Origin of the isostructural electronic states of the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$

The novel physics, such as the pressure-induced electronic topological transition (ETT), topological superconductivity and Majorana fermions in the isostructural $R\text{−}3m$ phase of three-dimensional topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_3$, holds considerable interest in condensed-matter physics. We carried out a combined investigation of single-crystal x-ray diffraction, high-quality x-ray absorption fine structure, and first-principles theoretical calculations to decipher the puzzling origin of the intriguing electronic states in the isostructural $R\text{−}3m$ phase of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ at high pressure. Three distinct regions with two isostructural phase transitions (IPTs) in the $R\text{−}3m$ phase have been identified. The first IPT, which is known as the ETT, occurs at the boundary of region I (0–2 GPa) and region II (2–5 GPa) with a sharp minimum in the $c/a$ ratio of $R\text{−}3m$ structure, while the second IPT happens as pressure increases from region II (2–5 GPa) to region III (5–7 GPa). The positions of the Bi ($6c$) and Se ($6c$) sites in the unit cell change rapidly in region II (2–5 GPa), but there is little change at these sites in region III (5–7 GPa). The band-gap closure in region I reflects the pressure-induced metallization. At higher pressures, the band gap opens in region II but remains almost constant after the second IPT in region III, which agrees well with the topological superconductivity of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. Our results demonstrate that the combination of local structure, long-range crystal structure, and first-principles calculation is critically important for understanding the isostructural electronic states and the connection between the structure and function as in ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ at high pressure.

Figure 1

(a) The selected angle-dispersive single-crystal XRD patterns of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ under various pressures at room temperature from 1.0 to 14.1 GPa. The vertical bars are the expected diffraction lines of $R\text{−}3m$ ($\alpha -{\mathrm{Bi}}_2{\mathrm{Te}}_3$), $C2/m$ ($\beta -{\mathrm{Bi}}_2{\mathrm{Te}}_3$), and $C2/c$ ($\gamma -{\mathrm{Bi}}_2{\mathrm{Te}}_3$) phases at 1.0, 12, and 14 GPa, respectively. (b)–(f) The combined 2D XRD images at various pressures, which were collected with 1° step over the 33° cell open at the 13-BM-C beamline, APS.

Figure 2

(a) The representative LeBail refinement using jana2006 software [29] and (b) Rietveld refinement using gsas [30] with spherical harmonics in the preferred orientation model for the single-crystal x-ray data of ${\mathrm{Bi}}_2{\mathrm{Te}}_3 R\text{−}3m$ phase at 1 GPa.

Figure 3

(a) Lattice parameters of the $R\text{−}3m$ phase of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ at high pressure by LeBail refinement (black symbols) and gsas Rietveld refinement (red symbols), respectively. (b) The corresponding $c/a$ ratio together with the published data [22, 24] for comparison. The vertical dashed line denotes the boundary between I ($c$-axis dominant compression) and II ($a$-axis dominant compression) of $R\text{−}3m$ phase. Linear fit (red) highlights the steadiest range of the $c/a$ ratio at 3.2–5.4 GPa.

Figure 4

Representative Bi L3-edge XAFS spectra of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ as a function of pressure using NPD anvils. (a) Normalized x-ray absorption $\mu$($E$) free of DAC-imposed glitches. (b) Corresponding $k2$-weighted XAFS spectra, $k^2\chi \left(k\right)$. (c) XAFS Fourier transform, $\left|\chi \left(R\right)\right|$, obtained in the range from 2.5 to $12.4{\AA }^{-1}$ with a Hanning window. Note the significant changes (red arrows) at 7, 12, and 14 GPa as pressure increases.

Figure 5

(a) First-shell modeling of Bi L3-edge XAFS data, $\left|\chi \left(R\right)\right|$, at 1.5 GPa, as an example. The quintuple layer contains five atomic planes with Te2-Bi-Te1-Bi-Te2 stacking along the $c$ axis. (b) Pressure dependence of the Bi-Te1 and Bi-Te2 bonds of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ QL with respect to the values at the lowest pressure (0.29 GPa). (c) Pressure dependence of DW factor, ${\sigma }^2$. The $R\text{−}3m$ phase can be divided into region I (0–2 GPa), region II (2–6 GPa), and region III (6–9 GPa).

Figure 6

(a) Representative structural modeling of Bi L3-edge XAFS data, $\left|\chi \left(R\right)\right|$, at 3.5 GPa within 4.7 Å; (b) Pressure dependence of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ QL bonds: Bi-Te2, Bi-Te1, Bi-Bi1 (in Bi plane), and Bi-Bi2 (inter-Bi plane) bonds; (c) and (d) Deviations of the Bi-Te and Bi-Bi bonds from the ambient structure, respectively.

Figure 7

Pressure dependence of Bi-Te2 and the Bi-Te1 bond lengths obtained by XAFS experiment (black symbols) in comparison with the results of Rietveld refinements of SC-XRD experiment using gsas (blue symbols) and jana2006 (red symbols).

Figure 8

Left panel: High-pressure behavior of (a) Bi ($6c$) site and (b) Te2 ($6c$) site of $R\text{−}3m$ phase from the XAFS/SCXRD optimized structure (black symbols) in comparison with the structural parameters of Rietveld refinements by jana2006 (red symbols) and gsas (blue symbols). Right panel: The expanded plots of the XAFS/SCXRD optimized Bi ($6c$) site (c) and Te2 ($6c$) site ($d$) in $R\text{−}3m$ structure, respectively. Red lines are the linear fits in the pressure range of 2–5 GPa to highlight the rapid changing range.

Figure 9

(a) Calculated band structure of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ including SOC. Notation for the high-symmetric points in the Brillouin zone is the same as in Ref. [2]; (b) Expanded region at Γ point. The valence band is slightly asymmetric at $\mathrm{\Gamma }$ point. The average direct band gap at $\mathrm{\Gamma }$ point is calculated by $E_{\mathrm{\Gamma }}=\left({E_1}^{\mathrm{\Gamma }}+{E_2}^{\mathrm{\Gamma }}\right)$.

Figure 10

Calculated band gap of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ as a function of pressure: (a) direct band gap at $\mathrm{\Gamma }$ point and (b) the smallest band gap together with the values of previous studies of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ by Hong et al. [19], Segura et al. [17], and Bera et al. [14] for comparison.

Figure 11

The calculated band gap of ${\mathrm{Bi}}_2{\mathrm{Te}}_3 R\text{−}3m$ phase including SOC. Values of unoptimized structure and experimental data of infrared spectroscopy [50], optical experiments [51], and resistivity measurements [15] are shown for comparison. Right axis shows the superconducting transition temperature of the $R\text{−}3m$ phase from Ref. [10]. Vertical lines mark the boundaries of the three regions in the $R\text{−}3m$ phase of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$.