The following is based on Apsley's work [19]. We will derive a formula for both the conductivity's temperature as well as its field dependence in the case of a Gaussian density of states mirroring the molecular disorder. We assume that (i) the localized states are distributed randomly in both space and energy, (ii) the states are occupied according to the Fermi-Dirac statistics, (iii) both hops upwards and hops downwards are regarded, (iv) the state energies are uncorrelated, and (v) the electric field may assume any value. Finally, the mobility's concentration dependence is discussed.
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(2.20) |
In various disordered systems, a Gaussian density of states has been used to describe the hopping transport in band tails.
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(2.22) |
Let
be the normalized Fermi-Dirac distribution
function. Then the carrier concentration can be written as
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(2.23) |
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According to Mott, will be the value of the radius in the hopping
space for which only one available vacant site is enclosed [72]. In
other words,
can be obtained by solving the equation
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(2.26) |
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(2.27) |
We investigated the mobility's temperature dependence in a three-dimensional hopping lattice. The crucial system parameters were set to the following values:
These results are in accordance with the measurements reorted in [79].
As the presented model expresses, only in the regime
an
approximately linear relation can be observed.
The mobility versus electric field characteristics predicted by the presented model is shown in
Fig 2.16. The parameters are
Å,
,
cm
and
.
, where
V/m, the mobility remains constant. At higher fields,
it increases with the field. Therefore,
the simple empirical relation between mobility and electric field of the
form
is not valid for all electric fields.
In addition to the temperature and
electric field dependence, mobility also depends on the carrier
concentration. Experiments show that
for a hole-only diode and a field effect transistor fabricated from the same
-conjugated polymer, the mobility can differ up to three orders of
magnitude [79]. Empirically, the mobility's dependence on the concentration
of localized states is written in the form
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(2.29) |
With the parameter
Å, we compare the presented mobility
model and this empirical
formula, as shown in Fig 2.17. The agreement is quite good when we use parameters
and
. We notice that the value of
is different from
given in [69] and
given in
[76]. Baranovskii [69] has stated that the parameters
and
are temperature dependent.
In Fig 2.18, we also show the values of parameters
and
that provide the
best fit for the solution of our model with the empirical expression (2.28). The input
parameters are
,
cm
,
and
Å. As illustrated in Fig 2.18, the parameter value of
is
less than
for temperatures low enough. The value of
is decreasing with increasing temperature, a result which coincides with
[10]. Here
is not constant, since the variable
range hopping (VRH)
transport mechanism is based on the interplay between the spacial and energy
factors in the exponent of transition probability, as given by (2.21). However,
assuming nearest neighbor-hopping (NNH) regime, which does not consider the effect of
energy dependent terms in (2.21) [69], leads to the values
.
Next, we discuss the effect of the
electric field on the parameters values and
. The results are shown
in Fig 2.19 and Fig 2.20. Input parameters are
Å,
kT,
cm
and
. From these figures we can see
that the values
and
are nearly constant in the low electric
field regime (
).
We have shown that, as expected in the variable range hopping picture, (2.24) with
is only approximately valid for restricted ranges of temperature
and electric field strength. So we consider the effect of the material parameter
on the values of
and
in Fig 2.21. The input parameters are
,
and
cm
.
Remarkably, both parameter values
and
are not constant in the
given range of
. With increasing
, the values of
will
decrease and the ones of
will increase.