Multi-scale Heat Conduction

Phonon Dispersion and Scattering

Hong goo, Kim1st year of M.S. course

Nov. 29th , 2011

ContentsI. Introduction

II. Phonon Dispersion 1-D Diatomic Chain Phonon Branch Real Crystals

III. Phonon Scattering Phonon-Phonon Process Anharmonic Effects Phonon-Defect Scattering Phonon-Electron Scattering Phonon-Photon-Electron Scattering Phonon-Photon Scattering

Phonon Concept

- Quantized energy of lattice vibration- Phonon is a boson with energy of ħω with to respect to the vibra-

tional mode with frequency of ω

I. Introduction

Bose-Einstein Distribution

Specific Heat and Thermal Conductivity

- Indistinguishable, unlimited # of particle per quantum state

1)/exp(

1

,

Tkg

Nf

BKPBE

m

m

dDT

fv

dDT

fkTc

BEg

P

BEBv

0

2

0

3

1

)(

Group Velocity- For superposition of two waves with k1 ≈ k2 , ω1 ≈ ω2

)cos()sin(2

)22

cos()22

sin(2

)sin()sin(

21212121

2211

tkxtkx

txkk

txkk

txktxk

vg = Δω/Δkvp = ω/k

- Group velocity is the speed of energy propagation

Modulation envelope

Harmonic WaveI. Introduction

Phase Velocity 0 ),exp(exp

dtkdxdtt

dxx

dconstitkxi kdtdxv constp /)/(

1-D Diatomic Chain Assumptions

- Displacement is sufficiently small → Linearity of atomic forces- Only the nearest neighbor atoms interact each other

nnnnnnnn uvvCuvCuvC

dt

udm 212122122122

22

1 2

nnnn vuuC

dt

vdm 22222

122

2 2

Motion of the Atoms: F = ma = kx

u2n u2n+2u2n-2 u2n+4

v2n-1 v2n+1 v2n+3 v2n+5

m1 m2C

II. Phonon Dispersion

1-D Diatomic Chain Harmonic Wave Solution: A exp( i(kx−ωt) )

a

na x(n+0.5)a (n+1)a (n+2.5)a (n+3)a (n+3.5)a

u2n

v2n+1

(n−0.5)a

v2n−1 u2n+2

tknaiAu n exp12

tankiAv n 5.0exp212 tankiAu n 1exp122

tankiAv n 5.0exp212

nnnn uvvC

dt

udm 212122

22

1 2 nnnn vuuC

dt

vdm 22222

122

2 2

Substitute

II. Phonon Dispersion

1-D Diatomic Chain Dispersion Relation

- For nontrivial solution of A1 and A2 , determinant should be zero

0

0

22cos

2cos2

2

1

22

21

A

A

mCkaC

kaCmC

tiiknan Amdt

udm ee1

212

22

1

tiiknaikaikannn AACuvvC

ee2ee2 122

221212

tiikaiknan Amdt

vdm eee 2

22

2212

2

2

tiikaiknaikaikannn AACvuuC eee2ee2 2

222

112222

02cos2 212

1 AkaCAmC

022cos 22

21 AmCAkaC Unknown : A1 and A2

II. Phonon Dispersion

1-D Diatomic Chain Dispersion Relation

02cos142 2221

421 kaCmmCmm

21

22

2121

2 2sin41111

mm

ka

mmC

mmC

ω

k

- Periodicity 2π/a of the reciprocal lattice space

sin2(ka/2)={1−cos(ka)}/2

- Only the 1st Brillouin zone is needed

- Two branches are formed because of the difference between m1 and m2

Acoustic branch

Optical branch

II. Phonon Dispersion

Dispersion Relation Physical Meaning of Dispersion Relation

- Relation between frequency(ω) and wavevector(k)

- In the presence of dispersion, phase velocity and group velocity is distinguished

- Characteristic of a material

- If dispersion relation is known, specific heat can be calculated (vg = dω/dk is known)

- Relation between energy(ħω) and momentum(ħk)

