Mg3Sb2基热电化合物固溶特性与热输运性能的理论研究

Mg3Sb2基热电化合物具有热电性能好以及组成元素含量丰富、无毒、价格低廉等环境友好型特点，近年来，引起研究者的广泛关注。但大多数研究集中在Mg3Sb2基材料的热电性能，很少有人关注其固溶性能。材料的热输运性能的模拟在科学研究和工业领域都很重要，但固溶体的热输运模拟是当前理论计算研究的一个挑战问题。本文以Mg3Sb2基热电化合物为研究对象，结合密度泛函理论（Density Functional Theory, DFT）、团簇展开（Cluster Expansion, CE）、蒙特卡罗（Monte Carlo, MC）以及机器学习等方法，开展同构固溶和异构固溶体系相关的热力学问题研究，并以同构固溶Mg3Sb2−Mg3Bi2和异构固溶Mg3Sb2−Mg2Sn体系的固溶特性为例展开应用；在此基础上，探索利用机器学习原子间势（Machine Learning Interatomic Potentials, MLIPs）开展复杂固溶体Mg2-δ(Sn1-xSbx)热输运模拟的可行性。研究内容和主要结论如下：

在同构固溶体系Mg3Sb2−Mg3Bi2中，采用第一性原理方法计算Mg3(Sb1-xBix)2固溶体的形成能，基于固溶体的Gibbs自由能得到Mg3Sb2-Mg3Bi2的相图，发现Mg3(Sb1-xBix)2固溶体存在混溶间隙。结合第一性原理计算以及与缺陷相关热力学模型分析，发展了适用于多元合金体系中溶质活度（Activity）的计算方法，并在Mg3(Sb1-xBix)2体系中应用此模型计算了溶质Mg的活度，结果与文献报道的实验值相吻合。根据相变平衡条件，估算了Mg在Mg3(Sb1-xBix)2中的固溶极限，并给出了Mg-Mg3Sb2-Mg3Bi2的等温截面相图。揭示了Mg对Mg3(Sb1-xBix)2热电性能的影响，在Mg过量的条件下，发现间隙Mg是Mg3(Sb1-xBix)2固溶体的主要缺陷，也是产生n型热电的原因。

由不同晶体结构的母相化合物形成的异构固溶体在探索新材料时已经引起人们的关注，但系统深入的理论研究缺乏。本文结合第一性原理计算、团簇展开和蒙特卡罗模拟，克服了异构固溶体系中能量和构型熵计算的困难，发展了研究复杂异构固溶体系的能量和构型演化的系统性的通用方法。在针对异构固溶体系Mg3Sb2−Mg2Sn的实际应用中，Gibbs自由能的结果表明，Mg2Sn基结构的Mg2-0.5xSn1-xSbx相在0 < x ≤ ~0.47成分范围内是稳定的，在~0.47 < x ≤ 0.75成分范围是亚稳的，然而Mg3Sb2基相在0.75 < x < 1.0成分范围内是非稳定的，这一结果与实验观测相吻合。大尺度的MC模拟发现Mg2-0.5xSn1-xSbx固溶体具有本征的层级微结构，主要由贫Mg富Sb的纳米级团簇和均匀的Mg2Sn基的基体共同组成。微结构的成分分析显示纳米团簇区域的Mg/(Sn+Sb)成分配比接近于Mg3Sb2的原子比3:2，基体区域的接近于Mg2Sn的2:1，这揭示了异构固溶体成分分布的不均匀性。因此，Mg2-0.5xSn1-xSbx微结构的电子结构表现出p型纳米团簇区域和n型基体区域的不均匀共存。选择性掺杂可以实现微结构电性能p型/n型的转变，这为异构固溶材料的优化设计提供了理论基础和方法。

机器学习原子间势函数（MLIPs）的应用已经很广泛，但是对于微结构复杂多样的固溶特别是异构固溶体系，可靠机器学习势函数的获取一直存在困难。本文提出，可以基于DFT-CE-MC模拟的层级微结构结果，同时兼顾成分涨落和层级微结构变化的宽覆盖范围取样，产生合理的结构训练数据集并获取可靠势函数。基于DFT和MD计算，采用双重自适应的取样方法在宽温度范围区间（50-800 K），宽成分范围区间（0 ≤ x < 0.8）以及复杂微结构的构型空间产生训练数据集，构建了Mg2-δSn1-xSbx异构固溶体的MLIPs。MLIPs预测的能量、力、原子的运动轨迹、晶格常数、形成能以及Mg2Sn和合金Mg1.875Sn0.75Sb0.25的热输运性质都与DFT计算的结果吻合很好，表明训练的MLIPs具有很好的可靠性和准确性。最后，采用基于MLIPs的Green−Kubo方法，初步估算了Mg2-δSn0.8Sb0.2固溶体的晶格热导率及其温度相关性。

In recent years, Mg₃Sb₂-based thermoelectric (TE) materials have gained attention due to their high TE performance and environmentally friendly features, as they comprise abundant, nontoxic, and low-cost constituent elements. Although the thermoelectric properties of Mg₃Sb₂-based TE materials have been extensively studied, reports on their solid-solution properties are limited. In scientific research and industry, the simulation of the thermal transport properties of materials is crucial, but the thermal transport simulation of solid solutions is a challenge in current theoretical research. In this dissertation, we investigated the thermodynamic properties of isostructural and hetero-structural solid-solution systems of Mg₃Sb₂-based TE materials using the approaches of density functional theory (DFT), cluster expansion (CE), Monte Carlo (MC), and machine learning. We used the solid-solution properties of isostructural Mg₃Sb₂−Mg₃Bi₂ and hetero-structural Mg₂Sn−Mg₃Sb₂ systems as examples for application. Furthermore, the feasibility of using machine learning interatomic potentials (MLIPs) in the thermal transport simulation of complex Mg_2-δ(Sn_1-xSb_x) solid solutions were studied. The research contents and main conclusions are listed as follows:

In the isostructural Mg₃Sb₂−Mg₃Bi₂ system, the formation energies of Mg₃(Sb_1-xBi_x)₂ solid solutions were calculated by first-principles method. Based on the Gibbs free energy of the solid solutions, we obtain the phase diagram for the Mg₃Sb₂−Mg₃Bi₂ system. Our findings reveal the presence of a miscibility gap in the Mg₃(Sb_1-xBi_x)₂ solid solutions. A method of calculating solute activity in multi-component alloys by combining first-principles calculations and thermodynamic analysis associated with defects is developed. Based on the model, the activities of Mg in Mg₃(Sb_1-xBi_x)₂ alloy were estimated. Our estimated activities reproduce the available experimental results well. According to the criterion of phase equilibrium, we estimate the solubility limits of Mg in Mg₃(Sb_1-xBi_x)₂ and construct the isothermal section phase diagram of Mg-Mg₃Sb₂-Mg₃Bi₂. We discuss the effect of Mg on the thermoelectric performance of Mg₃(Sb_1-xBi_x)₂ alloy and find that Mg interstitials are the dominant defects in Mg₃(Sb_1-xBi_x)₂ alloy, responsible for the n-type thermoelectric under Mg excess conditions.

