Chapter 4: Dynamics of Crystal Lattices

Question 1

Lattice Dynamics

(4 points)

Answer 1

• Lattice is not rigid
• Lattice vibrations due to finite temperature
• Even at T = 0, quantum fluctuations are possible
• Approximations are needed to explain lattice dynamics

Question 2

Adiabatic Approximation

(Assumptions: 4 points)

Answer 2

• Electrons instantaneously follow nuclei
  
  • Follows from Newton’s 3rd Law
    
    • m_e/m_p = 1/1836
    • Heavy nuclei oscillate slower than electrons

Question 3

Adiabatic Approximation

(Take-Away: 2 points)

Answer 3

• Can track motion of ion first and then electron
• Energy of electron corresponds to rigid lattice with momentary position of nuclei

Question 4

Harmonic Approximation

(Assumptions: 6 points)

Answer 4

• Displacement from equlibrium small
• Ion-ion potential considered harmonic
  
  • Comes trom Taylor expansion of potential between ions φ(r_n − r_m)
    
    • First term: Equilibrium energy → set to zero
    • Second term: Forces exerted by all other atoms → zero at equilibrium
    • Third term: Second derivative of potential → generalized Hooke’s Law

Question 5

Harmonic Approximation

(Take-Away: 3 points)

Answer 5

• Can consider crystal as masses conneced to all other masses by springs
• Usually only consider nearest-neighbor interactions
  
  • Valid because of Coulomb interactions

Question 6

Source of Lattice Vibrations

Answer 6

Thermal fluctuations

Question 7

Dispersion Relation

(3 points)

Answer 7

• Relates the wavenumber k to its frequency ω(k)
• Derive from solving equation of motion
  
  • N unit cells with r’ atoms → 3r’N differential equations

Question 8

Transverse vs Longitudinal Vibrations

(2 points)

Answer 8

• transverse → displacement perpendicular to propagation
• longitudinal → displacement parallel to propagation

Question 9

Monoatomic Dispersion Derivation

(Assumptions: 5 points)

Answer 9

• Pure longitudinal wave in 1D
• Force exerted in plane u_n by atoms in plane u_n+p
  
  • _​Harmonic approximation
• Ansatz: u_n+p = A e^i(qpa-ωt)
• Nearest-neighbor interactions only

Question 10

Monoatomic Dispersion Derivation

(Results)

Answer 10

Dispersion relation:

Question 11

Monoatomic Dispersion Relation

(Properties: 4 points)

Answer 11

• Periodic
• Symmetric
• 2πn/a is length of reciprocal lattice vector
• Domain −π/a ≤ q ≤ π/a → first Brillouin zone

Question 12

Group Velocity

Answer 12

Velocity of wave packet propagation through lattice

Question 13

Monoatomic Group Velocity

Answer 13

Question 14

Monoatomic Dispersion Extreme Cases

(9 points)

Answer 14

• q = π/a → v_g = 0 → standing wave
  
  • Edge of first Brillouin Zone
  • Maximum dispersion ω_max
• _​qa << 1 → long wavelength
  
  • dispersion approximately linear
  • v_g = constant
    
    • Velocity of longitudinal sound wave
  • Near Γ-point
  • Crystal lattice considered continuum

Question 15

First Brillouin Zone and Dispersion Relation

(3 points)

Answer 15

• ω(q) has periodicity q = 2π/a
  
  • Shortest possible reciprocal lattice vector
• Can just consider dispersion in first Brillouin Zone

Question 16

Diatomic Dispersion Relation

(Assumptions: 5 points)

Answer 16

• Diatomic basis with mass M₁, M₂
• Each plane contains one species
• 1D longitudinal wave
• Nearest-neighbor interactions only
• Same coupling constant C

Question 17

Diatomic Dispersion Relation

(Take-Away: 4 points)

Answer 17

• DIspersion relation (see below)
• Two branches
  
  • Optical (+): Atoms oscillate out of phase
  • Acoustic (-): Atoms oscillate in-phase

Question 18

Lattice Vibrations in 3D & Branches

Answer 18

Consider r’ atoms per unit cell

• 3 acoustic branches
• 3r’-3 optical branches
• 3r’ total branches

Question 19

Density of Phonons

(Overview: 5 points)

Answer 19

• Finite crystal leads to restrictions of allowable q
  
  • λ > L ( or q < 2π/L) not possible
• Finite number of eigenfrequencies
  
  • N masses ≤ 3N eigenfrequencies
    
    • Depends on boundary conditions

Question 20

Fixed Boundary Conditions - 1D

(Assumptions: 5 points)

Answer 20

• Atomic position: x_n = na for n = 0,…,N
• Displacement: u_n
• Chain length: L = Na
• Number of atoms: N+1
• Ansatz: u_n = A₁e^{i(qna - ωt)} + A₂e^{-i(qna + ωt)}​

Question 21

Fixed Boundary Conditions - 1D

(Boundary Conditions)

Answer 21

• u_n(0) = u_L(Na) = 0

Question 22

Fixed Boundary Conditions - 1D

(Take-Away)

Answer 22

• N - 1 oscillatory modes

Question 23

Periodic Boundary Conditions - 1D

(Assumptions: 5 points)

Answer 23

• Atomic position: x_n = na for n = 0,…,N
• Displacement: u_n
• Chain length: L = Na
• Number of atoms: N+1
• Ansatz: u_n = Ae^{i(qR_n - ωt)}; R_n = na

Question 24

Periodic Boundary Conditions - 1D

(Boundary Conditions)

Answer 24

• u_n = u_n+N

Question 25

Periodic Boundary Conditions - 1D

(Take-Away: 4 points)

Answer 25

• N oscillatory modes
• For r-atom basis
  
  • rN oscillatory modes
  • N Bravais lattice points

Question 26

Density of Phonon States

Answer 26

the number of q-vectors per volume element in momentum space

Question 27

Density of Phonons in 1D Momentum Space

Answer 27

Question 28

Periodic Boundary Conditions - 3D

(Assumptions: 7 points)

Answer 28

• Crystal with lattice vectors a₁, a₂, a₃
• Crystal dimension N₁a₁, N₂a₂, N₃a₃
• Number of lattice points N = N₁N₂N₃
• Monoatomic basis
  
  • Harmonic oscilations
• Ansatz: u(R) = Ae^{i(<strong>qR</strong>-ωt)}
  
  • q is reciprocal lattice vector

Question 29

Periodic Boundary Conditions - 3D

(Boundary Conditions)

Answer 29

• u(R) = u(R + N_ia_i) for i = 1,2,3

Question 30

Periodic Boundary Conditions - 3D

(Take-Away: 4 points)

Answer 30

• 3N oscillation modes
• For r-atom basis
  
  • 3rN modes
  • N Bravais lattice points

Question 31

Density of Phonon States in 3D Momentum Space

Answer 31

Question 32

Density of Phonons in 3D Frequency Space

Answer 32

Question 33

Properties of Phonons

(4 points)

Answer 33

• Energy: E = (h-bar)ω
• Momentum: p = (h-bar)q
• ω,q related via dispersion relation
• High amplitude of classic oscillation means larger number of phonons

Question 34

Phonon Momentum

(4 points)

Answer 34

• Momentum only conserved up to reciprocal lattice vector G
  
  • ​(h-bar)k’ = (h-bar)k + G
• Inelastic scattering conservation (see attached)
  
  • Phonons do not carry true momentum, but they behave as if they carry quasi-momentum (h-bar)q in scattering process