热电材料中的晶格热导率

doi:10.15541/jim20180320

摘要/Abstract

摘要：

随着可再生能源及能源转换技术的快速发展, 热电材料在发电及制冷领域的应用前景受到越来越广泛的关注。发展具有高热电优值材料的重要性日益突出, 如何获得低晶格热导率是热电材料的研究重点之一。本文阐述了热容、声速及弛豫时间对晶格热导率的影响, 介绍了本征低热导率热电材料所具有的典型特征, 如强非谐性、弱化学键、本征共振散射及复杂晶胞结构等, 并分析了通过多尺度声子散射降低已有热电材料晶格热导率的方法, 其中包括点缺陷散射、位错散射、晶界散射、共振散射、电声散射等多种散射机制。此外, 总结了几种预测材料最小晶格热导率的理论模型, 对快速筛选具有低晶格热导率的热电材料具有一定的理论指导意义。最后, 展望了如何获得低热导率热电材料的有效途径。

关键词: 热电材料, 晶格热导率, 热容, 弛豫时间, 综述

Abstract:

With rapid development of sustainable energies and energy conversion technologies, application prospect of thermoelectric (TE) materials in power generation and cooling has received increasing attention. The requirement of improving TE materials with high figure of merit becomes much more important. How to obtain the low lattice thermal conductivity is one of the main concerns in TE materials. In this review, the influences of specific heat, phonon group velocity and relaxation time on the lattice thermal conductivity are discussed, respectively. Several typical features of TE materials with intrinsic low lattice thermal conductivity are introduced, such as strong anharmonicity, weak chemical bonds and complex primitive cells. Introducing multiscale phonon scatterings to reduce the lattice thermal conductivity of known TE materials is also presented and discussed, including but not limited to point defect scattering, dislocation scattering, boundary scattering, resonance scattering and electron-phonon scattering. In addition, some theoretical models of the minimum lattice thermal conductivity are analyzed, which has certain theoretical significance for rapid screening of TE materials with low lattice thermal conductivity. Finally, the efficient ways to obtain the low lattice thermal conductivity for TE property optimization are proposed.

Key words: thermoelectric materials, lattice thermal conductivity, specific heat, relaxation time, review

中图分类号:

TB34

沈家骏, 方腾, 傅铁铮, 忻佳展, 赵新兵, 朱铁军. 热电材料中的晶格热导率[J]. 无机材料学报, 2019, 34(3): 260-268.

SHEN Jia-Jun, FANG Teng, FU Tie-Zheng, XIN Jia-Zhan, ZHAO Xin-Bing, ZHU Tie-Jun. Lattice Thermal Conductivity in Thermoelectric Materials[J]. Journal of Inorganic Materials, 2019, 34(3): 260-268.

图/表 9

图1 (a) Cu2-xSe化合物的热容与温度关系图[11]和(b) 室温晶格热导率与原胞中原子数关系[12,13,14,15,16]

Fig.1 (a) Temperature dependence of the lattice thermal conductivity for Cu2-xSe[11] and (b) number of atoms in the primitive unit cell versus room temperature lattice thermal conductivity[12,13,14,15,16]

图2 (a)Ba8Ga16Ge30晶体结构示意图, (b)未填充及填充笼式结构的弹簧模型及(c)色散关系[29]

Fig. 2 (a) Schematic diagram of crystal structure for Ba8Ga16Ge30, (b) a simple spring model and (c) the corresponding dispersion relation of filled and unfilled clathrate[29]describing interaction between the host cages with a spring constant K1 and the guest atoms attached to the cages with a spring constant K2

图3 格林内森常数与室温晶格热导率的关系图[6, 22, 39-53]

Fig. 3 Grüneisen parameter versus room temperature lattice thermal conductivity[6, 22, 39-53]

图4 (a)(Nb0.6Ta0.4)0.8Ti0.2FeSb和Nb0.8Ti0.2FeSb的晶格热导率与声子频率的依赖关系和(b)Ta掺杂量与无序散射因子及晶格热导率的关系图[70]

