Semiconductors Sensitive in the Infrared Region of the Spectrum*

R. A. Smith

Contents

§ 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
§ 2. Semiconductors of the lead sulfide group; thin layers and other polycrystalline formations . . . . . . . . . . . . . . . . . . . . . . 434
§ 3. Semiconductors of the lead sulfide group; single crystals . . . . . . 451
§ 4. Other semiconductors sensitive in the infrared region . . . . . . . . 463
§ 5. Comparison of substances of the lead sulfide group with germanium and silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . 466
§ 6. Theory of photoelectric phenomena in substances of the lead sulfide group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
§ 7. Photoresistors as receivers of infrared radiation . . . . . . . . . . 474

§ 1. Introduction

In the field of infrared spectroscopy, over the last thirty years—and especially after its significance for organic chemistry was understood—remarkable advances have been achieved. Nevertheless, from the beginning of this century until very recently, very little was done with regard to the development of receivers of infrared radiation for the wavelength range from \(\sim 1.5\) to \(\sim 10\,\mu\) (which is of greatest interest to us here). During this period, receivers suitable for the near infrared region (\(\sim 0.75 \div 1.5\,\mu\)) were created and were also widely used for recording visible radiation (thallous sulfide photoresistors, silver–oxygen–cesium photocells), and the sensitivity limit of photographic materials was shifted somewhat into the long-wavelength part of the spectrum. But until the appearance of lead sulfide photoresistors, for wavelengths exceeding \(\sim 1.5\,\mu\), there existed no other

* Advances in Physics 2, 321, 1953, translated by N. S. Khlebnikova.

methods of detecting and measuring radiation, apart from thermal ones, with their inherent shortcomings of low output and great inertia.

Lead sulfide photoresistance substantially, approximately to \(4\mu\), extended the region of the spectrum accessible to photoelectric recording, and quite recently lead telluride and lead selenide photoresistances have shifted this limit almost to \(10\mu\). The use of new receivers has given a strong impetus to the development of infrared spectroscopy.

A review of recent work on the solar spectrum was given by Goldberg[^1], whose article well illustrates the application of new methods using PbS photoresistances.

The recently published works of Thompson and his collaborators[^2],[^3] on the fine structure of infrared spectra clearly show what can be achieved with the aid of PbTe receivers. In addition to reducing the setting time to a few microseconds, the use of these receivers makes it possible to obtain at least a hundredfold increase in sensitivity in comparison with thermal indicators (see § 7.2). It is precisely these circumstances—an increase in sensitivity and a decrease in the time constant—that have led to major successes in the field of infrared spectroscopy in recent times.

A popular review of the development of apparatus for infrared spectroscopy over the last fifty years and, in particular, of new photoresistances was given by Strong[^4]. Detailed reviews of the new measuring technique and the types of spectra investigated up to 1947 were published in 1948 by Williams[^5], and also by Sutherland and Lee[^6]. The present review is devoted to new receivers as such, and to the physics of semiconductors, but not to questions of their application.

§ 2. SEMICONDUCTORS OF THE LEAD SULFIDE GROUP; THIN LAYERS AND OTHER POLYCRYSTALLINE FORMATIONS

Semiconductors possessing photosensitivity in the infrared region of the spectrum, such as PbS, form a very interesting group of semiconductors and deserve thorough study irrespective of the fact that they are used as materials for making photoresistances. They present an interesting contrast to elemental semiconductors (for example, germanium and silicon), although in many respects they are similar to them; in particular, they possess a comparatively narrow band of forbidden energy values between the filled band and the conduction band. Like elemental semiconductors, they may have \(p\)- and \(n\)-type conductivity, but they differ from the former in their optical properties.

Until recently these substances could be studied only in polycrystalline form, if one does not count very impure specimens of natural lead glance (galena). Since these

materials were used in photocells in the form of sublimed or chemically deposited thin layers, most early works examined precisely polycrystalline specimens. One of the main advantages of thin layers as objects of study is that they readily allow the introduction and removal of impurities by exposing the material to an appropriate gaseous medium.

The principal characteristics of any semiconductor are the width \(\Delta E\) of the forbidden energy gap between the filled band and the conduction band, and also the nature, density, and distribution of impurity levels between these bands as a function of temperature. In addition, it is of great interest to know the mobility of electrons and holes, as well as their effective mass. The measurements that can be carried out to obtain this information in the case of sublimed layers are very limited, and it must be said that very little had been done in this direction until it became possible to obtain pure single crystals. Certain features established in the study of polycrystalline specimens are of interest, which is why we shall briefly consider the corresponding measurements.

Until very recently, the principal method of investigation was the study of the change in resistance and thermoelectromotive force as functions of temperature. Assuming that the mobility of carriers does not vary too rapidly with temperature \(T\), one should expect that the resistance \(R\) will vary as

\[ R \simeq A \exp\left(\frac{\varepsilon}{2kT}\right). \tag{1} \]

In the case of intrinsic conductivity, \(\varepsilon\) corresponds to the width of the forbidden gap \(\Delta E\) between the filled band and the conduction band; in the case of impurity conductivity, \(\varepsilon\) approximately corresponds to the depth of the impurity levels below the conduction band (conductivity of \(n\)-type) or above the filled band (conductivity of \(p\)-type). Early measurements of the dependence of resistance on temperature had the purpose of determining \(\Delta E\) in the indicated manner. The sign of the thermoelectromotive force was usually regarded as an indication of the predominant type of conductivity. A number of difficulties are associated with such a procedure. At present it is well known that the sign of the thermoelectric emf is not a reliable indicator of the character of the conductivity; consequently, there is no convincing way to determine from the data of earlier measurements exactly what the quantity \(\varepsilon\), obtained from an equation of type (1), means. There is also a sufficient amount of evidence that, in the main, the change in the resistance of sublimed layers with temperature is due to the presence of barriers between the small crystallites forming such a layer (this question will be considered below). To avoid this difficulty, experiments were carried out not only with thin films, but also on massive polycrystalline specimens

and with crystals of natural PbS. The crystals available to the first investigators apparently contained a large amount of impurities and gave sharply differing results. When the data of such measurements were plotted as \(\ln R\) as a function of \(1/T\), the straight line expected according to equation (1) was generally not obtained. Curves with a positive slope were obtained, corresponding to values of \(\varepsilon\) in the range from \(0.3\) eV to zero; moreover, for some samples the slope even became negative. The slope could be varied over wide ranges by various treatments, including annealing in oxygen or in vacuum. These early works, especially those of Bauser\(^7\), Eisenmann\(^8\), and Hintenberger\(^9\), established that the substances of the group formed by PbS, PbTe, and PbSe are semiconductors that can exist as samples belonging both to \(n\)- and to \(p\)-types, that it is possible to effect a transition from one type of conductivity to the other by suitable treatment, and also that they can behave as semimetals. However, no reliable data on the magnitude of \(\Delta E\) were obtained, although it was believed that for PbS it is about \(0.3\) eV (see 3.2).

In addition, by measuring the Hall constant at room temperature, Hintenberger\(^9\) obtained values for the mobility of the carriers. These quantities ranged from 50 to 1 cm/sec per V/cm. As is now known, these values are 20–1000 times smaller than the mobility values found for these substances in single-crystal form (see 3.2). These early works were considered by Chesmer and Putley\(^ {10}\), who suggested that the change of resistance with temperature in polycrystalline samples, and especially in thin layers, is predominantly a consequence of changes in the resistance of the contacts between individual crystallites, whereas the change in the resistance of the crystals themselves is relatively small. These authors also found, and this was later confirmed by studies carried out on single crystals, that less pure (massive) samples behave as semimetals rather than as semiconductors, i.e., they possess a positive temperature coefficient of resistance.

2.1. Structure of thin layers

The most recent studies of sublimed layers have led to a better understanding of their nature. When examined under a powerful microscope, the layers used in photoresistors (having an average thickness of about \(1\ \mu\)) appear as a series of “islands” formed by small crystals whose size in most cases ranges from 1 to \(0.1\ \mu\). This was confirmed by electron-diffraction measurements carried out by Wilman\(^ {11}\). The crystals forming the layer have the lattice parameters of pure PbS, etc., within the accuracy of the measurements. An interesting observation, which may be of great importance for

of the photoconductivity theory of such layers consists in the fact that, in layers treated with oxygen, traces of the compound PbO·PbSO₄ appear. Measurements carried out at Purdue University confirm this and indicate that the formation of PbO·PbSO₄ takes place only on the surface of the crystals, since this compound is detected on electronograms and is absent on X-ray photographs*).

2.2. Resistance of Thin Layers

In establishing the polycrystalline structure of such layers, it is desirable to determine what role, in determining the dependence of the resistance of the layers on temperature, is played by the resistance of the contacts between crystals. For this purpose, Chesmer¹² carried out measurements of the resistance of layers of various types at frequencies up to 60 Mc, which showed that at frequencies above 100 kc the resistance of the layers rapidly decreases, reaching a more or less steady value at a frequency of about 60 Mc. Similar measurements were performed on PbS by Kolomiets¹³. These results were interpreted to mean that at high frequencies the resistance of the intercrystalline contacts is shunted by their own capacitance, so that the residual resistance is the resistance of the massive material forming the layer. If this is so, then it follows that the material of the layer as such accounts for less than 1% of the total direct-current resistance of the layer. It further turned out that the temperature coefficient of resistance decreases with increasing frequency and can even change sign and become positive, whereas in direct-current measurements it is negative. All this seems to indicate that intercrystalline barriers are responsible for the greater part of the temperature changes in resistance. From this, in turn, it follows that equation (1), together with the corresponding graphs, in the case of such layers cannot provide information concerning the magnitude of \(\Delta E\), nor about the distribution of impurity levels in the massive material.

The interpretation indicated, however, was criticized by Rittner and Grays¹⁴, who showed that a considerable decrease in the resistance of such layers at high frequencies should already be expected as a consequence of the presence of the distributed capacitance of the film. Recent measurements by Tempry, Lemmis, and Scanlon¹⁵ at frequencies up to 200 Mc showed that, in order to explain the change of resistance with frequency, it is necessary to take into account both distributed and intercrystalline capacitances.

There are, however, other data as well that testify to the presence of intercrystalline barriers. Thus, for example, Simpson¹⁶ showed that if an evaporated layer is compressed or stretched by 0.01%, a reversible decrease or increase in resistance of the instrument is obtained—

*) Private communication from Dr. Lark-Horovitz.

approximately by 5%. Assuming that the distance between crystallites is a linear function of temperature and that the passage of electrons through the intercrystalline barriers is due to a tunneling effect, he obtained a theoretical expression for the change in the resistance of the layer with temperature, in very good agreement with his observations. A similar model, in which conduction between crystallites occurs as a result of thermal excitation, was discussed by Cesmer and Putley¹⁰. In this way they were able to explain the temperature changes of resistance for a large number of evaporated layers.

There can hardly be any doubt that most of the resistance of evaporated films, at least at room and lower temperatures, is due to intercrystalline contacts, since, in addition to the data cited, there are many other data supporting this view. For example, Starkiewicz, Sosnowski, and Simpson¹⁷ observed distinct rectification effects as a result of the “polarization” of evaporated layers by passing a current through them at temperatures of about 250°C. When such layers are examined with a light probe, clearly pronounced photovoltaic effects are also observed, varying arbitrarily from point to point (Sosnowski, Suł, and Starkiewicz¹⁸). There is also convincing evidence obtained from the study of the photoconductivity of such layers (see below). As to the nature of these contacts, it has still not been clarified. This question is discussed in 6.1.

When a layer of pure PbS about \(1\mu\) thick is obtained by evaporation in a high vacuum, it has a relatively low resistance, corresponding to the resistivity of the bulk material, i.e. about \(0.01\ \Omega/\mathrm{cm}\) (at room temperature). When traces of oxygen are introduced, the resistance increases extremely rapidly, up to a factor of 1000, which may be due to the formation of intercrystalline contacts with high resistance. Further annealing of the film in oxygen usually leads to a further increase in resistance, and then, under certain conditions, to a decrease in it. This process was investigated by Sosnowski, Starkiewicz, and Simpson¹⁹, as well as by Schwartz²⁰. The former showed that, under certain conditions, a layer which initially had \(n\)-type conductivity (according to thermoelectric-power measurements) gradually acquires \(p\)-type conductivity; in this case the resistance of the layer passes through a maximum when the thermoelectric power passes through zero. This does not always occur, and oxidation of the layer may be accompanied by a continuous increase in resistance. The influence of oxygen on the resistance of PbS films was also investigated by Ehrenberg and Hirsch²¹ with the aid of an apparatus permitting such a rapid change in the temperature of the specimens that changes in their structure during the measurements could not occur. In interpreting the results of their measurements, these authors completely ignored the influence of intercrystalline barriers. Quite recently, by means of unsteady-

of direct comparison of single crystals and sintered polycrystalline PbS samples, Putley^22 showed that, with the exception of the high-temperature region (above 600° K), measurements of the temperature variation of resistance give very different results. This work apparently establishes definitively that (except in the high-temperature region) intercrystalline contacts play an important role in the temperature changes of the resistance of the layers under consideration.

