Problems of Photoconductivity

Problems of Photoconductivity

.

P GORLICH Institute for Optics and Spectroscopy. German Academy of Sciences. Berlin. and Friedrich Schiller University. Jena. Germany

Page I . Introductory Considerations on Photoconductivity . . 37 I1. Photoconduction in the Base Lattice and Tail Absor 39 I11. Theoretical Problems in Photoconductivity. . . . . . . . 40 42 A. Lifetime: Theoretical Considerations . . . . . . . . . . . . 43 B. Saturated and Unsaturated Photocurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 C. Advantages of the Concept of Lifetime . . . . . . . . D. Reaction Kinetic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 E. Steady State Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 . . . . . . . . . . . . . . . . . . 50 F Rise and Decay Processes sity . . . . . . . . . . . . . . . 50 G. Photocurrent Dependence H . Demarcation Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 I. Wave Vectors and Crystal Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 I V . Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 V . Negative Photoconductio .................................... 58 VI . Surface Conditions ....... ......................... 60 V I I . Ohmic and Unidirectional ns . . . . . . . . . . . . . . . . . . . . . 64 A . Unidirectional and Isotropic Contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 B. pn-Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 C. Photo-emf in Boundary Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 VIII . Photoelectromagnetic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 IX . Application of Photoconductors . . . . . . . . . . . . . . . '70 . . . . . . . . . . . . . . . . . . . '70 A . Tabular Survey of Photoconductors . . . . . . . . . 73 B. Frequency Dependence and Amplification F a 78 C . Statistical Fluctuations in Photoconductors . . X . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

.

I . INTRODUCTORY CONSIDERATIONS ON PHOTOCONDUCTIVITY Except for some special cases which are not sufficiently clear. semiconductors with completely filled valence bands. empty conduction bands. and lacking ionization of the defect states. would not exhibit any conductivity without electron excitation . Of the four possibilities of energy input for electron excitation. we will treat in this paper the thermal and optical excitations. excluding the excitation through particle bombardment and 37

38

P. GORLICH

electrical excitation which is essentially the problem of electrical breakdown. Thermal excitation interests us only insofar as a semiconductor at a given temperature exhibits a dark field conductivity. That is to say, that even prior to photon injection, the semiconductor exhibits the so-called dark current. The electron concentration n in the conduction band can be calculated by

where Ei is the energy of the Fermi level, m the effective mass of the electron, and EL the energy of the lowest level in the conduction band. If we assume the validity of Ohm's law in a homogeneous solid, the relation between the current density i and field strength E is given by

i

=

U E = nepnE,

(2)

where u is the conductivity of the n electrons with mobility p n . If, in addition, p holes with mobility p p participate in the production process then Eq. (2) is changed to

i = UE =

+ pepp)E.

(mpn

(3)

One might increase the current density by an amount A i by injecting photons into the lattice or tail absorption regions. By increasing the carrier concentrations ( A n and A p ) , the increased current density ( A i ) is given by Ai

=

(Amp,, - Apepp)E1

(4)

where it is assumed that neither the mobilities nor the field strength are changed. I n general, the observed changes of the mobilities are of no or, a t best, of little consequence. On the other hand the second assumption requires a homogeneous excitation of the photoconductors as well as a homogeneous distribution of the charge carriers during their migration in the electric field. The latter assumption is not always fulfilled a t high field strengths (insulators) so that deviations from Ohm's law are exhibited and saturation of the photocurrents is observed. Equation ( 4 ) describes the positive photocurrent conductivity assuming an increase of the carrier concentration, that is, an increase of electrical conductivity under illumination. Under certain conditions [for example, bombardment of germanium with fast electrons ( I ) ] one can bring about a negative photoconductivity, that is, a decrease of electrical conductivity under illumination.

PROBLEMS OF PHOTOCONDUCTIVITY

39

11. PHOTOCONDUCTION IN THE BASE LATTICE AND TAILABSORPTION REGIONS Although one should be able to excite electrons from the valence band t,o higher levels through photon injection throughout the complete base lattice absorption region, it is well known that not all semiconductors (and insulators) exhibit an inner photoelectric effect or, a t best, exhibit it only a t the band edge (except for some possible effects in the tail absorption region). There is a very good and interesting explanation for this phenomenon : that is, the photoelectric excitation is vanishingly small compared with the thermal excitation or the recombination rate of the photoelectrically formed charge carriers is extremely large (often brought about through recombination processes in the surfaces of the photoconductors, see Sec. VI). Also, excitons (2) may be formed. The observed fine structure of the yield of the inner photoelectric effect of cadmium sulfide near the band edge shows in an impressive manner the possibility for exciton formation (3). The investigations of photoconductivity in cadmium sulfide appear to be particularly suited to clarify extensively the exciton states. One then comes to the problem, if and in what manner the structure of the yield curve which appears more distinctly at low temperatures and under application of polarized light is related to the structure of the optical absorption. Apparently a relation exists (4),and the question arises whether the structural form of the absorption may be understood in terms of a base lattice effect or in terms of a n effect of the structure of the real lattice. The experiments indicate that the real structure of the crystals exerts a decisive effect ( 5 ) . Furthermore, the structural form of the absorption was already recognized earlier in the case of evaporated photoconductive CdSe layers (8). It has long been recognized that there is no photoconductivity observed in the region of the base lattice absorption of uncolored alkali halides. One has recently been led to the conclusion that the absorption of suitable radiation leads to exciton formation. The investigation of inner photoconductivity in the tail absorption region opens the possibility of relating certain absorption regions to known lattice defects. The excitation of electrons takes place from the forbidden levels lying between the valence and the conduction band to the conduction band or from levels in the valence bands to the levels in the forbidden zone (the latter case being defect (hole) conduction). If i t is possible to establish these correspondences in a clear-cut manner, preferentially in crystals with heteropolar binding (e.g. the colored alkali halides), it appears possible to establish methods to relate the lattice defects in homopolar crystals to their washed-out structureless tail absorption. For example, additional monochromatic irradiation in the region of the absorption edge

40

P. GORLICH

causes a change in the photoconductivity of weakly excited photoconducting cadmium sulfide with increasing wavelength ( 7 ) . We may expect clarification of the above-mentioned exciton mechanism at least in the cases of cadmium sulfide and zinc sulfide if we irradiate with light whose wavelength is longer than that corresponding to the absorption edge (8). If we take these experiments to be an indication of the existence of exciton excitations in cadmium sulfide and zinc sulfide then they lead to consideration of energy transport. That is, we take the diffusion parameter of the excitons to be dependent upon the real structure, the diffusion constant to lie between lo3and lo4 cmZ/sec and the lifetime to be of the order of to 10-6 sec. However, there exists experimental evidence contrary to the concept of exciton excitation (9) and work in this direction is in progress. The lifetime of the excitons increases with decreasing temperature. Based upon analyses of the changes in line widths, we are able to take 3X sec as the lifetime for excitons in silicon and 1.5 X 10-l' sec for those in germanium for temperatures below 100°K (10).

111. THEORETICAL PROBLEMS IN PHOTOCONDUCTIVITY Note that the theoretical treatment of the inner photoelectric effect requires as its basis that the process inverse to excitation of electrons by light be represented by recombination, which leads to a lowering of the concentration of charge carriers. A steady state between excitation and recombination is obtained if we maintain a steady illumination of the photoconductors. An interruption of the illumination causes a decay of the additionally excited charge carriers through recombination. The decay process is observed until one again reaches thermal equilibrium. Recombinations are, in principle, possible (1) from conduction band to valence band, (2) from conduction band to a defect state, (3) from a defect state to the valence band, (4)from a defect state to a defect state, and (5) from an exciton state to the ground state. One must distinguish between a radiative and a nonradiative recombination. Radiative recombination is called luminescence. I n general, only a small part of the excited electrons recombine with emission of light quanta. In order to be able to describe the conduction processes in a photoconductor, one must determine experimentally the chemical nature of the lattice defects and the influence of other parameters in addition to the spacings of the bands and the positions of the defect states and their excitations. The usual experimental methods used for this purpose in semiconductor physics, which are, of course, in part based on the well-known laws of photoconduction, are supplemented with the bombardment experiments that were recently related to the changes in photoconductivity

41

PROBLEMS O F PHOTOCONDUCTIVITY

brought about by bombardment. For example, this latter method mas used to bring about a decision regarding the absorption processes of injected electron hole pairs in germanium and silicon through charge carriers produced photoelectrically (11). The bombardment of germanium with fast electrons brings about a change in photoconduction which sheds some light on the properties of newly formed defect states formed through electron bombardment (12). These measurements on the changes of photoconduction brought about by electron bombardment which manifest themselves as conduction up to X = 6 p (cf. Fig. 1) suggest additions to the known Wavelength ( p )

4

0.8

0.6

5

0.4

Photon energy (ev)

FIG. 1. Spectral distribution of photoconduction for germanium. Curve a: before electron irradiation; curve b: after long irradlation. The specimen was still n-conducting. Curve C: after transition. The radiation intensity was the same for all wavelengths (approximately 10-6 watts/cmz).

facts regarding the defect state properties derived from measurements 011 the Hall effect and conduction processes. For instance, measurements 011 the time dependence of the rise and decay of the photoconduction permit us to conclude whether the defect states act as trapping or recombination centers (the state of the surface must not be neglected, cf. See. VI). 011 the other hand, germanium doped with gold or with gold antimonide has a long wavelength limit a t about 6 p (in particular, germanium doped with gold exhibits a sensitivity up to 9 . 5 ~if prepared in a special way) (12). This brings up the interesting question of whether defect states brought about through doping can act in a manner equivalent to those brought about through electron bombardment. Work on photoconductivity in doped germanium is in full progress. The latest investigations show that

42

P . GORLICH

germanium doped with zinc with an impurity in a concentration 4 3 X 1016 zinc atoms/cm3 is photoconducting up to wavelength Xo = 40p. However, in the bombardment experiments one must consider how much the state of the surface is changed by the bombardment. The electron bombardment seems t o influence strongly the state of the surface in the case of cadmium sulfide. Investigations of the change in photoconductivity of nondoped cadmium sulfide through neutron bombardment should be just as interesting (13). Neutron bombardment causes a small part of cadmium to change into a radioactive isotope which then decays to a stable indium isotope. I n this way cadmium sulfide is activated with indium. The photoconductivity is increased by a factor of 10 to 20 after being radiated with a flux of about 1OI6 thermal neutrons/cm2, and the spectral distribution is changed through the appearance of a new maximum a t 620 mp. Especially significant are the possible changes in the photoconductive process brought about through the Frenkel defects produced by the bombardment of fast neutrons.

