Quantum spin Hall effect in rutile-based oxide multilayers

Dirac points in two-dimensional electronic structures are a source for topological electronic states due to the $\pm \pi$ Berry phase that they sustain. Here we show that two rutile multilayers [namely ${\left({\mathrm{WO}}_2\right)}_2/{\left({\mathrm{ZrO}}_2\right)}_n$ and ${\left({\mathrm{PtO}}_2\right)}_2/{\left({\mathrm{ZrO}}_2\right)}_n$], where an active bilayer is sandwiched by a thick enough ($n=6$ is sufficient) band insulating substrate, show semimetallic Dirac dispersions with a total of four Dirac cones along the $\mathrm{\Gamma }-M$ direction. These become gapped upon the introduction of spin-orbit coupling, giving rise to an insulating ground state comprising four edge states. We discuss the origin of the lack of topological protection in terms of the valley spin-Chern numbers and the multiplicity of Dirac points. We show with a model Hamiltonian that mirror-symmetry breaking would be capable of creating a quantum phase transition to a strong topological insulator, with a single Kramers pair per edge.

Figure 1

(a) Sketch of the multilayer structure, based on the rutile unit cell, consisting of a bilayer of ${\mathrm{WO}}_2$ or ${\mathrm{PtO}}_2$ sandwiched between insulating ${\mathrm{ZrO}}_2$. In the absence of spin-orbit coupling, the low-energy electronic properties are dominated by four Dirac equations located along the $\mathrm{\Gamma }-M$ path, as shown in the sketch (b). When SOC is introduced, the band structure develops a gap for both the W-based (c) and Pt-based (d) multilayers. The solid lines in (c),(d) correspond to the DFT band structure, whereas the dashed lines correspond to the Wannier interpolation. In (d), the anticrossing between $\mathrm{\Gamma }$ and $X$ is already present without SOC.

Figure 2

Edge (01) $k$-resolved density of states for the W-based (a) and Pt-based multilayer (b). A set of edge states appear within the gap, but due to the lack of a strong topological index, their existence is not protected, developing a small gap.

Figure 3

Band structure (a) along the (1,1) direction for the model in Eq. (5), in the quantum anomalous $\left(C=2\right)$ regime $\left(\alpha =0,\lambda \ne 0\right)$, Berry curvature along the path (b) and in the whole Brillouin zone (c). Dashed lines in (a) show the bands for $\left(\alpha =\lambda =0\right)$, the gapless regime. Band structure (d), and Berry curvatures (e),(f) in the QAH regime with $\alpha \ne 0,\lambda \ne 0$, with total Chern number $C=1$. Parameters are $\epsilon =-2,t=1,t^{\prime }=2,\lambda =0.5$, and $\alpha =2$.

Figure 4

Sketch of the edge states for the Hamiltonian Eq. (8), showing the crystalline insulating state with spin Chern number $C_S=4$ (a), and the strong phase with $C_S=2$. The transition from (a) to (b) is driven by the mirror symmetry parameter $\alpha$. In a real material, such transition can be induced by strain in the (11) direction. The DFT results of Fig. 2 correspond to the symmetric case (a).