Impact of stoichiometry and disorder on the electronic structure of the ${\mathrm{PbBi}}_2{\mathrm{Te}}_{4-x}{\mathrm{Se}}_x$ topological insulator

Detailed comparative theoretical and experimental study of electronic properties and spin structure was carried out for a series of Pb-based quaternary compounds ${\mathrm{PbBi}}_2{\mathrm{Te}}_{4-x}{\mathrm{Se}}_x$. For all values of $x$, these compounds are theoretically predicted to be topological insulators, possessing at high Se content a remarkably large band gap and a Dirac point isolated from bulk states. Using spin- and angle-resolved photoemission spectroscopy, it was shown that the ${\mathrm{PbBi}}_2{\mathrm{Te}}_2{\mathrm{Se}}_2$ and ${\mathrm{PbBi}}_2{\mathrm{Te}}_{1.4}{\mathrm{Se}}_{2.6}$ compounds are characterized by well-defined spin-polarized topological surface state in the bulk gap. To define the probable distribution of atoms over the atomic sites for these samples, we performed ab initio calculations in ordered and disordered configurations of the unit cell. We found that theoretical calculations better reproduce photoemission data when Te atoms are placed in the outermost layers of the septuple layer block.

Figure 1

Crystal structure in rhombohedral (a) and hexagonal (b) representations of the ${\mathrm{PbBi}}_2{\mathrm{Te}}_{4-x}{\mathrm{Se}}_x$ compound. [(c), (d)] Crystal structure of the ordered ${\mathrm{PbBi}}_2{\mathrm{Te}}_2{\mathrm{Se}}_2$ compound in its two possible configurations.

Figure 2

(a) 3D Brillouin zone for rhombohedral elementary cell; [(b)–(f)] bulk electronic spectra of the disordered ${\mathrm{PbBi}}_2{\mathrm{Te}}_{4-x}{\mathrm{Se}}_x$ alloy at various $x$.

Figure 3

(a) Dependence of band-gap value on stoichiometry and (b) evolution of the bulk band maxima in ${\mathrm{PbBi}}_2{\mathrm{Te}}_{4-x}{\mathrm{Se}}_x$ alloy. The inset shows the high-symmetry points of the surface Brillouin zone.

Figure 4

[(a)–(e)] Surface electronic structure of ${\mathrm{PbBi}}_2{\mathrm{Te}}_{4-x}{\mathrm{Se}}_x$ along $\overline{\mathrm{K}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{M}}$ at various $x$. Circles represent weights of the states, localized in the outermost SL, multiplied by value of in-plane spin components (red and blue colors denote positive and negative values of spin, respectively). (f) Superimposed Dirac-cone states in the $\overline{\mathrm{\Gamma }}-\overline{\mathrm{M}}$ direction; the region under consideration is shown on panel (e).

Figure 5

Spatial charge density distribution of the Dirac state at $x=0$, 2, 4.

Figure 6

XRD patterns for ${\mathrm{PbBi}}_2{\mathrm{Te}}_2{\mathrm{Se}}_2$ ($x=2.0$) (a) and ${\mathrm{PbBi}}_2{\mathrm{Te}}_4$ ($x=0$) (b) from a powder of single crystals.

Figure 7

(Left) bulk electronic spectra and (right) surface electronic spectra of three configurations of ${\mathrm{PbBi}}_2{\mathrm{Te}}_2{\mathrm{Se}}_2$. Circles represent weights of the states, localized in the outermost SL, multiplied by value of in-plane spin components (red and blue colors denote positive and negative values of spin, respectively).

Figure 8

(a) ARPES spectra of ${\mathrm{PbBi}}_2{\mathrm{Te}}_2{\mathrm{Se}}_2$ (left column) and ${\mathrm{PbBi}}_2{\mathrm{Te}}_{1.4}{\mathrm{Se}}_{2.6}$ (right column) samples measured in the $\overline{\mathrm{K}}-\overline{\mathrm{\Gamma }}-\overline{\mathrm{K}}$ direction at various photon energies. (b) Full 3D mapping of the dispersion relations in $k$ space and (c) spin-resolved ARPES spectra taken for two emission angles, corresponding to the wave vectors of $-0.1{\AA }^{-1}$ and 0.1 ${\AA }^{-1}$, for ${\mathrm{PbBi}}_2{\mathrm{Te}}_2{\mathrm{Se}}_2$ measured at 25-eV photon energy.

Figure 9

ARPES measurement at 25-eV photon energy on ${\mathrm{PbBi}}_2{\mathrm{Te}}_2{\mathrm{Se}}_2$ with overlaid surface band structure obtained from DFT calculations (red lines).