Electron-phonon coupling and superconductivity in the doped topological crystalline insulator ${\left({\mathrm{Pb}}_{0.5}{\mathrm{Sn}}_{0.5}\right)}_{1-x}{\mathrm{In}}_x\mathrm{Te}$

We present a neutron-scattering study of phonons in single crystals of ${\left({\mathrm{Pb}}_{0.5}{\mathrm{Sn}}_{0.5}\right)}_{1-x}{\mathrm{In}}_x\mathrm{Te}$ with $x=0$ (metallic, but nonsuperconducting) and $x=0.2$ (nonmetallic normal state, but superconducting). We map the phonon dispersions (more completely for $x=0$) and find general consistency with theoretical calculations, except for the transverse and longitudinal optical (LO) modes at the Brillouin-zone center. At low temperature, both modes are strongly damped but sit at a finite energy ($\approx 4\mathrm{meV}$ in both samples), shifting to higher energy at room temperature. These modes are soft due to a proximate structural instability driven by the sensitivity of Pb-Te and Sn-Te $p$-orbital hybridization to off-center displacements of the metal atoms. The impact of the soft optical modes on the low-energy acoustic modes is inferred from the low thermal conductivity, especially at low temperature. Given that the strongest electron-phonon coupling is predicted for the LO mode, which should be similar for both studied compositions, it is intriguing that only the In-doped crystal is superconducting. In addition, we observe elastic diffuse (Huang) scattering that is qualitatively explained by the difference in Pb-Te and Sn-Te bond lengths within the lattice of randomly distributed Pb and Sn sites. We also confirm the presence of anomalous diffuse low-energy atomic vibrations that we speculatively attribute to local fluctuations of individual Pb atoms between off-center sites.

Figure 1

Left: Phonon dispersions for ${\left({\mathrm{Pb}}_{0.5}{\mathrm{Sn}}_{0.5}\right)}_{1-x}{\mathrm{In}}_x\mathrm{Te}$ with $x=0.3$ using density functional perturbation theory and the virtual crystal approximation [12]. The area of symbols is proportional to the calculated electron-phonon scattering rate. Right: Curves showing the calculated phonon density of states (PDOS, blue), the Eliashberg function ${\alpha }^{2}F\left(E\right)$ for $E=\hslash \omega$ (green), and the electron-phonon coupling $\lambda \left(E\right)$, corresponding to a running integral of $2{\alpha }^{2}F\left(E\right)/E$ (red).

Figure 2

(a) Elastic neutron-scattering spectra for ${\mathrm{Pb}}_{0.5}{\mathrm{Sn}}_{0.5}\mathrm{Te}$ measured with an incident energy $E_{\mathrm{i}}=20\mathrm{meV}$ at $T=5\mathrm{K}$. Peaks enclosed by white circles and rectangles are from the secondary grains present in the sample. (b) Same data as in (a) but with the coordinate grid rotated by $\approx {33}^{\circ }$ to highlight the indexing of peaks from the secondary grain indicated by circles. The weak peak enclosed by a rectangle appears to be from a third grain.

Figure 3

Color contour plots of inelastic-scattering intensity for ${\mathrm{Pb}}_{0.5}{\mathrm{Sn}}_{0.5}\mathrm{Te}$, measured at $T=5\mathrm{K}$ with either $E_{\mathrm{i}}=20$ (a, g) or $27\mathrm{meV}$ (others), illustrating the phonon dispersions along three high-symmetry directions, $(0,0,\zeta ), (\xi ,\xi ,0)$, and $(\xi ,\xi ,\xi )$. Lines display the calculated phonon dispersions [12]. Black, magenta, white, and red lines correspond to transverse acoustic (TA), transverse optical (TO), longitudinal acoustic (LA), and longitudinal optical (LO) modes. Same color dotted and dashed lines represent two branches, that are otherwise degenerate. The Bragg point corresponding to $\mathrm{\Gamma }$ is listed in the upper right of each panel. Labels at the top of each column (TA, TO, etc.) highlight the dominant phonon branches in the corresponding panels.

