lambda = 5400\ \text{Å}\).

Fig. 11. Theoretical and experimental dependence of \(\dfrac{\alpha'}{\pi}\) on \(t\) for silver at \(\lambda = 5400\ \text{Å}\).

Fig. 10. Theoretical and experimental dependences of \(R\) and \(R'\) on \(t\) for silver at \(\lambda = 5400\ \text{Å}\). The dash-dotted curve gives Rouard’s data for \(\lambda = 5720\ \text{Å}\).

Fig. 12. Theoretical and experimental dependence of \(\dfrac{\alpha'}{\pi}\) on \(t\) for silver at \(\lambda = 5400\ \text{Å}\).

of the curve of the dependence of \(T\) on \(t\) and the value of \(T\) at the minimum; moreover, for convenience of calculation the values of \(a_1\) and \(b_1\) obtained in this way were rounded off, and it was assumed that

\[ a_1=0.50,\qquad b_1=0.25. \]

As for the parameter \(\xi\), which characterizes the dimensions of the layer, it was assumed that, in the first approximation, the particles have a spherical shape, as a result of which \(\xi\) could be taken equal to the radius of the particles, i.e.,

\[ k\xi=\left(\frac{3k}{2\eta}\,t\right)^{1/3}. \]

Fig. 13. The theoretical (at \(\lambda=5400\ \text{Å}\)) and experimental (at \(\lambda=5900\ \text{Å}\)) dependences of \(\dfrac{\beta}{\pi}\) on \(t\) for silver.

The refractive index of the underlying surface, in accordance with Fost’s data, was taken to be \(n_0=1.56\). The subsequent calculations were carried out without any additional assumptions. The results obtained are shown in Figs. 9–14 by solid lines.

Comparison of the theoretical and experimental curves reveals quite satisfactory agreement—not only qualitative, but also quantitative—for all, without exception, optical properties of the films, especially if one takes into account that the value of \(N\) was taken on the basis of an electron microphotograph relating to a film prepared by another author and, consequently, under somewhat different conditions. To this it should be added that Fost’s measurements were made with broad-band light filters and that the values of \(t\) indicated by Fost and Isiguro do not coincide with \(t_e\). Sharp discrepancies begin at \(t>80\div100\ \text{Å}\), which is quite natural, since at these thicknesses coalescence of the granules begins and, thereby, the basic premises of the theory lose their force.

Of considerable interest is the presence of a “hump” on the theoretical curve of the dependence of \(R\) on \(t\), in the absence of such a hump on Fost’s experimental curve. Such a hump was observed on the curves of the dependence of \(R\) on \(t\) at \(\lambda\sim5000\ \text{Å}\) by almost all authors, including Sennett and Scott, from whose work Fig. 6 was borrowed. (The optical-measurement data of Sennett and Scott were evidently subjected by the authors to some recalculation and therefore are unsuitable for quantitative comparison with the theory.) It should therefore be supposed that, in the present case, Fost simply did not notice the “hump” because of the broad-band character of the light filters, since at larger wavelengths the “hump” disappears. For comparison, in Fig. 10 the dot-dashed lines show the re-

results of measuring the dependence of \(R\) on \(t\) according to Rouard’s data \(^{31}\), indicating the presence of a “hump.”

Thus, we are entitled to conclude that the peculiar optical anomalies inherent in thin metallic films, very diverse in character and different for different properties (\(R, R', T, a, a', \beta\)), find their explanation in the features of light scattering by a two-dimensional colloid, and by no means in the variability of the optical properties of the substance itself. We thereby return to the question posed at the beginning of the article—what information can be obtained about the properties of the granular substance itself by studying the optical properties of granular films?

