Electron Gas in Magnetic Fields - Energy Levels and Degeneracy
Explore the behavior of an electron gas in magnetic fields, focusing on energy levels in two and three dimensions, densities of states, and the effects of uniform magnetic fields on electron motion. The lecture covers the effective Hamiltonian, potential, eigenstates, and degeneracy of Landau levels, providing insights into the magnetic field effects on a three-dimensional electron gas.
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Presentation Transcript
PHY 752 Solid State Physics 11-11:50 AM MWF Olin 103 Plan for Lecture 32: Chapter 15 in GGGPP: Electron Gas in Magnetic Fields 1. Energy levels for 2-d electron gas 2. Energy levels for 3-d electron gas Lecture notes prepared with materials from GGGPP textbook 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 1
11/13/2015 PHY 752 Fall 2015 -- Lecture 32 2
Digression densities of states for one, two, or three dimensional free electron gas 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 3
Consider a system of electrons confined to move in the x-y plane. In the absence of an external magnetic field, the energy levels are given by: Consider the effects of a uniform magnetic field along the z-axis Effective Hamiltonian: 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 4
Effective potential -- continued Taking into account constants of the motion Eigenstates: 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 5
= S L L x y 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 6
Degeneracy of Landau levels # occupied Landau levels 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 7
B>>0 n=2 B>0 n=4 B=0 n=3 EF EF EF n=2 n=1 n=1 n=0 n=0 E=0 E=0 E=0 Degeneracy of each Landau level e B BL hc = ( ) N L L x y 1 0 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 8
Eigenstates: 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 9
Magnetic field effects on 3-dimensional electron gas 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 10
11/13/2015 PHY 752 Fall 2015 -- Lecture 32 11
Density of states for 3-dimensional system 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 12
Optical absorption in presence of Magnetic field; transitions between conduction and valence bands 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 13
11/13/2015 PHY 752 Fall 2015 -- Lecture 32 14
Next lecture de Haas-van Alphen effect Oscillator 1/B behavior of free energy of free electron gas 11/13/2015 PHY 752 Fall 2015 -- Lecture 32 15
