混合失配模型预测金属/半导体界面热导

宗志成; 潘东楷; 邓世琛; 万骁; 杨哩娜; 马登科; 杨诺

doi:10.7498/aps.72.20221981

摘要

声学失配模型和漫散射失配模型被广泛应用于界面热导的计算, 两种模型分别建立在极端光滑和粗糙界面的假设基础上. 由于实际界面结构与两种假设的区别较大, 造成两种模型预测结果与实际界面热导偏差较大. 近期提出的混合失配模型考虑了界面结构对声子镜面透射和漫散射透射比例的影响, 预测的准确度有所提高. 但该模型需要通过分子动力学模拟获取界面声子信息较为复杂. 为此, 本文通过引入测量的粗糙度数值简化混合失配模型, 并增加考虑界面结构对接触面积的影响, 实现对界面热导简单快捷、准确地预测. 基于该模型, 计算预测了金属(铝、铜、金)和半导体(硅、碳化硅、砷化镓、氮化镓)的界面热导. 并将铝/硅界面的结果与实验测量结果对比, 数据吻合较好. 该模型不仅有助于界面导热机理的理解, 而且利于与测量结果对比.

关键词:

Abstract

The acoustic mismatch model and diffuse mismatch model are widely used to calculate interfacial thermal conductance. These two models are respectively based on the assumption of extremely smooth and rough interfaces. Owing to the great difference between the actual interface structure and the two hypotheses, the predictions of these two models deviate greatly from the actual interfacial thermal conductance. The recently proposed mixed mismatch model considers the effect of interface structure on the ratio of phonon specular transmission to diffuse scattering transmission, and the prediction accuracy is improved. However, this model requires molecular dynamics simulation to obtain phonon information at the interface. In this work, the mixed mismatch model is simplified by introducing the measured roughness value, and the influence of interface structure on the contact area is taken into account to achieve a simple, fast and accurate prediction of interface thermal conductance. Based on this model, the interfacial thermal conductances of metals (aluminum, copper, gold) and semiconductors (silicon, silicon carbide, gallium arsenide, gallium nitride) are calculated and predicted. The results of Al/Si interface are in good agreement with the experimental results. This model is helpful not only in understanding the mechanism of interface heat conduction, but also in comparing with the measurement results.

Keywords:

作者及机构信息

1.
华中科技大学能源与动力工程学院, 武汉　430074

2.
北京理工大学宇航学院, 北京　100081

3.
南京师范大学物理科学与技术学院, 南京　210000

通信作者: 杨诺, nuo@hust.edu.cn

基金项目: 国家重点研发计划政府间联合项目(批准号: 2018YFE0127800 )资助的课题.

Authors and contacts

1.
School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

2.
School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

3.
School of Physics and Technology, Nanjing Normal University, Nanjing 210000, China

Corresponding author: Yang Nuo, nuo@hust.edu.cn

Funds: Project supported by the National Key Research and Development Project of China (Grant No. 2018YFE0127800).

