Chapter 8: Dynamics of Crystal Electrons

Question 1

Semi-Classical Treatment

(3 points)

Answer 1

• Classical description of external fields
• Quantum description of electron dynamics
  
  • TDSE

Question 2

Semi-Classical Motion of Electron

(Velocity, Acceleration Equations)

Answer 2

Question 3

Semi-Classical Wave Function

(3 points)

Answer 3

• Change from Block waves → superposition of Bloch waves
  
  • ε_n(k) ≡ dispersion
  • ∆k small compared to Brillouin zone

Question 4

Semi-Classical Momentum

(3 points)

Answer 4

• Well-defined momentum → delocalization in x-direction
  
  • Propagation from group velocity
  • Different ω(k) propagate at different rates → broadening over time

Question 5

Basics of Semi-Classical Model

(3 points)

Answer 5

• Describes motion of band electrons in presence of external fields
• Assign each electron postion r, wavenumber k, band index n (dispersion ε_n(k) known)
• External forces cause parameters to change according to dynamics rules

Question 6

Semi-Classical Model: Dynamics Rules

(6 points)

Answer 6

• Band index constant n
• Equations of motion (see below)
• Effective mass (own flash cards)
• Momentum only conserved up to G
• Thermal equilibrium, electron states in nth-band determined by Fermi statistics
  
  • D(k)F(ε_n, T)d³k

Question 7

Semi-Classical Model: Effective Mass

Answer 7

• time-derivative of group velocity (see below)
• Can read-off inverse of effective mass, because a = m^-1F

Question 8

Effective Mass Importance

(3 points)

Answer 8

• Can describe electrion in band as free electron with effective mass
• Most useful in upper-valance, and lower-conduction, band where bands approximation parabolic ε(k) = ε_o ± \hbar²/(2m)(k_x² + k_y² + k_z² )
• Effective mass inversely proportional to concavity of ε(k)

Question 9

Motion of Crystal Electrons: Filled Bands

(2 points)

Answer 9

• No electric current
• No heat current

Question 10

Motion of Crystal Electrons: Partially Filled Bands

(7 points)

Answer 10

• Electric Field
  
  • Non-zero current
  • Periodic potential leads to Bloch Oscillations
• _​_Magnetic field
  
  • Electrons move on surfaces of constant energy with trajectories perpendicular to B-Field
    
    • Open, and closed, trajectories
    • Cyclotron frequency of closed trajectory ω_c = eB/m

Question 11

Quantization of Electron Paths in B-Field

(6 points)

Answer 11

• Solving Schrodingers equation yields results similar to quantum harmonic oscillator
• Energy eigenvalues (see below)
  
  • E = (n + ¹/₂)(\hbar)ω_c + (\hbar k_z)²/(2m)
• Quantization of trajectories in xy-plane leads to Landau levels
  
  • Area between adjacent circles constant and ∆S ∝ B
  • Number of electrons per level finite and p ∝ B

Question 12

De Haas-van Alphen Effect

(Overview)

Answer 12

• Oscillation of magnetization of metals as function of high B-field and low temperature

Question 13

De Haas-van Alphen Effect

(Assumptions: 6 points)

Answer 13

• N electrons
• |B| > 0
• Landau level S fully filed and S+1 partially filled
  
  • E_F ∈ S+1
• ​Increasing B increases E_F, because E_S+1 ∝ B
• Degeneracy of S, p_S ∝ B, so eventually S+1 becomes empty and E_F ∈ S

Question 14

De Haas-van Alphen Effect

(Take-Away)

Answer 14

• Can see oscilation of Fermi energy in U vs 1/B