Understanding the Intrinsic P-Type Behavior and Phase Stability of

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Understanding The Intrinsic P-type Behavior and Phase Stability of Thermoelectric #-Mg3Sb2 XiaoYu Chong, Pinwen Guan, Yi Wang, Shun-Li Shang, Jorge Paz Soldan Palma, Fivos Drymiotis, Vilupanur Ravi, Kurt Star, Jean-Pierre Fleurial, and Zi-Kui Liu ACS Appl. Energy Mater., Just Accepted Manuscript • DOI: 10.1021/acsaem.8b01520 • Publication Date (Web): 08 Oct 2018 Downloaded from http://pubs.acs.org on October 12, 2018

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ACS Applied Energy Materials

Understanding The Intrinsic P-type Behavior and Phase Stability of Thermoelectric α-Mg3Sb2 XiaoYu Chong1, *, Pin-Wen Guan1, Yi Wang1, Shun-Li Shang1, Jorge Paz Soldan Palma1, Fivos Drymiotis2, Vilupanur A. Ravi2,3, Kurt E. Star2, Jean-Pierre Fleurial2 and Zi-Kui Liu1, * 1Department

of Materials Science and Engineering, The Pennsylvania State University,

University Park, PA 16802, USA 2Jet

Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive,

Pasadena, CA 91109, USA 3Department

of Chemical and Materials Engineering, California State Polytechnic University,

Pomona, CA 91768, USA

* Corresponding

Author: Dr. XiaoYu Chong, Dr. Zi-Kui Liu, E-mail address: [email protected]

(X.Y. Chong); [email protected] (Z.-K. Liu) 1

ACS Paragon Plus Environment

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Abstract α-Mg3Sb2 is an excellent thermoelectric material through excess-Mg addition and n-type impurity doping to overcome its persistent p-type behavior. It is generally believed that the role of excss-Mg is to compensate the single Mg vacancy to realize n-type carrier conduction. In contrary to this belief, the present work indicates that the role of excess-Mg is to compensate electronic charge of defect complex (VMg(2)+MgI)1-. The Mg solubility in α-Mg3+xSb2 is quite small when only consider single defect, but enlarged up to x = 0.011 with the defect complex (VMg(2)+MgI)1-, which are more reasonable as supported by experiments. Under Mg-poor condition, VMg(1)2- and VMg(2)2- are the dominant defects and their concentrations can reach (1.05-1.18)×1019 cm-3 at 1200 K. Under Mg-rich condition, (VMg(2)+MgI)1- is found to be the dominant reason for strong p-type behavior and their concentrations can reach as high as 3.5×1020 cm-3, which shifts the Fermi level closer to the valence band maximum. The predicted carrier concentrations in the range of 1017−1020 cm-3 are in the same range found experimentally for pure p-type α-Mg3Sb2.

Keyword: Thermoelectricity; Intrinsic defect complex; Phase boundary; Carrier concentration; Hybrid density functional

2


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1

Introduction Thermoelectric technology provides a unique opportunity to harvest thermal energy to

produce electric power in a renewable manner 1,2. Mg3Sb2-based compounds have been known as persistent p-type thermoelectric materials with moderate performance to exhibit a ZT value of 0.21 at 725 K without impurity doping as a result of the intrinsically low carrier concentration

3–5.

n-

type Mg3+xSb1.5Bi0.49Te0.01, first reported by Tamaki et al. 6, were successfully reached a high ZT value 1.5 and followed by multiple groups to demonstrate that Mg3Sb2-based n-type Zintls have much higher thermoelectric properties than the p-type Zintls 7–16. A defect chemistry approach by incorporating the excess Mg is crucial to realize n-type carrier conduction in the Mg3Sb2-based compounds and take advantage of the multi-valley conduction band. It is generally viewed that the role of extra Mg is to compensate the easily formed Mg vacancies as the dominant p-type character 6.

In previous research, Tamaki et al. reported that the excess-Mg compositions as high as 0.2 were essential to realize n-type character and achieve multivalley n-type carriers to reach a peak ZT of ∼1.5 combined with Te dopant, but exrtra Mg with x ≤ 0.1 can not overcome the persistent p-type behavior in Mg3+xSb1.5Bi0.49Te0.01 synthesized using Mg powder 6. While Zhang et al. claimed 0.04 Te doping alone in Mg3Sb1.5-0.5xBi0.5-0.5xTex or 0.02 Se doping alone in Mg3.07Sb1.5Bi0.5-xSex was enough to achieve similar n-type thermoelectric performance by arc melting without much extra Mg

7,8.

Recently, Shuai et al. found that a little bit extra Mg in the

initial composition can suppress those easily formed Mg vacancies to achieve n-type transport properties in Mg3+xSb1.5Bi0.5. Too much extra Mg would form a separate Mg-rich phase, which has a detrimental effect on further improving the transport performance

13.

Similar trend was

obtined by Ohno et al. that the Mg3+xSb1.99Te0.01 went through a p-to-n transition after 3



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Mg3+xSb1.99Te0.01 was synthesized using Mg-slug and the thermopower remains almost constant even with addition of extra Mg added 15. The above study showed that different content of excess Mg was needed to achieve the p-to-n transition according to different sample preparation methods. In this work, the energetics of intrinsic defects, electronic structure and their effect on phase stability are systematically investigated by first-principles caculations based on the hybrid density functional approach. Besides the single defects, various defect complexes are studied. This work shed light on the actual role of extra Mg and the essence of p-type behavior of α-Mg3Sb2 under Mg-rich condition, which is helpful to design α-Mg3Sb2-based system with high thermoelectric properties by defect chemistry. 2

Computational method All first-principles calculations based on the density functional theory (DFT) were performed

within the Vienna ab initio Simulation Package (VASP) using the projector augmented wave (PAW) method 17. Because the (semi) local functional such as local density approximation (LDA) and generalized gradient approximation (GGA) usually underestimate the band gap of semiconductor and cannot correctly predict the energetics of point defects

18,

all the supercell

calculations (with and without defects) in the present work were carried out using the Heyd– Scuseria–Ernzerhof (HSE06) hybrid functional 19. The HSE06 functional mixes 25% of screened Hartree–Fock exchange energy with the screening parameter being 0.2 Å-1

20.