[1]	Wei N, Zhou C, Li Z, Ou B, Zhao K, Yu P, Li S, Zhao J 2022 Mater. Today Commun. 30 103147 Google Scholar
[2]	Cui Y, Li M, Hu Y 2020 J. Mater. Chem. C 8 10568 Google Scholar
[3]	Deng S, Xiao C, Yuan J, Ma D, Li J, Yang N, He H 2019 Appl. Phys. Lett. 115 101603 Google Scholar
[4]	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
[5]	Peng X, Jiang P, Ouyang Y, Lu S, Ren W, Chen J 2021 Nanotechnology 33 035707 Google Scholar
[6]	Monachon C, Weber L, Dames C 2016 Annu. Rev. Mater. Res. 46 433 Google Scholar
[7]	Scott E A, Gaskins J T, King S W, Hopkins P E 2018 APL Mater. 6 058302 Google Scholar
[8]	Yang L, Wan X, Ma D, Jiang Y, Yang N 2021 Phys. Rev. B 103 155305 Google Scholar
[9]	Liu B, Khvesyuk V I, Barinov A A, Wang M 2022 Int. J. Mech. Sci. 218 106993 Google Scholar
[10]	Yang N, Luo T, Esfarjani K, Henry A, Tian Z, Shiomi J, Chalopin Y, Li B, Chen G 2015 J. Comput. Theor. Nanosci. 12 168 Google Scholar
[11]	Rustam S, Schram M, Lu Z, Chaka A M, Rosenthal W S, Pfaendtner J 2022 ACS Appl. Mater. Interfaces 14 32590 Google Scholar
[12]	Zhang P, Yuan P, Jiang X, Zhai S, Zeng J, Xian Y, Qin H, Yang D 2018 Small 14 1702769 Google Scholar
[13]	Giri A, Hopkins P E 2020 Adv. Funct. Mater. 30 1903857 Google Scholar
[14]	Ren W, Ouyang Y, Jiang P, Yu C, He J, Chen J 2021 Nano Lett. 21 2634 Google Scholar
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[16]	Hopkins P E, Duda J C, Petz C W, Floro J A 2011 Phys. Rev. B 84 035438 Google Scholar
[17]	Hopkins P E, Duda J C, Clark S P, Hains C P, Rotter T J, Phinney L M, Balakrishnan G 2011 Appl. Phys. Lett. 98 161913 Google Scholar
[18]	Hopkins P E, Phinney L M, Serrano J R, Beechem T E 2010 Phys. Rev. B 82 085307 Google Scholar
[19]	Lee E, Zhang T, Yoo T, Guo Z, Luo T 2016 ACS Appl. Mater. Interfaces 8 35505 Google Scholar
[20]	Park W, Sood A, Park J, Asheghi M, Sinclair R, Goodson K E 2017 Nanoscale Microscale Thermophys. Eng. 21 134 Google Scholar
[21]	Liang Y, Zhang B, Liu Z, Liu W 2021 Int. J. Heat Mass Transfer 174 121306 Google Scholar
[22]	Xu Y, Kato R, Goto M 2010 J. Appl. Phys. 108 104317 Google Scholar
[23]	Nylander A, Hansson J, Nilsson T, Ye L, Fu Y, Liu J 2021 ACS Appl. Mater. Interfaces 13 30992 Google Scholar
[24]	Zhang Y, Ma D, Zang Y, Wang X, Yang N 2018 Front. Energy Res. 6 00048 Google Scholar
[25]	Singh P, Seong M, Sinha S 2013 Appl. Phys. Lett. 102 181906 Google Scholar
[26]	Hamaoui G, Horny N, Hua Z, Zhu T, Robillard J F, Fleming A, Ban H, Chirtoc M 2018 Sci. Rep. 8 11352 Google Scholar
[27]	Giri A, Gaskins J T, Donovan B F, Szwejkowski C, Warzoha R J, Rodriguez M A, Ihlefeld J, Hopkins P E 2015 J. Appl. Phys. 117 105105 Google Scholar
[28]	Chen J, Xu X, Zhou J, Li B 2022 Rev. Mod. Phys. 94 025002 Google Scholar
[29]	Swartz E T, Pohl R O 1989 Rev. Mod. Phys. 61 605 Google Scholar
[30]	Kazan M, Bruyant A, Royer P, Masri P 2010 Surf. Sci. Rep. 65 111 Google Scholar
[31]	De Bellis L, Phelan P E, Prasher R S 2000 J. Thermophys. Heat Transfer 14 144 Google Scholar
[32]	Ziman J M 1960 Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford: Oxford University Press)
[33]	Gale J D, Rohl A L 2003 Mol. Simul. 29 291 Google Scholar
[34]	Mei J, Davenport J W 1992 Phys. Rev. B 46 21 Google Scholar
[35]	Foiles S M, Baskes M I, Daw M S 1986 Phys. Rev. B 33 7983 Google Scholar
[36]	Johnson R A 1989 Phys. Rev. B 39 12554 Google Scholar
[37]	Stillinger F H, Weber T A 1985 Phys. Rev. B 31 5262 Google Scholar
[38]	Tersoff J 1988 Phys. Rev. B 38 9902 Google Scholar
[39]	Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169 Google Scholar
[40]	Togo A, Tanaka I 2015 Scr. Mater. 108 1 Google Scholar
[41]	Mulakaluri N, Persson B N J 2011 EPL 96 66003 Google Scholar
[42]	Farber D L, Krisch M, Antonangeli D, Beraud A, Badro J, Occelli F, Orlikowski D 2006 Phys. Rev. Lett. 96 115502 Google Scholar
[43]	Trampenau J, Petry W, Herzig C 1993 Phys. Rev. B 47 3132 Google Scholar
[44]	Dal Corso A 2013 J. Phys. Condens. Matter 25 145401 Google Scholar
[45]	Koh Y R, Shi J, Wang B, Hu R, Ahmad H, Kerdsongpanya S, Milosevic E, Doolittle W A, Gall D, Tian Z, Graham S, Hopkins P E 2020 Phys. Rev. B 102 205304 Google Scholar
[46]	Debernardi A, Alouani M, Dreyssé H 2001 Phys. Rev. B 63 064305 Google Scholar
[47]	Serrano J, Manjón F J, Romero A H, Ivanov A, Cardona M, Lauck R, Bosak A, Krisch M 2010 Phys. Rev. B 81 174304 Google Scholar
[48]	Ghosh K, Singisetti U 2016 Appl. Phys. Lett. 109 072102 Google Scholar
[49]	Fritsch J, Pavone P, Schroder U 1995 Phys. Rev. B 52 11326 Google Scholar
[50]	Qi R, Shi R, Li Y, Sun Y, Wu M, Li N, Du J, Liu K, Chen C, Chen J, Wang F, Yu D, Wang E G, Gao P 2021 Nature 599 399 Google Scholar
[51]	Cheng Z, Li R, Yan X, Jernigan G, Shi J, Liao M E, Hines N J, Gadre C A, Idrobo J C, Lee E, Hobart K D, Goorsky M S, Pan X, Luo T, Graham S 2021 Nat. Commun. 12 6901 Google Scholar

