Theory of the Topological Anderson Insulator

We present an effective medium theory that explains the disorder-induced transition into a phase of quantized conductance, discovered in computer simulations of HgTe quantum wells. It is the combination of a random potential and quadratic corrections $\propto p^2{\sigma }_z$ to the Dirac Hamiltonian that can drive an ordinary band insulator into a topological insulator (having an inverted band gap). We calculate the location of the phase boundary at weak disorder and show that it corresponds to the crossing of a band edge rather than a mobility edge. Our mechanism for the formation of a topological Anderson insulator is generic, and would apply as well to three-dimensional semiconductors with strong spin-orbit coupling.

Figure 1

Computer simulation of a HgTe quantum well [10], showing the average conductance $⟨G⟩$ as a function of disorder strength $U_0$ (logarithmic scale) and Fermi energy $E_F$, in a disordered strip of width $W=100a$ and length $L=400a$. The TAI phase is indicated. Curves $A$ and $B$ are the phase boundaries resulting from the effective medium theory. Curve $A$ separates regions with positive and negative renormalized topological mass $\overline{m}$, while curve $B$ marks the crossing of the renormalized chemical potential $\overline{\mu }$ with the band edge ($|\overline{\mu }|=-\overline{m}$). Both curves have been calculated without any adjustable parameter. The phase boundary of the TAI at strong disorder is outside of the regime of validity of the effective medium theory.

Figure 2

Black curves, left axis: Average density of states $\rho$ as a function of $E_F$ for $U_0=100\mathrm{meV}$, calculated by computer simulation (solid curve, for a disordered $100a\times 100a$ square with periodic boundary conditions) or by effective medium theory (dashed curve). Red curve, right axis: Average conductance $⟨G⟩$, calculated by computer simulation in a disordered strip ($U_0=100\mathrm{meV}$, $W=100a$, $L=400a$). The TAI phase of quantized conductance lines up with the band gap.

Figure 3

Red dashed curve: Average conductance $⟨G⟩$ as a function of disorder strength ($E_F=25\mathrm{meV}$, $W=L=100a$). Black solid curve: Localization length $\xi$, showing the peak at the strong-disorder edge of the conductance plateau—characteristic of a localization transition. The scaling with system size $W$ of the width $\delta U_0$ of the peak is shown in the inset (double-logarithmic plot).