/
Classical Theory Expectations Classical Theory Expectations

Classical Theory Expectations - PowerPoint Presentation

jane-oiler
jane-oiler . @jane-oiler
Follow
377 views
Uploaded On 2018-03-14

Classical Theory Expectations - PPT Presentation

Equipartition 12k B T per degree of freedom In 3D electron gas this means 32k B T per electron In 3D atomic lattice this means 3k B T per atom why So one would expect C V du ID: 651239

heat lattice conductivity phonon lattice heat phonon conductivity thermal high scattering capacity frequency debye model atomic electron wave energy metals amp phonons

Share:

Link:

Embed:

Download Presentation from below link

Download Presentation The PPT/PDF document "Classical Theory Expectations" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

Slide1

Classical Theory Expectations

Equipartition: 1/2kBT per degree of freedomIn 3-D electron gas this means 3/2kBT per electronIn 3-D atomic lattice this means 3kBT per atom (why?)So one would expect: CV = du/dT = 3/2nekB + 3nakBDulong & Petit (1819!) had found the heat capacity per mole for most solids approaches 3NAkB at high T

32

Molar heat capacity @ high T

 25 J/mol/KSlide2

Heat Capacity: Real Metals

So far we’ve learned about heat capacity of electron gasBut evidence of linear ~T dependence only at very low TOtherwise CV = constant (very high T), or ~T3 (intermediate)Why?33

C

v

=

bT

due

to

electron gas

due

to

atomic latticeSlide3

Heat Capacity: Dielectrics vs. Metals

Very high T: CV = 3nkB (constant) both dielectrics & metalsIntermediate T: CV ~ aT3 both dielectrics & metalsVery low T: CV ~ bT metals only (electron contribution)34

C

v

=

bTSlide4

35

Phonons: Atomic Lattice VibrationsPhonons = quantized atomic lattice vibrations ~ elastic wavesTransverse (u ^ k) vs. longitudinal modes (u || k), acoustic vs. optical“Hot phonons” = highly occupied modes above room temperature

CO2 moleculevibrations

transverse

small k

transverse

max k=2

p

/a

k

Graphene Phonons [100]

200 meV

160 meV

100 meV

26

meV

=

300 K

Frequency

ω

(cm

-1

)

~63

meV

Si optical

phononsSlide5

A Few Lattice Types

Point lattice (Bravais)1D2D3D36Slide6

Primitive Cell and Lattice Vectors

Lattice = regular array of points {Rl} in space repeatable by translation through primitive lattice vectorsThe vectors ai are all primitive lattice vectorsPrimitive cell: Wigner-Seitz37Slide7

Silicon (Diamond) Lattice

Tetrahedral bond arrangement2-atom basisEach atom has 4 nearest neighbors and 12 next-nearest neighborsWhat about in (Fourier-transformed) k-space?38Slide8

Position

 Momentum (k-) SpaceThe Fourier transform in k-space is also a latticeThis reciprocal lattice has a lattice constant 2π/a39k

Sa(k)Slide9

Atomic Potentials and Vibrations

Within small perturbations from their equilibrium positions, atomic potentials are nearly quadraticCan think of them (simplistically) as masses connected by springs!40Slide10

Vibrations in a Discrete 1D Lattice

Can write down wave equationVelocity of sound (vibration propagation) is proportional to stiffness and inversely to mass (inertia)41

See C.

Kittel

, Ch. 4

or G. Chen Ch. 3 Slide11

Two Atoms per Unit Cell

42

Lattice Constant,

a

x

n

y

n

y

n-1

x

n+1

Frequency,

w

Wave vector, K

0

p

/a

LA

TA

LO

TO

Optical

Vibrational

ModesSlide12

Energy Stored in These Vibrations

Heat capacity of an atomic latticeC = du/dT =Classically, recall C = 3Nk, but only at high temperatureAt low temperature, experimentally C  0Einstein model (1907)All oscillators at same, identical frequency (ω = ωE)Debye model (1912)Oscillators have linear frequency distribution (ω = vsk)43Slide13

The Einstein Model

All N oscillators same frequencyDensity of states in ω (energy/freq) is a delta functionEinstein specific heat44

Frequency,

w

0

2p

/a

Wave vector,

kSlide14

Einstein Low-T and High-T Behavior

High-T (correct, recover Dulong-Petit):Low-T (incorrect, drops too fast)45

Einstein model

OK for

optical phonon

heat capacitySlide15

The Debye Model

Linear (no) dispersion with frequency cutoffDensity of states in 3D: (for one polarization, e.g. LA) (also assumed isotropic solid, same vs in 3D)N acoustic phonon modes up to ωDOr, in terms of Debye temperature46

Frequency,

w

0

Wave vector,

k

2p

/a

k

D

roughly corresponds to

max lattice wave vector (2

π

/a)

ω

D

roughly corresponds to

max acoustic phonon frequencySlide16

47

Annalen der Physik 39(4)p. 789 (1912)Peter Debye (1884-1966)Slide17

Website Reminder

48Slide18

The Debye Integral

Total energyMultiply 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 upC = du/dT49

