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Subsections
Recently it has been realized that the carrier concentration
plays an important role for the mobility. Experiments show that for a holeonly diode and
a FET fabricated from the same conjugated polymer, the mobility could differ up
to three orders of the magnitude [41]. This
difference can only be explained by taking into account the dependence of mobility on the
carrier concentration. Rubel [42] analyzed this problem with the
concept of a transport energy , but there is no direct proof for
the existence of such transport energy in organic systems. In this work we will focus
on extending the percolation model based on VRH theory by Vissenberg
[43] to explain the discrepancy of mobilities measured in OLEDs
and OFETs.
In this section, an analytical mobility model with a Gaussian DOS function has been obtained. It can explain the relation between the
mobility and carrier concentration. Results are in good agreement with
experimental data.
To calculate the mobility of an organic semiconductor, one can use
percolation theory, regarding such system as a random resistor network (network
of Miller and Abrahams) [7,44]. The current flows through the
bonds connecting the sites in the network. The conductance between the states
and can be described as
where is a prefactor,
is the Bohr radius of the
localized wave functions, is the temperature, and denote the position and energy
of site . In theory the value of
is determined by the threshold or critical conductance , at
which the first infinite cluster will form, given by the relation

(2.1) 
Here is a prefactor. To describe the fieldeffect mobility in organic
transistors, Vissenberg assumed an exponential density of localized states
[43].

(2.2) 
is the number of states per unit volume and specifies the width of the
exponential distribution. Connecting (2.1) and (2.2), the conductivity can be
described as [43]

(2.3) 
Here is the critical number
of bonds per site and is the fraction of occupied states, defined as
is the gamma function. Then an expression for the mobility as a function of
the carrier concentration can be obtained.

(2.4) 
However, this expression
can not account for the carrier concentration independent mobility when
the carrier concentration is very low (LED regime).
To overcome this problem, we derive another mobility model assuming a Gaussian DOS [9]
and VRH theory. In this model, the DOS function is given as

(2.5) 
Here is the energy
measured relative to the center of the DOS and
indicates the width of the DOS. The
value of the Fermi energy can be determined by the
equation for the carrier concentration .

(2.6) 
At low concentration, the exponential
function is large compared to one (the nondegenerate case) [45], and
we obtain the Fermi energy as

(2.7) 
According to percolation theory [17], at the onset of
percolation, the critical number can be written as

(2.8) 
for a threedimensional amorphous system, and are, respectively, the density of bonds and density of sites in a
percolation system, which can be calculated as [43,46]
and
Here
denotes the distance vector between sites and ,
is the exponent of the conductance given by the relation
[13] and is step function.
Substituting (2.5) and (2.7) into (2.8), we obtain a new
percolation criterion for an organic system as
This equation has to be solved for and an expression for mobility can be
obtained.

(2.9) 
where
is the Lambert function [47].
Equation (2.9) is obtained assuming
 that the site positions are random,
 the energy barrier for the critical hop is large,
 and the charge carrier concentration is very low.
Figure 2.1:
Comparison between the analytical model (2.9) and empirical model
for different temperature.

Figure 2.4:
The calculated mobility versus carrier occupation at different
temperature.

Figure 2.5:
Comparison between calculation and typical experimental results
[41].

So far, much attention has been devoted to explain the temperature
dependence of the mobility [48,49,50].
As shown in Fig 2.1, the model (2.9) gives a nonArrheniustype
temperature dependence of the form
, which has also been supported
by numerical simulations [51] and analytical calculations
[53]. The model (2.9) shows good agreement for a value
.
This value is close to
given in
[52] and in [53].
In Fig 2.2, the mobility is plotted as a function of
.
When plotted in this way, there exists the regime with a linear relation
between and . This indicates that the
variablerange hopping effect has to be taken into account
[54,55].
To obtain (2.7), a Boltzmann distribution function has been used. The degenerate limit of organic
semiconductors has been studied in [56,57]. In Fig 2.3 (a) we show the
Fermi energy for Boltzmann and FermiDirac distributions assuming some typical
values
of the parameter
as 1.5, 3.5 and 6.0
[48].
Fig 2.3 (b) is a comparison especially for the higher carrier occupation regime.
The analytical result
(2.7) agrees well with the numerically calculated result for decreasing
carrier occupation and increasing
. Therefore, for the LED regime with low charge
carrier concentration, (2.7) is a good approximation of the solution of
(2.6).
The mobility as a function of the carrier concentration is presented in Fig 2.4, where
is in the range , corresponding to some typical values for organic
semiconductors. The mobility stays constant
until a certain threshold value of the carrier occupation. Above this threshold, the mobility can increase
about four orders of magnitude at
9. These effects have also been
observed experimentally [41,58].
However, (2.9) is valid only in the LED regime with very low
carrier concentration. As it is difficult to get an analytical expression for the
mobility at higher carrier concentration, we use (2.4) as the
mobility model for the higher carrier concentration. The
combined model can explain the experimental data in [41,58],
as shown in Fig 2.5.
Next: 2.3 Temperature and Electric
Up: 2. Mobility Models for
Previous: 2.1 Introduction
Ling Li: Charge Transport in Organic Semiconductor Materials and Devices