On the Einstein relation for hopping electrons

Journal of Non-Crystalline Solids 227–230 Ž1998. 158–161

On the Einstein relation for hopping electrons S.D. Baranovskii a

a,)

, T. Faber a , F. Hensel a , P. Thomas

b

FB Physikalische Chemie und Zentrum fur ¨ Materialwissenschaften der Philipps-UniÕersitat ¨ Marburg, D-35032 Marburg, Germany b FB Physik und Zentrum fur ¨ Materialwissenschaften der Philipps-UniÕersitat ¨ Marburg, D-35032 Marburg, Germany

Abstract Transport properties of disordered semiconductors at low temperatures are determined by hopping of electrons via localized band tail states. Much attention has been paid recently in both experimental and theoretical studies to the relation between the diffusion coefficient of carriers, D, and their mobility, m , in the hopping regime. Rather controversial results have been reported. In particular, some computer simulations have been claimed to show that the conventional Einstein relation m s eDrkT is violated in the hopping regime even in the case of thermal equilibrium. We study the relation between m and D by a computer simulation and show that such statements were based on a wrong interpretation of the simulation results. In thermal equilibrium, the Einstein relation between m and D must hold, although in a nonequilibrium system this relation can be violated. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Hopping transport; Einstein relation

1. Introduction Transport properties of disordered semiconductors such as amorphous and polycrystalline materials, polymers, and mixed crystals have been attracting much attention in recent years due to various applications of these materials in electronic devices. At low temperatures Ž- 300 K. transport in disordered semiconductors is dominated by hopping of electrons through localized band tail states. To analyze the hopping transport theoretically, one usually calculates first a diffusion coefficient, D, of the hopping

)

Corresponding author. Tel.: q49-6421-285582; fax: q496421-288916; e-mail: [email protected].

electrons and then expresses the carrier mobility, m , via the Einstein relation

ms

e kT

D,

Ž 1.

where e is the elementary charge, k is the Boltzmann constant and T is the temperature. This calculation has already become a routine procedure. However, some recent reports claim that the Einstein relation does not hold for hopping electrons in various cases. When discussing the validity of the Einstein relation, it is important to distinguish between equilibrium and non-equilibrium conditions. In this report we discuss the case of thermal equilibrium. Richert et al. w1x have performed a Monte Carlo computer simulation of hopping drift and diffusion of charge carriers within an array of hopping states subject to a Gaussian distribution of site energies.

0022-3093r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 Ž 9 8 . 0 0 0 3 1 - 3

S.D. BaranoÕskii et al.r Journal of Non-Crystalline Solids 227–230 (1998) 158–161

They have found deviations from Einstein’s law with increasing disorder and field. We discuss below the low field Ž- 100 V cmy1 . regime only. Borsenberger et al. w2x have measured the ratio eDrm in films of 1,1-bisŽdi-4-tolylamionophenyl.cyclohexane ŽTAPC.. They have found that in the low field regime, this ratio, being field independent, is larger than kT expected from Eq. Ž1.. The authors have also carried out a computer simulation of the transport properties measured in their experiments and have found no deviation from Eq. Ž1.. In the simulation, a system with a Gaussian density of localized states ŽDOS. has been studied. As the most likely explanation of the discrepancy between the experimental data and the simulation results, Borsenberger et al. w2x suggested that probably in the chosen experimental set up there could be some additional reasons for the spatial spreading of carriers with respect to the spreading expected from just a thermal contribution. For example, some spatial distribution can be generated in the course of carrier injection into the sample. This distribution can lead to the overestimation of D, and hence, to the overestimation of eDrm. However, a more recent computer simulation of the hopping transport, carried out by Casado and Mejias w3x, has shown deviations from Eq. Ž1., which the authors consider as a support to the experimental results of Borsenberger et al. w2x. Moreover, Casado and Mejias w3x have found that the diffusivity to mobility ratio increases with the increase of the amplitude of the energy disorder. The algorithm used by Casado and Mejias seems correct. The results however, seem to be in contradiction to basic physical concepts, namely, to that of the validity of the Einstein relation described by Eq. Ž1. for a system in thermal equilibrium Žsee, e.g., Ref. w4x.. The aim of the present report is to resolve this contradiction. To this purpose, a Monte Carlo computer simulation has been performed. The simulation algorithm and results are given in Section 2 and discussed in Section 3.

