Conditions for superlinear intrinsic photoconductivity

J. Phys. Chem. Solids

Pergamon

Press 1964. Vol. 25, pp. 977-984.

CONDITIONS

Printed in Great Britain,

FOR SUPERLINEAR

INTRINSIC

PHOTOCONDUCTIVITY F. N. HOOGE

and

D. POLDER

Philips Research Laboratories, N. V. Philips Eindhoven-Netherlands (Received

Gloeilampenfabrieken,

8 October 1963)

Abstract-In this paper the precise and complete conditions are derived for the occurrence of superlinear intrinsic photoconductivity in a model substance having two different recombination levels in a forbidden energy zone. The analysis confirms the conclusion of Duboc and Klasens, that in this model, there are two superlinear situations, one in which the concentration of the majority charge carriers is proportional to the square of the light intensity and another one with a much stronger intensity dependence. The physical mechanism producing the latter situation is described.

1. INTRODUCTION

the experimentally observed conductivity o in semiconductors is plotted, on a double logarithmic scale, as a function of the light intensity U, d ln o/d In U is between 3 and 1 in most cases. The value 1 corresponds to a linear relationship between c and U. Only very rarely are values greater than 1 found, and this is called superlinear photoconductivity. ROSE(]) and BuBE@), in connexion with the interpretation of Bube’s extensive experimental material, have described how superlinear photoconductivity can arise, when recombination levels of different types are operative. DuBOC(~) explicitly considered the kinetics of the electron and hole generation and recombination processes. In a model, consisting of a semiconductor with two different levels between the conduction and the valence band the consequences of the mathematical equations are quite involved. Duboc’s technique is to distinguish different cases according to the relative orders of magnitude of the different terms in the equations. He points out U.O. that in case of superlinear intrinsic photoconductivity 71 may be proportional to Us or to Urn. In order to get a more complete picture of all the different typical cases contained in this model, KLASENS@) introduced a systematic scheme of classification. While in each typical case, there is some simple WHEN

977

relationship between concentration of free carriers or trapped carriers and the light intensity or the temperature, the scheme also tells when the transition from one typical case to another one occurs if the light intensity or temperature is varied. In particular the possible occurrence of superlinear photoconductivity can be extracted from the scheme. The aim of our paper is two-fold. With the aid of a precise analysis of the kinetics in a specific model, first of all we wish to obtain the complete conditions for the occurrence of superlinearity and secondly we wish to gain a closer insight into the characteristic features of a superlinear situation. We believe that such an analysis may be useful as it leads to a set of conclusions which are not well known in their totality, even though a number of them have been previously formulated by other authors. From the very beginning, we shall focus our attention on the question of superlinearity, i.e. we restrict our analysis to the question; when can d In c/d In U be greater than l? In fact, we shall restrict our problem a little more by neglecting any variation of carrier mobilities with U and by assuming that in the superlinear situation the conductance is due to the most abundant free carrier. The concentration of this carrier will be called 12 (the choice of electrons is of course arbitrary) so

978

F.

N.

HOOGE

we have n > p. The thermal equilibrium concentrations of n and p are no and ps. Our model will be essentially the same as that of Klasens. There are two levels A and B, A and B will indicate their concentrations; fc is the fraction of A levels occupied by electrons, f,” = 1 -f$. The quantities n$, andpf, introduced by Shockley and Read, are the electron and hole concentrations in thermal equilibrium with A levels that are just half occupied. Obviously n;‘p$ = neps. The kinetics of the process is shown in Fig. 1. The arrow U indicates the number of electrons raised per set from the valence into the conduction band by the action of the light. It is assumed that the light does not produce any other effects. Direct recombination of holes and electrons is neglected. The other arrows indicate transition rates of electrons, the kinetic constants C$, C$ and the constants Ct, C,” are proportional to the concentrations A and B respectively.

and

D.

POLDER

recombination centres can be formally treated with this procedure.* Precisely because we do not want as yet to specify the nature of the A and B centre there is a point of nomenclature we want to draw attention to here. We shall later use the symbols A+ = f;A and A0 = f $A. We use this notation without prejudice as to the charge or effective charge of the centre. 2. GENERAL

PROPERTIES

OF THE

EQUATIONS

U will be considered as being built up by two quantities UA and UB, given by the equations (cf. Fig. 1):

ctf

3

c;f

,“p = c;f

cff 3

= ctf$++ = c:f

;p: +

UA,

(1)

UA,

(2)

En?-+ U&j,

C;f ;p = C;f p”pf+

UB.