II. Phonon Dispersion

Phonon Branch Long wavelength limit : k → 0

02cos2 212

1 AkaCAmC

II. Phonon Dispersion

- Medium can be treated as a continuum

2121

22

2121

2 112

2sin41111

mmC

mm

ka

mmC

mmCOptical

0

2sin41111

21

22

2121

2

mm

ka

mmC

mmCAcoustic

Out-of-phase

In-phase

0211

222 1

2

211

212

1

m

m

mmCmC

C

mC

C

A

A

OpOptical

0

2

1

22

2cos2

12

12

1

AcAcAcousticmC

C

mC

kaC

A

A

Long wavelength limit : k → 0 - Acoustic branch : In-phase• No change in relative motions between neighboring atoms

- Optical branch : Out-of-phase• Restoring force acts within unit cells → high energy • If atoms have different charges, oscillating electric dipole is produced

Phonon BranchII. Phonon Dispersion

ω

k

Optical

Acoustic

Optical Branch- Vibration within a cell

- k → 0, vg = 0 ; standing wave, out-of-phase

- Interacts with EM waves

- Vibration of center of mass of a cell

Acoustic Branch

vg > 0

vg = dω/dk = 0

- k → 0, vg > 0 ; running wave, in-phase, acoustic wave

Phonon BranchII. Phonon Dispersion

• From radiation theory, oscillating dipole scatters radiation

Number of Branches : q-atom unit cell

- Longitudinal: atoms vibrate in the direction of wave propagation- Transverse: atoms vibrate perpendicular to wave propagation- Each cell has one LA branch and two TA branches - For each additional atom, one LO branch and two TO branches

are added- Symmetry leads to degeneracy of transverse modes

Acoustic Optical

Longitudinal 1 q − 1

Transverse 2 2(q − 1)

II. Phonon Dispersion

Phonon Branch

Real Crystals Silicon- Si monatomic diamond-like

structure

- LA meets LO (m1 = m2)

- TA, TO : degenerate

Silicon Carbide

II. Phonon Dispersion

B. N. Brockhouse et al.(1959)

TA

LA

LOTO

D. W. Feldman et al.(1968)

TOLO

LA

TA

- Si & C diatomic structure- Frequency gap exists

Frequency gap

Real Crystals Silicon

II. Phonon Dispersion

B. N. Brockhouse (1959)

TA

LA

LO

TO

LO

R. Tubino et al.(1971)

Brillouin zone for silicon

Optical Phonon- vg is small: slow propagation of phonons →

less contribution on heat conduction- Interaction with acoustic phonons at high temperature

→ reduction of thermal conductivity- Significant contribution on heat capacity at high temperature

Acoustic Phonon

Real CrystalsII. Phonon Dispersion

• With BE distribution, optical phonons (high frequency) get excited at higher tem-perature

- Longitudinal acoustic (LA)• More important at higher temperatures because upper limit of ω is higher than TA

- Transverse acoustic (TA)

TA

LA

LO

TO

• Dominant mode at low temperature because low frequency modes are numerous

Zeolite (Alx Siy Oz) - Nano-porous crystalline alumino-silicates- Applications• Sorption based heat exchanger: cooling of micro-elec-

tric devices• Catalyst, molecular sieves for chemical separations• Dielectric material

- MFI zeolite film • 288 atoms per unit cell• 864(=288×3) dispersion branches (polarization)

- Summation over all polarizations and wavevectors

P K BKP

BKP

B

KPB

P K

BEKPv Tk

Tk

Tkk

T

fTC

1)1)/(exp(

)/exp()(

2,

,

2

,,

Real CrystalsII. Phonon Dispersion

II. Phonon Dispersion

Real Crystals: Measurement Neutron-Phonon Scattering

- Neutron beam is incident on the target material- Emergent angle and energies of scattered neutron is measured- Energy lost by neutron = absorption of phonon

s

ssk

kk nEE

- Conservation of crystal momentum

s

sk

k Gkpp ns

EM(Photon-Phonon) Scattering- Same conservation laws for neutron scattering holds- X-ray scattering- Visible: Raman(optical phonon), Brillouin(acoustic) scattering- Very small frequency shift

Phonon dispersion can be derived

Interaction of PhononsIII. Phonon Scattering

Phonon Scattering- Phonon is a convenient concept in describing thermal transport

by lattice vibrational waves

- Phonons are treated as particles (wave → particle)

- Describes interaction of phonon with phonon/electron/defects and boundaries

- Anharmonic effect of phonon scattering governs the thermal transport properties of dielectric and semiconductor materials

