Unprotected edge modes in quantum spin Hall insulator candidate materials

The experiments in quantum spin Hall insulator candidate materials, such as HgTe/CdTe and InAs/GaSb heterostructures, indicate that in addition to the topologically protected helical edge modes, these multilayer heterostructures may also support additional edge states, which can contribute to scattering and transport. We use first-principles calculations to derive an effective tight-binding model for HgTe/CdTe, HgS/CdTe, and InAs/GaSb heterostructures, and we show that all these materials support additional edge states which are sensitive to edge termination. We trace the microscopic origin of these states back to a minimal model supporting flat bands with a nontrivial quantum geometry that gives rise to polarization charges at the edges. We show that the polarization charges transform into additional edge states when the flat bands are coupled to each other and to the other states to form the Hamiltonian describing the full heterostructure. Interestingly, in HgTe/CdTe quantum wells the additional edge states are far away from the Fermi level so that they do not contribute to the transport, but in the HgS/CdTe and InAs/GaSb heterostructures they appear within the bulk energy gap, giving rise to the possibility of multimode edge transport. Finally, we demonstrate that because these additional edge modes are nontopological it is possible to remove them from the bulk energy gap by modifying the edge potential, for example, with the help of a side gate or chemical doping.

Figure 1

Schematic illustration of the hopping matrices $h_{\alpha }$ ($\alpha =A,B,C,D$) between the nearest-neighbor lattice sites in zinc-blende crystals. Each cation (green) and anion (red) supports one $s$ orbital and three $p$ orbitals so that $h_i$ are $4\times 4$ matrices.

Figure 2

Energy gap $E_{\mathrm{gap}}$ and topological invariant $\nu$ as a function of the quantum well thickness $W_{\mathrm{HgTe}}$ in a ${\mathrm{CdTe}}_{10}/{\mathrm{HgTe}}_{W_{\mathrm{HgTe}}}/{\mathrm{CdTe}}_{10}$ heterostructure. The transition from the topologically trivial $\nu =0$ to nontrivial $\nu =1$ phase takes place at $W_{\mathrm{HgTe},\mathrm{c}}=12$ unit cells.

Figure 3

Edge state spectrum of a topologically nontrivial ${\mathrm{CdTe}}_{10}/{\mathrm{HgTe}}_{16}/{\mathrm{CdTe}}_{10}$ heterostructure with width $W^{\prime }=300$ unit cells. The colors (normalized to maximum absolute values) indicate the projection of the eigenstates onto 20 unit cells located at the left (blue) and right (red) edges of the system. Two pairs of topological helical edge states connect the conduction and valence band through the bulk gap. Additionally, there exists a large number of nontopological edge states far away from the Fermi level.

Figure 4

Edge state spectrum of a topologically trivial ${\mathrm{CdTe}}_{10}/{\mathrm{HgS}}_8/{\mathrm{CdTe}}_{10}$ heterostructure with width $W^{\prime }=300$ unit cells. In the presence of spin-orbit coupling we have used the same colors as in Fig. 3 (states localized at the right edge are red), whereas in the absence of spin-orbit coupling (${\lambda }_{1,2}=0$) the projection on the right-hand side is indicated with black. Inset: Local density of states (LDOS) as a function of position ${\mathbf{n}}_2$ close to the right edge for the edge states at $k_1=\pi$ (green dots).

Figure 5

Edge state spectrum of a topologically trivial ${\mathrm{AlSb}}_{10}/{\mathrm{InAs}}_{10}/{\mathrm{GaSb}}_{10}/{\mathrm{AlSb}}_{10}$ heterostructure with width $W^{\prime }=300$ unit cells.

Figure 6

(a) Buckled honeycomb lattice of anions (red) and cations (green) describing the minimal model. The directions in this two-dimensional system correspond to $n_1$ and $n_3$ in the original three-dimensional model [Eq. (1)]. (b) The Zak phases of the flat bands of the model give rise to an accumulated end charge proportional to the width of the ribbon $W$ because of an additional symmetry that allows for the decomposition of the Hamiltonian into diagonal blocks. Each dashed rectangle represents four states, labeled as $p_1, p_2, p_3$, and $s$ due to their orbital contents, of a single zigzag chain. The states represented by purple (blue) disks form $3\times 3$ ($1\times 1$) diagonal blocks after decomposition. The black lines indicate that the states connected by them go to the same block. The dashed black line denotes a coupling, which is present only in the cylinder geometry with periodic boundary conditions in the transverse direction.

Figure 7

(a) Band structure of minimal model ribbon of width $W=8$ unit cells in the ${\mathbf{n}}_3$ direction. The flat bands have a degeneracy 16, whereas the highest and lowest dispersive bands $E_{1,2}\left(k_1\right)$ are sevenfold degenerate and the other dispersive bands $E_{\parallel 1,2}\left(k_1\right)$ are nondegenerate. (b) LDOS of flat bands for a single zigzag chain of length $L=100$ unit cells as a function of position ${\mathbf{n}}_1$ with the accumulated charge quantized at both ends. (c) Accumulated charge as a function of thickness $W$ along the ${\mathbf{n}}_3$ direction. We have used $V_{sc}=-1.5535, V_{pa}=-1.1909, V_{s_c{p}_a}=1.1517, E_f=-1.33027$ (in meV) corresponding to HgS.

Figure 8

(a), (b) Edge state spectra of ${\mathrm{CdTe}}_{10}/{\mathrm{HgS}}_8/{\mathrm{CdTe}}_{10}$ and ${\mathrm{HgS}}_8$ of width $W^{\prime }=300$ unit cells in the ${\mathbf{n}}_2$ direction, respectively. (c) Evolution of eigenenergies at $k_1=\pi$ as a function of $x$ interpolating between coupled ($x=0$) and uncoupled ($x=1$) systems of ${\mathrm{HgS}}_8$ and ${\mathrm{CdTe}}_{10}$ as described in Eq. (9). (d) Evolution of eigenenergies at $k_1=\pi$ from ${\text{HgS}}_8$ ($x^{\prime }=0$) to the minimal model ($x^{\prime }=1$).

Figure 9

Edge state spectra of ${\mathrm{CdTe}}_{10}/{\mathrm{HgS}}_8/{\mathrm{CdTe}}_{10}$ of width $W^{\prime }=300$ unit cells with an on-site potential $\delta$ applied on the lattice sites at the right edge of the system. The values of the on-site potential are (a) $\delta =-0.1\text{eV}$, (b) $\delta =-0.15\text{eV}$, (c) $\delta =-0.2\text{eV}$, and (d) $\delta =-0.3\text{eV}$.

Figure 10

Crystal structure of the 2D QWs grown along the (001) orientation with (a) unsaturated dangling bonds on the top and bottom surface with stoichiometric concentration. (b) Dangling bond saturated with I and Na on the anion and cation surfaces, respectively.

Figure 11

Band structure of the 2D QWs with (a) unsaturated dangling bonds on the top and bottom surface. (b) Dangling bond saturated with I and Na on the anion and cation surface, respectively. After saturation the DFT band structure resembles the tight-binding Hamiltonian band structure.

Figure 12

Edge state spectra for heterostructures grown in the (110) direction: (a) ${\mathrm{CdTe}}_{10}/{\mathrm{HgTe}}_{16}/{\mathrm{CdTe}}_{10}$ and (b) ${\mathrm{CdTe}}_{10}/{\mathrm{HgS}}_8/{\mathrm{CdTe}}_{10}$. The widths are $W^{\prime }=150$ unit cells in the ${\mathbf{n}}_2^{\prime }$ direction.