Topological Anderson Insulator

Disorder plays an important role in two dimensions, and is responsible for striking phenomena such as metal-insulator transition and the integral and fractional quantum Hall effects. In this Letter, we investigate the role of disorder in the context of the recently discovered topological insulator, which possesses a pair of helical edge states with opposing spins moving in opposite directions and exhibits the phenomenon of quantum spin Hall effect. We predict an unexpected and nontrivial quantum phase termed “topological Anderson insulator,” which is obtained by introducing impurities in a two-dimensional metal; here disorder not only causes metal-insulator transition, as anticipated, but is fundamentally responsible for creating extended edge states. We determine the phase diagram of the topological Anderson insulator and outline its experimental consequences.

Figure 1

Band structure of $\mathrm{HgTe}/\mathrm{CdTe}$ quantum wells with finite width. (a) The “inverted” band structure case with $M=-10\mathrm{meV}$. Edge states (red lines) cross the bulk band gap and merge into bulk states (gray area) at a maximum energy in the upper band. The green dashed lines mark the boundary of bulk states. (b) The transition point between an inverted band structure and a “normal” band structure with $M=0\mathrm{meV}$. (c) The normal band structure with $M=2\mathrm{meV}$. In all figures, the strip width $L_y$ is set to $100\mu \mathrm{m}$. The sample-specific parameters are fixed for all calculations in this Letter to be $A=364.5\mathrm{meV}\mathrm{nm}$, $B=-686\mathrm{meV}{\mathrm{nm}}^2$, $C=0$, $D=-512\mathrm{meV}{\mathrm{nm}}^2$.

Figure 2

Conductance of disordered strips of $\mathrm{HgTe}/\mathrm{CdTe}$ quantum wells. The upper panels (a) to (c) show results for an quantum well inverted with $M=-10\mathrm{meV}$, and the lower panels (d) to (f) for a normal quantum well with $M=1\mathrm{meV}$. (a) The conductance $G$ as a function of disorder strength $W$ at three values of Fermi energy. The error bars show standard deviation of the conductance for 1000 samples. (b) Band structure calculated with the tight-binding model. Its vertical scale (energy) is same as in (c) and the horizontal lines correspond to the values of Fermi energy considered in (a). (c) Phase diagram showing the conductance $G$ as a function of both disorder strength $W$ and Fermi energy $E_f$. The panels (d),(e), and (f) are same as (a),(b), and (c), but for $M>0$. The TAI phase regime is labeled. In all figures, the strip width $L_y$ is set to 500 nm; the length $L_x$ is 5000 nm in (a) and (d), and 2000 nm in (c) and (f).

Figure 3

Width-dependence of the conductance in disordered strips of $\mathrm{HgTe}/\mathrm{CdTe}$ quantum wells with several values of strip width $L_y$ and a length $L_x=2000\mathrm{nm}$. In the inset, the conductance traces prior to the TAI phase (left-hand-side of the dashed line) are scaled with the width of the strips as $\sigma =G{L}_x/L_y$. The formation of the edge states is indicated by the presence of conductance quantization $2e^2/h$. In this figure, $M=2\mathrm{meV}$, and the Fermi energy $E_f=20\mathrm{meV}$.

Figure 4

Three independent spin-resolved transmission coefficients, $T_{21}^{\uparrow }$, $T_{31}^{\uparrow }$, and $T_{41}^{\uparrow }$, are plotted as functions of disorder strength $W$. Standard deviations of the transmission coefficients for 1000 samples are shown as the error bars. In the shadowed range of disorder strength, all bulk states are localized and only chiral edge states exist, which is schematically shown in the inset (for spin-up component only). The width of leads is 500 nm and $M=1\mathrm{meV}$ and $E_f=20\mathrm{meV}$.