Quantum Hall Effects in Physics
Explore the concepts of integer and quantum Hall effects, classical Hall effect, and quantum mechanics of an electron under a magnetic field. Discover the origin of integers, physical pictures of IQHE, and more. Dive into the fascinating world of quantum phenomena.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Integer Quantum Hall Effect Sankalpa Hazra @ Gopalan group 09/12/2022
Classical Hall effect Linear on B Independent of B Drude model is sufficient:
Quantum Hall effect Spikes ( xx) and plateaus ( xy) Quantum Hall regime: High magnetic field Low temperature High conductivity
Quantum mechanics of an electron under a magnetic field Step 1: Writing the Hamiltonian of the electron Hamiltonian : Assumptions: Spinless electrons Non-interacting electrons P.E. K.E. Vector potential of B Mechanical and canonical momentum become different.
Quantum mechanics of an electron under a magnetic field Defining the vector potential (not unique): Alternative (symmetric guage): Landau guage. x and y directions are not symmetric anymore
Quantum mechanics of an electron under a magnetic field Step 2: Solving the Hamiltonian Plane waves in the y - direction Schrodinger equation Simple Harmonic oscillator in the x- direction.
Quantum mechanics of an electron under a magnetic field Step 3: Plotting the solutions n: SHO modes in the x-direction Infinite degeneracy at each n (due to y- direction) Landau Levels
Origin of the integers Infinite degeneracy is broken in a finite dimensional sample: Electron density needed to fill Landau levels: V = LxLy Drude resistance:
Physical picture of IQHE Increasing the magnetic field Energy Energy k k If Landau Levels are filled: no states (in k-space) for electrons to move to No current under electric field ( xx = 0) Scattering time, also becomes infinitely large as a result no dissipation No current under electric field ( xx = 0)
Edge states and origin of plateaus Extende d Localise d Disorder in the system smears out each level, giving localized and extended states. Plateaus result from localized states which do not contribute to transport.
Further Fractional QHE regime arising from electron-electron interactions (many body systems) Electron charges fractionalize, and the resulting quasi particles themselves follow anyonic statistics. Edge modes give phenomena like, superconducting states, quantized heat transport, counter propagation of electronic or heat current opposite to their intuitive directions. Photonic crystals demonstrating dispersion less EM modes have also been demonstrated following the same underlying physics.
