Electron in Magnetic Field: Quantum Hall Effect

Back in 1980, when Klaus von Klitzing and coworkers cooled the silicon field-effect transistor at low temperature and subjected to a high magnetic field, they found that the Hall conductance was precisely quantized in an integer multiple of the \(e^2/h\) (Klitzing et al. 1980), where h is Planck’s constant and e is the elementary electronic charge. This phenomenon is known as the integer quantum Hall (IQH) effect. Within five years of the discovery, Klaus von Klitzing received the Nobel Prize in Physics in 1985 for his discovery of the quantum Hall effect, which was the first experimentally observed topological phase of electrons in a solid. It further helps to understand the other topological phases of the matter realized in various systems. In this chapter, we will mainly discuss the quantum Hall (QH) phase realized in ‘graphene’ and the experimental observable which can be used to detect the topological order of these QH phases.