Exploring the nontrivial band edge in the bulk of the topological insulators ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ and ${\mathrm{Bi}}_{2}{\mathrm{Te}}_{3}$

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Exploring the nontrivial band edge in the bulk of the topological insulators ${Bi}_{2} {Se}_{3}$ and ${Bi}_{2} {Te}_{3}$

Robin Guehne and Vojtěch Chlan

Phys. Rev. Research 6, 013214 – Published 28 February 2024

Abstract

${Bi}_{2} {Se}_{3}$ and related compounds are prototype three-dimensional topological insulators with a single Dirac cone in the surface band structure. While the topological surface states can be characterized with surface-sensitive methods, the underlying bulk energy band inversion has not been investigated in detail. Here, a study is presented that combines density-functional theory and nuclear magnetic resonance to explore the nontrivial band edge of ${Bi}_{2} {Se}_{3}$ and ${Bi}_{2} {Te}_{3}$ . It is found that the topological band inversion is not a discrete reversal of the order of the valence and conduction band at the $Γ$ point. Rather, the bands closest to the Fermi level become well mixed and spread evenly below and above the band gap, such that the characters of the valence- and conduction-band edges become indistinguishable. Beside those bands relevant for the band inversion, i.e., Bi and Se $p_{z}$ , also Bi $p_{x}$ and $p_{y}$ states are involved. As a part of this mixture of states, the band inversion shows no edges in $k$ space.

Received 31 August 2023
Accepted 6 February 2024

DOI:https://doi.org/10.1103/PhysRevResearch.6.013214

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Charge Doping effects Electronic structure First-principles calculations Topological materials

Topological insulators

Density functional theory Nuclear magnetic resonance

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Robin Guehne ^1,2,* and Vojtěch Chlan ³

¹Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany
²Felix Bloch Institute for Solid State Physics, Leipzig University, Linnéstraße 5, 04103 Leipzig, Germany
³Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 180 00, Prague 8, Czech Republic

^*robin.guehne@cpfs.mpg.de

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Vol. 6, Iss. 1 — February - April 2024

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Figure 1
The two cases of ${Bi}_{2} {Se}_{3}$ without (a) and with band inversion (b) induced by SOC are confronted. In the center the band structures are shown with schematic closeups of the Fermi edges to the respective left and right. The character of the bands involved in the band inversion, i.e., Bi (green) and ${Se}_{out}$ (orange), defines the location of Fermi-level states (red) in the crystal structure. The calculations reveal, in agreement with NMR experiments, that the band inversion is embedded in a mixture of states below and above the Fermi level. In particular, the relevant conduction band states (Bi $p_{z}$ ) are partially moved to the valence band (and vice versa for ${Se}_{out} p_{z}$ ) as illustrated in the closeup Fermi edge in panel (b).
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Figure 2
DOS without (grey) and with (red) SOC, as well as their difference (green) from $- 25$ to $5 eV$ . Below, the corresponding EFGs for a given bunch of bands are shown. The small EFG arising from the top valence band [orange in Fig. 1] is highlighted for the bunch of valence bands closest to the Fermi level (between $- 5$ and $0 eV$ ). The narrow blue slice covers the first $150 meV$ of the conduction band, representing occupied states due to the Fermi-level shift as a consequence of doping $0.005$ extra electrons per formula unit (concentration of free carriers of $\sim 4 \times 10^{19} {cm}^{- 3}$ ).
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Figure 3
Panels (a) and (d) depict the $p$ -state occupancy for the two cases NoSOC and SOC upon doping ${Bi}_{2} {Se}_{3}$ and ${Bi}_{2} {Te}_{3}$ , respectively, represented in extra charge per formula unit (extra charge of $- 1 (+ 1)$ then means doping 1 hole (electron) per formula unit). The corresponding EFGs as a function of doping are shown in panels (b) and (e). Solid squares (diamonds) represent the total $V_{ZZ}$ , including all valence and conduction bands, while open circles denote the $V_{ZZ}$ as stemming from the lowermost conduction band only (zero for hole doping). Dashed lines denote Bi $p$ -states population anisotropy, which connects state occupation and EFG. In the panels (c) and (f) close ups of the band edges of ${Bi}_{2} {Se}_{3}$ and ${Bi}_{2} {Te}_{3}$ with SOC are shown, including the position of the Fermi level (pink solid lines) for the doping cases of $\pm 0.005 e^{-} / FU$ . Note the different paths in the Brillouin zone, i.e., along the $F \to Γ \to L$ ( ${Bi}_{2} {Se}_{3}$ ) and the $Γ \to Z \to F$ ( ${Bi}_{2} {Te}_{3}$ ) direction (see text for details).
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Figure 4
(a) ${}^{209}{Bi}$ quadrupole spectra at 11.74 T room temperature with the quadrupole splitting proportional to $V_{ZZ}$ . With increasing carrier concentration, the splitting reduces measurably. In panel (b), the corresponding shift of the Fermi level into the conduction band is illustrated. (c) Calculated $V_{ZZ}$ for ${Bi}_{2} {Se}_{3}$ and ${Bi}_{2} {Te}_{3}$ without (blue) and with (red) SOC (band inversion). The experimental data points (triangles) agree well with the calculations when SOC is enabled and thus, band inversion emerges.
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Physical Review Research

Exploring the nontrivial band edge in the bulk of the topological insulators Bi2Se3 and Bi2Te3

Robin Guehne and Vojtěch Chlan

Phys. Rev. Research 6, 013214 – Published 28 February 2024

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Exploring the nontrivial band edge in the bulk of the topological insulators ${Bi}_{2} {Se}_{3}$ and ${Bi}_{2} {Te}_{3}$