Correlation and Dispersion of Vibrational and Electron Excitation Energy Spectra in Graphite and Graphene
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.
2. O.V. Kovalev. Irreducible and Induced Representations and Co-Representations of Fedorov groups (Nauka, 1986) (in Russian).
3. J.D. Bernal. The structure of graphite. Proc. Roy. Soc. London A 106, 749-773 (1924).
4. C. Herring. Effect on time-reversal symmetry on energy band of crystals. Phys. Rev. 52, 361 (1937).
5. E.A. Wood. The 80 diperiodic groups in three dimensions. Bell System Tech. J. 43, 541 (1964).
6. G.L. Bir, G.E. Pikus, Symmetry and Strain-Induced Effects in Semiconductors (Wiley, 1974).
7. H.A. Bethe. Termaufspaltung in Kristallen. Ann. Physik 395, 133 (1929).
8. 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).
9. E.P.Wigner. Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (Academic Press, 1959).
10. 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).
11. 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).
12. 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).
13. M.S. Dresselhaus, G. Dresselhaus, A. Jorio. Group Theory. Application to the Physics of Condensed Matter (Springer, 2008).
14. E. Doni, G. Pastori Parravicini. Energy bands and optical properties of hexagonal boron nitride and graphite. Nuovo Cimento B 64, 117 (1969).
15. F. Bassani, G. Pastori Parravicini. Electronic States and Optical Transitions in Solids (Pergamon Press, 1975).
License to Publish the Paper
The corresponding author and the co-authors (hereon referred to as the Author(s)) of the paper being submitted to the Ukrainian Journal of Physics (hereon referred to as the Paper) from one side and the Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, represented by its Director (hereon referred to as the Publisher) from the other side have come to the following Agreement:
1. Subject of the Agreement.
The Author(s) grant(s) the Publisher the free non-exclusive right to use the Paper (of scientific, technical, or any other content) according to the terms and conditions defined by this Agreement.
2. The ways of using the Paper.
2.1. The Author(s) grant(s) the Publisher the right to use the Paper as follows.
2.1.1. To publish the Paper in the Ukrainian Journal of Physics (hereon referred to as the Journal) in original language and translated into English (the copy of the Paper approved by the Author(s) and the Publisher and accepted for publication is a constitutive part of this License Agreement).
2.1.2. To edit, adapt, and correct the Paper by approval of the Author(s).
2.1.3. To translate the Paper in the case when the Paper is written in a language different from that adopted in the Journal.
2.2. If the Author(s) has(ve) an intent to use the Paper in any other way, e.g., to publish the translated version of the Paper (except for the case defined by Section 2.1.3 of this Agreement), to post the full Paper or any its part on the web, to publish the Paper in any other editions, to include the Paper or any its part in other collections, anthologies, encyclopaedias, etc., the Author(s) should get a written permission from the Publisher.
3. License territory.
The Author(s) grant(s) the Publisher the right to use the Paper as regulated by sections 2.1.1–2.1.3 of this Agreement on the territory of Ukraine and to distribute the Paper as indispensable part of the Journal on the territory of Ukraine and other countries by means of subscription, sales, and free transfer to a third party.
4.1. This Agreement is valid starting from the date of signature and acts for the entire period of the existence of the Journal.
5.1. The Author(s) warrant(s) the Publisher that:
– he/she is the true author (co-author) of the Paper;
– copyright on the Paper was not transferred to any other party;
– the Paper has never been published before and will not be published in any other media before it is published by the Publisher (see also section 2.2);
– the Author(s) do(es) not violate any intellectual property right of other parties. If the Paper includes some materials of other parties, except for citations whose length is regulated by the scientific, informational, or critical character of the Paper, the use of such materials is in compliance with the regulations of the international law and the law of Ukraine.
6. Requisites and signatures of the Parties.
Publisher: Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine.
Address: Ukraine, Kyiv, Metrolohichna Str. 14-b.
Author: Electronic signature on behalf and with endorsement of all co-authors.