A . Lifetime: Theoretical Considerations The concept of the lifetime T of the free charge carrier enables us to obtain an over-all view of the theoretical situation. The rates of production of the free charge carriers per unit volume and per unit time are gn and gp. Then we can define the lifetimes rn and T~ by

As in Eq. (4) A n and A p are the densities of the electrons and holes respectively which exist in the stationary state during bombardment. If the photoconductor does not have any defect states, then in the case of band-band excitation or recombination g, is equal to g p and A n is equal to Ap. Therefore, T,, is equal to T ~ On . the other hand, in the case of a perturbed photoconductor A n is not equal to A p and T,, is not equal to rP. Usually, either A n is very much less than A p or A p is very much less than An. We define a time T O by

which gives the time dependence of the photocurrent after the irradiation has been concluded. Thus there exists a relationship between T O and r of the type given by Eqs. 7 (in the case A p << An, T = T,, in the case A n << Ap, 7

=

Tp)

PROBLEMS OF PHOTOCONDUCTIVITY

43

This is so because there always exists a steady state condition between a group of defect states S with the free charge carriers in the bands. A loss of free charge carriers through recombination is compensated by charge carriers from the defect states. The quantity r0,which can be determined by the decay of the photocurrents after irradiation, is not in itself sufficient without further considerations to permit u s to determine the lifetimes of the charge carriers T,, and rP (14). As we will underscore in Sec. VI, the conditions of the surfaces of semiconductors can more or less influence the photoconductive processes. The form of the decay curve is modified if fast recombinations in the surface are possible. The volume lifetime r0 is defined in Eq. ( 6 ) for a photoconductor without considering the state of the surface. An effective lifetime re for a photoconductor where the state of the surface has been considered can be expressed in the form

-1 - -_ 1 Te

+ - 1

70

x 70

where X is a correction function which can be calculated approximately (15).

B. Saturated and Unsaturated Photocurrents Equations (4) and ( 5 ) help us to understand the saturated photocur-

rents in insulators a t high field strengths and also the amplification factors in photoconductors with large photoconductivity (16). I n general, one speaks of a conventional semiconductor if the forbidden zone EL - EV < 1.5 ev. If EL - Ev > 1.5 ev., the materials are called high-energy gap semiconductors or insulators. We define BD as the cross section of the current, L as the distance between the electrodes, and T , and T , as the times that the charge carriers travel the distance L. We employ the simplification that gn = g, = g, so the total photocurrent AT is represented by

AI

=

gBDLe(x,

+ x,,)/L,

(9b)

where xn and x, are the displacements of the free charge carriers in the direction of the field during their lifetimes. The mean distances z and the times T are related by

lcor small field strengths z remains much smaller than L and r much smaller than T . Equations (9) are equivalent to Eq. (4). For field strengths

44

P. GORLICH

+

in which xn x p 2 L, the photocurrent increases if we exclude insulators. The photocurrent in insulators does not increase but saturates. Photoconductors with large conductivities behave in a different manner. The motion of the primary charge carriers in opposite directions would cause the formation of a space charge if secondary charge carriers were not introduced from the electrodes as soon as charge carriers left the semiconductor (secondary photoeffect). The ratio T / T in Eq. (9a) becomes > 1. Therefore, we obtain a n amplification factor F which cannot become arbitrarily large for there exist phenomena which occur at high field strengths that limit the photocurrent. From a theoretical point of view we may expect certain types of saturated photocurrents in certain semiconductors, which may be described as being space charge limited (17). The injection of the secondary electrons from the cathode act, as already mentioned, to compensate the space charge. The compensation of the space charge requires, of course, a finite time, the dielectric relaxation time, TR

=

€€O/U,

(10)

where E is the dielectric constant, is equal to 8.86 X A sec/V cm. As soon as the time of passage T becomes less than the relaxation time r ~ , the mechanism of injection of secondary electrons becomes ineffective. The result is a “space charge limited photocurrent.” The search for semiconductors which exhibit that type of saturation condition (yield > 1, saturation currents greater than that in a n insulator by a factor 7 / 7 8 ) ) should lead us to the clarification of the unknown behavior in semiconductors which investigations in hexagonal selenium already leads us t o surmise. Those dielectric relaxation phenomena with a time constant T B are completely determined by the T R given in Eq. (10). Substances with small conductivity and with a time constant T R of the order of magnitude of seconds (for instance, cadmium sulfide) are especially suited for the investigation of conductivity inhomogeneities. Germanium with a conductivity of lo-’ cm-’ exhibits a T R of the order of sec.

C. Advantages of the Concept of Lifetime The lifetime T comprises a number of functions of the free charge carriers, with various parameters. The considerations of the laws of photoconduction with the aid of r should not be considered as complete. The improvement of these considerations with the concept of r should give a better over-all view on different processes in certain photoconductors. For instance, the photoelectric yields or quantum yields (and also their maximum values) can be expressed in a simple fashion with the aid of the lifetime T (18).

PROBLEMS O F PHOTOCONDUCTIVITY

45

\.there G is the gain and I?' the applied difference of poteiitial. The time of passage T is given by

Combining Eqs. (11)) (la), and (7) leads to extensive statements regarding the performance of a photoconductor, in particular, their limitations by injection of space charge. We are also able to find methods of determining the lifetime T itself (19). I n the experimental setup, shown in Fig. 2 . a photocurrent flows Ultrovtolet Light

FIG.2. Measurement of field effect in ZnO powder.

through an insulator (photoconducting insulator zinc oxide powder, n-type) between the ohmic contacts. By combining Eqs. (4) and ( 5 ) , the current is found to be proportional to 7% (in the case T~ << 7%).If we apply an electric field perpendicular to the surface of the photoconductor by means of an unidirectional contact, the electrons are driven towards ground and no new electrons are supplied from the unidirectional electrode. I n addition, free electrons are collected by the anode and are lost from the photocurrent. We denote the transit time of the electrons, while in the applied field between the unidirectional electrode and anode, by TR ,so that the lifetime T of the electrons is given by 1

-

T

=

1

7,

1 + TR'

We must realize that Eq. (13) is not strictly valid but only a good approximation because the loss of the free electrons in the applied field is not a relaxation process. Let d be the distance between the unidirectional elect8rodeand anode so we can express T Rwith the aid of Eq. (12),

46

P.

GGRLICH

and so obtain the relative change of the photocurrent

+ P*U, __

7%

Therefore, the lifetime T,, can be determined from a knowledge of the mobility pn,from the amplitude and time constant of the field effect modulation of the photocurrents, without information on the gain gn, and the incident intensity. I n addition the trap concentration may also be calculated. D. Reaction Kinetic Models An exact theory of photoconduction and luminescent phenomena can only be built up on the basis of a quantum mechanical or quantum electrodynamical defect state theory whereby all significant excitation and radiative as well as nonradiative recombinations are treated. Most of the theories formulated up until now may be considered plienomenological. This is so

FIG.3. Excitation in tail absorption region.

regardless of whether these theories are based on the band model of conductivity or on the law of mass action, according to which dissociation and association steady state processes of charge carriers in defect states, defect state-band transitions, etc., are expressed in terms of “electronic reaction equations.” The problems of the theoretical treatment already appear in the simplest case, i.e., where two types of defect terms S, trapping term H , and activator term A are present (see Fig. 3). Of course, this simple energy level scheme is valid only in a few real cases. I n those cases the reaction

PROBLEMS OF PHOTOCONDUCTIVITY

47

kinetic differential equation system for photoconduction in the tail region might be formulated in the form

I%

=

s3

-

s4.

The excitation s1 occurs with the frequency a per second per cubic centimeter. I n this way we obtain per unit volume n free electrons which go over to the conduction band and A+ ionized activators. The formed free electrons can recombine with the ionized activators with a probability Q! (process s2) or may be trapped by the empty trapping states H with the probability @ (process s3). The trapped electrons (concentration na) may be thermally excited to the conduction band with a probability y. We then obtain for the individuaI processes s1 through s4 the following equations.

I

s1 = a,

sz = m A + ,

Pn(H - m),

s3

=

84

= ynE.

(17)

In a similar manner we may obtain a system of differential equations for photoconduction in the region of the base lattice absorption in which electrons and holes are excited from the valence band (see Fig. 4). We must

FIG.4. Excitation in lattice absorpt~onregion.

48

P. GORLICH

consider the processes s1 (excitation in the valence band), s2 (recombination with holes in the valence band), s3 (capture in a trapping state), s4(thermal excitation and transition to the conduction band), SS, and s6 (transition to the valence band through a n activator term as .a recombination center). The solution of such a system of differential equations yields the quantities n and p . In the case of large field strengths, the theoretical considerations become complicated. Under the influence of an electric field, the bands are curved and the electron may then penetrate into the forbidden zone. The probability of finding electrons in the zones near the band edge is proportional to the field strength (Franz-Keldysch effect). Photoconduction may then appear a t lower excitation energies. Figure 5 depicts a concrete example of a suggested model based on

FIG.5. Suggested model for Ag-activated CdS.

experimental data (20). I n silver-activated cadmium sulfide we find a level 1 ev above the normal valence band of cadmium sulfide. On the other hand, silver activation brings forth an additional 0.4-ev level below the conduction band. The photoconduction and luminescence measurements require transitions of the type illustrated in the model in Fig. 5: s1 represents the excitation as a band-band transition; s2) a n infrared transition of captured electrons in the 3-p region; s3, a luminescent recombination of captured electrons with free holes; s4, a capture of electrons from the conduction band; SS, capture of holes from the valence band; and s g , the liberation of captured holes through excitation in the 1-p region. This model represents in a satisfactory manner the experimental findings in regard to the photoconduction in the base lattice region and in the infrared region as well as the luminescent phenomena. It can be treated by a system of differential equations according to the above-mentioned method.

