Effect of onsite Coulomb repulsion on thermoelectric properties of full-Heusler compounds with pseudogaps

Electronic structure calculations using local density and generalized gradient (LDA/GGA) approximations for the full Heusler compound Fe${}_2$VAl show that it is a pseudogap (negative gap) system with very small density of states (DOS) at the Fermi level but rapidly rising DOS away from it, a feature that makes this compound a promising thermoelectric material. Thermopower ($S$) measurements in nominally pure and $n$-doped Fe${}_2$VAl give indeed large values of $S$ ($\sim$$-$150 $\mu$V/K at 200 K). To improve on the inadequacy of LDA/GGA in handling $d$-electron systems and to understand the origin of large thermopowers measured, we have carried out electronic structure calculations using the $\mathrm{GGA}+U$ method with several values of the on-site Coulomb interaction parameter $U$, including the ones calculated using constrained density functional theory. For the latter, we found Fe${}_2$VAl to be a narrow band-gap semiconductor with a gap of 0.55 eV. With the calculated band structures, we have studied the carrier concentration and temperature dependence of $S$ using Boltzmann transport equation in constant relaxation time approximation for both the pseudogap and the gapped cases. Comparison between theory and experiment suggests that neither the pseudogap nor the finite gap (0.55 eV) model can explain all the transport properties consistently. Therefore, treatment of $U$ beyond simple mean-field approach (done in $\mathrm{GGA}+U$) and/or inclusion of defect-induced changes in the host electronic structure might be important in understanding the experiments.

Figure 1

Band structure of Fe${}_2$VAl showing $d$ characters of Fe and V (left) and DOS (right). $E_F$ denotes the Fermi level set to be zero. The size of the symbols represents the strength of orbital characters.

Figure 2

Band structure of Fe${}_2$VAl in absolute scale for $U_{\mathrm{Fe}}$ of (a) 0 eV, (b) 1 eV, (c) 2 eV, and (d) 4 eV. The Fe-$e_g$ and Fe-$t_{2g}$ bands are pointed out by arrows.

Figure 3

Band structure of Fe${}_2$VAl in absolute scale for $U_V$ of (a) 0 eV, (b) 1 eV, (c) 2 eV, and (d) 4 eV. The V-$e_g$ bands are pointed out by an arrow.

Figure 4

Band structure of Fe${}_2$VAl obtained in different methods: (a) $\mathrm{GGA}-\mathrm{PBE}$ (Ref. 27), (b) $\mathrm{GGA}+U$ (Ref. 17), (c) mBJ (Ref. 19), and (d) PBE0 (Ref. 38).

Figure 5

Thermopower ($S$) and chemical potential ($\mu$) versus carrier concentration ($n_e,n_h$) in the absence of $U$ (the pseudogap case) for $n$- and $p$-type doping. The Fermi level is chosen to be 0.

Figure 6

$S$ as functions of temperature obtained using the GGA band structure ($U=0$ eV, the pseudogap case) at different concentrations: for $n$-type (closed symbols) and $p$-type (open symbols) doping.

Figure 7

$S$ as functions of $T$ for $\mathrm{GGA}+U$ calculation ($U_{\mathrm{Fe}}=4$ eV and $U_{\mathrm{V}}=1$ eV) at different concentration: for $n$-type (closed symbols) and $p$-type (open symbols) doping.

Figure 8

Calculated $p$-type $S$ as a function of $n$ for the cases of pseudogap (GGA calculation) (solid line) and gapped ($\mathrm{GGA}+U$ calculation), $E_g=0.55$ eV (dashed line), at 300 K.

Figure 9

Experimental thermopower obtained by doping Si on Al site (symbols) [done by Skoug et al. (Ref. 14)] and theoretical curve (lines) with $U=1$ eV ($E_g=0.07$ eV). Carrier concentrations are given in units of cm${}^{-3}$.