Energy Spectra of Electron Excitations in Graphite and Graphene and Their Dispersion Making Allowance for the Electron Spin and the Time-Reversal Symmetry
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 high-symmetry points in the corresponding Brillouin zones. With the help of theoretical symmetry-group methods for the spatial symmetry groups of crystalline graphite and single-layer graphene (in particular, the splitting of п-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.
V.O. Gubanov, A.P. Naumenko, M.M. Bilyi, I.S. Dotsenko, O.M. Navozenko, M.M. Sabov, L.A. Bulavin. Energy spectra correlation of vibrational and electronic excitations and their dispersion in graphite and graphene. Ukr. J. Phys. 63, 431 (2018 ). https://doi.org/10.15407/ujpe63.5.431
M.I. Katsnelson. Graphene: Carbon in Two Dimensions (Cambridge Univ. Press, 2012 ). https://doi.org/10.1017/CBO9781139031080
J.D. Bernal. The structure of graphite. Proc. Roy. Soc. London A 106, 749 (1924 ). https://doi.org/10.1098/rspa.1924.0101
T. Hahn. International Tables for Crystallography. Vol. A. Space Group Symmetry (D. Reidel, 1983 ).
C. Herring. Effect on time-reversal symmetry on energy bands of crystals. Rhys. Rev. 52, 361 (1937 ). https://doi.org/10.1103/PhysRev.52.361
C. Herring. Accidental degeneracy in the energy bands of crystals. Rhys. Rev. 52, 365 (1937 ). https://doi.org/10.1103/PhysRev.52.365
E.A. Wood. The 80 diperiodic groups in three dimensions. Bell System Techn. J. 43, 541 (1964 ). https://doi.org/10.1002/j.1538-7305.1964.tb04077.x
M.S. Dresselhaus, G. Dresselhaus, A. Jorio. Group Theory. Application to the Physics of Condensed Matter (Springer, 2008 ).
G.L. Bir, G.E. Pikus. Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974 ).
D.S. Balchuk, M.M. Bilyi, V.P. Gryschuk, V.O. Gubanov, V.K. Kononov. Symmetry of vibrational modes, invariance of energy states to time inversion, and Raman scattering in 4H- and 6H-SiC crystals. 1. Classification of energy states in Brillouin zones. Ukr. Fiz. Zh. 41, 146 (1996) (in Ukrainian ).
E. Doni, G. Pastori Parravicini. Energy bands and optical properties of hexagonal boron nitride and graphite. Nuovo Cimento B 64, 117 (1969 ). https://doi.org/10.1007/BF02710286
F. Bassani, G. Pastori Parravicini. Electronic States and Optical Transitions in Solids (Pergamon Press, 1975).