J. SINGH and H. BAESSLER: Solution of the Onsager Ion Recombination Model 425

phys. stat. sol. (b) 63, 425 (1974)

Subject classification: 14.3.2 and 16; 22.9

Fachbereich Physikalische Chernie der Universitat Marburgllahn

Solution of the Onsager Ion Recombination Model in One Dimension

BY J. SINGH and H. BAESSLER

Applying Onsager’s theory in one dimension it is shown that there is a finite probability for a carrier to escape geminate recombination with its image charge a t the interface of a molecular crystal and metal even in the absence of an external field. In contrast to the three dimensional case, however, the escape probability is not a strictly linear function of the field and depends on the initial pair separation, even a t low fields. Good agreement between theory and experiment is obtained.

Die Onsagersche Theorie fur Paar-Rekombination von Ladungstragern wird fur den ein- dimensionalen Fall abgeleitet. Sie wird angewandt zur Deutung der Feldstarkenabhangig- keit der Ausbeute bei der photoelektrischen Ladungstragererzeugung an der Grenzflache zwischen einem Molekulkristall und einem Metallkontakt. Es zeigt sich, daB auch bei ver- schwindendem Feld ein Trager eine endliche Wahrscheinlichkeit besitzt, aus dem coulombi- schen Einfangbereich seiner Bildladung zu entkommen. Im Gegensatz zum dreidimensio- nalen Fall ist jedoch im gesamten Feldstarkebereich die Dissoziationswahrscheiqlichkeit keine streng lineare Funktion der Feldstarke und hangt auBerdem vom anfanglichen Paar- abstand ab. Die Vbereinstimmung zwischen Theorie und Experiment ist gut.

1. Introduction The generation of charge carriers in crystalline anthracene due to an intrinsic

single-photon process was discovered several years ago [l]. It is now generally accepted that this involves autoionization followed by a diffusive temperature and field assisted separation of the positive and negative charges [2]. Northrop and Simpson [3] had suggested a similar mechanism. Later on, several workers observed that the efficiency of charge carrier production depends upon temper- ature and applied electric field and suggested that this is due to the recombina- tion of ionized states. Kepler and Coppage [4] concluded that electron and hole recombine unless the electron reaches a mutual distance & 1200 i% before complete thermalization. Subsequently geminate recombination following photoionization has been discussed in a number of publications [5 to 71 in terms of Onsager’s recombination theory [8]. The experimental data support the application of this theory in bulk crystalline anthracene since the carrier quan- tum yield depends linearly on field with the correct Onsager slope/intercept- ratio and apparently the theoretical temperature dependence.

Recently Iiillesreiter and Baessler [9] and Geacintov and Pope [lo] sug- gested that the exciton dissociation phenomenon a t the interface of a metal contact in molecular crystals is equivalent t o autoionization of a bound mole- cular state in the bulk crystal. One may therefore expect that the influence of an external field on carrier generation rates is principally the same for both processes except that the former one is essentially in one dimension and the latter one is in three dimensions.

426 J. SINQH and H. BAESSLER

The purpose of this paper is to investigate under which conditions the three- dimensional Onsager treatment is valid also’ in one dimension. The result will be compared with the recent theoretical study by Haberkorn and Michel- Beyerle [ 111.

2. Formulation of the Problem

For better understanding i t seems necessary to review briefly the results of the original Onsager theory [8]. It considers the Brownian motion of a charged particle under the combined action of an external electric field E parallel to the z-axis and the coulombic field of its counter charge. The three-dimensional total potential (w) acting on the particle can thus be written in spherical polar coordinates as

Lo(?-, 6 ) = - er cos 0 - Dr where #? = eE/(BkT) and q = e2/2D l c l ’ . D is the dielectric constant of the me- dium. Onsager found that in the case of lowapplied fields the probabilityp, that the particle does not take part in geminate recombination, satisfies the dif- ferential equation

div{exp( --$)gradp} = 0 .

It may here be noted that (2 ) takes this form because the divergence of the field of potential co given by (1) is zero. The solution of ( 2 ) has been obtained by Onsager in the form

y ( r , 0) = e-Br(l+coW J J, (2[-pr (1 + cos 0) s ] ~ / ~ } ed8 ds = W

s =2p/r c o r n

- - e--2p/r--g~(i+cose) 2 gm+n (1 + cos e ) m + y 2 q ) m rn [m!(m + TL)!]’~ m=o n=o

(3)

p(r, 0) = e-2q/r for p = 0 . (4)

for p* 0 ’ and

Batt et al. [6] applied Onsager’s theory to explain the field and temperature dependence of the charge-carrier quantum yield in the bulk of crystalline anthracene. For small values of the applied field they expanded (3) as

p(r, e ) = exp -r + 2 ~ q (1 + cos 0) + p w q - Y ) (1 + cos e ) z i . (5) ( ““1 If one neglects the second and higher order terms of #?, (5) gives a linear depend- ence of p on the field E with a slope over intercept ratio S / i = eql2lcT which depends on temperature as T-2. The experimental results of Bat,t et al. [6] are in excellent agreement with these two predictions of the theory.

