Three-dimensional phase diagram of disordered HgTe/CdTe quantum spin-Hall wells

We compute the phase diagram of HgTe/CdTe quantum wells in the three-dimensional parameter space of Dirac mass, Fermi level, and disorder strength. The phase diagram reveals the quantum spin-Hall, the metallic, and the normal insulating phases. The phase boundary of the quantum spin-Hall state is shown to be strongly deformed by the disorder. Taking specific cuts into this three-dimensional phase diagram, we recover the so-called topological Anderson insulator (TAI) phase, but now we can demonstrate explicitly that the TAI is not a distinct phase; instead, it is part of the quantum spin-Hall phase. Calculations are performed with $S_z$-conserving and $S_z$-nonconserving Hamiltonians.

Figure 1

Numerical values of the spin-Chern number $C_s$ as a function of $\Delta$, when the other parameters are fixed at $A=B=a=1$, $\Lambda =0$, $W=5.71$, and $E_F=0$. $C_s$ was computed for three lattices of increasing size: $30\times 30$ (triangles), $40\times 40$ (squares), and $50\times 50$ (circles). The shaded (red) region indicates the region where $\left|C_s\right|⩾0.5$, which is being identified with the QSH phase.

Figure 2

Global view of all our numerical data for $\Lambda =0$ (see Sec. 4a), obtained on a $50\times 50$ lattice. $A$, $B$, and $a$ take the same values as in Fig. 1. $C_s$ is represented by a surface plot, with $E_F$ and $\Delta$ on the horizontal axes and $C_s$ on the vertical axis. There is one surface plot for each $W$ value (shown at the right) considered in our study. Surface plots are shown from three angles: sideways from the $E_F$ direction (first column), sideways from the $\Delta$ direction (second column), and from above (third column).

Figure 3

Contour levels $C_s=0.5$ obtained from the surface plots in Fig. 2 and from similar data obtained on $30\times 30$ and $40\times 40$ lattices. When the calculations are converged with the size of the lattice, the region inside these contours contains the QSH phase.

Figure 4

The 3D phase diagram of the model with $\Lambda =0$, as derived from the contours in Fig. 3. The 3D region delimited by the contours and the dotted line contains the QSH phase. The shaded region indicates the region that contains the so-called topological Anderson insulating phase. Dotted lines were obtain by extrapolation.

Figure 5

Numerical values of the spin-Chern number $C_s$ as a function of $\Delta$, when the other parameters are fixed at $A=B=a=1$, $\Lambda =0.5$, $W=5.71$, and $E_F=0$. $C_s$ was computed for three lattices of increasing size: $30\times 30$ (triangles), $40\times 40$ (squares), and $50\times 50$ (circles). The various shaded regions indicate the regions where (a) $1.5⩽\left|C_s\right|⩽0.9$, which is being identified with the metallic phase, and (b) $\left|C_s\right|⩾0.9$, which is being identified with the QSH phase.

Figure 6

Global view of all our numerical data for $\Lambda =0.5$ (see Sec. 4b), obtained on a $50\times 50$ lattice. $A$, $B$, and $a$ take the same values as in Fig. 5. $C_s$ is represented by a surface plot, with $E_F$ and $\Delta$ on the horizontal axes. There is one surface plot for each $W$ value (shown at the right) considered in our study. Surface plots are shown from three angles: sideways from the $E_F$ direction (first column), sideways from the $\Delta$ direction (second column), and from above (third column).

Figure 7

Contour levels $\left|C_s\right|=0.15$ and $\left|C_s\right|=0.9$ obtained from the surface plots in Fig. 6 and from similar data obtained on $30\times 30$ and $40\times 40$ lattices. The contours are easy to distinguish because the $0.9$-level contours are always inside the $0.15$-level contours. The regions inside the $0.9$-level contours give an approximate representation of the QSH phase, and the regions between the $0.15$- and the $0.9$-level contours give an approximate representation of the metallic phase.

Figure 8

The 3D phase diagram of the model with $\Lambda =0.5$, as derived from the contours in Fig. 7. The 3D region delimited by the inside contours and the dotted line contains the QSH phase. The 3D region between the contours and the dotted lines contains the metallic phase. The shaded region indicates the region that contains the so-called topological Anderson insulating phase. Dotted lines were obtained by extrapolation.

Figure 9

Contour levels $\left|C_s\right|=0.9$ for the HgTe/CdTe QSH wells computed on $30\times 30$, $40\times 40$, and $50\times 50$ lattices and for various disorder strengths. When the calculations are converged with the size of the lattice, the region inside these contours contains the QSH phase.

Figure 10

Filled circles and solid lines represent 2D slices of the 3D phase diagram shown in Fig. 5 (corresponding to the $50\times 50$ lattice), taken at (a) $\Delta =-13.5$ meV and (b) $\Delta =5.75$ meV. (a) The dashed contour represents the boundary of the QSH phase, which we traced from Fig. 2c in Ref. 5. (b) The dotted contour represents the boundary of the TAI phase, which we traced from Fig. 2f in Ref. 5.