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The dispersion dependences of electron excitations in crystalline graphite and single-layer graphene have been studied taking the electron spin into consideration. The correlations of the energy spectra of electron excitations and, for the first time, the compatibility conditions for two-valued irreducible projective representations characterizing the symmetry of spinor excitations in the indicated structures are determined, as well as the distributions of spinor quantum states over the projective classes and irreducible projective representations for all highsymmetry points in the corresponding Brillouin zones. With the help of theoretical symmetrygroup methods for the spatial symmetry groups of crystalline graphite and single-layer graphene (in particular, the splitting of π-bands at the Dirac points), the spin-dependent splittings in their electron energy spectra are found. The splitting magnitude can be considerable, e.g., for dichalcogenides of transition metals belonging to the same spatial symmetry group. But it is found to be small for crystalline graphite and single-layer graphene because of a low spin-orbit interaction energy for carbon atoms and, as a consequence, carbon structures.

ukrainian journal of physics

Energy Spectra of Electron Excitations in Graphite and Graphene and Their Dispersion Making Allowance for the Electron Spin and the Time-Reversal Symmetry