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 cos2πLnx

!

"#

$

%&+bn sin

2πLnx

!

"#

$

%&

'

()

*

+,

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 i2πanx

"

#$

%

&'

(

)*

+

,-

n∑ = φn ⋅exp iKnx( )(

)+,

n∑

φ x+ a( ) = φn ⋅exp i2πanx

"

#$

%

&'⋅exp i

2πana

"

#$

%

&'

(

)*

+

,-

n∑ = φ x( )


Phonon Dispersion Relationhttp://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

2aλ: wavelength

Group velocity

Dispersion Relation


Lattice Constant, a

xn ynyn-1 xn+1

Two Atoms Per Unit Cell


Optical Branch: Electromagnetic Wave


Freq

uenc

y, 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 OscillatorFrictionlessm

• 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=2m=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

F igure 1.6. Variations in the electronic and phononic densities of states in low-dimensional

structures.

R. Wang, Ph.D. dissertation


Debye vs Einstein ApproximationEinstein approximation

ωD

Debye approximation

kD

ωD: cutoff frequency

ωE



DOS based on the Debye approximation

DOS based on the Einstein approximation

D ω( )

ωω E

D ω( )

ωωD

ω 2



Reddy et al. APL 87, 211908 (2005)

Real DOS DOS based on the Debye approximation


Debye Model

Freq

uenc

y, 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

TθD

"

#$%

&'

3x4ex

ex −1( )2 dx0

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

Dispersionω = 0 ~ k ~ ∞

ω = Under Debye

Polarization 2 transverse2 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 ≈3kBhT

λmaxT ≈hc3kB

clight = 3×108 ms

csound = 3−10 ×103 ms

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

TipLaser Reflector

SiNx Cantilever

X-Y-Z Actuator

Sample

Temperature sensor

Laser

CantileverDeflectionSensing

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 kvalues of Si nanowires, but not to the extent observed here20,21. Thepeak k of the EE nanowires is shifted to a much higher temperaturethan that of VLS nanowires, and both are significantly higher thanthat of bulk Si, which peaks at around 25K (ref. 5). This shift suggeststhat the phonon mean free path is limited by boundary scattering asopposed to intrinsic Umklapp scattering.

While the above wires were etched from high-resistivity wafers, thepeak ZT of semiconductor materials is predicted to occur at highdopant concentrations (,13 1019 cm23; ref. 22). To optimize the

ZT of EE nanowires, lower resistivity nanowires were synthesizedfrom 1021V cm B-doped p-Si Æ111æ and 1022V cm As-doped n-SiÆ100æ wafers by the standard method outlined above. Nanowiresetched from the 1022V cm and less resistive wafers, however, didnot produce devices with reproducible electrical contacts, probablyowing to greater surface roughness, as observed in TEM analysis.Consequently, more optimally doped nanowires were obtained bypost-growth gas-phase B doping of wires etched from 1021V cmwafers (see Supplementary Information). The resulting nanowireshave an average r5 36 1.4mV cm (as compared to ,10V cm forwires from low-doped wafers).

Figure 2c shows the k of small-diameter nanowires etched from 10,1021, and 1022V cm wafers. The post-growth doped nanowire(52 nm diameter) etched from a 1021V cm wafer has a slightly lowerk than the lower-doped wire of the same diameter. This smalldecrease in k may be attributed to higher rates of phonon-impurityscattering. Studies of doped and isotopically purified bulk Si haverevealed a reduction of k as a result of impurity scattering6,23,24. Owingto 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 1022V cm wafer have a much lower k than theother nanowires, probably as a result of the greater surface roughness.

In the case of the 52 nm nanowire, k is reduced to 1.660.13Wm21 K21 at room temperature. For comparison, the temper-ature-dependent k of amorphous bulk SiO2 (data points used fromhttp://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 theplot, k of these single-crystalline EE Si nanowires is comparable tothat of insulating glass. Indeed, k of the 52 nm nanowire approachesthe minimum k predicted and measured for Si: ,1Wm21 K21

(ref. 26). The resistivity of a single nanowire of comparable diameter(48 nm) was measured (see Supplementary Information) and theelectronic contribution to thermal conductivity (ke) can be estimatedfrom the Wiedemann–Franz law16. For measured r5 1.7mV cm,ke5 0.4Wm21 K21, meaning that the lattice thermal conductivity(kl5 k2 ke) is 1.2Wm21 K21.

