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Phonon cross-plane transport and thermal boundary resistance: effect of heat source size and thermal boundary resistance on phonon characteristics

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Abstract

Phonon cross-plane transport across silicon and diamond thin films pair is considered, and thermal boundary resistance across the films pair interface is examined incorporating the cut-off mismatch and diffusive mismatch models. In the cut-off mismatch model, phonon frequency mismatch for each acoustic branch is incorporated across the interface of the silicon and diamond films pair in line with the dispersion relations of both films. The frequency-dependent and transient solution of the Boltzmann transport equation is presented, and the equilibrium phonon intensity ratios at the silicon and diamond film edges are predicted across the interface for each phonon acoustic branch. Temperature disturbance across the edges of the films pair is incorporated to assess the phonon transport characteristics due to cut-off and diffusive mismatch models across the interface. The effect of heat source size, which is allocated at high-temperature (301 K) edge of the silicon film, on the phonon transport characteristics at the films pair interface is also investigated. It is found that cut-off mismatch model predicts higher values of the thermal boundary resistance across the films pair interface as compared to that of the diffusive mismatch model. The ratio of equilibrium phonon intensity due to the cut-off mismatch over the diffusive mismatch models remains >1 at the silicon edge, while it becomes <1 at the diamond edge for all acoustic branches.

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Abbreviations

C k :

Frequency-dependent volumetric specific heat capacity of the dielectric material

D :

Density of state

f k :

Frequency-dependent probability distribution function in phase space

\({f_k^0}\) :

Equilibrium probability distribution function in phase space

\({\hbar}\) :

Reduced Planck’s constant

I k :

Frequency-dependent phonon intensity

\({I_k^{++}}\) :

Frequency-dependent phonon intensity in the first quadrant

\({I_k^{-+}}\) :

Frequency-dependent phonon intensity in the second quadrant

\({I_k^{+-}}\) :

Frequency-dependent phonon intensity in the third quadrant

\({I_k^{- -}}\) :

Frequency-dependent phonon intensity in the fourth quadrant

\({I_k^0}\) :

Frequency-dependent equilibrium phonon intensity

k :

Wavenumber

L x :

Thickness of the film

L z :

Width of the film

\({{q}''_x}\) :

Heat flux in the x-direction

\({{q}''_z}\) :

Heat flux in the z-direction

t :

Time

T :

Equivalent equilibrium phonon temperature

u :

Internal energy density of phonon

v k :

Frequency-dependent group velocity of phonons

\({{\rm w}_s}\) :

Heat source size

x :

Cartesian coordinate x-direction

z :

Cartesian coordinate z-direction

\({\Delta x}\) :

Grid spacing in the x-direction

\({\Delta z}\) :

Grid spacing in the z-direction

\({\Lambda_k}\) :

Frequency-dependent phonon mean free path

\({\theta}\) :

Polar angle

\({\phi}\) :

Azimuthal angle

\({\tau_k}\) :

Frequency-dependent relaxation time

\({\omega}\) :

Frequency

LA:

Longitudinal acoustic

LO:

Longitudinal optical

TA:

Transverse acoustic

TO:

Transverse optical

x :

x-axis

z :

z-axis

References

  1. Wang Z.L., Mu H.T., Liang J.G., Tang D.W.: Thermal boundary resistance and temperature dependent phonon conduction in CNT array multilayer structure. Int. J. Therm. Sci. 74, 53–62 (2013)

    Article  Google Scholar 

  2. Hida S., Hori T., Shiga T., Elliott J., Shiomi J.: Thermal resistance and phonon scattering at the interface between carbon nanotube and amorphous polyethylene. Int. J. Heat Mass Transf. 67, 1024–1029 (2013)

    Article  Google Scholar 

  3. Bin Mansoor S., Yilbas B.S.: Phonon transport in silicon–silicon and silicon–diamond thin films: consideration of thermal boundary resistance at interface. Phys. B Condens. Matter 406(11), 2186–2195 (2011)

    Article  ADS  Google Scholar 

  4. Hamian S., Yamada T., Faghri M., Park K.: Finite element analysis of transient ballistic–diffusive phonon heat transport in two-dimensional domains. Int. J. Heat Mass Transf. 80, 781–788 (2015)

    Article  Google Scholar 

  5. Mansoor S.B., Yilbas B.S.: Phonon radiative transport in silicon–aluminum thin films: frequency dependent case. Int. J. Therm. Sci. 57, 54–62 (2012)

    Article  Google Scholar 

  6. Guo R., Huang B.: Thermal transport in nanoporous Si: Anisotropy and junction effects. Int. J. Heat Mass Transf. 77, 131–139 (2014)

    Article  MathSciNet  Google Scholar 

  7. Liang Z., Tsai H.-L.: Reduction of solid–solid thermal boundary resistance by inserting an interlayer. Int. J. Heat Mass Transf. 55(11–12), 2999–3007 (2012)

    Article  Google Scholar 

  8. Duda J.C., Hopkins P.E., Beechem T.E., Smoyer J.L., Norris P.M.: Inelastic phonon interactions at solid–graphite interfaces. Superlattices Microstruct. 47(4), 550–555 (2010)

