Spin and Charge Transport of Multiorbital Quantum Spin Hall Insulators

The fabrication of bismuthene on top of SiC paved the way for substrate engineering of room temperature quantum spin Hall insulators made of group $V$ atoms. We perform large-scale quantum transport calculations in these two-dimensional (2D) materials to analyze the rich phenomenology that arises from the interplay between topology, disorder, valley, and spin degrees of freedom. For this purpose, we consider a minimal multiorbital real-space tight-binding Hamiltonian and use a Chebyshev polynomial expansion technique. We discuss how the quantum spin Hall states are affected by disorder, sublattice resolved potential, and Rashba spin-orbit coupling. It is also shown that these materials can be driven to a topological Anderson insulator phase by sufficiently strong disorder.

Figure 1

(a) Schematic representation of the $p_x\text{−}{p}_y$ orbitals in the honeycomb lattice. (b) Comparison between the first principles energy bands of planar bismuthene (black dotted line) and our effective $p_x$, $p_y$ tight-binding approximation (red solid line). (c) Numerical calculations of the spin Hall conductivity (red line) and of the density of states (black line).

Figure 2

Numerical calculation of the density of states [(a),(b), and (c)] and spin Hall conductivity [(d) and (e)], of a $p_x$, $p_y$ honeycomb system with Anderson disorder strengths: $W=0.05$ (black line), $W=0.8$ (red line), $W=1.2$ (blue line), and $W=1.6\mathrm{eV}$ (green line).

Figure 3

Left panels: Energy bands for a 30 nm wide zigzag nanoribbon described by a $p_x\text{−}{p}_y$ Hamiltonian on a honeycomb lattice. Blue and red curves represent the majority and minority spin bands, respectively. The calculations were performed with $V_{pp\sigma }=1.815$, $V_{pp\pi }=-0.315$, ${\lambda }_I=0.435$, and ${\lambda }_R=0\mathrm{eV}$, for different values of the staggered potential: (a) $V_{AB}=0.1$, (c) $V_{AB}=0.435$, and (e) $V_{AB}=0.8\mathrm{eV}$. Right panels: spin-Hall conductivity (black line) and normalized longitudinal conductivity (green line) calculated as functions of energy for $V_{pp\sigma }=1.815$, $V_{pp\pi }=-0.315$, ${\lambda }_I=0.435$, ${\lambda }_R=0$, $W=0.05\mathrm{eV}$, and different values of the staggered potential: (b) $V_{AB}=0.1$, (d) $V_{AB}=0.435$, and (f) $V_{AB}=0.8\mathrm{eV}$.

Figure 4

(a) DOS calculated as a function of the energy for $V_{AB}=0.5\mathrm{eV}$ for weak (black line) and strong disorder (red line), $W=0.05$ and $W=1.95\mathrm{eV}$, respectively. (b) Spin-Hall conductivity for $V_{AB}=0.5\mathrm{eV}$ for the same values of weak (black line) and strong disorder (red line). The inset highlights the DOS in the vicinity of $E=0$.