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Nano-to-Macroscale Transport Processes (Microscale Heat Transfer) A Quick Review. Ronggui Yang Department of Mechanical Engineering ECME 136, 427 UCB University of Colorado Boulder, CO 80309-0427 Tel: (303) 735-1003, Fax: (303) 492-3498 Email: Ronggui.Yang@Colorado.Edu

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slide1

Nano-to-Macroscale Transport Processes (Microscale Heat Transfer)A Quick Review

Ronggui Yang

Department of Mechanical Engineering

ECME 136, 427 UCB

University of Colorado

Boulder, CO 80309-0427

Tel: (303) 735-1003, Fax: (303) 492-3498

Email: Ronggui.Yang@Colorado.Edu

http://spot.colorado.edu/~yangr

slide2

Nano-to-Macroscale Transport Processes (Microscale Heat Transfer)

This course focuses on understanding thermal energy transport across all scales and particularly when device dimensions approaches the fundamental lengths-scales 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:

Bi-weekly homework 30%, midterm 25%, final exam 45%

Learning Goals

Understand, Analyze, Innovate

course objectives
Course Objectives
  • Students in this course will:
    • Gain an understanding of the fundamental elements of solid-state physics.
    • Develop skills to derive continuum physical properties from sub-continuum principles.
    • Apply statistical and physical principles to describe energy transport in modern small-scale materials and devices.
slide4

No Pains, No Gains!

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 Nano-to-Macroscale 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

length scales

red blood cell

~5 m (SEM)

diatom

30 m

DNA

proteins

nm

Simple molecules

<1nm

bacteria

1 m

nm

m

mm

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

SOI transistor

width 0.12m

semiconductor

nanocrystal (CdSe)

5nm

Circuit design

Copper wiring

width 0.2m

Nanometer memory element

(Lieber) 1012 bits/cm2 (1Tbit/cm2)

IBM PowerPC750TM Microprocessor

7.56mm×8.799mm

6.35×106 transistors

Length Scales
slide7

Biology

Nano-Bio

Interface

Nanoelectronics

Conventional

Spintronics

Molectronics

Nanotubes

NANO

Nanosciences

Nanostructures

Nanomaterials

Nanophotonics

Photonic Crystals

Plasmonic Photonics

Energy

Conversion, Storage, and Transportation

Nanotechnology Landscape

characteristic lengths of energy carriers

carriers

wavelength

electrons

phonons

photons

air molecules

10-100 nm

1 nm

0.1-10

0.01 nm

Characteristic Lengths of Energy Carriers

Room Temperature

slide9

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

simple kinetic theory

qx

Hot

Cold

x

vxt

x

Simple Kinetic Theory

Taylor Expansion:

,

local thermodynamics equilibrium: u=u(T)

slide11

Heat Conduction in Solids

  • Heat is conducted by electrons and phonons.
  • k is determined by electron-electron, phonon-phonon, and electron-phonon collisions.

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?

microstructure of solids
Microstructure of solids
  • Atomic bonds
  • Crystalline, polycrystalline, amorphous materials
  • Bravais lattice, reciprocal lattice, Miller indices
slide13

U: Potential Energy

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

vibrations in solids
Vibrations in solids
  • Crystal vibrations, dispersion relations
  • Quantization and phonons
  • Phonon branches and modes
  • Lattice specific heat
  • Thermal expansion
  • Phonon scattering
  • Heat conduction
1 d lattice with diatomic basis

Lattice Constant, a

xn+1

yn-1

xn

yn

1-D Lattice with Diatomic Basis

Consider a linear diatomic chain of atoms (1-D model for a crystal like NaCl):

In equilibrium:

Applying Newton’s second law and the nearest-neighbor approximation to this system gives a dispersion relation with two “branches”:

1 d lattice with diatomic basis results
1-D Lattice with Diatomic Basis: Results

-(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

slide17

3-D Solids: Phonon Dispersion

In a real 3-D solid the dispersion relation will differ along different directions in k-space. In general, for a p atom basis, there are 3 acoustic modes and p-1 groups of 3 optical modes, although for many propagation directions the two transverse modes (T) are degenerate.

slide18

Density of States

wD

Phonon Frequency w

wE

3-D Solids: Heat Capacity

Einstein

Model

Debye

Model

(a)

Optical

Phonons

Acoustic Phonons

Density of States

Phonon Frequency w

thermal expansion
Thermal Expansion
  • If the curve is not symmetric, the average position in which the atom sits shifts with temperature.
  • Bond lengths therefore change (usually get bigger for increased T).
  • Thermal expansion coefficient is nonzero.
thermal conductivity

k

=

C

v

l

1

3

Thermal Conductivity

C ~ 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 :

electrons in solids
Electrons in solids
  • Free electron theory of metals
  • Fermi-Dirac statistics
  • Electron structure and quantization
  • Band structures of metals, semiconductors, and insulators
  • Electron scattering and transport
the free electron gas model
The Free Electron Gas Model

Plot U(x) for a 1-D crystal lattice:

Simple and crude finite-square-well model:

U

U = 0

Can we justify this model? How can one replace the entire lattice by a constant (zero) potential?

properties of the feg
Properties of the FEG

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 k-space per solution is:

And thus the density of states in k-space is:

Since the FEG is isotropic, the surface of constant E in k-space 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

density of states n e
Density of States N(E)

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:

heat capacity of the quantum mechanical feg
Heat Capacity of the Quantum-Mechanical FEG

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

heat capacity of metals theory vs expt at low t
Heat Capacity of Metals: Theory vs. Expt. at low T

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 electron-ion interactions

energy bands and energy gaps in a periodic potential
Energy Bands and Energy Gaps in a Periodic Potential

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 insulators and semiconductors
Metals, Insulators, and Semiconductors

• 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).

slide29

Energy Bands in 3-D Solids

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

dynamics of electrons in a band

v

v

-

d

k

e

E

=

dt

h

Dynamics of Electrons in a Band

Now 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 re-emerges at k = -/a!!

kx

Thus, an AC current is predicted to result from a DC field! (Bloch oscillations)

physical meaning of the band effective mass
Physical Meaning of the Band Effective Mass

The effective mass is inversely proportional to the curvature of the energy band.

