PHYSICS

V. M. FRIDKIN, N. T. KASHUKEEV, and I. S. ZHELUDЕV

ON THE THEORY OF PHOTOELECTRETS

(Presented by Academician A. V. Shubnikov, 27 VII 1957)

The photoelectret state in polycrystalline sulfur was discovered and investigated by G. Nadzhakov in 1937 (¹–³). In the process of photoconductivity, a volume charge accumulated in a layer of polycrystalline sulfur which, after the polarization was discontinued, could persist for a long time. A dielectric polarized under photoconductivity was an analogue of the electrets discovered by Eguchi in 1921, and was named a photoelectret by G. Nadzhakov.

The scheme of electron energy levels in a sulfur single crystal proposed by P. S. Tartakovskii and G. Rekalova (⁴), and the investigation of volume charges under photoconductivity carried out by P. S. Tartakovskii and D. M. Kaminker (⁵), make it possible to draw the following conclusions concerning the mechanism of formation of a photoelectret. Under the action of light, electrons pass from the normal band into the conduction band, where the electrons undergo a displacement in the direction of the field and, at the end of the displacement, falling out of the conduction band, become fixed at certain local levels (sticking levels) situated below the conduction band (Fig. 1). In sulfur the local levels may be associated with inhomogeneities of the crystal structure. As shown by the investigations of G. Nadzhakov and N. T. Kashukeev (⁶), the dark depolarization of a photoelectret can be explained by the thermal transition of electrons from the normal band to free sticking levels, which at the same time causes an increase in the dark hole conductivity. The activation energy obtained in that work, corresponding to the transition of electrons from the normal band to sticking levels, proved to be equal to 0.48 eV, which is in agreement with the results of work (⁴).

Fig. 1. Scheme of electron energy levels in sulfur

In the present work an attempt is made to explain certain experimental results obtained in the investigation of photoelectrets (⁷).

Let $d_1$ be the number of electrons passing, under the action of light, per unit volume and per unit time from the normal band into the conduction band, and $d_2$ the number of electrons passing, under the action of light, into the conduction band from filled sticking levels.

\[ d_1 = s_1 E, \tag{1} \]

\[ d_2 = s_2 N E = kN. \tag{2} \]

Here $E$ is the intensity of light (illumination), $N$ is the concentration of sticking levels filled with electrons, and $s_1$ and $s_2$ are coefficients depending on the absorption of light and on the quantum yield. If

if one assumes for sulfur an excitonic character of light absorption and admits that the transition of electrons to the conduction band from the normal band and from local levels occurs as a result of “excitonic impacts of the second kind” \((^8)\), then for \(d_1\) and \(d_2\) one can obtain expressions analogous to those given in \((^8,\,^9)\).

The kinetic equations describing the process of filling trapping levels with electrons and corresponding to the scheme shown in Fig. 1 have the following form:

\[ \frac{dn}{dt}=d_1+kN-\beta n(M-N)-\alpha nP, \tag{3} \]

\[ \frac{dN}{dt}=Q+\beta n(M-N)-kN-\gamma NP, \tag{4} \]

\[ \frac{dP}{dt}=d_1+Q-\alpha nP-\gamma NP, \tag{5} \]

\[ P=N+n, \tag{6} \]

where \(M\) is the concentration of trapping levels, \(N\) is the concentration of electrons at trapping levels, \(n\) is the concentration of electrons in the conduction band, \(P\) is the concentration of holes in the normal band, \(Q\) is the number of electrons passing per unit volume and per unit time, under the action of thermal motion, from the normal band to the trapping levels, and \(\alpha\), \(\beta\), and \(\gamma\) are recombination coefficients, whose meaning is clear from the scheme shown in Fig. 1.

The problem reduces to determining the dependence of \(N\) on time, since the concentration of electrons at the trapping levels should be regarded as proportional to the charge of the photoelectret. In doing so, naturally, the kinetics of electron motion in the conduction band is not taken into account and, consequently, neither is the influence of the field strength on the magnitude of the photoelectret charge.

Since at room temperature the thermal transition of electrons from the normal band to the levels may be neglected \((^6)\), we put \(Q=0\). Assuming that \(n/M\) and \(N/M\) are small parameters, we restrict ourselves to the linear terms in equations (3) and (4). Bearing in mind, moreover, that \(d_2 \ll d_1\), we obtain, for zero initial conditions, the following solution of the system of equations (3) and (4):

\[ N=N_c(1-e^{-s_2Et}), \tag{7} \]

where the concentration of electrons in the stationary state is equal to:

\[ N_c=s_1/s_2. \tag{8} \]

This solution, as was noted above, was obtained for the case when, in the polarization process, only an insignificant fraction of the free levels is filled with electrons.

The expression (7) obtained for the dependence of the concentration density of electrons at trapping levels on time conveys the saturation effect found in \((^7)\) in the study of the dependence of the photoelectret charge on polarization time and illumination.