Fig.4 (a) Phonon frequency dependence of spectral lattice thermal conductivity for (Nb0.6Ta0.4)0.8Ti0.2FeSb and Nb0.8Ti0.2FeSb, and (b) relationship between Ta content and lattice thermal conductivity/disorder parameter for (Nb0.6Ta0.4)0.8Ti0.2FeSb[70]

图5 (a) Mg2Si0.5Sb0.5中位错的IFFT图及相应的应力扫描图, (b) Mg2Si1-xSbx及Mg2Si1-zSnz的室温晶格热导率对比图[84]

Fig. 5 (a) Inverse FFT images and strain mapping of dislocations in the Mg2Si0.5Sb0.5, and (b) lattice thermal conductivity comparison between Mg2Si1-xSbx and Mg2Si1-zSnz at room temperature[84]

图6 BiSe晶体结构示意图(a)和Bi2Se3及BiSe的晶格热导率对比图(b)[88]

Fig. 6 (a) Schematic diagram of crystal structure for BiSe and (b) lattice thermal conductivity comparison between Bi2Se3 and BiSe[88]

图7 电声散射示意图(a)和硅样品晶格热导率的实验值与Callaway模型计算值的对比图(b)[93]

Fig. 7 (a) Schematic diagram of electron-phonon scattering and (b) comparison of experimental and calculated lattice thermal conductivities by Callaway Model for the silicon sample[93]

图8 (a)扩散子模型及声子模型的差别示意图和(b)Cahill模型及扩散子模型预测的最小晶格热导率对比图

Fig. 8 (a) Schematic diagram of the difference between diffusion model and phonon model, and (b) comparison of calculated minimum lattice thermal conductivities by Cahill model and diffuson model

