On the Photoconductivity of Crystals

A. L. Hughes, Washington, USA *

I. Introduction

The special case of photoconductivity discussed in this article concerns the electrical conductivity of certain insulating crystals under appropriate illumination. It should be noted that the attention of researchers working in the field of photoelectricity has been concentrated almost exclusively on the external photoelectric effect; photoconductivity, however, a no less interesting aspect of photoelectricity, has remained neglected. Much of what is known about photoconductivity in crystals must be attributed to the brilliant and systematic work of Pohl and his collaborators in Göttingen.

Photoconductivity touches on many other areas of physics. It is closely connected with certain optical properties of crystals. As an important source of data on crystal structure, photoconductivity acquires special significance in the classification of the properties of crystals and their division into “structure-sensitive” and “structure-insensitive” properties. This was confirmed by frequent references to photoconductivity at the London Conference on the Solid State, held in 1934[^1]. A colored crystal is a model of a “dilute metal,” which may be represented as a solid containing a sufficient number of electrons to conduct current, but at the same time not so many as to be opaque in optical measurements. The study of photoconductivity has yielded many results that have not yet received a satisfactory explanation in the light of our present knowledge of the crystalline state. Finally, in photoconductivity we have an example of a continuous change in the development of a particular branch of science: ten years ago the principal aim in the study of photoelectric phenomena was the question of the quantum yield, i.e., the number of electrons liberated by one quantum of absorbed light; now photoconductivity should be regarded differently—namely,

* A. L. Hughes, Rev. Modern. Phys. 8, 294, 1936. Translation and adaptation by A. S. Toporets. Additions made by the translator are enclosed in square brackets.

as one of the means of approaching the solution of the problem of the nature of the crystal.

II. Review of Results Obtained up to 1932

The most complete review of the facts established up to 1932 is given in our monograph on photoconductivity ², ³. Here, however, a review will be given only of those results which are necessary for acquaintance with the terminology and methods described in the present article. The conductivity of crystals is measured in the following way: a voltage is applied to two electrodes (graphite or platinum) in contact with the crystal, and the resulting current is recorded by a fast-acting electrometer. (A single electrometer can measure a wide range of currents if it is shunted by a set of high-ohmic resistances from \(10^{6}\) to \(10^{11}\ \Omega\).) The current can be studied as a function of one or several variables: the applied voltage, the intensity of the light, its frequency, the temperature, and the previous history of the crystal. Gudden and Pohl made a significant step forward when they found that, together with the primary photoelectric current, there can exist a secondary current. In contrast to the primary photoelectric current, the secondary current is characterized by a strong hysteresis effect. The primary photoelectric current is the direct result of the absorption of a quantum inside the crystal; the secondary current, however, is the result of an increase (usually progressive) in the conductivity of the crystal, arising from the primary photoelectric current. In some cases it may exceed the primary photoelectric current by a thousand or more times; it ceases in the dark (a very instructive report on secondary currents is given by Lehfeldt.⁴) Usually the primary photoelectric current increases with the applied voltage; in some crystals, at a sufficiently high voltage, a saturation current can be reached. When saturation is attained, the charges separated photoelectrically traverse the entire path between the electrodes; before saturation, the field is insufficient to drive them along the whole path to the electrodes. In this way a quantum yield was found in certain crystals of diamond and zinc blende; it turned out that for each absorbed quantum there is one electron and one positive charge, separated from one another and moving toward the corresponding electrodes. The primary photoelectric current consists of two parts: an electronic primary current, which corresponds to the motion of electrons in the direction toward the anode, and a positive primary current, which carries out the displacement toward the cathode of the positive charges remaining where the electron was liberated upon absorption of the quantum. When both flow simultaneously, we have the full primary photoelectric current. Under some conditions (low temperature, an almost ideal lattice) the primary current is formed only by electrons; the positive charges remain immobile and form a spatial positive charge. It is said that the crystal is excited, as may be judged from the change in its properties. The absorption spectrum

of such a crystal changes in such a way that its absorption on the long-wavelength side of the band is greater than before. It temporarily acquires sensitivity to red and infrared rays; in the case of a large electron current this sensitivity is retained for a sufficiently long time. The positive charges remaining after the detachment of electrons can be made to move by heating the crystal. To avoid disturbances arising from the formation of a space charge, it is customary to illuminate the crystal simultaneously with photoelectrically active light, with red and infrared light. One cannot assume that the positive ions, as such, move in the direction of the cathode. It is more correct to think that this is the neutralization of a positive charge by an electron moving toward it from the nearest site in the rest of the lattice; in this process the new positive charge slips farther toward the cathode by a repetition of this process. A complete absence of positive primary current is observed only under special conditions; usually it is assumed that there is a certain spontaneous flow of it together with the primary electron current. In such cases the current continues even after the light is switched off, when the primary electron current has already ceased, until the positive space charges disappear. At a sufficiently high temperature, or when the crystal is illuminated with red or infrared rays, both parts of the primary photoelectric current flow simultaneously. This somewhat unusual way of representing ion transport is often convenient when considering the mechanism of conductivity.

Hilsh and Pohl5 studied photoconductivity in naturally colored rock salt at various temperatures. In some pieces, at \(100^\circ\), the current began at values of \(4 \cdot 10^{-9}\ \mathrm{A}\) and then, over the course of 10 sec, fell to \(2 \cdot 10^{-9}\ \mathrm{A}\). The initial value of the photoelectric current depends on the applied voltage, whereas the final value is determined by polarization, since the positive charges remain behind the centers. The final value of the current corresponds to an equilibrium between the electrons liberated photoelectrically and moving in the direction of the anode, and other electrons replacing them, liberated thermally or photoelectrically. The “replacement” of electrons constitutes a positive primary current. That the polarization acts in the opposite direction for an appreciable time is shown by the following experiments: after the illumination of the crystal is ended, the field is removed, and if after 30 sec the light is switched on again, a current arises in the opposite direction. If the experiment is carried out at \(220^\circ\), the value of the photoelectric current remains constant. At this temperature the electric current that flows even in the dark is quite noticeable. In this case polarization is not formed, because the large ionic conductivity allows the positive primary current to proceed at full strength. Polarization arises only when the mechanism for compensating the positive charge is insufficient.

De Boer6 proposes the following picture of the mechanism of the positive primary current. In a photoconducting crystal there is a certain number of centers from which electrons can be detached. The first result of illuminating the crystal with suitable light is the liberation of a certain number of electrons. They move in the direction of the electric field until they become stuck at distorted and irregular places in the lattice. In doing so, their displacements are very small in comparison with the dimensions of the crystal and large in comparison with the mean distance between centers. The displacement of the electrons constitutes the primary electron current. The electrons are bound more weakly to the lattice irregularities—let us call them “excitation” centers—where they become stuck, than to the true centers. They can therefore be freed from the “excitation” centers by infrared quanta, whose energy is insufficient to liberate electrons from the true centers. If one now assumes that, on the average, an electron freed from an “excitation” center by an absorbed infrared quantum must travel to one of the true centers the same distance as a true electron freed from this center, then it is clear that the positive primary current and the negative electron current must be equal. This assumption contradicts the case of saturation of the primary electron current, in which all the free electrons traverse the whole path to the anode and do not remain in the crystal, which is necessary for de Boer’s explanation of the positive primary current.

Even if one disregards the case of saturation, i.e., if the electrons do not leave the crystal, it is by no means clear why the mean displacement of a photoelectron from a true center should be the same as the subsequent displacement from an “excitation” center to a true center, which has remained charged.

Light produces conductivity of an electronic character in a crystal. Good “insulating” crystals conduct in the dark when the temperature is raised to a sufficient degree, but their conductivity is then purely electrolytic. The latter is connected with the transport of matter to the electrodes, which can be observed at a sufficiently large current7.

Gudden and Pohl found photoconductivity in two very different types of crystals, which they called idiochromatic and allochromatic. In the former, photoconductivity appears when the crystal is chemically pure and when it is a perfect, or sufficiently perfect, single crystal. Such crystals are diamond and zinc blende. In allochromatic crystals photoconductivity cannot be detected when the crystal contains no impurity. Their photoconductivity is the result of the introduction of foreign atoms, i.e., atoms not belonging to the lattice, or the result of physical changes produced by certain agents. Alkali-halide crystals are the most suitable illustration of this type. The photoconductivity of many allochromatic crystals has been investigated with great completeness during the last five

for years; on the contrary, very little attention has been paid to idiochromatic crystals.

III. Absorption Bands in Alkali-Halide Crystals

Crystals of pure alkali-halide salts are characterized by an almost complete absence of absorption in the near ultraviolet, in the visible part of the spectrum, and in the far infrared[^8],[^9]. In the far ultraviolet, each alkali-halide salt has a very strong characteristic absorption band, with an opacity almost like that of metals. Chlorides have one terminal peak near 1650 Å; bromides—near 1880 Å and another in the shorter-wavelength part of the spectrum (Fig. 1). When light is absorbed in this characteristic band, there is no trace of photoconductivity.

Fig. 1. Intrinsic absorption bands of alkali-halide crystals

The high value of the absorption coefficient shows that a very large number of atoms in the lattice are capable of absorbing light in this region, and the view exists that such absorption is connected with the transfer of an electron from a negatively charged halide ion in the regular lattice to the nearest positively charged alkali-metal ion. The details of this process lie beyond the scope of this article; it is immaterial that this type of absorption is not accompanied by any electrical conductivity. This is in agreement with the view that, as a result of absorption, the electron moves only from one ion to the next; the distances involved are too small to yield any measurable current. The absence of any noticeable permanent changes in the crystal indicates that the electron returns to its former

would return to its original position in a very short time; otherwise a noticeable number of neutral atoms could form in the crystal and, in this way, its absorption spectrum would change.

