Phonon Self-Energy and Origin of Anomalous Neutron Scattering Spectra in SnTe and PbTe Thermoelectrics

The anharmonic lattice dynamics of rock-salt thermoelectric compounds SnTe and PbTe are investigated with inelastic neutron scattering (INS) and first-principles calculations. The experiments show that, surprisingly, although SnTe is closer to the ferroelectric instability, phonon spectra in PbTe exhibit a more anharmonic character. This behavior is reproduced in first-principles calculations of the temperature-dependent phonon self-energy. Our simulations reveal how the nesting of phonon dispersions induces prominent features in the self-energy, which account for the measured INS spectra and their temperature dependence. We establish that the phase space for three-phonon scattering processes, combined with the proximity to the lattice instability, is the mechanism determining the complex spectrum of the transverse-optic ferroelectric mode.

Figure 1

${\chi }^{\prime \prime }(\mathbf{Q},\mathrm{\Omega })$ measured with INS and computed with first-principles anharmonic lattice dynamics, showing anomaly of TO mode at the zone center, $\mathrm{\Gamma }$ (arrows). (a,b) ${\chi }^{\prime \prime }(\mathbf{Q},\mathrm{\Omega })$ of SnTe measured with CNCS, at $T=50$ and 300 K, for $\mathbf{Q}$ along [$H$, $H$, 3] (intensities are integrated over $\pm 0.1\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}$ along [0, 0, 1] and [1, $\overline{1}$, 0] directions, and plotted on a logarithmic-scale). (d,e) Same for PbTe. Note that some data are unavailable for PbTe. Part of the data in (e) were shown in Ref. [8]. White lines are renormalized phonon dispersions calculated at $T=300\mathrm{K}$. (c,f) Anharmonic ${\chi }^{\prime \prime }(\mathbf{Q},\mathrm{\Omega })$ along [$H$, $H$, 3], computed from the phonon self-energy (see text).

Figure 2

Evolution of the TO phonon spectrum at the $\mathrm{\Gamma }$ point. (a),(b) Phonon spectra of SnTe (a) and PbTe (b) at (1,1,1), measured with HB-3 at a series of temperatures. The purple dots are CNCS measurements at (1,1,3) with intensities integrated over $\pm 0.1\mathrm{r}\mathrm{.}\mathrm{l}\mathrm{.}\mathrm{u}\mathrm{.}$ along all $\mathbf{Q}$ directions. Data in (b) were shown in Ref. [8]. (c),(d) Phonon spectra (light red: SnTe, dark blue: PbTe) computed from the self-energy, showing the more pronounced split-peak structure in PbTe. Dashed black dash lines are spectra convoluted with instrumental resolution.

Figure 3

Relation between the phonon self-energy and the complex features in INS spectra. (a) Schematic of nesting between acoustic and optical branches in phonon dispersions of PbTe. (b) Imaginary part of the phonon self-energy, $\mathrm{\Gamma }$ (shaded area), and joint density of states (lines). Solid (dashed) lines are coalescence (decay) contributions to the joint–DOS. (c) Real part of the phonon self-energy, $\mathrm{\Delta }$. Light red lines are for SnTe and dark blue lines for PbTe. (d) Minima in $D(\mathrm{\Omega })$ [Eq. (3)] determine the position of peaks in INS spectra.