图9 获得低晶格热导率的几种途径

Fig. 9 Several strategies to obtain low lattice thermal conductivity

参考文献 105

[1]	BELL L E.Cooling, heating, generating power, and recovering waste heat with thermoelectric systems. Science, 2008, 321(5895): 1457-1461.
[2]	SNYDER G J, TOBERER E S.Complex thermoelectric materials. Nature Materials, 2008, 7:101-110.
[3]	XIN J Z, TANG Y L, LIU Y T, et al.Valleytronics in thermoelectric materials. npj Quantum Materials, 2018, 3(1): 9.
[4]	LI W, ZHENG L L, GE B H, et al. Promoting SnTe as an eco-friendly solution for p-PbTe thermoelectric via band convergence and interstitial defects. Advanced Materials, 2017, 29(17): 1605887-1-8.
[5]	BISWAS K, HE J Q, BLUM I D, et al.High-performance bulk thermoelectrics with all-scale hierarchical architectures. Nature, 2012, 489: 414-418.
[6]	ZHAO L D, LO S H, ZHANG Y S, et al.Ultralow thermal conductivity and high thermoelectric figure of merit in SnSe crystals. Nature, 2014, 508: 373-377.
[7]	CHEN Z W, JIAN Z Z, LI W,et al. Lattice dislocations enhancing thermoelectric PbTe in addition to band convergence. Advanced Materials, 2017, 29(23): 1606768-1-8.
[8]	KIM S I, LEE K H, MUN H A, et al.Dense dislocation arrays embedded in grain boundaries for high-performance bulk thermoelectrics. Science, 2015, 348(6230): 109-114.
[9]	CHEN Z W, ZHANG X Y, PEI Y Z. Manipulation of phonon transport in thermoelectrics. Advanced Materials, 2018, 30(17): 1705617-1-12.
[10]	HE Y, DAY T, ZHANG T S, et al.High thermoelectric performance in non-toxic earth-abundant copper sulfide. Advanced Materials, 2014, 26(23): 3974-3978.
[11]	LIU H L, SHI X, XU F F, et al.Copper ion liquid-like thermoelectrics. Nature Materials, 2012, 11: 422-425.
[12]	VINING C B, LASKOW W, HANSON J O, et al.Thermoelectric properties of pressure-sintered Si0.8Ge0.2 thermoelectric alloys. Journal of Applied Physics, 1991, 69(8): 4333-4340.
[13]	PEI Y Z, LALONDE A, IWANAGA S, et al.High thermoelectric figure of merit in heavy hole dominated PbTe. Energy & Environmental Science, 2011, 4(6): 2085-2089.
[14]	ZEVALKINK A, TOBERER E S, ZEIER W G., et al.Ca3AlSb3: an inexpensive, non-toxic thermoelectric material for waste heat recovery. Energy & Environmental Science, 2011, 4(2): 510-518.
[15]	MAY A F, TOBERER E S, SARAMAT A, et al. Characterization and analysis of thermoelectric transport in n-type Ba8Ga16-xGe30+x. Physical Review B, 2009, 80(12): 125205-1-12.
[16]	COX C A, TOBERER E S, LEVCHENKO A A, et al.Structure, heat capacity, and high-temperature thermal properties of Yb14Mn1-xAlxSb11. Chemistry of Materials, 2009, 21(7): 1354-1360.
[17]	SLACK G A.The thermal conductivity of nonmetallic crystals. Solid State Physics, 1979, 34: 1-71.
[18]	TOBERER E S, ZEVALKINK A, SNYDER G J.Phonon engineering through crystal chemistry. Journal of Materials Chemistry, 2011, 21(40): 15843-15852.
[19]	WANG Y, HU Y J, FIRDOSY S A, et al. First-principles calculations of lattice dynamics and thermodynamic properties for Yb14MnSb11. Journal of Applied Physics, 2018, 123(4): 045102-1-10.
[20]	BROWN S R, KAUZLARICH S M, GASCOIN F, et al.Yb14MnSb11: new high efficiency thermoelectric material for power generation. Chemistry of Materials, 2006, 18(7): 1873-1877.
[21]	CHEN Z, LI D C, DENG S P, et al.Thermoelectric properties and thermal stability of Bi-doped PbTe single crystal. Physica B: Condensed Matter, 2018, 538: 154-159.
[22]	YING P J, LI X, WANG Y C, et al. Hierarchical chemical bonds contributing to the intrinsically low thermal conductivity in α-MgAgSb thermoelectric materials. Advanced Functional Materials, 2016, 27(1): 1604145-1-8.