Absorption in this characteristic band belongs to “structure-insensitive” phenomena, in contrast to that absorption which may be assigned to “structure-sensitive” phenomena.

The band of characteristic absorption has a long-wavelength branch extending into the near ultraviolet. It is not shown in Fig. 1, since the absorption in this branch is almost a million times smaller than in the band itself. When the crystal contains, for example, 1 part of \(KNO_3\) per 100 thousand parts of \(KBr\), the long-wavelength branch changes in shape and appears more strongly; thus in this case it is natural to ascribe its presence to foreign atoms, probably caught in imperfections of the lattice. The absorption here may be regarded as a displacement of an electron weakly bound to an impurity atom. If we consider the salt to be quite pure, we cannot suppose that the long-wavelength branch is connected with foreign atoms; another explanation for it is that electrons can more easily be detached from atoms located in lattice imperfections, in cracks or at intermediate surfaces, than from atoms within those elements in which the lattice is perfect. Such a view is supported by a number of experiments, among which the following may be noted \(^{10, 11, 12}\). A stress is applied to the crystal for a certain time and is then removed. The absorption coefficient of the long-wavelength branch of the pure crystal thereby increases. In this case there can be no question of an increase in the number of foreign atoms within the crystal; but it is easy to imagine that after compression of the crystal it will contain internal fissures, cracks, etc., to a greater degree than before.

Crystals of alkali-halide salts may have various colors, depending on the state of the metal incorporated in the lattice. Thus, rock salt may be yellow, red, violet, and blue \(^{13}\). The first of these colors is attributed to the atomic state of the excess metal, whereas the latter are attributed to colloidal particles of various degrees of dispersion. Upon absorption of light in such colored crystals a photoelectric current is observed, which should be regarded as the motion of electrons liberated by light from atomic or colloidal particles.

Perhaps the most interesting among colored photoconducting crystals, and the most studied, are those which contain no colloidal particles, but only atomic ones. We shall not go into the question of the nature of this coloration here and, following Pohl, shall call the carriers of this type of color “coloring centers,” or \(F\)-centers. Below, the methods of obtaining these centers will be described. It is remarkable that, by whatever method a crystal is colored, a simple band is always obtained, the so-called \(F\)-band (Fig. 2), whose position

of the photoconductivity of crystals

is determined by the lattice constant according to the formula:

\[ \nu_{\max} d^{2} = \mathrm{const}. \tag{1} \]

The maxima can be brought into coincidence; then it turns out that the curves for different alkali-halide salts are superposed very well on one another (Fig. 3). The bands become narrower when the temperature is lowered, but their total area remains the same, which shows that the number of absorbing centers remains unchanged. Upon absorption of light in this band, the crystal becomes conducting.

Fig. 2. \(F\) bands in alkali-halide crystals

Fig. 3. Superposition of \(F\) bands for various alkali-halide crystals and the influence of temperature on the band shape

It will be shown below that \(F\)-centers are associated with a stoichiometric excess of the alkali metal in the crystal. The \(F\)-bands of sodium halides occur in the spectrum at different positions in accordance with formula (1), i.e., the determining factor is the lattice constant. This is in sharp contradiction with the absorption spectra of Na dissolved in molten NaCl, NaBr, or NaJ, for which it has been found that the absorption band-

tion is not characteristic of the kind of molten salt in which the metal is dissolved; its position is determined exclusively by the nature of the metal[^14]. The center of the absorption band, whose width is approximately \(2V\), is situated on the long-wavelength side at a distance of \(0.5V\) from the resonance line of the metal in the vapor state. This circumstance indicates that the spectrum of an excess metal dissolved in a molten salt is indeed the same as that of the metal vapor, but shifted and much broader because of the proximity of the solvent molecules. But when the molten salt is cooled and passes into the crystalline state, \(F\)-bands are obtained with their characteristic differences, depending directly on the lattice constant and not at all on the nature of the metal.

Fig. 4. The \(U\) band in KBr at different temperatures

Recently Pohl and his collaborators[^19] discovered, under certain conditions in alkali-halide crystals, a new absorption band, this time in the ultraviolet. They called it the \(U\)-band (Fig. 4), and its carriers—\(U\)-centers. When light is absorbed in this band, photoconductivity is not observed; but it is interesting that absorption of light in the \(U\)-band transforms \(U\)-centers into \(F\)-centers; in other words, the crystal becomes colored and, when illuminated with visible light, conducts current. The \(F\) and \(U\) bands are similar in that both have a bell-shaped form and shift reversibly when the temperature changes. The position of the maximum of the \(U\)-band for different salts is also determined by the simple relation:

\[ \nu_{\max} d = \mathrm{const}. \tag{2} \]

The lattice constant, as the determining factor, enters here only in the first power, and the numerical constant has a different value*.

The altered absorption bands that were found in alkali-halide salts are presented in Fig. 5.

IV. Production of \(F\)-centers

Let us enumerate the methods of obtaining the \(F\) band: 1) first, crystals with this type of coloration, i.e. with already formed centers, are often found in nature. Secondly, they can be obtained by various artificial methods; 2) by irradiating the crystal with X-rays or radium rays; 3) by irradiating with ultraviolet rays (the ultraviolet light must be from the long-wave branch of the absorption curve, since within the band the light cannot penetrate into the interior of the crystal; it is strongly absorbed in the surface layer);

Fig. 5. Relative positions of the absorption bands: intrinsic, \(U\) centers, \(F\) centers, and colloidal particles

4) by exposing the crystal at a sufficiently high temperature to the vapor of an alkali metal; here it is immaterial which alkali metal is used; 5) by the penetration of electrons into the crystal from a sharp cathode*, pressed into the crystal; 6) by the absorption of ultraviolet light in the region of the \(U\) band, if the crystal contains \(U\)-centers.

It is very interesting and remarkable that the bands obtained by such different methods are identical in form and position.

The absorption of light within the \(F\) band in a colored crystal may have the following result: a) “excitation” (Erregung); b) bleaching; or c) formation of colloidal particles. Cases are known in which a) and b) occur simultaneously. In “excitation”

* This may include the method of S. A. Arzybyshev \(^{16}\), consisting in the fact that instead of a sharp source of electrons an alkali metal is used. A piece of this metal is placed in a hole made in the crystal and closed by a tightly pressed cover. This method makes it possible to color the crystal at a lower voltage.

Translator’s note.

the \(F\) band extends into the long-wavelength part of the spectrum. Illumination with infrared light destroys the “excitation,” and the band is restored in its former original form. One may think that “excitation” is a process in which an electron, freed from an \(F\)-center upon absorption of a quantum, moves a short distance until it is trapped at some imperfection of the lattice. It can then be regarded as part of an excited center, from which it can easily be liberated by infrared light and return to the original or to some other \(F\)-center.

Bleaching occurs when the liberated electrons, being captured by neutral halide atoms, turn them into ions. The electrons are more strongly bound to these latter, which is confirmed by the fact that bleaching is an irreversible process, whereas excitation is a reversible process. In photochemical coloring (methods 2 and 3), bleaching of crystals in scattered daylight occurs very rapidly. At the same time, “excitation” may also be observed. In strongly colored NaCl, absorption of light within the \(F\) band leads to coagulation of the \(F\)-centers into colloidal particles \(^{17,18}\).

V. Conductivity in crystals containing \(F\)-centers

We shall now return to the consideration of photoconductivity associated with the \(F\) band. The electrical arrangement for the investigation is very simple (Fig. 6). The crystal is placed between two electrodes. The potential difference is supplied by a suitable battery, and the current

Fig. 6. Experimental arrangement for measuring photoconductivity

is measured by a fast-acting electrometer shunted by a high-ohmic resistance. In the dark, the crystal is a good insulator, provided only that the temperature is not too high; but when it is illuminated with light absorbed in the \(F\) band, it becomes conducting. In many cases the current decreases with time, since the growing polarization inside the crystal tends to annul the applied field. There is an assumption that somewhere

in the \(F\) band a quantum of light is absorbed and liberates an electron, which moves in the direction of the field over short distances until it is trapped in some crack, gap, or intermediate surface in the crystal. With a sufficiently long current the crystal becomes electrically polarized as a result of the separation of electrons from their original sites, where positive charges remain. The \(F\) band is now lowered and broadened, chiefly in its long-wavelength branch. It is said that the crystal is “excited.” The “excitation” can be destroyed by heating the crystal or by illuminating it with red or infrared rays.

These results indicate that when electrons are liberated from \(F\)-centers, they are trapped in other places where the binding energy is so small that infrared light or thermal motion at moderate temperatures can liberate them. In the absence of an electric field they return again to the original \(F\)-centers, thereby destroying the state that had existed.

