Observation of a Warped Helical Spin Texture in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ from Circular Dichroism Angle-Resolved Photoemission Spectroscopy

A differential coupling of topological surface states to left- versus right-circularly polarized light is the basis of many optospintronics applications of topological insulators. Here we report direct evidence of circular dichroism from the surface states of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ using laser-based time-of-flight angle-resolved photoemission spectroscopy. By employing a novel sample rotational analysis, we resolve unusual modulations in the circular dichroism photoemission pattern as a function of both energy and momentum, which perfectly mimic the predicted but hitherto unobserved three-dimensional warped spin texture of the surface states. By developing a microscopic theory of photoemission from topological surface states, we show that this correlation is a natural consequence of spin-orbit coupling. These results suggest that our technique may be a powerful probe of the spin texture of spin-orbit coupled materials in general.

Figure 1

(a) Schematic of a time-of-flight based angle-resolved photoemission spectrometer. PSD: position-sensitive detector. (b) A typical isointensity surface in ($E$, $k_x$, $k_y$) space from ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ collected simultaneously using linearly polarized photons. (c) Difference of TOF-ARPES data measured using right- and left-circularly polarized light. $h\nu$ denotes the incident photon direction. (d) $E\mathrm{\text{−}}k$ cut through the CD data volume in (c) along the angle denoted by the green dash in (e). (e)–(h) Constant energy slices through (c) at the Fermi level and at 0.1 eV above the DP (insets) for sample rotation angles of $\varphi =0°$, 30°, 60°, and 90°, respectively. Green hexagons represent the Brillouin zone of ${\mathrm{Bi}}_2{\mathrm{Se}}_3(111)$, red lines are the mirror planes and the arrow denotes the photon incident direction.

Figure 2

(a) $x$, (b) $y$, and (c) $z$ components of the spin texture over the complete surface states obtained by summing or subtracting $\Delta I$ data volumes at different $\varphi$ (see text). Color maps were created by integrating the data radially in $k$ space over a $\pm 0.015{\AA }^{-1}$ window about the surface state contours at each energy. Constant energy cuts at the Fermi level for each spin component are shown directly below in (d) through (f).

Figure 3

The evolution of the (a) $S_x$, (b) $S_y$ and (c) $S_z$ components around constant energy contours in the vicinity of the Dirac point, obtained by taking cuts through the data in Figs. 2b, 2c, 2d. Curves are fits to $\sin (\theta )$ and $\cos (\theta )$ varying functions for $S_x$ and $S_y$ respectively. (d) and (e) show the spin vectors (each normalized by its magnitude $|S|$) of all positive and negative energy states within $\pm 0.1\mathrm{eV}$, respectively, projected onto the $k_x\mathrm{\text{−}}{k}_y$ plane.

Figure 4

Evolution of (a) $S_z$ and (b) $S_x$ around the Fermi surface. Red line in (a) is a fit to $k·p$ theory. The blue and pink lines in (b) are fits to phenomenological models [21] that, respectively, do and do not account for an in-plane angular deviation ($\delta$) away from $\overrightarrow{S}\perp \overrightarrow{k}$ locking behavior (inset). (c) The energy dependence of the amplitude of $S_z$ modulation (green symbols), its fit (magenta line), and $\delta$ (red symbols). Inset shows the spin-integrated Fermi surface together with a fit to $k·p$ theory. (d) Spin-orientation around the hexagonally deformed Fermi surface constructed from CD-ARPES data. 3D view shows spins plotted around Fermi contour and top-down view shows spins projected onto a segment of the unit circle in momentum space for ease of visualizing $\delta$.