Frequency,

w

0

Wave vector,

k

2p

/a

people like to write:

(note, includes 3x)Slide19

Debye Model at Low- and High-T

50At low-T (< θD/10):At high-T: (> 0.8 θD)“Universal” behavior for all solidsIn practice: θD ~ fitting parameter to heat capacity dataθD is related to “stiffness” of solid as expectedSlide20

Experimental Specific Heat

51ClassicalRegime

Each atom has a thermal energy

of 3

K

B

T

Specific Heat (J/m

3

-K)

Temperature (K)

C  T3

3NkBT

Diamond

In general, when T <<

θ

D

Slide21

Phonon Dispersion in Graphene

52Maultzsch et al., Phys. Rev. Lett. 92, 075501 (2004)OpticalPhonons

Yanagisawa

et al.

,

Surf.

Interf

. Analysis

37, 133 (2005)Slide22

Heat Capacity and Phonon Dispersion

Debye model is just a simple, elastic, isotropic approximation; be careful when you apply itTo be “right” one has to integrate over phonon dispersion ω(k), along all crystal directionsSee, e.g. http://www.physics.cornell.edu/sss/debye/debye.html53Slide23

Thermal Conductivity of Solids

54how do we explain the variation?thermal conductivity spans ~105x(electrical conductivity spans >1020x)Slide24

Kinetic Theory of Energy Transport

55z

z - z

z +

z

u(z-

z

)

u(z+

z)

λ

q

q

z

Net

Energy

Flux / # of Molecules

through Taylor expansion of

u

Integration over all the solid angles

 total energy flux

Thermal conductivity:Slide25

Simple Kinetic Theory Assumptions

Valid for particles (“beans” or “mosquitoes”)Cannot handle wave effects (interference, diffraction, tunneling)Based on BTE and RTAAssumes local thermodynamic equilibrium: u = u(T)Breaks down when L ~ _______ and t ~ _________Assumes single particle velocity and mean free pathBut we can write it a bit more carefully:56Slide26

Phonon MFP and Scattering Time

Group velocity only depends on dispersion ω(k)Phonon scattering mechanismsBoundary scatteringDefect and dislocation scatteringPhonon-phonon scattering57Slide27

Temperature Dependence of Phonon K

TH58

C

λ

low T

 T

d

n

ph

 0, so

λ

 , but then

λ

 D (size)

 T

d

high T

3Nk

B

 1/T

 1/TSlide28

Ex: Silicon Film Thermal Conductivity

59Slide29

Ex: Silicon Nanowire Thermal Conductivity

Recall, undoped bulk crystalline silicon k ~ 150 W/m/K (previous slide)60

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

Nanowire diameterSlide30

Ex: Isotope Scattering

61

~T3

~1/T

isotope

~impuritySlide31

Why the Variation in K

th?A: Phonon λ(ω) and dimensionality (D.O.S.)Do C and v change in nanostructures? (1D or 2D)Several mechanisms contribute to scatteringImpurity mass-difference scatteringBoundary & grain boundary scatteringPhonon-phonon scattering62Slide32

Surface Roughness Scattering in NWs

What if you have really rough nanowires?Surface roughness Δ ~ several nm!Thermal conductivity scales as ~ (D/Δ)2Can be as low as that of a-SiO2 (!) for very rough Si nanowires63

smooth

Δ

=1-3 Å

rough

Δ

=3-3.25 nmSlide33

Data and Model From…

64

Data…

Model…

(Hot Chips

class project)Slide34

What About Electron Thermal Conductivity?

Recall electron heat capacityElectron thermal conductivity65at most Tin 3D

Mean

scattering time:

t

e

=

_______

Electron Scattering

Mechanisms

Defect or impurity scattering

Phonon scattering

Boundary scattering (film thickness, grain boundary

)Slide35

Ex: Thermal Conductivity of Cu and Al

Electrons dominate k in metals66 Matthiessen Rule:Slide36

Wiedemann-Franz Law

Wiedemann & Franz (1853) empirically saw ke/σ = const(T)Lorenz (1872) noted ke/σ proportional to T67

where

recall electrical

conductivity

taking the ratioSlide37

Lorenz Number

68This is remarkable!It is independent of n, m, and even  !

L = /T 10

-8

W

/K

2

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

Agreement with experiment is quite good, although

L ~ 10x lower when T ~ 10 K… why?!

ExperimentallySlide38

Amorphous Material Thermal Conductivity

69Amorphous (semi)metals: bothelectrons & phonons contribute

Amorphous dielectrics:

K saturates at high T (why?)

a-Si

a-SiO

2

GeTeSlide39

Summary

Phonons dominate heat conduction in dielectricsElectrons dominate heat conduction in metals (but not always! when not?!)Generally, C = Ce + Cp and k = ke + kpFor C: remember T dependence in “d” dimensionsFor k: remember system size, carrier λ’s (Matthiessen)In metals, use WFL as rule of thumb70