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The Crystal Lattice

simple cubic

body-centered cubic

hexagonal

a

Ga4Ni3

tungsten carbide

NaCl

cst-www.nrl.navy.mil/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)

The Crystal Lattice

potential energy

- 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

Phonons Overview

- 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

- A phonon is a quantized lattice vibration that transports energy across a solid
- Phonon properties
- frequency ω
- energy ħω
- ħis the reduced Plank’s constant ħ= h/2π (h = 6.6261 ✕ 10-34Js)
- wave vector (or wave number) k =2π/λ
- phonon momentum = ħk
- the dispersion relation relates the energy to the momentum ω = f(k)

Phonons – Energy Carriers

approximate the potential energy in each bond as parabolic

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

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

Phonon – Dispersion Relation

- it can be shown using periodic boundary conditions that

smallest wave supported by atomic structure

- this is the first Brillouinzone 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

Phonon – Modes

M-1

M

a

1

0

λmin= 2a

λmax= 2L

note: k = Mπ/L is not included because it implies no atomic motion

- 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 phononmodes

Phonon: Density of States

more available seats (N states) in this energy level

fewer available seats (Nstates) in this energy level

The density of states does not describe if a state is occupiedonly if the state existsoccupation is determined statistically

simple view: the density of states only describes the floorplan & number of seats not the number of tickets sold

- 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/

Phonon – Density of States

more available modes k

(Nstates) in this dωenergy level

fewer available modes k

(N states) in this dωenergy level

chain

rule

Density of States:

For 1-D chain: modes (k) can be written as 1-D chain in k-space

Phonon - Occupation

The total energy of a single mode at a given wave vector kin 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

Phononsare bosons and the number available is based on Bose-Einsteinstatistics

Phonons – Occupation

The thermodynamic probability can be determined from basic statistics but is dependant on the type of particle.

Maxwell-Boltzmann

distribution

Maxwell-Boltzmann statistics

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

Phonons – Specific Heat of a Crystal

total energy in the crystal

specific heat

- 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

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

Phonons – Thermal Transport

G. Chen

- Now that we understand, fundamentally, how thermal energy is stored in a crystal structure, we can begin to look at how thermal energy is transportedconduction
- 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

Phonons – Thermal Conductivity

Fourier’s Law

what is the mean time between collisions?

Recall from kinetic theory we can describe the heat flux as

Leading to

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
- 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 1stBrillouin zone and transformed into 1stBrillouin zone
- impede phonon momentum dominate thermal conductivity

Phonons – Scattering Processes

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)

- Collision processes are combined using Matthiesen rule effective relaxation time
- Effective mean free path defined as

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

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