Lecture Notes on Quantum Hall Effect

Introduction

• Discussing previous lecture on quantization of the Hall plateaus.
• Quantum Hall Effect is characterized by quantized Hall conductivities.

Key Concepts

• Hall Resistivity ( (\rho_{xy})): Transverse resistivity.
• Longitudinal Resistance ( (\rho_{xx})): Also called magnetoresistivity, varies with magnetic field.
• Green plots represent (\rho_{xx}), which vanish except during jumps in the red curves (Hall conductivity).
• Robustness of plateaus against disorder and impurities._

Quantum Mechanical Nature

• Electrons in a magnetic field lead to quantization and plateaus.
• Plateaus set resistance benchmarks: (\frac{h}{e^2}) = 25.81 (\Omega)
• Hall voltage develops across transverse edges when longitudinal current flows.

Classical Trajectory of Charged Particles

Case 1: No Electric Field, Constant Magnetic Field

• Energy remains constant due to zero power delivered, described by Lorentz force.
• Motion is independent in longitudinal (X) and transverse (Y) directions.
• Equations of motion:
  • (m \frac{d^2 x}{dt^2} = Q v_y B)
  • (m \frac{d^2 y}{dt^2} = -Q v_x B)

Perpendicular Motion

• Derived equations yield harmonic motion.
• Final results for trajectories:
  • (x(t) = x_0 + \frac{v_{\perp}}{\omega_B} \sin(\omega_B t))
  • (y(t) = y_0 - \frac{v_{\perp}}{\omega_B} \cos(\omega_B t))

Circular Motion

• Superposition of perpendicular motions leads to circular motion with a guiding center.

Case 2: Electric Field Perpendicular to Magnetic Field

• Total force equation: (F = m \frac{dV}{dt} = Q (E + v \times B))
• Guiding center drifts perpendicular to both electric ( (E)) and magnetic fields ( (B)).

Drift Velocity and Lorentz Force

• Drift velocity (V_D = \frac{E \times B}{B^2})
• Net velocity is a combination of drift velocity and Lorentz velocity.
• Overall trajectory results in helical motion in the presence of electric and magnetic fields.

Quantum Mechanical Treatment

• For a charged particle in a magnetic field:
  • Schrödinger equation needs to be solved considering vector potential because of the magnetic field.
• Landau gauge is used to describe the system, leading to quantized energy levels.

Landau Levels

• Energy levels are equidistant and defined by:
  • (E_n = \left(n + \frac{1}{2}\right) \hbar \omega_B)
• High degeneracy for each Landau level, corresponding to many electrons occupying each level.

Properties of Landau Levels

• Degeneracy depends on the magnetic field and sample area:
  • (G \propto B \cdot A)
• Infinite degeneracy due to independence from quantum number.
• Flux quantum (\Phi_0 = \frac{h}{e}) is critical for understanding degeneracy.

Summary of Observations

• Hall resistivity shows plateaus at integral multiples of (\frac{h}{e^2}).
• Changes in resistivity occur based on electron density and flux quantization.

Conclusion

• Next discussion will provide references for further reading on the topic.
• Students encouraged to familiarize themselves with the materials.