In course of the measurements of photocurrents due to dissociation of Frenkel excitons in anthracene crystals a t the interface of a metal contact Killesreiter and Baessler [9] obtained results similar to those for charge carrier generation in the bulk. Therefore they suggested the validity of Onsager theory also in one dimension and explained their results in terms of dissociation of a singlet exciton localized on a molecule in the molecular crystal a t the interface of a

Solution of the Onsager Ion Recombination Model in One Dimension 427

metal producing a free-charge carrier in the crystal together with its image charge in the metal. In the course of correlated Brownian motion both charges can either recombine or get separated against the attractive coulombic force. The important difference from charge separation in the bulk is that due to charge and image charge correlation, the orientation of the charge-image charge dipole is always parallel to this direction of the applied field vector. This makes t,he process essentially an unidimensional one.

The total potential given in (3) becomes

where z is the separation of one of the carriers from the interface. It is important t o note that the divergence of the field of the potential given by (6 ) is no longer zero. Thus the reduction of dimensionality brings a basic difference from the original Onsager treatment,. Following Onsager's approach [8] and using the potential given by (6)) the differential equation satisfied by the probability of dissociation (9) is changed from (2) to

However, ( 7 ) is also a self-adjoint second order differential equation in one dimension like (2) in three dimensions. A complete solution of (7) is

where A and C are the integration constants. Applying the boundary conditions [8] 9 = 0 for z = 0 and pl = 1 for z+ KI one obtains A = 1 and C = 2g. Then (8) can be written as

(8) pl = Ae-qBZ-WZ + Ce--2@--9/2Z j e28z+@ & )

(9) ) 00

p(z) = e-zBz-Sl"z 1 $. 2,9 j e28ZW-2z dz ( 20

for B $; 0 and ~ ( z ) = e-q/2z for = 0. The lower limit of z is the initial intra- pair separation z,. In most cases the inequality 2/32 < 1 holds. One can therefore expand the exponential within the integral of (9) and express i t as Bessel function J,:

which is similar to (3). One may possibly think of expanding (10) in terms of a Laurent series as in (3). But in contrast to the three dimensional case the Laurent expansion of exp (q /2z) is not possible, since this function is not analytic between any two concentric circles. Instead it is analytic between any two planes in between z = 0 and z = co because the process is unidirectional. Here geminate recombination can take place only a t the interfacial plane along a single direction. In three-dimensional case, on the other hand, it can take place at any central point from all possible radial directions.

It is not possible to solve the integration in (9) analytically. However, it can be simplified without introducing any serious error. Photocurrent measure- ments are always performed a t finite electric fields. A carrier can be regarded as free whenever i t has reached a distance from the electrode where the potential due to the applied field dominates over the coulombic potential. This is the case

428 J. SINQH and H. BAESSLER

for z > z , where zm indicates the position of the potential maximum in front of the electrode. From (6) z, = (q/4/3)lI2 follows. Consequently the upper limit of the integration can be replaced by z,. It is obvious that for z 5 z, the equality 2gz 5 q/22 is fulfilled ( ( 2 / 3 2 ) / ( 4 / 2 ~ ) 5 1 for z 5 zm). Therefore the term 2/32 can be neglected in the exponential and then (9) becomes

Integration of (11) by substituting t ==- 4/22 yields

where p(t) = exp (-4 (1 + P I ) 9 ( 1 2 )

p = - exp ( t ) t '

I ( t ) = P - (In t + &),

3. Discussion It is obvious from (12) that p is not linear with the applied field because z ,

arid consequently the factor I depends on the applied field. The variation of p with the applied field is shown in Figs. 1 and 2. These curves have been drawn for initial separations zo = 5 A (Fig. l ) , 11 and 20 A (Fig. 2 ) . In all cases p starts decreasing after having reached a maximum a t a field of typically lo5 V/cm. However, this is due to the basic approximation 2gz 5 q/2z used to obtain (12)) which is no longer valid a t high fields. Instead, the first term will have a dominant effect. This yields an exponential increase of p with E.

