Effect of thermal disorder on high figure of merit in PbTe

With ab initio molecular dynamics we observe thermal disorder and find band convergence with increased temperature and close relation between thermal disorder and thermoelectric (TE) properties of $p$-doped PbTe. Lack of short-range order causes local overlap of valence orbitals and increase in density-of-states near the Fermi level. Effective mass becomes temperature-dependent peaking in the converged-band regime. With classical molecular dynamics (MD) and the Green-Kubo autocorrelation decay we find reduction in lattice thermal conductivity (suppression of short- and long-range acoustic phonon transports). The described thermal-disorder roles lead to high TE figure-of-merit ($ZT$), and good agreement with the experimental results.

Figure 1

(a) Conventional cell of PbTe showing simple cubic structure. (b) The first Brillouin zone for the primitive cell of PbTe and its high-symmetry $k$ points.

Figure 2

Time-dependent evolution of radial distribution functions $g$($r$) of PbTe supercell at $T=700$ K. Each snapshot is an average of 64 displaced coordinates obtained from (a) initial and (b) a well-converged AIMD step. Average $g$($r$) of all snapshots (second step, 0 to 11 ps) are also shown.

Figure 3

Calculated radial distribution functions of PbTe supercell obtained from (a) AIMD and (b) classical MD.

Figure 4

Variation of the rms atomic local off-centering in PbTe as a function of temperature. The Debye harmonic model prediction is also shown, with markers for differences with AIMD results. The inset image marks this rms displacement.

Figure 5

The charge densities and atomic positions of PbTe for (a) $T=0$ K, (b) 300 K, and (c) 700 K. A slice (101) illustrates the electron-density distribution (a distance from origin of 16.8 Å). The charge density contours are for 0 (blue) to 0.289 (red) $e$Å${}^{-3}$.

Figure 6

(a) Electronic density-of-states for PbTe using static DFT (0 K) and AIMD calculations (300 and 700 K). (b) Projected electronic density-of-states for PbTe, showing the $s$, $p$, and $d$ orbital contributions. Frozen ($T=0$ K) and thermal-disordered structure at 300 and 700 K are shown.

Figure 7

Calculated band structures of PbTe supercell (3 $\times$ 3 $\times$ 3 primitive cell) at (a) $T=0$ K, (b) 300 K, and 700 K.

Figure 8

Variations of PbTe band gap energy as a function of temperature. The experimental results and the proposed empirical model results are also shown.

Figure 9

Temperature dependent, the density-of-states effective masses $m_{i,e,\alpha }$($T$) obtained from DFT-AIMD. Two regimes, single and converged band, are also defined. The band-alignment evolution with temperature is also illustrated. $m_e$ is electron mass.

Figure 10

Calculated energy-dependent electron-phonon relaxation times for 300 and 700 K. The subscripts A, OD, and OP refer to acoustic, optical with deformation potential couplings, and optical with polar coupling.

Figure 11

Predicted TE properties of PbTe and comparison with experiments.[3] (a) Temperature dependence of the Seebeck coefficient, (b) electrical conductivity, and (c) total thermal conductivity for three different carrier concentrations $n_p$. (d) Temperature variations of lattice thermal conductivity and its short- and long-range acoustic and optical components. Cutoff frequency of 1.5 THz is used.[36, 37, 38] The amorphous-phase minimum lattice conductivity is also shown.

Figure 12

(a) Variations of the DFT-AIMD predicted $ZT$ as a function of temperature for the $p$-type PbTe, at three different carrier concentrations $n_p$. The corresponding experimental results[3] are also shown. (b) Variation of $ZT$($T$) obtained from the analytic model using constant $m_{i,e}^{\circ }$ and temperature-dependent $m_{i,e}$($T$) for the 9.0 $\times$ 10${}^{19}$ cm${}^{-3}$.