Advertisement

Nonequilibrium Energy Transfer in Nanostructures

  • Zhuomin M. ZhangEmail author
Chapter
  • 163 Downloads
Part of the Mechanical Engineering Series book series (MES)

Abstract

This chapter begins with a description of the phenomenological theories in which the energy transport processes are represented by a single differential equation or a set of differential equations that can be solved with appropriate initial and boundary conditions. These equations are often called non-Fourier heat equations, which can be considered as extensions of the conventional heat diffusion equation based on Fourier’s law. The limitations of the phenomenological theories are discussed. While the BTE, Monte Carlo method, and MD simulations have been presented in previous chapters, this chapter stresses the application in solid nanostructures, including thermal boundary resistance (TBR) and multilayer structures. The equation of phonon radiative transfer (EPRT) is introduced and used to delineate the diffusive and ballistic heat conduction regimes in thin films. A heat conduction regime with respect to length and time scale is presented, followed by a summary of the contemporary methods for measuring thermal transport properties of solids, thin films, and nanostructures.

Keywords

Non-Fourier heat equation Nonequilibrium heat conduction Hyperbolic heat equation Two-temperature model Short laser pulse Dual-phase lag Electron–phonon coupling constant Equation of Phonon Radiative Transfer (EPRT) Boundary thermal resistance Acoustic mismatch Atomistic Green’s function Heat conduction regimes Thermal metrology Measurement techniques 

References

  1. 1.
    H.S. Carslaw, J.C. Jaeger, Conduction of Heat in Solids, 2nd edn. (Clarendon Press, Oxford, 1959)zbMATHGoogle Scholar
  2. 2.
    M.N. Özişik, Heat Conduction, 2nd ed., Wiley, New York, 1993; also D.W. Hahn and M.N. Özişik, Heat Conduction, 3rd ed., Wiley, New York, 2012Google Scholar
  3. 3.
    T.J. Bright, Z.M. Zhang, Common misperceptions of the hyperbolic heat equation. J. Thermophys. Heat Transfer 23, 601–607 (2009)Google Scholar
  4. 4.
    D. D. Joseph, L. Preziosi, Heat waves. Rev. Mod. Phys., 61, 41–73 (1989)Google Scholar
  5. 5.
    D.D. Joseph, L. Preziosi, Addendum to the paper ‘heat waves’. Rev. Mod. Phys. 62, 375–391 (1990)Google Scholar
  6. 6.
    D.Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior, 2nd edn. (Wiley, New York, 2015)Google Scholar
  7. 7.
    M.N. Özişik, D.Y. Tzou, On the wave theory in heat conduction. J. Heat Transfer 116, 526–535 (1994)Google Scholar
  8. 8.
    W.K. Yeung, T.T. Lam, A numerical scheme for non-Fourier heat conduction, Part I: one-dimensional problem formulation and applications. Numer. Heat Transfer B 33, 215–233 (1998)Google Scholar
  9. 9.
    A. Haji-Sheikh, W.J. Minkowycz, E.M. Sparrow, Certain anomalies in the analysis of hyperbolic heat conduction. J. Heat Transfer 124, 307–319 (2002)Google Scholar
  10. 10.
    J. Gembarovic, J. Gembarovic Jr., Non-Fourier heat conduction modeling in a finite medium. Int. J. Thermophys. 25, 1261–1268 (2004)zbMATHGoogle Scholar
  11. 11.
    C.A. Bennett, R.R. Patty, Thermal wave interferometry: a potential application of the photoacoustic effect. Appl. Opt. 21, 49–54 (1982)Google Scholar
  12. 12.
    A. Mandelis (ed.), Photoacoustic and Thermal Wave Phenomena in Semiconductors (Elsevier, Amsterdam, 1987)Google Scholar
  13. 13.
    M.B. Rubin, Hyperbolic heat conduction and the second law. Int. J. Eng. Sci. 30, 1665–1676 (1992)MathSciNetzbMATHGoogle Scholar
  14. 14.
    C. Bai, A.S. Lavine, On hyperbolic heat conduction and the second law of thermodynamics. J. Heat Transfer 117, 256–263 (1995)Google Scholar
  15. 15.
    A. Barletta, E. Zanchini, Hyperbolic heat conduction and local equilibrium: a second law analysis. Int. J. Heat Mass Transfer 40, 1007–1016 (1997)zbMATHGoogle Scholar
  16. 16.
    D. Jou, G. Lebon, J. Casas-Vázquez, Extended Irreversible Thermodynamics, 4th edn. (Springer, Berlin, 2010)zbMATHGoogle Scholar
  17. 17.
    Z.M. Zhang, T.J. Bright, G.P. Peterson, Reexamination of the statistical derivations of Fourier’s law and Cattaneo’s equation. Nanoscale Microscale Thermophys. Eng. 15, 220–228 (2011)Google Scholar
  18. 18.
    J. Tavernier, Sur l’équation de conduction de la chaleur. Comptes Rendus Acad. Sci. 254, 69–71 (1962)MathSciNetGoogle Scholar
  19. 19.
    A. Majumdar, Microscale heat conduction in dielectric thin films. J. Heat Transfer 115, 7–16 (1993)Google Scholar
  20. 20.
