Ab initio calculations of the thermoelectric figure of merit, within the relaxation time approximation

TL;DR

The paper establishes a first-principles workflow to compute the thermoelectric figure of merit $ZT$ by explicitly treating electron–phonon and phonon–phonon interactions via finite-displacement methods in supercells, integrated with VASP and phono3py. By deriving electron and phonon transport within the Boltzmann formalism and employing relaxation-time approximations, it computes the energy-dependent transport function $\sigma(\epsilon)$ and Onsager coefficients $\mathcal{L}_{ij}$ to obtain $\sigma$, $S$, and $\kappa_e$, while $\kappa_L$ is obtained from phonon-phonon scattering. The method is applied to a set of doped thermoelectric materials (PbTe, PbSe, Mg$_2$X, Ni-based half-Heuslers), showing qualitative agreement with experiments and revealing sensitivities to exchange–correlation functionals, band gaps, and impurity scattering. The work demonstrates a workflow that avoids Wannierization, enabling straightforward application to large material sets, but highlights the need for improved electronic structure descriptions and impurity/band-structure-temperature effects for quantitative accuracy. Overall, this approach provides a rigorous, scalable route to predict thermoelectric performance from fundamental interactions, with clear avenues for refinement.

Abstract

In this paper, we propose a computational framework, based on the VASP and phono3py computer codes, to obtain the thermoelectric figure of merit from the electron-phonon and phonon-phonon interactions using finite displacements in supercells. Several numerical techniques are developed for efficiency. The method is applied to several thermoelectric materials. We found different behaviors for the lifetimes of the electrons in PbTe, PbSe, and in compounds of the half-Heusler and magnesium silicide family. This is traced back to the different frequencies of the phonons involved in the scattering around the Fermi level. They have a lower frequency in PbTe and PbSe. The magnitude of the thermoelectric figures of merit we computed compare well with experiments, but the agreement is far from perfect. The role of the defects, not explicitly considered in our calculations, but abundant in thermoelectric materials, is discussed as a possible explanation. It is also shown that the choice of the exchange-correlation functional can strongly impact the results.

Figures (27)

• Figure 1: Gauss-Legendre integration grids: (a) $\ln [(x+1)/(x-1)]$ the range of the energy window in units of $k_BT$ around the chemical potential $\mu$, for $N=21$ and $N=2001$, (b) $\ln [(x_N+1)/(x_N-1)]$, with $x_N$ the largest root, as a function of the number of roots $N$, (c) function in the integrand of Eq. \ref{['Lij_x']}, $(\ln [(1+x_k)(1-x_k)])^{n} w_k$, as a function of the $x_k$, for $n=0,1,2$.
• Figure 2: Flow chart for the calculation of electronic transport properties using VASP and phelel. (a) Preparatory steps are performed using the VASP code. (c) Supercell structures are generated with phelel. (d) Each displaced structure is then run as an individual VASP calculation. (e) The results are gathered by phelel to compute the potential derivatives and transferred to the primitive cell. (f) The final VASP calculation in the primitive cell then calculates the electron-phonon interactions and transport properties.
• Figure 3: Convergence of the (a) electronic conductivity ($\sigma$), (b) thermopower ($S$), (c) electronic part of the thermal conductivity ($\kappa_e$) and (d) lattice part of the thermal conductivity ($\kappa_L$), at $300$ K, with respect to the sampling of the first Brillouin zone.
• Figure 4: Reciprocal lifetimes, $\hbar/\tau$, at 300 K, as function of the band energies, are shown as black dots for (a) PbTe and (b) PbSe. The PBEsol exchange correlation functional is used. The transport function at constant relaxation time, $\sigma_0(\epsilon)$, and the density of states, $\rho(\epsilon)$, in arbitrary units, are shown as green and blue curves, respectively. The chemical potential is shown using a red dotted line. PbTe and PbSe are $p$ doped compounds in our calculations (see Tab. \ref{['roomT']}), therefore it is the lifetime of hole which is plotted.
• Figure 5: $(ST)^{2}$ as a function of temperature. The calculations are shown as continuous lines and experimental measurementsJood2020Zhang2012TANI2007GAO201533Saito2020renxie_2014yu2009 as filled circles. The $T^4$ law is shown as a black dotted line.