Ag-Mg antisite defect induced high thermoelectric performance of α-MgAgSb

Engineering atomic-scale native point defects has become an attractive strategy to improve the performance of thermoelectric materials. Here, we theoretically predict that Ag-Mg antisite defects as shallow acceptors can be more stable than other intrinsic defects under Mg-poor‒Ag/Sb-rich conditions. Under more Mg-rich conditions, Ag vacancy dominates the intrinsic defects. The p-type conduction behavior of experimentally synthesized α-MgAgSb mainly comes from Ag vacancies and Ag antisites (Ag on Mg sites), which act as shallow acceptors. Ag-Mg antisite defects significantly increase the thermoelectric performance of α-MgAgSb by increasing the number of band valleys near the Fermi level. For Li-doped α-MgAgSb, under more Mg-rich conditions, Li will substitute on Ag sites rather than on Mg sites and may achieve high thermoelectric performance. A secondary valence band is revealed in α-MgAgSb with 14 conducting carrier pockets.

In this work, the chemical potentials and defect formation energies of native point defects and Li doping in α-MgAgSb at all possible charge states are studied by using density functional theory. We found that the defect formation energies strongly depend on the chemical potentials. Ag vacancies and Ag-Mg antisites (Ag on Mg sites) are the dominant defects that act as shallow acceptors, which determine the p-type conduction. Moreover, the Ag Mg point defect in α-MgAgSb may have higher ZT than the Ag vacancy. For Li-doped α-MgAgSb, the doping formation energies strongly depend on the chemical potentials. Under more Mg-rich conditions, Li will substitute on Ag sites (Li Ag ) rather than on Mg sites (Li Mg ), and a larger ZT can be achieved by Li Ag doping than by Li Mg doping. By reasonably controlling the chemical potential, both the antisite defect Ag Mg and Li substitution on Ag sites of α-MgAgSb can be obtained, and the products may be promising thermoelectric materials for low temperature power generation.

Results and Discussion
Chemical potentials and formation energies of native point defects. Engineering intrinsic defects may be an effective way to improve the thermoelectric performance of α-MgAgSb. Due to the complex phase transitions and the appearance of secondary phases, previous experimental works have shown that it is difficult to synthesize pure phase α-MgAgSb. Different types of native point defects may easily appear in α-MgAgSb. Thus, it is necessary to first explore the conditions for forming intrinsic point defects in α-MgAgSb.
The defect formation energy ∆H ( ) f is defined as is the total energy of the supercell with the incorporated defect, E perfect is the total energy of the supercell without the incorporated defect, n i is the number of atoms being removed or added, and µ i is the corresponding chemical potential, E F is the Fermi energy, E V is the energy with respect to the valence band maximum (VBM), and ΔV is the average difference between the local potentials far from the defect in the defective supercell and the corresponding ones in the perfect supercell 47 .
We calculated the accessible range of chemical potentials for the equilibrium growth conditions of α-MgAgSb. Under equilibrium conditions for the crystal growth, the steady production of the host material, α-MgAgSb, should satisfy the following equations: MgAgSb Mg Ag Sb MgAgSb Mg Ag Sb f where µ Mg , µ Ag , and µ Sb are the chemical potentials of Mg, Ag, and Sb, respectively, and ∆H MgAgSb ( ) f is the formation energy for α-MgAgSb. To avoid the precipitation of source elements, µ ∆ Mg , µ ∆ Ag , and µ ∆ Sb should satisfy:

