Energy Spectra Correlation of Vibrational and Electronic Excitations and Their Dispersion in Graphite and Graphene

Abstract

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.K e y w o r d s: spinor representation of symmetry groups, factor-system, dispersion of elementary excitations.

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

“…At points Γ, the factor groups of the wave-vector groups with respect to the subgroups of trivial translations are isomorphic to the same point symmetry group 6/ , ( 6ℎ ) for both crystalline graphiteand single-layer graphene ,1 . This point group is a symmetry group of equivalent directions in the both structures: in crystalline graphite -, where it is a group of the crystalline class, and in single-layer graphene ,1 , where it characterizes the symmetry of the macromolecular class [1].…”

“…the bands of 2hybridized -electrons). In work [1], when determining the representations Γ for states neglecting the spin (at a weak spin-orbit interaction), we used the following formula, which, in addition to the representation Γ eq , includes only the representation Γ :…”

“…In turn, + 1/2 is an even two-dimensional (spinor) representation of the rotation group for the quantum number of total electron angular momentum = 1 2 . Its characters in the case of the rotation by the angle are equal to [9] ( ) = sin…”

“…For high-symmetry points in the Brillouin zone of crystalline graphite, irreducible projective representations were constructed according to which the wave functions of elementary excitations in this substance are transformed. In work [1], correlations between phonon and electron excitations in graphite and single-layer graphene were also demonstrated. For both structures, only -bands -i.e.…”

“…In work [1], a theoretical symmetry-group description was presented for the dispersion of vibrational and electron excitations in crystalline graphite. The analysis was carried out on the basis of projective classes of representations following from the spatial symmetry of crystalline graphite structure and determined at various points of the corresponding Brillouin zone.…”