Electronic, optical and thermoelectric properties of Fe2ZrP compound determined via first-principles calculations

In this study, based on the density functional theory and semi-classical Boltzmann transport theory, we investigated the structural, thermoelectric, optical and phononic properties of the Fe2ZrP compound. The results of the electronic band structure analysis indicate that Fe2ZrP is an indirect band gap semiconductor in the spin-down state with the band gap of 0.48 eV. Thermoelectric properties in the temperature range of 300–800 K were calculated. Fe2ZrP exhibits the high Seebeck coefficient of 512 μV K−1 at room temperature along with the huge power factor of 19.21 × 1011 W m−1 K−2 s−1 at 800 K, suggesting Fe2ZrP as a potential thermoelectric material. The Seebeck coefficient decreased with an increase in temperature, and the highest value was obtained for p-type doped Fe2ZrP when the optimum carrier concentration was 0.22 × 1023 cm−3; the n-type doped Fe2ZrP had high electrical conductivity than the p-type doped Fe2ZrP. Thermal conductivity increased with an increase in chemical potential. Optical calculations illustrated that there was a threshold in the imaginary dielectric function for the spin-down channel. Spin-dependent optical calculations showed that the intraband contributions affected only the spin-up optical spectra due to the free-electron effects. Generally, the results confirmed that the intraband contribution had the main role in the optical spectra in the low energy infra-red and visible ranges of light. We also presented the phononic properties and found that these materials were dynamically stable.


Introduction
Thermoelectric (TE) effect involves direct energy conversion by electrons in materials and is thus considered an alternative and "green" energy source. The TE effect has various advantages in industrial applications. [1][2][3] The Peltier and Seebeck effects are the main TE effects. Using the Peltier effect, the TE device can cool materials. On the other hand, via the Seebeck effect, thermal energy can be transformed into electric energy, and this phenomenon is called TE power generation. 4 The performance of a thermoelectric material is described by the gure of merit ZT ¼ S 2 sT/k, where S is the Seebeck coefficient, s is the electrical conductivity, k is the thermal conductivity, and T is the absolute temperature. Thus, to realize efficient energy conversion, a favorable thermoelectric material should possess high ZT, which indicates that a high Seebeck coefficient, high electrical conductivity, and low thermal conductivity are required for achieving efficient energy conversion. [5][6][7][8] In recent years, Heusler compounds have been theoretically investigated, and their TE properties have attracted signicant attention from researchers. [9][10][11][12][13][14][15][16][17] Generally, Heusler compounds have the stoichiometric composition XYZ or X 2 YZ and crystallize in the L2 1 structure, where X and Y are transition or rareearth metals and Z is the main group element. 18 These materials are half-metallic, where one spin channel shows metallicity, whereas the other spin channels are completely semiconducting. Because of this feature, half-metallic Heusler alloys can be considered as the most important class of spintronic materials. [19][20][21][22][23] Half-metallic material alloys have been found in some kinds of materials such as full 16,[24][25][26][27][28][29] and half Heusler alloys, 15,30,31 binary compounds [32][33][34] and 2D materials. [35][36][37] Recently, in a theoretical study, the effect of Ge substitution on the thermoelectric properties of the Heusler-type alloy Fe 2 -MnSi x Ge 1Àx has been investigated by Reshak. 13 It has been reported that the Seebeck coefficient (S) for Fe 2 MnGe exhibits an n-type behavior over the entire concentration range. In contrast, Fe 2 MnSi has a positive S of up to 250 mV K À1 . Comtesse et al. 14 have reported the spin polarization TE properties of Co-based half-metallic Heusler compounds using the fully relativistic screened Korringa-Kohn-Rostoker theory. The transport coef-cients of Co-based half-metallic Heusler materials are strongly inuenced in spin polarization cases. The thermoelectric properties of the CrVNbZn Heusler compound have been investigated by Kara et al. based on the Boltzmann transport theory. 9 It has been reported that a unique sharp electronic band, with highest contribution from valence electronic states, increases the TE gure of merit. Bhat et al. have focused on the TE performance of the ferromagnetic CoFeCrAs Heusler alloy. This material presents high S and huge power factor at room temperature. 11 The thermoelectric behaviors of Ru 2 VZ (Z ¼ Si, Ge and Sn) half-metallic full-Heusler compounds have been investigated by Yalcin. 12 The TE parameters, such as Pauli magnetic susceptibility, electrical conductivity, S, thermal conductivity and power factor, were obtained by the Boltzmann transport theories; moreover, in recent years, the thermoelectric properties of Heusler compounds have been investigated via experimental studies. [38][39][40][41] In this context, Chauhan et al. produced Zr 1Àx Hf x CoSb 0.9 Sn 0.1 Heusler alloys by employing high-energy ball-milling processes and investigated the thermoelectric properties of these alloys. 