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Introduction

  • Hai-Dong WangEmail author
Chapter
Part of the Springer Theses book series (Springer Theses)

Abstract

With the rapid development of femtosecond laser heating, integrated circuit, micro/nano electromechanical systems, the research on heat transfer in nanomaterials under high heat flux conditions has attracted increased attention. The traditional Fourier’s law is found to be broken under these extreme conditions. It is in urgent need to develop a general heat conduction model to replace Fourier’s law and give precise predictions for thermal analysis in practical applications. This thesis reports on the theoretical and experimental studies of non-Fourier heat conduction using a novel thermomass theory as basis. A femtosecond laser thermoreflectance system and a direct current electrical measurement system at liquid helium temperature have been established for experimental investigations. The heat transfer behaviors under the extreme conditions have been studied in-depth and the experimental data were utilized to verify the theoretical models. This chapter introduces the background of non-Fourier heat conduction and the recent research on the material properties of metallic nanofilms.

Keywords

Femtosecond Laser Thermal Wave Boltzmann Transport Equation Interfacial Thermal Resistance Heat Diffusion Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    J.B. Biot, Memoire sur la propagation de la chaleur. Bibliotheque Brittanique 37, 310–329 (1804)Google Scholar
  2. 2.
    V. Peshkov, “second sound” in helium II. J Physics-USSR 8, 381 (1944)Google Scholar
  3. 3.
    L. Landau, Theory of the superfluidity of helium II. Phys. Rev. 60(4), 356–358 (1941)ADSzbMATHGoogle Scholar
  4. 4.
    J.C. Ward, J. Wilks, The velocity of second sound in liquid helium near the absolute zero. Phil. Mag. 42(326), 314–316 (1951)Google Scholar
  5. 5.
    M. Chester, Second sound in solids. Phys. Rev. 131(5), 2013–2015 (1963)ADSGoogle Scholar
  6. 6.
    V. Narayana, R.C. Dynes, Observation of second sound in bismuth. Phys. Rev. Lett. 28(22), 1461–1465 (1972)ADSGoogle Scholar
  7. 7.
    S.D. Brorson, J.G. Fujimoto, E.P. Ippen, Femtosecond electronic heat-transport dynamics in thin gold-films. Phys. Rev. Lett. 59(17), 1962–1965 (1987)ADSGoogle Scholar
  8. 8.
    C. Cattaneo, Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ. Modena 3, 83–101 (1948)MathSciNetGoogle Scholar
  9. 9.
    P. Vernotte, Paradoxes in the continuous theory of the heat equation. C. R. Acad. Sci. 246, 3154–3155 (1958)MathSciNetGoogle Scholar
  10. 10.
    P.M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)zbMATHGoogle Scholar
  11. 11.
    A. Barletta, E. Zanchini, Hyperbolic heat conduction and local equilibrium: a second law analysis. Int. J. Heat Mass Transf. 40(5), 1007–1016 (1997)zbMATHGoogle Scholar
  12. 12.
    H.S. Chu, W.B. Lor, Hyperbolic heat conduction in thin-film high \(t_{c}\) superconductors with interface thermal resistance. Cryogenics 39, 739–750 (1999)ADSGoogle Scholar
  13. 13.
    M.A. Al-Nimr, A.F. Khadrawi, M. Hammad, A generalized thermal boundary condition for the hyperbolic heat conduction model. Heat Mass Transfer 39, 69–79 (2002)ADSGoogle Scholar
  14. 14.
    M. Lewandowska, L. Malinowski, An analytical solution of the hyperbolic heat conduction equation for the case of a finite medium symmetrically heated on both sides. Int. Commun. Heat Mass Transfer 33, 61–69 (2006)Google Scholar
  15. 15.
    D.W. Tang, N. Araki, Analytical solution of non-fourier temperature response in a finite medium under laser-pulse heating. Heat Mass Transfer 31, 359–363 (1996)ADSGoogle Scholar
  16. 16.
    B. Pulvirenti, A. Barletta, E. Zanchini, Finite-difference solution of hyperbolic heat conduction with temperature-dependent properties. Numer. Heat Transfer, part A: Applications. 34(2), 169–183 (1998)Google Scholar
  17. 17.
    Z.M. Tan, W.J. Yang, Heat transfer during asymmetrical collision of thermal waves in a thin film. Int. J. Heat Mass Transfer 40(17), 3999–4006 (1997)zbMATHGoogle Scholar
  18. 18.
