Anharmonicity and atomic distribution of SnTe and PbTe thermoelectrics

The structure and lattice dynamics of rock-salt thermoelectric materials SnTe and PbTe are investigated with single-crystal and powder neutron diffraction, inelastic neutron scattering (INS), and first-principles simulations. Our first-principles calculations of the radial distribution function in both SnTe and PbTe show a clear asymmetry in the first nearest-neighbor (1NN) peak, which increases with temperature, in agreement with recent experimental reports. We show that this peak asymmetry for the 1NN Sn-Te or Pb-Te bond results from large-amplitude anharmonic vibrations (phonons). No atomic off centering is found in our simulations. In addition, the atomic mean-square displacements derived from our diffraction data reveal stiffer bonding at the anion site, in good agreement with the partial phonon densities of states from INS and first-principles calculations. These results provide clear evidence for large-amplitude anharmonic phonons associated with the resonant bonding leading to the ferroelectric instability.

Figure 1

Phonon density of states of SnTe and PbTe. Generalized phonon DOS of (a) SnTe and (b) PbTe at several temperatures, obtained from powder inelastic neutron scattering on ARCS (incident energy $E_i=30$ and 25 meV). The curve labeled quasiharmonic approximation (QHA) corresponds to data at 100 K, shifted according to the quasiharmonic approximation, with the measured volumes at 100 and 500 K and an average Grüneisen parameter $\gamma =2$ from Ref. [44]. Panels (c) and (d) show the total and partial phonon DOS obtained from first-principles calculations. The results after applying neutron weighing and instrument resolution are compared with the neutron DOS at 100 K.

Figure 2

Mean-square atomic displacements for (a) SnTe and (b) PbTe, measured with neutron diffraction (markers) and calculated (lines) from the theoretical partial phonon DOS curves of Fig. 1. Open color markers correspond to MSD values measured with neutron single-crystal diffraction (HB-3A and TOPAZ), whereas filled color markers are values from powder diffraction (POWGEN). Black markers correspond to values in the literature. Blue lines and markers correspond to Sn or Pb, and red lines correspond to Te. The vibration amplitudes of the cations are systematically larger than those of Te anions.

Figure 3

SnTe: slices of dynamical structure factor $S(Q,E)$ along the $[0,0,L]$ direction measured on CNCS at (a) 50 and (b) 300 K, compared with (c) first-principles simulations and (d) phonon dispersion. $S(Q,E)$ along the $[-2,-2,L]$ direction showing same phonon dispersion branches but stronger transverse acoustic phonons, owing to polarization factor $\left(\mathbf{Q}·{\epsilon }_{ds}\right)$ in Eq. (1), measured at (e) 50 and (f) 300 K, compared with (g) first-principles simulations and (h) phonon dispersion. $S(Q,E)$ along the $[H,H,4]$ direction measured on CNCS at (i) 50 and (j) 300 K, compared with (k) first-principles simulations and (l) phonon calculation. [Intensities are integrated over $\pm 0.1\mathrm{r}.\mathrm{l}.\mathrm{u}.$ (where r.l.u. represents reciprocal lattice unit) along the $[0,0,1]$ and $[1,\overline{1},0]$ directions and plotted on a logarithmic scale.]

Figure 4

Longitudinal acoustic phonon dispersion relation and linewidth near the zone center, measured near 002 on CTAX and near 004 on CNCS along the (a) $HH0$ and $00L$ directions at 300 K. The error bars indicate the measured phonon linewidth after instrument resolution correction. The dashed lines show the results from sound velocity measurements [49]. The measurements are resolution limited for CNCS, and no resolution is shown. The reslib fit for the TA mode at $q=[0.12,0.12,0]$ is shown in (b).

Figure 5

Radial distribution functions of (a) SnTe and (c) PbTe calculated from first-principles molecular dynamics and harmonic classical molecular dynamics (using effective harmonic force constants extracted from the former). Note that full anharmonic calculation results in asymmetric peaks, especially for the 1NN bond, whereas the harmonic model shows symmetric peaks. In addition, softening the harmonic force constants by 30% does not induce asymmetry, as shown by the results for SnTe in (b).

Figure 6

The probability distribution functions of the Sn/Pb atom in SnTe/PbTe, calculated from first-principles molecular dynamics at 300 K, show no sign of off centering. The origin is the average crystallographic position. The integration range along the axis perpendicular to the visualized plane is $\pm 0.1\AA$. The color scale is linear. For one-dimensional cuts, the integration ranges along the two integrated axes are both $\pm 0.1\AA$.