Phonons lecture

National Leading Research Lab. Nanoengineered Energy Conversion Devices Lab.
Phonons

National Leading Research Lab. Nanoengineered Energy Conversion Devices Lab.National Leading Research Lab. Nanoengineered Energy Conversion Devices Lab.
google image
Lattice
vibration
http://socs.berkeley.edu/
~murphy/Movies/movie.html

How to model this vibration...?
x
y
atomic
displacement
at time t
equilibrium
position
Fourier Analysis!!
f x( ) = an
cos
2π
L
nx
!
"
#
$
%
&+bn
sin
2π
L
nx
!
"
#
$
%
&
'
(
)
*
+
,
n=−∞
∞
∑
Flexural
mode

Logitudinal
mode
Torsional
mode

Thermal Transport in a Crystal
atom
Electron (or hole)
Phonon
(lattice vibration)
Reciprocal Lattice and k-space
k-space
0 2π/a 4π/a 6π/a
K
First Brillouin zone
k-space in three
dimensional representation
Reciprocal
lattice vector
Class Note
φ x( ) = φn
⋅exp i
2π
a
nx
"
#
$
%
&
'
(
)
*
+
,
-
n
∑ = φn
⋅exp iKnx( )(
)
+
,
n
∑
φ x+ a( )= φn
⋅exp i
2π
a
nx
"
#
$
%
&
'⋅exp i
2π
a
na
"
#
$
%
&
'
(
)
*
+
,
-
n
∑ = φ x( )

Phonon Dispersion Relation
http://www.ioffe.ru/SVA/NSM/Semicond/GaN/figs/fmd28_1.gif
Electronic Band Structure
http://www.ioffe.ru/
SVA/NSM/
Semicond/GaN/figs/
fmd28_1.gif

Harmonic Approximation
Dispersion Relation
Class Note

Interatomic Bonding
1-D Array of Spring
& Mass System
Equation of motion with the
nearest neighbor interaction
Solution
Dispersion Relation
k = 2π/λ
λmin
= 2a
kmax
= π/a
-π/a<k< π/a
2a
λ: wavelength
Group velocity
Dispersion Relation

Lattice Constant, a
xn yn
yn-1 xn+1
Two Atoms Per Unit Cell
Optical Branch: Electromagnetic Wave

Frequency,ω
Wave vector, K0 π/a
LA
TA
LO
TO
Optical
Vibrational
Modes
LA & LO
TA & TO
Total 6 polarizations
Longitudinal and Transverse
Polarization
LA is higher than TA
Real Dispersion in GaAs

Classical Oscillator
Frictionlessm
• Displacement:
• Potential energy:
• The state of a particle at time t is
speciﬁed by location x(t) and
momentum p(t)
•Allowed energy states
n = 0, 1, 2,…
Quantum Oscillator
• The state of the particle is associated
with a wave function ψ, whose modulus
squared |ψ(x)|2 gives the probability of
ﬁnding the particle at x
Energy is quantized, and ħω is a quantum of energy
•Schrodinger equation:
• Newton’s 2nd law:
En = n +
1
2
!
"#
$
%& !ω
Classical vs Quantum Oscillator
Total Energy of a Quantum
Oscillator in a Parabolic Potential
n = 0, 1, 2, 3, 4…; !ω/2: zero point energy
Phonon: A quantum of vibrational energy,
!ω, which travels through the lattice
Phonons follow Bose-Einstein statistics.
Phonon
momentum
Phonon
energy
Energy Quantization: Phonon

1s
2s
2p
Excited state
Phonon Hydrogen atom
nth excited state -> n phonons
Physically, this relation dictates that a normal mode with
frequency ω is nth excited state. Another way of saying
this, which is more widely used, is that there are n
phonons in the normal mode.
Equilibrium properties
• Speciﬁc heat
• Thermal expansion
• Melting
Transport properties
• Superconductivity
• Thermal conductivity
• Speed of sound
Equilibrium vs Transport Properties

