Classical Theory Expectations - [email protected]/files/ece598/L5L6L7_Phonons.pdfClassical Theory Expectations ... Heat Capacity: Real Metals ... Experimental Specific

© 2008 Eric Pop, UIUC ECE 598EP: Hot Chips

Classical Theory Expectations

• Equipartition: 1/2kBT per degree of freedom

• In 3-D electron gas this means 3/2kBT per electron

• In 3-D atomic lattice this means 3kBT per atom (why?)

• So one would expect: CV = du/dT = 3/2nekB + 3nakB

• Dulong & Petit (1819!) had found the molar heat capacity

of most solids approaches 3NAkB at high T

32

Molar heat capacity @ high T 25 J/mol/K


Heat Capacity: Real Metals

• So far we’ve learned about heat capacity of electron gas

• But evidence of linear ~T dependence only at very low T

• Otherwise CV = constant (very high T), or ~T3 (intermediate)

• Why?

33

Cv = bT

3

VC bT aT

due to

electron gas

due to

atomic lattice


Heat Capacity: Dielectrics vs. Metals

• Very high T: CV = 3nkB (constant) both dielectrics & metals

• Intermediate T: CV ~ aT3 both dielectrics & metals

• Very low T: CV ~ bT metals only (electron contribution)

34

Cv = bT

© 2008 Eric Pop, UIUC ECE 598EP: Hot Chips 35

Phonons: Atomic Lattice Vibrations

• Phonons = quantized atomic lattice vibrations

• Transverse (u ^ k) vs. longitudinal modes (u || k), acoustic vs. optical

• “Hot phonons” = highly occupied modes above room temperature

CO2 moleculevibrations

)](exp[),( tiit rkAru

transversesmall k

transversemax k=2p/a

k

Graphene Phonons [100]

200 meV

160 meV

100 meV

26 meV =

300 K

Fre

qu

en

cy ω

(cm

-1)


A Few Lattice Types

• Point lattice (Bravais)

– 1D

– 2D

– 3D

36


Primitive Cell and Lattice Vectors

• Lattice = regular array of points Rl in space repeatable

by translation through primitive lattice vectors

• The vectors ai are all primitive lattice vectors

• Primitive cell: Wigner-Seitz

37


Silicon (Diamond) Lattice

• Tetrahedral bond arrangement

• 2-atom basis

• Each atom has 4 nearest neighbors and 12 next-nearest

neighbors

• What about in (Fourier-transformed) k-space?

38


Position Momentum (k-) Space

• The Fourier transform in k-space is also a lattice

• This reciprocal lattice has a lattice constant 2π/a

39

k

Sa(k)


Atomic Potentials and Vibrations

• Within small perturbations from their equilibrium

positions, atomic potentials are nearly quadratic

• Can think of them (simplistically) as masses connected

by springs!

40


Vibrations in a Discrete 1D Lattice

• Can write down wave equation

• Velocity of sound (vibration

propagation) is proportional to

stiffness and inversely to mass

(inertia)

41

Fre

quency,

Wave vector, K0 p/a


Two Atoms per Unit Cell

42

Lattice Constant, a

xn ynyn-1 xn+1

2

1 12

2

2 12

2

2

nn n n

nn n n

d xm k y y x

dt

d ym k x x y

dt

Fre

quency,

Wave vector, K0 p/a

LATA

LO

TO

OpticalVibrational

Modes


Energy Stored in These Vibrations

• Heat capacity of an atomic lattice

• C = du/dT =

• Classically, recall C = 3Nk, but only at high temperature

• At low temperature, experimentally C 0

• Einstein model (1907)

– All oscillators at same, identical frequency (ω = ωE)

• Debye model (1912)

– Oscillators have linear frequency distribution (ω = vsk)

43


The Einstein Model

• All N oscillators same frequency

• Density of states in ω

(energy/freq) is a delta function

• Einstein specific heat

44

Fre

quency,

0 2p/a

E

Wave vector, k

(3 )Eg N

( )

E

du dfC g d

dT dT


Einstein Low-T and High-T Behavior

• High-T (correct, recover Dulong-Petit):

• Low-T (incorrect, drops too fast)

45

2

2

1( ) 3 3

1 1

E

E

E

T

E B BT

T

C T Nk Nk

/2

2/

2/

( ) 3

3

E B

E

BE B

E E B

B

k T

E B k Tk T

k T

B k T

eC T Nk

e

Nk e

Einstein modelOK for optical phonon

heat capacity


The Debye Model

• Linear (no) dispersion

with frequency cutoff

• Density of states in 3D:

(for one polarization, e.g. LA)

(also assumed isotropic solid, same vs in 3D)

• N acoustic phonon modes up to ωD

• Or, in terms of Debye temperature

46

Fre

quency,

0

sv k

Wave vector, k 2p/a

2

2 32 s

gv

p

kD roughly corresponds to max lattice wave vector (2π/a)

ωD roughly corresponds tomax acoustic phonon frequency

1/3

26sD

B

vN

k p

© 2008 Eric Pop, UIUC ECE 598EP: Hot Chips 47

Annalen der Physik 39(4)p. 789 (1912)

Peter Debye (1884-1966)


The Debye Integral

• Total energy

• Multiply by 3 if assuming all

polarizations identical (one LA,

and 2 TA)

• Or treat each one separately

with its own (vs,ωD) and add

them all up

• C = du/dT

48

Fre

quency,

0 Wave vector, k 2p/a

sv k

0

( ) ( ) ( )D

u T f g d

3 / 4

2

0

( ) 9( 1)

D T x

D B x

D

T x e dxC T Nk

e

people like to write:(note, includes 3x)


Debye Model at Low- and High-T

49

• At low-T (< θD/10):

• At high-T: (> 0.8 θD)

• “Universal” behavior for all solids

• In practice: θD ~ fitting parameter

to heat capacity data

• θD is related to “stiffness” of solid

as expected

3412

( )5

D B

D

TC T Nk

p

( ) 3D BC T Nk


Experimental Specific Heat

50

104

103

102

101

101

102

103

104

105

106

107

Temperature, T (K)

Sp

ecif

ic H

ea

t, C

(J

/m

-K)

3

C T3

C 3kB 4.7 106 J

m3 K

D 1860 K

Diamond

ClassicalRegime

Each atom has a thermal energy of 3KBT

Specific

Heat

(J/m

3-K

)

Temperature (K)

C T3

3NkBT

Diamond

In general, when T << θD

1,d d

L Lu T C T


Phonon Dispersion in Graphene

51

Maultzsch et al., Phys. Rev. Lett. 92, 075501 (2004) Optical

Phonons

Yanagisawa et al.,Surf. Interf. Analysis37, 133 (2005)


Heat Capacity and Phonon Dispersion

• Debye model is just a simple, elastic, isotropic approximation; be

careful when you apply it

• To be “right” one has to integrate over phonon dispersion ω(k),

along all crystal directions

• See, e.g. http://www.physics.cornell.edu/sss/debye/debye.html

52


Thermal Conductivity of Solids

53

how do we explain this mess?

http://www.physics.cornell.edu/sss/debye/debye.html


Kinetic Theory of Energy Transport

54

z

z - z

z + z

u(z-z)

u(z+z)

λqz zzzz zuzuvq 2

1'

Net Energy Flux / # of Molecules

2' cosz z z

du duq v v

dz dz

through Taylor expansion of u

1

3z

du dT dTq v k

dT dz dz

Integration over all the solid angles total energy flux

1

3k CvThermal conductivity:


Simple Kinetic Theory Assumptions

• Valid for particles (“beans” or “mosquitoes”)

– Cannot handle wave effects (interference, diffraction, tunneling)

• Based on BTE and RTA

• Assumes local thermodynamic equilibrium: u = u(T)

• Breaks down when L ~ _______ and t ~ _________

• Assumes single particle velocity and mean free path

– But we can write it a bit more carefully:

55


l

Temperature, T/D

Boundary

Phonon

Scattering

Defect

Decreasing Boundary Separation

Increasing

Defect

Concentration

0.01 0.1 1.0

Phonon MFP and Scattering Time

• Group velocity only depends on dispersion ω(k)

• Phonon scattering mechanisms– Boundary scattering

– Defect and dislocation scattering

– Phonon-phonon scattering

56

21 1

3 3k Cv Cv

Temperature, T/D

0.01 0.1 1.00.01 0.1 1.0

kl

dl Tk

Boundary

Phonon

ScatteringDefect

Increasing Defect

Concentration


Temperature Dependence of Phonon KTH

57

C λ

low T Td

nph 0, so

λ , but then

λ D (size)

Td

high T 3NkB 1/T 1/T

C /11kT

ph ph

ph

en

3

d

B

T lowT

Nk high T

ThighkT

Tlow


Ex: Silicon Film Thermal Conductivity

58

Bulk single-crystal silicon:

Touloukian et al. (1970)

d = 0.44 cmsingle

crystal

110 100

Temperature (K)

10

100

1000

Th

erm

al C

on

du

cti

vit

y (

W m

-1K

-1)

104

bulk

1

21

3

1),(

i

G

G nAd

ACvndk

Doped polysilicon film:

McConnell et al. (2001)

d = 1 m

dg = 350 nm

n = 1.6·1019 cm-3 boron

Undoped polysilicon film:

Srinivasan et al. (2001)

d = 1 m

dg = 200 nm

undoped poly-

crystal

doped

films

undoped

doped

Undoped single-crystal film:

Asheghi et al. (1998)

d = 3 m

Doped single-crystal film:

Asheghi et al. (1999)

d = 3 m

n = 1·1019 cm-3 boron

McConnell, Srinivasan, and Goodson, JMEMS 10, 360-369 (2001)

size

effect


Ex: Silicon Nanowire Thermal Conductivity

• Recall, undoped bulk

crystalline silicon k ~ 150

W/m/K (previous slide)

59

Li, Appl. Phys. Lett. 83, 2934 (2003)

Nanowire diameter


Ex: Isotope Scattering

60

~T3

~1/T

isotope~impurity


Why the Variation in Kth?

• A: Phonon λ(ω) and dimensionality (D.O.S.)

• Do C and v change in nanostructures? (1D or 2D)

• Several mechanisms contribute to scattering

– Impurity mass-difference scattering

– Boundary & grain boundary scattering

– Phonon-phonon scattering

61

224

2

1

4

i

ph i s

nV M

v M

p

1 s

ph b

v

D

1exp

ph ph B

A T Bk T


What About Electron Thermal Conductivity?

• Recall electron heat capacity

• Electron thermal conductivity

62

0

e

du dfC E g E dE

dT dT

2

2

Be e B

F

k TC n k

E

p

at most Tin 3D

ek Mean scattering time:e = _______

e

Temperature, T

Defect

Scattering

Phonon

Scattering

Increasing

Defect Concentration

Bulk Solids

eElectron Scattering Mechanisms

Grain Grain Boundary

• Defect or impurity scattering

• Phonon scattering

• Boundary scattering (film

thickness, grain boundary)


Ex: Thermal Conductivity of Cu and Al

• Electrons dominate k in metals

63

103

102

101

100

100

101

102

103

Temperature, T [K

Th

erm

al

Co

nd

uc

tiv

ity

, k

[W

/cm

-K]

Copper

Aluminum

Defect Scattering Phonon Scattering

1

1

Matthiessen Rule:

1 1 1 1

1 1 1 1

e defect boundary phonon

e defect boundary phonon


Wiedemann-Franz Law

• Wiedemann & Franz (1853) empirically saw ke/σ = const(T)

• Lorenz (1872) noted ke/σ proportional to T

64

22 21

3 2e B F

F

Tk n v

E

p

FE where

2qq n n

m

recall electrical

conductivity

taking the ratio ek


Lorenz Number

65

This is remarkable!

It is independent of n,

m, and even !

2 2

23

e BkL

T q

p

L = /T 10-8 WΩ/K2

Metal 0 ° C 100 °C

Cu 2.23 2.33

Ag 2.31 2.37

Au 2.35 2.40

Zn 2.31 2.33

Cd 2.42 2.43

Mo 2.61 2.79

Pb 2.47 2.56

8 22.45 10 WΩ/KL

Agreement with experiment is

quite good, although L ~ 10x

lower when T ~ 10 K… why?!

Experimentally


Amorphous Material Thermal Conductivity

66

Amorphous (semi)metals: bothelectrons & phonons contribute

Amorphous dielectrics:K saturates at high T (why?)

a-Si

a-SiO2

GeTe


Summary

• Phonons dominate heat conduction in dielectrics

• Electrons dominate heat conduction in metals

(but not always! when not?!)

• Generally, C = Ce + Cp and k = ke + kp

• For C: remember T dependence in “d” dimensions

• For k: remember system size, carrier λ’s (Matthiessen)

• In metals, use WFL as rule of thumb

67

Classical Theory Expectations - [email protected]/files/ece598/L5L6L7_Phonons.pdfClassical Theory Expectations ... Heat Capacity: Real Metals ... Experimental Specific

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