2.3. Measurements of the Hall constant for evaporated layers

Since Hintenberger^9 first determined the Hall constant by an electrometric method and established that the mobility of electrons and holes in evaporated PbS layers and other substances is of the order of 1–50 cm/sec per V/cm, very few investigations in this direction have been carried out up to the present time. The method used by Hintenberger is poorly suited to high-resistance layers and, moreover, when using it, it is difficult to avoid other thermomagnetic effects. An alternating-current method, which makes it possible to eliminate some of these difficulties, has recently been used by Loferski^23 and Halvorsen^24. Another method was developed by Russell and Uehling^25. It is based on the use of different frequencies for the magnetic field and for the current passing through the specimen, the Hall potential difference being observed at the sum or difference frequency. These new measurements showed that films can be converted from \(n\)-type to \(p\)-type and back by treatment in oxygen and in vacuum, and also that they can change the type of conductivity depending on the temperature regime. Determination of the mobilities gave values of the same order as in Hintenberger’s measurements. Interpretation of these measurements is difficult because of the influence of intercrystalline barriers. However, Putley^26 recently showed that polycrystalline sintered samples give Hall-constant values close to those obtained for single crystals of comparable purity. There is reason to hope that measurements of the Hall constant for evaporated layers may provide valuable information concerning the properties of the crystals forming the layer, whereas measurements of resistance, as we have seen above, are unlikely to be useful in this respect.

2.4. Measurements of resistance and the Hall constant for massive polycrystalline samples

Since the time of the first measurements of resistance and the Hall constant, carried out by Baer^7, Eisenmann^8, and Hintenberger^9 on polycrystalline samples obtained by cooling a melt, very little has been done up to the present time. Early measurements

showed that such specimens have a much lower specific resistance at room temperature than evaporated layers, namely of the order of \(0.1 \div 0.001\ \Omega\cdot\text{cm}\). In some cases these specimens have negative temperature coefficients of resistance and may possess conductivity of both the \(n\)- and \(p\)-types, as follows from measurements of the Hall constant.

Along with this, positive temperature coefficients of resistance were also often observed, so that for a time the question arose whether these substances as such are indeed bulk semiconductors (since the observed cases of negative temperature coefficients could be due to the properties of intercrystalline barriers). At present it is known that such behavior is characteristic of specimens with impurities and that pure specimens actually behave as semiconductors. The fact that, from the slope of the curves of the dependence of \(\ln R\) on \(T^{-1}\) obtained in such measurements, it was not possible to obtain clear data on the intrinsic conductivity, as is now understood, was also a consequence of the insufficient purity of the specimens investigated. The first clear data on the intrinsic conductivity of materials of this group were obtained by Czemerys and Putley\(^{10}\) for PbTe. These authors studied the temperature variations of the resistance and determined the Hall constant for a number of highly pure polycrystalline specimens in the temperature range from \(100^\circ\) to \(300^\circ\) K, and specimens were investigated which had (at room temperature) both one and the other type of conductivity. At temperatures below \(600^\circ\) K, the resistance and the Hall constant varied greatly from specimen to specimen, but at higher temperatures almost identical values were obtained for all specimens. The curves of the temperature variations of the conductivity \(\sigma\) and the Hall constant \(R\) are shown in Figs. 1 and 2. The portions of the curves pertaining to high temperatures were interpreted as due to intrinsic conductivity. In this temperature range both the resistance and the Hall constant are inversely proportional to \(n_e\), the number of conduction electrons, which is practically equal to the number of holes. Since, according to Mott and Gurney\(^{27}\),

\[ n_e \simeq \mathrm{const}\cdot T^{3/2}\exp\left(-\frac{\Delta E}{2kT}\right), \tag{2} \]

the slope of the logarithmic straight line gives \(\Delta E\). From Figs. 1 and 2 it can be seen that the slopes of both curves in the high-temperature region are indeed almost constant. The curve for the Hall constant leads to the value \(\Delta E = 0.6\ \text{eV}\). That this value is approximately correct was confirmed by measurements on single crystals (see 3.2). In addition, Putley\(^{28}\) recently showed that, in the temperature region corresponding to intrinsic conductivity, sintered PbS specimens give approximately the same value of \(\Delta E\) as single crystals. However, the resistance values at lower temperatu-

...are very different for these two types of specimens. Thus, it turns out that resistance measurements carried out on polycrystalline specimens may give misleading results. Fortunately, measurements of the Hall constant for both types of specimens lead to almost identical results, and one may

Fig. 1. Conductivity of PbTe ($\sigma$ is given in $\Omega^{-1}\,\mathrm{cm}^{-1}$, $T$ in °K).

Fig. 2. Hall constant for PbTe (given in $\mathrm{cm}^3/\mathrm{coul}$, $T$ in °K).

think that they give a more reliable value of $\Delta E$. It should be noted that the value of $\Delta E$ obtained from Fig. 1 differs somewhat from the value obtained from Fig. 2, which is probably due to the variation of mobility with temperature (which enters into the variation of resistance as a first-order effect). This circumstance gives yet another reason to expect a more reliable value of $\Delta E$ from measurements of the Hall constant. For PbS, the $\Delta E$ found by the indicated method is about $1.2\,\mathrm{eV}$ (for more detail on measurements on single crystals see 3. 2). We may also note here that for PbSe $\Delta E = 0.5\,\mathrm{eV}$.

The Hall constant for the temperature region corresponding to intrinsic conductivity proves to be negative. This shows that the conductivity is due predominantly to electrons, whence it follows that the mobility of electrons is greater than the mobility of holes, since in this temperature region the numbers of the former and the latter are almost equal. The mobility values found from these experiments are of the order of 1000 cm/sec per V/cm, i.e., they prove to be 100–1000 times greater than those obtained from experiments with layers produced by sublimation.

2.5. Absorption of thin films in the infrared region of the spectrum

Another method for obtaining data concerning the magnitude of the forbidden energy interval \(\Delta E\) and, possibly, the distribution of impurity levels is the observation of the absorption spectrum of the corresponding semiconductor. This method, although useful for confirming data obtained by electrical measurements (see above), cannot by itself give unambiguous results, owing to the impossibility of determining exactly which electronic transition is associated with a given absorption band or a given absorption edge. As in most semiconductors, the optical absorption of substances of the PbS group in the visible and near-infrared regions is very large (absorption coefficient \(10^5—10^6\ \mathrm{cm}^{-1}\)). Therefore most measurements for these regions of the spectrum have been carried out on sublimed or chemically deposited films with thicknesses of the order of \(0.1—10\ \mu\). The most extensive measurements were made by Gibson\(^{29}\), who investigated absorption in films of PbS, PbSe, and PbTe prepared by various methods. He also investigated the dependence of absorption on temperature.

The absorption spectrum of a typical PbS film at room temperature is shown in Fig. 3. It can be seen that, as the wavelength decreases in the region around \(1.0\ \mu\), the absorption rises rapidly and reaches values of the order of \(10^6\ \mathrm{cm}^{-1}\) in the near ultraviolet. This “edge” of the absorption band is associated with transitions of electrons from the filled band to the conduction band. If this is indeed so—and the large values of the absorption coefficient confirm this—then the value of \(\Delta E\) should correspond to a quantum energy of about \(1.2\ \mathrm{eV}\). It is always difficult to obtain an exact value of \(\Delta E\) from measurement results of this kind, since the absorption edge is rarely sharp. Nevertheless, the indicated value agrees well with the value \(1.2\ \mathrm{eV}\), found by Putley from measurements of the Hall constant and conductivity (see 2.4).

Adjoining the long-wavelength end of the absorption edge is a long “tail,” extending to \(6\ \mu\), with a relatively high coeffi-

with an absorption coefficient of the order of \(10^4\ \mathrm{cm}^{-1}\). Beyond \(6\,\mu\) the absorption is much smaller, but experiments of this kind cannot give its magnitude. It must be admitted that, up to now, no really satisfactory explanation has been given for absorption in this spectral region, which, as we shall see below, cannot be connected either with transitions leading to photoconductivity or with any other known transitions. Moreover, it turns out that such absorption is almost absent in single crystals (see 3.3). It is possible that it is caused by scattering of the radiation by the microcrystals forming the layer.

Fig. 3. Absorption spectrum of a chemically deposited PbS layer.

Upon lowering the temperature, the absorption edge lying near \(1\,\mu\) is shifted somewhat toward longer wavelengths, the magnitude of the shift corresponding to an energy change of about \(5 \cdot 10^{-4}\ \mathrm{eV}/^\circ\mathrm{K}\). At the same time the boundary of the “tail” shifts toward shorter wavelengths and at a temperature of \(77^\circ\mathrm{K}\) approaches almost \(3\,\mu\) (Fig. 4).

Very similar curves of optical absorption and of its temperature changes were found for PbSe and PbTe layers. In the case of these materials the beginning of the absorption edge is not as sharply expressed as in PbS. It lies in the region of about \(1.5\)—\(2\,\mu\), which corresponds to quantum energies of \(0.8\)—\(0.6\ \mathrm{eV}\), i.e. it proves to be somewhat higher than the values found by Putley (see 3.2), but still sufficiently close to the latter for it to be considered that

optical absorption associated with the edges of the bands, and in these cases corresponds to transitions from the filled band into the conduction band. For PbSe and PbTe the long-wavelength “tails” extend beyond 12 μ; moreover, for the purest specimens the absorption in the tails is characterized by very high coefficients—up to \(10^5\ \mathrm{cm}^{-1}\). When oxygen is introduced into such layers (by annealing in the appropriate atmosphere), the absorption in the “tails” decreases, but not by more than approximately one-half of its maximum value. This treatment, however, has no effect on the absorption in the region of the short-wavelength side of the absorption edge. It should be noted here that, in the indicated layers, at wavelengths corresponding to the long-wavelength boundary of photoconductivity, no changes in absorption were observed (see 2.6).

Fig. 4. Change in the absorption of a PbS layer with temperature.

2.6. Photoconductivity of thin layers

Because of the great practical importance of the photoconductivity of substances of the PbS group under the action of infrared radiation, this property has been studied more than all the others. Nevertheless, it must be admitted that, as a result of such investigations, we have learned very little about the fundamental properties of this group of semiconductors.

Early work carried out in Germany and the USA showed that these substances are unique in that they exhibit quite distinct photoconductivity at much longer wavelengths than any other of the known photoconductors. It was known, for example, that PbS is photoconducting approximately up to 3.5 μ at room temperature and approximately up to 4.5 μ at the temperature of liquid air. It was assumed that PbTe and PbSe are photosensitive approximately up to 6 μ (at the temperature of liquid air). In recent years, a considerably more detailed study of the photoconductivity of these substances has been carried out over a wide temperature interval. The results found for PbS, PbTe, and PbSe will be given below.

2.6.1. Photoconductivity of PbS.

The most extensive recent investigations of the photoconductivity of PbS were carried out by Mos-

son \(^{30,31}\). Figure 5 shows the course of the spectral-sensitivity curve for an isoenergetic spectrum. The main feature of the curves presented is that, at wavelengths exceeding a certain value, the response falls rapidly; nevertheless, it is impossible to indicate a value of \(\lambda\) at which the response could be regarded as equal to zero. In practice, for some value of the wavelength the photoconductivity becomes undetectable; however, this is determined only by the properties of the measuring device, whose sensitivity limit is determined by fluctuation noise and can be lowered by narrowing the band of transmitted frequencies.

Fig. 5. Spectral response of PbS at different temperatures.

If the dependence of sensitivity on wavelength is plotted on a semilogarithmic scale, as was done in Fig. 5, then the falling part of the curve proves to be almost linear. This circumstance is very characteristic of such curves, and if it is absent, this usually means that the sensitivity of the device is insufficient for correct recording of the curve. It is desirable to be able to establish a definite “long-wavelength limit” of sensitivity for photoconductivity. Various methods have been proposed for this purpose, but the most widespread at present is the determination of this quantity proposed by Moss \(^{32}\), according to which the long-wavelength limit is taken to be that wavelength at which the response falls to one half of its maximum value (before the beginning of the fall of the curve). This wavelength is denoted by \(\lambda_{1/2}\), and, according to Moss, has a definite physical meaning from the point of view of comparing values

of the different kinds of energies associated with transitions leading to photoconductivity.

In Fig. 5 one can also see another very characteristic feature of the dependence of the yield on wavelength. As the temperature is lowered, the long-wavelength limit shifts toward longer wavelengths, which is opposite to the corresponding changes observed for most other photoconductors. It has been established that, over a wide temperature interval, this shift corresponds to a change in energy equal to \(4.7 \times 10^{-4}\) eV/°K.

