Spin and orbital angular momentum structure of Cu(111) and Au(111) surface states

We performed angle-resolved photoemission studies on Cu(111) and Au(111) surface states with circularly polarized light to investigate local orbital angular momentum (OAM) structures. Existence of OAM is confirmed, as predicted, to exist in systems with an inversion symmetry breaking. Cu(111) surface state bands are found to have chiral OAM in spite of very small spin-orbit coupling, consistent with the theoretical prediction. As for Au(111), we observe split bands for which OAM for the inner and outer bands are parallel, unlike the Bi${}_2$Se${}_3$ case. We also performed first-principles calculations and the results are found to be consistent with experimental results. Moreover, the majority of OAM is found to have $d$-orbital origin while a small contribution comes from $p$ orbitals. An effective Hamiltonian that incorporates the role of OAM is derived and is used to extract the spin and OAM structures. We discuss the evolution of angular momentum structures from a pure OAM system to a strongly spin-orbit-entangled state. We predict that the transition occurs through a reversal of the OAM direction at a $k$ point in the inner band if the system has a proper spin-orbit coupling strength.

Figure 1

ARPES results from Cu(111) surface states. (a) Fermi surface and (b) the cut along the $k_x=0$ line [dashed line in panel (a)] taken with RCP light. (c),(d) The same for LCP light. (e),(f) $\mathrm{RCP}-\mathrm{LCP}$ data. (g) Normalized CD defined as $\mathrm{NCD}=(\mathrm{RCP}-\mathrm{LCP})/(\mathrm{RCP}+\mathrm{LCP})$ as a function of the azimuthal angle defined in the inset.

Figure 2

ARPES results from Au(111) surface states. (a) Fermi surface and (b) the cut along the $k_x=0$ line [dashed line in panel (a)] taken with RCP light. (c),(d) The same for LCP light. (e),(f) $\mathrm{RCP}-\mathrm{LCP}$ data. (g) NCD as a function of the azimuthal angle. Data from the two bands are summed in the estimation to have averaged NCD.

Figure 3

DFT results on the angular momentum structures of Au(111) surface states. DFT results for (a) inner (dashed arrows) and (b) outer bands (solid arrows). OAM (spin) of a state is represented by blue (red) arrows. Shown in the bottom panels are contributions from $p$ state (violet) and $d$ state (green) for (c) inner and (d) outer bands.

Figure 4

Spin and OAM structures calculated by using the effective Hamiltonian for (a),(d) small, (b),(e) intermediate, and (c),(f) large atomic SOC parameter $\alpha$. All the parameters are fixed except $\alpha$. We chose 0.195 Å${}^{-1}$ as the Fermi momentum $k_F$ of the outer band, similar to the case of Au(111). The red and blue arrows represent spin and OAM, respectively. Top (bottom) panels are plot spin and OAM configurations for the inner (outer) band. Dashed (solid) arrows are used for the inner (outer) band to be consistent with the notation in Fig. 3. Also marked are the approximate locations of Cu, Au, and Bi${}_2$Se${}_3$ cases on the $\alpha$ asis.