38 Their method led to the production of nanoparticles, with low thermal conductivity and high gure of merit, suitable for thermoelectric applications. The thermoelectric properties of the Zr 0.5 Hf 0.5 Co 0.4 Rh 0.6 Sb 1Àx -Sn x (0.15 # x # 0.5) half-Heusler alloys synthesized using a hardened steel jar and balls have been investigated by Maji et al. 40 Their team found materials with a high power factor (800 mW K À2 ) and a low thermal conductivity (2.2 W m À1 ). An n-type half Heusler compound (HfZrCoSnSb) has been synthesized experimentally by Poon et al. 41 They succeeded in achieving a high gure of merit (1.05) at 900 C. This material was tested for application in p-n couple devices, and it showed good power generation efficiencies reaching 8.7% for the hot-side temperatures of about 700 C.
A recent study based on the density functional theory and semi-classical Boltzmann transport theory was aimed at providing more detailed information about the electrical, optical, phononic and thermoelectric behaviors of the Fe 2 ZrP half-metallic ferromagnetic full-Heusler compounds. Due to the novelty of this material, only one theoretical study has been conducted on this compound by Canko et al. 42 The electrical and magnetic properties of this material were studied by them. They have concluded that due to its high Curie temperature and sufficient chemical stability, this compound can be a suitable magnetic intermetallic material; 42 moreover, although the spinup electronic band structure is metallic, the spin-down band structure has a semiconductor behavior with the gap of 0.593 eV, and the spin-ip gap is 0.129 eV; due to this property, this compound exists in nature as well as can be synthesized experimentally. Their theoretical study indicates that the Fe 2 ZrP compound may exhibit signicant promise for application in spintronic devices. To complete their study, the thermoelectric and optical properties of this material were examined for the rst time in the present study.

Computational details
In this study, calculations were performed using density functional theory plane waves and pseudopotentials via the Quantum ESPRESSO package. 43 The exchange-correlation term was considered by the Perdew-Burke-Ernzerhof (PBE) functional. 44 Moreover, generalized gradient approximation (GGA) and ultraso pseudopotentials (US PPs) were utilized. The energy cut-off for the expansion of the wave-functions was set to 30 Ry (due to ultraso pseudopotentials and lattice symmetry, this energy was suitable for achieving base-state energy). The electronic wave function was expanded with the energy cutoff value of 300 Ry for charge density. Brillouin zone integration was performed over the Monkhorst-Pack 45 10 Â 10 Â 10 meshes. The lattice constant of the Fe 2 ZrP compound was optimized until the total energy converged to at least 10 À8 Ry. Structure optimization was performed based on variable-cell (vc-relax) calculations. Considering the symmetry structure of this compound, four atoms were used in the simulation, which has been discussed in more detail in the next section. The TE properties of the Fe 2 ZrP compound were investigated with the BoltzTraP code. 46 The denser k-mesh of 24 Â 24 Â 24 was used for the calculations of the TE properties such as Seebeck coef-cient, electrical conductivity, thermal conductivity, specic heat and magnetic susceptibility. The Seebeck coefficient S is related to carrier concentration via the Mott formula as follows: 47 where E F is the Fermi energy, q is the electron charge, n is the carrier concentration and k B is Boltzmann constant. Electrical conductivity is related to carrier concentration as follows: 48 where m is the electron mass, s is the relaxation time and n is the carrier concentration. Electronic specic heat is related to carrier concentration and chemical potentials as follows: 49 Pauli magnetic susceptibility is related to carrier concentration and chemical potentials as follows: 49 where m B and m 0 are the Bohr magneton and vacuum permeability, respectively. To obtain the phonon spectrum and the phonon density of states (PhDOS), herein, eight dynamical matrices were calculated using the (4 Â 4 Â 4) q-point mesh. To calculate the optical properties of the compound, random phase approximation (RPA) was used to derive the imaginary part of the dielectric function: 50 where |iki represents the state vector for the initial position and |fki represents the state vector for the nal position. f i k and f f represent the Fermi distribution functions of the occupied and unoccupied states, respectively. The real part of the complex dielectric function was expanded from the imaginary part using the Kramers-Kronig relations as follows: 51-54 where Pr denotes the Cauchy principal part of the integral. To achieve accurate optical spectra, it is necessary to perform optical calculations with a highly dense rst Brillouin zone. 55 Thus, the 58 Â 58 Â 58 highly dense k-mesh was considered in our optical calculations.