For the defect

calculations, a 3 × 3 × 3 supercell (constructed from the conventional unit cell) was used that consists of 135 atoms when the supercell was defect-free. An energy cutoff of 360 eV was used for plane-wave basis expansion. A 2×2×2 mesh is adopted for Monkhorst–Pack k-points 21. The

4


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effects of spin polarization are included for all charged states. Relaxations are performed for atomic positions with a fixed cell volume until the residual force reaches less than 0.02 eV Å-1. 3

Results and Discussion

3.1 Crystal and Electronic Structure Figure 1 (a) shows the supercell of α-Mg3Sb2. Its prototype is inverse α-La2O3-type (space —

group: P3m1) crystal structure. It can be described as two-dimensional Mg2+ layers separated by the covalently bonded [Mg2Sb2]2− layers. The electrons were donated by ionic Mg2+ layers to the covalently bonded [Mg2Sb2]2− layers 22,23. Mg (1) represents Mg atoms in the ionic Mg2+ layers, while Mg (2) denotes Mg atoms in the [Mg2Sb2]2− layers. The conventional unit cell of α-Mg3Sb2 is shown in Figure 1 (b). Among five atoms in the unit cell, the Mg atoms occupy 1a and 2d Wyckoff sites and Sb atoms occupy 2d Wyckoff site. There is an obvious interstitial site at (0, 0, 0.5). Table 1 summarizes the lattice constants, formation enthalpy and band gaps calculated using the HSE06 and Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, compared with experimental values 7,24–27. The calculations indicate that PBE overestimates lattice constants and severely underestimates the band gap. The band gap obtained from HSE06 is 0.824 eV, larger than 0.402 eV from PBE and closer to the experimental data of 0.960 eV from optical transmission spectra by Kim, et al 27. The HSE06 calculations also give better consistency with experimental lattice constants. Figure 2 shows the calculated band structures and electronic density of states (eDOS) for α-Mg3Sb2 from PBE and HSE06 functional, respectively. A valence band maximum (VBM) located at the Г point and a conduction band minimum (CBM) at the K point are observed for both PBE and HSE06, which indicates the indirect semiconductor characteristics of α-Mg3Sb2. The multi-valley band feature contributes to the excellent thermoelectric properties of this material. 5



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In contrast to the multi-valley bands at the conduction band minimum, there is only one near-edge valence band at the Г point. Besides the conduction band minimum at the K point, the dashed vertical lines in Figure 2 indicate the accurate conduction band minimum with high valley degeneracy along M-L line. The relative energies of the conduction band edges at K point and along M-L lines are almost the same in both PBE and HSE06, which indicates that it suffices to use PBE for band convergence. It should also be noted that the overall features of the electronic density of states structure are similar in both PBE and HSE06, except for the magnitude of the energy band gap and formation energy.

3.2 Point Defect Structure and Defect Formation Energy The formation energy of a native defect D with charged state q is defined as 28 E𝑓(𝐷𝑞) = 𝐸𝑡𝑜𝑡(𝐷𝑞) ― 𝐸𝑡𝑜𝑡(𝑀𝑔3𝑆𝑏2) ― ∑𝑖𝑛𝑖𝜇𝑖 +𝑞(𝐸𝑉 + 𝐸𝐹) + Ecor

Eq. 1

where Etot(Dq) is the total energy of the supercell with defect D in charged state q, Etot(Mg3Sb2) the total energy of the supercell without defects, ni the number of atoms of a species i (Mg or Sb) that have been added to (ni ＞ 0) or removed from (ni ＜ 0) the supercell, μi a chemical potential representing the energy for defect D that is taken from the reservoir, EF the Fermi energy (level) that takes the value in the range over the bandgap, and EV the valance band maximum (VBM). Note that EF is the energy of the electron reservoir with respect to EV. Ecor is the finite-size correction in a defect-containing supercell by aligning the electrostatic potential in a region far away from the defect site with the corresponding potential in a perfect crystal. As shown in Figure 3, there are two polymorphs (α-Mg3Sb2 to β-Mg3Sb2) in the Mg-Sb system and the allotropic reaction from α-Mg3Sb2 to β-Mg3Sb2 is at 1198.15±5 K 29. For α-Mg3Sb2, the 6


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Mg-rich region corresponds to the presence of Mg and α-Mg3Sb2, whereas the Sb-rich region corresponds to the presence of α-Mg3Sb2 and Sb. The lower and upper boundaries of µi are determined by the crystal-growth conditions and the stability against precipitation of elemental Mg and Sb 28,30. 1 3∆𝐻𝑓𝑜𝑟𝑚

< ∆𝜇𝑀𝑔 < 0

Eq. 2

1 2∆𝐻𝑓𝑜𝑟𝑚

< ∆𝜇𝑆𝑏 < 0

Eq. 3

∆𝐻𝑓𝑜𝑟𝑚 is the formation enthalpy of Mg3Sb2 per formula, which can be expressed as: Eq. 4