    A.A. Joshi, A. Majumdar, Transient ballistic and diffusive phonon heat transport in thin films. J. Appl. Phys. 74, 31–39 (1993)Google Scholar
  21. 21.
    S. Volz, J.-B. Saulnier, M. Lallemand, B. Perrin, P. Depondt, M. Mareschal, Transient Fourier-law deviation by molecular dynamics in solid argon. Phys. Rev. B 54, 340–347 (1996)Google Scholar
  22. 22.
    J. Xu, X.W. Wang, Simulation of ballistic and non-Fourier thermal transport in ultra-fast laser heating. Phys. B 351, 213–226 (2004)Google Scholar
  23. 23.
    M. Chester, Second sound in solids. Phys. Rev. 131, 2013–2015 (1963)Google Scholar
  24. 24.
    M.E. Gurtin, A.C. Pipkin, A general theory of heat conduction with finite wave speeds. Arch. Ration. Mech. Anal. 31, 113–126 (1968)MathSciNetzbMATHGoogle Scholar
  25. 25.
    P.J. Antaki, Solution for non-Fourier dual phase lag heat conduction in a semi-infinite slab with surface heat flux. Int. J. Heat Mass Transfer 41, 2253–2258 (1998)zbMATHGoogle Scholar
  26. 26.
    D.W. Tang, N. Araki, Wavy, wavelike, diffusive thermal responses of finite rigid slabs to high-speed heating of laser-pulses. Int. J. Heat Mass Transfer 42, 855–860 (1999)zbMATHGoogle Scholar
  27. 27.
    D.Y. Tzou, K.S. Chiu, Temperature-dependent thermal lagging in ultrafast laser heating. Int. J. Heat Mass Transfer 44, 1725–1734 (2001)zbMATHGoogle Scholar
  28. 28.
    L.Q. Wang, X.S. Zhou, X.H. Wei, Heat Conduction: Mathematical Models and Analytical Solutions (Springer-Verlag, Berlin, 2008)zbMATHGoogle Scholar
  29. 29.
    W.J. Minkowycz, A. Haji-Sheikh, K. Vafai, On departure from local thermal equilibrium in porous media due to a rapid changing heat source: the Sparrow number. Int. J. Heat Mass Transfer 42, 3373–3385 (1999)zbMATHGoogle Scholar
  30. 30.
    W. Kaminski, Hyperbolic heat conduction equation for materials with a nonhomogeneous inner structure. J. Heat Transfer 112, 555–560 (1990)Google Scholar
  31. 31.
    J. Callaway, Model for lattice thermal conductivity at low temperatures. Phys. Rev. 113, 1046–1951 (1959)zbMATHGoogle Scholar
  32. 32.
    R. A. Guyer, J. A. Krumhansl, Solution of the linearized phonon Boltzmann equation. Phys. Rev. 148, 766–778 (1966); Thermal conductivity, second sound, and phonon hydrodynamic phenomena in nonmetallic crystals. Phys. Rev. 148, 778–788 (1966)Google Scholar
  33. 33.
    J. Shiomi, S. Maruyama, Non-Fourier heat conduction in a single-walled carbon nanotube: Classical molecular dynamics simulations. Phys. Rev. B 73, 205420 (2006)Google Scholar
  34. 34.
    D.H. Tsai, R.A. MacDonald, Molecular-dynamics study of second sound in a solid excited by a strong heat pulse. Phys. Rev. B 14, 4714–4723 (1976)Google Scholar
  35. 35.
    X.W. Wang, X. Xu, Thermoelastic wave induced by pulsed laser heating. Appl. Phys. A 73, 107–114 (2001)Google Scholar
  36. 36.
    X.W. Wang, Thermal and thermomechanical phenomena in picosecond laser copper interaction. J. Heat Transfer 126, 355–364 (2004)Google Scholar
  37. 37.
    S.I. Anisimov, B.L. Kapeliovich, T.L. Perel’man, Electron emission from metal surfaces exposed to ultrashort laser pulses. Sov. Phys. JETP 39, 375–377 (1974)Google Scholar
  38. 38.
    J.G. Fujimoto, J.M. Liu, E.P. Ippen, N. Bloembergen, Femtosecond laser interaction with metallic tungsten and nonequilibrium electron and lattice temperatures. Phys. Rev. Lett. 53, 1837–1840 (1984)Google Scholar
  39. 39.
    S.D. Brorson, J.G. Fujimoto, E.P. Ippen, Femtosecond electronic heat-transport dynamics in thin gold films. Phys. Rev. Lett. 59, 1962–1965 (1987)Google Scholar
  40. 40.
    T.Q. Qiu, C.L. Tien, Short-pulse laser heating on metals. Int. J. Heat Mass Transfer 35, 719–726 (1992)Google Scholar
  41. 41.
    T.Q. Qiu, C.L. Tien, Size effect on nonequilibrium laser heating of metal films. J. Heat Transfer 115, 842–847 (1993)Google Scholar
  42. 42.
    T.Q. Qiu, T. Juhasz, C. Suarez, W.E. Bron, C.L. Tien, Femtosecond laser heating of multi-layer metals—II. Experiments. Int. J. Heat Mass Transfer 37, 2799–2808 (1994)Google Scholar
  43. 43.