There is an essential difference in the behavior of \(F\)-centers in an electric field, according to the method of their production. When a crystal is colored by methods 2 or 3, the application of an electric field to the crystal produces a current, tearing electrons away and causing them to move. This occurs only if the temperature of the crystal is sufficiently high (about \(250^\circ\) for NaCl, \(180^\circ\) for KCl). At the same time the color becomes paler, and the crystal, in the end, becomes colorless. Here we are dealing with decoloration. Other results are obtained in the case of crystals colored by methods 4, 5, or 6. In these cases the application of an electric field (it is understood that the temperature is sufficiently high) produces a current that reveals the motion of electrons; but now the colored region moves as a whole along the crystal toward the anode, leaving the crystal behind it colorless \(^{19,20}\). In case 6 the intensity of the colored region decreases as it moves, revealing the transformation of some \(F\)-centers into invisible \(U\)-centers, while the remaining electrons move toward the anode. The difference in the behavior of \(F\)-centers obtained by methods 2–3 or by methods 4, 5, and 6 can be explained as follows. In cases 2 and 3 the centers are created in the crystal by illumination with ultraviolet light or X-rays—a process which does not disturb the “one-to-one” ratio between metal and halide atoms, but which, it may be supposed, separates electrons from halide ions, leaving them neutral. The liberated electrons are weakly bound at various suitable places, forming \(F\)-centers. If the temperature is sufficiently high, the electric field forces the electrons to move, and sooner or later they are again captured by neutral halide atoms, thereby restoring the conditions that existed before coloration. In cases 4 and 5 such an effect cannot occur, since there are no neutral halide atoms with which the moving electrons could bind. The electrons execute migrating motion in the di-

toward the anode, and in all those places where they temporarily stop, they form \(F\)-centers. Thus the coloration moves along the crystal toward the anode. Such an explanation is consistent with the fact that in cases 4, 5, and 6 there is a stoichiometric excess of metal atoms over halide atoms, which is not present in cases 2 and 3.

The method of coloring a crystal by the penetration of electrons into it (designated as No. 5) deserves special attention. The crystal is mounted between a flat anode and a sharp cathode. A potential difference of several hundred volts is applied to the crystal, heated in a small furnace; a cloud develops from the cathode, as shown in Fig. 7, which moves through the crystal in the direction of the anode and in a few seconds passes through the whole crystal. When the crystal is cooled to room temperature, the cloud is yellow in NaCl, violet in KCl, blue in KBr. In all cases the absorption band is the same in shape and position as the \(F\) band described above. If the field is reversed, the cloud moves away from the new cathode, and the boundary moves toward the new anode. We now have something really very similar to the cathode dark space, but, unlike the dark space in a gas, it can be increased and decreased simply by passing current. Measurement of the potential also shows some similarities. The field is greater in the transparent region than in the colored one, which is the analogue of the positive column in a gas discharge.

Fig. 7. A colored cloud propagating from a point cathode

Fig. 8. Superposition of the electronic current on the ionic current

According to Pohl’s hypothesis, the colored cloud is associated with the penetration of electrons into the crystal. \(F\)-centers are formed where the electrons become bound at interfaces and lattice defects.

Not every type of cathode will produce an electron cloud. A heated transparent alkali-halide crystal conducts electrolytically; we may assume that the positive metal ions move toward the cathode and are neutralized there by electrons. The conditions change when a special type of electrode is used. A sharp cathode creates a strongly localized field, enabling electrons to pass directly into the crystal, so that the current in the crystal is carried by electrons moving from the cathode just as well as the positive ions move toward the cathode and the negative ions away from it.^21 An “anodically polarized” cathode will act in the same way as a source of electrons penetrating into the crystal. This is illustrated-

is characterized by the following experiment. A transparent crystal is provided with two electrodes: one of platinum foil, pressed tightly against the crystal, and the other a layer of graphite. From the graphite (as cathode) to the platinum (as anode) a current—usually electrolytic—flows through the heated crystal, which remains transparent. When the current is reversed, and consequently when the platinum is the cathode, a colored cloud spreads into the crystal.

Let us assume that the coloration is produced by the motion of electrons, which in some way form \(F\)-centers. One may suppose that the visible cloud is associated with an electron density of the order of \(10^{17}\) per \(\text{cm}^3\). With such a number, the repulsive forces should correspond to a million atmospheres or something of that order. Therefore there must be some compensation of the motion of positive and negative charges. While the cloud moves toward the anode, there must occur a compensating motion of halide ions, such that the number of moving electrons is equal to the number of moving Cl-ions. The conductivity associated with the electron cloud is illustrated by experiment \(^{22}\), in which a voltage of 2 thousand V is applied to a transparent KCl crystal, heated to \(675^\circ\), between a point anode and a plane cathode. In this case only an electrolytic current flows, and the crystal does not become noticeably colored. When the direction of the field is reversed, the point becomes negative, and a visible cloud spreads through the crystal. At the same time the current increases, and if the direction of the current is changed, the current decreases; in the end, when the cloud contracts and disappears, the initial value of the electrolytic current is obtained (Fig. 8). At lower temperatures (about \(390^\circ\)) the electrolytic current is much smaller, of the order of \(25\,\mu\text{A}\). As the cloud spreads, the current increases and, after passing through the whole crystal and with no change in coloration, reaches a value of \(40\) thousand \(\mu\text{A}\). We must suppose that while the front of the cloud advances, a sufficient number of Cl-ions moves forward to make room for the advancing electrons and thereby keep the crystal neutral or nearly neutral. As soon as the cloud has passed uniformly over the entire path, the ions no longer play a role, since exactly as many electrons leave the crystal through one electrode as enter through the other. We now have a picture that very much resembles what occurs in a metal: the colored crystal in this case can indeed be regarded as a “diluted” metal. In a transparent crystal of rock salt there are as many positive Na ions as negative Cl ions. In a colored crystal this balance is disturbed, i.e., there is a stoichiometric excess of Na atoms.* There must be \(N\) Na ions, \(n\) electrons, and \(N-n\) Cl ions. If the Cl-ions are repla—

* Joos believes that it is possible to distinguish two ways of obtaining a stoichiometric excess of metal: coloration in vapors and coloration from the cathode. In the first case, according to Joos, metal atoms penetrate inward; in the second, electrons do. This contradicts the circumstance that in additive coloration the band is determined only by the metal whose ions ...

...completely, then we must have \(N\) positive Na ions arranged in the lattice with \(N\) free electrons, which may be called ordinary metallic Na. Thus the colored crystal appears to be something intermediate between two extremes: a pure metal with positive ions and electrons, on the one hand, and a rock-salt crystal containing an equal number of positive Na ions and negative Cl ions, on the other. There is no doubt that the colored crystal is much closer to the transparent crystal than to the metal, since even under the most favorable conditions it is impossible to introduce into the crystal more than one electron per hundreds of thousands of ions present there. In this sense it is understandable why Pohl regards the crystal as a diluted metal, since at insufficiently high temperatures it gives only an electrolytic current, while the conductivity of the colored crystal is electronic, as in metals.

Let us now return to the case when the crystal is completely colored and a voltage is applied; electrodes are chosen such that electrons cannot penetrate from the cathode into the crystal. The colored region will move toward the anode until it disappears. In order to assert that the crystal is electrically neutral, we must assume that within the colored part there is an excess of positive Na ions or a deficiency of Cl ions, or, more precisely, a combination of both. This assumption agrees with the laws of electrolysis. Therefore the colored cloud cannot move faster than the compensation of ions takes place. This view is well confirmed by measurements of the mobility of the colored part of the electron cloud in a pure KCl crystal.^26 At temperatures somewhat below \(680^\circ\) the cloud becomes diffuse; its front boundary corresponds to rapidly moving electrons, the rear boundary to slowly moving ones. Above \(680^\circ\) the cloud preserves its shape, so that one may think that all the electrons forming it move with the same velocities (Fig. 9). The study of the electrolytic con—

Fig. 9. Comparison of the mobility of electrons with electrolytic conductivity at various temperatures

^26 They are contained in the lattice, regardless of the vapors of which metal are used to color the given crystal. It is more correct to say that in both cases electrons penetrate into the crystal.

Transl. note.

...conductivity shows a decisive break at the very same temperature \(^{23}\). Above this point all ions of the lattice participate equally well in the conduction process; below it—only ions more weakly bound at internal boundary surfaces. There is probably a wide dispersion in the velocities of the ions sliding along these surfaces, because the form of the latter is not the same and varies within wide limits. Conductivity at high temperature may be regarded as a “structure-insensitive” phenomenon; at low temperatures, as a “structure-sensitive” phenomenon \(^{23}\). Electrons, potentially capable of moving with considerable velocities, cannot move through the crystal faster than the ions bound to them, and the observed mobility of electrons is due to the motion of ions. At high temperatures the velocities of individual ions differ relatively little; therefore the electron cloud is preserved; at low temperatures the velocities of the individual participating ions are distributed over a wide range; therefore a broad dispersion of electron velocities also occurs.

If the electron cloud spreads shapelessly from one electrode to the other, and if the cathode is of such a type that electrons leave it directly into the crystal, pass through it, and arrive at the anode, then no compensating motion of ions is required, just as in a metal. But if the cloud is concentrated in part of the crystal, then, in order to keep the crystal electrically neutral, there must exist a compensating motion of ions.

Fig. 10. Mobilities of electrons in various alkali-halide crystals at different temperatures
\[ \left(\text{in } \frac{\mathrm{cm}}{\mathrm{sec}} \text{ under a field of } 1\,\frac{V}{\mathrm{cm}}\right) \]

The mobility of the electrons constituting the cloud in a pure crystal can be measured by observing the velocity with which the cloud moves in a given field \(^{25,26,27}\). The mobility of electrons in various alkali-halide crystals is shown in Fig. 10. The value of the mobility is given by the formula

\[ v_T = v_0 e^{-\frac{\varepsilon}{kT}}, \tag{3} \]

where \(v_T\) is the mobility at temperature \(T\); \(v_0\) is a constant, of the order of the mobility of electrons in metals; \(\varepsilon\) has a value slightly less than \(1\ \mathrm{eV}\).

