Energy Spectra Correlation of Vibrational and Electronic Excitations and Their Dispersion in Graphite and Graphene
Keywords:spinor representation of symmetry groups, factor-system, dispersion of elementary excitations
The correlation between the vibrational and electron excitation modes in the energy spectra of single-layer graphene and crystalline graphite, as well as the dispersion dependences of those modes, has been studied. The methods of the theory of projective representations of the point and spatial symmetry groups are used for the first time in order to interpret those correlations. The correlations of vibrational and electron excitation spectra and the compatibility conditions for irreducible projective representations in the descriptions of quantum states of graphene and crystalline graphite at various points of their Brillouin zones are determined. For the projective representations of all projective classes belonging to the hexagonal system, standard factor-systems are constructed for the first time. In particular, the factor-systems for electron states are first determined. The results obtained are used to calculate, also for the first time, the correct spinor multiplication tables, i.e. the multiplication tables for elements in double symmetry groups. The developed method is applied to classify all high-symmetry points in the Brillouin zones of single-layer graphene and crystalline graphite with respect to the symmetry type of vibrational excitations.
<li>O.V. Kovalev. Irreducible Representations of the Space Groups (Gordon and Breach, 1965).
<li>O.V. Kovalev. Irreducible and Induced Representations and Co-Representations of Fedorov groups (Nauka, 1986) (in Russian).
<li>J.D. Bernal. The structure of graphite. Proc. Roy. Soc. London A 106, 749-773 (1924).
<li>C. Herring. Effect on time-reversal symmetry on energy band of crystals. Phys. Rev. 52, 361 (1937).
<li>E.A. Wood. The 80 diperiodic groups in three dimensions. Bell System Tech. J. 43, 541 (1964).
<li>G.L. Bir, G.E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).
<li>H.A. Bethe. Termaufspaltung in Kristallen. Ann. Physik 395, 133 (1929).
<li>V.O. Gubanov, L.N. Ovander. Development of the Bethe method for the construction of two-valued space group representations and point groups. Ukr. J. Phys. 60, 950 (2015).
<li>E.P.Wigner. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959).
<li> D.S. Balchuk, M.M. Bilyi, V.P. Gryshchuk, V.O. Gubanov, V.K. Kononov. Symmetry of vibrational modes, time-inversion invariance of energy states, and Raman light scattering in 4H- and 6H-SiC crystals. 1. Classification of energy states in Brillouin zones. Ukr. Fiz. Zh. 41, 146 (1996) (in Ukrainian).
<li> T.L. Botte, V.A. Gubanov, L.A. Komarova, N.M. Belyi, S.V. Kovrigin, I.A. Petrusha, A.A. Shulzhenko. Correlation of vibrational modes and dispersion of phonon excitations in layered crystals of graphite and graphite-like boron nitride. 1. Classification and correlation of phonon states with zero quasimomentum. Nanosist. Nanomater. Nanotekhnol. 5, 287 (2007) (in Russian).
<li> V.A. Gubanov, L.A. Komarova, N.M. Belyi, S.V. Kovrigin. Correlation of vibrational modes and dispersion of phonon excitations in layered crystals of graphite and graphite-like boron nitride. 2. Dispersion of phonon states with nonzero quasi-momentum and the width of phonon bands. Nanosist. Nanomater. Nanotekhnol. 5, 307 (2007) (in Russian).
<li> M.S. Dresselhaus, G. Dresselhaus, A. Jorio. Group Theory. Application to the Physics of Condensed Matter (Springer, 2008).
<li> E. Doni, G. Pastori Parravicini. Energy bands and optical properties of hexagonal boron nitride and graphite. Nuovo Cimento B 64, 117 (1969).
<li> F. Bassani, G. Pastori Parravicini. Electronic States and Optical Transitions in Solids (Pergamon Press, 1975).</li>
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