The elementary step of charge carrier transport in polymeric systems studied by the irreversible stochastic transition theory

SYUiITH|TII[ I-IiII|TRII.S ELSEVIER

Synthetic Metals 64 (1994) 171-175

The elementary step of charge carrier transport in polymeric systems studied by the irreversible stochastic transition theory Yuri A. Berlin a'l, Dmitri O. Drobnitsky b, Valdimir V. Kuzmin c aInstitute for Molecular Sciences, Myodaiji, Okazaki 444, Japan bN.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, ul. Kossygina 4, Moscow 117334, Russian Federation ffnstitute for Nuclear Safety, Russian Academy of Sciences, B. Tulskaja 52, Moscow 113191, Russian Federation

Received 5 January 1994; in revised form 27 January 1994

Abstract

The influence of external stochastic impacts caused by configurational changes of charge carrier surroundings on the kinetics of the trap release process has theoretically been studied. To take into account these impacts, any single step of charge carrier hopping has been treated as a randomly affected monomolecular reaction rather than as a conventional first-order decay process. It has been shown that the combination of the irreversible stochastic transition theory with the concept of diffusion perpendicular to the reaction coordinate allows us to obtain the exact and general analytical result for key kinetic characteristics of a single hop. These results have been used to consider two physically important cases, when the stochastic impact can be approximated by the hopping-independent diffusion or, alternatively, by diffusion coupled with hopping. In both cases transient kinetics has been found to be essentially non-exponential. However, decay curves differ in their forms depending on the type of diffusion process. These findings suggest that in addition to site and bound disorders, conformational changes can be considered as another origin of the dispersive carrier transport in polymeric systems.

I. Introduction

Charge transport phenomena in organic materials remain a subject of intensive studies since the early 1960s, when several investigators employed the 'timeof-flight' technique to measure the drift mobility of charge carriers in organic solids (for a review see, e.g., Refs. [1-6]). The essential contribution to progress in this field has been made by Professor H. Inokuchi who was among the first to perceive the important role of intermolecular ~--electron overlapping in the mechanism of electric conductivity of organic semiconductors with conjugated bonds [7,8]. As was understood later, the intermolecular orbital overlap is of fundamental significance for charge transport in solids, in general, since in many cases it governs the elementary step of a charge carrier motion in such

IOn leave from N.N. Semenov Institute of Chemical Physics, Russian Academy of Sciences, ul. Kossygina 4, Moscow 117334, Russian Federation.

0379-6779/94/$07,00 © 1994 Elsevier Science S.A. All rights reserved SSDI 0379-6779(94)02102-5

media. In particular, it is widely accepted that in molecularly doped polymers charge transport occurs by carrier hopping between adjacent dopant molecules embedded in a polymer matrix serving as a spacer (see, e.g. Ref. [9] and references therein). For such a mechanism the frequency, v, of a carrier transition between dopant species separated by some distance, r, increases with the degree of overlapping following the familiar expression: u= 1:o(7) exp(-r/y)

(1)

Here y is the so-called overlapping parameter, while uo(T) is the frequency factor depending on temperature T. A similar consideration is appropriate for undoped polymers when hopping occurs due to a sequence of carrier transitions between adjacent pendants (see, e.g. Ref. [10]) or even for charge transfer in complex biomolecules with donor and acceptor sites incorporated in a long polymeric chain [11].

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Yu.A. Berlin et aL / Synthetic Metals 64 (1994) 171-175