VI. Concentration of Colored Centers in the Crystal

The next point on which we shall dwell is the relation between optical absorption in a colored crystal and the number of \(F\)-centers when they are produced by the penetration of electrons. Each electron penetrating inside is connected with one \(F\)-center or constitutes an \(F\)-center, if we consider it as bound inside the crystal.

If the colored crystal is placed between electrodes from which electrons do not enter the crystal, then the applied field causes the cloud slowly to leave the crystal for the anode. The measured current decreases with time and, when the cloud is exhausted, reaches a constant value, as is shown in Fig. 11.

Fig. 11. Decrease of conductivity upon removal of the electron cloud from the crystal

Fig. 12. True separation of the observed current between ions and electrons

Fig. 13. Voltage drop in the colored part of the crystal

The constant value represents the electrolytic current, while the shaded area represents the total charge carried by the electrons forming the cloud. The total number of electrons can be found by dividing by \(e\) (the value of the elementary charge). The number of electrons in the crystal can also be found by an optical method. Suppose that these electrons, weakly bound to the lattice, constitute \(F\)-centers. If each of them is regarded as a simple oscillator, then the usual dispersion formula leads to the result\(^{28,30}\) that the number of oscillators, or, what is the same thing, the number of electrons participating in the absorption of the band, is

\[ N_0 = 1.31 \cdot 10^{17} \frac{n}{(n^2 + 2)^2} KH, \tag{4} \]

where \(n\) is the refractive coefficient at the maximum of the band; \(K\) is the absorption coefficient at the maximum of the band, \(H\) is the half-width of the band.

It was thus found that the number of electrons obtained by the electrical method is comparable with the number obtained optically,

with an accuracy of up to a few percent. Probably this agreement is too good, since the exact value of the numerical constant in the optical formula depends on what assumptions were adopted in deriving it. Without entering into a detailed analysis of this question, we shall merely point out that the fact of agreement between these two values gives grounds for considering the electrons participating in conduction to be also responsible for the type of coloration. Concentrations between \(10^{17}\) and \(10^{18}\) electrons per \(\mathrm{cm}^3\) were obtained in alkali-halide crystals in vapors of K and Na at temperatures from \(400^\circ\) to the melting point.

Quite recently Kleinschrod\(^{29}\) determined the number of \(F\) and \(U\) centers in KCl by optical and chemical methods. It turned out that the numbers obtained by chemical analysis exceed, on the average, the corresponding numbers from optical measurements by a factor of 1.24. From this Kleinschrod concludes that the constant \(A\) in the dispersion formula should be corrected for KCl from \(1.1\cdot10^{16}\) to \(1.31\cdot10^{16}\). The same experiments for \(U\)-centers gave a different ratio, namely: 0.95. It is noteworthy that Kleinschrod’s experiments are the first attempt to determine this constant by a chemical method*.

Closer examination by Mott and Gurney\(^{28}\) shows that the described method of determining the number of electrons, or centers, from the shaded area in Fig. 11 requires correction. In this method it was tacitly assumed that the electrolytic current in the transparent part of the crystal has the same value as in the colored part (Fig. 13). Let \(V\) be the total potential drop along the crystal. The currents \(i_i\) and \(i_f\), carried by ions and electrons when the crystal is uniformly colored, are equal to \(V\) multiplied by the corresponding conductivities \(c_i\) and \(c_f\). When the whole crystal is transparent, the current is carried only by ions, and we have \(i_i=c_iV\). When the crystal is uniformly colored between the cathode and the anode, the current is equal to:

\[ I=i_i+i_f=(c_i+c_f)V. \tag{5} \]

(Here it is assumed that the presence of color centers does not affect the ionic conductivity.)

Hence the ratio of the current carried by ions to the total current is equal to:

\[ \frac{c_i}{c_i+c_f}. \]

This means that in the colored crystal \(OA\) corresponds to the electrolytic current and \(AB\) to the electronic current (Fig. 11). Now, when the field also includes the colored part adjacent to the anode,

\[ \text{* The constant } A \text{ is equal to} \]

\[ A=1.31\cdot10^{17}\frac{n}{(n^2+2)^2}. \]

Continued.

and transparent behind it (Fig. 13), the division of the current in the colored part between the ions and the electrons will remain in the ratio \(c_i\) to \(c_f\). Consequently, Fig. 11 must be replaced by Fig. 12.

Here the ordinates of any point on \(AC\) are always \(\dfrac{c_i}{c_i+c_f}\) times smaller than the ordinates of the corresponding point on \(BD\). The area representing the charge carried by the electrons is \(ABDC\); it is almost twice as large as the area \(ABD\), which was considered earlier as corresponding to the charge carried by the electrons. Consequently, we have a more exact method for calculating the number of electrons in the colored cloud from measurements of the charge associated with them. The expression for \(i_i\) and \(i_f\) may be obtained as follows. The total potential drop \(V\) consists of two parts: \(V_1\)—the drop along the transparent part of the crystal, and \(V_2\)—the drop along the colored part. Hence,

\[ V=V_1+V_2 \]

and it is clear that

\[ \frac{V_1}{V_2}=\frac{\dfrac{x}{c_i}(c_i+c_f)}{d-x}. \tag{6} \]

From this we obtain:

\[ V_1=\frac{(c_i+c_f)x}{c_i d+c_f x}\,V \quad \text{and} \quad V_2=\frac{c_i(d-x)}{c_i d+c_f x}\,V . \tag{7} \]

Obviously, the field is larger in the transparent part of the crystal. The current \(I\), flowing in the transparent part of the crystal, in the colored part consists of two parts, \(i_i\) and \(i_f\). Then we have:

\[ i_i=\frac{c_i^{\,2}A}{c_i d+c_f x}\,V \quad \text{and} \quad i_f=\frac{c_i c_f A}{c_i d+c_f x}\,V, \tag{8} \]

where \(A\) is the cross-sectional area. These two equations will give the two components of the current, decreasing as the transparent part of the crystal lengthens. Moreover, it is seen that

\[ \frac{i_i}{i_f}=\frac{c_i}{c_f}, \]

as was assumed at the outset. The fact that the front boundary of the colored cloud becomes diffuse, while the rear boundary remains sharp, follows only from the equations just derived or from the fact that the field is weaker in the colored part than in the nonconducting transparent part. If any electron tends to remain behind the rear boundary of the colored cloud, it enters a region where the field is stronger, and moves forward into the cloud. Therefore the boundary remains sharp. In an equal manner, by the very same reasoning, one can explain why the front boundary tends to blur. It is interesting here to note two formal analogies. The first is the sharpness or diffuseness of the boundary between two differently colored electrolytes ...

covered in a similar way. The second—the transparent region and the colored one—correspond to the sharply dark space and the positive column, since the relative magnitudes of the fields are important.

The interpretation of the phenomenon described may apply if one assumes that inside the cloud there are electrons capable of moving in the field. But these electrons are not as free as electrons in a metal. In the first place, there is a broad infrared region for which the crystal is completely transparent; no such region has been found in metals and cannot be expected even in an idealized finely divided metal. Likewise, the existence of a colored band in a definite region of the spectrum indicates selective bonds. The picture is then as follows. At some moment an electron is bound at some place in the lattice—probably on internal surfaces, cracks, etc. Thermal motion in the crystal liberates it, and the applied field displaces it over short distances in the direction of the anode until it becomes trapped in some other lattice irregularity. Sooner or later thermal motion knocks it out again, and thus it moves step by step toward the anode. During a halt, when it is bound to the lattice, it is capable of absorbing a quantum of light. Therefore the part of the crystal that contains electrons is colored. It is clear that at high temperature these electrons move more easily. At room temperature light liberates electrons, but because there is no ionic current, the polarization that arises tends to destroy the applied field.

Thus the conception of color centers, or \(F\)-centers, is not at variance with the conception of electrons, here and there weakly bound in the crystal. If one takes a transparent rock-salt crystal and a second one—colored by the method just described—and dissolves them both in water, the solution of the first will be neutral, while that of the second will be alkaline\(^{29}\). This is clear proof that in the colored crystal there are more Na atoms than Cl. Or, if a well-colored crystal is heated to a sufficiently high temperature in a vacuum, the color will disappear, and a Na condensate will be found on the walls of the vacuum vessel. Further, when the \(F\) band is obtained by “additive” coloring, i.e., in metal vapors at high temperature, it is natural to think that neutral atoms penetrate into the crystal. Such consideration from all three sides leads to the conclusion that the \(F\) band is associated with an excess of metal atoms in the crystal. But it does not follow definitely from this that the excess metal atoms, as such, are the color centers. With equal justification one might suppose that the excess atoms dissociate into positive ions and electrons, and regard the electrons as having a direct relation to the color centers; the function of the positive ions is secondary: to preserve the electrical neutrality of the crystal.

One piece of evidence that color centers may be regarded as electrons, and not as separate neutral atoms, is the fact that the band always turns out to be of exactly the same color, irrespective of whether the crystal was kept in K vapor or in Na vapor. The function of the metal, apparently, is to compensate the positive charges; the bond in the colored band is a bond between the electrons and the internal surfaces, and not with those metal atoms by means of which the electron is introduced into the lattice.