    S. Torii, W.J. Yang, Heat transfer mechanisms in thin film with laser heat source. Int. J. Heat Mass Transfer 48, 537–544 (2005)zbMATHGoogle Scholar
  19. 19.
    A. Vedavarz, K. Mitra, S. Kumar, Hyperbolic temperature profiles for laser surface interactions. J. Appl. Phys. 76(9), 5014–5021 (1994)ADSGoogle Scholar
  20. 20.
    D.W. Tang, N. Araki, The wave characteristics of thermal conduction in metallic films irradiated by ultra-short laser pulses. J. Phys. D: Appl. Phys. 29, 2527–2533 (1996)ADSGoogle Scholar
  21. 21.
    Z.M. Tan, W.J. Yang, Propagation of thermal waves in transient heat conduction in a thin film. J. Franklin Inst. 336B, 185–197 (1999)Google Scholar
  22. 22.
    K.C. Liu, Numerical simulation for non-linear thermal wave. Appl. Math. Comput. 175, 1385–1399 (2006)zbMATHMathSciNetGoogle Scholar
  23. 23.
    D.Y. Tzou, The generalized lagging response in small-scale and high-rate heating. Int. J. Heat Mass Transf. 38(17), 3231–3240 (1995)Google Scholar
  24. 24.
    S.J. Su, W.Z. Dai, P.M. Jordan, R.E. Mickens, Comparison of the solutions of a phase lagging heat transport equation and damped wave equation. Int. Heat Mass Transfer 48, 2233–2241 (2005)zbMATHGoogle Scholar
  25. 25.
    S.J. Su, W.Z. Dai, Comparison of the solutions of a phase-lagging heat transport equation and damped wave equation with a heat source. Int. Heat Mass Transfer 49, 2793–2801 (2006)zbMATHMathSciNetGoogle Scholar
  26. 26.
    A. Majumdar, Microscale heat conduction in dielectric thin films. J. Heat Transfer 115(1), 7–16 (1993)Google Scholar
  27. 27.
    G. Chen, Ballistic diffusive heat conduction equations. Phys. Rev. Lett. 86(11), 2297–2300 (2001)ADSGoogle Scholar
  28. 28.
    S.I. Anisimov, B.L. Kapeliovich, T.L. Perelman, Electron emission from surface of metals induced by ultrashort laser pulses. Sov. Phys. JETP 39, 375–377 (1974)ADSGoogle Scholar
  29. 29.
    T.Q. Qiu, C.L. Tien, Femtosecond laser heating of multi-layer metals - I. analysis. Int. J. Heat Mass Transfer 37(17), 2789–2797 (1994)Google Scholar
  30. 30.
    T.Q. Qiu, C.L. Tien, Femtosecond laser heating of multi-layer metals - II. experiments. Int. J. Heat Mass Transfer 37(17), 2799–2808 (1994)Google Scholar
  31. 31.
    T.Q. Qiu, C.L. Tien, Heat transfer mechanisms during short-pulse laser heating of metals. J. Heat Transfer 115, 835–841 (1993)Google Scholar
  32. 32.
    R.A. Guyer, J.A. Krumhansl, Solution of the linearized phonon boltzmann equation. Phys. Rev. 148(2), 766–778 (1966)ADSGoogle Scholar
  33. 33.
    D.Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior (Taylor & Francis, Washington D C, 1996)Google Scholar
  34. 34.
    G. Khitrova, P.R. Berman, M. Sargent, Theory of pump-probe spectroscopy. J. Opt. Soc. Am. B 5(1), 160–170 (1988)ADSGoogle Scholar
  35. 35.
    G.L. Eesley, Observation of nonequilibrium electron heating in copper. Phys. Rev. Lett. 51(23), 2140–2143 (1983)ADSGoogle Scholar
  36. 36.
    G.L. Eesley, Generation of nonequilibrium electron and lattice temperatures in copper by picoseconds laser pulses. Phys. Rev. B 33(4), 2144–2151 (1986)ADSGoogle Scholar
  37. 37.
    J.G. Fujimoto et al., Femtosecond laser interaction with metallic tungsten and nonequilibrium electron and lattice temperatures. Phys. Rev. Lett. 53(19), 1837–1840 (1984)ADSGoogle Scholar
  38. 38.
