Symmetry rules shaping spin-orbital textures in surface states

Strong spin-orbit coupling creates exotic electronic states such as Rashba and topological surface states, which hold promise for technologies involving the manipulation of spin. Only recently has the complexity of these surface states been appreciated: they are composed of several atomic orbitals with distinct spin textures in momentum space. A complete picture of the wave function must account for this orbital dependence of spin. We discover that symmetry constrains the way orbital and spin components of a state coevolve as a function of momentum, and from this, we determine the rules governing how the two degrees of freedom are interwoven. We directly observe this complexity in spin-resolved photoemission and ab initio calculations of the topological surface states of Sb(111), where the photoelectron spin direction near $\overline{\mathrm{\Gamma }}$ is found to have a strong and unusual dependence on photon polarization. This dependence unexpectedly breaks down at large $\left|k\right|$, where the surface states mix with other nearby surface states. However, along mirror planes, symmetry protects the distinct spin orientations of different orbitals. Our discovery broadens the understanding of surface states with strong spin-orbit coupling, demonstrates the conditions that allow for optical manipulation of photoelectron spin, and will be highly instructive for future spintronics applications.

Figure 1

The surface states of Sb(111), as measured with $p$-polarized (linear horizontal) light. (a) Spin-integrated ARPES map along the $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$ ($-k_x$) and $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$ ($+k_x$) directions. (b) Schematic of the surface states, color-coded to indicate spin polarizations measured previously [20]. Blue indicates $P_y>0$, as defined in Figs. 3 and 3, and red indicates $P_y<0$. (c) and (d) Representative spin-resolved energy distribution curves (EDCs) taken at momenta shown in (a). The spin polarization, defined as $P_y=(I_{\uparrow }-I_{\downarrow })/(I_{\uparrow }+I_{\downarrow })$, is shown below corresponding EDCs.

Figure 2

Spin-resolved measurements with $p$-polarized light. (a) Schematic of band structure along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$ with positions of measurements marked by dashed lines. (b) Spin-resolved EDCs at momenta marked in (a). Peaks are marked corresponding to bands in (a). (c) Spin polarization corresponding to EDCs in (b). (d)–(f) Same as (a)–(c) but along the $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$ direction.

Figure 3

Spin-resolved measurements with $s$-polarized light. (a) Experimental geometry for measurement along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$. The angle $\alpha$ of photon polarization can be rotated from $p$ polarized to $s$ polarized. (b) Spin-resolved EDCs at momenta marked in (a). Peaks are marked corresponding to bands in (a). (c) Spin polarization corresponding to EDCs in (b). (d)–(f) Same as (a)–(c) but along the $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$ direction. In (e), arrows mark the peaks that are not reversed relative to Fig. 2 with $p$-polarized light.

Figure 4

Spin-resolved maps of the $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$ and $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$ directions of Sb(111). (a) and (b) Spin-resolved maps of the $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$ direction, taken with (a) $p$-polarized and (b) $s$-polarized light. The two-dimensional color scale displays the total photoelectron intensity by relative darkness and the spin polarization by the balance of red and blue. (c) Spin polarization of the two surface bands as a function of photon polarization angle. The bands are labeled in (a), and spin polarizations were extracted at a fixed $\mathbf{k}$, as indicated by the small regions marked with arrows. (d) and (e) Similar to (a) and (b) but with sample azimuth rotated to cut along $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$. (f) Spin polarization along the left branch of the lower band, as measured with both $p$- and $s$-polarized light. The stretch of $\mathbf{k}$ space plotted here is indicated by dispersive lines in (d) and (e).

Figure 5

Calculated spin-orbital textures and simulated spin-ARPES plots. (a) Left: spin texture of $p_x$ orbitals in the lower surface band, with $\widehat{x}$ (horizontal dashed line) oriented along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$. These states can be photoemitted by $p$-polarized light. The black vectors are in-plane expectation values $\langle S_{x,y}\rangle$. Right: simulated spin-ARPES measurement along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$ using $p$-polarized light. Spin polarization of the $p_x$ component of the bands is shown by the color from blue to red, while the band energies are indicated by gray lines. The spin polarization of the lower band (with arrows) is associated with the spin texture in the left panel. (b) Left: spin texture of $p_y$ orbitals in the lower surface band, with $\widehat{x}$ oriented along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$. These states can be photoemitted by $s$-polarized light. Right: simulated spin-ARPES measurement along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$ using $s$-polarized light. (c) and (d) Same as (a) and (b), but now with $\widehat{x}$ oriented along $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$ and therefore the simulated measurements along $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$. Note that in the lower band, the orbital dependence of the spin texture ceases at large $\left|k\right|$ for $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$ but remains in $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$, consistent with experimental results.

Figure 6

Band structure evolution with spin-orbit coupling strength $\alpha$. (a) Calculated 90-bilayer Sb band structures with full SOC ($\alpha =1$). The darkly shaded area is the projection of bulk states onto the surface Brillouin zone. The violet color indicates the surface states that would be degenerate in the absence of SOC ($\alpha =0$). (b) Detailed band structures along two directions ($\overline{\text{M}}-\overline{\mathrm{\Gamma }}-\overline{\text{K}}$), with different SOC strengths $\alpha$. Along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}$, the two surface bands always couple to each other and stay within the gap; along $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$, the lower surface band couples to the upper surface band near $k=0$, then switches to a new surface band closer to the valence bulk continuum at larger $\left|k\right|$. (c) Projected $p$ orbital character at various SOC strengths along the lower topological surface band indicated by dashed lines in (b). A rapid change in the orbital character is seen along the $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$ direction. (d) Similar to (c), showing the two limits of full SOC ($\alpha =1$, solid lines) and no SOC ($\alpha =0$, dotted lines). Note that along $\overline{\mathrm{\Gamma }}-\overline{\text{K}}$, all orbitals have a finite projection even in the absence of SOC, whereas along $\overline{\mathrm{\Gamma }}-\overline{\text{M}}, p_y$ is finite only with SOC. The missing parts of the curves in the $0.06<k<0.12$ range (shaded area) shown in (c) and (d) in the small-$\alpha$ cases represent the fact that the band of interest disperses into the bulk continuum, as can be seen in (b).