Orbital Hall effect in transition metals from first-principles scattering calculations

We use first-principles scattering calculations based upon wave-function matching and implemented with a tight-binding MTO basis to evaluate the orbital Hall conductivity $\sigma_{\rm oH}$ for Ti, V, Cr, Cu and Pt metals with temperature-induced lattice disorder. Only interatomic fluxes of orbital angular momentum are included in these estimates; intraatomic fluxes which do not contribute to the transfer of angular momentum are explicitly excluded. The resistivity and orbital Hall angle are both found to be linear in temperature so $\sigma_{\rm oH}$ is at most weakly temperature dependent. The value of $\sigma_{\rm oH}$ we obtain for bulk Cr is $\approx 2 \times 10^3 (\hbar/e) \, (\Omega \, {\rm cm})^{-1}$ which is substantially lower than previously obtained theoretical results but agrees well with experiment. In units of $10^3 (\hbar/e) (\Omega \, {\rm cm})^{-1}$, the values obtained for Ti, V and Pt are $5$, $6$ and $7$, respectively.