Electrical Conductivity of Electronic Semiconductors

B. Gudden*

Contents

1. Introduction.
2. Formulation and limits of the problem.
3. Electronic and ionic conductivity.
4. Ohm’s law and the determination of specific electrical conductivity.
5. Is the specific electrical conductivity a constant of the substance?
6. Temperature dependence of the specific electrical conductivity and its significance.
7. Some data on the temperature dependence of electrical conductivity.
8. Photoelectric conductivity in semiconductors.
9. Chemical structure and electronic conductivity.
10. Theory of electronic semiconductors.
11. Conclusion.

1. Introduction

Approximately ten years ago, the attention of researchers was centered on ionic conductivity in crystals and photoelectric conductivity in insulators.** Views on the nature of so-called semiconductors were based mainly on indications made by Königsberger in various survey reports.^30 Since the appearance in 1924 of our review of electrical conductivity in crystalline substances, an enormous number of works have appeared devoted to the question of the passage of electric current in nonmetallic crystalline substances, so that in what follows only a certain part of them will be considered.

As regards ionic conductivity in solids, one should point here to Tubandt’s views, set forth in Handb. d. Experimentalphysik (vol. XII/1, 1932; “Conductivity and transport numbers in solid electrolytes”). Photoelectric conductivity in crystals is examined in detail in B. Gudden’s book Photoelectric Phenomena (Lichtelektrische Erscheinungen, 1928).

* B. Gudden, Elektrische Leitfähigkeit elektronischer Halbleiter. Ergebn. d. Exakt. Naturwiss., 13, 223–256, 1934.

** In 1924 Gudden published a review, “Electrical conductivity in crystalline substances with the exception of metals.” See Ergebnisse d. Exakt. Naturwiss., 3.

However, an exposition of the properties and theory of electronic semiconductors that corresponds at least somewhat to modern physical conceptions is still absent both from textbooks and from major handbooks. By electronic semiconductors we mean solid substances whose electrical conductivity is wholly or to a considerable extent due to the motion of electrons, and whose special features—for example, the increase of electrical conductivity with increasing temperature—are determined by the internal (spatial), and not only by the external (surface), properties of these substances. Thus we exclude from consideration all those cases in which a high resistance and the corresponding temperature dependence are connected with the existence of an insulating boundary layer between metallic conducting crystalline grains or flakes. Metals such as, for example, graphite, silicon, titanium, zirconium, etc., should in no case be confused with electronic semiconductors, since the causes of their outwardly similar behavior are essentially quite different.^15

The problem of semiconductors has, over the last decade, been approached from quite different sides and with quite different aims. It is evident that only by generalizing the various points of view can one succeed in clarifying the question. The starting points and the principal directions are as follows.

A special case of electronic semiconduction was photoelectric conduction in insulating crystals (Gudden and Pohl). From this follow the attempts by Gudden and his collaborators to elucidate the mechanism of electronic conduction in nonmetallic crystals and to relate it to the energy levels of the lattice. These works are also closely connected with the new investigations of Pohl, Hilsch, and their collaborators, who studied the motion of electrons in ionic crystals.

Quite independently of these questions, attempts were made, from the side of inorganic and physical chemistry, to establish a definite relation between electrical conductivity and the chemical structure of a substance. Here, first of all, one should note the investigations, originally carried out for technical purposes, by Friedrich and his collaborators, and then the purely scientific works of Le Blanc and Sachs.

The two just-mentioned lines of work are linked by a third—the thermodynamic one, whose development was begun with great success by Wagner and his collaborators.

Finally, it should be noted that a major step forward has also been made in the theoretical interpretation of electrical conductivity in the crystalline lattice. The ideas developed in this direction by Sommerfeld (initially without connection with the new experimental data) have at present advanced so far that they not only make it possible to understand the experimental data, but also allow

make it possible to set definite tasks for experimentalists to investigate. The following exposition will show how little, despite the abundance of work, has so far been clarified.

We see our task as presenting the account without embellishing anything, and even at the risk that the reader may conclude in the end that the field under consideration does not belong among the “achievements of the exact natural sciences.”

However, while the investigator may allow himself a certain measure of optimism, the reporter, on the contrary, is obliged to have doubt and indecision if he wishes his report to contribute to research.

2. Formulation and limits of the question

The question of electrical conductivity in electronic semiconductors is only a small part of the question of the physical properties of these solid bodies, and will probably later be considered once more within a more general framework.

Moreover, it is closely connected with questions concerning the electrical properties of the surfaces of the section: rectifying action, the valve photoelectric effect, the Becquerel effect, the crystal photoelectric effect, contact potential, thermoelectric effect, etc.

All these phenomena ought to have been considered in a review devoted to electronic semiconductors; however, the number of works is so great and the results so unclear and contradictory that in the present review we shall refrain from considering them.

Until greater clarity has been achieved in understanding the nature of internal electrical conductivity in semiconductors, one cannot hope to understand the nature of the phenomena occurring in boundary layers. It is possible that within a few years it will be possible to unite and present, in the form of a few regularities, what at present, owing to the existence of numerous contradictions and uncertainties, can be set forth only very diffusely, if one tries to avoid a false impression of certainty.

3. Electronic or ionic conductivity

The distinction between electronic and ionic electrical conductivity, as well as the determination of the relation of these two types of electrical conductivity when they coexist, is much more difficult than it may appear at first glance; even now, in the literature of the subject, one may encounter a considerable number of erroneous statements. The difficulties are especially great at high temperatures and low currents. These questions are thoroughly considered by Tubandt in volume XII/1 of Handbuch der Experimentalphysik.

It is significant that individual criteria by themselves cannot be considered decisive, and only consideration of the question from various points of view can help to eliminate error. Let us illustrate this with one remarkable example.

Despite numerous attempts by experimental investigators, 100 years passed before final agreement was reached on the nature of electrical conductivity in such an ordinary substance as silver sulfide. As early as the work of Tubandt mentioned above, published in 1932, it was definitely stated that the correct α-form of Ag₂S (stable above 179° C) is a purely ionic conductor; the properties that contradict this conclusion Tubandt does not regard as sufficiently conclusive.

The grounds pro and contra such a conclusion are the following. The fact that, at a silver anode brought into contact with α-Ag₂S, Faraday’s law is satisfied with an accuracy up to 1% speaks first of all in favor of pure ionic electrical conductivity; no ions are liberated at the cathode (Tubandt and his collaborators in 1921; confirmed by a number of other investigators). Against the existence of purely ionic, and even of predominantly ionic, conductivity the following facts speak:

1. The specific conductivity is approximately 50 times greater than in other good electrolytic conductors; the temperature coefficient of conductivity has a metallic character; there is a Hall effect, in magnitude and sign corresponding to the presence of predominant electronic conductivity (Kleiber²⁹, Tubandt and Reinhold⁵⁵˒⁵⁶).

2. The mobility of Ag ions, calculated on the basis of the study of diffusion, is, in order of magnitude, much smaller than that calculated from the electrical conductivity on the assumption that the latter has an ionic character. For other solid electrolytes, however, the relation \(D = ukT\), on the contrary, is well fulfilled (\(D\) is the diffusion constant, \(u\) is the mobility of ions, \(k\) is Boltzmann’s constant) (Tubandt and Reinhold⁵⁵˒⁵⁶).

3. The electromotive force of a cell in which α-Ag₂S is the electrolyte is also considerably smaller than it should be on the basis of calculation under the assumption of pure ionic conductivity of α-Ag₂S. Instead of 0.2 V, as follows from the magnitude of the chemical affinity in the formation of Ag₂S from the elements (4550 cal per equivalent), the magnitude of the electromotive force reaches only 0.002–0.005 V and corresponds, in the best case, to only a few percent of electrolytic conductivity (Wagner⁶⁷ᵃ).

The indisputable fulfillment of Faraday’s law seemed to such an experienced and successful investigator in this field as Tubandt so convincing that, in order to remove the existing contradictions, he tried to introduce a series of additional and very artificial concepts. In reality everything seems considerably simpler. The credit for unraveling this confusion belongs to Wagner⁶⁷ᵃ. He proposed that the fulfillment of Faraday’s law is due—

CONDUCTIVITY OF ELECTRONIC SEMICONDUCTORS

is caused by the presence of some secondary process, and not by the direct action of the current. At first Tubandt and Reinhold \(^{58}\) tried to refute this hypothesis with new experimental data, but in the end they themselves became convinced that, with the appropriate arrangement of the investigation \(^{57}\), the apparent ionic conductivity of \(\alpha\)-\(\mathrm{Ag_2S}\) almost completely disappears. At the present time it may be considered fully proved that the conductivity of \(\alpha\)-\(\mathrm{Ag_2S}\) is 99% electronic in type, and only 1% of the current is carried by \(\mathrm{Ag}\) ions; observations of diffusion and electromotive force are in complete agreement with the presence of so small an ionic conductivity.

Thus, it would seem, a final solution of this disputed question has been reached; however, anyone wishing to become more closely acquainted with the nature of electrical conductivity in solids and striving for a physical worldview is advised to reread the contradictory and highly instructive works on \(\mathrm{Ag_2S}\), beginning with the time of Faraday and down to our own day. Various questions may still find their solution there even now. Incidentally, it would be desirable to reexamine, from a new point of view, the earlier data \(^{54}\) on \(\mathrm{Ag_3SbS_3}\), \(\mathrm{Ag_3AsS_3}\), \(\mathrm{AgSbS_2}\), and \(\mathrm{CuS_2}\).

Such difficult cases, like the example with silver sulfide, are exceptions; in general, however, the applicability of Faraday’s law should still be regarded as proof of the existence of pure ionic conductivity. Deviations from Faraday’s law make it possible to determine the degree of participation of electrons in the total conductivity of the substance. Less conclusive are slight material changes at the electrodes, even when they are observed only during the passage of current, as well as slight deviations from Ohm’s law at small voltages, caused by the presence of decomposition potentials and polarization. The signs of electronic conductivity are, above all: an entirely linear dependence of current on voltage, persisting down to the very smallest values of the voltage, and especially the Hall effect. While the presence of the Hall effect indicates the participation of electrons in the conductivity, it says nothing about the magnitude of the electronic part of the conductivity; and the absence of the Hall effect by no means excludes the existence of electronic conductivity (for example, \(\mathrm{Cu_2O}\) at \(400^\circ\mathrm{C}\) \(^{50}\)).

