Non-Continuum Energy Transfer: Phonons D. B. Go Slide 1
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 Ga 4 Ni 3 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 (~k B T), covalent (~1-10 ev), ionic (~1-10 ev), metallic (~1-10 ev) D. B. Go Slide 2
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 V ( r) = Z nl e2 r As the atoms come together to form a crystal structure, these potential energies overlap è hybridize forming different, quantized energy levels è bonds This bond is not rigid but more like a spring potential energy D. B. Go Slide 3
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 è different wavelengths of propagation (wave vector) - polarization è direction of vibration (transverse/longitudinal) - branches è 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 è scattering D. B. Go Slide 4
Phonons Energy Carriers Because phonons are the energy carriers we can use them to determine the energy storage è 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 u( x) = 1 2 gx 2 x = r r o g spring constant D. B. Go Slide 5
Phonon Dispersion Relation - we can sum all the potential energies across the entire chain U = 1 2 g N n=1 [ x na x (n +1)a ] - equation of motion for an atom located at x na is F = m d 2 x na dt 2 = U = g[ 2x na x (n 1)a x (n +1)a ] x na 2 nearest neighbors - this is a 2 nd order ODE for the position of an atom in the chain versus time: x na (t) - solution will be exponential of the form x na ( t) ~ e i kna ωt ( ( )) form of standing wave - plugging the standing wave solution into the equation of motion we can show that ω( k) = 2 g m sin ( ka 2 ) dispersion relation for an acoustic phonon D. B. Go Slide 6
Phonon Dispersion Relation - it can be shown using periodic boundary conditions that k = 2π λ λ min = 2a ω( k) smallest wave supported by atomic structure - 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 v g = dω dk a g m speed of sound in a solid - at k = π/a, v g = 0 è the atoms are vibrating out of phase with there neighbors D. B. Go Slide 7
Phonon Real Dispersion Relation D. B. Go Slide 8
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 0 1 a M-1 M λ min = 2a k max = π a = Mπ L λ max = 2L k min = π L modes : k = π L, 2π L, 3π L,...,(M 1)π L note: k = Mπ/L is not included because it implies no atomic motion D. B. Go Slide 9
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 è 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
Phonon Density of States ω k ( ) more available modes k (N states) in this dω energy level k fewer available modes k (N states) in this dω energy level Density of States: D( ω) = 1 dn dω chain rule D( ω) = 1 dn dk dk dω = 1 dn dk 1 v g For 1-D chain: modes (k) can be written as 1-D chain in k-space dk dn = π L D( ω) = 1 L L π 1 = 1 v g πv g D. B. Go Slide 11
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 " E = n k,p,b + 1 % $ ' ω # 2& k,p,b number of phonons energy of phonons 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). Phonons are bosons and the number available is based on Bose-Einstein statistics 1 n k,p,b = # exp ω & k,p,b % ( 1 $ k B T ' k B Boltzmann constant =1.3807 10 23 J K D. B. Go Slide 12
Phonons Occupation The thermodynamic probability can be determined from basic statistics but is dependant on the type of particle. boltzons: gas distinguishable particles bosons: phonons indistinguishable particles Maxwell-Boltzmann statistics Ω MB = N! i= 0 g i N i N i! Bose-Einstein statistics Ω BE = i= 0 ( g i + N i 1)! ( g i 1)!N i! Maxwell-Boltzmann distribution f ( ε) = Bose-Einstein distribution 1 $ exp ε µ ' & ) 1 % k B T ( f ( ε) = 1 $ exp ε µ ' & ) % k B T ( fermions: electrons indistinguishable particles and limited occupancy (Pauli exclusion) Fermi-Dirac statistics Ω FD = i= 0 g i! ( g i N i )!N i! Fermi-Dirac distribution 1 $ exp ε µ ' & ) +1 % k B T ( f ( ε) = D. B. Go Slide 13
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) è total energy stored in the crystal! è SPECIFIC HEAT total energy in the crystal specific heat ω " U = n p,b + 1 % $ ' ω # 2& p,b D ω p,b p,b C = U T 0 ( ) D. B. Go Slide 14 dω
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 è good for optical phonons, but not acoustic phonons gives good high temperature behavior Debye approximation C ~ T 3 assume dispersion curve ω(k) is linear cuts of at Debye temperature C = 3Nk B! T C ~ 234N A k B # " recovers high/low temperature behavior but not intermediate temperatures not appropriate for optical phonons T D $ & % 3 D. B. Go Slide 15
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 è 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
Phonons Thermal Conductivity Recall from kinetic theory we can describe the heat flux as q x = v x τ dnev x dx = v x 2 τ dne dx = v 2 xτ du dx Leading to q x = 1 3 v 2 τ du dt dt dx = k dt dx Fourier s Law k = 1 3 v g 2 τc what is the mean time between collisions? D. B. Go Slide 17
Phonons Scattering Processes There are two basic scattering types è collisions elastic scattering (billiard balls) off boundaries, defects in the crystal structure, impurities, etc energy & momentum conserved τ d = 1 ασρ defect v g τ b = L v g 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 1 st Brillouin zone and transformed into 1 st Brillouin zone impede phonon momentum è dominate thermal conductivity τ u A T ' exp θ * D ), ωθ D ( γt + D. B. Go Slide 18
Phonons Scattering Processes Collision processes are combined using Matthiesen rule è effective relaxation time 1 τ = 1 + 1 + 1 τ d τ b τ u Effective mean free path defined as = τv g Molecular description of thermal conductivity k = 1 3 v 2 gτc = 1 3 v g C 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
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) è 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