Hetero-structural solid solutions, composed of parent compounds with distinct crystal structures, have garnered significant interest in the exploration of new materials, but a comprehensive and systematic theoretical research is still lacking. To overcome the calculation difficulties of energy and configurational entropy, we developed a general approach that combines first-principles calculations, CE, and MC simulations to evaluate the energetics and configuration evolution of complex hetero-structural solid solutions. We applied this approach to the Mg₂Sn−Mg₃Sb₂ system and found that the Gibbs free energy results indicate that Mg_2-0.5xSn_1-xSb_x is stabilized at the Mg₂Sn-based phase in the range of 0<x≤~0.47 and becomes metastable in the range of ~0.47 < x ≤ 0.75, while the Mg₃Sb₂-based phase is unstable in the range of 0.75 < x < 1.0, which is consistent with experimental observations. Large-scale MC simulations reveal that Mg_2−0.5xSn_1−xSb_x solid solutions have natural hierarchical microstructures, which contain primarily nanoscale Mg-deficiency and Sb-rich clusters and a homogeneous Mg₂Sn-based matrix. Compositional analysis of the microstructures shows that the composition ratio of Mg/(Sn+Sb) for the nanocluster region is close to the atomic ratio of 3:2 in Mg₃Sb₂, and the ratio for the matrix region is close to 2:1 in Mg₂Sn, which reveals the overall inhomogeneous nature of the hetero-structural Mg_2-0.5xSn_1-xSb_x solid solutions. The electronic structures of the microstructures of Mg_2-0.5xSn_1-xSb_x manifest an inhomogeneous coexistence of p-type nanoclusters and n-type matrix. Selective doping can realize the n-type/p-type alteration of microstructures, providing a theoretical basis and method for the optimization and design of hetero-structural solid solutions.

The application of machine learning interatomic potentials (MLIPs) is widespread, but obtaining reliable MLIPs for solid solutions with complex and diverse microstructures, particularly hetero-structural solid solution systems, has always been challenging. We propose that, based on the hierarchical microstructure results from DFT-CE-MC simulations, a reasonable structure training dataset can be established, and reliable MLIPs can be obtained by taking into account the wide sampling space containing both compositional fluctuations and hierarchical microstructure changes. Using DFT and MD calculations, we constructed MLIPs of hetero-structural Mg_2-δSn_1-xSb_x solid solutions by employing a dual adaptive sampling method to generate training datasets in the configuration space with a wide temperature range of 50-800 K, a wide composition range of 0 ≤ x < 0.8, and complex microstructures. The MLIPs predicted energies, forces, atomic motive trajectories, lattice constants, formation energy, and thermal transport properties of Mg₂Sn and Mg_1.875Sn_0.75Sb_0.25 agree well with the results of DFT calculations, indicating the MLIPs' good reliability and accuracy. Finally, we evaluated the lattice thermal conductivity and its temperature dependence for Mg_2−δSn_0.8Sb_0.2 solid solutions using the Green-Kubo method based on the MLIPs.

[1] Rowe D M. CRC handbook of thermoelectrics[M]. Boca Raton, FL: CRC Press, 1995: 1–3.
[2] Stockholm J G. Applications in thermoelectricity[J]. Materials Today: Proceedings, 2018, 5(4): 10257–10276.
[3] Bell L E. Cooling, Heating, Generating Power, and Recovering Waste Heat with Thermoelectric Systems[J]. Science, 2008, 321(5895): 1457–1461.
[4] Shi X-L, Zou J, Chen Z-G. Advanced Thermoelectric Design: From Materials and Structures to Devices[J]. Chemical Reviews, 2020, 120(15): 7399–7515.
[5] Shakouri A. Recent Developments in Semiconductor Thermoelectric Physics and Materials[J]. Annual Review of Materials Research, 2011, 41(1): 399–431.
[6] Uher C. Materials aspect of thermoelectricity[M]. Boca Raton: CRC Press, 2017: 39–42.
[7] He R, Schierning G, Nielsch K. Thermoelectric Devices: A Review of Devices, Architectures, and Contact Optimization[J]. Advanced Materials Technologies, 2018, 3(4): 1700256.
[8] Moshwan R, Yang L, Zou J, et al. Eco-Friendly SnTe Thermoelectric Materials: Progress and Future Challenges[J]. Advanced Functional Materials, 2017, 27(43): 1703278.
[9] Zhang X, Zhao L-D. Thermoelectric materials: Energy conversion between heat and electricity[J]. Journal of Materiomics, 2015, 1(2): 92–105.
[10] Ren Z F, Lan Y C, Zhang Q Y. Advanced Thermoelectrics: Materials, Contacts, Devices, and Systems[M]. Boca Raton, FL: CRC Press, 2017: 3–407.
[11] Liu Z-Y, Zhu J-L, Tong X, et al. A review of CoSb3-based skutterudite thermoelectric materials[J]. Journal of Advanced Ceramics, 2020, 9(6): 647–673.
[12] Huang L, Zhang Q, Yuan B, et al. Recent progress in Half-Heusler thermo-electric materials[J]. Materials Research Bulletin, 2016, 76: 107–112.
[13] Xia K, Hu C, Fu C, et al. Half-Heusler thermoelectric materials[J]. Applied Physics Letters, 2021, 118(14): 140503.
[14] Shuai J, Mao J, Song S, et al. Recent progress and future challenges on thermoelectric Zintl materials[J]. Materials Today Physics, 2017, 1: 74–95.
[15] Liu K-F, Xia S-Q. Recent progresses on thermoelectric Zintl phases: Structures, materials and optimization[J]. Journal of Solid State Chemistry, 2019, 270: 252–264.
[16] Wu Y. Metal Oxides in Energy Technologies[M]. Elsevier, 2018: 49–72.
[17] Minnich A J, Lee H, Wang X W, et al. Modeling study of thermoelectric SiGe nanocomposites[J]. Physical Review B, American Physical Society, 2009, 80(15): 155327.
[18] Poudel B, Hao Q, Ma Y, et al. High-Thermoelectric Performance of Nanostructured Bismuth Antimony Telluride Bulk Alloys[J]. Science, 2008, 320(5876): 634–638.
[19] Mamur H, Bhuiyan M R A, Korkmaz F, et al. A review on bismuth telluride (Bi2Te3) nanostructure for thermoelectric applications[J]. Renewable and Sustainable Energy Reviews, 2018, 82: 4159–4169.
[20] Xiao Y, Wu H, Cui J, et al. Realizing high performance n-type PbTe by synergistically optimizing effective mass and carrier mobility and suppressing bipolar thermal conductivity[J]. Energy & Environmental Science, 2018, 11(9): 2486–2495.
[21] You L, Liu Y, Li X, et al. Boosting the thermoelectric performance of PbSe through dynamic doping and hierarchical phonon scattering[J]. Energy & Environmental Science, 2018, 11(7): 1848–1858.
[22] Shu R, Zhou Y, Wang Q, et al. Mg3+δSbxBi2−x Family: A Promising Substitute for the State-of-the-Art n-Type Thermoelectric Materials near Room Temperature[J]. Advanced Functional Materials, 2019, 29(4): 1807235.
[23] Mao J, Zhu H, Ding Z, et al. High thermoelectric cooling performance of n-type Mg3Bi2-based materials[J]. Science, 2019, 365(6452): 495–498.
[24] Jiang G, He J, Zhu T, et al. High Performance Mg2(Si,Sn) Solid Solutions: a Point Defect Chemistry Approach to Enhancing Thermoelectric Properties[J]. Advanced Functional Materials, 2014, 24(24): 3776–3781.
[25] Pei Y, Zheng L, Li W, et al. Interstitial Point Defect Scattering Contributing to High Thermoelectric Performance in SnTe[J]. Advanced Electronic Materials, 2016, 2(6): 1600019.
[26] Hu L, Zhu T, Liu X, et al. Point Defect Engineering of High-Performance Bismuth-Telluride-Based Thermoelectric Materials[J]. Advanced Functional Materials, 2014, 24(33): 5211–5218.
