How Do Landau Levels Function in Solid State Band Theory?

In summary, Landau levels refer to the energy levels of free electron gas in a magnetic field and are frequently used in solid state physics. In band theory, this term is used to describe systems with certain types of interactions that can be classified as Fermi liquids. The de Haas-van Alphen period is related to the extremal orbit of the Fermi surface, which is proportional to the electron density. The quasiparticles near the surface behave like free electrons, with their charge, spin, and momentum remaining the same but other quantities being renormalized. Landau levels are obtained from the Hamiltonian of electrons coupled to a vector potential, which can be manipulated to resemble a harmonic oscillator and obtain the eigenvalues. In a
  • #1
taishizhiqiu
63
4
Landau level is the energy levels of free electron gas in a magnetic field. However, this term is also frequently used in solid state physics. I have the following questions:
1. what does this term exactly mean in band theory? After all, electrons are not free here.
2. why is de Haas-van Alphen period related to extremal orbit of the Fermi surface?
 
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  • #2
A lot of metallic systems can be modeled as a free electron gas. If they have a certain type of interactions they can be classified as Fermi liquid meaning there are long lived quasiparticles near the surface. The volume enclosed by the FS is proportional to the electron density. The charge, spin and momentum of the electrons remain the same but other quantities are renormalized. So basically the quasiparticles behave like free electrons.

Landau levels are obtained from the Hamiltonian of electrons coupled to a vector potential. It turns out you can manipulate the expression to make it look like a harmonic oscillator to obtain the eigenvalues.

A semiclassical picture in 2d (like the QHE) has the electrons in cyclotron orbits with the radius eB/m. The centers drift in an E field.
 

What is the Landau level in band theory?

The Landau level in band theory is a concept in condensed matter physics that describes the energy levels of electrons in a magnetic field. It was first proposed by Lev Landau in 1930 and is used to understand the behavior of electrons in a solid material.

What is the significance of the Landau level in band theory?

The Landau level plays a crucial role in explaining the behavior of electrons in materials with a strong magnetic field. It helps to understand the formation of energy bands and the quantization of energy levels in a solid material.

How does the Landau level affect electronic properties of materials?

The Landau level affects electronic properties of materials by causing the formation of energy bands and the quantization of energy levels. This leads to the emergence of unique electronic properties such as the quantum Hall effect and the anomalous Hall effect.

What is the relationship between the Landau level and the Fermi level?

The Landau level and the Fermi level are both concepts used to describe the energy levels of electrons in a material. The Fermi level represents the highest energy level occupied by electrons at absolute zero temperature, while the Landau level describes the allowed energy states of electrons in a magnetic field.

How does the Landau level explain the behavior of electrons in a magnetic field?

The Landau level explains the behavior of electrons in a magnetic field by quantizing their energy levels. This means that the energy of an electron can only take certain discrete values, leading to the formation of energy bands. It also explains the movement of electrons in a material under the influence of a magnetic field, such as the circular motion of electrons in the quantum Hall effect.

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