PHYSICS OF THIN METALLIC AND SEMICONDUCTOR LAYERS

STRUCTURE AND ELECTRICAL PROPERTIES OF METALS AND SEMICONDUCTORS IN THIN LAYERS

I. D. Konozenko

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561
2. Formation and structure of thin metallic layers . . . . . . . . . . . . . . 564
3. Electrical properties of metals in thin layers . . . . . . . . . . . . . . . 577
4. Formation and structure of thin layers of semiconductors . . . . . . . . . 583
5. On the electrical properties of semiconductors in thin layers . . . . . . . 585
6. Surface levels and their influence on the properties of semiconductors in thin layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
7. On theoretical studies of thin metallic layers . . . . . . . . . . . . . . . 594
8. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597

1. INTRODUCTION

At the present time the problem of thin layers, from both the practical and the theoretical point of view, is highly topical. Metallic and semiconductor layers are successfully used for the manufacture of various kinds of instruments and devices: photocells and photoresistors, thermoelements and thermopiles, bolometers and other receivers of thermal rays. Thin layers are used as reflecting surfaces in interference, astronomical, and optical apparatus in general; they have also found application in magnetic sound recording. From the theoretical point of view, thin layers are of great interest if only because a freshly prepared layer, if observed under

...with the aid of an electron microscope or an electron diffraction camera, reveals to us a picture of the structure of matter. Under these conditions the layer is relatively little subject to any deformations or external influence. Therefore electron-diffraction investigations help us solve many questions connected with the problem of metallography.

Over the last two decades in our country many interesting investigations have been carried out in this field. It was therefore quite natural that there arose the need, to some extent, to summarize these research results and to clarify the state of our knowledge in the field of the properties of thin metallic and semiconductor layers.

Investigations in this direction have, first of all, made it possible to discover sharply expressed anomalies where massive metals behave normally.

It is usually considered that for metals such physical quantities as electrical conductivity (or specific resistance) and the thermal coefficient of resistance (T.C.R.) are constants. As it turned out, in thin layers not only metals but also semiconductors behave in a somewhat peculiar manner. The anomalous properties are manifested especially clearly in the behavior of the indicated constants of a metal, and also in changes of optical constants. At thicknesses of the order of several mμ, a decrease in electrical conductivity is observed. With decreasing layer thickness the thermal coefficient of resistance not only decreases, but also changes its sign.* Table 1 gives data for some metals.

On the basis of investigations of the electrical properties of thin layers of the above-indicated thicknesses, one can draw the general conclusion that the change in the properties of metals with decreasing layer thickness occurs in such a way that, in their properties, they become closer to semiconductors and dielectrics than to metals. These anomalies in the behavior of metals in thin layers were discovered long ago, but an explanation of them was given considerably later, when sufficient experimental material had been accumulated. The anomalies may be due to a number of causes, and not only in thin layers but also in massive metal. It is known that the most insignificant impurity introduced into any metal causes a decrease in its electrical conductivity even in the case when the conductivity of the impurity is greater than that of the metal (Matthiessen’s rule). A change in electrical conductivity can also be observed in the case when deformation has produced a disturbance of the crystal lattice, and in general with any lattice disturbances. But in the case of semiconductors, the introduction of an impurity may also cause an increase

* Thicknesses of the order of \(10^{-7}\)—\(5\cdot 10^{-6}\) cm are meant.

electrical conductivity. Therefore Matthiessen’s rule is not applicable to them.

Those methods by means of which the experimenter prepares the thinnest layers (of the order of ten mμ), namely cathodic sputtering, thermal sublimation, electrolytic deposition, etc., knowingly provide conditions for violation of the ideality of the crystalline lattice. It is therefore not surprising that precisely in thin layers the anomalies manifest themselves especially sharply. The point is that to obtain a thin metal layer in pure form by means of thermal sublimation or cathodic sputtering is almost impossible. The process of depositing a layer is to a greater or lesser degree connected with its contamination. Even if sputtering is carried out at a vacuum of the order of \(10^{-6}\) mm Hg, there is in the bulb a sufficient number of gas molecules that can get into the layer. In this case foreign atoms can enter the interstices of the condensate lattice. In addition, in the process of crystallization defects may arise in the crystal lattice owing to the appearance of vacancies at lattice sites, or to the presence of condensate atoms in the interstices of the lattice.

Table I

Substances used in bolometric technology	Specific resistance at \(0^\circ\)C \((\Omega \cdot \mathrm{cm} \cdot 10^6)\)	Specific resistance at \(18^\circ\)C \((\Omega \cdot \mathrm{cm} \cdot 10^6)\)	Thermal coefficient of resistance at \(18^\circ\)C \((1/\text{degree})\)	Value of t.c.r. for thin layers of these substances
Cu			\(+0.0044\)	\(+0.0024\)
Pt	9.8	10.8	\(+0.0039\)	\(+0.0037\)
Pt	9.8	10.8	\(+0.0039\)	\(+0.0027\)
Au	2.06	2.81	\(+0.0040\)	\(+0.0014\)
Au	2.06	2.81	\(+0.0040\)	\(+0.0036\)
Ni	6.6	7.85	\(+0.00634\)	\(+0.0048\)
Ni	6.6	7.85	\(+0.00675\)	\(+0.0059\)
Fe	8.9	10.9	\(+0.00657\)	—
W	4.89	6.32	\(+0.00464\)	—
Bi	109.0	118.0	\(+0.0045\)	\(-0.0025\)
Bi	109.0	118.0	\(+0.0045\)	\(-0.0021\)
Sb	36.3	39.8	\(+0.00473\)	\(-0.012\)
Cd	6.7	13.4	\(+0.00428\)	\(-0.0014\)

Thus, a source of impurity or contamination is always present. Gas molecules that enter the metal layer will behave differently. In some cases a chemical compound will be formed and then the properties of the layer will be one thing; in others, the gas molecules will be present in the layer in the form of a solid solution, and then the properties of the layer will be different.

Sometimes the formation of a layer is not accompanied by absorption of gas. Thus, for example, an iron layer does not absorb nitrogen3 and CO₂, if the corresponding evaporation rate is selected. On the other hand, oxygen very actively enters into combination with metals.

In the case of producing a layer from metals with a complex crystal lattice, these contaminations may create conditions for the formation of an amorphous state at the initial stage of layer formation. As the thickness of the layer increases, the amorphous phase spontaneously transforms into the crystalline phase. In this case there should be no definite transformation temperature, such as that indicated by Kramer4, since the process of transformation of the amorphous phase into the crystalline one in a thin layer, where surface-tension forces and other factors act, will not occur instantaneously.

In this review we do not claim to give an exhaustive treatment of all questions connected with the physics of thin layers, nor to present all the works on this question available in the foreign literature. We have devoted more attention to an analysis of scientific works that have appeared in our domestic literature over the last 10–15 years and that, in our opinion, are of important scientific significance for the problem of studying thin metallic and semiconductor layers. In this connection, in the bibliography given we have tried to present more fully the works of Soviet scientists on this question and to note only the most important works of foreign authors. In writing this work, use was made of certain materials from the reviews of A. Asheulov5, G. Rosenberg6, and the works of S. Vekshinsky7.

We shall not dwell here on the question of the technique of obtaining thin layers and determining their thickness, but shall point to other literary sources7, 8, 9.

2. FORMATION AND STRUCTURE OF THIN METALLIC LAYERS

At the present time the experimenter has at his disposal various methods for investigating the structure of thin layers. The principal ones here are X-ray and electron-diffraction methods. It is true that X-rays are weakly scattered by thin layers and, consequently, their application is not very effective. On the other hand, the scattering of electron beams is large, and therefore they can successfully be used—

are suitable for the study of extremely thin layers. The range of thicknesses from \(10^{-7}\) to \(10^{-5}\) cm, accessible to electron-diffraction investigation, includes precisely those thicknesses in which the anomalous behavior of metals in thin layers is mainly observed. The importance of the electron-diffraction method of investigation lies not only in the fact that, on the basis of the data obtained with its help, it becomes possible to determine the structure of the layer. It gives indications of the possible orientation of the crystallites, their size, and the character of the surface of the specimen. Many investigations of this kind have been carried out by Z. G. Pinsker\(^{11}\) and by other authors\(^{7,12,13,14,15}\).

In the investigation of layers whose thickness is \(<0.1\mu\), the latter are deposited on thin celluloid or formvar films, and the investigation is carried out in transmission. At greater thicknesses the investigation is carried out in reflection. In this case the layers may be deposited on glass plates. A comprehensive treatment of questions of electron diffraction, a description of the electron-diffraction method, and the method of calculating electron-diffraction patterns may be found in Z. G. Pinsker’s book Electron Diffraction. Questions of electron-diffraction investigations are also discussed in a number of other works\(^{16,17}\).

It should be noted that recently, for the investigation of the structure of the process of formation of a thin layer and for the study of its structure, the electron microscope has begun to be used with great success\(^{23,24,25,6,32}\).

As will be shown below, the structure of a metallic layer can also be determined by studying the change in the electrical conductivity of the layer as a function of its temperature and thickness. Sometimes this method reveals certain features of the layer that cannot be detected by the electron-diffraction method.

At the present time there are sufficient theoretical and experimental data to form some conception of the process of formation and the structure of thin metallic layers.

1. As early as the dawn of the development of experimental investigations of the structure and electrical properties of thin metallic layers, Ya. I. Frenkel\(^{18}\) considered, from the theoretical point of view, the question of the process of formation of a thin layer. The author represents the surface of a dielectric substrate in the form of a certain “potential relief,” i.e. in the form of potential pits and hillocks.

