Orbital Magnetism in graphene-like materials
Translated title
Rigorous Aspects of Orbital Magnetism in Graphene-like Materials
Author
Brynildsen, Mikkel Haggren
Term
4. term (FYS10)
Education
Publication year
2011
Submitted on
2011-06-30
Pages
51
Abstract
Vi undersøger det off-diagonale element i den todimensionelle ledningstensor for en krystal, som er udsat for et konstant, homogent magnetfelt. Ledningstensoren beskriver, hvordan elektrisk strøm reagerer på et elektrisk felt; det off-diagonale led måler responsen i én retning forårsaget af et felt i den vinkelrette retning. Med en diskret model af krystallen, baseret på en Harper-type Hamiltonian og Peierls-substitution, viser vi, at dette off-diagonale element, σ21, har en asymptotisk udvidelse i magnetfeltstyrken b. Vi viser også, at σ12 ved nul felt, σ12(b=0), er nul, og vi udleder en formel for den første afledning ved nul, σ12′(b=0). Denne koefficient er vigtig for forståelsen af Faraday-rotation, hvor et magnetfelt roterer polariseringen af lys, der passerer gennem et materiale. Vores metode bygger på spektralteori for Harper-typer af operatorer, Combes–Thomas-lokalisering, magnetisk perturbationsteori og Bloch–Floquet-teori.
We study the off-diagonal element of the two-dimensional conductivity tensor for a crystal placed in a constant, uniform magnetic field. The conductivity tensor describes how electric current responds to an applied electric field; the off-diagonal term measures the response in one direction caused by a field in the perpendicular direction. Using a discrete crystal model with a Harper-type Hamiltonian and the Peierls substitution, we show that this off-diagonal element, σ21, has an asymptotic expansion in the magnetic field strength b. We also show that σ12 at zero field, σ12(b=0), is zero, and we derive a formula for its first derivative at zero, σ12′(b=0). This coefficient is central to understanding the Faraday rotation effect, in which a magnetic field rotates the polarization of light passing through a material. Our analysis relies on spectral theory for Harper-type operators, Combes–Thomas localization, magnetic perturbation theory, and Bloch–Floquet theory.
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