[23]	LI J Q, LI L F, SONG S H, et al.High thermoelectric performance of GeTe-Ag8GeTe6 eutectic composites. Journal of Alloys and Compounds, 2013, 565: 144-147.
[24]	FUJIKANE M, KUROSAKI K, MUTA H, et al.Thermoelectric properties of Ag8GeTe6. Journal of Alloys and Compounds, 2005, 396(1): 280-282.
[25]	HOU Y H, CHANG L S.Optimization on the figure-of-merit of p-type Ba8Ga16Ge30 type-I clathrate grown via the Bridgman method by fine tuning Ga/Ge ratio. Journal of Alloys and Compounds, 2018, 736: 108-114.
[26]	YAN X L, IKEDA M, ZHANG L, et al.Suppression of vacancies boosts thermoelectric performance in type-I clathrates. Journal of Materials Chemistry A, 2018, 6(4): 1727-1735.
[27]	BEEKMAN M, VANDERGRAAFF A.High-temperature thermal conductivity of thermoelectric clathrates. Journal of Applied Physics, 2017, 121(20): 205105.
[28]	GONZALEZ-ROMERO R L, ANTONELLI A. Estimating carrier relaxation times in the Ba8Ga16Ge30 clathrate in the extrinsic regime. Physical Chemistry Chemical Physics, 2017, 19(4): 3010-3018.
[29]	CHRISTENSEN M, ABRAHAMSEN A B, CHRISTENSEN N B, et al.Avoided crossing of rattler modes in thermoelectric materials. Nature Materials, 2008, 7: 811-815.
[30]	CALLAWAY J.Model for lattice thermal conductivity at low temperatures. Physical Review, 1959, 113(4): 1046-1051.
[31]	CHUNG J D, MCGAUGHEY A J H, KAVIANY M. Role of phonon dispersion in lattice thermal conductivity modeling. Journal of Heat Transfer, 2004, 126(3): 376-380.
[32]	SLACK G A, GALGINAITIS S.Thermal conductivity and phonon scattering by magnetic impurities in CdTe. Physical Review, 1964, 133(1A): A253-A268.
[33]	HEREMANS J P.Thermoelectric materials: the anharmonicity blacksmith. Nature Physics, 2015, 11: 990-991.
[34]	QIU W J, XI L L, WEI P, et al.Part-crystalline part-liquid state and rattling-like thermal damping in materials with chemical-bond hierarchy. Proceedings of the National Academy of Sciences, 2014, 111(42): 15031-15035.
[35]	TYAGI K, GAHTORI B, BATHULA S, et al.Thermoelectric properties of Cu3SbSe3 with intrinsically ultralow lattice thermal conductivity. Journal of Materials Chemistry A, 2014,2(38): 15829-15835.
[36]	DELAIRE O, MA J, MARTY K, et al.Giant anharmonic phonon scattering in PbTe. Nature Materials, 2011, 10: 614-619.
[37]	LEE S, ESFARJANI K, LUO T F, et al.Resonant bonding leads to low lattice thermal conductivity. Nature Communications, 2014, 5: 3525-1-8.
[38]	MURPHY R M, MURRAY ÉD, FAHY S, et al. Ferroelectric phase transition and the lattice thermal conductivity of Pb1-xGexTe alloys. Physical Review B, 2017, 95(14): 144302-1-8.
[39]	CHEN Y, HE B, ZHU T J, et al.Thermoelectric properties of non-stoichiometric AgSbTe2 based alloys with a small amount of GeTe addition. Journal of Physics D: Applied Physics, 2012, 45(11): 115302.
[40]	ZHANG Y, KE X Z, CHEN C F, et al. Thermodynamic properties of PbTe, PbSe,PbS: first-principles study. Physical Review B, 2009, 80(2): 024304-1-12.
[41]	MILLER A J, SAUNDERS G A, YOGURTCU Y K.Pressure dependences of the elastic constants of PbTe, SnTe and Ge0.08Sn0.92Te. Journal of Physics C: Solid State Physics, 1981, 14(11): 1569-1584.
[42]	ALEXANDER F Z, VOLKER L D, RALF P S, et al.Ab initio lattice dynamics and thermochemistry of layered bismuth telluride (Bi2Te3). Journal of Physics: Condensed Matter, 2016, 28(11): 115401-1-7.
[43]	RINCÓN C, VALERI-GIL M L, WASIM S M. Room-temperature thermal conductivity and grüneisen parameter of the I-III-VI2 chalcopyrite compounds. Physica Status Solidi (A), 1995, 147(2): 409-415.
