Calculating branching ratio and spin-orbit coupling from first principles: A formalism and its application to iridates

We present a simple technique to calculate spin-orbit coupling, $\langle \mathbf{L}·\mathbf{S}\rangle$, and branching ratio measured in x-ray absorption spectroscopy. Our method is for first-principles electronic structure calculation, and its implementation is straightforward for any of the standard formulations and codes. We applied this technique to several different large spin-orbit coupling iridates. The calculated $\langle \mathbf{L}·\mathbf{S}\rangle$ and branching ratio of a prototype $j_{\mathrm{eff}}=1/2$ Mott insulator, ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$, are in good agreement with recent experimental data over the wide range of Rh doping. Three different double-perovskite iridates (namely, ${\mathrm{Sr}}_2{\mathrm{MgIrO}}_6, {\mathrm{Sr}}_2{\mathrm{ScIrO}}_6$, and ${\mathrm{Sr}}_2{\mathrm{TiIrO}}_6$) are also well described. This technique can serve as a promising tool for studying large spin-orbit coupling materials from first principles and for understanding experiments.

Figure 1

The unit-cell structure of (a) ${\mathrm{Sr}}_2{\mathrm{IrO}}_4$ and (b) double perovskite. The green, light blue, light brown, and red spheres represent Sr, $X$, Ir, and O, respectively ($X$ = Mg, Sc, Ti). The $X{\mathrm{O}}_6$ and ${\mathrm{IrO}}_6$ cage is shaded in blue and brown, respectively. (c) The expectation value of spin-orbit coupling calculated in the $d^5$ atomic limit as a function of $10Dq$. The blue (${\langle \mathbf{L}·\mathbf{S}\rangle }_J$) and red (${\langle \mathbf{L}·\mathbf{S}\rangle }_{j_{\text{eff}}}$) line correspond to the result obtained by using both $t_{2g}$ and $e_g$ orbitals and only $t_{2g}$ orbitals, respectively.

Figure 2

The calculated values (filled or half-filled symbols) of (a) $\langle \mathbf{L}·\mathbf{S}\rangle$ and (b) branching ratio in comparison with experiments (open symbols). The upper (corresponding to the dark blue symbols) and lower axis (magenta symbols) refer to ${\mathrm{Sr}}_2{\mathrm{Rh}}_x{\mathrm{Ir}}_{1-x}{\mathrm{O}}_4$ and ${\mathrm{Sr}}_{2}X{\mathrm{IrO}}_6$ ($X$ = Mg, Sc, Ti), respectively. The error bars for experimental data are taken from the original papers. The error bars in the calculation results reflect the ambiguity related to the numerical parameters and other computation details (see the main text for more details). Inset: The effect of oxygen vacancy has been simulated for ${\mathrm{Sr}}_2{\mathrm{MgIrO}}_{6-\delta }$. The horizontal dashed lines refer to the experimental values. The filled (blue) symbols represent the result of rigid band shift, and the half-filled (magenta) symbols are the result of the supercell calculation with oxygen removal.

Figure 3

The calculated PDOS for (a) ${\mathrm{Sr}}_2{\mathrm{MgIrO}}_6$, (b) ${\mathrm{Sr}}_2{\mathrm{ScIrO}}_6$, and (c) ${\mathrm{Sr}}_2{\mathrm{TiIrO}}_6$, corresponding to Ir valence of 6+, 5+, and 4+, respectively. The blue and red lines represent $j_{\mathrm{eff}}=3/2$ and 1/2 states, respectively.