[27] Xin J, Zhang Y, Wu H, et al. Multiscale Defects as Strong Phonon Scatters to Enhance Thermoelectric Performance in Mg2Sn1–xSbx Solid Solutions[J]. Small Methods, 2019, 3(12): 1900412.
[28] He J, Girard S N, Kanatzidis M G, et al. Microstructure-Lattice Thermal Conductivity Correlation in Nanostructured PbTe0.7S0.3 Thermoelectric Materials[J]. Advanced Functional Materials, 2010, 20(5): 764–772.
[29] Xin J, Wu H, Liu X, et al. Mg vacancy and dislocation strains as strong phonon scatterers in Mg2Si1−xSbx thermoelectric materials[J]. Nano Energy, 2017, 34: 428–436.
[30] Gayner C, Amouyal Y. Energy Filtering of Charge Carriers: Current Trends, Challenges, and Prospects for Thermoelectric Materials[J]. Advanced Functional Materials, 2020, 30(18): 1901789.
[31] Tan G, Zeier W G, Shi F, et al. High Thermoelectric Performance SnTe−In2Te3 Solid Solutions Enabled by Resonant Levels and Strong Vacancy Phonon Scattering[J]. Chemistry of Materials, 2015, 27(22): 7801–7811.
[32] Zhu Y, Han Z, Jiang F, et al. Thermodynamic criterions of the thermoelectric performance enhancement in Mg2Sn through the self-compensation vacancy[J]. Materials Today Physics, 2021, 16: 100327.
[33] Nolas G S, Wang D, Beekman M. Transport properties of polycrystalline Mg2Si1−ySby (0≤y<0.4)[J]. Physical Review B, 2007, 76(23).
[34] Nolas G S, Wang D, Lin X. Synthesis and low temperature transport properties of Mg2Ge1–ySby[J]. physica status solidi (RRL) – Rapid Research Letters, 2007, 1(6): 223–225.
[35] Kato D, Iwasaki K, Yoshino M, et al. High carrier concentration in Mg2Si1−xSbx (0≤x≤0.10) prepared by a combination of liquid-solid reaction, ball milling, and spark plasma sintering[J]. Intermetallics, 2017, 81: 47–51.
[36] Pei Y, Shi X, LaLonde A, et al. Convergence of electronic bands for high performance bulk thermoelectrics[J]. Nature, 2011, 473(7345): 66–69.
[37] Liu W, Tan X, Yin K, et al. Convergence of Conduction Bands as a Means of Enhancing Thermoelectric Performance of n-Type Mg2Si1−xSnx Solid Solutions[J]. Physical Review Letters, 2012, 108(16).
[38] Zhang J, Liu R, Cheng N, et al. High-Performance Pseudocubic Thermoelectric Materials from Non-cubic Chalcopyrite Compounds[J]. Advanced Materials, 2014, 26(23): 3848–3853.
[39] Condron C L, Kauzlarich S M, Gascoin F, et al. Thermoelectric properties and microstructure of Mg3Sb2[J]. Journal of Solid State Chemistry, 2006, 179(8): 2252–2257.
[40] 余冠廷. Mg2XⅣ(X=Si,Ge,Sn)与Mg3X2V(XV=Sb,Bi)基材料的制备及热电性能[D]. 杭州, 浙江大学, 2019: 67–100.
[41] Li A, Fu C, Zhao X, et al. High-Performance Mg3Sb2-xBix Thermoelectrics: Progress and Perspective[J]. Research, 2020, 2020: 1934848.
[42] Shang H, Liang Z, Xu C, et al. N-Type Mg3Sb2-xBix Alloys as Promising Thermoelectric Materials[J]. Research, 2020, 2020: 1219461.
[43] Bhardwaj A, Rajput A, Shukla A K, et al. Mg3Sb2-based Zintl compound: a non-toxic, inexpensive and abundant thermoelectric material for power generation[J]. RSC Advances, 2013, 3(22): 8504.
[44] Kim S H, Kim C M, Hong Y-K, et al. Thermoelectric properties of Mg3Sb2-xBix single crystals grown by Bridgman method[J]. Materials Research Express, 2015, 2(5): 055903.
[45] Shuai J, Wang Y, Kim H S, et al. Thermoelectric properties of Na-doped Zintl compound: Mg3−xNaxSb2[J]. Acta Materialia, 2015, 93: 187–193.
[46] Tamaki H, Sato H K, Kanno T. Isotropic Conduction Network and Defect Chemistry in Mg3-δSb2-Based Layered Zintl Compounds with High Thermoelectric Performance[J]. Advanced Materials, 2016, 28(46): 10182–10187.
[47] Zhang J, Song L, Iversen B B. Insights into the design of thermoelectric Mg3Sb2 and its analogs by combining theory and experiment[J]. npj Computational Materials, 2019, 5(1): 1-17.
[48] Sun X, Li X, Yang J, et al. Achieving band convergence by tuning the bonding ionicity in n‐type Mg3Sb2[J]. Journal of Computational Chemistry, 2019, 40(18): 1693–1700.
[49] Zhang J, Song L, Sist M, et al. Chemical bonding origin of the unexpected isotropic physical properties in thermoelectric Mg3Sb2 and related materials[J]. Nature Communications, 2018, 9: 4716.
[50] Li J, Zhang S, Wang B, et al. Designing high-performance n-type Mg3Sb2-based thermoelectric materials through forming solid solutions and biaxial strain[J]. Journal of Materials Chemistry A, 2018, 6(41): 20454–20462.
[51] Peng W, Petretto G, Rignanese G-M, et al. An Unlikely Route to Low Lattice Thermal Conductivity: Small Atoms in a Simple Layered Structure[J]. Joule, 2018, 2(9): 1879–1893.
[52] Morelli D T, Jovovic V, Heremans J P. Intrinsically Minimal Thermal Conductivity in Cubic I-V-VI2 Semiconductors[J]. Physical Review Letters, 2008, 101(3): 035901.
[53] Zhu Y F, Xia Y, Wang Y C, et al. Violation of the T−1 Relationship in the Lattice Thermal Conductivity of Mg3Sb2 with Locally Asymmetric Vibrations[J]. Research, 2020, 2020: 4589786.
[54] Agne M T, Imasato K, Anand S, et al. Heat capacity of Mg3Sb2, Mg3Bi2, and their alloys at high temperature[J]. Materials Today Physics, 2018, 6: 83–88.
[55] Xin J, Li G, Auffermann G, et al. Growth and transport properties of Mg3X2 (X = Sb, Bi) single crystals[J]. Materials Today Physics, 2018, 7: 61–68.
[56] Gorai P, Ortiz B R, Toberer E S, et al. Investigation of n-type doping strategies for Mg3Sb2[J]. Journal of Materials Chemistry A, 2018, 6(28): 13806–13815.
[57] Gorai P, Toberer E S, Stevanović V. Effective n-type doping of Mg3Sb2 with group-3 elements[J]. Journal of Applied Physics, 2019, 125(2): 025105.
[58] Zhang J, Song L, Borup K A, et al. New Insight on Tuning Electrical Transport Properties via Chalcogen Doping in n-type Mg3Sb2-based Thermoelectric Materials[J]. Advanced Energy Materials, 2018, 8(16): 1702776.
[59] Imasato K, Wood M, Kuo J J, et al. Improved stability and high thermoelectric performance through cation site doping in n-type La-doped Mg3Sb1.5Bi0.5[J]. Journal of Materials Chemistry A, 2018, 6(41): 19941–19946.
