The accurate prediction of the properties of materials paves the way for the possible experimental simulation of such materials. The multilateral needs for materials in a technology-driven world are nearly insatiable, and the rate at which novel viable materials are discovered seems to be lagging behind the demand. The need for more innovative materials in electronic devices, agriculture, medicine, the environment, and energy is growing astronomically. Among these needs, energy is like a lifeline without which it will be impossible to meet other needs effectively. This background justifies the need for continued research for affordable, available, environmentally friendly, and easily fabricated materials to supply the growing energy demand. Many materials have competed in the energy supply chain [1,2]. However, most of these materials present one flaw or the other that leaves much to be desired.

An energy source that has proved to be sustainable is solar energy. One of the materials of interest in solar energy generation is the semiconductor materials that facilitate electron transport in solar systems [3,4]. Therefore, a lot of semiconductor materials have been reviewed in previous literature, among which are silicon and its alloys [5], germanium and its alloys [6], chalcogenides [7], pnictides [8], carbon nanotube and its variants [9,10], MXenes [11-13], and Heusler alloys [14-18].

Heusler alloy [19] and has since presented remarkable properties that have found economic applications on a commercial scale as energy harvesters [20], magnetic read access memories (MRAM) [21], sensors [22], spintronics, actuators [23], thermoelectric materials [24-26], solar cells [27], and magnetic cooling devices [28]. The Heusler family of alloys is attractive because a simple valence count can give valuable details of the possible property of such alloys; in addition, it is easy to achieve desired properties of the alloy by tuning. Heusler alloys comprise two transition metals and an element from the main group. There are three fundamental variants of Heusler alloys: the full Heusler alloys, Half Heusler alloys and the quaternary Heusler alloys. The half Heusler alloys are ternary alloy that leaves one interpenetrating layer vacant; they combine the zinc-blende and the rocksalt structure. There are reports in the literature on the use of half Heusler alloys ranging from the magnetic, ferromagnetic and non-magnetic. The 18-valence electrons are predicted to be semiconductors. These semiconductors find application in electronic devices as piezoelectric materials [29], thermoelectric materials and generators [30], and many more.

Surucu et al. [31] reported on the structural, electronic, magnetic and lattice dynamics of XCoBi (X=Ti, Hf, Bi) half Heusler alloys from first principles in the MgAgAs crystal structure using the Vienna Abinitio Simulation Package (VASP) with the generalised gradient approximation (GGA) and the Purdue-Burke-Enhzerhof (PBE) exchange-correlation functional. They reported the -phase of the three alloys to be stable, non-magnetic semiconductors with indirect bandgaps. Ma et al. [32] also reported the first principles' phonon dispersions, electronic properties, and the thermoelectric properties of MCoBi (M = Ti, Hf, Zr) alloys. They used the semi-classical Boltzmann transport theory with the relaxation time approximation and Slack model. They reported that the M transition metal affects the alloy's lattice thermal conductivities and electronic transport properties. They also posited that the p-type semiconductors exhibited a higher figure of merits (ZT) compared to their n-type counterparts at 800 K. The mass numbers of Ti and Co compared to that of Bi element plays a vital role in understanding the behaviour of the alloys. The spin-orbit coupling was, however, not taken into cognisance in the work of Sucuru et al. and Ma et al. To consider this effect, the spin-orbit coupling (SOC) must be factored in. Hence, in this work, we have computed the structural, electronic, and thermoelectric properties of TiCoBi with and without spin-orbit coupling.

Computational Method

In this study, we used the quantum espresso code (QE) [33] to study the structural, electronic, and thermoelectric properties of TiCoBi 18 valence electron half Heusler alloy from the first principles. All properties were computed using QE and its post processors. The potentials for calculations were constructed using the projector augmented wave (PAW) [34,35] method, and we used the generalised gradient approximation (GGA) and the Perdew-Burke-Erzhenhof exchange and correlation between electrons [36]. We used the Monkhorst-Pack scheme [37] to construct an 8×8×8 grid for the electronic structure calculation. A converged value of 70 eV was used as the kinetic energy cutoff of the plane-wave basis function. We relaxed each atom in the unit-cell using the Broyden–Fletcher–Goldfarb–Shanno (BFGS) to obtain the equilibrium configuration for the atomic positions. We used a denser k-point mesh of (20×20×20) grid with a tetrahedra occupation to calculate the electronic density of states. We also sampled the alloys' behaviour at equilibrium temperature as functions of the volume and pressure by fitting the result obtained from the total energy calculation to the Birch-Murnaghan equation of state [38-40]. As reported in the literature, we treated the compound as a ferromagnetic system in the fcc crystallised phase. We selected dense high symmetry k-points for the FCC structure using (X-window) Crystalline Structures and Densities (XCrySDen) package [41] for the band structure calculations. We used the Fermi-Dirac distribution to obtain the Fermi level from the electronic density of states distribution and computed the transport coefficients according to the Boltzmann equation. The relaxation time approximation is used, which assumes that the relaxation time is constant regardless of energy.