    J.L. Hostetler, A.N. Smith, D.M. Czajkowsky, P.M. Norris, Measurement of the electron-phonon coupling factor dependence on film thickness and grain size in Au, Cr, and Al. Appl. Opt. 38, 3614–3620 (1999)Google Scholar
  44. 44.
    S. Link, C. Burda, Z.L. Wang, M.A. El-Sayed, Electron dynamics in gold and gold-silver alloy nanoparticles: The influence of a nonequilibrium electron distribution and the size dependence of the electron-phonon relaxation. J. Chem. Phys. 111, 1255–1264 (1999)Google Scholar
  45. 45.
    A.N. Smith, P.M. Norris, Influence of intraband transition on the electron thermoreflectance response of metals. Appl. Phys. Lett. 78, 1240–1242 (2001)Google Scholar
  46. 46.
    R.J. Stevens, A.N. Smith, P.M. Norris, Measurement of thermal boundary conductance of a series of metal-dielectric interfaces by the transient thermoreflectance techniques. J. Heat Transfer 127, 315–322 (2005)Google Scholar
  47. 47.
    D.G. Cahill, K.E. Goodson, A. Majumdar, Thermometry and thermal transport in micro/nanoscale solid-state devices and structures. J. Heat Transfer 124, 223–241 (2002)Google Scholar
  48. 48.
    D.G. Cahill, W.K. Ford, K.E. Goodson et al., Nanoscale thermal transport. J. Appl. Phys. 93, 793–818 (2003)Google Scholar
  49. 49.
    J. Zhu, D.W. Tang, W. Wang, J. Liu, K.W. Holub, R. Yang, Ultrafast thermoreflectance techniques for measuring thermal conductivity and interface thermal conductance of thin films. J. Appl. Phys. 108, 094315 (2010)Google Scholar
  50. 50.
    D.M. Riffe, X.Y. Wang, M.C. Downer et al., Femtosecond thermionic emission from metals in the space-charge-limited regime. J. Opt. Soc. Am. B 10, 1424–1435 (1993)Google Scholar
  51. 51.
    A.N. Smith, J.L. Hostetler, P.M. Norris, Nonequilibrium heating in metal films: An analytical and numerical analysis. Numer. Heat Transfer A 35, 859–874 (1999)Google Scholar
  52. 52.
    M. Li, S. Menon, J.P. Nibarger, G.N. Gibson, Ultrafast electron dynamics in femtosecond optical breakdown of dielectrics. Phys. Rev. Lett. 82, 2394–2397 (1999)Google Scholar
  53. 53.
    L. Jiang, H.-L. Tsai, Energy transport and nanostructuring of dielectrics by femtosecond laser pulse trains. J. Heat Transfer 128, 926–933 (2006)Google Scholar
  54. 54.
    L. Jiang, H.-L. Tsai, Plasma modeling for ultrashort pulse laser ablation of dielectrics. J. Appl. Phys. 100, 023116 (2006)Google Scholar
  55. 55.
    Y. Ma, A two-parameter nondiffusive heat conduction model for data analysis in pump-probe experiments. J. Appl. Phys. 116, 243505 (2014); ibid, Hotspot size-dependent thermal boundary conductance in nondiffusive heat conduction. J. Heat Transfer 137, 082401 (2015)Google Scholar
  56. 56.
    G. Chen, Ballistic-diffusion heat-conduction equations. Phys. Rev. Lett. 86, 2297–2300 (2001); ibid, Ballistic-diffusive equations for transient heat conduction from nano to macroscales. J. Heat Transfer 124, 320–328 (2002)Google Scholar
  57. 57.
    T. Klitsner, J.E. VanCleve, H.E. Fischer, R.O. Pohl, Phonon radiative heat transfer and surface scattering. Phys. Rev. B 38, 7576–7594 (1988)Google Scholar
  58. 58.
    R.B. Peterson, Direct simulation of phonon-mediated heat transfer in a Debye crystal. J. Heat Transfer 116, 815–822 (1994)Google Scholar
  59. 59.
    E.T. Swartz, P.O. Pohl, Thermal boundary resistance. Rev. Mod. Phys. 61, 605–668 (1989)Google Scholar
  60. 60.
    W.A. Little, The transport of heat between dissimilar solids at low temperatures. Can. J. Phys. 37, 334–349 (1959)Google Scholar
  61. 61.
    G. Chen and C.L. Tien, “Thermal conductivity of quantum well structures,” J. Thermophys. Heat Transfer, 7, 311–318, 1993Google Scholar
  62. 62.
    G. Chen, Size and interface effects on thermal conductivity of superlattices and periodic thin-film structures. J. Heat Transfer 119, 220–229 (1997)Google Scholar
  63. 63.
    G. Chen, Thermal conductivity and ballistic-phonon transport in the cross-plane direction of superlattices. Phys. Rev. B 57, 14958–14973 (1998)Google Scholar
  64. 64.
    G. Chen, T. Zeng, Nonequilibrium phonon and electron transport in heterostructures and superlattices. Microscale Thermophys. Eng. 5, 71–88 (2001)Google Scholar
  65. 65.
    T. Zeng, G. Chen, Phonon heat conduction in thin films: impacts of thermal boundary resistance and internal heat generation. J. Heat Transfer 123, 340–347 (2001)Google Scholar
  66. 66.