By assuming the mean free path due to boundary scattering‘b~Fd, where F. 1 is a multiplier that accounts for the specularityof phonon scattering at the nanowire surface and d is the nanowirediameter, a model based on Boltzmann transport theory was able toexplain27 the diameter dependence of thermal conductivity in VLSnanowires, as observed in ref. 14. Because the thermal conductivity ofEE nanowires is lower and the surface is rougher than that of VLSones, it is natural to assume ‘b~d (F5 1), which is the smallestmeanfree path due to boundary scattering. However, this still cannotexplain why the phonon thermal conductivity approaches theamorphous limit for nanowires with diameters ,50 nm. In fact,theories that consider phonon backscattering, as recently proposedby ref. 21, cannot explain our observations either. The thermalconductivity in amorphous non-metals26 can be well explained by

50

b

a

40

30

20

10

0

0

4

8

0Temperature (K)

k (W

m–1

K–1

)

c

k (W

m–1

K–1

)

100 200

50 nm98 nm115 nm

115 nm

98 nm

50 nm

150 nm

75 nm52 nm

37 nm

10 Ω cm10–1 Ω cm

56 nm

115 nm

Vapour–liquid–solid nanowiresElectroless etching nanowires

300

0Temperature (K)

100 200 300

10–2 Ω cmAmorphous SiO2

Figure 2 | Thermal conductivity of the rough silicon nanowires. a, An SEMimage of a Pt-bonded EE Si nanowire (taken at 52u tilt angle). The Pt thinfilm loops near both ends of the bridgingwire are part of the resistive heatingand sensing coils on opposite suspendedmembranes. Scale bar, 2 mm. b, Thetemperature-dependent k of VLS (black squares; reproduced from ref. 14)and EE nanowires (red squares). The peak k of the VLS nanowires is175–200K, while that of the EE nanowires is above 250K. The data in thisgraph are from EE nanowires synthesized from low-doped wafers.c, Temperature-dependent k of EE Si nanowires etched from wafers ofdifferent resistivities: 10V cm (red squares), 1021V cm (green squares;arrays doped post-synthesis to 1023V cm), and1022V cm (blue squares).For the purpose of comparison, the k of bulk amorphous silica is plottedwithopen squares. The smaller highly doped EE Si nanowires have a kapproaching that of insulating glass, suggesting an extremely short phononmean free path. Error bars are shown near room temperature, and shoulddecrease with temperature. See Supplementary Information for kmeasurement calibration and error determination.

NATURE |Vol 451 | 10 January 2008 LETTERS

165Nature ©2007 Publishing Group

Phonon Boundary Scattering

Nature 451, 163 (2008)

the interfaces. These processes have been predicted to affect the kvalues of Si nanowires, but not to the extent observed here20,21. Thepeak k of the EE nanowires is shifted to a much higher temperaturethan that of VLS nanowires, and both are significantly higher thanthat of bulk Si, which peaks at around 25K (ref. 5). This shift suggeststhat the phonon mean free path is limited by boundary scattering asopposed to intrinsic Umklapp scattering.

While the above wires were etched from high-resistivity wafers, thepeak ZT of semiconductor materials is predicted to occur at highdopant concentrations (,13 1019 cm23; ref. 22). To optimize the

ZT of EE nanowires, lower resistivity nanowires were synthesizedfrom 1021V cm B-doped p-Si Æ111æ and 1022V cm As-doped n-SiÆ100æ wafers by the standard method outlined above. Nanowiresetched from the 1022V cm and less resistive wafers, however, didnot produce devices with reproducible electrical contacts, probablyowing to greater surface roughness, as observed in TEM analysis.Consequently, more optimally doped nanowires were obtained bypost-growth gas-phase B doping of wires etched from 1021V cmwafers (see Supplementary Information). The resulting nanowireshave an average r5 36 1.4mV cm (as compared to ,10V cm forwires from low-doped wafers).

