Is linear E-k dispersion relationship possible in semiconductor?

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Peter Schindler

Yes, this is famously observed for Graphene and is called a Dirac cone. Graphene is a semi-metal and similar behavior can be observed in other 2d materials as well. Also, you can look up "Dirac semimetal" to find further materials and information, as well as this paper: Article Weyl and Dirac Semimetals in Three Dimensional Solids

Yes, of course. However, there are cases where a small gap (~100 meV) is observed/predicted for these type of materials. For example here: https://www.nature.com/articles/nphys3856 and here: https://www.nature.com/articles/srep18869