[60] Song S W, Mao J, Bordelon M, et al. Joint effect of magnesium and yttrium on enhancing thermoelectric properties of n-type Zintl Mg3+δY0.02Sb1.5Bi0.5[J]. Materials Today Physics, 2019, 8: 25–33.
[61] Imasato K, Kang S D, Snyder G J. Exceptional thermoelectric performance in Mg3Sb0.6Bi1.4 for low-grade waste heat recovery[J]. Energy & Environmental Science, 2019, 12(3): 965–971.
[62] Kim S, Kim C, Hong Y-K, et al. Thermoelectric properties of Mn-doped Mg–Sb single crystals[J]. Journal of Materials Chemistry A, 2014, 2(31): 12311–12316.
[63] Han Z, Gui Z, Zhu Y B, et al. The Electronic Transport Channel Protection and Tuning in Real Space to Boost the Thermoelectric Performance of Mg3+δSb2-yBiy near Room Temperature[J]. Research, 2020, 2020: 1672051.
[64] Jiang G, He J, Zhu T, et al. High Performance Mg2(Si,Sn) Solid Solutions: a Point Defect Chemistry Approach to Enhancing Thermoelectric Properties[J]. Advanced Functional Materials, 2014, 24(24): 3776–3781.
[65] 徐祖耀. 材料热力学[M]. 北京: 高等教育出版社, 2009: 105.
[66] Shi X, Zhou Z, Zhang W, et al. Solid solubility of Ir and Rh at the Co sites of skutterudites[J]. Journal of Applied Physics, 2007, 101(12): 123525.
[67] Hildebrand J H. Solubility XII: regular solutions[J]. Journal of the American Chemical Society, 1929, 51(1): 66–80.
[68] Hillert M, Staffansson L I. The regular solution model for stoichiometric phases and ionic melts[J]. Acta Chemical Scandinavic, 1970, 24: 3618–3626.
[69] Shang H, Liang Z, Xu C, et al. N-type Mg3Sb2-Bi with improved thermal stability for thermoelectric power generation[J]. Acta Materialia, 2020, 201: 572–579.
[70] Dong E, Tan S, Wang J, et al. Thermodynamic activity of solute in multicomponent alloy from first-principles: Excess Mg in Mg3(Sb1-xBix)2 as an example[J]. Calphad, 2021, 74: 102318.
[71] Hillert M, Jansson B, Sundman B, et al. A two-sublattice model for molten solutions with different tendency for ionization[J]. Metallurgical Transactions A, 1984, 16A: 261-266.
[72] Viennois R, Colinet C, Jund P, et al. Phase stability of ternary antifluorite type compounds in the quasi-binary systems Mg2X–Mg2Y (X, Y = Si, Ge, Sn) via ab-initio calculations[J]. Intermetallics, 2012, 31: 145–151.
[73] Ektarawong A, Alling B. Stability of SnSe1-xSx solid solutions revealed by first-principles cluster expansion[J]. Journal of Physics: Condensed Matter, 2018,30: 1–5.
[74] Ohno S, Imasato K, Anand S, et al. Phase Boundary Mapping to Obtain n-type Mg3Sb2-Based Thermoelectrics[J]. Joule, 2018, 2(1): 141–154.
[75] Chong X, Guan P-W, Wang Y, et al. Understanding the Intrinsic P-Type Behavior and Phase Stability of Thermoelectric α-Mg3Sb2[J]. ACS Applied Energy Materials, 2018, 1(11): 6600–6608.
[76] Gaskell D R, Laughlin D E. Introduction to the Thermodynamics of Materials[M]. 6th Edition. CRC Press, 2017: 277, 288.
[77] Paliwal M, Jung I-H. Thermodynamic modeling of the Mg–Bi and Mg–Sb binary systems and short-range-ordering behavior of the liquid solutions[J]. Calphad, 2009, 33(4): 744–754.
[78] Jiang Y, Smith J R, Evans A G. Temperature dependence of the activity of Al in dilute Ni(Al) solid solutions[J]. Physical Review B, 2006, 74(22): 224110.
[79] Gao X, Ren H, Wang H, et al. Activity coefficient and solubility of yttrium in Fe-Y dilute solid solution[J]. Journal of Rare Earths, 2016, 34(11): 1168–1172.
[80] Srivastava G P. The Physics of Phonons[M]. New York: Routledge, 2019: 123–130.
[81] Li W, Carrete J, A. Katcho N, et al. ShengBTE: A solver of the Boltzmann transport equation for phonons[J]. Computer Physics Communications, 2014, 185(6): 1747–1758.
[82] Schelling P K, Phillpot S R, Keblinski P. Comparison of atomic-level simulation methods for computing thermal conductivity[J]. Physical Review B, 2002, 65(14): 144306.
[83] Müller-Plathe F. A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity[J]. The Journal of Chemical Physics, 1997, 106(14): 6082–6085.
[84] Kubo R. Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems[J]. Journal of the Physical Society of Japan, 1957, 12(6): 570–586.
[85] Green M S. Markoff Random Processes and the Statistical Mechanics of Time‐Dependent Phenomena. II. Irreversible Processes in Fluids[J]. The Journal of Chemical Physics, 1954, 22(3): 398–413.
[86] McQuarrie D A, Statistical Mechanics[M], New York: Harper & Row, 2000: 467–495.
[87] Dickey J M, Paskin A. Computer Simulation of the Lattice Dynamics of Solids[J]. Physical Review, 1969, 188(3): 1407–1418.
[88] Ladd A J C, Moran B, Hoover W G. Lattice thermal conductivity: A comparison of molecular dynamics and anharmonic lattice dynamics[J]. Physical Review B, 1986, 34(8): 5058–5064.
[89] Howell P C. Comparison of molecular dynamics methods and interatomic potentials for calculating the thermal conductivity of silicon[J]. The Journal of Chemical Physics, 2012, 137(22): 224111.
[90] Fan Z, Pereira L F C, Wang H-Q, et al. Force and heat current formulas for many-body potentials in molecular dynamics simulations with applications to thermal conductivity calculations[J]. Physical Review B, 2015, 92(9).
[91] Plimpton S. Fast Parallel Algorithms for Short-Range Molecular Dynamics[J]. Journal of Computational Physics, 1995, 117(1): 1–19.
[92] Zhang J, Song L, Pedersen S H et al. Discovery of high-performance low-cost n-type Mg3Sb2-based thermoelectric materials with multi-valley conduction bands[J]. Nature Communications, 2017, 8: 13901.
[93] Shuai J, Ge B, Mao J, et al. Significant Role of Mg Stoichiometry in Designing High Thermoelectric Performance for Mg3(Sb,Bi)2-Based n-Type Zintls[J]. Journal of the American Chemical Society, 2018, 140(5): 1910–1915.
[94] Imasato K, Ohno S, Kang S D, et al. Improving the thermoelectric performance in Mg3-xSb1.5Bi0.49Te0.01 by reducing excess Mg[J]. APL Materials, 2018, 6(1): 016106.
[95] Wang Z, Safarkhani S, Lin G, et al. Uncertainty quantification of thermal conductivities from equilibrium molecular dynamics simulations[J]. International Journal of Heat and Mass Transfer, 2017, 112: 267–278.
[96] Lennard-Jones J E. Cohesion[J]. Proceedings of the Physical Society, 1931, 43(5): 461–482.
[97] Stillinger F H, Weber T A. Computer simulation of local order in condensed phases of silicon[J]. Physical Review B, 1985, 31(8): 5262–5271.
[98] Tersoff J. New empirical approach for the structure and energy of covalent systems[J]. Physical Review B, 1988, 37(12): 6991–7000.
