energy spectra of electron excitations in graphite โฆ
TRANSCRIPT
V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
https://doi.org/10.15407/ujpe65.4.342
V.O. GUBANOV, A.P. NAUMENKO, M.M. BILYI, I.S. DOTSENKO, M.M. SABOV,M.S. IAKHNENKO, L.A. BULAVINTaras Shevchenko National University of Kyiv, Faculty of Physics(64/13, Volodymyrsโka Str., Kyiv 01601, Ukraine)
ENERGY SPECTRA OF ELECTRONEXCITATIONS IN GRAPHITE AND GRAPHENEAND THEIR DISPERSION MAKINGALLOWANCE FOR THE ELECTRON SPINAND THE TIME-REVERSAL SYMMETRY
The dispersion dependences of electron excitations in crystalline graphite and single-layergraphene have been studied taking the electron spin into consideration. The correlations ofthe energy spectra of electron excitations and, for the first time, the compatibility conditionsfor two-valued irreducible projective representations characterizing the symmetry of spinor ex-citations in the indicated structures are determined, as well as the distributions of spinor quan-tum 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 ๐-bands at the Dirac points), the spin-dependent splittings intheir electron energy spectra are found. The splitting magnitude can be considerable, e.g., fordichalcogenides of transition metals belonging to the same spatial symmetry group. But it isfound to be small for crystalline graphite and single-layer graphene because of a low spin-orbitinteraction energy for carbon atoms and, as a consequence, carbon structures.K e yw o r d s: crystalline graphite, single-layer graphene, spinor representations, factor-systems, dispersion of electron excitations, projective classes, two-valued irreducible projectiverepresentations.
1. Introduction
In work [1], a theoretical symmetry-group descrip-tion was presented for the dispersion of vibrationaland electron excitations in crystalline graphite. Theanalysis was carried out on the basis of projectiveclasses of representations following from the spatialsymmetry of crystalline graphite structure and deter-mined at various points of the corresponding Bril-louin zone. For high-symmetry points in the Bril-louin zone of crystalline graphite, irreducible pro-jective representations were constructed according towhich the wave functions of elementary excitationsin this substance are transformed. In work [1], cor-relations between phonon and electron excitations ingraphite and single-layer graphene were also demon-strated. For both structures, only ๐-bands โ i.e. the
cโ V.O. GUBANOV, A.P. NAUMENKO, M.M. BILYI,I.S. DOTSENKO, M.M. SABOV, M.S. IAKHNENKO,L.A. BULAVIN, 2020
electron bands of ๐-orbitals producing ๐-electronsand ๐-holes, whose wave functions are orthogonal tothe functions of ๐-zones of ๐ ๐2-hybridized ๐-orbitals โwere considered as electron ones. In so doing, we didnot consider the spin-orbit interaction for electronstates, because it is insignificant for ๐-bands in car-bon structures [2].
The symmetry of the crystal lattice of Bernalgraphite [3] is described by the spatial group๐63/๐๐๐ (๐ท4
6โ), which is also the spatial symmetrygroup of the crystal lattices of hexagonal boron ni-tride (BN) and hexagonal polytypes 2H ๐ and 2H ๐
of the dichalcogenides of transition metals (MoS2,MoSe2, WS2, WSe2, TeS2, and TeSe2). Therefore,it was important for us to determine the qualita-tive character of the influence of an electron spin onthe structure of ๐-bands in graphite and other com-pounds, whose crystal lattice is described by the spa-tial symmetry group ๐63/๐๐๐ (๐ท4
6โ). Another issue,also important for us, was to consider the influence of
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a bFig. 1. Structure of a standard unit cell of graphite crystals ๐พ-๐ถ (๐) and arrangement and orientation of the elements of thepoint symmetry group 6/๐๐๐ (๐ท6โ) (๐). Circles indicate the positions of carbon atoms (reproduced from work [1])
the time-reversal symmetry on elementary excitationsin lattices with the indicated spatial symmetry.
2. Standard Unit Cells, Brillouin Zones,and Basic Symmetry Elements of Graphiteand Single-Layer Graphene
In Fig. 1, ๐, a standard unit cell of the crystal lat-tice of Bernal graphite, ๐พ-๐ถ, is shown [3]. It corre-sponds to the standard diagram of its spatial sym-metry group ๐63/๐๐๐ (๐ท4
6โ) [4]. In Fig 1, ๐, the ar-rangement and orientation of the symmetry elementsfor the point group 6/๐๐๐ (๐ท6โ) are demonstrated.
Figure 2 illustrates the Brillouin zone in ๐พ-๐ถ crys-tals and its symmetry points. The points are denotedby letters corresponding to Herringโs notation forhexagonal structures [5, 6].
The WignerโSeitz unit cell and the Brillouin zonefor single-layer graphene ๐ถ๐ฟ,1 are depicted in Fig. 3, ๐and Fig. 3, ๐, respectively. Solid lines are used inFig. 3, ๐ to schematically mark the unit cell ofgraphene ๐ถ๐ฟ,1. The figure also illustrates the corre-sponding primitive translation vectors ๐1 and ๐2, aswell as the orientation of cell symmetry elements inthe three-dimensional space, which were used in cal-culations. The dashed lines in Fig. 3, ๐ demonstratethe reciprocal lattice vectors ๐1 and ๐2 on an arbi-
Fig. 2. Brillouin zone of graphite ๐พ-๐ถ crystals and its sym-metry points (reproduced from work [1])
trary scale and the positions of reciprocal lattice sitesin the reciprocal space. In Fig. 3, ๐, on the contrary,solid lines are used to show the reciprocal lattice vec-tors, and the dashed ones to demonstrate the directlattice vectors. The unit cells (the WignerโSeitz cells)of the graphene layer in the coordinate (Fig. 3, ๐) andreciprocal (Fig. 3, ๐) spaces (in the latter case, thiscell coincides with the first Brillouin zone) are col-ored grey. In Fig. 3, ๐, the high-symmetry points ฮ ,๐พ, and ๐ in the Brillouin zone of graphene are also
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V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
a bFig. 3. WignerโSeitz unit cell (๐) and Brillouin zone of single-layer graphene ๐ถ๐ฟ,1 (b) (reproduced from work [1])
shown. The equivalent points are marked by one ortwo primes.
The spatial symmetry group of the crystalline lat-tice of graphite, ๐63/๐๐๐ (๐ท4
6โ), is nonsymmor-phic. It is determined by the basic (main) elements,which can be chosen as follows:
โ1 = (0|๐), โ2 = (0|๐3), โ3 = (0|๐23), โ4 = (0|(๐ข2)1),
โ5 = (0|(๐ข2)2), โ6 = (0|(๐ข2)3), โ7 =(๐1
2
๐2
),
โ8 =(๐1
2
๐56
), โ9 =
(๐1
2
๐6
), โ10 =
(๐1
2
(๐ขโฒ
2)1
),
โ11 =(๐1
2
(๐ขโฒ
2)2
), โ12 =
(๐1
2
(๐ขโฒ
2)3
), โ13 = (0|๐),
โ14 = (0|๐๐3), โ15 = (0|๐๐23), โ16 = (0|๐(๐ข2)1),
โ17 = (0|๐(๐ข2)2), โ18 = (0|๐(๐ข2)3), โ19 =(๐1
2
๐๐2
),
โ20 =(๐1
2
๐๐56
), โ21 =
(๐1
2
๐๐6
), โ22 =
(๐1
2
๐(๐ขโฒ
2)1
),
โ23 =(๐1
2
๐(๐ขโฒ
2)2
), โ24 =
(๐1
2
(๐ขโฒ
2)3
),
where ๐1 is a primitive vector of the crystal latticedirected along the axis ๐๐ (๐๐ง). At the same time,
the spatial symmetry group of the crystal lattice ofsingle-layer graphene, ๐6/๐๐๐ (๐ท๐บ80) [7], whosediagram coincides with that of the tri-periodic spatialgroup ๐6/๐๐๐ (๐ท1
6โ), is symmorphic, and all itsโrotationalโ elementsโthe symmetry elements of thepoint group 6/๐๐๐ (๐ท6โ)โdo not contain nontrivial(partial) translations.
