The nature of critical phenomena of the scaling theory of quantum Hall transitions in two-dimensional electron systems

The nature of the transitions that occur between the quantized plateaux observed in the quantum Hall effect (QHE) have been classified as second order quantum phase transitions. These transitions occur between the localized and the extended states found within a Landau level band of energies. The theory of the critical phenomena associated with these quantum Hall transitions (QHTs) predicts a universal behaviour irrespective of any microscopic detail of the two-dimensional system (2DES) within which they are observed such as carrier concentration or mobility. This proposed universality of QHTs can be verified by measuring the value of certain critical exponents governing the transitions. If valid, these critical exponents should be measured as a universal constant in all instances. This thesis investigates the universality of QHTs using a finite-size scaling theory and attempts to address disagreements that exist in the literature on the critical exponents associated with QHTs. The scaling theory of QHTs presented here involves experimental studies based on varying either the temperature of the 2DES or the frequency of the applied electric field. It was found that the critical exponents of QHTs are not universal across all systems investigated. It is shown that changing the nature of disorder within the system influences the value of the critical exponent measured. In general, it was found that the experimental observation of quantum criticality, as expected from the critical phenomena theory of QHTs, depends on the competition between three key length scales characterizing the 2DES; the size of system, the phase coherent length and the typical size of the electron clusters forming within the system. A study on the limit of the observation of the QHE is also undertaken in the millimetre wave regime. It was found that localization within the 2DES, and as a result the QHE, is destroyed at frequencies below the millimetre wave regime for a GaAs based 2DES.