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
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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
Ω
/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