ABOUT THE COURSE:Quantum Hall effect is undoubtedly the greatest discovery in material science in the last century. The quantized Hall plateaus shows the occurrence of quantum effects at macroscopic scales and sets the standard of resistance from a lab experiment on a dirty two dimensional electron gas. The Hall effect is used in magnetic field sensors that are present in a large number of devices. It has already earned 3 noble prizes, one each of Integer Qauntum Hall effect (1985) , Fractional Quantum Hall effect (1998) and its connections with topology (2016). The course will introduce from the scratch the behavior of electrons at low temperatures and large magnetic fields, formation of Landau levels, Quantization of Hall plateaus from a physical perspective and hence derive Kubo formula. The connections with topology are shown via computing the Chern number. Moreover the fractional quantum Hall effect is introduced with a brief description of fractional statistics and Anyons. Finally the course introduces Spin Hall effect and its relevance to the field of spintronics.INTENDED AUDIENCE: M.Sc/PhD/Project students of different institutes and universities, researchers in Condensed Matter Physics.PREREQUISITES: Quantum MechanicsINDUSTRY SUPPORT: The applied component is the spintronic materials. Thus people from the industry who are engaged in computer storage, communication may be interested.
Overview
Syllabus
Week 1: Transport in mesoscopic systems, Historical introduction of Hall effect, Classical Hall effect, Discovery of Quantum Hall Effect (QHE), Two dimensional electron gases (2 DEG), Band diagramsWeek 2:Hall Resistivity and Conductivity, Resistivity measurements, QHE and Metrology, quantization of resistivity, experimental plot of quantized resistivity, Hall and longitudinal resistivity, plateaus, Integer QHE (IQHE)Week 3:2DEG in strong magnetic field: Solution of Schroedinger equation, Gauge invariance, Landau gauge, Landau levels degeneracy, quantum of flux, Shubnikov de Haas (SdH) oscillationsWeek 4:Laughlin’s argument: Corbino ring, Origin of the plateaux, effect of electric field, inclusion of spins of electrons, Role of disorder, edge states, topological insulatorWeek 5:Hall conductivity: Linear response theory, derivation of Kubo formula, choice of gauge potentials, Hall conductivity and Chern numberWeek 6:Topological considerations: Gauss-Bonnet theorem, Berry connection and Berry curvature, IQHE revisited in Graphene and 2D Dirac materials, quantization of the plateaus, Graphene nanoribbon.Week 7:Fractional Quantum Hall effect (FQHE): QHE at non-integer filling fractions, Role of Coulomb interactions, Laughlin wave function, Symmetric gauge, Lowest Landau level, Fractional Statistics, Fractional charge, Anyons, Braiding statisticsWeek 8:Spin Hall effect: Spin Orbit coupling, Spin Hall effect, application to spintronics, real materials, HgTe-CdTe quantum wells
Taught by
Prof. Saurabh Basu