∆𝐻𝑓𝑜𝑟𝑚 = 3∆𝜇𝑀𝑔 +2∆𝜇𝑆𝑏

In the equilibrium condition, ∆𝜇𝑀𝑔 and ∆𝜇𝑆𝑏 are the difference of chemical potentials defined as: ∆𝜇𝑀𝑔 = 𝜇𝑀𝑔 ― 𝜇0𝑀𝑔

Eq. 5

∆𝜇𝑆𝑏 = 𝜇𝑆𝑏 ― 𝜇0𝑆𝑏

Eq. 6

where 𝜇Mg and 𝜇Sb are the chemical potentials of Mg and Sb; 𝜇0𝑀𝑔 and 𝜇0𝑆𝑏 are the calculated total energies per atom in pure Mg and Sb crystals, respectively. To obtain the accurate formation enthalpy of α-Mg3Sb2, we compare the values between the theory and the experiment to assess the applicability of the PBE and HSE06 functional. In Table 1, the calculated formation enthalpy of α-Mg3Sb2 using PBE is -35.91 kJ/mol-atom, while the value using HSE06 -53.44 kJ/mol-atom (-0.56 eV/atom) which are in better agreement with experimental data -60.04 kJ/mol-atom25, likely originated from the improved description of the localized Sb p electrons. Combined the calculated formation enthalpy, the Mg-poor condition was set to ∆𝜇𝑀𝑔= -0.93 eV and ∆𝜇𝑆𝑏= 0 eV from Eq. 2-3 in terms of the equilibrium between pure Sb and stoichiometric α-Mg3Sb2, which means the chemical potential 𝜇𝑀𝑔= -2.47 eV and 𝜇𝑆𝑏= -5.12 eV from Eq. 5-6 with the reference states being -1.54 and -5.12 eV of 𝜇0𝑀𝑔 and 𝜇0𝑆𝑏 from DFT 7



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calculations, respectively. The Mg-rich condition was set to ∆𝜇𝑀𝑔= 0 eV and ∆𝜇𝑆𝑏= − 1.39 eV from Eq. 2-3, in terms of the equilibrium between pure Mg and stoichiometric α-Mg3Sb2, which makes the chemical potential 𝜇𝑀𝑔= − 1.54 eV and 𝜇𝑆𝑏= − 6.51 eV from Eq. 5-6 with the same reference states of the Mg-poor condition. To identify the dominant native defects, six typical single point defects are considered in αMg3Sb2, i.e. interstitials (MgI, SbI), antisites (MgSb, SbMg), and vacancies (VMg, VSb). Various charged states are calculated for all defects in the crystal structure. The formation energy of a single defect as a function of EF both in Mg-rich and Sb-rich regions is shown in Figure 4. The slope of the curve reflects the charge state of each defect. In the Mg-poor region, the vacancy on Mg (1) has slightly lower formation energy than the vacancy on Mg (2). VMg (1) with -2 charge state has the lowest formation energy among the defects examined over a wide range of Fermi energies, implying that VMg (1) should be the dominant p-type defect. The higher formation energy of SbMg(1), SbMg(2) and VSb leads to a low number of carriers and almost have no influence on the electronic characteristic of α-Mg3Sb2. However, under Mg-rich condition, the MgI with +2 charge state has the lowest formation energy near the VBM among all the single defects, which acts as an n-type shallow donor, introducing electrons into the system and raising the Fermi energy. The predominance will be overturned at a higher Fermi energy. The formation energy of VMg (1) and VMg (2) with below -2 charge state is slightly lower than that of MgI2+ when Fermi energy exceeds 0.18 eV. Besides the single defects, the formation of complex defects is considered in the α-Mg3Sb2. For the Schottky-type, the VMg and VSb all bear the negative charge, which cannot form the defect complex because of the strong Coulomb repulsion. For the antisite-type, the formation energy of SbMg and MgSb are all high as shown in Figure 4. For the Frenkel-type, various SbI defects are not 8


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stable over a wide range of Fermi energies. So only the Mg Frenkel defect complex consisting of VMg and MgI is considered in this work. In view of the opposite charge, there are two types of Mg Frenkel defect complexes with the shortest distance of single defects as shown in Figure 5. The stability of a complex defect with respect to its constituent single defects is defined as binding energy, i.e. 31,32 E𝑏 (𝐷𝐴 + 𝐷𝐵) = 𝐸𝑓(𝐷𝐴 + DB) ― 𝐸𝑓(𝐷𝐴) + 𝐸𝑓(𝐷B)

Eq. 7

where 𝐸𝑓(𝐷𝐴) and 𝐸𝑓(𝐷B) are the formation energies of single defects DA and DB from Eq. 1, respectively. 𝐸𝑓(𝐷𝐴 + DB) is the formation energy of a DA+DB complex from Eq. 1, and E𝑏

(𝐷𝐴 + 𝐷𝐵) is the binding energy of a DA+DB complex. The negative binding energy indicates that the complex defect is energetically more favorable than the single defects. The calculated results for VMg(1)+MgI and VMg(2)+MgI are listed in Table 2. The VMg(2)+MgI defect complex has negative binding energy in both Mg- and Sb-rich regions with the neutral and -1 charge states, suggesting this defect complex may form instead of VMg(2) and MgI individually. This is probably due to the short distance between MgI and VMg (2), i.e. 2.812 Å, as shown in Figure 5, much shorter than the distance of 3.620 Å between MgI and VMg (1), resulting strong Coulomb attraction between MgI and VMg (2), which will be further discussed below in terms of charge density. As shown in Figure 4, the VMg(2)+MgI defect complex with −1 charge state has lower formation energy than that of the neutral VMg(2)+MgI defect complex when crossing the thermodynamic transition level at 0.35 eV in both Mg- and Sb-rich regions. In Mg-rich conditions, (VMg(2)+MgI)1- becomes the dominated ptype defect besides the n-type MgI2+. The atomic structure and the square wavefunctions of the defect states for VMg(1)2-, VMg(2)2-, MgI2+ and (VMg(2)+MgI)1- are plotted in Figure 6. The acceptor defect states of VMg(1)2- and VMg(2)2are unoccupied by holes, which are released to VBM and become free holes moving around in the 9