Recently attempts have been made to connect the character of the conductivity with the magnitude of the internal potential of the crystal lattice, calculated from observations of electron diffraction. These attempts, naturally, still require further development before a conclusion can be drawn about their applicability. Metals should be assigned a positive potential of the order of \(10\text{–}16\ \mathrm{V}\), ionic crystals a negative one from \(0\text{–}5\ \mathrm{V}\). Semiconductors should lie between them; the values of the internal potential obtained by different observers are not yet in satisfactory agreement. Rupp \(^{43}\) for slowly-

electron potentials gives the following values of the potential: FeS₂ 6.5; Fe₂O₃ 3.5; PbS 2.6; CuJ 2.8; dark-blue CaF₂ 4.6; colorless CaF₂ 3.2; colorless NaCl 3.5; yellow NaCl (in bleaching light)—2.8 V.

Dixit⁷, with fast electrons (from 20 to 44 kV), gives other values for the internal potential, namely: for Fe₂S 5.1; Fe₂O₃ 12.6; PbS 12.5. It is quite possible that the difference in the data for PbS and Fe₂O₃ is explained by the circumstance that in the first case only the surface layer was effective, playing a special role precisely in PbS, whereas Dixit investigated the crystal lattice proper.

Attention should be paid to the small magnitude of the internal potential of potassium (according to Rupp, \(+7.3\) V) and the high negative potential obtained by both investigators for ZnS (\(-5.8\) and \(-4.8\) V), which can hardly be regarded as an ionic crystal, but rather as a good insulator. From consideration of the data cited above one may draw the conclusion, also reached by Rupp, that it is scarcely possible to judge the character of conductivity on the basis of the magnitude of the internal potential.

A connection between ionic conductivity and chemical structure does indeed exist, but it is not so unambiguous that in every individual case the character of the conductivity could be predicted. The inverse problem can be solved much better: to infer the structure of a compound from the type of conductivity²², ²⁴.

Guden and Pohl, Pohl and Hilsh and their co-workers²³, ⁴¹, ⁵¹ pointed out the fact that in pure ionic conductors additional electron currents may flow. Further, Tubandt and co-workers established several cases of mixed electrical conductivity in which both parts have approximately the same magnitude. The ratio of the ionic and electronic parts of the conductivity can hardly be regarded as a constant of the substance, since it depends not only on temperature but also on other circumstances determining the electronic conductivity. However, the dependence of this ratio on the current density or on the field strength, found by chance, is not reliable.

In general, all agree with Hevesy²² that, as a rule, mixed conductivity exists, and only the mutual ratio of the two kinds of conductivity is subject to large fluctuations. Knowledge of the character of the conductivity, or, in mixed electrical conductivity, of the ratio of its ionic and electronic parts, is a necessary condition for every investigation of an electronic semiconductor, and therefore this question is considered here in greater detail.

4. Ohm’s Law and the Determination of Specific Electrical Conductivity

An indication of the specific electrical conductivity of a substance has meaning only in the case of complete proportionality of the current through it to

...applied to it. However, one still encounters the assertion that the deviation from Ohm’s law in semiconductors is for them regular and essential, but that nevertheless a specific resistance is defined.

In reality one can often observe that at low voltages the current does not increase in direct proportion to the voltage, but much more rapidly, and, moreover, the current strength sometimes depends on its direction. At present it may be regarded as established that such a nonlinear dependence of current on voltage is due to incidental causes and does not depend on the nature of the semiconductor \(^{8a}\); this nonlinearity is due either to the presence of barrier layers (Sperrschicht), whose resistance depends strongly on the voltage and which may also exist at the internal boundaries of individual grains of the substance, or else is due to insufficient constancy of the temperature of the specimen under investigation during the experiment, which plays an extremely important role in view of the mostly strong temperature dependence of the electrical conductivity of semiconductors. For judging the character of the true dependence of electrical conductivity on field strength there are still not enough experimental data. One could, of course, calculate this dependence for very high electric fields, taking into account, on the one hand, the decrease in the number of carriers of electric charge (saturation), and, on the other, their increase as a result of impact ionization; however, experimental proof of such phenomena in actual semiconductors is very difficult because of the presence of the Joule effect. It remains only to transfer meaningfully to semiconductors the results of investigations of the electronic conductivity of insulating crystals. It is quite conceivable that, with far-reaching correspondence between electronic conductivity caused by thermal dissociation or by the photoeffect, such incidental phenomena as secondary photoelectric currents \(^{15e, 34}\), with their strong deviation from Ohm’s law, may also occur in semiconductors. Up to the present time, however, they have not been observed.

Since the barrier layers lie only at the electrodes themselves, under known conditions it is possible to eliminate them by appropriate treatment of the semiconductor specimen under investigation. Metal layers deposited by evaporation in a high vacuum prove very suitable as electrodes, and for evaporation one should take the metal entering into the composition of the semiconductor under investigation \(^{13}\). The surfaces of the specimen on which the electrodes are deposited must be mechanically isolated. The deposition of electrodes by cathode sputtering easily leads to the formation of barrier layers*. In individual cases the barrier layers attain thicknesses of \(10^{-3}—10^{-4}\) cm, for example, in carborundum. Grinding of this layer eliminates rectification \(^{6,35}\). The linearity of the volt—

* On the influence of surface treatment see, among others, the fundamental works of Schottky and his collaborators devoted to barrier layers.

the current-voltage characteristic and independence of the direction of the current do not yet prove, unfortunately, the absence of barrier layers. This circumstance should especially be borne in mind in measurements with finely crystalline aggregates (crystallites) and specimens pressed from powder; very many measurements for this reason lose much of their evidential force. Under all circumstances it is recommended to use the probe method and to measure statically the voltage drop between the electrodes, or at least to check other measurements by this method. This method also does not give complete reliability. On the contrary, owing to the fact that the boundary layers between the individual crystallites making up a crystallite often have a conductivity different from the conductivity of the lattice itself, in calculating the specific electrical conductivity completely incorrect values are obtained. It should also be borne in mind that the electrical conductivity of such boundary layers in directions parallel and perpendicular to the plane of the layer agrees only in order of magnitude.

As was emphasized at the beginning of this review, we shall not discuss in detail the entangled questions of barrier layers (Sperrschichten) and boundary layers (Grenzschichten), and shall confine ourselves merely to pointing out that in practical measurements they may be of very great importance. A particular difficulty in investigations of electronic semiconductors lies in obtaining specimens suitable for measurements. Whole crystals (single crystals) of a size sufficient for investigation are much more difficult to obtain than in the case of ionic crystals; the cohesion of the individual crystallites upon pressing is also considerably worse than in the case of ionic semiconductors. If it is still possible, after a fashion, to take into account and calculate the space filling in a pressed or sintered specimen, then with regard to transition resistances and their temperature dependence we are still quite in the dark.

Gudden and his collaborators proposed a method for determining, albeit not quite accurately, the electrical conductivity of powders. The method consists in measuring the damping in an electrical oscillatory circuit when a slight admixture of semiconducting powder is added to the dielectric of a capacitor included in it. The results of such measurements, carried out by Falkenhagen⁶⁰ and Gillery¹⁶, show that in the case of individual substances the values of the specific electrical conductivity obtained by the damping method and those determined on specimens pressed from powder coincide (for example, for Cu₂O), whereas in other cases a very considerable discrepancy is obtained (even by an order of magnitude) (for example, for CuO and ZnO). We believe that, on the basis of all measurements with powders, conclusions about the properties of an individual (single) crystal can be made only very conditionally. Microscopic grains may have a severely distorted crystal lattice, and electronic electrical conductivity is very sensitive to disturbances

...to violations of lattice regularity[^15e]. Le Blanc and Sachs, summarizing experiments with Cu₂O[^33b], have at their disposal significantly more reliable measurements made with powders. Further investigations are necessary. It should also be pointed out here that the crystallographic direction has a great influence (in determining electrical conductivity), especially in a layered lattice; in the study of powders and pressed specimens this influence disappears completely. As is known, for metals one can determine from the temperature dependence of the resistance that part which is called the residual resistance (Restwiderstand) and is due to disturbance of the regularity of the crystal lattice; for semiconductors we know of no corresponding method, since in this case the temperature dependence of the resistance is determined by various causes.

How difficult it is to interpret the results of determinations of the electrical conductivity of semiconductors, even when made on large crystallites, is shown by Schönwald’s measurements[^49]; despite the hourly heating of a polycrystalline Cu₂O plate at 1000°C that preceded the measurements, the nonuniformity of the field strength in the specimen was such that its fluctuations from point to point reached 50%. Hence we conclude that, although the result of determining the specific electrical conductivity of a given specimen is meaningful, the sources of error are very large and difficult to recognize.

5. Is the Specific Electrical Conductivity a Constant of the Substance?

Until recent years it was generally thought that to each crystalline compound there should be assigned a quite definite value of the specific electrical conductivity in order to characterize it unambiguously, approximately as is done for a metal. The large scatter in the results of measurements of electrical conductivity for most mineral substances was attributed to the presence of impurities. At the same time, Bädeker’s important experiments[^3],[^52] were known, showing that the specific electrical conductivity of CuI depends unambiguously on the vapor pressure of iodine in the atmosphere surrounding the specimen; Tubandt and co-workers[^55] additionally showed in 1927 that in all three copper halide compounds (CuCl, CuBr, and CuJ), which in pure form possess ionic conductivity, electronic conductivity appears in the presence of an excess of the corresponding halide.

In 1930 the view indicated above underwent a change, and since then, despite a whole series of proposed hypotheses, no unity of opinion has been reached. Gudden was perhaps the first to express the idea that the entire electrical conductivity of an electronic semiconductor is not a property of the substance, but is wholly determined by the presence of insignificant impurities[^15b]. On the basis of experiments carried out at that time with CuJ, Cu₂O, NiO, and UO₂, he believed that for the appearance of electronic conductivity in semiconduct-

decisive is the excess content of the electronegative constituent of the compound. All such semiconductors, when the composition of the crystal lattice is strictly stoichiometric, should in his opinion be insulators. He leaves open the question whether, alongside the above-mentioned type of electronic semiconductors, there are semiconductors in which the conduction electrons are formed through thermal dissociation of the undamaged lattice of the compound. Up to now this question has not been resolved in a definitive form.

Wagner[^67b] developed the above hypothesis more deeply, using for this purpose the theory of “lattice disorder” (Fehlordnungsthehre) developed by him jointly with Schottky[^63]. According to Wagner, electronic conductivity in polar compounds is caused by deviation from stoichiometric composition or from the ideal regularity of the lattice. He agrees with Gudden that without such a deviation electronic conductivity is absent, but goes further than the latter, pointing out that not only an excess of the electronegative component of the compound plays a role, but also a deficiency (deficit) of it, and possibly also a disorder of the crystal lattice caused by thermal motion, when even without deviation from stoichiometric composition individual lattice sites are distorted. As an example of the second case Wagner points to Ag₂S, ZnO, and CdO, since the specific electrical conductivity of these substances decreases with increasing partial pressure of the electronegative constituents; as an example of the third case he points to CuO, for which, in contrast to Leblanc and Sachse,[^33a] he was unable to establish any influence of oxygen pressure on the specific electrical conductivity[^66]. In individual cases, on the basis of the conception he developed, Wagner succeeded in giving a satisfactory quantitative interpretation of the observed dependence of electrical conductivity on partial pressure[^64],[^65],[^66].