(3) (4)

The centre over which the main part of the recombination takes place, we shall now call the A centre. So by definition VA

%

UB.

(5)

Following Shockley and Read, the relations (2), (3) and (4) can be combined into (np-mpoJ(g

+ ..5J$L)

=

J1

UA 43 FIG. 1. Transitions of electrons considered in the model. The relative positions of A and B are arbitrary.

(l),

u=

uA+ UB,

(1)

where UA contains the term with index A and UFJ the term with index B. The denominators are defined by (6)

We shall further adhere to the assumption that under illumination every volume eIement will remain electrically neutral. The precise form of the resulting charge neutrality equation depends on the nature of the centres (for instance A and B can be donors or acceptors). In fact we shall allow in our model the presence of further centres which do not take part in the kinetics, but which give a constant term in the charge neutrality equation by being always full or empty of electrons. The presence of coactivators can be described in this way many level conditions, certain and, under

MB

=

C~(p+pf)+Cf(n+af).

(7)

Besides equation (I) describing the transitions, there is the charge neutrality equation:

n+f tA+f

:B = p+K.

(11)

* Though it is not explicitly mentioned the possibility of the presence of further centres is also implied in Klasens’ treatment, however with the restriction that their charge contribution equals the concentration of either the A or the B centres. This situation is a common one in ZnS-type phosphors, e.g. coactivator concentration equals activator concentration.

CONDITIONS

FOR

SUPERLINEAR

INTRINSIC

In the case that there are only two types of centres

K K K K

= = = =

0 when A is an acceptor and B is an acceptor A when A is a donor and B is an acceptor B when A is an acceptor and B is a donor A+ B when A is a donor and B is a donor

A donor is a centre which does not contribute to the space charge when occupied by an electron. When a third centre is present, which is always filled or empty, K remains a constant, but it may be positive or negative. The following general properties of the system of equations will prove useful: 1. For any positive value of U there exists one and only one positive value of UA, UB, n, p, ji andfi. 2. For any positive value of U dUA/dU, dUB/dU, dn/dU and dpjdU are positive. The proof which requires some algebra goes along the following lines. As a consequence of (I): au/an > 0 and aU/ ap > 0. From the charge neutrality condition (II) follows dp/dn > 0. Thus dU/dn and dU[dp > 0. So n and p are both monotonically increasing with U, from which it follows that there is a one-to-one relation between n and U and also between p and U. The other given properties for f 2, f E, UA and UE then immediately follow from (1) (Z), (3) and (4). Note that df :jdU and df $dU may be positive or negative. Now our problem is to find the conditions under which the given relations will lead to d In n/d In U > 1

PHOTOCONDUCTIVITY

brackets, the denominator is not constant. Thus, keeping in mind our definition of A and B, JI/“A cannot be constant. When in NA the term Ci p is the dominant one, (9) results in n CC U. So in HA the term &n must be the dominant one and we have

C+z 9 Ctnfi + C;(p+pf). A consequence

FOR

(11) i.e. in superlinear situations the A centre must be practically full. A further consequence of (10) is that p cc U(cf. equation (I)) or d lnp/d In U = 1, so that dlnn/dln

dlnn/dlnp.

(12)

Condition (10) is a necessary condition for superlinearity; only if (10) is satisfied is it at all possible to have a large relative variation in n for a small relative variation in U. In order to see whether such a large relative variation can then actually occur, we need to use the charge neutrality equation. With the help of this equation we can find the conditions for d In n] d lnp being greater than one. 3.2 Conditions derived from equation (II) Without that

any approximation

it follows from (II)

n0po.

+B~~zC~~(C~~+c~~~).(13)

-=

SUPERLINEARITY

d lnp

Inspection of this equation shows that when np 2: nope, d In n/d In U cannot be greater than one over a considerable range of n-values, since dn/dU > 0 and dpjdU > 0. Thus for superlinearity over a considerable range, we must have

Under this condition,

U=

when n % p.

