Recent advances in thermal transport theory of metals

Wang Ao; Sheng Yu-Fei; Bao Hua

doi:10.7498/aps.73.20231151

Abstract

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.

Keywords:

Authors and contacts

1.
Global Institute of Future Technology, Shanghai Jiao Tong University, Shanghai 200240, China
2.
University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai 200240, China

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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References

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[12]	Lang H N D, Kempen H V, Wyder P 1978 J. Phys. F 8 L39 Google Scholar
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Cited By

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

Figure 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.

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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

Figure 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.

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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

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

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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]

Figure 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].

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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]

Figure 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].

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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]

Figure 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.

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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]

Figure 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].

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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/%
金属	总热导率	电子热导率	声子热导率	声子导热占比/%
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

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[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 118497 Google 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. 165 497 Google Scholar
[8]	Lorenz L 1881 Ann. Phys. 249 422 Google Scholar
[9]	Drude P 1900 Ann. Phys. 306 566 Google Scholar
[10]	Sommerfeld A 1928 Z. Phys. 47 43 Google Scholar
[11]	Butler W H, Williams R K 1978 Phys. Rev. B 18 6483 Google Scholar
[12]	Lang H N D, Kempen H V, Wyder P 1978 J. Phys. F 8 L39 Google Scholar
[13]	Williams R K, Yarbrough D W, Masey J W, Holder T K, Graves R S 1981 J. Appl. Phys. 52 5167 Google Scholar
[14]	Williams R K, Graves R S, Hebble T L, McElroy D L, Moore J P 1982 Phys. Rev. B 26 2932 Google Scholar
[15]	Williams R K, Butler W H, Graves R S, Moore J P 1983 Phys. Rev. B 28 6316 Google 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. 31 197 Google Scholar
[19]	Gregg J F, Haar D t 1996 Eur. J. Phys. 17 303 Google Scholar
[20]	White G K, Tainsh R J 1960 Phys. Rev. 119 1869 Google Scholar
[21]	Wang J L, Wu Z Z, Mao C K, Zhao Y F, Yang J K, Chen Y F 2018 Sci. Rep. 8 4862 Google Scholar
[22]	Poncé S, Margine E R, Verdi C, Giustino F 2016 Comput. Phys. Commun. 209 116 Google 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. 115 879 Google Scholar
[24]	Chen Y, Ma J, Li W 2019 Phys. Rev. B 99 020305 Google Scholar
[25]	Jain A, McGaughey A J H 2016 Phys. Rev. B 93 081206 Google Scholar
[26]	Li S, Tong Z, Zhang X, Bao H 2020 Phys. Rev. B 102 174306 Google Scholar
[27]	Bao H, Chen J, Gu X, Cao B 2018 ES Energy Environ. 1 16 Google Scholar
[28]	Broido D A, Malorny M, Birner G, Mingo N, Stewart D A 2007 Appl. Phys. Lett. 91 231922 Google Scholar
[29]	Walker C T, Pohl R O 1963 Phys. Rev. 131 1433 Google Scholar
[30]	Bardeen J, Pines D 1955 Phys. Rev. 99 1140 Google Scholar
[31]	Rice T M 1965 Ann. Phys. 31 100 Google Scholar
[32]	Langer J S 1960 Phys. Rev. 120 714 Google Scholar