施引文献

图 1 金属的声子色散关系和DOS　(a) Al; (b) Cu; (c) Au

Fig. 1. Phonon dispersion relations and DOS of metals: (a) Al; (b) Cu; (c) Au.

下载: 全尺寸图片幻灯片

图 2 半导体的声子色散关系和DOS　(a) Si; (b) SiC; (c) GaAs; (d) GaN

Fig. 2. Phonon dispersion relations and DOS of semiconductors: (a) Si; (b) SiC; (c) GaAs; (d) GaN.

下载: 全尺寸图片幻灯片

图 3 AMM, DMM和MMM (粗糙度: 0.28, 1.38, 2.16 nm)三种模型计算界面声子透射率频谱对比　(a) Al/Si界面; (b) Al/SiC界面; (c) Al/GaAs界面; (d) Al/GaN界面

Fig. 3. Comparison of phonon transmittance spectra calculated by AMM, DMM and MMM (roughness: 0.28, 1.38, 2.16 nm): (a) Al/Si interface; (b) Al/SiC interface; (c) Al/GaAs interface; (d) Al/GaN interface.

下载: 全尺寸图片幻灯片

图 4 界面接触系数S和粗糙度之间的关系, 其中η_max为界面两侧材料的原子间经验势函数截止半径, 当界面粗糙度大于这个值时, 认为界面接触系数趋近于0, 对于Al/Si界面, 该值为4.7 nm

Fig. 4. Relationship between interface contact coefficient S and roughness, where η_max is the cutoff radius of the interatomic empirical potential function of the materials on both sides of the interface. When the roughness of the interface is greater than this value, the interface contact coefficient is considered to approach 0, which is 4.7 nm for Al/Si interface.