At one time it was thought that the value of \(\lambda_{1/2}\) depends strongly on the degree of purity of the photoconducting layer\({}^{33}\). However, the experience accumulated in the field of sensitive layers used as photoresistances shows that the purity of the material has little effect on \(\lambda_{1/2}\). It is now accepted that certain sharp changes in \(\lambda_{1/2}\) observed earlier occurred because the photoconductivity near this wavelength value falls below the level accessible to detection, while remaining high at shorter wavelengths. This phenomenon is observed in layers that were either insufficiently sensitized with oxygen (or by another treatment), or strongly oxidized. Thus, for example, photosensitivity curves recently published for very different layers, including those obtained by sublimation (Clark and Cashman\({}^{34}\)) and chemically deposited ones (Eastman Kodak Co.\({}^{35}\), Miller and Uots\({}^{36}\)), show great similarity to Moss’s results. At present the usual values of \(\lambda_{1/2}\) for PbS are \(2.9\,\mu\) at room temperature and \(3.75\,\mu\) at the temperature of liquid air.

An interesting feature of the spectral-sensitivity curves of modern photoresistances is that they fall from the maximum toward shorter wavelengths continuously and slowly. If such a curve is plotted by referring the yield not to unit energy, but to a single quantum, then this part of the curve shows an almost horizontal course, beginning at wavelengths of about \(1\,\mu\) and up to the very beginning of the rapid fall beyond the maximum. There are also some data (see 7.4) indicating that the quantum yield for this region is close to unity. At present it has been reliably established that layers exhibiting the properties indicated above, especially high photosensitivity within the specified limits, can be obtained only after appropriate treatment, for example by annealing in oxygen at low pressure. Very little information has been published on the details of the process of making photoresistances. It is known, however, that in the manufacture of photoconducting layers in various laboratories and in industry, methods substantially different from one another are used.

In making photosensitive layers, two principal methods are currently used: sublimation in vacuum or in ki-

SEMICONDUCTORS SENSITIVE IN THE INFRARED REGION OF THE SPECTRUM

oxygen at low pressure, followed by annealing in oxygen at the same low pressure (Cashman[^37], Sosnovskii, Starkiewicz and Simpson[^19]) and chemical deposition (Kitsinskii[^38]) in the presence of an oxidizer. In sublimation, an electric discharge was also used (Schwarz[^20]).

If a PbS layer of the highest possible purity is sublimed in a vacuum higher than \(10^{-6}\) mm Hg, then its sensitivity proves to be small or entirely absent. This indicates that the high photosensitivity of such layers is due to secondary processes. Nevertheless, the fact that the long-wavelength limit of sensitivity is characteristic of the principal material itself apparently means that the primary process also plays a role. This question will be considered in 6.1. It appears that the role of treating the layer with oxygen consists, as it were, in revealing the principal effect.

The role of secondary processes in photoconductivity is also indicated by the fact that the rate of change of photoconductivity under abrupt changes in the illumination intensity of the layer varies over wide limits depending on the treatment. For layers deposited by evaporation, the time constants at room temperature may vary, depending on treatment, from one or two to several hundred microseconds. For chemically deposited layers the time constants are usually larger and lie within the range from several hundred to a thousand microseconds (see § 7). A number of experiments on the dependence of the time constant on various factors for PbS layers were carried out by Gibson[^29]. He found that the course of the decay of photoconductivity is usually determined by two exponentials, giving, respectively, a rapid initial decrease and a subsequent slower decay. The practically important one is the first of the corresponding time constants. Gibson found that it changes quite noticeably with temperature. Below a certain temperature, characteristic for each layer but usually close to 200°K, the initial time constant changes little. At higher temperatures, however, it rapidly decreases with increasing temperature, almost following an exponential law. The significance of this fact for the theory of photoconductivity will be indicated below (6.1). Gibson also established that the time constant begins to decrease rapidly when the current flowing through the layer exceeds some definite value, just as under intense illumination of the layer.

Gibson[^39], and also Chasmar and Gibson[^40], observed a number of other interesting phenomena connected with photoconductivity. It was found, for example, that if a PbS layer acquired a strongly pronounced \(p\)-type conductivity as a result of heating in sulfur vapor or in oxygen, owing to which it lost its normal photosensitivity to infrared radiation, then it could again be made photocon-

conducting under the action of infrared rays, illuminating it for some time with visible light at a temperature of 90°K. With such illumination the resistance of the layer continuously decreases. The light necessary to produce this effect must have wavelengths shorter than the corresponding absorption edge (\(\sim 1.1\,\mu\)), which, as is believed, corresponds to the transition of electrons from the filled band into the conduction band. If such a layer is at a temperature of 90°K, then this induced photosensitivity decreases slowly with time, but if it is heated to room temperature, it disappears rapidly.

2.6.2. Photoconductivity of PbTe. By treatment with oxygen, the resulting thin layers of PbTe, as well as layers of PbS, can be made photoconducting. The spectral sensitivity of such layers was also investigated by Moss\(^{30,41}\). The curves

Fig. 6. Spectral response of PbTe at different temperatures.

of spectral sensitivity for them are similar to those obtained for PbS, with the difference that the value of \(\lambda_{1/2}\) at a given temperature is shifted toward longer wavelengths (Fig. 6). Substantially different results were obtained by Simpson, Sutherland, and Blackwell\(^{42}\), as well as by Simpson and Sutherland\(^{43}\), who attempted to remove oxygen from the layers they studied. This, however, is impossible without the use of very advanced vacuum technique. Extensive experience of work at the NIIIR*) with sensitive photoresistances made of PbTe

*) Scientific Research Institute of Radio Communications, Ministry of Supply (Telecommunications Research Establishment, Ministry of Supply).

showed that for well-sensitized layers treated with acid, curves analogous to Moss’s curves are always obtained. Curves recently published by Clark and Cashman^34 have the same form. Like the curves for PbS, they give a sharp linear decrease (on a semilogarithmic scale) beginning at a certain wavelength. If the spectral response is plotted for an equal-energy spectrum, then toward short wavelengths the curve shows a slow decline. However, the quantum yield remains almost unchanged up to the onset of the rapid fall of the curve. The value of \(\lambda_{1/2}\) for PbTe at \(90^\circ\mathrm{K}\) is about \(4.75\,\mu\), and the rate of shift toward long wavelengths as the temperature is lowered corresponds to an energy change of about \(5.0\times 10^{-4}\,\mathrm{eV}/^\circ\mathrm{K}\). PbTe differs from PbS in that at room temperature it has a very low yield, and in order to obtain appreciable sensitivity it is necessary to cool the layer with solid carbon dioxide. PbTe has been investigated only in the form of evaporated layers; no chemical methods for producing them have been described. The time constants for photoconductivity in PbTe layers have not been studied as extensively as in the case of PbS. However, the behavior of the two is very similar. Measurements by Scanlon, Petriz, and Lemmis^44 showed the presence of a double time constant, just as was established by Gibson^45 for PbS.

2.6.3. Photoconductivity of PbSe. Both earlier work (see Sazerland and Lee^6) and some measurements made in recent years have shown that, apparently, the long-wavelength limit of the photoconductivity of PbSe is the same as for PbTe, or lies at somewhat shorter wavelengths. This is just what should have been expected, since selenium is located in the periodic table between S and Te. Experiments with PbSe were carried out both on evaporated layers (Blackwell, Simpson, and Sazerland^46, Moss and Chasmar^46a, Starkiewicz^47, Moss^31, ^41) and on chemically deposited layers (Milner and Watts^48). However, Gibson, Lawson, and Moss^49 showed that the long-wavelength limit of the photosensitivity for PbSe lies considerably farther in the long-wavelength region than for PbTe (Fig. 7). This unexpected result was first obtained on single crystals (see 3.3 and 6.4), but was then also established for evaporated layers. The values of \(\lambda_{1/2}\) for PbSe are \(4.7\,\mu\) at room temperature, \(7.1\,\mu\) at \(90^\circ\mathrm{K}\), and \(8.2\,\mu\) at \(20^\circ\mathrm{K}\). This corresponds to a rate of shift of about \(4\times 10^{-4}\,\mathrm{eV}/^\circ\mathrm{K}\). As regards the discrepancy between the earlier and the new results for PbSe, one can only suppose that the layers with which the earlier measurements were made were not sufficiently sensitized to reveal the long-wavelength photoconductivity which, as work with single crystals has shown, is characteristic of this material. It may be noted that PbSe, like PbS (but not PbTe), exhibits appreciable photoconductivity at room temperature.

The shape of the spectral-response curves in the case of PbSe, according to data from different investigators, is quite varied and gives maxima in different regions of the spectrum. For well-sensitized layers, for example those used in NIIR photoresistors, the characteristic curve shape is that typical of PbS and PbTe (see 7.3). This means an almost complete constancy of the quantum yield up to a certain wavelength value and an exponential fall-off at greater wavelengths. The time constants for PbSe are close to those for PbTe, but somewhat smaller than them.

Fig. 7. Spectral response of PbSe (curve A corresponds to room temperature, curve B to a temperature of 90° K).

2.7. Measurements of thermoelectromotive force

It was established by most earlier investigators (for example, Hintenberger^9) that in all substances of the PbS group the thermoelectric emf can be either positive or negative, and that its sign can be changed by applying different methods of heat treatment. For example, if a tellurium-rich PbTe sample is heated in vacuum, the sign of the thermoelectric emf changes from positive to negative. If the same sample is then heated in oxygen, a change of sign occurs.

Observations by Sosnovsky, Starkevich, and Simpson^18 on evaporated PbS layers showed that annealing in oxygen of a lead-rich PbS film first causes an increase and then a decrease in resistance, the maximum value of the resistance occurring at the moment when the sign of the thermoelectric emf changes. Similar results were obtained by Uffrick and Levinstein^50 for PbTe layers. However, experience in producing photoconductive elements at NIIR showed that such behavior is not observed under all conditions and that the sign of the thermoelectric emf cannot always serve as a reliable indicator of the sign of the predominant carriers.

§ 3. SEMICONDUCTORS OF THE LEAD SULFIDE GROUP; SINGLE CRYSTALS

Some of the early work on PbS was carried out with natural crystals of galena. It is unclear whether the specimens studied were single crystals. Natural crystals in general have too many impurities for intrinsic conductivity to be detected. The first convincing evidence for the existence of intrinsic conductivity in this group of substances, as we saw (2.4), was obtained by Putley and Chesmer\(^ {10}\) on polycrystalline but very pure PbTe specimens. Natural crystals of PbTe and PbSe of sufficient size and purity cannot be obtained, although these substances occur in nature in crystalline form (altaite and clausthalite). Later experiments by Dunaev and Maslakovets\(^ {51}\) with natural PbS crystals did not give convincing evidence for the presence of intrinsic conductivity in this substance. However, Putley and Arthur\(^ {52}\) found it in galena (from Sardinia) of exceptionally high purity.

3.1. Growth of single crystals

Since PbS was available only in the form of natural crystals, whose purity cannot be controlled, an attempt was made at the NIIR to grow artificial crystals of all three substances of this group. Since PbS crystals often occur in nature in the form of galena, it was thought that PbS crystals would be the easiest to obtain. However, the first attempts to grow crystals of substantial size—first by condensation from the vapor state, and then by the method of slowly passing the melt through the solidification plane—proved unsuccessful. Subsequently, Lawson\(^ {53}\) succeeded, by the second of the methods mentioned, in growing large single crystals of PbTe. At first it turned out that, despite differences in the ratio between Pb and Te in the starting mixture, these crystals were always obtained as \(p\)-type. It was then discovered that this circumstance was due to the presence of oxygen in the crystals. After careful removal of oxygen (reduction with hydrogen before melting), Lawson succeeded in growing good crystals of both \(n\)- and \(p\)-types. Applying this improved technique to PbS, he was able to grow good single crystals of this substance as well, and later also PbSe (details concerning the apparatus and methods of these investigations may be found in Lawson’s papers\(^ {53}\)). Clark and Cashman\(^ {34}\) have also recently grown PbTe crystals. Success in growing single crystals led to some very interesting new observations on optical absorption and photoconductivity.

3.2. Measurements of Conductivity and the Hall Constant on Single Crystals

Measurements of the temperature dependence of the conductivity and Hall constant for several natural crystals of galena, carried out by Putley and Arthur\(^{52}\), gave a series of curves very similar in form to those obtained by Cezair and Putley\(^{10}\) on pure polycrystalline PbTe samples (see Figs. 1 and 2). In this case the slope of the Hall-constant curves at high temperatures gave for the width of the forbidden band in PbS \(1.17\ \text{eV}\) (for PbTe \(0.6\ \text{eV}\)). Using single crystals grown by Lawson\(^{53}\), Putley\(^{22,25,28}\) in general confirmed these results and obtained accurate values of \(\Delta E\) for PbS, PbSe, and PbTe at temperatures above \(500^\circ\text{C}\). He found that, in the region of intrinsic conductivity, the conductivity of polycrystalline samples and of single crystals of comparable purity is approximately the same. This, however, is not the case at lower temperatures, where the values of the extrinsic conductivity are always higher for single crystals (which is undoubtedly due to the influence of intercrystalline barriers). Measurements of the Hall constant give analogous results for both types of samples, which was also verified by direct comparison of single crystals and sintered PbS samples, as indicated in 2.4.

