Electrical Conductivity of Solid Insulators and Semiconductors

A. F. Ioffe, Leningrad

1. Electric Current in Solid Dielectrics

An electric field causes in conductors a continuous displacement of charges (ions or electrons); in insulators, a quasi-elastic displacement of them or a rotation of dipoles. The motion of charges in a conductor continues as long as the field exists; displacement in an insulator takes place with the speed of light, and rotation with the speed at which thermal equilibrium is established. It would seem that there is sufficient data for separating conductivity from dielectric polarization; however, in real dielectrics the two phenomena are so intertwined that it is often very difficult to distinguish them. In many cases we still do not know to which of these phenomena dielectric losses should be attributed (heating of a dielectric in an alternating field).

Pierre and Jacques Curie were the first to carry out a systematic investigation of the properties of dielectric crystals in an electric field and to establish the principal laws. Here the current usually does not remain constant in time, but decreases after the application of a potential difference to the dielectric, at first rapidly and then ever more slowly. When the dielectric is short-circuited, a current of the opposite direction appears, gradually falling to zero. Figure 1 shows the general course of the phenomenon. Here \(I_0\) denotes the initial current at the moment the voltage \(V\) is applied; \(I_r\), the residual current established after a sufficiently long interval of time; \(I_p\), the initial polarization current. If the properties of the crystal do not change as a result of the passage of current, then \(I_0 = I_r + I_p\), and this equality is valid not only for \(t = \infty\), but also for any instant of time \(t\). Investigations of the distribution of potential in a dielectric during the passage of direct and reverse current showed that the cause of the decrease of the current is the accumulation of a volume positive charge near

of the cathode and negative near the anode. If one measures the field strength \(E\) in the middle part, where the potential distribution is rectilinear (Fig. 2), then at any moment the Ohm law holds here:

\[ \Sigma=\frac{I}{E}, \]

where \(\Sigma\) retains a constant value for any \(t\) in Fig. 1.

The field strength \(E\), however, is not equal to \(\frac{V}{D}\), where \(D\) is the thickness of the crystal, but must be expressed as follows:

\[ E=\frac{V-P}{D}. \]

Here \(P\) is the electromotive force of polarization, which I was able to measure directly[^1].

If the cross-sectional area of the current is denoted by \(S\), then we obtain the following value of the specific electrical conductivity:

\[ \sigma=\frac{I}{S}\cdot\frac{D}{V-P}. \tag{1} \]

Fig. 2.

Figs. 1 and 2 and equation (1) encompass the properties of a tremendous number of dielectrics. In some cases (mica, salt, calcite, quartz), especially at low temperatures and small field strengths,

\[ \frac{I_r}{I_0}\approx 0 \]

—all charges participating in the passage of the current accumulate at the electrodes, without giving up their charge to them until

\[ \frac{P}{V}\approx 1. \]

The areas under the curve \(\int_{0}^{t} I\,dt\) for the direct and reverse current in these cases are equal to one another and express the amount of volume charge accumulated at the electrodes. In other cases

\[ 1>\frac{I_r}{I_0}\gg 0 \]

—part of the charges transported by the current is delivered to the electrodes. The higher the temperature \(T\) and the stronger the electric field \(E\), the greater is \(\frac{I_r}{I_0}\). However, even in those cases when \(I_r\) is almost equal to \(I_0\), we often observe a polarization current \(I_p\), indicating the accumulation of volume charges. The point is that it is quite incorrect to divide the direct-current curve (Fig. 1) into two parts in this way: from the very first moment \(t=0\), the current \(I_r\) is delivered to the electrodes, while the difference \(I-I_r\) goes to the formation of volume charge. In Fig. 3 the shaded area of the polarization current, measuring the volume charge, is much larger than the corresponding area of the direct current after subtraction of the residual current. As direct measurements have shown, the current passing through is determined by the dotted curve. Consequently,[^1]

the number \(n_p\) leading to the formation of space charge is determined by the field strength at the electrode itself, being expressed by a formula of the form:

\[ \frac{n}{n+n_p}=Ae^{\chi E_0}. \tag{2} \]

A curve entirely analogous to Fig. 1 can also be obtained when orientation of dipoles, accompanied by high viscosity, occurs in the dielectric. The dependence of the current on time does not make it possible to determine the mechanism of the phenomenon; one cannot assert, for example, that \(I_r\) is a conduction current, and that the fall of the current from \(I_0\) to \(I_r\) is caused by dielectric polarization. Only the data of Fig. 2 make it possible to decide the question of the cause of the fall of the current.

Fig. 3.

A decrease of the current with time may also be caused by an actual decrease in electrical conductivity as a result of the removal of currents of impurities. If the purification of the dielectric occurs only in thin near-electrode layers, whose specific resistance increases, then we obtain the potential distribution shown in Fig. 2. If, however, the entire thickness of the dielectric has undergone purification, then the field will be almost homogeneous and the polarization part of the curve in Fig. 3 will be absent. The reverse current or diffusion may then again increase the electrical conductivity.

Finally, a curve similar to Fig. 1 may also be the result of the method of observation. Usually the dielectric is given the form of a plate (Fig. 4). To prevent charges from passing along the surface, one of the electrodes \(A\) is surrounded by a grounded guard ring \(B\). A voltage is applied to electrode \(C\). The current between \(A\) and \(C\) is measured by an electrometer \(E\) or a galvanometer. At the moment the voltage is switched on, electrode \(A\) is connected to ground and is insulated only after a certain time \((0.001—1\ \text{sec.})\) for observation of the current. However, on the insulated ring \(D\) between electrode \(A\) and the guard ring \(B\), there remains a free charge caused by induction from electrode \(C\). This charge then slowly passes from ring \(D\) to electrode \(A\) and is added to the current passing through from \(C\) to \(A\). Many measurements of currents in dielectrics are distorted by this source of error. When electrode \(C\) is connected to ground, we obtain on \(D\) a charge of the opposite sign, which, passing to electrode \(A\), creates the impression of a polarization current. In order to avoid this error, it is necessary to give the dielectric such a shape that electrode \(A\) is electrostatically shielded from charges spreading between \(C\) and \(B\), and the insulating ring \(D\) from electrode \(C\). Fig. 5 gives three arrangements satisfying these conditions.