3. Qualitative Characterof the Influence of Electron Spin andTime-Reversal Symmetry on the EnergySpectra of Elementary Excitationsin Crystalline Graphite, Their Dispersionat the Points Along the Lines ฮโฮโA,KโPโH, and MโUโL of Its Brillouin Zone,and the Energy Spectra and Dispersionof Electron ๐-Bands of Single-LayerGraphene at Points ฮ , K, and M
3.1. Line ฮโฮโA of crystalline graphite andpoint ฮ of single-layer graphene
3.1.1. Points ฮ
At points ฮ, the factor groups of the wave-vectorgroups with respect to the subgroups of trivial trans-lations are isomorphic to the same point symmetry
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group 6/๐๐๐, (๐ท6โ) for both crystalline graphite ๐พ-๐ถ and single-layer graphene ๐ถ๐ฟ,1. This point groupis a symmetry group of equivalent directions in theboth structures: in crystalline graphite ๐พ-๐ถ, where itis a group of the crystalline class, and in single-layergraphene ๐ถ๐ฟ,1, where it characterizes the symmetryof the macromolecular class [1].
The wave functions of vibrational elementary ex-citations at points ฮ โ ฮvib (ฮlat.vib) are trans-formed, for both crystalline graphite and single-layergraphene, by the single-valued vector irreducible rep-resentations of the point group 6/๐๐๐ (๐ท6โ), whichare irreducible representations of the projective class๐พ0 of this group. The representations of those exci-tations are determined by the formula [8]
ฮvib = ฮeq โ ฮvector, (1)
where ฮeq is the atomic equivalence representation 1
at point ฮ , and ฮvector is the representation of a polarvector r with the components ๐ฅ, ๐ฆ, and ๐ง.
Among the electron excitations at points ฮ , wewill consider only the excitations of electron ๐-bands,whose wave functions are orthogonal to those ofthe ๐ ๐2-hybridized ๐-bands (i.e. the bands of ๐ ๐2-hybridized ๐-electrons). In work [1], when determin-ing the representations ฮ๐ for states neglecting thespin (at a weak spin-orbit interaction), we used thefollowing formula, which, in addition to the represen-tation ฮeq, includes only the representation ฮ๐ง:
ฮ๐ = ฮeq โ ฮ๐ง, (2)
where ฮ๐ง is a representation that is an irreduciblerepresentation of the group 6/๐๐๐ (๐ท6โ) for a vec-tor directed along the ๐ง-axis, because the electron๐-bands in graphite and graphene are formed by thenondegenerate electron orbitals ๐๐ง.
In order to determine the electron representation of๐-bands making allowance for the electron spin, ฮโฒ
๐,we have to use the formula
ฮโฒ๐ = ฮeq โ ฮโฒ
๐ง. (3)
Here, ฮโฒ๐ง is the representation of the electron ๐-orbital
taking the spin into account. It is determined using
1 A technique used for determining the character of the atomicequivalence representations and the results of correspondingcalculations for the high-symmetry points in the Brillouinzones of graphite and single-layer graphene were presentedin work [1].
the formula
ฮโฒ๐ง = ฮ๐ง โ๐ท+
1/2. (4)
In turn, ๐ท+1/2 is an even two-dimensional (spinor) rep-
resentation of the rotation group for the quantumnumber of total electron angular momentum ๐ = 1
2 .Its characters in the case of the rotation by the angle๐ are equal to [9]
๐๐(๐๐) =sin
[(๐ + 1
2
)๐]
sin(๐2
) . (5)
In Table 1, the irreducible representations of theprojective class ๐พ0 for the group 6/๐๐๐ (๐ท6โ) aregiven. They describe the symmetry of vibrationaland electron excitations at points ฮ of crystallinegraphite ๐พ-๐ถ and single-layer graphene ๐ถ๐ฟ,1 mak-ing no allowance for the spin. They are identical tothe ordinary single-valued vector irreducible repre-sentations. In addition, Table 1 contains the irre-ducible representations of the projective class ๐พ1 forthe group 6/๐๐๐ (๐ท6โ). They characterize the sym-metry of electron states making allowance for thespin. They are two-valued spinors.
In Table 2, the characters of equivalence repre-sentations, ฮeq, and the characters of representa-tions that characterize the symmetry of electron ๐-bands making allowance for the electron spin (๐โฒ-bands), ฮโฒ
๐ = ฮeq โ ฮโฒ๐ง, are presented for points ฮ
in the Brillouin zones of crystalline graphite ๐พ-๐ถ andsingle-layer graphene ๐ถ๐ฟ,1. The table also presentsthe characters of the representations ฮ๐ง, ๐ท+
1/2, andฮโฒ๐ง = ฮ๐ง โ๐ท+
1/2, as well as the corresponding projec-tive representations, for other high-symmetry pointsin the Brillouin zones of those structures.
The electron excitation distributions at the high-symmetry points in the Brillouin zones of crystallinegraphite and single-layer graphene with respect toirreducible two-valued spinor projective representa-tions are shown in Table 3. For the sake of compari-son, the distributions of electron excitations with re-spect to irreducible projective representations for ๐-bands without taking the electron spin into consider-ation are also included.
In Table 3, the following notations for the irre-ducible projective representations are applied in orderto clearly distinguish two-valued spinor representa-tions in various projective classes for various points
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V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
Table 1. Characters of the one- and two-valued irreducible projectiverepresentations for points ฮ in the Brillouin zones of crystalline graphite ๐พ-๐ถ and single-layer graphene ๐ถ๐ฟ,1
Projec-tiveclass
Notation forirreducibleprojective
representation
6/๐๐๐(๐ท6โ)
๐ ๐3 ๐23 3๐ข2 ๐2 ๐56 ๐6 3๐ขโฒ2 ๐ ๐๐3 ๐๐23 3๐๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ
2
๐พ0 ฮ+1 ๐ด+
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1ฮโ1 ๐ดโ
1 1 1 1 1 1 1 1 1 โ1 โ1 โ1 โ1 โ1 โ1 โ1 โ1
ฮ+2 ๐ด+
2 1 1 1 1 โ1 โ1 โ1 โ1 1 1 1 1 โ1 โ1 โ1 โ1
ฮโ2 ๐ดโ
2 1 1 1 1 โ1 โ1 โ1 โ1 โ1 โ1 โ1 โ1 1 1 1 1ฮ+3 ๐ด+
3 1 1 1 โ1 1 1 1 โ1 1 1 1 โ1 1 1 1 โ1
ฮโ3 ๐ดโ
3 1 1 1 โ1 1 1 1 โ1 โ1 โ1 โ1 1 โ1 โ1 โ1 1ฮ+4 ๐ด+
4 1 1 1 โ1 โ1 โ1 โ1 1 1 1 1 โ1 โ1 โ1 โ1 1ฮโ4 ๐ดโ
4 1 1 1 โ1 โ1 โ1 โ1 1 โ1 โ1 โ1 1 1 1 1 โ1
ฮ+5 ๐ธ+
1 2 โ1 โ1 0 2 โ1 โ1 0 2 โ1 โ1 0 2 โ1 โ1 0ฮโ5 ๐ธโ
1 2 โ1 โ1 0 2 โ1 โ1 0 โ2 1 1 0 โ2 1 1 0ฮ+6 ๐ธ+
2 2 โ1 โ1 0 โ2 1 1 0 2 โ1 โ1 0 โ2 1 1 0ฮโ6 ๐ธโ
2 2 โ1 โ1 0 โ2 1 1 0 โ2 1 1 0 2 โ1 โ1 0
๐พ1 ฮ+7 (๐ธโฒ
1)+ 2 1 โ1 0 0
โ3 โ
โ3 0 2 1 โ1 0 0
โ3 โ
โ3 0
ฮโ7 (๐ธโฒ
1)โ 2 1 โ1 0 0
โ3 โ
โ3 0 โ2 โ1 1 0 0 โ
โ3
โ3 0
ฮ+8 (๐ธโฒ
2)+ 2 1 โ1 0 0 โ
โ3
โ3 0 2 1 โ1 0 0 โ
โ3
โ3 0
ฮโ8 (๐ธโฒ
2)โ 2 1 โ1 0 0 โ
โ3
โ3 0 โ2 โ1 1 0 0
โ3 โ
โ3 0
ฮ+9 (๐ธโฒ
3)+ 2 โ2 2 0 0 0 0 0 2 โ2 2 0 0 0 0 0
ฮโ9 (๐ธโฒ
3)โ 2 โ2 2 0 0 0 0 0 โ2 2 โ2 0 0 0 0 0
in the Brillouin zones: letters denote points in theBrillouin zone of the structure; primed letters de-note two-valued spinor representations, whereas non-primed letters denote ordinary single-valued vectorones; parenthesized figures in the superscript indi-cate projective classes; figures in the internal sub-script mean the ordinal number of an irreducible rep-resentation in the given projective class; and the signsโ+โ and โโโ in the external superscript indicate therepresentation parity. It is evident that in those no-tations, ((ฮโฒ)
(1)1 )+ โก ฮ+
7 and ((ฮโฒ)(1)2 )โ โก ฮโ
8 , whereฮ+7 and ฮโ
8 are the spinor representations (writtenin the conventional notation system) that were used,e.g., to denote two-valued spinor representations inTable 1. An additional external subscript, if any, in-dicates the ordinal number of the representation, ifthere are several ones.
