For conventional solids, the Pauli exclusion principle. What you mean by "disturbance"?

 Dear Sergei ,

excuse me , my meaning is perturbation theory in quantum mechanic.

Abbas, I very strongly recommend you to study an introductory book on solid state physics, such as the one by Ashcroft and Mermin. The nature of your question clearly signifies that you are not sufficiently versed in the theory of solid state. I could answer your present question, however that would not solve your problem; you will have to master the basics first.

Dear Prof.farid,

thanks.ok.

You are welcome Abbas. For orientation you may begin with Sec. 5.3 (Solids), in particular Subsection 5.3.2 (Band Structure), of the book Introduction to Quantum Mechanics, by David J. Griffiths. The one-dimensional model that he calls 'Dirac comb' should be very instructive.

Dear Prof.farid,

very very  thanks due to your attention and guidance.

All the best,A

You are welcome Abbas.

I do not know if you already took Solid State courses but in any case, a good reading on the references cited here will help. Anyway, I found this nice website that can help with your readings: http://ecee.colorado.edu/~bart/book/book/chapter2/ch2_3.htm

There is also the website of Dr. Goss (https://www.staff.ncl.ac.uk/j.p.goss/) and you go to "EEE1001 / PHY1002 Lecture notes and revision material" and then click on "Band theory of solids; metals, semiconductors and insulators". There is a very nice set of slides there.

The hint is: Pauli exclusion principle states that two identical electrons cannot occupy the same energy quantum state. So how this will affect the energy profile of systems containing e.g. 1,2, ..., N electrons?

Have a nice studying and do not hesitate in asking.

 Dear Claudia ,

thanks.you are very kind to me.

Regards,A

Dear Abbas, I think I finally understood your question. It's a bit tricky because you are using a phrase "perturbation theory" instead of "perturbation potential". Any "perturbation theory" is just a mathematical method used to treat a physical problem (if you have a small parameter). It does not have any physical meaning. The origin of the simplest band structure in solids is based on the following two physical concepts: (i) the free (non-interacting) electrons, and (ii) periodicity of the atomic lattice. The latter condition results in a periodic structure of the electronic wave functions (Bloch theorem). This simple (or better, simplified) picture is exact (at least in 1D) and hence it does not require any perturbation theory. In a more realistic situation, you have to account for some kind of interaction between electrons (because it always exists in any real material). In traditional textbook band theory (which is a bit more sophisticated than the above-mentioned simplified picture), instead of free electrons one considers the so-called "nearly free electrons" approximation which introduces a periodic potential into the total Hamiltonian.  The effects of this potential on the formation (or better, modification) of band structure (obtained within the exact free-electron picture) can be treated using the perturbation theory provided of course that the electron interactions are weak enough (this method is not applicable to treat the band structure in the so-called "strongly correlated" materials). Now, getting back to your question. Both, the "perturbation potential" and the exclusion principle are important for band formation. More precisely, the lower energy bands are normally completely filled by the electrons since the electrons always tend to occupy the lowest available energy states. While the higher energy bands may be completely empty or may be partly filled by the electrons. Pauli’s exclusion principle restricts the number of electrons that a band can accommodate. A partly filled band appears when a partly filled energy level produces an energy band or when a totally filled band and a totally empty band overlap.

For further details, please check the following link.

http://www.newagepublishers.com/samplechapter/001410.pdf

 Dear Sergei ,

Thanks so much.

Regards,A

Note that hard-core bosons don't obey to the Pauli exclusion principle and they also exhibit a band structure in 1D. But this is a special case, because in 1D free hard-core bosons map exactly onto free spinless fermions, so the Pauli exclusion principle is still there.

The interction of electrons with lattice atoms, especially in a case of its strong ordering...  Naturally, the Pauli principal should be taken into account.

The energy bands in solids appear because a) the potential energy for electrons in a crystal is periodic in space and b) because electrons obey quantum mechanics (in some sense they behave like waves). Similar bands will exist for light or sound waves if the media is periodic in space. The Pauli principle explains how the bands are filled by electrons (fermions). Bosons would all sit in the lowest energy band (at low temperature).

The reason for band structure is the potential. The gradual increase of complexity while studying Condensed matter Physics goes from 'Free electron model' (where we do not see band gaps) to 'Nearly free electron model' (where we see band gaps).

When we study free electron model there is no potential part in the Hamiltonian and the electron is free to move, electron-electron interactions are short and negligible and so on. However, when we add the potential we start seeing gaps in the dispersion relation.

You will ask me how can I backup my answer. I recommend you to read this paper

https://www.photonics.ethz.ch/fileadmin/user_upload/novotny10a.pdf

In this paper they have considered classical harmonic oscillators which are coupled. We can see in this paper when there is no coupling (i.e. there is no potential in the Hamiltonian part, which represents the interaction) there is no "avoided crossing"(band gap) of the dispersion curves, but as we increase the coupling we start seeing that the curves start avoiding each other. This shows the origin of band gap is the potential or interaction inside the system.

As Ashcroft and Mermin seems to show that symmetry also plays a role in the origin of band gap, I beg to differ from their viewpoint. Let us consider that the potential is not symmetric as given in usual models of band theory of solids. There will still be band gaps. So if I summarize, neither symmetry (try taking a small asymmetric potential), nor the many particle nature(band gap phenomenon is independent of number of particles if you see any expression), nor any quantum phenomenon like Pauli exclusion principle (we consider Pauli exclusion even in free electron model) cause the band gaps, interaction plays the role of creating what we call as "Band gaps".