[99] Yang H, Zhu Y, Dong E, et al. Dual adaptive sampling and machine learning interatomic potentials for modeling materials with chemical bond hierarchy[J]. Physical Review B, 2021, 104(9): 094310.
[100] Korotaev P, Novoselov I, Yanilkin A, et al. Accessing thermal conductivity of complex compounds by machine learning interatomic potentials[J]. Physical Review B, 2019, 100(14): 144308.
[101] 杨宏亮. 机器学习原子间势与复杂材料体系热输运研究[D]. 上海, 中国科学院大学（上海硅酸盐研究所）, 2022: 68–75.
[102] Sholl D S, Steckel J A. Density Functional Theory [M]. Hoboken: Wiley, 2009: 1–28.
[103] Cohen M L, Louie S G. Fundamentals of Condensed Matter Physics[M]. UK: Cambridge University Press, 2016:142–149.
[104] Hohenberg P, Kohn W. Inhomogeneous Electron Gas[J]. Physical Review, 1964, 136(3B): B864–B871.
[105] Kohn W, Sham L J. Self-Consistent Equations Including Exchange and Correlation Effects[J]. Physical Review, 1965, 140(4A): A1133–A1138.
[106] Perdew J P, Zunger A. Self-interaction correction to density-functional approximations for many-electron systems[J]. Physical Review B, 1981, 23(10): 5048–5079.
[107] Perdew J P, Chevary J A, Vosko S H, et al. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation[J]. Physical Review B, 1992, 46(11): 6671–6687.
[108] Langreth D C, Mehl M J. Beyond the local-density approximation in calculations of ground-state electronic properties[J]. Physical Review B, 1983, 28(4): 1809–1834.
[109] Becke A D. Density-functional exchange-energy approximation with correct asymptotic behavior[J]. Physical Review A, 1988, 38(6): 3098–3100.
[110] Perdew J P, Wang Y. Accurate and simple analytic representation of the electron-gas correlation energy[J]. Physical Review B, 1992, 45(23): 13244–13249.
[111] Perdew J P, Burke K, Ernzerhof M. Generalized Gradient Approximation Made Simple[J]. Physical Review Letters, 1996, 77(18): 3865–3868.
[112] Perdew J P, Kurth S, Zupan A, et al. Accurate Density Functional with Correct Formal Properties: A Step Beyond the Generalized Gradient Approximation[J]. Physical Review Letters, 1999, 82(12): 2544–2547.
[113] Heyd J, Scuseria G E. Efficient hybrid density functional calculations in solids: Assessment of the Heyd–Scuseria–Ernzerhof screened Coulomb hybrid functional[J]. The Journal of Chemical Physics, 2004, 121(3): 1187–1192.
[114] Vydrov O A, Scuseria G E. Assessment of a long-range corrected hybrid functional[J]. The Journal of Chemical Physics, 2006, 125(23): 234109.
[115] Kikuchi R. A Theory of Cooperative Phenomena[J]. Physical Review, 1951, 81(6): 988–1003.
[116] Sanchez J M, de Fontaine D. The fee Ising model in the cluster variation approximation[J]. Physical Review B, 1978, 17(7): 2926–2936.
[117] Sanchez J M, Ducastelle F, Gratias D. Generalized cluster description of multicomponent systems[J]. Physica A: Statistical Mechanics and its Applications, 1984, 128(1): 334–350.
[118] Kikuchi R. CVM Entropy Algebra[J]. Progress of Theoretical Physics Supplement, 1994, 115: 1–26.
[119] Mohri T, Sanchez J M, De Fontaine D. Overview no. 43: Binary ordering prototype phase diagrams in the cluster variation approximation[J]. Acta Metallurgica, 1985, 33(7): 1171–1185.
[120] Laks D B, Wei S-H, Zunger A. Evolution of alloy properties with long-range order[J]. Physical Review Letters, 1992, 69(26): 3766–3769.
[121] van de Walle A, Asta M, Ceder G. The alloy theoretic automated toolkit: A user guide[J]. Calphad, 2002, 26(4): 539–553.
[122] Kadkhodaei S, Muñoz J A. Cluster Expansion of Alloy Theory: A Review of Historical Development and Modern Innovations[J]. JOM, 2021, 73(11): 3326–3346.
[123] Fedorov M, Wróbel J S, Fernández-Caballero A, et al. Phase stability and magnetic properties in fcc Fe-Cr-Mn-Ni alloys from first-principles modeling[J]. Physical Review B, 2020, 101(17).
[124] van de Walle A. Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the Alloy Theoretic Automated Toolkit[J]. Calphad, 2009, 33(2): 266–278.
[125] Pomrehn G S, Toberer E S, Snyder G J, et al. Predicted Electronic and Thermodynamic Properties of a Newly Discovered Zn8Sb7 Phase[J]. Journal of the American Chemical Society, 2011, 133(29): 11255–11261.
[126] Tan S, Nan P, Xia K, et al. Sublattice Short-Range Order and Modified Electronic Structure in Defective Half-Heusler Nb0.8CoSb[J]. The Journal of Physical Chemistry C, 2021, 125(1): 1125–1133.
[127] Fernández-Caballero A, Fedorov M, Wróbel J, et al. Configurational Entropy in Multicomponent Alloys: Matrix Formulation from Ab Initio Based Hamiltonian and Application to the FCC Cr-Fe-Mn-Ni System[J]. Entropy, 2019, 21(1): 68.
[128] Žguns P A, Ruban A V, Skorodumova N V. Ordering and phase separation in Gd-doped ceria: a combined DFT, cluster expansion and Monte Carlo study[J]. Physical Chemistry Chemical Physics, 2017, 19(39): 26606–26620.
[129] Wróbel J S, Nguyen-Manh D, Kurzydłowski K J, et al. A first-principles model for anomalous segregation in dilute ternary tungsten-rhenium-vacancy alloys[J]. Journal of Physics: Condensed Matter, 2017, 29(14): 145403.
[130] van de Walle A, Ceder G. Automating first-principles phase diagram calculations[J]. Journal of Phase Equilibria, 2002, 23(4): 348–359.
[131] Landau D P, Binder K. A guide to Monte Carlo simulations in statistical physics[M]. 4th edition. UK: Cambridge University Press, 2015: 125.
[132] van de Walle A, Asta M. Self-driven lattice-model Monte Carlo simulations of alloy thermodynamic[J]. Modelling and Simulation in Materials Science and Engineering, 2002, 10(5): 521–538.
[133] Dünweg B, Landau D P. Phase diagram and critical behavior of the Si-Ge unmixing transition: A Monte Carlo study of a model with elastic degrees of freedom[J]. Physical Review B, 1993, 48(19): 14182–14197.
[134] Laradji M, Landau D P, Dünweg B. Structural properties of Si1−xGex alloys: A Monte Carlo simulation with the Stillinger-Weber potential[J]. Physical Review B, 1995, 51(8): 4894–4902.
[135] Terakura K, Akai H. Interatomic Potential and Structural Stability[M]. Berlin, Heidelberg: Springer Berlin Heidelberg, 1993, 133–142.
[136] Kohan A F, Tepesch P D, Ceder G, et al. Computation of alloy phase diagrams at low temperatures[J]. Computational Materials Science, 1998, 9(3): 389–396.
[137] Woodward C, Asta M, Kresse G, et al. Density of constitutional and thermal point defects in L12 Al3Sc[J]. Physical Review B, 2001, 63(9): 094103.
[137] Alder B J, Wainwright T E. Phase Transition for a Hard Sphere System[J]. The Journal of Chemical Physics, 1957, 27(5): 1208–1209.
[139] Haile J M. Molecular Dynamics Simulation[M]. New York: Wiley, 1997: 225–310.