3.1.2. Point ๐ด
The wave-vector star of point ๐ด in the Brillouin zoneof crystalline graphite ๐พ-๐ถ, similarly to that of pointฮ , is composed of a single vector ๐๐ด = โ 1
2 ๐1 [1]. The
factor group of the wave-vector group with respect tothe invariant translation subgroup for graphite crys-tals is, as it takes place for point ฮ , also isomorphicto the group 6/๐๐๐ (๐ท6โ).
It was shown in work [1] that the two-valued (spi-nor) irreducible projective representations at point ๐ดin the Brillouin zone of crystalline graphite ๐พ-๐ถ be-long to the projective class ๐พ4 of the point symme-try group of equivalent directions ๐น๐, 6/๐๐๐ (๐ท6โ),coinciding with the crystal class group. It is so be-cause the single-valued (vector) projective represen-tations for point ๐ด, which are determined by theproperties of the spatial symmetry group of graphite,๐63/๐๐๐ (๐ท6โ), at this point, belong to the pro-jective class ๐พ5 [1], the transformation of spinorsat symmetry operations of the directional groupsof wave-vector groups to the projective class ๐พ1,and the product of the projective classes ๐พ5 and๐พ1, which is determined by the pairwise multipli-cation of the values of the coefficients ๐ผ, ๐ฝ, and ๐พ(i.e. ๐ผ(5)๐ผ(1), ๐ฝ(5)๐ฝ(1), and ๐พ(5)๐พ(1)), equals ๐พ5๐พ1 =๐พ4 in the system of notations used for projectiveclasses [1].
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Table 2. Characters of the equivalence representations ๐ทeq, the representations ๐ท๐ง
characterizing the spatial symmetry of ๐-orbitals, the spinor representations ๐ท+1/2
, and the spinor
representations ๐ทโฒ๐ง = ๐ท๐ง ร ๐ท
+1/2
characterizing the symmetry of ๐-orbitals taking the spininto account (the spin ๐-orbitals); and the characters of the representations ๐ทโฒ
๐
that characterize the symmetry of electron ๐-bands taking the spin into account for variouspoints of Brillouin zones in crystalline graphite ๐พ-๐ถ and single-layer graphene ๐ถ๐ฟ,1
Points ฮPoint groups 6/๐๐๐(๐ท6โ)
Projective classes ๐พ1 except for the representations ฮeq and ฮ๐ง
6/๐๐๐(๐ท6โ) ๐ ๐3 ๐23 3๐ข2 ๐2 ๐56 ๐6 3๐ขโฒ2 ๐ ๐๐3 ๐๐23 3๐๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ
2
๐พ โ ๐ถ ๐ = 0 ฮeq 4 4 4 0 0 0 0 4 0 0 0 4 4 4 4 0ฮ๐ง 1 1 1 โ1 1 1 1 โ1 โ1 โ1 โ1 1 โ1 โ1 โ1 1
๐ท+1/2
2 1 โ1 0 0 โโ3
โ3 0 2 1 โ1 0 0 โ
โ3
โ3 0
ฮโฒ๐ง 2 1 โ1 0 0 โ
โ3
โ3 0 โ2 โ1 1 0 0
โ3 โ
โ3 0
ฮโฒ๐ 8 4 โ4 0 0 0 0 0 0 0 0 0 0 4
โ3 โ4
โ3 0
๐ถ๐ฟ,1 ๐ = 0 ฮeq 2 2 2 0 0 0 0 2 0 0 0 2 2 2 2 0ฮโฒ๐ 4 2 โ2 0 0 0 0 0 0 0 0 0 0 2
โ3 โ2
โ3 0
Point ๐ดPoint group 6/๐๐๐(๐ท6โ)
Projective class ๐พ4 except for the representation ๐ดeq and ๐ดโฒ๐ง
6/๐๐๐(๐ท6โ) ๐ ๐3 ๐23 3๐ข2 ๐2 ๐56 ๐6 3๐ขโฒ2 ๐ ๐๐3 ๐๐23 3๐๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ
2
๐พ โ ๐ถ ๐๐ด =
= โ(1/2)๐1
๐ดeq 4 4 4 0 0 0 0 0 0 0 0 4 0 0 0 0๐ดโฒ
๐ง 2 1 โ1 0 0 โโ3
โ3 0 โ2 โ1 1 0 0
โ3 โ
โ3 0
๐ดโฒ๐ 8 4 โ4 0 0 0 0 0 0 0 0 0 0 0 0 0
Points ๐พPoint groups 6๐2(๐ท3โ)
Projective classes ๐พ1 except for the representations ๐พeq, ๐พ๐ง and ๐พ๐
6๐2(๐ท3โ) ๐ ๐3 ๐23 3๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ2
๐พ โ ๐ถ (๐๐พ)1 = โ(1/3)(2๐2 โ ๐3) ๐พeq 4 1 1 0 4 1 1 0(๐๐พ)2 = (1/3)(2๐2 โ ๐3) ๐พ๐ง 1 1 1 โ1 โ1 โ1 โ1 1
๐พ๐ 4 1 1 0 โ4 โ1 โ1 0๐ท+
1/22 1 โ1 0 0 โ
โ3
โ3 0
๐พโฒ๐ง 2 1 โ1 0 0
โ3 โ
โ3 0
๐พโฒ๐ 8 1 โ1 0 0
โ3 โ
โ3 0
๐ถ๐ฟ,1 (๐๐พ)1 = โ(1/3)(2๐1 โ ๐2) ๐พeq 2 โ1 โ1 0 2 โ1 โ1 0(๐๐พ)2 = (1/3)(2๐1 โ ๐2) ๐พ๐ 2 โ1 โ1 0 โ2 1 1 0
๐พโฒ๐ 4 โ1 1 0 0 โ
โ3
โ3 0
Point ๐ปPoint group 6๐2(๐ท3โ)
Projective classes ๐พ1 (๐ปeq, ๐ป๐ , ๐ท+1/2
and ๐ปโฒ๐ง) and ๐พ0(๐ปโฒ
๐ง)
6๐2(๐ท3โ) ๐ ๐3 ๐23 3๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ2
๐พ โ ๐ถ (๐๐ป)1 = โ(1/2)๐1 โ (1/3)(2๐2 โ ๐3) ๐ปeq 4 1 1 0 0โ3 โ
โ3 0
(๐๐ป)2 = โ(1/2)๐1 + (1/3)(2๐2 โ ๐3) ๐ป๐ง 1 1 1 โ1 โ1 โ1 โ1 1๐ป๐ 4 1 1 0 0 โ
โ3๐
โ3๐ 0
๐ท+1/2
2 1 โ1 0 0 โโ3
โ3 0
๐ปโฒ๐ง 2 1 โ1 0 0
โ3 โ
โ3 0
๐ปโฒ๐ 8 1 โ1 0 0 3๐ 3๐ 0
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V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
Point ๐Point group 3๐(๐ถ3๐)Projective class ๐พ0
3๐(๐ถ3๐) ๐ ๐3 ๐23 3๐๐ขโฒ2
๐พ โ ๐ถ (๐๐ )1 = โ๐๐ง โ (1/3)(2๐2 โ ๐3) ๐eq 4 1 1 0(๐๐ )2 = โ๐๐ง + (1/3)(2๐2 โ ๐3) ๐๐ง 1 1 1 1(๐๐ )3 = ๐๐ง โ (1/3)(2๐2 โ ๐3) ๐๐ 4 1 1 0(๐๐ )4 = ๐๐ง + (1/3)(2๐2 โ ๐3) ๐ท+
1/22 1 โ1 0
๐ โฒ๐ง 2 1 โ1 0
๐ โฒ๐ 8 1 โ1 0
Points ๐Point groups ๐๐๐(๐ท2โ)
