1. Phonon Scattering & Thermal Conductivity
ME 381R Lecture 7:
Phonon Scattering & Thermal Conductivity
Dr. Li Shi
Department of Mechanical Engineering
The University of Texas at Austin
Austin, TX 78712
Reading: 1-3-3, in Tien et al
References: Ch5 in Kittel

2. 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)
Phonon Scattering
Defect
Boundary
0.01
0.1
1.0
Temperature, T/qD

3. Specular Phonon-boundary Scattering
Phonon Reflection/Transmission
TEM of a thin film superlattice
Acoustic Impedance
Mismatch (AIM)
= (rv)1/(rv)2

4. Phonon Bandgap Formation in Thin Film Superlattices
min
=50
100 50
n=2,
n=1,
=100
n=3,
=66
=200
n=4, n = 2d
cosq
(i)
(ii)
wavevector, K
frequency, w
(A)
(B)
Courtesy of A. Majumdar

5. Diffuse Phonon-boundary Scattering
Specular
Diffuse
Acoustic Mismatch Model (AMM)
Khalatnikov (1952)
Diffuse Mismatch Model (DMM)
Swartz and Pohl (1989)
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

6. SixGe1-x/SiyGe1-y Superlattice Films
Period
AIM = 1.15
Alloy limit
With a large AIM, k 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).

7. Effect of Impurity on Thermal Conductivity
Why the effect of impurity is negligible at low T?

8. 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/(sr)
where r is the impurity concentration, and the scattering cross section
=  R2 [4/(4+1)]
R: radius of lattice imperfaction
l: phonon wavelength
= 2R/l
-> 0: s ~ 4 (Rayleigh scatttering that is
responsible for the blue sky and red sunset)
 -> : s ~  R2

9. Effect of Temperature u(w)= s  (R/l)4 for l >> R
s  R2 for l << R
l: phonon wavelength
R: radius of lattice imperfection l
Temperature, T/qD
Boundary
Phonon Scattering
Defect
Decreasing Boundary
Separation
Increasing
Defect/impurity
Concentration
0.01
0.1
1.0
u(w)=
Increasing T wD w

10. Bulk Materials: Alloy Limit of Thermal Conductivity
k [W/m-K] A B
Alloy Limit
Impurity and alloy atoms scatter only short- l phonons that are absent at low T!

11. Phonon Scattering with Imbedded Nanostructures
Atoms/Alloys
Nanostructures
Frequency, w
wmax eb v
Spectral distribution of phonon energy (eb) & group velocity (v) @ 300 K
Phonon Scattering
Long-wavelength or low-frequency phonons are scattered by imbedded nanostructures!

12. Imbedded Nanostructures
5x1018 Si-doped InGaAs
Si-Doped ErAs/InGaAs SL (0.4ML)
Undoped ErAs/InGaAs SL (0.4ML)
Nanodot Superlattice
Data from A. Majumdar et al.
AgPb18SbTe20
ZT = 800K
AgSb rich
Hsu et al., Science 303, 818 (2004)
Bulk materials with embedded nanodots
Images from Elisabeth Müller Paul Scherrer Institut Wueren-lingen und Villigen, Switzerland

13. 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).

14. Phonon-Phonon Scattering (Normal Process)
Anharmonic Effects: Non-linear spring K1 K2
K3 = K1+K2
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: w3= w1 + w2

15. Phonon-Phonon Scattering (Umklapp Process)
that is outside
the first Brillouin Zone K1
What happens if
K3 = K1+K2
Then
(Bragg Condition as shown
in next page) K2 K1 K2 G
U-Process K3
The propagating direction is changed.

16. Reciprocal Lattice Vector (G)
G = 2p/a
l: wavelength
K = 2/l
lmin = 2a
Kmax = /a
-/a<K< /a 2a

17. Normal Process vs. Umklapp Process
Selection rules: K1 K2 K3
Normal Process: G =0
Umklapp process:
G = reciprocal lattice vector
= 2p/a 0 Ky Ky K1 K3 K3 Kx K1 K2 Kx K2
1st Brillouin Zone
Cause zero thermal resistance directly
Cause thermal resistance

18. Effect of Temperature phonon ~ exp(D/bT) l  phonon ~ exp(D/bT)
Decreasing Boundary
Separation l
Increasing
Defect
Concentration
 phonon ~ exp(D/bT)
phonon ~ exp(D/bT)
Phonon Scattering
Defect
Boundary
0.01
0.1
1.0
Temperature, T/qD

19. Phonon Thermal ConductivityCl
Kinetic Theory
Decreasing Boundary
Separation T l
Increasing
Defect
Concentration
Phonon Scattering
Defect
Boundary
0.01
0.1
1.0
Temperature, T/qD

20. Thermal Conductivity of Bulk Crystals3 k