hopping $\gamma_3$, but there is also noticeable lowering of the $d_{x^2-y^2}$ orbital energy level.

![\[fig:am\] Angular momentum texture from first-principles calculations in close vicinity of the high-symmetry points in BZ. Orbital and spin angular momenta are presented for the lower Rashba band of the (a) $d_{xy}$ and (b) lower $d_{xz/yz}$ (c) upper $d_{xz/yz}$ bands near $\Gamma$, and the lower Rashba-Dresselhaus band of the (d) $d_{yz}$ bands near $\mathrm{X}=(\pi,0)$. ](fig4.pdf){width="45.00000%"}

The $t_{2g}$-$e_g$ coupling also plays a key role in determining the angular momentum (AM) texture of the $t_{2g}$ bands in close vicinity of $\Gamma$ (Fig. \[fig:am\]). The tetragonal crystal field and the SOC determine the spin-orbital entanglement of the band manifolds; the spin-up and spin-down are in nearly the same orbital state for the $d_{xy}$ bands while they are in nearly orthogonal orbital states for the $d_{xz/yz}$ bands. This, in turn, affects the AM character; the spin AM is dominant in the $d_{xy}$ bands whereas the orbital AM is dominant [-@SPark2011; -@BKim2012; -@PKim2014] in the $d_{xz/yz}$ bands (see Appendix A). The $t_{2g}$-$e_g$ coupling is important in that it gives rise to finite AM in the lower $d_{xz/yz}$ bands and quantitatively changes the AM in the upper $d_{xz/yz}$. In the $t_{2g}$-only TB 