Semimetal and Topological Insulator in Perovskite Iridates

The two-dimensional layered perovskite Sr${}_2$IrO${}_4$ was proposed to be a spin-orbit Mott insulator, where the effect of Hubbard interaction is amplified on a narrow $J$${}_{\mathrm{eff}}=1/2$ band due to strong spin-orbit coupling. On the other hand, the three-dimensional orthorhombic perovskite (Pbnm) SrIrO${}_3$ remains metallic. To understand the physical origin of the metallic state and possible transitions to insulating phases, we construct a tight-binding model for SrIrO${}_3$. The band structure possesses a line node made of $J$${}_{\mathrm{eff}}=1/2$ bands below the Fermi level. As a consequence, instability toward magnetic ordering is suppressed, and the system remains metallic. This line node, originating from the underlying crystal structure, turns into a pair of three-dimensional nodal points on the introduction of a staggered potential or spin-orbit coupling strength between alternating layers. Increasing this potential beyond a critical strength induces a transition to a strong topological insulator, followed by another transition to a normal band insulator. We propose that materials constructed with alternating Ir- and Rh-oxide layers along the (001) direction, such as Sr${}_2$IrRhO${}_6$, are candidates for a strong topological insulator.

Figure 1

Phase diagram as a function of the staggered potential between layers $m$. The product of parity eigenvalues for the filled bands are denoted by ($\pm$) for each time-reversal invariant-momentum (TRIM) point. When $m$ is finite, the line node becomes a nodal point located between $U$ and $R$. This node gets shifted toward $R$ (as indicated by the red arrow) as $m$ increases until it reaches $m$${}_{c1}$, after which a full gap appears. Beyond $m$${}_{c1}$, the system turns into a strong topological insulator with indices ${\nu }_0;\left({\nu }_1{\nu }_2{\nu }_3\right)=1;\left(001\right)$. At $m$${}_{c2}$, the gap closes at the $Z$ point, and the system becomes a topologically trivial band insulator for $m>m_{c2}$.

Figure 2

Crystal structure for SrIrO${}_3$ from a side view. The different colors of blue (B), red (R), yellow (Y), and green (G) represent Ir atoms with different oxygen environment. These four Ir atoms form a unit cell.

Figure 3

LDA calculation with Hubbard $U=2$ eV and SOC tuning parameter $\alpha =0$ (a) and $\alpha =2.0$ (b). The band structure remains qualitatively similar from $U=0$ up to $U=2$ eV while SOC has a significant effect. See the main text for the comparison to the tight-binding model.

Figure 4

(a) Underlying band structure of orthorhombic perovskite SrIrO${}_3$ obtained by a tight-binding model including only Ir-Ir direct hoppings. Note that the band structure is qualitatively similar the the LDA calculation of Fig. 3, especially near the Fermi surface. (b) Brillouin zone with all TRIM points labeled along with the product of the parity eigenvalues of the filled bands. The red line in the XURS plane corresponds to the line node in the system.

Figure 5

The edge state calculation with a finite slab of material having periodic boundary conditions along the $x$ and $y$ directions and open boundary condition along the $z$ direction. From the Fu-Kane criteria, we expect to see an odd number of crossing of the edge state between $\Gamma$ and any of the other three TRIM points in the surface Brillouin zone. Here we compute the band structure of such a system between $\Gamma =(0,0)$ and $S=(\pi ,0)$ for $m=0.5$.