S. D. Baranovskii et al.: On the Einstein Relation for Hopping Electrons
87
phys. stat. sol. (b) 205, 87 (1998)
Subject classification: 72.20.Dp
On the Einstein Relation for Hopping Electrons
S. D. Baranovskii (a), T. Faber (a), F. Hensel (a), and P. Thomas (b)
(a) Institut f
ur Physikalische Chemie und Zentrum f
ur Materialwissenschaften
der Philipps-Universit
at Marburg, Hans-Meerwein-Straû e, D-35032 Marburg, Germany
(b) Fachbereich Physik und Zentrum f
ur Materialwissenschaften
der Philipps-Universit
at Marburg, Mainzer Gasse 33, D-35032 Marburg, Germany
(Received August 19, 1997)
Diffusion coefficient of carriers, D, and their mobility, m, in disordered semiconductors at very low
temperatures are temperature-independent, being determined by the energy-loss hopping of carriers
through localized band-tail states. In such a hopping relaxation in a system with exponential density of tail states, the relation between m and D has the form m eD=e0 , where e0 is the energy
scale of the exponential band tail. With rising temperature, thermally-activated hopping transitions
increase their contribution to transport processes and the model of the energy-loss hopping is not
applicable. We study by a Monte Carlo computer simulation how the relation between m and D
evolves with increasing temperature from its temperature-independent form at T ˆ 0 to the conventional Einstein relation m ˆ eD=kT:
Introduction. In order to analyze theoretically the hopping transport in disordered
semiconductors, one usually calculates first the diffusion coefficient D of carriers and
then expresses their mobility m via the Einstein relation
e
mˆ
D;
1†
kT
where e is the elementary charge, k is the Boltzmann constant and T is the temperature.
However, such approach cannot be correct in non-equilibrium conditions. For instance,
at extremely low temperatures, transport is provided by the energy-loss hopping [1], i.e.,
by the process in which neither carrier mobility, m, nor their diffusion coefficient, D,
depend on temperature and the relation in Eq. (1) cannot be valid. The relation between
m and D in the energy-loss hopping via randomly distributed sites has been recently
shown to have the form [2]
e
D;
2†
mˆC
e0
where C 2:3 is a numerical coefficient and e0 is the energy scale of the density of
localized states (DOS) in the band tail,
N0
e
:
3†
exp ÿ
g e† ˆ
e0
e0
Here energy e is measured positive from the mobility edge e ˆ 0† towards the gap center; N0 is the total concentration of localized tail states.
Recently Gu et al. [3] have tested the diffusivity-to-mobility ratio for a non-Gaussian
and hence non-equilibrium transport of carriers in amorphous silicon. They claim that
88
S. D. Baranovskii, T. Faber, F. Hensel, and P. Thomas
the hole diffusion coefficient in their experiments is not more than twice as large as
predicted by the Einstein relation and the mobility measurements. This was considered
as the upper bound for any true failure of the Einstein relation. It is worth noting, however, that the measurements of the drift mobility were carried out by Gu et al. for
temperatures above 200 K, while Eq. (2) has been derived by Baranovskii et al. [2] for
the limit of the infinitesimal temperature. To judge whether the results of Gu et al. are
in agreement with the theoretical concepts, it is necessary to find the diffusivity-to-mobility ratio in non-equilibrium conditions at finite temperatures.
To solve this problem, a computer simulation has been performed. The simulation
algorithm is described in the second section, while the results of the simulation are discussed in the third section.
Simulation Details. To simulate the behavior of electrons which hop via exponentially distributed localized states under the influence of a weak electric field, we choose
an algorithm similar to that suggested by Casado and Mejias [4], though with essential
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 F , which is supposed to be directed
along the x-axis. 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 sites of the lattice by means of the exponential distribution function described by Eq. (3).
Hopping of rather a large number of electrons n 103 to 104 ) is simulated simultaneously. Each of them 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 this sphere are calculated according to
the expression
2rij ei ÿ ej ‡ jej ÿ ei j
:
4†
Wij ˆ n0 exp ÿ
ÿ
a
2kT
Here Wij is the rate of the hop between the occupied site i and an empty site j separated
by distance rij ; a is the decay length of the wave function in the tail states; n0 is the
attempt-to-escape frequency. Then the hopping rate ni for each electron is calculated as
P
ni ˆ
Wij :
5†
j
Using the quantities ni 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 calculated anew for all its neighboring sites in the sphere of radius R.
At each step the mean value hxi of the coordinates in the field direction along with the
quantity hz2 i averaged over all electrons are calculated. The ratio eD=m is given as [4]
eD F Dhz2 i
ˆ
;
m
2 Dhxi
6†
On the Einstein Relation for Hopping Electrons
89
where Dhxi and Dhz2 i are the changes in the mean position of carriers in x direction and
the associated variance of their position in z direction. Typically the program made
more than 107 steps. After a very short transient regime (about 103 to 104 steps), the
quantity given by Eq. (6) did not depend on the number of steps and hence was accepted as a value of the diffusivity-to-mobility ratio for the chosen set of parameters.
Results and Discussion. In Fig. 1 the results of the simulation are presented for
e0 ˆ 0:025 eV and different values of F , a, R. The data for eD=m do not depend on the
strength of the electric field F for the chosen fields and also on the choice of a and R.
By the method developed in [2] it is easy to show that for a lattice model the coefficient C
in Eq. (2) is equal to 2, which predicts for T ! 0 the relation
eD e0
ˆ :
2
m
7†
For e0 ˆ 25 meV this value is 12.5 meV in good agreement with the simulation data in
Fig. 1. To check the agreement between the simulation results and Eq. (7), we also
performed the simulation at e0 ˆ 35 meV. The data in Fig. 2 show a very good agreement with Eq. (7).
At high temperatures, the results of the simulation agree well with the conventional
Einstein relation given by Eq. (1) and represented by straight lines in Figs. 1 and 2.
This very good agreement of the simulation results with the predictions of the analytical
theories at very low and very high temperatures allows one to hope that the results at
intermediate temperatures are also reliable.
The first question is whether these results are consistent with the experimental data.
The only available data for the non-equilibrium regime are those of Gu et al. [3]. These
data were obtained at T 200 K. It is well seen in Fig. 1 that at such high temperatures the relation between D and m differs from the Einstein's formula very little. Therefore it is not surprising that no essential deviations from Eq. (1) were manifested in the
experiments of Gu et al. [3]. In order to check any considerable deviation from the conventional Einstein relation, one should perform experiments in the non-equilibrium regime at much lower temperatures T < 50.
A challenging problem is to develop an analytical theory to describe the diffusivity-tomobility ratio for the non-equilibrium transport in the whole temperature range. The
Fig. 1. Temperature dependences of the
diffusivity-to-mobility ratio at e0 ˆ 0:025 eV
for different values of a, R (in units of the lattice constant) and F
90
S. D. Baranovskii et al.: On the Einstein Relation for Hopping Electrons
Fig. 2. Temperature dependence of the diffusivity-to-mobility ratio at e0 ˆ 0:035 eV
simulation data suggest that the relation between D and m can be generally described as
eD
ˆ f e0 ; kT † :
m
Trying to fit the data in Figs. 1 and 2 by the power function of the form
1=b
e0 b
f e0 ; kT † ˆ
‡ kT †b
;
2
8†
9†
we come to the value b ˆ 4:5 0:5. We are not aware of any arguments in support of
Eq. (9).
Acknowledgement. Financial support by the Deutsche Forschungsgemeinschaft
through the Sonderforschungsbereich 383 is gratefully acknowledged.
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