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NanotoMacroscale Transport Processes (Microscale Heat Transfer) A Quick Review. Ronggui Yang Department of Mechanical Engineering ECME 136, 427 UCB University of Colorado Boulder, CO 803090427 Tel: (303) 7351003, Fax: (303) 4923498 Email: Ronggui.Yang@Colorado.Edu
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NanotoMacroscale Transport Processes (Microscale Heat Transfer)A Quick Review
Ronggui Yang
Department of Mechanical Engineering
ECME 136, 427 UCB
University of Colorado
Boulder, CO 803090427
Tel: (303) 7351003, Fax: (303) 4923498
Email: Ronggui.Yang@Colorado.Edu
http://spot.colorado.edu/~yangr
NanotoMacroscale Transport Processes (Microscale Heat Transfer)
This course focuses on understanding thermal energy transport across all scales and particularly when device dimensions approaches the fundamental lengthsscales of the charge and energy carriers in nanostructures. The course will address size effects on thermal and fluid transport in nanostructures and how to possibly engineer novel effective transport properties. Moreover, the current state of the art developments in the microscale thermal transport field will be reviewed. Topics include the energy levels, the statistical behavior and internal energy, energy transport in the forms of waves and particles, scattering and energy conversation processes, Boltzmann equation and derivation of classical laws, deviation from classical laws at nanoscale and their appropriate descriptions, with applications in nanotechnology and microtechnology.
Textbook
Gang Chen, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. New York: Oxford University Press, 2005. ISBN: 019515942X.
Grading:
Biweekly homework 30%, midterm 25%, final exam 45%
Learning Goals
Understand, Analyze, Innovate
Required Textbook
Gang Chen, Nanoscale Energy Transport and Conversion: A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. New York: Oxford University Press, 2005. ISBN: 019515942X.
Recommended Books
C. Kittel, Introduction to Solid State Physics, 7th Ed., Wiley, 1996.
N.W. Ashroft and N.D. Mermin, Solid State Physics, Brooks Cole, 1976
C. Kittel and H. Kroemer, Thermal Physics, 2nd Ed., Freeman and Company, 1980.
M. Lundstrom, Fundamentals of Carrier Transport, 2nd Ed, Cambridge University Press, 2000.
Z.M. Zhang, Nano/Microscale Heat Transfer, Wiley, 2007.
Lecture Notes References
MIT Courses: 2.57 NanotoMacroscale Transport Processes
6.728 Quantum Mechanics and Statistical Mechanics
6.730 Solid State Physics
6.720 Physics of Semiconductor Devices
6.772 Physics of Semiconductor Compounds and Devices
Lecture Notes by colleagues in other universities
~5 m (SEM)
diatom
30 m
DNA
proteins
nm
Simple molecules
<1nm
bacteria
1 m
nm
m
mm
1010
109
108
107
106
105
104
103
102
SOI transistor
width 0.12m
semiconductor
nanocrystal (CdSe)
5nm
Circuit design
Copper wiring
width 0.2m
Nanometer memory element
(Lieber) 1012 bits/cm2 (1Tbit/cm2)
IBM PowerPC750TM Microprocessor
7.56mm×8.799mm
6.35×106 transistors
Length ScalesNanoBio
Interface
Nanoelectronics
Conventional
Spintronics
Molectronics
Nanotubes
NANO
Nanosciences
Nanostructures
Nanomaterials
Nanophotonics
Photonic Crystals
Plasmonic Photonics
Energy
Conversion, Storage, and Transportation
Nanotechnology Landscape
wavelength
electrons
phonons
photons
air molecules
10100 nm
1 nm
0.110
0.01 nm
Characteristic Lengths of Energy CarriersRoom Temperature
Microscopic Picture of Thermal Transport
Th
Tc
Air molecules
Phonon gas
ε = nhν
Free electron model
Th
Th
Tc
Tc
Free electron
Atom core
Conduction  random motion of energy carriers
Gas Molecules
Electrons
Phonons
Hot
Hot
Cold
p

Cold
To understand transport and energy conversion, we need to know:
How much energy/momentum can a particle have?