Near the bottom of a nearly-free electron band m* is approximately constant, but it increases dramatically near the inflection point and even becomes negative (!) near the zone edge.

wave vs particle

Photoelectric Effect

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?
slide33

qx

Hot

Cold

x

Q1->2

vxt

x

1

2

Q2->1

x

How wave-like 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:

slide34

Reflection and Refraction: EM Waves

n2<0?

The Snell’s law indicated momentum conservation in x-direction

If n2<0, n1>0, the refraction light will be bended as the right figure.

Metamaterials

Reflectivity

Transmissivity

What happens if n1>n2?

low dimensional electrons and phonons
Low-Dimensional Electrons and Phonons

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

characteristic lengths of energy carriers36

carriers

wavelength

electrons

phonons

photons

air molecules

10-100 nm

1 nm

0.1-10

0.01 nm

Characteristic Lengths of Energy Carriers
transition from quantum to classical nanowires

Transition from Quantum to Classical: Nanowires

Wavelength vs. Diameter

Typical

Nanowire

Specular

Wavelength vs. Roughness

Specularity (Ziman)

Diffuse scattering  Incoherence

Diffuse

10-3

10-2

10-1

100

10+1

Roughness / Wavelength

statistical transport theories
Statistical Transport Theories
  • Time and length scales
  • Boltzmann transport equation
  • Carrier scattering
  • Moments of the BTE
boltzmann transport equation

From Equilibrium

To Nonequilibrium

Boltzmann Transport Equation
  • The distribution function can change by…
    • a spatial inflow/outflow of carriers (recall that the distribution function describes the probability of finding a carrier at a particular spatial location r, among other things)
    • an inflow/outflow of carriers in momentum space by means of a force (recall also that the distribution function describes the probability of finding a carrier with a particular momentum p)
    • scattering of carriers into or out of a particular location in position-momentum space
    • the presence of a source or sink of carriers
forms of the boltzmann transport equation bte
Forms of the Boltzmann Transport Equation (BTE)
  • One dimension
  • General, multi-dimensional
  • PDE in 7 dimensions (3 spatial, 1 time, 3 momentum—but wait! there’s more)
scattering term
Scattering Term
  • Scattering can increase the distribution function f(p,…) by in-scattering from p’ to p
  • It can also decrease the distribution function by out-scattering from p to p’
  • Net result
  • Normally, f(p)<<1. Thus,
  • S(p’,p) represents the probability per unit time that a carrier of momentum p’ will scatter in a state with momentum p
  • The scattering term’s sum can be converted to an integral
    • Thus, the BTE is a 7-dimensional integro-differential equation!
slide43

1.0

0.01

0.1

Phonon Scattering and Mean Free Path

Phonon Scattering Mechanisms

  • Boundary Scattering
  • Defect & Dislocation Scattering
  • Phonon-Phonon Scattering

Decreasing Boundary

Separation

l

Increasing

Defect

Concentration

PhononScattering

Defect

Boundary

Temperature, T/qD

slide44

Carrier Scattering

  • Carrier Scattering Mechanisms
  • Defect Scattering
  • Phonon Scattering
  • Boundary Scattering (Film Thickness,
  • Grain Boundary)

Electronic Bandstructure

  • Intra-valley
  • Inter-valley
  • Inter-band
momentum and energy relaxation time
Momentum and Energy Relaxation Time
  • Some scattering events only slightly perturb the incident particle’s direction
    • Thus, several collisions may be necessary to randomize the particle’s direction
    • where  is the polar angle between incident and scattered momentum vectors
  • Some scattering events (called elastic) alter momentum without altering energy
    • Thus, many scattering events may be needed to relax a particle’s energy
moments of the distribution function
Moments of the Distribution Function
  • We generally seek to determine macroscopic quantities from the statistical distribution function
    • carrier number density
    • electric current
    • average energy
  • These averages are obtained via weighting the distribution function and summing over all states
    • where n is a general averaged quantity that depends on the form of 
balance equations
Balance Equations
  • Carrier concentration
  • Carrier momentum
  • Carrier energy
thermoelectrics
Thermoelectrics

COLD SIDE

I

I

Phonons

L=10-100 nm

l=1 nm

-

Electrons

L=10-100 nm

l=10-50 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

  • Interfaces scatter phonons to reduce thermal conductivity
  • Quantum effects to improve electron behaviors

carrier concentration

thermal characterization in nanostructures
Thermal Characterization in Nanostructures

3w Method

Optical Pump-Probe Method

Thermal Measurement of 1-D Nanostructures

Nano-Resolution Thermal Microscopy

PROBE

HEATING

“PUMP”

FILM

GENERATION

REFLECTION

SUBSTRATE

x

+