Structural Properties

TiCoBi is a non-magnetic semiconductor which crystallises in the face-centred cubic structure in the space group  and space number 216. The Three possible atomic positions for a half Heusler alloy were considered in this investigation. The configurations are termed the  phases for the alloy. Of the three configurations, the γ - phase proves to have the lowest ground-state energy making it the most stable state, as can be seen in Fig. 1. Hence, the alloys crystallize in the MgAgAs structure with X, Y, Z atoms occupying the atomic positions (0, 0, 0), (0.5, 0.5, 0.5) and (0.25, 0.25, 0.25) respectively. TiCoBi obeys the Slater-Pauling rule for hH alloys and crystallises as a non-magnetic semiconductor. We confirmed the rule using the relation for 18 valence electrons as M_t = Z_t – 18 as proposed by Slater & Pauling, where Z_t is the alloy's valence number, and M_t is the magnetic moment per unit formula of the alloy. In our case, Z_t is 18.

The electronic configuration of the elements investigated is 3d²4s² and 3d⁷4s² for titanium and cobalt, respectively, while bismuth is 6s²6p³. The equilibrium lattice parameter in Angstrom is 6.02. The lattice parameter agrees with the value reported by Surucu et al. [31]. However, there is no experimental report for comparison.

We fitted the lattice optimisation for energy and volume to the Murnaghan equation of state using the relation in Eqn. 1: 

Where  equilibrium values of energy and volume, respectively, without pressure, while B and B denote the bulk modulus and its derivative. The relationship between the energy and volume is shown in Fig. 1. The bulk modulus value, pressure derivative of bulk modulus, volume, ground state energy, energy bandgap, and lattice constant using PBE-GGA are presented in Table 1 for computations with and without spin-orbit coupling. The compound's magnetic moments are zero for both cases among the interacting elements; this confirms that TiCoBi is non-magnetic.

We investigate the alloy's formation energy to ascertain an alloy's structural stability and establish the possibility of experimentally simulating. Negative formation energy supports simulation, while positive formation energy prohibits experimental simulation. We calculated the formation energy by subtracting the energy of each element from the energy of the bulk compound using the equation; The formation energy obtained for TiCoBi is -0.27 eV and -0.31 eV without SOC and with SOC, respectively, establishing both stability and the possibility of  experimental simulation of the alloy.

Figure 1: Lattice optimisation for the minimum energy versus volume as fitted to the  Murnaghan equation of state for the  phases of TiCoBi half Heusler alloy.

Table 1:   Lattice constant (a₀), ground state energy E, bulk modulus  pressure derivative of the bulk modulus  volume V₀, formation energy  and energy bandgap (E_g) of TiCoBi compound in the most stable state using PBE-GGA.

compound	a0  (Å)	E (Ry)	(GPA)		V0 (a.u.)3	(eV)	Eg (eV)
TiCoBi   TiCoBiSOC	6.02 6.033 31 6.06	-1252.20   -1261.43	125.40   127.30	7.47   7.88	367.57   351.41	-0.27   -0.31	0.903   0.9337

Electronic Properties

We present results for the electronic structure with and without SOC of the TiCoBi compound. The results include the density of states (DOS) and electronic band structure calculations using PBE-GGA. The FCC structure's band structure calculation is along the high symmetry path Γ→X→W→K→Γ→L with a dense k-point. From the band structure and DOS in Figure 2, there is a superposition of the L symmetry on the Γ symmetry. The L symmetry terminates the band structure without SOC; meanwhile, the band structure with SOC is shifted by one more band to the next L. We also observe that there is a downward shift in the bands in the valence band maximum and conduction band minimum compared to the position of the bands computed without SOC. Furthermore, the computation of the SOC gives rise to a minimal increase in the band gap from 0.903 eV to 0.9337 eV. The flatness of the bands at Γ and X suggests covalent bonding between the orbitals of Ti and Co.

