Quantum anomalous Hall phase in (001) double-perovskite monolayers via intersite spin-orbit coupling

Using tight-binding models and first-principles calculations, we demonstrate the possibility to achieve a quantum anomalous Hall (QAH) phase on a two-dimensional square lattice, which can be realized in monolayers of double perovskites. We show that effective intersite spin-orbit coupling between $e_g$ orbitals can be induced perturbatively, giving rise to a QAH state. Moreover, the effective spin-orbit coupling can be enhanced by octahedral rotations. Based on first-principles calculations, we propose that this type of QAH state could be realized in ${\mathrm{La}}_2{\mathrm{MnIrO}}_6$ monolayers, with the size of the gap as large as 26 meV in the ideal case. We observe that the electronic structure is sensitive to structural distortions, and that an enhanced Hubbard $U$ tends to stabilize the nontrivial gap.

Figure 1

(a) Oblique view of the crystal structure of an ${\mathrm{La}}_2{\mathrm{MnIrO}}_6$ (LMIO) monolayer (ML) sandwiched between ${\mathrm{LaAlO}}_3$ layers. (b) Sketch defining the parameters of the tight-binding model of Eq. (1) that describes the LMIO ML. Only the local $e_g$ orbitals on Mn atoms (purple, at origin) are shown; orbitals on Ir (brown, at the center) are suppressed. The intersite hoppings between Mn $e_g$ orbitals on nearest neighbors ($t_{1i}$) and next-nearest neighbors ($t_{2i}$) are shown by arrows (see main text for details). The octahedral rotations, denoted by angle $\theta$ in (b), are exaggerated for clarity of illustration.

Figure 2

Electronic structure of the tight-binding model of Eq. (1) with parameters as given in the text. (a) Band structure along the high-symmetry $\Gamma -X-M-\Gamma$ $k$ path. Inset shows the projected bulk band structure (shaded region) and edge states for an 80-unit-cell-wide ribbon in the reduced 1D BZ along $\tilde{\Gamma }-\tilde{X}-\tilde{\Gamma }$, color coded to distinguish the contributions from the two edges; dashed line denotes $E_F$. (b)–(d) Color map showing the distribution of negative (blue) and positive (red) Berry curvature in the BZ for three cases: (b) without octahedral rotations ($t_{2c}$ and ${\lambda }^{\left(1\right)}$ nonzero); (c) with octahedral rotations ($t_{1c}$ and ${\lambda }^{\left(2\right)}$ nonzero) but $t_{2c}$ and ${\lambda }^{\left(1\right)}$ artificially set to zero; and (d) with octahedral rotations, including all four terms. For guidance, nodes of $h_1$, $h_2$, and $h_0+h_3$ in Eq. (5) are shown as dotted blue, dashed black, and solid green curves, respectively.

Figure 3

Electronic structures of the LMIO/LAO superlattices. Red (blue) color coding highlights the character of the $d_{z^2}$ ($d_{x^2-y^2}$) orbitals of the Mn atoms, and black indicates the Ir-$5d$ states. All calculations are done with $U=5.0$ eV and $J=1.0$ eV on Mn unless stated otherwise. (a) Hypothetical structure without octahedral rotations (see main text for details). (b) Hypothetical structure with ${15}^{\circ }$ rotations about the $z$ axis in the LMIO layers only. (c) Calculated anomalous Hall conductivity for case (b). (d) Relaxed structure at zero epitaxial strain. (e) Relaxed structures at $2\%$ tensile epitaxial strain. (f) Same as (e) but with $U$ on Mn sites increased to $7.0$ eV. Insets in (e) and (f) zoom in on the region around the $M$ point. Dashed lines denote the Fermi energies.

Figure 4

Symmetry operations without octahedral rotations. $M^{\prime }$ means a combined mirror and time-reversal operation.

Figure 5

Symmetry operations for ${\mathrm{La}}_2{\mathrm{MnIrO}}_6$ monolayers with finite octahedral rotations. Red (blue) lattices of oxygen denote octahedral rotations with angle $\theta$ ($-\theta$), which are related to each other by combined mirror and time-reversal operation $M^{\prime }$.