Dear Vk Singh,

Generally, the bandstructure of a solid is not parabolic. The E(k) relation is parabolic for free particle E = p²/2m. Furthermore, this approximation can be applied for weak bounded states. In such cases you describe the influence of the crystal potential by an effective mass - deviating from the free electron mass. A parabolic approximation is valid near the local (in k-space) extrema of the bandstructure. You can carry out an expansion of series with a dominating parabolic term. This extrema you have at high symmetric points (cf. Gamma (0,0,0) or like in your figures K and M-point of the Brillouin zone). Due to the fact, that you have much maxima or minima, you have the region between them, which cannot be parabolic.

The parabolic approximation is important in different materials to describe the conductivity. In semiconductors, carriers are excited into the band minima to have a conductance. As an example, silicon. In metals you must discuss the form of the fermi surface. In silver it is nearly a sphere. So you could use well the parabolic approximation.

With Regard

R. Mitdank

The same question, albeit in different terms, was presented and responded to earlier here:

https://www.researchgate.net/post/Why_in_Topological_Insulators_TIs_bands_are_not_parabolic_type

Usually, k = -pi/a -> pi/a refers to a one dimensional system and the parabolic dispersion is characteristic of free fermions E = p^2/2m , that is, no interaction is considered.

When you go to higher dimensions, say 2d, you should describe your energy as a surface instead of a curve as it will depend on kx and ky , E(kx,ky). To avoid ploting a complex surface in the case when you have interaction and the surface is not parabolic, you just calculate a path of some values in your domain passing by some high symmetry points, in which we can by this especial names, Gamma, X, M , etc.

Usually Gamma represents the center of your Brillouin Zone. For exemple, Gamma = (kx=0, ky=0) in 2d, and Gamma = (kx= 0, ky=0, kz=0) in 3D.