1. Multi-scale Heat Conduction Phonon Dispersion and Scattering
Nov. 29th , 2011
Multi-scale Heat Conduction Phonon Dispersion and Scattering
Hong goo, Kim
1st year of M.S. course

2. Contents Introduction Phonon Dispersion Phonon Scattering
1-D Diatomic Chain
Phonon Branch
Real Crystals
Phonon Scattering
Phonon-Phonon Process
Anharmonic Effects
Phonon-Defect Scattering
Phonon-Electron Scattering
Phonon-Photon-Electron Scattering
Phonon-Photon Scattering

3. Phonon Concept Bose-Einstein Distribution
Introduction
Phonon
Concept
Quantized energy of lattice vibration
Phonon is a boson with energy of ħω with to respect to the vibrational mode with frequency of ω
Bose-Einstein Distribution
Indistinguishable, unlimited # of particle per quantum state
Specific Heat and Thermal Conductivity

4. Harmonic Wave Phase Velocity Group Velocity
Introduction
Harmonic Wave
Phase Velocity
Group Velocity
For superposition of two waves with k1 ≈ k2 , ω1 ≈ ω2
vp = ω/k
vg = Δω/Δk
Modulation envelope
Group velocity is the speed of energy propagation

5. 1-D Diatomic Chain Assumptions Motion of the Atoms: F = ma = kx
Phonon Dispersion
1-D Diatomic Chain
Assumptions
Displacement is sufficiently small → Linearity of atomic forces
Only the nearest neighbor atoms interact each other m1 m2 C
v2n-1
v2n+1
v2n+3
v2n+5
u2n-2
u2n
u2n+2
u2n+4
Motion of the Atoms: F = ma = kx

6. 1-D Diatomic Chain Harmonic Wave Solution: A exp( i(kx−ωt) )
Phonon Dispersion
1-D Diatomic Chain
Harmonic Wave Solution: A exp( i(kx−ωt) ) a
v2n−1
v2n+1 na x
(n+0.5)a
(n+1)a
(n+2.5)a
(n+3)a
(n+3.5)a
u2n
u2n+2
(n−0.5)a
Substitute

7. 1-D Diatomic Chain Dispersion Relation Unknown : A1 and A2
Phonon Dispersion
1-D Diatomic Chain
Dispersion Relation
Unknown : A1 and A2
For nontrivial solution of A1 and A2 , determinant should be zero

8. 1-D Diatomic Chain Dispersion Relation Phonon Dispersion ω
Two branches are formed because of the difference between m1 and m2
Acoustic branch
Optical branch
Periodicity 2π/a of the reciprocal lattice space
sin2(ka/2)={1−cos(ka)}/2
- Only the 1st Brillouin zone is needed k

9. Dispersion Relation Physical Meaning of Dispersion Relation
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)

10. Phonon Branch Long wavelength limit : k → 0
Phonon Dispersion
Phonon Branch
Long wavelength limit : k → 0
Medium can be treated as a continuum
Out-of-phase
In-phase

11. Phonon Branch Long wavelength limit : k → 0 Acoustic branch : In-phase
Phonon Dispersion
Phonon Branch
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
Long wavelength가 대표하는 것은?

12. Phonon Branch Optical Branch Acoustic Branch Vibration within a cell
Phonon Dispersion
Phonon Branch
Optical Branch ω k
Optical
Acoustic
vg = dω/dk = 0
Vibration within a cell
k → 0, vg = 0 ; standing wave, out-of-phase
Interacts with EM waves
vg > 0
From radiation theory, oscillating dipole scatters radiation
Acoustic Branch
Vibration of center of mass of a cell
k → 0, vg > 0 ; running wave, in-phase, acoustic wave

13. Phonon Branch Number of Branches : q-atom unit cell Acoustic Optical
Phonon Dispersion
Phonon Branch
Number of Branches : q-atom unit cell
Acoustic
Optical
Longitudinal 1
q − 1
Transverse 2
2(q − 1)
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

14. Real Crystals Silicon Silicon Carbide
Phonon Dispersion
Real Crystals
Silicon
Silicon Carbide
Si monatomic diamond-like structure
LA meets LO (m1 = m2)
TA, TO : degenerate
Si & C diatomic structure
Frequency gap exists
D. W. Feldman et al.(1968) TO LO LA TA
B. N. Brockhouse et al.(1959) TA LA LO TO
Frequency gap

15. Real Crystals Silicon Phonon Dispersion TO LO LO LA TA
Brillouin zone for silicon
B. N. Brockhouse (1959) TA LA LO TO LO
R. Tubino et al.(1971)

16. Real Crystals Optical Phonon Acoustic Phonon
Phonon Dispersion
Real Crystals
Optical Phonon TA LA LO TO
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
With BE distribution, optical phonons (high frequency) get excited at higher temperature
Acoustic Phonon
Transverse acoustic (TA)
Dominant mode at low temperature because low frequency modes are numerous
Longitudinal acoustic (LA)
More important at higher temperatures because upper limit of ω is higher than TA

17. Real Crystals Zeolite (Alx Siy Oz)
Phonon Dispersion
Real Crystals
Zeolite (Alx Siy Oz)
Nano-porous crystalline alumino-silicates
Applications
Sorption based heat exchanger: cooling of micro-electric 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

