Exchange interaction in the ${\mathrm{YbCrBr}}_9^{3-}$ mixed dimer:  The origin of a strong ${\mathrm{Yb}}^{3+}-{\mathrm{Cr}}^{3+}$ exchange anisotropy

The superexchange interaction between ${\mathrm{Yb}}^{3+}$ and ${\mathrm{Cr}}^{3+}$ ions in the mixed ${\mathrm{YbCrBr}}_9^{3-}$ bioctahedral face-sharing dimer is quantitatively analyzed using a modified kinetic exchange theory, which is adapted to a realistic description of the electronic structure of lanthanide ions in solids. The general procedure of the calculation of the $4f-3d$ anisotropic exchange spin Hamiltonian is presented and applied to the ${\mathrm{YbCrBr}}_9^{3-}$ dimer. The spin-Hamiltonian of the ${\mathrm{Yb}}^{3+}-{\mathrm{Cr}}^{3+}$ exchange interaction is found to be extremely anisotropic, ${H=J}_z{S}_{\mathrm{Yb}}^z{S}_{\mathrm{Cr}}^z{+J}_{\perp }{(S}_{\mathrm{Yb}}^x{S}_{\mathrm{Cr}}^x{+S}_{\mathrm{Yb}}^y{S}_{\mathrm{Cr}}^y),$ with the antiferromagnetic $J_z$ and ferromagnetic $J_{\perp }$ parameters, where $S_{\mathrm{Yb}}^{\mu }$ and $S_{\mathrm{Cr}}^{\mu }$ $(\mu =x,y,z)$ are the components of the effective spin $S_{\mathrm{Yb}}=\frac{1}{2}$ of the ${\mathrm{Yb}}^{3+}$ ion (corresponding to the ground ${\Gamma }_6$ Kramers doublet) and the true spin $S_{\mathrm{Cr}}=\frac{3}{2}$ of the ${\mathrm{Cr}}^{3+}$ ion, respectively. The calculated exchange parameters are quite consistent with the experimental data ${(J}_z=-5.16{\mathrm{cm}}^{-1}$ and $J_{\perp }=+4.19\mathrm{}{\mathrm{cm}}^{-1})$ at reasonable values of the $\mathrm{Yb}\to \mathrm{Cr}$ and $\mathrm{Yb}\leftarrow \mathrm{Cr}$ charge transfer energies. The contributions to the $J_z$ and $J_{\perp }$ exchange parameters from the individual states of the ${4f}^{12}{-3d}^4$ and ${4f}^{14}{-3d}^2$ charge transfer configurations are analyzed in detail and general regularities are established. Our results indicate that a very strong $4f-3d$ exchange anisotropy can appear even in the absence of the crystal-field anisotropy on the lanthanide ion.