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

Abstract

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.

Fine Spin-Dependent Splitting of Electronic Excitations and Their Dispersion in Single-Layer Graphene and Graphite

“…Therefore, we decided to consider the qualitative character of the electron spin influence on the structure of electronic -bands and the dispersion of -electronic states in graphite and other compounds, whose crystal lattices are characterized by this spatial symmetry, in more details. Among the electronic excitations at various highsymmetry points in the corresponding Brillouin zones of crystalline graphite -and single-layer graphene ,1 , which were analyzed in works [1,2], we will consider only the excitations of electronic -bands. Their wave functions are orthogonal to the wave functions of the 2 -hybridized bands of -electrons.…”

“…When determining the characters of the projective representations , which describe the symmetry of electronic states (without taking the electron spin into account) at the point of the corresponding Brillouin zone, we used formulas (1)-(18) from work [1]. At the same time, in order to find the characters of the projective representations ′ , which do the same but making allowance for the electron spin, formulas (2)-(8) from work [2] were applied. Whether the projective representations of the wave-vector groups of spatial symmetry groups belong to the -th projective class or not was determined by constructing the factor systems ( 2 , 1 ) and reducing them to the standard form with the help of the functions on groups, the coefficients ( ).…”

Симетрія Коливальних Станів Та Електронних Π-Орбі-Талей Молекули Бензену C6H6. Тонка Структура Спінза-Лежних Розщеплень

“…As was said above in connection with the other symmetry types, it is easy to obtain the components of the patterns for the doubly degenerate normal vibrations of the benzene molecule with the symmetry − 2 : they are the sums and the differences of the analytically identical components of the patternss for the doubly degenerate normal vibrations of the carbon and hydrogen atoms. The symmetry of the electronic states of -orbitals without taking and taking the electron spin into account can be determined following the method of works [7,8], which corresponds to a practical application of the LCAO method. According to this method, the characters of the equivalence representation oforbitals in the benzene molecule are determined at first, i.e.…”

“…This means that if the electron spin is not taken into account, the -orbitals are divided into two nondegenerate or-bitals, − 3 and + 4 , and two doubly degenerate ones, − 1 and + 2 . It was shown in works [7,8] (see Table 4 in work [7] and Table 1 in work [8]) that if the electron spin is taken into account, the electronic states can be classified according to the irreducible projective representations of the projective class 1 of the 6/ ( 6ℎ ) group. Those representations are quoted in Table 5.…”

“…As was done in work [7], to determine the representation of the electronic -orbitals taking the electron spin into account (the representation of ′ -orbitals), Γ ′ , the formula Γ ′ = ( eq ) ⊗ Γ ′ will be used. Here, the representation Γ ′ describing the symmetry of the -orbital taking the electron spin into account (the spinor ′ -orbital) is given by the formula…”