Field dependence of hopping mobility: Lattice models against spatial disorder

The theoretical description of the effect of the electric field $F$ on the hopping mobility $\mu$ belongs to the not-yet-resolved problems related to charge transport in disordered materials. An often proposed solution is to simulate hopping transport via sites placed on regular grids and to fit the results by phenomenological equations. This approach currently dominates the theoretical research of hopping transport in organic disordered semiconductors. We show that the dependence $\mu \left(F\right)$ in the case of regular grids can drastically differ from that in systems with spatial disorder. While $\mu$ increases with $F$ on lattices, it can decrease in random systems with the same material parameters. Moreover, the material parameters responsible for the dependence $\mu \left(F\right)$ on lattices differ from those responsible for $\mu \left(F\right)$ in spatially disordered systems, which makes lattice models inappropriate for studying the field dependence of the hopping mobility.

Figure 1

Comparison of $\mu \left(F\right)/\mu \left(0\right)$ at $\sigma /kT=3$ and $\sigma /kT=4$ for hopping on a cubic lattice (squares, triangles) and on random sites (circles) at $N^{-1/3}/a=10$. The solid lines are drawn in accord with Eq. (8) for the corresponding values of the ratio $\sigma /kT$.

Figure 2

Comparison of $\mu \left(F\right)$ at $\sigma /kT=3$ between cubic lattice and random sites at different values $a/N^{-1/3}$ as a function of $F$ in units of $\sigma /e{N}^{-1/3}$.

Figure 3

Comparison of $\mu \left(F\right)$ at $\sigma /kT=4$ between cubic lattice and random sites at different values $a/N^{-1/3}$ as a function of $F$ in units of $\sigma /e{N}^{-1/3}$.

Figure 4

A simplified model with only two energy levels: transport states and traps.

Figure 5

Normalized mobility $\mu \left(F\right)/\mu \left(0\right)$ on the cubic lattice at $b/a=10$ for the nearest-neighbor hopping (NNH) (squares) and for the variable-range hopping (VRH) (circles) at $\sigma /kT=3$ and $\sigma /kT=4$.

Figure 6

Normalized $\mu \left(F\right)/\mu \left(0\right)$ on the cubic lattice at $\sigma /kT=4$ for different values $a/N^{-1/3}\equiv a/b$ as functions of $F$ in units of $\sigma /e{N}^{-1/3}\equiv \sigma /eb$.

Figure 7

Normalized $\mu \left(F\right)/\mu \left(0\right)$ for the lattice at $\sigma /kT=4$ for different values $a/N^{-1/3}\equiv a/b$ as functions of $F$ in units of $\sigma /ea$.

Figure 8

Normalized mobility $\mu \left(F\right)/\mu \left(0\right)$ for the system of random sites at $\sigma /kT=4$ and different values $a/N^{-1/3}$ as functions of $F$ in units $\sigma /e{N}^{-1/3}$.

Figure 9

Normalized mobility $\mu \left(F\right)/\mu \left(0\right)$ for the system of random sites at $\sigma /kT=4$ and different values $a/N^{-1/3}$ plotted vs $Fa/(\sigma /e)$. The solid line is plotted in accord with Eqs. (5) and (4) with $\gamma =0.6$.