For single crystals the general course of the temperature variation of the conductivity is as follows: in relatively pure samples (having about \(10^{16}\) effective donors or acceptors per \(1\ \text{cm}^3\)) the conductivity rises rapidly when the temperature is increased above \(500^\circ\text{K}\), is almost constant at room temperature, and usually increases upon further lowering of the temperature. From measurements of the Hall constant it follows that the number of carriers increases rapidly with increasing temperature above \(500^\circ\text{K}\) and is fairly constant at temperatures below room temperature. For \(p\)-type samples, reversal of the sign of the Hall constant usually occurs within the range between the indicated temperatures. The temperature above which the conductivity becomes intrinsic is determined by the impurity content, and for impure samples (say, with a content of more than \(10^{18}\) impurity atoms per \(1\ \text{cm}^3\)) the conductivity may decrease with increasing temperature up to \(900^\circ\text{C}\), at which noticeable and irreversible changes of the characteristics occur. For the region of intrinsic conductivity, according to theory, we have:

\[ R = AT^{-3/2}\exp\left(\frac{\Delta E}{2kT}\right), \tag{3} \]

and if \(RT^{3/2}\) is plotted as a function of \(1/T\), the graph usually represents an almost (but not exactly) straight line, from whose slope

which can be used to determine \(\Delta E\). In this way the following values were obtained:

\[ \begin{aligned} \mathrm{PbS}\quad \Delta E &= 1.17\ \mathrm{eV},\\ \mathrm{PbSe}\quad \Delta E &= 0.5\ \mathrm{eV},\\ \mathrm{PbTe}\quad \Delta E &= 0.63\ \mathrm{eV}. \end{aligned} \]

These quantities may be regarded as the most reliable values of \(\Delta E\) for the region of intrinsic conductivity (temperature above \(500^\circ\mathrm{C}\))*). The value of \(\Delta E\) for PbSe is probably less accurate than the others, owing to the difficulty of obtaining sufficiently pure crystals of this substance; nevertheless it is undoubtedly considerably lower than \(\Delta E\) for PbTe—an entirely unexpected result, which, however, could have been predicted on the basis of absorption data (see 3.3). The values given above are in general agreement with the rapid decrease in absorption found for sublimed films by Gibson\({}^{29}\) (see 2.5).

From these measurements, values of the carrier mobilities were also found. Usually it turns out that the mobilities for single crystals are several times greater than for polycrystalline specimens, which again indicates the influence of intercrystalline barriers. For single crystals, remarkably high mobility values are obtained. The highest values found for room temperature are given in the following table:

		PbS	PbSe	PbTe
Maximum observed mobility at \(290^\circ\mathrm{K}\), in \(\mathrm{cm/sec}\) per \(\mathrm{V/cm}\)	Electrons . . .	640	1400	2100
Maximum observed mobility at \(290^\circ\mathrm{K}\), in \(\mathrm{cm/sec}\) per \(\mathrm{V/cm}\)	Holes . . . . .	800	1400	840

(The values indicated for electrons and holes refer, of course, to different crystals.)

Since the value of the Hall constant in the region of intrinsic conductivity is always negative, it follows, consequently, that the electron mobility is higher than the hole mobility. It is probably at least twice as high. Therefore, judging from the maximum given

*) Note added in proof: the quoted values of \(\Delta E\), obtained from the slope of the curves analogous to those shown in Fig. 2, may be overestimated. If a more complicated expression for \(R\) is used (obtained by taking account of mixed conductivity), together with the measured values of the hole and electron mobilities (obtained for different temperatures on very impure specimens of \(n\)- and \(p\)-types), then the values of \(\Delta E\) prove to be somewhat smaller.

values, considerably higher electron mobilities should be expected. The mobility, apparently, does not change significantly as a function of impurity content, and Putley concludes that there is no evidence for electron scattering by impurity imperfections in the lattices of these compounds. At lower temperatures a strong increase in the mobility is observed. Thus, for example, the mobilities at a temperature of \(77^\circ\mathrm{C}\) are about \(10^4\ \mathrm{cm/sec}\) per \(\mathrm{V/cm}\), and at \(20^\circ\mathrm{K}\) about \(10^5\ \mathrm{cm/sec}\) per \(\mathrm{V/cm}\). In the temperature interval from 700 to \(100^\circ\mathrm{K}\) the mobility \(\mu\) is well described by the equation

\[ \mu=\mu_0 T^{-5/2}. \]

At present there are no theoretical grounds for such a course of the variation of the mobility.

We may note that these mobility values are much higher than those observed for sublimed layers \((0.5—50\ \mathrm{cm/sec}\) per \(\mathrm{V/cm})\); moreover, the mobility in the case of layers obtained by sublimation usually decreases with decreasing temperature.

3.3. Absorption measurements in single crystals

Until recently, measurements of the optical absorption of bulk specimens of all three substances in the group under consideration seemed to indicate that they were completely opaque in the ultraviolet, visible, and infrared regions of the spectrum. Gibson’s measurements\(^{29}\) for thin films appeared to confirm this, at least up to wavelengths of the order of \(6\ \mu\), where absorption coefficients greater than \(10^4\ \mathrm{cm}^{-1}\) were observed. The first observations showing that the absorption in some single crystals may be much smaller than had previously been supposed were made by Paul, Jones, and Jones\(^{54}\), who showed that, in one crystal of natural galena, the absorption coefficient for wavelengths of about \(4\ \mu\) proved to be very small—about \(5\ \mathrm{cm}^{-1}\). At wavelengths shorter than \(3\ \mu\), a very rapid increase of absorption was established. This boundary of optical absorption corresponds almost exactly to the long-wavelength boundary of photoconductivity. Although attempts were made to detect absorption of this character also in thin films, no indication of its existence was found (Gibson\(^{29}\)). For films of the type investigated by Gibson\(^{29}\), much greater absorption is measured (\(\sim 10^4\ \mathrm{cm}^{-1}\)), the cause of which has not yet been established and which, apparently, is absent in single crystals. The existence of the boundary of optical absorption was also found by Paul, Jones, and Jones\(^{54}\) for a synthetic PbSe crystal prepared at the NIIR. Following

After this interesting discovery it was found (Gibson, Lawson, and Moss \(^{49}\)) that all three substances of the group under consideration—PbS, PbSe, and PbTe in the form of single crystals (grown by Lawson)—have considerable transparency at wavelengths lying beyond the absorption edge, whose position coincides very well with the long-wavelength limit of photoconductivity observed for photosensitive layers. It should be noted here that, up to the present time, bulk photoconductivity in natural specimens has not been observed at all. The indicated connection between optical absorption and photoconductivity was also observed by Clark and Cashman \(^{34}\) for natural specimens of galena and for a PbTe crystal grown in the laboratory. The latter authors showed, moreover, that the temperature shift of the absorption edge has the same magnitude as the displacement of the long-wavelength limit of photoconductivity.

A detailed investigation of this absorption and of its changes as a function of impurity content was carried out by Gibson \(^{55}\) on artificial single crystals of PbS, PbSe, and PbTe grown at the NIIR, and also by Paul and Jones \(^{56}\) on crystals of galena. These experiments confirm in every detail, over a wide temperature interval, the correspondence between the long-wavelength limit of photoconductivity (both in photoresistors with thin layers and in photocells with point contacts made from single crystals—see 3.4) and the optical absorption edge for PbS and PbTe. The first results for PbSe, however, proved to be quite unexpected. For the position of the optical absorption edge a value of about \(5\,\mu\) was found at room temperature and about \(7\,\mu\) at the temperature of liquid air, whereas from measurements of photoconductivity for thin layers it followed that the corresponding photosensitivity limits lie at about \(3.5\) and \(5\,\mu\). Soon, however, it was shown both on single-crystal photocells and on evaporated layers (Gibson, Lawson, and Moss \(^{49}\)) that these values were erroneous and that in the case of PbSe as well there is exact coincidence of the two limits. Most surprising is that these wavelength values substantially exceed the corresponding values for PbTe.

For crystals grown in the laboratory, such low values of the minimum absorption coefficient as those found by Paul, Jones, and Jones \(^{54}\) were not detected. The minimum value lay approximately in the range from \(30\) to \(300\ \mathrm{cm}^{-1}\). Many natural galena specimens show no noticeable transmission at all, from which it follows that the specimen found by Paul, Jones, and Jones \(^{56}\) was exceptional (Paul and Jones \(^{56}\)).

Typical absorption spectra obtained by Gibson over a wide temperature interval for PbS, PbSe, and PbTe are given in

Fig. 8, 9, and 10, respectively. Common to them is the following: a very rapid fall of absorption, occurring at a definite wavelength (the absorption edge), which shifts toward the side

Fig. 8. Absorption in an \(n\)-type PbS crystal (sample thickness \(1.25\ \text{mm}\); the numbers indicate the temperature in \({}^{\circ}\mathrm{K}\)).

of longer wavelengths as the temperature is lowered (excluding the temperature range above \(500^{\circ}\mathrm{K}\), where these changes are small); the rate of displacement of the edge for each of the substances is—

Fig. 9. Absorption in a \(p\)-type PbSe crystal (sample thickness \(0.68\ \text{mm}\)).

almost the same as the rate of shift of the long-wave edge of photoconductivity, amounting to about \(4 \times 10^{-4}\ \mathrm{eV}/^{\circ}\mathrm{C}\). These changes are shown in Fig. 11.

At wavelengths somewhat exceeding the corresponding boundary, absorption passes through a minimum and then slowly, but

Fig. 10. Absorption in an \(n\)-type PbTe crystal (sample thickness \(0.35\) mm).

continuously increases with increasing wavelength. In this region it is approximately proportional to the square of the wavelength and increases with increasing temperature. As has already been said, the minimum value of the absorption varies strongly from sample to sample, but Gibson finds no relation whatsoever between the number of carriers and this minimum value. \(p\)-type samples absorb somewhat more strongly than \(n\)-type samples, but this difference is not distinct. On the contrary, the position of the absorption edge is very characteristic of the material and, apparently, is the same for all samples of both conductivity types. This circumstance is very important, since it apparently indicates that, contrary to earlier assumptions (see 2.6), the long-wavelength boundary of photoconductivity is a characteristic of the principal material and is not subject to the influence of such impurities as oxygen. This follows

Fig. 11. Position of the absorption boundary as a function of temperature for \(n\)-type PbS (curve 1), \(n\)- and \(p\)-type PbTe (curve 2), and \(n\)-type PbSe (curve 3).

from the observed agreement between the absorption edge and the long-wavelength limit of photoconductivity.

The absorption coefficient on the short-wavelength side of the edge is difficult to measure. It rapidly increases to values of the order of \(10^4\ \mathrm{cm}^{-1}\), which is another convincing indication that the absorption is a characteristic of the basic substance and not of impurities. For such values of impurity absorption it would be necessary for their concentration to exceed \(0.1\%\), which is highly improbable. Until now it has proved impossible to investigate the change in the magnitude of absorption for this region of the spectrum as a function of carrier concentration. The absorption edges observed at substantially shorter wavelengths (\(\sim 1\,\mu\) for PbS) in thin films cannot be observed on single-crystal specimens, since these cannot be made thinner than \(0.05\ \mathrm{mm}\). Lawson grew several very thin (\(\sim 1\,\mu\)) small PbTe crystals by the sublimation method used by Gibson \(^{55}\) for measuring absorption down to approximately \(\lambda = 1\,\mu\). In this case, again, a rapid rise of absorption was found in the region between 1 and \(2\,\mu\), as observed for sublimed layers, and for the value of the absorption coefficient at wavelengths around \(1\,\mu\) a value of about \(10^5\ \mathrm{cm}^{-1}\) was again found. Therefore it should be thought that up to approximately \(2.5\,\mu\) the behavior of thin layers and single crystals is the same. However, in the region of longer wavelengths the two kinds of specimens differ very substantially: polycrystalline layers have an additional high absorption of the order of \(10^4\ \mathrm{cm}^{-1}\), extending rather far beyond the long-wavelength photoconductivity edge corresponding to room temperature. Sublimed layers were made with thicknesses down to \(0.05\ \mathrm{mm}\); it turned out that they behave quite differently from single-crystal layers of the same thickness.

Another approach to measuring absorption in single crystals of the substances under consideration was used by Avery \(^{57}\). Measuring the coefficient of reflection from a plane crystalline surface at different angles of incidence for two directions of polarization, he was able to calculate the real and imaginary parts of the dielectric constant, and from this to determine the absorption. This method is not applicable for small values of the absorption coefficient, which are typical for wavelengths exceeding the photoconductivity edge, but it is quite suitable in the region of shorter wavelengths, in which direct measurement of transmission does not give reliable results because of excessively strong absorption.

These measurements also show that for PbS single crystals the absorption coefficient rises rapidly from several tens of thousands \(\mathrm{cm}^{-1}\) to more than \(4\times 10^5\ \mathrm{cm}^{-1}\) on going to wavelengths shorter than \(1\,\mu\). As the wavelength increases it remains of the order of \(10^4\ \mathrm{cm}^{-1}\) at least up to \(2.5\,\mu\). The values of the absorption coefficients turn out to be somewhat higher than those obtained by Gibson \(^{55}\), which does not-

is surprising, taking into account the difficulties associated with obtaining homogeneous layers. Similar results are found for PbTe.