The Gudden–Wagner view found many adherents and easily formed the basis of new theoretical works, since in them it is assumed that the sources of conduction electrons are defective sites (Störstellen) of the crystal lattice. However, this point of view is not the only one and not indisputable. Thus, first of all, Leblanc and Sachse[^33a] recently expressed the conviction that a definite intrinsic electrical conductivity exists in the above-mentioned compounds; Juse and Kurchatov suppose that they have succeeded in detecting the intrinsic electrical conductivity of pure Cu₂O[^27].

In their investigations of the influence of gases on electrical conductivity, Wagner and his co-workers attach great importance to the possibility of working with samples that are in a state of true equilibrium with the atmosphere surrounding them; therefore they limited their investigations to the region of high temperatures (600–1000°C) and used, for study, very thin layers of substance or porous bodies, thereby refraining to a certain extent from determi-

of absolute values of the electrical conductivity and its temperature dependence. The same is true of the measurements of Leblanc and Sachs. The works of the Physical Institute in Erlangen\(^{8, 13, 16, 31, 60, 61, 70}\), on the contrary, strive in every way for an exact determination of that part of the conductivity which is caused by disturbance of the regularity of the crystal lattice; they were carried out under conditions of so-called “frozen equilibrium” and in that temperature region in which no appreciable change of this kind occurs. Similar conditions are found in some works by Russian investigators and in works carried out in the laboratory of the Siemens concern. This group of works has another (not chemical) aim and is undertaken with the purpose of determining the number of conduction electrons, the laws of their motion, and their energy relations; but these works too indicate that the specific electrical conductivity of the semiconductors investigated, at almost identical chemical composition, varies within many orders of magnitude and therefore is not a substance constant. It is noteworthy that indifferent gases have no effect on the magnitude of the electrical conductivity; contrary indications cannot be regarded as confirmed.

Below we give some factual data from the literature of the question: the electrical conductivity of Cu\(_2\)O in the temperature range from 800 to 1000°C and in equilibrium with oxygen increases approximately proportionally to \(p_{\mathrm{O}_2}^{1/6}\) (Wagner\(^{64}\)). Within the limits of accuracy of this relation (0.3—100 mm Hg) this amounts, approximately, to an increase in the electrical conductivity of only 2 times. Leblanc and Sachs\(^{33b}\) observed at low temperatures on powders a considerably larger, but irregular, effect. The electrical conductivity of powdered Cu\(_2\)O, kept in a vacuum at room temperature, increases, according to their statement, upon admission of oxygen by several thousand times and can again be lowered to the former value by heating in vacuum at 250°C. Cf. also their measurements given in Fig. 2, in which the excess of oxygen reached several atomic %. A much greater difference is indicated in the work of Juse and Kurchatov\(^{27}\). Plates of Cu\(_2\)O with an analytically determined excess of oxygen reaching 0.1% have at room temperature approximately \(10^6\) times greater electrical conductivity than the same plates without excess oxygen (Fig. 2). That this kind of change in electrical conductivity depends not on the stoichiometric excess of oxygen, but on the nature of its bonding with the crystal lattice, is convincingly evident from the investigation of Weybel\(^{68}\). The specific electrical conductivity of the Cu\(_2\)O plates investigated by him lies between \(10^{-2}\) and \(10^{-7}\ \Omega^{-1}\ \mathrm{cm}^{-1}\), and the method of heat treatment and cooling has no less influence than gas treatment. Juse and Kurchatov also mention that, depending on the rate of cooling, the excess oxygen is distributed within the lattice in entirely different ways.

Wagner\(^{66}\), in investigating Cu\(_2\)O at temperatures above

at \(500^\circ\mathrm{C}\) was unable to establish the influence of the gaseous atmosphere surrounding the specimen, whereas according to Leblanc and Sacks, at \(500^\circ\mathrm{C}\) oxygen is taken up, and the value of the electrical conductivity at room temperature increases by approximately a factor of 30 in comparison with a specimen treated in vacuum. For CuJ in contact with iodine in a solution of \(\mathrm{CS}_2\), the electrical conductivity is proportional to the fifth root of the iodine concentration\({}^3\).

On the basis of the data of Bohlender and Idaszewski (1905), one should think that the increase in the electrical conductivity of \(\mathrm{CuS}_2\), by many orders of magnitude with a very small excess of sulfur, is due to the formation of through conducting paths of metallically conducting CuS.

Leblanc and Sacks\({}^{32a}\) observed in NiO an increase of the content of excess oxygen up to five atomic percent; in this case the initially colorless powder became black and the electrical conductivity increased by a factor of \(10^5\). Heating in vacuum returns the specimen to its initial state. Wagner\({}^{66a}\) finds that, at equilibrium between NiO and oxygen in the temperature range from 800 to \(1000^\circ\mathrm{C}\), the electrical conductivity is proportional to the fourth root of the oxygen pressure (this relation holds at pressures greater than \(2\cdot 10^{-4}\) mm Hg).

The electronegative components—ZnO, CdO, AgS—produce the opposite effect. For ZnO in equilibrium with oxygen at a temperature of about \(900^\circ\mathrm{C}\), Wagner\({}^{65}\) found a decrease in electrical conductivity with increasing oxygen pressure; the electrical conductivity is proportional to the fourth root of the pressure. At pressures below \(10^{-2}\) atm the dependence is manifested to a lesser degree. At \(650^\circ\mathrm{C}\) the electrical conductivity in the absence of oxygen in the surrounding atmosphere is approximately \(2\ \Omega^{-1}\ \mathrm{cm}^{-1}\), whereas in air it reaches \(6\cdot 10^{-3}\). It is possible that boundary surfaces already play a role here, because the corresponding difference in electrical conductivity indicated for room temperature: \(1\ \Omega^{-1}\ \mathrm{cm}^{-1}\) and \(6\cdot 10^{-8}\ \Omega^{-1}\ \mathrm{cm}^{-1}\), is due only to these boundary layers, as follows from measurements with ZnO single crystals.

In the limiting cases investigated by Wagner, the difference in the contents of excess oxygen was less than 0.2 atomic ‰. The extremely difficult measurements of Fritsch\({}^{13}\), carried out with artificially prepared ZnO crystals (needles several tenths of a millimeter thick and 2–3 mm long), gave reproducible values in the range between \(1\ \Omega/\mathrm{cm}\) and \(10^{-4}\ \Omega/\mathrm{cm}\), with the cooling rate exerting a greater influence than the gaseous atmosphere.

In investigating CdO, Wagner\({}^{65}\) did not find any appreciably regular relation; the decrease in electrical conductivity with increasing oxygen pressure is smaller than in the case of ZnO and very inconstant. The electrical conductivity of \(\alpha\)-\(\mathrm{Ag}_2\mathrm{S}\) at \(200^\circ\mathrm{C}\),

when \(Ag_2S\) is saturated with sulfur, may fall to \(\frac{1}{20}\); at higher temperatures the influence of excess sulfur is considerably smaller\(^{55—56}\).

In none of the cases so far investigated is there a simple proportionality between electrical conductivity and the deviation from stoichiometric composition. However, other dependences, too, can hardly yet be regarded as more or less probable. In any case, it must be assumed that the deviation from stoichiometric composition (stöchiometrische Fehlbetrag) is not in itself a sufficient measure of the irregularity of the lattice.

6. Temperature dependence of specific electrical conductivity and its significance

The investigation of the influence of a gas or other external conditions, such as, for example, definite impurities, on the electrical conductivity at one definite temperature can give only a very imperfect picture of the phenomenon. A considerable deepening in the study of the phenomenon is achieved by investigating the temperature dependence of electrical conductivity. It is known that the specific conductivity of a semiconductor increases with temperature and, for the most part, over a rather wide range of temperatures is satisfactorily represented by the expression:

\[ \sigma = ae^{-\frac{b}{T}} \quad \text{or} \quad \sigma = AT^n e^{-\frac{E_0}{kT}} \tag{0} \]

In what follows the constant \(E_0\) will always be expressed in electron-volts eV (conversion of \(b\) into eV is carried out as follows: \(1\ \mathrm{eV} = \frac{bk}{\varepsilon} = 8.6 \cdot 10^{-5} b\)).
(\(k\) is Boltzmann’s constant, \(\varepsilon\) is the elementary charge).

Since the temperature dependence of the quantity \(a\) is not taken into account, it may be considered that the logarithm of the specific electrical conductivity is a linear function of the reciprocal absolute temperature. Exactly the same dependence is also found in the case of solid ionic conductors. Whether there exist temperature dependences with some limiting value of the specific conductivity in the region from low to medium temperatures (below \(1000^\circ\mathrm{C}\)), which Königsberger regarded as a rule, has remained doubtful up to the present time. Sometimes “logarithmic straight lines” exhibit a more or less sharp bend (Knicke). To relate these bends, as has already repeatedly been done, to transformation points would apparently be incorrect\(^{98}\). Transformation points should have affected not only the change in slope but should also have led to a jump-like change of the ordinate; this occurs in the case when, as is accepted by most investigators, the slope of the logarithmic straight line

connected with the work of detachment. The nonlinear dependence of \(\lg \sigma\) on \(\frac{1}{T}\) generally indicates that a change is taking place in the composition of the substance. The cause may also be lag phenomena. Thus, depending on the rate and the direction in which the measurements are carried out (with increasing or decreasing temperature), different values of the electrical conductivity are observed\(^{61}\). Further discussion was given to the dependence of such a curvature of the “logarithmic straight line” on the temperature dependence of boundary layers (at the boundaries of individual grains, semiconductor crystallites), which, for example, in the case of Si and other metals may lead to the erroneous conclusion that they are semiconducting in nature\(^{15a,b,44}\). Generally speaking, the slope of the “logarithmic straight line” is the greater, the smaller the electrical conductivity of the substance at room temperature (Fig. 1). However, for this rule, recently once again confirmed by V. Meyer\(^{37b}\), there are rather numerous exceptions. Perhaps one cannot say anything other than that the concentration of lattice irregularities (Störstellenkonzentration) has one order of magnitude for all semiconductors, and the difference in the magnitude of the electrical conductivity is determined by the value of \(E\).

