Spin mapping of surface and bulk Rashba states in ferroelectric $\alpha$-GeTe(111) films

The breaking of bulk inversion symmetry in ferroelectric semiconductors causes a Rashba-type spin splitting of electronic bulk bands. This is shown by a comprehensive mapping of the spin polarization of the electronic bands in ferroelectric $\alpha$-GeTe(111) films using a time-of-flight momentum microscope equipped with an imaging spin filter that enables a simultaneous measurement of more than 10 000 data points. The experiment reveals an opposite spin helicity of the inner and outer Rashba bands with a different spin polarization in agreement with theoretical predictions, confirming a complex spin texture of bulk Rashba states. The outer band has about twice larger spin polarization than the inner one, giving evidence of a spin-orbit effect being related to the orbital composition of the band states. The switchable inner electric field of GeTe implies new functionalities for spintronic devices.

Figure 1

(a) Bulk and surface Brillouin zone of GeTe with marked high symmetry points and the projection of the hexagonal face of the Brillouin zone encircled in green. (b) Constant energy slice $I(E_F,k_x,k_y)$ of the photoelectron intensity distribution for a photon energy of $h\nu =18.5$ eV covering the complete bulk Brillouin zone projection. (c) Constant energy slice $I(E_b,k_x,k_y)$ with $E_b=0.4$ eV for $h\nu =22$ eV with larger magnification. The two surface bands $\left(S_{1,2}\right)$ and the bulk band $B_1$ are marked. Full lines at constant $k_x$ indicate positions of cuts shown in (d)–(f). (d)–(f) Photoelectron intensity $I(0.4\mathrm{eV},k_x,k_y,k_z)$ for fixed values $k_x=0$ and $k_x=\pm 0.2\AA {}^{-1}$. $k_z$ is measured relative to the $Z$ point in $k$ space, which is probed for $h\nu =19.5$ eV. Dash-dotted lines indicate bulk bands.

Figure 2

(a)–(g) Energy isosurfaces of the photoelectron spin-polarization component $P_x(E_b,k_x,k_y)$ combined with the corresponding intensity distribution measured without spin resolution for $h\nu =22$ eV. Each slice averages over an energy interval of 43 meV. The corresponding color code for $P_x$ and $I$ is indicated in (h). Arrows in (a) and (f) mark the deduced spin direction assuming a negligible perpendicular spin component.

Figure 3

(a), (b) Constant momentum cut $P_x(E_b,k_x,k_y)$ for $k_x=0$ and $k_x=+0.1\AA {}^{-1}$ combined with the corresponding intensity distribution. For the color scale see Fig. 2. The cuts are averaged over a momentum interval of $\mathrm{\Delta }{k}_x=0.05\AA {}^{-1}$. Red (dashed) and blue (dotted) lines are guides to the eye; arrows and the dotted rectangle denote hybridization regions of bulk (BR) and surface states (SR). (c), (d) Calculations of the photoemission spin polarization and intensity shown in a similar representation as in (a) and (b): constant momentum cut $P_x(E_b,k_x,k_y)$ for $k_x=0$ (c) and $k_x=+0.1\AA {}^{-1}$ (d). Red (dashed) and blue (dotted) lines are guides to the eye; arrows and dotted rectangle (d) denote the energy region of the surface resonance. (e) Intensity line profile $I(E_b,k_x,k_y)$ for $k_x=0.11\AA {}^{-1}$ and $E_b=0.565$ eV along the line indicated in (b). (f) Corresponding line profile of the spin polarization $P_x$. (g) Resulting intensity for spin-up and spin-down electrons. Arrows in (f) and (g) mark the extrema positions.