Orbital Angular Momentum Induced Spin Polarization of 2D Metallic Bands

The electrons in 2D systems with broken inversion symmetry are spin-polarized due to spin-orbit coupling and provide perfect targets for observing exotic spin-related fundamental phenomena. We observe a Fermi surface with a novel spin texture in the 2D metallic system formed by indium double layers on Si(111) and find that the primary origin of the spin-polarized electronic states of this system is the orbital angular momentum and not the so-called Rashba effect. The present results deepen the understanding of the physics arising from spin-orbit coupling in atomic-layered materials with consequences for spintronic devices and the physics of the superconducting state.

Figure 1

(a) Top view and (b) side view of ($\sqrt{7}\times \sqrt{3}$)-In. The red solid line in (a) represents the unit cell. Green circles in (a) and (b) represent Si atoms, light gray circles represent first layer In atoms, and dark gray circles represent In of the second layer. (c) The FS of ($\sqrt{7}\times \sqrt{3}$)-In. The theoretical calculation (green, red, and purple solid lines) is overlaid by the experimental data (blue). The Brillouin zone is indicated by black broken lines. $\overline{\mathrm{\Gamma }}-\overline{Y}$ corresponds to the [$11\overline{2}$] direction and $\overline{X}-\overline{\mathrm{\Gamma }}$ to the [$1\overline{1}0$] direction.

Figure 2

(a) Spin-resolved energy distribution curves measured at point $A$ in the FS shown in (f). (b)–(e) Spin-resolved momentum distribution curves measured at the cuts $B$, $C$, $D$, and $E$ points. The $x$, $y$, and $z$ axes correspond to the $[1\overline{1}0]$, $[11\overline{2}]$, and $[111]$ directions of the substrate, respectively. Red data indicate spins in the positive direction of each axis, and blue data are spins in the negative direction of each axis. The red and blue bars indicate the peak position in each spectrum and the gray dashed lines in (b)–(e) are the $k_x$ value of the FS obtained by SARPES. Error bars are smaller than the symbol sizes and not shown. The directions of the spin estimated from the results in (a)–(e) are schematically represented in the experimental FS in (f).

Figure 3

(a) Theoretically obtained spin polarization (arrows) of the FS (inner side only). (b) Theoretical OAM distribution (arrows) in absence of SOC. (c)–(k) Real space electron probability isosurfaces ($0.008{\AA }^{-3}$) corresponding to selected $k$ points on the FS. Isosurfaces (c)–(e), (f)–(h), and (i)–(k) correspond, respectively, to the $k$ points marked by a circle, square, and triangle in (b). (d) [(g), (j)] is the cross sectional view of $A-A^{\prime }$ ($C-C^{\prime }$, $E-E^{\prime }$) seen from direction of the arrow marked with the same labeling in (c), and (e) [(h), (k)] is that of $B-B^{\prime }$ ($D-D^{\prime }$, $F-F^{\prime }$). In (d), (e), (g), (h), (j), and (k), the blue areas are the cross sections at the plane and the yellow ones are the isosurfaces behind the planes.

Figure 4

Band dispersions (ARPES intensity map) of (a) the butterfly FS at $k_y=-0.15{\AA }^{-1}$ close to the point marked by a square in Fig. 3, and 3 the circular FS at $k_y=-0.1{\AA }^{-1}$.