Changes in the logarithm of the electrical conductivity were observed within the range from 0.05 to 4 when the reciprocal of the absolute temperature was changed by \(10^{-3}\); this corresponds to \(E = 0.01\text{–}0.8\ \mathrm{eV}\), or, more graphically: when the temperature is lowered from room temperature to \(-50^\circ\mathrm{C}\), the electrical conductivity falls by 90% to \(\sim 0.0001\). The following considerations may serve as a first interpretation of the temperature dependence. Independently of any model representation, the specific electrical conductivity is determined as the product of the elementary charge by the number of particles carrying electric charge per unit volume and by their mobility. The increase of electrical conductivity with temperature may be due to an increase both in the number of particles \(n\) and in their mobility \(u\). For the separate determination of both these quantities one may use the Hall effect, as was already done in its time by Königsberger. On the basis of the simplest conception of the origin of the Hall electromotive force, it follows that

\[ u=\frac{P_h}{P_0}\cdot\frac{l}{b}\cdot\frac{1}{H}, \tag{1} \]

where \(P_h\) is the Hall electromotive force, measured by the static method, \(P_0\) is the potential difference applied to the specimen under investigation in the form of a plate of length \(l\), width \(b\), and thickness \(d\), and \(H\) is the flux density of the lines of force of the transverse magnetic field (if \(H\) is measured in \(\mathrm{V\,sec/cm^2}\), then \(u\) is expressed in \(\frac{\mathrm{cm}}{\mathrm{sec}}\) per \(\frac{\mathrm{V}}{\mathrm{cm}}\)).

The number of conduction electrons per unit volume is

\[ n=\frac{1}{R_{\mathrm{e}}} \tag{2} \]

Here \(R\) is the so-called Hall constant. \(P_i = R \dfrac{iH}{d}\), where \(i\) is the current flowing in the plate in the longitudinal direction, \(e\) is the elementary electric charge; \(R\) is obtained in \(\dfrac{\mathrm{cm}^3}{\mathrm{A}\cdot\mathrm{sec}}\).

Fig. 1. Relation between the constant \(E\left(\sigma = ae^{-\frac{E\varepsilon}{kT}}\right)\) and the specific electrical conductivity \(\sigma\).

1. CuO (foil), Wagner’s data, extrapolated from 800 to 20°C.
2. CuO (powder), Leblanc and Sachs.
3. CuO (foil), Kapp and Tre.
4. Cu\(_2\)O (crystalline), Nasledov and Nemenov.
5. Cu\(_2\)O (crystalline), Engelhard, Vogt (selected from more than 100 observations).
6. Cu\(_2\)O (crystalline), Zhuse and Kurchatov.
7. Cu\(_2\)O (powder), Leblanc and Sachs.
8. Cu\(_2\)O (crystalline), Kröger.
9. WO\(_3\) (sintered body), V. Meyer.
10. UO\(_2\) (sintered body), V. Meyer.
11. ZnO (sintered body), V. Meyer.
12. ZnO (crystals), O. Fritsch.
13. Cr\(_2\)O\(_3\), V. Meyer.
14. CdS, V. Meyer.

Expression (1) is equivalently, and more commonly, written as

\[ u = R\sigma, \tag{3} \]

where \(\sigma\) is the specific electrical conductivity.

When calculating the mobility on the basis of the determination of the Hall effect, the following should be borne in mind: as is known, in very numerous cases the sign of the Hall effect corresponds to the deflection of positive carriers; regardless of whether or not one accepts the explanation of the anomalous sign of the Hall effect[^21] in terms of so-called hole conduction (Ersatzleitung), it must be borne in mind that the observed Hall effect may be the result of the action of two opposite processes, and consequently the mobility calculated on the basis of experimental data may prove to be too small. Indeed, Weibel points out that the Hall constant of Cu\(_2\)O, as the temperature is raised, gradually decreases, falls to zero at about \(400^\circ\) C, and then at a higher temperature changes sign[^50]. In other respects as well there are a number of difficulties. It is still not entirely clear why, thanks to boundary layers that raise the resistance of a metallic specimen by several orders of magnitude and thereby completely distort the value of the specific electrical conductivity, the Hall voltage increases to such an extent that the mobility calculated on this basis has the correct value. Königsberger[^30] proposed, using the normal value \(R\sigma\) for silicon and a number of other substances, to eliminate the confusing action of insulating intermediate layers. The sign and magnitude of the Hall effect have so far been determined only for a small number of cases.

Ag\(_2\)S in the temperature range \(160\)—\(225^\circ\) C[^29]; Mo\(_2\)S[^19] and ZnO[^13] have been studied at room temperature; Cu\(_2\)O above \(500^\circ\) C[^50] has a Hall effect of normal (negative) sign, as, for example, does bismuth; from the point of view of the theory[^21] this means that the conduction has a predominantly electronic character (Überschussleitung).

The value \(R\sigma\) for ZnO is so small that one may assume the presence of considerable “hole” conduction (Ersatzleitung, Lochleitung). CuJ[^52], in contrast to the initial observations, like Cu\(_2\)O[^13], below approximately \(400^\circ\) C has a Hall effect of anomalous (positive) sign; from the point of view of the theory, the conduction in this case also has a predominantly “hole” character.

We point here to the remarkable results of Hips[^19], obtained in the investigation of PbS (a reversal of the sign of the Hall effect from a large positive to a large negative value depending on impurities), only because we are doubtful about the semiconducting properties of PbS that are ascribed to this compound on the basis of electrical-conductivity measurements made by Frey[^14]; moreover, it is completely unclear why the value \(R\sigma\) is so large (270 cm/sec per V/cm for “hole” conduction and 290 for electronic conduction).

Extensive measurements for determining the mobility were carried out with Cu\(_2\)O by Engelhard[^8b]. According to his data, the “mobility” at room temperature is 50 cm/sec per V/cm and increases with

upon lowering the temperature approximately proportional to \(\frac{1}{T}\). At the same time it was assumed, following Koenigsberger, that in all semiconductors the number of carriers increases exponentially as the temperature is raised. In this general explanation there seem to be no disagreements; on a whole series of particular questions, however, agreement has not yet been reached. W. Meyer, in the relation \(\sigma = a e^{-\frac{b}{T}}\), is inclined to see merely an interpolation formula; namely, he believes that the scatter of the constant \(a\) is devoid of any regularity, and in particular is not in any way connected with changes in stoichiometric ratios; at the same time the constant \(b\) is a monotonic function of the resistance and may even take a negative value when the specific resistance becomes less than \(1\ \Omega\cdot\text{cm}\).

Despite these objections, which undoubtedly deserve attention, the majority of investigators adhere to the view that the constant \(b\) is closely related to the work of detachment (Ablösearbeit).

In any case, the matter is not at all as simple as Koenigsberger imagined and as later attempts were made to explain it. We must, of course, take into account a certain distribution of the detachment works, their temperature dependence, and, moreover, in individual cases even the temperature dependence of the concentration of local disturbances of the regularity of the crystal lattice (\(N_0\)). In the following paragraph some experimental data are presented; here it is necessary to bear the following in mind: the constant \(E\) cannot be calculated from the temperature dependence of the constant \(a\), since the mobility entering into the latter is not known with sufficient accuracy (with the exception of \(\mathrm{Cu_2O}\)), and at the same time measurements of the Hall effect are lacking. This calculation should have led to a slight increase in the value of the constant \(E\). In all measurements with powders, especially at low temperatures (room temperature and below), the influence of boundary layers remains in question; at higher temperatures one cannot vouch for the constancy of stoichiometric and other disturbances of the regularity of the crystal lattice. The fact that we do not give values of the constant \(E\) greater than \(0.9\ \mathrm{eV}\) is explained by the fact that at room temperature bodies possessing a constant \(E\) of greater magnitude behave as insulators.

7. Some Data on the Temperature Dependence of Electrical Conductivity

The temperature dependence of the electrical conductivity of \(\mathrm{Cu_2O}\) has been studied best of all; however, the results are not very satisfactory (Fig. 2). Wagner carried out\({}^{64}\) measurements at the upper limit of the temperature range of existence of \(\mathrm{Cu_2O}\). The temperature dependence

at constant excess oxygen content was calculated on the basis of measurements at constant partial pressure of oxygen; independently of the oxygen content, the value

Fig. 2. Temperature dependence of the specific electrical conductivity of \(\mathrm{Cu_2O}\)

\(a\) — Wagner (recalculated for constant oxygen content; crystalline specimen; oxygen pressure \(1\) mm Hg; the straight line \(a\) should be shifted 4 mm upward).

\(b\) — Juse and Kurnatov (excess oxygen not detected analytically; polycrystalline specimen).

\(c, d\) — same; analytically determined oxygen content is \(0.1\) and \(0.06\) wt.%; polycrystalline specimen.

\(e\) — Leblanc and Sachse, powder with composition \(\mathrm{Cu_2O}_{1.0035}\).

\(f\) — same, powder with composition \(\mathrm{Cu_2O}_{1.000}\).

\(g, h\) — same, polycrystalline specimens.

\(l—q\) — Engelhard (agrees with Fortin); crystalline specimens with different preliminary treatments, selected from more than 100 samples.

\(r\) — Engelhard; the measurements indicate a slight bend at \(-70^\circ\mathrm{C}\).

\(s\) — Nasledov and Nemenov; crystalline specimen. The slope persists down to \(-183^\circ\mathrm{C}\) \(\left(\dfrac{1}{T}\cdot 10^3 = 11\right)\).

\(o\) — Weibell, crystalline specimens with different preliminary treatment.

\(x\) — Kapp and Tre; thin layer of \(\mathrm{CuO}\) on glass (the oxide has the highest value for electrical conductivity).