2. Simulation details and results Let us assume, following Borsenberger et al. w2x and Casado and Mejias w3x, that the density of local-

159

ized states, g Ž e ., which the carriers use for hopping is given by a Gaussian function of the form gŽe. s

No

Ž 2p .

1r2

s

½

exp y

e2 2s 2

5

Ž 2.

where the energy parameter, s , describes the width of the distribution and No is the total concentration of localized states. To simulate the behavior of electrons which hop via such distributed localized states under the influence of an electric field, we choose an algorithm similar to that suggested by Casado and Mejias w3x, though with modifications. In this algorithm, electrons can hop between the sites of a cubic lattice. When a carrier is situated on a particular lattice site, its energy is a sum of two different contributions. The first is provided by the electric field, E, which is supposed to be directed along the x-axes. This contribution varies linearly with the x coordinate as the electric field is taken to be uniform. The second contribution to the electron energy comes from the disorder. It is assigned to the site by means of the Gaussian distribution function described by Eq. Ž2.. Hopping of a rather large number of electrons Ž n f 10 4 . is simulated simultaneously. Each is moved as an independent entity. When an electron is located at some site, i, the energies of all its neighboring sites in the sphere of radius R are calculated taking into account both contributions described above. Then the transition rates of the electron to each of the neighbors, j, in the sphere are calculated according to the Miller–Abrahams expression:

½

Wi j s no exp y

2 ri j

a

y

ei y e j q < e j y ei < 2 kT

5

.

Ž 3.

Here, Wi j is the rate of the hop between the occupied site i and an empty site j separated by distance ri j ; a is the decay length of the wave function in the tail states; yo is the attempt-to-escape frequency. All distances are measured in the units of the lattice constant chosen as 0.6 nm. The hopping rate, y i , for each electron is calculated as

n i s Ý Wi j . j

Ž 4.

S.D. BaranoÕskii et al.r Journal of Non-Crystalline Solids 227–230 (1998) 158–161

160

Using the quantities y i for all electrons and a random number generator, the program finds which electron hops. Then, using the hopping rates of this electron to all its neighbors and a random number generator, the program finds to which neighbor the hopping occurs. Then the electron is transferred and energies are recalculated for all its neighboring sites in the sphere of radius R. At each step, the mean value, ² x :, of the coordinates in the field direction along with the quantity ² z 2 : averaged over all electrons are calculated. The ratio, eDrm, is given as w3x ED² z 2 :

eD s

m

2D² x :

,

They did not depend on the choice of a for a G 1. To make a comparison with the previous simulations more transparent, we measure below the amount of the energy disorder by the quantity, d ' srkT. It is well seen in Fig. 1 that ratio eDrm for d F 1.5 agrees well with the Einstein relation described by Eq. Ž1. and represented in Fig. 1 by a straight line; the data at d s 3.5 demonstrate deviations from Eq. Ž1.. These are in agreement with the results of Ref. w3x. On the basis of these data, the authors of Ref. w3x came to the conclusion that the Einstein relation between D and m described by Eq. Ž1. is violated at larger values of the energy disorder d.

Ž 5.

where D² x : and D² z 2 : are the changes in the mean position of carriers in x direction and the associated variance of their position in z direction, respectively. Typically, the program made more than 10 7 steps. After a very short transient regime Žabout 10 4 steps., the quantity given by Eq. Ž5. did not depend on the number of steps, and hence, was accepted as the value of the diffusivity to mobility ratio for the chosen set of parameters. In the simulation of Casado and Mejias w3x, electrons were allowed to hop to the nearest neighbors on the lattice only. In our notations this corresponds to R s 1. First, we made a run of the program with different parameters, though with fixed R s 1. In Fig. 1, the results are shown for such a simulation.

Fig. 1. Temperature dependences of the diffusivity to mobility ratio at Rs1 for different amplitudes of the energy disorder d ˚ .. The straight and different field strengths E Žin the units VrA line corresponds to Eq. Ž1..

