Thermopower of thermoelectric materials with resonant levels: PbTe:Tl versus PbTe:Na and ${\mathrm{Cu}}_{1-x}{\mathrm{Ni}}_x$

Electronic transport properties of thermoelectric materials containing resonant levels are discussed by analyzing the two best known examples: copper-nickel metallic alloy (Cu-Ni, constantan) and thallium-doped lead telluride (PbTe:Tl). As a contrasting example of a material with a nonresonant impurity, sodium-doped PbTe is considered. Theoretical calculations of the electronic structure, Bloch spectral functions, and energy-dependent electrical conductivity at $T=0$ K are done using the Korringa-Kohn-Rostoker method with the coherent potential approximation and the Kubo-Greenwood formalism. The effect of a resonance on the residual resistivity and electronic lifetimes in PbTe is analyzed. By using the full Fermi integrals, room-temperature thermopower is calculated, confirming its increase in PbTe:Tl versus PbTe:Na, due to the presence of the resonant level. In addition, our calculations support the self-compensation model, in which the experimentally observed reduction of carrier concentration in PbTe:Tl against the nominal one is explained by the presence of $n$-type Te vacancies.

Figure 1

(a)–(c) Calculated density of states (DOS) of ${\mathrm{Cu}}_{1-x}{\mathrm{Ni}}_x$ alloys. (d) Bloch spectral function of ${\mathrm{Cu}}_{0.60}{\mathrm{Ni}}_{0.40}$ at the $\mathrm{\Gamma }=(0,0,0)$ k point.

Figure 2

The two-dimensional projections of Bloch spectral functions of ${\mathrm{Cu}}_{0.60}{\mathrm{Ni}}_{0.40}$ in high-symmetry directions. Here and in the following figures, the BSFs are given in atomic units $\left({\mathrm{Ry}}^{-1}\right)$; the black color corresponds to BSF values greater than 300 a.u., i.e., $22{\mathrm{eV}}^{-1}$.

Figure 3

Top panel: DOS of ${\mathrm{Cu}}_{0.60}{\mathrm{Ni}}_{0.40}$ near the Fermi energy. Middle panel: Energy-dependent electric conductivity $\sigma \left(E\right)$ at $T=0$ K. Bottom panel: computed thermopower $S$ at $T=300$ K. The inset in the middle panel shows the power factor (PF) normalized by the maximum PF, which was found at $E\simeq E_F$.

Figure 4

Thermopower of ${\mathrm{Cu}}_{0.60}{\mathrm{Ni}}_{0.40}$ as a function of temperature. Experimental data after Ho et al. [57].

Figure 5

Calculated densities of states (DOS) of ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}\mathrm{Te}$: total DOS (per formula unit) and partial DOS of Tl $6s$ and $6p$ states (per Tl atom) with two resonant DOS peaks, around $-5.5$ eV and near the valence band edge.

Figure 6

(a) Calculated densities of states (DOS) of ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}\mathrm{Te}$; (b) ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}{\mathrm{Te}}_{0.9934}{\mathrm{Vac}}_{0.0066}$, i.e., containing 0.66% of Te vacancies; (c),(d) 2% Tl-doped compared to 1% Na-doped PbTe, in which the zero of energy is set to (c) the valence band edge $\left(E_V\right)$ and (d) the Fermi energy $\left(E_F\right)$. The partial Tl $6s$ DOS in (a) and (b) is given per one impurity atom.

Figure 7

The two-dimensional projection of Bloch spectral functions of ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}{\mathrm{Te}}_{0.9934}{\mathrm{Vac}}_{0.0066}$ (top panel) and ${\mathrm{Pb}}_{0.99}{\mathrm{Na}}_{0.01}\mathrm{Te}$ (bottom panel), between the $L$ and $\mathrm{\Sigma }$ points.