\(y\) — same, \(\mathrm{Cu_2O}\) remelted from \(\mathrm{CuO}\) in an oxygen atmosphere.

of the constant \(E\) was approximately \(0.4\ \mathrm{eV}\). In the region of intermediate temperatures, measurements were carried out by Juse and Kurnatov\(^{27}\), and also by Leblanc and Sachse\(^{33}\). A considerable deviation from

logarithmic straight line, especially in the measurements of Leblanc and Saxe, which can be explained by a change in the content of excess oxygen. The Russian investigators explain their observations in the following simple way: the logarithmic straight line with a slope of \(0.72\ \mathrm{eV}\) should belong to the lattice of “pure \(\mathrm{Cu_2O}\),” whereas samples with an excess oxygen content have a constant \(E\) of the order of \(0.13\)—\(0.15\ \mathrm{eV}\), i.e., the same as had previously been found by Leblanc and Saxe for “pure \(\mathrm{Cu_2O}\).” The constant \(a\) of \(\mathrm{Cu_2O}\), as the excess oxygen content increases from \(0.04\) to \(0.1\) weight percent, increases by a factor of 70; in “pure \(\mathrm{Cu_2O}\)” it is, however, another 100 times larger. At high temperatures the straight lines of impurity conductivity coincide with that of “lattice conductivity,” precisely as occurs in the case of solid ionic conductors. Unfortunately, the number of measurements seems to us insufficient for such a conclusion; moreover, the authors say nothing about the reproducibility of the results. It may be thought that in this case, during the measurements, the constancy of the oxygen content was not maintained to a sufficient degree, and the measurements were made in the absence of equilibrium with the gas phase. This is important both for the amount of absorbed oxygen and for the nature of its bonding. Such changes, depending on the initial state, may lead to error and give values of the constant \(E\) that are too high or too low.

From Fort’s work \(^{64}\) it can be seen that at temperatures above room temperature internal redistributions of excess oxygen occur, and above \(20^\circ\mathrm{C}\) the specific electrical conductivity changes slowly; other investigators \(^{27}\) also point to the same phenomenon. The latter indicate that within slowly cooled \(\mathrm{Cu_2O}\) samples with a high content of excess oxygen one can detect the presence of CuO needles; at the same time, when the sample is cooled very rapidly, these needles are absent. Excess oxygen contained in the form of CuO needles, in contrast to a molecular solution, can affect the electrical conductivity of \(\mathrm{Cu_2O}\) in the same way as an oxygen deficiency in the form of crystals of metallic copper.

Leblanc and Saxe worked in the same temperature region as the Russian investigators. The agreement of the results is not particularly good. They also consider it possible to determine the properties of “pure \(\mathrm{Cu_2O}\),” but the value of the electrical conductivity of “pure \(\mathrm{Cu_2O}\)” obtained by them is approximately 50 times greater than \(10^{-10}\ \Omega^{-1}\ \mathrm{cm}^{-1}\) at \(0^\circ\mathrm{C}\)—the value given in the work of Zhuse and Kurchatov. This discrepancy between the results indicates that the constant of the substance has not yet been found. Further considerations will be given below.

In order that neither the amount nor the nature of the bonding of excess oxygen should change during the measurements, Engelhard \(^{86}\) worked exclusively in the low-temperature region. He determined only the electrical conductivity and the Hall constant. His prolonged and extensive investigations clearly indicate that in this case there occur

very confused relationships. The values of the constant \(E\) for various specimens lie around \(0.3\ \mathrm{eV}\) (the determination was made from Hall’s constant; distortions introduced by the temperature dependence of the mobility are thereby excluded). No clear connection was found between the constant \(E\) and the preliminary treatment of \(\mathrm{Cu}_2\mathrm{O}\) or the magnitude of the electrical conductivity. The mean value (25 specimens) of the constant \(E\) for cuprous-oxide specimens prepared by oxidizing metallic copper in air and cooled comparatively slowly was \(0.32\ \mathrm{eV}\); for specimens heated for one hour at \(1000^\circ\mathrm{C}\) in oxygen and then rapidly cooled, a value of \(0.34\ \mathrm{eV}\) was obtained; for specimens heated for one hour at \(1000^\circ\mathrm{C}\) in vacuum, a value of \(0.31\ \mathrm{eV}\) was obtained. In 24 cases, when excess oxygen was removed, the constant \(E\) decreased; in 10 cases it increased. The corresponding values of the constant \(E\) in 20 cases, when equilibrium was established at \(500^\circ\mathrm{C}\), were \(0.31\), \(0.30\), \(0.32\ \mathrm{eV}\). The same investigator established that the value of the constant \(A\)* increased when \(\mathrm{Cu}_2\mathrm{O}\) specimens were treated in an oxygen atmosphere at \(1000^\circ\mathrm{C}\); removal of oxygen, however, lowered the value of the constant (only one exception in 35 experiments). In fact, the value of the constant \(A\) varied within the limits from 1.3 to 2300. This enormous difference can be explained in part, on the one hand, by the fact that even after several hours equilibrium in \(\mathrm{Cu}_2\mathrm{O}\) plates sometimes more than 1 mm thick was still not fully established, and, on the other hand, by the fact that the cooling of the specimens was not always carried out sufficiently rapidly and uniformly, so that the form of oxygen bonding in the \(\mathrm{Cu}_2\mathrm{O}\) lattice could have been different in different experiments. The “mobility” calculated from the Hall effect in all cases without exception (35 specimens) decreases when oxygen is removed. For specimens with a high content of excess oxygen the value of the mobility at \(0^\circ\mathrm{C}\) lies approximately within the limits from 50 to \(100\ \mathrm{cm/sec}\) per \(\mathrm{V/cm}\); for oxygen-poor specimens, within the limits from 1 to \(50\ \mathrm{cm/sec}\) per \(\mathrm{V/cm}\). This result, surprising at first sight, may be connected with the results of the work of Hecht\(^{20}\), who found that the “mean displacement” (mittlere Schubweg) of photoelectrons in AgCl is the greater, the more strongly colored the crystal. Here one may also point to the notion developed by Schottky\(^{50}\) of the presence in the \(\mathrm{Cu}_2\mathrm{O}\) lattice of so-called acceptors (Akzeptor), serving as places of electron attachment in the case where \(R\sigma\) is regarded as the mobility.

The data of Engelhard’s work warn against any generalization from individual measurements. Unfortunately, in this work it was not possible to carry out an analytical determination of stoichiometric inhomogeneities and of the liberated \(\mathrm{CuO}\). With regard to the slight change of the constant \(E\) from excess oxygen, the conclusions

* Determined from the Hall effect and therefore independently of the mobility.

Engelhard’s agree with the data of Juse and Kurchatov and contradict the results of Waibel\(^ {68}\), who gives, however, only the temperature dependence of several differently treated specimens in the region of room temperature. Up to now it has not been possible to give a satisfactory explanation of the circumstance that the values obtained by Engelhard are twice as large as those of the Russian investigators, in the event that the experiments of the latter were carried out at a genuinely constant content of excess oxygen. Engelhard likewise finds little basis for attributing the value of the \(E\)-constant found at low temperatures to “pure \(\mathrm{Cu_2O}\)”; Wagner holds the same opinion with regard to the \(E\)-constant at high temperature. It is also essential that below \(-80^\circ\mathrm{C}\) Engelhard finds a somewhat smaller value of the constant \(E\). In connection with the study of \(\mathrm{Cu_2O}\), \(\mathrm{CuO}\), of course, deserves special attention, since cupric oxide is formed from cuprous oxide by the addition of excess oxygen, and the high electrical conductivity of \(\mathrm{Cu_2O}\) with a large content of excess oxygen is explained by the presence of \(\mathrm{CuO}\). Unfortunately, extremely little is known about this substance. Leblanc and Sachse\(^ {33b}\), from measurements on powdered \(\mathrm{CuO}\), found for the constant \(b\) first a value equal to \(0.13\ \mathrm{eV}\), and later \(0.3\ \mathrm{eV}\). These measurements were made in the temperature range from 20 to \(400^\circ\mathrm{C}\). Although the results of the measurements were well reproduced, it is nevertheless difficult to decide, as the author himself also indicates, whether the results obtained should be regarded as characteristic of \(\mathrm{CuO}\) in vacuum. What is already puzzling, first of all, is the circumstance that, with different initial material, the electrical conductivity of two specimens at room temperature differs by several orders of magnitude. Above all there is an irreconcilable contradiction with Gilley’s data\(^ {16}\); for compact material (from Shukhard) he found an electrical conductivity at room temperature more than \(10^5\) times greater; at the temperature of liquid air, however, the conductivity measured by Gilley had a value equal to that which Leblanc and Sachse determined only at \(+300^\circ\mathrm{C}\). Gilley’s specimens were not analyzed and were not measured exactly, but it is noteworthy that after grinding very hard specimens the electrical conductivity retained its value, characteristic of pressed specimens, and corresponded to the data of Leblanc and Sachse. The correctness of their data—irrespective of whether or not we are dealing here with a material constant of copper oxide—is indicated, on the other hand, by the circumstance that Wagner, for the electrical conductivity of copper foil oxidized through and through, of thickness \(10\ \mu\), and also for pressed specimens at high temperatures, obtained approximately identical values of the electrical conductivity.

Recently Kann and Treu\(^ {70}\) made an attempt at a more detailed study of \(\mathrm{CuO}\). Since it is extremely difficult to oxidize Cu in bulk to \(\mathrm{CuO}\), they used for their investigation thin layers deposited on glass. The specific electrical conductivity

obtained CuO was, at room temperature, within the range from \(1.5 \cdot 10^{-1}\) to \(1.6 \cdot 10^{-3}\ \Omega^{-1}\,\text{cm}^{-1}\). The constant \(E\) varied within the range from 0.12 to 24 eV, without, however, revealing any simple dependence on the electrical conductivity. It was shown by X-ray analysis that, in CuO layers obtained by oxidation of copper layers deposited on glass, cuprous oxide was not present in appreciable quantity. On the basis of these measurements one may conclude that CuO conducts, on the average, \(10^2\)—\(10^3\) times better than \(\mathrm{Cu_2O}\), and that the constant \(E\) is approximately 2 times smaller. However, the transition from cuprous oxide to cupric oxide with respect to electrical conductivity occurs quite smoothly. What remains unsatisfactory in these experiments is the fact that the internal cohesion of the oxide layers cannot in any way be vouched for. The formation of the oxide from the metal passes through an intermediate stage of cuprous-oxide formation; at first there is an increase in volume, and then a decrease of it. This decrease does not exclude the possibility that the CuO layer itself consists of more or less isolated microparticles. Single crystals of CuO (the mineral tenorite), of more than microscopic dimensions, have apparently not yet been made. The question is still not at all clear with regard to ZnO. Skaupy \(^{45}\), Jander and Stamm \(^{24}\), and also Wagner and Baumbach \(^{66}\) point to an increase of electrical conductivity with rising temperature; the temperature coefficient is the greater, the smaller the electrical conductivity at room temperature. The exponential dependence (0) is poorly satisfied. On the contrary, Baх \(^{2}\) found that the characteristic dependence first indicated by Königsberger \(^{30}\) occurs: the electrical conductivity reaches its highest value at \(-100^\circ\text{C}\); at temperatures above room temperature, the temperature coefficient of electrical conductivity is of the same order as that for a pure metal. On the other hand, in investigating both artificial single crystals and polycrystalline specimens in the range from \(-180^\circ\text{C}\) to \(+20^\circ\text{C}\), Frisch \(^{13}\) found that equation (0) is accurately fulfilled. The value of the constant \(E\) for samples with specific conductivity at room temperature of the order of \(1\)—\(5\ \Omega^{-1}\,\text{cm}^{-1}\) is, on the average, 0.01 eV and increases to 0.4 eV for samples with electrical conductivity \(10^{-5}\ \Omega^{-1}\,\text{cm}^{-1}\). In this case there is a connection between the value of \(\sigma\) and the constant \(E\) in the sense indicated by Meyer \(^{37}\). It should also be noted that precisely in the case of ZnO the question of contact resistances inside a polycrystalline specimen must be given greater attention \(^{16}\).

