The electronic and thermoelectric properties of a d²/d⁰ type tetragonal half-Heusler compound, HfSiSb: a FP-LAPW method

HfSiSb is a hypothetical compound and we have assumed that it has the same structure as Ge based HfGeSb. HfGeSb crystallizes in a ZrSiS-type structure possessing tetragonal-phase with P4/nmm space group. A unit cell of HfSiSb consists of six atoms occupying three two-fold positions: Hf on 2c (1/4, 1/4, 0.2420), Si on 2a (3/4, 1/4, 0) and Sb on 2c (1/4, 1/4, 0.6112) [11], see figure 1. Structural optimization based on Murnaghan's equation of state [29] was performed to obtain the relax structure with minimum energy.

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The output of the volume optimization gives the equilibrium lattice constant, bulk modulus, its pressure derivative, etc. The smooth energy versus volumes curve is obtained by fitting the data in Murnaghan's equation of state (see figure 1). Unfortunately, to the best of our knowledge, there are no experimental results for the comparison of the calculated lattice constant, bulk modulus, etc. Thus we have compared it with that of the available theoretical ones, as shown in table 1. The change in the parameters are given by Δ. Our calculated value of the bulk modulus is 111.223 GPa, its pressure derivative is 4.644 and the optimized lattice constants are, a  =  3.746 Å and c  =  8.618 Å. It is seen that the present structural parameters of HfSiSb compound are in excellent agreement with the theoretical values of [11].

The optimal lattice constants obtained were used to study the electronic properties. To determine the thermoelectric properties it is necessary to obtain a good description of the electronic structure. The band structure, along with the DOS, is plotted in figure 2. The contribution of different electronic states in the valance and the conduction band determines the electronic property of the material. This is shown with the help of the total and partial DOS in figure 3.

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It can be clearly seen that no band gap is observed, implying that HfSiSb is a metal. In the Fermi level E_F, there is no hybridization between the Hf-d² and Si/Sb-d⁰ states and, hence, a band gap is missing. This is because of dispersed bands that are a result of the overlap between p-states of Hf with that of Si and Sb. The band structure is related to the Seebeck coefficient according to the formula ${{E}_{{\rm Gap}}}=2eT{{S}_{{\rm max}}}$ (${{E}_{{\rm Gap}}}$ is the band gap energy, e is the electronic charge, T is the absolute temperature and S_max is the maximum Seebeck coefficient) [30]. HfSiSb should have zero Seebeck coefficient as the observed band gap is zero. However, our calculated value of the Seebeck coefficient is of the order of 10⁻⁵. The band curves near the Fermi level are quite sharp in almost all directions except in the XM direction, where the band curves are flat in the valance band region. Sharp curves are beneficial for the Seebeck coefficient and this may be the reason for non-zero value of the Seebeck coefficient. From the partial DOS plots (figure 3) the maximum contribution to the total DOS is due to the Hf-d² states (~4.2 eV) and there is an appreciable contribution from the p states of Si and Sb. Since Hf has the highest contribution near the Fermi energy, therefore the sharp peaks in the DOS are mainly due to d-state electrons. The DOS near the Fermi energy of HfSiSb is about 8 electrons eV⁻¹, which is the main reason for high electrical conductivity. Thus, the electrons at the top of the valance band and those at the bottom of the conduction band mostly come from the d-orbital and some from the p-orbital. The sharp peak at 0–2 eV below the Fermi energy is due to the occupied dt_2g (d_xy, d_xz, d_yz) states while the peaks above the Fermi energy at around 2–6 eV are due to the unoccupied d_eg (d_z2, d_x2+y2) states. In comparison to the DOS results, it can be seen from the band structure results that the bands from 1.4 eV to 8 eV are due to the d-state electron of the Hf atom. The dense band at the vicinity of the Fermi energy (E_F) is mostly due to the d-states of Hf atoms, whereas the thin bands away from E_F are mainly due to the Sb-p and Si-p states. In the valance band region, from about  −7.2 eV to 1.6 eV, the bands are dominated by the d-states of Hf atoms; the p-states of Si and Sb have a very low contribution. In the conduction band region, the contribution to DOS by all Hf, Si and Sb atoms is very small and is almost equal.