PROBLEMS O F PHOTOCONDUCTIVITY

49

The most general form, the system of differential equations for t,he calculation of charge carrier concentration are of the type

In the above, a k represents the excitation through photoninjection, b k the possible loss in the k-term group, and S, and the term densities. The summation indices run through the above-considered term groups; n j and n k represent the carrier concentrations in the j - and k-term groups respectively. The transition probabilities between these groups are given by Y j k and 8 j ~ . One recognizes in the example (Fig. 5) that setting up such differential equations is justified in principle. One starts with simplifications which limit the validity of the model. Besides this, the large number of unknown parameters is in most cases not uniquely and exactly determinable. This requires that we must introduce a further simplified reaction kinetic model. One ascribes to every lattice defect in the forbidden zone only a single term and assumes that every trap can accept oiily a single electron and every activator level only a single hole. This excludes, therefore, bivalent lattice imperfections. These, however, appear in germanium with an impurity of the order of 1015 atoms/cm3 of transition metal a t liquid air temperatures. These form centers with double negative charge ( 2 1 ) . One also assumes that holes are not bound by unoccupied trapping levels and electrons are not bound by unexcited activator levels. Further, no direct transitions of charge carriers between lattice defects are allowed. In general, it is postulated that the lattice defects are statistically distributed, independent of one another, and that this distribution is stable during the photoconduction process; i.e., that there is no diffusion of lattice defects. It should be especially noted that the equations do not take into account the influence of the surface. One knows, however, from a multitude of experiments, that it is just these properties of the surface which are important in the photoconduction processes and, so much so, that in certain cases they completely influence these processes.

E. Steady State Conditions

In order to utilize the many experimental results one often considers only the stationary processes; that is, the stationary state which sets in after excitation by a constant radiation intensity. This causes the elimination of the derivatives with respect to time in the equations. I n order to calculate the charge carrier concentrations, for instance, the concentration of conduction electrons, it is sufficient then to solve second, third, or higher

50

P. GORLICH

order equations (22).One can, in principle, determine the number and type of defect term groups from the dependence of the photocurrent upon the incident radiation intensity. However, one should not underestimate the difficulties of this approach in many cases.

F. Rise and Decay Processes If one investigates experimentally the nonstationary states as well as the stationary states (that is the rise and decay processes in photoconduction), one should be able to utilize the curves of the solutions of the reaction kinetic system of differential equations to compare with the experimental results. However, here we experience difficulties. One must consider that in many cases we cannot obtain analytic representations of the solutions. This is in addition to the simplifications in the models mentioned above that we utilized in order to derive the system of differential equations and their limits of validity for applications. In these cases one is naturally forced to employ various sorts of approximation procedures (23).However, one is able to obtain a more precise knowledge of, for instance, the positions of the traps and their concentrations through the investigations of the fine structure of, for instance, the rise curves (24).This leads to a more exact knowledge of the parameters of a model. It is found, for instance, in the case of cadmium sulfide that the time dependence of the rise of the photoconduction is very strongly dependent upon the intensity of the incident light. One can see this in the rise curves plateaus and infiection points. These can be explained if one assumes that distinct groups of traps are filled with electrons successively; and after each filling of a group of traps, the photocurrent again rises. In such, and comparable, ways one is a t least able to obtain certain ideas regarding the reaction kinetic parameters. G. Photocurrent Dependence u p o n the Radiation Intensity For all photoconductors (those with small as well as with large dark currents io), we obtain experimentally a power law dependence of the stationary photocurrent Ai upon the radiation intensity IB, which can be understood in terms of reaction kinetic considerations. One tries to find the proper reaction kinetic model to fit the measured dependence of the photocurrent upon the intensity of radiation in order to recognize the particular photoconduction process. One can readily show in an example how the simplest reaction kinetic model leads to a power law relation between the photocurrent and the radiation intensity. The current increase a t the beginning of irradiation, (di/dt)A, is proportional to the radiation intensity Ie. Thus,

(di/dt)A

-

In.

(19a>

51

PROBLEMS OF PHOTOCONDUCTIVITY

One assumes that the primary process produces pairs, that is, electrons and holes. The production rate of free charge carriers is naturally proportional to I B :

(di/dt)A

-

(dn/dt)A

-

(19b)

TB.

If one simply takes as a basis a bimolecular recombination of free electrons and holes, one then obtains a recombination velocity proportional to the product of the concentrations of both kinds of charge carriers. If is the concentration of free electrons in the case of dark field photoconduction, then the concentration of electrons in the bright field case is no An. The concentration of the holes must be A n because the material is neutral. At the end of the irradiation of the photoconductor, the result is therefore

+

-

( d n / d t ) ~ - (no

+ An)An

For the stationary case it is necessary that (di/dt)A

-

-

(dildt)~.

(20)

= - (di/dt)E.

Because io is proportional to no and Ai is proportional to An,

IB

io Ai

+ (Ai)'.

For the special case (a) that A i is much greater than io,

~i

-

1~34;

(22a 1

for the case (b) that Ai is much smaller than io, Ai

IB/~o.

(22b)

H . Demarcation Levels A special type of stationary consideration has been brought about advantageously through the introduction of the so-called demarcation levels (14). A demarcation level is an energy level from which transitions to higher as well as to lower levels are equally probable. The introduction of the demarcation levels is based upon the experimental fact that not all levels in the forbidden zone act as recombination centers. This is, for instance, the case €or the levels in the neighborhood of the band edge. If we assume that a photoconductor contains a n arbitrary number of defect states, then among the processes between excitation and recombination one must naturally be the slowest. This slowest process determines the speed of the total process, and, with respect to this process, all other processes also under irradiation are in approximate equilibrium. Equilibrium must be assumed for two groups of defect states. First, for defect states which are in occupation equilibrium with the free electrons in the

52

P. C ~ R L I C H

conduction bands, and, second, for those defect states which are in occupation equilibrium with the holes in the valence band. The equilibrium conditions under the assumption of a Fermi energy Enfor electrons and E, for holes are

I

n

=

(-

),

EL - En ~ ~ e x p kT E, - E v

where N L is the effective term density for the conduction band and N v is that. for the valence band. Both stationary Fermi levels (quasi-Fermi levels) are determined through the concentration of charge carriers in the conduction and valence bands. The photoconductor with a band spacing EL- E v becomes, under irradiation, a semiconductor with a band spacing reduced by t,he amount E, - E,. The group of defect states which determine the course of the total process lies between the demarcation levels D, and D , (see Fig. 6 ) . All defect states S belong to the first group if the process s2 E

FIG.6. Demarcation and quasi-Fermi levels in an irradiated photoconductor.

(transition of the electron in X to the conduction band by a thermal excitation) is faster than the process s3 (recombination in the valence band with a hole). Both processes are equally probable from the demarcation level D,. The probabilities for the processes s2 and s3 are each half as large as the probability for the process sl. We obtain an expression for the position of the demarcation level, for instance, the position of D,,by utilizing the kinetic Ansatz. It is

D, = En - kT In (2S-/S).

(24)

PROBLEMS O F PHOTOCONDUCTIVITY

53

With the aid of the concept of the demarcation levels, we can understand very well certain processes in photoconductors. This can be accomplished without more exact calculations which would otherwise be necessary if we were to base them upon a kinetic model. The stationary Fermi levels and, therefore, also the demarcation levels approach the band edges under increasing irradiation intensity. I n this manner, the number of recombination possibilities increases. The photocurrent exhibits superlinearity if the defect states get into the recombination zone thereby slowing down the recombination process. Indeed, we are able to observe superlinear photocurrents (Zb), that is, photocurrents which increase more rapidly than the radiation intensity. Above all, the dependence of the position of the stationary Ferrni levels (and, therefore, the demarcation levels by which it is possible to include every type of recombination center) upon the temperature allows, in principle, every type of variation of the dependence of the photocurrents upon the illumination intensity.

I . Wave Vectors and Crystal Momenta It is known that the electronic levels in the allowed energy bands of a crystal can be determined by the wave vector k of the electron waves. The energy can be expressed through the components of the wave vector k,, k,, and k,. The wavelength of the electron wave is given by Xn

=

2n

The product hk (with the dimensions of a momentum) is called the crystal momentum,

M

=

Xk.

(26)

The energy can be considered to be a function of this vector M. When in thermal equilibrium, most of the electrons of the conduction band occupy energy levels which correspond to the values of k in the neighborhood of the energy minimum while the holes in the valence band occupy, in a similar manner, levels near the energy maximum. The most clear-cut situation is shown in Fig. 7. This is the case where the bottom of the conduction band and the top of the valence band are at M = 0. The energy of a n electron in the conduction band may be interpreted upon the basis of the motion of the electron in a crystal as kinetic energy. The energy EL, given by E L = E L k ( M ) ,is greater than or equal to 0 for M = 0. In a similar fashion, the energy of an electron in the valence band is f i V = -AE - Evk(M),where Evk(M) is greater than or equal to 0 when M = 0. An electron in the valence band with a crystal momentum M may

54

P. GORLICH

Conduclon

Valence

M

FIQ.7. Simplest case: minimum of the conduct.ionband and maximum of the valence band for (M)= 0.

be raised to a state in the conduction band with a crystal momentum M* by absorbing a quantum with sufficient energy. This is expressed by

hv

=

AE

+ E ~ ~ ( M *+) E ~ ~ ( M ) .

(274

From this we can calculate the smallest as well as the largest energy of the incident radiation with frequency Y necessary for such a transition. This is given by

hv

=

AE.