3.1 Conditions derived from equation (I)

np %

(10)

of this relation is that

d In n 3. CONDITIONS

979

(8)

relation (I) reads

n + AM-J

C$J + C+z”,)C+z + B.h’“$( C’:p + C~n~)C$

Because of n $ p

and CA nn +c;pi’. ?P 2: CA

It is necessary-although

not sufficient-in to obtain d In n/d In p > 1, that

(9) d In njd In U > 1 over a long n-range is only possible if in the greater of the two terms between

C:p(Cfn+C:pf)

9 Cfn(C,Bp+C$f).

order

(14)

In view of the fact that np $ nope, the only possibility for (14) to be valid over a considerable

F. N.

980

HOOGE

and D. POLDER

range of n-values is cp”p $ c;nl”

(15)

Cipf

(16)

and conditions, np 4 nOpOtlead to These

B

C$.

together

with

(3),

(4)

and

At this stage it is of interest to investigate whether d In nldlnp can be greater than 1 over a considerable range of p-values, while the relation n Cc py exists (V is a constant). If so: n a Uy, p cc U. Substitution of these expressions for n and p into the expressions for B+, B* and A+ (18-20) leads to the condition that the only possibility for (21) to give d In n/d lnp = Y over a considerable U- or n-range is

(17) Equation (17) describes the relative magnitudes of the transitions to and from the B-level, and this is depicted in Fig. 2. The approximate equality of the first two expressions in (17) simply means that there is thermal equilibrium between free holes and the holes captured in the B-level. Using this information we can now write:

(18) (19) GO) while

(13) can be rewritten d In n -=-= dlnU

as

d In n

p + aA+ + BQB+lB

dlnp

n + A+ +SBOB+/B

where

FIG. 2. Conditions

for transition

v=2

(22)

OtZ1

(23)

with

A+EB”B+/B$=~~,

(24)

B+ 21 B.

(25)

This type of superlinearity where n oc Us will be called quadratic superlinearity. In Fig. 2 the situation can be visualized. The relative magnitude of the arrows to and from the A-level is a consequence of a N 1 or

the emptiness gives

of 3 follows from (29,

while (24)

A+ 21 BQ.

(21)

(26)

We may now ask ourselves whether there still exist superlinear situations for a considerable n-range in which n is not proportional to some power of U. Returning to equation (21) we observe that if d In n,ld lnp is to be greater than one,p cannot be the dominant term in the numerator of (21). Also aA+ on its own can never dominate, since u < 1. The quadratic superlinear case, just

rates in the superlinear situations. Left: Extreme Quadratic superlinearity.

superlinearity.

Right:

CONDITIONS

discussed, corresponds

FOR

SUPERLINEAR

INTRINSIC

to the situation

aA+ 2: BoB+/B 9 n. We are still left with the possibility that the term BoB+JB on its own is dominating. In that case we must have BOB-!-/B$ n + A+.

(27)

We shall call this case the extremely superlinear situation. We shall now show that in a first approximation this case of superlinearity will occur around a fixed value of U = Us, where n jumps over a considerable range. From (27) one can deduce B+ > B”B+/B & n+A+

3 p

(28)

BO > B”B+/B B n+A+

B p.

(29)

The charge neutrality condition (II) will be written now in two forms Kl+n = p+A++B+, because of (28): B+ N K1 n+BO = p+A++Kz, because of (29): BO 21 Kz So both Bs and B+ are virtually constant, i.e. they do not change by an order of magnitude. The larger of these two is of course virtually constant, because it is nearly equal to B. We have proved now that the smaller one is virtually constant too. From this follows

PHOTOCONDUCTIVITY

981

both BO and B+ are virtually constant it follows from (18, 19) that p and hence U is virtually constant. The fixed value of Us is easily found, using (I), (10) and (17). U, N CAfAp P n

N CAp = C,p;B+/BO, P

which is entirely determined by the constants, thus Us = C$fB+/Bo.

(31)

From our simple model one may conclude that there are two types of superlinearity, one in which II cc Us and the one in which n rapidly varies at a fixed value U = US. So if one observes a superlinear variation of the photoconductivity over a long U-range, the majority charge carrier concentration must be proportional to U2, if our model is at all applicable. Deviations of dln o/din U from the value 2 must then be due to other effects, such as mobility variations or inhomogeneous illumination. All conditions necessary for superlinearity in n (n 9 p) are shown in Table 1. Important conclusions derived from certain conditions are also included in this table. The conditions for transition rates are also depicted in Fig. 2.

