classical theory expectations

© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 1

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 heat capacity per mole for most solids approaches 3NAkB at high T

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?

Cv = bT

3VC bT aT

due toelectron gas

due toatomic 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)

Cv = bT


Phonons: Atomic Lattice Vibrations

• Phonons = quantized atomic lattice vibrations ~ elastic waves

• 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

Freq

uenc

y ω

(cm

-1)

~63 meVSi opticalphonons


A Few Lattice Types• Point lattice (Bravais)

– 1D

– 2D

– 3D


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


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?


Position Momentum (k-) Space

• The Fourier transform in k-space is also a lattice• This reciprocal lattice has a lattice constant 2π/a

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!


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)

Freq

uenc

y,

Wave vector, K0 p/a

2 2

2 2x x

Yu uEt x

2 2

2 2 2

1x x

s

u ux v t

Y

sEv

sv k See C. Kittel, Ch. 4or G. Chen Ch. 3


Two Atoms per Unit Cell

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 xdtd ym k x x ydt

Fr

eque

ncy,

Wave vector, K0 p/a

LATA

LOTO

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)


The Einstein Model

• All N oscillators same frequency• Density of states in ω

(energy/freq) is a delta function

• Einstein specific heat

Freq

uenc

y,

0 2p/a

E

Wave vector, k (3 )Eg N

( )E

du dfC g ddT dT


Einstein Low-T and High-T Behavior

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

• Low-T (incorrect, drops too fast)

2

2

1( ) 3 3

1 1

E

E

E

TE B BT

T

C T Nk Nk

/2

2/

2 /

( ) 3

3

E BE

BE B

E E B

B

k T

E B k T k T

k TB k T

eC T Nke

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

Freq

uenc

y,

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

1/326sD

B

v Nk

p


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

Peter Debye (1884-1966)


Website Reminder


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/dTFr

eque

ncy,

0 Wave vector, k 2p/a

sv k

0

( ) ( ) ( )D

u T f g d

3 / 4

20

( ) 9( 1)

D T x

D B xD

T x e dxC T Nke

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


Debye Model at Low- and High-T• 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( )5D B

D

TC T Nkp

( ) 3D BC T Nk


Experimental Specific Heat

10 410 310 210 110 1

10 2

10 3

10 4

10 5

10 6

10 7

Temperature, T (K)

Spec

ific

Hea

t, C

(J/

m -

K)3

C T 3

C 3kB 4.7106 Jm3 K

D 1860 K

Diamond

ClassicalRegime

Each atom has a thermal energy of 3KBT

Spec

ific

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

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

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


Thermal Conductivity of Solids

how do we explain the variation?thermal conductivity spans ~105x(electrical conductivity spans >1020x)


Kinetic Theory of Energy Transport

z

z - z

z + z

u(z-z)

u(z+z) λqz zzzz zuzuvq

21'

Net Energy Flux / # of Molecules

2' cosz z zdu duq v vdz dz

through Taylor expansion of u

13z

du dT dTq v kdT dz dz

Integration over all the solid angles total energy flux

13

k 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:


l

Temperature, T/D

Boundary

PhononScattering

Defect

Decreasing Boundary Separation

IncreasingDefectConcentration

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

21 13 3

k Cv Cv

Temperature, T/D

0.01 0.1 1.00.01 0.1 1.0

kl

dl Tk

Boundary

PhononScatteringDefect

Increasing DefectConcentration


Temperature Dependence of Phonon KTH

C λ

low T Td

nph 0, soλ , but then

λ D (size) Td

high T 3NkB 1/T 1/T

C /1 1kTph ph

ph

en

3

d

B

T low TNk high T

Thigh

kT

Tlow


Ex: Silicon Film Thermal Conductivity

Bulk single-crystal silicon:Touloukian et al. (1970)d = 0.44 cm

singlecrystal

110 100Temperature (K)

10

100

1000

Ther

mal

Con

duct

ivity

(W m

-1K

-1)

104

bulk

1

21

31),(

i

GG nA

dACvndk

Doped polysilicon film:McConnell et al. (2001)d = 1 mmdg = 350 nmn = 1.6·1019 cm-3 boron

Undoped polysilicon film:Srinivasan et al. (2001)d = 1 mmdg = 200 nm

undoped poly-crystal

doped

films

undoped

doped

Undoped single-crystal film:Asheghi et al. (1998)d = 3 mm

Doped single-crystal film:Asheghi et al. (1999)d = 3 mmn = 1·1019 cm-3 boron

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

sizeeffect


Ex: Silicon Nanowire Thermal Conductivity

• Recall, undoped bulk crystalline silicon k ~ 150 W/m/K (previous slide)

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

Nanowire diameter


Ex: Isotope Scattering

~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

224

2

14

i

ph i s

nV Mv M

p

1 s

ph b

vD

1 expph ph B

A T Bk T


Surface Roughness Scattering in NWsWhat if you have really rough nanowires?• Surface roughness Δ ~ several nm!• Thermal conductivity scales as ~ (D/Δ)2• Can be as low as that of a-SiO2 (!) for very rough

Si nanowires

smooth Δ=1-3 Å

rough Δ=3-3.25 nm


Data and Model From…

Data…

Model…(Hot Chips

class project)


What About Electron Thermal Conductivity?

• Recall electron heat capacity

• Electron thermal conductivity

0

edu dfC E g E dEdT dT

2

2B

e e BF

k TC n kE

p

at most Tin 3D

ek Mean scattering time: e = _______

e

Temperature, T

Defect Scattering

PhononScattering

IncreasingDefect Concentration

Bulk Solidse

Electron 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

10 310 210 110 010 0

10 1

10 2

10 3

Temperature, T [K

Ther

mal

Con

duct

ivity

, k [

W/c

m-K

]

Copper

Aluminum

Defect Scattering Phonon Scattering

11

Matthiessen Rule:1 1 1 1

1 1 1 1e 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

22 21

3 2e B FF

Tk n vE

p

FE where

2qq n nm m recall electrical

conductivity

taking the ratio ek


Lorenz Number

This is remarkable!It is independent of n, m, and even !

2 2

23e BkLT 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

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

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