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.

(2.20) |

where is the Miller Abrahams rate. In the presence of an electric field , the actual energy differences will be modified from to , where is the angle enclosed by the jump and the field direction. Using the reduced coordinates and , the hopping range may be re-written to

where . Since the hopping probability depends on both the spacial and the energetic difference between the hopping sites, it is natural to describe the hopping processes in a four-dimensional hopping space, which is spanned by three spacial and one energy coordinate. The hopping range , as given by (2.21), defines a metric on this space.

In various disordered systems, a Gaussian density of states has been used to describe the hopping transport in band tails.

(2.22) |

where is the center of Gaussian function and .

Let be the normalized Fermi-Dirac distribution function. Then the carrier concentration can be written as

(2.23) |

with denoting the normalized chemical potential. The conductivity can be written as

Here is the forward hopping distance in the direction of the electric field and, stands for the nearest neighbor hopping range in the hopping space. To calculate the conductivity, we need to calculate . First, the number of unoccupied states within a radius in the hopping space is calculated [19].

Here . The factor arises from the reduced coordinate system.

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

Similarly, the expression for can be calculated as [19]

(2.26) |

where

depends only weakly on [19]. Therefore, for , we can obtain the value for by solving (2.25) numerically. Then the mobility for electrons at energy amounts to

(2.27) |

Finally, the total conductivity for the organic semiconductor can be calculated numerically according to (2.24). The mobility can be determined from

We investigated the mobility's temperature dependence in a three-dimensional hopping lattice. The crucial system parameters were set to the following values: Å, , cm, V/cm and . Fig 2.14 depicts the mobility as a function of the lattice temperature . A linear dependence is observed between 3 and 8. Fig 2.15 displays the mobility as a function of . The range with linear dependence of mobility on is not as broad as the one for the dependence of mobility on . This can be used to test the validity of Arrhenius law and the empirical model , where is the activation energy.

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

(2.29) |

with constant and [59,69,74,75].

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.

Ling Li: Charge Transport in Organic Semiconductor Materials and Devices