4.3 Quantum Hall effect and fractional quantum Hall effect

The quantum Hall effect reveals fascinating behavior of electrons in 2D systems under strong magnetic fields. It shows how quantum mechanics leads to unexpected phenomena like quantized resistance and fractional charges, pushing our understanding of condensed matter physics.

These effects have practical applications too, from precise resistance standards to potential quantum computing. They highlight the rich physics that emerges when electrons interact strongly in confined spaces, a key theme in nanoelectronics.

Quantum Hall Effect Fundamentals

Landau Levels and Magnetic Fields

• Landau levels represent quantized energy states of electrons in a magnetic field
• Magnetic field applied perpendicular to a 2D electron system creates Landau levels
• Energy spacing between Landau levels increases with stronger magnetic fields
• Landau level degeneracy determines the number of electrons that can occupy each level
• Cyclotron frequency relates to the energy spacing between Landau levels, given by $\omega_c = \frac{eB}{m}$

Hall Resistance and Filling Factor

• Hall resistance measures the voltage perpendicular to current flow in a magnetic field
• Quantized Hall resistance occurs at specific magnetic field strengths
• Filling factor ν indicates the number of filled Landau levels
• Integer filling factors correspond to fully occupied Landau levels
• Hall resistance quantization follows the formula $R_H = \frac{h}{e^2\nu}$

Edge States and Current Flow

• Edge states form at the boundaries of a 2D electron system in a magnetic field
• Electrons in edge states propagate along sample edges without scattering
• Edge states contribute to the quantized Hall conductance
• Bulk states remain localized and do not contribute to conductance
• Current flows through edge states, creating a dissipationless transport channel

Integer Quantum Hall Effect

Quantized Hall Resistance

• Integer quantum Hall effect occurs when the filling factor is an integer
• Hall resistance exhibits plateaus at precise quantized values
• Quantized resistance values given by $R_H = \frac{h}{e^2\nu}$ where ν is an integer
• Longitudinal resistance drops to zero at plateau centers
• Precision of quantization allows for resistance standard definition

Topological Order and Robustness

• Integer quantum Hall effect exhibits topological order
• Topological order manifests as the quantization of Hall conductance
• Quantized Hall conductance remains robust against small perturbations
• Topological protection arises from the gap between Landau levels
• Chern number characterizes the topological nature of the quantum Hall state

Experimental Observations and Applications

• High-mobility 2D electron systems (GaAs/AlGaAs heterostructures) used for observations
• Low temperatures and strong magnetic fields required for clear quantization
• Integer quantum Hall effect utilized in metrology for precise resistance measurements
• Quantum Hall resistance standard defined as $R_K = \frac{h}{e^2}$
• Applications in quantum computing and topological quantum computation

Fractional Quantum Hall Effect

Fractional Filling Factors and Electron Correlations

• Fractional quantum Hall effect occurs at fractional filling factors
• Observed at filling factors like 1/3, 2/5, 3/7, etc.
• Strong electron-electron interactions crucial for fractional quantum Hall states
• Laughlin wavefunction describes the ground state for certain filling factors
• Fractional filling factors correspond to partially filled Landau levels

Composite Fermions and Effective Magnetic Field

• Composite fermions consist of electrons bound to an even number of magnetic flux quanta
• Composite fermion theory explains fractional quantum Hall effect in terms of integer quantum Hall effect
• Effective magnetic field experienced by composite fermions differs from applied field
• Composite fermions fill effective Landau levels, leading to fractional filling factors
• Jain series of filling factors explained by composite fermion theory (2/5, 3/7, 4/9, etc.)

Quasiparticles and Fractional Charge

• Fractional quantum Hall effect supports quasiparticles with fractional charge
• Laughlin quasihole carries a fraction of an electron charge (e/3 for ν = 1/3 state)
• Quasiparticles obey fractional statistics (anyons)
• Braiding of quasiparticles proposed for topological quantum computation
• Experimental detection of fractional charges through shot noise measurements