The electron forming the \(F\) center is probably located on an internal surface or crack of the crystal. These surfaces are themselves filled with ions of the lattice, and, if desired, one may

Fig. 14. Concentration of color centers, or electrons, in NaCl and CaF\(_2\) at different temperatures

Fig. 15. Diagram of the apparatus for measuring thermoelectromotive forces

think that the electron is weakly bound to the nearest positive ion, thereby forming a neutral atom, as an \(F\)-center. The two points of view found in the literature do not conflict. Which of them to adopt is no more than a question of terminological convenience.

In Fig. 14 are given the results of an investigation with the greatest possible concentration of \(F\)-centers in NaCl, colored in vapors, and in fluorite (CaF\(_2\)), colored by electrons penetrating from a point cathode\(^ {35}\). As is seen from Fig. 14, the concentration is an exponential function of temperature. In the case of fluorite the temperature may reach \(1500^\circ\), and the concentration then up to \(10^{22}\) per cm\(^3\). A crystal with such a concentration is distinguished by a noticeably lower density than that of a normal one, which agrees with the view that penetration of electrons causes the departure from the crystal of the same number of fluorine ions.

The diffusion of color centers in the crystal was investigated by Stasiw\(^ {34}\). The experiment is as follows: a flat crystal, uniformly colored, is kept at a suitable tempera-

...ture for a certain interval of time and is then cooled. As a result of diffusion the homogeneous distribution is replaced by one in which the concentration decreases from the center to the surface. The concentration gradient is measured as follows: the crystal is split into pieces, and in each piece the absorption coefficient is measured. For KCl the diffusion coefficient varies from \(9.4 \cdot 10^{-5}\ \mathrm{cm^2/sec}\) at \(755^\circ\) to \(5.7 \cdot 10^{-7}\ \mathrm{cm^2/sec}\) at \(490^\circ\). Stasov showed that at temperatures above \(700^\circ\) the mobility \(v\) and the diffusion coefficient \(D\) are connected by the simple formula:

\[ v=\frac{De}{kT}. \tag{9} \]

This formula has long been known in the theory of the motion of ions in gases. However, at lower temperatures this cannot be expected. Apparently, such a relation will hold only so long as nothing hinders the motion of the electrons. If the motion of the positive charges, which must move together with the electrons in order to preserve the electrical neutrality of the crystal, becomes slow, then the theory shows that the above equation must be replaced by the following:

\[ v=\frac{D v_{+}}{\dfrac{v_{+}kT}{e}-D}, \tag{10} \]

where \(v_{+}\) is the mobility of the positive charges moving in the crystal. This is not the mobility of an individual metal ion; \(v_{+}\) is equal to that quantity if it is multiplied by the ratio of the number of metal ions moving in the crystal to the number of electrons\(^{22}\). This dependence can be used to calculate \(v_{+}\), the mobility of the positive charges, from the experimental values of \(v\) and \(D\), which was done by Stasov. When the temperature is lowered below \(700^\circ\), the advance of the electrons is impeded by the slowing of the positive charges. Above \(700^\circ\) the electrons are only briefly detained by the positive charges, and the speed with which they move now depends simply on the ratio of the interval of time during which they remain free to that during which they are temporarily bound somewhere to the \(F\) centers.

Thermoelectric potentials in crystals containing color centers were measured by Stasov\(^{36}\). The color centers were introduced into the middle part of a long KCl crystal, cut so as to have a rectangular cross-section. In contact with platinum electrodes, which were pressed against the two ends, are two water-cooled crystals (Fig. 15). “Junctions” 1 and 2 are maintained at different temperatures by means of two small furnaces \(O_1\) and \(O_2\). If the temperature of one junction is maintained at \(700^\circ\), and that of the other is varied from \(700\) to \(400^\circ\), then an e.m.f. arises which, for a temperature difference of \(270^\circ\), at a concentration of centers of \(1.2 \cdot 10^{18}\ \mathrm{cm^3}\), reaches \(0.4\ \mathrm{V}\).

This result can be interpreted from the point of view of Nernst’s thermodynamic theory of diffusion potentials, according to which the measured emf \(V\) is the difference of the emfs \(V_1\) and \(V_2\) arising at each junction. Each of them depends on the difference between the mobility of the electrons and the mobility of the positive charges. When the temperature of the junction is below \(700^\circ\), a difference between the mobilities exists, as we saw in the preceding paragraph, and accordingly a thermoelectric emf develops. But if both junctions have a temperature above \(700^\circ\), the thermoelectric emf is absent, in agreement with the fact that, when positive charges move as rapidly as electrons, no diffusion potential arises.

VII. Positive Primary Photoelectric Current

In Section II it was mentioned that the positive charge remaining at the place from which an electron has departed after absorbing a quantum cannot always move toward the cathode. If conditions are favorable for its motion, then, as has already been noted, we may conceive not an actual motion of ions through the crystal, but rather a motion of electrons in the sequence 1, 2, 3..., i.e. a displacement of the localization site of the positive charge (Fig. 16).

We have seen that one of the methods of introducing an electron cloud, whose motion may be regarded as an electronic primary current, consists in introducing into the crystal an excess of metal atoms by heating in metal vapor. Molwo^37 succeeded in showing that the introduction of an excess of halogen atoms makes visible the primary positive current, i.e. the current compensating or “replacing” it. A NaCl crystal was kept in iodine vapor at \(600^\circ\), and in the process became deeply colored. If, after this, it was cooled slowly, the coloration was turbid, which revealed the presence of colloidal particles that could be destroyed by heating to \(600^\circ\) and subsequent instantaneous cooling. The coloration of a crystal not containing colloidal particles is pure yellow. Just as there is no connection between the absorption spectrum of a crystal containing an excess of metal atoms and the spectrum of metal vapor, so the absorption spectrum of free iodine and that of KJ with an excess of iodine have nothing in common. If such a crystal is heated to \(400\)—\(600^\circ\) and an electric field is applied to it, the colored cloud moves toward the cathode; i.e. its behavior is similar to that of a cloud of \(F\)-centers, with the sole exception that it moves in the opposite direction. The greatest conductivity is observed when the cloud fills the whole crystal, and the smallest when it is driven out completely by the field. The mobility at \(450^\circ\) is of the order of \(2.6 \cdot 10^{-4}\ \text{cm/sec}\) in a field of \(1\ \text{V/cm}\).

Electrical and optical methods (if they are used exactly as described above) give \(1.2 \cdot 10^{17}\) and \(1.7 \cdot 10^{17}\) centers per \(\text{cm}^3\) for a crystal at \(455^\circ\). In normal

in the crystal, iodine atoms are always encountered as negative ions in a number equal to the number of positive metal ions. If there is some excess of iodine, it must exist in the form of neutral atoms. We saw earlier, when considering an alkali-halide crystal \(MH\) containing an excess of metal, that this excess metal can be conceived both in the form of neutral atoms and in the form of positive ions bound to the same number of electrons, according to the following scheme:

\[ (N-n)\ \text{atoms } M^+,\quad (N-n)\ \text{atoms } H^- \]
\[ \text{and } n\ \text{atoms } M\ (M=M^+ + e). \]

In a similar way we can conceive the very same crystal with an excess of halide, thus:

\[ (N-n)\ \text{atoms } M^+,\quad (N-n)\ \text{atoms } H^- \]
\[ \text{and } n\ \text{atoms } H\ (H=H^- - e). \]

Here \(e\) represents the missing electron, i.e., it plays the role of a fictitious positive charge, the charge that moves toward the cathode. This is shown in Fig. 16, where we see,

Fig. 16. Illustration of the mechanism of the positive primary current: the site of positive charge slides from \(A\) to \(B\).

that the mechanism is in reality carried out by the motion of electrons. It is not positively charged particles as such that move. In recent years, work with semiconductors has led to the concept of semiconductors with an “excess” and a “deficiency.” Crystals with an excess of metal are typical conductors with an “excess”; their conductivity is determined by almost free electrons bound to \(F\)-centers. Mollwo’s experiments showed quite clearly that the conductivity of a crystal with an excess of halide is the result not of a direct motion of electrons toward the anode, but rather of so-called “electron holes,” or regions with a deficiency of electrons, which act like a positive charge. The positive primary current, whose mechanism is essentially identical with that studied by Mollwo, may be regarded as an example of “defect” conductivity.

VIII. Formation of \(U\)-Centers

We shall now consider a new type of absorption band, discovered three years ago in Pohl’s laboratory[^38]. This is the same \(U\) band already mentioned. It is located in the ultraviolet, on the long-wavelength branch of the intrinsic absorption curve of the crystal. We shall first describe the methods of obtaining the \(U\) band, and then consider its properties.