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crystal. It can be seen that the square wavefunctions of the defect states of VMg(1)2- and VMg(2)2- are very similar, and both of them are mainly distributed around the Sb atoms in a “dumbbell” shape, since they are majorly contributed by the Sb p-orbitals. The donor defect state of MgI2+ is unoccupied by electrons, which are released to CBM and become free electrons moving around in the crystal. It can be seen that the defect state of MgI2+ is mainly distributed around the interstitial Mg atom nearly spherically, due to a large portion of Mg s-orbitals in its character. The defect state of (VMg(2)+MgI)1- is occupied by electrons, leaving holes in VBM. It is distributed around both the Mg atoms in the perfect site and in the interstitial site.

3. 3 Defect and Carrier Concentration

The concentration of a defect D with charge state q in the dilute limit is given by33,34

(

c(Dq) = Nsite𝑒𝑥𝑝 ―

𝐸𝑓(𝐷𝑞) 𝑘𝐵𝑇

)

Eq. 8

where Nsite is the concentration of possible defect sites determined by the multiplicity of the defect’s Wyckoff site, kB is the Boltzmann’s constant, and T is the absolute temperature. The electron concentration (n) in the conduction band and hole concentration (p) in the valence band are obtained from the Fermi-Dirac distribution as 35,36: 1

+∞

n = ∫𝐸 𝑔(𝐸) 𝐶

𝑒𝑥𝑝

Eq. 9

𝑑𝐸

Eq. 10

𝑘𝐵𝑇

1

𝐸

𝑉 p = ∫ ―∞ 𝑔(𝐸)

𝑑𝐸

( )+1 𝐸 ― 𝐸𝐹

( )+1

𝑒𝑥𝑝

𝐸𝐹 ― 𝐸 𝑘𝐵𝑇

where g (E) is the density of states of ideal host cell in bulk, EC is the conduction band minimum and EV is the valence band maximum. The defect and carrier concentrations are constrained by the condition of the charge neutrality, 10


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∑𝑖𝑞𝑖𝑐𝑖(𝐷𝑞) +𝑝 ― 𝑛 = 0

Eq. 11

The index i represents the charged defect. From the above equations, the Fermi level (EF), defect and free carrier concentrations can be determined by iterations and are plotted in Figure 79. Figure 7 (a) and (b) show the temperature-dependent Fermi levels without or with defect complex under Mg-rich and Mg-poor conditions, respectively. The results indicate the p-type behavior of α-Mg3Sb2 under both Mg-poor and Mg-rich conditions because EF is close to VBM or in the valence band. Without defect complex (only single defects), the dominant defect is the VMg(1)2- and VMg(2)2- in Mg-poor region, which are the shallow acceptors, leading to more holes than free electrons in the p-type α-Mg3Sb2. EF decreases from -0.046 to -0.260 eV from 300 to 1200 K and is deeply located into the valence band even at room temperature, and keeps locating at the valence band in Figure 7 (a). In Mg-rich region, as observed in Figure 4, formation energy of MgI with +2 charge is lower than that in Mg poor region. However, EF is always close to valence band maximum and decreases from 0.257 to 0.150 eV in Figure 7 (a) because of the Mg vacancy. When considering the defect complex, the neutral VMg(2)+MgI is more stable than charged state under Mg-poor condition as shown in Figure 4, which cannot effect the Fermi level. EF decreases from -0.046 to -0.270 eV when temperature increases from 300 K to 1200 K in Figure 7 (b), which is almost the same with the only single point situation. Under Mg-rich condition, the EF considering the VMg(2)+MgI complex is decreasing from 0.094 to 0.051 eV, lower than that without VMg(2)+MgI. This is because VMg(2)+MgI with -1 charge state is more stable than MgI2+ when it is near the valence band maximum, which acts as a shallow acceptor to enhance the p-type behavior of α-Mg3Sb2. Based on the above discussion, α-Mg3Sb2 is an intrinsic p-type semiconductor both without and with defect complex. Moreover, (VMg(2)+MgI)1- leads the Ef under Mg-rich condition closer to 11



