Theoretical Study on Thermoelectric Properties and Doping Regulation of Mg 3 X 2 (X = As, Sb, Bi)

: For searching both high-performances and better ﬁts for near-room temperature thermoelectric materials, we here carried out a theoretical study on thermoelectric properties and doping regulation of Mg 3 X 2 (X = As, Sb, Bi) by the combined method of ﬁrst principle calculations and semi-classical Boltzmann theory. The thermoelectric properties of n -type Mg 3 As 2 , Mg 3 Sb 2 , and Mg 3 Bi 2 were studied, and it was found that the dimensionless ﬁgures of merit, zT , are 2.58, 1.38, 0.34, and the p -type ones are 1.39, 0.64, 0.32, respectively. Furthermore, we calculated the lattice thermal conductivity of doped structures and screened out the structures with a relatively low formation energy to study the phonon dispersion and thermal conductivity in Mg 3 X 2 (X = As, Sb, Bi). Finally, high thermoelectric zT and ultralow thermal conductivity of these doped structures was discussed.


Introduction
Thermoelectric (TE) materials are non-polluted and can directly convert heat energy and electric energy to each other, and have a wide range of applications in areas such as waste heat recovery for power generation, car manufacturing and space probes. TE materials attract much attention. The TE efficiency can be measured by the dimensionless figure of merit, zT = S 2 σT κ e +κ l , where S is the Seebeck coefficient, σ is the electrical conductivity, T is the absolute temperature, κ e and κ l are the electrical and lattice thermal conductivity, respectively. Power factor (PF) can be defined as the product of the square of Seebeck coefficient and the electrical conductivity, that is, PF = S 2 σ. Therefore, it is essential to achieve a high power factor and a low thermal conductivity for a high zT value. As a result, discovering TE materials with outstanding properties is of great practical significance for many applications. Recently, many thermoelectric materials have been explored for power generation applications, such as GeTe [1], PbTe [2], half-Heusler [3] and skutterudites [4]. Among the reported TE materials, the Zintl phase, a class of intermetallic compounds, has been paid attention to because they meet the character of phonon-glass electron-crystal (PGEC) [5]. As the typical Zintl phase, Mg 3 Sb 2 have been investigated experimentally and theoretically. Condron et al. [6] prepared Mg 3 Sb 2 by a direct reaction of the elements and obtained the maximum zT value of 0.21 at 873 K. Chen et al. [7] got the maximum zT valuẽ 0.6 at 773 K in Mg 2.975 Li 0.025 Sb 2 (p-type Li-doped Mg 3 Sb 2 ). Wang et al. [8] carried out p-type Li-doping on Mg 3 Sb 2 and got a higher zT value of 0.59 at 723 K. Xu et al. [9] explored theoretically the relation between the electronic structure and the TE properties of Mg 3 Sb 2 . Meng et al. [10] studied anisotropic thermoelectric in Mg 3 Sb 2 and confirmed anisotropic thermoelectric of p-type Mg 3 Sb 2 . The low thermoelectric properties of thermoelectric materials limit their further application under a medium-low temperature; even when the ultralow thermal conductivity in MgSb-based materials was reported, the search for high thermoelectric materials and great potential as candidates for near-room temperature thermoelectric generators seems still to be more urgent. As and Bi belongs to the same main V group as Sb, by contrast, few studies are conducted on Mg 3 Bi 2 and Mg 3 As 2 .
In this study, we investigated the thermoelectric properties of n-type and p-type Mg 3 X 2 (X = As, Sb, Bi) systematically using the combined method of the first principles calculations and the semi-classical Boltzmann theory. A higher peak zT of 2.58 was obtained. In addition, we discussed the n-type and p-type doping effect in Mg 3 X 2 (X = As, Sb, Bi), and the calculated doping formation energy and phonon dispersion curves proved that the doped structures are dynamically stable.