Projective classes ๐พ0 (๐eq, ๐๐ง and ๐๐) and ๐พ1 (๐ท+1/2
, ๐ โฒ๐ง and ๐ โฒ
๐)
๐๐๐(๐ท2โ) ๐ (๐ข2)1 ๐2 (๐ขโฒ2)1 ๐ ๐(๐ข2)1 ๐๐2 ๐(๐ขโฒ
2)1
๐พ โ ๐ถ (๐๐ )1 = โ(1/2)๐3, ๐eq 4 0 0 4 0 4 4 0(๐๐ )2 = (1/2)๐2 ๐๐ง 1 โ1 1 โ1 โ1 1 โ1 1
(๐๐ )3 = โ(1/2)(๐2 โ ๐3), ๐๐ 4 0 0 โ4 0 4 โ4 0๐ท+
1/22 0 0 0 2 0 0 0
๐ โฒ๐ง 2 0 0 0 โ2 0 0 0
๐ โฒ๐ 8 0 0 0 0 0 0 0
๐ถ๐ฟ,1 (๐๐ )1 = โ(1/2)๐2 ๐eq 2 0 0 2 0 2 2 0(๐๐ )2 = (1/2)๐1 ๐๐ 2 0 0 โ2 0 2 โ2 0(๐๐ )3 = โ(1/2)(๐1 โ ๐2) ๐ โฒ
๐ 4 0 0 0 0 0 0 0
Point ๐ฟPoint group ๐๐๐(๐ท2โ)
Projective classes ๐พ5 (๐ฟ๐) and ๐พ4 (๐ฟโฒ๐)
๐๐๐(๐ท2โ) ๐ (๐ข2)1 ๐2 (๐ขโฒ2)1 ๐ ๐(๐ข2)1 ๐๐2 ๐(๐ขโฒ
2)1
๐พ โ ๐ถ (๐๐ฟ)1 = โ(1/2)(๐1 + ๐3) ๐ฟeq 4 0 0 0 0 4 0 0(๐๐ฟ)2 = โ(1/2)(๐1 โ ๐2) ๐ฟ๐ง 1 โ1 1 โ1 โ1 1 โ1 1(๐๐ฟ)3 = โ(1/2)(๐1 + ๐2 โ ๐3) ๐ฟ๐ 4 0 0 0 0 4 0 0
๐ท+1/2
2 0 0 0 2 0 0 0
๐ฟโฒ๐ง 2 0 0 0 โ2 0 0 0
๐ฟโฒ๐ 8 0 0 0 0 0 0 0
The factor-system for point ๐ด with making al-lowance for the spin, ๐2,๐ด(๐2, ๐1), is the prod-uct of the factor-systems ๐1,๐ด(๐2, ๐1) and ๐2(๐2, ๐1),i.e. ๐2,๐ด(๐2, ๐1) = ๐1,๐ด(๐2, ๐1)๐2(๐2, ๐1). The formeris determined by the structure of the spatial group ofa graphite crystal, ๐63/๐๐๐ (๐ท4
6โ), at point ๐ด ne-glecting the spin, and the latter describes the trans-formations of spinors at point ฮ (in the point symme-try group 6/๐๐๐ (๐ท6โ)). They and their structureare described in details in work [1] (see Tables 1 and6 in the cited work). The factor-system ๐2,๐ด(๐2, ๐1) ispresented in Table 4.
The standard factor-system for point ๐ด with mak-ing allowance for the spin, ๐โฒ
2,๐ด(๐2, ๐1), which belongsto the projective class ๐พ4, coincides with the standardfactor-system of this class, ๐โฒ
(4)(๐2, ๐1), and is theproduct of the standard factor-systems ๐โฒ
1,๐ด(๐2, ๐1) โกโก ๐โฒ
(5)(๐2, ๐1) and ๐โฒ2(๐2, ๐1), i.e. ๐โฒ
2,๐ด(๐2, ๐1) โกโก ๐โฒ
(4)(๐2, ๐1) = ๐โฒ(5)(๐2, ๐1)๐
โฒ2(๐2, ๐1). This factor-
system is presented in Table 5. The reductioncoefficients ๐ข2,๐ด(๐) of the factor-system ๐2,๐ด(๐2, ๐1)to the standard form ๐โฒ
2,๐ด(๐2, ๐1) โก ๐โฒ(4)(๐2, ๐1) are
determined as the products of the correspondingreduction coefficients ๐ข1,๐ด(๐) and ๐ข2(๐) of the factor-
348 ISSN 2071-0194. Ukr. J. Phys. 2020. Vol. 65, No. 4
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Table 3. Distributions of electron excitations over the irreducible projectiverepresentations of corresponding projective classes for electron ๐-bands at high-symmetrypoints in the Brillouin zones of the crystalline graphite ๐พ-๐ถ and single-layer graphene๐ถ๐ฟ,1 structures not taking (๐) and taking (๐) the electron spin into account
Crystallinegraphite ๐พ-๐ถ
Single-layergraphene ๐ถ๐ฟ,1
Points ฮPoint groups 6/๐๐๐(๐ท6โ)
a) Projective classes ๐พ0
ฮ๐ = 2(ฮ(0)2 )+ + 2(ฮ
(0)3 )โ(ฮ๐ = 2ฮ+
2 + 2ฮโ3 ) ฮ๐ = (ฮ
(0)2 )+ + (ฮ
(0)3 )โ(ฮ๐ = ฮ+
2 + ฮโ3 )
b) Projective classes ๐พ1
ฮโฒ๐ = 2((ฮโฒ)
(1)1 )+ + 2((ฮโฒ)
(1)2 )โ(ฮโฒ
๐ = 2ฮ+7 + 2ฮโ
8 ) ฮโฒ๐ = ((ฮโฒ)
(1)1 )+ + (ฮโฒ)
(1)2 )โ(ฮโฒ
๐ = ฮ+7 + ฮโ
8 )
Points ๐ดPoint group 6๐๐๐(๐ท6โ)
a) Projective class ๐พ5
๐ด๐ = 2๐ด(5)1 [(๐ด
(5)1 )1 + (๐ด
(5)1 )2]
b) Projective class ๐พ4
๐ดโฒ๐ = 2(๐ดโฒ)
(4)3 [((๐ดโฒ)
(4)3 )1 + ((๐ดโฒ)
(4)3 )2]
Points ๐พPoint groups 6๐2(๐ท3โ)Projective classes ๐พ0
๐พ๐ = ๐พ(0)2 +๐พ
(0)4 +๐พ
(0)6 ๐พ๐ = ๐พ
(0)6
Projective classes ๐พ1
๐พโฒ๐ = 2(๐พโฒ)
(1)1 + (๐พโฒ)
(1)2 + (๐พโฒ)
(1)3 ๐พโฒ
๐ = (๐พโฒ)(1)2 + (๐พโฒ)
(1)3
Points ๐ปPoint group 6๐2(๐ท3โ)
a) Projective class ๐พ1
๐ป๐ = ๐ป(1)1 +๐ป
(1)3
b) Projective class ๐พ0
๐ปโฒ๐ = ((๐ปโฒ)
(0)1 + (๐ปโฒ)
(0)3 ) + (๐ปโฒ)
(0)5 + 2(๐ปโฒ)
(0)6
Points ๐Point group 3๐(๐ถ3๐)
a) Projective class ๐พ0
๐๐ = ๐(0)1 + ๐
(0)2 + ๐
(0)3
b) Projective class ๐พ0
๐ โฒ๐ = ((๐ โฒ)
(0)1 + (๐ โฒ)
(0)2 ) + 3(๐ โฒ)
(0)3
Points ๐Point groups ๐๐๐(๐ท2โ)a) Projective classes ๐พ0
๐๐ = 2(๐(0)2 )+ + 2(๐
(0)3 )โ ๐๐ = (๐
(0)2 )+ + (๐
(0)3 )โ
b) Projective classes ๐พ1
๐ โฒ๐ = 2((๐ โฒ)(1))+ + 2((๐ โฒ)(1))โ ๐ โฒ
๐ = ((๐ โฒ)(1))+ + ((๐ โฒ)(1))โ
Point ๐ฟPoint group ๐๐๐(๐ท2โ)
ะฐ) Projective class ๐พ5
๐ฟ๐ = 2๐ฟ(5)1
b) Projective class ๐พ4
๐ฟโฒ๐ = 2((๐ฟโฒ)
(4)1 ) + (๐ฟโฒ)
(4)2 )
ISSN 2071-0194. Ukr. J. Phys. 2020. Vol. 65, No. 4 349
V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
Ta
ble
4.F
act
or-
syst
em
2,A(r2,r1)
for
poin
t A
inth
e B
rill
ou
in z
on
e of
cry
stall
ine
gra
ph
ite
-(t
he
spati
al
sym
metr
y g
rou
p
4
36h
P6/mmc(D
),
the
pro
ject
ive
class
4).
