Finite-size scaling of charge carrier mobility in disordered organic semiconductors

Simulations of charge transport in amorphous semiconductors are often performed in microscopically sized systems. As a result, charge carrier mobilities become system-size dependent. We propose a simple method for extrapolating a macroscopic, nondispersive mobility from the system-size dependence of a microscopic one. The method is validated against a temperature-based extrapolation [A. Lukyanov and D. Andrienko, Phys. Rev. B 82, 193202 (2010)]. In addition, we provide an analytic estimate of system sizes required to perform nondispersive charge transport simulations in systems with finite charge carrier density, derived from a truncated Gaussian distribution. This estimate is not limited to lattice models or specific rate expressions.

Figure 1

(a) System-size extrapolation for energetic disorder of $\sigma =0.1eV$, external field of $F={10}^{6}V/m$, lattice spacing of 1 nm, transfer integral of $J={10}^{-3}eV$, and reorganization energy of $\lambda =0.3eV$. (b) Validation using the temperature extrapolation. Large symbols denote the extrapolated mobilities ${\mu }_{\infty }$, summarized in Table 1.

Figure 2

Convergence of the carrier energy with system size $N$ for different carrier densities $n$. Symbols are the results of random-number experiments, Eq. (28), while solid lines are the predictions of our analytic model, Eq. (35). The dashed lines are the exact values for ${\epsilon }_{\mathrm{c}}$ as $N\to \infty$ obtained from Eq. (5).

Figure 3

The occupational density of states (ODOS) is the product of the Gaussian density of states (DOS) and the Fermi-Dirac distribution (red). For finite systems the DOS is approximated by a truncated Gaussian distribution. The light blue part is the missing contribution to the ODOS in a finite system, leading to increased energy values. An increase of the Fermi energy (larger density) or a decrease of ${\epsilon }_{\min }\left(N\right)$ (larger box size) reduces this part and thus the finite-size error.