    H.E. Elsayed-Ali et al., Time-resolved observation of electron-phonon relaxation in copper. Phys. Rev. Lett. 58(12), 1212–1215 (1987)ADSGoogle Scholar
  39. 39.
    C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Methods (Cambridge University Press, New York, 1987)Google Scholar
  40. 40.
    P.B. Allen, Theory of thermal relaxation of electrons in metals. Phys. Rev. Lett. 59(13), 1460–1463 (1987)ADSGoogle Scholar
  41. 41.
    H.E. Elsayed-Ali et al., Femtosecond thermoreflectivity and thermotransmissivity ofpolycrystalline and single-crystalline gold films. Phys. Rev. B 43(5), 4488–4491 (1991)ADSGoogle Scholar
  42. 42.
    T. Juhasz et al., Time-resolved thermoreflectivity of thin gold films and its dependence on the ambient temperature. Phys. Rev. B 45(23), 13819–13822 (1992)ADSGoogle Scholar
  43. 43.
    C.K. Sun et al., Femtosecond-tunable measurement of electron thermalization in gold. Phys. Rev. B 50(20), 15337–15348 (1994)ADSGoogle Scholar
  44. 44.
    J. Hohlfeld et al., Time-resolved thermoreflectivity of thin gold films and its dependence on film thickness. Appl. Phys. B: Lasers and Optics 64(3), 387–390 (1997)ADSGoogle Scholar
  45. 45.
    N. Taketoshi, T. Baba, O. Akira, Development of a thermal diffusivity measurement system- for metal thin films using a picoseconds thermoreflectance technique. Meas. Sci. Technol. 12, 2064–2073 (2001)ADSGoogle Scholar
  46. 46.
    S.D. Brorson, M.K. Kelly, U. Wenschuh, R. Buhleier, J. Kuhl, Femtosecond pump-probe investigation of electron dynamics in idid \(c_{60}\) films. Phys. Rev. B 46(11), 7329–7332 (1992)ADSGoogle Scholar
  47. 47.
    W.L. McMillan, Transition temperature of strong-coupled superconductors. Phys. Rev. 167, 331–344 (1968)ADSGoogle Scholar
  48. 48.
    P.B. Allen, R.C. Dynes, Transition temperature of strong-coupled superconductors reanalyzed. Phys. Rev. B 12, 905–922 (1975)ADSGoogle Scholar
  49. 49.
    P.E. Hopkins, J.L. Kassebaum, P.M. Norris, Effects of electron scattering at metal-nonmetal interfaces on electron-phonon equilibration in gold films. J. Appl. Phys. 105, 023710 (2009)ADSGoogle Scholar
  50. 50.
    P.E. Hopkins, P.M. Norris, L.M. Phinney, S.A. Policastro, R.G. Kelly, Thermal conductivity in nanoporous gold films during electron-phonon nonequilibrium. J. Nanomater. 418050 (2008)Google Scholar
  51. 51.
    P.E. Hopkins, P.M. Norris, Substrate influence in electron-phonon coupling measurements in thin au films. Appl. Surf. Sci. 253, 6289–6294 (2007)ADSGoogle Scholar
  52. 52.
    R.J. Stevens, A.N. Smith, P.M. Norris, Signal analysis and characterization of experimental setup for the transient thermoreflectance technique. Rev. Sci. Instrum 77, 084901 (2006)ADSGoogle Scholar
  53. 53.
    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 technique. J. Heat Transfer 127, 315–322 (2005)Google Scholar
  54. 54.
    P.M. Norris, A.P. Caffrey, R.J. Stevens, J.M. Klopf, J.T. McLeskey Jr, A.N. Smith, Femtosecond pump-probe nondestructive examination of materials. Rev. Sci. Instrum. 74(1), 400–406 (2003)ADSGoogle Scholar
  55. 55.
    A.N. Smith, P.M. Norris, Influence of intraband transitions on the electron thermoreflectance response of metals. Appl. Phys. Lett. 78(9), 1240–1242 (2001)ADSGoogle Scholar
  56. 56.
    J.T. McLeskey Jr, P.M. Norris, Femtosecond transmission studies of a-si:h, a-sige:h and a-sic:h alloys pumped in the exponential band tails. Sol. Energy Mater. Sol. Cells 69, 165–173 (2001)Google Scholar
  57. 57.