the VBM. The details and mechanism can be explored more from the concentration of the major defects as shown in Figure 8. Without defect complex, the concentration of VMg(1)2- and VMg(2)2- is higher than other single defects under Mg-poor condition as shown in Figure 8 (a). The major defect VMg(1)2- is higher than VMg(2)2- at low temperature, but becomes almost equal with temperature increasing and reaches (1.05-1.18)×1019 cm-3 at 1200 K. The concentration of MgI2+ is pretty low under Mg-poor condition especially at low temperature, but becomes higher than the Mg vacancies under Mg-rich condition in Figure 8 (b) at high temperature due to the formation energy of MgI2+ lower than VMg(1)2- and VMg(2)2- when Ef is near VBM in Figure 4 under Mg-rich condition. Per MgI2+ introduces two electrons into the system, making the Fermi energy locate above the VBM as shown in Figure 7 (a). But the electron carriers induced by the donor-type defect MgI2+ are incapable of compensating the effects of holes induced by the acceptor-type defects (VMg(1)2-, VMg(2)2-, VMg(1)3- and VMg(2)3-) under Mg-rich condition, which has been reflected in Figure 8 (b) showing that the concentrations of the major defects are all close. The primary reason is that the EF ranges from 0.257 to 0.150 eV, and when EF exceeds 0.18 eV, the formation energy of MgI2+ is higher than VMg(1)2- and VMg(2)2- near the VBM in Figure 4 in Mg-rich region. With the defect complex, the concentration of (VMg(2)+MgI)1- is relative small under low temperature but can reach 1.7×1019 cm-3 at 1200 K as shown in Figure 8 (c). This is because the Fermi level moves to -0.27 eV at 1200 K, where the formation energy of (VMg(2)+MgI)1- is close to VMg(1)2- and VMg(2)2as shown in Figure 4 in Mg-poor region. However, the defect complex (VMg(2)+MgI)1- only bear 1 charge state and the concentration is lower than the sums of two kinds of Mg vacancies, which makes it rarely affect the electronic structure of α-Mg3Sb2. That’s why EF is almost the same without and with defect complex with the increasing temperature under Mg poor condition in Figure 7 (a) and (b). Under Mg-rich condition, the concentration of MgI2+ become much larger 12


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than all the p-type Mg vacancy defects as shown in Figure 8 (d), because the formation energy of MgI2+ is much lower than others with temperature increasing when EF shifts from 0.094 to 0.051 eV. However, the concentration of (VMg(2)+MgI)1- overturn the ascendancy of MgI2+, which introduces the hole into the system and compensates for electron carriers induced by the MgI2+ defects. The role of the defect complex (VMg(2)+MgI)1- further enhances the intrinsic p-type behavior of α-Mg3Sb2 and moves EF much closer to VBM in Figure 7 (b) than that only considering single defects in Figure 7 (a). The root cause should also attribute to the Fermi-energy dependent stability of defects that formation energy of (VMg(2)+MgI)1- is much lower than MgI2+ within almost the entire energy range of the band gap shown in Figure 4 under Mg-rich region. The above discussion indicates that the intrinsic p-type of α-Mg3Sb2 is so persistent that the intrinsic n-type conductivity can never be realized under any equilibrium growth condition in MgSb system. n-type transition only can be realized by heavy impurity doping due to the existence of defect complex (VMg(2)+MgI)1-. Figure 9 (a) and (b) show the calculated hole and electron carriers concentration as a function of temperature without defect complex under Mg-poor and Mg-rich conditions, respectively. The hole carrier concentration in the Mg-poor region is higher than that in the Mg-rich region because of the lower Fermi level in the Mg-poor region compared with that in the Mg-rich region. The electron carrier concentration in the Mg-poor region is lower than that in the Mg-rich region because of the higher formation energy of MgI2+ in the Mg-poor region than that in the Mg-rich region. The experimental data of carrier concentration showed in Figure 9 is measured from Hall Effect. The Hall carrier concentration (nH) is obtained via nH = 1/eRH, where RH is the Hall coefficient and e is the electron charge. It is evident that the calculated holes concentration is much lower than the experimental carrier concentration in Figure 9 (b). When considering the defect complex under Mg-poor and Mg-rich conditions, the intrinsic free carriers 13



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are found to be 1017−1020 cm−3 as shown in Figure 9 (c) and (d), which is the same range found experimentally for pure p-type α-Mg3Sb2 3–5,26,27,37–39. The concentration of holes under Mg-rich condition in Figure 9 (d) is larger and closer to the experimental Hall carrier concentration than that without defect complex (VMg(2)+MgI)1- in Figure 9 (b).

3.4 Phase boundary of α-Mg3Sb2 With an equilibrium set of chemical potentials, the composition x of element i in α-Mg3Sb2 is given by 40 𝑁𝑖0 + ∆𝑁𝑖

Eq. 12

x = ∑ (𝑁𝑖 + ∆𝑁𝑖) 𝑖

0

where 𝑁𝑖0 is the nominal stoichiometry of element i (i = Mg, Sb) in α-Mg3Sb2, and ∆N𝑖 is the change in the composition due to the point defect at specific temperature T. The change in composition of α-Mg3Sb2 is calculated by summing over the carriers of each defect j, weighted by the change in composition due to the defect 31 ∆𝑁𝑖 = ∑𝑗𝑁𝑖𝑗𝑐𝑗

Eq. 13

Following the above procedure, the binary phase diagram of α-Mg3Sb2 is obtained as a function of temperature. The calculated solvus boundaries between Sb and α-Mg3Sb2 as blue line (the Mgpoor region) and between α-Mg3Sb2 and Mg as red line (the Mg-rich region) are shown in Figure 10. Without defect complex, because of lower formation energy and higher concentration of VMg(1)2- and VMg(2)2- in the Mg-poor region compared with that of MgI2+ in the Mg-rich region, the solubility limit of Sb－α-Mg3Sb2 boundary is much higher than that of α-Mg3Sb2－Mg boundary. The Mg solubility in α-Mg3+xSb2 is as low as x = 0.0003, which is consistent with the results from Ohno, et al

15.