Fig. 4.

A. F. IOFFE

In practice, the measurement of the electrical conductivity of dielectrics encounters great difficulties when the current decreases extremely rapidly with time and when the very passage of the current changes the properties of the dielectric. For very many crystals there is no doubt about extrapolating the curve \(I=f(t)\) to the instant \(t=0\).

Then

\[ \sigma=\frac{I_0}{S}\cdot\frac{D}{V}. \]

Such are NaCl, calcite, quartz, and others. In the case of glass, however, the current falls so rapidly even in thousandths of a second that it is impossible to establish the value of the initial current \(I_0\); on the other hand, the measurement of the electric field \(E\) and the current \(I\) at some later instant \(t\), in addition to the difficulties associated with measuring the polarization \(P\), determines the properties of a substance already altered by the current. Therefore, instead of the true electrical conductivity, one often measures the so-called residual electrical conductivity:

Fig. 5.

\[ \sigma_r=\frac{I_r}{S}\cdot\frac{D}{V}, \]

although it has no clear physical meaning. Since the ratio \(\frac{I_r}{I_0}\) increases with temperature and with the field strength \(E\), replacing \(\sigma\) by \(\sigma_r\) often leads to incorrect conclusions. Thus, for many insulators (mica, NaCl, NaNO\(_3\), and others) \(\sigma\) does not depend on \(E\) up to fields of \(10^6\ \frac{\text{volts}}{\text{cm}}\), whereas, according to equation (2),

\[ \sigma_r=Be^{kE}_0 \simeq Ce^{kE}, \]

it increases rapidly with the field. This same formula was found by a purely empirical route and is known as Poole’s formula. Whether a dependence of the true electrical conductivity \(\sigma\) on the field exists in those cases where the initial current \(I_0\) could not be determined (for example, in glass) cannot be said with certainty. The experiments of Gubman and Aleksandrov, however, suggest that in glass \(\sigma\) begins to increase only in fields greater than \(6\cdot10^5\ \frac{\text{volts}}{\text{cm}}\).

Likewise, the determination of the temperature coefficient of the “residual” electrical conductivity \(\sigma_r\) is erroneous:

\[ \frac{1}{\sigma_r}\frac{d\sigma_r}{dT}>\frac{1}{\sigma}\frac{d\sigma}{dt}. \]

Whereas \(\sigma\) is usually expressed with great accuracy by the formula

where \(B\) denotes the heat of dissociation (at constant volume); for \(\sigma\) one obtains very complicated dependences, which have no simple physical meaning.

The question of the nature of the charges that carry the current in dielectrics cannot always be resolved. On the one hand, the currents are often so weak that it is impossible to observe the electrolysis processes caused by them. Even if, at high temperatures, when the electrical conductivity is sufficiently large, we convince ourselves of the ionic character of the current, this result cannot be transferred to low temperatures, where the relative role of ions and electrons is often quite different. In better-conducting solid dielectrics we encounter another difficulty: the metal liberated as a result of electrolysis does not cover the cathode \(a\) with an even layer, but grows there in the form of needles, which, branching, penetrate through the entire dielectric and form metallic bridges connecting the electrodes. From this moment the liberation of electrolysis products drops sharply, no longer satisfying Faraday’s law; moreover, a considerable part of the cations is discharged and gives up its charge inside the dielectric, forming dendrites. To prevent this, Tubandt interposes between the crystal and the cathode a layer of AgJ, BaCl\(_2\), etc. It can be said with certainty only that where Faraday’s law has been confirmed with sufficient accuracy, we are dealing with ionic conduction; where a Hall effect of corresponding magnitude has been found, there is transport of electrons by the current. Cases of mixed conduction have also been established, with the most diverse ratios of the electronic and ionic components. NaCl, KCl, NaF, AgCl, AgBr, AgNO\(_3\), NaNO\(_3\), the halide salts of Pb and Ba, and glasses possess purely ionic conductivity. Purely electronic conductivity is possessed by oxides, sulfur compounds of Cu, Pb, Mo, Sn, and carbides and nitrides of heavy metals. Mixed conductivity has been established in CuCl. The \(\beta\) modifications of Ag\(_2\)S, Ag\(_2\)Se, Ag\(_2\)Te possess mixed conductivity, whereas the \(\alpha\) modifications of the same salts are purely ionic conductors.

The ionic or electronic character of the conductivity is determined by the work of dissociation \(U\). The number of \(n\)-charges that are in a state of free motion and possess an energy, in \(U\) units, exceeding the energy of their equilibrium position in the crystal lattice is determined by the expression:

\[ n = Ne^{-\frac{U}{kT}}, \]

where \(N\) is the total number of particles of the given kind, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature. The quantity \(kT\) at room temperatures has the value \(0.028\) electron-volts. Therefore, for values of \(U\) exceeding \(1\,\text{V}\), \(\frac{n}{N}<10^{-17}\), and for \(U = 1.4\,\text{V}\), \(\frac{n}{N}=10^{-22}\), which denotes 1 charge in \(1\ \text{cm}^3\). In comparison

...compared with the difference in the degree of dissociation, the difference in the mobilities of ions and electrons plays a secondary role. The value of \(U\) for ions can be determined from the temperature dependence of diffusion and electrical conductivity, expressed with great accuracy by the formula:

\[ \sigma=\sigma_0 e^{-\frac{\alpha}{kT}}, \tag{3a} \]

However, the quantity \(\alpha\) may fail to coincide with \(U\), if the change of mobility with temperature is of the same order as the increase of dissociation. Experimental separation of \(\alpha\) and \(U\) is possible only in special cases when, for example, the ions recombine very slowly. Then, by rapid cooling (quenching), one can obtain a dielectric with an increased dissociation corresponding to a high temperature, but with a mobility corresponding to a low temperature. In such cases (quartz) it turns out that \(U\) approaches \(\frac{\alpha}{2}\). For typical ionic conductors \(\alpha\) has a value from 1 to 0.3 V. Consequently, \(U\) is from \((0.5—1)\) to \((0.15—0.3)\) volts. For rock salt, for example, \(\alpha=1\ \mathrm{V}\); \(u\approx 0.5\ \mathrm{V}\). In this case

\[ \frac{n}{N}\approx 10^{-9}, \qquad n=10^{13}\ \frac{\text{ions}}{\mathrm{cm}^3}. \]

On the other hand, the work of dissociation of the electron belonging to the chlorine ion is determined by the greatest wavelength of light that still produces the internal photoelectric effect; in rock salt this boundary is \(\lambda\leq 240\,\mu\mu\), and the corresponding \(h\nu\geq 5\) electron volts. The degree of thermal dissociation of the electrons should therefore be \(10^{-87}\). Even in X-rayed rock salt, the removal of an electron from a sodium atom requires work \(h\nu>1.5\ \mathrm{V}\). Therefore X-rayed rock salt does not possess dark electronic conductivity at ordinary temperatures.

