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**Non-Continuum Energy Transfer: Phonons**

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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-www.nrl.navy.mil/lattice

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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

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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 = ✕ Js) 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

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**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

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**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

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**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

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**Phonon – Real Dispersion Relation**

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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

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**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

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**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

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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

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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

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**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

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**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

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**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

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**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?

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**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

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**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)

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**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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