(30)

4. MECHANISM OF THE TRANSITIONS IN THE EXTREMELY SUPERLINEAR SITUATION

The values of K1, K2 and Ks are of no interest here, only their constancy is of importance. When

In this section we shall try to describe the mechanism leading to a sudden increase in free

BOB+/B 2: Ks.

Table 1 Conditions for superlinearity in n, (n > p) Equation (I)

(5)

UA

>

UB

--f

C”p(P+p,“, > CBpn

(8) nP > nope (10) c;?z > CA(p+pA)+C;n: 1 P

Equation (II)

--tf$N

(17) C,f ,“p N C,“f ,“pl” > Cif F” -

Extreme superlinearity

1,p cc u

Ue > C,f ,BnB, Quadratic superlinearity

(30, 27) BOB+/B = const. > n+A+

(23) a N 1 + C;f ,“p > C;f ;n;l

(31) + U, = C,“pfB+/B”

(25) BEB++p>p; (24, 26) A+ N B” > n -_____

_

F. N. HOOGE

982

electron concentration when U 21 Us and all further conditions for extreme superlinearity are satisfied. Because of the special situation at the A level (A = A”, Cip = U) p increases proportionally with U. Thus Ap -=-_

AU

P

u

P

An

B

(33)

BO

Pl from which follows

ABf

AU

BOB+IB

U

1

(34)

-=-_

P Because of (BOB+/B)

lAA+I

AA0

ABf

> p we also have

AB+

AU,

BOB+IB

U

g-=-,

n

B+

_=-

Afi

the increased recombination via A. Indeed, the expression U+C~f$z~=C~f$z [equation (l)] which described the recombination of electrons through A, shows that f gn (or A+n) must increase, so that ->A+=AfrAf

The conditions at the B-level imply thermal equilibrium between free holes and the holes in the B-level. Thus

-=

and D. POLDER

(37) the latter inequality being again a consequence of (27). Thus we see that a superlinear increase of n with U is either a direct result of the compensation of the charge in the B levels by free electrons or is due to the frustration of the recombination through A, this in itself being a result of charge compensation by A levels. Which of these two mechanisms actually operates is determined by the ratio of A+ and n, as can be seen in the following way. Differentiation of equation (1) leads to

AB+ B Ap i.e. the increase of positive charge in the B-levels is far greater than that due to the free holes. This increase must be compensated by an increase of negative charge provided by additional free electrons and additional electrons captured in the A-levels (An+ AA0 2: AB+). If charge compensation takes place mainly by additional free electrons (An 1: AB+), we can immediately see that a small relative increase of the light intensity produces a large relative increase in free majority carriers, viz. An _N- n

AB+ n

’

AB+ --_=BOB+IB

AU U

(36)

the inequality being a consequence of the condition (27). If the number of electrons in A-levels were in fact decreasing (which is not so), An would be even greater and (36) would hold a fortiori. However, if charge compensation takes place mainly by additional electrons in A levels (AA0 N AB+), the rapid increase of n is not so obvious at first sight. It is then a result of the fact that further filling of the already nearly full A levels tends to block the recombination of electrons through A centres, so that a rapid increase in n is necessary to maintain

In the extremely superlinear AU/U and we have An lAf:l _N _-=-=-. n f;

situation

-AA+

AA0

A+

A+

An/n $

(39)

Thus, so long as A+ > n, charge compensation is mainly by A levels, while if n > A+ the free electrons take care of the required compensation of charge. As a matter of fact it is quite possible that if the former mechanism is operating at a given light intensity U 2: Us, the latter mechanism takes over at a slightly higher U-value, i.e. as soon as n outnumbers A+. Up till now we have discussed the nature of the extreme superlinear situation. We wish to proceed with a consideration of the limits of this situation i.e. we want to see why the variation of n with U ceases to be superlinear for somewhat higher U-values as well as for somewhat lower U-values. With increasing U, n increases and one of the conditions (5), (17) or (27) will be violated, and with decreasing U one of the conditions (8), (10) or (27). Depending on the actual values of the constants (which are functions of the temperature) at the