Furthermore, the indirect band gap at Γ→X shows that Ti's d states are delocalised in the region of the Fermi energy. On the other hand, we observe shallow or flat bands between W→K, indicating ionicity and low group velocity resulting from the s and p orbitals' localisation between Bi and Co, respectively.

Figure 2 also shows the electronic density of states for TiCoBi alloy in the non-magnetic state. The shift arising from the effect of the spin-orbit coupling is seen in the electronic density of states. The SOC splits the s, p, and d orbitals of Ti and Co into two orbitals, each giving rise to 12 orbitals compared to 6 seen in calculations without SOC. On the other hand, the s orbital is not split between Bi but the p and d orbital again split in two resulting in 5 orbitals compared to the 3 seen in computations without SOC. The size of the bandgap supports possible charge mobility.

Figure 2: Electronic band structure and density of states of TiCoBi with and without SOC using PBE-GGA.

Thermoelectric Properties

We have calculated the power factor per unit relaxation time, electrical conductivity per unit relaxation time, and the Seebeck coefficient at temperatures of 300 K, 600 K, 900 K, and 1200 K versus chemical potential dependence of TiCoBi hH alloy with and without SOC. The efficiency of a TEM material is computed using the dimensionless figure of merit described by the relation  S and T represent the Seebeck coefficient and absolute temperature, respectively, while σ and K are the electrical conductivity and lattice conductivity, respectively. The mutual dependence of the parameters requires that the material have a high-power factor and a low thermal conductivity for a TEM to  perform optimally. We obtain the relation for the power factor using  retain their usual meaning.

In figure 3, we show results for the power factor per unit relaxation time, electrical conductivity per unit relaxation time, and Seebeck coefficients as functions of the free carrier concentration of and respectively for both n-type and p-type phases. The Seebeck coefficient peaks at  with a free carrier coefficient of -0.03 cm^-3 for both n and p types at a temperature of 300 K, at the same temperature, the electrical conductivity per unit relaxation time also attains a peak of  for the hH alloy with SOC at -0.1 cm^-3. For the power factor, however, the highest peak of for calculation with SOC is obtained at a high temperature of 1200 K and free carrier concentration of -0.075 cm^-3. The Seebeck coefficient attenuates to zero with the increase in carrier concentration.

We, however, observe a different behaviour for calculations without SOC. The peak of the electrical conductivity per unit relaxation time, as shown in Figure 4, is shifted from -0.1 cm^-3 to -0.025 cm^-3 at 300 K, indicating that at moderate temperature, a lower concentration is required to increase the electrical conductivity. The maximum value of the electrical conductivity is reduced to about On the other hand, the peak of the Seebeck coefficient is shifted from -0.03 cm^-3 to 0.04 cm^-3, hence promoting the p-type semiconductor, with the Seebeck coefficient increasing from  The power factor is shifted to around the Fermi level at zero, and we see the peak around zero concentration to be . From the results, we can conclude that consideration of the SOC enhances the transport properties of the alloy because there is a marked increase in the power factor per relaxation time from in calculations without SOC to in calculations with SOC.

Figure 3: The (a) power factor per unit relaxation time, (b) electrical conductivity per unit relation time and (c) Seebeck coefficient at temperatures of 300 K, 600 K, 900 K, and 1200 K versus chemical potential dependence of TiCoBi hH alloy with SOC.

Figure 4: The (a) power factor per unit relaxation time, (b) electrical thermal conductivity per unit relation time and (c) Seebeck coefficient at temperatures of 300 K, 600 K, 900 K, and 1200 K versus chemical potential dependence of TiCoBi hH alloy without SOC.

We have explored the structural, electronic, and transport properties of TiCoBi half Heusler alloy with and without the spin-orbit coupling. We investigated the structural and electronic properties using the generalised gradient approximation based on the density functional theory. The results we obtained for the fundamental lattice parameter for the  phase align with results from previous work for calculations without SOC. Still, the lattice bandgap is wider with SOC calculation. The results of the transport properties calculated using the Bolztrap code suggest that the n-type TiCoBi is a better thermoelectric material than the p-type. The n-type semiconductor result is obtained when the spin-orbit coupling is considered in the calculation; otherwise, the p-type semiconductor is promoted. However, the transport property is enhanced in the n-type with spin-orbit coupling considered in the calculation. This conclusion is based on the marked increase in the power factor per relaxation time from in the calculation without SOC to in calculation with SOC.

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