Th
e b
ott
om
part
of
the
tab
le c
on
tain
s
the
valu
es o
f th
efu
nct
ion
2,A
u(r)
that
tra
nsf
orm
th
e fa
ctor-
syst
em
2,A
21
(r,r)
to t
he
sta
nd
ard
form
2,A
21
(4)
21
(r,r)
(r, r)
350 ISSN 2071-0194. Ukr. J. Phys. 2020. Vol. 65, No. 4
Energy Spectra of Electron Excitations
21
4 6h
Ta
ble
5.S
tan
da
rd f
act
or-
syst
em
2,A(r,r)
(th
e sp
ati
al
sym
met
ry g
rou
p P63/mmc(D
),th
e p
roje
ctiv
e cl
ass
4)
for
po
int A
in t
he
Bri
llo
uin
zo
ne
of
cryst
all
ine
gra
ph
ite
-ta
kin
g t
he
elec
tro
n s
pin
in
to a
cco
un
t, w
hic
h c
oin
cid
es w
ith
th
e st
an
dard
(4)
21
fact
or-
syst
em
(r,r)
of
the
pro
ject
ive
cla
ss
4of
the
po
int
sym
met
ry g
rou
p2,A
21
6h
(4)
21
(r,r)
6/mmm(D
):
(r,r)
ISSN 2071-0194. Ukr. J. Phys. 2020. Vol. 65, No. 4 351
V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
Table 6. Characters of the irreducible projective representationsof the projective class ๐พ4 corresponding to the standard factor-system ๐โฒ
(4)(๐2, ๐1) of this class
Projec-tiveclass
Notationfor irreducible
projectiverepresentation
6/๐๐๐(๐ท6โ)
๐ ๐1 ๐23 3๐ข2 ๐2 ๐56 ๐6 3๐ขโฒ2 ๐ ๐๐3 ๐๐23 3๐๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ
2
๐พ4 ๐(4)1 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0
๐(4)2 2 2 2 โ2 0 0 0 0 0 0 0 0 0 0 0 0
๐(4) 4 โ2 โ2 0 0 0 0 0 0 0 0 0 0 0 0 0
๐ข2,๐ด(๐) 1 โ1 1 ๐ ๐ โ๐ โ๐ โ1 1 โ1 1 ๐ โ๐ ๐ ๐ 1
Table 7. Characters of the two-valued (spinor) irreducibleprojective representations of point ๐ด in the Brillouin zone of crystalline graphite ๐พ-๐ถ
Projec-tiveclass
Notationfor irreducible
projectiverepresentation
6/๐๐๐(๐ท6โ)
๐ ๐3 ๐23 3๐ข2 ๐2 ๐56 ๐6 3๐ขโฒ2 ๐ ๐๐3 ๐๐23 3๐๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ
2
๐พ4 (๐ดโฒ)(4)1 2 โ2 2 2๐ 0 0 0 0 0 0 0 0 0 0 0 0
((๐ดโฒ)(4)1 + (๐ดโฒ)
(4)2 )<
(๐ดโฒ)(4)2 2 โ2 2 โ2๐ 0 0 0 0 0 0 0 0 0 0 0 0
(๐ดโฒ)(4)3 4 2 โ2 0 0 0 0 0 0 0 0 0 0 0 0 0
systems ๐1,๐ด(๐2, ๐1) and ๐2(๐2, ๐1) to the standardform, i.e. ๐ข2,๐ด(๐) = ๐ข1,๐ด(๐)๐ข2(๐) [1]. The values ofthe coefficients ๐ข2,๐ด(๐) are shown in the bottom partof Table 4.
The characters of the irreducible projective rep-resentations of the projective class ๐พ4, which cor-respond to the standard factor-system of this class,๐โฒ(4)(๐2, ๐1), are given in Table 6, and the characters
of the two-valued (spinor) irreducible projective rep-resentations of point ๐ด are quoted in Table 7. In thebottom part of Table 6, the values of the coefficients๐ข2,๐ด(๐) are shown. As it has to be, the equalities
(๐ดโฒ)(4)๐ = ๐ข2,๐ด(๐)๐
(4)๐ (6)
are satisfied.The characters of the projective equivalence repre-
sentation at point ๐ด, i.e. the representation ๐ดeq andthe two-valued representations ๐ดโฒ
๐ง โก ฮโฒ๐ง and ๐ดโฒ
๐ aregiven in Table 2. The distributions of electron exci-tations at point ๐ด in the Brillouin zone of crystallinegraphite ๐พ-๐ถ for the ๐-bands neglecting the electronspin and taking it into account are presented in Ta-ble 3. It is of interest that electron excitations of ๐-bands at point ๐ด in the Brillouin zone of crystalline
graphite ๐พ-๐ถ making allowance for the spin are four-fold degenerate, because their states are transformedaccording to the four-dimensional irreducible projec-tive representations of the projective class ๐พ4. In theabsence of external magnetic fields, additional condi-tions associated with the time-reversal invariance areimposed on the wave functions of states and, accord-ingly, on the representations. In this case, some of thestates may become additionally degenerate.
Let us account for the time-reversal invariance ofthe states at points ฮ and ๐ด in the Brillouin zone ofcrystalline graphite and point ฮ in the Brillouin zoneof single-layer graphene with the help of the Herringcriterion [5, 6, 9]. The corresponding calculation pro-cedure is described in works [9,10] in detail. In partic-ular, the summation is carried out over the elements๐โฒ = (๐ผ|๐โฒ) of the wave-vector group ๐บ๐ that satisfythe condition ๐โฒ๐ = โ๐ (๐โฒ๐ = โ๐).