For \(\mathrm{WO_3}\), \(\mathrm{Cr_2O_3}\), CdS, NiO, \(\mathrm{UO_2}\), and some others, measurements made only on powders have so far been published, so that the values of the electrical conductivities and their temperature dependence, in our opinion, are not sufficiently reliable. Here there remains a large field for investigation. It seems to us highly desirable to carry out further, as far as possible comprehensive, investigations of CuO, in order convincingly to refute the supposition, expressed up to now, of the possibility of the existence

...formation in this case of a metallic conductor with insulating blocking layers between the individual crystalline grains.

8. Photoelectric Conductivity in Semiconductors

The change in the electrical conductivity of semiconductors under illumination has been known for a long time (selenium, stibnite, cuprous oxide, molybdenite, silver sulfide, etc.^{15c}). The importance of the photoelectric properties of semiconductors for understanding the mechanism of conductivity in the unilluminated state was first pointed out by Gudden^{15b}. He assumed that electrons bound in various ways in the lattice can dissociate either as a result of the absorption of light or as a result of the energy of thermal motion and, in this way, increase the electrical conductivity. The product of the frequency of the absorbed, photoelectrically active light and Planck’s quantum of action gives the magnitude of the energy threshold that must be overcome by an electron in order to enter the conduction band, and is expressed by the constant \(E\), or corresponds to the latter (in the case of dark conductivity caused by thermal dissociation of electrons, this threshold is overcome at the expense of thermal energy). We shall return to this question in more detail below; first let us consider the factual material.

The detection of the photoelectric effect in a semiconductor is the easier the lower the dark electrical conductivity is or, if one is speaking of a definite substance, the lower the temperature is, provided that strongly interfering space charges or layered polarization are not thereby formed. As far as we know, semiconductors behave in the same way as insulating crystals, in which the laws of photoconductivity have been well studied by Gudden and Pohl, Pohl and Hiltsch and their collaborators. Conversely, at high temperatures insulating crystals acquire the properties of semiconductors. In investigating insulators Gudden and Pohl established that electrons torn out of the lattice upon the absorption of light move in an electric field from the places of their origin toward the anode, while “holes” (Fehlstelle) move in the direction of the cathode. They designated both these phenomena as the “positive” and “negative” parts of the primary photoelectric current; at present they prefer to use more convenient expressions, namely: electronic conductivity (Überschussleitung) and hole conductivity (Ersatzleitung, Defektleitung, Löchleitung). An electron liberated upon the absorption of light, after traversing a more or less short or long path, which may consist of many “free paths,” is again captured by one of the inhomogeneities and no longer participates in the process of electrical conduction^{41}.

The mean projection, on the direction of the electric field, of such a zigzag displacement of the electron is called the “mean dis-

…placement” (mittlere Schubweg). The magnitude of the “mean displacement” is proportional to the acting field strength and varies from substance to substance and, probably, even from sample to sample. In a field of 500 V/cm the magnitude of the “mean displacement” in NaCl is approximately \(10^{-5}\) cm, while in AgCl\(^{20}\) it reaches 1 cm. So long as the “mean displacement” is small in comparison with the distance between the electrodes, Ohm’s law is fully obeyed in a photoelectrically conducting insulator; but as soon as its magnitude becomes greater than the distance between the electrodes, the current ceases to depend on the voltage (saturation). If only part of the crystal is illuminated, namely a strip parallel to the electrodes, then a current nevertheless passes through the insulating crystal; in this case the magnitude of the current is considerably smaller than it would be if the entire surface of the crystal between the electrodes were uniformly illuminated by the same amount of light. The main reason for this phenomenon is that the field strength in the illuminated strip decreases, as a result of which the magnitude of the “mean displacement” decreases and space charges are formed. This circumstance disappears in the case of additional ionic or thermally induced electronic conductivity. All experiments performed with insulating crystals can be reproduced at sufficiently low temperatures and on semiconductors (antimony glance\(^{62}\), Cu\(_2\)O\(^{8b, 28, 31}\)). Engelhard\(^{8b}\), studying Cu\(_2\)O at the temperature of liquid air, found an inertia-free increase of the current upon illumination, its proportionality to the voltage and to the light intensity, and, moreover, noted that the “mobility” of the electrons corresponds to the low temperature of the lattice, and not to the conductivity achieved.

Ioffe and his co-workers\(^{25,26}\) expected to find a difference between electrical conductivity caused by the motion of electrons in the conduction band (Überschussleitung) and electrical conductivity due to the replacement of empty places or free energy levels (Ersatzleitung or Löchleitung—“hole conductivity”), by illuminating a CuO sample with a narrow strip of light alternately at the cathode and at the anode. It turned out that the current strength at the temperature of liquid air does not depend on the sign of the field; moreover, the currents within the limits of Ohm’s law, as well as the saturation currents in both directions, are identical. On this basis it was concluded that not only is the number of electrons transferred by light into the conduction band equal to the number of “holes” formed thereby, but their mobilities are also completely identical. This conclusion seems to us unnecessary, since the work was carried out with steady-state values of the current, and the field strengths established were those that equalized any possible difference in mobility.

Even less convincing is the experiment performed with the aim of proving the existence of diffusion of electrons and “holes” from the illuminated part of the crystal into the unilluminated part. It was shown that the electrical conductivity of that part of the plate which, at \(-180^\circ\)C, was situated where light could in no case reach, increa…

increased, under illumination for about half an hour with strongly absorbed light, by approximately a factor of 100 in the part of the specimen remote from the illuminated part. Since a similar effect produces a rise in temperature of 5–10°C (when a plate situated in vacuum is illuminated), it seems to us that, most probably, the increase in the electrical conductivity of the unilluminated part of the specimen is thermal in character. It is also difficult to understand, taking into account the assumption put forward by the authors, why electrons that enter the unilluminated part of the crystal by diffusion are not immediately removed by the electric field, as happens, for example, when the illuminated part of the specimen is darkened.

At room temperature the photocurrent in Cu₂O has the same absolute value as at the temperature of liquid air, but is relatively very small in comparison with the dark current. A very thorough investigation of the photoconductivity of Cu₂O was carried out by Schönwald⁴⁹, who illuminated the specimen with modulated light and amplified the alternating photoelectric current arising in this way. It turned out that even in the presence of a large dark current the photocurrent is practically inertia-free, and its magnitude is proportional to the voltage applied to the specimen and to the intensity of the light; in this connection it is immaterial whether, with the given quantity of light, the entire distance between the electrodes is illuminated or whether it is concentrated only on some part.

Gudden assumes that the thermal dissociation of electrons and the photoelectric detachment take place in the same way. Experiment shows that the photoelectric effect constantly appears on the long-wavelength side of the lattice’s intrinsic absorption band. In the semiconductors investigated up to the present, the photoelectrically active portions of the spectrum lie in the long-wavelength ultraviolet, in the visible part of the spectrum, and in the short-wavelength infrared region approximately down to 1.5 μ. Correspondingly, the work functions have values of 4 (ZnO, SiC), 2 (Cu₂O), and approximately 0.8 eV (MoS₂, Ag₂S), i.e. considerably larger than the observed constants \(E\).

With the exception of substances for which the intrinsic absorption band of the lattice lies in the long-wavelength part of the spectrum, such as, for example, MoS₂ and Ag₂S, in other semiconductors, on the basis of what has been said above, dark conductivity ought not to be observed at all. Gudden¹⁵ᵇ therefore also suggested that such semiconductors must possess a further weak absorption in the infrared part of the spectrum, caused by the presence of impurities or inhomogeneities in the lattice. In fact this was discovered by Schönwald⁴⁹ in the investigation of the photoelectric properties of Cu₂O. In other substances there is probably likewise a region of photoelectrically active absorption lying outside the band of intrinsic absorption of the lattice. Proof of the existence of absorption caused by impurities or inhomogeneities in the lattice is much easier to obtain by the photoelectric method than directly

optical, because with such weak absorption it would hardly be possible to obtain an optically transparent specimen on which differential measurements could be carried out with any hope of success. Schenwald’s measurements, moreover, indicate a close connection between the intensity of absorption in the infrared region of the spectrum near \(1.7\,\mu\) and the dark conductivity.

Schenwald’s data confirmed Gudden’s assumption; however, the numerical value of the absorption frequency agrees poorly with the value of the constant \(E\), determined from the temperature dependence of the electrical conductivity. On this basis Gudden\(^{15c}\) suggested that the phenomenon of photoconductivity is observed not in the absorption band itself, but at its long-wave end, similarly to what Flesch and Bonner\(^{10}\) observed in studying the phenomena of phosphorescence. From the point of view of the theory\(^{69}\) it is assumed that the value of the constant \(E\) is only one half of the energy difference of the two electronic energy states (i.e. the energy for transferring an electron from the filled normal band into the conduction band). The same result is also obtained by calculation if, in this case, the electron concentration in the conducting state is regarded as the result of dynamic equilibrium between dissociation and recombination of electrons (assuming, at the same time, one band of normal levels and one band of conduction levels). Meanwhile it is known from experiment that the additional photoconductivity produced by light is directly proportional to the intensity of the light, and not to its square root, as would follow on the basis of the first assumption. In reality, in the phenomenon of photoconductivity there is nothing resembling equilibrium of this character, and the removal of electrons from the conduction band, at least in the photoeffect, occurs in some other way, and not by a direct transition to the free levels of the normal band.

Ioffe\(^{25}\) assumes that different mechanisms should be ascribed to the thermal and photoelectric dissociation of electrons. We, on the contrary, on the basis of what has been said above, conclude that the scheme of dynamic equilibrium under thermal excitation is not as simple as has been assumed up to now. If the expected relation between the position of the absorption band and the constant \(E\) actually exists, it would be very important to clarify the relation between their temperature dependences. From the work of the Göttingen school\(^{41}\) it is known that the “center of gravity” of the absorption band shifts toward shorter waves as the temperature is lowered; at the same time the width of the band also decreases. As Engelhard showed, the existence of a definite temperature dependence of the value of the observed constant \(E\) will be understandable even in the absence of a change of the absorption band with temperature; the actual change probably has still greater significance.

