Microscopic mechanism for asymmetric charge distribution in Rashba-type surface states and the origin of the energy splitting scale

The microscopic mechanism for Rashba-type band splitting is examined in detail. We show how an asymmetric charge distribution is formed when the local orbital angular momentum (OAM) and crystal momentum get interlocked due to surface effects. An electrostatic energy term in the Hamiltonian appears when such an OAM- and crystal-momentum-dependent asymmetric charge distribution is placed in an electric field produced by inversion-symmetry breaking. Analysis by using an effective Hamiltonian shows that, as the atomic spin-orbit coupling (SOC) strength increases from weak to strong, the originally OAM-quenched states evolve into well-defined chiral OAM states and then to states of total angular momentum $J$. In addition, the energy scale of the band splitting changes from the atomic SOC energy to electrostatic energy. To confirm the validity of the model, we study OAM and spin structures of the Au(111) system by using an effective Hamiltonian for the $d$-orbital case. As for the strong-SOC regime, we choose Bi${}_2$Te${}_2$Se as a prototype system. We performed circular dichroism angle-resolved photoemission spectroscopy experiments as well as first-principles calculations. We find that the effective model can explain various aspects of the spin and OAM structures of the system.

Figure 1

Mechanism behind the formation of asymmetric charge distribution. Each atom has a finite OAM in the $z$ direction. The helical arrow on each atom represents the phase variation due to the OAM and the crystal momentum. $\theta$ is the angle between the $y$ axis and the point of interest. ${\xi }_n^A$ is the phase at the point $A$ of the orbital of the $n$th atom. Also plotted in color are electron densities for (a) zero and (b) nonzero crystal momenta. $\delta$ is an additional phase due to the crystal momentum.

Figure 2

Schematic of energy levels and angular momentum configurations of $p$ states with (a) crystal field only. Configurations when additional (b) weak ${\vec{E}}_S$, (c) strong ${\vec{E}}_S$, (d) strong ${\vec{E}}_S$+small SOC, and (e) strong ${\vec{E}}_S$+large SOC are applied. $\Delta$ represents the crystal-field energy between in- and out-of-plane orbitals, and $\alpha$ the atomic SOC strength. Red (blue) arrows represent spin (OAM). For simplicity, we assume that the energy scale of ${\widehat{H}}_{ES}$ is larger than $\Delta$. Dashed box represents Rashba-type band splitting in each case.

Figure 3

Expectation values of different terms in the Hamiltonian of the $p$-orbital case as a function of atomic SOC strength and crystal momentum. Plotted are contributions from (a) total energy, (b) atomic SOC energy, and (c) electrostatic energy.

Figure 4

(a) Au(111) surface state band structure obtained from the $d$-state effective Hamiltonian. Blue (red) symbols mark the OAM (spin) direction. (b) Contributions from various terms in the Hamiltonian as a function of momentum. Plotted in (c) and (d) are the spin and OAM textures, respectively, on a constant-energy surface.

Figure 5

ARPES results along the $\Gamma$-$M$ direction from Bi${}_2$Te${}_2$Se surface states using (a) RCP and (b) LCP light. (c) Circular dichroism (CD) data obtained as the difference between (a) and (b). (d) Fermi-surface CD data. (e) Cumulative CD data from the Dirac point to the Fermi level.

Figure 6

DFT calculation result from 4 QLs of Bi${}_2$Te${}_2$Se. (a) Band structure along the $K$-$\Gamma$-$M$ direction. In-plane components of (b) spin and (c) OAM from surface states at the Fermi energy. $\Gamma$-$K$ direction is the $k_x$ direction.