First of all, let us note that the form of the curves of the dependence of the optical characteristics of a film on its thickness is very sensitive to changes in the parameters \(a\) and \(b\). Therefore, if the particle concentration \(N\) is determined sufficiently well by means of an electron microscope, and \(t_{\mathrm{в}}\) by weighing, then it is possible to determine \(a_j\) and \(b_j\) with sufficiently high accuracy. In particular, in the case described above, deviations of \(a_1\) and \(b_1\) from the adopted values by more than \(0.05\) are absolutely excluded. Further, the factors \(f_1\) and \(f_2\) can be determined by studying the angular dependence of the quantities \(a_j\) and \(b_j\). Thus estimates of the particle shape can be obtained; these estimates, in turn, can be checked by an electron-microscopic investigation of the profile of the film. All this gives grounds to hope that a comprehensive study of a series of films of one and the same metal, having different \(t_{\mathrm{в}}\), can provide reliable information on the optical constants of the metal in the finely dispersed—

Fig. 14. Theoretical and experimental diagrams of the dependence of the reflectivity from the side of the underlying surface (\(R'\)) on \(t\) for silver. The length of the radius vector drawn from the origin to the curve is proportional to \(R'\). The angle of rotation of the radius vector is equal to the phase shift \(\alpha'\). The numbers on the curve indicate the corresponding values of the film thickness \(t\).

states—a problem that is of great interest from the standpoint of metal physics. In particular, the validity of formula (9) must be checked, since it remains unclear to what extent such small metal particles may be assigned optical constants (anomalous skin effect!). This problem is also of interest in connection with testing the Mie theory, since, as is known, numerous experiments on its verification in colloidal suspensions of metals could give only qualitative confirmation of it, owing to the uncertainty in particle sizes both as a consequence of the imperfection of the methods for determining them and as a consequence of the polydispersity of the suspensions themselves.

The values \(a_1\) and \(b_1\) obtained above for silver particles make it possible, using (26), to calculate the optical constants of the metal as functions of \(f_1\), i.e., of the particle shape. The corresponding curves and the maximum error limits, depending on the limiting inaccuracy in the determination of \(a_1\) and \(b_1\) from the given experimental curves, are shown in Fig. 15.

From consideration of this figure it follows first of all that, without knowledge of the particle shape, the choice among the various possible values of \(n\) and \(\varkappa\) is immaterial. Further, it is obvious that for any acceptable particle shape the optical constants of silver in the granulated state have nothing in common with the optical constants of the massive metal. The latter, as can be shown, cannot in any way be reconciled with the optical properties of a real silver film, if, of course, one trusts the values of \(t\) reported by Fost and regards them as equal to \(t_{\mathrm{B}}\).

At the same time, the assumption that these constants remain unchanged when the particle diameter varies within the limits of approximately 50 to 150 Å is apparently confirmed. Both conclusions are, as we have seen, in complete agreement with theoretical expectations.

Attention should be drawn to the fact that formulas (20) cannot in any way be reduced to a form corresponding to a homogeneous single-layer coating. This thereby proves the inconsistency, noted above and long since apparent to experimentalists, of the single-layer model and the impossibility of characterizing the layer by effective parameters \(n_{\mathrm{eff}}\), \(\varkappa_{\mathrm{eff}}\), and \(t_{\mathrm{eff}}\). At the same time, in accordance with the general theorem of Herpin\(^7\), formulas (20) can be reduced to a form corresponding to a certain two-layer coating. In this case, however, in accordance with the same theorem, the parameters of such a two-layer model prove to depend on the angle of incidence of the light beam. At the same time, formulas (20), describing the behavior of all optical characteristics of the film at arbitrary angles of incidence and wavelengths, contain only four parameters to be determined and characterizing the shape and properties of the particles.

In conclusion, we note that for \(t \gtrsim 100\) Å, i.e., from the moment when coalescence of the granules begins, the theory of the two-dimensional colloid loses its force. Attempts to construct a theory of the optical properties

Fig. 15. Dependences of the optical constants of the substance forming the granules on the shape factor \(f\), for fixed values of \(a\) and \(b\).

two-dimensional and three-dimensional gel \(^{20}\) have not yet led to any notable successes and, evidently, should serve as the next stage in the development of the theory of the optical properties of metallic films. However, for a three-dimensional gel, i.e., at \(t \gtrsim 200\text{--}300\,\text{\AA}\), the inverted version of the Maxwell–Garnett theory may already become applicable (pores or air bubbles in a continuous body of metal \(^{20}\)), and it becomes possible to characterize the layer by the effective optical constants of the substance forming it.