Conversely, in dielectrics with electronic conductivity the work of dissociation of an electron is less than 1 V, whereas the dissociation of an ion requires greater energy. Thus, for example, in pure \(\mathrm{Cu}_2\mathrm{O}\) \(U=(0.8—1.5)\ \mathrm{V}\), and in regions where there is excess oxygen, \(U=(0.3—0.6)\ \mathrm{V}\), whereas for \(\mathrm{Cu}^{+}\) ions one may take \(U>1\ \mathrm{V}\).

2. Ionic conductors

The principal fact that must be taken into account in the study of the electrical conductivity of solid electrolytes is the enormous influence of comparatively insignificant impurities. It is manifested, first, in an increase of the electrical conductivity and, second, in the accumulation of space charges. Different “chemically pure” specimens of ammonium alum gave, for example, when measured, the specific electrical conductivity: \(8000;\ 1300;\ 4200;\ 36\,000\cdot 10^{-18}\ \frac{1}{\Omega}\).

After three- and fourfold crystallization with the use of [[unclear: text cut off at bottom of page]]

removal of the surface layer, we succeeded in obtaining several tens of crystals with an electrical conductivity of \((21 \pm 1)\cdot 10^{-16}\ \frac{1}{\Omega}\).

Mechanical inclusions in this case do not affect the electrical conductivity appreciably. The greatest influence is exerted by an impurity entering in the form of a solid disordered solution. The increase in dissociation then exceeds the number of introduced ions, so that each ion introduced into the lattice lowers the work of dissociation in some region encompassing hundreds of atoms. The nonuniform distribution of impurities in a crystal often leads to a number of irreversible phenomena on heating and on deformation of it. Diffusing and redistributing themselves, the impurities change the total electrical conductivity of the specimen. These phenomena were studied by B. Hochberg^4 in rock salt and made it possible to explain a number of anomalies.

It is well known that a real crystal differs from the mathematical scheme of a crystal lattice. Disturbances of the regularity of the lattice arising in the course of the crystallization process can be detected by a number of indications. They have a great influence on such properties as the mechanical elastic limit and the coloration of the crystal. Smekal attempted to ascribe to these inhomogeneities the decisive role in the creation and motion of ions. However, detailed analysis shows that mechanical irregularities have practically no effect on electrical conductivity, but that it is very sensitive to chemical impurities. One of the grounds for asserting that the principal mass of ions is created at low temperatures in distorted regions of the lattice was the behavior of the electrical conductivity of alkali-halide salts with temperature

Fig. 6.

\[ \sigma = A_1 e^{-\frac{B_1}{T}} + A_2 e^{-\frac{B_2}{T}} . \tag{4} \]

The first term was referred by Smekal to the ions of distorted places, the second to the entire remaining lattice. Experimental determination of the transport numbers at different temperatures in NaCl, carried out by T. D. Tazulakhova, however, gave an entirely different interpretation of this formula: the first term expresses the current carried by sodium ions, the second—by chlorine ions. Already at \(700^\circ\mathrm{C}\) the second term exceeds the first, whereas at room temperature it plays no role—the entire current is carried by sodium ions. The same explanation holds for \(\mathrm{PbCl_2}\). In the following table are given the values of \(B_1\) and \(B_2\) for a number of solid electrolytes. The corresponding curves are shown in Fig. 6.

Curve I refers to NaF, II—NaCl; III—NaBr, and IV—NaJ.

For a number of other dielectrics, current transfer is produced throughout the measured temperature interval by ions of one sign. In these cases the second term in formula (4) drops out, and the entire curve is expressed with greater accuracy by a one-term formula. True, in the literature one can find data of a more complicated character, but all of them are quite explainable by experimental errors: failure to observe the above-mentioned measurement conditions, inhomogeneous chemical impurities, and the use of fine-crystalline aggregates. Wherever these experimental errors were excluded, for one kind of ion the simple formula (3) is obtained:

I—quartz,
II—saltpeter,
III—calcite,
IV—ammonium alum.

Fig. 7.

In Fig. 7 some of these straight lines are given. I refers to quartz, II—NaNO$_3$; III—CaCO$_3$, IV—alums. Particularly convincing are the measurements of the electrical conductivity of NaNO$_3$ by V. Gochberg and N. Usataya$^{5}$ (straight line II), carried out up to the melting temperature. The logarithm of the electrical conductivity over the entire extent is expressed by a linear function of $\frac{1}{T}$, according to formula (3a). Here there are no traces whatsoever of the participation of the ions of smaller dissociation work from the lattice sites, as assumed by Smeckal. Measurements of pressed powders or the replacement of the true conductivity by the ratio of the residual current to the potential difference lead to distortion of the curve and, in particular, to the appearance of two kinds of ions. Above 200°C there is no accumulation of volume charges by the current, and we obtain the true temperature coefficient. Below 200°C, however, there is polarization, increasing with decreasing temperature, and therefore the temperature coefficient obtained is different (usually greater than the true one).

