Quantum spin Hall effect in antiferromagnetic topological heterobilayers

The quantum spin Hall (QSH) insulator, ensured by the time-reversal ($\mathcal{T}$) symmetry, has been a conceptual milestone of complex topological phases and well known to realize the quantum anomalous Hall effect when $\mathcal{T}$ symmetry is broken by introducing ferromagnetism. Here, based on first-principles calculations and a tight-binding model, we show that, unlike previous $\mathcal{T}$-symmetry breaking, the QSH phase can survive under the antiferromagnetic long-range order in a RbCuSe/CsMnP heterobilayer consisting of a nonmagnetic QSH insulator RbCuSe and an antiferromagnetic insulator CsMnP. The calculated spin Chern number, ${\mathcal{Z}}_2$ invariant, and gapless edge state confirm the topological nontrivial phase clearly. Moreover, the role of effective and Rashba spin-orbit coupling and the magnitude of antiferromagnetism are discussed to reveal the underlying physical mechanism. Our results may lead to further scientific and technological advances in topological magnetism and antiferromagnetic spintronics.

Figure 1

(a) Perspective top and (b) side view of the RbCuSe/CsMnP heterobilayer. The yellow and cyan isosufaces in (b) represent electron accumulation and depletion, respectively. (c) Phonon spectra of the CsMnP QLs show that the QLs are dynamically stable. (d) Potential profiles of the RbCuSe/CsMnP heterobilayer.

Figure 2

Band structures of (a) RbCuSe and (b) CsMnP QLs without (solid lines) and with (dashed lines) spin-orbit coupling (SOC). The Fermi level $E_F$ is indicated with a dashed horizontal line. (c) Evolution of the Wannier charge center (WCC) for RbCuSe QLs along $k_y$, confirming the topologically nontrivial nature of the gap with ${\mathbb{Z}}_2=1$. (d) Spin Hall conductivity ${\sigma }_{xy}^S$ of RbCuSe QLs as a function of the position of the Fermi level $E_F$, showing a quantized value within the energy window of the SOC gap. (The inset) $K$-space distribution of spin Berry curvature within the SOC gap. (e) Topological edge states of the semi-infinite RbCuSe nanoribbon with SOC.

Figure 3

Band structures of the RbCuSe/CsMnP heterobilayer (a) without and (b) with SOC, weighted with the contribution of RbCuSe and CsMnP. (c) Evolution of WCC for the RbCuSe/CsMnP heterobilayer along $k_y$, confirming the topologically nontrivial nature of the gap with ${\mathbb{Z}}_2=1$. (d) Topological edge states of the semi-infinite RbCuSe/CsMnP nanoribbon with SOC.

Figure 4

(a) Band structure of an AFM QSH insulator in the square lattice with $t_{\mathrm{in}}=-0.8t$ and $t_{\mathrm{ra}}=-0.8t$. (b) Phase diagram with respect to intrinsic SOC $t_{\mathrm{in}}$ and Rashba SOC $t_{\mathrm{ra}}$ where colors represent the value of the bulk band gap. Dashed lines mark the boundary between different insulating phases for which the band gap closes. The inset shows the sketch of the square lattice with AFM ordering. (c) Evolutions of the WCC for (left) nontrivial and (right) trivial insulating phases. (d) Dispersion of the edge states, which reveal the topologically nontrivial nature.