Energy and temperature dependence of relaxation time and Wiedemann-Franz law on PbTe

Recent revival of interest in high-temperature $\left(T\right)$ thermoelectrics has made it necessary to understand in detail the $T$ dependence of different transport coefficients, and different processes contributing to this temperature dependence. Since PbTe is a well-studied prototypical high-temperature thermoelectric, we have carried out theoretical studies to analyze how different physical sources contribute to electronic transport coefficients in this system over a wide $T$ and concentration $\left(n\right)$ range; $300\text{K}<T<900\text{K}$ and $1<n/n_o<10$, where $n_o={10}^{19}{\text{cm}}^{-3}$, extending earlier works on this problem. We have used Boltzmann equation within energy-dependent relaxation time approximations. Although the $T$ dependence of the electrical conductivity $\sigma$ comes from several sources (band structure parameters, chemical potential $\mu$, relaxation time $\tau$), we find that the $T$ dependence of $\tau$ dominates. We fit the $T$ and the energy $\left(\epsilon \right)$ dependence of the total relaxation time ${\tau }_{tot}$ by a simple function $\tau \sim a{T}^{-p}/\left(b+c{\epsilon }^r\right)$, where $a$, $b$, $c$, $p$, and $r$ are $T$ and $\epsilon$ independent parameters but depend on $n$. Using this function, we find that for concentration range of interest, changing $r$ which governs the energy dependence of scattering does not appreciably affect the $T$ dependence of $\sigma$. Electronic thermal conductivities both at constant current $J$ and constant electric field $E$ were calculated using this $\tau$ to reexamine the validity of Wiedemann-Franz (WF) law in PbTe, extending the earlier work of Bhandari and Rowe to higher temperatures. We find that using standard WF law to obtain the electronic contribution of the thermal conductivity $\left({\kappa }_{el}\right)$ usually overestimates this contribution by more than $0.5{\text{WK}}^{-1}{\text{m}}^{-1}$. Therefore the value of the lattice thermal conductivity obtained by subtracting this ${\kappa }_{el}$ from the total thermal conductivity is underestimated roughly by the same amount.

Figure 1

Energy dependence of the inverse of the relaxation time of the three dominant scattering mechanisms (${\tau }_a$, ${\tau }_o$, and ${\tau }_{po}$) in PbTe for electron carrier density $n=5\times {10}^{19}{\text{cm}}^{-3}$ at different temperatures (a) 300 K, (b) 600 K, and (c) 900 K. The smooth curves gives the $\left(-\partial f_0/\partial \epsilon \right)$ indicating the energy region contributing to the transport at different temperatures.

Figure 2

Temperature dependence of the chemical potential $\mu$ for $n$-type PbTe at different concentrations $n=5\times {10}^{19}{\text{cm}}^{-3}$ and $n=5\times {10}^{20}{\text{cm}}^{-3}$.

Figure 3

Temperature dependence of the electrical conductivity $\sigma$ and thermopower $S$ for $n$-type PbTe at concentration $n=5\times {10}^{19}{\text{cm}}^{-3}$. The experimental values are shown as solid points.

Figure 4

Energy dependence of the total relaxation time ${\tau }_{tot}$, calculated from Eq. 5.80, for $n$-type PbTe at concentration $n=5\times {10}^{19}{\text{cm}}^{-3}$ for different temperatures.

Figure 5

Energy dependence of the scaled (to 300 K) total relaxation time ${\tau }_{tot}$ at different temperatures.

Figure 6

Energy dependence of the total relaxation time ${\tau }_{tot}$ for $T=300\text{K}$.

Figure 7

Temperature dependence of the total relaxation time ${\tau }_{tot}$ at $\epsilon =0.1\text{eV}$.

Figure 8

A comparison between the electrical conductivity $\sigma$ using the scattering mechanisms of the relaxation time with the scaled formula.

Figure 9

A comparison between the thermopower $S$ using the scattering mechanisms of the relaxation time with the scaled formula.

Figure 10

(a) Temperature dependence of $\sigma$, using the scaling formula of the total relaxation time for different $r$ parameter. (b) Temperature dependence of the scaled $\sigma$ (to $r=1.74$).

Figure 11

Temperature dependence of the electronic thermal conductivity $\left({\kappa }_{el}\right)$ at constant electric field $E\left({\kappa }_{el,E}\right)$ and at constant current density $J\left({\kappa }_{el,J}\right)$ using Boltzmann transport equations. We also give their values using WF law.

Figure 12

Temperature dependence of $ZT$ for $n$-type PbTe at concentration $n=5\times {10}^{19}{\text{cm}}^{-3}$ assuming ${\kappa }_l=0$.

Figure 13

Temperature dependence of the scaled effective Lorenz number $\left(L/L_0\right)$ for $n$-type PbTe at different concentration $n$ $\left(L_0=2.45\times 10-8\text{W}\Omega {\text{K}}^{-2}\right)$.