Mg Ag Sb
To maintain the stability of MgAgSb during growth and avoid any competing phases (such as MgAg, Mg 3 Sb 2 , and Ag 3 Sb), the chemical potential µ ∆ s Mg , µ ∆ Ag , and µ ∆ Sb must satisfy the following limits: All calculated heats of formation of ternary and binary compounds in this work are given per formula unit. Equations (4)-(8) can be projected onto the two-dimensional plane with two independent variables, µ ∆ Mg and µ ∆ Ag , as shown in Fig. 1. The shaded region represents the area for the equilibrium growth conditions of α-MgAgSb. Figure 1 asserts that α-MgAgSb is only thermodynamically stable within a narrow Mg-Ag compositional range. The thermodynamically stable ranges of chemical potentials for the elements in α-MgAgSb are obtained by excluding the regions of chemical potentials in which competing phases are thermodynamically stable. Here, we present the calculated values at two representative chemical potential points labeled as A (−0.69 eV, 0, 0) and B (−0.469 eV, −0.032 eV, −0.162) in Fig. 1 for µ ∆ Mg , µ ∆ Ag , and µ ∆ Sb , respectively. To predict the conductivity type of MgAgSb with intrinsic defects, we calculated the Fermi level pinning positions. Figure 2(a) and (b) shows the calculated formation energies of native point defects as a function of the Fermi levels at chemical potential points A and B, respectively. The calculated transition energies for these defects are shown in Fig. 3. From the single-particle energy point of view, V Mg , V Ag , Sb I , and Ag Mg should be acceptor-like defects, whereas V Sb , Mg I , Ag I , and Mg Ag should be donor-like defects. Under Mg-poor-Ag/Sb-rich conditions (point A in Fig. 1), the formation energy of the Ag Mg antisite defect is very low, meaning that it is the dominant acceptor, and p-type conductivity can be realized by forming Ag Mg antisite defects. The Ag Mg antisite defect is thermodynamically stable. This suggests that Ag Mg may stably exist in an Mg poor environment. Under more Mg-rich conditions (point B in Fig. 1), the V Ag defect has the lowest formation energy, indicating that it is now the dominant type of acceptor, which is consistent with the results reported by Liu et al. 30 . Thus, our calculation results for the formation energy can explain why α-MgAgSb often exhibits p-type conductivity. From Fig. 3, it is seen that the transition energies of V Ag and Ag Mg are 0.036 eV and 0.068 eV above the VBM, respectively, indicating that V Ag and Ag Mg are shallow acceptors. On the other hand, all the defects that create deep levels, such as Mg I and V Mg , have higher formation energies. Thus, the formation energies of the native point defects strongly depend on the chemical potentials, and Ag Mg antisites and Ag vacancies are the dominant acceptor defects in α-MgAgSb. The calculated formation energy using chemical potentials is close to the real preparation environment. Under the different circumstances, we can compare the types of doping with which conditions are easier or more difficult to achieve, which can explain the experimental phenomena and provide a reference for controlling the defect type.
Effects of native defects on electronic structure. Miao et al. have calculated the band structure of α-MgAgSb using the Perdew-Burke-Ernzerhof generalized gradient approximation (GGA) (GGA-PBE) exchange-correlation functional in the Vienna ab-initio simulation package (VASP), and predicted that α-MgAgSb is a semimetal 37 . Using the local density approximation (LDA) exchange-correlation potential as Transition energy level (eV)

