Calculated $g$-factors of $5d$ double perovskites ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ and ${\mathrm{Ba}}_2{\mathrm{YOsO}}_6$

Using Wannier functions to represent the density functional results we calculate the hybridization corrections to the orbital momentum operator in the Os $5d$ shell of the Mott insulators ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$ and ${\mathrm{Ba}}_2{\mathrm{YOsO}}_6$. The $g$-factors are obtained by evaluating the spin and orbital momentum operators in the atomic ground states of the Os ion. While the hybridization corrections play a minor role in the $d^3$ ion of ${\mathrm{Ba}}_2{\mathrm{YOsO}}_6$ with a dominant spin moment, they are instrumental for the observation of the nonzero $g$-factor of the $d^1$ ions of ${\mathrm{Ba}}_2{\mathrm{NaOsO}}_6$. In addition, we analyze the exchange interactions in ${\mathrm{Ba}}_2{\mathrm{YOsO}}_6$ and find them to be consistent with the reported magnetic structure.

Figure 1

$d_{xy}$ WF plot of (a) SVO and (b) BNOO. The latter is identical to that of BYOO. The isovalues are 2.0 for SVO and 0.5 for BNOO.

Figure 2

Convergence of the weight of the Wannier function (${\left|\psi \right|}^2$) and the one-particle orbital $g$-factor ($g_l$) with the spherical harmonic order $l$ for BNOO. Note that there is no contribution of the odd $l$ terms because of the WF parity. These data are nearly identical for each $t_{2g}$ orbital of BNOO and BYOO.