下载: 全尺寸图片幻灯片

图 5 AMM, DMM和MMM (粗糙度: 0.28, 1.38, 2.16 nm)预测界面热导随温度的变化　(a) Al/Si界面; (b) Al/SiC界面; (c) Al/GaAs界面; (d) Al/GaN界面; 实验值来源于Hopkins等^[16]的测量

Fig. 5. Curves of interfacial thermal conductance predicted by AMM, DMM and MMM (roughness: 0.28, 0.53, 1.38 nm) models as a function of temperature: (a) Al/Si interface; (b) Al/SiC interface; (c) Al/GaAs interface; (d) Al/GaN interface. The experimental values were obtained from measurements made by Hopkins et al.^[16]

下载: 全尺寸图片幻灯片

图 6 MMM模型预测300 K时的金属(Al, Cu, Au)和半导体(Si, SiC, GaAs, GaN)界面热导, 其中粗糙度取值为0.28, 1.38, 2.16 nm

Fig. 6. Interfacial thermal conductance of metal (Al, Cu, Au) and semiconductor (Si, SiC, GaAs, GaN) interfaces predicted by MMM model at 300 K, roughness values: 0.28 nm, 1.38 nm, 2.16 nm.

下载: 全尺寸图片幻灯片

[1]	Wei N, Zhou C, Li Z, Ou B, Zhao K, Yu P, Li S, Zhao J 2022 Mater. Today Commun. 30 103147 Google Scholar
[2]	Cui Y, Li M, Hu Y 2020 J. Mater. Chem. C 8 10568 Google Scholar
[3]	Deng S, Xiao C, Yuan J, Ma D, Li J, Yang N, He H 2019 Appl. Phys. Lett. 115 101603 Google Scholar
[4]	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
[5]	Peng X, Jiang P, Ouyang Y, Lu S, Ren W, Chen J 2021 Nanotechnology 33 035707 Google Scholar
[6]	Monachon C, Weber L, Dames C 2016 Annu. Rev. Mater. Res. 46 433 Google Scholar
[7]	Scott E A, Gaskins J T, King S W, Hopkins P E 2018 APL Mater. 6 058302 Google Scholar
[8]	Yang L, Wan X, Ma D, Jiang Y, Yang N 2021 Phys. Rev. B 103 155305 Google Scholar
[9]	Liu B, Khvesyuk V I, Barinov A A, Wang M 2022 Int. J. Mech. Sci. 218 106993 Google Scholar
[10]	Yang N, Luo T, Esfarjani K, Henry A, Tian Z, Shiomi J, Chalopin Y, Li B, Chen G 2015 J. Comput. Theor. Nanosci. 12 168 Google Scholar
[11]	Rustam S, Schram M, Lu Z, Chaka A M, Rosenthal W S, Pfaendtner J 2022 ACS Appl. Mater. Interfaces 14 32590 Google Scholar
[12]	Zhang P, Yuan P, Jiang X, Zhai S, Zeng J, Xian Y, Qin H, Yang D 2018 Small 14 1702769 Google Scholar
[13]	Giri A, Hopkins P E 2020 Adv. Funct. Mater. 30 1903857 Google Scholar
[14]	Ren W, Ouyang Y, Jiang P, Yu C, He J, Chen J 2021 Nano Lett. 21 2634 Google Scholar
[15]	Duda J C, Hopkins P E 2012 Appl. Phys. Lett. 100 111602 Google Scholar
[16]	Hopkins P E, Duda J C, Petz C W, Floro J A 2011 Phys. Rev. B 84 035438 Google Scholar
[17]	Hopkins P E, Duda J C, Clark S P, Hains C P, Rotter T J, Phinney L M, Balakrishnan G 2011 Appl. Phys. Lett. 98 161913 Google Scholar
[18]	Hopkins P E, Phinney L M, Serrano J R, Beechem T E 2010 Phys. Rev. B 82 085307 Google Scholar
[19]	Lee E, Zhang T, Yoo T, Guo Z, Luo T 2016 ACS Appl. Mater. Interfaces 8 35505 Google Scholar
[20]	Park W, Sood A, Park J, Asheghi M, Sinclair R, Goodson K E 2017 Nanoscale Microscale Thermophys. Eng. 21 134 Google Scholar
[21]	Liang Y, Zhang B, Liu Z, Liu W 2021 Int. J. Heat Mass Transfer 174 121306 Google Scholar
