Landau Quantization - Landau Levels

Landau Levels

Each set of wave functions with the same value of is called a Landau level. Effects of Landau levels are only observed when the mean thermal energy is smaller than the energy level separation, meaning low temperatures and strong magnetic fields.

Each Landau level is degenerate due to the second quantum number . If periodic boundary conditions are assumed, can take the values, where is an integer. The allowed values of are further restricted by the condition that the center of the oscillator must physically lie within the system, . This gives the following range for :

For particles with charge, the upper bound on can be simply written as a ratio of fluxes:

where is the fundamental quantum of flux and is the flux through the system (with area ). Thus for particles with spin, the maximum number of particles per Landau level is

The above gives only a rough idea of the effects of finite-size geometry. Strictly speaking, using the standard solution of the harmonic oscillator is only valid for systems unbounded in the x-direction (infinite strips). If the size is finite, boundary conditions in that direction give rise to non-standard quantization conditions on the magnetic field, involving (in principle) both solutions to the Hermite equation. The filling of these levels with many electrons is still an active area of research.

The apparent "oscillator center" is in this sense spurious, as the system has no reference point on the x-axis. It is, however, an indication of the very real issue of translational symmetry breaking: orbits in a magnetic field are circles, so how to choose their centers? Related issues on a lattice have also been discussed at length.

Generally, Landau levels are observed in electronic systems, where and . As the magnetic field is increased, more and more electrons can fit into a given Landau level. The occupation of the highest Landau level ranges from completely full to entirely empty, leading to oscillations in various electronic properties (see De Haas-van Alphen effect and Shubnikov-De Haas effect).

If Zeeman splitting is included, each Landau level splits into a pair, one for spin up electrons and the other for spin down electrons. Then the occupation of each spin Landau level is just the ratio of fluxes . Zeeman splitting will have a significant effect on the Landau levels because their energy scales are the same, . However, the Fermi energy and ground state energy stay roughly the same in a system with many filled levels since pairs of split energy levels cancel each other out when summed.