Assuming no correlations between the occupation probabilities of different localized sates, the net electron flow between two states is given as

with denoting the occupation probability of site and the electron transition rate of the hopping process between the occupied state to the empty state . The probabilities (5.11) are then employed in a master equation for describing charge transport. With the electrochemical potential at the position of state the occupation probability is described by a Fermi-Dirac distribution as

(5.12) |

For the metal electrode we assume a fixed electron concentration and a Fermi-level of zero. All injected carriers are assumed to hop from the metal Fermi-level. Under the effect of a constant electric field and the Coulomb field binding the carrier with its image charge on the electrode the energy and the electrochemical potential of a localized state are given by

where denotes the distance of state from the interface, the angle between and , the barrier height, and the energy at state without electric field. According to Mott's formalism [44], the transition rate from the metal Fermi-level to state reads as

(5.13) |

Connecting with a Gaussian DOS, the net current across the metal-organic contact can be written as

(5.14) |

where is the attempt-to-jump frequency and

where , is the distance from the electrode to the first hopping site in the bulk and . and describe the charge injection from the electrode downwards and upwards, respectively. and describe the backflow of charge to the electrode. The net current can be calculated by evaluating , , and numerically.

Fig 5.6 shows a semilogarithmic plot of the current versus with the same parameters as used in Fig 5.5. This presentation is appropriate for testing RS behavior as . Since the dependence of versus is not linear, a deviation from the RS characteristics is observed.

Fig 5.7 shows the current-field characteristics for different and s, the other parameters are the same as in Fig 5.5. The injection current increases with decreasing barrier height and with electric field. The comparison between calculation and experimental data of DASMB sandwiched between ITO and Al electrodes [112] is given in Fig 5.8. The parameters are eV and K, the other parameters are the same as in Fig 5.5. The agreements is quite good at low electric fields. The discrepancy between calculation and experimental data comes from the resistance of the ITO contact at high electric field [112].

Ling Li: Charge Transport in Organic Semiconductor Materials and Devices