Thermopower of correlated semiconductors: Application to ${\text{FeAs}}_2$ and ${\text{FeSb}}_2$

We investigate the effect of electronic correlations onto the thermoelectricity of semiconductors and insulators. Appealing to model considerations, we study various many-body renormalizations that enter the thermoelectric response. We find that, contrary to the case of correlated metals, correlation effects do not per se enhance the Seebeck coefficient or the figure of merit, for the former of which we give an upper bound in the limit of vanishing vertex corrections. For two materials of current interest, ${\text{FeAs}}_2$ and ${\text{FeSb}}_2$, we compute the electronic structure and thermopower. We find ${\text{FeAs}}_2$ to be well described within density-functional theory and the therefrom deduced Seebeck coefficient to be in quantitative agreement with experiment. The capturing of the insulating ground state of ${\text{FeSb}}_2$, however, requires the inclusion of many-body effects, in which we succeed by applying the GW approximation. Yet, while we get qualitative agreement for the thermopower of ${\text{FeSb}}_2$ at intermediate temperatures, the tremendously large Seebeck coefficient at low temperatures is found to violate our upper bound, suggesting the presence of decisive (e.g., phonon mediated) vertex corrections.

Figure 1

(Color online) The symmetric, coherent, and large gap semiconductor. (a) coefficient of the $1/T$ low-temperature behavior of the Seebeck coefficient as a function of the chemical potential and for different temperatures. (b) temperature evolution of the extremal value of the $1/T$ coefficient of the thermopower.

Figure 2

(Color online) Thermoelectric properties of the model semiconductor with parameters inspired from ${\text{FeAs}}_2$. (a) from top to bottom: chemical potential, Seebeck coefficient, and figure of merit $ZT$ as a function of temperature for different impurity concentrations. (b) ingredients to the figure of merit $ZT=S^2\sigma T/\kappa$ at $T=220\text{K}$ as a function of carrier density. The inset shows the general setup of asymmetric valence and conduction dispersions, with an impurity level $E_D$ in the gap $\Delta$. (c) effect of the scattering amplitude $\Gamma$ (top) and the impurity level position $E_D$ (bottom) onto the figure of merit $ZT$, and the thermopower (inset). (d) top: comparison of the figure of merit from (a) with the purely electronic figure of merit $\left({\kappa }_L=0\right)$, bottom left: carrier density that maximizes $\left|S\right|$ and $ZT$ (for the chemical potential in the upper half of the gap), bottom right: $ZT$ as a function of the donor concentration for several temperatures, and a fixed impurity level $E_D=95\text{meV}$. We assume a unit-cell volume $V=80{\text{Å}}^3$ and a thermal lattice conductivity (scaled with the quasiparticle weight $Z$, see text) of ${\kappa }_L/Z^2=250\text{W}/\left(\text{Km}\right)$.

Figure 3

(Color online) Band structures of ${\text{FeSb}}_2$ (a) GGA, (b) Hybrid functional, and ${\text{FeAs}}_2$ (c) GGA, and (d) Hybrid functional.

Figure 4

(Color online) (a) ${\text{FeSb}}_2$, (b) ${\text{FeAs}}_2$. Band structure in the GW approximation (full lines), in comparison with LDA (dashed).

Figure 5

(Color online) Thermopower of ${\text{FeAs}}_2$. Shown are the theoretical Seebeck coefficient for $x$ polarization, using a constant self-energy, and the GGA band structure, with the gap scissored to 0.2 eV. Experimental results are of Sun et al. (Ref. 14) ($x$ polarization). Also displayed is a simple 1/T fit corresponding to the large gap semiconductor, Eq. (17), yielding $\Delta /2\delta \lambda -\mu \approx 86\text{meV}$. Further indicated is the largest possible purely electronic Seebeck coefficient for ${\text{FeAs}}_2$.

Figure 6

(Color online) Thermopower of ${\text{FeSb}}_2$. Shown are our theoretical together with experimental results from Ref. 4 for measurements along the crystallographic orientations x, y, z, as indicated. We can expect reasonable agreement in the temperature range indicated by the gray area. See text for details.