The quantity of electricity $Q$, accumulated inside the dielectric in the process of passage of current, can be with sufficient accu-

\[ \int_{0}^{\infty} I_p\,dt. \]
Indeed, if in the dielectric we have \(n\)-charges \(e\), displaced in the direction of the current by an amount \(x\) at a distance between the electrodes \(D\), then

\[ D_\infty \int_{0}^{\infty} I_p\,dt = xne. \tag{5} \]

Usually the entire middle part of the dielectric remains neutral and the charges accumulate in thin layers near the electrodes: in \(\mathrm{CaCO_3}\) at distances \(10^{-4}—10^{-3}\) cm, and in \(\mathrm{NaNO_3}\)—\(10^{-3}—10^{-2}\) cm. Therefore, in practice \(x\) differs little from \(D\), and we may take:

\[ \int_{0}^{\infty} I_p\,dt = ne = Q. \tag{5a} \]

Having determined, under this assumption, the distribution of volume charges, one may then correct the value \(Q\) on the basis of equation (5).

Assuming that the volume charges formed earlier do not mix during the further passage of the current, we can determine the distance \(d\) from the electrode at which the center of gravity of the charges \(dQ\), carried by the current over some interval of time, becomes fixed, from the rise in the polarization potential \(dP\) which it produces:

\[ I\,dt \approx dQ = C\,dP, \]

where \(C\) is the capacitance relative to the electrode, equal to \(C=\varepsilon \dfrac{S}{4\pi d}\).

The quantity \(dP\) can be determined from the decrease of the current during the time \(dt\):

\[ \frac{dP}{V} = \frac{dI}{I_0}, \]

whence

\[ d=\frac{\varepsilon S}{4\pi}\frac{V}{I_0}\frac{d\log I}{dt}. \tag{6} \]

The values of \(d\) calculated in this way have a very simple physical meaning. In the case of \(\mathrm{CaCO_3}\), where the through current may be neglected, \(d\) retains a constant value from the first moment to the end. The charge density decreases with distance from the electrode according to an exponential law. The theory of this case was developed by G. Paffrе\({}^{6}\) and gives good agreement with experiment. However, it does not explain the striking constancy of the quantity \(d\), which depends neither on temperature, nor on field, nor on current density.

In the case of \(\mathrm{NaNO_3}\), where, in addition to the accumulation of volume charge, there is a considerable through current, Paffre’s theory does not give the proper—

of a proper representation of the course of the phenomenon. Here \(d\) is determined by the arrangement of the impurities introduced into the crystal. If the impurities were introduced by diffusion from the electrodes, then the volume charges are distributed in this layer at some constant mean distance \(d\), with a density gradually decreasing from the electrode inward. If, however, the impurities are uniformly distributed in the bulk of the crystal, then the charges also are distributed uniformly; moreover, the filling of the crystal with volume charge always proceeds gradually, beginning at the electrodes. The charge density is the greater, the greater the concentration of impurities.

The mechanism of the ionic conductivity of a crystal may be pictured in the following way. Fluctuations of thermal motion remove some ion from its position of stable equilibrium in the lattice into one of the neighboring cells, where its charge is excessive. Together with the other ions of the given region of the crystal, the ion that has entered it forms a new system in which the number of ions of a given sign is greater by one than the number of ions of the opposite sign. The distortion of the lattice decreases with distance from the place where the excess charge is located.

In the case of diffusion, this entire picture changes chaotically; in an electric field, however, it is displaced in the direction of the field, with some lattice ions leaving the distorted region and others entering it. The work of friction, released in the form of Joule heat, is performed by these boundary ions. In an easily deformable lattice, with a very smooth transition from the distorted region to the normal one, the resistance may be very small. In this way, it will probably be possible to explain the exceptionally large values of electrical conductivity (reaching \(1000 \frac{1}{\Omega}\)), observed in certain crystals with purely ionic conductivity (\(\mathrm{Ag_2S}\), \(\mathrm{AgCl}\)). Conversely, it seems to me incorrect, even as a first approximation, to imagine the crystal lattice as unchanging and the ion as forcing its way through energy barriers between the individual cells of the lattice. In the case of \(\mathrm{Ag_2S}\), such a conception leads to incredible conclusions, for example, that the mean free path of an ion in the lattice is hundreds of times greater than the path of an electron in silver. The magnitude of the energy barrier, calculated from the structure of the lattice for the transition of an ion from one unchanged lattice into another, likewise does not agree with the temperature dependence of mobility and electrical conductivity.

Can a certain degree of dissociation be ascribed to the crystal lattice, i.e., can it be assumed that at every moment some small number of ions are in a state of mobility, distinct from the rest of the mass of the crystal?

In the picture set forth above, one would have to speak of a series of distorted regions with excess or deficient

in the fact of “hardening” of dielectrics (for example, quartz): after rapid cooling they retain an increased electrical conductivity, which disappears with the same slowness as the increase in electrical conductivity caused by an artificial ionizer (radium rays, ultraviolet light). It is natural to suppose that in all these cases we have a definite number of special ions in a state of dissociation, produced by heat, light, or radium rays, whereas the mobility, if judged by the rate of return to normal conductivity, in all these cases corresponds to room temperature and may be regarded as a single-valued function of temperature. The mobility is restored rapidly, whereas the dissociation of the ions very slowly approaches its equilibrium value.

3. Electronic Conductors

One may distinguish three cases of electronic conductivity: 1) in many ionic conductors we observe electronic conductivity if, in addition to thermal motion, there are other sources of ionization—ultraviolet light, X-rays, $\alpha$- and $\beta$-rays; 2) in those cases where thermal motion alone is sufficient for electronic dissociation, it is increased by illumination; and 3) a group of cases in which colloidal inclusions of metals make dielectrics photosensitive and create an electronic current. In NaCl, very short ultraviolet waves or X-rays, tearing electrons from chlorine ions, return them to sodium ions. To tear an electron from the sodium atoms formed inside the lattice requires much less energy, and therefore previously X-rayed rock salt becomes photosensitive in visible light, whereas before X-raying the salt did not change its resistance under illumination. I first noticed this phenomenon in 1904, while studying the influence of X-rays on the electrical conductivity of crystals. X-rayed salt changed its resistance when clouds passed, and the transition from darkness to bright light increases the electrical conductivity by a factor of $10^6$.

Other alkali-halide salts, as well as silver halide salts, possess the same properties. A detailed investigation of the electric current in such crystals, of the absorption spectra and of the phosphorescence observed in them, carried out by P. Pohl and B. Gudden, has to a considerable degree clarified the mechanism of the phenomenon.