3. Discussion It is worth noting that the conventional derivation of Eq. Ž1. is carried out only under the assumption of thermal equilibrium. In a Gaussian DOS, a system of electrons is supposed to be in thermal equilibrium. It seems therefore puzzling that the deviations from Eq. Ž1. appear in the simulation. To solve this puzzle, let us consider at which energy, e m , the distribution function of electrons has its maximum in thermal equilibrium. To find this energy, one just has to multiply the Fermi equilibrium distribution function with the DOS function and to find the maximum position of the product. It can be easily shown w5x that this maximum corresponds to the energy, e m s ys 2rkT ' yd 2 kT. In thermal equilibrium, the electrons must be found most frequently in the vicinity of e m . Let us now estimate whether the energies in the vicinity of e m can be found among those six neighbors, which are available for electron hopping in each event. It is clear that sites with energies close to the maximum of the DOS function described by Eq. Ž2., i.e., those with smaller absolute values < e <, have a higher probability to be found than those with largest absolute values < e <, which belong to the wings of the DOS distribution. Typical sites among the six neighbors, available for electron hopping in each event, have energies e ) e lim where e lim is determined by the equation e lim

Hy` g Ž e . d e s

No 6

.

Ž 6.

S.D. BaranoÕskii et al.r Journal of Non-Crystalline Solids 227–230 (1998) 158–161

161

and d s 3.5 are in excellent agreement with Einstein’s formula.

4. Conclusions

Fig. 2. Temperature dependences of the diffusivity to mobility ˚ . and ratio at ds 3.5 for different field strengths Žin the units VrA different R. The straight line corresponds to Eq. Ž1..

Using Eq. Ž2., one obtains e lim f y0.97 s s y0.97 d kT. We assume the system to be in thermal equilibrium if e lim - e m only. This inequality cannot hold for the larger d, because e lim is proportional to yd, while e m is proportional to yd 2 . The inequality breaks at d f 1. It is clear now that for d F 1, the simulated system is in thermal equilibrium, whereas at d G 1, equilibrium conditions cannot be established in the simulation due to the artificial restriction on the number of sites, which an electron can use for hopping. To determine if this idea is correct, we increased the number of neighbors to which electrons can hop by increasing R. In Fig. 2 the results of the simulation are presented for d s 3.5 and R s 5 along with those for R s 1. The value R s 5 corresponds to 515 sites available. This number shifts the limiting energy, e lim , well below the energy, e m . The a in this run of the program was chosen as 10 just to provide easy hopping to more distant sites in the sphere of radius R s 5. Under such conditions, the thermal equilibrium is well reproduced by the simulation. It is seen in Fig. 2 that the simulation results at R s 5

It is shown by a Monte Carlo computer simulation that in the hopping regime in a system with energy disorder, the conventional Einstein relation described by Eq. Ž1. holds in thermal equilibrium. This result contradicts the conclusion of the previous simulations w3x, that the energy disorder leads to the violation of the Einstein relation. We show that the parameters chosen for the simulation in Ref. w3x bring the system artificially out of thermal equilibrium. For non-equilibrium conditions, it is well known w6x that the relation between D and m can differ from Einstein’s formula, and the result of Ref. w3x is not therefore surprising, although it should not be used for an equilibrium case.

Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 383 is gratefully acknowledged.

References w1x R. Richert, L. Pautmeier, H. Bassler, Phys. Rev. Lett. 63 ¨ Ž1989. 547. w2x P.M. Borsenberger, L. Pautmeier, R. Richert, H. Bassler, J. ¨ Chem. Phys. 94 Ž1991. 8276. w3x J.M. Casado, J.J. Mejias, Phil. Mag. B 70 Ž1994. 1111. w4x R.P. Feynman, R.B. Leighton, M. Sands, 1964. The Feynman Lectures on Physics. Addison-Wesley Publishing, Palo Alto. w5x B. Movaghar, M. Grunewald, B. Ries, H. Bassler, D. Wurtz, ¨ ¨ ¨ Phys. Rev. B 33 Ž1986. 5545. w6x S.D. Baranovskii, T. Faber, F. Hensel, P. Thomas, G.J. Adriaenssens, J. Non-Cryst. Solids 198–200 Ž1996. 214.