Unfortunately, using the Gudden—Engelhard conception

We lose the simple connection of the constant \(A\) with the number of violations of lattice regularity, since in this case the width of the absorption band has a greater influence than the total surface, proportional to the number of lattice irregularities; on the other hand, this picture clarifies a number of existing discrepancies.

Pohl and his collaborators\(^{41}\) determined the number of centers in crystals of halide compounds of the alkali metals which are sources of electrons upon absorption of photoelectrically active light, by two methods that gave excellently coinciding results, namely: 1) on the basis of the classical theory of dispersion, using the absorption constant and the half-width of the absorption band, and 2) directly by an electrical method, measuring the quantity of electricity released when the absorption disappeared. The number of centers proved to be approximately \(10^{16}\) in \(1\ \mathrm{cm}^3\). Corresponding measurements on \(\mathrm{Cu}_2\mathrm{O}\) have not so far been possible, since it was not possible to prepare a pure, transparent single crystal of this substance of the necessary thickness (approximately \(1\ \mathrm{cm}\)). In any case, one should not think that the number \(N_0\), determined by Engelhard\(^{8b}\) from the temperature dependence of the Hall effect, actually corresponds to the number of lattice irregularities; here, probably, something similar occurs to what is observed in investigations of thermionic phenomena, when the temperature dependence of the work function and the covering of the surface by atoms of a foreign substance in most cases distort the universal value of the Richardson constant \((A)\). There is hope that by using the results of experimental investigations carried out in Göttingen on various absorption bands in crystals of halide compounds of the alkali metals, as well as experiments with crystalline phosphors, it will be possible gradually to elucidate the question, still obscure at present, of energy levels in semiconductors. Of course, the need for extensive experimental work has not passed.

9. Chemical Structure and Electronic Conductivity

Up to now it has been indicated that, with great probability, electronic conductivity is not in itself a property of the substance, but is conditioned by stoichiometric or other violations of lattice regularity. On the other hand, the fact is also indisputable that only certain compounds exhibit the properties of semiconductors, while others do not. It is therefore necessary to strive to explain the different properties of individual chemical compounds by making use of the concept of inhomogeneities in the lattice. In 1925 Friederich\(^{42}\) attempted to reduce the electronic conductivity of inorganic compounds to free chemical valences. Saturated compounds should be insulators, while unsaturated ones should possess metallic conductivity. At the same time

Fig. 3. Schemes of energy levels in metals, semiconductors, and insulators.

1. Metallic conductivity (intrinsic). The uppermost band, containing electrons, band \(A\), is filled only halfway; the sum of the valence electrons is an odd number. Electronic conductivity (Überschussleitung) in \(A\). Fermi–Dirac statistics; example—alkali metals.

2. Metallic conductivity (intrinsic). The uppermost band containing electrons, band \(A\), is completely filled; the sum of the valence electrons is an even number; the nearest unfilled band \(B\) overlaps. Electronic conductivity (Überschussleitung) in \(B\); “hole” conductivity (Ersatzleitung) in \(A\). Fermi–Dirac statistics; example—alkaline-earth metals (?).

3. Metallic conductivity caused by the presence of violations of the regularity of the crystal lattice (Metallische Störleitung). The band \(A\) is completely filled; the nearest unfilled band \(B\) is separated by a band of forbidden energy values. Within band \(B\), energy levels \(N_0\) are occupied by embedded impurity atoms (inhomogeneities, etc.). In band \(B\) there is electronic conductivity. Classical statistics:

\[ \sigma = \sigma_0 T^{-\frac{3}{4}}. \]

The Hall effect gives \(n = N_0\); an example may be, perhaps, \(\alpha\)-Ag\(_2\)S.

4. Intrinsic semiconductivity of the lattice. The band \(A\) is completely filled; the empty band \(B\) is separated by an energy difference equal to \(E_g\). At very low temperature or a large value of \(E_g\)—an insulator. If \(n\) electrons out of the number \(N_0\), completely filling band \(A\), are transferred into band \(B\), then electronic conductivity appears in \(B\), and “hole” conductivity in \(A\). Since \(E_g\) is much greater than \(kT\), classical statistics hold, and

\[ \sigma = \sigma_0 T^{-\frac{3}{4}} \cdot e^{-\frac{E_g}{kT}}. \]

The Hall effect has a differential character; an example may be, perhaps, Cu\(_2\)O of stoichiometric composition. If \(E_g\) is not very large in comparison with \(kT\), then Fermi–Dirac statistics are applicable.

5. Semiconductivity with disturbed lattice regularity (Störhalbleitung). The band \(A\) is completely filled; the empty band \(B\) is separated from it by an energy difference \(E_g\); \(N_0\) embedded impurity atoms are situated at the energy level \(E_s\) below \(B\). At very low temperatures or large values of \(E_s\) and \(E_g\)—an insulator. Thermal excitation transfers \(n\) electrons from \(N_0\) impurity atoms into band \(B\); electronic conductivity appears in it. If \(E_s\) is not very large in comparison with \(kT\), classical statistics are applicable;

\[ \sigma = \sigma_0 T^{-\frac{3}{4}} e^{-\frac{E_s}{2kT}}; \]

\(n\) is determined from the Hall effect, and \(N_0\) from the temperature dependence. Example—Cu\(_2\)O with excess oxygen.

6. Schottky’s attempt to explain both signs of the Hall effect in Cu\(_2\)O by introducing the concept of so-called “catchers” (Fänger, Akzeptoren) and “donors” (Donatoren). At high temperature, owing to the transition of electrons from the completely filled band \(A\), “hole” conductivity appears in it, exceeding the electronic conductivity in band \(B\).

7. Energy levels in NaCl established by photoelectric and optical methods. Conductivity takes place only in band \(B\)

with his collaborators—Zittig and Meyer—he carried out a large number of measurements of electrical conductivity, especially on compounds that had previously been inaccessible. Recently Meyer^37 expressed the opinion that the electrical conductivity is substantially affected by an excess or deficiency of metalloid atoms, and formulated Friederich’s rule in the following form: in unsaturated compounds of the lowest valences the electrical conductivity increases with increasing excess of the metalloid \((\mathrm{CuJ}, \mathrm{Cu}_2\mathrm{O}, \mathrm{NiO}, \mathrm{CoO}, \mathrm{UO}_2)\). In saturated compounds the electrical conductivity increases with decreasing metalloid content \((\mathrm{WO}_3, \mathrm{ZnO}, \mathrm{CdO}, \mathrm{TiO}_2)\). Exceptions are also substances such as \(\mathrm{V}_2\mathrm{O}_5\), \(\mathrm{PbO}_2\), \(\mathrm{BiO}_3\), \(\mathrm{MnO}_2\), with their very high conductivity. With regard to these so-called exceptions, we adhere to the view that in this case we are hardly dealing with high conductivity caused by large disturbances of the stoichiometric composition, as Meyer assumes, but rather with a metallic conductor whose conductivity is strongly disturbed by boundary layers, as, for example, in \(\mathrm{CuS}\) and, possibly, \(\mathrm{PbS}\); in this case there may also be conductivity due to impurities and other inhomogeneities of the lattice (Fig. 3). The metallic nature of \(\mathrm{CuS}\) was discovered by Fischbeck and Dörner and confirmed by Meissner^36 on the basis of the discovery of superconductivity in it. It is precisely the presence of superconductivity that we regard as a sufficiently convincing indication of metallic conductivity, as distinct from electronic semiconductivity. Since the data concerning superconductivity of \(\mathrm{PbS}\) are contradictory^36, our assignment of \(\mathrm{PbS}\) to metallic conductors is based above all on the high value of the electrical conductivity and its temperature dependence^14. Leblanc and Sachs point out that both the highest and the lowest metal oxides do not exhibit electronic conductivity, and on this basis conclude that “strong electronic conductivity appears either when a part of the valence electrons remains chemically unbound, or when, by chemical addition of oxygen in an amount exceeding the normal valence state (Normalvalenzstufe), electrons of one of the inner groups become bound. Only in the case when all electrons of the inner group are chemically bound does strong electronic conductivity disappear again. In this case a saturated compound is formed.” On comparison with § 10 (Fig. 3) we shall see that chemists and theoreticians are speaking in completely different languages, although both conceptions are probably closely connected with one another. Here one may cite some examples given by Leblanc and Sachs: the specific conductivity was measured on powders in vacuum at \(20^\circ\mathrm{C}\) (in \(\Omega^{-1}\,\mathrm{cm}^{-1}\)) (see the table on p. 732).

Our doubts as to whether we are here dealing with a material constant have already been expressed above. Before comparing individual cases with theoretically predicted possibilities, one should await more accurate determinations of the specific electrical conductivity and of its temperature dependence.

10. Theory of Electronic Semiconductors

Independently of the experimental investigations, during the last 5 years a theory of electronic semiconductors has been developed which, in the opinion of the researchers who took part in its creation, may be regarded as formally complete, if phenomena at boundary surfaces are left aside. In reality, however, this theory gives only rough outlines, requiring completion in each individual case. In what follows we shall consider it only to the extent that it can be used for comparison with experiment. Experience will show us how broad the framework of this theory is and what changes must be made in it. For all justifications and calculations we shall refer to already published works⁴, ⁵, ¹¹, ¹⁸, ³², ⁴⁰, ⁴⁷, ⁶⁹, and above all to Pauli’s review in volume 11 of Ergebnisse der Exakten Naturwissenschaften on metallic conductivity.

Every atom possesses a definite series of possible energy values. Part of the energy levels is occupied by electrons ($K$, $L$, and other shells), while another part of them remains free. In each energy state, according to the Pauli principle, there can be no more than two electrons with oppositely directed axes of their mechanical and magnetic moments. When $N$ atoms are combined into a regular crystal lattice, each of the $N$ identical energy levels splits into $N$ distinct levels lying extremely close to one another. Thus, instead of each level there arises a band of levels, the energy width of which is the greater, the more wavelike (welligen) the course of the lattice potential. With a constant potential, any energy values would be possible. These distinct bands of energy levels, corresponding to the discrete energy levels of individual atoms, will, generally speaking, be separated by zones of forbidden energy values, but may also overlap one another; the latter, however, will not occur for energy values corresponding to the $K$, $L$, etc. shells,

but possibly at higher energy values. In each band, with its \(N\) energy levels, \(2N\) electrons can be accommodated. The transition of an electron from one state to a neighboring one, for example under the action of an electric field, is possible in an ideal lattice irrespective of the fact that to do this a higher or lower potential threshold must be overcome; the mutual distance must only not be too large, since otherwise the probability of transition will be very small. It is assumed here, on the basis of Pauli’s principle, that the level to which the electron passes is not yet occupied. Consequently, a \(K\)-electron can move in the lattice if such an electron is absent at another place in it. Since, however, all the lowest energy bands are filled (with 2 electrons at each level), conductivity there is impossible. The decisive circumstance in this respect is the degree of filling of the higher bands of energy levels, in which there are electrons, and also the position of the further energy bands.

