As regards graphene, consult Sec. 2.3.1 (p. 52) of [1]. More generally, consider the original and very readable publication by van Hove [2] and the subsequent relevant publications by Phillips, of which [3] is one. There is some topological significance associated with the van Hove singularities (the first Brillouin zone being a closed topological manifold), leading to their classification in terms of the Betti number [4] and the determination of their numbers on the basis of the celebrated Morse theorem [5].

[1] H Aoki, and MS Dresselhaus, editors, Physics of Graphene (Springer, Heidelberg, 2014).

[2] L van Hove, Phys. Rev. 89, 1189 (1953).

[3] JC Phillips, Phys. Rev. 104, 1263 (1956). Erratum: Phys. Rev. 105, 1933 (1957).

[4] http://mathworld.wolfram.com/BettiNumber.html

https://en.wikipedia.org/wiki/Betti_number

[5] http://mathworld.wolfram.com/MorseTheory.html

https://en.wikipedia.org/wiki/Morse_theory

Dear Pandey,

The following article may answer your request:

Li, G., Luican, A., Dos Santos, J.L., Neto, A.C., Reina, A., Kong, J. and Andrei, E.Y., 2010. Observation of Van Hove singularities in twisted graphene layers. Nature Physics, 6(2), p.109.

In two dimensions, a saddle point in the electronic band structure leads to divergence in the density of states, also known as a Van Hove-singularity (VHS). When the Fermi energy is close to the VHS, interactions, however weak, are magnified by the enhanced density of states (DOS), resulting in instabilities. In twisted graphene layers, both the position of fermi energy and that of the VHSs can be controlled by gating and rotation respectively, providing a powerful toolkit for manipulating electronic phases.

However, although the band structure of graphene contains a VHS, its large distance from the Dirac point makes it prohibitively difficult to reach by either gating or chemical doping. By introducing a rotation between stacked graphene layers, it is possible to induce VHSs that are within the range of fermi energy EF achievable by gate tuning.