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Non-Continuum Energy Transfer: Phonons Non-Continuum Energy Transfer: Phonons

Non-Continuum Energy Transfer: Phonons - PowerPoint Presentation

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Non-Continuum Energy Transfer: Phonons - PPT Presentation

The Crystal Lattice The crystal lattice is the organization of atoms andor molecules in a solid The lattice constant a is the distance between adjacent atoms in the basic structure 4 Å ID: 365206

phonons energy states phonon energy phonons phonon states wave crystal thermal level number acoustic density specific dispersion relation heat momentum atoms lattice

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Slide1

Non-Continuum Energy Transfer: PhononsSlide2

The Crystal Lattice

The

crystal lattice is the organization of atoms and/or molecules in a solidThe 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 atomsVan 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

Ga

4

Ni3

simple cubic

body-centered cubic

tungsten carbide

hexagonal

aSlide3

The Crystal Lattice

Each electron in an atom has a particular potential energy

electrons inhabit quantized (discrete) energy states called orbitalsthe potential energy V is related to the quantum state, charge, and distance from the nucleusAs the atoms come together to form a crystal structure, these potential energies overlap 

hybridize forming different, quantized energy levels 

bondsThis bond is not rigid but more like a spring

potential energySlide4

Phonons Overview

A phonon is a

quantized lattice vibration that transports energy across a solidPhonon propertiesfrequency ωenergy ħωħ is the reduced Plank’s constant ħ

= h/2π (

h = 6.6261 ✕

10-34

J

s)wave vector (or wave number) k =2π/λ

phonon momentum =

ħkthe

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

scatteringSlide5

Phonons – Energy Carriers

Because phonons are the

energy carriers we can use them to determine the energy storage  specific heatWe must first determine the dispersion relation which relates the energy of a phonon to the mode/wavevectorConsider 1-D chain of atoms

approximate the potential energy in each bond as parabolicSlide6

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 2

nd

order ODE for the position of an atom in the chain versus time:

x

na

(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 phononSlide7

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

,

v

g

= 0

the atoms are vibrating

out of phase

with there neighborsSlide8

Phonon – Real Dispersion RelationSlide9

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 structurethese allowable wave vectors are the phonon modes

0

1

M-1

M

a

λ

min

= 2

a

λ

max

= 2

L

note:

k

=

Mπ/L

is not included because it implies no atomic motionSlide10

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 occupiedsimple view: think of an auditorium where each tier represents an energy levelhttp://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 soldSlide11

Phonon – Density of States

fewer available modes

k

(

N

states

) in this

energy level

more available

modes

k

(

N

states) in this

energy

level

Density of States:

chain

rule

For 1-D chain: modes (

k

) can be written as 1-D chain in

k

-

spaceSlide12

Phonon - Occupation

The 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).Slide13

Phonons – Occupation

The thermodynamic probability can be determined from basic statistics but is

dependant on the type of particle.

boltzons

: gas

distinguishable particles

bosons:

phonons

indistinguishable particles

fermions:

electrons

indistinguishable

particles and limited occupancy (Pauli exclusion)

Maxwell-Boltzmann statistics

Bose-Einstein statistics

Fermi-Dirac statistics

Fermi-Dirac

distribution

Bose-Einstein

distribution

Maxwell-Boltzmann

distributionSlide14

Phonons – Specific Heat of a Crystal

Thus far we understand:

phonons are quantized vibrationsthey 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 limitedthe number of phonons (occupation) is governed by Bose-Einstein statisticsIf 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 heatSlide15

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 approximationassume all phonon modes have the same energy  good for optical phonons, but not acoustic phononsgives good high temperature behaviorDebye approximationassume dispersion curve ω(

k) is linear

cuts of at “Debye temperature”recovers high/low temperature behavior but not intermediate temperaturesnot appropriate for optical phonons Slide16

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  conductionWe will use the kinetic theory approach to arrive at a relationship for thermal conductivityvalid for any energy carrier that behaves like a particleTherefore, we will treat phonons as particlesthink of each phonon as an

energy packet moving along the crystal

G. ChenSlide17

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?Slide18

Phonons – Scattering Processes

elastic scattering

(billiard balls) off boundaries, defects in the crystal structure, impurities, etc …energy & momentum conservedinelastic scattering between 3 or more different phononsnormal processes: energy & momentum conserveddo not impede phonon momentum directlyumklapp processes: energy conserved, but momentum is not – resulting phonon is out of 1st

Brillouin

zone and transformed into 1st

Brillouin zoneimpede phonon momentum

 dominate thermal conductivity

There are two basic scattering types

collisionsSlide19

Phonons – Scattering Processes

Collision processes are combined using

Matthiesen rule  effective relaxation timeEffective 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)Slide20

Phonons – What We’ve Learned

Phonons are

quantized lattice vibrationsstore and transport thermal energyprimary energy carriers in insulators and semi-conductors (computers!)Phonons are characterized by theirenergywavelength (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