Native point defects and low $p$-doping efficiency in $Mg_2 (Si,Sn)$ solid solutions: A hybrid-density functional study

We perform hybrid-density functional calculations to investigate the charged defect formation energy of native point defects in $Mg_2 Si$, $Mg_2 Sn$, and their solid solutions. The band gap correction by hybrid-density functional is found to be critical to determine the charged defect density in these materials. For $Mg_2 Si$, $Mg$ interstitials are dominant and provide unintentional $n$-type conductivity. Additionally, as the $Mg$ vacancies can dominate in $Mg$-poor $Mg_2 Sn$, $p$-type conductivity is possible for $Mg_2 Sn$. However, the existence of low formation energy defects such as $Mg_{Sn}^{1+}$ and $I_{Mg}^{2+}$ in $Mg_2 Sn$ and their diffusion can cause severe charge compensation of hole carriers resulting in low $p$-type doping efficiency and thermal degradation. Our results indicate that, in addition to the extrinsic doping strategy, alloying of $Mg_2 Si$ with $Mg_2 Sn$ under $Mg$-poor conditions would be necessary to enhance the $p$-type conductivity with less charge compensation.

Mg2Si is a potential semiconductor for thermoelectric and optoelectronic devices owing to its high power factor and high optical absorption coefficient, respectively [1][2][3]. By alloying with Mg2Sn, the performance of the materials can be tuned to achieve high thermoelectric performance via conduction band convergence [4] or control the optical-absorption spectra by the band gap (Eg) engineering [3]. For such applications, the symmetric doping nature is required to form a p-n dual-leg in thermoelectric devices or a p-n junction in electrical devices [1,3].

There have been a lot of experimental investigations on the electrical properties of
Mg2Si-based alloys. Mg2Si has been reported to have an unintentional n-type conductivity [5][6][7][8]. By alloying Mg2Si with Mg2Sn, with the narrower Eg and the increased electrical conductivity, conduction band convergence, and enhanced phonon scattering by point disorders, the solid solutions can achieve higher n-type thermoelectric performance [4,9]. Although Mg2Sn can show p-type conductivity [10,11], both Mg2Si and Mg2(Si,Sn) solid solutions are difficult to be doped as p-type even under Mg-poor conditions [2,12]. The hole carrier density in doped Mg2(Si,Sn) is lower than the optimal doping concentration for an optimal power factor; thus, there is strong desires to reach high p-type doping efficiency for high thermoelectric performance.

Previous first-principles calculations reveal that native point defects in Mg2Si and
Mg2Sn may play a critical role in determining their electrical properties. The Mg interstitials are a dominant defect and are responsible for unintentional n-type conductivity [13][14][15]. The origin of the p-type doping difficulty in Mg2(Si,Sn) has been explained by the interplay between acceptors and intrinsic defects [2,[15][16][17][18]. However, while the Eg underestimation in density functional theory (DFT) is challenging in defect physics [19], many of these recent defect computations in thermoelectric materials are merely based on this scheme using conventional local-density approximation or generalized-gradient approximation (GGA). For example, as the Mg2Sn is negative in DFT calculations [20], the detailed quantitative role of intrinsic defects in Mg2Sn and Mg2(Si,Sn) solutions are not clearly understood. Furthermore, there have been many attempts outside the thermoelectric community, to calculate the defect properties using Eg correction methods such as hybrid-DFT and quasi-particle GW calculations to reveal the device instability or doping asymmetricity for Si, ZnO, and HfO2 [21,22,23]. Unfortunately, in contrast to other materials, there are few hybrid-DFT studies of native defects in Mg2Si, Mg2Sn, and their solid solutions. Moreover, to the best of our knowledge, there has been no hybrid-functional study on native defects in Mg2Sn.
In this study, we investigated the native point defects in Mg2Si, Mg2Sn, and their solid solutions using hybrid-DFT within the Heyd-Scuseria-Ernzerhof (HSE) non-local exchangecorrelation functional. The Eg correction is found to be critical to determine the stability of positively charged defects especially for p-type conditions. Mg2Si is unintentional n-type due to the Mg interstitial (IMg 2+ ). Mg2Sn can be either por n-type depending on the Mg chemical potential.
Although the Mg vacancies (VMg 2-) are dominant defects as a shallow acceptor for Mg2Sn, many free hole carriers can be compensated by the Sn-substitutional defect at the Mg site (SnMg 1+ ), leading to low p-doping efficiency. When Mg2Si is alloyed with Mg2Sn, p-type conductivity can be achieved in Mg-poor conditions. However, the overall free hole densities are not high enough to reach the optimal carrier concentration (10 20 cm -3 ) to maximize the power factor. Our result suggests that, in addition to the material alloying between Mg2Si and Mg2Sn, external impurity point defect doping with very low formation energy might be necessary to achieve high p-type doping concentration with less charge compensation.
Our first principles calculations of native point defects in Mg2Si and Mg2Sn are based on hybrid-DFT within the HSE hybrid exchange-correlation functional (HSE06) [24], to overcome the band gap problem in DFT [24]. We used the exact-exchange mixing parameter of 25% and the screening parameter of 0.208 Å -1 with the GGA parameterized by Perdew, Burke, and Ernzerhof [25]. Experimental lattice parameters were used for Mg2Si (6.35 Å) and Mg2Sn (6.75 Å) [8]. In conventional DFT the band gap problem is severe: for Mg2Sn the DFT Eg is negative (-0.341 eV) whereas the experimental gap is about 0.36 eV [8,20]. As the charge state of the defects are sensitive to the Fermi level (EFermi), the positions of band edge states are critical for the defect stability. In this study, by adopting hybrid-DFT, reliable Egs were obtained as 0.570 and 0.145 eV for Mg2Si and Mg2Sn respectively. We modeled the native point defects in Mg2Si and Mg2Sn using a 96atom cubic supercell. We considered six defect configurations as native point defects: Mg interstitial where the subscript i indicates the atomic element consisting pristine Mg2X, δni is the change of number of i element atoms in the defective supercell compared to the pristine supercell, and ECBM is the energy of the conduction band minimum (CBM) state [26,27]. Note that the EFermi can range from near the valence band maximum (VBM) to near the CBM.