Figure 8

Bloch spectral function of (a) ${\mathrm{Pb}}_{0.99}{\mathrm{Na}}_{0.01}\mathrm{Te}$ and (b) ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}{\mathrm{Te}}_{0.9934}{\mathrm{Vac}}_{0.0066}$, computed for k $\simeq (0.406,0.406,0.125)2\pi /a$, i.e., point located in 0.75 of the distance from $L$ to $\mathrm{\Sigma }$.

Figure 9

Electronic lifetime $\tau$ of the valence band states in ${\mathrm{Pb}}_{0.99}{\mathrm{Na}}_{0.01}\mathrm{Te}$ and ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}{\mathrm{Te}}_{0.9934}{\mathrm{Vac}}_{0.0066}$ along the $L-\mathrm{\Sigma }$ direction, as a function of the (a) $k$ vector and (b) energy. In (b), the energy scale was shifted relative to the valence band edge, $E_V$.

Figure 10

Transport functions of 2% Tl-doped PbTe with Te vacancies or Te/S substitution. (b) The zoom of (a) near the VB edge, set as zero of energy.

Figure 11

Transport function of 1% Na-doped PbTe compared to 2% Tl-doped PbTe. Inset shows the zoom near the VB edge, set as zero of energy.

Figure 12

Thermopower at $T=300$ K as a function of hole concentration (Pisarenko plot) of 2% Tl-doped PbTe containing Te vacancies or Te/S substitution, compared to 1% Na-doped PbTe. Points show experimental results, as described in the legend, from Heremans et al. [7], Jaworski et al. [60], Keiber et al. [65], Pei et al. [75], Airapetyants et al. [77], Crocker and Rogers [78], and Chernik et al. [79]. As it is rather impossible to obtain 2% Tl-doped PbTe samples with $p<{10}^{19}{\mathrm{cm}}^{-3}$, this part of the plot does not describe any real material, and is left only for the completeness of the plot.

Figure 13

Calculated DOS effective mass $m_{\mathrm{D}}^{*}$ as a function of hole concentration $p$ for (a) ${\mathrm{Pb}}_{0.995}{\mathrm{Na}}_{0.005}\mathrm{Te}$ and ${\mathrm{Pb}}_{0.99}{\mathrm{Na}}_{0.01}\mathrm{Te}$ and (b) ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}{\mathrm{Te}}_{0.9934}{\mathrm{Vac}}_{0.0066}$. Points are based on the experimental measurements of Giraldo-Gallo et al. [82], Jensen et al. [83], Heremans et al. [7], and Jaworski et al. [60]. For PbTe:Na, an increase in the computed $m_{\mathrm{D}}^{*}$ marks the concentration $p=6\times {10}^{-19}{\mathrm{cm}}^{-3}$, where the second valence band maximum at the $\mathrm{\Sigma }$ point is located in our LDA calculations. Above this point, the single-band effective mass picture does not apply.

Figure 14

Logarithm of the transport function $\sigma \left(E\right)$ as a function of reduced energy $E-\mu$ in ${\mathrm{Pb}}_{0.99}{\mathrm{Na}}_{0.01}\mathrm{Te}$ and ${\mathrm{Pb}}_{0.98}{\mathrm{Tl}}_{0.02}{\mathrm{Te}}_{0.9934}{\mathrm{Vac}}_{0.0066}$. Chemical potential $\mu$ is computed at $T=300$ K for the nominal carrier concentrations: $1.5\times {10}^{20}{\mathrm{cm}}^{-3}$ (Na) and $1.0\times {10}^{20}{\mathrm{cm}}^{-3}$ (Tl).

Figure 15

Calculated Pisarenko curve for ${\mathrm{Pb}}_{0.99}{\mathrm{Na}}_{0.01}\mathrm{Te}$ and ${\mathrm{Pb}}_{0.995}{\mathrm{Na}}_{0.005}\mathrm{Te}$, compared with the experimental results (see Fig. 12 for references). In the lower doping regime, calculations for 0.5% Na-doped PbTe better reproduce the experimental trend and correct the deviation seen at lower $p$ in Fig. 12.