Quantum Hall Effect

Definition

The quantum Hall effect is a quantum phenomenon observed in two-dimensional electron systems under low temperatures and strong magnetic fields, where the Hall conductivity becomes quantized in integer or fractional values. This effect is crucial for understanding electron behavior in low-dimensional systems and has deep connections to topological phases of matter and various advanced materials.

5 Must Know Facts For Your Next Test

1. The quantum Hall effect was first observed experimentally by Klaus von Klitzing in 1980, leading to the discovery of quantized Hall conductance.
2. In the integer quantum Hall effect, the Hall resistance exhibits plateaus at quantized values of $$R_H = \frac{h}{e^2} \nu$$, where $$\nu$$ is an integer representing the filling factor.
3. The fractional quantum Hall effect occurs at specific fractional values of the filling factor and is associated with emergent collective excitations known as anyons.
4. The robustness of the quantum Hall effect arises from its topological nature, meaning it is stable against perturbations like disorder and impurities in the material.
5. Graphene has shown unique properties related to the quantum Hall effect, exhibiting both integer and fractional effects due to its linear dispersion relation near the Dirac point.

Review Questions

• How does the quantization observed in the quantum Hall effect relate to the Fermi surface and electronic states?
  • The quantization in the quantum Hall effect directly connects to how electrons occupy states in a two-dimensional system. The Fermi surface's characteristics dictate which electronic states are filled at low temperatures and how they respond to external magnetic fields. This leads to discrete levels of conductivity known as plateaus, as electrons fill Landau levels which are a direct consequence of the underlying Fermi surface structure.
• Discuss how topological concepts enhance our understanding of the quantum Hall effect and relate this to Chern numbers.
  • Topological concepts play a crucial role in understanding the quantum Hall effect through the notion of Chern numbers, which characterize different phases of matter. In this context, each quantized value of Hall conductance corresponds to a different Chern number that reflects the topology of the underlying band structure. This relationship emphasizes that physical properties such as conductivity can arise from global geometric features rather than local details, thereby showcasing how topology influences electronic behavior.
• Evaluate the significance of graphene's realization of both integer and fractional quantum Hall effects in advancing our knowledge of condensed matter physics.
  • Graphene's ability to exhibit both integer and fractional quantum Hall effects highlights its unique electronic properties and challenges conventional understanding in condensed matter physics. Its linear dispersion relation allows for a high mobility of charge carriers, enabling detailed exploration of quantized conductance under varied conditions. The fractional quantum Hall effect observed in graphene provides insights into exotic quasiparticles like anyons, pushing forward research into new phases of matter and potential applications in quantum computing, thus broadening our comprehension of fundamental physics.

Related terms

Fermi Surface: The Fermi surface is the collection of points in momentum space that separates occupied from unoccupied electron states at absolute zero temperature, playing a critical role in determining the electronic properties of metals and semiconductors.

Topological Insulator:

A topological insulator is a material that behaves as an insulator in its interior but supports conducting states on its surface, a property closely related to the quantum Hall effect and topological phases.

Chern Number: The Chern number is an integer that characterizes the topology of the electronic band structure in a system, playing a key role in determining the quantized Hall conductivity in the quantum Hall effect.