Theoretical studies of integer quantum hall transitions

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Title:	Theoretical studies of integer quantum hall transitions
Authors:	Ho, C. M.
Contributors:	淡江大學物理學系
Date:	1998-10
Issue Date:	2012-05-30 22:57:51 (UTC+8)
Publisher:	University of Oxford
Abstract:	The topics covered in this thesis involve properties of electron states in the regime between integer quantum Hall plateaus, both in the bulk and in mesoscopic systems. The first three chapters discuss issues concerning bulk or macroscopic quantum Hall systems with strong disorder. Scaling behaviours of the longitudinal and Hall conductivities observed in experiments can be described as a continuous quantum critical phenomenon: electrons in the bulk undergo a localisation-delocalisation transition when the Fermi energy is swept across the centre of each broadened Landau level. Firstly, work examining mappings between different models of the integer plateau transition is presented. In particular, the network model based on the quantum percolation picture is mapped, in its continuum limit, to the two-dimensional Dirac fermion model coupled with randomness. Random Dirac particles therefore have a delocalisation transition in the same universality class as the integer quantum Hall plateau transition. Comparison is also made bet\veen the discrete unitary operator associated with the network model and some tight-binding Hamiltonians. Furthermore, the model of two-dimensional massless Dirac particles moving in a random magnetic field is studied around zero energy. This energy, which can be identified as the centre of a Landau level, turns out to be a mobility edge and states are critical. The critical wavefunction can be exactly written down for any realisation of randomness. Previously, this model has only been studied field-theoretically. We, by contrast, extract the Green's function associated with the square of the original Dirac Hamiltonian on a finite system. The power-law exponent of the density of states, behaviours of states at different energies, and conductance are investigated. Mesoscopic conductance fluctuations in the integer quantum Hall regime are studied in the last two chapters. Universal behaviour of the conductance fluctuations in weak magnetic fields can be understood in terms of the quantum phase coherence of the diffusive electron. However, recent measurements on the two-terminal conductance of small Silicon MOSFETs in high fields clearly exhibits no compatibility with the quantum interference picture. \Ve propose a new mechanism for the occurrence of fluctuations. Coulomb interactions are incorporated into the picture of edge-state transport through a single saddle-point. The occupancies of classical localised states change due to the interaction between electrons when the gate voltage on top of the device is varied. The energy of the saddle-point and therefore the transmission probability of edge states fluctuate because of the electrostatic potential between the localised states and the saddle-point. This model is studied numerically by .\lonte Carlo methods and exact enumeration, \yhich produces fluctuations ranging from 0 to (;2/ h between two conductance plateaus.
Appears in Collections:	[Graduate Institute & Department of Physics] Thesis

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