(27b)

In order that the transition have a large probability, the condition

M* = M

+ Mphi

(28)

must be fulfilled. This is based upon the quantum mechanical expression for the transition matrix element. In Eq. 28, Mph is equal to h/Xph for the absorbed photon with wavelength X p h which is moving in the direction of the unit vector i. In the transition, the sum of the crystal momentum and the photon momentum is a constant. In the case where Mph is much less As a consequence of than /MI or IM*l we can assume IM*l is equal to [MI. this we have a vertical or direct transition which is shown, for example, in Fig. 7 by a n arrow. This means that the transition takes place with constant M. The probability for a direct transition is approximately independent of M and the probability for absorption depends only upon the number of available states if the magnitudes of M are not too large, if the transition is an allowed one, and if the electron hole interaction is negligible. The number of states in the conduction band for energies between E and E dE is of the form CE% where C is a constant. The number of possible

+

PROBLEMS O F PHOTOCONDUCTIVITY

55

+

traiisitioiis between v and v dv is, therefore, of the same form. The absorption coefficient can, therefore, be expressed by (Y

=

cl(hv - A E ) ~ ~ ,

where hv is greater than AE, and if hv is less to or equal to AE, a = 0. In the range where hv - AE is small, Eq. (29a) is not valid; for in this case, IM[is also small. The electron and hole move separately and slowly and are subject to a strong Coulomb attraction which strongly influences the shape of the absorption spectrum. If M = 0, the transition probability vanishes and the transition is forbidden. For small deviations of M from 0 the transition probability is proportional to /MI2and, therefore, proportional to hv - AE. Thus, a =

cz(hv - AE)%,

(29b)

if hv is greater than AE.If hv is less than or equal to AE, a = 0. If we consider the possibility that the Coulomb interaction between electron and hole can lead to the formation of a n exciton state, we obtain for the absorption of a quanta, whose energy is not sufficient to cause a direct transition, the formation of an exciton such that

hv

=

AE - Eexcl,

where Eexczis the binding energy of the exciton. The direct transitions therefore cause a hydrogen-like line absorption spectrum which extends to the long wavelength side of the edge of the base lattice as already mentioned (2, 3, 6). The simplest case for the simple situation described in Fig. 7 is only valid for a few semiconductors (for instance InSb). In most semiconductors the minimum of the conduction band and the maximum of the valence band generally do not have the same value for the crystal momentum M. One must note that the range of values for M is not arbitrarily large because it can be shown that the energy is a periodic function of the crystal momentum. The maximum value of M is of the order of h/2& where d~ is the lattice constant; therefore, the smallest electron wavelength is of the order of 2 dc. Figure 8 is a representation of the energy band structure for germanium, based upon experimental information. One sees clearly that the direct transition with M = 0 which is depicted in Fig. 8 as s1 is not the one which requires the smallest amount of energy, but the one with the smallest energy requirement is shown as s2. The indirect transition requires a large change of M so that the transition is not possible by the absorption of only a single photon. This latter transition can only take place if a photon is absorbed a t the same time as another photon is either absorbed or emitted. Let

56

P. G ~ R L I C H

E

I

M2

0

-

M

FIG.8. Energy band structure of germanium in (111) direct’ioo.

be the momentum a t the minimum in the conduction band and Ma be the momentum of the phonon which is necessary for the conservation law, then we have M L

M L = AMc

(31)

where the sign depends upon whether the phonon is absorbed or emitted. The energy of the phonon in a crystal can be expressed in the analogy with the electron as a function of its wave vector k’,also the momentum of the phoiion can be given analogously to Eq. (26) Mc

=

fik’.

(32)

Let EG be the energy of a phonon with momentum G = ML, then the minimum frequency for the transition s2 (see Fig. 8) is given by hv = AE f Ec.

(33)

Again, the sign depends upon emission or absorption of phonons. This simplified representation by no means exhausts the phonon problem. The problems become more complicated because of the different types of vibrations possible in a crystal, each with its own kind of phonons. Finally we must consider that indirect exciton transitions are possible. Whereas the direct exciton transitions result in a line absorption as mentioned earlier, it is not true for the indirect transitions. We obtain a continuous absorption spectrum for the excitons. The lowest absorption

PROBLEMS OF PHOTOCONDUCTIVITY

57

frequency which results in the formation of excitons through absorption or emission of phonons is given by hv = AE f Eo - Eex$.

(34)

IV. DISLOCATIOXS Dislocations are becoming more important in the field of photoconductivity. Step dislocations as, for instance, in germanium act as electron traps (26), which are able to decrease the lifetime T~ of the liberated photoelectrons. Similar conditions may be obtained in cadmium sulfide ( 2 7 ) . Figure 9 shows the lattice configuration of a step dislocation in cadmium

(110)

FIG.9. Structure of step dislocation in CdS

sulfide which clarifies the action of a dislocation line as an electroii trap. The data shown in Fig. 10 are a direct proof of the influence of the dislocations on photoconductivity. The photocurrent is plotted as a function of the distance from the grain boundary. The material was illuminated by sranning with a movable slit. One may expect similar influences upon the photoconductivity in other semiconductors. When, for instance, alumiiium migrates from silicon, it is preferentially along the dislocation lines. Consequently the A1 concentration is dependent upon the perpendicular distance from the dislocation line. Upon suitable doping, the dislocation line is enclosed by a tubular pn-junction (28). Such treatment cannot but influence the photoconduction process. One may expect that complete planes of step dislocations, which exist where there is twinning, should show exceptional photoconduction effects. This is so because in such lattice defects the activation energy may take on exceptionally small values. We would like to mention investigations on the increase of noise and change in photoconduction in evaporated lead sulfide layers which were mechani-

58

P. G ~ R L I C H

E 0

E e a a n .c

I

- Distance (mm)

FIG.10. Photocurrent in CdS as a function of distance from the grain boundary.

cally treated. It is suggested that lead sulfide is a suitable substance for investigations on dislocations (29).

V. NEGATIVE PHOTOCONDUCTION In spite of the great interest shown in the normal inner photoelectric effect as manifested by the large number of investigations of this phenomenon, there are still a large number of unsolved problems. This also holds for the negative photoelectric effect even to a much larger extent. This is so in spite of the fact that both effects have been known for a long time. We consider negative photoconduction to apply not only to those effects which are characterized by a decrease of the conductiv;ty while under irradiation to a level below the dark conductivity (for instance, selenium, germanium (I), and Ag2S (SO), under certain conditions of preparation which can give positive or negative effects) but aIso to such cases where the photoconductivity exhibits first a rise and after having reached a maximum decreases to a lower level which may in some cases be below the dark conductivity. According to this, the appearance of fstigue in photoconductors, photoelements, and photocells, where the changes in conductivity are exhibited in this or in similar manners, may be considered to be a negative photoeffect. Fatigue has been the subject of investigations for many years because of the great importance of constant photo devices in industry. Photocells based on photoemission are also mentioned here because all compound photocathodes exhibit semiconductor characteristics and therefore what applies to photoconduction (positive or negative)

PROBLEMS O F PHOTOCONDUCTIVITY

59

should also apply to this type of photoeffect (31).The conditions for which negative photoconduction occurs are very complicated and are difficult to understand from a common point of view. This is so because both in natural crystals and artifically produced photoconductors, negative and positive effects are reversible depending upon the wavelength of the incident light and depending upon temperature region and applied voltage. Examples of materials in which these phenomena occur are MoS2, SbzSi, single crystal CUZO, single crystal CdS (S2), pressed powder of ZnO, and doped Ge (33).

FIG.11. Explanation of negative photoconduction.

There is no dirth of kinetic models to explain the negative photoeffect (34). One model (35) is based essentially upon an exciton mechanism. Another model which was extensively discussed because of its clarity has the advantage of being readily checked by experiment and may lead to new considerations regarding the doubly-charged traps (1, 36). The abnormal negative photoconduction is usually excited by light on the long wavelength side of the absorption edge if we exlcude all the negative photoeffects which can be explained through the change of the adsorption equilibrium or through change of the stoichiometric proportions. I n this way we exclude a large part of the fatigue effects; therefore, we can assume (see Fig. 11) that the excitation process s1 consists of the liberation of an electron from the valence band and its transition to a trap (or the liberation of a hole from a trap and its transitions to the valence band). The processes sz and s3 are recombination processes of a type where there already exists in the nonirradiated condition an electron in the conduction band which combines with a hole in the valence band a t a recombination center. This must occur

60

P. GORLICH

a t the recombination center because the direct transition of the conduction electron to the valence band is fairly improbable. If the recombination processes s2 and s3 are faster than the thermal transition of the trapped electrons to the conduction band, then the stationary electron concentration is smaller than in the dark field case. One must therefore observe a decrease in the conductivity if the free minority carriers are split off from a defect state by photon injection. This holds when the defect state has a recombination coefficient for free majority carriers which is orders of magnitude smaller than that of other defect states acting as recombination centers. If the very small recombination coefficients are interpreted as being a result of the repulsive Coulomb forces, then one has to assume that the defect state from which the minority carriers are split off is able to trap two electrons (n-type conductor) or to yield two electrons (p-type). This holds, of course, in the nonirradiated condition. Such circumstances seldom occur and, therefore, one must conclude that the negative photoeffect can be observed less often than the positive photoeffect. If the small recombination coefficients are a consequence of a small transition probability, then one may disregard the assumption of doubly-charged defect states.

VI. SURFACE CONDITIONS

It has already been noted twice in this report that the photoconduction processes are strongly or completely influenced by the properties of the surface of a semiconductor. Figure 12 illustrates an example of this (37). I t depicts the spectral distribution of the photocurrent of cadmium sulfide as a function of the humidity. One recognizes that both the shape of the spectral distribution and the magnitude of the photocurrent change noticeably as a function of the humidity. The photocurrent decreases in the region of the characteristic absorption at small wavelengths with increasing humidity. From this, one must conclude that the absorbed moisture on the surface forms traps. The traps clearly exhibit a large binding energy for the electrons so that the electrons can easily recombine with the holes. It is clear that the part of the photocurrent which flows in the neighborhood of the surface is essentially decreased. This is so because the moisture is absorbed only on the surface and so the traps are formed only on the surface. Since short-wavelength light in the region of self-absorption is strongly absorbed in the region of the surface and light of longer wavelength is absorbed less in the surface region but much more within the volume of the crystal, it is understandable that the part of the photocurrent in the region of self-absorption decreases strongly. It was probably in germanium that the first scheme to explain the surface recombination was set up (38). In Fig. 13 we have a simple example of the usual current type of consideration. There we have shown the

PROBLEMS OF PHOTOCONDUCTIVITT

61

100

90

-

80

._ 70 3 c

,x 6 0

e $

._ &

-@ -

50 40

L

V 3

0

30

2

20 I0

-

n

3000

4000

5000

6000

Wovelength, A

FIG.12. Dependence of the spectral distribution of CdS crystal upon the moisture in surrounding at,mosphere. Curve 1, 10 mm Hg; curve 2, average vacuum; curve 3, high vacuum.

loyer

E"

FIG.13. Energy level diagram of a surface of Ge.

equilibrium condition for the surface and semiconductor. As usual, E L and Ev are the boundaries of the forbidden zone and Ei the position of the Fermi level; Ei, is the value of E , in an intrinsic sample. The presence of the surface states within the forbidden zone permits space charges near the surface and so the formation of a potential barrier. Let n, be t,he inversion density of the carriers in an intrinsic sample then we can as usual state,

62

P. GORLICH

n

=

ni exp (e+/kT),

p = n, exp ( - e + / k T ) ,

(35)

where e.9 = Ei - Ei,. I n Figure 13 cpi is for the value in the bulk semiconductor and q h b for the value in the surface of the semiconductor. The contamination density within the bulk semiconductor determines & whereas &b depends upon the barrier height and is given by e+ob = Ei

- Ei,

+ eVob.