    W.S. Capinski, H.J. Maris, T. Ruf, M. Cardona, K. Ploog, D.S. Katzer, Thermal conductivity measurements of gaas/alas superlattices using a picosecond optical pump and probe technique. Phys. Rev. B 59(12), 8105–8113 (1999)ADSGoogle Scholar
  58. 58.
    T. Nakamiya, T. Ueda, T. Ikegami, F. Mitsugi, K. Ebihara, R. Tsuda, Pulsed laser heating process of multi-walled carbon nanotubes film. Diam. Relat. Mater. 17, 1458–1461 (2008)ADSGoogle Scholar
  59. 59.
    S. Wolltersen, U. Emmerichs, H.J. Bakker, Femtosecond mid-ir pump-probe spectroscopy of liquid water: evidence for a two-component structure. Science 278(24), 658–660 (1997)ADSGoogle Scholar
  60. 60.
    A. Tokmakoff, B. Sauter, M.D. Fayer, Temperature-dependent vibrational relaxation in polyatomic liquids: picosecond infrared pump-probe experiments. J. Chem. Phys. 100(12), 9035–9043 (1994)ADSGoogle Scholar
  61. 61.
    P. Han, D.W. Tang, L.P. Zhou, Numerical analysis of two-dimensional lagging thermal behavior under short-pulse-laser heating on surface. Int. J. Eng. Sci. 44, 1510–1519 (2006)Google Scholar
  62. 62.
    K. Ramadan, Treatment of the interfacial temperature jump condition with non-fourier heat conduction effects. Int. Commun. Heat Mass Transfer 35(9), 1177–1182 (2008)Google Scholar
  63. 63.
    Y.M. Lee, T.W. Tsai, Ultra-fast pulse-laser heating on a two-layered semi-infinite material with interfacial contact conductance. Int. Commun. Heat Mass Transfer 34, 45–51 (2007)Google Scholar
  64. 64.
    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
  65. 65.
    Y. Yamashita, T. Yokomine, S. Ebara, A. Shimizu, Heat transport analysis for femtosecond laser ablation with molecular dynamics two temperature model method. Fusion Eng. Des. 81, 1695–1700 (2006)Google Scholar
  66. 66.
    B.H. Christensen, K. Vestentoft, P. Balling, Short-pulse ablation rates and the two temperature model. Appl. Surf. Sci. 253, 6347–6352 (2007)ADSGoogle Scholar
  67. 67.
    M.E. Povarnitsyn, T.E. Itina, K.V. Khishchenko, P.R. Levashov, Multi-material two-temperature model for simulation of ultra-short laser ablation. Appl. Surf. Sci. 253, 6343–6346 (2007)ADSGoogle Scholar
  68. 68.
    H.J. Wang, W.Z. Dai, L.G. Hewavitharana, A finite difference method for studying thermal deformation in a double-layered thin film with imperfect interfacial contactexposed to ultrashort pulsed lasers. Int. J. Therm. Sci. 47, 7–24 (2008)Google Scholar
  69. 69.
    H.J. Wang, W.Z. Dai, R. Melnik, A finite difference method for studying thermal deformation in a double-layered thin film exposed to ultrashort pulsed lasers. Int. J. Therm. Sci. 45, 1179–1196 (2006)Google Scholar
  70. 70.
    H.J. Wang, W.Z. Dai, R. Nassar, R. Melnik, A finite difference method for studying thermal deformation in a thin film exposed to ultrashort-pulsed lasers. Int. J. Heat Mass Transfer 49, 2712–2723 (2006)zbMATHGoogle Scholar
  71. 71.
    A. Saidane, S.H. Pulko, High-power short-pulse laser heating of low dimensional structures: a hyperbolic heat conduction study using tlm. Microelectron. Eng. 51–52, 469–478 (2000)Google Scholar
  72. 72.
    B.S. Yilbas, A.F.M. Arif, Laser short pulse heating: influence of pulse intensity ontemperature and stress fields. Appl. Surf. Sci. 252, 8428–8437 (2006)ADSGoogle Scholar
  73. 73.
    J. Xu, X.W. Wang, Simulation of ballistic and non-fourier thermal transport in ultra-fast laser heating. Phys. B 351, 213–226 (2004)ADSGoogle Scholar
  74. 74.
    J.C. Wang, C.L. Guo, Effect of electron heating on femtosecond laser-induced coherent acoustic phonons in noble metals. Phys. Rev. B 75, 184304 (2007)ADSGoogle Scholar
  75. 75.
