Magnetoresistance in organic semiconductors: Including pair correlations in the kinetic equations for hopping transport

We derive kinetic equations for polaron hopping in organic materials that explicitly take into account the double occupation possibility and pair intersite correlations. The equations include simplified phenomenological spin dynamics and provide a self-consistent framework for the description of the bipolaron mechanism of the organic magnetoresistance. At low applied voltages, the equations can be reduced to those for an effective resistor network that generalizes the Miller-Abrahams network and includes the effect of spin relaxation on the system resistivity. Our theory discloses the close relationship between the organic magnetoresistance and the intersite correlations. Moreover, in the absence of correlations, as in an ordered system with zero Hubbard energy, the magnetoresistance vanishes.

Figure 1

The effect of the applied magnetic field on the hyperfine mechanism of spin relaxation.

Figure 2

The cartoon for A-type and B-type sites. A-type sites have the single occupation states in the energy band of width $k_{B}T$ around the Fermi level (shown with light grey color). B-type sites have the double-occupied states in this band. To provide both types of sites, the density of states $g\left(E\right)$ should be wider than $E_{\mathrm{hub}}$.

Figure 3

Hopping between different kinds of sites. The AB hop (b) is allowed only for opposite spin directions of the hopping electron and of the electron on “free” $B$-type site. Other hops are always possible. The $BA$ hop (c) always results in an antiparallel spin configuration on the sites.

Figure 4

The dependence of the magnetoresistance on magnetic field. Here we assume that the spin relaxation rate is determined by Eq. (1) with ${\omega }_s^2/{\omega }_0^2=1/15$.

Figure 5

The dependence of the magnetoresistance on disorder for different electric fields.

Figure 6

The dependence of the magnetoresistance on the Hubbard energy. All the sites are considered to have the same single-occupation energy, $E_0=-E_{\mathrm{hub}}/2$.

Figure 7

The dependence of the magnetoresistance on the external electric field.