    S. Sinha, K.E. Goodson, Review: multiscale thermal modeling in nanoelectronics. Int. J. Multiscale Comp. Eng. 3, 107–133 (2005)Google Scholar
  67. 67.
    R.A. Escobar, S.S. Ghai, M.S. Jhon, C.H. Amon, Multi-length and time scale thermal transport using the lattice Boltzmann method with application to electronics cooling. Int. J. Heat Mass Transfer 49, 97–107 (2006)zbMATHGoogle Scholar
  68. 68.
    E.M. Sparrow, R.D. Cess, Radiation Heat Transfer, Augmented edn. (McGraw-Hill, New York, 1978)Google Scholar
  69. 69.
    M.F. Modest, Radiative Heat Transfer, 3rd edn. (Academic Press, New York, 2013)Google Scholar
  70. 70.
    T.J. Bright, Z.M. Zhang, Entropy generation in thin films evaluated from phonon radiative transport. J. Heat Transfer 132, 101301 (2010)Google Scholar
  71. 71.
    H.B.G. Casimir, Note on the conduction of heat in crystal. Physica 5, 495–500 (1938)Google Scholar
  72. 72.
    R.G. Deissler, Diffusion approximation for thermal radiation in gasses with jump boundary condition. J. Heat Transfer 86, 240–245 (1964)Google Scholar
  73. 73.
    A. Malhotra, K. Kothari, M. Maldovan, Cross-plane thermal conduction in superlattices: Impact of multiple length scales on phonon transport. J. Appl. Phys. 125, 044304 (2019)Google Scholar
  74. 74.
    K. Kothari, A. Malhotra, M. Maldovan, Cross-plane heat conduction in III–V semiconductor superlattices. J. Phys. Condens. Matter 31, 345301 (2019)Google Scholar
  75. 75.
    M.M. Yovanovich, Four decades of research on thermal contact, gap, and joint resistance in microelectronics. IEEE Trans. Compon. Packag. Technol. 28, 182–206 (2005)Google Scholar
  76. 76.
    R.J. Stoner, H.J. Maris, Kapitza conductance and heat flow between solids at temperatures from 50 to 300 K. Phys. Rev. B 48, 16373–16387 (1993)Google Scholar
  77. 77.
    R.S. Prasher and P.E. Phelan, “Review of thermal boundary resistance of high-temperature superconductors,” J. Supercond., 10, 473–484, 1997Google Scholar
  78. 78.
    P.E. Phelan, Application of diffuse mismatch theory to the prediction of thermal boundary resistance in thin-film high-Tc superconductors. J. Heat Transfer 120, 37–43 (1998)Google Scholar
  79. 79.
    L. De Bellis, P.E. Phelan, R.S. Prasher, Variations of acoustic and diffuse mismatch models in predicting thermal-boundary resistance. J. Thermophys. Heat Transfer 14, 144–150 (2000)Google Scholar
  80. 80.
    A. Majumdar, Effect of interfacial roughness on phonon radiative heat conduction. J. Heat Transfer 113, 797–805 (1991)Google Scholar
  81. 81.
    A. Majumdar, P. Reddy, Role of electron–phonon coupling in thermal conductance of metal–nonmetal interfaces. Appl. Phys. Lett. 84, 4768–4770 (2004)Google Scholar
  82. 82.
    A. Giri, P.E. Hopkins, A review of experimental and computational advances in thermal boundary conductance and nanoscale thermal transport across solid interfaces. Adv. Func. Mater. 2019, 1903857 (2019)Google Scholar
  83. 83.
    S. Mazumdar, A. Majumdar, Monte Carlo study of phonon transport in solid thin films including dispersion and polarization. J. Heat Transfer 123, 749–759 (2001)Google Scholar
  84. 84.
    Q. Hao, G. Chen, M.-S. Jeng, Frequency-dependent Monte Carlo simulations of phonon transport in two-dimensional porous silicon with aligned pores. J. Appl. Phys. 106, 114321 (2009)Google Scholar
  85. 85.
    J.-P.M. Péraud, C.D. Landon, N.G. Hadjiconstantinou, Monte Carlo methods for solving the Boltzmann transport equation. Annu. Rev. Heat Transfer 17, 205–265 (2014)Google Scholar
  86. 86.
    A. Nabovati, D.P. Sellan, C.H. Amon, On the lattice Boltzmann method for phonon transport. J. Comput. Phys. 230, 5864–5876 (2011)MathSciNetzbMATHGoogle Scholar
  87. 87.
    S.R. Phillpot, P.K. Schelling, P. Keblinski, Phonon wave-packet dynamics at semiconductor interfaces by molecular-dynamics simulation. Appl. Phys. Lett. 80, 2484–2486 (2002); ibid, Interfacial thermal conductivity: Insights from atomic level simulation. J. Mater. Sci. 40, 3143–3148 (2005)Google Scholar
  88. 88.
    C.-J. Twu, J.-R. Ho, Molecular-dynamics study of energy flow and the Kapitza conductance across an interface with imperfection formed by two dielectric thin films. Phys. Rev. B 67, 205422 (2003)Google Scholar
  89. 89.
    H. Zhong, J.R. Lukes, Interfacial thermal resistance between carbon nanotubes: Molecular dynamics simulations and analytical thermal modeling. Phys. Rev. B 74, 125403 (2006)Google Scholar
  90. 90.
