System of correlation kinetic equations and the generalized equivalent circuit for hopping transport

We derive the system of equations that allows to include nonequilibrium correlations of filling numbers into the theory of the hopping transport. The system includes the correlations of arbitrary order in a universal way and can be cut at any place relevant to a specific problem to achieve the balance between rigor and computation possibilities. In the linear-response approximation, it can be represented as an equivalent electric circuit that generalizes the Miller-Abrahams resistor network. With our approach, we show that nonequilibrium correlations are essential to calculate conductivity and distribution of currents in certain disordered systems. Different types of disorder affect the correlations in different applied fields. The effect of energy disorder is most important at weak electric fields while the position disorder by itself leads to nonzero correlations only in strong fields.

Figure 1

Part of equivalent circuit representing five sites and pair correlations between them (a). The connection between pair correlation layer and charge layer is shown with blue triangles. The triangles are explained in (b).

Figure 2

Results of the calculation in $10\times 10$ lattice. (a) The structure of the system. Overlap integrals $t_{ij}$ are shown as the width of grey lines connecting the sites. Color gradient corresponds to site energy in units of temperature. (b) The dependence of total current on the correlations included into the computation. (c) The distribution of currents in the sample. (d) The difference in current distributions when correlations are neglected and are taken into account.

Figure 3

The dependence of correlations on distance and order. (a) Averaged squared potential ${\overline{u}}_I$ and (b) averaged squared covariations. The $x$ axis (range) on both plots corresponds to the maximum distance between sites in the correlation along the lattice bonds. Different curves (ord) correspond to different orders of correlations.

Figure 4

The system with two bridges where one bridge is sensitive to correlations and one does not. This system display a strong effect of correlations on currents. (a) the structure of the system. Overlap integrals $t_{ij}$ are shown as the width of grey lines connecting the sites. Color gradient corresponds to site energy in units of temperature. (b)–(d) The current distributions calculated in different approximations as described in the text.

Figure 5

(a) The sample composed by the polymer chains. (b) The results for this sample.

Figure 6

Current in the Mott law regime calculated in three approximations: the Miller-Abrahams approximation (“MA”), approximation where pair correlations up to the distance $2r_c$ are taken into account (“cor1”) and with correlations up to the third order and up to the distance $r_c$ (“cor2”). (a) the currents for two localization lengths compared to the Mott law, (b) the difference between “MA” approach and “cor1” approach, and (c) the difference between “cor1” and “cor2” approaches. In (b) and (c), points correspond to disorder-averaged values and bars to the standard deviation of this averaging.

Figure 7

Distribution of current in a system in Mott law regime calculated in different approximations: Miller-Abrahams approximation (a), “cor1” approximation (b), and “cor2” approximation (c). (d) and (e) show zoom of blue (bottom-left) rectangles in (b) and (c) correspondingly.

Figure 8

The correlations of filling numbers in high electric field. (a) The structure of the considered system. (b) The ratio of current calculated with correlation to the current calculated in the mean-field approximation as a function of electric field. The results are averaged over 100 disordered samples. (c) The dependence of a covariation of filling numbers in a close pair of sites on the applied field. Different curves in (b) and (c) correspond to different degrees of disorder. $\mathrm{\Delta }\varepsilon$ controls the disorder in site energies while $\alpha$ controls the disorder in overlap integrals $t_{ij}$ as described in the text.

Figure 9

The comparison between CKE calculations and Monte Carlo algorithm. (a) The dependence of conductivity on the applied field calculated in the Miller-Abrahams (MA) approximation, in “cor1” approximation and with Monte Carlo algorithm. (b) The current distribution calculated in “cor1” approximation. (c) The current distribution calculated with Monte Carlo algorithm. (d) the averaged current between sites and their standard deviations in Monte Carlo algorithm.

Figure 10

The chain with resistance dominated by the hop between sites with different signs of energy. Blue arrows show the hops that dominate the conductivity and generate the correlations. Green arrows correspond to the hops that allow effective transfer of correlations between different sites. Red arrows show the hops that block this transfer.