If an atomic beam is directed at such a surface, collisions of the atoms with the surface will occur. Having given up their excess kinetic energy, the metal atoms will execute thermal motion together with the atoms of the dielectric surface. Some of them, because of their weak bond with the atoms of the dielectric, will leave the surface, while another part, owing to the existence of the potential

of the relief will be retained on the surface. This part will perform oscillations normal to the surface in the places of potential wells. The adsorbed particle, being in a state of migrating over the surface, collides with other atoms. Such a collision may lead to the formation of doublets and triplets, and since the process of formation of the latter is connected with the release of crystallization energy, reverse evaporation will be impeded and the process of layer formation will begin.

Thus, an essential feature in the theory of Ya. I. Frenkel is the presence of migration of adsorbed atoms. Therefore, in order to understand the entire mechanism of layer formation it is necessary to elucidate the energetic conditions of migration\(^{1,2}\), in which atoms must overcome potential barriers. In work\(^{19}\) it is shown that migration of atoms takes place as a result of repeated collisions, owing to which an activation energy arises that is sufficient for overcoming the barriers. A detailed study of questions concerning the migration of gold and palladium atoms on the surface of zinc was carried out by Echistova et al.\(^{20}\), who found that the order of magnitude of possible displacements of migrating atoms is from \(10^{-6}\) to \(10^{-5}\) cm.

Ya. I. Frenkel, in his work, showed that for the formation of a metallic layer on an underlying surface during thermal evaporation, the presence of a certain critical density of the atomic beam is necessary. The magnitude of the critical density must depend on the nature of the underlying surface, its purity and temperature, and also on the nature of the metal being deposited.

All these factors must exert a substantial influence on the process of layer formation and on its structure.

Such were some theoretical predictions on which the experimental studies of our scientists were based.

Soon N. N. Semenov and Yu. Khariton\(^{21}\) experimentally confirmed the existence of a critical density of metallic vapor in the formation of a layer. The authors showed that when the density of the metal vapors reaches a certain critical value, a sudden formation of metal particles occurs.

On the basis of experimental studies of the electrical properties of thin cadmium layers, A. Shalnikov\(^{22}\), as early as 1938, expressed the following point of view on the process of formation of thin metallic layers. At the initial moment of condensation, the atoms of the evaporated metal settling on the metal substrate are separated from one another. In this case the resistance of the layer is infinitely large. At this moment there are as yet no crystals, and the deposited atoms may be regarded as a gas adsorbed by the surface of the substrate. (A number of authors take this state to be an amorphous phase.) But because of the existence of migration of atoms over the surface, atoms become grouped into certain centers and ob-

develops a cluster of particles (granules). With time these condensation centers increase in size and contacts—bridges—appear between them, which corresponds to a layer thickness of \(\sim 8 \cdot 10^{-7}\) cm (effective thickness).

Upon reaching this thickness, the resistance begins to fall rapidly, as shown by the curves in Fig. 1, \(a\) (in the case of nickel) and 1, \(b\) (in the case of cesium). In Fig. 1, \(b\), the dotted line shows Thomson’s theoretical curve\(^{23}\). The course of the change in the resistance of the layer with thickness indicates that in this case a crystalline structure, and not an amorphous one, is present.

Fig. 1.

Studies by many authors\(^{24,25,26,33}\) have shown that in the thinnest layers the deposited substance is not distributed uniformly over the entire surface of the substrate, but condenses in the form of islands isolated from one another. At the same time, on the basis of electron-diffraction studies\(^{27,23}\) of the structure of thin layers of gold, silver, bismuth, antimony, and other metals\(^{14}\), conclusions were drawn that metallic layers possess a crystalline structure in the very thinnest layers, when the absorption of light amounts to only 2% of the incident power of the light beam. Observations showed that the electron-diffraction rings in this case are strongly blurred, which indicates very small sizes of the crystals forming the layer. With increasing layer thickness the rings become sharper, which indicates an increase in the crystals. This crystalline structure of the layer proves to be unstable; recrystallization is observed even at room temperature, causing a change in the resistance of the layer.

According to Haas’s investigations[^29], even layers several tens of atoms thick did not possess an amorphous phase.

Prince[^30] investigated thin tin layers by the electron-diffraction method and came to the conclusion that, when a layer of white tin is deposited, crystallites of gray tin are initially formed as smaller ones. The thinnest tin layers are thus layers of gray tin, and, according to Prince’s conclusions, they may be regarded as nuclei for obtaining gray tin in a thin layer. Goryunova and Konozenko[^31] verified this assumption and, when spraying white tin onto a cooled plate, obtained layers containing gray tin. However, experimental data obtained recently compel one to be somewhat cautious in judging the structure of a layer in its initial phase of formation.

At present the experimental technique has reached such a degree of progress that it is possible, in an electron microscope, to observe the change in the structure of a layer during the process of its deposition, without subjecting the layer to the action of atmospheric air and other similar factors. Thus, in the works[^32][^33] results are presented of electron-microscopic observations of the process of growth of layers of silver, gold, cadmium, and zinc. On the basis of the photographs obtained, the authors indicate that particles of silver and gold become visible beginning with such sizes as the resolving power of the microscope permits one to detect. This corresponds to an effective layer thickness equal to approximately 6–7 atomic layers.

The atomic formations differ in magnitude and have the semblance of a spherical form. At the same time the authors do not detect any traces of crystalline structure in the form of the particles. With increasing layer thickness the sizes of these particles grow without an increase in their number and without any special changes in form, until the process of their coalescence into a continuous layer begins.

The fact that in this entire process the authors did not observe the appearance of a crystalline form indicates that the process of crystallization does not occur instantaneously and that it depends on the conditions of migration of atoms and on the acting forces of surface tension. In some cases the crystallization process may proceed more rapidly, in others more slowly. And indeed, if one turns to electron-microscopic photographs of the process of formation of a cadmium and zinc layer (see Fig. 2a and b), borrowed by us from[^32], one may note that the particle formations that have already initially appeared have a regular crystalline form, which they retain with increasing thickness up to coalescence into agglomerates. In this work attention is drawn to the circumstance that particles of this kind appear when the density of cadmium vapor reaches a certain critical value, as was stated above in considering the theoretical works of Ya. I. Frenkel.

For relatively thicker layers, the process of formation of a silver layer as a function of thickness is presented in Fig. 3, which we have borrowed from \({}^{34}\) (see also \({}^{6}\)). It is clearly seen that, as the thickness of the layer increases, the crystals gradually grow in size and then, merging, pass into a continuous layer.

Fig. 2. The process of formation of a layer deposited by evaporation in vacuum onto a formvar substrate. (Electron-microscopic photograph, magnification 30 thousand.) \(a\)—cadmium layer; \(b\)—zinc layer.

Electron-microscopic investigations \({}^{34,35,36,37}\) have made it possible to observe directly the features of the granular structure of the thinnest metallic layers and have made it possible to establish that the thinnest metal deposits are an accumulation of very small crystalline particles. The picture observed in this case depends substantially on the nature of the condensate; as is seen from Fig. 4, under the same conditions for obtaining the layers, the character of the surface relief of different metals is completely different even at the same layer thickness.

1. As early as 1933, Gehn, Shchalnikov, and Zelman \(^{21, 38}\) showed that the size of crystallites is influenced by the temperature

Fig. 3. The process of formation of a silver layer with increasing thickness.

Fig. 4. Films of various metals: \(a\) — silver, thickness \(85 \ \text{Å}\); \(b\) — gold, thickness \(75 \ \text{Å}\); \(c\) — antimony, thickness \(180 \ \text{Å}\); \(d\) — copper, thickness \(120 \ \text{Å}\); \(e\) — nickel, thickness \(75 \ \text{Å}\). Film deposition time: 2 seconds.

of the substrate onto which the layer is deposited. The crystalline structure of the layer changes upon heating; recrystallization occurs in the layer \(^{36, 39, 40}\), which leads to an increase in the size of the crys-

crystals, which can be observed even from the change in the resistance of the layer. This characteristic change in the resistance of a layer upon heating is clearly shown in the work of P. Zuurman[^41]. The results presented in Fig. 5 were obtained in the study of a nickel layer 140 Å thick, which had been deposited on a substrate cooled to 60° K. During the measurement the layer was heated to 182.5° C. The influence of the annealing temperature of a layer on its resistance was also investigated with thin cadmium layers by Shalnikov[^22]. The results are presented in Fig. 6. From the curves in Figs. 5 and 6 it is seen how the resistance of a heated layer gradually changes with time. This change can be explained only by internal structural changes occurring in the layer. It was also shown that the influence of temperature depends on the thickness of the layer and on the rate of deposition. These points are illustrated by the experimental data presented in Figs. 7 and 8, borrowed from work[^6] and requiring no special explanation.

Fig. 5.

Fig. 6.

Krautkrämer[^42] investigated the dependence of the change in resistance on the thickness of the layer at various substrate-heating temperatures. In Fig. 9 the results of his observations on thin gold layers are presented. Similar regularities were also obtained for thin silver layers. From the disposition of the curves in Fig. 9 it is clearly seen that the higher the temperature, the greater the thicknesses at which the resistance of the layer becomes measurable. These experimental facts become understandable if one proceeds from the fact that in a finely dispersed layer contacts between crystallites arise more rapidly, i.e., at smaller thicknesses. At a higher substrate temperature (at the moment

Fig. 7. Dependence of the film structure on the temperature of the underlying surface: a—silver films of various thicknesses at a temperature of \(300^\circ\text{C}\), b—aluminum films \(500\,\text{Å}\) thick at various temperatures. The cellular structure formed at high temperature is clearly visible. The scale in Figs. a and b is the same.