    R.J. Stevens, L.V. Zhigilei, P.M. Norris, Effects of temperature and disorder on thermal boundary conductance at solid-solid interfaces: non-equilibrium molecular dynamics simulations. Int. J. Heat Mass Transfer 50, 3977–3989 (2007)zbMATHGoogle Scholar
  91. 91.
    E.S. Landry, A.J.H. McGaughey, Thermal boundary resistance predictions from molecular dynamics simulations and theoretical calculations. Phys. Rev. B 80, 165304 (2009)Google Scholar
  92. 92.
    Y. Chalopin, K. Esfarjani, A. Henry, S. Volz, G. Chen, Thermal interface conductance in Si/Ge superlattices by equilibrium molecular dynamics. Phys. Rev. B 85, 195302 (2012)Google Scholar
  93. 93.
    S. Merabia, K. Termentzidis, Thermal conductance at the interface between crystals using equilibrium and nonequilibrium molecular dynamics. Phys. Rev. B 86, 094303 (2012)Google Scholar
  94. 94.
    Z. Liang, M. Hu, Tutorial: Determination of thermal boundary resistance by molecular dynamics simulations. J. Appl. Phys. 123, 191101 (2018)Google Scholar
  95. 95.
    F. VanGessel, J. Peng, P.W. Chung, A review of computational phononics: the bulk, interfaces, and surfaces. J. Mater. Sci. 53, 5641–5683 (2018)Google Scholar
  96. 96.
    S. Datta, Nanoscale device modeling: the Green’s function method. Superlattices Microstruct. 28, 253–278 (2000)Google Scholar
  97. 97.
    N. Mingo, L. Yang, Phonon transport in nanowires coated with an amorphous material: an atomistic Green’s function approach. Phys. Rev. B 68, 245406 (2003)Google Scholar
  98. 98.
    W. Zhang, T.S. Fisher, N. Mingo, Simulation of interfacial phonon transport in Si–Ge heterostructures using an atomistic Green’s function method. J. Heat Transfer 129, 483–491 (2007); ibid, The atomistic Green’s function method: an efficient simulation approach for nanoscale phonon transport. Numerical Heat Transfer B 51, 333–349 (2007)Google Scholar
  99. 99.
    S. Sadasivam, Y. Che, Z. Huang, L. Chen, S. Kumar, T.S. Fisher, The atomistic Green’s function method for interfacial phonon transport. Annu. Rev. Heat Transfer 17, 89–145 (2014)Google Scholar
  100. 100.
    A. Ozpineci, S. Ciraci, Quantum effects of thermal conductance through atomic chains. Phys. Rev. B 63, 125415 (2001)Google Scholar
  101. 101.
    Z.-Y. Ong, G. Zhang, Efficient approach for modeling phonon transmission probability in nanoscale interfacial thermal transport. Phys. Rev. B 91, 174302 (2015)Google Scholar
  102. 102.
    L. Yang, B. Latour, A.J. Minnich, Phonon transmission at crystalline-amorphous interfaces studied using mode-resolved atomistic Green’s functions. Phys. Rev. B 97, 205306 (2018)Google Scholar
  103. 103.
    D.A. Young, H.J. Maris, Lattice-dynamical calculation of the Kapitza resistance between fcc lattices. Phys. Rev. B 40, 3685–3693 (1989)Google Scholar
  104. 104.
    H. Zhao, J.B. Freund, Lattice-dynamical calculation of phonon scattering at ideal Si–Ge interfaces. J. Appl. Phys. 97, 024903 (2005)Google Scholar
  105. 105.
    S. Sadasivam, N. Ye, J.P. Feser, J. Charles, K. Miao, T. Kubis, T.S. Fisher, Thermal transport across metal silicide-silicon interfaces: First-principles calculations and Green’s function transport simulations. Phys. Rev. B 95, 085310 (2017)Google Scholar
  106. 106.
    Z. Tian, K. Esfarjani, G. Chen, Green’s function studies of phonon transport across Si/Ge superlattices. Phys. Rev. B 89, 235307 (2014)Google Scholar
  107. 107.
    Z. Yan, L. Chen, M. Yoon, S. Kumar, Phonon transport at the interfaces of vertically stacked graphene and hexagonal boron nitride heterostructures. Nanoscale 8, 4037 (2016)Google Scholar
  108. 108.
    J. Lai, A. Majumdar, Concurrent thermal and electrical modeling of sub-micrometer silicon devices. J. Appl. Phys. 79, 7353–7361 (1996)Google Scholar
  109. 109.
    P.G. Sverdrup, Y.S. Ju, K.E. Goodson, Sub-continuum simulation of heat conduction in silicon-on-insulator transistors. J. Heat Transfer 123, 130–137 (2001)Google Scholar
  110. 110.
    S. Sinha, E. Pop, R.W. Dutton, K.E. Goodson, Non-equilibrium phonon distribution in sub-100 nm silicon transistors. J. Heat Transfer 128, 638–647 (2006)Google Scholar
  111. 111.
    C.D.S. Brites, P.P. Lima, N.J.O. Silva, A. Millán, V.S. Amaral, F. Palacio, L.D. Carlos, Thermometry at the nanoscale. Nanoscale 4, 4799–4829 (2012)Google Scholar
  112. 112.
