Non-Continuum Energy Transfer: Phonons

AME 60634 Int. Heat Trans.

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Non-Continuum Energy Transfer: Phonons

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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Phonon – Real Dispersion Relation

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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).

AME 60634 Int. Heat Trans.

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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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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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?

AME 60634 Int. Heat Trans.

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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

AME 60634 Int. Heat Trans.

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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)

AME 60634 Int. Heat Trans.

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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