However, when consider the defect complex (VMg(2)+MgI)1-, the Fermi level is 14


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shifted close to the VBM. Although (VMg(2)+MgI)1- make no contribution to the chemistry of αMg3Sb2, it increases the concentration of single MgI2+ and suppresses the concentration of VMg(1)2and VMg(2)2- under Mg-rich condition, which enlarges the solubility of Mg in α-Mg3+xSb2 up to x = 0.011 and solubility limit of α-Mg3Sb2 －Mg boundary. In the experiments of Shuai et al., the semiconductor character turn from p-type to n-type when the extra initial Mg content increases to x = 0.025 in Mg3+xSb1.5Bi0.5 system 13. It is expected that the solubility of excess-Mg is larger than 0.01 so that the excess-Mg can enter into the lattice and alter its electronic structure. Our predicted results are more reasonable as supported by the experiments. The slight underestimation of solubility is from the dilute-limit approximation, which does not take into account defect interactions and change in vibrational entropy. 4

Conclusions The intrinsic defect chemistry based on hybrid density functional theory of α-Mg3Sb2 shows

that VMg(1)2- and VMg(2)2- are the dominant defects and the concentration reaches (1.05-1.18)×1019 cm-3 at 1200 K under Mg-poor condition. A defect complex (VMg(2)+MgI)1- is found to be the dominant reason for p-type character under Mg-rich condition and the concentration is as high as 3.5×1020 cm-3, which shifts the Fermi level close to the VBM. It is predited that the excess-Mg used in experiments is to compensate the (VMg(2)+MgI)1- rather than the single Mg vacancy. Considering the defect complex (VMg(2)+MgI)1-, the predicted carrier concentration is 1017−1020 cm−3, which is the same range found experimentally for pure p-type α-Mg3Sb2 and more closer to the experimental data than that without the defect complex. Although the defect complex (VMg(2)+MgI)1- cannot change the chemistry of α-Mg3+xSb2, it tailors the electronic structure and defect concentrations, which enlarges the solubility of Mg in α-Mg3+xSb2 up to x = 0.011 and 15



Page 16 of 34

changes the α-Mg3Sb2－Mg boundary, more reasonable as supported by experiments. Acknowledgments This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, and The Pennsylvania State University, under a contract with the National Aeronautics and Space Administration. Thanks to the Scholarship from the China Scholarship Council (201608530171). First-principles calculations were carried out partially on the LION clusters at the Pennsylvania State University, partially on the resources of NERSC supported by the Office of Science of the U.S. Department of Energy under contract No. DE-AC02-05CH11231, and partially on the resources of XSEDE supported by NSF with Grant No. ACI-1053575.

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References (1)

Disalvo, F. J. Thermoelectric Cooling and Power Generation. Science. 1999, 285 (5428), 703–706.

(2)

Snyder, G. J.; Toberer, E. S. Complex Thermoelectric Materials. Nat. Mater. 2008, 7, 105– 114.

(3)

Song, L.; Zhang, J.; Iversen, B. B. Simultaneous Improvement of Power Factor and Thermal Conductivity via Ag Doping in P-Type Mg 3 Sb 2 Thermoelectric Materials. J. Mater. Chem. A 2017, 5, 4932–4939.

(4)

Bhardwaj, A.; Chauhan, N. S.; Misra, D. K. Significantly Enhanced Thermoelectric Figure of Merit of P-Type Mg

3

Sb

2

-Based Zintl Phase Compounds via Nanostructuring and

Employing High Energy Mechanical Milling Coupled with Spark Plasma Sintering. J. Mater. Chem. A 2015, 3, 10777–10786. (5)

Ren, Z.; Shuai, J.; Mao, J.; Zhu, Q.; Song, S.; Ni, Y.; Chen, S. Significantly Enhanced Thermoelectric Properties of P-Type Mg 3 Sb 2 via Co-Doping of Na and Zn. Acta Mater. 2018, 143, 265–271.

(6)

Tamaki, H.; Sato, H. K.; Kanno, T. Isotropic Conduction Network and Defect Chemistry in Mg3+δSb2-Based Layered Zintl Compounds with High Thermoelectric Performance. Adv. Mater. 2016, 28, 10182–10187.

(7)

Zhang, J.; Song, L.; Pedersen, S. H.; Yin, H.; Hung, L. T.; Iversen, B. B. Discovery of HighPerformance Low-Cost N-Type Mg3Sb2-Based Thermoelectric Materials with MultiValley Conduction Bands. Nat. Commun. 2017, 8, 1–8.

17



(8)

Page 18 of 34

Zhang, J.; Song, L.; Mamakhel, A.; Jørgensen, M. R. V.; Iversen, B. B. High-Performance Low-Cost N-Type Se-Doped Mg3Sb2-Based Zintl Compounds for Thermoelectric Application. Chem. Mater. 2017, 29, 5371–5383.

(9)

Shuai, J.; Mao, J.; Song, S.; Zhu, Q.; Sun, J.; Wang, Y.; He, R.; Zhou, J.; Chen, G.; Singh, D. J.; Ren, Z. Tuning the Carrier Scattering Mechanism to Effectively Improve the Thermoelectric Properties. Energy Environ. Sci. 2017, 10, 799–807.

(10)

Mao, J.; Wu, Y.; Song, S.; Zhu, Q.; Shuai, J.; Liu, Z.; Pei, Y.; Ren, Z. Defect Engineering for Realizing High Thermoelectric Performance in N-Type Mg3Sb2-Based Materials. ACS Energy Lett. 2017, 2, 2245–2250.

(11)

Mao, J.; Shuai, J.; Song, S.; Wu, Y.; Dally, R.; Zhou, J.; Liu, Z.; Sun, J.; Zhang, Q.; dela Cruz, C.; Wilson, S.; Pei, Y.; Singh, D. J.; Chen, G.; Chu, C.-W.; Ren, Z. Manipulation of Ionized Impurity Scattering for Achieving High Thermoelectric Performance in N-Type Mg3 Sb2 -Based Materials. Proc. Natl. Acad. Sci. 2017, 114, 10548–10553.

