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AME 60634 Int. Heat Trans.
D. B. Go Slide 1
Non-Continuum Energy Transfer: Phonons
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AME 60634 Int. Heat Trans.
D. B. Go Slide 2
The Crystal Lattice• The crystal lattice is the organization of atoms and/or molecules in
a solid
• The lattice constant ‘a’ is the distance between adjacent atoms in the basic structure (~ 4 Å)
• The organization of the atoms is due to bonds between the atoms– Van der Waals (~0.01 eV), hydrogen (~kBT), covalent (~1-10 eV), ionic
(~1-10 eV), metallic (~1-10 eV)
cst-www.nrl.navy.mil/lattice
NaCl Ga4Ni3
simple cubic body-centered cubic
tungsten carbide
hexagonal
a
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AME 60634 Int. Heat Trans.
D. B. Go Slide 3
The Crystal Lattice
• Each electron in an atom has a particular potential energy– electrons inhabit quantized (discrete) energy states called orbitals– the potential energy V is related to the quantum state, charge, and
distance from the nucleus
• As the atoms come together to form a crystal structure, these potential energies overlap hybridize forming different, quantized energy levels bonds
• This bond is not rigid but more like a spring
potential energy
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AME 60634 Int. Heat Trans.
D. B. Go Slide 4
Phonons Overview• A phonon is a quantized lattice vibration that transports energy
across a solid
• Phonon properties– frequency ω– energy ħω
• ħ is the reduced Plank’s constant ħ = h/2π (h = 6.6261 ✕ 10-34 Js)
– wave vector (or wave number) k =2π/λ– phonon momentum = ħk– the dispersion relation relates the energy to the momentum ω = f(k)
• Types of phonons- mode different wavelengths of propagation (wave vector)- polarization direction of vibration (transverse/longitudinal)- branches related to wavelength/energy of vibration (acoustic/optical)
heat is conducted primarily in the acoustic branch
• Phonons in different branches/polarizations interact with each other scattering
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AME 60634 Int. Heat Trans.
D. B. Go Slide 5
Phonons – Energy Carriers
• Because phonons are the energy carriers we can use them to determine the energy storage specific heat
• We must first determine the dispersion relation which relates the energy of a phonon to the mode/wavevector
• Consider 1-D chain of atoms
approximate the potential energy in each bond as parabolic
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AME 60634 Int. Heat Trans.
D. B. Go Slide 6
Phonon – Dispersion Relation
- we can sum all the potential energies across the entire chain
- equation of motion for an atom located at xna is
nearest neighbors
- this is a 2nd order ODE for the position of an atom in the chain versus time: xna(t)- solution will be exponential of the form
form of standing wave
- plugging the standing wave solution into the equation of motion we can show that
dispersion relation for an acoustic phonon
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AME 60634 Int. Heat Trans.
D. B. Go Slide 7
Phonon – Dispersion Relation- it can be shown using periodic boundary conditions that
smallest wave supported by atomic structure
- this is the first Brillouin zone or primative cell that characterizes behavior for the entire crystal
- the speed at which the phonon propagates is given by the group velocity
speed of sound in a solid
- at k = π/a, vg = 0 the atoms are vibrating out of phase with there neighbors
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AME 60634 Int. Heat Trans.
D. B. Go Slide 8
Phonon – Real Dispersion Relation
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AME 60634 Int. Heat Trans.
D. B. Go Slide 9
Phonon – Modes
• As we have seen, we have a relation between energy (i.e., frequency) and the wave vector (i.e., wavelength)
• However, only certain wave vectors k are supported by the atomic structure– these allowable wave vectors are the phonon modes
0 1 M-1 Ma
λmin = 2a
λmax = 2L
note: k = Mπ/L is not included because it implies no atomic motion
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AME 60634 Int. Heat Trans.
D. B. Go Slide 10
Phonon: Density of States
• The density of states (DOS) of a system describes the number of states (N) at each energy level that are available to be occupied– simple view: think of an auditorium where each tier represents an
energy level
http://pcagreatperformances.org/info/merrill_seating_chart/
more available seats (N states) in this energy level
fewer available seats (N states) in this energy level
The density of states does not describe if a state is occupied only if the state exists occupation is determined statistically
simple view: the density of states only describes the floorplan & number of seats not the number of tickets sold
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AME 60634 Int. Heat Trans.
D. B. Go Slide 11
Phonon – Density of States
fewer available modes k(N states) in this dω energy level
more available modes k(N states) in this dω energy level
Density of States:chainrule
For 1-D chain: modes (k) can be written as 1-D chain in k-space
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AME 60634 Int. Heat Trans.