    H.D. Wang, W.G. Ma, X. Zhang, W. Wang, Measurement of thermal wave in metal films using femtosecond laser thermoreflectance system. Acta Physica Sinica 59(6), 3856–3862 (2010). in ChineseMathSciNetGoogle Scholar
  76. 76.
    Z.X. Li, X.B. Luo, Z.Y. Guo, Mems technology status and development trend. J. Sens. Technol. 20(9), 58–60 (2001). in ChineseGoogle Scholar
  77. 77.
    J.P. Uyemura, Introduction to Visi Circuits and System Uyemura, 1st edn. (Wiley, New york, 2001)Google Scholar
  78. 78.
    J.J. Thomson, On the theory of electric conduction through thin metallic films. Proc. Camb. Phil. Soc 11, 120 (1901)Google Scholar
  79. 79.
    A.C.B. Lovell, Proc. Roy. Soc. (London) 157, 311 (1936)ADSGoogle Scholar
  80. 80.
    K. Fuchs, The conductivity of thin metallic films according to the electron theory of metals. Proc. Camb. Phil. Soc. 34, 100–108 (1938)ADSGoogle Scholar
  81. 81.
    E.H. Sondheimer, The mean free path of electrons in metals. Advan. Phys. 1, 1–42 (1952)ADSGoogle Scholar
  82. 82.
    A.F. Mayadas, M. Shatzkes, J.F. Janak, Electrical resistivity model for polycrystalline films: the case of specular reflection at external surfaces. Appl. Phys. Lett. 14(11), 345–347 (1969)ADSGoogle Scholar
  83. 83.
    A.F. Mayadas, M. Shatzkes, Electrical-resistivity model for polycrystalline films: the case of arbitrary reflection at external surfaces. Phys. Rev. B 1(4), 1382–1389 (1970)ADSGoogle Scholar
  84. 84.
    C.L. Tien, B.F. Armaly, P.S. Jagannathan, Thermal conductivity of thin metallic films and wires at cryogenic temperatures. Thermal conductivity. (Plenum, New york, 1969), pp. 13–19Google Scholar
  85. 85.
    J. Bass, W.P. Pratt, P.A. Schroeder, The temperature dependent electrical resistivities of the alkali metals. Rev. Mod. Phys. 62(3), 645–744 (1990)ADSGoogle Scholar
  86. 86.
    Z.S. Chen, X.S. Ge, Y.Q. Gu, Calorimetry and Determination of Thermal Properties (University of Science and Technology of China Press, China, 1990). in ChineseGoogle Scholar
  87. 87.
    Y.Z. Cao, X.G. Qiu, Experimental Heat Transfer (National Defense Industry Press, China, 1998). in ChineseGoogle Scholar
  88. 88.
    X.S. Wang, X.P. Wu, J. Qin et al., Experimental study of the infrared thermal imaging method for measuring the temperature of the flame. Laser and Infrared 3, 101–104 (2001). in ChineseGoogle Scholar
  89. 89.
    Z.Q. Yu, C.A. Moore, Y. Hu et al., Measurement of the surface temperature using the laser raman method. Chinese Laser 12(8), 492–494 (1985). in ChineseGoogle Scholar
  90. 90.
    S. Paoloni, H.G. Walther, Photothermal radiometry of infrared translucent materials. J. Appl. Phys. 82(1), 101–106 (1997)ADSGoogle Scholar
  91. 91.
    G.B. Zhang, J.Y. Shi, C.S. Shi et al., Photoacoustic technology in thermal diffusivity measurements of solid materials. Physics 29(7), 616–619 (2000). in ChineseGoogle Scholar
  92. 92.
    E. Doebelin, Measurement Systems: Application and Design, 3rd edn. (McGraw-Hill, New York 1985)Google Scholar
  93. 93.
    D.W. Pohl, W. Denk, M. Lanz, Optical stethoscopy: image recording with resolution \(\lambda \)/20. Appl. Phys. Lett. 44, 651–653 (1984)ADSGoogle Scholar
  94. 94.
    A. Majumdar, Scanning thermal microscopy. Ann. Rev. Mater. Sci. 29, 505–585 (1999)ADSGoogle Scholar
  95. 95.