Figure 2c shows the k of small-diameter nanowires etched from 10,1021, and 1022V cm wafers. The post-growth doped nanowire(52 nm diameter) etched from a 1021V cm wafer has a slightly lowerk than the lower-doped wire of the same diameter. This smalldecrease in k may be attributed to higher rates of phonon-impurityscattering. Studies of doped and isotopically purified bulk Si haverevealed a reduction of k as a result of impurity scattering6,23,24. Owingto 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 1022V cm wafer have a much lower k than theother nanowires, probably as a result of the greater surface roughness.

In the case of the 52 nm nanowire, k is reduced to 1.660.13Wm21 K21 at room temperature. For comparison, the temper-ature-dependent k of amorphous bulk SiO2 (data points used fromhttp://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 theplot, k of these single-crystalline EE Si nanowires is comparable tothat of insulating glass. Indeed, k of the 52 nm nanowire approachesthe minimum k predicted and measured for Si: ,1Wm21 K21

(ref. 26). The resistivity of a single nanowire of comparable diameter(48 nm) was measured (see Supplementary Information) and theelectronic contribution to thermal conductivity (ke) can be estimatedfrom the Wiedemann–Franz law16. For measured r5 1.7mV cm,ke5 0.4Wm21 K21, meaning that the lattice thermal conductivity(kl5 k2 ke) is 1.2Wm21 K21.

By assuming the mean free path due to boundary scattering‘b~Fd, where F. 1 is a multiplier that accounts for the specularityof phonon scattering at the nanowire surface and d is the nanowirediameter, a model based on Boltzmann transport theory was able toexplain27 the diameter dependence of thermal conductivity in VLSnanowires, as observed in ref. 14. Because the thermal conductivity ofEE nanowires is lower and the surface is rougher than that of VLSones, it is natural to assume ‘b~d (F5 1), which is the smallestmeanfree path due to boundary scattering. However, this still cannotexplain why the phonon thermal conductivity approaches theamorphous limit for nanowires with diameters ,50 nm. In fact,theories that consider phonon backscattering, as recently proposedby ref. 21, cannot explain our observations either. The thermalconductivity in amorphous non-metals26 can be well explained by

50

b

a

40

30

20

10

0

0

4

8

0Temperature (K)

k (W

m–1

K–1

)c

k (W

m–1

K–1

)

100 200

50 nm98 nm115 nm

115 nm

98 nm

50 nm

150 nm

75 nm52 nm

37 nm

10 Ω cm10–1 Ω cm

56 nm

115 nm

Vapour–liquid–solid nanowiresElectroless etching nanowires

300

0Temperature (K)

100 200 300

10–2 Ω cmAmorphous SiO2

Figure 2 | Thermal conductivity of the rough silicon nanowires. a, An SEMimage of a Pt-bonded EE Si nanowire (taken at 52u tilt angle). The Pt thinfilm loops near both ends of the bridgingwire are part of the resistive heatingand sensing coils on opposite suspendedmembranes. Scale bar, 2 mm. b, Thetemperature-dependent k of VLS (black squares; reproduced from ref. 14)and EE nanowires (red squares). The peak k of the VLS nanowires is175–200K, while that of the EE nanowires is above 250K. The data in thisgraph are from EE nanowires synthesized from low-doped wafers.c, Temperature-dependent k of EE Si nanowires etched from wafers ofdifferent resistivities: 10V cm (red squares), 1021V cm (green squares;arrays doped post-synthesis to 1023V cm), and1022V cm (blue squares).For the purpose of comparison, the k of bulk amorphous silica is plottedwithopen squares. The smaller highly doped EE Si nanowires have a kapproaching that of insulating glass, suggesting an extremely short phononmean free path. Error bars are shown near room temperature, and shoulddecrease with temperature. See Supplementary Information for kmeasurement calibration and error determination.