Computational Details
In this study, the structure of Mg 3 X 2 (X = As, Sb, Bi) was built in VESTA (version 2.90.0b, Tohoku University, Sendai, Japan). The structure relaxations, the total density of states, projected density of states, and band structure were carried out in Vienna Ab-initio Simulation Package (VASP, vasp. 5.3.5.neb, University of Vienna, Vienna, Austria) based on density functional theory (DFT). The exchange-correlation functionals was described by the generalized gradient approximation (GGA) with Perdew, Burke, and Ernzerhof (PBE) functional [11]. The plane-wave cutoff energy was set to 400 eV and the energy convergence criterion was set up to 10 −5 eV. The ionic relaxation was interrupted at −0.02 eV and was calculated by using 9 × 9 × 6 Monkhorst-Pack grid meshes in irreducible Brillouin Zone. To get the phonon dispersion curves of these structures, the primitive cell of Mg 3 X 2 (X = As, Sb, Bi) was expanded to a 3 × 3 × 2 supercell in Phonopy package (version 1.9.6.1, National Institute for Material Science, Tsukuba, Japan) by the density functional perturbation theory (DFPT) [12]. The calculation of the TE transportation properties was based on the semi-classical Boltzmann theory and the rigid-band approach in the BoltzTrap code [13]. To verify the feasibility of doping on Mg 3 X 2 (X = As, Sb, Bi), the formation energy of the doped structure was calculated using a 3 × 3 × 2 supercell of Mg 3 X 2 (X = As, Sb, Bi). We replaced one Mg atom with one Hf (Sn, Zr) atom for n-type doping, and with one Ag (Li, Na) atom for p-type doping, respectively. The doping concentration was 5.56%. The phonon dispersion curves, the total density of states, and projected density of states were calculated, where the K-Points were set as 3 × 3 × 3. Figure 1 shows the structure of Mg 3 X 2 (X = As, Sb, Bi). As is shown, the structure of Mg 3 X 2 (X = As, Sb, Bi) belongs to the hexagonal system with the space group P-3m1 [14,15]. The relaxed lattice constants of them are shown in Table 1. The parameter we obtained was not more than 1.5% higher than that of the previous study and the parameter c we got was not more than 0.6% higher than that of the early research, which agrees with those in the previous study [16][17][18]. orbital plays a major role in the total density of states at the conduction band minimum (CBM) of Mg3As2 and Mg3Sb2, while the peak of TDOS is mainly contributed by As porbital in Mg3As2 and Sb p-orbital in Mg3Sb2 at the valence band maximum (VBM), respectively. In Mg3Bi2, the TDOS at CBM was mainly controlled by the hybrid orbital of Mg sorbital and Bi p-orbital, while Bi p-orbital contributes mainly at VBM.   The band structures, total density of states (TDOS), and the projected density of states (PDOS) of Mg 3 X 2 (X = As, Sb, Bi) are calculated and displayed from left to right in Figure 2. From the figure of the band structures, it can be seen that Mg 3 Bi 2 may be a semimetal, which is consistent with the published researches [18,19]. Besides, Ferrier et al. [20,21] speculated that Mg 3 Bi 2 is a semimetal on the basis of the conductivity-composition result they obtained, whereas Mg 3 As 2 and Mg 3 Sb 2 are both semiconductors. Mg 3 As 2 has a direct band gap of 0.863 eV, and Mg 3 Sb 2 has an indirect band gap of 0.205 eV, which is close to the band gap of 0.303eV of Mg 3 Sb 2 by Yu et al. [22] by a different interactive correlation function in PW91. Obviously, it is seen from the figure of TDOS and PDOS that Mg s-orbital plays a major role in the total density of states at the conduction band minimum (CBM) of Mg 3 As 2 and Mg 3 Sb 2 , while the peak of TDOS is mainly contributed by As p-orbital in Mg 3 As 2 and Sb p-orbital in Mg 3 Sb 2 at the valence band maximum (VBM), respectively. In Mg 3 Bi 2 , the TDOS at CBM was mainly controlled by the hybrid orbital of Mg s-orbital and Bi p-orbital, while Bi p-orbital contributes mainly at VBM. orbital plays a major role in the total density of states at the conduction band minimum (CBM) of Mg3As2 and Mg3Sb2, while the peak of TDOS is mainly contributed by As porbital in Mg3As2 and Sb p-orbital in Mg3Sb2 at the valence band maximum (VBM), respectively. In Mg3Bi2, the TDOS at CBM was mainly controlled by the hybrid orbital of Mg sorbital and Bi p-orbital, while Bi p-orbital contributes mainly at VBM.    Figure 3a,c,e are the phonon dispersion curves and Figure 3b,d,f are the phonon density of states. Clearly, there are no imaginary frequencies in the three phonon dispersion curves, demonstrating that the three structures are all dynamically stable. The mixing of low frequency optical and acoustic modes demonstrates their strong phonon scattering, which benefits low lattice thermal conductivity.  Figure 3a,c,e are the phonon dispersion curves and Figure 3b, Figure 3d,f are the phonon density of states. Clearly, there are no imaginary frequencies in the three phonon dispersion curves, demonstrating that the three structures are all dynamically stable. The mixing of low frequency optical and acoustic modes demonstrates their strong phonon scattering, which benefits low lattice thermal conductivity.  The vibration of atoms at their equilibrium position can be described in as Debye temperature, which can be defined as,