Fig. 8. Difference in the structure of silver films of approximately the same thickness, prepared over the course of 20 minutes (a) and 75 minutes (b).

application of the layer), larger crystallites are formed and contacts between them arise at a greater layer thickness.

In the case of a heated substrate (approximately \(200^\circ\) C), migration of the atoms of the deposited metal does not lead to the formation of a continuous thin layer; on the contrary, with increasing layer thickness the granular structure acquires a clearly pronounced cellular character (Fig. 7). As is seen from Fig. 4, the structure of the layer depends on the nature of the metal being sputtered. As the layer thickness increases, recrystallization occurs—a rearrangement of the structure of the substances \(^{24,36,43}\). This conclusion is also confirmed by the electron-diffraction investigations of G. Hass \(^{44}\) on layers of antimony and silver.

1. The structure of the layer is influenced not only by the temperature, but also by the nature of the substrate, as was shown by N. Andryushchenko, V. Tyapkina, and P. Dankov \(^{45}\), who established that a mica substrate exerts an orienting action in the process of formation of crystals of silver and potassium iodide. The same was found by Lassen and Brück \(^{46}\), who obtained oriented single-crystal layers of silver on NaCl crystals at \(100^\circ\) C, whereas on a celluloid substrate no such orientation of crystal growth was detected. At the same time Z. G. Pinsker \(^{47}\) observed the formation of highly oriented crystallites of sodium chloride (from a 20.5% solution), as well as the formation of lamellar crystals of \(\mathrm{CdJ_2}\) and \(\mathrm{PbJ_2}\) (from an aqueous solution) with thickness up to \(10^{-6}\) cm and length up to 2 mm on celluloid films. M. M. Umanskii and V. A. Krylov \(^{48}\) observed orientation of crystals in layers of various metals (Mg, Cd, Zn) on a celluloid film and of Cd on glass.

Fig. 9. Dependence of the specific resistance of a gold layer on its thickness at different substrate-heating temperatures at the moment of deposition of the layer.

In studying the influence of the nature of the substrate and its temperature, as follows from the works considered above, anomalous effects are usually attributed to the formation of a new structure of the layer. At the same time, attention is not paid to the fact that in studying the temperature dependence, a difference in the values of the thermal expansion coefficients of the layer and the substrate may also have an effect. In one case a compaction of the layer may occur, and in another its decrease, which will likewise have a noticeable influence on the properties of the layer. (In this case an exception may be...

be granular layers.) This also applies to some extent to the dependence of the electrical conductivity on the temperature of the layer.

1. Kramer⁴ expressed another point of view on the nature of the formation of a thin layer. On the basis of his studies of the dependence of the electrical resistance of the layer on its temperature, Kramer concludes that thermal distillation onto a cooled substrate leads to the formation of an amorphous layer. With an increase in temperature the amorphous phase passes into the crystalline one. This transition occurs, in each individual case, at a different temperature for different metals. He calls this temperature the transformation temperature. The fact that electronographic studies do not reveal an amorphous phase is explained by the author as being due to the fact that, under the action of the electron beam (in an electronographic study), instantaneous ionization of the atoms takes place and the amorphous phase is transformed into the crystalline one.

This point of view on the formation of the layer, expressed only by Kramer, was not approved by a number of authors, although there were no sufficiently convincing arguments against it. The only argument was the absence of a sufficient quantity of experimental data. At present there is a series of experimental data testifying against this theory ³², ³³, ⁴⁹. In addition, a new possibility is appearing for checking Kramer’s point of view.

As early as 1938, E. Pavlova and A. Shalnikov⁵⁰ developed a method for studying the structure of the surface layer with the aid of a Geiger–Müller counter and in this way investigated the surface of an aluminum cathode. This method was also applied by Kramer⁵¹ for the study of certain questions of metallography and, in particular, of the surface states of metals after their mechanical treatment. It is now already an indisputable fact that, as a result of mechanical treatment of a surface, equilibrium is disturbed in the surface layer. On returning to the equilibrium state such a surface emits electrons. This emission can be detected with the aid of a counter. Besides Kramer, similar electron emission by the surfaces of various metals was also detected by other authors⁵². According to Kramer⁵¹, mechanical treatment leads to the formation on the metal surface of an amorphous layer, which, after the treatment is discontinued, again passes into the crystalline state. At the moment of transformation of the amorphous layer into the crystalline one, electron emission occurs at the expense of the energy released in the phase transformation. If Kramer’s assertions are correct, then they can be tested on thin layers and the fact of a phase transformation can be detected. Indeed, electron emission from freshly prepared thin films has been observed experimentally. But the point is that other authors⁵³ observed electron emission in the indicated

the above experiments of Kramer are explained differently. It is assumed that mechanical treatment destroys the oxide layer present in almost all cases on the surface of a metal. Therefore, when after the treatment this oxide layer is formed, the adsorption of an oxygen atom by a metal atom may cause the emission of one electron, and, consequently, the emission of electrons by the surface.

Thus, the question of what causes the emission of electrons in these experiments has not yet been resolved unambiguously. But it seems to us that this question can be resolved unambiguously. In the case of thin layers, the formation and influence of an oxide film can be substantially weakened or even completely eliminated by carrying out the experiment under conditions of a sufficiently high vacuum \((10^{-9}\ \mathrm{mm}\ \text{Hg})\), with careful preparation of the evaporated substance and of the entire vacuum apparatus.

1. A major contribution to the physics of thin layers was made by the work of S. A. Vekshinskii \(^{7}\). Vekshinskii’s studies of allotropic transformations in metallic layers after their deposition made it possible to reveal the profound structural changes that occur in them. These transformations are, in essence, the primary cause of those anomalies that are observed in thin layers.

As is known, some metals can exist in two or more modifications, each of which is stable in a certain temperature interval. S. A. Vekshinskii observed similar allotropic transformations in thin layers of manganese at room temperature. During thermal sublimation, manganese crystallizes in a thin layer in the \(\beta\)-Mn modification, the unit cell of whose crystal consists of 20 atoms. This modification is unstable and spontaneously transforms into \(\alpha\)-Mn, with a crystal unit cell consisting of 58 atoms. It is clear that in the thinnest layers such transformations entail appreciable changes in the electrical properties of the layer.

In the thinnest layers, antimony was observed by S. A. Vekshinskii in the amorphous state, but it turned out that this state is unstable—there occurs a spontaneous transformation of the amorphous phase into the crystalline one. In work \(^{39}\) it is indicated that rhodium may possibly exist in a thin layer in the amorphous phase.

These experimental data, as well as the data of recent electron-microscopic studies \(^{32,33}\), indicate that initially condensed metal vapors may be in an amorphous phase, which then passes into the crystalline phase. In this case, the question of how long the condensed metal atoms can remain in the amorphous state is of interest. Obviously, for

the case of mercury, which stands closer to metalloids than to metals, this time will be different from the transition time of the amorphous phase into the crystalline one for gold and silver (if such a phase actually exists). It is possible that the transition time of the amorphous phase into the crystalline one for such metals as zinc and cadmium^33 is so short that modern experimental technique is not capable of recording it.

Unfortunately, in published works on the structure of thin films the element of time, as an essential factor influencing the structure of a metastable medium, is not reflected, which is a substantial shortcoming of such investigations.

As a summary of the works considered above, we shall cite the conclusions of S. A. Vekshinskii, who, on the question of the formation of thin metallic films, writes the following^7:

“1. Individual metal atoms deposited on a neutral substrate must be regarded as an adsorbed gas capable of surface migration.

1. Beginning with a certain minimum surface concentration, different for different metals, the layer of adsorbed atoms spontaneously passes into the crystalline state; the rate of this transition increases as the layer thickness grows and, within certain limits, as the temperature rises.

2. For metals of the cubic system the rate and the number of crystallization centers are very high, and therefore, even at very low temperatures, condensed pure metals cannot be preserved in thick layers in the atomic state.

3. Contamination of the condensate by impurities hampers migration and lowers the probability of the formation of nuclei.

4. Raising the temperature of the condensate within certain limits increases the migration rate of the condensed metal atoms; as a result, both the probability of nucleus formation and the rate of crystallization increase.”

These conclusions, drawn by S. A. Vekshinskii on the basis of extensive and systematic investigations carried out by him and his collaborators, do not contradict the most recent data.

To what has been said it should be added that: a) the dispersion of the layer, as we saw in Figs. 3 and 4, depends on its thickness. It increases when the temperature of the substrate is lowered and when the rate of deposition of the layer is decreased; b) the nature of the substrate affects the orientation of crystal growth in the layer; c) the structure of the layer depends on many causes: the nature of the condensate, the nature of the substrate and its temperature, the rate of deposition of the layer and its thickness; d) in the thinnest layers the process of change in layer thickness is accompanied by complex transformations of the lattice structure, allotropic transformations, and other phenomena.

3. ELECTRICAL PROPERTIES OF METALS IN THIN LAYERS

1. As early as 1933, Yu. P. Maslakovets[^54], studying the electrical properties of thin metallic layers which he obtained by cathode sputtering, found that in thin layers the specific resistance of platinum increases anomalously as the layer thickness decreases. The character of the variation of the specific resistance with thickness depends substantially on whether or not the layer had been subjected to preliminary heating in vacuum. In Fig. 10 three curves are presented which depict this dependence.