Speciﬁc heat (or heat capacity)
• Phonon density of states
• Debye vs. Einstein model
• Phonon heat capacity
Outline: Equilibrium Properties
Phonon Density of States (DOS)
a
A linear chain of M atoms
with two ends jointed
(periodic boundary condition)
DOS: the number of phonon
modes per unit frequency
m=1
m=2
m=3
um
Solution
Allowed values of k This periodic boundary condition leads to
one allowed mode per mobile atom
kL = 2nπ

Phonon Density of States
Only M wavevectors (k) are allowed (one per mobile atom):
k= -Maπ/L -6π/L -4π/L -2π/L 0 2π/L 4π/L 6π/L π/a=Maπ/L
Only 1 k state lies within a dk interval of 2π/L

# of states between k and k + dk is: (L/2π)dk

N: total number of modes with wavevector less than k.
D(ω): density of states (# of k-vibrational modes between ω and ω+dω) :
1 dimensional
2 dimensional
3 dimensional

Density of States
22
1.4 Thermoelectric Energy Conversion
Thermoelectric devices exploit the Seebeck coefficient to turn voltage
gradients into thermal gradients and vice versa. A schematic of a thermoelectric
device is shown in Figure 1.7. If one supplies a thermal current, a corresponding
electrical current is generated by the device (power generation). Similarly, if one
supplies an electrical current, a temperature gradient is generated by the device
(refrigeration). A thermoelectric device usually consists of many n-p couples that are
connected electrically in series and thermally in parallel. Thermoelectric devices have
Figure 1.6. Variations in the electronic and phononic densities of states in low-dimensional
structures.
R. Wang, Ph.D. dissertation
Debye vs Einstein Approximation
Einstein
approximation
ωD
Debye
approximation
kD
ωD: cutoﬀ frequency
ωE

DOS based on the Debye
approximation
DOS based on the Einstein
approximation
D ω( )
ωωE
D ω( )
ωωD
ω2
Reddy et al. APL 87, 211908 (2005)
Real DOS
DOS based on the Debye
approximation

Debye Model
Frequency,ω
Wave vector, k0 π/a
Debye Approximation:
Debye Density of States
Number of Atoms:
Debye Wave Vector
Debye Cut-off Freq.
Debye Temperature:
!
Temperature where all phonon modes are excited
Higher speed of sound -> higher Debye temperature
Debye Temperature [K]
Mode Counting
Mode counting in D dimensions
M: number of unit cells
s: number of atoms per unit cell
!
1. Total number of modes: sMD
2. Number of branches (mode for each k): sD
3. Number of acoustic branches: D
4. Number of optical braches: Ds-D

Total Energy of Lattice Vibration
Total Energy of Lattice Vibration
Debye approximation: ω=csk

Phonon Heat Capacity under Debye
Approximation
Debye temperature
[J/K]
[J/m3-K]
cv
=
∂U
∂T
"
#$
%
&'
v
= 9NkB
T
θD
"
#
$
%
&
'
3
x4
ex
ex
−1( )
2
dx
0
xD
∫
Phonon Heat Capacity
Heat capacity
When T << θD,
Quantum
Regime
Classical
Regime
When T >> θD,
Dulong-Petit’s law
D: dimensionality

Principle of equipartition of energy
에너지는 자유도 사이에 똑같이 나누어지며, 자유도 한 개 당의 평균에너지는 1/2kT와 같다.
Monatomic molecule
x, y, z kinetic E
Diatomic molecule
2 rotational E vibrational E
(1 kinetic & 1 potential)
Crystal solid
x,y,z vibrational E (3) * (1 kinetic & 1 potential)
Dulong-Petit’s law
Dulong - Petit’s Law
Phonon Radiation
For comparison, photon radiation
Stefan Boltzmann constant

Photon Phonon
Distribution Bose-Einstein Bose-Einstein
Radiation
!
Under Debye
Dispersion
ω =
0 ~ k ~ ∞
ω =
Under Debye
Polarization 2 transverse
2 transverse
1 longitudinal
Scattering Photon-photon (no) Phonon-phonon (yes)
Wave Electromagnetic wave Elastic wave
Wien’s Displacement Law
u(ω)
ω
Increasing T
ωmax
Blackbody Phonon Radiation
For comparison, photon
ωmax ≈
3kB
h
T
λmaxT ≈
hc
3kB
clight = 3×108 m
s
csound = 3−10 ×103 m
s
λmaxT = 2898µm ⋅ K