(12)

Zhang, J.; Song, L.; Borup, K. A.; Jørgensen, M. R. V.; Iversen, B. B. New Insight on Tuning Electrical Transport Properties via Chalcogen Doping in N-Type Mg 3 Sb 2 -Based Thermoelectric Materials. Adv. Energy Mater. 2018, 1702776, 1702776.

(13)

Shuai, J.; Ge, B.; Mao, J.; Song, S.; Wang, Y.; Ren, Z. Significant Role of Mg Stoichiometry in Designing High Thermoelectric Performance for Mg3(Sb,Bi)2-Based N-Type Zintls. J. Am. Chem. Soc. 2018, 140, 1910–1915.

(14)

Imasato, K.; Kang, S. D.; Ohno, S.; Snyder, G. J. Band Engineering in Mg 3 Sb 2 by Alloying with Mg 3 Bi 2 for Enhanced Thermoelectric Performance. Mater. Horizons 2018, 5, 59–64.

18


Page 19 of 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(15)

Ohno, S.; Imasato, K.; Anand, S.; Tamaki, H.; Kang, S. D.; Gorai, P.; Sato, H. K.; Toberer, E. S.; Kanno, T.; Snyder, G. J. Phase Boundary Mapping to Obtain N-Type Mg3Sb2-Based Thermoelectrics. Joule 2018, 2, 141–154.

(16)

Kuo, J. J.; Kang, S. D.; Imasato, K.; Tamaki, H.; Ohno, S.; Kanno, T.; Snyder, G. J. Grain Boundary Dominated Charge Transport in Mg 3 Sb 2 -Based Compounds. Energy Environ. Sci. 2018, 11, 429–434.

(17)

Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953–17979.

(18)

Krukau, A. V.; Vydrov, O. A.; Izmaylov, A. F.; Scuseria, G. E. Influence of the Exchange Screening Parameter on the Performance of Screened Hybrid Functionals. J. Chem. Phys. 2006, 125, 224106.

(19)

Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865–3868.

(20)

Guan, P.-W.; Liu, Z.-K. A Hybrid Functional Study of Native Point Defects in Cu 2 SnS 3 : Implications for Reducing Carrier Recombination. Phys. Chem. Chem. Phys. 2018, 20, 256– 261.

(21)

Monkhorst, H.; Pack, J. Special Points for Brillouin Zone Integrations. Phys. Rev. B 1976, 13, 5188–5192.

(22)

Toberer, E. S.; May, A. F.; Snyder, G. J. Zintl Chemistry for Designing High Efficiency Thermoelectric Materials. Chemistry of Materials. 2010, pp 624–634.

(23)

Shuai, J.; Mao, J.; Song, S.; Zhang, Q.; Chen, G.; Ren, Z. Recent Progress and Future Challenges on Thermoelectric Zintl Materials. Mater. Today Phys. 2017, 1, 74–95.

(24)

Ahmadpour, F.; Kolodiazhnyi, T.; Mozharivskyj, Y. Structural and Physical Properties of Mg3-xZnxSb2 (x=0-1.34). J. Solid State Chem. 2007, 180, 2420–2428. 19



Page 20 of 34

(25)

Metallurgical Thermochemistry. Phys. Today 1959, 12, 50–51.

(26)

Bhardwaj, A.; Rajput, A.; Shukla, A. K.; Pulikkotil, J. J.; Srivastava, A. K.; Dhar, A.; Gupta, G.; Auluck, S.; Misra, D. K.; Budhani, R. C. Mg3Sb2-Based Zintl Compound: A NonToxic, Inexpensive and Abundant Thermoelectric Material for Power Generation. RSC Adv. 2013, 3, 8504.

(27)

Kim, S. H.; Kim, C. M.; Hong, Y.-K.; Sim, K. I.; Kim, J. H.; Onimaru, T.; Takabatake, T.; Jung, M.-H. Thermoelectric Properties of Mg 3 Sb

2− X

Bi

X

Single Crystals Grown by

Bridgman Method. Mater. Res. Express 2015, 2, 55903. (28)

Van De Walle, C. G.; Neugebauer, J. First-Principles Calculations for Defects and Impurities: Applications to III-Nitrides. J. Appl. Phys. 2004, 95, 3851–3879.

(29)

Balakumar, T.; Medraj, M. Thermodynamic Modeling of the Mg-Al-Sb System. Calphad Comput. Coupling Phase Diagrams Thermochem. 2005, 29, 24–36.

(30)

Freysoldt, C.; Grabowski, B.; Hickel, T.; Neugebauer, J.; Kresse, G.; Janotti, A.; Van De Walle, C. G. First-Principles Calculations for Point Defects in Solids. Rev. Mod. Phys. 2014, 86, 253–305.

(31)

Li, G.; Aydemir, U.; Wood, M.; Goddard, W. A.; Zhai, P.; Zhang, Q.; Snyder, G. J. DefectControlled Electronic Structure and Phase Stability in Thermoelectric Skutterudite CoSb 3. Chem. Mater. 2017, 29, 3999–4007.

(32)

Duan, X. M.; Stampfl, C. Vacancies and Interstitials in Indium Nitride: Vacancy Clustering and Molecular Bondlike Formation from First Principles. Phys. Rev. B - Condens. Matter Mater. Phys. 2009, 79, 174202.

20


Page 21 of 34 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(33)

Li, G.; Bajaj, S.; Aydemir, U.; Hao, S.; Xiao, H.; Goddard, W. A.; Zhai, P.; Zhang, Q.; Snyder, G. J. P-Type Co Interstitial Defects in Thermoelectric Skutterudite CoSb3Due to the Breakage of Sb4-Rings. Chem. Mater. 2016, 28, 2172–2179.

