Enhancement of hopping conductivity by spontaneous fractal ordering of low-energy sites

Variable-range hopping conductivity has long been understood in terms of a canonical prescription for relating the single-particle density of states to the temperature-dependent conductivity. Here we demonstrate that this prescription breaks down in situations where a large and long-ranged random potential develops. In particular, we examine a canonical model of a completely compensated semiconductor, and we show that at low temperatures hopping proceeds along self-organized, low-dimensional subspaces having fractal dimension $d=2$. We derive and study numerically the spatial structure of these subspaces, as well as the conductivity and density of states that result from them. One of our prominent findings is that fractal ordering of low energy sites greatly enhances the hopping conductivity and allows Efros-Shklovskii type conductivity to persist up to unexpectedly high temperatures.

Figure 1

Single-particle DOS as a function of energy. (a) and (b) show the typical situation, where electron-hole correlations arising from the ES stability criterion dig a narrow Coulomb gap out of a roughly constant DOS. If the total width of the DOS is $\sim 2W$, then the Coulomb gap width is $\sim 1/\sqrt{W}$. (c) and (d) show the results from our numeric simulation for $\mathrm{\Delta }=10$. The red dashed line shows the 3D Coulomb gap, Eq. (4). Shaded regions in these plots show the band of energies, as measured by our numeric simulations, that are used at the highest temperature at which we observe ES VRH [Eq. (5)]. Units of energy in this plot are $e^2/a_0$ and $g\left(\varepsilon \right)$ is plotted in units of ${\left(e^2{a}_0^2\right)}^{-1}$.

Figure 2

Schematic illustration of the random potential. The upper (lower) meandering thin lines show the energy of donor (acceptor) sites as a function of spatial coordinate. Screening occurs when these levels cross the Fermi level $\left(\varepsilon =0\right)$, at which point donor or acceptor sites become neutralized (indicated by red and blue shaded areas, respectively) and further growth of the potential is truncated. $R_g$ denotes the correlation length of the potential, and horizontal thin lines show $\varepsilon =\pm \mathrm{\Delta }/2$.

Figure 3

Fractal ordering of sites with $\left|\varepsilon \right|<1$ for a system with size $L=80$ and gap $\mathrm{\Delta }=20$. (a) The function $M\left(s\right)$ is plotted, as measured by our numeric simulation (thick gray line) and as derived theoretically in Eq. (13) (red dashed lines). The dashed lines use the derived values $d=2$ and $A\sim 1$ and have no free parameters. The vertical dotted lines indicate the length scales $R_g$ and $L$. (b) An example of the spatial arrangement of sites with $\left|\varepsilon \right|<1$. Points show the locations of such sites in a typical pseudoground state. (Only a subvolume of the total simulated system is shown.) The points are colored according to their distance from the front left face of the system.

Figure 4

(a) Density of states as a function of energy for different values of $\mathrm{\Delta }$, as measured by our numeric simulations. From highest to lowest, the curves correspond to $\mathrm{\Delta }=10,12,15$, and 20. The dashed line shows the 3D Coulomb gap equation. (b) When the DOS is plotted as ${\mathrm{\Delta }}^{2}g\left(\varepsilon \right)$ versus energy, all the data collapses onto a single curve $g\left(\varepsilon \right)\sim \left|\varepsilon \right|/{\mathrm{\Delta }}^2$ within the energy range $1/R_g\ll \varepsilon \ll 1$.

Figure 5

Temperature-dependent conductivity for $\mathrm{\Delta }=10$, as measured by our numeric simulation. (a) The dimensionless logarithm of the conductivity, ${(ln\sigma )}^{*}$, as a function of ${\left(T^{*}\right)}^{-1/2}$. The thick gray curve shows our numerical result, and the dashed red line is a linear fit. The linear dependence at ${\left(T^{*}\right)}^{-1/2}\gtrsim 1$ implies that the conductivity follows Eq. (5) at all $T^{*}\lesssim 1$. (b) Zabrodskii-Zinov'eva analysis of conductivity at different temperature regimes [34, 35]. The vertical axis shows the reduced activation energy $w=d{(ln\sigma )}^{*}/d(ln{T}^{*})$, which has a power-law dependence on $T^{*}$ with an exponent that reflects the VRH mechanism. The thick gray curve is our numerical result, and dashed lines indicate the results of Eq. (16).

Figure 6

Activation energy for nearest-neighbor hopping ${\varepsilon }_a$ as a function of the band gap $\mathrm{\Delta }$. Black points are our numerical results, and the solid line is a linear fit. The dash-dotted line is the result reported in Ref. [25].