[22]	Xu Y, Kato R, Goto M 2010 J. Appl. Phys. 108 104317 Google Scholar
[23]	Nylander A, Hansson J, Nilsson T, Ye L, Fu Y, Liu J 2021 ACS Appl. Mater. Interfaces 13 30992 Google Scholar
[24]	Zhang Y, Ma D, Zang Y, Wang X, Yang N 2018 Front. Energy Res. 6 00048 Google Scholar
[25]	Singh P, Seong M, Sinha S 2013 Appl. Phys. Lett. 102 181906 Google Scholar
[26]	Hamaoui G, Horny N, Hua Z, Zhu T, Robillard J F, Fleming A, Ban H, Chirtoc M 2018 Sci. Rep. 8 11352 Google Scholar
[27]	Giri A, Gaskins J T, Donovan B F, Szwejkowski C, Warzoha R J, Rodriguez M A, Ihlefeld J, Hopkins P E 2015 J. Appl. Phys. 117 105105 Google Scholar
[28]	Chen J, Xu X, Zhou J, Li B 2022 Rev. Mod. Phys. 94 025002 Google Scholar
[29]	Swartz E T, Pohl R O 1989 Rev. Mod. Phys. 61 605 Google Scholar
[30]	Kazan M, Bruyant A, Royer P, Masri P 2010 Surf. Sci. Rep. 65 111 Google Scholar
[31]	De Bellis L, Phelan P E, Prasher R S 2000 J. Thermophys. Heat Transfer 14 144 Google Scholar
[32]	Ziman J M 1960 Electrons and Phonons: The Theory of Transport Phenomena in Solids (Oxford: Oxford University Press)
[33]	Gale J D, Rohl A L 2003 Mol. Simul. 29 291 Google Scholar
[34]	Mei J, Davenport J W 1992 Phys. Rev. B 46 21 Google Scholar
[35]	Foiles S M, Baskes M I, Daw M S 1986 Phys. Rev. B 33 7983 Google Scholar
[36]	Johnson R A 1989 Phys. Rev. B 39 12554 Google Scholar
[37]	Stillinger F H, Weber T A 1985 Phys. Rev. B 31 5262 Google Scholar
[38]	Tersoff J 1988 Phys. Rev. B 38 9902 Google Scholar
[39]	Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169 Google Scholar
[40]	Togo A, Tanaka I 2015 Scr. Mater. 108 1 Google Scholar
[41]	Mulakaluri N, Persson B N J 2011 EPL 96 66003 Google Scholar
[42]	Farber D L, Krisch M, Antonangeli D, Beraud A, Badro J, Occelli F, Orlikowski D 2006 Phys. Rev. Lett. 96 115502 Google Scholar
[43]	Trampenau J, Petry W, Herzig C 1993 Phys. Rev. B 47 3132 Google Scholar
[44]	Dal Corso A 2013 J. Phys. Condens. Matter 25 145401 Google Scholar
[45]	Koh Y R, Shi J, Wang B, Hu R, Ahmad H, Kerdsongpanya S, Milosevic E, Doolittle W A, Gall D, Tian Z, Graham S, Hopkins P E 2020 Phys. Rev. B 102 205304 Google Scholar
[46]	Debernardi A, Alouani M, Dreyssé H 2001 Phys. Rev. B 63 064305 Google Scholar
[47]	Serrano J, Manjón F J, Romero A H, Ivanov A, Cardona M, Lauck R, Bosak A, Krisch M 2010 Phys. Rev. B 81 174304 Google Scholar
[48]	Ghosh K, Singisetti U 2016 Appl. Phys. Lett. 109 072102 Google Scholar
[49]	Fritsch J, Pavone P, Schroder U 1995 Phys. Rev. B 52 11326 Google Scholar
[50]	Qi R, Shi R, Li Y, Sun Y, Wu M, Li N, Du J, Liu K, Chen C, Chen J, Wang F, Yu D, Wang E G, Gao P 2021 Nature 599 399 Google Scholar
[51]	Cheng Z, Li R, Yan X, Jernigan G, Shi J, Liao M E, Hines N J, Gadre C A, Idrobo J C, Lee E, Hobart K D, Goorsky M S, Pan X, Luo T, Graham S 2021 Nat. Commun. 12 6901 Google Scholar

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