    X.W. Wang, Experimental Micro/Nanoscale Thermal Transport (Wiley, New York, 2012)Google Scholar
  113. 113.
    A.J. McNamara, Y. Joshi, Z.M. Zhang, Characterization of nanostructured thermal interface materials—a review. Int. J. Thermal Sci. 62, 2–11 (2012)Google Scholar
  114. 114.
    G. Chen, Probing nanoscale heat transfer phenomena. Annu. Rev. Heat Transfer 16, 1–8 (2013)Google Scholar
  115. 115.
    D. Zhao, X. Qian, X. Gu, S.A. Jajja, R. Yang, Measurement techniques for thermal conductivity and interfacial thermal conductance of bulk and thin film materials. J. Electron. Package 138, 040802 (2016)Google Scholar
  116. 116.
    Z.M. Zhang, Surface temperature measurement using optical techniques. Annu. Rev. Heat Transfer 11, 351–411 (2000)Google Scholar
  117. 117.
    B. Abad, D.-A. Borca-Tasciuc, M.S. Martin-Gonzalez, Non-contact methods for thermal properties measurement. Renew. Sustain. Energy Rev. 76, 1348–1370 (2017)Google Scholar
  118. 118.
    A.C. Jones, B.T. O’Callahan, H.U. Yang, M.B. Raschke, The thermal near-field: coherence, spectroscopy, heat-transfer, and optical forces. Prog. Sur. Sci. 88, 349–392 (2013)Google Scholar
  119. 119.
    K.E. Goodson, Y.S. Ju, Heat conduction in novel electronic films. Annu. Rev. Mater. Sci. 29, 261–293 (1999)Google Scholar
  120. 120.
    K. Park, G.L.W. Cross, Z.M. Zhang, W.P. King, Experimental investigation on the heat transfer between a heated microcantilever and a substrate. J. Heat Transfer 130, 102401 (2008)Google Scholar
  121. 121.
    D. G. Cahill and R. O. Pohl, “Thermal conductivity of amorphous solids above the plateau,” Phys. Rev. B, 35, 4067–4073, 1987Google Scholar
  122. 122.
    D.G. Cahill, H.E. Fischer, T. Klitsner, E.T. Swartz, R.O. Pohl, Thermal conductivity of thin films: measurements and understanding. J. Vac. Sci. Technol. A 7, 1259–1266 (1989)Google Scholar
  123. 123.
    D.G. Cahill, Thermal conductivity measurement from 30 K to 750 K: the 3-omega method. Rev. Sci. Instrum. 61, 802–808 (1990)Google Scholar
  124. 124.
    C. Dames, Measuring the thermal conductivity of thin films: 3 omega and related electrothermal methods. Annu. Rev. Heat Transfer 16, 7–49 (2013)Google Scholar
  125. 125.
    S. Kommandur, S.K. Yee, A suspended 3-omega technique to measure the anisotropic thermal conductivity of semiconducting polymers. Rev. Sci. Instrum. 89, 114905 (2018)Google Scholar
  126. 126.
    L. Shi, D. Li, C. Yu, W. Jang, D. Kim, Z. Yao, P. Kim, A. Majumdar, Measuring thermal and thermoelectric properties of one-dimensional nanostructures using a microfabricated device. J. Heat Transfer 125, 881–888 (2003)Google Scholar
  127. 127.
    P. Kim, L. Shi, A. Majumdar, P.L. McEuen, Thermal transport measurements of individual multiwalled nanotubes. Phys. Rev. Lett. 87, 215502 (2001)Google Scholar
  128. 128.
    C. Yu, L. Shi, Z. Yao, D. Li, A. Majumdar, Thermal conductance and thermopower of an individual single-wall carbon nanotube. Nano Lett. 5, 1842–1846 (2005)Google Scholar
  129. 129.
    A. Mavrokefalos, M.T. Pettes, F. Zhou, L. Shi, Four-probe measurements of the in-plane thermoelectric properties of nanofilms. Rev. Sci. Instrum. 78, 034901 (2007)Google Scholar
  130. 130.
    A. Weathers, L. Shi, Thermal transport measurement techniques for nanowires and nanotubes. Annu. Rev. Heat Transfer 16, 101–134 (2013)Google Scholar
  131. 131.
    M. Fujii, X. Zhang, H. Xie, H. Ago, K. Takahashi, T. Ikuta, H. Abe, T. Shimizu, Measuring the thermal conductivity of a single carbon nanotube. Phys. Rev. Lett. 95, 065502 (2005)Google Scholar
  132. 132.
    J. Kim, E. Ou, D.P. Sellan, L. Shi, A four-probe thermal transport measurement method for nanostructures. Rev. Sci. Instrum. 86, 044901 (2015)Google Scholar
  133. 133.
    J. Kim, D.A. Evans, D.P. Sellan, O.M. Williams, E. Ou, A.H. Cowley, L. Shi, Thermal and thermoelectric transport measurements of an individual boron arsenide microstructure. Appl. Phys. Lett. 108, 201905 (2016)Google Scholar
  134. 134.
    A. Majumdar, Scanning thermal microscopy. Annu. Rev. Mater. Sci. 29, 505–585 (1999)Google Scholar
  135. 135.