(36)

Investigations on the surface states in germanium have resulted in the appearance of surface terms with different reaction times (39).One is able to differentiate between slow and fast surface terms. The number of the fast terms is probably quite smaller than that of the slow ones. The fast terms are probably continuously distributed around the band center and some others of great,er width are found a t a distance approximately of the order of 0.1 ev from the band center. The scheme that is represented in Fig. 13 is a n extremely simplified case. The influence of the surface states in zinc oxide single crystals upon the photoconduction was investigated in some detail (40). One can obtain a very large surface conductivity in zinc oxide crystals by allowing atomic hydrogen to activate the surface of the crystal. This brings about an enriched layer of n-type conductivity. One can again reduce the surface conductivity by adsorption of oxygen or through heating. One can determine the lifetime of the excited electrons by measuring the surface photoconduction. The lifetimes of these electrons are between 0.5 see and 7 X see under a n irradiation intensity increase of four orders of magnitude. If the surface conductivity is changed by four orders of magnitude, the lifetime remains constant. We can conclude therefrom that the lifetime is independent of the amount of the adsorbed gas on the surface. Figure 14 illustrates a scheme of the surface of the zinc oxide crystal with a n enriched layer and with surface states. This scheme is sufficient to explain the surface processes. One assumes a continuum of electron-trapping states near the lower edge of the valence band. One is then able to get information regarding the density of the fast terms. By changing the surface terms and also the field (change of carrier density due to transverse electric fields), the edge of the conduction band moves toward El at the surface. Under absorption of light, the quasi-Fermi level for electrons moves toward the conduction band and the quasi-Fermi level for holes moves toward the valence band with increasing irradiation intensity. Lead sulfide is a particularly interesting but obviously a more complicated photoconductor. Chemically deposited or evaporated polycrystalline lead sulfide films on insulated substrates exhibit, as is well known,

03

PROBLEMS O F PHOTOCONDUCT1VITY

large photosensitivity in the visible region and in the near infrared region. PbSe and PbTe films exhibit similar behavior but with greater infrared sensitivity. The polycrystalline PbS films with thicknesses from 0 . 1 ~to 1p have a large surface-to-volume ratio. I n addition, the minority carriers can easily diffuse towards the surface so that one may already expect the surface conditions to exert a great influence upon the photoconduction processes. The surface conditions are not only determined by the outer

1

Recombination

I -3ev

“Surface

FIG.14. Energy scheme of the surface of a ZnO crystal in thermal equilibrium. surface of the films but also by the intercrystalline grain boundaries. Experiments indicate that the crystallites are surrounded by intercrystalline oxide barriers and the crystallites are essentially of the p-type. This should not exclude the possibility of the existence of small n-type regions within the crystallites. The appearance of photo-emf’s in micro regions confirms the assumption that p-type regions can surround n-type regions ( / t l )The . conductivity can be expressed through 0

= ePPP*,

(37)

and effective mobility of the holes is limited through the intercrystalline barriers pP* = p P

exp (-EB/kT)7

(38)

where En is the height of intercrystalline barriers. The energy schemes for the intercrystalline barriers and for the outer surface barrier can be illustrated on the basis of Fig. 13 (42). Investigations with the aid of the field effects permit us to state that the surfaces of the crystallites have ‘‘fast” states and the outer surfaces have states. The various results for the different semiconductors can be in principle understood in terms of the discussed surface states. In summary, one should

64

P. GORLICH

note that there are still a large number of questions regarding the complicated photoconduction behavior in the surfaces of semiconductors. It seems to be of general validity that the photocurrents from pure surface photoconduction have laws which are of the same type as those for photocurrents in a homogeneous volume (@). Oiie is therefore not able to conclude from the measured dependence of the photocurrents upon the radiation in tensity whether one is dealing with a photocurrent from within the volume or from a surface.

VII. OHMIC

AXD

UNIDIRECTIONAL

CONTACTS,

pn-JUNCTIONS

In principle, we can explain the strong and sometimes extreme influence on the photoconduction process, as shown by experiment, by the boundary layers on the electrodes and the pn-junctions (as in the previous section, the boundary layers have surface states). We need to note however that in surface state photoconduction there is, in general, an assumed recomhination of the produced charge carriers in the surface states.

A . Unidirectional and Isotropic Contacts One knows that in metal-semiconductor boundaries, the contacts generally exhibit rectifying properties. An electron which goes from the metnl to the semiconductor must overcome a work function which may be approximately the difference between the metal-vacuum and semiconductor -vacuum work functions. At the contact, the work function causes a concentration nR in the semiconductor which is temperature dependent only and independent of any change in the electron concentration by photon injection into the interior of the semiconductor. Within the boundary layer of the semiconductor which is in contact with the metal, there is a concentration gradient of charge csrriers and so a space charge. If the thickness of the boundary layer is small compzred to the mean free path of the electrons, the rectification process may be described by the diode theory where one assumes the boundary layer is idealized as a vacuum. From theory we obtain

{

isp = A 1 - exp

(-

%)I,

where is, is the current in the blocking direction, iD, is the current in the forward direction, and A is the saturation current which depends upon the thermal velocity of the electrons and nR. For boundary layers which are large compared to the mean free path of the electrons, it is necessary

PROBLEMS O F PHOTOCOKDUCTIVITY

65

to take into account many collisions of the electrons with defect states and phonons, etc. The above considerations lead to a diffusion theory where is, and iD, are given by

where E R is the magnitude of the boundary layer field strength. If a photoconductor has a unidirectional contact, one should expect a photocurrentvoltage characteristic nhich is similar to that of the dark field case of a semiconductor with the same unidirectional contact. The experiments verify these expectations (44). In order that the semiconductor or the photoconductor exhibit rectifier characteristics, they must be bound by a t least two boundary layers of which one must be a unidirectional contact (i.e., depleted layer). The depleted layer is a layer in which the charge carrier concentration is less than that within the interior of the semiconductor. An enriched boundary layer is one in which the charge carrier concentration is greater than that in the interior of the semiconductor, and, in contrast to the previous case, it is a necessary condition for an isotropic contact. The investigations of photoconductors with negligible dark current, that is insulators (insulating ZnO), have gained importance for the study of the influence of rectifying contacts on the photoconduction processes (45). Rectifying contacts for insulating ZnO may be produced in various ways; for instance, by negative ions from a corona discharge in air, by negative ions of an electrolyte, and by a p-type semiconductor. It is difficult to obtain ohmic contacts for ZnO. Therefore “quasi-ohmic” contacts (Hg contact) are regarded as being sufficient substitutes. Experiments show that rectifying contacts on ZiiO lead to saturated photocurrents, whereas ohmic contacts result in secondary photoconduction. Ohmic contacts for CdS can be produced by causing Gu or In to be melted on the crystals or by evaporating these substances. Silver, gold, and graphite also result in ohmic contacts with CdS if the crystal surface is subjected to a cleaning procedure (electron and ion bombardment) prior to connecting the electrodes.

B. pn-Junctions It is known that one is able to excite isotropic transitions within a semiconductor by doping it in such a manner that one part with an excess of charge carriers is bounded by another part with a lack of charge carriers

66

P. GORLICH

(pn-junction). The electron and hole concentrations in a pn-junction are illustrated in Fig. 15 for the stationary case. By applying a field in the flow direction, electrons and holes are then concentrated in the pn-junction (diminishing of the total resistance of semiconductors). If the field is applied p- concentration

p-conduction

n-concentration

' pn- junction '

n-conduction-

x

FIG.15. Electron and hole concentrations in pn-junction.

in the blocking direction, then both types of charge carriers are taken away from the junction (increase of total resistance of semiconductors). The influence of pn-junctions on the photoconductive processes can be expected especially in those cases where an apparently homogeneous material becomes inhomogeneous through diffusion processes causing pn-junctions (see Sec. IV, Dislocations).

C . Photo-emf in Boundary Layers I n addition to the photoconductivity changes of photo-semiconductors through irradiation [which is described by Eq. (4)], there may appear

photo-emf's in pn-junctions or in a metal-photoconductor junction where there is a depleted layer. For example, Fig. 16 illustrates schematically the formation of a photo-emf through a depleted layer in a metal-defect-semiconductor boundary (for example, Se or CuzO). In the unidirectional layer there is a depletion of holes. If process s (formation of electron hole pairs) is caused by irradiation, then electrons are driven by the action of the boundary potentials into the metal and the holes are driven into the semiconductor before a recombination is possible. The unidirectional electrode is negatively charged by the electrons with respect to the photoconductor. Similar conditions exist for pair production through irradiation in a pn-junction (46). The space charge field a t a pn-junction is directed so that the electrons drift into the n-region and holes into the p-region. If there is pair production in the direct immediate neighborhood of the space charge region, (that is in the field free space) and if no recombination takes place during the time

PROBLEMS OF PHOTOCONDUCTIVITY

67

“mox

!

