Quantum anomalous Hall and quantum spin-Hall phases in flattened Bi and Sb bilayers

Abstract

Discovery of two-dimensional topological insulator such as Bi bilayer initiates challenges in exploring exotic quantum states in low dimensions. We demonstrate a promising way to realize the Kane-Mele-type quantum spin Hall (QSH) phase and the quantum anomalous Hall (QAH) phase in chemically-modified Bi and Sb bilayers using first-principles calculations. We show that single Bi and Sb bilayers exhibit topological phase transitions from the band-inverted QSH phase or the normal insulator phase to Kane-Mele-type QSH phase upon chemical functionalization. We also predict that the QAH effect can be induced in Bi or Sb bilayers upon nitrogen deposition as checked from calculated Berry curvature and the Chern number. We explicitly demonstrate the spin-chiral edge states to appear in nitrogenated Bi-bilayer nanoribbons.

Figure 1. The lattice geometry for Bi and Sb based structure.

Atomic structure of two-dimensional single Bi (111) and Sb (111) bilayers (a) without and (b) with adatom deposition. Blue balls denote Bi or Sb, and white balls the chemicals (H, F or N).

Figure 2. Structural stability of bismuthane: MD simulation and phonon spectrum.

(a) Molecular dynamics (MD) simulation using the NVT ensemble for a 3 × 3 bismuthane at 300K; (upper panel) the temperature variation in time with blue dots denoting the time steps at which the snap shots (lower panel) are taken. Atomic deformation is insignificant, supporting the structural stability at room temperature. (b) Calculated phonon band structure without the SOC for bismuthane. No imaginary frequency is observed, again supporting the structural stability of bismuthane.

Figure 3. Electronic properties of chemically modified bilayer system.

Calculated Dirac-cone band structures of (a) bismuthane and (b) anitmonane near the Fermi level without SOC (left) and with SOC (right). (c) The energy gaps at Γ or K(K') point for pristine or hydrogenated Bi and Sb (111) bilayers upon varying the spin-orbit coupling strength in first-principles calculations from zero to the true value set as 1. (d) Wannier orbitals constructed from the Dirac states near the Fermi level. Bi p_y (p_x) orbitals represented at A (B) site of the hexagonal lattice. The sign of the wave function is represented in purple (+) or yellow (−) color.

Figure 4. Edge states of QSH phase Bismuthane.

Evolution of the edge states as the SOC strength (x) is changed from 0 to 50 (0.5), and to 100% (1.0) of the true value in bismuthane nanoribbons (a) with the zig-zag edge without passivation and (b) with the hydrogen-passivated zigzag edge. The edge states (depicted in red) have a linear dispersion with the Dirac point at Γ (without passivation) or at X point (hydrogen passivation). The 2D bulk bands projected on the 1D Brillouin zone are depicted in green.

Figure 5. Edge states of armchair-edged Bismuthane nanoribbons.

A similar plot as in Fig. 4 with the SOC strength (x) set to 0, 50 (0.5), and to 100% (1.0) of the true value. The armchair-edged bismuthane nanoribbons have a linear dispersion with the Dirac point at Γ regardless of the edge passivation; (a) no passivation and (b) the hydrogen passivation. The 2D bulk-bands projected on the 1D Brillouin zone are depicted in green.

Figure 6. Realization of QAH phase in Bi bilayer.

(a) Calculated band structures of Bi(HN) (left panel). Color (red/blue) of the bands represents the spin (up/down) with detailed spin texture near the Dirac point on the right panel. (b) Calculated Berry curvature of the valence band with the hexagon in red denoting the first Brillouin zone. (c) Calculated edge band-structures of armchair-edged Bi(HN) nanoribbon. The dark region represents the bulk bands projected onto the 1D Brillouin zone of the edges. Left panel shows the site projection to the left (blue) or right (red) edges and right panel the spin z-component with up (red) or down (blue). We observe only one linear band inside the bulk gap with the same spin, which is a direct proof of the quantum anomalous Hall effect.

Figure 7. Nitrogen coverage for the band inversion and QSH-QAH phase transition.

(a) Calculated band structures of Bi bilayer nitrogenated with coverage ρ_N using 2 × 2 supercell. As the nitrogen coverage is increased, the exchange field is enhanced to split the spin up and down states. (b) The band gap as a function of ρ_N. It closes at about 0.65ML where the topological phase transition from QSH to QAH phases occurs.