Metal-insulator transition in three-dimensional Anderson superlattice with rough interfaces

We study the electronic properties of superlattice with rough interfaces in two and three dimensions using the transfer-matrix method and direct diagonalization of the Anderson Hamiltonian. The system consists of layers with an average constant width, but with stochastic roughness added to the interfaces between the layers. The numerical results indicate that, in the thermodynamic limit, the two-dimensional superlattice is an insulator in the presence of even small roughness. In three-dimensional systems, however, the superlattice exhibits a metal-insulator transition with a well-defined mobility edge located at an energy $E_c$ that we compute numerically. For three-dimensional superlattice, the localization length follows a power law near the mobility edge $\xi \left(E\right)\sim {(E_c-E)}^{-\nu }$, where the exponent is $\nu \simeq 1.6$. We also show that the existence of the extended states in three-dimensional superlattices gives rise to a finite conductivity in the limit $M/L\to \infty$, where $L$ is the length and $M$ the width of the bar.

Figure 1

The geometry of a superlattice in 2D. Different colors relate to a different medium in the superlattice. The average width of each layer in the superlattice is $W$. In each layer we add a randomly selected value $\delta (x,y)$ to the average position of the $n$th layer, i.e., $x(n,y)=nW+\delta (x,y)$, where $\delta (x,y)$ is a random and uncorrelated (white noise) stochastic variable with zero mean, distributed uniformly in the interval $[-\sigma ,\sigma ]$. The on-site energy in the $n$th layer is ${\epsilon }_A=1$, and in the $(n+1)$st layer it will be ${\epsilon }_B=-1$.

Figure 2

Rescaled localization length of a two-dimensional superlattice as a function of energy for two values of the strength of roughness.

Figure 3

Variation of the rescaled localization length with respect to the width of the system $M$ for two energies and for a superlattice with the roughness strength $\sigma =5$ in two dimensions.

Figure 4

Rescaled localization length as a function of the strength of roughness for two values of energies and for two widths $M$ in a two-dimensional superlattice.

Figure 5

Rescaled localization length of a three-dimensional superlattice as a function of energy for two widths $M$ and for the strength of roughness $\sigma =20$.

Figure 6

(a) Conductivity as a function of the aspect ratio of the system for a vanishing chemical potential $\mu$. The thermodynamic limit is defined as $M/L\to \infty$. We note that this limit is reached for moderate aspect ratios $M/L\ge 6$. The dc conductivity is given for the strength of disorder $\sigma =20$. The conductivity approaches $\simeq 0.13(e^2/h)$ in the thermodynamic limit. (b) The $\mu$ dependence of the conductivity for a fixed aspect ratio $M/L=6$.