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
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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
dω
energy level
more available
modes
k
(
N
states) in this
dω
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