Equilibrium properties
• Speciﬁc heat
• Thermal expansion
• Melting
Transport properties
• Superconductivity
• Speed of sound
Equilibrium vs Transport Properties
Outline: Transport Properties
Ballistic transport
Diﬀusive transport
✓ phonon heat capacity
✓ phonon group velocity
✓ phonon mean free path

Scanning Thermal Microscopy
Pt-Cr

Junction
10 µm
Pt Line
Cr Line
Tip
Laser Reflector
SiNx Cantilever
X-Y-Z

Actuator
Sample
Temperature sensor
Laser
Cantilever
Deflection

Sensing
Thermal
x
T
Shi, Kwon, Miner, Majumdar, JMEMS 10, 370 (2001)
Ballistic versus Diﬀusive Transport
Topographic Thermal
1 µm
A B C D
Low bias:
Ballistic
High bias:
Dissipative
ΔTtip
2 K
0
Courtesy of Li Shi

Wave Packet: Wave to Particle!!
http://www.astro.ucla.edu/~wright/anomalous-dispersion.html
Phonon Thermal Conductivity
Atom Spring
A phonon is a quantum of crystal vibration energy.
Energy transport can be regarded as phonon transport
(Diffusive transport)

k : Bond strength

m : Mass
Phonon Scattering
Mode counting in D dimensions
s: number of atoms per unit cell
✓ Number of acoustic branches: D
✓ Number of optical branches: Ds-D

TC vs Temperature: Scattering Mechanisms
Boundary
scattering
Defect scattering Phonon-phonon
scattering
Phonon Scattering
Phonon-
Defect
Scattering
Phonon-
Phonon
Scattering
Phonon-
Electron
Scattering
Phonon-
Boundary
Scattering
Λ = phonon mean free path
Vg = phonon group velocity
τ = phonon mean free time
Λ = Vg τ
Boundary (Interface) scattering
important at small length scales!

Phonon Boundary Scattering
Ashcroft & Mermin (text book)
Heat Capacity (Boundary Scattering)
CVD SWCN
• An individual nanotube has a high k ~ 2000-11000 W/m-K at 300 K
• k of a CN bundle is reduced by thermal resistance at tube-tube junctions
• Potential applications as heat spreading materials for electronic packaging applications
CNT
Courtesy of C. Yu

