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ME 381R Lecture 7: Phonon Scattering & Thermal ConductivityPowerPoint Presentation

ME 381R Lecture 7: Phonon Scattering & Thermal Conductivity

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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

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

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

Phonon Reflection/Transmission

TEM of a thin film superlattice

Acoustic Impedance

Mismatch (AIM)

= (rv)1/(rv)2

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 SuperlatticesCourtesy of A. Majumdar

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

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).

Effect of Impurity on Thermal Conductivity

Why the effect of impurity is negligible at low T?

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

- = 2R/l
- -> 0: s ~ 4 (Rayleigh scatttering that is
responsible for the blue sky and red sunset)

-> : s ~ R2

Separation

l

Increasing

Defect/impurity

Concentration

PhononScattering

Defect

Boundary

1.0

0.01

0.1

Temperature, T/qD

Effect of Temperatures (R/l)4forl >> R

s R2forl << R

l: phonon wavelength

R: radius of lattice imperfection

u(w)=

Increasing T

wD

w

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!

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!

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

- 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).

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

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.

Normal Process vs. Umklapp Process

Selection rules:

K1

K2

K3

Normal Process: G =0

Umklapp process:

G = reciprocal lattice vector

= 2p/a0

Ky

Ky

K1

K3

K3

Kx

K1

K2

Kx

K2

1st Brillouin Zone

Cause zero thermal resistance directly

Cause thermal resistance

0.01

0.1

Effect of TemperatureDecreasing Boundary

Separation

l

Increasing

Defect

Concentration

phonon~ exp(D/bT)

phonon~ exp(D/bT)

PhononScattering

Defect

Boundary

Temperature, T/qD

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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