It is evident from Fig, 1 and 2 that the initial separation zo between charge and image charge carrier has a remarkable influence on the field dependence of p and the photocurrent i,, which in the absence of space charge effect is propor- tional to p. zo can be calculated from the activation energy E, = e2/4Dz0 of

E(103V/cm) - E(103V/cm) - Fig. 1 Fig. 2

Pig. 1. Variation of e, - (1 + pal) with the applied field a t an initial thermalized separa- tion zo = 5 A

Fig. 2. Variation of e, - (1 + p a l ) with the applied field calculated for initial pair separa- tions zo = 11 and 20 A. The experimental curve is taken from [9]. It shows the field de- pendence of the photocurrent originating a t an anthracene/aluminum contact (ordinate

scale in arbitrary units). -x-x- experimental, ~ theory

Solution of the Onsager Ion Recombination Model in One Dimension 429

the photocurrent a t low fields. The agreement between theory and experiment is tested for the values of z, = 11 A and z, = 5 A which had been obtained under different carrier generation conditions :

1. For singlet exciton-induced photocurrents originating a t anthracene- metal contacts E, is typically (0.08 f 0.02) eV [12] yielding z,, = (11 3) A. Comparison between the measured photocurrent-field dependence and the field dependence predicted by (12) (see Fig. 2) shows good quantitative agreement for E & lo6 V/cm. It is apparent that only in a medium field range, 2 x lo4

E & lo6 V/cm an almost linear relationship between i, and E exists. At lower fields, E lo4 V/cm, the apparent slope/intercept ratio ([aip/aE)/(ip(E=O)]) obtained from (12) approaches a value which can be as much as one order of magnitude higher than the value pq/E predicted by the three dimensional theory. Such a behaviour has also been observed by Michel-Beyerle e t al. [13] with chloranil crystals for photocurrents generated by exciton dissociation a t a gold contact.

2. Photocurrents generated by direct photoemission of electrons or holes from an optically excited metal contact into an anthracene crystal are associated with an activation energy of (0.2 f 0.05) eV corresponding to z, = (5 1) A [14]. Fig. 1 shows that the predicted photocurrent field characteristic is almost ohmic, i.e. the calculated slope over intercept ratio is >j3q/E. This is confirmed by experiments showing that photoemission currents fulfil practically Ohm’s law [la].

It is important to note that the present treatment yields a finite escape probability of a carrier from the attractive coulombic field due to its image charge also in the absence of an external field. This is in agreement with measure- ments of exciton-induced photocurrents in metal contacted crystals (see e.g. [12]). At very low fields the photocurrent becomes space-charge limited. This proves that an injected carrier has a finite chance to escape geminate recombination with its image charge even if the external field near the contact is completely screened by a bulk space charge. On the theoretical ground and, as they say, following the Onsager model, Haberkorn and Michel-Beyerle recently questioned [ll] the existence of a finite carrier escape probability a t zero field. This discrepancy is due to their particular approach which does not exactly follow the Onsager treatment.

4. Conclusion In the present paper a rigorous treatment of Onsager’s theory of geminate

recombination in one dimension is presented. The theory is applied to explain the field-dependence of photocurrents generated a t the interface between a molecular crystal and a metal. The results obtained are in good agreement with the experiment.

The main results of the calculations are : 1. The current-field relationship is not strictly linear as in the three dimen-

sional case, even a t low fields. 2. There is a finite dissociation probability, given by Boltzmann’s factor, of

the charge-image charge pair a t vanishing external field where the activation energy for pair dissociation is identical with its coulombic binding energy.

3. The current-field slope normalized to the current a t zero field (“slope- intercept” ratio), which is a function of the field, depends on the initial pair separation. It increases with decreasing z,. Since z, depends on the particular

430 J. SINGR and H. BAESSLER: Solution of the Onsager Ion Recombination Model

carrier generation mechanism, i t can therefore not be expected that contact limited photocurrents follow a common current-field relationship. Further complication can arise if an interfacial layer, e.g. an oxide, exists between contact and crystal, which may affect carrier generation and possibly hinder geminate recombination and thus change the charge distribution in the contact zone.

Acknomledgement

The authors are indebted to Dr. H. Killesreiter and Dr. M. E. Michel-Beyerle for several discussions. Financial support by the Deutsche Forschungsgemein- schaft is gratefully acknowledged.

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[5] M. SILVER and R. C. JARNAGIN, Mol. Cryst. 3,461 (1968). [6] R. H. BATT, C. L. BRAUN, and J. F. HORNIG, J. chem. Phys. 49, 1967 (1968); Appl.

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(Received February 21, 1974)