Conduction band
Valence band  valence band has a stronger dispersion than the bottom of the conduction band. The band dispersion relationship determines the effective mass, and the band mass of a single valley can be obtained by the following: where k is the wave vector, E F is the Fermi energy, and ħ is the reduced Planck's constant. According to Eq. (9), we know that the band effective mass at the top of the valence band is smaller than that at the bottom of the conduction band. Such large band dispersion of the valence band is conducive to the transmission of electrons. The small effective mass of top valence bands is helpful for increasing the electrical conductivity of p-type α-MgAgSb, although electrical conductivity is also determined by the carrier concentration.
As is well known, the maximum ZT of a material depends on the dimensionless thermoelectric quality factor 48 , where μ is the mobility of the carrier and ⁎ m DOS is the density-of-states (DOS) effective mass. The relationship between the density of states effective mass, the band degeneracy, N V , and the band effective mass, ⁎ m b , is given by . If acoustic phonon scattering dominates the carrier transport, then µ ∝ where ⁎ m I is the inertial mass. Thus, a large N V is beneficial to a large ⁎ m DOS without deterioration of μ 6 . The band degeneracy N V is based on the effective total number of independent carrier pockets or valleys in the Brillouin zone, including both orbital and symmetry related degeneracy. We adopted the strategy of increasing N V for a high ZT as an example, as was well demonstrated for PbTe 6 . As a result of heavy hole doping and relatively light bands at the VBM, the Fermi level quickly moves down into the valence band, allowing a large population of holes to form in the secondary valence band. The calculations show that the secondary VBM is located at about −0.11 eV below the VBM. The Fermi surface calculations for a Fermi level −0.11 eV below the VBM of the red valence band and of the blue valence band are shown in Fig. 4(b) and (c), respectively. Figure 4(b) shows 8 half-pockets along Z-A, 4 quarter pockets at the M point, and 4 half-pockets at the X point so that the full number of valleys is 7. Figure 4(c) also shows that the full number of valleys is 7. Therefore, the iso-energy Fermi surface for an energy level at −0.11 eV has a high degeneracy with 14 isolated pockets. The large band degeneracy N V may contribute to the high Seebeck coefficient at relatively high carrier concentrations. Based on above analysis, the large band dispersion of the valence band, together with the high band degeneracy with N V = 14, may be the most significant feature that contributes to the good thermoelectric performance of p-type heavily doped α-MgAgSb. A high carrier concentration may help to increase the electrical conductivity. Therefore, the native defects V Ag and Ag Mg may play an important role in achieving a higher Seebeck coefficient and higher electrical conductivity, which will lead to a large ZT for α-MgAgSb. α-MgAgSb with the Ag Mg defect has a larger number of band valleys near the Fermi level than with the V Ag defect, which may lead to a larger ZT than with V Ag .   39 . It is valuable to explore how the chemical potential affects the doping sites in Li-doped α-MgAgSb. We calculated the formation energies of Li-doped α-MgAgSb as a function of chemical potential. For Li doping, the chemical potentials of impurities should satisfy other constraints to avoid the formation of impurity-related phases (such as Li source element, LiAg, Li 2 Sb, or Li 3 Sb): Based on the representative chemical potential points, we can calculate the chemical potential of Li, and the values of the chemical potentials at points A and B are −0.8857 eV and −0.8163 eV for µ ∆ Li , respectively. Then, the chemical potential is used for calculating the formation energy for Li-related defects. The impurities can either be at interstitial sites or substitute for Mg, Ag, or Sb. Therefore four different point defects, Li Mg , Li Ag , Li Sb , and Li I , have been included in our calculation. Because of the large formation energy for Li Sb , we only show the Li Sb with zero charges.
The calculated impurity formation energies of the doping systems are plotted in Fig. 6. As shown in Fig. 6, formation energies strongly depend on the chemical potentials. The thermodynamic transition level between Li I  Fig. 7(a) and (b), respectively. As shown in Fig. 7(a) valley degeneracy is helpful for the thermoelectric material 6 . As can be seen in Fig. 7(a), the Fermi level moves down into the valence band by 0.11 eV because of Li doping on Mg sites in α-MgAgSb, and the energy of the Γ point becomes higher towards the Fermi level so that the number of band valleys near the Fermi level increases.
To explain the reason why the Γ point becomes higher and moves toward the Fermi level, we calculated the partial charge densities near the Fermi level at the Γ point using VASP, as shown in Fig. 7(c). Because there is little charge density distribution around the Mg atoms, we do not display the Mg atoms. From the shape of the charge density, we can see that the states near the Fermi level at the Γ point mainly come from the Sb p orbitals. The large band degeneracy N V and heavy band effective mass can jointly contribute to the high Seebeck coefficient. Moreover, for p-type α-MgAgSb, the carrier concentration largely depends on the number of band valleys near the Fermi level, which is mainly due to the fact that more carriers can be activated across the band gap. High carrier concentration may help to increase the electrical conductivity. Therefore, Li doping may play an important role in achieving a higher Seebeck coefficient and electrical conductivity, which will lead to a large ZT for Mg 47 Fig. 7(b). As can be seen in Fig. 7(b), the number of band valleys near the Fermi level increases because the energy along Z-R becomes higher towards the Fermi level. Multiple degenerate valleys may produce a large ⁎ m DOS , and a large ⁎ m DOS may lead to a large Seebeck coefficient.
Elastic and thermal properties. Ying et al. found that the appearance of three-centered Mg-Ag-Sb bonds in α-MgAgSb results in low intrinsic lattice thermal conductivity 50 . To investigate the elastic properties of α-MgAgSb with intrinsic defects and Li doping, the stress-strain method was used to calculate the elastic constants and other elastic properties 51 . A small finite strain is applied on the optimized structure, and then the atomic positions are optimized. The elastic constants are obtained from the stress of the strained structure. The calculated elastic constants of MgAgSb and Mg 0.98 Li 0.02 AgSb are listed in Table 1. It is clearly seen that all the studied compounds satisfy the mechanical stability criteria 52 , indicating that they are elastically stable. On the other hand, the positive eigenvalues of the elastic constant matrix for each compound further prove that they are elastically stable. From the calculated elastic constants C ij , the polycrystalline bulk modulus B and shear modulus G were estimated using the Voigt-Reuss-Hill approximation 53 . A high (low) B/G ratio of a material indicates that it is ductile (brittle), and the critical value is about 1.75 54 . The calculated B/G ratios for MgAgSb and Mg 0.98 Li 0.02 AgSb are larger than the critical value (1.75), indicating that they are all ductile materials.
Thermal conductivity of a material includes both electronic and lattice thermal conductivity. The electronic contribution to the thermal conductivity is described by the Wiedemann-Franz relation, κ σ = LT e , where L is the Lorenz number. Above the Debye temperature, the lattice thermal conductivity is generally limited by Umklapp scattering, which leads to κ ∝ T 1/ l . This 1/T decay can only continue, however, until the minimum lattice thermal conductivity (κ min ) is reached, as defined by Cahill 55,56 . At high temperature (T > Θ D ), κ min can be approximated by the following formula: Scientific RepoRts | 7: 2572 | DOI:10.1038/s41598-017-02808-8 min B s l where V is the average volume per atom, and v s and v l are the shear and longitudinal sound velocities, respectively. As a fundamental parameter, the Debye temperature is connected with many physical properties of solids, such as the specific heat, melting point, and elastic constant. At low temperatures, the vibrational excitations arise solely from acoustic vibrations. One of the methods used to calculate the Debye temperature is based on the elastic constant data. The Debye temperature is given by: v s and v l can be obtained using the polycrystalline shear modulus G and the bulk modulus B from Navier's equation as follows 58 : s l 4 3 B and G can estimate using the Voigt-Reuss-Hill approximation from the calculated elastic constant data, which were obtained by the stress-strain method 53 . The calculated elastic constants and the minimum lattice thermal conductivity are listed in Table 1. As shown in Eq. 14, the minimum lattice thermal conductivity is strongly affected by the shear sound velocity. Table 1 shows that the V Ag and Li Mg defects induce an obviously decreasing shear modulus in α-MgAgSb, which indicates that V Ag and Li Mg defects weaken the resistance against shear deformation of α-MgAgSb. Thus, the shear sound velocity decreases due to Ag vacancy and Li Mg doping. Consequently, the minimum lattice thermal conductivity values are reduced due to Ag vacancy and Li Mg doping. For Ag Mg and Li Ag defects, the decrease in the shear modulus is not so large compared with V Ag and Li Mg defects. Thus, the change in the minimum lattice thermal conductivity due to Ag Mg and Li Ag defects is smaller than that due to V Ag and Li Mg defects.