Figure 2 presents the experimental curves obtained in \((^7)\) for the dependence of the initial depolarization current of a photoelectret (proportional to the magnitude of the polarization of the photoelectret) on the polarization time for different illuminations (a sulfur single crystal). From the curves presented it is seen that (7) satisfactorily conveys the exponential character of the dependence of the photoelectret charge on illumination and polarization time.

From (7) and (8) it also follows that, in the formation of a photoelectret, ful-

the reciprocity law is satisfied. This means that for any polarization time \(t\) and any illumination \(E\), the charge of the photoelectret is only a function of the product \(Et\). With the aid of the curves shown in Fig. 2, an experimental verification of the reciprocity law was carried out for a sulfur single crystal. Fig. 3 shows the dependence of the exposure \(Et\), required to obtain a definite charge of the photoelectret, on the illumination \(E\). It is easy to see that the reciprocity law is fulfilled for all illuminations used in polarizing the sulfur single crystal.

Fig. 2. Curves of the dependence of the initial depolarization current, proportional to the charge of the photoelectret, on the polarization time (sulfur single crystal). The polarizing-field strength is \(U = 1.5\) kV/cm. Illuminations used in polarization:
\(1\)—\(110 \cdot 10^{-6}\), \(2\)—\(55 \cdot 10^{-6}\), \(3\)—\(33 \cdot 10^{-6}\), \(4\)—\(17 \cdot 10^{-6}\), \(5\)—\(7 \cdot 10^{-6}\) W/cm\(^2\).

In this case the illumination was varied within the limits from \(110 \cdot 10^{-6}\) W/cm\(^2\) to \(7 \cdot 10^{-6}\) W/cm\(^2\).

The theoretical derivation and experimental verification of the reciprocity law for photoelectrets are of independent interest. This law can be made the basis of electrophotography on photoelectrets \((^{10,11})\), by analogy with the reciprocity law for photochemical processes in silver halides. Taking into account the kinetics of electron motion in the conduction band during the formation of polarization should lead to deviations from the reciprocity law. This may serve as a possible explanation of certain violations of the reciprocity law that were observed during short-time polarization of sulfur single crystals in work \((^{7})\).

As was assumed above, the saturation effect observed in studying the dependence of the photoelectret charge on the polarization time and illumination may be due to filling by electrons of only a small part of the free levels. The correctness of this assumption is confirmed by an experimental study of the dependence of the photoelectret charge on the strength of the polarizing field. Fig. 4 presents the dependence of the charge density of a photoelectret (sulfur single crystal), measured by integrating the depolarization current over time, on the strength of the polarizing field. From the results obtained it is seen that for sulfur there is a direct proportionality between the charge of the photoelectret and the strength of the polarizing field, and saturation is absent. Saturation was not observed even when fields of the order of 20 kV/cm were used. In this case the polarization conditions were chosen so that the charge of the photoelectret did not change when the illumination and polarization time were varied over wide limits, i.e., saturation due to the mechanism discussed above took place. The results obtained confirm the assumption made above that, in the process of formation of the photoelectret, partial filling of the levels by electrons occurs, which, in turn, makes it possible to abandon the earlier \((^{7})\) assumption of the filling by electrons of the entire finite number of traps in the crystal. On the other hand, the theoretical dependence of the photoelectret charge on the strength of the polarizing field can be given only on the basis of the general mechanism

Fig. 3. Verification of fulfillment of the reciprocity law in the formation of the charge of a photoelectret (sulfur single crystal)

polarization, including the kinetics of electron motion in the conduction band.

Taking into account in the kinetic equations \(^{4,5}\) the effect, investigated in \(^{6}\), of the transition of electrons under the action of thermal motion from the normal band to sticking levels leads to a dependence of the stationary electron concentration \(N_c\) and, consequently, of the photoelectret charge on temperature.

In this case the charge of the photoelectret, proportional to the stationary concentration of electrons on the levels, decreases with increasing temperature. This effect is due to the fact that the thermal transition of electrons from the normal band to the levels reduces the total polarization of the crystal, caused by the external field and due to the nonuniform distribution of charges. Investigations of the dependence of the photopolarizability of sulfur single crystals on temperature, carried out by G. Nadjakov and N. T. Kashukeev \(^{6}\), confirm the results obtained.

Fig. 4. Dependence of the charge density of the photoelectret \(\sigma\) (sulfur single crystal) on the strength of the polarizing field. Polarization time \(t = 10\) min.; illumination during polarization \(E = 100 \cdot 10^{-6}\) watt/cm\(^2\).

Taking the temperature effect into account leads to a violation of the reciprocity law. At high temperatures the deviations from the reciprocity law should increase as the illumination used for polarizing the crystal decreases.

The authors express their deep gratitude to Academician A. V. Shubnikov and Academician G. Nadjakov for their constant attention to the work.

Received
27 VII 1957

CITED LITERATURE

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