Percolating states in the topological Anderson insulator

We investigate the presence of percolating states in disordered two-dimensional topological insulators. In particular, we uncover a close connection between these states and the so-called topological Anderson insulator, which is a topologically nontrivial phase induced by the presence of disorder. The decay of this phase could previously be connected to a delocalization of bulk states with increasing disorder strength. We identify this delocalization to be the result of a percolation transition of states that circumnavigate the hills of the bulk disorder potential.

Figure 1

Sketch of an uncorrelated disorder potential (left panel) in a system of width and length $W=L=1000\mathrm{nm}$ and a correlated potential (right panel) with correlation length $\xi =35\mathrm{nm}$. The color code ranges from blue (strong negative potential) to red (strong positive potential).

Figure 2

Average conductance $\langle G\rangle$ as a function of disorder strength $U$ through different HgTe/CdTe quantum wells. All systems are of width and length $W=L=1000\mathrm{nm}$ and the Fermi energy is set to $E_F=16\mathrm{meV}$. The ribbon geometries are shown by red solid curves while the cylinder geometries are shown by the blue dashed lines. (a) Average conductance through a system of topological mass $m=-10\mathrm{meV}$ (topological insulator) in an uncorrelated potential. The disorder average is taken over 1000 configurations. The ribbon geometry features the TAI conductance plateau of $\langle G\rangle =1e^2/h$ for $80\mathrm{meV}\lesssim U\lesssim 280\mathrm{meV}$. In this region the conductance through the cylinder is almost entirely suppressed, followed by a delocalization transition of the bulk states starting at $U\approx 280\mathrm{meV}$. (b) Average conductance through systems with the same parameters as in (a) but in a correlated potential with correlation length $\xi =35\mathrm{nm}$. The cylinder geometry (blue dashed curve) clearly shows the bulk delocalization transition, which occurs here for weaker disorder strengths $U$ than in the uncorrelated case. (c) Average conductance through systems with positive topological mass $m=10\mathrm{meV}$ (ordinary insulator) in a correlated potential of correlation length $\xi =35\mathrm{nm}$. The disorder average is taken over 200 configurations. The cylinder geometry shows a less pronounced bulk delocalization transition than in (b), followed here also by the results for the ribbon due to the absence of edge states.

Figure 3

Comparison of the scattering wave function $|\psi (x_i,y_i){|}^2$ (shown on a logarithmic scale in the left column) and the potential itself (right column) at the delocalization transition ($U=220\mathrm{meV}$) for systems with $W=L=1000\mathrm{nm}, E_F=16\mathrm{meV}$, and $m=-10\mathrm{meV}$. We only colored the potential values between $26\mathrm{meV}\le V(x_i,y_i)\le 78\mathrm{meV}$ for which the Fermi energy is effectively shifted into the valence band of the clean band structure (the remaining potential values are left in white). Two different correlated disorder configurations for cylinder and ribbon geometry are considered in the top and bottom row, respectively (the correlation length $\xi =35$ nm is the same for both). The similarity of the wave functions and these truncated potentials illustrates that the delocalizing bulk states are percolating around the hills of the potential landscape and that these percolating states have their origin in the bulk band structure of the clean sample.

Figure 4

Probability density distribution $P\left(V\right)$ for disorder values $V(x_i,y_i)$ at grid points $(x_i,y_i)$ weighted by the absolute value $|\psi (x_i,y_i){|}^2$ of the scattering wave function at $E_F=16\mathrm{meV}$ with injection in the first lead mode in a system of width and length $W=L=1000\mathrm{nm}$ and topological mass $m=-10\mathrm{meV}$. A disorder average is taken over 1000 configurations of random potentials with disorder strength $U=220\mathrm{meV}$ and correlation length $\xi =35\mathrm{nm}$. The distribution shows an enhancement of the wave functions at disorder values $V$ situated between $V_{\min }\approx 25\mathrm{meV}$ and $V_{\max }\approx 75\mathrm{meV}$. These values correspond nicely to the distance of Fermi energy $E_F$ to the flat valence bulk band states at the BZ edges.

Figure 5

(a) Band structure $E\left(k_x\right)$ of a system of width $W=1000\mathrm{nm}$ and topological mass $m=-10\mathrm{meV}$ obtained by directly solving the Bloch-eigenvalue problem numerically for the Hamiltonian in Eq. (1). (b) Density of states $\rho \left(E\right)$ in a closed and clean system of width and length $W=L=1000\mathrm{nm}$, topological mass $m=-10\mathrm{meV}$, and cylindrical geometry. (c) Weighted probability density distribution $P(E_F-V)$ for systems with $m=-10\mathrm{meV}, W=L=1000\mathrm{nm}$ in a correlated potential of correlation length $\xi =35\mathrm{nm}$. The wave functions are calculated in the region of the bulk delocalization at disorder strength $U=220\mathrm{meV}$ and for 1000 random disorder configurations. The red solid and blue dashed lines show $P(E_F-V)$ at $E_F=16\mathrm{meV}$ in the ribbon and the cylinder geometry. The green dashed-dotted and black dotted lines represent $P(E_F-V)$ for ribbon and cylinder at $E_F=30\mathrm{meV}$. The similarity of all four curves shows that the distribution is a very fundamental system property. The shape of the distributions closely resembles the density of states at the BZ boundary shown in (b). This indicates the bulklike nature of the states responsible for the delocalization transition. (d) Weighted probability density distribution $P(E_F-V)$ with the same system parameters as in (c) but in an uncorrelated disorder potential. The calculations were performed in the region of the bulk delocalization at disorder strength $U=370\mathrm{meV}$ and for 1000 disorder configurations.