D. B. Go Slide 12
Phonon - OccupationThe total energy of a single mode at a given wave vector k in a specific polarization (transverse/longitudinal) and branch (acoustic/optical) is given by the probability of occupation for that energy state
number of phonons
energy of phonons
Phonons are bosons and the number available is based on Bose-Einstein statistics
This in general comes from the treatment of all phonons as a collection of single harmonic oscillators (spring/masses). However, the masses are atoms and therefore follow quantum mechanics and the energy levels are discrete (can be derived from a quantum treatment of the single harmonic oscillator).
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AME 60634 Int. Heat Trans.
D. B. Go Slide 13
Phonons – OccupationThe thermodynamic probability can be determined from basic statistics but is dependant on the type of particle.
boltzons: gas distinguishable particles
bosons: phononsindistinguishable particles
fermions: electronsindistinguishable particles and limited occupancy (Pauli exclusion)
Maxwell-Boltzmann statistics
Bose-Einstein statistics
Fermi-Dirac statisticsFermi-Diracdistribution
Bose-Einsteindistribution
Maxwell-Boltzmanndistribution
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AME 60634 Int. Heat Trans.
D. B. Go Slide 14
Phonons – Specific Heat of a Crystal
• Thus far we understand:– phonons are quantized vibrations– they have a certain energy, mode (wave vector), polarization (direction),
branch (optical/acoustic)– they have a density of states which says the number of phonons at any
given energy level is limited– the number of phonons (occupation) is governed by Bose-Einstein
statistics
• If we know how many phonons (statistics), how much energy for a phonon, how many at each energy level (density of states) total energy stored in the crystal! SPECIFIC HEAT
total energy in the crystal
specific heat
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AME 60634 Int. Heat Trans.
D. B. Go Slide 15
Phonons – Specific Heat
• As should be obvious, for a real. 3-D crystal this is a very difficult analytical calculation– high temperature (Dulong and Petit):– low temperature:
• Einstein approximation– assume all phonon modes have the same energy good for optical
phonons, but not acoustic phonons– gives good high temperature behavior
• Debye approximation– assume dispersion curve ω(k) is linear– cuts of at “Debye temperature”– recovers high/low temperature behavior but not intermediate
temperatures– not appropriate for optical phonons
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AME 60634 Int. Heat Trans.
D. B. Go Slide 16
Phonons – Thermal Transport
• Now that we understand, fundamentally, how thermal energy is stored in a crystal structure, we can begin to look at how thermal energy is transported conduction
• We will use the kinetic theory approach to arrive at a relationship for thermal conductivity– valid for any energy carrier that behaves like a particle
• Therefore, we will treat phonons as particles– think of each phonon as an energy packet moving along the crystal
G. Chen
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AME 60634 Int. Heat Trans.
D. B. Go Slide 17
Phonons – Thermal Conductivity
• Recall from kinetic theory we can describe the heat flux as
• Leading to
Fourier’s Law
what is the mean time between collisions?
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AME 60634 Int. Heat Trans.
D. B. Go Slide 18
Phonons – Scattering Processes
• elastic scattering (billiard balls) off boundaries, defects in the crystal structure, impurities, etc …– energy & momentum conserved
• inelastic scattering between 3 or more different phonons– normal processes: energy & momentum conserved
• do not impede phonon momentum directly
– umklapp processes: energy conserved, but momentum is not – resulting phonon is out of 1st Brillouin zone and transformed into 1st Brillouin zone
• impede phonon momentum dominate thermal conductivity
There are two basic scattering types collisions
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AME 60634 Int. Heat Trans.
D. B. Go Slide 19
Phonons – Scattering Processes• Collision processes are combined using Matthiesen rule effective
relaxation time
• Effective mean free path defined as
Molecular description of thermal conductivity
When phonons are the dominant energy carrier:• increase conductivity by decreasing collisions (smaller size) • decrease conductivity by increasing collisions (more defects)
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AME 60634 Int. Heat Trans.
D. B. Go Slide 20
Phonons – What We’ve Learned
• Phonons are quantized lattice vibrations– store and transport thermal energy– primary energy carriers in insulators and semi-conductors (computers!)
• Phonons are characterized by their– energy– wavelength (wave vector)– polarization (direction)– branch (optical/acoustic) acoustic phonons are the primary thermal
energy carriers
• Phonons have a statistical occupation, quantized (discrete) energy, and only limited numbers at each energy level – we can derive the specific heat!
• We can treat phonons as particles and therefore determine the thermal conductivity based on kinetic theory