the interfaces. These processes have been predicted to affect the k
values of Si nanowires, but not to the extent observed here20,21
. The
peak k of the EE nanowires is shifted to a much higher temperature
than that of VLS nanowires, and both are significantly higher than
that of bulk Si, which peaks at around 25 K (ref. 5). This shift suggests
that the phonon mean free path is limited by boundary scattering as
opposed to intrinsic Umklapp scattering.
While the above wires were etched from high-resistivity wafers, the
peak ZT of semiconductor materials is predicted to occur at high
dopant concentrations (,1 3 1019
cm23
; ref. 22). To optimize the
ZT of EE nanowires, lower resistivity nanowires were synthesized
from 1021
V cm B-doped p-Si Æ111æ and 1022
V cm As-doped n-Si
Æ100æ wafers by the standard method outlined above. Nanowires
etched from the 1022
V cm and less resistive wafers, however, did
not produce devices with reproducible electrical contacts, probably
owing to greater surface roughness, as observed in TEM analysis.
Consequently, more optimally doped nanowires were obtained by
post-growth gas-phase B doping of wires etched from 1021
V cm
wafers (see Supplementary Information). The resulting nanowires
have an average r 5 3 6 1.4 mV cm (as compared to ,10 V cm for
wires from low-doped wafers).
Figure 2c shows the k of small-diameter nanowires etched from 10,
1021
, and 1022
V cm wafers. The post-growth doped nanowire
(52 nm diameter) etched from a 1021
V cm wafer has a slightly lower
k than the lower-doped wire of the same diameter. This small
decrease in k may be attributed to higher rates of phonon-impurity
scattering. Studies of doped and isotopically purified bulk Si have
revealed a reduction of k as a result of impurity scattering6,23,24
. Owing
to the atomic nature of such defects, they are expected to predomi-
nantly scatter short-wavelength phonons. On the other hand, nano-
wires etched from a 1022
V cm wafer have a much lower k than the
other nanowires, probably as a result of the greater surface roughness.
In the case of the 52 nm nanowire, k is reduced to 1.6 6
0.13 W m21
K21
at room temperature. For comparison, the temper-
ature-dependent k of amorphous bulk SiO2 (data points used from
http://users.mrl.uiuc.edu/cahill/tcdata/tcdata.html agree with mea-
surement in ref. 25) is also plotted in Fig. 2c. As can be seen from the
plot, k of these single-crystalline EE Si nanowires is comparable to
that of insulating glass. Indeed, k of the 52 nm nanowire approaches
the minimum k predicted and measured for Si: ,1 W m21
K21
(ref. 26). The resistivity of a single nanowire of comparable diameter
(48 nm) was measured (see Supplementary Information) and the
electronic contribution to thermal conductivity (ke) can be estimated
from the Wiedemann–Franz law16
. For measured r 5 1.7 mV cm,
ke 5 0.4 W m21
K21
, meaning that the lattice thermal conductivity
(kl 5 k 2 ke) is 1.2 W m21
K21
.
By assuming the mean free path due to boundary scattering
‘b~Fd, where F . 1 is a multiplier that accounts for the specularity
of phonon scattering at the nanowire surface and d is the nanowire
diameter, a model based on Boltzmann transport theory was able to
explain27
the diameter dependence of thermal conductivity in VLS
nanowires, as observed in ref. 14. Because the thermal conductivity of
EE nanowires is lower and the surface is rougher than that of VLS
ones, it is natural to assume ‘b~d (F 5 1), which is the smallest mean
free path due to boundary scattering. However, this still cannot
explain why the phonon thermal conductivity approaches the
amorphous limit for nanowires with diameters ,50 nm. In fact,
theories that consider phonon backscattering, as recently proposed
by ref. 21, cannot explain our observations either. The thermal
conductivity in amorphous non-metals26
can be well explained by
50
b
a
40
30
20
10
0
0
4
8
0
Temperature (K)
k(Wm–1K–1)
c
k(Wm–1K–1)
100 200
50 nm
98 nm
115 nm
115 nm
98 nm
50 nm
150 nm
75 nm
52 nm
37 nm
10 Ω cm
10–1 Ω cm
56 nm
115 nm
Vapour–liquid–solid nanowires
Electroless etching nanowires
300
0
Temperature (K)
100 200 300
10–2 Ω cm
Amorphous SiO2
Figure 2 | Thermal conductivity of the rough silicon nanowires. a, An SEM
image of a Pt-bonded EE Si nanowire (taken at 52u tilt angle). The Pt thin
film loops near both ends of the bridging wire are part of the resistive heating
and sensing coils on opposite suspended membranes. Scale bar, 2 mm. b, The
temperature-dependent k of VLS (black squares; reproduced from ref. 14)
and EE nanowires (red squares). The peak k of the VLS nanowires is
175–200 K, while that of the EE nanowires is above 250 K. The data in this
graph are from EE nanowires synthesized from low-doped wafers.
c, Temperature-dependent k of EE Si nanowires etched from wafers of
different resistivities: 10 V cm (red squares), 1021
V cm (green squares;
arrays doped post-synthesis to 1023
V cm), and1022
V cm (blue squares).
For the purpose of comparison, the k of bulk amorphous silica is plotted with
open squares. The smaller highly doped EE Si nanowires have a k
approaching that of insulating glass, suggesting an extremely short phonon
mean free path. Error bars are shown near room temperature, and should
decrease with temperature. See Supplementary Information for k
measurement calibration and error determination.
165
Nature©2007 Publishing Group
Phonon Boundary Scattering
Nature 451, 163 (2008)
wires etched from a
other nanowires, pr
In the case of
0.13 W m21
K21
at
ature-dependent k
http://users.mrl.uiu
surement in ref. 25)
plot, k of these sing
that of insulating gl
the minimum k p
(ref. 26). The resisti
(48 nm) was measu
electronic contribut
from the Wiedema
ke 5 0.4 W m21
K2
(kl 5 k 2 ke) is 1.2 W
By assuming th
‘b~Fd, where F .
of phonon scatterin
diameter, a model b
explain27
the diame
nanowires, as obser
EE nanowires is low
ones, it is natural to
free path due to b
explain why the
amorphous limit f
theories that consid
by ref. 21, cannot
conductivity in am
50
b
40
30
20
10
0
0
4
8
0
Temperature (K)
k(Wm–1K–1)
c
k(Wm–1K–1)
100 200
50 nm
98 nm
115 nm
115 nm
98 nm
50 nm
150 nm
75 nm
52 nm
37 nm
10 Ω cm
10–1 Ω cm
56 nm
115 nm
Vapour–liquid–solid nanowires
Electroless etching nanowires
300
0
Temperature (K)
100 200 300
10–2 Ω cm
Amorphous SiO2
Figure 2 | Thermal co
image of a Pt-bonded
film loops near both e
and sensing coils on o
temperature-depende
and EE nanowires (re
175–200 K, while that
graph are from EE na
c, Temperature-depen
different resistivities:
arrays doped post-syn
For the purpose of com
open squares. The sm
approaching that of i
mean free path. Error
decrease with temper
measurement calibra
Nature©2007 Publishing Group
Thermal conductivity of bulk Si at room temperature ~ 140 W/m-K
Phonon mean free path at room temperature ~ 300 nm (Goodson et al.)
Si Nanowires
Phonon Defect Scattering
Temperature, T/θD
0.01 0.1 1.0
kl
Boundary
Phonon-phonon 
ScatteringDefect
Increasing Defect 
Concentration