    C.Y. Bao, W.Y. Feng, X.M. Liu, Laser fluorescence measurement of the gas temperature. J. Tsinghua Univ. (Sci. Technol.) 36(6), 40–43 (1999). in ChineseGoogle Scholar
  96. 96.
    B.K. You, Temperature measurement and instrumentation: thermocouples and thermal resistance (Science and Technology Literature Publishing House, China, 1990) in ChineseGoogle Scholar
  97. 97.
    V.P. Duggal, V.P. Nagpal, Size effect in thin single-crystal silver films. Appl. Phys. Lett. 13(6), 206–207 (1968)ADSGoogle Scholar
  98. 98.
    V.P. Duggal, V.P. Nagpal, Geometrical size effect in resistivity and hall coefficient in single-crystal silver films. J. Appl. Phys. 42(11), 4500–4502 (1971)ADSGoogle Scholar
  99. 99.
    L.R. Kirkland, R.L. Chaplin, Electrical size effect of aluminum single crystals. J. Appl. Phys. 42(8), 3054–3057 (1971)ADSGoogle Scholar
  100. 100.
    L.A. Moraga, J. Caballero, G. Kremer, Electrical resistivity of very thin single-crystal titanium films as a function of temperature. Thin Solid Films 117, 1–8 (1984)Google Scholar
  101. 101.
    G. Kästle, H.G. Boyen, A. Schröder et al., Size effect of the resistivity of thin epitaxial gold films. Phys. Rev. B 70(16), 165414 (2004)ADSGoogle Scholar
  102. 102.
    G. Ramaswamy, A.K. Raychauhuri, J. Goswami et al., Scanning tunneling microscope study of the morphology of chemical vapor deposited copper films and its correlation with resistivity. J. Appl. Phys. 82(8), 3797–3807 (1997)ADSGoogle Scholar
  103. 103.
    M. Fenn, G. Akuetey, P.E. Donovan, Electrical resistivity of cu and nb thin films. J. Phys.: Condens. Matter 10, 1707–1720 (1998)ADSGoogle Scholar
  104. 104.
    C. Durkan, M.E. Welland, Size effects in the electrical resistivity of polycrystalline nanowires. Phys. Rev. B 61(20), 14215–14218 (2000)ADSGoogle Scholar
  105. 105.
    W. Wu, S.H. Brongersma, M. Van Hove et al., Influence of surface and grain-boundary scattering on the resistivity of copper in reduced dimensions. Appl. Phys. Lett. 84(15), 2838–2840 (2004)ADSGoogle Scholar
  106. 106.
    H. Maroma, M. Eizenberg, The effect of surface roughness on the resistivity increase in nanometric dimensions. J. Appl. Phys. 99(12), 123705 (2006)ADSGoogle Scholar
  107. 107.
    B.T. Boiko, A.T. Pugachev, V.M. Bratsychin, Method for the determination of the thermophysical properties of evaporated thin films. Thin Solid Films 17, 157–161 (1973)ADSGoogle Scholar
  108. 108.
    P. Nath, K.L. Chopra, Thermal conductivity of copper films. Thin Solid Films 20, 53–62 (1974)ADSGoogle Scholar
  109. 109.
    F. Kelemen, Pulse method for the measurement of the thermal conductivity of thin films. Thin Solid Films 36, 199–203 (1976)ADSGoogle Scholar
  110. 110.
    C.A. Paddock, G.L. Eesley, Transient thermoreflectance from thin metal films. J. Appl. Phys. 60(1), 285–290 (1986)ADSGoogle Scholar
  111. 111.
    M. Rohde, Photoacoustic characterization of thermal transport properties in thin films and microstructures. Thin Solid Films. 238, 199–206, (1994)Google Scholar
  112. 112.
    T. Yamane, Y. Mori, S. Katayama et al., Measurement of thermal diffusivities of thin metallic films using the ac calorimetric method. J. Appl. Phys. 82(3), 1153–1156 (1997)ADSGoogle Scholar
  113. 113.
    S.M. Lee, D.G. Cahill, Heat transport in thin dielectric films. J. Appl. Phys. 81(6), 2590–2595 (1997)Google Scholar
  114. 114.
    W.V. Houston, The temperature dependence of electrical conductivity. Phys. Rev. 34, 279–283 (1929)ADSGoogle Scholar
  115. 115.