Dynamics Stability
is Boltzmann constant, n is the number of the unit cell, Ω is the volume of the unit cell, v m is the average velocity of sound. Then v m can be described by the transverse velocity of sound (v t ) and the longitudinal velocity of sound (v l ), the expression is as , where ρ is the density of materials, B and G are volume modulus and shear modulus. The elastic constants of c 11 , c 33 , c 44 , c 12 , c 13 and B, G are calculated [14] of Mg 3 As 2 , Mg 3 Sb 2 , Mg 3 Bi 2 , respectively. According to the Debye theory, when the temperature is higher than the Debye temperature, the lattice thermal conductivity is proportional to 1/T, which can be approximately equal to the minimum lattice thermal conductivity. The minimum lattice thermal conductivity is where V is the average volume of every atom in the unit cell. All the above results are shown in Table 2. The elastic constants we calculated meet the Born stability criterion [23], demonstrating that the results are credible. The Debye temperature of Mg 3 As 2 , Mg 3 Sb 2 and Mg 3 Bi 2 is 332 K, 237 K and 175 K. Table 2. Calculated elastic constants (c ij ), bulk modulus (B), shear modulus (G), density (ρ), volume (V), the transverse velocity of sound (v t ) and the longitudinal velocity of sound (v l ), the average velocity of sound (v m ), Debye temperature (Θ D ) and the minimum lattice thermal conductivity (κ min ).