For points ฮ and ๐ด of crystalline graphite ๐พ-๐ถand point ฮ of single-layer graphene ๐ถ๐ฟ,1, each wave-vector star has one ray. At those points, the wave vec-tors โ๐ and ๐ are equivalent (โ๐ โก ๐), and the con-dition ๐โฒ๐ = โ๐ is satisfied, for crystalline graphite๐พ-๐ถ, by the elements ๐โฒ1 = (0 |๐ ), ๐โฒ2 = (0 |๐3 ),
352 ISSN 2071-0194. Ukr. J. Phys. 2020. Vol. 65, No. 4
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๐โฒ3 = (0๐23 ), ๐โฒ4 = (0 |(๐ข2)1 ), ๐โฒ5 = (0 |(๐ข2)2 ), ๐โฒ6 =
= (0 |(๐ข2)3 ), ๐โฒ7 = (๐1
2 |๐2 ), ๐โฒ8 = (๐1
2
๐56 ), ๐โฒ9 =
= (๐1
2 |๐6 ), ๐โฒ10 = (๐1
2 |(๐ขโฒ2 )1), ๐โฒ11 = (๐1
2 |(๐ขโฒ2 )2),
๐โฒ12 = (๐1
2 |(๐ขโฒ2 )3), ๐โฒ13 = (0 |๐ ), ๐โฒ14 = (0 |๐๐3 ), ๐โฒ15 =
= (0๐๐23 ), ๐โฒ16 = (0 |๐(๐ข2)1 ), ๐โฒ17 = (0 |๐(๐ข2)2 ), ๐โฒ18 =
= (0 |๐(๐ข2)3 ), ๐โฒ19 = (๐1
2 |๐๐2 ), ๐โฒ20 = (๐1
2
๐๐56 ), ๐โฒ21 =
= (๐1
2 |๐๐6 ), ๐โฒ22 = (๐1
2 |๐(๐ขโฒ2 )1), ๐โฒ23 = (๐1
2 |๐(๐ขโฒ2 )2),
and ๐โฒ24 = (๐1
2 |๐(๐ขโฒ2 )3) and, for single-layer graphene
๐ถ๐ฟ,1, by the same values, for which the nontrivialtranslation ๐1
2 = 0, because the spatial group ofsingle-layer graphene is symmorphic and, unlike thespatial group of crystalline graphite, does not containnontrivial translations.
It is easy to calculate the squares of those ele-ments, (๐โฒ)2 = (๐๐ผ+๐ผ
๐2 ). In particular, for points
ฮ and ๐ด of crystalline graphite, (๐โฒ1)2 = (0 |๐ ),
(๐โฒ2)2 = (0
๐23 ), (๐โฒ3)
2 = (0 |๐๐3 ), (๐โฒ4)2 = (0 |๐ ),
(๐โฒ5)2 = (0 |๐ ), (๐โฒ6)
2 = (0 |๐ ), (๐โฒ7)2 = (๐1 |๐ ),
(๐โฒ8)2 = (๐1
๐๐23 ), (๐โฒ9)
2 = (๐1 |๐3 ), (๐โฒ10)2 = (0 |๐ ),
(๐โฒ11)2 = (0 |๐ ), (๐โฒ12)
2 = (0 |๐ ), (๐โฒ13)2 = (0 |๐ ),
(๐โฒ14)2 = (0
๐23 ), (๐โฒ15)
2 = (0 |๐๐3 ), (๐โฒ16)2 = (0 |๐ ),
(๐โฒ17)2 = (0 |๐ ), (๐โฒ18)
2 = (0 |๐ ), (๐โฒ19)2 = (0 |๐ ),
(๐โฒ20)2 = (0
๐๐23 ), (๐โฒ21)
2 = (0 |๐3 ), (๐โฒ22)2 = (๐1 |๐ ),
(๐โฒ23)2 = (๐1 |๐ ), (๐โฒ24)2 = (๐1 |๐ ) where ๐ is the rota-
tion by the angle 2๐ around the corresponding axis;and, for point ฮ of single-layer graphene ๐ถ๐ฟ,1, theseare the same values, for which the trivial translationvector ๐1 = 0.
The stages of calculations according to the Her-ring criterion and the results obtained are presentedin Table 18 (see Appendix). From the values of theHerring criterion, it is easy to see that both thesingle- and two-valued projective representations atpoints ฮ and ๐ด of crystalline graphite ๐พ-๐ถ andpoint ฮ of single-layer graphene ๐ถ๐ฟ,1, except forthe representations (๐ดโฒ)
(4)1 and (๐ดโฒ)
(4)2 for crystalline
graphite, are related to the case ๐1 [9], where thereis no additional degeneration of the states providedthat their invariance with respect to the time re-versal is taken into account. At the same time, theprojective representations at point ๐ด for crystallinegraphite โ these are the representations (๐ดโฒ)
(4)1 and
(๐ดโฒ)(4)2 โ are related to the case ๐1 [9] and, be-
ing representations with complex-conjugated char-acters, group together to increase the degeneracyorder of electron states to four. This grouping ofcomplex-conjugated representations is illustrated inTable 7.
3.1.3. Point ฮ
The group of equivalent directions of the wave-vectorgroup at point ฮ of crystalline graphite ๐พ-๐ถ is thegroup 6๐๐(๐ถ6๐ฃ). The wave-vector star at this pointcontains two rays. The irreducible single- and two-valued projective representations for point ฮ aregiven in work [1]. The condition ๐โฒ๐ = โ๐ at thispoint is satisfied by the elements ๐โฒ๐ = โ๐: ๐โฒ4 == (0 |(๐ข2)1), ๐โฒ5 = (0 |(๐ข2)2), ๐โฒ6 = (0 |(๐ข2)3), ๐โฒ10 == (๐1
2 |(๐ขโฒ2)1), ๐โฒ11 = (๐1
2 |(๐ขโฒ2)2), ๐โฒ12 = (๐1
2 |(๐ขโฒ2)3),
๐โฒ13 = (0 |๐), ๐โฒ14 = (0 |๐๐3), ๐โฒ15 = (0๐๐23), ๐โฒ19 =
= (๐1
2 |๐๐2), ๐โฒ20 = (๐1
2
๐๐56) and ๐โฒ21 = (๐1
2 |๐๐6).The squares of those elements can be easily calcu-lated: (๐โฒ4)
2 = (0 |๐), (๐โฒ5)2 = (0 |๐ ), (๐โฒ6)
2 = (0 |๐),(๐โฒ10)
2 = (0 |๐), (๐โฒ11)2 = (0 |๐), (๐โฒ12)
2 = (0 |๐),(๐โฒ13)
2 = (0 |๐), (๐โฒ14)2 = (0
๐23), (๐โฒ15)
2 = (0 |๐๐3),(๐โฒ19)
2 = (0 |๐), (๐โฒ20)2 = (0๐๐23) and (๐โฒ21)
2 = (0 |๐3).After calculating the Herring criteria, it is easy to
see that all, both single- and two-valued, irreducibleprojective representations at this point belong to thecase ๐2 [9] (subscript 2 means that ๐ is not equiva-lent to โ๐, but the space group contains an element๐ that transforms ๐ into โ๐), when there is no addi-tional degeneration of the states provided that theirinvariance with respect to the time reversal is takeninto account. The consistency conditions for the irre-ducible projective representations along the directionฮ โฮโ๐ด, which is the direction of the highest sym-metry in the Brillouin zone of a crystal structure withthe spatial symmetry group ๐63/๐๐๐(๐ท4
6โ), wereshown in Fig. 5 of work [1].
3.2. Line KโPโH of crystallinegraphite and point K of single-layer graphene
3.2.1. Redesignation
First of all, let us make an important clarifica-tion. The matrices of the irreducible representationsof the wave-vector groups, ๐ท๐(โ), and their charac-ters ๐๐ท๐(โ) contain the phase factor ๐โ๐๐(๐ผ+๐), where๐ is the wave vector, ๐ผ the vector of nontrivial trans-lation, and ๐ the vector of trivial translation forthe element ๐ = (๐ผ + ๐|๐) of the spatial symmetrygroup. This factor was not taken into consideration,when constructing the characters of irreducible rep-resentations in work [1] (see formulas (6) and (7) inwork [1]). For the basic elements of the spatial group,one can choose a = 0, so that this phase factor takes
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V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
the form ๐โ๐k๐ผ, where the nontrivial translation vec-tor ๐ผ corresponds to the โrotationalโ element ๐, i.e. tothe basic element โ = (๐ผ|๐) of the spatial symme-try group. The account for the phase factor ๐โ๐๐๐ผ
for the elements ๐โs is necessary, when constructinga correct irreducible projective representation of thewave-vector group in the cases where the scalar prod-uct (๐,๐ผ) = 0. Note that in the case of point ๐ด con-sidered above, the nontrivial translation vector ๐ผ wasalways equal to zero for the nonzero characters of โro-tationalโ elements ๐โs.
