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The rapid development of nanotechnology makes it possible to further understand nanoscale heat conduction. Theoretical analysis and experimental measurement have demonstrated the size-dependence of thermal conductivity on a nanoscale. As dielectric material (such as silicon), phonons are the predominant carriers of heat transport. Phonon ballistic transport and boundary scattering lead to the significant reduction of thermal conductivity. Various models, in which only one geometrical constraint of phonon transport is considered, have been proposed. In engineering situations the phonon transport can be influenced by multiple geometrical constraints, especially for material with long intrinsic phonon mean free path. However, at present a phonon thermal conductivity model in which the multiple geometrical constraints of phonon transport are taken into account, is still lacking. In the present paper, a multi-constrained phonon thermal conductivity model is obtained by using the phonon Boltzmann transport equation and modifying the phonon mean free path. The geometrical constraints are dealt with separately, and the effects of these constraints on thermal conductivity are then combined by the Matthiessen's rules. Different boundary conditions can lead to different influences on the phonon transport, so different methods should be used for different boundary constraints. The differential approximation method is utilized for the constraint in the direction of heat flux, while phonon scatterings on side surfaces are characterized by modifying the phonon mean free path. The model which characterizes various nanostructures including nanofilms(in-plane and cross-plane) and finite length rectangular nanowires, can well agree with the Monte Carlo simulations of different Knudsen numbers. The model with the Knudsen number Knx equal to 0 can well predict the experimental data for the in-plane thermal conductivity of nanofilm. When the Knudsen numbers Kny and Knz vanish, the model corresponds to the cross-plane thermal conductivity of nanofilm. Moreover, with Knx=0 and Kny=Knz, the model corresponds to the square nanowires of infinite length, and the similar slopes between the model and the experimental data of nanowires can be achieved.
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[10] Flik M, Tien C 1990 J. Heat Trans. 112 872
[11] Majumdar A 1993 J. Heat Trans. 115 7
[12] Alvarez F, Jou D 2007 Appl. Phys. Lett. 90 083109
[13] Alvarez F, Jou D 2008 J. Appl. Phys. 103 094321
[14] Lü X, Shen W, Chu J 2002 J. Appl. Phys. 91 1542
[15] Dong Y, Cao B Y, Guo Z Y 2015 Physica E 66 1
[16] McGaughey A J, Landry E S, Sellan D P, Amon C H 2011 Appl. Phys. Lett. 99 131904
[17] Hua Y C, Cao B Y 2014 Int. J. Heat Mass Trans. 78 755
[18] Kaviany M 2008 Heat Transfer Physics (Cambridge: Cambridge University Press) pp154-264
[19] Siegel R, Howell J R 1990 Thermal Radiation Transfer (Beijing: Science Press) pp385-397 (in Chinese) [R.西格尔, J. R.豪厄尔 著, 曹玉璋, 黄素逸 等 译 1990 热辐射传热 (北京: 科学出版社)第385-397页]
[20] Peraud J P M, Hadjiconstantinou N G 2012 Appl. Phys. Lett. 101 153114
[21] Hua Y C, Dong Y, Cao B Y 2013 Acta Phys. Sin. 62 244401 (in Chinese) [华钰超, 董源, 曹炳阳 2013 62 244401]
[22] Pop E, Sinha S, Goodson K E 2006 Proceedings of the IEEE 94 1587
[23] Chen G 1998 Phys. Rev. B 57 14958
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[1] Toberer E S, Baranowski L L, Dames C 2012 Annu. Rev. Mater. Res. 42 179
[2] Ju Y, Goodson K E 1999 Appl. Phys. Lett. 74 3005
[3] Liu W, Asheghi M 2004 Appl. Phys. Lett. 84 3819
[4] Asheghi M, Leung Y, Wong S, Goodson K E 1997 Appl. Phys. Lett. 71 1798
[5] Ju Y 2005 Appl. Phys. Lett. 87 153106
[6] Hopkins P E, Reinke C M, Su M F, Olsson III R H, Shaner E A, Leseman Z C, Serrano J R, Phinney L M, El-Kady I 2010 Nano Lett. 11 107
[7] Li D, Wu Y, Kim P, Shi L, Yang P, Majumdar A 2003 Appl. Phys. Lett. 83 2934
[8] Boukai A I, Bunimovich Y, Tahir-Kheli J, Yu J K, Goddard III W A, Heath J R 2008 Nature 451 168
[9] Ziman J M 2001 Electrons and Phonons: the Theory of Transport Phenomena in Solids (UK: Clarendon Press Oxford) pp451-482
[10] Flik M, Tien C 1990 J. Heat Trans. 112 872
[11] Majumdar A 1993 J. Heat Trans. 115 7
[12] Alvarez F, Jou D 2007 Appl. Phys. Lett. 90 083109
[13] Alvarez F, Jou D 2008 J. Appl. Phys. 103 094321
[14] Lü X, Shen W, Chu J 2002 J. Appl. Phys. 91 1542
[15] Dong Y, Cao B Y, Guo Z Y 2015 Physica E 66 1
[16] McGaughey A J, Landry E S, Sellan D P, Amon C H 2011 Appl. Phys. Lett. 99 131904
[17] Hua Y C, Cao B Y 2014 Int. J. Heat Mass Trans. 78 755
[18] Kaviany M 2008 Heat Transfer Physics (Cambridge: Cambridge University Press) pp154-264
[19] Siegel R, Howell J R 1990 Thermal Radiation Transfer (Beijing: Science Press) pp385-397 (in Chinese) [R.西格尔, J. R.豪厄尔 著, 曹玉璋, 黄素逸 等 译 1990 热辐射传热 (北京: 科学出版社)第385-397页]
[20] Peraud J P M, Hadjiconstantinou N G 2012 Appl. Phys. Lett. 101 153114
[21] Hua Y C, Dong Y, Cao B Y 2013 Acta Phys. Sin. 62 244401 (in Chinese) [华钰超, 董源, 曹炳阳 2013 62 244401]
[22] Pop E, Sinha S, Goodson K E 2006 Proceedings of the IEEE 94 1587
[23] Chen G 1998 Phys. Rev. B 57 14958
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