Over the last few years, we have noticed importance of orbital degree of freedom in many spin-orbitronic phenomena such as spin Hall effect and spin-orbit torque [1,2]. It is exemplified by precedence of orbital dynamics to spin dynamics when the electronic system is driven by an external electric field. Thus, the spin dynamics is governed by the orbital dynamics, which are entangled by the spin-orbit coupling. However, experimental verification of the orbital transport effects are yet to be made. Thus, theoretical prediction of real material systems and comparison with experiments are necessary, in which first-principles calculation based on the density functional theory can play a decisive role. In this talk, I introduce methods for describing orbital transport effects based on the density functional theory calculation and suggest a few material systems that are expected to exhibit significant orbital dynamic effects. As a main method, I show that equations of motion of the spin and orbital angular momentum based on the continuity equation enable systematic tracking of angular momentum flow, not only in space but also between different degrees of freedom [3]. Then I present detailed analysis of real materials by applying the developed formalism: (i) Fe/W(110) and Ni/W(110) [3], (ii) Fe/Pt/Cr(001) [4], and (iii) surface oxidized Cu(111) [5]. In (i), I show that the directions of the spin-orbit torque are opposite between Fe/W(110) and Ni/W(110) as a result of the competition between spin Hall and orbital Hall currents from W. In (ii), I show that insertion of Pt layer between Fe and Cr(001) provides a way to harness gigantic orbital Hall current arising from Cr, leading to sign change of the spin-orbit torque. In contrast to (i) and (ii), where bulk orbital Hall current plays significant role,  in (iii), I show that surface oxidization of Cu(111) leads to significantly pronounced orbital polarization of the surface states via orbital Rashba effect. Considering substantial on-going efforts in these materials from the experimental side, I not only compare the calculation with experimental data but also discuss further implications of the theoretical prediction.

I acknowledge funding under SPP 2137 “Skyrmionics” (Project No. MO 1731/7-1) and TRR 173-268565370 (Project No. A11) of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation).

[1] D. Go, D. Jo, C. Kim, and H.-W. Lee, Phys. Rev. Lett. 121, 086602 (2018).
[2] D. Go and H.-W. Lee, Phys. Rev. Res. 2, 013177 (2020).
[3] D. Go et al. Phys. Rev. Res. 2, 033401 (2020).
[4] S. Lee, M.-G. Kang, D. Go et al. in preparation.
[5] D. Go et al. in preparation.