Quantum Hall Effects

Quantum Hall Effects
This series of lectures is meant to yield an introduction to the physics of two-dimensional (2D) electron systems in a strong magnetic field. Most saliently, such systems display the quantum Hall effect, i.e. the quantisation of the Hall resistance at low temperatures. Whereas the integer quantum Hall effect is due to the quantisation of the one-particle kinetic energy into highly-degenerate Landau levels and localisation effects, the fractional quantum Hall effect is intimately related to electron-electron interactions, which become relevant when a Landau level is only partially filled. From a technical point of view, 2D electron systems may be realised at the interface in semiconductor heterostructures, such as the widely used GaAS/AlGaAs compound. More recently, a one-atom thick layer of graphite (graphene) has been discovered. Graphene, as opposed to conventional 2D electron systems, has very particular electronic properties due to a linear dispersion of the energy band in the vicinity of the Fermi level. The electron dynamics thus needs to be described in the framework of a relativistic Dirac rather than a Schrödinger equation. Exposed to a strong magnetic field, graphene shows a very particular form of the quantum Hall effect, which may be viewed as a relativistic quantum Hall effect. In the framework of theses lectures, the basic quantum-mechanical treatment of 2D electrons in a strong magnetic field will be reviewed, which is the basis for the understanding of the quantum Hall effect. The case of relativistic electrons in graphene will also be introduced. The discussion of the integer quantum Hall effect comprises the physics of edge states and more generally the electron dynamics in the presence of both a strong magnetic field and a weak impurity potential, which localises the electrons in a particular manner. Finally, basic aspects of the fractional quantum Hall effect will be presented, such as Laughlin's wave function and quasi-particles with fractional charge.

This series of lectures is meant to yield an introduction to the physics of two-dimensional (2D) electron systems in a strong magnetic field.

Most saliently, such systems display the quantum Hall effect, i.e. the quantisation of the Hall resistance at low temperatures. Whereas the integer quantum Hall effect is due to the quantisation of the one-particle kinetic energy into highly-degenerate Landau levels and localisation effects, the fractional quantum Hall effect is intimately related to electron-electron interactions, which become relevant when a Landau level is only partially filled. From a technical point of view, 2D electron systems may be realised at the interface in semiconductor heterostructures, such as the widely used GaAS/AlGaAs compound.

More recently, a one-atom thick layer of graphite (graphene) has been discovered. Graphene, as opposed to conventional 2D electron systems, has very particular electronic properties due to a linear dispersion of the energy band in the vicinity of the Fermi level. The electron dynamics thus needs to be described in the framework of a relativistic Dirac rather than a Schrödinger equation.

Exposed to a strong magnetic field, graphene shows a very particular form of the quantum Hall effect, which may be viewed as a relativistic quantum Hall effect. In the framework of theses lectures, the basic quantum-mechanical treatment of 2D electrons in a strong magnetic field will be reviewed, which is the basis for the understanding of the quantum Hall effect.

The case of relativistic electrons in graphene will also be introduced.

The discussion of the integer quantum Hall effect comprises the physics of edge states and more generally the electron dynamics in the presence of both a strong magnetic field and a weak impurity potential, which localises the electrons in a particular manner.

Finally, basic aspects of the fractional quantum Hall effect will be presented, such as Laughlin's wave function and quasi-particles with fractional charge.