6. MONOMOLECULAR COATING LAYERS

The study of the optical properties of monomolecular coating layers is of great interest in connection with a number of physicochemical problems, including the measurement of the geometrical dimensions of molecules, and also from the point of view of using the known deviations from the Fresnel laws in reflection from the surfaces of pure liquids for the investigation of surface phenomena. At present, following Langmuir and Blodgett \(^{32}\), Drude’s theory, referring to a homogeneous three-dimensional layer, is used to measure molecular dimensions, while deviations from the Fresnel laws in reflection from the surfaces of pure liquids are considered on the basis of the theory of D. V. Sivukhin \(^{27}\). The theory of a two-dimensional colloid developed above makes it possible to approach both cases from a unified point of view and to introduce certain refinements \(^{20}\).

Considering a surface monomolecular layer as a two-dimensional colloid, we can directly use formulas (20), which, owing to the smallness of \(P_s\), \(P_{p1}\), and \(P_{p2}\), take in the first approximation the form

\[ \left. \begin{aligned} r_s &= r_{s0} + (1+r_{s0})^2 P_s,\\ r'_s &= -r_{s0} + (1-r_{s0}^2)P_s,\\ r_p &= r_{p0} + (1+r_{p0})^2 P_{p1} - (1-r_{p0})^2 P_{p2},\\ r'_p &= -r_{p0} + (1-r_{p0}^2)(P_{p1}-P_{p2}). \end{aligned} \right\} \tag{28} \]

Next, expression (19) for \(P_s\), \(P_{p1}\), and \(P_{p2}\) in the first approximation can be written in the form \(^{20}\)

\[ \left. \begin{aligned} P_s &= -\,i3\,\frac{k\xi}{\cos\vartheta}\,\frac{u}{1-u},\\ P_{p1} &= -\,i3k\xi\,\frac{\sin^2\vartheta}{\cos\vartheta}\,\frac{w u}{1+2w u},\\ P_{p2} &= -\,i3k\xi\,\cos\vartheta\,\frac{u}{1-u}, \end{aligned} \right\} \tag{29} \]

where

\[ u=\frac{3}{w+2}\,\frac{n_c^2-1}{n_c^2+2} \quad \text{and} \quad w=\frac{a_z}{a_x} \tag{30} \]

is the mean anisotropy of the molecules in the layer, and \(n_c\) is the refractive index-

THEORY OF THE OPTICAL PROPERTIES OF SEMITRANSPARENT METALLIC COATINGS

of a bulk substance formed from the same particles as the layer and having the same density, but under the condition of a chaotic orientation of the particles in space (here it is taken into account that \(\xi=\dfrac{2}{3\sqrt{\pi N}}\), where \(\xi\) is the distance of the center of a particle of the coating layer from the surface of the underlying medium). It is not difficult to see that expressions (28) for \(w=1\) (an isotropic layer) coincide, in the first approximation, with the expressions following from the Drude theory, which is connected with the possibility of neglecting the higher terms in the expansion of the exponential factors. In higher approximations (for example, for granular metallic layers, and also as applied to quadratic effects) this equivalence is lost.

A monomolecular layer has a noticeable effect only on the reflectivity of the surface for the \(p\)-component in the vicinity of the Brewster angle \((\vartheta_{\mathrm{Br}})\). Expanding expressions (28) in a series in \(\vartheta\) and retaining only the first nonvanishing terms, we have:

\[ \left. \begin{aligned} r_p&=-i\alpha_p-\frac{1-n_0^4}{2n_0^3}(1-2i\alpha_p)(\vartheta-\vartheta_{\mathrm{Br}}),\\[6pt] r_s&=\frac{1-n_0^2}{1+n_0^2},\\[6pt] \alpha_p&=\frac{3k\xi}{\sqrt{\,n_0^2+1\,}} \left(\frac{n_0^2wu}{1+2wu}-\frac{u}{1-u}\right). \end{aligned} \right\} \tag{31} \]