Total energies versus lattice constant were calculated for both the L2 1 and the X a structures, and the results for the ferromagnetic (FM) and non-magnetic (NM) states are shown in Fig. 2. According to this gure, ferromagnetic order conguration in the L2 1 structure was found to be the most stable ground state phase as compared to other phases. The calculated values of lattice parameter (a 0 ), bulk modulus (B 0 in GPa) and the pressure derivative of bulk modulus ðB 0 0 Þ at the equilibrium lattice constant are presented in Table 1. In the previous study, the total energies were plotted in terms of the lattice volume, and it was concluded that the ferromagnetic order conguration in the L2 1 structure was the most stable phase; this conrmed the results of the present study. 42

Vibrational properties
The thermoelectric properties of materials are oen due to the movement of phonons. Materials that have a positive phonon frequency are thermally stable. In previous theoretical studies, to ensure thermoelectric properties, phonon properties were investigated rst. [58][59][60][61][62] The calculated phonon dispersion curve along the W-L-G-X-W-K directions and phonon density of states (PhDOS) are shown in Fig. 3. The calculated results show that the Fe 2 ZrP crystal is dynamically stable at zero pressure as no negative frequencies (imaginary modes) exist in the entire Brillouin zone. As observed from Fig. 3a, three vibrational modes below 0.5 THz are acoustic branches, and the remaining vibrational modes are optical modes. The number of optical modes is 3N-3. Therefore, the Fe 2 ZrP crystal with N ¼ 4 atoms in the primitive cell exhibits three acoustic and nine optical modes. The acoustic bands are contributed by the Zr element due to its larger atomic mass. 63 The highest frequency at G point   is about 1.4 THz. According to Fig. 3b, there is no phonon anomaly in the phonon density of states. The heavier atoms are at low frequencies in the range of 0-0.5 THz.

Electronic band structure and density of states
In this subsection, the band structure and density of states of the Fe 2 ZrP compound have been discussed. Fig. 4 illustrates the      (Fig. 4b). We found the direct band gap of about 0.485 eV near the high symmetry direction G point. It is obvious that Fe 2 ZrP is metallic and semiconductive in the majority and minority spin channels, respectively. This suggests that the Fe 2 ZrP compound exhibits half-metallic ferromagnetic properties. In addition, Canko et al. have shown that the Fermi level is located within the band gap of the spin-down channel but crosses the valence band of the spin-up channel; this is in accordance with the ndings of the present study. They found a direct band gap near the high symmetry direction G. 42 To understand the nature of the electronic states of the Fe 2 ZrP compound at its equilibrium lattice constant, the spinpolarized total density of states (DOS) and partial density of states (PDOS) are displayed in Fig. 5-8. In the previous study on this material, only DOS was examined; on the other hand, in the present theoretical study, in addition to DOS, PDOS was studied in more detail for the better understanding of the electronic structure of this compound. 42 The energy with respect to the Fermi level is signied by a dashed line. As shown in Fig. 5, in the valence band near the Fermi level, the minority (spin-down) and majority (spin-up) spins are semiconductor and metallic, respectively. This conrms that the compound has a halfmetallic behavior. According to Fig. 6a, both spin channels mainly originate from the Fe-3d states, with a small contribution from the Zr and P atoms with s and p orbitals. The Fe-4s, Zr-5p, Zr-4s and P-1s states have a slight effect on the formation of the half-metallic band gap. The transition metal Fe and Zr-3dstates make the main contributions to both spin congurations in the energy range from À4 eV to 4 eV. The electrons at E F are fully polarized as the density of spin-up or spin-down channels equals zero. For the Fe 2 ZrP compound, the energy gap located at E F leads to a 100% spin polarization.