[33]	Giri A, Tokina M V, Prezhdo O V, Hopkins P E 2020 Mater. Today Phys. 12 100175 Google Scholar
[34]	Tong Z, Li S, Ruan X, Bao H 2019 Phys. Rev. B 100 144306 Google Scholar
[35]	Wang Y, Lu Z, Ruan X 2016 J. Appl. Phys. 119 225109 Google Scholar
[36]	Wen S, Ma J, Kundu A, Li W 2020 Phys. Rev. B 102 064303 Google Scholar
[37]	Li S, Tong Z, Shao C, Bao H, Frauenheim T, Liu X 2022 The J. Phys. Chem. Lett. 13 4289 Google Scholar
[38]	Su C, Li D, Luo A A, Ying T, Zeng X 2018 J. Alloys Compd. 747 431 Google Scholar
[39]	Tong Z, Bao H 2018 Int. J. Heat Mass Transf. 117 972 Google Scholar
[40]	Liu Y Z, Zheng B C, Jian Y X, Zhang L, Yi Y L, Li W 2020 Intermetallics 124 106880 Google 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 117671 Google Scholar
[42]	Wang A, Li S, Ying T, Zeng X, Bao H 2023 J. Appl. Phys. 133 015101 Google Scholar
[43]	Li C, Ravichandran N K, Lindsay L, Broido D 2018 Phys. Rev. Lett. 121 175901 Google Scholar
[44]	Kundu A, Ma J, Carrete J, Madsen G K H, Li W 2020 Mater. Today Phys. 13 100214 Google Scholar
[45]	Li S, Wang A, Hu Y, Gu X, Tong Z, Bao H 2020 Mater. Today Phys. 15 100256 Google Scholar
[46]	Yang J Y, Zhang W, Xu C, Liu J, Liu L, Hu M 2020 Int. J. Heat Mass Transf. 152 119481 Google 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 115901 Google Scholar
[48]	Liu Z, Luo T 2021 Appl. Phys. Lett. 118 043102 Google Scholar
[49]	Moore A L, Shi L 2014 Mater. Today 17 163 Google Scholar
[50]	Cahill D G, Braun P V, Chen G, Clarke D R, Fan S, Goodson K E, Keblinski P, King W P, Mahan G D, Majumdar A, Maris H J, Phillpot S R, Pop E, Shi L 2014 Appl. Phys. Rev. 1 011305 Google Scholar
[51]	Chen G 2005 Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons (Oxford: Oxford University Press
[52]	Mayadas A F, Shatzkes M 1970 Phys. Rev. B 1 1382 Google Scholar
[53]	Nath P, Chopra K L 1974 Thin Solid Films 20 53 Google Scholar
[54]	Kelemen F 1976 Thin Solid Films 36 199 Google Scholar
[55]	Kumar S, Vradis G C 1994 J. Heat Transfer 116 28 Google Scholar
[56]	Feng B, Li Z, Zhang X 2009 Thin Solid Films 517 2803 Google Scholar
[57]	Zhang Q G, Cao B Y, Zhang X, Fujii M, Takahashi K 2006 Phys. Rev. B 74 134109 Google Scholar
[58]	Lin H, Xu S, Li C, Dong H, Wang X 2013 Nanoscale 5 4652 Google Scholar
[59]	Sawtelle S D, Reed M A 2019 Phys. Rev. B 99 054304 Google Scholar
[60]	Mason S J, Wesenberg D J, Hojem A, Manno M, Leighton C, Zink B L 2020 Phys. Rev. Mater. 4 065003 Google Scholar
[61]	Stojanovic N, Maithripala D H S, Berg J M, Holtz M 2010 Phys. Rev. B 82 075418 Google Scholar
[62]	Wang H, Liu J, Zhang X, Takahashi K 2013 Int. J. Heat Mass Transf. 66 585 Google Scholar
[63]	Cheng Z, Liu L, Xu S, Lu M, Wang X 2015 Sci. Rep. 5 10718 Google Scholar
[64]	Kojda D, Mitdank R, Handwerg M, Mogilatenko A, Albrecht M, Wang Z, Ruhhammer J, Kroener M, Woias P, Fischer S F 2015 Phys. Rev. B 91 024302 Google Scholar
[65]	Zhao Y, Fitzgerald M L, Tao Y, Pan Z, Sauti G, Xu D, Xu Y-Q, Li D 2020 Nano Lett. 20 7389 Google Scholar
[66]	Avery A D, Mason S J, Bassett D, Wesenberg D, Zink B L 2015 Phys. Rev. B 92 214410 Google Scholar
[67]	Hu Y, Li S, Bao H 2021 Phys. Rev. B 103 104301 Google Scholar
[68]	Dong L, Wu X, Hu Y, Xu X, Bao H 2021 Chin. Phys. Lett. 38 027202 Google Scholar
[69]	Huang Y, Zhou J, Wang G, Sun Z 2019 J. Am. Chem. Soc. 141 8503 Google Scholar
[70]	Wang A, Li S, Zhang X, Bao H 2022 Phys. Rev. Mater. 6 014009 Google Scholar
[71]	Lavasani A, Bulmash D, Das Sarma S 2019 Phys. Rev. B 99 085104 Google Scholar
[72]	Yao M, Zebarjadi M, Opeil C P 2017 J. Appl. Phys. 122 135111 Google Scholar
[73]	Mao H K, Chen X J, Ding Y, Li B, Wang L 2018 Rev. Mod. Phys. 90 015007 Google Scholar
[74]	Bohlin L 1976 Solid State Commun. 19 389 Google Scholar
[75]	Giri A, Gaskins J T, Li L, Wang Y S, Prezhdo O V, Hopkins P E 2019 Phys. Rev. B 99 165139 Google Scholar
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Vol. 73, Issue 3 - 2024-02-05

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