Solid State in tight binding approximation, Brillouin zone, Fermi

[Plasmon1212]

Hello,

In few days, I have an examination, and I still have some dark zone in my head! If somebody could help me by giving me some advices/answers/way of reflexion/books to consult, it could be very great!

Here is my questions:
How to determine energy levels and wavefunction of the localized electronic states on
the defect for a periodic lattice?

Electrons on a two dimensional square lattice in tight-binding approximation
-how to obtain band structure?
-how to obtain E(kx; ky) in the first Brillouin zone?
-how to draw the Fermi surface with one and two electrons per site?
-how to determine if it is a metal or insulator?

Thank's a lot

(I'm sorry, but I'm not English speaker, so mistakes are possible;))

 

[BigJDubs]

The best way to answer these questions is to consult a textbook on solid state physics. Some key topics to look for are: tight-binding approximation, band structure, Brillouin zone, Fermi surface, metal vs insulator. Many of these topics will also have associated solved problems that can help you better understand the concepts. Additionally, there are many online resources available that may be able to answer any specific questions you may have. Good luck with your examination!

 

[charng]

Hello,

I am happy to provide some guidance and resources to help you with your examination on the topic of solid state in tight binding approximation, Brillouin zone, and Fermi. The following is a brief explanation of these concepts and some suggestions for further reading.

In solid state physics, the tight binding approximation is a method used to describe the electronic structure of a solid material. It assumes that the electrons in a crystal are localized around individual atoms and interact with each other through their overlapping wavefunctions. This allows us to simplify the problem of calculating the electronic properties of a solid to one of solving for the energy levels and wavefunctions of these localized electronic states.

The Brillouin zone is a concept used to describe the periodicity of a crystal lattice in reciprocal space. It is the first Brillouin zone that is of particular interest, as it contains all the information about the electronic properties of the material. The boundaries of the first Brillouin zone are determined by the reciprocal lattice vectors of the crystal.

To determine the energy levels and wavefunctions of localized electronic states on defects in a periodic lattice, you can use the tight binding approximation. This involves solving the Schrödinger equation for the localized states at the defect, taking into account the interactions with the surrounding lattice. Some recommended resources for further reading on this topic are "Introduction to Solid State Physics" by Charles Kittel and "Solid State Physics" by Neil W. Ashcroft and N. David Mermin.

To obtain the band structure of electrons on a two-dimensional square lattice in tight binding approximation, you can use the Bloch theorem, which states that the wavefunction of an electron in a periodic potential can be written as a product of a plane wave and a periodic function. This allows us to express the energy levels as a function of the wavevector k in reciprocal space.

To draw the Fermi surface, which represents the boundary between occupied and unoccupied energy levels at absolute zero temperature, you can plot the energy levels as a function of the wavevector k. The Fermi surface will be a closed curve in the first Brillouin zone. If there is one electron per site, the Fermi surface will be a single point, whereas if there are two electrons per site, the Fermi surface will be a closed curve.

To determine whether a material is a metal or insulator, you can use the concept of the band gap. If there is a gap between the