How many particles have the specified energy E?
How fast do they move?
How far can they travel?
How do they interact with each other?
a+b
x
Energy
Repulsion
=
¥
U
Harmonic Force
Approximation
Interatomic
Distance
Attraction
Equilibrium Position
Transmission wave
Reflection wave
Energy barrier U0
ENERGY AND
Incoming wave
WAVEFUNCTION
n=3
n=2
n=1
x
U=0
u
r
Quantum Mechanics 101
Problem 2.10
Next Lecture
xn+1
yn1
xn
yn
1D Lattice with Diatomic BasisConsider a linear diatomic chain of atoms (1D model for a crystal like NaCl):
In equilibrium:
Applying Newton’s second law and the nearestneighbor approximation to this system gives a dispersion relation with two “branches”:
(k) 0 as k 0 acoustic modes (M1 and M2 move in phase)
+(k) max as k 0 optical modes (M1 and M2 move out of phase)
These two branches may be sketched schematically as follows:
optical
gap in allowed frequencies
acoustic
In a real 3D solid the dispersion relation will differ along different directions in kspace. In general, for a p atom basis, there are 3 acoustic modes and p1 groups of 3 optical modes, although for many propagation directions the two transverse modes (T) are degenerate.
wD
Phonon Frequency w
wE
3D Solids: Heat Capacity
Einstein
Model
Debye
Model
(a)
Optical
Phonons
Acoustic Phonons
Density of States
Phonon Frequency w
=
C
v
l
1
3
Thermal ConductivityC ~ constant
l
Phonon mfp
k
lum ~eQ/ T
Specific heat
Phonon group velocity
C ~ T d
l= vt
Matthiessen Rule:
lst ~ lum
lboundary ~ constant
limpurity ~ weak dependance on T
T
Low T:
High T: Umklapp phonon scattering:
lum ~ eQ/ T
If T > Q, C ~ constant
If T << Q, C ~ T d (d: dimension)
Specific heat :
Plot U(x) for a 1D crystal lattice:
Simple and crude finitesquarewell model:
U
U = 0
Can we justify this model? How can one replace the entire lattice by a constant (zero) potential?
By using periodic boundary conditions for a cubic solid with edge L and volume V = L3, we define the set of allowed wave vectors:
This shows that the volume in kspace per solution is:
And thus the density of states in kspace is:
Since the FEG is isotropic, the surface of constant E in kspace is a sphere. Thus for a metal with N electrons we can calculate the maximum k value (kF) and the maximum energy (EF).
ky
kx
kz
Fermi sphere
We often need to know the density of electron states, which is the number of states per unit energy, so we can quickly calculate it:
The differential number of electron states in a range of energy dE or wavevector dk is:
This allows:
Now using the general relation:
we get:
Quantum mechanics showed that the occupation of electron states is governed by the Pauli exclusion principle, and that the probability of occupation of a state with energy E at temperature T is:
where = chemical potential EF for kT << EF
Very low temperature measurements reveal:
Results for simple metals (in units mJ/mol K) show that the FEG values are in reasonable agreement with experiment, but are always too high:
The discrepancy is “accounted for” by defining an effective electron mass m* that is due to the neglected electronion interactions
Recall the electrostatic potential energy in a crystalline solid along a line passing through a line of atoms:
bare ions
solid
Along a line parallel to this but running between atoms, the divergences of the periodic potential energy are softened:
U
x
A simple 1–D mathematical model that captures the periodicity of such a potential is:
• Metals are solids with incompletely filled energy bands
• Semiconductors and insulators have a completely filled or empty bands and an energy gap separating the highest filled and lowest unfilled band. Semiconductors have a small energy gap (Eg < 2.0 eV).