In all these cases the application of Eq. (1) to the description of the elementary step of charge transport implies that the probability of the carrier transition between two adjacent traps exponentially decreases with time, t, regardless of the type of centers initially involved in the charge localization (dopant molecules, donor chemical groups, structural traps, etc.). Therefore, from a kinetic standpoint, the elementary step of charge transport is seemingly assumed to be the monomolecular (first-order) reaction with the rate constant given by Eq. (1). However, as has been shown in our recent publication [12], the latter assumption means that one ignores the existence of external stochastic impacts exerting influence on the rate of a monomolecular process. In the case of charge transport in polymeric systems such impacts may include different types of a molecular motion of polymer chains leading to fluctuations in the separation distance between trapping sites or conformational changes in dopant molecules inducing variations of the frequency factor. Nevertheless, current theory is not sufficiently advanced to analyze the role of these factors in the mechanism of charge transfer in polymers. In the present paper we propose the approach that may partially fill this gap. To take into account impacts mentioned above, any single step of charge carrier hopping is treated here as a randomly affected monomolecular reaction rather than as a conventional firstorder decay process. Our consideration is based on the combination of the irreversible stochastic transition (IST) theory [12] with the concept of diffusion perpendicular to the reaction coordinate (DPRC) [13]. Following this concept, we approximate the stochastic process responsible for the action of an external random impact on a trap release process by diffusion along the coordinate different from that associated with a charge transfer event. The mathematical formulation of our theoretical model is given in section 2. The subsequent section contains the derivation of key kinetic characteristics of a single hop. Results obtained are applied to the consideration of two physically important cases, when the stochastic impact can be approximated by the hopping-independent diffusion or, alternatively, by the diffusion coupled with hopping. In both cases transient kinetics is found to be essentially non-exponential. However, decay curves differ in their forms depending on the type of stochastic process along the conformational coordinate. Typical examples of the trap release influenced by the hopping-independent diffusion and by the diffusion coupled with hopping are given in section 4. The discussion of our theoretical findings performed in this section allows us to conclude that conformational changes in the surroundings of trapping sites may be considered as the possible origin of dispersive transport in polymeric systems.

2. Model for randomly affected single hop The description of the elementary step of charge transport as a randomly affected monomolecular reaction in the framework of the DPRC concept implies the diffusion-like 'motion' of carrier surroundings along some degree of freedomx, other than the charge transfer coordinate, r. Unlike the conventional approach [14--18] this motion proceeds in the x-direction 'perpendicular' to the r-axis and can be associated with changes in the conformational state of traps due to physical reasons discussed in section 1. In addition, the probability of the trap release at any time instant, t, is assumed to be dependent on x. We will seek the kinetic characteristics of this process. They involve the total probability, pxo(t), of a carrier hop as a function of t and the distribution of the probability, p~o(x, t), that a trap remains occupied by a charge carrier at time t (the survival probability distribution). Using notations of the general probability theory these quantities can be written as

pxo(t)-P{A

-, B, t~c(0)=Xo)

px,,(x, t) dx=P{x<~x(t)<<.x+dx, A / , B, t~c(0)=Xo} In principle, such a formulation of the problem takes into account various stochastic influences (effects of surroundings, external noise, thermal fluctuations, etc.) on the elementary step of charge transport through the dependence of the trap release probability on x. However, kinetically, it seems to be important whether or not a charge transfer event and diffusion are coupled due to the physical mechanism of a particular affection. The first possibility means that the distribution of traps over conformational states at time instant t > 0 (if at t = 0 they have been in the initial conformation Xo) is governed by both charge transfer and diffusion, while in the second case only the latter process determines the probability mentioned above. In the next section both situations are analyzed in terms of the approach based on the IST theory. The analysis is preceded by the derivation of general expressions for quantities describing transient reaction kinetics.

3. The IST approach

3.1. Evaluation of kinetic characteristics The starting point for the application of the IST theory to the problem under consideration is the expression for the conditional probability p(t, x, At) of a trap release as a function ofx. Following general theoretical results obtained in our previous publication [12,19] one can write:

Yu.A. Berlin et al. / Synthetic Metals 64 (1994) 171-175

p(t, x, A t ) - P { A

, B, At~c(t)=x; A

exp[H t

/ , B, t}

=k(x)At + o( At)

173

(2)

0

where k(x)>0 and depends only on x. The function k(x) accounts for random impacts on the charge transfer process and can be regarded as a 'fluctuating rate constant' [19]. The x-motion is described by the probability distribution P;od~f't'x t) defined by pj,,difZ~x,t)=P{x <~x(t) <~x+dxlA

/ , B, t; x(O)=xo}

On the other hand, the conditional probability that a trapping site in some conformational state between x and x + dx is still occupied by a charge carrier at time t may be expressed in terms of the evident relationship:

pxo(x, t) dr=P{x
/> B, t~c(0)=Xo}

3.2. Hopping-independent diffusion If the external stochastic impact can be approximated by this diffusion-like process, p~odif~(X,t) should satisfy the equation of the form:

P{A / , B, t~c(0)=Xo}= 1-Pxo(t)

dif 0p;,) (x, t)

from Eq. (3) we obtain

Ot

(4)