The on-site spin-polarized valence charge density was calculated and is illustrated in Fig. 9 to discuss the origin of magnetic properties in more details. In addition, we selected a crystal direction and made all atoms to lie in this direction to compare charge accumulation between different atoms. As can be observed from Fig. 9, there is an exchange splitting in the valence charge density for all atoms. However, the main difference between spin-up and -down channels is related to the Fe atom. Therefore, similar to other full Heusler alloys X 2 YZ, the main contribution to the magnetic properties is provided by the X (herein, Fe) atoms. The spin-polarized total and atom- Fig. 8 Spin-polarized partial density of states for the P atom in the Fe 2 ZrP compound. Fig. 9 The on-site spin-polarized valence charge density of Fe 2 ZrP elements.
projected DOS of the Fe 2 ZrP compound are in agreement with a previous study. 42

Thermoelectric properties
The TE properties were calculated in the constant relaxation time approximation within the semi-classical Boltzmann theory using the Boltztrap package. 46 The calculated properties were plotted for three considered temperatures: 300, 600, and 800 K. Fig. 10 presents the S of the Fe 2 ZrP compound as a function of chemical potential (m) in the range from À2 eV to 2 eV (Fig. 10a) and carrier concentration (Fig. 10b). Fig. 10a shows two peaks, which are located at the chemical potentials of À0.65 and À0.55 eV. The Seebeck coefficient inclined rapidly to zero outside this range. As the temperature increased, S decreased because of the increase in thermal energy. This indicates that this material has a good thermoelectric performance. The maximum value of S is 512 mV K À1 at 300 K. For a higher temperature (800 K), S is slightly decreased to 260 mV K À1 . The negative and positive S peaks are À606 and 512 mV K À1 at 300 K, À324, and 301 mV/K at 600 K and À253 and 260 mV K À1 at 800 K, respectively. The positive and negative values of the chemical potential m indicate that the dopants are electrons (n-type) and   holes (p-type), respectively. According to Fig. 10b, the maximum value of S is obtained for p-type doping, and the optimum carrier concentration is 0.22 Â 10 23 cm À3 . According to eqn (1), S at different temperatures is higher for smaller concentrations. In Table 2, a comparison between the present theoretical study and previous studies is made. [64][65][66][67] According to this table, as the temperature increases, S decreases, and the Fe 2 ZrP compound has suitable values for thermoelectric application. The experimental results of other studies related to the Seebeck coefficient of Heusler compounds are presented in Table 3. 38,39,41 By comparing the Tables 2 and 3, we concluded that the Fe 2 ZrP compound had good potential for experimental production. Fig. 11 shows electrical conductivity (s/s) as a function of chemical potential (Fig. 11a) and carrier concentration (Fig. 11b) at different temperatures. Unlike the Seebeck coefficient, the electrical conductivity displays similar behavior at all temperatures.