Phonon Defect Scattering: The Alloy Limit
The alloy limit
k[W/m-K]
A BAxB1-x
Si GeSixGe1-x
~ 140 W/m-K
~ 60 W/m-K
~ 5 - 10 W/m-K
e experimental
erefore, only a
this method is
sink method,16
al conductivity
n nitride and a
tion containing
nes. A Si1−xGex
evaporation. To
NW and mem-
ta 3D dual fo-
ged and placed
he membranes
il. The Pt coil
on of the gen-
nd reached the
f the NW was
ed through the
ere determined
All the thermal
conductivity measurements of the Si1−xGex NWs were car-
ried out over the temperature range of 40–420 K and at
ϳ6ϫ10−6
Torr.
Figure 3 shows the temperature dependence of the ther-
mal conductivities of the Si1−xGex NWs with different Ge
concentrations and diameters. The diameters, d, of the NW
samples were: d=140 and 147 nm for Si0.996Ge0.004, d=229,
330, and 344 nm for Si0.96Ge0.04, and d=160 and 205 nm for
Si0.91Ge0.09. The thermal conductivities of Si NWs and a
Si0.85Ge0.15 thin film17
are shown for reference. As shown in
the figure, the thermal conductivity of the Si1−xGex NWs
decreases with decreasing NW diameter, which is consistent
with the results of a previous study by Li et al.12
However,
based on the thermal conductivity of the Si0.96Ge0.04 and
Si0.91Ge0.09 NWs, this diameter dependence is not as signifi-
cant as in the case of Si NWs. We suspect that this is because
alloy scattering is more dominant in this case. For Si1−xGex
NWs, alloy scattering and phonon boundary scattering
should be two dominant scattering mechanisms in the tem-
perature range of 40–400 K considering that the Debye tem-
peratures of Si and Ge are 640 K and 374 K, respectively.18
FIG. 4. Thermal conductivities of the Si1−xGex NWs at 300 K at the differ-
ent Ge concentrations. The thermal conductivities of Si NWs, a Si0.85Ge0.15
thin film, and Si0.8Ge0.2 bulk ͑the thin film and bulk data was from Ref. 17͒
are shown for reference.
Si1−xGex NWs: ͑a͒
trate, ͓͑b͒ and ͑c͔͒
gle-crystalline and
, Si0.96Ge0.04, and
n taken along the
NW with the ͗110͘
d on the suspended
FIG. 3. Thermal conductivities of the Si1−xGex NWs with x=0.004, 0.04,
and 0.09, Si NWs, and a Si0.85Ge0.15 thin film ͑the thin film data was from
Ref. 17͒.
Thin film & bulk
Kim et al, Appl. Phys. Lett. 96, 233106 (2010)
122nm
92nm
147nm
140nm 344nm
330nm
229nm
205nm
160nm
k[W/m-K]
A BAxB1-x
Phonon Defect Scattering: Alloy Scattering