[140] Rapaport D C. The Art of Molecular Dynamics Simulation[M]. 2nd edition. UK: Cambridge University Press, 2004: 1–2.
[141] Allen M P, Tildesley D J. Computer simulation of liquids[M]. 2nd edition. Oxford, UK: Oxford University Press, 2017: 4.
[142] Hollingsworth S A, Dror R O. Molecular Dynamics Simulation for All[J]. Neuron, 2018, 99(6): 1129–1143.
[143] Verlet L. Computer "Experiments" on Classical Fluids. I. Thermodynamical Properties of Lennard-Jones Molecules[J]. Physical Review, 1967, 159(1): 98–103.
[144] Swope W C, Andersen H C, Berens P H, et al. A computer simulation method for the calculation of equilibrium constants for the formation of physical clusters of molecules: Application to small water clusters[J]. The Journal of Chemical Physics, 1982, 76(1): 637–649.
[145] Jones J E, Chapman S. On the determination of molecular fields. —I. From the variation of the viscosity of a gas with temperature[J]. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1924, 106(738): 441–462.
[146] Jones J E, Chapman S. On the determination of molecular fields. —II. From the equation of state of a gas[J]. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1924, 106(738): 463–477.
[147] Daw M S, Baskes M I. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals[J]. Physical Review B, 1984, 29(12): 6443–6453.
[148] Daw M S, Foiles S M, Baskes M I. The embedded-atom method: a review of theory and applications[J]. Materials Science Reports, 1993, 9(7): 251–310.
[149] Brommer P, Gähler F. Potfit: effective potentials from ab initio data[J]. Modelling and Simulation in Materials Science and Engineering, 2007, 15(3): 295–304.
[150] Tersoff J. Empirical interatomic potential for silicon with improved elastic properties[J]. Physical Review B, 1988, 38(14): 9902–9905.
[151] Berendsen H J C, Postma J P M, van Gunsteren W F, et al. Molecular dynamics with coupling to an external bath[J]. The Journal of Chemical Physics, 1984, 81(8): 3684–3690.
[152] Martyna G J, Klein M L, Tuckerman M. Nosé–Hoover chains: The canonical ensemble via continuous dynamics[J]. The Journal of Chemical Physics, 1992, 97(4): 2635–2643.
[153] Nosé S. A molecular dynamics method for simulations in the canonical ensemble[J]. Molecular Physics, 1984, 52(2): 255–268.
[154] Hoover W G. Canonical dynamics: Equilibrium phase-space distributions[J]. Physical Review A, 1985, 31(3): 1695–1697.
[155] Attig N, Binder K, Grubmüller H. Computational Soft Matter: From Synthetic Polymers to Proteins, Lecture Notes[M]. Jülich: NIC, 2004: 10.
[156] Behler J, Parrinello M. Generalized neural-network representation of high-dimensional potential-energy Surfaces[J]. Physical Review Letters, 2007, 98(14): 146401.
[157] Gubaev K, Podryabinkin E V, Hart G L W, et al. Accelerating high-throughput searches for new alloys with active learning of interatomic potentials[J]. Computational Materials Science, 2019, 156: 148–156.
[158] Grabowski B, Ikeda Y, Srinivasan P, et al. Ab initio vibrational free energies including anharmonicity for multicomponent alloys[J]. npj Computational Materials, 2019, 5(1): 1–6.
[159] Novoselov I I, Yanilkin A V, Shapeev A V, et al. Moment tensor potentials as a promising tool to study diffusion processes[J]. Computational Materials Science, 2019, 164: 46–56.
[160] Novikov I S, Suleimanov Y V, Shapeev A V. Automated calculation of thermal rate coefficients using ring polymer molecular dynamics and machine-learning interatomic potentials with active learning[J]. Physical Chemistry Chemical Physics, 2018, 20(46): 29503–29512.
[161] Novikov I S, Shapeev A V, Suleimanov Y V. Ring polymer molecular dynamics and active learning of moment tensor potential for gas-phase barrierless reactions: Application to S+H2[J]. The Journal of Chemical Physics, 2019, 151(22): 224105.
[162] Qiu W, Xi L, Wei P, et al. Part-crystalline part-liquid state and rattling-like thermal damping in materials with chemical-bond hierarchy[J]. Proceedings of the National Academy of Sciences, 2014, 111(42): 15031–15035.
[163] Yang J, Wang Y, Yang H, et al. Thermal transport in thermoelectric materials with chemical bond hierarchy[J]. Journal of Physics: Condensed Matter, 2019, 31(18): 183002.
[164] Zhang L, Lin D-Y, Wang H, et al. Active learning of uniformly accurate interatomic potentials for materials simulation[J]. Physical Review Materials, 2019, 3(2): 023804.
[165] Smith J S, Nebgen B, Lubbers N, et al. Less is more: Sampling chemical space with active learning[J]. The Journal of Chemical Physics, 2018, 148(24): 241733.
[166] Singraber A, Morawietz T, Behler J, et al. Parallel Multistream Training of High-Dimensional Neural Network Potentials[J]. Journal of Chemical Theory and Computation, 2019, 15(5): 3075–3092.
[167] Bartók A P, Kondor R, Csányi G. On representing chemical environments[J]. Physical Review B, 2013, 87(18): 184115.
[168] Artrith N, Behler J. High-dimensional neural network potentials for metal surfaces: A prototype study for copper[J]. Physical Review B, 2012, 85(4): 045439.
[169] Khorshidi A, Peterson A A. Amp: A modular approach to machine learning in atomistic simulations[J]. Computer Physics Communications, 2016, 207: 310–324.
[170] Smith J S, Isayev O, Roitberg A E. ANI-1: an extensible neural network potential with DFT accuracy at force field computational cost[J]. Chemical Science, 2017, 8(4): 3192–3203.
[171] Gastegger M, Schwiedrzik L, Bittermann M, et al. wACSF—Weighted atom-centered symmetry functions as descriptors in machine learning potentials[J]. The Journal of Chemical Physics, 2018, 148(24): 241709.
[172] Bartók A P, Payne M C, Kondor R, et al. Gaussian Approximation Potentials: The Accuracy of Quantum Mechanics, without the Electrons[J]. Physical Review Letters, 2010, 104(13): 136403.
[173] Deringer V L, Csányi G. Machine learning based interatomic potential for amorphous carbon[J]. Physical Review B, 2017, 95(9): 094203.
[174] Bartók A P, Kermode J, Bernstein N, et al. Machine Learning a General-Purpose Interatomic Potential for Silicon[J]. Physical Review X, 2018, 8(4): 041048.
[175] Deringer V L, Pickard C J, Csányi G. Data-Driven Learning of Total and Local Energies in Elemental Boron[J]. Physical Review Letters, 2018, 120(15): 156001.
[176] Braams B J, Bowman J M. Permutationally invariant potential energy surfaces in high dimensionality[J]. International Reviews in Physical Chemistry, 2009, 28(4): 577–606.
[177] Shapeev A V. Moment Tensor Potentials: A Class of Systematically Improvable Interatomic Potentials[J]. Multiscale Modeling & Simulation, 2016, 14(3): 1153–1173.
[178] Drautz R. Atomic cluster expansion for accurate and transferable interatomic potentials[J]. Physical Review B, 2019, 99(1): 014104.
[179] Lysogorskiy Y, Oord C van der, Bochkarev A, et al. Performant implementation of the atomic cluster expansion (PACE) and application to copper and silicon[J]. npj Computational Materials, 2021, 7(1): 1–12.
