Laser angle-resolved photoemission as a probe of initial state $k_z$ dispersion, final-state band gaps, and spin texture of Dirac states in the ${\mathbf{Bi}}_2{\mathbf{Te}}_3$ topological insulator

We have obtained angle-resolved photoemission spectroscopy (ARPES) spectra from single crystals of the topological insulator material ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ using a tunable laser spectrometer. The spectra were collected for 11 different photon energies ranging from 5.57 to 6.70 eV for incident light polarized linearly along two different in-plane directions. Parallel first-principles, fully relativistic computations of photointensities were carried out using the experimental geometry within the framework of the one-step model of photoemission. A reasonable overall accord between theory and experiment is used to gain insight into how properties of the initial- and final-state band structures as well as those of the topological surface states and their spin textures are reflected in the laser-ARPES spectra. Our analysis reveals that laser-ARPES is sensitive to both the initial-state $k_z$ dispersion and the presence of delicate gaps in the final-state electronic spectrum.

Figure 1

Top row: Experimental laser-ARPES spectra over the photon energy range of 5.57–6.70 eV from a ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ single crystal. Spectra are shown as a function of the initial-state binding energy and momentum taken along a direction which is misaligned by $5^{\circ }$ with respect to the $\overline{\mathrm{\Gamma }}\overline{M}$ direction. The incoming light is propagating along $\overline{\mathrm{\Gamma }}\overline{K}$, and it is $p$ polarized with a polar angle of ${50}^{\circ }$. All spectra are normalized to the maximum in each panel as shown. Bottom row: Corresponding theoretical intensities computed within the one-step model, normalized in each panel as shown; experimental conditions of light polarization and misalignment of the sample have been taken into account in the computations.

Figure 2

(a) Theoretical energy distribution curves (EDCs) for photon energies $\left(h\nu \right)$ varying from 5.5 to $12.9\mathrm{eV}$, giving photointensity as a function of the initial-state binding energy, with $k_{\parallel }$ held fixed at the Fermi momentum $k_F$ along the $\overline{\mathrm{\Gamma }}\overline{M}$ direction. The initial and final states associated with the peaks in the EDCs are clarified in panels (b)–(d) by considering the example of the peak marked by a red cross on the EDC for $h\nu =7.7\mathrm{eV}$. (b) $k_z$ dispersion of the initial (bulk) state associated with the spectral peaks in the EDC for $k_{\parallel }=k_F$. $k_z$ value corresponding to the peak marked with a cross is identified. (c) $k_z$ dispersion of the final (bulk) states. Note that $k_z$ is given on the horizontal scale, while the vertical scale gives the final-state energy. All final states accessed through the EDC peaks at various photon energies in (a) are marked by red crosses. Arrows on the vertical axis mark final-state band gaps. (d) The initial-state $k_z$ dispersion of (b) is replotted by interchanging the horizontal and vertical scales for ease of identifying the final states corresponding to EDC peaks in (a). Vertical line shows how the initial state related to the specific EDC peak marked by the red cross connects with the final state. (e) Spectral feature at zero binding energy due to the Dirac cone in (a) is shown on an enlarged scale in order to highlight details of intensity variations with photon energy.

Figure 3

Same as the caption to Fig. 1, except that the experimental (top row) and computational (bottom row) spectra are taken along the $\overline{\mathrm{\Gamma }}\overline{K}$ direction; properties of the incoming light are also similar, except that the propagation direction is along $\overline{\mathrm{\Gamma }}\overline{M}$.

Figure 4

Same as the caption to Fig. 1, except that instead of the energy vs momentum cuts of Fig. 1, here spectral intensities for photoemission from the Fermi energy are shown in the $\overline{\mathrm{\Gamma }}\overline{K}-\overline{\mathrm{\Gamma }}\overline{M}$ momentum plane for various photon energies. Experimental spectra (top row) have been rotated anticlockwise by $5^{\circ }$ to account for the misalignment of the experimental spectra with respect to $\overline{\mathrm{\Gamma }}\overline{K}$. Fermi energy in the computations has been chosen to obtain a reasonable accord between the sizes of the experimental and theoretical Fermi surfaces associated with the Dirac cone states.

Figure 5

(a)–(f) Theoretical photointensities for emission from the Fermi energy in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ for light incident normal to the surface $\left(h\nu =9.6\mathrm{eV}\right)$ with various directions of the in-plane polarization vector, shown by the double arrows; the out-of-plane component of the polarization vector is zero. The $k$ points marked with red and blue crosses in (a) and (d) are discussed in the text. (g) Top view of the crystal surface with Te atoms (filled circles) in the topmost layer. Bi atoms (stars) lie in the second layer, followed by another layer of Te atoms (crosses). (h) Reciprocal lattice points are shown along with the first Brillouin zone and a few high-symmetry points. (i) Same as (a)–(f), except that the light here is incident horizontally, so that the polarization vector lies normal to the surface with no in-plane component. (j)–(l) Spin-up component of the intensity for electron spin polarized along the $x,y$, and $z$ directions, respectively. Incident light polarization is the same as in (a). (m)–(o) Same as (j)–(l), except these panels give the corresponding spin-down intensities.