The first method consists in heating the crystal for a prolonged time at a high temperature in metal vapor^19. The crystal is soon colored all the way through, because \(F\)-centers are formed, but at the same time other changes also occur which are not immediately obvious. After prolonged heating a piece breaks off from the middle of the crystal. If it is examined in visible light, it proves to be characteristically colored (the \(F\) band), except for the edges, which are somewhat more transparent. If such a crystal is now illuminated with ultraviolet light, its center becomes transparent, whereas the edges are opaque. A new absorption band appears in the ultraviolet. It has been called the \(U\) band, and its carriers—\(U\)-centers. Suppose that the crystal is heated again to a high temperature, but now without metal vapor. The crystal becomes completely transparent in the visible region of the spectrum, which occurs as a result of the evaporation of the \(F\)-centers responsible for the coloration. Examination of the edges of the crystal in ultraviolet light reveals a slow diffusion of \(U\)-centers into the crystal. The second method^40,20, which gives a more uniform distribution of \(U\)-centers throughout the entire crystal, is as follows. A small crystal is kept on the surface of the molten salt. From a sharp cathode pressed into the solid crystal, electrons penetrate into the crystal. The bottom of the molten salt serves as the anode. Under controlled cooling the crystal grows, while at the same time a current passes through it. By means of such a procedure the entire crystal is filled with \(U\)-centers just as well as with \(F\)-centers. Probably the following occurs here: at the boundary between the solid crystal and the melt, many electrons moving out of the crystal combine with metal ions entering it and thus form, so to speak, frozen neutral atoms in the growing crystal.

From consideration of the methods of obtaining \(U\)-centers it is clear that they, like \(F\)-centers, give a stoichiometric excess of metal atoms in the crystal, which is confirmed by experiments showing that a completely transparent crystal containing \(U\)-centers, when dissolved, gives an alkaline reaction. However, the fact that the \(U\) band and the \(F\) band differ in position and that \(F\)-centers move in an electric field whereas \(U\)-centers do not move indicates that they must differ in some way. If \(U\)-centers can readily be obtained in a growing crystal into which electrons penetrate, then this is possible, although to a lesser extent, also in an ordinary crystal. In these cases one can see both the cloud of \(U\)-centers and the cloud of \(F\)-centers spreading from the cathode. (In order to see the cloud of \(U\)-centers, one should project ultraviolet light onto a fluorescent screen.) If, at \(700^\circ\), after the penetration of a certain number of electrons, the field is reversed and the \(F\)-centers are driven out, then only 60% of the electrons return. The remaining 40% are, in effect, immobilized \(U\)-centers. This, therefore, may be taken to mean that there is some definite number of “places” in the crystal suitable for \(U\)-centers to be formed in them.

Experiments by Kleinschrod showed that in this crystal the number of sites suitable for the formation of \(U\)-centers is limited. In KCl the concentration of \(U\)-centers, remaining almost unchanged, is at most \(5 \cdot 10^{16}\) per \(\text{cm}^3\), whereas the number of \(F\)-centers increases from \(9 \cdot 10^{16}\ \text{cm}^{-3}\) to \(115 \cdot 10^{16}\ \text{cm}^{-3}\) upon coloring in metal vapors. However, if the crystal is colored several times in succession, first driving \(F\)-centers in and then removing them from the crystal, then ultimately, with sufficient repetition of this procedure, it is possible to bring it about that the number of \(U\)-centers increases from \(19 \cdot 10^{16}\) per \(\text{cm}^3\) to \(120 \cdot 10^{16}\) per \(\text{cm}^3\). This indicates that the number of locations of \(U\)-centers can increase sixfold upon the repeated occurrence of \(F\)-centers. This process undoubtedly produces physical changes which increase the possible number of \(U\)-centers. From this it is clear that the formation of \(U\)-centers is a “structure-sensitive” phenomenon.

In one of his first papers, Pohl indicates that the centers may be thought of as electrons sitting at the sites of halide ions that have left the crystal. In another case he expresses the assumption that they can be identified with molecules of the alkali metal. Tartakovskii\({}^{44}\) studied the clarification of this question. Somewhat earlier he had proposed a scheme of energy levels in the crystal\({}^{49}\), according to which the difference in energies between the \(U\) level and the level of the corresponding absorption of the crystal is of the order of \(1\)—\(1.5\text{ V}\). Proceeding from this, one may seemingly expect that the energy of a light quantum corresponding to \(\lambda=\mu\) will be sufficient to transfer an electron from the fundamental level to the \(U\) level. In this case, of course, there must be free sites in the lattice on which \(U\)-centers could form.

If one adheres to the view that such sites are the sites of Cl ions, then it is clear that electron transitions are possible only at temperatures at which Cl ions already take a noticeable part in the conductivity of the crystal. According to Lehfeldt’s data, such a temperature for NaCl lies in the range from 500 to \(600^\circ\).

The experiment was arranged as follows. Filtered infrared rays with \(\lambda = 1.2\)—\(1.3\ \mu\) were passed through a crystal located in a furnace; these rays then fell on a thallofide photoelement. The absorption coefficient of these rays, \(\alpha\), was measured in the crystal at various temperatures. The experiments showed that, indeed, beginning at a temperature of 500—\(600^\circ\), a strong increase in the absorption coefficient is observed (Fig. 17).

Fig. 17. Change in the absorption coefficient \(\alpha\) for \(\lambda = 1\ \mu\) as a function of temperature.

Simultaneously with this, the current flowing through the crystal was measured. The change of current strength with temperature is approximately the same as for the absorption coefficient. Thus the results of these experiments seem to speak in favor of Poole’s first assumption, i.e., that the \(U\)-centers are electrons somehow bound at sites freed from Cl ions.

IX. Equilibrium between \(U\)- and \(F\)-centers

A crystal containing \(U\)-centers can be completely transparent in the visible part of the spectrum if the \(F\)-centers formed together with them are removed by the field or by evaporation. The crystal is transparent, but it differs substantially from an ordinary transparent crystal in the following respect. If it is heated, it becomes more and more colored, i.e., the \(U\)-centers transform into \(F\)-centers. If the crystal is cooled sufficiently slowly so as to guarantee equilibrium at all times, the crystal becomes bleached, which means that the \(F\)-centers disappear and transform back into \(U\)-centers. This process can be repeated several times.

It has been found that the relation governing the equilibrium \(U \rightleftarrows F\) is expressed by the simple formula: \(e^{-\frac{E}{kT}}\), where \(E\) is of the order of \(1.1\ \mathrm{V}\). These remarks are applicable only when the crystal is not in an electric field. If the crystal is placed in an electric field, then electrons, or \(F\)-centers, are displaced. It is interesting that \(U\)-centers cannot be either reproduced or restored by the penetration of new supplies of electrons. The transformation of \(U\)-centers into \(F\)-centers and the displacement of the latter in an electric field is an irreversible process. The \(U\) absorption band disappears in reality.

Fig. 18. Transformation of \(U\) centers into \(F\) as a function of temperature. Concentration of \(U\) centers: \(1.3 \cdot 10^{16}\) per \(\mathrm{cm}^{3}\) for curve 1, \(16 \cdot 10^{16}\) for curve 2.

Recently, Gilsch\(^{40}\) studied the thermal equilibrium between \(U\)- and \(F\)-centers in KBr. The number of \(U\)- and \(F\)-centers was determined from absorption. A crystal initially containing only \(U\)-centers, when heated to \(400^\circ\), remained colorless, but above this temperature coloration appeared, becoming more and more noticeable. The degree of dissociation of \(U\)-centers into \(F\)-centers for different concentrations of \(U\)-centers is shown in Fig. 18. The ratio of the number of \(F\)-centers to the number of \(U\)-centers was found in the form of the formula:

\[ \frac{N_F}{N_U}=Ae^{-\frac{E'}{kT}}, \tag{11} \]

where \(E' = 1.1\ \mathrm{V}\). If this is regarded as a bimolecular reaction, then \(E'\) is half the heat of reaction (this relation was established by Wilson and Fowler).

The formula can be derived in the following way. Suppose that the difference in energy between the bands \(F\) and \(U\) (\(F\)—higher) is \(E\). Let \(P_F\) and \(P_U\) be the number of places in which electrons can be located. Let \(N_F\) be the number of electrons located at the level \(F\) at temperature \(T\). They have appeared from the level \(U\), and if we assume that at zero temperature all places of the level \(U\) were occupied, then the number of electrons remaining at the level \(U\) at temperature \(T\) is equal to \(P_U - N_F\). Equilibrium occurs when the free energy \(\psi(N_F)\) is minimal. We have:

\[ \psi(N_F)=N_F E-TS, \tag{12} \]

where \(S\) is the entropy. For equilibrium

\[ \frac{\partial}{\partial N_F}\,[\psi(N_F)]=0=E-T\frac{dS}{dN_F}. \tag{13} \]

But

\[ S=k\lg w, \]

where \(w\) is the thermodynamic probability. The thermodynamic probability for the distribution of \(N_F\) electrons among \(P_F\) places and \(P_U-N_F\) electrons among \(P_U\) places is

\[ w=\frac{P_F!}{N_F!(P_F-N_F)!}\cdot \frac{P_U}{N_F!(P_U-N_F)!}. \]

Applying Stirling’s theorem, we obtain:

\[ \lg w=-\lg N_F^2(P_F-N_F)(P_U-N_F). \]

From substituting this into equation (10) we have:

\[ N_F=\bigl[(P_F-N_F)(P_U-N_F)\bigr]^{\frac12}\cdot e^{-\frac{E}{2kT}}. \tag{14} \]

This explains why the previously mentioned \(1.1\ \mathrm{V}\) constitutes only half the heat of reaction. Moreover, it is shown here that if \(\lg N_F\) is plotted as a function of \(1/T\), a straight line should be obtained, provided the expression under the radical sign may be considered constant. The latter is the case when \(N_F\) is small in comparison with the number of all possible places at this level. Fig. 17 shows that this condition is fulfilled for the number of \(U\)-centers equal to \(1.3\cdot10^{16}\ \mathrm{cm}^{-3}\), and is not fulfilled when the number of \(U\)-centers is less than \(16\cdot10^{16}\ \mathrm{cm}^{-3}\). Hence one could conclude that in the latter case the number of electrons transferred to the level \(F\) is not so small in comparison with the number of all possible places that it could be neglected.