Kim et al., Physical Review Letters 96, 045901 (2006)
In0.53Ga0.47AsIn0.53Ga0.47As
0.4 ML 40 nm
0.1 ML 10 nm
0.3 % ErAs/In0.53Ga0.47As0.3 % ErAs/In0.53Ga0.47As
In0.53Ga0.47As
0.3 % ErAs:In0.53Ga0.47As
In0.53Ga0.47As
0.3 % ErAs/In0.53Ga0.47As
In0.53Ga0.47As
Kim et al., Nano Letters 8, 2097 (2008)
In0.53Ga0.47AsIn0.53Ga0.47As
Phonon defect scattering: Nanoparticles
Phonon Defect & Boundary Scattering
SiSi/Si0.95Ge0.05
Li et al., APL 83, 3186 (2003)

Phonon Phonon Scattering (Anharmonic)
• The presence of one phonon causes a periodic elastic strain which modulates in
space and time the elastic constant (E) of the crystal. A second phonon sees the
modulation of E 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).
This is the only way that thermal
conductivity of a crystal decreases with
increasing temperature
Phonon Phonon Scattering
Temperature, T/θD
0.01 0.1 1.0
kl
Boundary
Phonon-phonon 
ScatteringDefect
Concentration
Why high T???

Phonon Phonon Scattering
Normal process Umklapp process
Umklapp process does not conserve crystal momentum and
restores equilibrium to Bose-Einstein distribution. It poses
resistance to phonon transport.
The propagating
direction is changed.
The ﬁrst Brillouin zone
Umklapp (in German): ﬂipped over
U-Process & Dispersion Relation

ℓl
Temperature, T/θD
Boundary
Phonon-phonon 
ScatteringDefect
Decreasing Boundary
Separation
Increasing
Defect
Concentration
• Boundary Scattering
• Defect Scattering
• Phonon-Phonon Scattering
0.01 0.1 1.0
Temperature, T/θD
0.01 0.1 1.0
kl
Boundary
Phonon-phonon 
ScatteringDefect
Concentration
Phonon Scattering Mechanisms
Elastic scattering: boundary & defect
Inelastic scattering: phonon-phonon scattering
Temperature, T/θD
0.01 0.1 1.0
kl
Boundary
Phonon-phonon 
ScatteringDefect
Concentration

Thin Film Superlattice
TEM of a thin film superlattice
S.T. Huxtable, Ph.D dissertation
Interfaces
Interface Scattering
Acoustic Mismatch Model (AMM)
Khalatnikov (1952)
Diffuse Mismatch Model (DMM)
Swartz and Pohl (1989)
E. Swartz and R. O. Pohl,“Thermal Boundary Resistance,”Reviews of Modern Physics 61, 605 (1989).
D. Cahill et al.,“Nanoscale thermal transport,”J. Appl. Phys. 93, 793 (2003).
Courtesy of A. Majumdar
Specular Diffuse

Si/SixGe1-x Superlattice
AIM = 1.15
Superlattice
Period
Huxtable et al., APL 80, 1737 (2002).
Alloy limit
Acoustic
impedance
E: elastic modulus
g: spring constant
ThermalConductivity[W/m-K]

Phonons lecture

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