Fig. 10. Influence of preliminary heating on the resistance of a layer as a function of its thickness.

Fig. 11. Influence of the technology of obtaining the layer: I—cathode sputtering, II—thermal evaporation.

The curves I and II refer to the case where the layers were subjected to preliminary heating. The dashed curve II refers to the case where the layer was cooled to the temperature of liquid air. Curve III refers to the case where the layer was not subjected to preliminary heating in vacuum. The slope of this curve differs sharply from the slope of the first two. The reason for this phenomenon was considered by us above (see Figs. 5, 6).

In Fig. 1 the dependence of the specific resistance on layer thickness was shown for nickel and cesium. These experimental results are also in agreement with G. Scott’s[^24] data for thin layers of silver (Fig. 11). Here curve I refers to a layer obtained by cathode sputtering, and curve II to a layer obtained by thermal evaporation.

When considering the question of the structure of thin layers of metals, we pointed out that such a form of the dependence is explained by the cha

…characteristic changes which are observed in the layer when its thickness is changed (see Fig. 2). A sharp decrease in the specific resistance is observed in all cases at layer thicknesses \(\sim 8\cdot 10^{-7}\) cm, i.e., when conductive bridges form between individual scattered crystallites. At thicknesses \(\sim 5\cdot 10^{-6}\) cm the specific resistance of the layer differs little from its value in a massive specimen.

Fig. 12. Dependence of the resistance of a platinum layer on temperature.

The second conclusion that should be drawn from the experimental data presented (Figs. 1, 10, and 11) is the following. Although the curves shown were obtained by two different methods of depositing the layer, no substantial discrepancies in the results obtained are found. This should be noted, since there are attempts to explain anomalies in the properties of metals in thin layers by different methods of obtaining thin layers.

1. In the work by Maslakovets cited above it was also shown that platinum in thin layers has an anomalous dependence on temperature. Fig. 12 presents the results obtained by Maslakovets for thin layers of platinum. The course of the change of resistance with temperature can be represented, as was done in work\(^4\), by an empirically selected logarithmic function of the form

\[ R = Ae^{\frac{B}{T}}, \tag{1} \]

where \(A\) is a constant equal to \(A = 2.24 \cdot 10^{6}\); \(B = 341^\circ\text{C}\). On the basis of (1) Maslakovets comes to the conclusion that the thermal coefficient of resistance of platinum in a thin layer is equal to \(\tilde{\alpha} = -B/T^{2}\) and is negative in sign.

Fig. 13.

But a negative sign is characteristic only of unheated layers, since after heating the temperature coefficient of resistance changes sign to positive. In Fig. 13 are presented the results of a study of the dependence of the temperature coefficient of resistance on heating of the layer. As was indicated above, upon heating the resistance of the layer decreases, and the temperature coefficient of resistance, as follows from Fig. 13, also decreases.

In 1939 A. Shal’nikov[^22], studying in exceptionally thoroughly degassed and evacuated flasks thin cadmium layers obtained by thermal evaporation, found that the temperature coefficient of resistance in the temperature interval from \(-180^\circ\mathrm{C}\) to \(+18^\circ\mathrm{C}\) decreases by a factor of 3 in comparison with its value for a bulk specimen. The resistivity of cadmium at a layer thickness of \(1.4\cdot 10^{-6}\ \mathrm{cm}\) exceeds its value for a bulk specimen by a factor of 2.

Fig. 14.

In the work of Zuurman and Barth[^53] the results of a study of the dependence of the temperature coefficient of resistance on temperature for a copper layer of thickness \(1.15\cdot 10^{-5}\ \mathrm{cm}\) are set forth. They obtained the following data:

At a temperature of \(78^\circ\mathrm{K}\), the value \(\alpha = 0.028\cdot 10^{-2}\ \mathrm{degree}^{-1}\)

» » \(192^\circ\mathrm{K}\) » \(\alpha = 0.061\cdot 10^{-2}\) »

» » \(297^\circ\mathrm{K}\) » \(\alpha = 0.141\cdot 10^{-2}\) »

» » \(416^\circ\mathrm{K}\) » \(\alpha = 0.24\cdot 10^{-2}\) »

whereas for a bulk copper specimen at room temperature \(\alpha = 0.44\cdot 10^{-2}\ \mathrm{degree}^{-1}\)[^5,^62].

Interesting results on the question under consideration were obtained for tungsten and platinum by N. Mostovich and G. Vordarom[^55,^57]. The data of these authors are in agreement with the investigations of Maslakovets described above. The authors conclude that the course of the change in the resistance of a layer with its temperature can be represented by the following exponential dependence:

\[ R = A_0 T^{-\frac{1}{2}} e^{\frac{B}{T}}. \tag{2} \]

According to (2), a linear dependence of \(\lg R\) on \(\frac{1}{T}\) should be observed. In Fig. 14 are presented the results of observations

of the authors on thin layers of platinum. Curves I depict the dependence of \(\lg R\) on \(\dfrac{10^3}{T}\), and curves II the dependence of \(\lg R + \dfrac{1}{2}\lg T\) on \(\dfrac{10^3}{T}\). These curves are almost parallel. From their slope the value of the constant \(B\) was calculated, using which the t.c.r. was determined. It turned out that at \(14^\circ\text{K}\) the t.c.r. reaches \(30\%\) per \(1^\circ\text{C}\), whereas at \(100^\circ\text{K}\) it decreases to \(5\%\). These data are very instructive, since they show that thin metallic layers possess the same properties as semiconductors and can successfully be used for making thermoelements and bolometers of high sensitivity.

1. Zhurman and Barth\(^ {55}\) investigated the electrical properties of many metals condensed at a low substrate temperature (from 30 to \(80^\circ\text{K}\)). They established that the electrical conductivity increased irreversibly with increasing temperature.

The authors explain the results of their investigations by assuming that layers condensed at low temperatures have an additional resistance due to disorder of the crystalline structure. With increasing temperature the crystalline structure becomes ordered and the resistance of the layer decreases.

Such an explanation is correct to some extent, but it requires supplementation. First of all, it should be noted that during heating a change in the granularity of the layer takes place. In addition, in the work of Maslakovets, mentioned above, it was shown that heating the layer also leads to some degassing of it, which sharply lowers the resistance of the layer. Oxygen has an especially strong influence.

In work\(^ {49}\) the authors investigated the electrical properties not only of platinum but also of other metals, including tantalum, tungsten, molybdenum, rhodium, nickel, copper, and aluminum. As a result of these investigations it was established that metallic layers are always partially oxidized: on the one hand, due to the residual gas—oxygen molecules—and on the other, as a result of contact of the layer with the glass of the substrate on which the layers were deposited. Contamination of the layer manifests itself in the fact that reversible and irreversible changes in the resistance of one and the same layer are observed experimentally on different substrates. Any impurities that enter the layer change the activation energy and, consequently, the properties of the layer.

In addition, an adsorption effect is also observed, manifested in a lowering of the potential barrier and in other phenomena.

The electrical conductivity of thin layers of Ag, Cu, Fe was investigated by Steinberg\(^ {58}\); Ni and Pt by Ride\(^ {59}\); Fe, Co, Ni, Pd, Ir were investigated

Pikokom[^60], Au and Ag by Berrleton[^61], and others. We shall not dwell on an analysis of these works.

Of interest is the work of a group of authors[^62] who investigated the influence of electric current on the resistance of thin metallic layers at various temperatures. They showed that, with an increase in the current passing through the layer, the resistance decreases, as is seen in Fig. 15. The decrease in the resistance of the layer is the greater, the lower the temperature of the layer. With

Fig. 15. \(R/R_{10\,\mu\text{A}}\) as a function of current at various temperatures.

an increase in temperature, the resistance of the layer changes only weakly with increasing current.

The results of these experiments are quite explicable from the standpoint of ideas about the formation of the layer and the influence of the temperature of annealing on its structure and properties.

It was indicated above that annealing of the layer leads to a decrease in its resistance. The thermal effect of the current in the present case may, to the same extent, act on the change in the resistance of the layer as does the annealing mentioned above. In addition, an influence of the electric field is not excluded. Unfortunately, the authors do not indicate the nature of the reversibility of the change in the layer resistance with current. It seems to us that in the present case one is dealing with an irreversible change in the resistance of the layer.

1. G. Scott\(^{24}\) investigated the influence of the rate of condensation of silver vapor on the electrical properties of a layer as a function of its thickness. In Fig. 16 three curves are shown, obtained respectively for layer-deposition times equal to 5, 20, and 120 minutes. From these curves it is clearly seen that the more rapidly the layer is condensed, the smaller its initial resistance at one and the same thickness. Apparently this is explained by the fact that, with increasing rate of deposition of the layer, the number of condensation centers increases and the formation of contact bonds between them is facilitated.

Fig. 16. Influence of the rate of condensation of a silver layer obtained by spraying molten metal.

Fig. 17. Photoelectric effect of a monatomic layer of an alkali metal on quartz as a function of thickness for wavelengths \(\lambda = 4000\text{–}2500\) Å (a) and the course of the change in specific resistance (b).

Mayer’s investigations\(^{43}\), as well as those of other authors\(^{63,64}\), made it possible to establish experimentally that the thinnest layers of alkali metals exhibit conductivity even in the case when there is not yet a continuous monatomic layer on the surface of the substrate. In this case the specific resistance of such a layer is \(> 10^6\ \Omega\cdot\text{cm}\). The photoelectric effect practically begins with the condensation of the first atoms of the alkali metal and reaches a maximum already at two or three atomic layers. The results of Mayer’s similar investigations are presented in Fig. 17.