NATURE |Vol 451 | 10 January 2008 LETTERS

165Nature ©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

BoundaryPhonon-phonon ScatteringDefect

Increasing DefectConcentration


Phonon Defect Scattering: The Alloy Limit

The alloy limitk [W

/m-K

]

A BAxB1-x

Si GeSixGe1-x

~ 140 W/m-K

~ 60 W/m-K

~ 5 - 10 W/m-K


two suspended membranes. The details of the experimentalprocedure are available in the literature.15 Therefore, only abrief explanation is provided here. Essentially, this method isbased on the standard heat source and heat sink method,16

which was scaled down to measure the thermal conductivityof the NWs. The membrane consists of silicon nitride and aplatinum coil. An isopropyl alcohol !IPA" solution containingSi1−xGex NWs was dropped onto the membranes. A Si1−xGexNW was put across the membranes after IPA evaporation. Toenhance the thermal contacts between the NW and mem-branes, Pt/C was deposited by the FEI Quanta 3D dual fo-cused ion beam. Then, this device was packaged and placedon a variable temperature cryostat. One of the membraneswas heated by applying current to the Pt coil. The Pt coilworks as a heater and a thermometer. A fraction of the gen-erated heat was transported through the NW and reached theother membrane. The thermal conductivity of the NW wasdetermined by calculating the heat transported through theNW. The temperatures of the membranes were determinedby measuring the resistance of the Pt heater. All the thermal

conductivity measurements of the Si1−xGex NWs were car-ried out over the temperature range of 40–420 K and at#6!10−6 Torr.

Figure 3 shows the temperature dependence of the ther-mal conductivities of the Si1−xGex NWs with different Geconcentrations and diameters. The diameters, d, of the NWsamples 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 forSi0.91Ge0.09. The thermal conductivities of Si NWs and aSi0.85Ge0.15 thin film17 are shown for reference. As shown inthe figure, the thermal conductivity of the Si1−xGex NWsdecreases with decreasing NW diameter, which is consistentwith the results of a previous study by Li et al.12 However,based on the thermal conductivity of the Si0.96Ge0.04 andSi0.91Ge0.09 NWs, this diameter dependence is not as signifi-cant as in the case of Si NWs. We suspect that this is becausealloy scattering is more dominant in this case. For Si1−xGexNWs, alloy scattering and phonon boundary scatteringshould 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 Si0.85Ge0.15thin film, and Si0.8Ge0.2 bulk !the thin film and bulk data was from Ref. 17"are shown for reference.

FIG. 1. !Color online" Structures and compositions of Si1−xGex NWs: !a"SEM image of Si1−xGex NWs grown on a Si !111" substrate, $!b" and !c"%HRTEM images of Si1−xGex NWs showing their single-crystalline anddefect-free nature, and !d" EDS spectra of Si0.91Ge0.09, Si0.96Ge0.04, andSi0.996Ge0.004 NWs. The inset in !c" is the SAED pattern taken along the$"111% zone axis showing the diamond structure of the NW with the &110'growth direction.

FIG. 2. A SEM image of individual Si1−xGex NWs placed on the suspendedMEMS device.

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 fromRef. 17".

233106-2 Kim et al. Appl. Phys. Lett. 96, 233106 !2010"

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

Thin film & bulk

Kim et al, Appl. Phys. Lett. 96, 233106 (2010)

122nm

92nm

147nm

140nm 344nm330nm229nm

205nm160nm

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

BoundaryPhonon-phononScatteringDefect

Increasing DefectConcentration 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 zoneUmklapp (in German): flipped over


U-Process & Dispersion Relation



ℓl

Temperature, T/θD


Decreasing Boundary Separation

Increasing Defect Concentration

• Boundary Scattering • Defect Scattering • Phonon-Phonon Scattering

0.01 0.1 1.0Temperature, T/θD

0.01 0.1 1.0

kl

Boundary

Phonon-phononScatteringDefect


Phonon Scattering Mechanisms


Elastic scattering: boundary & defect

Inelastic scattering: phonon-phonon scattering

Temperature, T/θD

0.01 0.1 1.0

kl




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 SuperlatticeAIM = 1.15

Superlattice Period

Huxtable et al., APL 80, 1737 (2002).

Alloy limit

Acoustic impedance

E: elastic modulus

g: spring constant

Ther

mal

Con

duct

ivit

y [W

/m-K

]

Documents

Phonons lecture