    A. Majumdar, J. P. Carrejo, J. Lai, Thermal imaging using the atomic force microscope. Appl. Phys. Lett. 62, 2501–2503 (1993)Google Scholar
  136. 136.
    A. Majumdar, J. Lai, M. Chandrachood, O. Nakabeppu, Y. Wu, J. Shi, Thermal imaging by atomic force microscopy using thermocouple cantilever probes. Rev. Sci. Instrum. 66, 3584–3592 (1995)Google Scholar
  137. 137.
    C.C. Williams, H.K. Wickramasinghe, Scanning thermal profiler. Appl. Phys. Lett. 49, 1587–1589 (1986)Google Scholar
  138. 138.
    A. Majumdar, J. Varesi, Nanoscale temperature distribution measured by scanning Joule expansion microscopy. J. Heat Transfer 120, 297–305 (1998)Google Scholar
  139. 139.
    S.P. Gurrum, W.P. King, Y.K. Joshi, K. Ramakrishna, Size effect on the thermal conductivity of thin metallic films investigated by scanning Joule expansion microscopy. J. Heat Transfer 130, 082403 (2008)Google Scholar
  140. 140.
    K.L. Grosse, M.-H. Bae, F. Lian, E. Pop, W.P. King, Nanoscale Joule heating, Peltier cooling and current crowding at graphene-metal contacts. Nat. Nanotech. 6, 287–290 (2011)Google Scholar
  141. 141.
    T. Borca-Tasciuc, Scanning probe methods for thermal and thermoelectric property measurements. Annu. Rev. Heat Transfer 16, 211–258 (2013)Google Scholar
  142. 142.
    S. Gomès, A. Assy, P.-O. Chapuis, Scanning thermal microscopy: a review. Phys. Status Solidi A 212, 477–494 (2015)Google Scholar
  143. 143.
    K. Kim, J. Chung, G. Hwang, O. Kwon, J.S. Lee, Quantitative measurement with scanning thermal microscope by preventing the distortion due to the heat transfer through the air. ACS Nano 11, 8700–8709 (2011)Google Scholar
  144. 144.
    H.F. Hamann, Y.C. Martin, H.K. Wickramasinghe, Thermally assisted recording beyond traditional limits. Appl. Phys. Lett. 84, 810–812 (2004)Google Scholar
  145. 145.
    W.P. King, T.W. Kenny, K.E. Goodson et al., Atomic force microscope cantilevers for combined thermomechanical data writing and reading. Appl. Phys. Lett. 78, 1300–1302 (2001)Google Scholar
  146. 146.
    J. Lee, T. Beechem, T. L. Wright, B. A. Nelson, S. Graham, W. P. King, Electrical, thermal, and, mechanical characterization of silicon microcantilever heaters. J. Microelectromech. Syst. 15, 1644 (2007)Google Scholar
  147. 147.
    J. Lee, T.L. Wright, M.R. Abel et al., Thermal conduction from microcantilever heaters in partial vacuum. J. Appl. Phys. 101, 014906 (2007)Google Scholar
  148. 148.
    K. Park, J. Lee, Z.M. Zhang, W.P. King, Frequency-dependent electrical and thermal response of heated atomic force microscope cantilevers. J. Microelectromech. Syst. 16, 213–222 (2007)Google Scholar
  149. 149.
    K. Park, A. Marchenkov, Z.M. Zhang, W.P. King, Low temperature characterization of heated microcantilevers. J. Appl. Phys. 101, 094504 (2007)Google Scholar
  150. 150.
    W.P. King, B. Bhatia, J.R. Felts, H.J. Kim, B. Kwon, B. Lee, S. Somnath, M. Rosenberger, Heated atomic force microscope cantilevers and their applications. Annu. Rev. Heat Transfer 16, 287–326 (2013)Google Scholar
  151. 151.
    A.J. Schmidt, X. Chen, G. Chen, Pulse accumulation, radial heat conduction, and anisotropic thermal conductivity in pump-probe transient thermoreflectance. Rev. Sci. Instrum. 79, 114802 (2008)Google Scholar
  152. 152.
    A.J. Minnich, Measuring phonon mean free paths using thermal conductivity spectroscopy. Annu. Rev. Heat Transfer 16, 183–210 (2013)Google Scholar
  153. 153.
    J. Zhu, H. Park, J.-Y. Chen et al., Revealing the origins of 3D anisotropic thermal conductivities of black phosphorus. Adv. Electron. Mater. 2, 1600040 (2016)Google Scholar
  154. 154.
    P. Jiang, X. Qian, R. Yang, Tutorial: time-domain thermoreflectance (TDTR) for thermal property characterization of bulk and thin film materials. J. Appl. Phys. 124, 161103 (2018)Google Scholar
  155. 155.
    Z. Cheng, T. Bougher, T. Bai et al., Probing growth-induced anisotropic thermal transport in high-quality CVD diamond membranes by multifrequency and multiple-spot-size time-domain thermoreflectance. ACS Appl. Mater. Interfaces. 10, 4808–4815 (2018)Google Scholar
  156. 156.
    S. Huxtable, D.G. Cahill, V. Fauconnier, J.O. White, J.-C. Zhao, Thermal conductivity imaging at micrometrescale resolution for combinatorial studies of materials. Nat. Mater. 3, 298–301 (2004)Google Scholar
  157. 157.