Figure 1 shows the charged defect formation energies of native point defects in Mg2Si
and Mg2Sn for Mg-rich and Mg-poor conditions. In these materials, the most dominant defects are    Figure 3 (Figure 4 and SI). We used the constant-relaxation time approximation given as τ = ( 300 ) × 10 −14 [30]. For Mg2(Si,Sn), the optimal effective doping density is 10 20 -10 21 cm -3 , consistent with other reports [31,32], which is much larger than the possible p-type hole densities [ Figure 3 IMg migrates via an interstitialcy diffusion mechanism. The diffusion coefficient of IMg is calculated using the EFORM, Emig, and the vibrational frequencies of IMg [27,33]. In Mg-rich n-type Mg2Si, the D is 3.7 × 10 -19 m 2 /s at 800 K when the EFermi was at the CBM. From the D and diffusion time t, we estimated the diffusion length L as = √6 [27]. For 1000 hours under the annealing temperature of 800 K, IMg can diffuse to 2.8 μm, which is comparable to the grain size of polycrystalline Mg2Si [32,34]. In this Mg-poor condition, the D can be highly enhanced to 1.9 × 10 -15 m 2 /s at 800 K with the lowered defect formation energy. The L of 6.4 μm is possible for 1 hour under at 800 K.
Our results imply a thermal instability of the thermoelectric properties due to formation of native defects at thermoelectric working temperature. Although we can make ptype solid solutions by controlling the μMg or by doping suitable p-type elements, the acceptors can be self-compensated by the generation and diffusion of IMg defects from outside. Thus, for stable ptype conduction, very low acceptor formation energies are required to overcome the defect compensation by native donor states (IMg 2+ and SnMg 1+ ).
In summary, we investigated the native point defects in Mg2Si, Mg2Sn, and their solid solutions. We adopted the hybrid-functional to overcome the band gap problem in DFT. There are abundant intrinsic defects in Mg2Si such as Mg interstitials and Mg vacancies. While Mg2Si is unintentional n-type, the Mg2(Si,Sn) can be tuned from n-type to p-type going from Mg-rich to Mgpoor conditions. As Mg2Sn has a low p-type doping efficiency due to the severe charge compensation effect between donors and acceptors, an alloying strategy between Mg2Si and Mg2Sn might be required to obtain a higher p-type doping efficiency. However, the possible p-type carrier density is still low compared to the optimal carrier concentration for thermoelectric power factor.
Moreover, the fast diffusion of Mg interstitial can degrade the thermal stability of p-type samples.
Based on the results, we suggest that an extrinsic impurity doping strategy is required for Mg2(Si,Sn) thermoelectric materials to stabilize the hole high doping concentration with high doping efficiency.
Further hybrid-density functional studies on impurity doped Mg2(Si,Sn) will help to find suitable ptype impurity dopants.

II. Calculation Method: effective doping density
The defect density of defect D q [n(D q )] in materials can be computed using the defect formation energy EFORM and the Boltzmann factor, considering the equilibrium conditions of defect generation temperature T. As the defect formation energies are a function of atomic chemical potential and the Fermi level during defect generation, the defect densities also depend on the atomic chemical potential, Fermi level, and T given as where nlatt is the number density of available lattice sites in materials, θdeg is the number of degrees of internal freedom of the defect on a lattice site, HFORM is the formation Enthalpy of the charged defect, and kB is the Boltzmann constant. Here we assume that the volume expansion effects for defects are negligible and thereby we use the defect formation energy instead of the formation Enthalpy. At last, the defect density of D q in Mg2X are determined when atomic chemical potential and Fermi level are determined.
At a finite and non-zero temperature, the electrons in materials can be thermally activated.
Thus, in gapped materials, the electron densities at the conduction bands (nC) and the hole densities at the valence bands (nV) are also determined when Fermi level and temperature are given as .
The g(E) is calculated using DFT, PAW, PBE level with spin-orbit interaction. At this moment, to overcome the severe bipolar nature, we correct the band gap using the HSE06 band gap. At this moment, the HSE06 band gap is calculated using the virtual crystal approximation (VCA). The left hand side of the equation is called "Effective doping density". Since the whole system should be charge-neutral, the total of charge densities from free carriers and defect densities should be zero.
From this charge neutrality condition, the charge-neutrality level (CNL) is determined from the Fermi level which satisfies the following equation

III. Calculation Method: doping efficiency
We also define the doping efficiency (e) of Mg2X as the ratio of the effective doping density over the whole defect density as = Σ , ⋅ ( ) Σ , ( ) .
Thus, as the major defects in Mg2X are double donors or acceptors, the doping efficiency of Mg2X will be between 0% and 200%.