To find px.(t) it is useful to note that definitions of total probability and the distribution p;,,,dif/"~.~,t) immediately yield

p~o(t, A t ) - P { A , B, AtlA /, B, t; x(0)=Xo} dif(x, t) dr = f p (x, t, At)pjo

(5)

x

Substituting Eq. (2) into Eq. (5), one finds

pxo(t, At)=

AtJk(x)p','(x,t) dr + o(At)

(6)

x

Consider next the probability

px,,(t+At)-P{A

, B, t+Atk(O)=xo }

By virtue of Eq. (6) this quantity can be rewritten as

x

(7)

In the limit, when At ~ 0, the latter expression reduces to the differential equation: dt

= [1-p,o(t)]fk(x)p~,~,f(x, t) dr

with the solution

x

02

0

dif

dif

&2 [A(x)p;,, (x, t ) ] - ~ [B(x)p;,, (x, t)] (10)

with p;,,dif'(x,O) = 6(X-Xo) and realistic boundary conditions (or sometimes with conditions at infinity). Here A(x) and B(x) are some functions which may take into account some additional factors, such as the dependence of the diffusion constant on x, the influence of the potential on the motion along the x-axis, etc. Once the functions k(x) and PxodifZlx,t) are defined, the solution of the problem under consideration is straightforward, since Pxo(t) can be found from Eq. (9) while px,,(x, t) is given by Eq. (4). Not that although the characteristics of a single hop d i f z(x, t ) the distribution pxo(X, t) is have no effect on p~,, affected by k(x) through px,,(t). Moreover, as follows from Eqs. (4), (9) and (10) px,,(x, t) has to be a solution of the equation:

-p.,,(x. t)f k(s, t)p;~, (s, t) ds dif

(11)

x

p,,,(t + At) = At[1 -px,,(t)lJk(x)p~',f(x, t) dr +px,,(t)+o(At)

--

Opx.(x, t) 02 0 Ot - ~c2 [A(x)pxo(x, t ) ] - ~x [B(x)px,(x, t)]

=pxo(t, At)[ 1 -pxo(t)] +p,,,(t)

dp~,,

Eqs. (4) and (9) allow us to find the sought-fur kinetic characteristics of the charge transfer process if pdxof(X, t) and k(x) are prescribed by physical models of a 'perpendicular' random motion and by a mechanism of carrier hopping in each particular case. Their derivation requires only probabilistic arguments rather than any mathematical assumptions. Therefore, the application of the IST theory to the problem under discussion furnishes the exact and general analytical result. This makes it possible to consider two physically important cases of the stochastic influence on transient kinetics which have already been mentioned in section 2.

(3)

Minding that

pxo(X, t) = p~d,~f(x,t)[1 -pxo(t)]

x

(8)

Therefore, the form of this distribution depends only on the mean 'rate constant' obtained by averaging k(x) dill over p~,, (x, t). To interpret Eq. (11), it seems useful to introduce the so-called reaction variable a(t) which takes the value zero (if charge transfer occurs in time t) or unity (if there is no charge transfer in time t). This quantity has a physical meaning similar to that usually attributed to the spin variable in atomic physics. Then a reactant is described by the joint probability:

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Yu.A. Berlin et al. / Synthetic Metals 64 (1994) 171-175

P(x, a, t) -P{x <~x(t) ~
pxo(X, t)=P(x, 1, t) Due to the definition given above, we conclude that the charge transfer along with diffusion represents a stochastic unified non-Markovian process with respect to the pair of variables x and a(t). This explains the appearance of the sink term in Eq. (11) which relates to the 'total loss' of the probability density for unreleased traps due to hopping. Nevertheless, the motion in conformational space remains to be Markovian by itself in conformity with Eq. (10).