    D.G. Cahill, Analysis of heat flow in layered structures for time-domain thermoreflectance. Rev. Sci. Instrum. 75, 5119–5122 (2004)Google Scholar
  158. 158.
    J. Jeong, X. Meng, A. K. Rockwell et al., Picosecond transient thermoreflectance for thermal conductivity characterization. Nanoscale Microscale Thermophys. Eng. 23, 211−221 (2019)Google Scholar
  159. 159.
    P.M. Norris, A.P. Caffrey, R.J. Stevens, J.M. Klopf, J.T. McLeskey, A.N. Smith, Femtosecond pump–probe nondestructive examination of materials. Rev. Sci. Instrum. 74, 400–406 (2003)Google Scholar
  160. 160.
    K.E. Goodson, M. Asheghi, Near-field optical thermometry. Microscale Thermophys. Eng. 1, 225–235 (1997)Google Scholar
  161. 161.
    D. Seto, R. Nikka, S. Nishio, Y. Taguchi, T. Saiki, Y. Nagasaka, Nanoscale optical thermometry using a time-correlated single-photon counting in an illumination-collection mode. Appl. Phys. Lett. 110, 033109 (2017)Google Scholar
  162. 162.
    M.E. Siemens, Q. Li, R. Yang, K.A. Nelson, E.H. Anderson, M.M. Murnane, H.C. Kapteyn, Quasi-ballistic thermal transport from nanoscale interfaces observed using ultrafast coherent soft X-ray beams. Nat. Mater. 9, 26–30 (2010)Google Scholar
  163. 163.
    T. Favaloro, J.-H. Bahk, A. Shakouri, Characterization of the temperature dependence of the thermoreflectance coefficient for conductive thin films. Rev. Sci. Instrum. 86, 024903 (2015)Google Scholar
  164. 164.
    C. Wei, X. Zheng, D.G. Cahill, J.-C. Zhao, Invited article: Micron resolution spatially resolved measurement of heat capacity using dual-frequency time-domain thermoreflectance. Rev. Sci. Instrum. 84, 071301 (2013)Google Scholar
  165. 165.
    P.E. Hopkins, C.M. Reinke, M.F. Su, R.H. Olsson III, E.A. Shaner, Z.C. Leseman, J.R. Serrano, L.M. Phinney, I. El-Kady, Reduction in the thermal conductivity of single crystalline silicon by phononic crystal patterning. Nano Lett. 11, 107–112 (2011)Google Scholar
  166. 166.
    M.R. Wagner, B. Graczykowski, J.S. Reparaz et al., Two-dimensional photonic crystals: Disorder matters. Nano Lett. 16, 5661–5668 (2016)Google Scholar
  167. 167.
    X. Wang, T. Mori, I. Kuzmych-Ianchuk, Y. Michiue, K. Yubuta, T. Shishido, Y. Grin, S. Okada, D.G. Cahill, Thermal conductivity of layered borides: The effect of building defects on the thermal conductivity of TmAlB4 and the anisotropic thermal conductivity of AlB2. APL Mater. 2, 046113 (2014)Google Scholar
  168. 168.
    J. Liu, G.-M. Choi, D.G. Cahill, Measurement of the anisotropic thermal conductivity of molybdenum disulfide by the time-resolved magneto-optic Kerr effect. J. Appl. Phys. 116, 233107 (2014)Google Scholar
  169. 169.
    A. J. Schmidt, R. Cheaito, M. Chiesa, A frequency-domain thermoreflectance method for the characterization of thermal properties. Rev. Sci. Instrum. 80, 094901 (2009); ibid, Characterization of thin metal films via frequency-domain thermoreflectance. J. Appl. Phys. 107, 024908 (2010)Google Scholar
  170. 170.
    K.T. Regner, D.P. Sellan, Z. Su, C.H. Amon, A.J.H. McGaughey, J.A. Malen, Broadband phonon mean free path contributions to thermal conductivity measured using frequency domain thermoreflectance. Nat. Commun. 4, 1640 (2013)Google Scholar
  171. 171.
    D. Rodin, S.K. Yee, Simultaneous measurement of in-plane and through-plane thermal conductivity using beam-offset frequency domain thermoreflectance. Rev. Sci. Instrum. 88, 014902 (2017)Google Scholar
  172. 172.
    J. Johnson, A. A. Maznev, J. Cuffe, J. K. Eliason, A. J. Minnich, T. Kehoe, C. M. Sotomayor Torres, G. Chen, K. A. Nelson, Direct measurement of room-temperature nondiffusive thermal transport over micron distances in a silicon membrane. Phys. Rev. Lett. 110, 025901 (2013)Google Scholar
  173. 173.
    J. Cuffe, J.K. Eliason, A.A. Maznev et al., Reconstructing phonon mean-free-path contributions to thermal conductivity using nanoscale membranes. Phys. Rev. B 91, 245423 (2015)Google Scholar
  174. 174.
    A. Vega-Flick, R.A. Duncan, J.K. Eliason et al., Thermal transport in suspended silicon membranes measured by laser-induced transient gratings. AIP Adv. 6, 120903 (2016)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.MariettaUSA

Personalised recommendations