[44]	WANG H F, JIN H, CHU W G, et al.Thermodynamic properties of Mg2Si and Mg2Ge investigated by first principles method. Journal of Alloys and Compounds, 2010, 499(1): 68-74.
[45]	BERNSTEIN N, FELDMAN J L, SINGH D J. Calculations of dynamical properties of skutterudites: thermal conductivity, thermal expansivity,atomic mean-square displacement. Physical Review B, 2010, 81(13): 134301-1-11.
[46]	BHASKAR A, PAI Y H, WU W M, et al.Low thermal conductivity and enhanced thermoelectric performance of nanostructured Al-doped ZnTe. Ceramics International, 2016,42(1, Part B): 1070-1076.
[47]	KATRE A, TOGO A, TANAKA I, et al. First principles study of thermal conductivity cross-over in nanostructured zinc-chalcogenides.Journal of Applied Physics, 2015, 117(4): 045102-1-6.
[48]	NUNES O A C. Piezoelectric surface acoustical phonon amplification in graphene on a GaAs substrate. Journal of Applied Physics, 2014, 115(23): 233715-1-7.
[49]	REEBER R R.Thermal expansion of some group IV elements and ZnS. Physica Status Solidi (a), 1975, 32(1): 321-331.
[50]	QIN L, TEO K L, SHEN Z X, et al. Raman scattering of Ge/Si dot superlattices under hydrostatic pressure. Physical Review B, 2001, 64(7): 075312-1-5.
[51]	SILPAWILAWAN W, KUROSAKI K, OHISHI Y, et al.FeNbSb p-type half-Heusler compound: beneficial thermomechanical properties and high-temperature stability for thermoelectrics. Journal of Materials Chemistry C, 2017, 5(27): 6677-6681.
[52]	BOSONI E, SOSSO G C, BERNASCONI M.Grüneisen parameters and thermal conductivity in the phase change compound GeTe. Journal of Computational Electronics,2017, 16(4): 997-1002.
[53]	LI W, LIN S, GE B, et al. Low sound velocity contributing to the high thermoelectric performance of Ag8SnSe6. Advanced Science, 2016, 3 (11): 1600196-1-7.
[54]	CALLAWAY J, VON B, HANS C.Effect of point imperfections on lattice thermal conductivity. Physical Review, 1960, 120(4): 1149-1154.
[55]	HAO F, QIU P F, TANG Y S, et al.High efficiency Bi2Te3-based materials and devices for thermoelectric power generation between 100 and 300 ℃. Energy & Environmental Science, 2016, 9(10): 3120-3127.
[56]	HU L P, ZHU T J, LIU X H, et al.Point defect engineering of high-performance bismuth-telluride-based thermoelectric materials. Advanced Functional Materials, 2014, 24(33): 5211-5218.
[57]	PEI Y Z, SHI X Y, LALONDA A, et al.Convergence of electronic bands for high performance bulk thermoelectrics. Nature, 2011, 473(7345): 66-69.
[58]	QIN Y T, QIU P F, SHI X, et al.Thermoelectric properties for CuInTe2-xSx(x = 0, 0.05, 0.1, 0.15) solid solution. Journal of Inorganic Materials, 2017, 32(11): 1171-1176.
[59]	JIANG G Y, HE J, ZHU T J, et al.High performance Mg2(Si,Sn) solid solutions: a point defect chemistry approach to enhancing thermoelectric properties. Advanced Functional Materials, 2014, 24(24): 3776-3781.
[60]	LIU X H, ZHU T J, WANG H, et al.Low electron scattering potentials in high performance Mg2Si0.45Sn0.55 based thermoelectric solid solutions with band convergence. Advanced Energy Materials, 2013, 3(9): 1238-1244.
[61]	TRIPATHI M N, BHANDARI C M.High-temperature thermoelectric performance of Si-Ge alloys. Journal of Physics: Condensed Matter, 2003, 15(31): 5359-5370.
[62]	FU C G, ZHU T J, PEI Y Z, et al. High band degeneracy contributes to high thermoelectric performance in p-type half-Heusler compounds. Advanced Energy Materials, 2014, 4(18): 1400600-1-6.
[63]	YU J J, XIA K Y, ZHAO X B, et al.High performance p-type half-Heusler thermoelectric materials. Journal of Physics D: Applied Physics, 2018, 51(11): 113001.
[64]	SHEN J J, FU C G, LIU Y T, et al.Enhancing thermoelectric performance of FeNbSb half-Heusler compound by Hf-Ti dual-doping. Energy Storage Materials, 2018, 10: 69-74.
