Boosting the thermoelectric performance of ${\mathrm{Fe}}_2\mathrm{VAl}\text{−}\mathrm{type}$ Heusler compounds by band engineering

Linking the fundamental physics of band structure and scattering theory with macroscopic features such as measured temperature dependencies of electronic and thermal transport is indispensable to a thorough understanding of thermoelectric phenomena and ensures more targeted and efficient experimental research. Regarding ${\mathrm{Fe}}_2\mathrm{VAl}$-based compounds, experimental work has seen mostly qualitative and often speculative interpretations, preventing this class of materials from tapping their full potential when it comes to applications. In this paper, the temperature-dependent Seebeck coefficient and electrical resistivity of a set of $p$-type and $n$-type samples with the composition ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{1-y}{\mathrm{Si}}_y$ are presented from 4 K up to 800 K as well as the Hall mobility and carrier concentration from 4 K to 520 K. We attempt a quantitative analysis of our data using a parabolic two- and three-band model and compare the model results with those from density functional theory calculations. Our findings show an increase of the band gap $E_{\text{g}}$ from almost zero in undoped ${\mathrm{Fe}}_2\mathrm{VAl}$ toward $E_{\text{g}}\approx 0.1\mathrm{eV}$ with increasing Ta substitution, consistent with results from first-principles calculations. Due to the resulting enhancement of the Seebeck coefficient, the maximum power factor is boosted up to 10.3 mW/${\mathrm{mK}}^2$, which is, to the best of our knowledge, the highest value among $n$-type bulk semiconductor systems reported near room temperature up until now. We further show that for the $p$-type ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x\mathrm{Al}$ compounds, the dominant scattering mechanism of electrons is intrinsically different compared to the $n$-type samples, for which acoustic phonon scattering can well describe the temperature-dependent Hall mobility in a broad range of temperatures.

Figure 1

Calculated electronic band structures at equilibrium volume (VASP-GGA) of (a) ${\mathrm{Fe}}_2\mathrm{VAl}$ and (b) ${\mathrm{Fe}}_2\mathrm{TaAl}$ with and without spin-orbit coupling (SOC). Energy-derivative of the Fermi-Dirac distribution is plotted for $T=300$ and 600 K.

Figure 2

Energy-dependent electronic density of states (DOS) of (a) ${\mathrm{Fe}}_2\mathrm{VAl}$ and (b) ${\mathrm{Fe}}_2\mathrm{TaAl}$ at equilibrium volume (VASP-GGA). V contributes delocalized states at the Fermi energy while the Ta conduction band states are located far above $E_{\text{F}}$.

Figure 3

XRD powder patterns of ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{1-y}{\mathrm{Si}}_y$ (upper panel) and concentration-dependent lattice parameters taken at room temperature (lower panel). The inset sketches the crystal structure of ${\mathrm{Fe}}_2\mathrm{VAl}$.

Figure 4

Energy-dependent DOS (KKR-CPA) of ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{1-y}{\mathrm{Si}}_y$ at the respective experimental volume. The first panel (a) demonstrates a rigid-band-like shift of the Fermi energy caused by the Al/Si substitution, while panels (b) and (c) show an opening of the pseudo band gap toward the conduction band side, which is caused by the V/Ta substitution.

Figure 5

Temperature-dependent Seebeck coefficient of (a) ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x\mathrm{Al}$ and (b) ${{\mathrm{Fe}}_2\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{0.9}{\mathrm{Si}}_{0.1}$ for various concentrations of $x$. Dashed and solid lines represent least-squares fits calculated within a parabolic two-band (2PB) and three-band (3PB) model, respectively. Details are explained in the text.

Figure 6

Temperature-dependent electrical resistivity of ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{1-y}{\mathrm{Si}}_y$. Dashed lines represent least-squares fits as explained in the text.

Figure 7

Temperature-dependent charge carrier concentration of (a) ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x\mathrm{Al}$ and (b) ${{\mathrm{Fe}}_2\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{0.9}{\mathrm{Si}}_{0.1}$.

Figure 8

Temperature-dependent Hall mobility of ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{1-y}{\mathrm{Si}}_y$.

Figure 9

Schematic sketch of the V/Ta substitution effect on the electronic band structure and band gap of ${\mathrm{Fe}}_2\mathrm{VAl}$-based (a) $p$-type and (b) $n$-type samples. The energy scaling is in arbitrary units.

Figure 10

Temperature and band gap dependence of the Seebeck coefficient for a fixed position of the Fermi energy $E_{\text{F}}$ inside the valence and conduction band; calculated within a parabolic two-band model.

Figure 11

Seebeck coefficient versus carrier concentration for different concentrations of Ta in ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x\mathrm{Al}$, compared with data from literature. The solid lines are calculated within a parabolic two-band model for varying band gaps and effective masses.

Figure 12

Concentration-dependent increase of the band gap of ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x\mathrm{Al}$ obtained from various phenomenological models and DFT calculations.

Figure 13

Temperature-dependent electrical resistivity above 300 K for (a) ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x\mathrm{Al}$ and (b) ${{\mathrm{Fe}}_2\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{0.9}{\mathrm{Si}}_{0.1}$ modeled with parabolic band parameters obtained from fitting the temperature-dependent Seebeck coefficient.

Figure 14

Temperature-dependent Hall mobility of (a) ${\mathrm{Fe}}_2{\mathrm{V}}_{1-x}{\mathrm{Ta}}_x\mathrm{Al}$ and (b) ${{\mathrm{Fe}}_2\mathrm{V}}_{1-x}{\mathrm{Ta}}_x{\mathrm{Al}}_{0.9}{\mathrm{Si}}_{0.1}$; the data was least squares fitted within a parabolic two-band model assuming a valence band mass of $|m_{\text{VB}}|=m_{\text{e}}$. Various scattering mechanisms were considered for the $p$-type samples (a). A quantitative analysis is only possible for the $n$-type samples (b), where a much simpler behavior of the mobility is observed that can be well described by acoustic phonon scattering in a wide temperature range $100<T<500$ K.

Figure 15

Temperature-dependent power factor $S^2\sigma$ times temperature for several high performance $n$-type materials from literature and the present paper.