Universal Size-Dependent Conductance Fluctuations in Disordered Organic Semiconductors

Numerically exact results of hopping charge transport in disordered organic semiconductors show for uncorrelated and dipole-correlated Gaussian energy disorder a universal, power-law, and non-power-law dependence, respectively, of the relative conductance fluctuations on the size of the considered region. Data collapse occurs upon scaling with a characteristic length having a power-law temperature dependence. Below this length, which can be as high as 100 nm for correlated disorder in a realistic case, fluctuations dominate and a continuum description of charge transport breaks down.

Figure 1

Bond current distribution (arrows, directions; line thicknesses, sizes) for a representative 2D case with uncorrelated disorder, for two disorder strengths. At the size $L_0$ of the red boxes, conductance fluctuations are comparable to the conductance itself.

Figure 2

(a) Normalized occurrence frequency of $\ln (G/G_0)$ of conductances of boxes with different size $L$ (offsets applied), for a representative 3D case with uncorrelated disorder. The mean-square fluctuation $\delta (L)$ is indicated for one case. The bin size was 0.04, and the numbers of disorder configurations considered are given. (b) Conductivities vs $a/L$ normalized to the macroscopic conductivity (dashed horizontal line), calculated from the data in (a): disorder averages of the conductance ($G_0^C$), resistance ($G_0^R$), and logarithm of the conductance ($G_0$). See the main text for the explanation of the functions $\exp [\delta (L{)}^2/2]$ and $\exp [-\delta (L{)}^2/2]$. At the vertical dashed line, $\delta (L)=\delta (L_0)=1$. Errors in the data are smaller than the symbols.

Figure 3

Mean-square fluctuation $\delta$ vs box size $L$ (left) and scaled size $L/L_0$ (right) for different disorder strengths and three different cases, corresponding to uncorrelated disorder in 3D [(a),(b)] and 2D [(c),(d)] and dipole-correlated disorder in 3D [(e),(f)]. Values of $L_0$ used in the size scaling are indicated and a parametrization of the curves according to Eqs. (1) and (2). Numbers of considered disorder configurations varied between 1000 (uncorrelated disorder) and 150 (correlated disorder) for the largest box to 50 000 (both cases) for the smallest box. Errors in the data are smaller than the displayed symbols.