Theoretical and Experimental Studies on Non-Fourier Heat Conduction Based on Thermomass Theory pp 1-20 | Cite as
Introduction
- 1 Citations
- 690 Downloads
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 EquationReferences
- 1.J.B. Biot, Memoire sur la propagation de la chaleur. Bibliotheque Brittanique 37, 310–329 (1804)Google Scholar
- 2.V. Peshkov, “second sound” in helium II. J Physics-USSR 8, 381 (1944)Google Scholar
- 3.L. Landau, Theory of the superfluidity of helium II. Phys. Rev. 60(4), 356–358 (1941)ADSzbMATHGoogle Scholar
- 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.M. Chester, Second sound in solids. Phys. Rev. 131(5), 2013–2015 (1963)ADSGoogle Scholar
- 6.V. Narayana, R.C. Dynes, Observation of second sound in bismuth. Phys. Rev. Lett. 28(22), 1461–1465 (1972)ADSGoogle Scholar
- 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.C. Cattaneo, Sulla conduzione del calore. Atti Semin. Mat. Fis. Univ. Modena 3, 83–101 (1948)MathSciNetGoogle Scholar
- 9.P. Vernotte, Paradoxes in the continuous theory of the heat equation. C. R. Acad. Sci. 246, 3154–3155 (1958)MathSciNetGoogle Scholar
- 10.P.M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)zbMATHGoogle Scholar
- 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.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.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.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.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.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.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.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.A. Vedavarz, K. Mitra, S. Kumar, Hyperbolic temperature profiles for laser surface interactions. J. Appl. Phys. 76(9), 5014–5021 (1994)ADSGoogle Scholar
- 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.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.K.C. Liu, Numerical simulation for non-linear thermal wave. Appl. Math. Comput. 175, 1385–1399 (2006)zbMATHMathSciNetGoogle Scholar
- 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.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.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.A. Majumdar, Microscale heat conduction in dielectric thin films. J. Heat Transfer 115(1), 7–16 (1993)Google Scholar
- 27.G. Chen, Ballistic diffusive heat conduction equations. Phys. Rev. Lett. 86(11), 2297–2300 (2001)ADSGoogle Scholar
- 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.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.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.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.R.A. Guyer, J.A. Krumhansl, Solution of the linearized phonon boltzmann equation. Phys. Rev. 148(2), 766–778 (1966)ADSGoogle Scholar
- 33.D.Y. Tzou, Macro- to Microscale Heat Transfer: The Lagging Behavior (Taylor & Francis, Washington D C, 1996)Google Scholar
- 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.G.L. Eesley, Observation of nonequilibrium electron heating in copper. Phys. Rev. Lett. 51(23), 2140–2143 (1983)ADSGoogle Scholar
- 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.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.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.C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Methods (Cambridge University Press, New York, 1987)Google Scholar
- 40.P.B. Allen, Theory of thermal relaxation of electrons in metals. Phys. Rev. Lett. 59(13), 1460–1463 (1987)ADSGoogle Scholar
- 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.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.C.K. Sun et al., Femtosecond-tunable measurement of electron thermalization in gold. Phys. Rev. B 50(20), 15337–15348 (1994)ADSGoogle Scholar
- 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.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.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.W.L. McMillan, Transition temperature of strong-coupled superconductors. Phys. Rev. 167, 331–344 (1968)ADSGoogle Scholar
- 48.P.B. Allen, R.C. Dynes, Transition temperature of strong-coupled superconductors reanalyzed. Phys. Rev. B 12, 905–922 (1975)ADSGoogle Scholar
- 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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.J.P. Uyemura, Introduction to Visi Circuits and System Uyemura, 1st edn. (Wiley, New york, 2001)Google Scholar
- 78.J.J. Thomson, On the theory of electric conduction through thin metallic films. Proc. Camb. Phil. Soc 11, 120 (1901)Google Scholar
- 79.A.C.B. Lovell, Proc. Roy. Soc. (London) 157, 311 (1936)ADSGoogle Scholar
- 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.E.H. Sondheimer, The mean free path of electrons in metals. Advan. Phys. 1, 1–42 (1952)ADSGoogle Scholar
- 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.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.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.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.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.Y.Z. Cao, X.G. Qiu, Experimental Heat Transfer (National Defense Industry Press, China, 1998). in ChineseGoogle Scholar
- 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.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.S. Paoloni, H.G. Walther, Photothermal radiometry of infrared translucent materials. J. Appl. Phys. 82(1), 101–106 (1997)ADSGoogle Scholar
- 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.E. Doebelin, Measurement Systems: Application and Design, 3rd edn. (McGraw-Hill, New York 1985)Google Scholar
- 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.A. Majumdar, Scanning thermal microscopy. Ann. Rev. Mater. Sci. 29, 505–585 (1999)ADSGoogle Scholar
- 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.B.K. You, Temperature measurement and instrumentation: thermocouples and thermal resistance (Science and Technology Literature Publishing House, China, 1990) in ChineseGoogle Scholar
- 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.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.L.R. Kirkland, R.L. Chaplin, Electrical size effect of aluminum single crystals. J. Appl. Phys. 42(8), 3054–3057 (1971)ADSGoogle Scholar
- 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.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.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.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.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.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.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.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.P. Nath, K.L. Chopra, Thermal conductivity of copper films. Thin Solid Films 20, 53–62 (1974)ADSGoogle Scholar
- 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.C.A. Paddock, G.L. Eesley, Transient thermoreflectance from thin metal films. J. Appl. Phys. 60(1), 285–290 (1986)ADSGoogle Scholar
- 111.M. Rohde, Photoacoustic characterization of thermal transport properties in thin films and microstructures. Thin Solid Films. 238, 199–206, (1994)Google Scholar
- 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.S.M. Lee, D.G. Cahill, Heat transport in thin dielectric films. J. Appl. Phys. 81(6), 2590–2595 (1997)Google Scholar
- 114.W.V. Houston, The temperature dependence of electrical conductivity. Phys. Rev. 34, 279–283 (1929)ADSGoogle Scholar
- 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.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.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.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.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.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.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.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.M.G. Vavilov, A.D. Stone, Failure of the wiedemann-franz law in mesoscopic conductors. Phys. Rev. B 72, 205107 (2005)ADSGoogle Scholar
- 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.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.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.A. Casian, Violation of the wiedemann-franz law in quasi-one-dimensional organic crystals. Phys. Rev. B 81, 155415 (2010)ADSGoogle Scholar
- 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.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.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.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.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