Delocalization by Disorder in Layered Systems

Motivated by anomalously large conductivity anisotropy in layered materials, we propose a simple model of randomly spaced potential barriers (mimicking stacking faults) with isotropic impurities in between the barriers. We solve this model both numerically and analytically by utilizing an exact solution for the conductivity of a one-dimensional disordered system. In the absence of bulk disorder, electron motion in the out-of-plane direction is localized. Bulk disorder destroys one-dimensional localization. As a result, the out-of-plane conductivity is finite and scales linearly with the scattering rate by bulk impurities until planar and bulk disorder become comparable. The ac out-of-plane conductivity is of a manifestly non-Drude form: the real part starts from finite value at zero frequency and has a maximum at the frequency corresponding to the scattering rate by potential barriers.

Figure 1

Left: a system of randomly spaced parallel potential barriers and randomly distributed isotropic impurities. Right: expected dependences of the in-plane and out-of-plane conductivities on bulk disorder.

Figure 2

Out-of-plane conductance versus the bandwidth of bulk disorder $W_B$ for a range of values of planar disorder $W$, as shown in the figure, and $L=30$.

Figure 3

Out-of-plane conductivity versus the bandwidth of bulk disorder $W_B$ on a double-logarithmic scale for a range of system sizes, as shown in the figure, and three values of planar disorder: $W=1.5$ (upper set), $W=2$ (middle set), and $W=2.5$ (lower set).

Figure 4

Diagrams for the out-of-plane conductivity to leading order in bulk disorder. Thin lines represent exact Green’s functions in the presence of both types of disorder; thick solid lines, Green’s functions in the presence of planar disorder only; dashed lines, Green’s functions averaged over planar disorder; zigzag, correlator of bulk disorder; solid and dashed brackets, averaging over bulk and planar disorder, respectively.