Topological phases in iridium oxide superlattices: Quantized anomalous charge or valley Hall insulators

We study topological phases in orthorhombic perovskite iridium (Ir) oxide superlattices grown along the $\left[001\right]$ crystallographic axis. Bilayer Ir oxide superlattices display topological magnetic insulators exhibiting quantized anomalous Hall effects due to strong spin-orbit coupling of Ir $5d$ orbitals and electronic correlation effects. We also find a valley Hall insulator with counterpropagating edge currents from two different valleys and a topological crystalline insulator with edge states protected by the crystal lattice symmetry based on stacking of two layers. In a single-layer superlattice, a topological insulator can be realized, when a strain field is applied to break the symmetry of a glide plane protecting the Dirac points. It turns into a topological magnetic insulator in the presence of magnetic ordering and/or in-plane magnetic field. We discuss essential ingredients for these topological phases and experimental signatures to test our theoretical proposals.

Figure 1

Left: ${\mathrm{IrO}}_6$ octahedron with the rotation $\theta$ along the $c$ axis and tilting $\varphi$ along the local $\left(110\right)$ axis. Right: Single-layer Ir oxide superlattice structure. ${\mathrm{IrO}}_2$ layer contains two different sites denoted by $A$ and $B$ representing different rotations and tiltings, $(\theta ,\varphi )$ and $(-\theta ,-\varphi )$ oxygen octahedra, and it is grown on a band insulator ${\mathrm{AMO}}_3$ with $Pbnm$ structure. The primitive lattice vectors are $\vec{a}=(\widehat{x}-\widehat{y})/2$ and $\vec{b}=(\widehat{x}+\widehat{y})/2$.

Figure 2

Band dispersion of single-layer Ir oxide (a) without tilting $\varphi$. It shows fourfold degeneracy along the $S=(\pi ,0)\to X=(\frac{\pi }{2},-\frac{\pi }{2})$ direction. (b) Finite rotation and tilting leaves two Dirac points at $X$ and $Y=(\frac{\pi }{2},\frac{\pi }{2})$. (c) When the $b$-glide symmetry is broken, the Dirac point acquires a finite gap at the $X$ and $Y$ points. The set of $(\theta ,\varphi )$ for both (b) and (c) is $(7^{\circ },{19}^{\circ })$.

Figure 3

Edge state calculation of (a) the topological insulator (TI) shown in Fig. 2, and 2 the quantized anomalous Hall insulator (QAHI) when the TR is broken due to a noncollinear magnetic ordering or an in-plane magnetic field. Gray lines represent the bulk state and red (blue) lines denote the edge state at $L=0 \left(L=N\right)$ plotted along $k_a=k_x-k_y-\pi$. The parameter set is the same with the band dispersion in Fig. 2. The two gapless edge modes at $L=0/L=N$ (red/blue) crossing at 1D TR invariant momentum indicate that the system belongs to a 2D TI. After breaking the TR, only one gapless edge state is left propagating along the boundary.

Figure 4

Left: ABAB stacking with $A=(\theta ,\varphi )$ and $B=(-\theta ,-\varphi )$ types of octahedra rotation and tilting. Right: ABCD bilayer stacking which contains $A$ and $B$ in the top layer, while $C=(\theta ,-\varphi )$ and $D=(-\theta ,\varphi )$ types of octahedra rotation and tilting are in the bottom layer.

Figure 5

Band structure for bilayer with no tilt effect (i.e., $\varphi =0$) in octahedron environment (a) has a degenerate line circling the $\Gamma$ point [47]. (b) ABAB: Finite tilting lifts the line node degeneracy but leaves one Dirac point protected by the $b$-glide symmetry along $S\to X$ for ABAB stacking. (c) Band structure for the ABCD bilayer with finite tilting $\varphi$. It has a band gap at $(k_0,\pm k_0)$ (circled out by band lines). The Fermi energy is ${\epsilon }_F=0$, indicated by gray solid lines.

Figure 6

Slab dispersion with (a) ferromagnetic (FM) ordering with $m_{\left(110\right)}=0.09t$ and (b) antiferromagnetic (AFM) ordering with strength $m_{\left(1\overline{1}0\right)}=0.06t$. Two gapless edge modes at the $L=0$ and $L=N$ boundaries are represented by red and blue, respectively.

Figure 7

Phase diagram when rotation degree is $\theta ={13}^{\circ }$ in (c) plotted as the $z$-direction exchange field $h_z$ in units of Tesla (T) vs tilting degree $\varphi$. Different phases has been characterized by different topological invariants $(C_{mv},C_m,C_v,C)$. The edge state for each phase has been displayed in (a) QAH, (b) quantized valley Hall (QVH), (d) mirror valley Hall (MVH), and (e) topological crystalline insulator (TCI). Two gapless edge modes in (a), (b), (d), and (e) at the $L=0$ and $L=N$ boundaries are represented by red and blue, respectively. Edge states are purple (mixed color of red and blue) in (d) and (e) because of the degeneracy between edge modes at $L=0$ and $L=N$. See the main text for finite $C_{mv}$ and $C_m$ related to these edge modes.