-

s

Deplelionboundary layer (Unidirectional layer)

c

FIG.16. Appearance of a photo-emf at a metal-defect-semiconductor boundary.

of diffusion of the carriers to the space charge zone, then these carriers participate in the formation of the photo-emf. Photoconduction processes and the formation of photo-emf’s are the basic processes which enable us to understand the technically important photodiodes and phototransitors. VIII. PHOTOELECTROMAGNETIC EFFECTS In addition to the production of photo emf’s in a semiconductor which has been described in See. VII, one can cause pair production through photoninjection in the surface of a semiconductor By separating the diffusing charge carriers, by the application of a magnetic field, as they move into the semiconductor, one causes another photo-emf. This effect is known as a photonelectromagnetic, or a photomagnetoelectric, or a photogalvanomagnetic effect. It was first observed in CuzO a t liquid air temperatures and under 11,000 gauss where the measured potential was up to 20 v (47). This effect was later also observed in Ge, PbS, InSb, and Si, and not only a t low temperatures (48). The possibIe arrangements in which the effect was originally measured are illustrated in Figs. 17a and 17b. One utilizes electrodes in the yz-plane when the magnetic field is applied in the z-direction and the radiation is in the m-plane as illustrated in Fig. 17a; or one may place probes in the positions as shown in Fig. 17b under the same illumination and magnetic field direction as in Fig. 17a. Between the electrodes, or between the corresponding probes in the various positions, there exists a photo-emf or at short circuit a photocurrent. I n the case of the photoelectromagnetic effect, the forces which separate the charges are the Lorentz forces in a magnetic field instead of the Coulomb forces in an electric field. The theo-

68

P. GORLICH Another possible setup

0 Yl

Y f

3

3 (bl

(0)

FIG.17. Photoelectromagnetic effect.

retical treatment of the photoelectromagnetic effect indicates that with the aid of this effect there is the possibility of a direct determination of the velocity of the surface recombination and the lifetime of the free charge carriers. Figure 18 illustrates schematically the basis of the theoretical treat,ment. The applied magnetic field causes circular currents which enable us to differentiate between a transverse and a longitudinal photoelectromagnetic effect (49).The longitudinal effect in the irradiated parts may be Light

FIG. 18. Schematic representation of the longitudinal and transverse photoelectromagnetic effects in a semi-infinite plate. Vt-potential difference of the transverse effect; Vi-potential difference of the longitudinal effect; I-total current; E-magnetic field strength, vl, vz, r8-surface recombination velocity of thc irradiated (index l ) , not irradiated (index 2 ) , and sides (index 3) of the plate.

classified as being either linear with the magnetic field or quadratic with the magnetic field. Under illumination and without applying the magnetic field one obtains the Dember effect (50).(Dember emf). There occurs across the plate an electric field which is proportional to the concentration gradient and is, therefore, largest in the region near the illuminated surface. The differential equation system on which the kinetic processes of the photoelectromagnetic effect are based, contains the Dember field strength in addition to the surface recombination rates which may be assumed constant. In order to neglect the changes in resistance in the magnetic field,

PROBLEMS O F PHOTOCONDUCTIVITY

69

one must assume first of all small Hall angles. The theory can be generalized to arbitrary Hall angles in order to interpret theoretically the results in InSb and InAs. Under certain approxiniatioris one obtains for the short circuit current Ais, Ais

=

~ T B M ~ B ~~LD

d1

+ pm2Bnt21 +

TnTOh

1

d1

+

(41) pn2Bm2

if one neglects the contribution of the holes because of their small mobility. In Eq. (41), is the surface recombination rate, B, is the magnetic induction, and LO is the diffusion length (the square root of the product of the lifetime and diffusion constant). The maximum photocurrent caused by the photoelectromagnetic effect corresponds to a quantum yield of 1 as in the case of the photoelements. From this, one may expect that the technical utilization of the photoelectromagnetic effect is of interest only in special cases except for application in the infrared region (51).This is illustrated in Fig. 19 where A is a single crystal plate of InSb (2 mm X 1 mm

FIG.19. Schematic arrangrmcnt of photoelectromagnetic cell.

X 0.1 mm), B refers to a pair of electrodes, and the semiconductor is located ill C a magnet with a field of about 10,000gauss. The radiation is iiicident on the front side of the plate (direction of the arrow in Fig. 19). The spectral

distribution is shown in Fig. 20.

Wovelength (microns)

FIG.20. Spcctrd senritivity of indium antimonide photo~.lcctromagneticcell.

70

P. GORLICH

The photoelectromagnetic effect is an important method utilized in determining the surface recombination rate in the inner photoelectric effect. It is also possible to have a combination of both effects. By utilizing different wavelengths of light, corresponding to different absorption coefficients, and applying a field, a superposition of both effects is then obtainable. The magnetic unidirectional layers in such cases may lead in addition to complicated conditions (52).

IX. APPLICATION OF PHOTOCONDUCTORS A . Tabular Survey of Photoconductors Solid state physics must in general determine the real nature of the recombination centers and trapping states and the imperfection terms. (Trapping centers for holes in CdS and CdSb can be, for example, cation holes). Furthermore, the question of trapping cross sections and a number of other questions must still be clarified. These similar problems need the methods of photoelectric investigations to help towards their solution and to clarify the general laws of solid state physics, as is already illustrated in the previous sections. One utilizes for such purposes model substances. That is, such materials for which it is convenient experimentally to combine electrical, in particular photoelectric, investigations with optical investigations, in particular optical absorption (for instance, CdS, colored and uncolored alkali halides as well as alkali earth halides). The investigations of the model substances naturally lead to an extensive specialized literature. Publications of the results of the investigations of t,he technically important semiconductors (for instance, Ge, Si, PbS) which naturally belong to the model substances are hopelessly impossible to survey meaningfully. Therefore, in order to study the special details we refer to certain communications (53) which are listed in the references. Here, we can only attempt to note the main properties of photoconductors in tabular form. The refractive index of photoconducting elements is given in the fifth column of Table I. According to older empirical considerations (Gudden 1928), a pure substance would only exhibit photoconduction if the refractive index were greater than two. More recent considerations (Moss 1950) suggest that there exists a relationship between the refractive index n B and the long wavelength limit Xo such that the form nB4/Xo = constant. As shown in Table I the requirement that nB > 2 for photoconducting elements is surely fulfilled. A calculation shows that the relationship between the refractive index and the long wavelength limit, taking into account the expreimental difficulties, is only approximately valid. The application of this relationship to binary semiconductors is useful only for making estimates.

71

PROBLEMS O F PHOTOCONDUCTIVITY

Table I1 shows a summary of photoconducting sulfides, selenides, and tellurides. Table I1 contains only the binary compounds. It is necessary to note the properties of the ternary sulfides, selenides, and tellurides (54),which are known to be photoconducting. These compounds are TlzSe-Sb2Sea, CdTe-ZnTe, BizS3Sb2S3,SbzSea-AszSee, Tl2S-Sb2S3, CdSe-InBe3, and T18e-As2Sea. These compounds are of Groups A I I I B V and AIIBVIwhich are able to form solid solutions. These compounds are characterized by the fact that the width of the forbidden zone varies monotonically with composition from one binary compound to another; in some cases linearly with composition (for instance CdTe-ZnTe) . The derivation of the spectral distribution with composition is shown in Fig. 21 in the case of two good I

23

4 5 6

789

Composition

O/O

-

2 100 ..-5 V

D

c

2 a.

50

0

0.5

06

07

Wavelength ( p )

0.8

0.9

FIG,21. Spectral distribution of CdS-CdSe.

fertile conductors (CdS and CdSe). I n this case we deal with the solid solution of CdS-CdSe. The ternary mixed crystal PgTe-CdTe seems likely to be photoconducting until -lop as appears to be experimentally established. The furthest limit towards the infrared can be expected to be about 120p in the case of ternary compounds with impurities. Another group of ternary compounds of the type A V B ~ I C V I Iappears also to be photoconducting (55). Depending upon the band separations of these compounds, the long wavelength limits are in the visible and in the infrared regions. Investigations have already been carried out on the combinations of Sb and Bi (Group V) with S, Se, and Te (Group VI) and C1, Br 1, and I (Group VII), where contacts were made with metallic Ga. With increasing temperature up to 100°C, one is able to bring about an increase in sensitivity. In addition to the photoconductors given in the tables, one can list as photoconductors the alkali halides (which do not exhibit the photoeffect i K I the h s e lattice region, on the other hand exhibit F-, 11-, and V1-center

72

P. GORLICH

TABLE I. Atomic number in Group merit periodic system

Allotropic modification of photoelectric element

Refractive index (extrapolated for

~

B

3

111

C

6

IV

Si

14

IV

3.43

Ge

32

IV

P

15 33 16 34

V V VI VI

As

S

8e

Te

52

VI

I

53

VII

Diamond, cuhic zinc blendestructure

Red Grey Crysta!line Amorphous monoclin (red) hexagonal (met.)

Long limit

(PI

(k)*

photoconduction (ev)

3.5

2.2-2.6

1.1

2.4

-0.241

0.234

5.3

1.48

1.08

1.15

4.0

2.2

1.7

0.73

2.6 3.35 2.0

-1.2 2 .O-2.4

0.85 1.03 0.5 0.5 1 and 0.8 and

1.3‘3 1.2 2.4 5-1.9 2.5

4.3

3.3

0.37

1 .0

0.96

1.30

2.45 4.8

-3

Activation

X J ~ energy from

0.96-1.27

* A;d is defined as one half of the value of the maximum sensitivity in the spectral distribu-

tion,

phot,oconduction) and the silver halides, the iodized thallium and mercury, and a large group of inorganic phosphors regarding which certain statements may be made about the radiative excitation mechanisms as a basis uf their photoconducting properties. Phosphors with bimolecular radiative excitation mechanisms (so-called recombiliation radiators) exhibit “good” photoconduction. On the other hand phosphors with monomolecular radiative

PROBLEMS O F PHOTOCONDUCTIVITY k’HOTOCONEUCTING

ELEMENTS

Method of investigation

Evaporated layer, decomposition layer of boron hydrid, hulk Crystal

Type of conduction

Insulator a t irradiation

Decomposition layer of silicon tetrachloride, bulk

p and n according to activation

Single rrj-stals

p and n according to activation (Group 111respectively Group V)

P

crystal

Ilistilled layer Thin plate, through melting

Remarks

Technical application

P

n

Layer Evaporated lager Sublimated layer Evaporated layer, pressed layer,

73

P

Additional irradiation with red light increases the sensitivity in the uv. “Semiconducting” diamond (p-type) shows photoconduction UP to -1.15~ n-type a t liquid helium temperature, photo. conducting up to 38p .4u- and Zn-doptpd nand p- Ge photoconducting up to ~ 6 p . Displacement of long wavelength limit to longer wavelengths at lower temperatures.

Photoresistor, photoelement (solar battery) photodiode Photoresistor, photodiode, phototransistor

Photoresistor, photoelement (xerography) Suitable for photoelectromagnetic purposes

clxcitation mechanisms (so-called configuration radiators) exhibit “poor” photoconduction.