If each atom has one weakly bound electron, then only half of all the places in the highest, electron-containing band \(A\) will be occupied, and the electric field will be able to transfer electrons into neighboring states (the most characteristic example of metallic conductivity) (Figs. 3, 4). If, conversely, the number of valence electrons is even, then metallic conductivity can appear only in the case when the next allowed energy band overlaps (Fig. 3, 2). In the case when it does not overlap, i.e., if above the highest energy band \(A\), occupied by \(2N\) electrons, there lies a forbidden energy zone, then conductivity is impossible; in this case we have a typical insulator. All this applies to the temperature of absolute zero. For the transfer of an electron by an electric field it is required that the corresponding energy be accumulated over the distance of one whole mean free path; with an energy difference of \(1\ \mathrm{eV}\) and a mean free path of \(10^{-6}\ \mathrm{cm}\) (at ordinary temperature) a field strength of \(10^6\ \mathrm{V/cm}\) would be required.

At a temperature different from absolute zero, the transition of electrons from the completely filled band \(A\) into the higher empty band \(B\) may occur at the expense of thermal energy; in this case an equilibrium is established which obeys the usual Maxwell statistics as long as the number of these electrons is small in comparison with the number \(N\) (Figs. 3, 4). The law of effective masses and the relations of classical statistics give:

\[ n = N^{1/2} T^{1/2} e^{-\frac{E_g}{2kT}}, \tag{4} \]

where \(E_g\) is the energy difference between the highest level of the filled band \(A\) and the lowest level of the next allowed empty band \(B\). If \(n\) is very small in comparison with \(N\), then to the motion of electro-

in band \(B\) one must apply classical statistics, and, as Sommerfeld, Bloch, Wilson, and others have shown, in this case the well-known equations of the Drude—Lorentz electron theory are obtained. In particular, the mean thermal velocity of the electrons is proportional to \(T^{1/2}\). On the basis of the ideas of wave mechanics, at not very low temperatures the mean free path should be proportional to \(T^{-1}\), and, moreover, it follows that the conductivity is due not only to the motion of electrons in the upper, almost empty band, but also to the displacement by unoccupied electrons of the levels—the motion of empty places—in the initially completely filled lower band of quantum states; this so-called hole conductivity behaves exactly as if it were due to the motion of positive electrons \(^{21}\). Therefore the “hole” conductivity should give a Hall effect of the opposite sign (anomalous), and the observed Hall electromotive force will be the difference of the electromotive forces due to the motion of electrons and of free levels (“holes”) in a magnetic field. In this case Peierls gives the following expression for the Hall constant:

\[ R=-\frac{3\pi}{8}\cdot \frac{1}{\varepsilon}\cdot \frac{u_1^{\,2} n_1-u_2^{\,2} n_2}{u_1 n_1+u_2 n_2}, \tag{5} \]

where \(u\) is the mobility; \(n_1\) and \(n_2\) are the numbers of electrons per unit volume and of free levels (“holes”), respectively. Such conductivity, which appears in a crystal lattice with an undisturbed structure only under the influence of thermal excitation, we call the “intrinsic semiconductivity of the lattice” (Eigenhalbleitung), in contrast to the conductivity considered below, which is due to the presence in the lattice of inhomogeneities and impurities (Störhalbleitung).

If the energy difference \(E_g\) is not very large in comparison with \(kT\), then \(n\) is comparable with \(N\), and in this case one must use Fermi—Dirac statistics. We shall not dwell here in greater detail on these complicated questions.

Wilson and others have considered theoretically, on the basis of Gudden’s assumptions, the case in which the conductivity is due to the presence of impurities and inhomogeneities in the crystal lattice.

Thus, let us suppose that within the crystal lattice there is distributed a small number of foreign atoms (impurities) \(N_0\). The arrangement of the energy levels of the lattice is not thereby changed, but now the electron of each of the impurity atoms will have such a value of energy that it falls into the zone of forbidden energy levels of the lattice (Fig. 3, b). Since the impurity atoms are situated at large distances from one another, no conductivity whatsoever is possible by itself even in the case when the energy level occupied by the electrons of the impurity is not completely filled; the energy of the impurity electrons is usually greater than the energy of the normal levels of the lattice; to transfer them from

it is necessary to expend considerably less work to transfer an impurity into the band of conduction levels of the crystal lattice. Since the number of electrons dissociating from impurity atoms, under all circumstances, remains insignificant (a small number of impurity atoms), in this case classical statistics is applicable, and for the number of conduction electrons we obtain the former relation (4), only with \(E_s\) in place of \(E_g\) and \(N_s\) in place of \(N\); here one must remember that \(E_s\) represents the work required to transfer an electron of an impurity atom from the highest level occupied by it into the band of conduction levels of the lattice itself (\(B\)). The case just considered differs from the preceding one in two respects: 1) there is no so-called hole conduction, and the Hall constant must give the correct value of \(n\); 2) even in the case when \(E_s\) is comparable with \(kT\), classical statistics is applicable, since \(N_0 \ll N\).

In the case when the levels filled by the electrons of impurity atoms are located in the band of allowed, but empty, levels of the lattice itself (Fig. 3, 3), they acquire the possibility of taking part in electrical conduction even without thermal excitation; we then observe an example of metallic conduction caused by the presence of impurity atoms and other disturbances in the lattice (metallische Störleitung), and in this case the number \(n\) does not depend on temperature, but is small in comparison with \(N\). Classical statistics is applicable, and the Hall effect has the correct sign. In Fig. 3 the various possibilities are presented in a clear comparison.

Considering now simultaneously all the possibilities that have been analyzed, one can imagine what variety of phenomena is accommodated within the framework of the developed theory. It seems to us, however, that reality is much more complicated, and that in a crystal lattice there exist considerably more energy steps than has been assumed up to the present. This can be seen, for example, from the detailed experimental works of the Göttingen physics school devoted to the study of energy levels in the halide compounds of the alkali metals (Fig. 3, 7). Schottky\(^{50}\) indicates that in \(\mathrm{Cu_2O}\), along with sources of electrons (Donatoren), one should suppose the existence also of so-called electron traps (Akzeptoren), and he explains the change of sign of the Hall effect in \(\mathrm{Cu_2O}\) at \(400^\circ\mathrm{C}\) by the ensuing predominance of “hole” conduction over electronic conduction. The arrangement of energy levels in \(\mathrm{Cu_2O}\) assumed by him is shown in Fig. 3, 6. At a higher temperature, electrons from impurity atoms and other inhomogeneities enter the conduction band \(B\) of the lattice; in addition, electrons from the levels of the filled band \(A\) of the lattice are torn out by thermal motion and captured by inhomogeneities; at the same time, in the normal band a portion of the levels is freed, and “hole” conduction appears in it (the displacement in an electric field of levels unoccupied by electrons, the motion of empty places in a band of quantum states completely filled with electrons, corresponding to the motion

positive charges; with respect to semiconductors these concepts were first introduced by M. P. Bronstein), which in this case outweighs the electronic conductivity in band \(B\). For further details we refer the reader to the original paper\(^ {50}\). That “traps” of this kind have a considerably greater significance and must necessarily be introduced into the theory of the phenomena of semiconductivity can also be concluded on the basis of the study of photoconductivity (the concept of the “mean displacement”—mittlere Schubweg). In the event that such “traps” play a role also in dark conductivity, the relation given by the theory,

\[ E=-\frac{E_s}{2} \]

must be replaced by

\[ E=-E_s. \]

This question can, of course, be resolved only by experiment.

In conclusion it is necessary to note that the theory (as it has been set forth here) has been developed for an infinitely large crystal lattice, whereas one of the decisive factors is undoubtedly the properties of the boundaries. Photoelectric conductivity is not observed in a microcrystalline pressed specimen, evidently only because the electronic and “hole” conductivities are impeded by the faces of the lattice. The poor agreement of measurements of the dark electrical conductivity of powders also confirms the great role of the faces and surface properties of the crystal.

11. Conclusion

From the present survey, it seems to us, it follows quite plainly that, with regard to a real understanding of the electronic semiconductor, we are only at the very beginning. This applies both to the experimental and to the theoretical aspects of the question. It will also apply to the theory even if one considers the solution of the problem in broad outline to be correct; before it lies the task, on the basis of the already known energy levels of the atom, the structure of the lattice, and the positions of the allowed and forbidden energy states, of predicting the type and magnitude of the electrical conductivity, and not merely of giving explanations after the fact. This also applies to the theory of electronic conductivity in a lattice containing impurities and inhomogeneities. With regard to the experimental side of the question, it is above all necessary to accelerate the accumulation of factual material, considerably more reliable than that which we possess at present, and for every case required by theory to find examples and compare them with the predicted ones. In doing so, all already known sources of error should be re-examined and new ones sought. Naturally, the investigator has every right to put forward a working hypothesis on the basis of individual observations; however, precisely in questions of semiconductivity, owing to the exceptionally complex interplay of causes, only extensive series of observations can help the matter. The experimental physicist, the theoretical physicist, the chemist, and the physical chemist must work in close

contact with one another; only in this case will success be assured. The study of semiconductors plays an enormous role in the understanding of the solid state.

Over the last 10 years our knowledge in the field of the phenomena of semiconductivity has considerably exceeded the level outlined in Königsberger’s review of 1914, but we are still very far from a final solution of the problem. At present it is more convenient to speak of the aim than of the results.

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10. — and H. v. Baumbach, ibid. 22, 226, 1933.

11. — and H. v. Baumbach, ibid. 22, 199, 1933.

66a. —, ibid. 24, 59, 1934.

67a. Wagner C., ibid. 21, 42, 1933.

67b. —, ibid. 22, 181, 1933.

1. Waibel F., Wiss. Veröff. Siemens-Konz. 10, 65, 1931.

2. Wilson A. H., Proc. Roy. Soc. Lond. (A) 133, 458; 134, 277, 1931; Actual. Sci. et Industr. 89, H. 2, 1933.

3. Kapp G. and M. Trou, Erlang. Ber. (in press), see also Ann. d. Physik, 22, 257—281, 1935.