Chapter 8: Dynamics of Crystal Electrons Flashcards

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1
Q

Semi-Classical Treatment

(3 points)

A
  • Classical description of external fields
  • Quantum description of electron dynamics
    • TDSE
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2
Q

Semi-Classical Motion of Electron

(Velocity, Acceleration Equations)

A
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3
Q

Semi-Classical Wave Function

(3 points)

A
  • Change from Block waves → superposition of Bloch waves
    • εn(k) ≡ dispersion
    • k small compared to Brillouin zone
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4
Q

Semi-Classical Momentum

(3 points)

A
  • Well-defined momentum → delocalization in x-direction
    • Propagation from group velocity
    • Different ω(k) propagate at different rates → broadening over time
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5
Q

Basics of Semi-Classical Model

(3 points)

A
  • Describes motion of band electrons in presence of external fields
  • Assign each electron postion r, wavenumber k, band index n (dispersion εn(k) known)
  • External forces cause parameters to change according to dynamics rules
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6
Q

Semi-Classical Model: Dynamics Rules

(6 points)

A
  • 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(εn, T)d3k
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7
Q

Semi-Classical Model: Effective Mass

A
  • time-derivative of group velocity (see below)
  • Can read-off inverse of effective mass, because a = m-1F
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8
Q

Effective Mass Importance

(3 points)

A
  • 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) = εo ± \hbar2/(2m)(kx2 + ky2 + kz2 )
  • Effective mass inversely proportional to concavity of ε(k)
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9
Q

Motion of Crystal Electrons: Filled Bands

(2 points)

A
  • No electric current
  • No heat current
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10
Q

Motion of Crystal Electrons: Partially Filled Bands

(7 points)

A
  • 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 ωc = eB/m
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11
Q

Quantization of Electron Paths in B-Field

(6 points)

A
  • Solving Schrodingers equation yields results similar to quantum harmonic oscillator
  • Energy eigenvalues (see below)
    • E = (n + 1/2)(\hbar)ωc + (\hbar kz)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
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12
Q

De Haas-van Alphen Effect

(Overview)

A
  • Oscillation of magnetization of metals as function of high B-field and low temperature
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13
Q

De Haas-van Alphen Effect

(Assumptions: 6 points)

A
  • N electrons
  • |B| > 0
  • Landau level S fully filed and S+1 partially filled
    • EFS+1
  • Increasing B increases EF, because ES+1 ∝ B
  • Degeneracy of S, pS ∝ B, so eventually S+1 becomes empty and EF ∈ S
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14
Q

De Haas-van Alphen Effect

(Take-Away)

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