Summary of Lec 11: Properties of Landau Levels

Scientific Concepts and Discoveries

• Integer Quantum Hall Effect: Discussed as a phenomenon that can be understood without invoking Coulomb interactions initially.
• Landau Levels:
  • Flat energy levels that arise in a magnetic field, resembling the harmonic oscillator's energy levels but with specific dependencies on the magnetic field (cyclotron frequency).
  • Each Landau level is infinitely degenerate, with degeneracy determined by the magnetic field strength and the area of the sample.
  • The energy of the Landau Levels is expressed as \( E_n = (n + \frac{1}{2}) \hbar \omega_B \), where \( \omega_B \) is the cyclotron frequency.
• Hall Resistivity and Plateaus:
  • The Hall Resistivity shows plateaus that correspond to integer values, related to the degeneracy of Landau Levels and the density of electronic states.
  • The relationship between the Hall Resistivity and the degeneracy is given by \( \rho_H = \frac{h}{ne^2} \), where \( n \) is the filling factor.
• Conductivity of Landau Levels:
  • Despite the flatness of Landau Levels leading to zero velocity of electrons, the Hall conductivity can still arise due to Edge States.
• Role of Spin:
  • Discussion on how including spin affects the Landau Levels and introduces a Zeeman term, which modifies the energy levels but does not significantly alter the quantum Hall effect.
• Effect of Electric Field:
  • The introduction of an electric field modifies the Hamiltonian and causes the Landau Levels to become dispersive, leading to a velocity for electrons and changing the nature of the wave functions.
• Edge States:
  • In a two-dimensional electron gas subjected to a strong magnetic field, Edge States allow for conduction while the bulk remains insulating, leading to the classification of these systems as topological insulators.

Methodology and Relationships

• Degeneracy Calculation: Degeneracy is calculated based on the area of the sample and the strength of the magnetic field.
• Chemical Potential Adjustment: As the magnetic field changes, the chemical potential adjusts to accommodate more electrons, leading to the observed plateaus in resistivity.
• Wave Function Modification: The wave functions of electrons are modified in the presence of an electric field, leading to new terms in the Hamiltonian.

Researchers and Sources Featured

• References:
  • Reviews of Modern Physics, Volume 58, Page 519 (1986)
  • Robert Llin, Physical Review B, Volume 20, Page 5632 (1981)
  • S.M. Gurin (preprint arXiv: 99702)
  • David Tong (preprint arXiv: 1606.0687)
  • Zahid Hasan (Reviews of Modern Physics, regarding topological insulators)

This lecture provides a comprehensive overview of the properties of Landau Levels and their significance in understanding the quantum Hall effect, emphasizing the interplay between magnetic fields, electronic properties, and Edge States.

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