• Phonon wave function can be localized by the uncertainty principle

Phonon-Phonon

- Simply a name for ħ times phonon wavevector - Similarity with physical momentum in terms of expression and scatter-

ing behavior- Crystal momentum is only conserved within the 1st Brillouin zone

Crystal Momentum ħk

Energy Conservation321 321

III. Phonon Scattering

Crystal Momentum Conservation Gkkk1 32 32 kkGk1

Three-Phonon Process- Dominant phonon-phonon scattering in terms of scattering probability- 3rd order anharmonic term of interatomic potential

Crystal Momentum Conservation

U Process

k1

k2

G

k3 k1 + k2

ky

kxa

a

N Process

a

a

kx

k1

k2

k3

ky

1st Brillouin zone

III. Phonon Scattering

a

a

a

a

Gkkk1 32 32 kkk1

Phonon-Phonon

Case 1 : N Process only

III. Phonon Scattering

- Net phonon momentum is conserved (G = 0)

- Nonzero phonon flux exists even without temperature gradient- Equilibrium cannot be reached by N Processes only• At Equilibrium, phonon momentum distribution is symmetric

→ Average of phonon momentum should be zero at equilibrium

- Thermal conductivity is infinite

- N process can be neglected in terms of thermal transport

• Phonon flux is the heat flow• Net phonon flux is conserved throughout the system

→ Nothing impedes the flow of phonon momentum (no resistance)

32 kkk1

Phonon-Phonon

Case 2 : Umklapp Process involved

III. Phonon Scattering

- U processes do not conserve net phonon momentum- U processes are more frequent at higher temperatures • U process must involve at least one phonon that has wavevector size comparable

to the Brillouin zone• At high temperature, high frequency modes are excited (BE distribution), result-

ing in more phonons available for U process

- U processes resists the phonon mo-mentum flux

- Scattering rate for U process deter-mines the thermal conductivity

k1

k2

G

k3 k1 + k2

ky

kx TBAU 2

23/exp TbTB DU

(G.P. Srivastava, 1990)

(G. Chen, 2005)

Phonon-Phonon

Phonon-Defect Interaction- Impurities, vacancies, dislocations- Defects influence the mean free path of phonons by altering lo-

cal acoustic impedance - Elastic scattering • Although magnitude of phonon wavevector does not change(elastic), the direction

of the wave propagation changes• As a consequence, net phonon momentum flux is not conserved

Phonon-DefectIII. Phonon Scattering

Scattering Rate- Independent of temperature- Contribution to heat resistance is significant at low temperature• Wavelength of phonons increases at low temperature, number of phonons with

wavelength comparable to the defect radius increases

- Rayleigh law4 dph

Thermal Conductivity vs. Temperature

Anharmonic EffectsIII. Phonon Scattering

- A ~ B : κ ~ T 3 at low temperatures• Most of the phonons have wavelength larger than the system size and defect size• Temperature independent scattering processes(defect / boundary) are dominant

→ Phonon mean free path is constant• Thermal conductivity is proportional to specific heat with T3 dependence

κ (log)

T (log)

A

B

C

D

• Point where scattering rate of U process is frequent enough to yield phonon mean free path shorter than the size parameters

• Scattering rate of U process increase exponentially

- B ~ D : U process significant

- C : Maximum point- D ~ : κ ~ T −x at high temperatures • Specific heat becomes constant (Dulong-Petit)• Number of phonons available for U processes pro-

portional to T 1 ~ T 2

UgvUgv vcvc 2

3

1

3

1

Thermal Conductivity vs. Temperature

Anharmonic EffectsIII. Phonon Scattering

κ

T (log) A B C D

1

1/

bTDe

cv

nU

3TCv UgvUgv vcvc 2

3

1

3

1

0Un

1TnU

.constCv

Phonon-Electron Scattering

Phonon-ElectronIII. Phonon Scattering

• Phonon absorption/emission takes place• Associated with Joule heating

- Energy transfer between electrons and phonons

phononif EEE phononif kGkkk • Momentum of electrons and phonons are crystal momentum

- Conservation of energy and crystal momentum

- Dominant scattering mechanism for electrons in metals

dephedepheeee

111111

• Electron-electron scattering is negligible compared to electron-phonon scattering• Electron-defect scattering is important at low temperatures