R. Frequency Dependence and Amplz$cation Factors of Photoconductors The frequency dependence of the photocurrent under sinusoidal excitatioil n a y be characterized by a relaxation process and so is given by

TABLE11. PHOTOCONDUCTING SULFIDES, SELENIDES AND TELLIJRIDES ~~

~~~

Long wavelength limit XO ( p ) at irradiation in the region of Binary compound CunTe ZnS ZnSe ZnTe

Atomic no. of the metallic component 29 30

33 42

Lattice absorption 1.5 0.38 0.5

0.63 -0.75 1.1

CdSe CdTe

2.2 1.3 1.4 -3.5

Technical application

Width of forbidden zone (ev)

1.3 0.55

0 .9 1.2

1.5 1.6

Remarks

n-type. Single crystals, polycrystalline specimens. Influence of specific contaminations (Cu+, Ag+, Al", Se3+, Ga3+,C1- , Bi-, etc.)

Model substances for luminescent and photoconduction processes Semiconductor image tube

2.0 2.6 1.35 1.8 3.0 0.9

47

48

Lattice defect absorption (longest wavelength observed)

Model substance, photoresistor Photoresistor Photoresistor

5

3:

2 h 9

*.I

0.6

Photoelement

7

0.9 Evaporated layers, Eingle crystals, and polycrystalline specimens. Structured absorption, even with CdSe. Doped with Cu, Ag, C1. Surface terms caused by influence of oxygen. CdTe becomes p-conducting by doping with Cu or Ag.

InSe

49

.o

1

0.9

HgS

HgSe

HgTe

TlZS

TLTe PbS PbSe PbTc

Bi&

Bizsea

BizTea

80

81 82

83

Excess of metallic component leads to higher infrarcd senssi tivity. Evaporated layers. Weak influence of Cu, Sb and Cd. Probably Sb203(cubic crystallites) on surface, also metallic Sb, strong influence of the phase of the surface on the photoconduction. n-type.

1.8

0.63

1.o 1. g 2.8 5.0 3.6

2.2 2.6

1.2 3.9 1.8 2.6

Semiconductor image tube

F

5.0

E m

0.4 Photoresistor Photoresis tor

1.6

m

0

r

0.41 0.26 0.32

Evaporated layers, chemically depofiited layers, single crystals. Also used a t lower temperatures. (Large displacement of the long wavelength limit to longer waveIengths). n- and p-type. I n polycrystalline unidirectional layer effects, probably also in PbSe and PbTe. Oxygcn influence. Evaporated layers, At cooling, displacement of the long wavelength limit to longer wavelengths. Strong oxygen influence on sensitivity.

0 ~3

80

3

2

$

-l 01

TABLE 111. PHOTOCONDUCTING OXIDES Long wavelength limit An ( p ) at irradiation in the region of Compound

M go

TiO?

Atomic no. of the metal

Lattice absorption

Lattice defect absorption

12

0.8

1.5

22

Technical application

Conduction type

Remarks

p

A t uv irradiation ( 0 . 3 1 ~ and ) neutron bombardment, An displaced to -1.8~. New long wave-

0.48

cuzo

29

0.63

I .5

ZnO

30

0.48

0.55

n

Inz03

49 52 56

1.8 1.6

p

0.33

TeOz BaO

0.55 0,53

Pho toeleinent

p

length max. appears. Gudden criterium not fulfilled, for n~g= 1.74. Rutile Sperial level diagram, for band-band transition docs not correspond with optical ahsorption cdge Single crystals and polgcrystalline specimens. Surface effects. Sintwed specimens, thin layers, single crystals. Photoconduction depends strongly on atmosphere (oxidizing or reducing). Evaporated layers Investigation of photoconduction to gain better understanding of thermal emission. BaO can change color in Ba-vapor analogously to the alkali halides. Anodic deposited lagers.

? G)

0:

!a t+

z

Long wavelength limit ( p ) a t irradiation in the region of Compound

Lattice defect at)sorption absorption

MgaSbz

1.5

ZIlJ?*

1.1 2.2 2.1

GaBb Inp InAs InSb

Condtiction type

7.85 4.8

7.7

3.5

2.8 1.6

Remarks Measured only a t low temperatures (85°K and lower) Evaporated layers are photosensitive only under special conditions Small single crystals

5.3

MgtSn

ZnSt) Cd3-4~2 GaAs

Technical application

May be suitable for solar batteries

3.0

Up tonow n-type

Width of the forbidden zone (ev) 0.22

0.8

Photosensitivity is higher in polycrystalline specimens than in single crystals

P

Photoresistor, photodiode, photoelectromagnetic cell

n

Very high carrier mobility. Low effective electron mass. Strong temperature dependence of photoconduct,ion.

8

0. 3 3 0.17

%

78

P . GORLICH

where A B is the amplitude of the alternating radiation, w is the circular frequency, and Ai(w) is the change in photocurrent as a function of frequency. In addition Ai(w) is proportional to the applied potential difference; 7 0 is the time constant of the photocurrent defined in Eq. (6). According to the proportionality relationship (42), Ai(w) should be essentially frequency independent for all radiation frequencies w << 1/70. For higher frequencies, Ai(w) decreases as l/o.The requirement of a slight frequency dependence implies a small time constant; T~ is of the order of sec for sec for CdSe and PbS. CdS and of the order of The amplification factor F for photoconductors with large conductivity may be defined by

F

=

AI/egBDL.

(43)

On the basis of Eqs. (9) and (7), we obtain an amplification factor proportional to 70. A characterization of the photoconductors in terms of a quality index may be made on the basis of the relationship between the frequency dependence and their amplification factors with the time constant. This index is set proportional to F and 1/70.

C . Statistical Fluctuations in Photoconductors The estimates on the statistical fluctuation phenomena in photoconductors, which are brought about by various causes, serve not only to gain information on the lower limit of detection of photoresistors, but also in its precise investigation, the dependence of fluctuation phenomena of semiconducting properties and electrode configurations is, by all means, suitable to contribute to the clarification of the conduction mechanism of the semiconductor itself (56).One should regard the photoresistor and the radiation source as one system. In order to analyze it, it is necessary to eliminate the individual sources of noise. One should take into account, first of all, the fluctuations of the radiation incident on the photoconductor. The mean square fluctuation per frequency interval of the emitted light quanta (number n p h ) from the surface Ob of the radiator is given by

which may be derived from the fluctuations of radiations of a black body a t a temperature T and volume V in the spectral region v1 to v2. Fluctuations in black-body radiation are given by

79

PROBLEMS OF PHOTOCONDUCTIVITY

I n considering the fluctuation phenomena in the photoconductor itself, we have statistical laws which are valid for the collision processes, excitation and recombination phenomena. The charge carrier concentrations and the velocities of the charge carriers fluctuate whether or not the photoconductor is irradiated. For simplicity let us consider an n-type semiconductor. If we introduce the electron velocity v instead of the electron mobility /* in Eq. (2), we obtain the current fluctuation which is given by 6 i = e(n&

+ van).

(46)

The first term of Eq. (46) expresses the Nyquist noise and is given by __

4k T 6inr2 = __ d

R

”

(47)

where R is the resistance of the semiconductor and df is the bandwidth. One must note that a t extremely low temperatures deviations from Eq. (47) may occur. The second term of Eq. (46) can be calculated if we assume that every charge carrier that is present in a given band causes a current pulse during the time 7 . According to Eqs. (2) and (3), this current pulse is proportional to the field strength E and the mobility p . One may call this term pure semiconductor noise. Considering the fact that 7 is proportional to 70, as given in Eq. (7), we obtain for the second term

therefore, the average fluctuation increases as &. If we assume that the lifetime T is statistically distributed then Eq. (48) is transformed to

Another cause for noise, brought about by the metal contact (electrodes), may be called contact or electrode noise. One must also add a boundary layer noise if there are no ohmic contacts and only unidirectional contacts. One may calculate the current fluctuations brought about by the contact noise from

The fuiiction F ( f ) in most cascs is proportioiial to (l/f).

80

P. GORLICH

Finally, photoconductors which exist in polycrystalline layers or as sintered and pressed layers exhibit an additional noise which may be called crystal noise. The traversal of charge carriers in such layers is not unhindered. There are difficulties in expressing the crystal noise in terms of a formula but it might be expressed in a form similar to Eq. (50). The l/f dependence plays a large role also for the noise caused by the surface conditions. The clarification of these is presently under investigation (57).

