Electron-phonon scattering and thermoelectric transport in $p$-type PbTe from first principles

We present a first-principles-based model of electron-phonon scattering mechanisms and thermoelectric transport at the $L$ and $\mathrm{\Sigma }$ valleys in $p$-type PbTe, accounting for their thermally induced shifts. Our calculated values of all thermoelectric transport parameters at room temperature are in very good agreement with experiments for a wide range of doping concentrations. Scattering due to longitudinal optical phonons is the main scattering mechanism in $p$-type PbTe, while scattering due to transverse optical modes is the weakest. The $L$ valleys contribute most to thermoelectric transport at 300 K due to the sizable energy difference between the $L$ and $\mathrm{\Sigma }$ valleys. We show that both scattering between the $L$ and $\mathrm{\Sigma }$ valleys and additional transport channels of the $\mathrm{\Sigma }$ valleys are beneficial for the overall thermoelectric performance of $p$-type PbTe at 300 K. Our findings thus support the idea that materials with high valley degeneracy may be good thermoelectrics.

Figure 1

Schematic electronic band structure of PbTe, illustrating the conduction and valence band states at $L$ and valence band states at $\mathrm{\Sigma }$ at 0 K (black) and 300 K (blue). The horizontal lines represent the energies of the valence band maxima and the conduction band minimum. The states at the $L$ ($\mathrm{\Sigma }$) valleys are shown in solid (dashed) lines.

Figure 2

The valence-band dispersion of PbTe calculated using density functional theory (DFT, solid blue lines) and the Kane model fitted to DFT (dashed red lines). Horizontal dashed lines refer to the Fermi level at 300 K for the doping concentration of ${10}^{18}$, ${10}^{19}$, and ${10}^{20}{\mathrm{cm}}^{-3}$.

Figure 3

Electron-phonon scattering rates at 300 K for the valence bands of PbTe vs hole energy (from the valence-band maxima at $L$ corresponding to zero energy and below). The scattering rates are calculated using the electronic band structure at 0 K and resolved by (a) acoustic, (b) transverse optical (TO), and (c) longitudinal optical (LO) phonon modes. Blue dots correspond to the total scattering rates computed using density functional perturbation theory and the electron-phonon Wannier (EPW) approach. Solid (dashed) lines refer to the scattering rates obtained using our first-principles-based model for the $L$ ($\mathrm{\Sigma }$) valleys. The intravalley and intervalley contributions to the total scattering rate (black lines) are given by orange and green lines, respectively. The gray vertical line corresponds to the energy difference between the $L$ and $\mathrm{\Sigma }$ valleys at $0\mathrm{K}$.

Figure 4

Room-temperature thermoelectric transport parameters of $p$-type PbTe: (a) mobility ($\mu$), (b) conductivity ($\sigma$), (c) Seebeck coefficient ($S$), (d) power factor (PF), (e) electrical thermal conductivity (${\kappa }_e$), and (f) figure of merit ($ZT$) as a function of doping concentration $n_h$. Solid black lines show the results obtained using our model and the electronic band structure at 300 K. Dashed blue and dotted orange lines represent the contributions from the $L$ and $\mathrm{\Sigma }$ valleys, respectively. Dash-dotted green lines correspond to the contribution from the $L$ valleys in the total absence of the $\mathrm{\Sigma }$ valleys (ignoring scattering between the $L$ and $\mathrm{\Sigma }$ valleys). Symbols represent measurements: blue circles from Ref. [71], orange stars from Ref. [55], purple hexagons from Ref. [72], red squares from Ref. [73], magenta diamonds from Ref. [74], brown plus signs from Ref. [75], and gray triangles from Ref. [76].

Figure 5

Intravalley acoustic electron-phonon matrix elements [scaled by the factor $\frac{k_{B}T{I}^2(\mathit{k},{\mathit{k}}^{\prime })}{V}$, where $I(\mathit{k},{\mathit{k}}^{\prime })$ is the wave function overlap] as a function of the angle between the phonon wave vector and the principal valley axis ($\theta$), averaged over the azimuthal angles, for the $L$ valleys (top panels) and the $\mathrm{\Sigma }$ valleys (bottom panels). Left and right panels correspond to longitudinal (LA) and transverse (TA) acoustic modes, respectively. Solid lines show the results obtained using our generalized method, while dots refer to a spherical harmonics interpolation technique obtained by Herring and Vogt for the $L$ valley [33].

Figure 6

Electron-phonon matrix elements due to acoustic modes between the states $\mathrm{\Sigma }$ (0.1875, 0.375, 0.1875) and $\mathrm{\Sigma }+\mathit{q}$ as a function of the phonon wave vector $\mathit{q}$ whose directions are indicated at the top of each panel. Orange and blue dots represent the matrix elements calculated using density functional perturbation theory and the electron-phonon Wannier (EPW) approach for transverse and longitudinal acoustic modes, respectively. Dashed lines are the fifth-order polynomial fits to the EPW matrix elements. Red (green) lines are obtained using the computed values of deformation potentials (DPs) given in Table 3 for transverse (longitudinal) acoustic modes in the long-wavelength limit.