Thermoelectric materials and their applications provide an option of improving the sustainability of electricity which is generated from the waste heat by means of thermoelectric generators that convert heat energy directly into electricity and offer many benefits like moderate efficiency, simple device structure and no moving parts. Thermoelectric-based generators and refrigerators are solid state devices in which heat energy is converted into electricity and vice versa. Thermoelectric generators have been used in niche applications such as power generation or remote power sources.^[1] Thermoelectric refrigerator is environment friendly and has long life, and potential to replace traditional air conditioner. Semiconducting alkaline earth metal compounds have been of immense interest for use in thermoelectric device applications.^[2,3] One important parameter, which evaluates the performance of thermoelectric material, is a dimensionless figure of merit ZT,^[4] which is defined as

Good thermoelectric material has high thermo power and low thermal conductivity. High performance thermoelectric material is usually considered to have ZT equal to or greater than one. Magnesium compounds Mg₂X (X = Si, Ge, Sn) and their solid solutions are of great interest as promising thermoelectric materials in a temperature range from 500 K to 900 K. These materials are semiconductors having antifluorite structure. The antifluorite semiconductor is important because it forms metal-semiconductor hybrid material.^[6] Mg₂Si and Mg₂Ge are good materials for technological applications as infrared detectors in a wavelength range of 1.2 μm–1.8 μm for optical fiber.^[7] Carbides, which have been intensively studied for more than half a century, still remain a major center of scientific and technological attention.^[8] Magnesium compounds containing Mg–C and C–C bonds are quite fascinating from both fundamental science and synthesis perspectives.^[9,10] The carbon has two Mg–C bonds and a lone pair. The two magnesium atoms are bonded by single Mg–Mg bond. The carbon center is much like an organic carbene. The nature of chemical bond presents and determines the properties of Mg₂C. There is no covalent C–C bond in Mg₂C. So these types of materials are ionic semiconductors.^[11] sp³ and sp² carbon network are intercalated with Mg^[12,13] and novel polymeric carbide.^[10] The nature of Mg–C chemical bond has great importance in polar organometallic compounds and in the covalent /ionic nature of carbanions.^[14,15] There are four magnesium carbides, which are well known as (i) tetragonal MgC₂, (ii) orthorhombic α-Mg₂C₃, (iii) monoclinic β-Mg₂C₃, and (iv) cubic Mg₂C. In the present paper, we will discuss the electronic and thermoelectric properties of cubic Mg₂C because this phase exists at high temperature and pressure. It was theoretically investigated by Cork ill and Cohen^[11] in 1993. And recently synthesized by Kurakevych et al.^[16] at high pressure conditions, this compound forms from the elements at pressure above 15 GPa and stable in ambient Fm3m conditions. Owing to small size of C atom, it has large elastic modulus, which is related to hardness. This material is good for high temperature and pressure applications.^[9] This compound is unique in the sense that carbon is stabilized in a rare C^4– ionic state, which is extremely rare for alkaline–earth metal carbides.^[16] Carbon is eight fold coordinated by magnesium, and magnesium has four fold coordinated with carbon. This compound is easily hydrolyzed by the moisture in air, so it must be handled carefully. Li et al.^[17] have investigated the electronic and lattice vibrational properties of Mg₂C by the pseudo potential method. They found that this compound is small gap semiconductor. Laref et al.^[18] reported the mechanical, electronic and optical properties of this compound. Recent study by Chernatynskiy and Phillpot^[19] revealed that the calculated thermal conductivity is 34 W/mK for Mg₂C at 300 °C. Theoretical and experimental studies of thermoelectric properties for Mg₂C system are not yet available in the literature. These calculations may serve as the guidance for researchers in future. The aim of the present work is to investigate the electronic properties by using density functional theory and thermoelectric properties by using phonon Boltzmann transport equation (pBTE) and Boltzmann transport equations.

Mg₂C is face centered cubic (FCC) structures with primitive translation vectors a = a(0,1/2,1/2), b = a(1/2,0,1/2), and c = a(1/2,1/2,0) where a is lattice parameter. The structure symmetry of this compound is , the corresponding space group is Fm-3m (group No. 225).^[20] Three in-equivalent sites can be specified in the irreducible unit cell, namely C : a(0,0,0), Mg : a(1/4,1/4,1/4), and Mg : a(1/4,1/4,3/4). This compound belongs to the antifluorite structure family and semiconductor in nature. We considered super cell 2 × 1 × 1 of Mg₂C conventional unit cell which contains 24 atoms (16 Mg atoms and 8 C atoms).

The present calculations were performed using density functional theory (DFT) with the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA)^[21] and using plane wave pseudo potential method as implemented in Quantum Espresso package.^[22] Norm-conserving pseudo potential based on the Troullier–Martins scheme were utilized to model core electrons. The cutoff for the kinetic energy was set to be 70 Ry (1 Ry = 13.6056923 eV) for the plane-wave expansion of the electronic wave function. The marzari-Vanderbilt smearing size was fixed to be 0.003 Ry. The internal lattice pressure was smaller than 0.2 kbar (1 bar = 10⁵ Pa). The Brillouin zone integration was performed over a Monkhorst–Pack^[23] 5 × 10 × 10 meshes. The lattice constant of Mg₂C was optimized until the total energy converged to at least 10⁻⁸ Ry, the forces between atoms became smaller than 10⁻⁵ Ry/bohr. A denser k-mesh of 16 × 16 × 16 was used for calculating the density of states. A very fine mesh of 58 × 58 × 58 points was used to determine the final electronic structure for use in the electronic thermoelectric properties calculations. BoltzTraP code^[24] was used to find the electronic thermoelectric properties (Seebeck coefficient (S), electrical conductivity (σ), and electronic thermal conductivity (k_e), which is based on constant time approximation and rigid band approximation [RBA].^[25,26] To calculate transport properties, the following Boltzmann semi classic equations are used

In RBA approximation the band structure remains unchanged as the Fermi level is moved up and down. Shifting the Fermi level only results in changes of electron and hole concentrations because the donor and acceptor doping levels are varied. With this code, the thermoelectric properties were calculated with respect to the relaxation time (τ). To calculate lattice thermal conductivity (k_l), ShengBTE code^[27] based on the phonon Boltzmann transport equation (pBTE) was used. This code is based on the second-order (harmonic) and third-order (an harmonic) interatomic force constants (IFCs) combined with a full solution of the pBTE and successfully predicts the lattice thermal conductivity (k_l).^[28,29] The lattice parameter and bulk modulus were calculated by computing the total energy for different volumes and fitted to Murnaghan’s equation of state:^[30]

By using first principle calculations, the electronic and thermoelectric properties of pure Mg₂C are investigated. The calculated result shows that Mg₂C exhibits the n-type conduction, i.e., the conduction is due to electrons, and also it is confirmed that Mg₂C is an indirect semiconductor. Thermoelectric studies show that Seebeck coefficient increases with increasing temperature. The electrical conductivity decreases with increasing temperature due to change of mobility. The electronic part of thermal conductivity (k_e) increases with increasing temperature and lattice contribution decreases with increasing temperature. The calculated result proves that the figure of merit (ZT) is very low.