Non-Continuum Energy Transfer: Phonons

1. Non-Continuum Energy Transfer: Phonons

2. 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)

3. 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

4. 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 ✕ 10-34Js) • wave vector (or wave number) k =2π/λ • phonon momentum = ħk • the dispersion relation relates the energy to the momentum ω = f(k)

5. 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

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 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

8. Phonon – Real Dispersion Relation

9. 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

10. 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 existsoccupation 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/

11. 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

12. 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

13. 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

14. 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

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 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 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

17. 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

18. 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

19. 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

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