Dominant electron-phonon scattering mechanisms in $n$-type PbTe from first principles

We present an ab initio study that identifies the main electron-phonon scattering channels in $n$-type PbTe. We develop an electronic transport model based on the Boltzmann transport equation within the transport relaxation time approximation, fully parametrized from first-principles calculations that accurately describe the dispersion of the electronic bands near the band gap. Our computed electronic mobility as a function of temperature and carrier concentration is in good agreement with experiments. We show that longitudinal optical phonon scattering dominates electronic transport in $n$-type PbTe, while acoustic phonon scattering is relatively weak. We find that scattering due to soft transverse optical phonons is by far the weakest scattering mechanism, due to the symmetry-forbidden scattering between the conduction band minima and the zone center soft modes. Soft phonons thus play the key role in the high thermoelectric figure of merit of $n$-type PbTe: they do not degrade its electronic transport properties although they strongly suppress the lattice thermal conductivity. Our results suggest that materials like PbTe with soft modes that are weakly coupled with the electronic states relevant for transport may be promising candidates for efficient thermoelectric materials.

Figure 1

The conduction band of PbTe near the L point at 0 K calculated from first principles using the local density approximation (LDA) without spin-orbit coupling (red dots) and the HSE03 hybrid functional with spin-orbit coupling (blue squares). The conduction band minima from the LDA and HSE03 calculations are aligned. The path L-$\mathrm{\Gamma }$ and L-W corresponds to the parallel and perpendicular direction, respectively. The fits to the HSE03 band using the generalized and two-band Kane model are shown in solid and dashed black lines, respectively. Fermi levels for different doping concentrations calculated with the HSE03 bands are also indicated by dotted black lines.

Figure 2

Electron-phonon scattering rates of $n$-type PbTe for different phonon modes versus electronic energy. Dots represent the scattering rates calculated using the electron-phonon Wannier approach and density functional perturbation theory (see text for explanation), and solid lines correspond to the scattering rates computed using our model parameterized from first principles. Dashed blue line shows the nonpolar contribution to longitudinal optical scattering from our model.

Figure 3

Relaxation times of $n$-type PbTe for different phonon modes versus electronic energy. Solid lines represent transport relaxation times (see text for explanation), while dashed lines correspond to self-energy relaxation times i.e. inverse scattering rates. The two relaxation times are identical for acoustic phonons.

Figure 4

Temperature dependence of the Hall mobility of $n$-type PbTe calculated using the transport relaxation time (TRT) approximation with the generalized Kane model (solid lines) and with the Kane two-band model (dashed lines). Black and red lines correspond to the doping concentrations of ${10}^{18}$ ${\mathrm{cm}}^{-3}$ and ${10}^{19}$ ${\mathrm{cm}}^{-3}$, respectively. Black circles and red triangles represent the experimental data from Ref. [5] for the doping concentrations of ${10}^{18}$ ${\mathrm{cm}}^{-3}$ and ${10}^{19}$ ${\mathrm{cm}}^{-3}$, respectively. Dotted black line shows the mobility calculated using the self-energy relaxation time (SERT) approximation and the two-band Kane model for the carrier concentration of ${10}^{18}$ ${\mathrm{cm}}^{-3}$. The additional experimental data at 300 K are shown in an open square and triangle (Ref. [12]), and a circle (Ref. [82]).

Figure 5

Drift mobility of $n$-type PbTe calculated using the generalized Kane model for the carrier concentration of ${10}^{18}$ ${\mathrm{cm}}^{-3}$ as a function of temperature (solid black line) and its contributions due to scattering with different phonon modes and ionized impurities (colored dashed lines).

Figure 6

Hall mobility of $n$-type PbTe limited by scattering due to phonons and ionized impurities versus doping concentration calculated using the generalized and two-band Kane model (solid and dash-dotted black lines, respectively). Experimental data are shown in circles (Ref. [44]), squares (Ref. [82]), and triangles (Ref. [12]).

Figure 7

Drift mobility of $n$-type PbTe calculated using the generalized Kane model including screening at 300 K as a function of doping concentration (solid black line) and its contributions due to scattering with different phonon modes and ionized impurities (dashed colored lines). Dotted black line shows the drift mobility without screening effect.