Electronic transport properties of $\mathrm{PbTe}$ and ${\mathrm{AgPb}}_m{\mathrm{SbTe}}_{2+m}$ systems

Transport calculations using the Boltzmann equation within energy-dependent relaxation time approximations were performed for $\mathrm{PbTe}$ and ${\mathrm{AgPb}}_m{\mathrm{SbTe}}_{2+m}$ (LAST-m) systems. We have used both the nonparabolic Kane model for the energy dispersion and the energy dispersion given by ab initio electronic structure calculations. For $\mathrm{PbTe}$ we find that the temperature dependence of the density of states effective mass $m_d$ is very important in order to have good agreement with experiment for electrical conductivity $\sigma$ and thermopower $S$. Transport calculations in $n$-type $\mathrm{PbTe}$ using the energy dispersion given by the ab initio electronic structure results in overestimation of $\sigma$ and underestimation of $S$ because the temperature dependence of $m_d$ cannot be incorporated in the calculation of the chemical potential. Transport calculations in $n$-type LAST-m systems using the nonparabolic Kane model for the energy dispersion show a small enhancement of the power factor $\left(\sigma S^2\right)$ in $0–500\mathrm{K}$ temperature range relative to $\mathrm{PbTe}$. The observed large ZT values of the LAST-12 and LAST-18 systems are a combination of a small enhancement of the power factor and a strong reduction in the thermal conductivity due to the formation of $\mathrm{Ag}\text{−}\mathrm{Sb}$ microstructures.

Figure 1

Band structure of $\mathrm{PbTe}$ in face centered cubic Brillouin zone.

Figure 2

Temperature dependence of (a) electrical conductivity, (c) thermopower, and (e) power factor for $n$-type $\mathrm{PbTe}$ at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ obtained using $E_{ac}=15\mathrm{eV}$ and $E_{oc}=26\mathrm{eV}$ values of the deformation potential coupling constants. Temperature dependence of (b) electrical conductivity, (d) thermopower, and (f) power factor for $n$-type $\mathrm{PbTe}$ at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ using $E_{ac}=15\mathrm{eV}$ and $E_{oc}=15\mathrm{eV}$. Theoretical values using temperature dependent $m_d$ and $E_g$ are shown as dashed lines and the experimental values are shown as black points from Ref. 20 and as gray points from Ref. 15.

Figure 3

Temperature dependence of (a) electrical conductivity, (c) thermopower, and (e) power factor for $n$-type $\mathrm{PbTe}$ at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ obtained using $E_{ac}=15\mathrm{eV}$ and $E_{oc}=26\mathrm{eV}$ values of the deformation potential coupling constants. Temperature dependence of (b) electrical conductivity, (d) thermopower, and (f) power factor for $n$-type $\mathrm{PbTe}$ at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ using coupling constant values $E_{ac}=15\mathrm{eV}$ and $E_{oc}=15\mathrm{eV}$. Theoretical values obtained using constant $m_d$ are shown as dashed lines. Experimental values are shown as black points from Ref. 20 and as gray points from Ref. 15.

Figure 4

Temperature dependence of (a) electrical conductivity, and (c) thermopower for $n$-type $\mathrm{PbTe}$ at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ using $E_{ac}=15\mathrm{eV}$ and $E_{oc}=26\mathrm{eV}$ values of the deformation potential coupling constants. Temperature dependence of (b) electrical conductivity and (d) thermopower for $n$-type $\mathrm{PbTe}$ at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ using coupling constant values $E_{ac}=15\mathrm{eV}$ and $E_{oc}=15\mathrm{eV}$. Theoretical values obtained using temperature dependent $E_g$ in the $0–950\mathrm{K}$ range are shown as dashed lines. Experimental values are shown as black points from Ref. 20 and as gray points from Ref. 15.

Figure 5

Temperature dependence of (a) electrical conductivity, (c) thermopower, and (e) power factor for $n$-type $\mathrm{PbTe}$ at concentration $n=2.4\times {10}^{19}{\mathrm{cm}}^{-3}$. Temperature dependence of (b) electrical conductivity, (d) thermopower, and (f) power factor for $n$-type $\mathrm{PbTe}$ at concentration $n=10\times {10}^{19}{\mathrm{cm}}^{-3}$. Theoretical values obtained using temperature dependent $E_g$ and $m_d$ and using $E_{ac}=15\mathrm{eV}$ and $E_{oc}=15\mathrm{eV}$ values for the deformation potential coupling constants are shown as dashed lines. Experimental values are shown as black points from Ref. 20 and as gray points from Ref. 15.

Figure 6

Temperature dependence of (a) electrical conductivity and (b) thermopower for $n$-type $\mathrm{PbTe}$ at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ and using $E_{ac}=15\mathrm{eV}$ and $E_{oc}=15\mathrm{eV}$ values of the deformation potential coupling constants. Theoretical values obtained using ab initio energy dispersion and temperature dependence of $E_g$ and $m_d$ are shown as dashed lines. Theoretical values obtained using nonparabolic Kane energy dispersion with constant $m_d$ in the expression of $n$ are shown as continuous lines. Experimental values are shown as black points from Ref. 20 and as gray points from Ref. 15.

Figure 7

Unit cell model for $\mathrm{Ag}\text{−}\mathrm{Sb}$ chain parallel to the [001] direction in ${\mathrm{AgSbPb}}_{30}{\mathrm{Te}}_{32}$. For the reason of clarity we show only Pb fcc lattices with Pb in gray, Ag in white, and Sb in black colors.

Figure 8

(a) Band structure in simple cubic Brillouin zone of (a) bulk $\mathrm{PbTe}$ and (b) $\mathrm{Ag}\text{−}\mathrm{Sb}$ chain model; orbital character of $\mathrm{Sb}p$. The size of the circles is proportional to the strength of the orbital character.

Figure 9

Transport coefficients at concentration $n=5\times {10}^{19}{\mathrm{cm}}^{-3}$ using temperature dependent $E_g$ and $m_d$ and $E_{ac}=15\mathrm{eV}$ and $E_{oc}=15\mathrm{eV}$ for the deformation potential coupling constants. Temperature dependence of (a) electrical conductivity, (c) thermopower, and (e) power factor for $m_d^{chain}$=1.5$m_d$. Temperature dependence of (b) electrical conductivity, (d) thermopower, and (f) power factor for $m_d^{chain}=2.5m_d$. Theoretical values for bulk $\mathrm{PbTe}$ are shown as dashed lines and for the chain model of LAST-30 as continuous lines.

Figure 10

Transport coefficients at concentration $n=1\times {10}^{20}{\mathrm{cm}}^{-3}$ using temperature dependent $E_g$ and $m_d$ and $E_{ac}=15\mathrm{eV}$ and $E_{oc}=15\mathrm{eV}$ values of the deformation potential coupling constants. Temperature dependence of (a) electrical conductivity, (b) thermopower, and (c) power factor for $m_d^{chain}=2.5m_d$. Theoretical values for bulk $\mathrm{PbTe}$ are shown as dashed lines and for the chain model of LAST-30 as continuous lines.