Summarizing the above, it may be considered experimentally established that the change in the electrical resistance of thin metallic layers obtained by cathodic sputtering or thermal sublimation on some substrate, as well as the course of the change in the T.C.R. as a function of the layer thickness, can

characterized with the aid of the curves shown in Fig. 18.

It has been established that the electrical properties depend substantially on the conditions under which the film is obtained (see Fig. 16), on its structure, on the nature of the condensate, and on the nature of the substrate.

Fig. 18. Variation of the resistance and temperature coefficient for thin metallic films.

Heating of the substrate onto which the film is deposited, or preliminary annealing of the film, has a substantial influence on the properties of the film (see Figs. 10, 12, and 15).

4. FORMATION AND STRUCTURE OF THIN SEMICONDUCTOR FILMS

Investigations of the structure and electrical properties of semiconductors in thin films are still at an initial stage; there are few published data on this question. Lead sulfide and lead selenide have been studied \(^{65,69}\), as well as copper sulfide \(^{66}\), copper oxide \(^{67}\), bismuth sulfide, and other semiconductors \(^{70}\).

In the case of semiconductors, the technique of investigation and the elucidation of the nature and physics of the state of the film are considerably more complicated than in the case of metals. At the same time, some conclusions can already be drawn from the available data.

Semiconductors that are difficult to obtain in strict stoichiometry without disturbance of the crystal lattice are an example of how negligible impurities cause a sharp change in properties. In thin films, a semiconductor impurity may turn out to have been introduced into the film during thermal evaporation as a result of partial dissociation of the molecules of the evaporated substance, even in the case where this substance was prepared in strict stoichiometry.

In thin layers of semiconductor substances one observes not only a decrease of the t.c.r. as a function of thickness, but sometimes, on the contrary, its increase, as occurs in tellurium (see Table II).

Table II

Substance	$\rho$ (ohm-cm) in a bulk specimen	$\rho$ (ohm-cm) in a thin layer	$\alpha$ (1/degree) in a bulk specimen	$\alpha$ (1/degree) in a thin layer	Layer thickness (in cm)
$\mathrm{Cu_2O}^{93}$	$10^4$	$10^4$	0.035	−0.033	$1.7\cdot10^{-3}$
Oxides of Mn, Co, and Ni$^{94}$	$10^4$	$10^4$	up to 0.04	up to −0.035	$1.5\cdot10^{-4}$
Oxides of Mn and Co *)	$10^3$	$10^4$	0.04	up to −0.04	$10^{-3}$
$\mathrm{Ag_2S}$ *)	—	$10^3$	up to −0.035	up to −0.03	$3\cdot10^{-5}$
$\alpha\ \mathrm{Sn}^{27}$	—	$3\cdot10^5$	—	up to −0.07	$2\cdot10^{-5}$
$\mathrm{Te}^{92}$	—	$3\cdot10^4$	0.006	−0.016	$2\cdot10^{-5}$
$\mathrm{Cu_2S}^{66}$	—	$5\cdot10^2$	—	—	$5\cdot10^{-5}$

*) Data obtained by the author.

The question of the structure of thin layers of semiconductors has been considered in a number of works$^{68,69,70,71,72}$; it has been established that the same semiconductors can be obtained in a thin layer both in the amorphous phase and in the crystalline phase. For example, it is known that lead sulfide can be obtained in a layer in crystalline form if the condensate substrate has been heated to 400°C. On less strongly heated substrates it is obtained in the amorphous phase. M. Dunoyer$^{68}$ investigated the electrical properties of thin germanium layers and showed that a crystalline layer of germanium can be obtained by thermal sublimation when the substrate is heated to 300°C. At temperatures below 300°C, amorphous layers are obtained. The amorphous phase of germanium was also observed by N. V. Rapp$^{70}$. To transform an amorphous germanium layer into a crystalline one, it is necessary to heat it to 375°C. The amorphous phase of semiconductors, unlike metals, is obtained at any layer thickness.

In addition, it has been shown$^{24}$ that silicon and silicon oxide, upon thermal sublimation, form clearly expressed homogeneous amorphous layers; moreover, a thin silicon layer becomes crystalline after it is heated to 700°C. The amorphous phase was also found in thin layers of other semiconductors$^{71,72,70}$, including lead sulfide. It should be noted that observations of the amorphous phase in thin layers of semiconductors

were performed with the aid of an electron-diffraction apparatus. It turns out that under the action of cathode rays no transformation of the amorphous phase into the crystalline one occurs, contrary to Kramer’s statement^4. Let us note that in works ^36 and ^73 the presence of an amorphous phase for certain semiconductors is not confirmed.

In 1948 there appeared a paper by Willman^69, who investigated the structure and photosensitivity of lead sulfide and lead selenide in layers with thicknesses from 0.05 to 0.1 μ. The layers were obtained in three ways: by thermal evaporation in vacuum, by chemical deposition, and by sensitization during sublimation and treatment in oxygen. It was established that all the tested specimens of thin layers obtained by different methods had a crystalline structure. The lattice proved to be of the sodium-chloride type, i.e. face-centered cubic. These substances possess only one modification of the crystal lattice. In the work of I. Kononenko and N. Goryunova^31 it was shown that the thinnest layers of a mixture of white and gray tin have a crystalline structure. It has also been established that other semiconductors as well, including copper sulfide, bismuth sulfide, bismuth telluride, lead telluride, and bismuth selenide, on thermal evaporation in a thin layer form a crystalline structure.

Interesting investigations were carried out by A. I. Frimer^99 and by A. I. Frimer and T. G. Sinitskaya^74 on the use of the electron microscope for studying the microstructure of antimony-cesium photocathodes. The authors showed that with the aid of the electron microscope one can directly observe the structure of complex cathodes in thin layers. These investigations showed that antimony layers, after their treatment in cesium vapor, sharply change their structure. Layers possessing high sensitivity are characterized by a particularly developed surface relief, with irregularities reaching the order of 200–600 Å; the more homogeneous the layers, the less sensitive they are.

The importance of this method of investigation is indisputable. Knowing the structure of the layer of highly sensitive cathodes, one can more precisely explain its photoelectric properties and indicate ways for further improving the technique of manufacturing highly sensitive photocells.

5. ON THE ELECTRICAL PROPERTIES OF SEMICONDUCTORS IN THIN LAYERS

First of all let us consider the most thoroughly studied semiconductor—lead sulfide, which has been investigated by many authors ^69, ^76, ^75, ^65, ^99.

According to the investigations of Maslakovets and Dunaev^76, lead sulfide retains a constant number of current carriers in the zone

conductivity in the temperature interval up to \(450^\circ\)C. In this temperature interval, lead sulfide behaves like a typical metal, and the change in resistance with temperature is entirely due to the change in the mobilities of the current carriers. With an increase in temperature above \(450^\circ\)C, the concentration of current carriers increases exponentially, and in this case lead sulfide behaves analogously to a semiconductor. The transfer of current carriers into the upper zone, evidently, occurs not from the lower filled zone, but from local levels. Measurements of the Hall effect do not indicate the presence of mixed conductivity. In thin layers anomalies are observed, especially as a result of temperature annealing. Lead sulfide in a thin layer, even at room temperature, behaves like a typical semiconductor. Investigations of the electrical properties of PbS and PbSe have shown that during temperature annealing oxygen is introduced into the lattice, where it reacts with the excess of Pb, S, or Se atoms arising during thermal sublimation and representing a disturbance of the crystal structure. In this process there may occur the formation of chemical compounds of the type PbO, PbO·PbSO\(_4\) \(^{69}\), which leads to an increase in the resistance of the layer during temperature annealing. In addition, in the PbS and PbSe lattice, molecular centers containing oxygen are probably formed, i.e. oxygen may be present in the form of an impurity or a solid solution.

At the initial moment after deposition the layers possess electronic conductivity, while after oxidation as a result of temperature annealing the conductivity becomes hole conductivity, which is associated with the introduction of oxygen into the lead sulfide lattice. Lead selenide behaves analogously.

Willman \(^{69}\) carried out studies of the electrical conductivity of lead selenide on layers with thicknesses from 0.1 to 0.05 \(\mu\). It was shown that the smallest value of the electrical conductivity corresponds to the stoichiometric ratio of the components lead and selenium. An excess of selenium, as a rule, increased the conductivity. In the range of thicknesses investigated, the electrical conductivity varied within the limits from 10 to 2300 \(\Omega^{-1}\cdot\text{cm}^{-1}\). The value of the conductivity with an excess of lead was \(100\ \Omega^{-1}\cdot\text{cm}^{-1}\). Investigation of the temperature dependence of the conductivity shows that in the case of small layer resistances \(R\) increases with temperature, whereas for large values of the resistance \(R\) decreases with \(T\); moreover, the decrease of \(R\) does not obey the exponential law (1) known for semiconductors.

Temperature annealing of layers with an excess of selenium in selenium vapor improves the conductivity by almost an order of magnitude. At the same time, vacuum temperature annealing of these layers leads to a decrease in conductivity, whereas layers with an excess of lead in this case increase their conductivity. The Hall constant

for layers with an excess of selenium it is positive, and for layers with an excess of lead it is negative.

