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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 [email protected] Reading: 1-3-3, 1-6-2 in Tien et al References: Ch5 in Kittel. 1.0. 0.01.

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Slide1 l.jpg

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

[email protected]

  • Reading: 1-3-3, 1-6-2 in Tien et al

  • References: Ch5 in Kittel


Slide2 l.jpg

1.0

0.01

0.1

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/qD


Specular phonon boundary scattering l.jpg
Specular Phonon-boundary Scattering

Phonon Reflection/Transmission

TEM of a thin film superlattice

Acoustic Impedance

Mismatch (AIM)

= (rv)1/(rv)2


Phonon bandgap formation in thin film superlattices l.jpg

w

w

w

w

l

l

=

=

n

n

2d

2d

cosq

frequency,

frequency,

frequency,

frequency,

100

100

l

l

(i)

(i)

(i)

(i)

=50

=50

min

min

50

50

wavevector, K

wavevector, K

wavevector, K

wavevector, K

l

l

l

n=1,

n=1,

=100

=100

n=1,

=100

w

w

w

w

(i)

(i)

l

l

l

n=2,

n=2,

=50

=50

n=2,

=50

frequency,

frequency,

frequency,

frequency,

l

l

l

n=1,

n=1,

=200

=200

n=1,

=200

l

l

l

n=2,

n=2,

=100

=100

n=2,

=100

(ii)

(ii)

(ii)

(ii)

(ii)

(ii)

l

l

l

n=3,

n=3,

=66

=66

n=3,

=66

l

l

l

n=4,

n=4,

=50

=50

n=4,

=50

wavevector, K

wavevector, K

wavevector, K

wavevector, K

(A)

(A)

(B)

(B)

Phonon Bandgap Formation in Thin Film Superlattices

Courtesy of A. Majumdar


Slide5 l.jpg

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 Physics61, 605 (1989).

D. Cahill et al., “Nanoscale thermal transport,” J. Appl. Phys. 93, 793 (2003).

Courtesy of A. Majumdar


Slide6 l.jpg

SixGe1-x/SiyGe1-y Superlattice Films

Superlattice

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).


Slide7 l.jpg

Effect of Impurity on Thermal Conductivity

Why the effect of impurity is negligible at low T?


Phonon impurity scattering l.jpg
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


Effect of temperature l.jpg

Decreasing Boundary

Separation

l

Increasing

Defect/impurity

Concentration

PhononScattering

Defect

Boundary

1.0

0.01

0.1

Temperature, T/qD

Effect of Temperature

s (R/l)4forl >> R

s R2forl << R

l: phonon wavelength

R: radius of lattice imperfection

u(w)=

Increasing T

wD

w


Slide10 l.jpg

k [W/m-K]

Alloy Limit

B

A

Bulk Materials: Alloy Limit of Thermal Conductivity

Impurity and alloy atoms scatter only short- lphonons that are absent at low T!


Slide11 l.jpg

Phonon Scattering with Imbedded Nanostructures

Phonon Scattering

v

eb

Nanostructures

Atoms/Alloys

wmax

Frequency, w

Spectral distribution of phonon energy (eb) & group velocity (v) @ 300 K

Long-wavelength or low-frequency phonons are scattered by imbedded nanostructures!


Slide12 l.jpg

Imbedded Nanostructures

5x1018 Si-doped InGaAs

Si-Doped ErAs/InGaAs SL (0.4ML)

Undoped ErAs/InGaAs SL (0.4ML)

Hsu et al., Science303, 818 (2004)

AgPb18SbTe20

ZT = 2 @ 800K

AgSb rich

  • 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


Phonon phonon scattering l.jpg
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).


Slide14 l.jpg

K1

K3 = K1+K2

K2

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: w3= w1 + w2


Slide15 l.jpg

G

K3

K2

K1

U-Process

Phonon-Phonon Scattering (Umklapp Process)

that is outside

the first Brillouin Zone

K1

What happens if

K3 = K1+K2

(Bragg Condition as shown

in next page)

Then

K2

The propagating direction is changed.


Slide16 l.jpg

Reciprocal Lattice Vector (G)

G = 2p/a

l: wavelength

K = 2/l

lmin = 2a

Kmax = /a

-/a<K< /a

2a


Slide17 l.jpg

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


Effect of temperature18 l.jpg

1.0

0.01

0.1

Effect of Temperature

Decreasing Boundary

Separation

l

Increasing

Defect

Concentration

 phonon~ exp(D/bT)

phonon~ exp(D/bT)

PhononScattering

Defect

Boundary

Temperature, T/qD


Slide19 l.jpg

Phonon Thermal Conductivity

1.0

0.01

0.1

Cl

Kinetic Theory

Decreasing Boundary

Separation

T

l

Increasing

Defect

Concentration

PhononScattering

Defect

Boundary

Temperature, T/qD



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