Double valley Dirac fermions for 3D and 2D Hg$_{1-x}$Cd$_x$Te with strong asymmetry

In this paper the possibility to bring about the double- valley Dirac fermions in some quantum structures is predicted. These quantum structures are: strained 3D Hg$_{1-x}$Cd$_x$Te topological insulator (TI) with strong interface inversion asymmetry and the asymmetric Hg$_{1-x}$Cd$_x$Te double quantum wells (DQW). The numerical analysis of the dispersion relation for 3D TI Hg$_{1-x}$Cd$_x$Te for the proper Cd ($x$)-content of in the Hg$_{1-x}$Cd$_x$Te-compound clearly show that the inversion symmetry breaking together with the unaxial tensile strain causes splitting each of Dirac nodes (two belonging to two interfaces) into two in the proximity of $\Gamma$-point. The similar effects can be obtained for asymmetric Hg$_{1-x}$Cd$_x$Te DQW with the proper content of Cd and proper width of the quantum wells. The aim of this work is to explore the inversion symmetry breaking in 3D TI and 2D DQWs mixed HgCdTe-systems. It is shown that this symmetry breaking leads to the dependence of carriers energy vs quasi-momentum similar to that of Weyl fermions.