In Dunoyer’s work, which has already been mentioned, the results of a study of the electrical properties of germanium in thin layers are described. Thin layers in the amorphous and crystalline states were investigated. Fig. 19 presents the results of a study of the dependence of the resistance of an amorphous layer on temperature. As is seen from the figure, up to a temperature of \(300^\circ\)C germanium remains in the layer in the amorphous state.

The course of the change in resistance in the forward and reverse directions forms a kind of hysteresis loop, which is difficult to explain on the basis of the above-stated point of view on the structure of the layer.

Fig. 19. Resistance of a thin amorphous germanium layer as a function of temperature.

The resistance of a thin crystalline germanium layer depends substantially not only on its thickness, but also on the temperature of the substrate at the moment of deposition of the layer, as is seen from Fig. 20. The higher the substrate temperature, the lower the resistance of the layer.

Fig. 20. Dependence of the resistance of a crystalline germanium layer on the temperature of the substrate during deposition of the layer.

Fig. 21. Dependence of the specific resistance of a cuprous sulfide layer on temperature.

L. Eizeman \(^{65}\) studied the electrical properties of cuprous sulfide. The dependence of the specific resistance of a layer on temperature is shown in Fig. 21. The course of the curve calls for no special remarks, but is characteristic of semiconductors, with a sharp increase in resistance as the temperature is lowered.

The author studied layers of bismuth sulfide with thicknesses from \(0.01\) to \(1\,\mu\), obtained by thermal evaporation in vacuum. In this case the presence of partial dissociation during evaporation was established, in consequence of which the properties of the layer depend on the conditions of its subsequent treatment. According to the investigations of Gokhberg and Sominskii \(^{77}\), bismuth sulfide in the fine-crystalline state does not change its electrical conductivity as a function

Fig. 22.

Fig. 23.

of temperature within the temperature interval they investigated, as is seen from Fig. 22. In thin layers the picture is somewhat different. When a layer is deposited by thermal evaporation, at the initial moment layers \(0.01\,\mu\) thick have a resistance of \(10^8\)—\(10^9\ \Omega\). At a layer thickness of \(0.1\,\mu\), the resistance drops sharply.

The independence of electrical conductivity from temperature in thick layers, according to Gokhberg and Sominskii (in the temperature interval \(200^\circ\) C), does not mean that this substance does not belong to the semiconductor type. The electrical conductivity of such semiconductors as \(\mathrm{Bi_2S_3}\), Ge, Si, PbS should be regarded as an example of additive complication of the semiconductor conduction mechanism with metallic \(^{78}\). In these semiconductors the forbidden band in the energy spectrum of the crystal is masked by electrons of impurity atoms. In the case of PbS, as was stated, the forbidden band can be detected at temperatures above \(400^\circ\) C. In a thin layer, bismuth sulfide behaves as a semiconductor. In this case the reproducibility of the dependence of \(\rho\) on \(T\) for different layers is characteristic (Fig. 23), in contrast to thin metallic

layers. Measurements of the Hall effect showed that, as a result of temperature annealing of a bismuth sulfide layer, a sharp change in the concentration of current carriers occurs in it. If immediately after deposition of the layer the concentration of current carriers was \(10^{20}\), then after annealing it was equal to \(10^{18}\).

It is assumed that when oxygen penetrates into the layer, the resistance of the layer increases as a result of the capture, by oxygen atoms, of electrons of impurity bismuth atoms, which determine the metallic character of the conductivity of bismuth sulfide in a bulk specimen.

The semiconductors investigated in thin layers also include the oxides of Cd, Sn, and Te \(^{79,80}\).

The indicated interpretation of the anomalous properties of thin semiconductor layers explains only to a certain extent such phenomena as changes in electrical conductivity, the thermal coefficient of resistance, etc.; much still remains unclear. This includes questions concerning the change in the sign and magnitude of the t.c.r., a number of anomalies of an optical nature, which will be discussed below, the change in the electron work function, shielding of an external electric field applied to a semiconductor layer, and other questions.

6. SURFACE LEVELS AND THEIR INFLUENCE ON THE PROPERTIES OF SEMICONDUCTORS IN THIN LAYERS

In seeking an answer to the questions posed at the end of the preceding section, let us consider a number of works on the investigation of electronic states at the surface of semiconductors, carried out mainly at the Institute of Physics of the Academy of Sciences of the Ukrainian SSR \(^{67,81,82,83}\). However, before proceeding to the presentation of these works, we shall briefly dwell on the history of the question.

In B. Davydov’s work \(^{84}\), as early as 1939 it was shown that the contact resistance \(R_k\) of a semiconductor with a metal, for not too large values of the contact potential difference \((V_k)\), must obey the following law:

\[ R_k = \frac{2}{\chi e u n}\left(e^{\frac{f}{2}} - 1\right), \tag{3} \]

where \(f = \dfrac{eV_k}{kT}\), \(\chi = \sqrt{\dfrac{8\pi p n_0 e^2}{\varepsilon kT}}\), and \(\dfrac{1}{\chi}\) is the screening length. This formula is valid for \(\lvert eV_k\rvert < \varepsilon\), where \(\varepsilon\) is the chemical potential. The electron levels are removed from the chemical potential by a distance \(\gg kT\). From this formula it is seen that, as \(V_k\) increases, the contact resistance must increase rapidly.

In 1945, in studying the contact resistance of a cuprous-oxide semiconductor with a metal, A. V. Ioffe^85 indicated that the magnitude of the contact resistance at large potential differences between the semiconductor and the metal is much smaller than that which would have been expected on the basis of Davydov’s theory (formula (3)).

In 1950, V. I. Lyashenko and A. Pavlenko^81, in studying the contact resistance between Cu₂O and a metal as a function of the contact potential difference, established that at small values of \(V_k\) the increase in contact resistance corresponds only approximately to Davydov’s theoretical conclusions, while at large values \((\mathrm{Cu}_2\mathrm{O}—\mathrm{Ae})\) sharp discrepancies are observed between the measured and calculated quantities. It was precisely this circumstance that drew the attention of these authors, who interpreted the results of their observations with the aid of the theory of surface levels, developed by I. E. Tamm^86 in 1932. According to this theory, any boundary of the surface of a crystal must lead to a violation of the periodicity of the lattice, of the periodicity of the field, and consequently to the appearance of special surface electronic states, whose wave function \(\psi\) rapidly decreases on passing from the surface into the interior of the crystal and with increasing distance from it. A consequence of this theory was that, alongside the three-dimensional energy system of electronic states in the crystal, there must also exist a surface energy system with its own bands and local levels, and both of these systems must be connected with one another by mutual electronic transitions. It is assumed that the surface energy system in the case of conducting crystals reaches a depth of \(10^{-7}\) cm, i.e. of the order of magnitude of the mean free path of electrons.

In the work of V. E. Lashkarev and V. I. Lyashenko^82 and of V. I. Lyashenko and Steĭko^67, the existence of surface levels on thin layers of cuprous oxide and oxides of other metals \((\mathrm{ZnO}; \mathrm{CdO}; \mathrm{MnO}_2)\) was shown experimentally. The authors carried out extensive studies of thin cuprous-oxide layers, which were prepared by thermal oxidation of copper on a quartz plate, followed by conversion of this layer into cuprous oxide in the usual way. The layer thickness was \(0.1\,\mu\), and the resistance reached \(10^8—10^9\ \Omega\) at room temperature. Experiments showed that such freshly prepared \(\mathrm{Cu}_2\mathrm{O}\) layers, when in air, considerably increase their resistance over the course of a month, after which these changes are scarcely noticeable. In such layers, in vacuum and in air, the layer resistance is different and reproducible. On these layers the dependence of the work function and of the electrical conductivity on the action of adsorbed dipole molecules at various pressures was investigated.

On Fig. 24 are presented the results of an investigation of the dependence of the relative change \(\dfrac{\Delta R_0}{R_0}\) (curve \(a\)) and of the work-function change \(\Delta f\) (curve \(b\)) for a thin layer on the vapor pressure of ethyl alcohol. From this figure it is seen that, at a low vapor pressure, the changes \(\dfrac{\Delta R_0}{R_0}\) and \(\Delta f\) are fairly large (at \(p = 1\) mm Hg, \(\dfrac{\Delta R_0}{R_0}\) reaches 50% of its value at \(p = 50\) mm Hg, while \(\Delta f\) reaches even 70%).

In all cases, adsorption on cuprous-oxide specimens—of alcohols as well as acetone and water—reduces the electron work function relative to vacuum; moreover, as is evident from the graph, parallelism is observed in the changes of the work function and of the conductivity. When the work function decreases, the conductivity also decreases, i.e., the resistance increases. Proceeding from this, the authors believe that adsorption of molecules on the surface of a \(\mathrm{Cu}_2\mathrm{O}\) layer causes a change in the degree of filling of the surface levels, which also leads to a change in the resistance and in the work function. The more electrons there are on the surface levels, the greater the thin-layer conductivity. This, apparently, is explained either by the appearance of a zone of surface conductivity, or by an increase in the concentration of holes in the layer of a hole semiconductor at the expense of holes compensating the surface charge.

Fig. 24.

In the case of thin layers of electron semiconductors, such as zinc oxide or manganese dioxide, upon adsorption of molecules of acetone, ethyl alcohol, etc., the work function also decreases, but the conductivity increases, i.e., the resistance decreases. In this case the adsorbed molecules reduce the degree to which the surface levels are filled with electrons, as a result of which there occurs a decrease in the number of screening holes and an increase in the concentration of electrons in the layer of the electron semiconductor. This leads to an increase in conductivity and a decrease in the work function.