X. CONCLUSION From the large number of investigations in the field of photoconduction, we have endeavored to sift out the most important and most definite laws. We are trying to supplement these with the most recent results and so to outline a survey of the present status of the problems in the field of photoconduction. I n a review article of this sort we are not able to deal with details. The choice of subject material is, therefore, somewhat arbitrary. In the field of solid state investigations the phenomenon of photoconduction occupies a position in solid state physics equivalent in value as an investigative method to, for instance, the temperature and conductivity measurements or the magnetic measurements which are necessary to understand solid state physics. One may expect future investigations of photoconducting processes in liquid solid boundary regions and in organic compounds. The methods of investigation of the interior and surfaces of these systems and substances will be at least equivalent and probably superior to those available a t present. List of Symbols

A A B

B B,

c

D Dindrx

F

G

H

IB

k L LD

M

Activator concentration (A+) Amplitude of alternating radiation Width of the electrodes Magnetic induction Constant Thickness of layer Demarcation level Amplification factor Gain Concentration of trapping states Intensity of illumination Wave vector Separation of electrodes Diffusion length Crystal momentum

PROBLEMS O F PHOTOCONDUCTIVITY

81

Surface Resistance Black-body radiation Concentratioii of defect states (8-, S+) Transit time Potential Unidirectional potential Rectification potential Volume Excitation Loss Constant Separation distaiice Lattice constant Elementary charge Production rate (gn, g p ) Dark current Wave vector Effective electron mass Concentration of electrons Inversion density Concentration of holes TTelocityof surface concentration Excitation and recombination processes Velocity of charge carriers Displacements (xn,2), Po ten tial Absorption coefficient Dielectric coiistaiit Mobility ( p n , p p ) Frequency of the light Conductivity Lifetime ( T ~T, ~ ) Decay time (time constant) Space charge relaxation time Circular frequency of irradiation RXFERENCES 1 . Stockmmn, F., Z.Physik. 143, 348 (1955). 2. Haken, H., in “Semiconductor Problems” (W. Schottky, ed.), p. 1. Vieweg, Braunschweig, 1958. 3. Gross, E. F., Kapljanski, A. A , , and Novikon-, B. V., Doklady Akad. Nnuk S.S.A’.R.

82

P. GORLICH

110, 761 (1956); J . Tech. Phys. (U.S.8.R.) 26, 697, 913 (1956); Broser, I. and Broser-Warminsky, R., 2. Elektrochem.. 61, 209 (1957). 4. Gross, E. F. and Rasbirin, B. S., J . Tech. Phys. (U.S.S.R.) 27, 1398 (1957). 5. Gross, E. F. and Rasbirin, B. S., J . Tech. Phys. (U.S.S.R.) 28, 237 (1958); Boer, K. W. and Gutjahr, H., 2. Physik 162, 203 (1958). 6. Gijrlich, P. and Heyne, I., Optik 4, 206 (1948). 7. Cf. e.g. Bube, R. H., Phys. Rev. 99, 1105 (1955). 8. Balkanski, M. and Waldron, R. D., Phys. Rev. 112, 123 (1958). 9. Broser, I. and Broser-Warminsky, R., Phys. and Chem. Solids 8, 177 (1959). 10. Macfarlane, G. G., Mclean, T. P., Quarrington, J. E. and Roberts, V., Phys. und Chem. Solids 8, 388 (1959). 11. Kessler, F. R., Phys. and Chem. Solids 8, 275 (1959). 12. Sttjckmann, F., Klontx, E. E., MacKay, J., Fan, H. Y. and Lark-Horovita, K., 2. Physik 163, 331 (1958); Cronemeyer, D. C., J . Appl. Phys. 29, 1730 (1958); Lasser, M. E., Cholet, P. and Wurst, E. C., J . Opt. Soc. Am. 48, 468 (1958); Beyen, W., Bratt, P., Davis, H., Johnson, L., Levinstein, H. and MacRae, A., ibid. 49, 686 (1959). 13. Kohler, V. and Hauser, O., Lecture, Ann. Meeting Phys. SOC.,Leipaig (1959). 14. Rose, A., Proc. I. R. E. 43, 1850 (1955); Phys. Rev. 97, 322 (1955). 15. Sim, A. C., J . Electronics and ContrgZ 6, 251 (1958). 16. Pohl, R. W. and Sttjckmann, F., Ann. Physik [6] 6, 89 (1949); Stockmann, I?., Z.Physik 147, 544 (1957). 1 7 . Cf. Sttjckmann, F., 2. Physilc 138, 404 (1954); Polke, M., Rtorch, G. and Stiickmann, F., ibid. 164, 51 (1959). 18. Cf. Rose, A., Helv. Phys. Actu 30, 242 (1957); Gorlich, P., Krohs, A. and Lang, W., Arch. Tech. Messen 394-1 (1958); Rose, A. and Lampert, M. A., Phys. Rev. 113, 1227 (1959); Redington, R. W., ibid. 116, 894 (1959). 18. Cf. e.g. Ruppel, W., Z. Physik 162, 235 (1958). 20. Lambe, J., Phys. Rev. 98, 985 (1955). % I . Newman, R., Woodbury, H. H. and Tyler, W. W., Pltys. Rev. 102, 613 (1956). 22. Cf. e.g. Schon, M., in “Semiconductor Problems” (W. Schott,ky, ed.), Vol. 4, p. 306. Vieweg: Braunschweig, 1958; Rittner, E. S., in “Photoconductivity Conference” (R. G. Breckenridge, B. R. Russell, and E. E. Hahn, eds.), p. 215. Wiley, New York, 1956. Gorlich, P., Krohs, A , , and Lang, W., Arch. tech. Messen 394-1 (1958); Muser, H., Z. Naturforsch. 78, 729 (1952). 63. Cf. e.g. Boer, K. W. and Vogel, H., 2. physik Chern. (Frankfurt) \?J.S.] 1, I , 17 (1956); Schijn, M., in “Semiconductor Problems” (W. Schottky, ed.), pp. 319, 333, 345. Vieweg, Braunschweig, 1958. $4. Boer, K. W. and Wantosch, H., Ann. Physik [6] 2, 406 (1959). 2.5. Cf. e.g. Bube, R. I%.,in “Photoconductivity Conference” (R. G. Rreckenridge, B. R. Russell, and E. E. Hahn, eds.), p. 575 VC’iley, New York, 1956; Tolstoi, N. A. and Sokolow, V. A., Optika i Spektroskopiya 3, 495 (1957). 26. Read, W. T., Phil. May. [7] 46, 775 (1954); Matarb, H. F., J . Appl. Phys. 30, 581 (1959). 27. Votava, E., 2. Natt~rfo~sch. 13a, 542 (1958). 28. Bullough, R., Newman, R. C. and Wakefield, J., Lecture, Intern. Convention on Transistors, London, May 21-27, 1959. 69. Cf. Gorlich, P., Optik 8, 512 (1951); Jenaer Jahrb. p. 237 (1951). SO. Miseljuk, E. and Martens, E., Lvest. Akad. Nauk S.S.S.R., Ser. Fiz. 14, 115 (1952). Sl. Cf. Snmmary, Gorlich, P., Advances in Electronics and Electroh Phys. 11, (1959).

PROBLEMS OF PHOTOCONDUCTIVITY

83

32. Boer, K. W., Borchardt, E. and Borchardt, W., 2. physik Chem. (heipzig) 203, 145 (1954). 33. Tyler, W. W., Newman, R. and Woodbury, H. €I., Phys. Reu. 97, 669 (1955); 98, 461 (1955). 34. Wolkenstein, F., “Conductivity of Semiconductors,” p. 340. Moscow-Leningrad 1947; Juse, W. and Rywkin, S., J . Exptl. Thedret. Phys. U.S.S.R. 20, 152 (1950). 35. Borissov, M. and Kanev, St., 2. physik. Chem. (Leipzig) 206, 56 (1955). 56. Rose, A.,R C A Rev. 12, 401 (1951). 57. Bube, R. H., J . Chem. Phys. 21, 1409 (1953). 58. Shockley, W. and Read, W. T., Phys. Rev. 87, 835 (1952); Brattain, W. H. aud Bardeen, J., Bell System Tech. J . 82, 1 (1953); Kingston, R. H., J. Apppl. Phys. 27, 101 (1956). 39. Cf. Brown, W. L., Brattain, W. H., Garrett, C. G. B. and Montgomery, H. C., in “Semiconductor Surface Physics” p. 111. 1956; Bardeen, J., Zntem. C m f . on Semiconductors and Phosphors, Garmisch-Partenkirchen p. 81 (1956); Wang, s. and Wallis, G., Phys. Rev. 107, 947 (1957). 40. Heiland, G., Phys. and Chem. SoZids 6, 155 (1958). 41. Sosnowski, L., Soole, B. W. and Starkicwicz, J., Nuture 160,471 (1947); Gorlich, P. and Krohs, A., Jsnaer Jahrb. p. 155 (1952). 42. Petrita, R. L., Lummie, F. L., Sorrows, H. E. and Woods, J. F., in “Semiconductor Surface Physics” p. 229 (1956). 43. Stiickmann, F.,2. Physik 146,407 (1956). 44. Batt, E. and Stockmann, F.,Z. Naturforsch. 13a, 352 (1958). 46. Gerritsen, H. J., Ruppel, W. and Rose, A., Helv. Phys. Acta 30,235,495,504, (1957). 46. Wiesner, R., in “Semiconductor Problems” (W. Schottky, ed.),Vol. 3, p. 59, 1956. 47. Kikoin, I. K. and Noskow, M. M., Physik. 2. Sowjetunlion 6, 586 (1934). 48. Aigrain, P. and Bulliard, H., Coompt. rend. acad. sci. 236, 672 (1953); Bulliard, H., Ann. phys. 9, 52 (1954); Moss, T. S., Pincherle, L. and Woodward, A. M., Proc. Phys. Soc. B66, 743 (1953); Moss, T. S., ibid. p. 993; Kurnick, S. W., Straws, A. J. and Zitter, R. N., Phys. Rev. 94, 1791 (1954); Bulliard, H., ibid. p. 1564. 49. van Roosbroeck, W., Phys. Rev. 101, 1713 (1956); Guro, G. M., J. Tech. Phys. (U.S.S.R.) 28, 1036 (1958); Walton, A. K. and Moss, T. S.,Proc. Phys. Soc. 73,399 (1959); Cardona, M. and Paul, W., Phys. und Chem. Solids 7, 127 (1958). 60. Dember, H., Physilz. 2. 32, 554, 856 (1931); 33, 207 (1932); Frenkel, J., Nature 132,312 (1933); Ph?/sik. 2. Sowjetunion 8, 185 (1935). 61. Rilsum, C . and Ross, I. M., Nature 179, 146 (1957). 56. Welker, H., Z. Naturforscli. 6a, 184 (1951); Msdelung, O., Tewordt, L. and Welker, H.,ibid. lo&, 476 (1955). 65. Smit,h, R. A,, Advances in Phys. 2, 321 (1953); Schulta, M. L. and Morton, G. A., Proc. I . R. B. 43, 1819 (1955); Bube, R. H., ibid. p. 1836; Moss, T. S., ibid. p. 1869; Kluge, W., 2. angew. Phys. 7, 302 (1955); Garlick, G. F. J., in “Handbuch der Physik” (S. Flugge, ed.), Vol. XIX, p. 316 (1956); GBrlich, P., Krohs, A. and Lang, W., Arch. Tech. Messen 394-1 (1958); Wright, D. A,, Brit. J . Appl. Phys. 9, 205 (1958). 64. Gorjunowa, N. A. and Kolomijez, B. T., Radiotechnika and Elektronika (U.S.S.R.) 1, 1155 (1956); Chansewarow, R., Rywkin, S. M. and Agejewa, I. N., J . Tech. Phys. (U.S.S.R.) 28, 480 (1958); Kolomijez, B. T. and Malkowa, A. A., ibid. p. 1662; Lawson, W. D., Nielsen, S., Putley, E. H. and Young, A. S., Phys. and Chem. Solids 9, 325 (1959). 55. Nitsche, R., Festkorpertagung, Balatonfured, Hungary, 1959, to be published.

84

P. G ~ ~ R L I C H

66. Cf. e.g. Gorlich, P., Jenaer Jahrh. p. 229 (1951); Optik 8, 512 (1951) (additional references); for complete treatment cf. Jones, R. C., Advances in Electronics 6, 1 (1953); Roberts, D. H. and Wilson, B. 1,.H., Brit. J . Appl. Phys. 9, 291 (1958). 67. McWhorter, A. L., in “Semiconductor Surface Physics” p. 207 (1956); Kingston, R. H., J . A p p l . P h p 27, 114 (1956).