Electrical transport properties.
A material with a large ZT needs to have a large S (found in low carrier concentration semiconductors or insulators) and a large σ (found in high carrier concentration metals). The carrier concentration dependence of the Seebeck coefficient and the electrical conductivity are shown in Eqs (18) and (19), respectively 2 . In these equations, T is the temperature, and μ is the charge carrier mobility. σ µ = ne (19) Equation (18) suggests that the Seebeck coefficient is proportional to the temperature and ⁎ m DOS , yet is inversely related to the carrier concentration. The electrical conductivity is proportional to the carrier concentration and inversely proportional to the effective mass. We calculated the Seebeck coefficient, S, the carrier concentration, n, the electrical conductivity relative to relaxation, σ/τ, the thermopower relative to relaxation, and the figure of merit, S 2 σ/τ, as a function of temperature, as shown in Fig. 8. As can be seen in Fig. 8 ) are all positive over the entire studied temperature range, indicating p-type transport for the four types of defects. The n and σ/τ, of Ag Mg are lower than those of V Ag , while Ag Mg has the larger S 2 σ/τ owing to its large S. Liu et al. reported that Ag vacancy can be rationally engineered by controlling the hot pressing temperature, and a high peak ZT of ~1.4 and an average ZT of ~1.1 can be achieved 30 . α-MgAgSb containing Ag Mg point defects may have higher ZT than with Ag vacancy because α-MgAgSb with Ag Mg has a larger σ τ S / 2 than with Ag vacancy. The S of Li Ag -doped α-MgAgSb is larger than for Li Mg -doped α-MgAgSb. Although Li Ag -doped α-MgAgSb has the lowest n and σ/τ, the S 2 σ/τ of Li Ag -doped α-MgAgSb is larger than that with Li Mg defects, due to the large S, as shown in Fig. 8

