Orbital Rashba effect and its detection by circular dichroism angle-resolved photoemission spectroscopy

We show, by way of tight-binding and first-principles calculations, that a one-to-one correspondence between an electron's crystal momentum $\mathbf{k}$ and nonzero orbital angular momentum (OAM) is a generic feature of surface bands. The OAM forms a chiral structure in momentum space much as its spin counterpart in Rashba model does, as a consequence of the inherent inversion symmetry breaking at the surface but not of spin-orbit interaction. This is the orbital counterpart of conventional Rashba effect and may be called the “orbital Rashba effect.” The circular dichroism (CD) angle-resolved photoemission (ARPES) method is an efficient way to detect this new order, and we derive formulas explicitly relating the CD-ARPES signal to the existence of OAM in the band structure. The cases of degenerate $p$- and $d$-orbital bands are considered.

Figure 1

(a) Band structure obtained from the Hamiltonian (3.11) with the parameter $(V_1,V_2,\gamma ,\alpha )=(1,0.1,1,0)$. (b)–(d) Calculated OAM in the Brillouin zone $[-\pi ,\pi ]\times [-\pi ,\pi ]$ for each band. $E_{1,k}$ through $E_{3,k}$ denoting the different bands are indicated with arrows in (a) and their corresponding OAM are shown in (b) through (d), respectively. The total OAM summed over all bands is zero for each $\mathbf{k}$ in the Brillouin zone.

Figure 2

OAM and CD from first-principles and tight-binding calculations of Bi monolayer without SOI. (a) LDA band structure for Bi monolayer with SOI turned off. A perpendicular electric field of 3 V/Å was imposed externally. Three dashed curves represent the tight-binding energy dispersions around the $\Gamma$ point. (b)–(d) OAM vectors (green arrows) and NCD signals (color backgrounds) for the three bands, $E_1$ (b), $E_2$ (c), and $E_3$ (d), over the whole Brillouin zone are marked by solid hexagons. The largest OAM has a magnitude $\approx$$1\hslash$ for bands $E_2$ and $E_3$. NCD is calculated with $k_{F,z}=2.27$ Å${}^{-1}$. (e),(f) NCD calculated with $k_{F,z}=0$ in Eq. (4.8) for $E_2$ (e) and $E_3$ (f) bands. The opposite color assignments between (c) and (e) is a consequence of photon energy dependence of the scattering intensity.

Figure 3

(a) First-principles electronic band structure of Cu(111) 30-layer slab. Red dashed lines indicate the two surface bands. (b) Calculated OAM and CD corresponding to outer energy surface bands. Inner surface band shows basically the same OAM and CD patterns. Maximum OAM vector is $\sim$$0.07\hslash$.