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金属导热理论的研究进展与前沿问题

王奥 盛宇飞 鲍华

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金属导热理论的研究进展与前沿问题

王奥, 盛宇飞, 鲍华

Recent advances in thermal transport theory of metals

Wang Ao, Sheng Yu-Fei, Bao Hua
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  • 金属是人类使用最广泛的材料之一. 相对于对金属力学性能的研究, 金属导热性能的相关研究较为匮乏. 对金属导热机制的理解往往还依赖于一百多年前建立的威德曼-弗朗兹定律. 金属导热和电子输运有密切联系, 同时又与晶格振动有关. 深入理解金属导热机制, 不但对材料应用意义重大, 而且有利于提高对导热基本理论的认知. 本文回顾了金属导热研究的历史, 并对最近十几年来金属导热的研究进行了总结, 特别是对基于第一原理电子-声子耦合模式分析的金属导热机理的研究进行了综述. 此外, 本文也对金属导热理论的未来发展方向进行了探讨.
    Metal is one of the most widely used engineering materials. In contrast to the extensive research dedicated to their mechanical properties, studies on the thermal conductivity of metals remain relatively rare. The understanding of thermal transport mechanisms in metals is mainly through the Wiedemann-Franz Law established more than a century ago. The thermal conductivity of metal is related to both the electron transport and the lattice vibration. An in-depth understanding of the thermal transport mechanism in metal is imperative for optimizing their practical applications. This review first discusses the history of the thermal transport theory in metals, including the Wiedemann-Franz law and models for calculating phonon thermal conductivity in metal. The recently developed first-principles based mode-level electron-phonon interaction method for determining the thermal transport properties of metals is briefly introduced. Then we summarize recent theoretical studies on the thermal conductivities of elemental metals, intermetallics, and metallic ceramics. The value of thermal conductivity, phonon contribution to total thermal conductivity, the influence of electron-phonon interaction on thermal transport, and the deviation of the Lorenz number are comprehensively discussed. Moreover, the thermal transport properties of metallic nanostructures are summarized. The size effect of thermal transport and the Lorenz number obtained from experiments and calculations are compared. Thermal transport properties including the phonon contribution to total thermal conductivity and the Lorenz number in two-dimensional metals are also mentioned. Finally, the influence of temperature, pressure, and magnetic field on thermal transport in metal are also discussed. The deviation of the Lorenz number at low temperatures is due to the different electron-phonon scattering mechanisms for thermal and electrical transport. The mechanism for the increase of thermal conductivity in metals induced by pressure varies in different kinds of metals and is related to the electron state at the Fermi level. The effect of magnetic field on thermal transport is related to the coupling between the electron and the magnetic field, therefore the electron distribution in the Brillouin zone is an important factor. In addition, this review also looks forward to the future research directions of metal thermal transport theory.
      通信作者: 鲍华, hua.bao@sjtu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 52122606)资助的课题.
      Corresponding author: Bao Hua, hua.bao@sjtu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 52122606).
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  • 图 1  (a)金属导热与导电的微物理观过程示意图, 包括三声子散射和电声相互作用; (b)金属导热主要研究对象, 包括金属单质、金属性化合物、金属纳米结构与二维金属的导热, 以及金属导热受到不同外界因素诸如温度、压强等的影响

    Fig. 1.  (a) Microscopic physical process of electron thermal transport and electrical transport, including three-phonon scattering and electron-phonon interaction; (b) the main research objects for thermal transport in metals, including the thermal transport in elemental metals, metallic compounds, metallic nanostructure, and two-dimensional metals; as well as the effect of external environments, such as temperature and pressure on thermal conductivity.

    图 2  (a) 300 K下部分金属中电子与声子导热对总热导率的贡献占比[34]; (b) 200—500 K下钨的声子热导率、电子热导率及总热导率与实验的对比(上图), 考虑总热导率与只考虑电子热导率下钨的洛伦兹数(下图)[24], 虚线所示L0为标准洛伦兹数

    Fig. 2.  (a) Percentage of electron and phonon thermal conductivity contributing to total thermal conductivity for several metals[34]. (b) The phonon, electron, and total thermal conductivity of tungsten compared with experiments from 200 to 500 K (upper panel); the Lorenz number of tungsten considering total thermal conductivity versus considering electronic thermal conductivity (lower panel) [24], the dashed line is the standard Lorenz number L0.

