Scaling Theory for Percolative Charge Transport in Disordered Molecular Semiconductors

We present a scaling theory for charge transport in disordered molecular semiconductors that extends percolation theory by including bonds with conductances close to the percolating one in the random-resistor network representing charge hopping. A general and compact expression is given for the charge mobility for Miller-Abrahams and Marcus hopping on different lattices with Gaussian energy disorder, with parameters determined from numerically exact results. The charge-concentration dependence is universal. The model-specific temperature dependence can be used to distinguish between the hopping models.

Figure 1

Normalized current (line thickness) and dissipated power (line opacity; see the legend) in bonds of a $15\times 15$ square lattice of sites with Gaussian energy disorder of width $\sigma$. The results shown are for Marcus hopping with reorganization energy $E_r\to \infty$, carrier concentration $c={10}^{-5}$, and three different temperatures $T$. A small electric field is applied from left to right. The arrow indicates the critical bond.

Figure 2

Dependence of mobility $\mu$ on $T$ for Marcus hopping with $E_r\to \infty$ on a simple cubic (SC) lattice, for $c={10}^{-2}$. Green triangles: ME. Green solid curve: Scaling ansatz, Eq. (2), with $A=1.8$ and $\lambda =0.85$. Blue dotted curve: Standard percolation theory, scaled to match the ME mobility at $\sigma /k_{\mathrm{B}}T=3$. Red dashed curve: Effective-medium theory [Eq. (5.4) in Ref. 12].

Figure 3

(a) Dependence of $\mu$ on $T$ for different hopping rates and lattices, for a typical $c$. (b) Dependence on $c$, for a typical $\sigma /k_{\mathrm{B}}T$. Symbols: ME. Curves: Scaling theory, Eq. (5), with values of $B$, $\lambda$, and $E_{\mathrm{crit}}$ as given in Table . For clarity, all mobilities for Marcus hopping are multiplied by 100.

Figure 4

(a) Dependence of $\mu$ on $E_r$ for Marcus hopping and different lattices. (b) Dependence of $\mu$ on transfer-integral disorder strength $\Sigma$ for Miller-Abrahams and Marcus hopping on a SC lattice. Symbols: ME. Curves: Scaling ansatz, Eq. (2), with values of $A$ and $\lambda$ as given in Table . Typical values for $\sigma /k_{\mathrm{B}}T$ and $c$ are chosen.