Figure 4

Color contour maps of ${\chi }^{\prime \prime }(\mathbf{Q},E)$ for (a) the superconducting $x=0.2$ sample at 1.7 K [measured at CNCS with $E_i=12\mathrm{meV}$ near $\mathbf{Q}=(1,1,-3)$], (b) the $x=0$ sample at 5 K, and (c) the $x=0$ sample at 300 K [measured at HYSPEC with $E_i=27\mathrm{meV}$ near $(1,1,-5)$]. [The strong feature near $H=0.5$ in (b) and (c) is a TA mode from the $(2,2,-4)$ peak of the second grain.] Corresponding constant-Q cuts at $\mathbf{q}=0$ are plotted in (d), (e), and (f). Here, the data are averaged over $\pm 0.05$ r.l.u. in each in-plane direction of Q and $\pm 0.07$ r.l.u. out of plane, and a constant background, determined from the energy-gain side, has been subtracted. Blue lines show fits to a damped-harmonic-oscillator function; fitting parameters are given in Table 1. Error bars represent one standard deviation.

Figure 5

Temperature dependence of the thermal conductivity for ${\mathrm{Pb}}_{1-y}{\mathrm{Sn}}_y\mathrm{Te}$ with $y=0,0.2$, and 0.5. Orange circles, our measurement on $y=0.5$; purple circles, results for $y=0.2$ from Fig. 2 of [52]; gray diamonds, results for $y=0$ from Fig. 2 of [51]. Error bars represent one standard deviation.

Figure 6

Transverse acoustic phonons for the $x=0$ (top) and $x=0.2$ (bottom) samples. Results for $x=0$ were done at HYSPEC with $T=5\mathrm{K}$ and $E_i=20\mathrm{meV}$; $x=0.2$ results are from CNCS with $T=1.7\mathrm{K}$ and $E_i=12\mathrm{meV}$. (a, b, e, f) TA phonon dispersions in zones $(0,0,-4), (2,2,0), (0,0,-2)$, and $(2,2,0)$, respectively. (c, d, g, h) Constant-Q cuts at the corresponding zone boundary points. Green dashed lines are fits to a Gaussian line shape plus background. Error bars represent one standard deviation.

Figure 7

(a) Elastic intensity showing diffuse scattering near Bragg peaks measured on the $x=0$ sample at $5\mathrm{K}$. (b) Model calculation, as discussed in the text; red dots indicate Bragg peak positions.

Figure 8

Illustration of the cluster used to calculate the diffuse scattering due to bond-length variations. Magenta, Sn or Pb; blue, Te. Smaller spheres are weighted as a fraction of the large spheres.

Figure 9

PDOS for nonsuperconducting sample $x=0$. (a) Map of elastic scattering with boxes (labeled A, B, C) indicating regions used to evaluate the effective PDOS, as discussed in the text. (b) $S\left(E\right)$ obtained from an average over box B for $T=5\mathrm{K}$; constant background estimated from the energy-gain side has been subtracted. The green solid line is a Gaussian fit to the elastic peak. (c) Phonon density of states (PDOS) obtained from $S\left(E\right)$ (b) using Eq. (2), after subtracting the fit to the elastic scattering. (d)–(f) Temperature dependence of effective PDOS evaluated for boxes B, C, and A, respectively; blue (red) symbols, $T=5\mathrm{K}$ (300 K). Error bars represent one standard deviation.

Figure 10

Inelastic-scattering results on the superconducting $x=0.2$ sample ($T_c=3.8\mathrm{K}$) measured at SPINS. (a, b) Constant-Q scans with Gaussian fits (green dashed lines) to the elastic scattering. Black dotted lines are background obtained from $-E$ (energy gain) data points. (c, d) $S(\mathit{Q},E)$ vs $E$ for $(0,0,2.5)$ and $(1.7,1.7,0)$, respectively, for several temperatures (legend at top). (e, f) ${\chi }^{\prime \prime }(\mathbf{Q},E)$ vs $E$ for $(0,0,2.5)$ and $(1.7,1.7,0)$, respectively. The indicated $2\mathrm{\Delta }\left(0\right)$ (dotted maroon lines) is twice the superconducting gap at zero temperature estimated from $T_c$. Error bars represent one standard deviation.