CONDITIONS

FOR

SUPERLINEAR

higher U-values superlinearity will end for one of the following reasons : 1. n + BOBfIB (27). The only reason for the disappearance of superlinearity is that 11 itself becomes so large that the relative increase in n now becomes equal to the relative in;re;se in U. 2. CnfPn + C~$p~ (17). This means that the communication between B-level and conduction band is no longer negligible, in other words the B-level ceases to be a “trap” for holes, which is in thermal equilibrium with the free holes. 3. Un --f UA (5). By definition the A level is the level over which the main part of the recombination proceeds. Let us assign the symbol X to the chemical centre, the level of which acts as the A level in the superlinear situation and the symbol Y to the other recombination centre. From (10) it follows that

for light intensities equal or greater than Us. Now UB + UA means that recombination no longer proceeds via the X-centre but that the Y-centre takes over with increasing U. If superlinearity were going to continue, the level of the X-centre would have acted as B-level and hence (16) would give

Cgpf $- Czn, which is clearly in contradiction with (40). In other words, once the X-centre has acted as the main recombination centre in a superlinear region, it can no longer act as a trap for holes at the higher light intensity. At the lower U-values superlinearity will end for one of the following reasons, when n g p in and before the superlinear situation. 1. In the superlinear range n decreases rapidly with decreasing U. Therefore A+ increases rapidly with U (cf. 39) and may reach the value (BOB+/B), violating (27). Though in this situation charge compensation still occurs by A levels, the relative change in A+ is no longer great and hence it causes no serious blocking effect on the electron recombination. 2. As soon as one of the conditions (8, 10) is

INTRINSIC

PHOTOCONDUCTIVITY

983

violated, equation (I) shows that we are out of the region where rapid variations in n are at al possible for small changes in U. One further point may be worth mentioning. We have seen that a centre, once it has acted as the main recombination centre in a superlinear region, cannot act as a trap for holes at a higher light intensity. Thus two or more superlinear regions for different intensities should all have the same centre as main recombination centre. In particular an extremely superlinear region can only occur once, as the intensity is fixed at the Us value of the corresponding centre.

5. FURTHER SPECIFICATION OF THE CENTRES So far we have investigated the conditions to be fulfilled for superlinearity to occur. The question arises to what extent these conditions can be realized in practice. Important in this respect are the nature of the centres, their concentrations, and their cross sections for holes and electrons. As a complete discussion of the question would be rather involved, we shall restrict ourselves in this paper to a few simple considerations, based on the confrontation of the charge neutrality equation with the condition A+ II B’J g n for quadratic superlinearity and with the condition BOB+/B 2: K B>n+ A+ for extreme superlinearity. It is clear that the condition A+ _NBO $ n, which must hold in the case of quadratic superlinearity, can be realized if the charge neutrality equation reads p + A+ = n + B”, i.e. if A is a donor and B an acceptor. The concentrations of A and B could then be arbitrary. The concentrations of the other levels that give a constant term in the neutrality equation (C-levels e.g. coactivators) should be negligible lest they should spoil the relation p+ A+ z n+ BO. If however the concentrations of A and B are very nearly equal, A could also be an acceptor and B a donor. For extreme superlinearity the relation 12+ A+ < BOB+jB < B’J excludes the possibility that A is a donor and B an acceptor if no C-levels are present. If A is an acceptor and B is a donor the charge neutrality equation reads p + B+ = n + A0 and this is compatible with the condition for extreme superlinearity, viz. A+ < A0 N B+ < BOB+/B < B. Apparently the concentration of A N A0 must then be smaller than that of B. If,

984

F. N.

HOOGE

however, C-levels are present in a sufficient concentration the charge neutralitv equation can always be made compatible with the condition for extreme superlinearity whatever the nature of the A and B centres. Apart from the foregoing considerations concerning concentrations and the nature of the centres, one also has to consider the positions of the levels and the capture cross-sections in order to see whether a particular superlinear situation is I

A

and

D. POLDER

feasible. We shall following paper.t5)

treat

these

questions

in

REFERENCES ROSEl., Proc. I.R.E. 43, 1850 (1955);

a

Phys. Rev. 97, 322 (1955). 2. BUBE R. M., Proc. I.R.E. 43, 1836 (1955); J. Phys. Chew Solids 1, 234 (1957). DUBOC C. A., Brit. J. Appl. Phys. Suppl. 4, S107

3’ (1955). 4. KLASENSH. A., J. Phys. Chem. Solids 7,175 (1958). 5. HOOGE F. N., Philips Res. Repts., to be published.