1. Non-Continuum Energy Transfer: Phonons

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)
simple cubic
body-centered cubic
hexagonal a
NaCl
Ga4Ni3
tungsten carbide
cst-

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

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 = ✕ 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

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

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

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

8. Phonon – Real Dispersion Relation

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 1 a
M-1 M
λmin = 2a
λmax = 2L
note: k = Mπ/L is not included because it implies no atomic motion

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
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

11. Phonon – Density of States
more available modes k
(N states) in this dω energy level
fewer available modes k
(N states) in this dω energy level
chain
rule
Density of States:
For 1-D chain: modes (k) can be written as 1-D chain in k-space

12. 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
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 phonons
energy of phonons
Phonons are bosons and the number available is based on Bose-Einstein statistics

13. Phonons – Occupation
The thermodynamic probability can be determined from basic statistics but is dependant on the type of particle.
Maxwell-Boltzmann statistics
Maxwell-Boltzmann
distribution
boltzons: gas distinguishable particles
Bose-Einstein
distribution
Bose-Einstein statistics
bosons: phonons
indistinguishable particles
Fermi-Dirac
distribution
fermions: electrons
indistinguishable particles and limited occupancy (Pauli exclusion)
Fermi-Dirac statistics

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

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

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

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?

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

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)

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