18. Real Crystals: Measurement
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
Conservation of crystal momentum
Phonon dispersion can be derived
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

19. Interaction of Phonons
Phonon Scattering
Interaction of Phonons
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

20. Phonon-Phonon Three-Phonon Process Energy Conservation
Phonon Scattering
Phonon-Phonon
Three-Phonon Process
Dominant phonon-phonon scattering in terms of scattering probability
3rd order anharmonic term of interatomic potential
Energy Conservation
Simply a name for ħ times phonon wavevector
Similarity with physical momentum in terms of expression and scattering behavior
Crystal momentum is only conserved within the 1st Brillouin zone
Crystal Momentum ħk
Crystal Momentum Conservation

21. Phonon-Phonon Crystal Momentum Conservation N Process U Process k1 k1
Phonon Scattering
Phonon-Phonon
Crystal Momentum Conservation
N Process
U Process ky ky
1st Brillouin zone k1 k1 k2 k2 kx kx k3 G k3
k1 + k2

22. Phonon-Phonon Case 1 : N Process only
Phonon Scattering
Phonon-Phonon
Case 1 : N Process only
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)

23. Phonon-Phonon Case 2 : Umklapp Process involved
Phonon Scattering
Phonon-Phonon
Case 2 : Umklapp Process involved
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), resulting in more phonons available for U process k1 k2 G k3
k1 + k2 ky kx
U processes resists the phonon momentum flux
Scattering rate for U process determines the thermal conductivity
(G.P. Srivastava, 1990)
(G. Chen, 2005)

24. Phonon-Defect Phonon-Defect Interaction Scattering Rate
Phonon Scattering
Phonon-Defect
Phonon-Defect Interaction
Impurities, vacancies, dislocations
Defects influence the mean free path of phonons by altering local 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
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 law

25. Anharmonic Effects Thermal Conductivity vs. Temperature
Phonon Scattering
Anharmonic Effects
Thermal Conductivity vs. Temperature
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
B ~ D : U process significant
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
κ (log)
T (log) A B C D
C : Maximum point
D ~ : κ ~ T −x at high temperatures
Specific heat becomes constant (Dulong-Petit)
Number of phonons available for U processes proportional to T 1 ~ T 2

26. Anharmonic Effects Thermal Conductivity vs. Temperature cv nU κ
Phonon Scattering
Anharmonic Effects
Thermal Conductivity vs. Temperature κ
T (log) A B C D cv nU

27. Phonon-Electron Phonon-Electron Scattering
Phonon Scattering
Phonon-Electron
Phonon-Electron Scattering
Lattice vibration distorts the electron wave function
Phonon absorption/emission takes place
Associated with Joule heating
Energy transfer between electrons and phonons
Momentum of electrons and phonons are crystal momentum
Conservation of energy and crystal momentum
Dominant scattering mechanism for electrons in metals
Electron-electron scattering is negligible compared to electron-phonon scattering
Electron-defect scattering is important at low temperatures

28. Phonon-Electron Scattering Rate
Phonon Scattering
Phonon-Electron
Scattering Rate
Electron-phonon scattering rate is inversely proportional to temperature at high temperatures
At temperature higher than Debye temperature, number of phonons is proportional to temperature
Number of electrons remain unchanged
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

29. Phonon-Electron Transport Properties (Metals) Electrical resistance
Phonon Scattering
Phonon-Electron
Transport Properties (Metals)
Electrical resistance
at low temperatures
at high temperatures
Electrical conductivity: proportional to T at high temperatures
Drude-Lorentz expression
Thermal conductivity: nearly constant at high temperatures
Kinetic theory

30. Phonon-Photon-Electron
Phonon Scattering
Phonon-Photon-Electron
Inter-band Transition (Indirect Semiconductors)
Electron excitation by incident radiation (photon)
Electron/photon/phonon interaction
Phonon is absorbed/emitted to provide sufficient momentum change for electron band transition
Conservation of energy and momentum k E
Direct semiconductor
Indirect semiconductor
kphonon Eg Eg

31. Phonon-Photon Raman Effect
Phonon Scattering
Phonon-Photon
Raman Effect
Frequency shift between incident photon and scattered photon induced by phonon-photon scattering
Spectroscopy : Position of Δωphoton for peak intensity depend upon temperature
Stokes shift: phonon is absorbed from the photon
Anti-stokes shift: phonon is emitted into the photon
Stokes
Anti-Stokes
intermediate energy
Scattered
photon
Incident
photon
Scattered
photon
Incident
photon
final energy
initial energy
Absorbed Phonon
Emitted Phonon
initial energy
final energy

32. Conclusion Phonon Dispersion Phonon Scattering
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
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

34. Template

35. Free electrons (Metals)
Phonon Scattering
Anharmonic Effects
Phonon Scattering and Thermal Resistance
Mechanism
Frequency
Temperature
Low (T < θD)
High (T > θD)
Boundary (Size) ω0
T −3
T 0
Defects ω4
T 1
U Process
ω1~2
T −3 exp(−αθD / T )
T 1~2
Free electrons (Metals) ω1
T −2
at low temperatures
at high temperatures

37. Dispersion Relation
Introduction
Phonon Scattering
Real Crystals