Stochastic modeling of molecular charge transport networks

We develop a stochastic network model for charge transport simulations in amorphous organic semiconductors, which generalizes the correlated Gaussian disorder model to realistic morphologies, charge transfer rates, and site energies. The network model includes an iterative dominance-competition model for positioning vertices (hopping sites) in space, distance-dependent distributions for the vertex connectivity and electronic coupling elements, and a moving-average procedure for assigning spatially correlated site energies. The field dependence of the hole mobility of the amorphous organic semiconductor, tris-(8-hydroxyquinoline)aluminum, which was calculated using the stochastic network model, showed good quantitative agreement with the prediction based on a microscopic approach.

Figure 1

Schematic representation of the iterative dominance-competition model: (a) thinning of homogeneous Poisson process according to the dominance-competition model; and (b) iterative addition of points in the complementary phase to achieve desired point density.

Figure 2

Characteristic distribution functions of the microscopic and stochastic point-process model: (a) nearest-neighbor distance distribution function; (b) pair correlation function; and (c) spherical contact distribution function.

Figure 3

Correlation function for the site energies estimated from the microscopic model and the fitted stochastic model.

Figure 4

(a) Distance-dependent probability of two hopping sites being connected. Histograms of (b) coordination number and (c) edge lengths. The mean (variance) of the coordination number is 11.43 (2.03) in the microscopic model compared to 11.37 (3.18) in the stochastic model. For the edge lengths, we obtain 0.99 nm ($0.03{\mathrm{nm}}^2$) and 0.97 nm ($0.03{\mathrm{nm}}^2$), respectively.

Figure 5

Mean (a) and variance (b) of distributions of ${\log }_{10}(J_{ij}^2/{\mathrm{eV}}^2)$ as a function of distance between hopping sites. (c) Distributions of logarithmic transfer rates ${\log }_{10}\left({\omega }_{ij}\mathrm{s}\right)$.

Figure 6

Hole mobilities $\mu$ as a function of applied field $F$, based on the microscopic (black) and stochastic (red) graph. (a) Results obtained without site-energy disorder, i.e., $\Delta {\mathcal{E}}_{ij}^{\mathrm{el}}=0$. (b) Poole-Frenkel plot. Error bars for the stochastic model have been determined by averaging over three independent realizations. Since respective errors cannot be calculated for the microscopic model due to the computational cost of generating and evaluating independent snapshots, relative errors identical to the stochastic model have been assumed.