In recent years a large number of investigations have been devoted to electronic currents in dielectrics and semiconductors and to the influence of light upon them, in connection with their rectifying and photoelectric properties, which have found wide technical application.

The literature on these questions is presented in a number of detailed monographs (B. Gudden^7, Joz and du Bridge^8, Nix^9, Arsen’eva and Bronstein^10). The theoretical conclusions based on these facts ...

considerations were set forth by A. G. Wilson and Bronstein; therefore I shall chiefly present the new experimental data of our laboratory.

Impurities play an even greater role in electronic conductors than in ionic ones. Thus, the specific electrical conductivity of \(\mathrm{Cu_2O}\) can be changed by absorption of oxygen from \(10^{-10}\ \frac{1}{\Omega}\) to \(10^{-2}\ \frac{1}{\Omega}\) at room temperature. The reason for this action should be considered to be the circumstance that excess oxygen creates regions with a reduced work of dissociation. Thus, in pure \(\mathrm{Cu_2O}\), from the temperature coefficient of resistance of formula (3) we obtain for \(B_0 = 0.72\) electron-volts, whereas with a content of \(0.1\%\mathrm{O_2}\), \(B = 0.18\) electron-volts, the same as in pure \(\mathrm{CuO}\). The study of the temperature dependence of the electrical conductivity of \(\mathrm{Cu_2O}\) with different contents of excess oxygen led to the following picture\({}^{11}\).

Fig. 8.

Pure \(\mathrm{Cu_2O}\) possesses its own conductivity \(\sigma_0\), expressed by the straight line \(I\) between 700 and \(0^\circ\mathrm{C}\) (Fig. 8). To this conductivity is added the conductivity of the regions altered by inclusions of oxygen. This additional conductivity \(\sigma_1\) is also expressed by a straight line with a slope that decreases somewhat with increasing concentration of \(\mathrm{CuO}\). The total conductivity of the specimen over the whole temperature interval can be obtained by simple addition,

\[ \sigma = \sigma_0 + \alpha_1 \cdot \sigma_1 \]

and increases with concentration; however, an exact quantitative relation \(\sigma_1 = f(c)\) has not yet been obtained, since the oxygen is distributed nonuniformly in the \(\mathrm{Cu_2O}\) specimens.

Fig. 9.

The removal of oxygen from \(\mathrm{Mo_2O_5}\) crystals has a somewhat different effect\({}^{12}\). Here, with decreasing O content, not only the electrical conductivity increases, but also its temperature coefficient. In this case, however, the measurement could be carried out only in a limited temperature interval (\(150^\circ\mathrm{C}\)); therefore it is not yet possible to establish the general law \(\sigma = f(T,c)\).

The simplest theory of electronic concentration as an equilibrium between the dissociation and recombination of electrons in the crystal lattice can be represented by the energy scheme of Fig. 9. The lower lines represent the energy of the [[unclear: text cut off at bottom of page]]

$J$ transfers some part of the total number $N$ of bound electrons into the band of unfilled levels. In the state of equilibrium there are $n$ electrons in this upper band. The same number of unfilled levels remains in the lower band. Equality between dissociation and recombination requires that

\[ \gamma N = \beta n^2 . \tag{7} \]

The electrical conductivity, determined by the number $n$ of free electrons, must increase as $\sqrt{N}$, which is not confirmed by experiment. Apparently $\sigma$ increases much faster than $\sqrt{N}$ and even than $N$. When the source of dissociation is thermal motion one may put:

\[ \frac{\gamma}{\beta}=AT^{\frac{3}{2}}e^{-\frac{U}{kT}}. \]

The temperature coefficient of electrical conductivity $\alpha$ [formula (3a)] must then, according to formula (7), be equal to $\frac{U}{2}$. The value of $U$ can be determined from the limiting wavelength still capable of causing dissociation of electrons in the lattice (internal photoelectric effect). Exact comparisons of $\alpha$ and $U$ have so far not been possible. The existing data do not contradict the relation $\alpha=\frac{U}{2}$, but neither do they prove it to any degree. The study of the photoeffect in dielectric crystals undoubtedly proves the incorrectness of the third conclusion from the same premises. When dissociation is produced by light of intensity $J_1=aJ$, the additional photoconductivity $\sigma_p$ should increase proportionally to $\sqrt{J}$, whereas it has been established with great accuracy that $\sigma_p \sim J$. This discrepancy is apparently connected with the fact that electrons dissociated by light do not return to their initial levels, but remain inside the lattice in an excited state with an energy intermediate between the initial energy and the energy of photoconduction electrons. The existence of such intermediate levels is proved by the fact that after illuminating a crystal with light that increases its conductivity, its absorption spectrum changes: in it there appears a region of absorption of longer wavelengths than those that caused the initial photoeffect. In such an excited state, light of considerably longer wavelength is capable of increasing the electrical conductivity; moreover, under the action of such illumination the excitation itself disappears, and the crystal again returns to its initial state, losing sensitivity to long wavelengths. The number of electrons released during the disappearance of the excited state is almost equal to the number that was removed from the crystal by the current during illumination with the short wavelength. The excitation also disappears as a result of thermal fluctuations, the faster the higher the temperature.

Another fact testifying to the adhesion of electrons at intermediate levels in the lattice is the formation of space charges from electrons during the passage of current. These charges, however, do not recombine in the dark, despite the field produced by them. Consequently, they are not in a continuous band of free lattice levels and cannot move within it. Illumination with red light is sufficient to transfer them again into a state of electronic conductivity. That we are dealing here precisely with electrons is proved by the presence of the Hall effect \(^{13}\).

From the excited state the electrons may be led both into the band of free conduction levels and into their normal position as bound electrons of the lattice. To return to the initial state, the adhering electrons must overcome a certain energy barrier, whose transparency rapidly increases with increasing energy of thermal motion. We shall therefore call electrons in this state metastable. The question of whether they are associated with local distortions of the lattice or are a property of correctly built lattices remains open.

Fig. 10.