[180] Zuo Y, Chen C, Li X, et al. Performance and Cost Assessment of Machine Learning Interatomic Potentials[J]. The Journal of Physical Chemistry A, 2020, 124(4): 731–745.
[181] Novikov I S, Gubaev K, Podryabinkin E V, et al. The MLIP package: moment tensor potentials with MPI and active learning[J]. Machine Learning: Science and Technology, 2021, 2(2): 025002.
[182] Wang H, Zhang L, Han J, et al. DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics[J]. Computer Physics Communications, 2018, 228: 178–184.
[183] Kresse G, Furthmüller J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[J]. Physical Review B, 1996, 54(16): 11169–11186.
[184] Kresse G, Furthmüller J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[J]. Computational Materials Science, 1996, 6(1): 15–50.
[185] Medeiros P V C, Stafström S, Björk J. Effects of extrinsic and intrinsic perturbations on the electronic structure of graphene: Retaining an effective primitive cell band structure by band unfolding[J]. Physical Review B, 2014, 89(4): 041407.
[186] Medeiros P V C, Tsirkin S S, Stafström S, et al. Unfolding spinor wave functions and expectation values of general operators: Introducing the unfolding-density operator[J]. Physical Review B, 2015, 91(4): 041116.
[187] Togo A, Tanaka I. First principles phonon calculations in materials science[J]. Scripta Materialia, 2015, 108: 1–5.
[188] Momma K, Izumi F. VESTA: a three-dimensional visualization system for electronic and structural analysis[J]. Journal of Applied Crystallography, 2008, 41(3): 653–658.
[189] Stukowski A. Visualization and analysis of atomistic simulation data with OVITO–the Open Visualization Tool[J]. Modelling and Simulation in Materials Science and Engineering, 2009, 18(1): 015012.
[190] Shuai J, Mao J, Song S, et al. Tuning the carrier scattering mechanism to effectively improve the thermoelectric properties[J]. Energy & Environmental Science, 2017, 10(3): 799–807.
[191] Pan Y, Yao M, Hong X, et al. Mg3(Bi,Sb)2 single crystals towards high thermoelectric performance[J]. Energy & Environmental Science, 2020, 6(13): 1717-1724.
[192] Bhardwaj A, Chauhan N S, Goel S, et al. Tuning the carrier concentration using Zintl chemistry in Mg3Sb2, and its implications for thermoelectric figure-of-merit[J]. Physical Chemistry Chemical Physics, 2016, 18(8): 6191–6200.
[193] Bhardwaj A, Misra D K. Enhancing thermoelectric properties of a p-type Mg3Sb2-based Zintl phase compound by Pb substitution in the anionic framework[J]. RSC Advance, 2014, 4(65): 34552–34560.
[194] Ponnambalam V, Morelli D T. On the Thermoelectric Properties of Zintl Compounds Mg3Bi2-xPnx (Pn = P and Sb)[J]. Journal of Electronic Materials, 2013, 42(7): 1307–1312.
[195] Xin H X, Qin X Y, Jia J H, et al. Thermoelectric properties of nanocrystalline (Mg1−xZnx)3Sb2 isostructural solid solutions fabricated by mechanical alloying[J]. Journal of Physics D: Applied Physics, 2009, 42(16): 165403.
[196] Pizzini S. Physical Chemistry of Semiconductor Materials and Processes[M].UK: Wiley, 2015.
[197] Tang Y, Hanus R, Chen S, et al. Solubility design leading to high figure of merit in low-cost Ce-CoSb3 skutterudites[J]. Nature Communications, 2015, 6(1): 7584.
[198] Tang Y, Qiu Y, Xi L, et al. Phase diagram of In–Co–Sb system and thermoelectric properties of In-containing skutterudites[J]. Energy & Environmental Science, 2014, 7(2): 812–819.
[199] Komisarchik G, Gelbstein Y, Fuks D. Solubility of Ti in thermoelectric PbTe compound[J]. Intermetallics, 2017, 89: 16–21.
[200] Jood P, Male J P, Anand S, et al. Na Doping in PbTe: Solubility, Band Convergence, Phase Boundary Mapping, and Thermoelectric Properties[J]. Journal of the American Chemical Society, 2020, 142(36): 15464–15475.
[201] Li X, Yang P, Wang Y, et al. Phase Boundary Mapping in ZrNiSn Half-Heusler for Enhanced Thermoelectric Performance[J]. Research, 2020, 2020: 1–9.
[201] Ohno S, Imasato K, Anand S, et al. Phase Boundary Mapping to Obtain n-type Mg3Sb2-Based Thermoelectrics[J]. Joule, 2018, 2(1): 141–154.
[203] Ohno S, Aydemir U, Amsler M, et al. Achieving zT>1 in Inexpensive Zintl Phase Ca9Zn4+xSb9 by Phase Boundary Mapping[J]. Advanced Functional Materials, 2017, 27(20): 1606361.
[204] Ortiz B R, Gordiz K, Gomes L C, et al. Carrier density control in Cu2HgGeTe4 and discovery of Hg2GeTe4 via phase boundary mapping[J]. Journal of Materials Chemistry A, 2019, 7(2): 621–631.
[205] Matsushita H, Hagiwara E, Katsui A. Phase diagram and thermoelectric properties of Ag3−xSb1+xTe4 system[J]. Journal of Materials Science, 2004, 39(20): 6299–6301.
[206] Blöchl P E. Projector augmented-wave method[J]. Physical Review B, 1994, 50(24): 17953–17979.
[207] Monkhorst H J, Pack J D. Special points for Brillouin-zone integrations[J]. Physical Review B, 1976, 13(12): 5188–5192.
[208] Togo A, Tanaka I. First principles phonon calculations in materials science[J]. Scripta Materialia, 2015, 108: 1–5.
[209] Zhang X, Jin L, Dai X, et al. Topological Type-II Nodal Line Semimetal and Dirac Semimetal State in Stable Kagome Compound Mg3Bi2[J]. The Journal of Physical Chemistry Letters, 2017, 8(19): 4814–4819.
[210] Watson L M, Marshall C A W, Cardoso C P. On the electronic structure of the semiconducting compounds Mg3Bi2 and Mg3Sb2[J]. Journal of Physics F: Metal Physics, 1984, 14(1): 113–121.
[211] van de Walle A, Tiwary P, de Jong M, et al. Efficient stochastic generation of special quasirandom structures[J]. Calphad, 2013, 42: 13–18.
[212] Fuks D, Dorfman S, Piskunov S, et al. Ab initio thermodynamics of BacSr(1−c)TiO3 solid solutions[J]. Physical Review B, 2005, 71(1): 014111.
[213] Liu Z-K. First-Principles Calculations and CALPHAD Modeling of Thermodynamics[J]. Journal of Phase Equilibria and Diffusion, 2009, 30(5): 517.
[214] Kaufman L, Ågren J. CALPHAD, first and second generation–Birth of the materials genome[J]. Scripta Materialia, 2014, 70: 3–6.
[215] Sundman B, Chen Q, Du Y. A Review of Calphad Modeling of Ordered Phases[J]. Journal of Phase Equilibria and Diffusion, 2018, 39(5): 678–693.
[216] Emsley J. The Elements [M]. 3rd edition. Oxford: Clarendon Press, 1998.
[217] Eckert C A, Irwin R B, Smith J S. Thermodynamic activity of magnesium in several highly-solvating liquid alloys[J]. Metallurgical Transactions B, 1983, 14(3): 451–458.
[218] Egan J J. Thermodynamics of liquid magnesium alloys using CaF2 solid electrolytes[J]. Journal of Nuclear Materials, 1974, 51(1): 30–35.
