Thermodynamic properties of PbTe, PbSe, and PbS: First-principles study

The recent discoveries of novel nanocomposite and doped lead chalcogenide-based thermoelectric materials have attracted great interest. These materials exhibit low thermal conductivity which is closely related to their lattice dynamics and thermodynamic properties. In this paper, we report a systematic study of electronic structures and lattice dynamics of the lead chalcogenides $\text{Pb}X$ ($X=\text{Te}$, Se, and S) using first-principles density-functional-theory calculations and a direct force-constant method. We calculate the structural parameters, elastic moduli, electronic band structures, dielectric constants, and Born effective charges. Moreover, we determine phonon dispersions, phonon density of states, and phonon softening modes in these materials. Based on the results of these calculations, we further employ quasiharmonic approximation to calculate the heat capacity, internal energy, and vibrational entropy. The obtained results are in good agreement with experimental data. Lattice thermal conductivities are evaluated in terms of the Grüneisen parameters. The mode Grüneisen parameters are calculated to explain the anharmonicity in these materials. The effect of the spin-orbit interaction is found to be negligible in determining the thermodynamic properties of PbTe, PbSe, and PbS.

Figure 1

(Color online) Electronic band structure and total electron DOS for (a) PbTe, (b) PbSe, and (c) PbS. The solid (in black) and dash lines (in red) are the band structures with and without SOI, respectively. The total electron DOS (normalized) in the right panels are results with SOI. The zero energy is set to be Fermi level.

Figure 2

Calculated phonon dispersions of PbTe, PbSe, and PbS. The phonons were calculated by using equilibrium volume.

Figure 3

Calculated total and projected phonon density of states for (a) PbTe, (b) PbSe, and (c) PbS.

Figure 4

Calculated phonon dispersions of PbTe at critical points. (a) Compressed PbTe with the lattice parameter of $6.10\text{Å}$ at 14.3 GPa pressure. (b) Expanded PbTe with the lattice parameter of $6.60\text{Å}$ at $-0.8\text{GPa}$ pressure.

Figure 5

Calculated phonon dispersions of PbSe at critical points. (a) Compressed PbSe with the lattice parameter of $5.70\text{Å}$ at 21.8 GPa pressure. (b) Expanded PbSe with the lattice parameter of $6.25\text{Å}$ at $-1.1\text{GPa}$ pressure.

Figure 6

Calculated phonon dispersions of PbS at critical points. (a) Compressed PbS with the lattice parameter of $5.50\text{Å}$ at 25.5 GPa pressure. (b) Expanded PbS with the lattice parameter of $6.05\text{Å}$ at $-1.5\text{GPa}$ pressure.

Figure 7

Calculated Helmholtz free energy $F\left(T\right)$ as a function of lattice parameter for (a) PbTe, (b) PbSe, and (c) PbS. There are seven free-energy curves (solid lines) plotted in each figure range from 0 to 300 K with $\Delta T=50\text{K}$. The diamond symbols represent the energy minima at different temperatures. The insets are fittings of linear thermal expansivities as a function of temperature.

Figure 8

(Color online) Calculated thermodynamic functions as a function of temperature for PbTe, PbSe, and PbS. The panel (a), (b), and (c) represents heat capacity, internal energy $U-U\left(20\right)$, and entropy $S-S\left(20\right)$, respectively. $U\left(20\right)$ and $S\left(20\right)$ are internal energy and entropy at 20 K, respectively. The diamond symbols are experimental data.

Figure 9

Calculated Grüneisen parameter with respect to the wave vector for (a) PbTe, (b) PbSe, and (c) PbS, respectively. The top branches including the peaks at $\Gamma$ points belong to the TO vibrational modes. For PbS, the Grüneisen parameters of LA and TA branches are found to be very large in the regions of $X\to W$ and $\Gamma \to K$, respectively.