Beyond the effective temperature: The electron ensemble at high electric fields in disordered organics

Hopping between localized polaronic states, with a Gaussian distribution in energy, is regarded as the main mechanism of electric conduction in disordered organics. Several authors have recently suggested that the hopping electrons, subjected to an electric field, can be described as a homogeneous ‘overheated’ gas, with its “effective temperature” sufficient for a parametrization of the ensemble and the current. It is not clear how such a picture could be reconciled with the observed strongly oriented filamentization of the flow. We show, through extensive numeric simulations, that the picture is misleading as it can overestimate the electron mobility by orders of magnitude. The reason lies in deviations of the average site occupancies from the effective Boltzmann distribution. The ensemble can be described by a distribution function with two parameters—the effective temperature and the variance of the occupancy deviations. The two are connected by a simple universal relation. The spatial structure of the occupancy deviations is found to be connected to the current filaments and its neglect is recognized as the cause for the failure of mobility calculations based on the overheated gas concept. Thus we identify those aspects, lying beyond the overheated gas picture, that are of a fundamental importance to a proper understanding of the transport in disordered organics.

Figure 1

(Color online) Logarithm of the mobility as a function of the square root of the electric field $F$. The points are results of a Monte Carlo calculation (Fig. 3 of Ref. 29) and the lines are solutions to our master equation. Both calculations use the Miller-Abrahams form of energy dependence [Eq. (5)], in a noncorrelated energy disorder with the variance ${\sigma }_E$ expressed in terms of the environment temperature $T_{real}$ set at 295 K. The lattice constant $a_0$ is 6 $\AA$ and the wave-function attenuation rate $\alpha$ is $5/a_0$. Our procedure for calculating the mobility is detailed in Sec. 3.

Figure 2

(Color online) (a) The distribution of carriers over energies $O\left(E\right)$ for several field strengths and the density of states $g\left(E\right)$. The width of the DOS is ${\sigma }_E=0.06\text{eV}$ and the correlation length $L_c=2$. (b) The effective temperature dependence on the field strength for correlated and noncorrelated disorder, for ${\sigma }_E=0.06\text{eV}$. (c) Dependence of the effective temperature on the width of the DOS ${\sigma }_E$, for $F=0.8\text{MV}/\text{cm}$. In all three graphs the environment temperature is $T_{real}=300\text{K}$.

Figure 3

(Color online) The main graphs: tests of Eq. (12) for (a) the Miller-Abrahams and (b) the symmetric form of the hopping law, with the parameter $\beta$ set to 1.54, the value found by Jansson et al. (Ref. 17). Different lines have different, constant values of the environment temperature $T_{real}$, with the field strength $F$ varied. If the Eq. (12) was valid, all lines would merge into one. The insets: logarithm of the mobility (in arbitrary units) as a function of the effective temperature of the ensemble. The full red line shows the mobility calculated for field strengths up to over 2 MV/cm in the Miller-Abrahams case and over 3.5 MV/cm in the symmetric case, at a constant environment temperature $T_{real}=300\text{K}$. The dashed green line show the mobility calculated for a constant weak field $F_0=0.01\text{MV}/\text{cm}$ at higher environment temperatures (equal to the value of $T_{eff}$ on the abscissa). In all graphs, the energy disorder is noncorrelated with ${\sigma }_E=0.06\text{eV}$.

Figure 4

(Color online) Logarithm of the mobility as a function of the square root of the field strength. (a) presents the noncorrelated disorder case and (b) the correlated case with $L_c=2$. The full red line is the real mobility obtained by solving the master equation. The green dashed line is the mobility calculated from the ansatz in Eq. (14). The blue doted line is mobility calculated from the same ansatz but with $T_{eff}$ replaced by $T_{real}=300\text{K}$. The upper set of lines is for ${\sigma }_E=0.06\text{eV}$ and the lower set for ${\sigma }_E=0.12\text{eV}$. The lower set was additionally pushed downwards by 2, to make the sets clearly distinguishable.

Figure 5

(Color online) Histogram of the distribution function $G\left(E,\text{ln}n\right)$ (purple) for ${\sigma }_E=0.06\text{eV}$, $F=0.8\text{MV}/\text{cm}$, and $L_c=2$. The green line (marked $T_{eff}$) corresponds to an equilibrium distribution given by Eq. (16) with $T=T_{eff}$. The magenta line (marked $T_{real}$) corresponds to $T=T_{real}$. The red arrows show the variance of the deviations from the equilibrium distribution at $T_{eff}$. The scattering visible at the edges is due to a poor statistic for sites whose energy is more than $3{\sigma }_E$ distant from the mean.

Figure 6

(Color online) Agreement of ${\sigma }_{\delta }$ and $T_{eff}$ with the relation in Eq. (18) (the red line) for several choices of parameter values. The inset shows dependence of the variance ${\sigma }_{\delta }$ on the field strength $F$ for the three correlation lengths with ${\sigma }_E=0.06\text{eV}$ and $T_{real}=300\text{K}$.

Figure 7

(Color online) [(a) and (b)] The current-current autocorrelation $C_{jj}\left(r\right)$ and [(c) and (d)] the overoccupancy-overoccupancy autocorrelation $C_{\delta \delta }\left(r\right)$ for several field strengths. The figures on the left [(a) and (c)] are for a noncorrelated disorder and those on the right [(b) and (d)] are for a correlated disorder with $L_c=2$. In each graph, the correlation in a direction perpendicular to the field is pictured left of the ordinate axis, and the correlation in the direction of the field is pictured right of the axis. The red, dashed-dotted line is the autocorrelation profile of the site energies $C_{EE}\left(r\right)$.

Figure 8

(Color online) [(a)–(c)] The current-energy correlation $C_{jE}\left(r\right)$ and [(d)–(f)] the overoccupancy-energy correlation $C_{\delta E}\left(r\right)$ for several field strengths. (a) and (d) The correlations in the field direction for a noncorrelated disorder. (c) and (f) The correlations in the field direction for a correlated disorder with $L_c=2$. (b) and (e) The correlations in a direction perpendicular to the field for a noncorrelated disorder (left of the ordinate axis) and a correlated disorder (right of the ordinate axis).

Figure 9

(Color online) The on-site correlation between the overoccupancy and the current, $C_{\delta j}\left(r=0\right)$, as a function of the field strength $F$, for the three energy correlation lengths $L_c$.