    Article  ADS  Google Scholar 

  9. Prasher R.: Thermal boundary resistance of nanocomposites. Int. J. Heat Mass Transf. 48(23–24), 4942–4952 (2005)

    Article  MATH  Google Scholar 

  10. Galkina T.I., Klokov A.Yu., Sharkov A.I., Bagaev V.S., Onishchenko E.E., Zaitsev V.V., Ralchenko V.G., Dravin V.A., Khmel’nitskii R.A., Gippius A.A.: Propagation of acoustic phonons across the interfaces in CdTe and Si/CVD-diamond and quasi-two-dimensional phonon wind in CdTe/ZnTe quantum wells. Phys. B Condens. Matter 316-317, 243–246 (2002)

    Article  ADS  Google Scholar 

  11. Warzoha R.J., Fleischer A.S.: Heat flow at nanoparticle interfaces. Nano Energy 6, 137–158 (2014)

    Article  Google Scholar 

  12. Stevens R.J., Zhigilei L.V., Norris P.M.: Effects of temperature and disorder on thermal boundary conductance at solid–solid interfaces: nonequilibrium molecular dynamics simulations. Int. J. Heat Mass Transf. 50(19–20), 3977–3989 (2007)

    Article  MATH  Google Scholar 

  13. Heino P.: Lattice-Boltzmann finite-difference model with optical phonons for nanoscale thermal conduction. Comput. Math. Appl. 59(7), 2351–2359 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Choi S-H., Maruyama S.: Thermal boundary resistance at an epitaxially perfect interface of thin films. Int. J. Therm. Sci. 44(6), 547–558 (2005)

    Article  Google Scholar 

  15. Ali S.S., Mazumder S.: Phonon heat conduction in multidimensional heterostructures: predictions using the boltzmann transport equation. J. Heat Transf. 137, 102401-1/11 (2015)

    Article  Google Scholar 

  16. Saaskilahti K., Oksanen J., Tulkki J., Volz S.: Role of anharmonic phonon scattering in the spectrally decomposed thermal conductance at planar interfaces. Phys. Rev. B 90, 134312-1/8 (2014)

    Article  ADS  Google Scholar 

  17. Minnich, A.J., Chen, G., Mansoor, S.B., Yilbas, B.S.: Quasi-ballistic heat transfer studied using the frequency-dependent Boltzmann transport equation. Phys. Rev. B 84(23), Art. no. 235207 (2011)

  18. Brockhouse B.N.: Lattice vibrations in silicon and germanium. Phys. Rev. Lett. 6(2), 256–258 (1959)

    Article  ADS  Google Scholar 

  19. Ward A., Broido D.A., Stewart D.A.: Ab initio theory of the lattice thermal conductivity in diamond. Phys. Rev. B 80, 125203-1–125203-8 (2009)

    Article  ADS  Google Scholar 

  20. Majumdar A.: Microscale heat conduction in dielectric thin films. ASME J. Heat Transf. 115, 7–16 (1993)

    Article  Google Scholar 

  21. Ziman J.M.: Electrons and Phonons. Oxford University Press, London (1960)

    MATH  Google Scholar 

  22. Pilon L., Katika K.M.: Modified method of characteristics for simulating microscale energy transport. ASME J. Heat Transf. 126, 735–743 (2004)

    Article  Google Scholar 

  23. Jeans J.H.: The Dynamical Theory of Gases, 4th edn. Dover, New York (1954)

    MATH  Google Scholar 

  24. Arhcroft N.W., Mermin N.D.: Solid State Physics. Harcourt College Publishers, San Diego (1976)

    Google Scholar 

  25. Ward A., Broido D.A., Stewart D.A.: Ab initio theory of the lattice thermal conductivity in diamond. Phys. Rev. B 80, 125203-1–125203-8 (2009)

    Article  ADS  Google Scholar 

  26. Asheghi M., Leung Y.K., Wong S.S., Goodson K.E.: Phonon-boundary scattering in thin silicon layers. Appl. Phys. Lett. 71(13), 1798 (1997)

    Article  ADS  Google Scholar 

  27. Yilbas, B.S., Mansour, S.B.: Influence of heat source size on phonon transport in thin silicon film. Transp. Theory Stat. Phys. 42(2–3), 65–84 (2013)

  28. Goodson K.E., Kading O.W., Rosner M., Zachi R.: Thermal conduction normal to diamond-silicon boundaries. Appl. Phys. Lett. 66(23), 3134–3136 (1995)

    Article  ADS  Google Scholar 

  29. Yilbas, B.S., Bin Mansoor, S.B.: Frequency dependent phonon transport in two-dimensional silicon and diamond films. Mod. Phys. Lett. B 26(17), Art. no. 1250104 (2012)

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  1. ME Department, KFUPM, Dhahran, Saudi Arabia

    H. Ali & B. S. Yilbas

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  1. H. Ali
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  2. B. S. Yilbas
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Correspondence to B. S. Yilbas.

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Communicated by Andreas Öchsner.

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Ali, H., Yilbas, B.S. Phonon cross-plane transport and thermal boundary resistance: effect of heat source size and thermal boundary resistance on phonon characteristics. Continuum Mech. Thermodyn. 28, 1373–1393 (2016). https://doi.org/10.1007/s00161-015-0480-z

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