By this the authors explain the coincidence of changes in the work function and in the sign of the change in conductivity in hole semiconductors, and their opposite character in electron semiconductors.

The observed phenomena, as is supposed, are connected with surface charges. As early as 1946, N. D. Morgulis established\({}^{83}\) that the work function of thermoelectrons from a semiconductor cathode is influenced not only by the applied external accelerating electric field, but also by the surface charge. This ex—

is explained by the fact that at the surface of a semiconductor there is a limited concentration of free electrons, as a result of which the external electric field is not screened at the very surface, as is the case in metals, but penetrates to some depth. All this leads to a decrease in the width of the forbidden band at the surface and, consequently, to an increase in the surface concentration. The surface charge that arises can substantially change the work function.

Naturally, in the presence of a surface charge \(\pm g\), in the case \(E = 0\), a change in the work function must also occur. According to \(^{83}\), this change in the work function can be calculated by means of the following formula:

\[ \Delta f = \frac{2kT}{e}\operatorname{arg\,sh}\left(\frac{1}{2\sqrt{2}}\,\frac{e}{kT}\,\frac{4\pi g\rho}{\varepsilon}\right), \tag{4} \]

where \(\varepsilon\) is the dielectric constant of the semiconductor and \(\rho\) is the screening radius.

In the work of V. E. Lashkarev and V. I. Lyashenko \(^{82}\), detailed theoretical substantiations of the above-mentioned experimental results are given, proceeding from the concept of band bending under the influence of surface charges. For calculating the change in the work function associated with a change in the surface charge, an analogous formula is derived, namely:

\[ \Delta f \simeq \frac{T^{1/2}\Delta N} {(enp)^{1/2}\operatorname{ch}\frac{e}{2kT}(f-f_0)}, \tag{5} \]

where \(f_0\) is the work function in the absence of surface charge, \(\Delta N\) is the change in the surface charge \((10^{11}—10^{12}\ \text{cm}^{-2})\), \(n\) is the concentration of holes in the absence of space charge, and \(p\) is the hole sticking factor, which is calculated by the formula of S. I. Pekar \(^{87}\). From the point of view of band theory, the change in the electron work function under the influence of surface charges is explained by the bending of the bands relative to the chemical potential.

The relation between the change in surface charges and the change in conductivity is given by the following formula:

\[ \frac{\Delta\sigma}{\sigma}=\frac{1}{p}\frac{\Delta N}{n_e}, \tag{6} \]

where \(n_e\) is the total number of current carriers falling on \(1\ \text{cm}^2\) of the surface of the layer.

From this theoretical point of view, many phenomena are satisfactorily explained and, in particular, the change in the work function in thin layers, changes in electrical conductivity, screening of an external field by surface charges, etc.

In the work of V. I. Lyashenko and I. I. Stenko, results are described from investigations of the change in the conductivity of a semiconductor under the influence of an external transverse electric field. It was found that if an external transverse electric field is applied to a semiconductor, it is screened in the upper layer, and the conductivity of the semiconductor changes as a result. The resistance \(\Delta R\) varied linearly with the field; moreover, the hole semiconductor \(\mathrm{Cu_2O}\) showed an increase in \(R\), while the electron semiconductor (zinc oxide) showed a decrease in \(R\), which is also in agreement with the conclusions of other authors\(^ {88}\).

The fact that, in the experimental works considered, pairs of polar molecules were used which do not enter into chemical interaction with the semiconductor makes these investigations convincing.

In order to eliminate possible attempts to explain these phenomena by an assumed diffusion of alcohol along the intercrystalline interlayers of the samples or by capillary condensation in intercrystalline pores, V. Lyashenko\(^ {89}\) carried out detailed investigations on a single crystal of molybdenum sulfide. The course of the change in the work function and conductivity with increasing pressure of the adsorbed vapors was obtained, similar to that observed earlier. This confirms the necessity of invoking electronic surface states to explain the indicated phenomena.

From the works considered one may apparently draw the following conclusion. It is necessary to recognize the real existence on the surface of cuprous oxide and other substances of numerous surface levels of acceptor type, creating a negative surface charge that substantially affects a whole series of phenomena. These investigations clearly indicate that in thin semiconductor layers, when the influence of surface charges begins to make itself felt, one may speak of a characteristic conductivity of a thin layer associated with the existence of surface levels—surface conductivity. In the work of Harris and Shafroth\(^ {99}\), anomalies in the electrical properties of thin metal layers are explained on the basis of the theory of surface energy levels. The authors arrive at this conclusion as a result of the following considerations: if one plots the dependence curve \(\rho = f\left(\frac{1}{d}\right)\) (where \(d\) is the layer thickness) for a thin antimony layer, two breaks can be observed: one at layer thickness \(d = 1.7 \cdot 10^{-5}\ \mathrm{cm}\) and the second at \(d = 4 \cdot 10^{-6}\ \mathrm{cm}\). The authors interpret these breaks as the presence of two mean free paths of conduction electrons. To verify the correctness of their interpretation, they calculate the dependence \(\dfrac{\Delta \rho}{\Delta T} = f\left(\dfrac{1}{d}\right)\) and again establish the presence of two breaks at the same thicknesses. On the basis of this fact

it is concluded that the first kink arises as a result of the influence of the layer thickness on the magnitude of the electron mean free path, and the second—as a consequence of the influence of surface levels. In the authors’ opinion, at a layer thickness of \(40\,m\mu\), all conduction electrons are captured by surface levels, which also leads to a sharp increase in \(\frac{\Delta \rho}{\Delta T}\). On the basis of the effect of capture of conduction electrons by surface levels, it turns out, one can explain the negative sign of the t.c.r. of thin metallic layers.

The development of the theory of surface levels, as well as further experimental investigations of these phenomena, will apparently make it possible to explain many questions connected with the physics of the surface of a thin layer.

At the present time it is known that the surface of a thin semiconductor layer is a kind of catalyst. Upon adsorption of gases their chemical properties may change, adsorbed molecules may dissociate into atoms, and new aggregate states may be formed\(^{90}\). Adsorbed molecules may also create additional surface levels, which are a kind of “traps” for electrons, leading to a change in the electrical properties.

Without a comprehensive consideration of all these factors, it is impossible to gain a deep understanding of the whole physics of thin layers.

7. ON THEORETICAL INVESTIGATIONS OF THIN METALLIC LAYERS

Thin metallic and semiconductor layers are finding ever wider application in science and technology. This has aroused interest in theoretical investigations aimed at elucidating the causes of those anomalies which are observed in thin metallic layers. Attempts to apply quantum theory have so far not yielded the desired results.

The first serious theoretical work should be considered the work of Ya. I. Frenkel, which has already been mentioned in part. About two decades ago, attempts were made to explain the anomalies of thin metallic layers on the basis of the assumption that, with decreasing layer thickness, the mean free path of the conduction electrons decreases. For the first time J. Thomson\(^{9,64}\), on the basis of these ideas, established a relation between the conductivity of the bulk metal \(\sigma_m\) and the conductivity of the layer \(\sigma_c\), the mean free path of the electrons \(\bar{l}\), and the thickness of the layer \(d\):

\[ \sigma_c = \sigma_m \left[\frac{d}{2\bar{l}}\left(\ln \frac{\bar{l}}{d} + \frac{3}{2}\right)\right]. \tag{7} \]

This formula proved to be in contradiction with the experimental data, if only because, in deriving it, the existence of the inevitable elastic scattering of conduction electrons was not taken into account. In the present case electron scattering occurs not at impurities, but at the boundaries of crystalline aggregates, which decrease as the thickness of the layer decreases. If the elastic scattering of electrons is taken into account, then for the case \(d > \bar l\) the following expression is obtained\(^ {64}\):

\[ \sigma_c=\sigma_M\left[\frac{d}{2l}\left(\ln\frac{\bar l}{d}+1\right)\right]. \tag{8} \]

On the basis of this formula Thomson calculated the theoretical curve (shown in Fig. 1, b) for a thin layer of cesium at \(T=64^\circ\). In this case the theoretical and experimental curves coincide\(^ {63}\).

The value of the mean free path of electrons \((\bar l)\) can be calculated with the aid of the following relation\(^ {91}\):

\[ \bar l=\frac{h}{\sqrt{2}\left(\frac{8\pi}{3}\right)^{2/3}N^{2/3}e^2\rho}\cong\frac{\lambda}{\sqrt{2}}, \tag{9} \]

where \(\lambda\) is the electron wavelength.

It follows from (9) that, as the thickness of the layer decreases, the electrical conductivity should decrease, which is indeed observed experimentally. For \(\bar l\), different authors have obtained different values. Thus, for platinum the value of \(\bar l\) was found to lie in the range from \(5\cdot10^{-5}\) to \(2.6\cdot10^{-7}\) cm, for silver—from \(3.2\cdot10^{-6}\) to \(6.25\cdot10^{-6}\) cm, for gold \(4.32\cdot10^{-6}\) cm, and for palladium from \(2\cdot10^{-6}\) to \(8\cdot10^{-6}\) cm.

In 1934 Mott and Zener established a connection between optical and electrical quantities\(^ {91}\). From these works it followed that the specific resistance of a thin layer is determined by the following formula:

\[ \rho=\frac{mv}{Ne^2}, \tag{10} \]

where \(m\) and \(e\) are, respectively, the mass and charge of the electron, \(N\) is the concentration of electrons in \(1\ \text{cm}^3\), and \(v\) is one-half of the relaxation frequency. It follows from (10) that, as the thickness of the layer decreases, the concentration of electrons should decrease, since the specific resistance increases as the thickness of the layer decreases. However, all these theoretical considerations cannot explain the negative sign of the thermal coefficient of resistance.