The rate at which \(F\)-centers change into \(U\)-centers when equilibrium is disturbed was measured in the following way. The crystal was heated to a suitable temperature and illuminated with ultraviolet light; in this process \(F\)-centers arose, thereby disturbing the thermal equilibrium between the \(U\)- and \(F\)-centers. The time during which the coloration faded to the equilibrium state gave the value of the rate at which the \(F\)-centers changed back into \(U\)-centers. The time required for the concentration of \(F\)-centers to fall to one half of its initial value changes from 90 sec. at \(375^\circ\) to 2.0 sec. at \(625^\circ\).

Fig. 19. Formation of \(F\) centers at the expense of \(U\) centers. The hatched band \(F\) grows at the expense of the hatched area of the \(U\) band.

There is another method of obtaining an \(F\)-band in a pure crystal, when it contains \(U\)-centers and is illuminated with ultraviolet light in the \(U\) band. The crystal becomes colored, since \(F\)-centers are formed, and measurements show that this occurs with a decrease in the number of \(U\)-centers (Fig. 19). Here we are dealing with an interesting phenomenon. If the crystal is kept at the temperature of liquid air, absorption of light in the \(U\) band does not lead to the formation of \(F\)-centers, but when the temperature is raised, the quantum yield coefficient of the transition of \(U\)-centers into \(F\)-centers also increases, reaching at \(500^\circ\), for KBr, the full quantum equivalent. Otherwise

speaking, for each quantum of ultraviolet light absorbed in the \(U\) band, one \(U\)-center changes into an \(F\)-center. It must be emphasized that the quantum equivalent is observed only at a sufficiently high temperature. One may think that the electron is liberated from the center by the absorbed quantum, but at low temperatures it immediately falls back, whereas at high temperatures the motion of the atoms in the lattice allows it to leave and become an \(F\)-center elsewhere.

Recently new measurements have been made[^41]. A transparent crystal containing nothing except \(U\)-centers was illuminated with ultraviolet light from the \(U\) band and thereby became colored (\(F\)-centers). The total number of quanta \(U\) absorbed by the \(U\)-centers was measured. Then the absorption of the \(F\) band formed was measured, and in this way the number of \(F\)-centers was determined. As a result, the ratio of the number of \(F\)-centers formed to the number of quanta absorbed in the \(U\) band was determined. This method is not suitable for KBr at temperatures above \(150^\circ\), since the \(F\)-centers change back into \(U\)-centers at a rate that does not permit measurements of the constants of the \(F\)-band.

We saw in the preceding paragraph that, when the \(U\)- and \(F\)-centers were in thermal equilibrium, the number of \(F\)-centers decreased with decreasing temperature. This refers to true equilibrium. One can disturb this equilibrium by illuminating the crystal with ultraviolet light, which increases the number of \(F\)-centers while simultaneously decreasing the number of \(U\)-centers. But the state immediately after illumination is not an equilibrium one, since the excess of \(F\)-centers begins to disappear. The time during which the centers return to the equilibrium state rapidly decreases with temperature (the values were given earlier in this section), so that it becomes clear why the number of \(F\)-centers formed at room temperature at the expense of \(U\)-centers is almost constant, even though there is no true equilibrium.

In order to obtain the quantum yield at high temperatures, Gilsch and Pohl applied another method. Light from the long-wavelength part of the \(U\) band fell on a limited part of the crystal. The number of absorbed quanta was determined by corresponding measurements. The number of \(F\)-centers was derived from the values of the saturation photoelectric current. The basic idea is that each absorbed quantum replaces a \(U\)-center by an \(F\)-center, which we may regard as an electron weakly bound at the internal surface. At sufficiently high temperatures the electrolytic current, which flows even in the dark, is considerably larger than the photoelectric current superposed on it when the crystal is illuminated. When these conditions are attained, it is found that the excess current obtained under illumination and which may be called the photoelectric current reaches saturation in a field of the order of 30 to 100 V/cm. The experiments were carried out as follows: a narrow strip of the crystal was illuminated at right angles to the direction of the field at a distance \(x\) from the cathode (Fig. 20). If \(N\) is the number of electrons liberated by photo-

electrically in 1 sec and reaching the anode (in the case of saturation current), then the current is expressed by the formula:

\[ i_s = Ne\left(1-\frac{d}{2l}-\frac{x}{l}\right), \tag{15} \]

where \(d\) is the width of the strip, and \(l\) is the length of the crystal. The quantity \(N\), obtained from the equation by substituting into it the experimentally determined values \(i_s\), \(d\), \(x\), and \(l\), is assumed to be identical with the number of \(F\)-centers formed in 1 sec. Thus the quantum yield can also be found at high temperatures. The method for determining the quantum yield was described earlier. The complete curve of the quantum yield for such photochemical formation of \(F\)-centers from \(U\)-centers is given in Fig. 20 and can be represented by the formula:

Fig. 20. Diagram of the setup for measuring the transformation of \(U\) centers into \(F\) centers upon absorption of light in the \(U\) band. The hatched strip was illuminated

\[ \eta = 1-\left[1-e^{-\frac{E}{kT}}\right]^A, \tag{16} \]

where \(A\) is equal to 7 or 8 and \(E\) is equal to 0.085 V. A practically complete quantum equivalent is reached for KBr at temperatures above \(500^\circ\).

Equation (16) is based on the following. Hilsch and Pohl postulate that an electron is liberated from a \(U\)-center by an absorbed quantum and forms an \(F\)-center at another place if one of the atoms near the \(U\)-center has an energy greater than \(E\), and thus can exchange places with the electron and release it. The probability that the atom has energy greater than \(E\) is equal to

\[ e^{-\frac{E}{kT}}, \]

and in this case the probability that it does not have this energy is equal to

\[ 1-e^{-\frac{E}{kT}}. \]

If there are \(A\) atoms around the \(U\)-center, then the probability that none of them has energy \(E\) is equal to

\[ \left[1-e^{-\frac{E}{kT}}\right]^A. \]

Hence the probability that one (or at least one) atom has energy greater than \(E\) is equal to:

\[ 1-\left[1-e^{-\frac{E}{kT}}\right]. \]

This is the probability that the electron will exchange places with an atom and form an \(F\)-center here or elsewhere.

When \(U\)-centers are introduced into a crystal, an interesting phenomenon can be observed. Suppose that we have a transparent crystal containing \(U\)-centers. Now let ultraviolet light from the \(U\) band fall on the crystal through a stencil. After a few seconds the part illuminated by ultraviolet light will become visibly colored: yellow for rock salt, blue for KBr. By means of an electric field the colored part can be driven out of the crystal, and it

will again become transparent. If the entire crystal is now illuminated with ultraviolet light, it will become colored only in that part which had not previously been illuminated. In the part that had previously been illuminated, the centers \(U\) were converted into centers \(F\), and these latter disappeared under the action of the field; thus, during the secondary illumination with ultraviolet light, no effect was obtained.

X. Photoconductivity in Diamond and Red Iodide of Mercury

Robertson, Fox, and Martin \(^{40}\) studied photoconductivity in many diamonds, and found that it differed greatly from specimen to specimen. It is possible, however, to divide them into two classes. In class I we have diamonds possessing unexpectedly low photoconductivity. These diamonds are characterized by an absorption band cut off in the ultraviolet at \(3\) thousand Å and in the infrared region at \(8\,\mu\), in addition to the usual infrared bands. Class II diamonds exhibit considerable photoconductivity. They do not have the infrared absorption band at \(8\,\mu\) and are transparent down to \(2250\) Å. The authors confirm the indications of Pohl and Gudden that, under illumination with light of wavelength between \(2400\) and \(5\) thousand Å, the quantum yield tends toward the quantum equivalent. However, this is observed only on condition that the diamonds have not previously been illuminated with light of \(\lambda = 2300\) Å; such treatment produces an effect not previously described by anyone. After a diamond had been illuminated with light \(\lambda = 2300\) Å, photoconductivity was detected in it in the dark, which decreased with time. If, after illumination at \(\lambda = 2300\) Å, the diamond is tested for photoconductivity in the region \(2400\)—\(5000\) Å, a much greater sensitivity can be found than in a diamond “not activated” by the wavelength \(2300\) Å. However, in a short time the quantum yield for the region \(2400\)—\(5000\) Å falls to its normal value, and the excited dark current disappears. But if, after illumination at \(\lambda = 2300\) Å, the diamond is illuminated with red and infrared light \((5000\)—\(9000\) Å), a very great sensitivity is obtained, which remains practically constant (the results are shown in Fig. 21). This is a remarkable result, since it turns out (although this is clearly not

Fig. 21. Quantum yield in the case of formation of centers \(F\) from centers \(U\) at various temperatures

it has been established), that the total charge carried under illumination by long waves may be many times greater than that associated with the primary electron current. This apparently argues against our attempt to identify the effect of infrared light with a positive primary current, which for these rays can never exceed the electronic primary current. There are as yet insufficient data to explain this phenomenon.

