AME 60634
Int. Heat Trans.
Non-Continuum Energy
Transfer: Phonons
D. B. Go
Slide 1
AME 60634
Int. Heat Trans.
The Crystal Lattice
• The crystal lattice is the organization of atoms and/or molecules in
a solid
simple cubic
body-centered cubic
hexagonal
a
NaCl
Ga4Ni3
tungsten carbide
cst-www.nrl.navy.mil/lattice
• 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)
D. B. Go
Slide 2
AME 60634
Int. Heat Trans.
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
-Z nl e 2
V ( r) =
r
• 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
D. B. Go
Slide 3
AME 60634
Int. Heat Trans.
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 J s)
– 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
D. B. Go
Slide 4
AME 60634
Int. Heat Trans.
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
D. B. Go
1 2
u( x ) = gx
2
x = r - ro
g º spring constant
Slide 5
AME 60634
Int. Heat Trans.
Phonon – Dispersion Relation
- we can sum all the potential energies across the entire chain
N
1
U = gå [ x na - x(n +1)a ]
2 n=1
2
- equation of motion for an atom located at xna is
d 2 x na
¶U
F=m
== -g[2x na - x(n-1)a - x(n +1)a ]
2
dt
¶x na
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
( i( kna-wt ))
form of standing wave
x na ( t ) ~ e
- plugging the standing wave solution into the equation of motion we can show that
w( k) = 2
D. B. Go
( )
g
sin ka 2
m
dispersion relation for an acoustic phonon
Slide 6
AME 60634
Int. Heat Trans.
Phonon – Dispersion Relation
- it can be shown using periodic boundary conditions that
k=
2p
l
lmin = 2a
smallest wave supported by
atomic structure
w( k)
- this is the first Brillouin zone or
primative cell that characterizes
behavior for the entire crystal
k
- the speed at which the phonon propagates is given by the group velocity
vg =
dw
g
»a
dk
m
speed of sound in a solid
- at k = π/a, vg = 0  the atoms are vibrating out of phase with there neighbors
D. B. Go
Slide 7
AME 60634
Int. Heat Trans.
D. B. Go
Phonon – Real Dispersion Relation
Slide 8
AME 60634
Int. Heat Trans.
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
a
1
0
kmax
λmin = 2a
p
Mp
= =
a
L
λmax = 2L
modes : k =
D. B. Go
p 2p 3p
,
L L
,
L
,...,
(M -1)p
L
M-1
kmin =
M
p
L
note: k = Mπ/L is not included
because it implies no atomic motion
Slide 9
AME 60634
Int. Heat Trans.
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
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
D. B. Go
http://pcagreatperformances.org/info/merrill_seating_chart/ Slide 10
AME 60634
Int. Heat Trans.
Phonon – Density of States
w( k)
more available modes k
(N states) in this dω energy level
fewer available modes k
(N states) in this dω energy level
k
Density of States:
D(w ) =
1 dN
" dw
chain
rule
D(w ) =
1 dN dk 1 dN 1
=
" dk dw " dk v g
For 1-D chain: modes (k) can be written as 1-D chain in k-space
dk p
=
dN L
D. B. Go
Þ D (w ) =
1L 1
1
=
L p vg p vg
Slide 11
AME 60634
Int. Heat Trans.
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
æ
1ö
E = ç n k, p,b + ÷ w k,p,b
è
2ø
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).
number of energy of
phonons
phonons
Phonons are bosons and the number available is based on Bose-Einstein statistics
n k, p,b =
1
æ w k, p,b ö
expç
÷ -1
è kB T ø
kB º Boltzmann constant =1.3807 ´10-23 J
D. B. Go
K
Slide 12
AME 60634
Int. Heat Trans.
Phonons – Occupation
The thermodynamic probability can be determined from basic statistics but is
dependant on the type of particle.
Maxwell-Boltzmann statistics
boltzons: gas
distinguishable particles
bosons: phonons
indistinguishable particles
fermions: electrons
indistinguishable particles
and limited occupancy
(Pauli exclusion)
D. B. Go
Maxwell-Boltzmann
distribution
f (e) =
Bose-Einstein statistics
Fermi-Dirac statistics
1
æe - mö
expç
÷
è kB T ø
Bose-Einstein
distribution
f (e) =
1
æe - mö
expç
÷ -1
k
T
è B ø
Fermi-Dirac
distribution
f (e) =
1
æe - mö
expç
÷ +1
k
T
è B ø
Slide 13
AME 60634
Int. Heat Trans.
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
æ
1ö
U = å ò ç n p,b + ÷ w p,b D(w p,b )dw
è
2ø
p,b 0
w
total energy in the crystal
specific heat
D. B. Go
¶U
C=
¶T
Slide 14
AME 60634
Int. Heat Trans.
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: C ~ T 3
C = 3NkB
• 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
æT ö
C ~ 234N A kB ç ÷
è TD ø
3
– 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
D. B. Go
Slide 15
AME 60634
Int. Heat Trans.
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
D. B. Go
Slide 16
AME 60634
Int. Heat Trans.
Phonons – Thermal Conductivity
• Recall from kinetic theory we can describe the heat flux as
dNEv x
2 dNE
2 dU
qx = -v xt
= -v x t
= -v x t
dx
dx
dx
• Leading to
1 2 dU dT
dT
qx = - v t
= -k
3
dT dx
dx
Fourier’s Law
1 2
k = vg t C
3
D. B. Go
what is the mean time
between collisions?
Slide 17
AME 60634
Int. Heat Trans.
Phonons – Scattering Processes
There are two basic scattering types  collisions
• elastic scattering (billiard balls) off boundaries, defects in the
crystal structure, impurities, etc …
– energy & momentum conserved
td =
1
asrdefect v g
L
tb =
vg
• 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
æ qD ö
tu » A
expç ÷
wq D
è gT ø
T
D. B. Go
Slide 18
AME 60634
Int. Heat Trans.
Phonons – Scattering Processes
• Collision processes are combined using Matthiesen rule 
effective relaxation time
1
t
=
1
td
+
1
tb
+
1
tu
• Effective mean free path defined as
= tv g
Molecular description of thermal conductivity
1
1
k = vg2t C = vg C
3
3
When phonons are the dominant energy carrier:
• increase conductivity by decreasing collisions (smaller size)
• decrease conductivity by increasing collisions (more defects)
D. B. Go
Slide 19
AME 60634
Int. Heat Trans.
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
D. B. Go
Slide 20