Since the phase factor ๐โ๐k๐ผ(๐) is inherent to thespatial symmetry of a periodic structure, it can be in-troduced into the definition of the atomic equivalencerepresentation by replacing formula (15) in work [1]by the formula๐eq(๐ ๐ผ) = ๐โ๐๐๐ผ(๐ ๐ผ)
โ๐
๐ฟ๐ ๐ผ๐๐ ,๐๐๐๐๐พ๐๐๐ , (7)
denoting the previously used ๐eq(๐ ๐ผ) without thephase factor as (๐eq)0(๐ ๐ผ), i.e. the new ๐ทeq will con-tain the phase factor ๐โ๐๐๐ผ(๐ ๐ผ), and also denot-ing ๐ทeq obtained for various points in the Brillouinzone in work [1] and not containing the phase factoras (๐ทeq)0.
At points ๐พ and ๐ป of crystalline graphite ๐พ-๐ถ andpoint ๐พ of single-layer graphene ๐ถ๐ฟ,1, the factor-
Table 8. Characters of the irreducibleprojective representations of the projective classes๐พ0 and ๐พ1 of the group 6๐2(๐ท3โ) correspondingto the standard factor-systems ๐โฒ
(0)(๐2, ๐1) (the
unity-containing factor-system for ordinary vectorrepresentations) and ๐โฒ
(1)(๐2, ๐1), respectively
Projec-tiveclass
Notationfor irreducible
projectiverepresentation
6/๐๐๐(๐ท6โ)
๐ ๐3 ๐23 3๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ2
๐พ0 ๐พ(0)1 ๐พ1 1 1 1 1 1 1 1 1
๐พ(0)2 ๐พ2 1 1 1 1 โ1 โ1 โ1 โ1
๐พ(0)3 ๐พ3 1 1 1 โ1 โ1 โ1 โ1 1
๐พ(0)4 ๐พ4 1 1 1 โ1 โ1 โ1 โ1 1
๐พ(0)5 ๐พ5 2 โ1 โ1 0 2 โ1 โ1 0
๐พ(0)6 ๐พ6 2 โ1 โ1 0 โ2 1 1 0
๐พ1 ๐(1)1 2 โ1 โ1 0 0
โ3๐ โ
โ3๐ 0
๐(1)2 2 โ1 โ1 0 0 โ
โ3๐
โ3๐ 0
๐(1)2 2 2 2 0 0 0 0 0
๐ข2,๐พ(๐) โก ๐ข2(๐) 1 โ1 1 ๐ ๐ โ๐ โ๐ โ1
groups of wave-vector groups are isomorphic to thesame point symmetry group 6๐2(๐ท3โ) with respectto the invariant translational subgroups.
Each of the stars of the wave-vector groups atpoints ๐พ in both crystalline graphite ๐พ-๐ถ and single-layer graphene ๐ถ๐ฟ,1 contains two vectors. These arethe vectors (๐๐พ)1 = โ 1
3 (2๐2 โ ๐3) and (๐๐พ)2 == 1
3 (2๐2 โ ๐3) for crystalline graphite, and the vec-tors (๐๐พ)1 = โ 1
3 (2๐1 โ ๐2) and (๐๐พ)2 = 13 (2๐1 โ ๐2)
for single-layer graphene. For crystalline graphite, thenontrivial translation vector ๐ผ = ๐1/2 is perpen-dicular to the wave vectors (๐๐พ)1 and (๐๐พ)2. The-refore, the phase factor ๐โ๐๐๐พ๐ผ equals 1 in allcases. This is also true for single-layer graphene aswell, where the nontrivial translation vector equalszero. This means that, for both structures, thefactor-systems ๐1,๐พ(๐2, ๐1) include the โ+1โ-valuesonly, i.e. they are completely unity-containing factor-systems, which coincide with the standard factor-systems of the projective class ๐พ0. At the sametime, since ๐2,๐พ(๐2, ๐1) = ๐1,๐พ(๐2, ๐1)๐2(๐2, ๐1) == ๐โฒ
(0)(๐2, ๐1)๐2(๐2, ๐1) = ๐2(๐2, ๐1), the factor-sys-tems for the projective representations at points ๐พtaking the spin into account, i.e. ๐2,๐พ(๐2, ๐1), coin-cide with the factor-system ๐2(๐2, ๐1) belonging tothe projective class ๐พ1.
The characters of the irreducible projective repre-sentations of the projective classes ๐พ0 (where theycoincide with the ordinary or vector representations)and ๐พ1 of the group 6๐2(๐ท3โ), which correspondto the standard factor-systems ๐โฒ
(0)(๐2, ๐1) (a unity-containing factor-system, whose coefficients equalonly to +1) and ๐โฒ
(1)(๐2, ๐1), respectively, are pre-sented in Table 8. The characters of two-dimensionalirreducible projective representations correspondingto the standard factor-system ๐โฒ
(1)(๐2, ๐1) are markedwith the symbol ๐ . The coefficients ๐ข2,๐พ(๐) โก ๐ข2(๐)are given in the bottom part of Table 8.
In Table 9, the characters of the two-valued(spinor) irreducible projective representations ofpoints ๐พ in the Brillouin zones of crystalline graphiteand single-layer graphene are given. They are identi-cal for those two structures.
It should be noted that, for the two-valued (spinor)irreducible projective representations of points ๐พin the Brillouin zones of crystalline graphite andgraphene, the following equations are satisfied:
(๐พ โฒ)(1)๐ = ๐ข2(๐)๐
(1)๐ . (8)
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In addition, for various points in the Brillouin zones โe.g., for points ๐พ โ we can also determine how thecharacters of the representations of the direct prod-ucts ๐พ๐ โ ๐ท+
1/2, where ๐พ๐ is the character of the ๐-th projective irreducible representation making no al-lowance for the electron spin, can be expanded in two-valued (spinor) irreducible projective representationswith regard for the electron spin. In other words, wecan determine which orbitals or the sum of orbitalsfound making allowance for the spin will correspondto an orbital obtained without regard for the spin, ifthe latter is taken into account.
Let us determine this correspondence proceedingfrom the distributions of the electron ๐-band rep-resentations at points ๐พ in crystalline graphite andsingle-layer graphene without taking the electron spininto account (see Table 12 in work [1]). In particu-lar, for the products ๐พ
(0)2 โ ๐ท+
1/2, ๐พ(0)4 โ ๐ท+
1/2, and
๐พ(0)6 โ ๐ท+
1/2, whose characters are presented in thebottom part of Table 9, it is easy to find the con-sistency conditions for the irreducible projective rep-resentations, which arise, when the electron spin istaken into account for the orbitals ๐พ๐ determinedwithout regard for the spin. In particular, the orbital(๐พ โฒ)
(1)1 , which makes allowance for the electron spin,
corresponds to the orbital ๐พ(0)2 , if the spin is taken
into account; the orbital (๐พ โฒ)(1)1 also corresponds to
the orbital ๐พ(0)4 ; the sum of orbitals (๐พ โฒ)
(1)2 +(๐พ โฒ)
(1)3
corresponds to the orbital ๐พ(0)6 , i.e. if the electron
spin is taken into account, the orbital ๐พ(0)6 , which is
doubly degenerate, if the electron spin is not takeninto account, becomes split into two doubly degener-ate spin orbitals (๐พ โฒ)
(1)2 and (๐พ โฒ)
(1)3 .
The characters of projective representations forpoints ๐พ, namely, the equivalence representations๐พeq, the representations of the spatial symmetry of๐-orbitals ๐พ๐ง, the representations of the symmetryof electron ๐-bands without regard for the electronspin ๐พ๐, two-valued representations ๐ท+
1/2, ๐พ โฒ๐ง, and
representations of the symmetry of electron ๐โฒ-bandswith regard for the electron spin ๐พ โฒ
๐, are presentedin Table 2. The distributions of electron excitationsfor ๐-bands (without taking and taking the electronspin into account) over the two-valued (spinor) irre-ducible projective representations of points ๐พ in theBrillouin zones of crystalline graphite ๐พ-๐ถ and single-layer graphene ๐ถ๐ฟ,1 are presented in Table 3.
