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