© 2010 eric pop, uiucece 598ep: hot chips classical theory expectations equipartition: 1/2k b t per...
TRANSCRIPT
© 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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 2
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 3
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 4
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
Fre
qu
en
cy ω
(cm
-1)
~63 meVSi opticalphonons
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 5
A Few Lattice Types
• Point lattice (Bravais)– 1D
– 2D
– 3D
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 6
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 7
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?
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 8
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)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 9
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!
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 10
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)
Fre
quen
cy,
Wave vector, K0 p/a
2 2
2 2x x
Y
u uE
t x
2 2
2 2 2
1x x
s
u u
x v t
Y
s
Ev
sv k See C. Kittel, Ch. 4or G. Chen Ch. 3
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 11
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 x
dt
d ym k x x y
dt
Fr
eq
uen
cy,
w
Wave vector, K0 p/a
LATA
LO
TO
OpticalVibrational
Modes
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 12
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)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 13
The Einstein Model
• All N oscillators same frequency
• Density of states in ω (energy/freq) is a delta function
• Einstein specific heat
Frequency
, w
0 2p/a
E
Wave vector, k
(3 )Eg N
( )E
du dfC g d
dT dT
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 14
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
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 T k T
k TB k T
eC T Nk
e
Nk e
Einstein modelOK for optical phonon
heat capacity
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 15
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
Frequency
, w
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
vN
k p
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 16
Annalen der Physik 39(4)p. 789 (1912)
Peter Debye (1884-1966)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 17
Website Reminder
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 18
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
equency
, w
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 Nk
e
people like to write:(note, includes 3x)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 19
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 BD
TC T Nk
p
( ) 3D BC T Nk
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 20
Experimental Specific Heat
10 410 310 210 110 1
10 2
10 3
10 4
10 5
10 6
10 7
Temperature, T (K)
Sp
ec
ific
He
at,
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
Speci
fic
Heat
(J/m
3-K
)
Temperature (K)
C T3
3NkBT
Diamond
In general, when T << θD 1,d d
L Lu T C T
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 21
Phonon Dispersion in Graphene
Maultzsch et al., Phys. Rev. Lett. 92, 075501 (2004) Optical
Phonons
Yanagisawa et al.,Surf. Interf. Analysis37, 133 (2005)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 22
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 23
Thermal Conductivity of Solids
how do we explain the variation?thermal conductivity spans ~105x(electrical conductivity spans >1020x)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 24
Kinetic Theory of Energy Transport
z
z - z
z + z
u(z-z)
u(z+z
) λqqz zzzz zuzuvq 21
'
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:
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 25
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:
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 26
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 1
3 3k Cv Cv
Temperature, T/D
0.01 0.1 1.00.01 0.1 1.0
kl
dl Tk
Boundary
PhononScatteringDefect
Increasing DefectConcentration
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 27
Temperature Dependence of Phonon KTH
C λ
low T Td
nph 0, so
λ , but thenλ D (size)
Td
high T 3NkB 1/T 1/T
C /11kT
ph phph
en
3
d
B
T low T
Nk high T
Thigh
kT
Tlow
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 28
Ex: Silicon Film Thermal Conductivity
Bulk single-crystal silicon:Touloukian et al. (1970)d = 0.44 cm
singlecrystal
110 100
Temperature (K)
10
100
1000
Th
erm
al C
on
du
cti
vity
(W
m-1
K-1
)
104
bulk
1
21
3
1),(
i
GG nA
d
ACvndk
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 29
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 30
Ex: Isotope Scattering
~T3
~1/T
isotope~impurity
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 31
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
1
4i
ph i s
nV M
v M
p
1 s
ph b
v
D
1exp
ph ph B
A T Bk T
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 32
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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 33
Data and Model From…
Data…
Model…(Hot Chips
class project)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 34
What About Electron Thermal Conductivity?
• Recall electron heat capacity
• Electron thermal conductivity
0
e
du dfC E g E dE
dT dT
2
2B
e e BF
k TC n k
E
p
at most Tin 3D
ek Mean scattering time: te = _______
e
Temperature, T
Defect Scattering
PhononScattering
IncreasingDefect Concentration
Bulk Solids
eElectron Scattering Mechanisms
Grain Grain Boundary
• Defect or impurity scattering• Phonon scattering• Boundary scattering (film
thickness, grain boundary)
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 35
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
Th
erm
al
Co
nd
uc
tiv
ity,
k
[W/c
m-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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 36
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 v
E
p
FE where
2qq n n
m
m recall electricalconductivity
taking the ratio ek
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 37
Lorenz Number
This is remarkable!It is independent of n, m, and even !
2 2
23e Bk
LT 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
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 38
Amorphous Material Thermal Conductivity
Amorphous (semi)metals: bothelectrons & phonons contribute Amorphous dielectrics:
K saturates at high T (why?)
a-Si
a-SiO2
GeTe
© 2010 Eric Pop, UIUC ECE 598EP: Hot Chips 39
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