    R.N. Gurzhi, A.I. Kopeliovich, Low-temperature electrical conductivity of pure metals. Sov. Phys. Usp. 24(1), 17–41 (1981)ADSGoogle Scholar
  116. 116.
    J.K. Hulm, The thermal conductivity of tin, mercury, indium and tantalum at liquid helium temperatures. Proc. R. Soc. Lond. A 204, 98–123 (1950)ADSGoogle Scholar
  117. 117.
    F.A. Andrews, R.T. Webber, D.A. Spohr, Thermal conductivities of pure metals at low temperatures. I. Alum. Phys. Rev. 84(5), 994–996 (1951)ADSGoogle Scholar
  118. 118.
    D.B. Poker, C.E. Klabunde, Temperature dependence of electrical resistivity of vanadium, platinum and copper. Phys. Rev. B 26(12), 7012–7014 (1982)ADSGoogle Scholar
  119. 119.
    Y. Nishi, A. Igarashi, K. Mikagi, Temperature dependence of electrical resistivity for gold and lead. J. Mater. Sci. Lett. 6, 87–88 (1987)Google Scholar
  120. 120.
    G.S. Kumar, G. Prasad, R.O. Pohl, Review, experimental determinations of the lorenz number. J. Mater. Sci. 28, 4261–4272 (1993)ADSGoogle Scholar
  121. 121.
    A. Houghton, S. Lee, J.B. Marston, Violation of the wiedemann-franz law in a large-\(n\) solution of the \(t-j\) model. Phys. Rev. B 65, 220503 (2002)ADSGoogle Scholar
  122. 122.
    A.V. Sologubenko, N.D. Zhigadlo, J. Karpinski, H.R. Ott, Thermal conductivity of al-doped mgb\(_{2}\): impurity scattering and the validity of the wiedemann-franz law. Phys. Rev. B 74, 184523 (2006)ADSGoogle Scholar
  123. 123.
    M.G. Vavilov, A.D. Stone, Failure of the wiedemann-franz law in mesoscopic conductors. Phys. Rev. B 72, 205107 (2005)ADSGoogle Scholar
  124. 124.
    G.Z. Liu, G. Cheng, Chiral symmetry breaking and violation of the wiedemann franz law in underdoped cuprates. Phys. Rev. B 66, 100505 (2002)ADSGoogle Scholar
  125. 125.
    M.F. Smith, R.H. McKenzie, Apparent violation of the wiedemann-franz law near a magnetic field tuned metal-antiferromagnetic quantum critical point. Phys. Rev. Lett. 101, 266403 (2008)ADSGoogle Scholar
  126. 126.
    K. Vafayi, M. Calandra, O. Gunnarsson, Electronic thermal conductivity at high temperatures: violation of the wiedemann-franz law in narrow-band metals. Phys. Rev. B 74, 235116 (2006)ADSGoogle Scholar
  127. 127.
    A. Casian, Violation of the wiedemann-franz law in quasi-one-dimensional organic crystals. Phys. Rev. B 81, 155415 (2010)ADSGoogle Scholar
  128. 128.
    K.S. Kim, C. Pépin, Violation of the wiedemann-franz law at the kondo breakdown quantum critical point. Phys. Rev. Lett. 102, 156404 (2009)ADSGoogle Scholar
  129. 129.
    A. Garg, D. Rasch, E. Shimshoni, A. Rosch, Large violation of the wiedemann franz law in luttinger liquids. Phys. Rev. Lett. 103, 096402 (2009)ADSGoogle Scholar
  130. 130.
    N. Stojanovic, D.H.S. Maithripala, J.M. Berg, M. Holtz, Thermal conductivity in metallic nanostructures at high temperature: electrons, phonons, and the wiedemann franz law. Phys. Rev. B 82, 075418 (2010)ADSGoogle Scholar
  131. 131.
    Q.G. Zhang, B.Y. Cao, X. Zhang, M. Fujii, K. Takahashi, Influence of grain boundary scattering on the electrical and thermal conductivities of polycrystalline gold nanofilms. Phys. Rev. B 74, 134109 (2006)ADSGoogle Scholar
  132. 132.
    Q.G. Zhang, B.Y. Cao, X. Zhang, M. Fujii, K. Takahashi, Size effects on the thermal conductivity of polycrystalline platinum nanofilms. J. Phys.: Condens Matter. 18, 7937–7950 (2006)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Engineering MechanicsTsinghua UniversityBeijingPeople’s Republic of China

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