    图 3  (a) NiAl[39], (b) MgZn2和Mg4Zn7[42]中的声子热导率、电子热导率和总热导率随温度的变化

    Fig. 3.  Variation of phonon thermal conductivity, electron thermal conductivity, and total thermal conductivity with temperature for (a) NiAl[39], (b) MgZn2 and Mg4Zn7[42].

    图 4  (a) 200—1000 K下只考虑声子-声子和声子-同位素散射(虚线), 与同时考虑声子-声子、声子-同位素散射和声电散射(实线)下NbC和TiN的声子热导率[43], 插图为二者的电子费米面; (b) 200—1000 K下TiN的声子热导率、电子热导率、总热导率随温度的变化[45]; (c) 200—1000 K下TiN的电导率与洛伦兹数随温度的变化[45]

    Fig. 4.  (a) Phonon thermal conductivity for NbC and TiC limited by phonon-phonon and phonon-isotope scattering (dashed curve), phonon-phonon, phonon-isotope, and phonon-electron scattering (solid curve) from 200 to 1000 K[43]. Inset: The Fermi surfaces of NbC and TiC. (b) The phonon thermal conductivity, electron thermal conductivity, and total thermal conductivity of TiN from 200 to 1000 K[45]. (c) The Lorenz number of TiN from 200 to 1000 K[45].

    图 5  (a)铜纳米薄膜理论计算与实验测量的热导率和薄膜厚度的关系[56]; (b)金纳米薄膜的理论计算与实验的归一化洛伦兹数随温度的变化[67]

    Fig. 5.  (a) Calculated and experimental thermal conductivity with respect to thickness of copper nanofilm[56]; (b) the calculated and experimental normalized Lorenz number for gold[67].

    图 6  考虑MRTA, ERTA计算的铜的(a)电子热导率和(b)电导率[26]与Allen模型结果的对比; (c) MRTA, ERTA, 常弛豫时间近似(Constant), Allen模型和布洛赫-格律乃森模型(BG)计算得到的铜的洛伦兹数的对比[26]; (d)外电场与温度梯度下电子的散射过程, 实心圆代表占据的电子态, 空心圆代表未占据的电子态[26]

    Fig. 6.  Calculated (a) electron thermal conductivity and (b) electrical conductivity of copper considering MRTA and ERTA[26], compared with the results from the Allen model; (c) the calculated Lorenz number by MRTA, ERTA, constant relaxation time approximation (Constant), Allen model, and the BG model for Cu[26]; (d) the electron scattering in an electric field and under a temperature gradient[26]. Note that the filled small spheres are occupied electron states and the open small spheres are unoccupied electron states.

    图 7  (a)铝、金刚石、立方氮化硼和砷化硼的热导率随压强的变化[77]; (b)铝、(c)钨的归一化电子热导率、平均速度、弛豫时间和总电子比热容随压强的变化[78]

    Fig. 7.  (a) Variation of thermal conductivity with pressure for aluminum, diamond, cubic boron nitride, and boron arsenide[77]; normalized electron thermal conductivity, averaged velocity, relaxation time, and total electron specific heat of (b) Al and (c) W as a function of pressure[78].

    表 1  金属中的电子与声子热导率

    Table 1.  Electron and phonon thermal conductivity in metals.