From Éveri’s measurement data, values of the refractive index for different wavelengths can be determined. In the interval from \(1.5\) to \(3.5\,\mu\), only small changes are found. For shorter wavelengths the refractive index begins to decrease. For a wavelength of \(3\,\mu\), the mean values of the refractive index for a number of samples are as follows:

Material	\(n\) at \(\lambda = 3\,\mu\)
PbS	\(4.01 \pm 0.06\)
PbSe	\(4.59 \pm 0.06\)
PbTe	\(5.35 \pm 0.10\)

It is evident from the table that the refractive index should be ordered according to the position of the elements in the periodic system, but not according to the order of the optical or photoelectric properties of the materials (absorption-band edges and long-wavelength limits of photosensitivity). The significance of these results for determining the band structure of the crystals under consideration will be discussed in § 6.

3.4. Photogalvanic effects and photoconductivity of single crystals

It has long been known that in a point contact between a metal and certain PbS crystals a photoelectromotive force may arise, and that the spectral characteristic of this effect is similar to the spectral characteristic of photoconductivity in layers (see Elliott\(^ {58}\), Sutherland and Lee\(^6\))*). It has also long been known that such contacts often exhibit a distinct rectifying action (see 3.5).

Under certain conditions such contacts also reveal photoconductivity, which is usually manifested in a decrease in the contact resistance and in the direction of “blocking” under the action of illumination. It is not easy to separate the photogalvanic and internal photoeffects, but such an attempt was made by Gibson\(^ {55}\) in his detailed study of the photoconducting properties of contacts between metals and single crystals of PbTe and PbS. For PbTe, a distinct rectifying action is found only in the case of \(p\)-type crystals, and then only at temperatures not higher than approximately \(190^\circ\)K. At \(90^\circ\)K such contacts exhibit a noticeable internal

*) Apparently, these facts were first established around 1930 by the Soviet scientist O. V. Losev. (Translator’s note.)

photoeffect, as well as the photovoltaic effect. The internal photoeffect is observed only for the direction of blocking. In the case of \(n\)-type crystals the effect is very small and, moreover, appears only after a treatment which may well create a surface layer of \(p\)-type. Similar results are obtained for crystals of natural galena, and also for artificial single crystals of PbS. In this case, too, only \(p\)-type crystals again give a noticeable internal photoeffect, but it is easily observed already at room temperature (which is in agreement with the results of experiments with photoconducting layers). The spectral characteristics for both materials prove to be almost identical with the spectral characteristics of well-sensitized photoresistors made of the same substances.

Gibson explained his observations on the basis of the idea of a Schottky-type barrier arising at the surface of a semiconductor in contact with a metallic point. The observed changes in resistance are attributed to a change in the effective height of the barrier under the action of the incident radiation. These ideas lead to a satisfactory explanation of the changes in photocurrent as a function of the applied voltage. For PbTe the effective barrier height at \(90^\circ\) K is of the order of \(0.1\) ev; it decreases to a very small value at room temperature. In the case of PbS the barrier height is about \(0.2\) and \(0.1\) ev at \(90^\circ\) K and at room temperature, respectively.

By the light-probe method the effective range of the charge carriers producing the photoeffect was determined. The range varied from sample to sample, but the distance at which the signal falls to one half proved on the average to be of the order of \(50 \mu\).

Similar results were later obtained for PbSe, and, as we saw (2.6.3), it was precisely measurements of photoconductivity in a point contact with a PbSe crystal that first made it possible to establish for this substance the correspondence between the long-wavelength limit of photoconductivity and the edge of the absorption band, and also showed that the long-wavelength limit for PbSe lies considerably beyond that for PbTe (Gibson, Lawson, and Moss\(^{49}\)*).

3.5. Rectification and transistor action in single crystals

It is well known that some crystals of galena possess a strong rectifying action. Such crystals were widely used in the detectors of the first radio receivers. Recent investigations have shown that only crystals

* Since this was written, Moss has observed photoconductivity in single crystals of PbS.

became p-type crystals (Legrand[^59], Arianova and Sokolskaya[^60]), which was also confirmed on synthetic crystals of PbS, PbSe, and PbTe (Gibson[^55]). The current–voltage characteristics for PbS were investigated by Henisch and Grunewald[^61]. Changes in rectifying properties as a function of surface treatment were considered by Hogarth and Grunewald[^62], and the effect of introducing various components into a massive crystal (including excess sulfur) by Reimann and Seliverstov[^63].

Proceeding from the similarity of the group of substances under consideration, in respect to their semiconductor properties, to germanium, it is natural to expect that they may exhibit transistor action, i.e., the effect of the current in a contact with a bias of one sign on the current in a nearby contact whose bias has the opposite sign (see, for example, Shockley[^64]). Such a phenomenon was first observed for p-type PbS samples by Gebbie, Banbury, and Hogarth[^65],[^65a]. Voltage gains of the order of 25 are usual; sometimes still higher values are observed, but current gain greater than unity has not yet been reported. The frequency characteristics of such transistors were investigated by Banbury and Henisch[^66], who showed that a rapid fall-off in response occurs at frequencies above approximately 200 kc/s. By placing the “emitter” and “collector” on opposite sides of a thin crystal, Banbury[^67] showed that the transistor effect is not merely a surface phenomenon. He also measured the mean free path of the injected charge carriers, which, for the materials used, proved to be of the order of 50 μ, in agreement with the value found by Gibson[^55] for optically introduced carriers in PbS crystals (see above).

Transistor action was also detected by Hogarth[^68] in artificial PbSe crystals. The phenomena in this case are less clearly expressed than in the best PbS samples, but this is almost certainly due to the fact that synthetic PbSe crystals are less free of impurities than the best samples of natural galena. Hogarth[^68] also found transistor action for PbTe (but only at low temperatures: 90° K). This is in agreement with Gibson’s observations[^55] on the photoconductivity of point contacts of metals with PbTe. The indicated effects are clearly expressed only for those samples of all three substances which possess p-type conductivity.

3.6. Other properties of semiconductors of the lead sulfide group

In this section we shall discuss other properties of semiconductors belonging to the group under consideration. They all have the sodium chloride crystal structure, with an edge length

of the elementary cube, equal respectively to

\[ \begin{array}{ll} \mathrm{PbS} & \ldots\ldots\ldots 5.97\ \text{\AA} \\ \mathrm{PbSe} & \ldots\ldots\ldots 6.14\ \text{\AA} \\ \mathrm{PbTe} & \ldots\ldots\ldots 6.45\ \text{\AA} \end{array} \]

Recently, precise determinations of the molecular heat capacity have been made for all three substances of this group in the temperature range from 14 to \(300^\circ\) K (Parkinson, NPL), showing that in this temperature interval there are no anomalies whatever in the course of the specific heat, and that the molecular heat capacity increases smoothly with temperature; moreover, the values of this quantity for PbSe everywhere lie between the values for PbS and PbTe. This is of the highest interest from the point of view of the electronic properties of the materials of this group considered above. To this it should be added that, from recent measurements of the heats of formation of the compounds under consideration, it is clear that in this respect too they also follow the position of the metalloid in the periodic system (National Bureau of Standards \(^{69}\)).

For some time it was believed that lead sulfide becomes superconducting at low temperatures, which was seemingly confirmed for all three substances by later measurements of Darby, Hatton, and Rollin \(^{70}\). This seems somewhat surprising in view of the other properties of these semiconductors. Using moderately pure samples of natural \(p\)-type galena (\(7 \cdot 10^{17}\) and \(2 \cdot 10^{16}\) carriers per \(1\ \text{cm}^3\) at room temperature), in subsequent measurements Hatton, Rollin, and Seymour \(^{71}\) found no superconductivity down to a temperature of \(1^\circ\) K. These authors conclude that the previous observations of superconductivity may have been caused by the formation of lead filaments at the boundaries between grains. Hudson \(^{72}\) came to the same conclusion, finding no signs of superconductivity except in samples for which X-ray examination showed the presence of metallic lead. In particular, no signs of superconductivity were found for a synthetic PbTe crystal on cooling to \(1.3^\circ\) K. It may therefore be concluded that these substances in pure crystalline form are not superconducting, at least in the region of “helium” temperatures.

Grenville and Hogarth \(^{73}\) measured the thermoelectromotive force on single crystals, and also attempted to establish a relation between the direction of rectification, the sign of the photovoltaic effect, and the sign of the thermoe.m.f. It turned out that a relation exists only for freshly cleaved surfaces, carefully etched after polishing. From this it is clear that great caution is necessary in interpreting the results of thermoe.m.f. measurements if one has in mind determining from these data the type of conductivity of the sample.

§ 4. OTHER SEMICONDUCTORS SENSITIVE IN THE INFRARED REGION

A large number of compounds of one or two elements are photoconducting in the near infrared region—from 1 to 2 μ. However, only very few of them have \(\lambda_{1/2}\) (see 2.6.1) greater than 2 μ.

4.1. Compounds containing sulfur, selenium, or tellurium

A number of such compounds were investigated by Moss \(^{74,75}\). None of them has \(\lambda_{1/2}\) greater than 2 μ, although some, in particular MoS\(_2\), for which \(\lambda_{1/2}=2\) μ, may exhibit photoconductivity at wavelengths greater than 2 μ. The photoconductivity of a number of such compounds was also studied by Breithaupt \(^{76}\), who used layers obtained by sublimation. The values he found for \(\lambda_{1/2}\) and \(\lambda_m\) (the wavelength at which the greatest photoconductivity was observed) are given in the table (for cases in which \(\lambda_m\) exceeds 2 μ).

It should be noted that the indicated values of \(\lambda_{1/2}\) and \(\lambda_m\) were measured at \(90^\circ\) K. As is readily seen, of all these substances only HgTe is at all comparable with the substances of the PbS group.

Compound	\(\lambda_{1/2}\) (μ)	\(\lambda_m\) (μ)
CuTe	1.3	2.2
AgTe	1.3	3.0
ZnTe	1.4	3.5
HgTe	3.1	3.9
TlTe	1.6	2.6
SbTe\(_3\)	1.6	2.6
MoTe\(_2\)	1.6	2.6
UTe\(_2\)	1.35	2.5

The photoconductivity of a series of bismuth compounds (which corresponds to the PbS group) was studied by Gibson and Moss \(^{77}\), using sublimed layers. These compounds behave in many respects like materials of the PbS group, and the value of \(\lambda_{1/2}\) for them shifts toward longer wavelengths as the layer is cooled. Their sensitivity is greatly increased by treatment with oxygen, but it does not reach the high values found in the PbS—PbSe—PbTe series. Bismuth sulfide has very low sensitivity at wavelengths greater than 2 μ, but the values of \(\lambda_{1/2}\) and \(\lambda_m\) for bismuth telluride at \(90^\circ\) K are respectively about 2.8 and 4 μ. As in the case of the lead series, bismuth telluride shows no sensitivity at room temperature, whereas its sulfide does.

A large number of compounds as possible materials for photocells was investigated by Schwartz \(^{20}\). Of these, only InTe shows sensitivity beyond 2 μ, and then only up to 2.2 μ. It should be noted, however, that CdSe has very high sensitivity, comparable with the sensitivity of the PbS group, but only in the wavelength region below 2 μ.

4.2. Intermetallic Compounds

Juze, Močan, and Ryvkin\(^{78}\), as well as Moss\(^{74}\), showed that the intermetallic compound \(\mathrm{Mg_3Sb_2}\) is photoconductive in the infrared region. Moss found that layers evaporated in vacuum do not exhibit photoconductivity at wavelengths greater than \(2.8\,\mu\), but if they are evaporated in air at low pressure, the value of \(\lambda_{1/2}\) shifts to \(3.5\,\mu\).

None of these substances has been investigated as thoroughly as the PbS group. In particular, no work has been done with single crystals. It would be very interesting to carry out equally detailed investigations of some of these substances, especially HgTe, \(\mathrm{Bi_2Te_3}\), and intermetallic compounds similar to \(\mathrm{Mg_3Sb_2}\), which exhibit photoconductivity in the wavelength region beyond \(2\,\mu\).

4.3. Photoconductivity of Elements

A number of elements that are semiconductors or insulators exhibit photoconductivity, and some of them have values of \(\lambda_{1/2}\) in the near-infrared region of the spectrum. These elements were described in detail by Moss\(^{75,79}\). Of the semiconductor elements, silicon and germanium have been studied most thoroughly; their \(\lambda_{1/2}\) values are \(1.1\) and \(1.7\,\mu\), respectively. Germanium can be obtained in the form of single crystals of extremely high purity. We shall make a comparison of the properties of this element with those of the PbS group compounds in the following section.

The only element that exhibits photoconductivity at ordinary temperatures and at wavelengths considerably greater than \(2\,\mu\) is tellurium. The properties of this substance, which in pure form behaves as a semiconductor, have been studied very extensively, and Moss\(^{41,75}\) observed its photoconductivity using evaporated layers. For layers cooled with liquid air, \(\lambda_{1/2}\) is about \(3.7\,\mu\), and \(\lambda_m\) is about \(4.3\,\mu\). At room temperature the photoconductivity is very small. As the temperature is lowered, the value of \(\lambda_{1/2}\) shifts toward longer wavelengths (Moss\(^{80}\)), the shift corresponding approximately to \(2 \cdot 10^{-4}\) eV/\(^{\circ}\)C.