[65]	ZHU T J, LIU Y T, FU C G, et al. Compromise and synergy in high-efficiency thermoelectric materials. Advanced Materials, 2017, 29(14): 1605884-1-26.
[66]	FU C G, WU H J, LIU Y T, et al. Enhancing the figure of merit of heavy-band thermoelectric materials through hierarchical phonon scattering. Advanced Science, 2016, 3(8): 1600035-1-6.
[67]	ZHU T J, FU C G, XIE H H, et al. High efficiency half-Heusler thermoelectric materials for energy harvesting. Advanced Energy Materials, 2015, 5(19): 1500588-1-7.
[68]	FU C G, BAI S Q, LIU Y T, et al.Realizing high figure of merit in heavy-band p-type half-Heusler thermoelectric materials. Nat. Commun., 2015, 6: 8144.
[69]	XIE H H, WANG H, PEI Y Z, et al.Beneficial contribution of alloy disorder to electron and phonon transport in half-heusler thermoelectric materials. Advanced Functional Materials, 2013, 23(41): 5123-5130.
[70]	YU J J, FU C G, LIU Y T, et al. Unique role of refractory ta alloying in enhancing the figure of merit of NbFeSb thermoelectric materials. Advanced Energy Materials, 2018, 8(1): 1701313-1-8.
[71]	XIA K Y, LIU Y T, ANAND S, et al. Enhanced thermoelectric performance in 18-electron Nb0.8CoSb half-heusler compound with intrinsic Nb vacancies. Advanced Functional Materials, 2018, 28(9): 1705845-1-7.
[72]	LI W, LIN S Q, ZHANG X Y, et al.Thermoelectric properties of Cu2SnSe4 with intrinsic vacancy. Chemistry of Materials, 2016, 28(17): 6227-6232.
[73]	KLEMENS P G.The scattering of low-frequency lattice waves by static imperfections. Proceedings of the Physical Society. Section A, 1955, 68(12): 1113-1128.
[74]	ZHANG S N, HE J, JI X H, et al.Effects of ball-milling atmosphere on the thermoelectric properties of TAGS-85 compounds. Journal of Electronic Materials, 2009, 38(7): 1142-1147.
[75]	LI Y, MEI D Q, WANG H, et al.Reduced lattice thermal conductivity in nanograined Na-doped PbTe alloys by ball milling and semisolid powder processing. Materials Letters, 2015, 140: 103-106.
[76]	HONG M, CHEN Z G, ZOU J. Fundamental and progress of Bi2Te3-based thermoelectric materials. Chinese Physics B, 2018, 27(4): 048403-1-46.
[77]	XIE J, OHISHI Y, ICHIKAWA S, et al. Naturally decorated dislocations capable of enhancing multiple-phonon scattering in Si-based thermoelectric composites. Journal of Applied Physics, 2018, 123(11): 115114-1-8.
[78]	YU Y, HE D S, ZHANG S Y, et al.Simultaneous optimization of electrical and thermal transport properties of Bi0.5Sb1.5Te3 thermoelectric alloy by twin boundary engineering. Nano Energy, 2017, 37: 203-213.
[79]	XIE W J, HE J, KANG H J, et al.Identifying the specific nanostructures responsible for the high thermoelectric performance of (Bi,Sb)2Te3 nanocomposites. Nano Letters, 2010, 10(9): 3283-3289.
[80]	YANG X Y, WU J H, REN D D, et al.Microstructure and thermoelectric properties of p-type Si80Ge20B0.6-SiC nanocomposite. Journal of Inorganic Materials, 2016, 31(9): 997-1003.
[81]	YU C, XIE H H, FU C G, et al.High performance half-Heusler thermoelectric materials with refined grains and nanoscale precipitates. Journal of Materials Research, 2012, 27(19): 2457-2465.
[82]	HU L P, WU H J, ZHU T J, et al. Tuning multiscale microstructures to enhance thermoelectric performance of n-type bismuth-telluride-based solid solutions. Advanced Energy Materials, 2015, 5(17): 1500411-1-13.
[83]	HE J Q, GIRARD S N, KANATZIDIS M G, et al.Microstructure-lattice thermal conductivity correlation in nanostructured PbTe0.7S0.3 thermoelectric materials. Advanced Functional Materials, 2010, 20(5): 764-772.