    金属 300 K下热导率/(W·m–1·K–1) 声子导热
    占比/%
    总热导率 电子
    热导率
    声子
    热导率
    Au 278[25] 276[25] 2[25] 0.7
    Ag 374[25] 370[25] 4[25] 1.1
    Al 252[25] 246[25] 6[25] 2.4
    Cu 378.7[34] 361.3[34] 17.4[34] 4.6
    Mn 8[34] 5[34] 3[34] 37.5
    Ti 30.6[34] 25.3[34] 5.3[34] 17.3
    W 186[24] 140[24] 46[24] 24.7
    Mo 162[36] 125[36] 37[36] 22.8
    hcp-Au (a-axis) 201.3[37] 199[37] 2.3[37] 1.1
    hcp-Ag (a-axis) 276.6[37] 274[37] 2.6[37] 0.9
    hcp-Cu (a-axis) 279.4[37] 270[37] 9.4[37] 3.4
    NiAl 71[39] 59[39] 12[39] 16.9
    Ni3Al 28[39] 22[39] 6[39] 21.4
    MgZn2 53.9[42] 52[42] 1.9[42] 3.5
    Mg4Zn7 21.9[42] 21.4[42] 0.5[42] 2.3
    WC (a-axis) 177[44] 46[44] 131[44] 74.0
    NbC 74[44] 43[44] 31[44] 41.9
    TiN 69[45] 49[45] 20[45] 29.0
    HfN 93[45] 69[45] 24[45] 25.8
    θ-TaN (a-axis) 1031[47] 36[47] 995[47] 96.5
    hcp-NbN 4.4[48] 1.5[48] 2.9[48] 65.9
    下载: 导出CSV
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    Shah R K, Sekulic D P 2003 Fundamentals of Heat Exchanger Design (John Wiley & Sons

    [2]

    Rajagopal M C, Chang H C, Man T, Kuntumalla G, Meng Y, Sundar S, Zhao H, Salapaka S, Shao C, Ferreira P, Miljkovic N, Sinha S 2019 Int. J. Heat Mass Transf. 143 118497Google Scholar

    [3]

    Lu D, Wong C 2009 Materials for Advanced Packaging (Vol. 181) (Springer

    [4]

    Kittel C 2005 Introduction to Solid State Physics (New York: John Wiley & Sons, inc

    [5]

    Jones W, March N H 1985 Theoretical Solid State Physics (Vol. 35) (Courier Corporation

    [6]

    Tritt T M ed 2005 Thermal Conductivity: Theory, Properties, and Applications (New York: Springer

    [7]

    Franz R, Wiedemann G 1853 Ann. Phys. 165 497Google Scholar

    [8]

    Lorenz L 1881 Ann. Phys. 249 422Google Scholar

    [9]

    Drude P 1900 Ann. Phys. 306 566Google Scholar

    [10]

    Sommerfeld A 1928 Z. Phys. 47 43Google Scholar

    [11]

    Butler W H, Williams R K 1978 Phys. Rev. B 18 6483Google Scholar

    [12]

    Lang H N D, Kempen H V, Wyder P 1978 J. Phys. F 8 L39Google Scholar

    [13]

    Williams R K, Yarbrough D W, Masey J W, Holder T K, Graves R S 1981 J. Appl. Phys. 52 5167Google Scholar

    [14]

    Williams R K, Graves R S, Hebble T L, McElroy D L, Moore J P 1982 Phys. Rev. B 26 2932Google Scholar

    [15]

    Williams R K, Butler W H, Graves R S, Moore J P 1983 Phys. Rev. B 28 6316Google Scholar

    [16]

    Klemens P G (Seitz F, Turnbull D ed) 1958 Solid State Physics (Academic Press) pp1–98

    [17]

    Slack G A (Ehrenreich H, et al. ed) 1979 Solid State Physics (Academic Press) pp1–71

    [18]

    Klemens P G, Williams R K 1986 Int. Metals Rev. 31 197Google Scholar

    [19]

    Gregg J F, Haar D t 1996 Eur. J. Phys. 17 303Google Scholar

    [20]

    White G K, Tainsh R J 1960 Phys. Rev. 119 1869Google Scholar

    [21]

    Wang J L, Wu Z Z, Mao C K, Zhao Y F, Yang J K, Chen Y F 2018 Sci. Rep. 8 4862Google Scholar

    [22]

    Poncé S, Margine E R, Verdi C, Giustino F 2016 Comput. Phys. Commun. 209 116Google Scholar

