Strain-induced topological insulator phase and effective magnetic interactions in Li${}_2$IrO${}_3$

We present an effective tight-binding Hamiltonian for Li${}_2$IrO${}_3$ based on maximally localized Wannier functions for states near the Fermi level as obtained from first-principles electronic structure calculations. The majority of the Wannier orbitals are positioned on the center site with dominant $j_{\mathrm{eff}}=1/2$ character, while relatively small $j_{\mathrm{eff}}=3/2$ tails lie on the three nearest-neighbor sites. Interestingly, the spin quantization axis of the $j_{\mathrm{eff}}=1/2$ components deviates from the local octahedral axis and points toward the nearest-neighbor Ir direction. In our tight-binding model, there are relatively strong next-nearest- and the third-nearest-neighbor hopping terms within the two-dimensional Ir honeycomb lattice in addition to the relatively small but significant interlayer hopping terms. The ratio between the nearest-neighbor and the third-nearest-neighbor hoppings, which can be controlled by the lattice strain, plays a critical role in determinating the $Z_2$-invariant character of Li${}_2$IrO${}_3$. From our tight-binding model, we also derive an effective Hamiltonian and its parameters for the magnetic exchange interactions. Due to the complex spin-dependent next-nearest-neighbor hopping terms, our pseudospin Hamiltonian includes significant next-nearest-neighbor antiferromagnetic Kitaev terms as well as Dzyaloshinskii-Moriya and Heisenberg interactions. From our model Hamiltonian we estimate classical energies of collinear magnetic configurations as functions of the Hund's coupling of the Ir atom, from which zigzag-type magnetic order gives the lowest energy.

Figure 1

A schematic drawing of the Li${}_2$IrO${}_3$ crystal structure, where ${\mathbf{a}}_1$, ${\mathbf{a}}_2$, and ${\mathbf{a}}_3$ are unit vectors of the unit cell adopted in our calculations. (a) Side view of the crystal structure. Each Li and Ir${}_{2/3}$Li${}_{1/3}$ layer is stacked along the ${\mathbf{a}}_3$ direction, on the three inequivalent positions in a cyclic order. (b) The three inequivalent positions, marked in the layer view of the Ir${}_{2/3}$Li${}_{1/3}$ lattice. (c) The Ir-O bond lengths and bond angles within an IrO${}_6$ octahedron, where the local $x^{\prime }$, $y^{\prime }$, and $z^{\prime }$ coordinates are along the Ir-O bonds stretching out of the plane.

Figure 2

Band structure of Li${}_2$IrO${}_3$, with and without SOC. (a) Bands without SOC; light blue, dark blue, and green colors represent Ir $e_g$, Ir $t_{2g}$, and oxygen $p$-type bands, respectively. The inset shows the first Brillouin zone and the special $k$ points used in the calculations and the band plots. (b) The $t_{2g}$ complex splits into $a_{1g}$ (yellow) and $e_g^{\prime }$ (blue) doublets. The separation of $a_{1g}$ and $e_g^{\prime }$ states appears clearly near $\Gamma$ but becomes unclear near the zone boundary. (c) The bands with SOC decompose into $j_{\mathrm{eff}}=1/2$ (red) and 3/2 (cyan) character.

Figure 3

(a) Wannier-interpolated bands for the top four bands near the Fermi level. (b) The direction and the amount of the shift of the center of the $j_{\mathrm{eff}}=1/2$ Wannier orbitals. (c)–(e) $|W_{\sigma =+}\rangle$ with its center on site 0, where the plots correspond to (c) the real part of spin-up, (d) the real part of spin-down, and (e) the imaginary part of spin-down components of $|W_{\sigma =+}\rangle$.

Figure 4

Major in-plane hopping terms from the tight-binding parameter from the Wannier function results: (a) nearest-neighbor terms, (b) complex and spin-dependent next-nearest-neighbor terms where $t_{2n;A}^{{\mathbf{a}}_{1,2,3}}={\left(t_{2n;B}^{{\mathbf{a}}_{1,2,3}}\right)}^{*}$, with $A$ and $B$ denoting two different sublattices represented as open and solid circles, respectively, and (c) the third-nearest-neighbor hopping terms.

Figure 5

The in-plane uniaxial strain and its effect on the band structure. In (a) one direction of compressive strain that is effective in achieving band inversion at the $M_2$ point is shown. (b) to (d) show the evolution of bands as the strain increases from $0\%$ to $4\%$, where the evolution of the two states at $M_2$ near the Fermi level are illustrated in (e). In (e), surface bands are drawn in a slab of 50 Ir${}_{2/3}$Li${}_{1/3}$O${}_2$ layers with the (001) surface in our unit cell, using the bulk hopping parameters of a $4\%$-compressed structure.

Figure 6

Three collinear spin configurations: (a) Neel, (b) stripy, and (c) zigzag. Red and blue dots represent spin up and spin down, respectively. (d) Classical energy of Neel, stripy, zigzag, and ferromagnetic order with respect to Hund's coupling. The vertical dotted line shows the value of Hund's coupling $J_{\mathrm{H}}=0.14$ eV from the constrained RPA calculation result.[43]

Figure 7

(a) Planar view of an Ir honeycomb lattice with the position index of each Ir atom in terms of the Bravais lattice vectors, with a single unit cell represented as dotted lines. (b) Side view of the stacking of Ir lattices. ${\mathbf{a}}_1$ to ${\mathbf{a}}_3$ are the Bravais lattice vectors of the unit cell used in this calculation. (c) Projected view of adjacent Ir layers onto the $xy$ plane; major interlayer hopping terms are illustrated as arrows with different colors and line types.