Taking into account the large number of metastable levels for photoelectrons in the lattice, we must modify the equilibrium scheme. Fig. 10 gives the distribution of the levels. We shall denote the total number of electrons by \(N\), that at metastable levels by \(n_1\), and that in the continuous band of free levels by \(n\). Let us consider two schemes \(I\) and \(II\) (Fig. 10). In doing so we shall assume that \(n+n_1 \ll N\), and that the number of metastable levels is very large in comparison with \(n_1\).

In the case of equilibrium the following relations hold:

\[ \begin{array}{cc} I & II\\[4pt] \alpha NJ+\beta n_1J=\gamma n, & \alpha NJ=\gamma n,\\ \alpha NJ=\delta n_1(n+n_1), & \alpha NJ=\delta(n_1+n). \end{array} \]

For scheme \(II\),

\[ n=\frac{\alpha N}{\gamma}J; \]

for \(I\) one obtains a quadratic expression, which can be simplified by assuming that \(n\ll n_1\), as follows from the metastable character of the level. Then

\[ n=\frac{\alpha}{\gamma}NJ+\beta\sqrt{\frac{\alpha}{\delta}\,J^{3/2}}. \tag{8} \]

It is obvious that scheme \(II\) is obtained from \(I\) by the assumption \(\beta=0\). Apparently \(\beta\) is not equal to zero if red light is also present, transferring excited electrons into the free state, or if we are dealing with light capable of simultaneously from normal and from meta-

stable levels. In these cases one may expect a considerable deviation from the proportionality of the current to the intensity of the light at very high intensities.

In the absence of light, \(\beta\) is apparently very close to zero, since no noticeable increase in conductivity is observed in the excited state. If the same scheme of levels could be extended to thermal equilibrium, then we would have \(a=U\), instead of \(a=\dfrac{U}{2}\). In thermal dissociation, however, we do not have electrons on metastable levels, and therefore the scheme of Fig. 9 is more probable.

The measurement of Hall’s constant will make it possible to determine the number of electrons both in the state of thermal equilibrium at different temperatures and those produced by light. Forth’s experiments showed that \(a\) in formula (3a) actually measures the temperature dependence of the number of electrons in the band of free levels. Measurements of electrical conductivity and thermoelectromotive force established that the mean free path of electrons in semiconductors is the same as in metals.

To express the current \(I\), two expressions are used

\[ I=Ne=\frac{x}{D}, \tag{9} \]

where \(N\) is the total number of electrons participating during one second in the passage of the current, \(e\) is the charge of the electron, \(x\) is the mean displacement of one electron along the field, and \(D\) is the distance between the electrodes. Ohm’s law is obtained from the assumption \(x=kE\), where \(E\) is the field strength.

Instead of this, one may also represent the current density \(i\) by the usual formula:

\[ i=neuE. \tag{10} \]

Here \(n\) is the number of electrons per unit volume, \(u\) is the mobility.

Obviously,

\[ n=\frac{k}{u}\cdot\frac{N}{DS}, \]

where \(S\) is the area of the electrodes. It would be erroneous to consider \(\dfrac{N}{DS}=n\).

Indeed, \(\dfrac{N}{DS}\) is the number of electrons arising in \(1\ \mathrm{cm}^{3}\) per \(1\ \mathrm{sec}\), whereas \(n\) is the number of electrons in \(1\ \mathrm{cm}^{3}\) in the state of stationary equilibrium; \(k\) expresses the projection of the displacement of the electron from the moment of its creation to the moment of its sticking, in the direction of the field for \(E=1\); \(u\) is the projection of the mean velocity of displacement of the electron in the same field. The relation between \(k\) and \(u\) is determined by the mean lifetime \(\tau\) of the electron in the band of free levels, or by the reciprocal magni-

with a certain \(w\)-probability of sticking and recombination; \(k=uw\), \(u=wk\). The mobility of \(u\)-electrons can be determined from the mean free path \(l\) and the mean electron velocity \(v\):

\[ u=\frac{1}{2}\frac{e}{m}\cdot\frac{l}{v}. \]

Both the mean free path \(l\) and the velocity \(v\) may be regarded as almost the same for all semiconductors:

\[ l\approx 10^{-6}\frac{\text{cm}}{\text{sec}};\quad \frac{e}{m}=1.76\cdot 10^{-7}\ \text{abs. el.-magn. units}; \]

\[ v\approx 6.7\cdot 10^{6}T^{1/2}\frac{\text{cm}}{\text{sec}};\quad v_{290^\circ}=1.1\cdot 10^{7}\frac{\text{cm}}{\text{sec}}; \]

\[ u\approx 1\cdot 10^{-6}\ \text{abs. units}\approx 10^{2}\frac{\text{cm}}{\text{sec}}\cdot\frac{\text{cm}}{\text{volt}}. \]

Thus \(n\) is determined from expression (10):

\[ n=\frac{1}{eu}\frac{i}{E}=\frac{1}{eu}\sigma\approx 6.3\cdot 10^{17}\sigma. \tag{11} \]

Conversely, the quantities \(k\), \(\tau\), or \(w\) depend strongly on the properties of the given crystal and vary by tens of times in passing from one specimen to another. As a general rule, electrons do not stick after each free path \(l\), but merely change the direction of their velocity and the energy accumulated along the free path. Owing to this, \(k\) sometimes reaches \(10^{-3}\frac{\text{cm}}{\text{volt}}\), \(\tau=10^{-5}\) sec., and since the mean interval of time between two collisions is \(\tau=\frac{l}{v}\approx 10^{-13}\) sec., this means that sticking occurs, on the average, only after \(10^{8}\) collisions. This should not surprise us if we recall that the sticking of an electron to the sharply electronegative oxygen molecule in a gas takes place only after \(10^{6}\) collisions.

For fields of the order of \(1000\frac{\text{volts}}{\text{cm}}\), \(x\) reaches values of several millimeters or tenths of a millimeter, i.e. the entire distance to the electrode. In these cases we obtain the saturation current and, from expression (9), knowing \(X\) and \(D\), determine \(N\)—the number of electrons created by light per second.

