Limit of $zT$ enhancement in rocksalt structured chalcogenides by band convergence

Rocksalt structured chalcogenides, such as PbTe, PbSe, and SnTe, are the top candidates for midtemperature thermoelectric applications, and their $p$-type thermoelectric efficiencies can be enhanced via aligning the valence bands. Here, we provided comprehensive numerical investigations on the effects of band convergence on electronic properties. We found that the extra valance band can indeed significantly enhance the power factor. Nevertheless, the extra valance band can also increase the electronic thermal conductivity, which partially offsets the enhanced power factor for the overall figure of merit. Finally, we predicted that the maximum figure of merit for PbTe, PbSe, and SnTe can reach 2.2, 1.8, and 1.6, respectively, without relying on the reduction in lattice thermal conductivity.

Figure 1

Calculated (a) total power factor $\left(S^2\sigma \right)$ using the three-band model to include the contributions of CB, $\mathrm{V}{\mathrm{B}}_L$, and, $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$, (b) the partial power factor $\left(S_{CL}^2{\sigma }_{CL}\right)$ using the two-band model to include the contributions of CB and $\mathrm{V}{\mathrm{B}}_L$, (c) and the partial power factor contributed by $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$ $\left(S^2\sigma -S_{CL}^2{\sigma }_{CL}\right)$, all as a function of reduced Fermi level (η) for SnTe at temperature ranging from 300 to 900 K. These three subfigures demonstrate the significant role of $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$ in increasing the total $S^2\sigma$. (d) Temperature-dependent weighed mobility ratio between $\mathrm{V}{\mathrm{B}}_L$ and $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}\left({B_L}_{/\mathrm{\Sigma }}\right)$ and energy separation between $\mathrm{V}{\mathrm{B}}_L$ and $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$ $\left(\mathrm{\Delta }E\right)$ for SnTe. The variations of ${B_L}_{/\mathrm{\Sigma }}$ and $\mathrm{\Delta }E$ interpret the increase of peak magnitude and the shift of peak for $S^2\sigma -S_{CL}^2{\sigma }_{CL}$ to low η with increasing temperature being, respectively, caused by the decreased ${B_L}_{/\mathrm{\Sigma }}$ and $\mathrm{\Delta }E$ at high temperature.

Figure 2

Calculated (a) electronic thermal conductivity $\left({\kappa }_e\right)$, and (b) bipolar thermal conductivity $\left({\kappa }_{bi}\right)$ as a function of reduced Fermi level (η) for SnTe using the three-band model to include the contributions of CB, $\mathrm{V}{\mathrm{B}}_L$, and, $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$ at temperature ranging from 300 to 900 K. These two subfigures present the effect of the energy separation between $\mathrm{V}{\mathrm{B}}_L$ and $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$ $\left(\mathrm{\Delta }E\right)$ on ${\kappa }_e$ and ${\kappa }_{bi}$.

Figure 3

Calculated (a) total figure of merit $\left(zT\right)$ using the three-band model to include the contributions of CB, $\mathrm{V}{\mathrm{B}}_L$, and, $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$, and (b) partial figure of merit contributed by $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}(zT-z{T}_{CL})$ with $z{T}_{CL}$ denoting the partial figure of merit calculated using the two-band model to include the contributions of CB and $\mathrm{V}{\mathrm{B}}_L$, both $zT$ and $zT-z{T}_{CL}$ as a function of reduced Fermi level (η) for SnTe at temperature ranging from 300 to 900 K. These two subfigures illustrate the effect of reducing the energy separation between $\mathrm{V}{\mathrm{B}}_L$ and $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$ $\left(\mathrm{\Delta }E\right)$ on increasing the total $zT$.

Figure 4

Determined temperature-dependent optimal Hall carrier concentration $\left(n_H^{opt}\right)$ for (a) total power factor $\left(S^2\sigma \right)$ and (b) total figure of merit $\left(zT\right)$ using the three-band model. (c) Calculated total power factor $\left(S^2\sigma \right)$ and (d) total figure of merit $\left(zT\right)$ using the three-band model as a function of temperature for SnTe, PbSe, and PbTe, respectively. These figures demonstrate the significant enhancement in $zT$ by only reducing the energy separation between $\mathrm{V}{\mathrm{B}}_L$ and $\mathrm{V}{\mathrm{B}}_{\mathrm{\Sigma }}$ $\left(\mathrm{\Delta }E\right)$, and the further enhancement in $zT$ by reducing the lattice thermal conductivity $\left({\kappa }_l\right)$ to the amorphous limits of these rocksalt structured materials.