- Lattice vibration distorts the electron wave function

Phonon-ElectronIII. Phonon Scattering

Scattering Rate- Electron-phonon scattering rate is inversely proportional to

temperature at high temperatures Dphe TT

1• At temperature higher than Debye temperature, number of phonons is proportional

to temperature• Number of electrons remain unchanged D

bTph

bT

en

D 1

1~ /

• Electron energy > phonon energy• Acoustic phonon has low energy compared to electron energy → can be neglected• Significant at high temperature where optical phonons are excited• Inelastic scattering (U process)

- Contribution of optical phonons are dominant

Phonon-ElectronIII. Phonon Scattering

Transport Properties (Metals)- Electrical resistance

50~ Trr phe

Trr phe 0~

at low temperatures

at high temperatures

- Electrical conductivity: proportional to T at high temperatures

eee

e

m

en ~2

• Drude-Lorentz expression

- Thermal conductivity: nearly constant at high temperatures• Kinetic theory

eee

BeeFv T

m

TknvC ~

33

1 22

Phonon-Photon-ElectronIII. Phonon Scattering

Inter-band Transition (Indirect Semiconductors)

- Conservation of energy and momentumphononphotonif EE phononphotonif kkkk

- Electron/photon/phonon interaction- Phonon is absorbed/emitted to provide sufficient momentum

change for electron band transition

- Electron excitation by incident radiation (photon)

EgEgk

E

k

EDirect semiconductor Indirect semiconductor

kphonon

Raman Effect- Frequency shift between incident photon and scattered photon

induced by phonon-photon scattering

Phonon-PhotonIII. Phonon Scattering

phononphoton

phononphoton

Incidentphoton

Scatteredphoton

Absorbed Phonon

Scatteredphoton

Incidentphoton

Emitted Phonon

intermediate energy

final energy

final energy

initial energy

initial energy

Stokes Anti-Stokes

- Anti-stokes shift: phonon is emitted into the photon

• Spectroscopy : Position of Δωphoton for peak intensity depend upon temperature

- Stokes shift: phonon is absorbed from the photon

Phonon Dispersion - Relation between ω(energy) vs. k(momentum) - Acoustic branch: significant contribution to thermal conductivity

- Optical branch: high frequency, slow vg , significant at high temperature

- Thermal properties can be calculated from dispersion relation

Conclusion

Phonon Scattering- Conservation of energy and crystal momentum- Phonon-Phonon (U Process) : impedes phonon momentum flux- Phonon-Defect : elastic, significant at low temperature- Phonon-Electron (metals) : dominant at high temperature, optical branch- Phonon-Photon : Raman scattering, Stokes shift

-

Template

•

Phonon Scattering and Thermal Resistance

Anharmonic EffectsIII. Phonon Scattering

Mechanism FrequencyTemperature

Low (T < θD) High (T > θD)

Boundary (Size) ω0 T −3 T 0

Defects ω4 T 1 T 0

U Process ω1~2 T −3 exp(−αθD / T ) T 1~2

Free electrons (Metals) ω1 T −2 T 1

50~ Trr phe

Trr phe 0~

at low temperatures

at high temperatures

0

0

22cos

2cos2

2

1

22

21

A

A

mCkaC

kaCmC

02cos2

02cos2

2eeee2

ee

02cos142

122

2

212

1

122

221212

12

122

2

1

2221

421

AkaCAmC

AkaCAmC

AACuvvC

Amdt

udm

kaCmmCmm

ikaikatiiknannn

tiiknan

02

1

2

0211

2222

2cos

02cos2

212

1

1

2

211

21

212

1

212

1

mC

C

A

A

m

m

mmCmC

C

mC

C

mC

kaC

A

A

AkaCAmC

Acoustic

Optical

02cos2 212

1 AkaCAmC

02

1

2 212

1

mC

C

A

A

Acoustic

0

21122

22

2cos

1

2

211

21

212

1

m

m

mmCmC

C

mC

C

mC

kaC

A

A

Optical

0

2

1

22

2cos2

12

12

1

mC

C

mC

kaC

A

A

Acoustic

II. Dispersion RelationIV. Phonon Scattering

III. Real Crystals

I. Introduction