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Landau levels. • Simplest case: “free” 2d electrons in a magnetic field (applies to electrons in a semiconductor 2DEG). • Hamiltonian. • Choose kx eigenstate. H = 1. 2m. (p + eA). 2. A = By?x. (x, y) = e ikxx. Y (y) metals. Recently an effect analogous to the AB effect, but for neutral particles, was predicted by Aharonov and Casher7 and has also been mea- sured.8. 2.4 Landau Levels. Consider 2D electron system in x ? y plane with field B ?z. Convenient to choose “Landau gauge” A = Bx?y, check that B = ? ? A = B?z. With this choice Harmonic Oscillator: Motion in a Magnetic Field. * The Schrodinger equation in a magnetic field. ? The vector potential. * Quantized electron motion in a magnetic field. ? Landau levels. * The Shubnikov-de Haas effect. ? Landau-level degeneracy & depopulation Landau Levels in Two and Three-Dimensional Electron. Gases in a Wide Parabolic Quantum Well. C.S. Sergio, G.M. Gusev, J.R. Leite,. Instituto de Fiisica da Universidade de Sгo Paulo, SP, Brazil. E.B. Olshanetskii, A.A. Bykov, N.T. Moshegov, A.K. Bakarov, A.I. Toropov,. Institute of Semiconductor Physics, Novosibirsk, INTRODUCTON TO QHE: INVERSION LAYERS AND LANDAU LEVELS. 1.1 D Landau level eigenstates. In a magnetic Bz, as you know, the non-interacting eigenstates of the Hamiltonian. (1.1.8) of charged particles are no longer plane waves, but rather Landau level states. From Hamiltonian to effective Hamiltonian. Sep 20, 2016 1.4 Landau Levels. 14. 1.4.1 Landau Gauge. 18. 1.4.2 Turning on an Electric Field. 21. 1.4.3 Symmetric Gauge. 22. 1.5 Berry Phase. 27. 1.5.1 Abelian Berry Phase and Berry Connection. 28. 1.5.2 An Example: A Spin in a Magnetic Field. 32. 1.5.3 Particles Moving Around a Flux Tube. 35. 1.5.4 Non-Abelian gradually and energy En will increase with B. One by one, all states with En < Eref will cross Eref. Define Bn as the B field when En < Eref. III. Landau's levels – energy degeneracy. 1. Assume the constant B field is along the z-axis. We need only to consider the kx- ky plane. The wave function of the electron can be written as. Free electrons in a magnetic field: Landau levels. But what happens when free (i.e. unbound) charged particles experience a magnetic field which influences orbital motion? e.g. electrons in a metal. ?. H = 1. 2m. (p ? qA(x,t)). 2. + q?(x,t), q = ?e. In this case, classical orbits can be macroscopic in extent, and there is no reason Landau Levels. 1 Classical Mechanics. We are interested in a charged particle (electron) in a uniform magnetic field. To simplify the setup, let us consider the electron to move only on xy plane under a magnetic field pointing along the z direction. You can always bring back the motion along the z direction, which is just a Landau Levels. Yen Lee Loh. Started 2012-2-8, touched 2012-7-24, 2013-2-1. 1. Setup. 2. Prelude: 3D hydrogen atom wavefunctions. 3. 2D Harmonic Oscillator using Cartesian Coordinates. Consider a 2D harmonic oscillator with M = k = — =W= 1. The Hamiltonian is. H. ` = 1. 2. Ip. ` x. 2 + p. ` y. 2 + x. `2 + y. `2M. This is the