Ferroelectric phase transition and the lattice thermal conductivity of ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ alloys

We show how tuning the proximity to the soft optical mode phase transition via chemical composition affects the lattice thermal conductivity $\kappa$ of ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ alloys. Using first-principles virtual-crystal simulations, we find that the anharmonic contribution to $\kappa$ is minimized at the phase transition due to the maximized acoustic-optical anharmonic interaction. Mass disorder significantly lowers and flattens the dip in the anharmonic $\kappa$ over a wide composition range, thus shifting the $\kappa$ minimum away from the phase transition. The total $\kappa$ and its anharmonic contribution vary continuously between the rocksalt and rhombohedral phases as expected for the second-order phase transition. The actual phase and its strength of resonant bonding play a less prominent role in reducing the $\kappa$ of ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ alloys than the proximity to the phase transition and the atomic mass. Our results show that alloys with soft optical mode transitions are promising materials for achieving low thermal conductivity and possibly high thermoelectric efficiency.

Figure 1

Rocksalt and rhombohedral structures of PbTe and GeTe, respectively. The differences between the two structures are the position of Te atom along the trigonal [111] axis and the angle between the primitive lattice vectors. The rocksalt phase is a special case of the rhombohedral phase with the relative Te position of $(0.5,0.5,0.5)$ within the primitive cell, and the angle of ${60}^{\circ }$.

Figure 2

Frequency of the lowest transverse optical mode at the zone center, $\mathrm{TO}(\mathrm{\Gamma }$), as a function of ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ alloy composition.

Figure 3

(a) Phonon band structures at 0 K for GeTe (solid black line) and an alloy in the rhombohedral structure near the phase transition, ${\mathrm{Pb}}_{\text{0.49}}{\mathrm{Ge}}_{\text{0.51}}\mathrm{Te}$ (dashed red line). The frequencies of the soft transverse optical phonon modes at the zone center, $\mathrm{TO}(\mathrm{\Gamma }$), are highlighted in black circles for GeTe and red squares for ${\mathrm{Pb}}_{\text{0.49}}{\mathrm{Ge}}_{\text{0.51}}\mathrm{Te}$. (b) Phonon band structures at 0 K for PbTe (solid black line) and an alloy in the rocksalt structure near the phase transition, ${\mathrm{Pb}}_{\text{0.51}}{\mathrm{Ge}}_{\text{0.49}}\mathrm{Te}$ (dashed red line). $\mathrm{TO}(\mathrm{\Gamma }$) mode frequencies are highlighted in black circles for PbTe and red squares for ${\mathrm{Pb}}_{\text{0.51}}{\mathrm{Ge}}_{\text{0.49}}\mathrm{Te}$. (c) Phonon band structures at 0 K for alloy compositions in the rocksalt and rhombohedral phases near the phase transition: ${\mathrm{Pb}}_{\text{0.51}}{\mathrm{Ge}}_{\text{0.49}}\mathrm{Te}$ (solid black line) and ${\mathrm{Pb}}_{\text{0.49}}{\mathrm{Ge}}_{\text{0.51}}\mathrm{Te}$ (dashed red line), respectively.

Figure 4

(a) Anharmonic (three-phonon) phonon lifetimes at 450 K as a function of frequency for GeTe (black pluses) and a rhombohedral alloy near the phase transition, ${\mathrm{Pb}}_{\text{0.49}}{\mathrm{Ge}}_{\text{0.51}}\mathrm{Te}$ (red crosses). Inset: averaged anharmonic lifetimes vs frequency at 450 K for GeTe (solid black line) and ${\mathrm{Pb}}_{\text{0.49}}{\mathrm{Ge}}_{\text{0.51}}\mathrm{Te}$ (dashed red line). (b) Anharmonic lifetimes at 450 K as a function of frequency for PbTe (black pluses) and a rocksalt alloy near the phase transition, ${\mathrm{Pb}}_{\text{0.51}}{\mathrm{Ge}}_{\text{0.49}}\mathrm{Te}$ (red crosses). Inset: averaged lifetimes vs frequency at 450 K for GeTe (solid black line) and ${\mathrm{Pb}}_{\text{0.51}}{\mathrm{Ge}}_{\text{0.49}}\mathrm{Te}$ (dashed red line). (c) Anharmonic lifetimes at 450 K as a function of frequency for rocksalt ${\mathrm{Pb}}_{\text{0.51}}{\mathrm{Ge}}_{\text{0.49}}\mathrm{Te}$ (black pluses) and rhombohedral ${\mathrm{Pb}}_{\text{0.49}}{\mathrm{Ge}}_{\text{0.51}}\mathrm{Te}$ (red crosses). Inset: averaged lifetimes vs frequency at 450 K for ${\mathrm{Pb}}_{\text{0.51}}{\mathrm{Ge}}_{\text{0.49}}\mathrm{Te}$ (solid black line) and ${\mathrm{Pb}}_{\text{0.49}}{\mathrm{Ge}}_{\text{0.51}}\mathrm{Te}$ (dashed red line).

Figure 5

Lattice thermal conductivity of ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ as a function of the alloy composition at 450 K. The unfilled symbols show the thermal conductivity due to anharmonic (three-phonon) processes only, and the filled symbols show the thermal conductivity due to anharmonic processes and mass disorder. Black circles show the thermal conductivity in the rocksalt phase and its isotropic average in the rhombohedral phase. Red up and blue down triangles show the thermal conductivity perpendicular and parallel to the trigonal [111] axis in the rhombohedral phase, respectively.

Figure 6

Anharmonic contribution to the lattice thermal conductivity of ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ as a function of the alloy composition at 450 K, where the group IV element mass of each alloy composition is artificially set to that of ${\mathrm{Pb}}_{\text{0.5}}{\mathrm{Ge}}_{\text{0.5}}\mathrm{Te}$. Black circles show the thermal conductivity in the rocksalt phase, while red up and blue down triangles show the thermal conductivity perpendicular and parallel to the trigonal [111] axis in the rhombohedral phase, respectively. Also shown in green crosses is the frequency of the lowest transverse optical mode at the zone center for the same mass of the group IV element as described above.

Figure 7

(a) Nearest neighbors within the rocksalt structure of PbTe along the [001] direction, and their counterparts for the rhombohedral structure of GeTe. (b) The trace of the harmonic interatomic force constant tensor vs atomic distance for several ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ alloy compositions in the rhombohedral phase. An alloy composition in the rocksalt phase near the phase transition, ${\mathrm{Pb}}_{0.51}{\mathrm{Ge}}_{0.49}\mathrm{Te}$, is also included for comparison. (c) The trace of the harmonic interatomic force constant tensor vs atomic distance for several ${\mathrm{Pb}}_{1-x}{\mathrm{Ge}}_x\mathrm{Te}$ alloy compositions in the rocksalt phase.