Giant Anisotropy of Spin-Orbit Splitting at the Bismuth Surface

We investigate the bismuth (111) surface by means of time and angle resolved photoelectron spectroscopy. The parallel detection of the surface states below and above the Fermi level reveals a giant anisotropy of the spin-orbit spitting. These strong deviations from the Rashba-like coupling cannot be treated in $\mathbf{k}·\mathbf{p}$ perturbation theory. Instead, first principles calculations could accurately reproduce the experimental dispersion of the electronic states. Our analysis shows that the giant anisotropy of the spin-orbit splitting is due to a large out-of plane buckling of the spin and orbital texture.

Figure 1

Photoelectron intensity map integrated in an energy window of 20 meV around the Fermi level. The data have been acquired with circularly polarized photons centered at 18 eV. The red and blue line stand for the wave vectors cut of Figs. 3c, 3d, 4a, 4b, respectively. Bright colors stand for high photoelectron intensity.

Figure 2

[(a)–(b)]: Photoelectron intensity map acquired along the $\Gamma \mathrm{\text{−}}\mathrm{M}$ direction [panel (a)] and $\Gamma \mathrm{\text{−}}K$ direction [panel (b)] is compared to the calculated band structure. The data have been acquired with circularly polarized photons centered at 18 eV. The solid lines stand for the two surface states whereas the dashed line marks the upper boundary of the projected bulk states that are below $E_F$.

Figure 3

(a): Photoelectron intensity map as a function of pump-probe delay for $k_{\parallel }=0.35{\AA }^{-1}$ along the $\Gamma \mathrm{\text{−}}M$ direction. (b): Temporal evolution of the effective electronic temperature. (c)–(d): Photoelectron intensity map acquired along the $\Gamma \mathrm{\text{−}}M$ direction (red line in Fig. 1) just before (c) and 200 fs after (d) the arrival of the pump beam. Solid lines stand for the calculated dispersion of the surface states while no projected bulk bands exist in between the two dashed lines. The data of this figure have been generated by ultrafast pulses of linearly polarized photons centred at 6.3 eV. The electric field polarization was nearly parallel to the surface plane and formed an angle $\alpha =20°$ with respect to the $\Gamma \mathrm{\text{−}}M$ direction.

Figure 4

[(a)–(b)]: Photoelectron intensity map as a function of azimuthal angle for $k_{\parallel }=0.35{\AA }^{-1}$ (blue line in Fig. 1) and pump-probe delay of 200 fs. The probing photons are polarized either parallel [panel (a)] or orthogonal [panel (b)] to the $\Gamma \mathrm{\text{−}}M$ direction. The white lines stand for the calculated dispersion of the surface states. (c)–(d): In-plane component of the spin (${\sigma }_y$, red dashed line), out-of-plane component of the spin (${\sigma }_z$, solid blue line) and of the orbital momentum ($L_z$, open circles) for the lower band [panel (c)] and upper band [panel (d)] as a function of azimuthal angle. The orbital momentum ${\mathrm{L}}_z$ has been multiplied by $-3.5$ for better comparison with the out-of-plane spin.