According to Fig. 11a, the electrical conductivity increases with an increase in chemical potential. With an increase in chemical potential, the carrier concentration increases, and an increase in mobility increases the conductivity. The inset image in Fig. 11a shows that electrical conductivity is zero in the range from À0.49 to À0.73 at 300 K. As shown in Fig. 11b, the n-type doped compound has higher electrical conductivity than the ptype doped compound. According to eqn (2), the electrical conductivity increases with an increase in carrier concentration. The inset image in Fig. 11b shows that s/s is zero at 300 K in the p-type doping area, where the carrier concentration is about 0.22 Â 10 23 cm À3 . Fig. 12 displays the electronic power factor values (S 2 s) as a function of chemical potential relative to the Fermi level (Fig. 12a) and carrier concentration (Fig. 12b) at different temperatures. This quantity investigates the efficiency of the thermoelectric materials. According to Fig. 12a, as the temperature increases, power factor also increases. The maximum value of power factor is 19.21 Â 10 11 W m À1 K À2 s À1 for negative chemical potential at 800 K. At room temperature, the power factor is slightly decreased to 4.43 Â 10 11 W m À1 K À2 s À1 . As shown in Fig. 12b, the maximum value of power factor is located in the p-type doping area (0.22 Â 10 23 cm À3 ), which is higher than that of the n-type doping area (8.61 Â 10 11 W m À1 K À2 s À1 ). Fig. 12 The power factor of the Fe 2 ZrP material as a function of (a) chemical potential and (b) carrier concentration. Fig. 13 The electronic thermal conductivity of the Fe 2 ZrP compound as a function of (a) chemical potential and (b) carrier concentration. Fig. 13 displays the electronic thermal conductivity (k/s) as a function of chemical potential (Fig. 13a) and carrier concentration (Fig. 13b) at three constant temperatures (300, 600 and 800 K). According to this gure, as the temperature increases, thermal conductivity also increases. To increase the thermoelectric properties, the materials must have large S, high electrical conductivity, and low thermal conductivity. 68 Therefore, the optimum temperature to obtain lower ke/s is 300 K. According to Fig. 13a, the thermal conductivity increases with an increase in chemical potential. The thermal conductivity is zero in the range from À0.68 to À0.55 at 300 K. As shown in Fig. 13b, the n-type doped compound has higher thermal conductivity than the p-type doped compound. Moreover, the electrical conductivity increases with an increase in carrier concentration. The inset image in Fig. 13b shows that k/s is zero at 300 K in the p-type doping area where the carrier concentration is about 0.22 Â 10 23 cm À3 . Fig. 14 displays the electronic specic heat (c) as a function of chemical potential (Fig. 14a) and carrier concentration (Fig. 14b) at different temperatures. According to this gure, as the temperature increases, specic heat also increases. According to eqn (3), the specic heat increases with an increase in carrier concentration and chemical potential. According to   This journal is © The Royal Society of Chemistry 2019 Fig. 14a, the maximum value of the specic heat is 11.22 J (mol K) À1 for negative chemical potential at 800 K. At room temperature, the specic heat is decreased to 4.81 J mol À1 K À1 . As shown in Fig. 14b, the maximum value of specic heat is obtained in the p-type doping area (11.19 Â 10 23 cm À3 ), which is higher than that of the n-type doping area (8.7 J mol À1 K À1 ). The inset image in Fig. 14b shows that the specic heat is zero at 300 K in the p-type doping area where the carrier concentration is about 0.22 Â 10 23 cm À3 . Fig. 15 exhibits the Pauli magnetic susceptibility (c) as a function of chemical potential (Fig. 15a) and carrier concentration (Fig. 15b) at different temperatures. According to Fig. 15a, the Pauli magnetic susceptibility displays an almost similar behavior at all temperatures except near the Fermi level. At this point, as the temperature increases, c decreases. The maximum value of c is 38.65 Â 10 À10 m 3 mol À1 at 300 K. For a higher temperature (800 K), c is slightly decreased to 27.12 Â 10 À10 m 3 mol À1 . As observed from the inset image shown in Fig. 15a, the Pauli magnetic susceptibility is zero in the range from À0.68 to À0.55 at 300 K. According to eqn (4) and Fig. 15b, the Pauli magnetic susceptibility increases with an increase in carrier concentration. As shown in Fig. 15b, the maximum value of specic heat is obtained in the n-type doping area (À0.08 Â 10 23 cm À3 ). The inset image in Fig. 15b shows that the Pauli magnetic susceptibility is zero at 300 K in the p-type doping area where the carrier concentration is about 0.22 Â 10 23 cm À3 . Fig. 16 displays the electronic gure of merit (ZT) values of the Fe 2 ZrP compound as a function of chemical potential at three constant temperatures (300, 600 and 800 K). This quantity investigates the efficiency of the thermoelectric materials. According to this gure, as the temperature increases, ZT decreases. At all temperatures, ZT is low where chemical potential is negative. According to this gure, the best temperature for thermoelectric applications is 300 K because ZT has a good value in the negative and positive elds of chemical potential. In Table 4, a comparison between the present theoretical study and previous studies is shown. [64][65][66]69 According to the table, the Fe 2 ZrP compounds are good thermoelectric materials. The experimental results of other studies on Heusler compounds based on the gure of merit are also presented in Table 5. 38,39,41 By comparing the Tables 4 and 5, we conclude that the Fe 2 ZrP compounds have a suitable gure of merit.