Tellurium has been studied as a semiconductor by many authors, but, as has usually been the case until quite recently, on insufficiently pure samples. Bottom\(^{81}\) established that, in the pure state, tellurium is an intrinsic semiconductor at room temperature. The value he obtained for the forbidden-band width \(\Delta E\) is \(0.38\) eV. Johnson\(^{82}\) found a close value for \(\Delta E\) and established that the mobilities at room temperature (deduced from measurements of the Hall constant) are about \(550\) cm/sec per V/cm for both holes and electrons. Numerous measurements using highly purified material in the form of mo—

single crystals, carried out by Fukuro, Tanuma, and Tobisawa ^83, led to values of 0.34 eV for $\Delta E$ and to room-temperature mobilities: for electrons $\mu_e = 1600\ \mathrm{cm/sec}$ per $\mathrm{V/cm}$ and for holes $\mu_h = 1100\ \mathrm{cm/sec}$ per $\mathrm{V/cm}$. The variation of mobility with temperature in Te proves to be complicated because of the anisotropic character of its structure, and for some temperature intervals $\mu_h$ is greater than $\mu_e$. This leads to a double reversal of the sign of the Hall constant.

Measurements by Moss ^75,84 showed that an increase in absorption occurs at wavelengths shorter than $3.5\ \mu$. In the longer-wavelength region the absorption coefficient is constant and is of the order of $180\ \mathrm{cm}^{-1}$. Loferski and Miller ^85 give the value $4.3\ \mu$ for the position of the absorption edge in their samples (of high purity). On the long-wavelength side of the absorption edge they found values of the absorption coefficient approximately ten times smaller than the values reported by Moss. Thus, the absorption edge coincides rather accurately with the long-wavelength limit of photoconductivity.

It is interesting to note that Moss ^32,75 found a high value of the refractive index in Te: for wavelengths greater than $4\ \mu$, $n$ is approximately equal to 5. From this it can be seen that in many respects tellurium is similar to substances of the PbS group.

Measurements made by Kendall ^86, and also by Busch, Wieland, and Zoller ^87 for gray tin, showed that in the pure state it is a semiconductor with a forbidden-band width $\Delta E$ of the order of 0.1 eV. Photoconductivity has not yet been observed for it, but if it is discovered, one may expect that it will extend beyond $10\ \mu$. Moss ^32,75 obtained sublimed antimony layers having the properties of a semiconductor with $\Delta E$ from 0.05 to 0.2 eV. A change in the resistance of these layers under the action of infrared radiation up to wavelengths of $16\ \mu$ was observed. It has not been established, however, what is the cause of these changes—the bolometric effect or photoconductivity.

4.4. Photoconductivity at Low Temperatures

At very low temperatures a very interesting effect is observed. Rollin and Simmons ^88 showed that massive crystalline silicon specimens become photoconducting at wavelengths in the range $2$–$14\ \mu$ at liquid-hydrogen temperatures. Measurements were not made for longer wavelengths, but data are available indicating that photoconductivity also exists in the region beyond $14\ \mu$. This phenomenon was interpreted as excitation of electrons from impurity levels, since the investigated

the samples contained \(10^{16}\)—\(10^{17}\) impurity centers at energy levels located approximately \(0.05\ \mathrm{eV}\) above the filled band. If this explanation is correct, then one may expect that the long-wavelength limit of photoconductivity lies near \(25\ \mu\). It seems probable that similar phenomena can also be observed in other semiconductors. They will probably occur only at low temperatures, since otherwise such acceptor centers must be completely filled.

§ 5. COMPARISON OF LEAD SULFIDE GROUP SUBSTANCES WITH GERMANIUM AND SILICON

A great deal of work has been done on the study of the electrical and optical properties of Ge and Si, especially germanium. At present Ge can be obtained in such a pure form that it is an intrinsic semiconductor at room temperature, and the content and character of impurities in it can be controlled. Therefore germanium is an ideal substance for use as a standard semiconductor. Moreover, since Ge is an element, one may expect its behavior to be simpler than that of complex semiconductors, for example PbS. The electrical and optical properties of Ge and Si have been described by Shockley\(^{64}\), and a review of work on their photoconductivity has been given by Moss\(^{75}\).

The electrical properties of Ge and Si, on the one hand, and of the PbS group, on the other, are very similar in many respects. The value \(\Delta E\) for Ge is \(0.74\ \mathrm{eV}\), and for Si, \(1.1\ \mathrm{eV}\). Therefore one should expect that the properties of Ge and Si should resemble the properties of PbTe, PbSe, and PbS (see 3.2). This is indeed so, but in addition there are important and interesting differences. PbTe and PbSe should be intrinsic semiconductors at room temperature, like Ge, but the artificial crystals presently available are still not sufficiently pure for this.

All these substances have high mobilities of electrons and holes, but for Ge the lifetime of minority carriers is much greater than for the substances of the PbS group, for which values exceeding several microseconds have not yet been found. This also may be largely connected with the degree of purity of the samples. Ge exhibits a strongly pronounced rectifying and transistor action when it has \(n\)-type conductivity, whereas Si and all three substances of the PbS group exhibit them only with \(p\)-type conductivity. However, the most important difference between these two groups of substances is that, whereas for Ge and Si the long-wavelength limit of strong absorption and the boundary of photoconductivity correspond to energies very close to the band gap width, determined

in terms of conductivity and the Hall constant in the intrinsic-conduction region, this does not hold for the PbS group. Thus there is no doubt that for Ge and Si, and also, for example, for Te, the photoconductivity threshold corresponds to the transition of an electron from the top of the filled band to the lower levels of the conduction band. This cannot be the case for substances of the PbS group, unless at room and low temperatures the forbidden band is not strongly narrowed. Indeed, for PbS the value of \(\Delta E\) in the intrinsic-conduction region is \(1.17\ \text{eV}\), whereas the quantum energy corresponding to \(\lambda_{1/2}\) is about \(0.4\ \text{eV}\). The latter value also corresponds to the quantum energy in the region of the long-wavelength absorption edge. For Ge and Si, as for most photoconductors, the long-wavelength limit of photoconductivity shifts, with decreasing temperature, toward shorter wavelengths, whereas for substances of the PbS group a shift in the opposite direction is observed. In this latter respect, however, Te behaves like the substances of the PbS group. We shall dwell on these differences in the following section.

§ 6. THEORY OF PHOTOELECTRIC PHENOMENA IN SUBSTANCES OF THE LEAD SULFIDE GROUP

In developing any theory of photoelectric phenomena in the infrared region characteristic of substances of the PbS group, it is necessary to start from one or another model of the basic semiconductor. We may assume that the width of the forbidden band is determined from Hall-constant measurements carried out by Putley on single crystals (see 3.2), and there is almost no doubt that this is correct for the temperature range in which intrinsic conduction occurs (above \(500^\circ\text{K}\)). However, many of those who considered the theory of photoconducting layers did not make this assumption. They considered that the energy corresponding to the long-wavelength limit of photoconductivity is equal to the energy corresponding to the width of the forbidden band. Unless one assumes that at temperatures below those corresponding to the intrinsic-conduction region there occur no changes whatever in the energy structure of PbS and of other substances of this group, it seems quite improbable that the width of the forbidden band should decrease to such an extent—for example, in the case of PbS from \(1.17\ \text{eV}\) at \(500^\circ\text{C}\) to approximately \(0.4\ \text{eV}\) at \(300^\circ\text{K}\) (it is known that no appreciable changes in the crystal structure occur in this case). Absorption measurements on single crystals, performed by Gibson \(^{55}\), showed that the rate of the temperature shift of the absorption edge at temperatures above room temperature decreases rather than increases (Fig. 11). In considering some theoretical works it is necessary to take this difficulty into account.

6.1. Theory of Photoelectric Phenomena in Thin Layers

The first attempt to give a theory of the extremely high photosensitivity of fired PbS layers was the theory of Sosnovskii, Starkiewicz, and Simpson[^19]. From their experimental work (see 2.4 and 2.6) they concluded that the layers are polycrystalline and that the principal part of the photoeffect is due to the action of radiation on intercrystalline barriers. The evidence in favor of this point of view is very weighty and has already been considered in 2.4. The authors mentioned assumed that the intercrystalline barriers are caused by \(p\)-\(n\) junctions between the microcrystals forming the layer. This conclusion was drawn from the fact that the resistance of the layer reaches a maximum when an \(n\)-type layer has just

p-type                  n-type

![diagram]

Conduction band
Filled band
Electron energy
\(\Delta E\)
\(\Delta E\)
\(F\)
\(e\)
\(p\)-\(n\) junction

• Electron
○ Hole

Fig. 12. \(p\)—\(n\) junction (• — electron, ○ — hole).

been converted into a \(p\)-type layer as a result of annealing in oxygen. This was interpreted to mean that, in the indicated state of the layer, part of the microcrystals will belong to the \(p\)-type and part to the \(n\)-type. It was assumed that donor centers in \(n\)-type crystals are caused by an excess of lead, while acceptor centers in \(p\)-type crystals are caused by excess oxygen. The theory of contact between \(n\)- and \(p\)-type crystals was considered by Sosnovskii[^89] and is at present well known, especially as applied to germanium (see Shockley[^64]). The photoelectric phenomena arising upon illumination of such a junction were considered in sufficient detail, in particular for germanium, by Becker and Fan[^90]. Fig. 12 illustrates the basic ideas. Electrons from donor centers near the surface of the \(n\)-type material pass to acceptor centers near the surface

material of the \(p\)-type. Thus, in the \(n\)-type material a positive space charge arises, and in the \(p\)-type material a negative one. This continues until the potential difference between the two materials rises enough for the top of the filled band in the \(p\)-type material to reach the bottom of the conduction band of the \(n\)-type material. In this case a potential barrier is established between the two materials, whose height is approximately equal to the width \(\Delta E\) of the forbidden band, and a strong electric field arises in the transition region. Thus, the pairs of holes and electrons created by the radiation near the transition turn out to be separated and alter the equilibrium charge distribution, giving rise to a photoelectromotive force. The occurrence of photoconductivity in this case is assumed to be caused by a change in the potential barrier produced by free charges. The influence of barriers created by volume charges was subsequently considered by Schwartz \(^{91}\), James \(^{92}\), and Rittner \(^{93}\), who extended the model described to the case of changes in the concentration of impurities causing a transition of the main material itself from \(p\)-type to \(n\)-type. These authors assumed that the primary process is the excitation of electrons from the filled band into the conduction band, and they considered that, in the case of PbS, an energy of about \(0.4\) eV is required for this. If, on the basis of Putley’s measurements, the required energy were taken to be \(1.17\) eV, the resulting potential barriers would be too high, and it would be necessary to seek another source of the primary photoelectrons.

In order to avoid this difficulty, Smith \(^{94}\) suggested that in this case one could apply the model previously proposed by Mitchell and Sillars \(^{95}\) for explaining certain phenomena in silicon carbide. Smith carried out the discussion with an \(n\)-type semiconductor in mind. It is now known that the photoelectric phenomena in substances of the PbS group are connected predominantly with \(p\)-type conductivity. Nevertheless, the essential features of the model have retained their significance. They are as follows. Surface states that capture holes can create in a \(p\)-type material space-charge barriers whose height may vary within wide limits depending on the density of the surface states and the properties of the material. This is illustrated in Fig. 13. Photoelectrons produced near such a barrier are drawn into it and release a certain number of trapped holes. This, in turn, lowers its height. Any asymmetry between two crystals in contact will, as above, produce a photo-emf. Thus, it is assumed that photoconductivity is to a large extent a “capture” effect: the primary electrons only lower the height of the barriers and thereby create the conditions for the flow of a larger current. This model was examined in detail by Gibson \(^{45}\), who provided the foundations of its theory and described a number of experiments that may be considered...

as confirmation of its correctness. In particular, it proves possible to find the effective height of the barriers. On average it does not exceed \(0.2\) eV, i.e. it is considerably smaller than what should be expected from the \(p\)—\(n\) junction itself. Barriers of such height were found for contacts between \(p\)-type crystals and metals (see 3.4). It therefore appears that, although \(p\)—\(n\) junctions may play some role, the more important ones are \(p\)—\(p\) junctions.

Fig. 13. \(p\)—\(p\) junction (\(\bullet\) — electron, \(\bigcirc\) — hole).

Gibson\(^{45}\) compared this theory with the theory which assumes that the incident radiation simply increases the number of electrons in the conduction band or the number of holes in the filled band. In particular, on the basis of the nature of the changes in the time constant for photoconductivity as a function of temperature and illumination intensity, he concluded that these phenomena can be explained only by the barrier theory.

