Short Notes K29

phys. stat. sol. (b) @, K29 (1975)

Subject classification: 1 4 . 3 . 2 and 3 6; 22.9

Fachbereich Physikalische Chemie d e r Universitat Marburg

On the Onsager Theory for Geminate Recombination in One Dimension

BY 1) J. SINGH andH. BAESSLER

Recently the present authors (1) (SB) calculated the probability that a charge

carrier generated in a molecular crystal at a distance z from a plane metal con-

tact escapes geminate recombination with i ts image charge. In contrast to other

workers (2 , 3) they found that a t zero external field the ca r r i e r escape probability

0

2 'p i s finite and given by Boltzmann's factor exp(-e / 4 & z kT). However, as op-

posed to the three-dimensional case treated by Onsager (4, 5) no field-independent

slope over intercept ratio of the resulting photocurrent could be defined.

The SB result is based on a solution of the differential equation

0

2 2

dz 2 dz dz

d r p - * d V - 'p? d V = o , dz

where V is the electrostatic potential. In a recent note (6) Haberkorn and Michel-

Beyerle (HMB) claim that the term cp(d V/dz ) is in e r r o r and should be omitted.

The solution of equation (1) simplified in this way leads to the result (2, 3)

2 2

The purpose of the present note is to derive equation (1) explicitly thereby showing

that i t is the only correct differential equation to solve the Onsager problem of

charge separation in one dimension. According to Onsager (equation (18) of reference

(4)) the three-dimensional problem of Brownian motion of two oppositely charged

particles under the combined action of their mutual coulombic field and an external

field E acting in z-direction can be solved starting with the continuity equation under

steady-state conditions

1) Present address: Department of Mathematics, The University of Southampton, Southampton SO9 5NH, UK.

K30 physica status solidi (b) 69

a t

where f is the particle distribution function and

V ( r , z) = -2 p z - 2q/r ( 3) 2

is the total potential in units of kT. The abbreviations p = eE/2kT and q = e /2C kT

a r e used. E is the dielectric constant of the medium. The mean velocity v of car-

riers in a given configuration is found to be (reference (4))

f v = fgradV - gradf , (4)

The problem becomes one-dimensional when motion of a charge carrier generated

at a distance z from a metallic interface by exciton dissociation o r injection is

considered. The countercharge i s then replaced by the image charge, which, a t the

beginning of ca r r i e r diffusion, is located in the metal a t z = -z . Consequently

equation (4) reduces to

0

0

dv df dz dz ’ fv(z) = f - - - (5)

where

V(z) = -2 p z - q/2z . The ca r r i e r escapes geminate recombination with i t s imagecharge if its mean

Minimization re- velocity component in a direction towards the interface is zero.

quires that

dz dz dz

Equation (6) is obtained from equation (5) by substituting z by -z and d/dz by -d/dz

and utilizing the fact that f(z) = f(-z). It is identical with the continuity equation

(equation (2)) in one dimension. Fo r z 4 00 the carrier inside the crystal is free,

hence f(z- 0 0 ) = const. The probability that the ca r r i e r escapes geminate recom-

bination can therefore be defined a s

q ( z ) =f(z)/f(z+-) . (7)

From equations (6) and (7) it is apparent that cp must fulfill the differential

equation

Short Notes K31

a s stated in reference (1). Its solution is

’p (z) = e (9)

The total escape probabilityofcarriers located in a zone z < z < z 1 2 replacing the indefinite by a definite integral with the limits z and z 1

is obtained by

2’ The HMB result (reference (6))

for the probability that a carr ier initially located in a region 0 < z < z reaches

the counterelectrode at z = L cannot be correct since it follows from an incomplete

differential equation. Nevertheless, it shall briefly be discussed in reference to the

problem of the boundary condition Q(L) = 1 which has been used to derive equation

(10). HMB consider the case V(z) = 0 which yields ~ ( z 0 0 ) = z /L. This i s nothing

else but the probability for finding a free particle within a region 0 < z < z in a

one-dimensional box of length L. It is not identical with the probability that the

particle reaches the point z = L. Therefore the fact that lim cp(zo) = 0 cannot be

used a s an argument that lim cp (z E) = 0. E d 0

0

0

L-00

0 ’

Furthermore it is necessary to distinguish between the probability cp that a car -

rier escapes coulombic attraction by i ts image charge and the probability that it

reaches an opposite electrode located at z = L. Cp approaches unity if the coulombic

binding energy is much less than kT. In a medium with t = 3 this condition is ful-

filled for z(E = 0) > 50 % in the case of zero field and for z > z (E) in the case of

finite field. z

(3) has a maximum. 2, For primary car r ie r escape to occur it is therefore not ne-

cessary, a s HMB claim (2) , that a constant diffusion flux of particles is maintained

for 0 < z $ L. However, in a macroscopic electrode system to which no external

voltage i s applied most of the car r ie rs which escaped geminate, i . e. monomolecular

recombination with their image charges will not reach the counterelectrode: A dif-

2) More correctly ‘p(z = z ) w 1/2 since a car r ie r located at z = z m m’ = 0, has equal chance to diffuse either back into the geminate recombination zone

or towards the bulk to be collected by the external field. This, however, is unim- Portant for calculation of the functional dependence ~ ( 2 ) .

m denotes the distance from the interface where V(z) given by equation m

where grad V =

K32 physica status solidi (b) 69

fusion potential is built up leading to stationary equilibrium between production of

llescaped'l charge ca r r i e r s and their annihilation at the injection contact in course

of a bimolecular reaction. It prevents flow of a stationary photocurrent. On the

other hand, observation of a photovoltage at an illuminated metal-anthracene con-

tact (7) clearly indicates that the ca r r i e r escape probability i s finite not only at

zero but even at negative fields. A stationary photocurrent is a measure for the

ca r r i e r escape probability only if an external voltage sufficient to overcompensate

any diffusion and space charge potentials is applied to the sample ensuring that

every escaped ca r r i e r i s extracted from the generation zone. A similar situation

is observed with geminate recombination in three dimensions: lim j > 0 is obtained E - 0

by extrapolation of finite field data (8). The real photocurrent approaches zero for

E - 0 since free ca r r i e r s recombine following second order kinetics. In view of

these arguments we believe that the SB treatment of Onsager's theory in one di-

mension is essentially correct.

Acknowledgement

This work was supported by the Deutsche Forschungsgemeinschaft.

References

(1) J. SNGH and H. BAESSLER, p h p . Stat. Sol. (b) gs 425 (1974).

(2) R. HABERKORN and M.E. MICHEL-BEYERLE, Chem. Phys. Let tersL3,

128 (1973).

(3) D.F. BLOSSEY, Phys. Rev. BZ, 5183 (1974).

(4) L. ONSAGER, J. chem. Phys. 3, 599 (1934).

(5) L. ONSAGER, Phys. Rev. 54, 554 (1938).

(6) R. HABERKORN and M.E. MCHEL-BEYERLE, phys. stat. sol. (b) 61, K61

(1975).

(7) P. THO?MA and G. VAUBEL, phys. stat . sol. Is, 663 (1966).

(8) R.R. CHANCE and C. L. BRAUN, J. chem. Phys. 59, 2269 (1973).

(Received March 1 0 , 1975)