Optical properties
Herein, the optical properties of the Fe 2 ZrP compound have been studied using a random phase approximation (RPA) method. To investigate the optical properties of a half-metallic material, it was necessary to consider both intraband and interband contributions in our calculations; due to their transitional nature, the intraband transitions affected only the infra-red and visible ranges of light in the optical spectra. 70 The spin-dependent imaginary and real parts of the dielectric function are illustrated in Fig. 17. The electronic band structure exhibits that the spin-up channel has a metallic behavior, whereas the spin-down channel has a semiconductive behavior. Therefore, intraband transitions occur only for the free electrons of the spin-up channel. As a result, the intraband, interband and total contributions have been plotted only for the spin-up channel. As can be observed, the intraband transitions have the main role in the range of 0-2 eV in the real and imaginary parts of the spin-up channel. This trend refers to the free electron effect in the spin-up channel. For metallic materials in low frequency range, the refractive index n(u) is lower than the extinction coefficient k(u); thus, the real part of dielectric function has a negative value, 3 1 ¼ n 2 À k 2 < 0. The imaginary part of the dielectric function refers to optical  absorption from the occupied states to the unoccupied states. In Fig. 17c, we can see a high value of absorption from zero energy to 2 eV (free electron absorption), whereas there is an absorption threshold in the spin-down imaginary part spectrum (Fig. 17d) that is according to the half-metallic band gap structure of the spin-down channel. Furthermore, the total spectra, i.e. spin up plus spin down spectra, of optical conductivity and reectivity were calculated and are plotted in Fig. 18 with and without intraband transitions. The results indicate that due to the partially occupied states in the spin-up band structure of Fe 2 ZrP around the Fermi level, the intraband contribution has the main role in the infrared range of optical spectra. This phenomenon leads to a high reectivity spectrum in the infra-red range of incident light.

Conclusion
Herein, the electronic, phononic and thermoelectric properties of the Fe 2 ZrP compound were calculated using the DFT and Boltzmann transport theory calculations. It was found that this material was half-metallic with the indirect band gap of 0.485 eV along the G V -G C symmetry line. The phonon density of states and phonon dispersion curves conrm that the Fe 2 ZrP compound is dynamically stable. The results of Boltzmann calculations showed that the Fe 2 ZrP compound exhibited better thermoelectric properties aer p-type doping than aer n-type doping; the highest S value was obtained at the temperature of 300 K upon p-type doping. The thermoelectric and phononic properties of the Fe 2 ZrP compound were considered for the rst time in this study. The maximum value of the power factor reaches 19.21 Â 10 11 W m À1 K À2 at the hole concentration of 0.22 Â 10 23 cm À3 and about 8.61 Â 10 11 W m À1 K À2 s À1 at the electron concentration at 800 K. The electrical and thermal conductivity increase with the increasing chemical potential. This study shows that the Fe 2 ZrP compound has a good potential for application in the thermoelectric eld. The optical calculations conrm that the intraband contribution has the main role in the low energy ranges (infra-red and visible) of optical spectra.

Conflicts of interest
There are no conicts to declare.