ME 381R Lecture 7:

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ME 381R Lecture 7 Phonon Scattering Thermal
Conductivity
Dr. Li Shi Department of Mechanical Engineering
The University of Texas at Austin Austin, TX
78712 www.me.utexas.edu/lishi lishi_at_mail.utexas.
edu

• Reading 1-3-3, 1-6-2 in Tien et al
• References Ch5 in Kittel

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Phonon Thermal Conductivity
Matthiessen Rule
Kinetic Theory
Phonon Scattering Mechanisms
Decreasing Boundary Separation

• Boundary Scattering
• Defect Dislocation Scattering
• Phonon-Phonon Scattering

?l
Increasing Defect Concentration

• Boundaries 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)?

PhononScattering
Defect
Boundary
Temperature, T/?D
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Specular Phonon-boundary Scattering
Phonon Reflection/Transmission
TEM of a thin film superlattice
Acoustic Impedance Mismatch (AIM)? (?v)1/(?v)2
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Phonon Bandgap Formation in Thin Film
Superlattices
Courtesy of A. Majumdar
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Diffuse Phonon-boundary Scattering
Specular
Diffuse
Diffuse Mismatch Model (DMM)? Swartz and
Pohl (1989)?
Acoustic Mismatch Model (AMM)? Khalatnikov
(1952)?
E. 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. Majumdar
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SixGe1-x/SiyGe1-y Superlattice Films
Superlattice Period
AIM 1.15
Alloy limit
With a large AIM, ? can be reduced below the
alloy limit.
Huxtable et al., Thermal conductivity of Si/SiGe
and SiGe/SiGe superlattices, Appl. Phys. Lett.
80, 1737 (2002).
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Effect of Impurity on Thermal Conductivity
Why the effect of impurity is negligible at low T?
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Phonon-Impurity Scattering

• Impurity? 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/(??)?
• where ? is the impurity concentration, and the
  scattering cross section
• ????R2 ?4/(?41)
• R radius of lattice imperfaction
• ? phonon wavelength
• 2?R/?
• ???????????4 (Rayleigh scatttering that is
• responsible for the blue sky and red sunset)?
• ??????????????R2

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Effect of Temperature

• ??????R/??4?for ? gtgt R
• ?????R2?for ? ltlt R
• ? phonon wavelength
• R radius of lattice imperfection

u(?)
Increasing T
?D
?
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Bulk Materials Alloy Limit of Thermal
Conductivity
Impurity and alloy atoms scatter only short-
??phonons that are absent at low T!
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Phonon Scattering with Imbedded Nanostructures
Spectral distribution of phonon energy (eb)
group velocity (v) _at_ 300 K
Long-wavelength or low-frequency phonons are
scattered by imbedded nanostructures!
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Imbedded Nanostructures

• Nanodot Superlattice

Data from A. Majumdar et al.

• Bulk materials with embedded nanodots

Images from Elisabeth Müller Paul Scherrer
Institut Wueren-lingen und Villigen, Switzerland
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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).

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Phonon-Phonon Scattering (Normal Process)?
Anharmonic Effects Non-linear spring
Non-linear Wave Interaction
Because the vectorial addition is the same
as momentum conservation for particles Phonon
Momentum ?K
Momentum Conservation ?K3 ?K1 ?K2 Energy
Conservation ??????????????
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Phonon-Phonon Scattering (Umklapp Process)?
that is outside the first Brillouin Zone
K1
What happens if
K3 K1K2
(Bragg Condition as shown in next page)?
Then
K2
The propagating direction is changed.
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Reciprocal Lattice Vector (G)?
G 2?/a
? wavelength
K 2?/? ?min????a Kmax ?/a -?/altKlt ?/a
2a
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Normal Process vs. Umklapp Process
Selection rules
K1
K2
K3
Normal Process G 0
Umklapp process G reciprocal lattice vector
2?/a ?0
Ky
Ky
K1
K3
K3
Kx
K1
K2
Kx
K2
1st Brillouin Zone
Cause zero thermal resistance directly
Cause thermal resistance
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Effect of Temperature
Decreasing Boundary Separation
?l
Increasing Defect Concentration
? ?phonon exp(?D/bT)?
?phonon exp(?D/bT)?
PhononScattering
Defect
Boundary
Temperature, T/?D
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Phonon Thermal Conductivity
Cl
Kinetic Theory
Decreasing Boundary Separation
T
?l
Increasing Defect Concentration
PhononScattering
Defect
Boundary
Temperature, T/?D
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Thermal Conductivity of Bulk Crystals
3
?