(34)

Boonchun, A.; Reunchan, P.; Umezawa, N. Energetics of Native Defects in Anatase TiO 2 : A Hybrid Density Functional Study. Phys. Chem. Chem. Phys. 2016, 18, 30040–30046.

(35)

Bjerg, L.; Madsen, G. K. H.; Iversen, B. B. Ab Initio Calculations of Intrinsic Point Defects in ZnSb. Chem. Mater. 2012, 24, 2111–2116.

(36)

Kumagai, Y.; Choi, M.; Nose, Y.; Oba, F. First-Principles Study of Point Defects in Chalcopyrite ZnSnP2. Phys. Rev. B - Condens. Matter Mater. Phys. 2014, 90, 1–12.

(37)

Bhardwaj, A.; Chauhan, N. S.; Goel, S.; Singh, V.; Pulikkotil, J. J.; Senguttuvan, T. D.; Misra, D. K. Tuning the Carrier Concentration Using Zintl Chemistry in Mg 3 Sb 2 , and Its Implications for Thermoelectric Figure-of-Merit. Phys. Chem. Chem. Phys. 2016, 18, 6191– 6200.

(38)

Ponnambalam, V.; Morelli, D. T. On the Thermoelectric Properties of Zintl Compounds Mg3Bi2−x Pn X (Pn = P and Sb). J. Electron. Mater. 2013, 42, 1307–1312.

(39)

Bhardwaj, A.; Misra, D. K. Enhancing Thermoelectric Properties of a P-Type Mg 3 Sb 2 Based Zintl Phase Compound by Pb Substitution in the Anionic Framework. RSC Adv. 2014, 4, 34552–34560.

(40)

Bajaj, S.; Wang, H.; Doak, J. W.; Wolverton, C.; Jeffrey Snyder, G. Calculation of Dopant Solubilities and Phase Diagrams of X–Pb–Se (X = Br, Na) Limited to Defects with Localized Charge. J. Mater. Chem. C 2016, 4, 1769–1775.

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Table 1 Calculated and experimental lattice parameters, unit cell volume, formation enthalpy and band gap of the stoichiometric α-Mg3Sb2 Method PBE HSE06 a=b (Å) 4.59 4.58 c (Å) 7.28 7.24 3 V (Å ) 133.09 131.59 ΔH (kJ/mol-atom) -35.91 -53.44 Band gap (eV) 0.402 0.824 aExp. by Ahmadpour, et al24 bExp. by Kubaschewski, et al25 cCal. by Bhardwaj, et al using mBJ-GGA26 dCal. by Zhang, et al using TB-mBJ7 eExp. by Kim, et al27

Other theoretical results

0.700c, 0.600d

Experiment 4.56a 7.23a 130.08a -60.04±3.35b 0.960e

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Table 2 Binding energies (in eV) calculated for Frenkel complex defect of Mg

Mg-poor Mg-rich

Eb (VMg(1)+MgI)0 0.07 0.07

Eb (VMg(2)+MgI)0 -1.40 -1.40

Eb (VMg(1)+MgI)1- Eb (VMg(2)+MgI)10.03 -1.37 0.03 -1.37

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Figure 1. (a) 3×3×3 supercell, (b) conventional unit cell of α-Mg3Sb2

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Figure 2. Band structure and partial and total electronic density of states for α-Mg3Sb2 using (a) PBE; (b) HSE06 functional. The dashed vertical lines indicate the accurate conduction band minimum along M-L line

25



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Figure 3. Phase diagram of Mg-Sb system29. Note that there is no solubility range in α-Mg3Sb2.

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Figure 4. Calculated defect formation energies as a function of Fermi level under both Mg-poor and Mg-rich conditions. The orange vertical lines represent the Fermi level at 300 K with the defect complex (VMg(2)+MgI)1-.

27



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Figure 5. Defect complex models in α-Mg3Sb2, (a) VMg (1) + MgI; (b) VMg (2) + MgI.

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Figure 6. Square wavefunction isosurfaces of the defect states of the four major defects in αMg3Sb2. (a) VMg(1)2-; (b) VMg(2)2-; (c) MgI2+; (d) (VMg(2)+MgI)1-. For (a) and (b), the values of the isosurfaces are from -7.18×10-4 to 1.75 ×10-3, and the isosurface level is 4.58×10-4. For (c), the values of the isosurfaces are from -8.57×10-5 to 2.52 ×10-3, and the isosurface level is 5.11× 10-4. For (d), the values of the isosurfaces are from -2.47×10-5 to 7.00×10-4, and the isosurface level is 4.53×10-4

29



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Figure 7. Calculated Fermi level (EF) as a function of temperature under both Mg-poor and Mgrich conditions. (a) Without defect complex; (b) with defect complex

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Figure 8. Calculated defect concentration as a function of temperature under Mg-poor and Mgrich conditions: (a) and (b) without defect complex; (c) and (d) with defect complex

31



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Figure 9. Calculated carriers concentration as a function of temperature under both Mg poor and Mg rich conditions without defect complex ((a) and (b)) and with defect complex ((c) and (d)), in comparison with experimental data of Kim et al. 27(: ■), Bhardwaj et al.4,26,37,39 (:●, : ▲, : ▼, and : ★), Ponnambalam et al.38 (: ◆), Ren et al.5 (:

), and Song et al.3 (:

).

32


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Figure 10. Calculated phase boundary of α-Mg3Sb2 as a function of temperature with and without defect complex.

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Graphical abstract

34


Understanding the Intrinsic P-Type Behavior and Phase Stability of

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