An attempt was also made to explain the negative value of the t.c.r. by applying the theory of contacts, based on the tunnel effect. On the basis of the assumption that рас-

the distance between individual crystallites of a thin layer changes as a function of temperature, an expression was obtained for determining the electrical conductivity of the thin layer, namely:

\[ \sigma = A e^{-BT}, \tag{11} \]

but such a regularity is not confirmed experimentally.

As early as 1946, Ya. I. Frenkel^92, in considering the theory of electrical contacts, established that the electrical conductivity of metallic powders and thin layers consisting of very small grains increases with rising temperature according to the same law as the electrical conductivity of semiconductors, namely:

\[ \sigma = \frac{aCeT}{k}\, e^{-\frac{af-\gamma e^{2}}{akT}}, \tag{12} \]

where \(f\) is the work function of an electron from the metal, \(a\) is the width of the contact gap, \(C\) is a constant proportional to \(f\), and \(\gamma\) is a coefficient taking into account the multifaceted character of the surface and equal to \(0.9\)—\(1\). Since \(f > \gamma \dfrac{e^{2}}{a}\), the electrical conductivity must increase with rising temperature. At constant temperature, \(\sigma\) will change with a change in the thickness of the layer.

From equation (12) one can find \(\Delta \rho\), and also the value of \(\alpha\), i.e. the thermal coefficient of resistance, which is equal to

\[ \alpha = \frac{\Delta \rho}{\rho \Delta T} \simeq - f - \frac{\dfrac{\gamma e^{2}}{a}}{kT}. \tag{13} \]

Naturally, when \(f > \gamma \dfrac{e^{2}}{a}\), the value \(\alpha < 0\), and with increasing temperature the t.c.r. must decrease, which is also observed experimentally. Thus, these theoretical investigations are in agreement with the experimental data given above.

Other authors^93 arrived at similar results; they resolved this question on the basis of the following assumptions: a) a metallic thin layer consists of small crystalline grains; b) the work required for an electron to pass between two grains is less than \(0.1\) ev, and the tunnel effect is insignificant; c) the conductivity of the layer is practically equal to the conductivity of the gaps between the grains; d) the electrons in the grains obey Maxwell statistics, not Fermi statistics. As a result, the authors obtained a formula for determining the electrical conductivity of thin metallic layers:

\[ \sigma_c = AaT^{-\frac{1}{4}} e^{-\frac{f-\Delta f}{kT}}, \tag{14} \]

where \(\Delta f\) is the change in the work function of the electrons under the influence of the applied field, and \(a\) is the barrier width. This formula yielded

experimental confirmation on thin layers of tungsten and platinum^56,57 and explained the negative value of the t.c.r. The derivation of the indicated formula by Mostovich was based on the assumption that, for a small field strength, the change in the electron work function \(\Delta f\) is proportional to this field \(F\), whereas for large values of it \(\Delta f \sim F^{1/2}\). Gerter^94 also arrived at similar results. It follows from this that in the case when a change of temperature has no substantial effect on the work function \(f\), we must obtain a linear dependence of \(\lg R\) on \(F^{1/2}\).

In this case \(R(F)\) is determined by means of the following formula^49:

\[ R(F)=A_0 T^{-\frac14} e^{-\frac{W_0-e^{3/2}F^{1/2}}{kT}} . \tag{15} \]

Here \(W_0\) is the magnitude of the potential barrier. The dependence \(R(F)\) established by formula (15) was confirmed experimentally in the work cited above^62; its results are presented in Fig. 25.

Fig. 25. \(\lg R\) as a function of \(F^{1/2}\) at various layer temperatures.

Linearity must also hold in the case of the dependence of \(\lg RT^{1/4}\) on \(\frac{1}{T}\), which also found confirmation in work^62. In addition, the slopes of the straight lines in Fig. 23, according to equation (15), must be proportional to \(\frac{1}{T}\). The indicated authors calculated, on the basis of their data, the value of \(a\) and obtained the following results: at \(T=20.3^\circ\mathrm{K}\), \(a=0.00316\); at \(T=2.10^\circ\mathrm{K}\), \(a=0.073\); and at \(T=1.31^\circ\mathrm{K}\), \(a=0.124\), i.e., in the region of low temperatures this regularity is also confirmed.

Thus, the fact that thin metallic layers, in their properties, stand closer to semiconductors than to metals has not only been shown experimentally but also substantiated theoretically.

8. CONCLUSIONS

Let us draw brief conclusions from the works considered on the question of the structure and electrical properties of metals and semiconductors in thin layers.

1. The electrical properties of thin layers of metals or semiconductors cannot be considered in isolation from the question of struc-

type of layer. Not only the electrical properties, but in general the physical properties of the layer depend on the conditions under which it is obtained, for the conditions of preparation ultimately determine the structure of the layer.

The structure of the thinnest layers of metals and semiconductors depends on many causes and, above all, on: a) the nature of the sputtered metal, b) the nature of the substrate, c) the temperature of the substrate, d) the rate of condensation of the layer, e) the thickness of the layer.

As the thickness of the layer increases, a rearrangement of the structure of the substance forming the layer takes place; one type of lattice may pass into another, and allotropic transformations are observed. The rate of thermal re-evaporation influences the structure of the layer.

1. At the present time, on the basis of electron-diffraction and electron-microscopic investigations, a clearer conception is being formed of the whole process of formation of a thin layer. It becomes possible not only to establish the general character of the course of the process of layer formation, but also to detect the distinctive features in the formation of thin layers of various metals (see, for example, works \(^{24, 25, 26, 32, 33, 36, 37, 69, 96}\)).

The views previously expressed by A. Shalnikov and S. Vekshinskii on the question of the formation of a thin layer are confirmed by new data from electron-microscopic investigations.

Contrary to the data of electron-diffraction investigations, which do not confirm the existence of an amorphous phase of metals in thin layers, one must take into account the new data of electron-microscopic investigations, on the basis of which the presence of an amorphous phase at the initial moment of layer formation is admitted for one group of metals and a crystalline phase for another. It may be supposed that, in the process of layer formation, metals are initially in an amorphous phase, which then passes into the crystalline phase. In some cases the process of crystallization may occur instantaneously, in others more slowly. In this case further experimental data are needed, taking account of the time factor, which may help to clarify this question more thoroughly. For the present, nothing definitive can be said about the amorphous phase of metals in thin layers.

1. The picture of the influence of the underlying surface on the electrical properties of the layer is insufficiently clear. It is indisputable that the substrate exerts an orienting effect on the growth of crystals, and also that the temperature of the underlying surface influences the dispersity of the layer.

But the influence of the difference between the t.c.r. of the substrate and that of the film has not been clarified.

1. The picture of the structure of thin layers of semiconductors is clearer. Despite the small amount of experimental data, it is already known with certainty that there exist semiconductors which—

... in thin films can be obtained both in the crystalline and in the amorphous states, and these substances do not belong to the group of substances with a complex crystal-lattice structure. At the same time there are semiconductors that do not possess this property. In thin films these substances are deposited only in crystalline form.

1. Up to the present time there are in fact still no experimental data obtained under completely “clean” conditions. If experiments are carried out in a vacuum at a rarefaction of from \(10^{-6}\) to \(10^{-7}\) mm Hg, then even under these conditions the films obtained prove to be saturated with molecules of residual gases. These gases may be both adsorbed and absorbed by the film, and the possibility is not excluded of the formation of active chemical compounds, which will predetermine the properties of the film. This will be especially pronounced in the properties of thin semiconductor films. In thin metallic films, gases (especially oxygen) may exert almost the same influence as oxygen exerts on lead sulfide, which becomes a semiconductor as a result of temperature conditioning.

2. The influence of gases is not limited to a change in the bulk properties of a thin film. Molecules adsorbed by the surface of a thin semiconductor film also substantially change the properties of the surface. It turns out that the surface of a thin film in some cases has the property of a catalyst. The study of these phenomena has in fact only just begun.

3. As a positive point, one should note the work begun on the investigation of surface levels in thin semiconductor films. This question is of great importance in understanding the physical properties of thin films. It is not excluded that further investigation will help to study more deeply and to understand the phenomena occurring not only in thin semiconductor films but also in thin metallic films.

4. It seems to us that the technique for investigating thin films should be made somewhat more complicated, in order, as far as possible, to eliminate saturation of the film by gas molecules. The use of a counter for the investigation of surface states, and wider use of electron-diffraction and electron-microscopic methods, will make it possible to obtain additional information about this interesting field, which has great practical significance.

5. The general picture of the dependence of the specific resistance and of the thermal coefficient of resistance can be satisfactorily represented by the curves in Fig. 15.

6. The theory proposed by Ya. I. Frenkel to explain the negative sign of the t.c.r. of metals in thin films, as well as

...of the influence of the field on electrical conductivity satisfactorily explains the new experimental data. The Mostovich–Gerter theory also agrees satisfactorily with the experimental data.

It is quite clear that the influence of an electric field on the electrical conductivity of thin layers cannot be explained on the basis of the theory of the Schottky effect, since the latter assumes a uniform distribution of the field’s influence on the particles, whereas in thin layers the field is distributed mainly between the gaps of crystallites or between their aggregates. In this connection, the observed activation energy must be attributed not to the width of the forbidden band, but to electron transitions between crystalline grains.

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