Formulas (31) make it possible to determine the dependence of the energy reflection coefficient for the \(p\)-component and the degree of ellipticity

\[ q=2\,\frac{|r_s|\cdot |r_p|}{|r_s|^2+|r_p|^2} \sin\left[\arg r_p-\arg r_s\right] \]

of the light reflected at an angle of incidence in the vicinity of the Brewster angle as a function of the properties and state of the molecules forming the surface layer. Thus they can be compared with experiment and serve as the basis for solving the inverse problem—determining the state of the molecules in the surface layer. In particular, if the molecules of the surface layer do not possess absorption, then after simple transformations we find:

\[ q=6\,\frac{\sqrt{\,n_0^2+1\,}}{n_0^2-1}\,k\xi\, \frac{3}{w+2}\, \frac{(n_c^2-1)}{(n_c^2+2)} \times \]

\[ \times \left[ \frac{n_0^2w} {1+2w\,\dfrac{3}{w+2}\,\dfrac{(n_c^2-1)}{(n_c^2+2)}} - \frac{1} {1-\dfrac{3}{w+2}\,\dfrac{(n_c^2-1)}{(n_c^2+2)}} \right]. \tag{32} \]

This expression differs from the expression obtained by D. V. Sivukhin[^27] by the factor not taken into account by him,

\[ \frac{4}{3\sqrt{\pi}}, \]

which is connected with the distinction between the quantities \(N'\) and \(N'^{3/2}\), and also by the fact that here \(n_c\) differs from \(n_0\), i.e., no assumption is made about the identity of the concentration and the nature of the molecules in the surface layer and in the underlying substance, an assumption made implicitly in D. V. Sivukhin’s theory. This makes it possible to apply expression (32) to analyze the influence on the ellipticity of the reflected light not only of the orientation of molecules in the surface layer, but also of such factors as changes in the density of the surface film, surface contamination, etc., and also to use it for measuring the dimensions \(2\xi\) of the molecules of the surface layer (for which it is necessary to know the refractive index \(n_c\) of the substance forming the film, the refractive index \(n_0\) of the underlying substance, and the anisotropy \(w\) of the molecules in the surface layer).

Let us note that D. V. Sivukhin also considers, as an alternative possibility, other expressions for \(q\), which are obtained if one takes into account an additional irregular intermolecular field which, in his opinion, should exist. However, it is not difficult to see that the existence of such an irregular field is inadmissible on the average, for otherwise it would entail the failure of the Lorentz–Lorenz formula for massive isotropic bodies, for which, as is known, it is satisfied quite well. It is easy to see that the appearance of this irregular field in D. V. Sivukhin’s theory is connected with neglecting the influence of the film on the field in the underlying substance. Thus the choice among the alternative variants proposed by D. V. Sivukhin becomes, for isotropic media, unambiguous and leads to the expression discussed above.

7. CONCLUDING REMARKS

The theoretical considerations we have examined still require experimental verification in many respects. However, even now there is no doubt about the need to abandon those primitive notions which for too long, and in spite of numerous experimental indications, continue to exist in works devoted to the study of thin films, and thereby hinder the correct analysis of experimental data and, at the same time, the choice of the proper direction for experimental research in this field. It seems to us that at the present time the most urgent task is a comprehensive and all-sided investigation of the structure and optical properties of a certain number of samples, so that, having sufficiently detailed data on the angular and spectral dependences of all six optical characteristics of the layer \((T, R, R', \alpha, \alpha'\) and \(\beta)\) for layers of different thicknesses, it would be possible to form a clear idea of the correspondence between theoretical pred—

tions and reality. At the same time this will make it possible to reveal those real possibilities which the study of metallic films opens up for investigating the properties of colloidal particles of a metal. Further, it seems desirable to develop a theory of the optical properties of metallic layers of intermediate thickness \((t \sim 100—300\ \text{Å})\), having the structure of a two-dimensional and three-dimensional gel.

On the other hand, a more detailed experimental investigation of the optical properties of monomolecular coatings is necessary from the standpoint of comparing them with theoretical concepts and of revealing the real possibilities and limits of applicability of this method of studying surface layers.

Finally, the questions of the theoretical metal optics of colloidal particles and the closely related questions of the theory of colloidal dyes remain completely unclear; in particular, investigations in the field of applying Mie theory to absorbing media are almost entirely lacking.

Thus, the range of unclear questions connected, to one degree or another, with the optics of thin metallic films is very extensive, and if the present article helps to revive interest in this field of knowledge, its aim will have been achieved.

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