Thermal conductivity in PbTe from first principles

We investigate the harmonic and anharmonic contributions to the phonon spectrum of lead telluride and perform a complete characterization of how thermal properties of PbTe evolve as temperature increases. We analyze the thermal resistivity's variation with temperature and clarify misconceptions about existing experimental literature. The resistivity initially increases sublinearly because of phase space effects and ultra strong anharmonic renormalizations of specific bands. This effect is the strongest factor in the favorable thermoelectric properties of PbTe, and it explains its limitations at higher $T$. This quantitative prediction opens the prospect of phonon phase space engineering to tailor the lifetimes of crucial heat carrying phonons by considering different structure or nanostructure geometries. We analyze the available scattering volume between TO and LA phonons as a function of temperature and correlate its changes to features in the thermal conductivity.

Figure 1

Phonon dispersion relation and Mode Grünesien parameters for $T=0$ K for a chosen path within the Brillouin zone.

Figure 2

The energy of the $\text{TO}\left(\mathrm{\Gamma }\right)$ mode as a function of temperature. The harmonic line has no temperature dependence at all, the quasiharmonic has temperature dependence through the volume, the quasiharmonic+$\mathrm{\Delta }$ is with the anharmonic shifts, which increases the disagreement. The experimental points are from Refs. [10] (circles) and [23] (triangles).

Figure 3

Cartoon of the potential felt by a given atom using the two different approaches to describe the materials thermal behavior. Left: QHA—the potential at a given volume is fixed for all $T$. Right: TDEP—the ions see a different average potential energy surface depending on the temperature and the present anharmonicity.

Figure 4

Mode Grünesien parameters as a function of temperature in the TDEP and quasiharmonic (QH) formalisms. The trends with increasing temperature are opposite, with the TDEP being the physically reasonable ones.

Figure 5

PbTe Neutron scattering cross section. Notice the double peaks close to $\mathrm{\Gamma }$ and the lifting of the optical branches above the acoustic captured within the TDEP method as compared to experiment. The quasiharmonic are included only for reference.

Figure 6

The energy and momentum conserving surfaces for scattering between the TO mode and LA and TA phonons. As temperature increases and the TO mode stiffens, the available scattering volume for this channel of decay disappears, which is the origin of the kink in thermal resistivity. The isosurface represents the nesting factor for scattering between bands 2, 3, and 4, while conserving energy and momentum, integrated over $q^{\prime }$ and $q^{\prime \prime }$.

Figure 7

Thermal resistivity of PbTe versus temperature. The red lines are QHA results with $T=0$ force constants and volumes corresponding to the experimental thermal expansion. The green lines are calculated using temperature-dependent force constants, with volumes following the experimental thermal expansion.

Figure 8

Electronic band structure average over a molecular dynamics trajectory.