However, Simpson and Sutherland\(^{43}\) explained their results for PbS from the standpoint of a simple process of bimolecular recombination and came to the conclusion that, if one assumes the existence of a “tail” in the distribution of energy levels in the filled band or in the conduction band, then barriers prove unnecessary. These authors believe that the concept of barriers is needed only in the case of layers treated with oxygen. They attempted, as far as possible, to remove oxygen from their layers.

In general, it is quite possible that there exist both changes in bulk conductivity and the action of barriers, as was recently proposed by Ewals\(^{56}\) to explain the photoconductivity of Tl\(_2\)S. However, it seems sufficiently clear that barriers play the predominant role in the case of layers sensitized with oxygen.

6.2. Source of Primary Photoelectrons

The theoretical considerations discussed above do not shed light on the question of the sources of primary photoelectrons. All theories in fact simply assume the generation of pairs of holes and electrons, independently of how these pairs are created.

Most authors, apart from Gibson[^45], believe that primary photoelectrons arise as a result of transitions from the filled band to the conduction band. We have already pointed out the difficulties connected with this assumption. Various impurity centers have also been considered as possible sources of primary photoelectrons. For example, Pincherle[^97] investigated \(F\)-, \(F'\)-, and \(F''\)-centers, i.e., lattice sites in which negative ions are absent and where one, two, or three electrons, respectively, are located. He concludes that \(F\)- and \(F'\)-centers in PbS are situated energetically too deeply, while \(F''\)-centers are unstable. He also considered the question of \(D\)-centers, i.e., pairs of missing adjacent ions of opposite signs. Such centers give energy levels (about \(0.25\ \text{eV}\)) below the conduction band and may play an important role as electron traps, but it is unlikely that they could serve as sources of primary photoelectrons.

From the high values of the absorption coefficient (\(\sim 10^4\ \text{cm}^{-1}\)) for single crystals, observed up to the short-wavelength side of the long-wavelength limit of photoconductivity, it follows that impurity centers cannot be the sources of primary photoelectrons. In that case two possibilities remain: 1) that the width of the forbidden band at room temperature is indeed of the order of \(0.4\ \text{eV}\) for PbS and still smaller for PbTe and PbSe, and 2) that the electrons are first excited into states with low energy, characteristic of the basic lattice, but not producing free holes, i.e., that excitons are initially formed (see Mott and Gurney[^27], p. 84). Pincherle (see 6.6) recently calculated the energy required to create such an exciton in PbS and obtained a value of about \(0.3\ \text{eV}\), which is correct as to order of magnitude. If this second possibility is adopted, it is necessary to explain how the exciton is destroyed with the formation of a hole–electron pair, for which an additional \(0.9\ \text{eV}\) must be expended (if the forbidden-band width \(\Delta E\) found by Putley is accepted).

Thus, it must be admitted that there is no satisfactory answer to the question of the origin of primary photoelectrons, and clarification of this circumstance is at present the most important problem in the theory of photoelectric phenomena in substances of the group under consideration.

6.3. Energy Structure of PbS

An attempt to solve the difficult problem of calculating the electron energy levels for semiconductors of the PbS group was undertaken by Pincherle and his collaborators at N.I.I.R. (Bell et al.\(^{98}\)). These authors used the so-called “cell method,” consisting in the calculation, for each elementary cell into which the lattice can be divided, of self-consistent “radial” wave functions satisfying the boundary conditions. The calculation of these functions is carried out only for the zero value of \(\mathbf{k}\) (the wave vector), and this turns out to be the principal shortcoming of this method. Preliminary results showed,

Fig. 14. Curves of \(E\) as a function of \(\mathbf{k}\) for the direction \((1,1,0)\) in a PbS crystal.

that the filled band and the conduction band are separated by a narrow band of forbidden energies, situated, however, not at the edge of the Brillouin zone, as had been expected. The width of the forbidden band proved to be about \(0.3\ \text{eV}\), but since these calculations in their present form can give only the order of magnitude of \(\Delta E\), this result cannot be regarded as convincing evidence against the higher value of \(1.17\ \text{eV}\) (see 3.2), found experimentally by Putley. The form of the energy levels as a function of \(\mathbf{k}\) for one direction in a PbS crystal is shown in Fig. 14. The curves presented here refer to the conduction band and to the highest level of the filled band, formed by two overlapping energy bands.

As can be seen from Fig. 14, the highest value of the energy in the filled band exhibits a maximum at a certain value of \(\mathbf{k}\). If this maximum had the character of a sharp peak, then it could have been

explain the low value of the energy associated with the long-wavelength limit of optical absorption in relation to the value \(\Delta E\), determined from measurements of the Hall constant. The form of this maximum is sensitive to small changes in the electronic wave functions, so that it would be very interesting to obtain the results of more accurate calculations.

Starting from the self-consistent field obtained by means of the calculations mentioned above, the authors determined the energy required for the formation of an exciton. It proves to be close to \(0.3\) eV. This quantity is much less sensitive to the form of the wave functions than \(\Delta E\), and it is probably not too far from the true value. It is easy to see that the indicated value is much larger than that which would be obtained by calculating the energy levels in a Coulomb field with a changed dielectric constant—a method sometimes used in the literature to determine the exciton formation energy. In this case the energy of the first excited state is equal to \(10 \cdot \dfrac{1}{n^{4}}\), where \(n\) is the corresponding refractive index. For \(n = 4\) this gives \(0.04\) eV.

6.4. Relation between the Long-Wavelength Limit of Photosensitivity and the Dielectric Constant

Moss \(^{74,75}\) showed that there is an approximate correspondence between the dielectric constant \(\varepsilon\) and the value \(\lambda_{1/2}\) for the long-wavelength limit of photoconductivity of many photoconducting compounds, including substances of the PbS group. This correspondence is expressed by the equality

\[ \frac{n^{4}}{\lambda_{1/2}} = 77, \tag{4} \]

where \(\lambda_{1/2}\) is expressed in microns. Moss also considered possible explanations for this type of dependence. If it is assumed that the long-wavelength limit of photoconductivity is due to impurity centers and a field of Coulomb type, then a dependence like that represented by equality (4) would occur if, in all photoconductors for which this dependence is valid, the corresponding centers had one and the same nature. Pincherle \(^{97}\) showed that for the case of \(F\)-centers the Coulomb field is a poor approximation. In any case it appears unlikely that photoconductivity should be caused by centers of a single kind in such a great variety of substances for which this dependence is approximately valid. Some dependence of this kind should exist as a consequence of the relation between the refractive indices and absorption, but it is unlikely that one could derive so ...

a simple dependence from the complex course of the actually observed absorption. At present it is most correct to regard relation (4) as an empirical dependence, useful for an approximate determination of the position of the long-wave limit of a substance’s photoconductivity from the known value of the refractive index. It should be emphasized that this dependence is only approximate and in fact gives incorrect values in the case of PbTe and PbSe.

§ 7. PHOTORESISTORS AS RECEIVERS OF INFRARED RADIATION

The properties of PbS photoresistors and their applications for the detection of infrared radiation were considered in the reviews by Elliott\(^{58}\), and also by Sze­kerland and Lee\(^{6}\). The first review covers chiefly work carried out in Germany during the war. The second also includes these works, but, in addition, considers work carried out in England and America up to 1947. The receivers described in these reviews have been considerably surpassed by the receivers available at present. In particular, much higher sensitivities have now been obtained for layers operating without cooling. This applies both to evaporated layers and to layers obtained by the method of chemical deposition. In photoresistors using evaporated layers, the photosensitive elements are usually placed in an evacuated glass bulb. Chemically deposited layers are protected from the action of the atmosphere by means of a thin film of transparent material. Placing them in vacuum is also practiced, however, for chemically deposited layers. Very little is known about the operation of photoresistors sensitive in the infrared region as receivers of infrared radiation. This also applies to the details of methods for their fabrication, which, apparently, are still to a considerable extent connected with the skill of the experimenter. Various methods for producing photoconducting layers were already considered in 2.6, where references to the corresponding articles were given. In a recently published review, Simpson and Sze­kerland\(^{99}\) give some data on recent American work. Milner and Watts\(^{36}\) and Moss\(^{75}\) reviewed recent English work. Here we shall first consider the available data on photoresistors made of PbS, PbTe, and PbSe, and then briefly discuss the principal factors limiting the sensitivity of such photocells. In doing so we shall deal only with photoresistors, since up to the present no other type of photocell can compare with them in threshold sensitivity.

Before speaking about the sensitivity of photoresistors, we must establish the distinction between two quantities, ot-

...relating to this concept. These quantities are: a) the sensitivity as the magnitude of the response of the photoresistor and b) the minimum energy that the photoresistor is capable of detecting. We shall call the first of these quantities “sensitivity,” and the second “threshold sensitivity.” One should not think that the photoelement with the greatest sensitivity necessarily also has the best threshold sensitivity. This is not so, because the latter depends on the level of its own noise. These noises usually have the character of current noises and are caused by the flow of current through the photosensitive layer of the photoresistor.

Let us suppose that we have a photoelement with resistance \(R\), connected to the input of an amplifier by means of a load with the same resistance and supplied by a voltage \(V\) (Fig. 15). Then the signal voltage \(V_s\), produced by a change in resistance \(\Delta R\) as a result of illumination, will be equal to

Fig. 15. Circuit for connecting a photoresistor.

\[ V_s=\frac{V\Delta R}{4R} \tag{5} \]

provided that \(\Delta R \ll R\). If the amount of incident energy \(W\) produces a change in the resistance \(R\) by \(\Delta R\), then for small values of \(W\) we may write \(\Delta R=aW\). Then

\[ V_s=\frac{VaW}{4R}=rW . \tag{6} \]

The quantity \(r\) is usually called the “sensitivity” and is expressed in volts per watt. As is seen from (6), it depends on the applied voltage \(V\).

If \(V_n\) is the effective noise voltage across the load, then the signal-to-noise ratio will be \(\dfrac{V_s}{V_n}\). In particular, when this...

if the ratio is equal to unity, then the corresponding value of the incident energy \(W_m\) will be:

\[ W_m=\frac{V_n}{r}. \tag{7} \]

This value of \(W\) is usually taken as the minimum detectable amount of radiant energy. Usually the current noises \(V_n\) are approximately proportional to \(V\), so that, to a first approximation, \(W_m\) does not depend on \(V\). This holds over a fairly wide interval of values of \(V\), which must be large enough for current noises to predominate, but at the same time so small that no appreciable heating of the layer occurs. For such noises in a layer of the given type with area \(A\), \(V_n\) is proportional to \(A^{1/2}\) for a given value of the current strength, so that one may expect that photoresistors with a small area will have a smaller value of \(W_m\) than similar photoresistors with a larger area of the photosensitive surface. This somewhat complicates the comparison of actual photoresistors, and therefore it is desirable to reduce the actual value of \(W_m\) to the value of this quantity for a standard area. As such a standard we shall take \(0.1\ \mathrm{cm}^2\), since this quantity is often used in practice \((1\ \mathrm{cm}\times 1\ \mathrm{mm})\).

The value of \(V_n\), besides the properties of the layer, also depends on the bandwidth \(\Delta f\) of the amplifier and the indicator device. For values of \(\Delta f\) so small that the noise spectrum may be regarded as uniform within the interval \(\Delta f\), \(V_n\) is proportional to \((\Delta f)^{1/2}\). However, for current noises over a wide range of frequencies (from several hertz to several kilohertz) the noise power per unit bandwidth is proportional to \(1/f\), as a result of which proportionality between \(V_n\) and \((\Delta f)^{1/2}\) at real values of \(\Delta f\) may fail to hold. Therefore it is convenient to reduce the value of \(W_m\) to a standard bandwidth (for example, \(1\ \mathrm{cps}\)), but when using values obtained in this way one must be cautious.

7.1. Photoresistors from PbS

Some authors (for example, Strong[^4], Simpson and Seeland[^99]) have pointed out that PbS photoresistors, with respect to threshold sensitivity, are approximately 100 times better than a good thermocouple. Therefore we might expect that, within the region of high sensitivity, we shall have a minimum detectable energy of the order of \(10^{-12}\ \mathrm{W}\) for monochromatic radiation with a bandwidth of \(1\ \mathrm{cps}\). This gives only an order of magnitude, since the area of the receiving element is not specified. According to Moss[^100], for a PbS photoresistor at room temperature only \(0.7\%\) of the energy of a source with a temperature of \(200^\circ\mathrm{C}\) is useful energy, so that the value of \(W_m\) for black-body radiation at tem-

temperature of 200° C will be about \(10^{-10}\) W for a bandwidth of 1 cps, i.e., it proves comparable with the threshold sensitivity of a thermocouple. Milner and Watts\(^{36}\) give values of only \(2.5 \cdot 10^{-8}\) W for radiation at a temperature of 200° C and \(2 \cdot 10^{-11}\) W for monochromatic radiation with wavelength \(2.2\,\mu\) for certain types of industrial photoresistors. Thus, their sensitivity proves to be somewhat lower than that which should have been expected on the basis of the above. The values cited improve to \(5 \cdot 10^{-9}\) and \(4 \cdot 10^{-12}\) W, respectively, when the layer is cooled with solid carbon dioxide, which brings them into the expected range of values. These values refer to a frequency of 800 cps. After cooling, the time constant of the photocells increases from 40 to 200 μsec.