3.3. Diffusion coupled with hopping In this case the definition of the survival probability given above makes it possible to use the condition of flux conservation for deriving the equation for

px,,(x, t): Op.,,(x, t)

8t

O2

-

U2 IA(x)pxo(x, t)] Ox [B(x)p.~,(x, t)]-k(x)pxo(x, t)

(12)

The substitution of Eqs. (4) and (9) into Eq. (12) allows us to deduce the differential equation for

px,,(x, t): dif(x, t) Op;,,

at

~

O

Ox2 [A(x)p~i,f(x, t ) ] - ~x [B(x)~°I(x' t)] - p,~,, dif,tx,

t)[k(x)-fk(s)p~ff(s, t ) d s ]

(13)

Evidently.Eq. (13) differs from Eq. (10) by the form of the sink term due to the dissimilar nature of the random motion in the conformation space. Contrary to the situation considered in the preceding subsection, here the process along thex-axis ceases to be Markovian. However, the evolution of the survival probability distribution for occupied traps is described in terms of the Markov process with respect to the pair of variables, x and a(t), and is governed by Eq. (12) postulated in Refs. [13,20].

4. D i s c u s s i o n

and conclusions

In the present paper we have developed an approach to the description of the elementary step of charge

transport affected by external random impacts. The latter are associated with stochastic configurational changes in carrier surroundings (e.g. different types of polymer chain motion, relaxation, conformational changes in dopant molecules, etc.) and in many cases can be approximated by 'perpendicular' diffusion along the coordinate other than separation distance between adjacent localization centers. The main advantage of the approach proposed is the possibility to obtain the exact analytical result for the probability of trap release without invoking any mathematical or additional physical assumptions. As has been shown, the probability of charge transfer from traps initially occupied by carriers depends on the distribution of these localization centers over conformational states, p~of(X,t), with the explicit form being determined by the physical nature of the stochastic process along the 'perpendicular' coordinate. In some cases this random process is independent of hopping (e.g. electron tunneling transfer between donor and acceptor sites imbedded in a flexible polymer chain as side groups). By contrast, in many other instances it should be treated as a diffusion-like process coupled with hopping (e.g. transfer of a carrier localized at a dopant molecule with the rate dependent on molecular conformations). Therefore p~f(x, t) cannot be evaluated using only the IST approach. However, since this distribution, the survival probability distribution for occupied traps and the probability of a carrier hop are related by Eq. (4), such an approach makes it possible to propose the appropriate equations for p~i,f(x, t) in both cases (see Eqs. (10) and (13)) by invoking the condition of flux conservation in combination with Eqs. (4) and (9). Equations for pxdli,f(x, t) (as well as Eqs. (11) and (12) for the survival probability distribution) differ by their sink terms due to distinct physical roles of the 'perpendicular' diffusion in the mechanisms of charge transfer in the two cases mentioned above. If the stochastic motion proceeds independently of hopping, the part played by diffusion consists in the preparation of the trapping site distribution along the 'perpendicular' coordinate for their release. If, however, the stochastic motion is coupled with the elementary step of charge transport, diffusion in conformational space ceases to be the only factor capable of shaping such a distribution since the charge transfer appears to be also involved in the latter process. The distinctions between two types of stochastic influences result in different forms of corresponding decay curves. However, according to Eq. (9), they remain non-exponential for both kinds of such random

Yu~. Berlin et al. / Synthetic Metals 64 (1994) 171-175

impacts2). Such behavior suggests that in addition to site and bound disorders [9,14,15,18], conformational changes can be considered as another origin of the dispersive carrier transport in polymeric systems. While this manuscript was being prepared, some experimental evidence [21,22] supporting our expectations appeared in the literature. In particular, it has been demonstrated that conformational changes in polymer chains affect charge transport in neutral poly(3-hexylthiophene) [21]. The detailed interpretation of these findings from the standpoint of the approach developed in the present paper is now in progress.

Dedication The authors would like to dedicate this article to Professor Hiroo Inokuchi whose contribution to the study of electronic properties of organic materials is well recognized by the world scientific community. His hearty encouragement and support stimulated the present investigation.

Acknowledgements Yu.A.B. would like to thank Professor Kalinowski, Drs L.B. Schein and M. Tachiya for fruitful discussions of some aspects of charge carrier transport in polymeric systems. He is also indebted to the Institute for Molecular Science for inviting him as a professor and to 2Note that in the absence of any stochastic influences, when k(x) becomes a constant, Eq. (9) reduces to the familiar exponential kinetic law, while Eqs. (10) and (11) coincide with Eqs. (12) and (13). This means that we come to the classical kinetic description of the elementary step of charge carrier transport in terms of conventional rate constants given by Eq. (1).

175

the staff of the Solid State Chemistry Division for the assistance in preparing the manuscript.

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