Time-reversal symmetry protected chiral interface states between quantum spin and quantum anomalous Hall insulators

We theoretically investigate the electronic properties of the interface between quantum spin Hall (QSH) and quantum anomalous Hall (QAH) insulators. A robust chiral gapless state, which substantially differs from edge states of QSH or QAH insulators, is predicted at the QSH/QAH interface using an effective Hamiltonian model. We systematically reveal distinctive properties of interface states between QSH and single-valley QAH, multivalley high-Chern-number QAH and valley-polarized QAH insulators based on tight-binding models using the interface Green's function method. As an example, first-principles calculations are conducted for the interface states between fully and semihydrogenated bismuth (111) thin films, verifying the existence of interface states in realistic material systems. Due to the physically protected junction structure, the interface state is expected to be more stable and insensitive than topological boundary states against edge defects and chemical decoration. Hence our results of the interface states provide a promising route towards enhancing the performance and stability of low-dissipation electronics in real environment.

Figure 1

Schematics of the evolution of edge states in momentum space (a) and real space (b) when putting the QSH and QAH insulators together to form an interface. Red and blue colors represent different spin polarizations.

Figure 2

Schematic illustration of a system consisting of two semi-infinite materials. “I” denotes the interface region for which we need to calculate the Green's function $G_I$. Each semi-infinite system can be split into principal layers with Hamiltonian submatrices labeled according to the distance to the interface.

Figure 3

(a) The schematic of the calculated system. (b)-(j) LDOS of the interface between semi-infinite QSH and single-valley QAH insulators vs the strength of Zeeman term ${\alpha }_A$ and ${\alpha }_B$.

Figure 4

Layer-resolved LDOS of the QSH/QAH interface state as a function of the principal layers index.

Figure 5

Spin-resolved electronic structures of the outmost principal layer (i.e., the $0\text{th}$ layer) of QSH (left panel) and QAH (right panel) parts in the interface system. The scaling factor $\kappa$ increases from top to bottom ($\kappa =0.0$ uncoupled; $\kappa =1.0$ fully coupled). Different colors represent $S_z$ along different directions (red and blue represent the $+z$ and $-z$ directions, respectively). The spin LDOS of $S_z$ is depicted as a color scale.

Figure 6

(a)–(e) LDOS of the QSH/HQAH interface region as a function of $\kappa$. The average spin LDOS of $S_z$ is marked for all gapless states in (a) and (e). (f) Schematic of the spatial distribution of the edge (interface) channels at $\kappa =0.0$ ($\kappa =1.0$).

Figure 7

(a)–(e) LDOS of the QSH/VQAH interface region as a function of $\kappa$. The average spin LDOS of $S_z$ is marked for each gapless states in (a) and (e). (f) Schematic figure of the spatial distribution of the edge (interface) channels at $\kappa =0.0$ ($\kappa =1.0$). The pair of gapless states which would vanish in the presence of intervalley scattering are labeled by a dashed-line arrow.

Figure 8

(a) A schematic illustration showing the QSH/QAH interface state between the fully and semihydrogenated Bi (111) films. (b) The helical edge state of H-Bi(111), (c) the QSH/QAH interface state between H-Bi(111) and sH-Bi(111), and (d) the chiral edge state of sH-Bi(111).

Figure 9

Schematic illustration of the simple example. (a) In the first case where $f\left(a\right)>0, f\left(b\right)=0$, and $\dot{f}\left(b\right)>0$, there must be at least one zero point. (b) In the second case where $f\left(a\right)>0, f\left(b\right)=0$, and $\dot{f}\left(b\right)<0$, the existence of zero points cannot be determined directly.