In these cases Gudden and Pohl succeeded in establishing that the number of electrons \(N\) liberated by light very closely approaches the number of absorbed photons \(Z\). However, this simple relation is valid only for the spectral region of weak absorption on the long-wavelength side of the absorption band. When we enter the absorption band itself, beginning with an absorption coefficient of \(1\ \text{mm}^{-1}\), the ratio \(\frac{N}{Z}\) begins sharply

[[unclear: beginning of paragraph]] the empty places in the bulk of the crystal. It is possible that in such cases the light, falling in a thin layer at the point of entry, increases its electrical conductivity so much that the field in the layer drops sharply in comparison with the field in the unilluminated part of the crystal, and the field strength inside the layer is insufficient not only for the saturation current, but also for extracting from it any appreciable part of the electrons. To this is further added a redistribution of the thermal equilibrium of the electrons, when regions of strong and weak dissociation border on one another in the same crystal.

In such electric fields, which extract all the electrons created by light, the saturation current measures their number. It is clear that this number is proportional, and in those cases when all absorption of light amounts to dissociation of electrons, is equal to the number of light quanta. More remarkable is that even in weak electric fields, where Ohm’s law still holds and where the photocurrent measures the number of electrons in a cubic centimeter in the state of equilibrium, the photocurrent is proportional to the intensity of the light, whereas the simplest equilibrium scheme (Fig. 9) would lead one to expect proportionality of \(n\) not to \(J\), but to \(\sqrt{J}\).

When, alongside the light conductivity, there is also a dark conductivity of the same order of magnitude, the equilibrium conditions become considerably more complicated. The electrical conductivity at a given place cannot be regarded as a function of the number of light quanta absorbed in the given region. The electric field and diffusion carry considerable numbers of electrons from neighboring, and sometimes from rather distant, more strongly illuminated regions. Thus, for example, in cuprite crystals illuminated from the side of one electrode, we observed \(^{14}\) a threefold or even tenfold stronger current when the illuminated electrode served as the cathode than for the reverse direction of the current (Fig. 11, straight lines \(I\)). Still more complex are the phenomena in strong fields, which carry photoelectrons over considerable distances and remove them from the crystal. Here we observe a decrease of the current upon illumination, reaching \(40\%\). In Fig. 11 the photocurrent is plotted, obtained by subtracting the dark current from the current under illumination. The straight lines \(I\) refer to fields up to \(100\ \dfrac{\text{volts}}{\text{cm}}\); the curves \(II\) to fields up to \(5000\ \dfrac{\text{volts}}{\text{cm}}\).

Fig. 11.

In liquid air, when the dark current is 1000 times weaker than the light current, there is observed neither a sharp asymmetry of the current with direction, nor a decrease of the current under illumination up to fields of \(12\,000\ \dfrac{\text{volts}}{\text{cm}}\). In exactly the same way, often the dark current, strictly obeying Ohm’s law, gives no anomalies. The latter

appear only when dark and light conductivity are combined, which compels one to assume that the energy levels of the electrons of the light conductivity differ from the levels under thermal equilibrium.

In addition to the increase of electrical conductivity in crystals, the appearance of an electromotive force under one-sided illumination is also observed. Between the point of entrance and the point of exit of the light there is established a potential difference of the order of several hundredths or tenths of a volt. Dember, who discovered this phenomenon, proposed two hypotheses for its explanation: 1) light pressure and the preferential ejection of electrons by light in the direction of its propagation, and 2) osmotic pressure caused by the difference in electron concentrations at the points of entrance and exit. The distribution of potential in an illuminated crystal measured by us¹⁴ is in agreement with the first phenomenon, but not with the second explanation and is poorly consistent with it: we deposited on a crystal by evaporation in vacuum four transparent gold electrodes (Fig. 12). The illuminated electrode always proves to be positive with respect to all the others, which is understandable from both points of view. But it is remarkable that the opposite electrode (3 in Fig. 12) proves to be negative not only with respect to 1, but also with respect to 4, although it cannot be less illuminated than 4.

Fig. 12.

Although, in this way, Dember’s first explanation gives a qualitatively correct picture, it is entirely insufficient for a quantitative explanation of the effect. Both the asymmetry of the photoeffect in visible light and the light pressure are too small to create electromotive forces of several tenths of a volt.

The other explanation is likewise not wholly satisfactory. To the electrons in a semiconductor one may apply the formulae of classical statistics. Therefore the photo-electromotive force \(P\), arising between two regions of a crystal with electron concentrations \(n_1\) and \(n_2\), may be represented in the form:

\[ P=\frac{RT}{F}\log\frac{n_1}{n_2},^{15} \tag{12} \]

where \(R\) is the gas constant, \(F\) the Faraday constant, and \(T\) the absolute temperature; \(\frac{RT}{F}\) has the value \(0.028\ \mathrm{V}\) at room temperature. Therefore, for \(\frac{n_1}{n_2}=2\), \(P=0.02\ \mathrm{V}\). For \(P=0.2\ \mathrm{V}\) it is necessary to have \(\frac{n_1}{n_2}=7000\). However, the electrical conductivity, which measures the concentration of free electrons \(n\), does not give such a strong increase as formula (12) would lead one to expect.

Thus, statistical equilibrium with one strip

free levels, which underlies formula (12), is insufficient for explaining photoelectromotive forces. It is necessary to take into account volume charges, the transition of electrons to metastable levels, and the associated displacements of electronic equilibrium. Nor can it be considered proven that the mean kinetic energy of the electrons is determined by the temperature of the crystal. Photoelectrons may be located in the band of free levels corresponding to higher energies than thermal electrons.

At the boundary between a semiconductor and a metal, certain special properties appear (rectification and the photoeffect), which have been well studied experimentally but have not yet been explained theoretically. Both properties are connected with the presence, at the boundary, of a rather thin layer of semiconductor of considerably lower conductivity, the so-called blocking layer (Sperrschicht). The exit of electrons from the semiconductor through this layer is impeded, whereas for electrons entering the semiconductor from the metal it offers substantially less resistance. At small potential differences in the blocking layer, of the order of less than \(10^{-2}\ \mathrm{V}\), the current is symmetric. As the potential difference increases to several tens and even hundreds of volts, the resistance of the blocking layer for the direction of current carrying electrons from the metal into the semiconductor falls, whereas for the opposite direction it rises, and the current in the first direction is many times greater than in the opposite direction. The ratio of these currents remains constant over a large voltage interval, provided only that care is taken to maintain a constant temperature in the blocking layer. With increasing temperature the rectification coefficient decreases and reaches zero at about \(200^\circ\mathrm{C}\).