Nix^43 studied a special effect in crystals of red mercuric oxide. These crystals lose their conductivity with time. At the same time there is a progressive change from the mono- to the polycrystalline fibrous state. Crystals that had strongly lost their photoconductivity regained it after a brief application of an electric field.

XI. Concluding Remarks

In the preceding sections the most important of what has been done over the last several years in the field of the photoconductivity of crystals has been summarized. Most of the results have been obtained chiefly in the study of the properties of \(F\)- and \(U\)-centers. However, despite the fact that much useful information has been obtained, it should be noted that we do not have a clear and distinct theory that precisely describes what occurs in photoconductivity. What we have, rather, are working hypotheses, as has repeatedly been emphasized by Pohl, the leader in this field. The characteristic properties of photoconductivity leave no doubt that it should be regarded as a “structure-sensitive” phenomenon. But the explanation of “structure-sensitive” phenomena is in a far more primitive state than that of “structure-insensitive” phenomena. For example, in the “structure-sensitive” domain we do not have a theory comparable in accuracy and completeness to the theory developed for the analysis of crystal structure by X-rays—a typical “structure-insensitive” domain. The absence of a comprehensive theory of photoconductivity is characteristic of the group of phenomena to which it belongs.

The presence of \(F\)- and \(U\)-centers, as we have seen, is connected with a stoichiometric excess of metal atoms in the crystal, i.e., the crystal contains unequal numbers of metal atoms and halide atoms (this is true for \(F\)-centers obtained by methods 4, 5, and 6, but has not been proved for \(F\)-centers obtained by methods 1, 2, and 3). It follows from this, however, that \(F\)-centers must necessarily be neutral atoms. Pohl regards the excess atoms as positive metal ions and electrons, dissociated from one another, at least in the case of \(F\)-centers. The behavior of the centers is compatible with the view that they are electrons bound more or less weakly at cracks, slits, and internal surfaces of the crystal with positive ions, which serve to remove the negative local charges arising from the electrons. How

as was noted above, \(U\)-centers are similar to \(F\)-centers in that they are associated with a stoichiometric excess of metal atoms, but differ from them in having a greater binding energy. Above, Pohl’s hypothesis concerning the nature of \(U\)-centers was indicated, as were Tartakovsky’s experiments connected with it. The data of these experiments seem to favor the view that \(U\)-centers are electrons sitting in the sites of departed \(\mathrm{Cl}\) ions. De Boer\(^6\) puts forward the supposition that \(F\)-centers are obtained when atoms of the alkali metal are adsorbed on negative ions located on internal surfaces in the crystal, whereas \(U\)-centers are obtained when atoms are adsorbed on positive ions situated on these surfaces.

In view of the successes of quantum mechanics, it is desirable briefly to formulate, in its terms, what has been established concerning photoconductivity. Since this article considers only experimental results, only a brief description will be given. For a fuller acquaintance with the theory we refer the reader to what has already been written on this subject\(^45, 46, 47, 48, 50, 51\).

The scheme of the energy levels of a crystal consists of broad bands within which electrons may be located. Each band may contain some definite maximum number of electrons: it may be filled completely, partially, or remain empty. The bands may be separated from one another by forbidden zones, concerning which it is said that electrons cannot exist inside the crystal with energy values lying within these zones. Conductivity is represented as follows. If an applied electric field moves an electron and in doing so performs work, then the energy increases continuously. If, however, the electron is inside a band that is only partially filled, then in the band there are empty levels that can accommodate electrons when their energy changes. Such a crystal conducts. If, however, the band is completely filled and there are no levels into which electrons could be transferred, then the crystal is an insulator. The electric field can impart energy to the electron only continuously; consequently, the electron cannot be transferred from a filled band into an empty one across a forbidden zone. A crystal with a forbidden zone between the filled and the free zones is also an insulator (Fig. 23). Electrons can pass through the forbidden zone when the temperature is raised or, with certain restrictions, when quanta of sufficient energy are absorbed. The number of electrons in the upper band, which remains free at \(0^\circ\mathrm{K}\), is proportional to

\[ e^{-\frac{E}{kT}}, \]

where \(E\) is half the energy difference of the two bands, and \(T\) is the temperature. Since there are now electrons in a band that still has many free places, the crystal becomes conducting. The conductivity depends on the number of electrons, and it is clear that, since \(E\) is large, the temperature must be sufficiently high before appreciable conductivity appears.

Along with the conductivity arising from the motion of electrons toward the anode, known under the name “excess” conductivity, we also have defect conductivity, resulting from the motion of “electron holes” remaining in the filled band. These electron holes behave like positive charges and, consequently, move toward the cathode. Their motion may be likened to a positive primary current.

When we have excess conductivity, directly involving the motion of electrons, we should expect the normal (negative) Hall effect predicted by simple classical theory. It is found in some conductors, but there are many cases in which an anomalous (positive) Hall effect occurs. The latter is inexplicable from the point of view of classical ideas, but it appears as a natural consequence when quantum mechanics is applied to defect conductivity. When a crystal is colored by the additive introduction of metal atoms, which, as we have seen, are obtained by the introduction of electrons, we have an example of excess conductivity. When it is colored by the introduction of halide atoms, which may be thought of as “electron holes,” we have a case of defect conductivity.

Fig. 22. Photoconductivity in diamond

In Fig. 22 a simplified scheme is presented of what occurs in a photoconducting crystal.

A pure alkali-halide crystal is not a conductor until a high temperature has been reached; under these conditions its conductivity is ionic. We have no evidence in favor of the existence of electronic conductivity in these crystals. This leads us to the conclusion that the energy of separation \(E\) of the filled band from the free band is considerable. Electronic conductivity has been found only in colored crystals, i.e. in crystals containing \(F\)-centers, and only when the temperature is moderately high, or when the crystal is illuminated by light absorbed in the \(F\)-band. Hence it follows that the electrons bound to the centers are situated in localized energy bands (it is understood that there is no continuous transition between them). These bands are located somewhat below

empty band. A scheme of the energy levels in a colored crystal is shown in Fig. 24.

The quantity \(E'\) is of the order of the energy of a quantum absorbed in the \(F\) band. Fig. 3 shows that, for KBr at low temperatures, the latter is of the order of 2 eV. As for \(U\)-centers, they, like \(F\)-centers, may be regarded as localized energy levels. Since for the conversion of a \(U\)-center

Fig. 23. Scheme of energy levels for an insulating crystal exhibiting excess and defect conductivity at \(T > 0^\circ K\) (Gudden and Schottky)

Fig. 24. Scheme of energy levels in a colored crystal (Gudden and Schottky)

into an \(F\)-center a quantum energy \(E''\) is required, in accordance with Fig. 18 approximately equal to 3.5 eV, the \(U\) levels are placed as shown in Fig. 23. Absorption in the long-wave branch of the intrinsic absorption band also results in the formation of \(F\)-centers, and therefore we must place the corresponding localized levels in the vicinity of the \(U\) levels.

Gurney proved that transitions between the \(U\) and \(F\) levels in either direction cannot occur directly; rather, a transition from one to the other must take place through a higher level. This is done in order to avoid contradiction with the assertions of thermodynamics, which establishes that there cannot simultaneously exist direct upward transitions (with greater probability) and direct downward transitions (with small probability), which, obviously, must occur in a crystal in which the coloration persists for a long time. (In reality Gurney considered

formation of centers upon absorption of light in the long-wavelength branch of the intrinsic-absorption band. At the time when he wrote the article, \(U\)-centers had not yet been discovered. Nevertheless, these same considerations can be applied to the same extent to \(U\)-centers as well.)

The best-developed scheme of energy levels for alkali-halide crystals is the scheme proposed by Tartakovsky, which compares data from various sources. It is presented in Fig. 25. According to this scheme, in a colored crystal, in addition to the ground level (I) and the conduction level (II),

Fig. 25. Scheme of energy levels in a colored crystal (Tartakovsky)

there also arise certain intermediate ones. One of them, \(F\), corresponds to electrons located near lattice imperfections (Lockerstellen), in which there are excess positive metal ions. In other words, the coloring centers are regarded as “quasi-atoms.” In the case, however, where the electrons sit in lattice irregularities that do not contain excess metal ions, a second level (\(I'\)) is obtained, situated several tenths of a volt above the level \(F\). Near the ground level, above it, lies the level \(U\), which appears in the crystal under additive coloration. When the crystal is illuminated with light from the intrinsic-absorption band, electrons from the ground level pass into the conduction band and from there can fall either to the levels \(F\) and \(I'\), or back to the ground level. A colored crystal, however, can be bleached if it is illuminated with light from the \(F\) band. In this case there is also possible an “excitation” (Erregung) of the crystal, corresponding to an increase in the long-wavelength branch of the \(F\) band. In the presence of the \(U\) band, the number of electron transitions increases. Here transitions from the \(U\) band to the conduction band are possible, and from there back to the \(F\) and \(U\) bands. According to Tartakovsky, the possibility is not excluded of transitions directly from the \(U\) band to \(F\) and back.

None of the schemes proposed so far can locate all the energy levels with reasonable certainty, nor

can indicate the reasons why transitions occur between some levels, but not between others. It must be admitted that the various attempts to describe photoconductivity in the language of quantum mechanics have not yet passed beyond the stage of attempts; we are still far from having a rational theory that quantitatively explains the principal phenomena and gives a new direction to research.

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1. In Seitz it is erroneously stated that the position of the maximum of the \(U\) band is determined by the same formula as in the case of the \(F\) band. — Translator’s note. ↩