Microscopic origin of the high thermoelectric figure of merit of $n$-doped SnSe

Excellent thermoelectric performance in the out-of-layer $n$-doped SnSe has been observed experimentally [Chang et al., Science 360, 778 (2018)]. However, a first-principles investigation of the dominant scattering mechanisms governing all thermoelectric transport properties is lacking. In the present paper, by applying extensive first-principles calculations of electron-phonon coupling associated with parameterized calculation of the scattering by ionized impurities, we investigate the reasons behind the superior figure of merit as well as the enhancement of $zT$ above 600 K in $n$-doped out-of-layer SnSe, as compared to $p$-doped SnSe with similar carrier densities. For the $n$-doped case, the relaxation time is dominated by ionized impurity scattering and increases with temperature, a feature that maintains the power factor at high values at higher temperatures and simultaneously causes the carrier thermal conductivity at zero electric current $\left({\kappa }_{\mathrm{el}}\right)$ to decrease faster for higher temperatures, leading to an ultrahigh-$zT=3.1$ at 807 K. We rationalize the roles played by ${\kappa }_{\mathrm{el}}$ and ${\kappa }^0$ (the thermal conductivity due to carrier transport under isoelectrochemical conditions) in the determination of $zT$. Our results show the ratio between ${\kappa }^0$ and the lattice thermal conductivity indeed corresponds to the upper limit for $zT$, whereas the difference between calculated $zT$ and the upper limit is proportional to ${\kappa }_{\mathrm{el}}$.

Figure 1

(a) Calculated carrier density ($n_{\mathrm{carr}}$) in comparison to experimental values inferred from Hall measurements, as reported by Chang et al. [34], (b) calculated ionized impurity concentration ($n_{\mathrm{ii}}$), and (c) the resulting figure of merit ($zT$) for $p$- and $n$-doped SnSe, along with experimental results reported by Chang et al. [34]. The quadratic fits to low temperature points only (300–600 K, blue dot-dashed line) and high temperature points only (600–807 K, red dot-dashed line) highlight the enhancement of $zT$ above 600 K for $n$-doped SnSe.

Figure 2

Calculated thermoelectric properties of $p$- and $n$-doped SnSe for $T=$ 300–807 K in comparison with available experimental data reported by Chang et al. [34] up to 773 K. (a) Seebeck coefficient (S), (b) electrical conductivity ($\sigma$), (c) power factor ($PF$), (d) thermal conductivity due to the carrier transport at zero electric current (${\kappa }_{\mathrm{el}}$), (e) Lorenz function ($\mathrm{\Lambda }$), and (f) electronic figure of merit ($z{T}_{\mathrm{el}}$). The fits to points in different temperature ranges (orange for 300–600 K, blue for 400–807 K and red for 600–807 K) highlight the enhancement (faster decrease) of $S$ and $z{T}_{\mathrm{el}}$ (${\kappa }_{\mathrm{el}}$ and $\mathrm{\Lambda }$) above 600 K for $n$-doped SnSe.

Figure 3

Temperature dependence (300–807 K) of relaxation times (RTs) as a function of carrier energies calculated for each scattering process in the out-of-layer direction of $p$- and $n$-doped SnSe: nonpolar scattering of acoustic and optical phonons (${\tau }_{\mathrm{npol}}$, magenta), screened polar scattering of optical phonons (${\tau }_{\mathrm{pol}}$, blue), scattering by ionized impurities (${\tau }_{\mathrm{imp}}$, red), and the total RT (${\tau }_{\mathrm{tot}}$, green) calculated using Mathiessen's rule. The light grey rectangles represent the gap region with an energy gap of 0.86 eV.

Figure 4

(a) Electronic velocity $v\left(\epsilon \right)$ and (b) the transport distribution function, $\mathrm{\Sigma }\left(\epsilon \right)$ at 300 and 807 K, as a function of the carrier energies for $n$- and $p$-doped SnSe, where both quantities are plotted in relation to their respective band edges placed at 0 eV. (c) Lattice (${\kappa }_{\mathrm{latt}}$) and total (${\kappa }_{\mathrm{tot}}$) thermal conductivities and (d) the contribution of carrier transport to the total thermal conductivity at zero electric current as a function of temperature for $n$- and $p$-doped SnSe. The red dashed linear fit (300–807 K) highlights the monotonic increase of ${\kappa }_{\mathrm{el}}$/${\kappa }_{\mathrm{tot}}$.

Figure 5

(a) Thermal conductivity due to carrier transport under isoelectrochemical conditions (${\kappa }^0$) and the ratio $\sigma S^{2}T/{\kappa }^0$, (b) calculated thermoelectric figure of merit ($zT$) and the upper limit of $zT$, namely ${\kappa }^0/{\kappa }_{\mathrm{latt}}$, as a function of temperature for $p$- and $n$-doped SnSe.