Advanced Solid State Physics

Magnetic effects and Fermi surfaces

This section begins with a discussion of free electrons in a magnetic field. Classically, the electrons move in a circle in a plane perpendicular to the magnetic field or they spiral along the magntic field lines. (See: Motion of a Charged Particle in a Constant Magnetic Field) Quanutm mechanically, the electron states arrange into Landau levels. (See: The Hamiltonian of a charged particle in a magnetic field and Solutions to the Schrödinger equation for a charged particle in a magnetic field). The presence of a magnetic field modifies the density of states of of the free elelctrons in such a way that all of the thermodynamic properties oscillate with $1/B$. (See: Thermodynamic properties of free electrons in a magnetic field) Oscillations of the magnetization with $1/B$ are called de Haas - van Alphen oscillations and the oscillations of the electrical resistivity are called the Shubnikov - de Haas oscillations. At high magnetic fields only a few Landau levels are occupied and the Shubnikov - de Haas oscillations develop into the Quantum Hall effect.

Reading
Kittel chapter 9: Fermi surfaces and metals
Kittel Appendix G: Vector potential, field momentum, and gauge transformations
or Gross and Marx: 9.7.1: Freie Ladungsträger

Additional references
• Seitz, Modern Theory of Solids
• Landau and Lifshitz, Statistical Physics, part 1
• Peter Young at UCSC: http://physics.ucsc.edu/~peter/231/magnetic_field/magnetic_field.html

Student presentations
• Quanten-Hall-Effekt - Michael Krenn, 2008

Resources
• Fermi surface database
• Fermi surfaces calculated by tight binding
• Fermi surface applet
• The Quantized Hall Effect, Klaus von Klitzing, Nobel Prize 1985 (pdf).
• Fractional Quantization, Robert B. Laughlin, Nobel Prize 1998 (pdf).
• The Fractional Quantum Hall Effect, Horst L. Störmer, Nobel Prize 1998 (pdf).
• Interplay of Disorder and Interaction in Two-Dimensional Electron Gas in Intense Magnetic Fields, Daniel C. Tsui, Nobel Prize 1998 (pdf).
• Seitz, Modern Theory of Solids
• Landau and Lifshitz, Statistical Physics, part 1
• R. Gross und A. Marx, Magnetismus