Conclusions
In this work, we investigated the defect formation energies, the electronic structure, and the thermoelectric performance of the host α-MgAgSb and the effects of substitutional Li doping of α-MgAgSb, by using density functional theory combined with semiclassical Boltzmann theory. We found that the formation energies strongly depend on the chemical potentials. Ag vacancy and Ag-Mg antisite defects are the dominant defects, acting as the shallow acceptors that determine the p-type conduction of experimentally synthesized α-MgAgSb. Moreover, for α-MgAgSb, the Ag Mg antisite defect may induce a higher ZT than Ag vacancy, due to the more numerous band valleys near the Fermi level than with Ag Mg in α-MgAgSb. α-MgAgSb has a secondary valence band with 14 carrier pockets, which indicates that heavily p-type doping may lead to a high thermoelectric performance in α-MgAgSb. For Li-doped α-MgAgSb, Li doping on Ag sites has a lower formation energy than on Mg sites under more Mg-rich conditions, and Li Ag may lead to a larger ZT than for Li doping on Mg sites. Thus, engineering atomic scale defects is an effective strategy for enhancing the thermoelectric properties of α-MgAgSb, and the achieved high ZT demonstrates that Ag Mg antisite defects and the substitution of Li on Ag sites in α-MgAgSb could lead to materials with good potential for future application in the thermoelectric area.

Computational Details
The electronic structure of α-MgAgSb was investigated using the full-potential linearized augmented plane wave method 59 , as implemented in WIEN2k [60][61][62] . The Tran and Blaha modified semi-local Becke-Johnson exchange correlation potential (TB-mBJ) 63 was used, which is known to give much more accurate band gaps than the

Lattice parameter
Atomic type x y z  Table 2. Lattice constants and atomic coordinates of α-MgAgSb. Ag1, Ag2, and Ag3 represent three crystallographically unique Ag sites.
standard Engel-Vosko generalized-gradient approximation (EV-GGA) 64 . The muffin-tin radii were chosen to be 2.5 a.u. for Mg, Ag, and Sb. The cut-off parameter R mt × K max = 9 (where K max is the magnitude of the largest k vector) was used, and the self-consistent calculations were performed with 2000 k-points in the irreducible Brillouin zone; the total energy was made to converge to within 1 mRy. The electrical transport properties were then calculated by using semiclassical Boltzmann theory 65,66 within the constant scattering time approximation, as implemented in the Boltzmann Transport Properties (Boltz-TraP) code 67 . This approximation has been used to calculate the transport coefficients of some known thermoelectric materials and very good agreement with experimental results was achieved 68,69 . We simulated various defects in α-MgAgSb, along with Li doping, using a supercell that contained 144 atoms. We considered three intrinsic point defects, vacancy, interstitial, and antisite. Because of their large formation energies, cation/anion antisites, such as Mg or Ag on the Sb site and Sb on the Mg or Ag sites, are not discussed in this study. The intrinsic defects considered in this study include V Ag (Ag vacancy), V Mg (Mg vacancy), V Sb (Sb vacancy), Mg I (Mg interstitial), Ag I (Ag interstitial), Sb I (Sb interstitial), Ag Mg (Ag on Mg site), and Mg Ag (Mg on Ag site). In the case of Li doping, we simulated interstitial doping (Li I ) and substitutional doping, including Li Mg (Li doping on Mg site), Li Ag (Li doping on Ag site), and Li Sb (Li doping on Sb site).
As shown in Fig. 9, there are 48 atoms in each unit cell of α-MgAgSb, which contains five crystallographically unique atomic sites: one Mg, three Ag, and one Sb. The structural parameters of α-MgAgSb are shown in Table 2. α-MgAgSb consists of a distorted Mg-Sb rock-salt lattice, rotated by 45° about the c axis, with half of the Mg-Sb pseudocubes filled with Ag, although the pseudocubes where silver atoms are located are quite different from those in half-Heusler compounds 70 . Such a complex lattice structure may lead to a relatively small thermal conductivity.
We also studied the electronic structure and thermoelectric properties of V Ag , Ag Mg , Li Mg , and Li Ag using the supercell (144 atoms in MgAgSb supercell), corresponding to a doping level of 2% for α-MgAgSb. We also fixed the lattice constants, only optimizing the internal coordinates. The electronic structures of Mg 48