Thermoelectric Transport Properties
Using the formula of zT = S 2 σT κ e +κ l and κ e = LσT, where L is the standard Lorenz number, L = 2.45 × 10 8 W Ω K 2 , we calculated the thermoelectric transport properties of Mg 3 As 2 , Mg 3 Sb 2 and Mg 3 Bi 2 by semi-classical Boltzmann theory and the rigid-band approach in the BoltzTrap code. Figure 4 presents the electrical conductivity of the n-type and p-type Mg 3 X 2 (X = As, Sb, Bi). The carrier concentration of Mg 3 Sb 2 is adopted as 1.47 × 10 19 cm −3 from the experimental data at room temperature [24], and the electrical resistivity of p-type Ag-doped Mg 2.995 Ag 0.005 Sb 2 is about 8 mΩ cm. Based on the data of the experiment and the calculated result, the relaxation time of τ = 1.398 × 10 −14 s is obtained. Due to the lack of experimental data of Mg 3 As 2 and Mg 3 Bi 2 , the same relaxation time as that of Mg 3 Sb 2 is adopted. At the same time, σ can be defined as, σ = neµ, where n, e and µ are the carrier concentrations, the charge of an electron and the carrier mobility, respectively. As shown in Figure 4, the σ of n-type materials increases with the carrier concentration, whereas the σ of p-type ones have the same change trend.   Figure 5 is the electrical thermal conductivity of the n-type and p-type Mg3X2 (X = As, Sb, Bi) with carrier concentration. As can be seen, the electrical thermal conductivity increases with the temperature at the same carrier concentration for all the n-type or p-type Mg3X2 (X = As, Sb, Bi), and increases when the carrier concentration rises at the same tem-  Figure 5 is the electrical thermal conductivity of the n-type and p-type Mg 3 X 2 (X = As, Sb, Bi) with carrier concentration. As can be seen, the electrical thermal conductivity increases with the temperature at the same carrier concentration for all the n-type or p-type Mg 3 X 2 (X = As, Sb, Bi), and increases when the carrier concentration rises at the same temperature. It is noted that the electrical thermal conductivity value of n-type in the same material is generally larger than that of the p-type. Figure 6 displays the Seebeck coefficient of the n-type and p-type Mg 3 X 2 (X = As, Sb, Bi). As shown, the S of n-type Mg 3 X 2 (X = As, Sb, Bi) is negative, while that of most p-type material curves is positive. For Mg 3 As 2 and n-type Mg 3 Bi 2 , the absolute value of S increases when the temperature increases. For Mg 3 Sb 2 and p-type Mg 3 Bi 2 at a low carrier concentration, the absolute value of the Seebeck coefficient decreases with the temperature increases, which is due to the influence of the bipolar effect. The bipolar effect usually occurs in wide band gap semiconductors at high temperatures and narrow band gap semiconductors at room temperature. With the higher carrier concentration, the absolute value of Mg 3 Sb 2 and p-type Mg 3 Bi 2 ' s S increases with the temperature increases.  Figure 5 is the electrical thermal conductivity of the n-type and p-type Mg3X2 (X = As, Sb, Bi) with carrier concentration. As can be seen, the electrical thermal conductivity increases with the temperature at the same carrier concentration for all the n-type or p-type Mg3X2 (X = As, Sb, Bi), and increases when the carrier concentration rises at the same temperature. It is noted that the electrical thermal conductivity value of n-type in the same material is generally larger than that of the p-type. Figure 6 displays the Seebeck coefficient of the n-type and p-type Mg3X2 (X = As, Sb, Bi). As shown, the S of n-type Mg3X2 (X = As, Sb, Bi) is negative, while that of most p-type material curves is positive. For Mg3As2 and n-type Mg3Bi2, the absolute value of S increases when the temperature increases. For Mg3Sb2 and p-type Mg3Bi2 at a low carrier concentration, the absolute value of the Seebeck coefficient decreases with the temperature increases, which is due to the influence of the bipolar effect. The bipolar effect usually occurs in wide band gap semiconductors at high temperatures and narrow band gap semiconductors at room temperature. With the higher carrier concentration, the absolute value of Mg3Sb2 and p-type Mg3Bi2' s S increases with the temperature increases.   Figure 7 is the power factor performance with the carrier concentration of the n-type and p-type Mg3X2 (X = As, Sb, Bi). For n-type one, PF increases first and then decreases when the carrier concentration increases. In addition, it can be found that, with the same carrier concentration, the power factor of n-type material and p-type Mg3As2 increases with the temperature gradually. However, it is an exception to that of p-type Mg3Sb2 and  Figure 7 is the power factor performance with the carrier concentration of the n-type and p-type Mg 3 X 2 (X = As, Sb, Bi). For n-type one, PF increases first and then decreases when the carrier concentration increases. In addition, it can be found that, with the same carrier concentration, the power factor of n-type material and p-type Mg 3 As 2 increases with the temperature gradually. However, it is an exception to that of p-type Mg 3 Sb 2 and p-type Mg 3 Bi 2 at a low concentration due to the influence of the negative Seebeck coefficient. Figure 6. The Seebeck coefficient of the n-type and the p-type Mg3X2 (X = As, Sb, Bi) with carrier concentration. (a) n-Mg3As2, (b) n-Mg3Sb2, (c) n-Mg3Bi2, (d) p-Mg3As2, (e) p-Mg3Sb2, (f) Figure 7 is the power factor performance with the carrier concentration of the n-type and p-type Mg3X2 (X = As, Sb, Bi). For n-type one, PF increases first and then decreases when the carrier concentration increases. In addition, it can be found that, with the same carrier concentration, the power factor of n-type material and p-type Mg3As2 increases with the temperature gradually. However, it is an exception to that of p-type Mg3Sb2 and p-type Mg3Bi2 at a low concentration due to the influence of the negative Seebeck coefficient.  The zT increases with temperature ranging from 300 K to 800 K. The n-type and p-type Mg3Bi2 have the maximum zT values of 0.34 and 0.32, respectively, and that of the n-type and p-type Mg3Sb2 can reach 1.38 and 0.64, respectively. It is noted that the maximum zT value of n-type and p-type Mg3As2 can achieve 2.58 and 1.39, which is competitive among most published results, indicating that Mg3As2 is a promising candidate for TE materials.  Figure 8 demonstrates the zT values of the n-type and p-type Mg 3 X 2 (X = As, Sb, Bi). The zT increases with temperature ranging from 300 K to 800 K. The n-type and p-type Mg 3 Bi 2 have the maximum zT values of 0.34 and 0.32, respectively, and that of the n-type and p-type Mg 3 Sb 2 can reach 1.38 and 0.64, respectively. It is noted that the maximum zT value of n-type and p-type Mg 3 As 2 can achieve 2.58 and 1.39, which is competitive among most published results, indicating that Mg 3 As 2 is a promising candidate for TE materials.