A closer look at figures 3(b) and (c) shows that the maximum contribution of Si-p states (mainly from the p_z orbital) to the total DOS is at  −1.6 eV, whereas Sb-(p_x  +  p_y) contributed at  −1.8 eV, below E_F. Still, the sharp peaks at (1–2 eV) below E_F are very small compared to the peaks due to Hf-d states. Thus, the hybridization between Hf-d² and Si/Sb-d⁰ states are missing, and this may be the reason for deformed and dispersed DOS at E_F. The band structure is directly related to the Seebeck coefficient, and from the Seebeck coefficient we can calculate the other transport coefficients. Unfortunately, there are no works reported on electronic properties of HfSiSb, either theoretical or experimental, for comparison.

The thermoelectric properties such as S, σ/τ, and κ/τ were calculated from the BTE in combination with the first-principles band-structure calculations using a simple rigid-band and the RTA. One of the demerit of BoltzTraP is that the relaxation time is not known. TiGeSb and TiSiSb compounds have negligible heat transport, hence it can be assumed that their relaxation time is constant [17]. Our calculation for electrical conductivity and electronic thermal conductivity are based upon the assumption that relaxation time is energy independent constant equal to 6.4  ×  10⁻¹⁵ [31]. Thermoelectric parameters were calculated along the XX (perpendicular) direction and along the ZZ (parallel) direction due to the tetragonal structure of the compound. Figures 4 and 5 gives the thermoelectric parameters of TiXSb with respect to the chemical potential (µ). The positive trend in the Seebeck coefficient represents p-type carriers, while the negative trend indicates the n-type carriers. Along both the perpendicular and parallel direction, the magnitude of the n-type trend is larger, hence making HfSiSb n-type material. In order to ensure high Seebeck coefficient, there should only be one type of carrier. Mixed conduction of n-type and p-type will cancel out the induced Seebeck voltage [32]. From figure 4, the Fermi energy of thermoelectric parameters are considered corresponding to each temperature. The Fermi energy (E_F) calculated was about 0.632 Ry. From figure 4(b) we see that at around the Fermi energy, the electrical conductivity values increase for increases in temperature along both the parallel and perpendicular directions for about 400 K temperature, after that the value decreases slightly with the increase in temperature. In figure 4(c), for the perpendicular direction, the electronic thermal conductivity increases with increasing temperatures; this is expected. For the parallel direction, the situation is a bit complex but can be assumed to have the same trend as that of the perpendicular direction. The complexity along the parallel direction may be due to lattice imperfections. The lattice thermal conductivities were estimated by the RTA–BTE, as implemented in the code Phono3py [28]. The thermal conductivity above 100 K is due to both the electronic as well as phononic contribution. Below 100 K, electronic thermal conductivity is negligible [33].

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A plot of thermoelectric parameters as a function of temperature is shown in figure 5. The calculated Seebeck coefficient is quiet low, only of the order of 10⁻⁴ at room temperature (300 K). The values being  −2.12  ×  10⁻⁴ (V K⁻¹) and -3.26  ×  10⁻⁴ (V K⁻¹) for the XX and ZZ directions, respectively. The negative sign in the Seebeck coefficient predicts the majority charge carriers are n-type. It is seen from figure 5(a) that the S value almost becomes constant for higher temperatures. This is because as the temperature increases, more electrons get excited, and the electrons tend to undergo mixed carrier concentration, thus cancelling the induced Seebeck voltage (see figure 4(a)). The electrical conductivity increases initially with the increase in temperature and then gradually decreases for high temperature values. This is because of high electronic thermal conductivity at higher temperatures. The highest ZT value is achieved at room temperature (200–300 K). The calculated ZT values at temperature range 200 K–300 K are ~0.25 and ~0.39 along the XX and ZZ directions, respectively. The calculated thermoelectric parameters at room temperature are presented in table 2.

Table 2. Calculated thermoelectric parameters at 300 K.

Thermoelectric parameters	HfSiSb (XX)	HfSiSb (ZZ)
S (V K−1)	−2.12  ×  10−4	−3.26  ×  10−4
Majority carriers	n-type	n-type
$\sigma {{(\Omega {\rm m})}^{-1}}$	7.13  ×  107	6.71  ×  106
${{\kappa }_{{\rm e}}}({\rm W}\,{{{\rm m}}^{-1}}\,{{{\rm K}}^{-1}})$	5.67  ×  102	0.59  ×  102
${{\kappa }_{{\rm p}}}({\rm W}\,{{{\rm m}}^{-1}}\,{{{\rm K}}^{-1}})$	17.65 [10]	6.56 [10]
ZT	0.31	0.25