Electronic excitations in $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$

We investigate the electronic properties of the three-dimensional stripyhoneycomb $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ via relativistic density functional theory calculations as well as exact diagonalization of finite clusters and explore the details of the optical conductivity. Our analysis of this quantity reveals the microscopic origin of the experimentally observed (i) optical transitions and (ii) anisotropic behavior along the various polarization directions. In particular, we find that the optical excitations are overall dominated by transitions between $j_{\text{eff}}=1/2$ and 3/2 states and the weight of transitions between $j_{\text{eff}}=1/2$ states at low frequencies can be correlated to deviations from a pure Kitaev description. We furthermore reanalyze within this approach the electronic excitations in the known two-dimensional honeycomb systems $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ and ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and discuss the results in comparison to $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$.

Figure 1

(a) Crystal structure of stripyhoneycomb $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ [16, 17]. Honeycomb rows alternate in orientation along $c$. The black axes $a$, $b$, and $c$ are the vectors of the unit cell. (b) Crystal structure showing only Ir atoms. The red, green, and blue bonds show the seven different types of bonds $X_A$, $X_B$, $Y_A$, $Y_B$, $Z_A$, $Z_B$, $Z_C$. $x$, $y$, $z$ are the cartesian coordinates for the $d$ orbitals. (c) Zigzag magnetic configuration used in our GGA+SO+$U$ calculations.

Figure 2

Density of states (DOS) for $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ in the nonmagnetic configuration obtained (a)–(c) within GGA and (d) GGA+SO.

Figure 3

Ir $5d{t}_{2g}$ band structure and relativistic DOS for $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ in zigzag magnetic order, obtained with GGA+SO+$U$ ($U_{\mathrm{eff}}=U-{\mathrm{J}}_{\mathrm{H}}=2.4$ eV).

Figure 4

Schematic diagrams of lowest-energy subspace ${\mathcal{S}}_1$ and one particle excitations ${\mathcal{S}}_2$, ${\mathcal{S}}_4$, and ${\mathcal{S}}_5$. Solid circles indicate electrons while empty circles are holes. ${\mathcal{S}}_1$ are all the states with ${\left(j_{3/2}\right)}^4{\left(j_{1/2}\right)}^1$, ${\mathcal{S}}_2$ are the states obtained from ${\mathcal{S}}_1$ by promoting an electron $j_{3/2}\to j_{1/2}$ on the same site. ${\mathcal{S}}_4$ are the states obtained from ${\mathcal{S}}_1$ by promoting an intersite $j_{1/2}\to j_{1/2}$ transition, and ${\mathcal{S}}_5$ are the states with promotion of an electron $j_{3/2}\to j_{1/2}$.

Figure 5

The investigated four-site cluster (inset) and spectral weight (SW) of projected excitations spectra for $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. $P_1$ includes all states with ${\left(j_{3/2}\right)}^4{\left(j_{1/2}\right)}^1$ (${\mathcal{S}}_1$), $P_2$ and $P_3$ include the states with an exciton on one site (${\mathcal{S}}_2$) or on more sites (${\mathcal{S}}_3$), respectively. $P_4$ includes states from ${\mathcal{S}}_1$ with promotion of an electron $j_{1/2}\to j_{1/2}$ to another site (${\mathcal{S}}_4$), and $P_5$ includes states with promotion of an electron $j_{3/2}\to j_{1/2}$ to another site (${\mathcal{S}}_5$). $P_6$ is for states that contain two-particle excited states for which the $d^4$ site contains occupancies ${\left(j_{3/2}\right)}^2{\left(j_{1/2}\right)}^2$ (${\mathcal{S}}_6$), while $P_7$ includes all other excitations with occupancy of $d^4-d^6-d^5-d^5$ (${\mathcal{S}}_7$). $\mathrm{\Delta }{E}_2\sim 0.6$ eV, $\mathrm{\Delta }{E}_4\sim 1.1 \mathrm{eV}$, and $\mathrm{\Delta }{E}_5\sim 1.6$ eV are the excitation energies for $P_2$, $P_4$, and $P_5$ respectively.

Figure 6

Optical conductivity components for $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$. (a) Results from DFT within GGA+SO+U. (b) Results from exact diagonalization and (c) results reported from experimental observations [23]. $\mathrm{\Delta }{E}_2\sim 0.6$ eV, $\mathrm{\Delta }{E}_4\sim 1.1$ eV, and $\mathrm{\Delta }{E}_5\sim 1.6$ eV are the characteristic excitation energy for subspaces ${\mathcal{S}}_2$, ${\mathcal{S}}_4$, and ${\mathcal{S}}_5$, respectively. The inset of (b) is the crystal structure projected in the $ab$ plane.

Figure 7

Optical conductivity component ${\sigma }_a$ for $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ and different $d\text{−}d$ transitions in the relativistic basis calculated (a) with DFT within GGA+SO+U and (b) with the ED method. The comparison to experiment is also shown [23].

Figure 8

Optical conductivity component ${\sigma }_c$ for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and different $d\text{−}d$ transitions in the relativistic basis calculated (a) with DFT within GGA+SO+U and (b) with the ED method. Comparison with experimental results from Ref. [37] and Ref. [36] is also shown. ${\sigma }_c$ of ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ corresponds to the ${\sigma }_{zz}$ component in Ref. [10].

Figure 9

Optical conductivity ${\sigma }_c$ for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$, $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$, and $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ calculated (a) with DFT within GGA+SO+U and (b) with the ED method. ${\sigma }_c$ for ${\mathrm{Na}}_2{\mathrm{IrO}}_3$ and $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ corresponds to ${\sigma }_{zz}$ in Ref. [10]. Please note that for $\alpha \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$ differences in the DFT optical conductivity with respect to results in Ref. [10] lie in the employed crystal structure.

Figure 10

Local octahedral environment of (a) type 1 and (b) type 2 in $\gamma \text{−}{\mathrm{Li}}_2{\mathrm{IrO}}_3$.

Figure 11

Optical conductivity tensor components with spin polarization (a) along $c$ and (b) along $a$ in the zigzag configuration.