- 1.3.1 Gaussian Disorder Model (GDM)
- 1.3.2 Percolation Theory
- 1.3.3 Transport Energy
- 1.3.4 Multiple Trapping Theory

(1.2) |

with denoting the occupation probability of state and the electron or hole transition rate of the hopping process between the occupied state to empty state . Defining as the chemical potential at the position of state and as the energy of state , the occupation probability is given by the Fermi-Dirac distribution function

(1.3) |

Assuming no correlations between the occupation probability of different localized states, the steady-state current between these two sites is given by

Substituting the Miller-Abrahams rate (1.1) in (1.4) the current becomes

(1.5) |

In the case of low electric field, resulting in a small voltage drop over a single hopping distance ( ), the following conductance is obtained

Here . This expression was introduced in 1960 by Miller and Abrahams [7] and is often referred as the Miller-Abrahams conductance. Equation (1.6) has an important implication. Even if the energies are moderately distributed, the exponential dependence of on these energies makes them enormously broadly distributed. This can be used to reduce the computations of the effective properties of the network, since the broadness of the distribution of implies that there are many small conductances that can be removed from the network. This resulting network is called the reduced network [13].

Miller and Abrahams [7] were the first to calculate the hopping conductivity of semiconductors using reduced networks. They assumed that the statistical distribution of the resistances depends only on and not on the site energies. This was justified because the experimental data for some semiconductors indicated that the impurity conduction exhibits a well-defined activation energy. But Mott [16] pointed out that the exponential dependence of the resistances on the site energies can not be ignored in most cases. When a carrier close to the Fermi energy hops away over a distance with an energy , it has sites to choose from, where is the site density function. In general, the carrier will jump to a site for which is as small as possible. The constraint to find a site within a range is given by . Substituting this relation into (1.6) yields

(1.7) |

The optimum conductance is obtained by maximizing with respect to the hopping distance as

(1.8) |

with .

where is the energy measured relative to the center of the density of states and is the standard deviation of the Gaussian distribution. Within this distribution, all the states are localized. The choice of such distribution was based on the Gaussian profile of the excitonic absorption band, as well as on the recognition that the polarization energy is determined by a large number of internal coordinates, which vary randomly by a small amount, so the central limit theorem of statistics holds.

The Gaussian disorder model has been treated by the Monte Carlo simulation technique based on the Miller Abraham equation [9]. In this simulation charge transport is described as an incoherent random walk. The carriers start their motion from randomly chosen sites at one of the boundaries of the system sample. Their trajectories are specified from the constraint that the probability for a carrier to jump between two transport sites is

With this technique, TOF measurements can be simulated [9], in which mobility is derived from the mean arrival time of the carrier at the end of the sample and from their mean displacement. The predictions made concern the temperature and electric field dependence of the mobility.

In the Gaussian disorder model, the strength of electronic coupling among sites is split into separate contributions from the relevant sites, each obtained from a Gaussian probability density. However, the choice for the off-diagonal disorder of a Gaussian distribution is not theoretically sustained unlike in the case of energy disorder, and a more realistic way of representing geometric disorder has been pursued. One of such attempt is described in [14], in which an alternative approach comprising positional and orientation disorder is introduced via fluctuations in the bonds adjoining the various transport sites rather than site fluctuations. This model gets rid of the unnecessary corrections between hops and results in overestimating the contribution of the log hops.

In particular, Gartstein and Conwell's [14] Monte Carlo simulations of hopping with the elementary jump rate described by 1.1, but in which

where is a uniformly distributed random variable. In this way, the different bonds of a given site with its neighbors are uncorrelated.

Another approach for the description of positional disorder was presented by Hartenstein [15], and was also based on Monte Carlo simulations of transport in a dilute lattice. This treatment employs the GDM, but without the need for defining a distribution function for the electronic coupling between different sites. In this case the hopping sites having the nearest neighbors were grouped in clusters whose size depends on the random intercluster distances, but ignores any contribution from the random orientation of the transporting elements. Nevertheless, the model is adequate for low dopant concentrations for which there are large fluctuations in the intersite distances.

The percolation subnetwork consists of conductances with . Using (1.6), this condition can be written as

with

is the maximum distance between any two sites between which a hop can occur, and is the maximum energy that any initial or final state can have. Thus the density of states that can be part of the percolating subnetwork is given by

(1.12) |

Since the sites in the subnetwork are linked only to sites within a range , this criterion has the form

(1.13) |

with being a dimensionless constant. A combination of (1.9) to (1.11) yields Mott's law (1.8), with .

(1.14) |

Here is the DOS function. The hopping distance can be calculated as

(1.15) |

So the corresponding hopping rate is

Maximization of (1.16) over energy gives the equation

(1.17) |

The finial energy that maximizes the hopping probability does not depend on the initial energy . This particular energy is called transport energy [10].

Arkhipov extended this theory to the effective transport energy [18]. In this theory, the Miller Abrahams equation is rewritten as

(1.18) |

with the hopping parameter and the unity step function . The average number of target sites for a starting site with energy , whose hopping parameters are not larger than can be calculated as

Neglecting the downward jumps and defining

(1.20) |

transform (1.19) into

(1.21) |

According to variable range hopping theory [19], a hop is possible if there is at least one such hopping neighbor, i.e. . This leads to the following equation

If the DOS distribution decreases with energy faster than then the integral on the left-hand side of (1.22) depends weakly upon the lower bound of integration for sufficiently deep starting sites, and (1.22) is reduced to

(1.23) |

where is the effective transport energy.

The multiple-trapping model for unipolar conduction is defined by the following equations [12]

(1.24) |

where

(1.25) |

and

(1.26) |

Here is the local photogeneration rate, is the flux of mobile charge carriers, the total carrier concentration is , is concentration of mobile carriers, is the carrier concentration temporarily immobilized in the th trap, is the capture rate by the th trap and is the release rate.

Later multiple trapping theory was extended for disordered organic semiconductors as:

(1.27) |

Here is the total hole concentration, is the hole concentration in extended states and is the energy distribution of localized (immobile) holes. Since carrier trapping does not change the total carrier concentration , the continuity equation can be written as

(1.28) |

with the mobility and the diffusion coefficient . This equation assumes two simplifications: no carrier recombination and constant electric field (no space charge). Substituting the trapping rate

(1.29) |

and release rate

(1.30) |

gives the following equation

(1.31) |

In equilibrium the energy distribution of localized carriers is established, and the function does not depend upon time

(1.32) |

Ling Li: Charge Transport in Organic Semiconductor Materials and Devices