[84]	XIN J Z, WU H J, LIU X H, et al.Mg vacancy and dislocation strains as strong phonon scatterers in Mg2Si1-xSbx thermoelectric materials. Nano Energy, 2017, 34: 428-436.
[85]	SHI X, BAI S Q, XI L L, et al.Realization of high thermoelectric performance in n-type partially filled skutterudites. Journal of Materials Research, 2011, 26(15): 1745-1754.
[86]	KEPPENS V, MANDRUS D, SALES B C, et al.Localized vibrational modes in metallic solids. Nature, 1998, 395: 876-878.
[87]	DUAN B, YANG J, SALVADOR J R, et al.Electronegative guests in CoSb3. Energy & Environmental Science, 2016, 9(6): 2090-2098.
[88]	SAMANTA M, PAL K, PAL P, et al.Localized vibrations of bi bilayer leading to ultralow lattice thermal conductivity and high thermoelectric performance in weak topological insulator n-type BiSe. Journal of the American Chemical Society, 2018, 140(17): 5866-5872.
[89]	UHER C, YANG J, HU S, et al.Transport properties of pure and doped MNiSn (M=Zr, Hf). Physical Review B, 1999, 59(13): 8615-8621.
[90]	LI J F, LIU W S, ZHAO L D, et al.High-performance nanostructured thermoelectric materials. NPG Asia Materials, 2010, 2(4): 152-158.
[91]	FANG T, ZHAO X B, ZHU T J.Band Structures and transport properties of high-performance half-heusler thermoelectric materials by first principles. Materials, 2018, 11(5): 847.
[92]	TANG Y L, LI X S, MARTIN L H J, et al. Impact of Ni content on the thermoelectric properties of half-Heusler TiNiSn. Energy & Environmental Science, 2018, 11(2): 311-320.
[93]	ZHU T J, YU G T, XU J, et al.The role of electron-phonon interaction in heavily doped fine-grained bulk silicons as thermoelectric materials. Advanced Electronic Materials, 2016, 2(8): 1600171.
[94]	ABELES B.Lattice thermal conductivity of disordered semiconductor alloys at high temperatures. Physical Review, 1963, 131(5): 1906-1911.
[95]	CLARKE D R.Materials selection guidelines for low thermal conductivity thermal barrier coatings.Surface and Coatings Technology, 2003, 163: 67-74.
[96]	CAHILL D G, POHL R O.Heat flow and lattice vibrations in glasses. Solid State Communications, 1989, 70(10): 927-930.
[97]	CAHILL D G, WATSON S K, POHL R O.Lower limit to the thermal conductivity of disordered crystals. Physical Review B, 1992, 46(10): 6131-6140.
[98]	ALLEN P B, DU X Q, MIHALY L, et al.Thermal conductiity of insulating Bi2Sr2YCu2O8 and superconducting Bi2Sr2CaCu2O8: failure of the phonon-gas picture. Physical Rview B, 1994, 49(13): 9073-9079.
[99]	FELDMAN J L, ALLEN P B, BICKHAM S R.Numerical study of low-frequency vibrations in amorphous silicon. Physical Review B, 1999, 59(5): 3551-3559.
[100]	AGNE M T, HANUS R, SNYDER G J.Minimum thermal conductivity in the context of diffuson-mediated thermal transport.Energy & Environmental Science, 2018, 11(3): 609-616.
[101]	POHL R O.Lattice vibrations of glasses. Journal of Non-Crystalline Solids, 2006, 352(32): 3363-3367.
[102]	FU C G, ZHU T J, LIU Y T, et al.Band engineering of high performance p-type FeNbSb based half-Heusler thermoelectric materials for figure of merit zT>1. Energy & Environmental Science, 2015, 8(1): 216-220.
[103]	WEI P, YANG J, GUO L, et al.Minimum thermal conductivity in weak topological insulators with bismuth-based stack structure. Advanced Functional Materials, 2016, 26(29): 5360-5367.
[104]	RASCHE B, ISAEVA A, RUCK M, et al.Stacked topological insulator built from bismuth-based graphene sheet analogues. Nature Materials, 2013, 12(5): 422-425.
[105]	PAULY C, RASCHE B, KOEPERNIK K, et al.Subnanometre-wide electron channels protected by topology. Nature Physics, 2015, 11(4): 338-343.