According to solid state texts in Brillouin zone borders where diffraction condition satisfies we have a band gap. However I usually see Band gaps which are located in the center of Brillouin zone. Please correct me if I am wrong.
In "solid state physics, Ashcroft - Mermin, chapter 9, electrons in a weak periodic potential", it is explained that due to weak perturbation we have energy gap at Brillouin zone border. Moreover we now for example that in GaAs the energy gap is at K=0 (Gamma point) which is not at the Brillouin zone border.
I'm sure where to start. Do you understand how a weak periodic potential gives rise to a bandgap? Liboff's quantum textbook has a nice derivation using perturbation theory. Alternatively, you could use Fermi's golden rule using the weak periodic potential at the perturbing hamiltonian.
What do you mean? Is there any direct relationship between "how a weak periodic potential gives rise to a bandgap" and "where a band gap is" ? In many cases, the band gap is indeed not at the high symmetry points or located in the boundaries of the Brillouin zones (using vasp or wien2k, for examples),is it ? Another, what is "Fermi's golden rule" ? Is it related to band gap ? http://en.wikipedia.org/wiki/Fermi's_golden_rule "In quantum physics, Fermi's golden rule is a way to calculate the transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into another energy eigenstate, due to a perturbation."
You are perfectly right, In a solid, you can find the bandgap at the center of the BZ. It is easy to find a tight-binding model with just two orbitals per unit cell having the gap at the center of the BZ. It is also true that, starting from free electrons, with a parabolic dispersion relation, and considering the periodic potential as a perturbation, the gap shows up necessarily at the border of the BZ.
Starting from a free electron picture, the only region where you would expect bands to cross is at the BZ. This degeneracy is usually lifted when a periodic potential is present. So if you define the band gap as the minimal distance between two bands, you will find a band gap only, if the degeneracy at the BZ is lifted. This does not preclude the situation that in real solids with strong potentials the minimal distance between the bands is not at the BZ, but e.g. at its center.