Phonon signatures of multiple topological quantum phase transitions in compressed $\mathrm{TlBi}{\mathrm{S}}_2$: A combined experimental and theoretical study

We report the hydrostatic pressure induced two topological phase transitions in strong spin-orbit coupled material $\mathrm{TlBi}{\mathrm{S}}_2$ at room temperature. Frequencies of the ${\mathrm{A}}_{1\mathrm{g}}$ and ${\mathrm{E}}_{\mathrm{g}}$ phonons are observed to increase monotonically up to $\sim 4.0\mathrm{GPa}$, but with a clear slope change in ${\mathrm{A}}_{1\mathrm{g}}$ mode at $\sim 1.8\mathrm{GPa}$. Interestingly, there are two clear anomalies noticed in phonon linewidths of ${\mathrm{E}}_{\mathrm{g}}$ mode at pressures $\sim 0.5$ and $\sim 1.8\mathrm{GPa}$. Such anomalies are evidence of isostructural electronic transitions associated with unusual electron-phonon coupling. The high-pressure synchrotron powder diffraction and Raman show a first-order phase transition above 4 GPa. First-principles density functional theory-based calculations of electronic band structure, topological invariant ${\mathbb{Z}}_2$ and mirror Chern number $n_M$ reveal that the phonon anomalies at $\sim 0.5$ and $\sim 1.8\mathrm{GPa}$ are linked to the band inversions at $\mathrm{\Gamma }$ and $F$ points of the Brillouin zone respectively. The first band inversion at $\mathrm{\Gamma }$ point at $\sim 0.5$ GPa changes the ${\mathbb{Z}}_2$ from 0 to 1 leading to the transition of $\mathrm{TlBi}{\mathrm{S}}_2$ system into a topological insulator. The second band inversion at $F$ point at $\sim 1.8\mathrm{GPa}$ results in $n_M=2$, revealing a transition to a topological crystalline insulating state. Therefore the applied pressure systematically tunes the electronic states of $\mathrm{TlBi}{\mathrm{S}}_2$ from a normal semiconductor to a topological insulator and finally into a topological crystalline insulator at two distinct pressures of $\sim 0.5$ and $\sim 1.8\mathrm{GPa}$ respectively, before undergoing a structural phase transition at $\sim 4\mathrm{GPa}$.

Figure 1

(a) Unit cell of the hexagonal (supercell) phase of $\mathrm{TlBi}{\mathrm{S}}_2$ (blue, violet, and red spheres represents the Tl, Bi, and S atoms, respectively) and (b) Rietveld refinement of the XRD pattern for $\mathrm{TlBi}{\mathrm{S}}_2$ at ambient conditions $(\lambda =0.50070\AA )$.

Figure 2

(a) Experimental Raman spectrum of $\mathrm{TlBi}{\mathrm{S}}_2$ at ambient conditions, and (b) visualization of the atomic displacement patterns for the ${\mathrm{E}}_{\mathrm{g}}$ and ${\mathrm{A}}_{1\mathrm{g}}$ modes obtained from calculations. The blue, violet, and red spheres represents the Tl, Bi, and S atoms, respectively.

Figure 3

Pressure dependence of the synchrotron XRD patterns $(\lambda =0.50070\AA )$ of $\mathrm{TlBi}{\mathrm{S}}_2$ at selected pressure values. The red color asterisk symbol represents the appearance of new peaks.

Figure 4

Le Bail fit to the synchrotron XRD patterns $(\lambda =0.50070\AA )$ of $\mathrm{TlBi}{\mathrm{S}}_2$ at (a) 1.55 and (b) 3.92 GPa.

Figure 5

(a) Pressure dependence of the lattice parameters (a and c) of $\mathrm{TlBi}{\mathrm{S}}_2$. The inset shows the P vs $c/a$ ratio. (b). Pressure vs volume of $\mathrm{TlBi}{\mathrm{S}}_2$. The red and blue solid line represents the Murnaghan EOS fit and guide to the eye, respectively.

Figure 6

Experimental Raman spectrum of $\mathrm{TlBi}{\mathrm{S}}_2$ at different selected pressures up to $\sim 5.9\mathrm{GPa}$.

Figure 7

Pressure dependence of the frequency of (a) ${\mathrm{E}}_{\mathrm{g}}$ and (b) ${\mathrm{A}}_{1\mathrm{g}}$ modes of $\mathrm{TlBi}{\mathrm{S}}_2$. (c) $P$ vs FWHM of ${\mathrm{E}}_{\mathrm{g}}$ mode and (d). $P$ vs FWHM of ${\mathrm{A}}_{1\mathrm{g}}$ mode. In (a)–(d), black (Expt-1) and blue (Expt-2) filled circles indicate data from two independent pressure runs. The error bars are obtained from instrumental error for these experiments. Solid red lines in (a) and (b) are linear fit to the first pressure run (Expt-1). Solid green lines in (c) and (d) are only guides to the eyes. Red arrows in (b) and (c) represent the isostructural electronic transitions. Red dashed line represents the extrapolation of the linear fit.

Figure 8

(a)–(f) Electronic band structure of $\mathrm{TlBi}{\mathrm{S}}_2$ are calculated with SOC at different hydrostatic pressures (from −1 to 4 GPa). Band inversion takes place at the $\mathrm{\Gamma }$ and $F$ points in the BZ as a function of pressure near $P_c^{\mathrm{\Gamma }}=-0.4\mathrm{GPa}$ and $P_c^F=3.6\mathrm{GPa}$, respectively.

Figure 9

(a) Evolution of band gap with hydrostatic pressure at $\mathrm{\Gamma }$ and $F$ points showing the opening and closing of gaps at the critical pressures ($P_C^{\mathrm{\Gamma }}$ and $P_C^{\mathrm{F}}$). (b) Electronic density of states of $\mathrm{TlBi}{\mathrm{S}}_2$ calculated with SOC at different hydrostatic pressures.

Figure 10

Electronic structures of rhombohedral $\mathrm{TlBi}{\mathrm{S}}_2$ around the (a) $\mathrm{\Gamma }$ point and the (c) $F$ point showing the band inversion between the top valence and lowest conduction bands as a function of hydrostatic pressures. (b) Isosurfaces of charge densities associated with VBM and CBM before and after the band inversion at the $\mathrm{\Gamma }$ point and the (d) $F$ point reveal that the band inversion at $\mathrm{\Gamma }$ takes place in between −0.3 and −0.5 GPa (i.e., $-0.5<P_c^{\mathrm{\Gamma }}<-0.3\mathrm{GPa}$, hence $P_c^{\mathrm{\Gamma }}\sim -0.4\mathrm{GPa}$ at $\mathrm{\Gamma }$ point), whereas the band inversion at the $F$ point occurs in between 3.5 and 3.7 GPa (i.e., $3.5<P_c^F<3.7\mathrm{GPa}$, $P_c^F\sim 3.6\mathrm{GPa}$ at $F$ point).