Doping Effect
The studies above show that Mg3X2 (X = As, Sb, Bi) have good TE properties and potential application prospects. The doping strategy is considered as an effective approach for improving TE performance; therefore, further investigations into doping were carried out in this work. To verify the feasibility of doping on Mg3X2 (X = As, Sb, Bi), elements of Hf (or Sn, Zr, Ag, Li and Na) atom substituting to the Mg atom are investigated. The formation energy of each doped structure is calculated by = ( ) − ( ) − ( ) + ( ), where X can be As, Sb or Bi. Y can be Hf, Sn, Zr, Ag, Li and

Doping Effect
The studies above show that Mg 3 X 2 (X = As, Sb, Bi) have good TE properties and potential application prospects. The doping strategy is considered as an effective approach for improving TE performance; therefore, further investigations into doping were carried out in this work. To verify the feasibility of doping on Mg 3 X 2 (X = As, Sb, Bi), elements of Hf (or Sn, Zr, Ag, Li and Na) atom substituting to the Mg atom are investigated. The formation energy of each doped structure is calculated by  Figure 9) and the corresponding density of states are shown in Figure 10. Figure 9 presents that all doped structures have no imaginary frequencies, indicating that all the doped structures are dynamically stable. Compared with the phonon dispersion curve of the un-doped structure, the slope of the acoustic mode of the doped one decreases, which proves that the doped structure has a lower phonon group velocity. The mixing amplitude of the low-frequency optical mode and the acoustic one increases, demonstrating the strong phonon scattering between them, especially in the doped Mg 54 Bi 36 . Figure 10 reveals TDOS and PDOS of the doped structures. The doping of an atom does not affect the dominant contribution orbitals to the density of states, but the band gaps move to the lower energy direction for n-type doped systems, while moving to the higher energy direction for p-type doped ones. In addition, we calculated the lattice thermal conductivity of doped structures shown in Table 4. As is shown, the lattice thermal conductivity and Debye temperature was reduced by doping except Mg 53 ZrBi 36 and Mg 53 SnBi 36 . Therefore, it can be safely concluded that implementing n-type and p-type doping on Mg 3 X 2 (X = As, Sb, Bi) by replacing an Mg atom can improve the thermoelectric properties.

Conclusions
In summary, the thermoelectric properties of Mg3X2 (X = As, Sb, Bi) were studied by first principles and semi-classical Boltzmann theory. The calculated results show that the maximum zT values of n-type Mg3Sb2 and p-type Mg3Sb2 are 1.38 and 0.64, respectively. The maximum zT values of n-type Mg3Bi2 and p-type Mg3Bi2 are 0.34 and 0.32 respectively, and these maximum values were obtained at a temperature of 800 K. It is noted that Mg3As2 is a direct bandgap semiconductor with a band gap of 0.8626 eV, and the maximum zT of n-type and p-type Mg3As2 can reach 2.58 and 1.39, respectively. Based on the calculation results, the element of Hf (or Sn, Zr, Ag, Li and Na) substituting to Mg atoms in doped Mg3X2 (X = As, Sb, Bi) is investigated, and the formation energy and stability of the p-type doping of Li for Mg3As2, p-type doping of Li and Na for Mg3Sb2, and p-type doping of Li and Na for Mg3Bi2 was discussed.

Conclusions
In summary, the thermoelectric properties of Mg 3 X 2 (X = As, Sb, Bi) were studied by first principles and semi-classical Boltzmann theory. The calculated results show that the maximum zT values of n-type Mg 3 Sb 2 and p-type Mg 3 Sb 2 are 1.38 and 0.64, respectively. The maximum zT values of n-type Mg 3 Bi 2 and p-type Mg 3 Bi 2 are 0.34 and 0.32 respectively, and these maximum values were obtained at a temperature of 800 K. It is noted that Mg 3 As 2 is a direct bandgap semiconductor with a band gap of 0.8626 eV, and the maximum zT of n-type and p-type Mg 3 As 2 can reach 2.58 and 1.39, respectively. Based on the calculation results, the element of Hf (or Sn, Zr, Ag, Li and Na) substituting to Mg atoms in doped Mg 3 X 2 (X = As, Sb, Bi) is investigated, and the formation energy and stability of the p-type doping of Li for Mg 3 As 2 , p-type doping of Li and Na for Mg 3 Sb 2 , and p-type doping of Li and Na for Mg 3 Bi 2 was discussed.