Unraveling the spin structure of unoccupied states in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$

The optical control of spin currents in topological surface states opens new perspectives in (opto-) spintronics. To understand these processes, a profound knowledge about the dispersion and the spin polarization of both the occupied and the unoccupied electronic states is required. We present a joint experimental and theoretical study on the unoccupied electronic states of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. We discuss spin- and angle-resolved inverse-photoemission results in comparison with calculations for both the intrinsic band structure and, within the one-step model of (inverse) photoemission, the expected spectral intensities. This allows us to unravel the intrinsic spin texture of the unoccupied bands at the surface of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$.

Figure 1

(a) Experimental setup. (b) Low-energy-electron-diffraction (LEED) pattern taken at $E_{\mathrm{kin}}=30\mathrm{eV}$, showing threefold symmetry. (c) Auger electron spectroscopy (AES) signal of an exfoliated ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ sample measured with a retarding-field analyzer at a beam energy of $E_{\mathrm{Beam}}=3\mathrm{keV}$. (d) IPE spectra for $\mathrm{\Theta }$ = $0^{\circ }$ and $\mathrm{\Theta }$ = $4^{\circ }$ in the $\overline{\mathrm{\Gamma }}\overline{M}$ direction. Up to an energy of 3 eV above the Fermi edge, four structures $A–D$ are observed. For higher energies, two more features plus the image-potential state (IS) are present. (e) Time-dependent energy positions of spectral features $A–D$: Second derivative of spectra taken at an angle of incidence of $\mathrm{\Theta }$ = $4^{\circ }$ where features are more clearly separated. Points mark the average peak positions of 20 spectra. (f) $E\left({\mathit{k}}_{\parallel }\right)$ diagram: Experimental peak positions in the IPE spectra together with a $GW$ calculation for six quintuple layers (QLs).

Figure 2

Band-structure calculations for ${\mathrm{Bi}}_2{\mathrm{Se}}_3$: (a) $GW$ calculation for six QLs, (b) LDA-DFT calculations with pseudopotentials for six QLs, and (c) atomic-sphere approximation for the half-space. All calculations have been performed for the experimental lattice parameters of Ref. [53].

Figure 3

Spectral densities $n(E,{\mathit{k}}_{\parallel })$ of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ calculated with the LDA-ASA method (a) for the entire half-space, (b) for the topmost quintuple layer, and (c) for the surface region weighted with a probing depth of $\lambda =2$ atomic layers (ALs). (e)–(g) Corresponding spin polarization (in-plane perpendicular to ${\mathit{k}}_{\parallel }$) for (a)–(c). The highest spin-polarization values ($\approx 70\%$) are obtained for the TSS in the calculation for the half-space in agreement with literature values [63, 64]. (d) and (h) reproduce (c) and (d) with experimental peak positions taken from the IPE spectra added. The data points in (h) are colored where one spin channel exceeds the other.

Figure 4

(a) Spin- and angle-resolved IPE spectra along ${\overline{M}}^{\prime }\overline{\mathrm{\Gamma }}\overline{M}$ (left: spin integrated; middle and right: spin resolved). (b) Corresponding calculated spectra within the one-step model for a photon energy of $\hslash \omega =9.2\mathrm{eV}$ and a probing depth of $\lambda =2$ atomic layers. Secondary processes leading to an inelastic-background intensity are not included.

Figure 5

Calculated IPE intensities $I(E,{\mathit{k}}_{\parallel })$ of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ along ${\overline{M}}^{\prime }\overline{\mathrm{\Gamma }}\overline{M}$ for (a) a photon energy of $\hslash \omega =9.2\mathrm{eV}$ and a probing depth of $\lambda =2$ atomic layers and (b) corresponding spin differences. The experimental data points are superimposed. In (b), they are colored where one spin channel exceeds the other.