Bulk Signatures of Pressure-Induced Band Inversion and Topological Phase Transitions in ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$

The characteristics of topological insulators are manifested in both their surface and bulk properties, but the latter remain to be explored. Here we report bulk signatures of pressure-induced band inversion and topological phase transitions in ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$ ($x=0.00$, 0.15, and 0.23). The results of infrared measurements as a function of pressure indicate the closing and the reopening of the band gap as well as a maximum in the free carrier spectral weight. The enhanced density of states near the band gap in the topological phase gives rise to a steep interband absorption edge. The change of density of states also yields a maximum in the pressure dependence of the Fermi level. Thus, our conclusive results provide a consistent picture of pressure-induced topological phase transitions and highlight the bulk origin of the novel properties in topological insulators.

Figure 1

(a,b) Electronic band structure of PbSe at various lattice parameter ratio $a/a_0$ along the $\mathrm{L}\text{−}\mathrm{\Gamma }$ and $\mathrm{L}\text{−}\mathrm{W}$ directions of the Brillouin zone. $a_0$ is the experimental lattice constant in the zero temperature limit and at ambient pressure [33]. The TPT occurs at $a/a_0\approx 1.0255$. The dashed lines indicate the Fermi level $E_F$ for a hole density of $N=10^{18}{\mathrm{cm}}^{-3}$. (c) The direct band gap $E_g$ at the $\mathrm{L}$ point, absolute value of the Fermi level $|E_F|$ relative to the top valence band, and $E_g+2|E_F|$ [roughly the energy threshold for direct interband transitions in the presence of free carriers, as indicated by the arrow in (a)] as a function of $a/a_0$. (d) Fermi velocity $v_F$ along the $\mathrm{L}\text{−}\mathrm{\Gamma }$ and $\mathrm{L}\text{−}\mathrm{W}$ directions as a function of $a/a_0$ for $N=10^{18}{\mathrm{cm}}^{-3}$.

Figure 2

(a–c) Pressure-dependent midinfrared reflectance of ${\mathrm{Pb}}_{1-x}{\mathrm{Sn}}_x\mathrm{Se}$ measured at the diamond-sample interface and at room temperature [34]. The dashed lines are guides to the eye for the shift of the plasma minimum. (d) Example fits (solid lines) to the experimental data of $x=0.23$ (dots). (e) Pressure dependence of ${\omega }_p$ extracted from the fit.

Figure 3

Pressure-dependent midinfrared absorbance of PbSe measured at (a) 298 K and (b) 70 K [34]. Data in the blank region between 200–300 meV are not shown because of unreliability caused by diamond absorption. (c) Real part of the interband optical conductivity ${\sigma }_1$ of intrinsic PbSe at various $a/a_0$ from first-principles calculations. The dashed line shows the result including holes with a density of ${10}^{18}{\mathrm{cm}}^{-3}$. (d) [inset to (b)] A diagram illustrating hybridization opens up a band gap (in the bands shown as solid lines) when the conduction band and valence band cross (dashed lines).