    [23]

    Liu T H, Zhou J, Li M, Ding Z, Song Q, Liao B, Fu L, Chen G 2018 Proc. Natl. Acad. Sci. U. S. A. 115 879Google Scholar

    [24]

    Chen Y, Ma J, Li W 2019 Phys. Rev. B 99 020305Google Scholar

    [25]

    Jain A, McGaughey A J H 2016 Phys. Rev. B 93 081206Google Scholar

    [26]

    Li S, Tong Z, Zhang X, Bao H 2020 Phys. Rev. B 102 174306Google Scholar

    [27]

    Bao H, Chen J, Gu X, Cao B 2018 ES Energy Environ. 1 16Google Scholar

    [28]

    Broido D A, Malorny M, Birner G, Mingo N, Stewart D A 2007 Appl. Phys. Lett. 91 231922Google Scholar

    [29]

    Walker C T, Pohl R O 1963 Phys. Rev. 131 1433Google Scholar

    [30]

    Bardeen J, Pines D 1955 Phys. Rev. 99 1140Google Scholar

    [31]

    Rice T M 1965 Ann. Phys. 31 100Google Scholar

    [32]

    Langer J S 1960 Phys. Rev. 120 714Google Scholar

    [33]

    Giri A, Tokina M V, Prezhdo O V, Hopkins P E 2020 Mater. Today Phys. 12 100175Google Scholar

    [34]

    Tong Z, Li S, Ruan X, Bao H 2019 Phys. Rev. B 100 144306Google Scholar

    [35]

    Wang Y, Lu Z, Ruan X 2016 J. Appl. Phys. 119 225109Google Scholar

    [36]

    Wen S, Ma J, Kundu A, Li W 2020 Phys. Rev. B 102 064303Google Scholar

    [37]

    Li S, Tong Z, Shao C, Bao H, Frauenheim T, Liu X 2022 The J. Phys. Chem. Lett. 13 4289Google Scholar

    [38]

    Su C, Li D, Luo A A, Ying T, Zeng X 2018 J. Alloys Compd. 747 431Google Scholar

    [39]

    Tong Z, Bao H 2018 Int. J. Heat Mass Transf. 117 972Google Scholar

    [40]

    Liu Y Z, Zheng B C, Jian Y X, Zhang L, Yi Y L, Li W 2020 Intermetallics 124 106880Google Scholar

    [41]

    Daeumer M, Sandoval E D, Azizi A, Bagheri M H, Bae I T, Panta S, Koulakova E A, Cotts E, Arvin C L, Kolmogorov A N, Schiffres S N 2022 Acta Mater. 227 117671Google Scholar

    [42]

    Wang A, Li S, Ying T, Zeng X, Bao H 2023 J. Appl. Phys. 133 015101Google Scholar

    [43]

    Li C, Ravichandran N K, Lindsay L, Broido D 2018 Phys. Rev. Lett. 121 175901Google Scholar

    [44]

    Kundu A, Ma J, Carrete J, Madsen G K H, Li W 2020 Mater. Today Phys. 13 100214Google Scholar

    [45]

    Li S, Wang A, Hu Y, Gu X, Tong Z, Bao H 2020 Mater. Today Phys. 15 100256Google Scholar

    [46]

    Yang J Y, Zhang W, Xu C, Liu J, Liu L, Hu M 2020 Int. J. Heat Mass Transf. 152 119481Google Scholar

    [47]

    Kundu A, Yang X, Ma J, Feng T, Carrete J, Ruan X, Madsen G K H, Li W 2021 Phys. Rev. Lett. 126 115901Google Scholar

    [48]

    Liu Z, Luo T 2021 Appl. Phys. Lett. 118 043102Google Scholar

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    Moore A L, Shi L 2014 Mater. Today 17 163Google Scholar

    [50]

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  • 被引次数: 0
出版历程
  • 收稿日期:  2023-07-17
  • 修回日期:  2023-09-19
  • 上网日期:  2023-11-09
  • 刊出日期:  2024-02-05

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