When the blocking layer is illuminated, a flow of electrons from the semiconductor into the metal is observed, the density \(I\) of which is, over wide limits, proportional to the illumination. This current creates a photoelectromotive force \(P\) of the order of several hundredths and tenths of a volt, which increases somewhat more slowly than the illumination. The sign of the photoeffect does not depend on the nature of the metal. Even in those cases where the metal is zinc or aluminum, whose contact potential is smaller than that of cuprous oxide, illumination nevertheless transfers electrons from cuprous oxide into zinc, and not conversely.

Rectification and the photoeffect at the semiconductor–metal boundary are usually connected with one another; however, cases have been established (selenium, cuprous oxide) in which the photoeffect is observed in the absence of rectification, when Ohm’s law is confirmed for the dark current. It is possible, however, that in these cases the blocking layer itself also does not obey Ohm’s law, but its resistance is so small in comparison with the resistance of the remaining part of the semiconductor that the properties of the blocking layer do not affect the total current.

Schottky’s experiments convincingly showed that the blocking layer in \(\mathrm{Cu_2O}\) is a real material, poorly

conducting layer, of thickness from \(10^{-6}\) to \(10^{-5}\) cm. Lossen finds in carborundum crystals semiconducting interlayers of the order of \(10^{-4}\) to \(10^{-3}\) cm, giving both strong rectification and a photoeffect. Both properties increase as the thickness of the interlayer decreases from \(10^{-3}\) to \(10^{-4}\) cm, and disappear completely when the interlayer is removed.

The considerable thickness of the blocking layer poses obstacles to its theoretical explanation. Joffe, Frenkel\({}^{16}\), Neimann\({}^{17}\), Wilson\({}^{18}\) proposed a theory based on the conditions for the passage of electrons from a metal through an energy barrier, and leading to the correct dependence of the current on voltage and temperature. However, this theory is not applicable to barriers of the order of \(10^{-5}\) cm. The other explanation of rectification, by the tearing of electrons out of the metal by the field, presents the same difficulties, since for a layer of \(10^{-4}\) cm and a potential difference of \(0.1\) V one would have to assume that a field of \(10^{3}\ \frac{\text{volts}}{\text{cm}}\) is already capable of producing a large electron current at the metal–semiconductor boundary, and moreover independently of their contact potentials.

The third explanation proposed by us leads to fewer contradictions. The blocking layer has a thickness of the order of the mean free path. A portion of the electrons undergoes collisions in it and creates a considerable potential difference. Some fraction of the electrons, however, passes through without collisions. This fraction is larger in the transition from the metal, where there is a Fermi distribution, than from the semiconductor with a classical Boltzmann distribution. To explain the photoeffect it is necessary to assume that at the boundary with an environment possessing a considerable dielectric constant, the work of extracting an electron from the surface of the metal decreases sharply and the photoeffect from the metal accordingly becomes unlikely, just as the extraction of bound electrons from the semiconductor occurs with the same probability as in the absence of the metal.

Rectification at the boundary between electronic semiconductors and a metal has features in common with the conditions at the boundary between ionic dielectrics and a metal. We have seen there that the probability of the escape of ions or of their transfer of their charge to the electrode is expressed by the formula:

\[ \frac{n}{N}=Ae^{kE}. \]

An analogous law has also been established for the passage of electrons through the blocking layer into the semiconductor. The emission of electrons encounters here, moreover, greater difficulties than their entry into the semiconductor. However, we still know very little about the conditions of charge transfer at the boundary of two dissimilar bodies. This constitutes one of the most interesting problems of contemporary electrochemistry.

4. Organic Substances

The electrical conductivity of organic crystals of the closed type—for example, pure naphthalene crystals—is extremely small, less than \(3\cdot 10^{-18}\,\frac{1}{\Omega}\). Likewise, glassy compounds below the glass-transition temperature \(T_g\), and strongly polymerized solid organic substances, possess extremely low electrical conductivity; for styrene at room temperature, \(\sigma < 10^{-21}\,\frac{1}{\Omega}\). Its resistance becomes measurable beginning at \(60^\circ\mathrm{C}\), and ebonite has a very high temperature coefficient \(B = 27000\).

A very interesting dependence was found by P. P. Kobeko and I. Yu. Neldov for the temperature range above \(T_g\), where the substance is plastic, with a viscosity coefficient from \(10^{13}\) at \(T_g\) to \(10^6\). The electrical conductivity is expressed by formula (3), with coefficients \(B\) lying between 6000 and 40000. At the temperature \(T_g\), all the substances investigated (alcohol, citric acid, brucine, sugar, rosin, phenolphthalein, glycerin, Rochelle salt) have the same conductivity

\[ \sigma_{T_g}=3\cdot 10^{-15}\,\frac{1}{\Omega}. \]

Fig. 13.

The coefficient \(B\) is the greater, the higher \(T_g\). Thus the electrical conductivity for all these substances can be expressed by the formula:

\[ \sigma = 3\cdot 10^{-15} e^{\frac{\alpha}{T_g}-\frac{\alpha}{T}} . \]

Fig. 13 gives the dependence of \(\log \sigma\) on \(\frac{1}{T}\) for various substances in the plastic state, with the temperature \(T_g\) of the given substance taken everywhere as the origin of coordinates.

With some approximation, \(B\) is proportional to \(T_g\), and therefore the formula can be rewritten in a form corresponding to the law of corresponding states. Taking \(\frac{B}{T_g}\simeq 85\), we obtain:

\[ \sigma \approx 3\cdot 10^{21} e^{-85\frac{T_g}{T}} . \tag{13} \]

This formula cannot lay claim to great accuracy, since the temperature \(T_g\) itself represents an interval of several degrees and has no exact definition. Moreover, substances are known in which sharp changes in the thermal properties and in the viscosity characterizing \(T_g\) do not all occur at the same temperatures, and in which the solidification of the substance does not coincide with the moment at which thermal rotations cease. In these cases the applicability of formula (13) must be considered separately.

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