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Studying of relativistic particles in the condensed matter system became possible after a discovery of graphene in 2004. What follows from graphene was search for other materials hosting relativistic particles, such as Dirac semimetals and Weyl semimetals. Low energy fermionic excitations of such materials show linear energy-momentum dispersion relationship, which can be described by the Dirac and Weyl Hamiltonians. In this dissertation, we utilize low temperature magneto-infrared spectroscopy to obtain detailed understanding of band dispersion of the Dirac and Weyl semimetals. Low energy electronic band structures of Weyl semimetals consist of two non-degenerated linear bands with crossing points, which is called Weyl bands. Weyl semimetals can be categorized into type-I and type-II depending on degree of tilting of the Weyl bands. In the first part of this dissertation, we present magneto-optical spectroscopy experiment on a archetypical type-I Weyl semimetal, NbP. We observed a complicated structure of inter-Landau level transitions, which only can be satisfactorily explained by a newly proposed Weyl Hamiltonian model, which includes tilting effect. Our work demonstrates that Weyl band of NbP is strongly tilted from ideal type-I case and a theoretical model with effect of tilting can give accurate information of Weyl band structures. A zinc-blende crystal of HgCdTe has been known to have highly tunable band structure as a function of cadmium content and temperature. At optimal concentration and temperature, band structure is predicted to show linear band dispersion. Band structures of HgCdTe have been successfully described by Kane model, thus, quasiparicles in the system are called Kane fermions. Here, in the second part of the thesis, we study chemical composition and temperature dependent band structure evolution of Hg₁−ₓCdₓTe through magneto-optical spectroscopy. Our observation proves that the Kane fermions in HgCdTe become true massless quasiparticles at a specific condition.