[219] Nayeb-Hashemi A A, Clark J B. The Bi-Mg (Bismuth-Magnesium) system[J]. Bulletin of Alloy Phase Diagrams, 1985, 6(6): 528–533.
[220] Palatnik L S, Fedorov G V, Bogatov P N, et al. Study of structure and electrical properties of the Mg-Sb, Mg-Bi and Mg-Bi-Sb systems [J]. Izv. Akad. Nauk SSSR, Neorganic Material, 1969, 5: 1377-1380.
[221] Wang H, LaLonde A D, Pei Y, et al. The Criteria for Beneficial Disorder in Thermoelectric Solid Solutions[J]. Advanced Functional Materials, 2013, 23(12): 1586–1596.
[222] Dung D D, Hung N T. Structural, Optical, and Magnetic Properties of the New (1-x)Bi0.5Na0.5TiO3+xMgCoO3-δ Solid Solution System[J]. Journal of Superconductivity and Novel Magnetism, 2020, 33(5): 1249–1256.
[223] Fu C, Zhu T, Liu Y, et al. Band engineering of high performance p-type FeNbSb based half-Heusler thermoelectric materials for figure of merit zT>1[J]. Energy & Environmental Science, 2015, 8(1): 216–220.
[224] Li L, Muckerman J T, Hybertsen M S, et al. Phase diagram, structure, and electronic properties of (Ga1-xZnx)(N1-xOx) solid solutions from DFT-based simulations[J]. Physical Review B, 2011, 83(13): 134202.
[225] Huang B, Kobayashi H, Yamamoto T, et al. Solid-Solution Alloying of Immiscible Ru and Cu with Enhanced CO Oxidation Activity[J]. Journal of the American Chemical Society, 2017, 139(13): 4643–4646.
[226] Zeng C, Huang H, Zhang T, et al. Fabrication of Heterogeneous-Phase Solid-Solution Promoting Band Structure and Charge Separation for Enhancing Photocatalytic CO2 Reduction: A Case of ZnXCa1−XIn2S4[J]. ACS Applied Materials & Interfaces, 2017, 9(33): 27773–27783.
[227] Hasebe M, Nishizawa T. Calculation of phase diagrams of the iron-copper and cobalt-copper systems[J]. Calphad, 1980, 4(2): 83–100.
[228] Biswas K, He J, Blum I D, et al. High-performance bulk thermoelectrics with all-scale hierarchical architectures[J]. Nature, 2012, 489(7416): 414–418.
[229] Sanchez J M, Ducastelle F, Gratias D. Generalized cluster description of multicomponent systems[J]. Physica A: Statistical Mechanics and its Applications, 1984, 128(1–2): 334–350.
[230] Fontaine D D. Cluster Approach to Order-Disorder Transformations in Alloys[A]. Solid State Physics[M]. Elsevier, 1994, 47: 33–176.
[231] Tran F, Blaha P. Accurate Band Gaps of Semiconductors and Insulators with a Semilocal Exchange-Correlation Potential[J]. Physical Review Letters, 2009, 102(22): 226401.
[232] Wang Y, Zhang X, Liu Y, et al. Optimizing the thermoelectric performance of p-type Mg3Sb2 by Sn doping[J]. Vacuum, 2020, 177: 109388.
[233] Li H, Li M, Wu Y, et al. Site occupation behavior of sulfur and phosphorus in NiAl, TiAl and FeAl[J]. Intermetallics, 2012, 28: 156–163.
[234] Gusev A I. Short-range order and diffuse scattering in nonstoichiometric compounds[J]. Uspekhi Fizicheskih Nauk, 2006, 176(7): 717.
[235] Cowley J M. Short- and Long-Range Order Parameters in Disordered Solid Solutions[J]. Physical Review, 1960, 120(5): 1648–1657.
[236] Oates W A. Configurational Entropies of Mixing in Solid Alloys[J]. Journal of Phase Equilibria and Diffusion, 2007, 28(1): 79–89.
[237] Qin J Y. A New Model for the Configurational Entropy of Mixing in Liquid Alloys Based on Short-Range Order[J]. Acta Physico-Chimica Sinica, 2012, 28(07): 1586–1592.
[238] Saito W, Hayashi K, Huang Z, et al. Enhancing the Thermoelectric Performance of Mg2Sn Single Crystals via Point Defect Engineering and Sb Doping[J]. ACS Applied Materials & Interfaces, 2020, 12(52): 57888–57897.
[239] Broido D A, Malorny M, Birner G, et al. Intrinsic lattice thermal conductivity of semiconductors from first principles[J]. Applied Physics Letters, 2007, 91(23): 231922.
[240] Ma T. First-principles Modeling of Thermal Transport in Materials: Achievements, Opportunities, and Challenges[J]. International Journal of Thermophysics, 2020: 41:9.
[241] Lindsay L, Katre A, Cepellotti A, et al. Perspective on ab initio phonon thermal transport[J]. Journal of Applied Physics, 2019, 126(5): 050902.
[242] Lindsay L, Hua C, Ruan X L, et al. Survey of ab initio phonon thermal transport[J]. Materials Today Physics, 2018, 7: 106–120.
[243] Lindsay L. First Principles Peierls-Boltzmann Phonon Thermal Transport: A Topical Review[J]. Nanoscale and Microscale Thermophysical Engineering, 2016, 20(2): 67–84.
[244] Behler J. Perspective: Machine learning potentials for atomistic simulations[J]. The Journal of Chemical Physics, American Institute of Physics, 2016, 145(17): 170901.
[245] Podryabinkin E V, Shapeev A V. Active learning of linearly parametrized interatomic potentials[J]. Computational Materials Science, 2017, 140: 171–180.
[246] Huang Y, Kang J, Goddard W A, et al. Density functional theory based neural network force fields from energy decompositions[J]. Physical Review B, 2019, 99(6): 064103.
[247] Liu Z, Yang X, Zhang B, et al. High Thermal Conductivity of Wurtzite Boron Arsenide Predicted by Including Four-Phonon Scattering with Machine Learning Potential[J]. ACS Applied Materials & Interfaces, 2021, 13(45): 53409–53415.
[248] Mortazavi B, Podryabinkin E V, Novikov I S, et al. Accelerating first-principles estimation of thermal conductivity by machine-learning interatomic potentials: A MTP/ShengBTE solution[J]. Computer Physics Communications, 2021, 258: 107583.
[249] Martin J J. Thermal conductivity of Mg2Si, Mg2Ge and Mg2Sn [J]. Journal of Physics Chemical Solids 1972, 33, 1139-1148.
[250] Chernatynskiy A, Phillpot S R. Anharmonic properties in Mg2X (X=C, Si, Ge, Sn, Pb) from first-principles calculations[J]. Physical Review B, 2015, 92(6): 064303.
[251] Li W, Lindsay L, Broido D A, et al. Thermal conductivity of bulk and nanowire Mg2SixSn1-x alloys from first principles[J]. Physical Review B, 2012: 86(17): 174307.
[252] Shit S P, Ghosh N K, Pal S, et al. Particle size and temperature effects on thermal conductivity of aqueous Ag nanofluids: modelling and simulations using classical molecular dynamics[J]. The European Physical Journal D, 2022, 76(12): 238.
[253] McGaughey A J H, Kaviany M. Thermal conductivity decomposition and analysis using molecular dynamics simulations: Part II. Complex silica structures[J]. International Journal of Heat and Mass Transfer, 2004, 47(8): 1799–1816.
[254] Wang H, Chu W. Thermal conductivity of ZnTe investigated by molecular dynamics[J]. Journal of Alloys and Compounds, 2009, 485(1): 488–492.