6
Si
Cu
5
GaAs
4
3
2
Energy (eV)
Direct
Band Gap
Indirect
Ban Gap
1
Eg=1.42 eV
Eg=1.12 eV
E
f
0
1
2
3
4
G
G
G
L
[111]
[100]
X
[111]
[100]
X
[111]
[100]
X
L
L
Fermi Level
v

d
k
e
E
=
dt
h
Dynamics of Electrons in a BandNow we see that the external electric field causes a change in the k vectors of all electrons:
E
If the electrons are in a partially filled band, this will break the symmetry of electron states in the 1st BZ and produce a net current. But if they are in a filled band, even though all electrons change k vectors, the symmetry remains, so J = 0.
When an electron reaches the 1st BZ edge (at k = /a) it immediately reappears at the opposite edge (k = /a) and continues to increase its k value.
kx
v
As an electron’s k value increases, its velocity increases, then decreases to zero and then becomes negative when it reemerges at k = /a!!
kx
Thus, an AC current is predicted to result from a DC field! (Bloch oscillations)
The effective mass is inversely proportional to the curvature of the energy band.
Near the bottom of a nearlyfree electron band m* is approximately constant, but it increases dramatically near the inflection point and even becomes negative (!) near the zone edge.
Blackbody Radiation
Emissive Power
Light
Continuum Theory
Metal
Electrodes
Experiments
Wavelength
.
Current
Gas Absorption Spectrum
de Broglie’s Materials Wave
656.5 nm
486.3 nm
364.7 nm
434.2 nm
Wave vs. Particle?Hot
Cold
x
Q1>2
vxt
x
1
2
Q2>1
x
How wavelike energy carriers transport energy?
Generalizing the simple kinetic theory
Landauer formulism
Net Energy Flux
Energy Storage (J/m3)
The key is how to calculate energy transmissivity:
Reflection and Refraction: EM Waves
n2<0?
The Snell’s law indicated momentum conservation in xdirection
If n2<0, n1>0, the refraction light will be bended as the right figure.
Metamaterials
Reflectivity
Transmissivity
What happens if n1>n2?
Reflection and Transmission
DENSITY OF STATES
n=2
r12f1
t12f1
n=1
f1
r21f2
t21f2
ENERGY
f2
(a) Electrons in Quantum Well
Single Interface
Multiple Interfaces: Wave Effects
(b) Phonons in Superlattice
Mode Coupling – Multiple Carriers
wavelength
electrons
phonons
photons
air molecules
10100 nm
1 nm
0.110
0.01 nm
Characteristic Lengths of Energy CarriersWavelength vs. Diameter
Typical
Nanowire
Specular
Wavelength vs. Roughness
Specularity (Ziman)
Diffuse scattering Incoherence
Diffuse
103
102
101
100
10+1
Roughness / Wavelength
To Nonequilibrium
Boltzmann Transport Equation0.01
0.1
Phonon Scattering and Mean Free Path
Phonon Scattering Mechanisms
Decreasing Boundary
Separation
l
Increasing
Defect
Concentration
PhononScattering
Defect
Boundary
Temperature, T/qD
Electronic Bandstructure
COLD SIDE
I
I
Phonons
L=10100 nm
l=1 nm

Electrons
L=10100 nm
l=1050 nm
+
N
P
I
INSULATOR
SEMICONDUCTOR
HOT SIDE
SEMIMETAL
S
METAL
s
ZT
k
Reducing k In Bulk Materials
Alloy, 1950s (Ioffe)
Nanostructured Materials
Wanted: Phonon Glass / Electron Crystal
carrier concentration
3w Method
Optical PumpProbe Method
Thermal Measurement of 1D Nanostructures
NanoResolution Thermal Microscopy
PROBE
HEATING
“PUMP”
FILM
GENERATION
REFLECTION
SUBSTRATE
x
+