Fig. 16. Spectral output of a lead-sulfide photoresistor at room temperature.

Data for industrial PbS photoresistors prepared by chemical deposition, published by Eastman Kodak Co.,\(^{35}\) give values of \(W_m\) for radiation at a temperature of 500° K of the order of \(3 \cdot 10^{-8}\) W at room temperature (when recalculated to an area of \(0.1\,\mathrm{cm}^2\)). However, the bandwidth is not specified. The time constant of these photocells is rather large (of the order of 200–300 μsec), and the values of \(W_m\) refer to a frequency of 90 cps.

From the experience of spectroscopists using PbS photoresistors, it follows that, with photocells fabricated under laboratory conditions,

under these conditions, values of \(W_m\) smaller than those indicated above are obtained. For example, Feldget\(^{101}\) gives a value of \(W_m\) in the region of maximum sensitivity equal to \(8 \cdot 10^{-13}\) W/Hz (reduced to an area of \(0.1\ \mathrm{cm}^2\)); this value was obtained with solid-carbon-dioxide cooling. His photoresistor had a sensitized light-sensitive layer and, apparently, with photocells of this type one can obtain somewhat lower values of \(W_m\), especially at room temperature, than in the case of chemically deposited layers. Evidently, experience with such laboratory-made photoresistors indicates that for the best photocells cooling produces only a very small increase in output (not to mention a shift of the long-wavelength limit). These photocells usually have time constants of the order of \(20\)—\(100\ \mu\mathrm{s}\) at room temperature. The spectral sensitivity of a modern PbS photoresistor is shown in Fig. 16. The changes of the spectral characteristic with temperature are similar to those indicated by Moss (see Fig. 5). These photoresistors have a remarkable constancy of yield per quantum in the wavelength region approximately from 1 to \(2.5\ \mu\) (at room temperature); then the yield falls, reaching half its value \((\lambda_{1/2})\) at about \(2.9\ \mu\), and decreases to 1% of its maximum value at about \(3.35\ \mu\).

7.2. PbTe photoresistors

The chief advantage of PbTe photoresistors over PbS photoresistors is that they can be used in the region of considerably longer wavelengths. There are also indications that with PbTe photoresistors somewhat larger values of maximum sensitivity can be obtained. Various authors (for example, Smith\(^{94}\), Simpson and Sutherland\(^{99}\)) give threshold sensitivities exceeding by 100—1000 times the sensitivity of a good thermocouple. Thus one should expect values of the minimum detectable energy for monochromatic radiation of the order of \(10^{-12}\)—\(10^{-13}\) W for a bandwidth of 1 Hz. Simpson, Sutherland, and Blackwell\(^{42}\) gave the figure \(2 \cdot 10^{-14}\), but this refers to a photoresistor whose maximum sensitivity lies near \(2\ \mu\), and Feldget expressed doubt as to the reliability of this figure. Feldget\(^{101}\) indicates a value of \(W_m\) (reduced to an area of \(0.1\ \mathrm{cm}^2\)) equal to \(4 \cdot 10^{-13}\) W.

The methods of making photoconducting layers of PbTe used by various authors have already been described in 2.6. A number of PbTe photoresistors made at the NPL were given to several investigators in the field of infrared spectroscopy. Although none of them reported numerical data on threshold sensitivity, from the published spectra (for example, Thompson and Williams\(^{2}\), Boyd and Thompson\(^{3}\)), obtained with the aid of these

photosensitivities, it is clear that the minimum energies detected by them must be of the order of \(10^{-13}\) W for a bandwidth of \(1\ \mathrm{cm}^{-1}\). In a recently published review of PbTe and PbSe photoresistors, Moss \(^{100}\) reports that Tompson obtained a resolution of \(0.15\ \mathrm{cm}^{-1}\) (with a 100-mm grating) at \(3\ \mu\), using NIER PbTe photoresistors. This indicates that the threshold sensitivity is approximately 500 times higher than that of a thermocouple. In addition, these photoresistors may have a very small time constant (of the order of several microseconds) and thus can be used for recording spectra at much higher speeds than are attainable with a thermocouple.

Fig. 17. Spectral response of a lead telluride photoresistor at \(90^\circ\mathrm{K}\).

NIER photocells are made by evaporating a PbTe layer in oxygen at low pressure (several variants of this method are described by Moss \(^{100}\)). The layers are obtained in a glass vessel, similar to a Dewar vessel, provided, for access of radiation to the sensitive layer, with a sapphire window sealed to the glass through transition seals. The photoresistors have very low sensitivity at room temperature, so that they are always operated at the temperature of liquid air. The spectral characteristic of a modern PbTe photoresistor (NIER) at liquid-air temperature is shown in Fig. 17. The value of \(\lambda_m\), approximately equal to \(4.75\ \mu\), and the sensitivity falls to 1% of the maximum value at \(\sim 5.75\ \mu\).

7.3. PbSe Photoresistors

The discovery by Gibson, Lawson, and Moss[^49], who showed that photoelements made of PbSe can have a long-wavelength limit at considerably greater wavelengths than in the case of PbTe, led to an unexpected step forward and opened the possibility of applying photoelectric methods in new regions of the spectrum. Various authors had previously described PbSe photoresistors, but these did not surpass PbTe photoresistors in long-wavelength sensitivity and, moreover, seemed inferior in threshold sensitivity (with the exception of the room-temperature region, for which PbSe has a considerably greater sensitivity than PbTe). The methods of preparing PbSe layers used by various authors were described in 2.6.

Fig. 18. Spectral response of a lead selenide photoresistor at 290, 90, and 20° K.

In his latest review Moss[^100] gives further information on PbSe photoresistors (prepared at the NPL), sensitive beyond 9 μ at the temperature of liquid air. The method of preparation is almost the same as for photoelements made of PbTe. In the case of PbSe, entrance windows of periclase (MgO) are used, since the transmission of sapphire in the long-wave region is insufficient for full utilization of the long-wavelength sensitivity of PbSe layers. The curves of spectral sensitivity of typical NPL PbSe photoelements (normalized to a single value at the maximum of the curve for three temperatures) are shown in Fig. 18. At tem-

at liquid-air temperature, the maximum sensitivity of modern PbSe photoresistors is lower than that of the best PbTe specimens. At the same time it is considerably higher than that of thermopiles, and, moreover, their time constant is much smaller—of the order of 10 μsec. With the aid of such a photoresistor, Roberts and Young\(^ {102}\) observed the center of the water-vapor absorption band in the region of 6 μ at a rate of 50 spectra per second. These authors also obtained a photoelectric recording of the entire absorption spectrum of water vapor in the interval from 5 to 7 μ at a normal recording speed, with a resolution of the order of \(1\ \mathrm{cm}^{-1}\). The resolution, however, was determined rather by the spectrometer than by the photoresistor.

It is interesting to note that photoresistors capable of operating at room temperature, like PbS photoresistors, can be made from PbSe. As yet, however, their sensitivity at room temperature is considerably lower. Starkiewicz\(^ {47}\) gives a value of \(W_m\) equal to \(2 \cdot 10^{-9}\) W for a bandwidth of 1 cps. This is a very old figure, and it is reasonable to suppose that, as for all other types of photoelements, later work has led to higher sensitivity values. When PbSe layers are cooled with liquid hydrogen instead of liquid air, no further increase in threshold sensitivity is observed, but the long-wavelength sensitivity increases and becomes appreciable near 10 μ. The values of \(\lambda_{1/2}\) for 290, 90, and \(20^\circ\) K are respectively 4.7, 7.1, and 8.2 μ; the fall of sensitivity to 1% occurs respectively at about 7.0, 9.3, and 10.2 μ (extrapolated).

7.4. The sensitivity limit of photoresistors

If the noises produced by the sensitive layer of a photoresistor were sufficiently small, then its threshold sensitivity would be determined by fluctuations of the light flux incident on the photoelement. This question was considered by Feldget\(^ {101}\) and Moss\(^ {103}\), who calculated the threshold sensitivity for a number of photoelements. These authors showed that if the quantum energy corresponding to the long-wavelength limit is less than \(5kT\) (which is true for the photoelements considered), then the influence of radiation fluctuations can be calculated simply from the fluctuations \(\Delta N\) in the number of active quanta \(N_e\) incident on the layer per second, by means of the simple “classical” formula for the mean-square value

\[ \overline{\Delta N^2} = N_e, \tag{8} \]

where \(N_e\) is given by the expression \(q_m n_e\), where

\[ n_e = \frac{1}{h\nu_m}\int_0^\infty s(\nu) W(\nu)\,d\nu, \tag{9} \]

and \(q_m\) is the quantum yield at the light frequency \(\nu_m\) corresponding to the maximum sensitivity, \(s(\nu)\) is the equal-energy spectral response of the photoresistor, reduced to unity for the frequency \(\nu_m\), and \(W(\nu)\) is the radiant energy incident on the layer in the frequency interval from \(\nu\) to \(\nu + d\nu\). \(W(\nu)\) is usually expressed simply by the Planck distribution of the radiation incident on the layer with room temperature (the receiving surface in this case will be one or both sides of the layer, depending on whether it is cooled or not).

Let us now assume that \(N_s\), the number of active quanta absorbed by the layer, produces a voltage signal \(aV_s\). Then the voltage fluctuation \(\Delta V\) corresponding to the fluctuation \(\Delta N\) will be:

\[ \overline{\Delta V^2} = a^2 \overline{\Delta N^2}. \tag{10} \]

For a bandwidth equal to \(\Delta f\), we have:

\[ \overline{\Delta V^2} = 2a^2 \overline{\Delta N^2}\Delta f. \tag{11} \]

The signal voltage \(V_m\), when energy \(W_m\) of frequency \(\nu_m\) is incident on the layer, is equal to

\[ V_m = aq_m \frac{W_m}{h\nu_m}. \tag{12} \]

The value \(W_m\), which is usually taken as the minimum detectable energy, is obtained by putting \(V_m^2 = \Delta V^2\), whence

\[ W_m = \left(\frac{2\Delta f}{q_m}\right)^{1/2} h\nu_m n_e^{1/2}. \tag{13} \]

Feldget \(^{101}\) and Moss \(^{103,104}\) calculated \(W_m\) by the indicated method for a number of photocells, taking \(q_m = 1\). For example, Feldget found \(W_m = 3.5 \cdot 10^{-13}\) W for the PbS photoresistor discussed in 7.1, whereas the measured value (reduced to an area of \(0.1\ \text{cm}^2\)) turned out to be \(8 \cdot 10^{-13}\) W, i.e. only 2.8 times greater than the sensitivity threshold due to radiation fluctuations. For a cooled PbTe photoresistor he found an even smaller coefficient, namely 1.9. Thus it is clear that the best photoresistors possess a threshold sensitivity very close to the ultimate limit and, moreover, that the value \(q_m\)—the quantum yield at the maximum of the spectral-sensitivity curve—cannot differ greatly from unity. As we have seen, for the majority of good modern photocells the yield per quantum is practically constant over a fairly wide wavelength interval. It is very tempting to suppose that in this range \(q_m\) is indeed equal to unity. The threshold in real photoresistors is usually determined by current noises, which have a spectrum

of the \(1/f\) type, whereas it should be expected that the spectrum of radiation fluctuations is uniform, at least at frequencies that are small compared with \(1/r\), where \(r\) is the time constant of the photoresistance.

Proceeding from the considerations given above, Simpson\(^{16}\) and Uot\(^{105}\) expressed the idea that if room-temperature radiation were completely screened from the photoconducting layer when it is cooled, say, to \(90^\circ\) K, then its threshold sensitivity should increase as a result of the decrease in radiation fluctuations. Moreover, they showed experimentally that such an increase does in fact occur. For PbTe photoresistances, and also for some PbS photoresistances, screening the room-temperature radiation can lead to an increase of the layer resistance by as much as 100 times, and also leads to a considerable lowering of the minimum detectable energy. A noticeable increase in sensitivity as a result of eliminating extraneous radiation was also observed by Simpson and Sutherland\(^{43}\). However, the lowering of the minimum detectable energy is not entirely due to the reduction of radiation fluctuations. An increase in resistance reduces the current at a given voltage across the layer, which in turn reduces the current noise. Therefore the indicated phenomenon should be observed even in photocells whose threshold sensitivity is limited by current noise. Unfortunately, with photocells used in spectroscopy it is not possible to make full use of the advantage of the considerable reduction in minimum detectable energy, which in some cases reaches a factor of ten. In spectroscopy, in order for all the observed radiation to fall on the receiving surface, it is usually necessary for the photoresistance to have a rather large entrance window, which exposes the photosensitive layer to the action of a certain amount of room-temperature radiation. In the case where it is possible to use a sufficiently narrow entrance window, a significant gain in threshold sensitivity can be obtained. Under such conditions PbTe photoresistances were used. They gave values of the minimum detectable energy smaller than that which theoretically should be observed if only radiation fluctuations are taken into account, assuming that the sensitive layer is not screened from room-temperature radiation on one side.

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