lecture05j thermal conductivity

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  • ME 381R Lecture 7:

    Phonon Scattering & Thermal ConductivityDr. Li ShiDepartment of Mechanical Engineering The University of Texas at AustinAustin, TX 78712www.me.utexas.edu/~lishi [email protected]: 1-3-3, 1-6-2 in Tien et alReferences: Ch5 in Kittel

  • *Phonon Thermal ConductivityKinetic Theory:lTemperature, T/qDBoundaryPhonon ScatteringDefectDecreasing BoundarySeparationIncreasingDefectConcentrationPhonon Scattering Mechanisms Boundary Scattering Defect & Dislocation Scattering Phonon-Phonon ScatteringBoundaries change the spring stiffness (acoustic impedance) crystal waves scatter when encountering a change of acoustic impedance (similar to scattering of EM waves in the presence of a change of an optical refraction index)

    Matthiessen Rule:

  • *Specular Phonon-boundary ScatteringPhonon Reflection/TransmissionAcoustic ImpedanceMismatch (AIM)= (rv)1/(rv)2TEM of a thin film superlattice

  • *Phonon Bandgap Formation in Thin Film SuperlatticesCourtesy of A. Majumdar

  • *Acoustic Mismatch Model (AMM) Khalatnikov (1952)Diffuse Mismatch Model (DMM) Swartz and Pohl (1989)Diffuse Phonon-boundary ScatteringE. Swartz and R. O. Pohl, Thermal Boundary Resistance, Reviews of Modern Physics 61, 605 (1989). D. Cahill et al., Nanoscale thermal transport, J. Appl. Phys. 93, 793 (2003). Courtesy of A. MajumdarSpecularDiffuse

  • *SixGe1-x/SiyGe1-y Superlattice FilmsAIM = 1.15SuperlatticePeriodHuxtable et al., Thermal conductivity of Si/SiGe and SiGe/SiGe superlattices, Appl. Phys. Lett. 80, 1737 (2002).Alloy limitWith a large AIM, k can be reduced below the alloy limit.

  • *Effect of Impurity on Thermal ConductivityWhy the effect of impurity is negligible at low T?

    EMBED Word.Picture.8

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  • *Phonon-Impurity ScatteringImpurity change of M & C change of spring stiffness (acoustic impedance) crystal wave scatter when encountering a change of acoustic impedance (similar to scattering of EM wave in the presence of a change of an optical refraction index)Scattering mean free time for phonon-impurity scattering:li ~ 1/(sr)where r is the impurity concentration, and the scattering cross section= R2 [4/(4+1)]R: radius of lattice imperfactionl: phonon wavelength= 2R/l-> 0: s ~ 4 (Rayleigh scatttering that is responsible for the blue sky and red sunset) -> : s ~ R2

  • *Effect of Temperatures (R/l)4 for l >> Rs R2 for l
  • *Bulk Materials: Alloy Limit of Thermal ConductivityImpurity and alloy atoms scatter only short- l phonons that are absent at low T!

  • *Phonon Scattering with Imbedded NanostructuresLong-wavelength or low-frequency phonons are scattered by imbedded nanostructures!Spectral distribution of phonon energy (eb) & group velocity (v) @ 300 K

  • *Imbedded NanostructuresImages from Elisabeth Mller Paul Scherrer Institut Wueren-lingen und Villigen, Switzerland Data from A. Majumdar et al.Nanodot SuperlatticeBulk materials with embedded nanodots

  • *Phonon-Phonon Scattering The presence of one phonon causes a periodic elastic strain which modulates in space and time the elastic constant (C) of the crystal. A second phonon sees the modulation of C and is scattered to produce a third phonon.

    By scattering, two phonons can combine into one, or one phonon breaks into two. These are inelastic scattering processes (as in a non-linear spring), as opposed to the elastic process of a linear spring (harmonic oscillator).

    EMBED Word.Picture.8

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  • *Phonon-Phonon Scattering (Normal Process)Anharmonic Effects: Non-linear springNon-linear Wave InteractionBecause the vectorial addition is the same asmomentum conservation for particles: Phonon Momentum = KMomentum Conservation: K3 = K1+ K2Energy Conservation: w3= w1 + w2

  • *Phonon-Phonon Scattering (Umklapp Process)K1K2K3 = K1+K2What happens if that is outsidethe first Brillouin Zone

    Then(Bragg Condition as shownin next page)The propagating direction is changed.

  • *Reciprocal Lattice Vector (G)K = 2/llmin = 2aKmax = /a-/a
  • *Umklapp process: G = reciprocal lattice vector = 2p/a 0Normal Process vs. Umklapp ProcessNormal Process: G =01st Brillouin ZoneSelection rules:KyKxK1K2K3KyKxK1K2K3K1K2K3Cause zero thermal resistance directlyCause thermal resistance

  • *Effect of Temperature phonon ~ exp(D/bT)lTemperature, T/qDBoundaryPhonon ScatteringDefectDecreasing BoundarySeparationIncreasingDefectConcentrationphonon ~ exp(D/bT)

  • Phonon Thermal ConductivityKinetic TheorylTemperature, T/qDBoundaryPhonon ScatteringDefectDecreasing BoundarySeparationIncreasingDefectConcentrationClT

    EMBED Word.Picture.8

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  • *Thermal Conductivity of Bulk Crystals3k