An important consequence of the account for theelectron spin for electron excitations at points ๐พ isthe splitting of the doubly degenerate spinless orbitals๐พ
(0)6 in the structures of both crystalline graphite
and single-layer graphene. This splitting is predictedby the theoretical-group analysis and occurs as a re-sult of the consideration of the spin-orbit interac-tion. But it is extremely weak for carbon structures(about 1.0รท1.5 meV [2]) and will not be noticeableagainst the state energies measured in electronvoltsand even tens of electronvolts. From the viewpoint ofa theoretical group description, this splitting of thespinless orbitals ๐พ
(0)6 into the spin orbitals (๐พ โฒ)
(1)2
and (๐พ โฒ)(1)3 , if the spin is taken into account, has a
principal character for noncarbon structures with thespatial symmetry group ๐63/๐๐๐(๐ท4
6โ) as well. Forinstance, it can be significant for dichalcogenides oftransition metals.
3.2.2. Point ๐ป
As was mentioned above, the wave-vector factor-group at point ๐ป in the Brillouin zone of crystallinegraphite ๐พ-๐ถ with respect to the infinite invarianttranslation subgroup is isomorphic to the point group6๐2(๐ท3โ). The star of the wave-vector group con-tains two vectors, (๐๐ป)1 = โ 1
2๐1 โ13 (2๐2 โ ๐3) and
(๐๐ป)2 = โ 12๐1 +
13 (2๐2 โ ๐3).
Table 9. Characters of the two-valued (spinor)irreducible projective representations of the group6๐2(๐ท3โ) (the projective class ๐พ1) of the spinorrepresentation ๐ท
+1/2
(๐) and the projective
representation products ๐พ(0)2 โ ๐ท
+1/2
,
๐พ(0)4 โ ๐ท
+1/2
, and ๐พ(0)6 โ ๐ท
+1/2
Projec-tiveclass
Notationfor irreducible
projectiverepresentation
6๐2(๐ท3โ)
๐ ๐3 ๐23 3๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ2
๐พ1 (๐พโฒ)(1)1 2 1 โ1 0 0
โ3 โ
โ3 0
(๐พโฒ)(1)2 2 1 โ1 0 0 โ
โ3
โ3 0
(๐พโฒ)(1)3 2 โ2 2 0 0 0 0 0
๐ท+1/2
(๐) 2 1 โ1 0 0 โโ3
โ3 0
๐พ1 ๐พ(0)2 โ๐ท+
1/22 1 โ1 0 0
โ3 โ
โ3 0
๐พ(0)4 โ๐ท+
1/22 1 โ1 0 0
โ3 โ
โ3 0
๐พ(0)6 โ๐ท+
1/24 โ1 1 0 0 โ
โ3
โ3 0
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V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
Table 10. Characters of (๐) the one-valued irreducible projective representations of the projectiveclass ๐พ1 of the group 6๐2(๐ท3โ) corresponding to its standard factor-system and (๐) the one-valued ๐-equivalentrepresentations describing the symmetry of vibrational and electron excitations without taking the spininto account at point ๐ป in the Brillouin zone of crystalline graphite for the spatial symmetry group (thewave-vector group of point ๐ป) that is a subgroup of the spatial symmetry group ๐63/๐๐๐ (๐ท4
6โ)
Projec-tiveclass
Notationfor irreducible
projectiverepresentation
6๐2(๐ท3โ)
๐ ๐3 ๐23 3๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ2
๐พ1 ๐ ๐(1)1 2 โ1 โ1 0 0
โ3๐ โ
โ3๐ 0
๐(1)2 2 โ1 โ1 0 0 โ
โ3๐
โ3๐ 0
๐(1)3 2 2 2 0 0 0 0 0
๐ข1,๐ป(๐) 1 1 1 1 ๐ ๐ ๐ ๐
๐๐๐๐ป๐(๐) 1 1 1 1 ๐ ๐ ๐ ๐
๐๐๐๐ป๐(๐)๐ข1,๐ป(๐) 1 1 1 1 โ1 โ1 โ1 โ1
๐พ1 b ๐ป(1)1 2 โ1 โ1 0 0 โ
โ3๐
โ3๐ 0
๐ป(1)2 2 โ1 โ1 0 0
โ3๐ โ
โ3๐ 0
๐ป(1)3 2 2 2 0 0 0 0 0
In work [1], it was shown that both the factor-system of the spatial symmetry group of crystallinegraphite at point ๐ป of its Brillouin zone, ๐1,๐ป(๐2, ๐1),with the coefficients ๐ข1,๐ป(๐) of reduction to thestandard form and the factor-system of the spinvariable transformations, ๐2(๐2, ๐1), with the coeffi-cients ๐ข2(๐) of reduction to the standard form be-long to the same projective class ๐พ1. This meansthat the two-valued (spinor) irreducible projectiverepresentations for point ๐ป in the Brillouin zoneof crystalline graphite ๐พ-๐ถ belong to the projec-tive class ๐พ0 (๐พ1 ยท ๐พ1 = ๐พ0), and the reductioncoefficients of the transformation factor-system tothe standard form taking the electron spin into ac-count, ๐2,๐ป(๐2, ๐1) = ๐1,๐ป(๐2, ๐1)๐2(๐2, ๐1), are de-termined at this point by the equality ๐ข2,๐ป(๐) == ๐ข1,๐ป(๐)๐ข2(๐). When constructing the charactersof the two-valued (spinor) irreducible projective rep-resentations of point ๐ป, it is also necessary toconsider the phase factor ๐โ๐๐๐ป๐ผ(๐), i.e. to deter-mine the resulting factors ๐โ๐๐๐ป๐ผ(๐)๐ข1,๐ป(๐)๐ข2(๐),by which the characters of the irreducible projec-tive representations of the projective class ๐พ0 (theseare the characters of the vector representations forthe point symmetry group 6๐2(๐ท3โ)) have to bemultiplied.
The characters of the single-valued irreducible pro-jective representations of the projective class ๐พ1 of
the point group 6๐2(๐ท3โ), which correspond to thestandard factor-system of this class, and the charac-ters of the single-valued irreducible projective repre-sentations characterizing the spatial symmetry of vi-brational and electron โ in the latter case, not takingthe electron spin into account โ excitations at point๐ป in the Brillouin zone of crystalline graphite aregiven in Tables 10, ๐ and 10, ๐, respectively 2. Thebottom part of Table 10, ๐ contains the values of thecoefficients ๐ข1,๐ป(๐) that transform the factor-systemof the projective class ๐พ1, which corresponds to thespatial symmetry of point ๐ป, to the standard form,as well as the values of the phase factors ๐โ๐k๐ป๐ผ(๐)and the coefficient products ๐โ๐k๐ป๐ผ(๐)๐ข1,๐ป(๐).
The characters of the two-valued (spinor) irre-ducible projective representations belonging to theprojective class ๐พ0 and characterizing the symme-try of electron excitations taking the electron spin
2 It is of interest to note that the characters of the single-valued irreducible projective representations for point ๐ป inthe Brillouin zone of crystalline graphite (the spatial sym-metry group of crystalline graphite is ๐63/๐๐๐(๐ท4
6โ), andthe factor-group of the wave-vector group for point ๐ป is iso-morphic to the point group 6๐2(๐ท3โ) with respect to theinvariant translation subgroup), which belong to the projec-tive class ๐พ1, exactly coincide with the characters of the firstthree irreducible representations presented in Table C.28 ofwork [8].
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Table 11. Characters of (๐) the two-valued (spinor) irreducible projective representationsat point ๐ป in the Brillouin zone of crystalline graphite ๐พ-๐ถ (the spatial symmetry group ๐63/๐๐๐ (๐ท4
6โ),the projective class ๐พ0) and (๐) the projective representation products ๐ป
(1)1 โ ๐ท
+1/2
and ๐ป(1)3 โ ๐ท
+1/2
Projec-tiveclass
Notationfor irreducible
projectiverepresentation
6/๐๐๐(๐ท6โ)
๐ ๐3 ๐23 3๐ข2 ๐๐2 ๐๐56 ๐๐6 3๐๐ขโฒ2
๐พ0 ๐ (๐ปโฒ)(0)1 1 โ1 1 ๐ โ๐ ๐ ๐ 1
(๐ปโฒ)(0)2 1 โ1 1 ๐ ๐ โ๐ โ๐ โ1
(๐ปโฒ)(0)3 1 โ1 1 โ๐ โ๐ ๐ ๐ โ1
(๐ปโฒ)(0)4 1 โ1 1 โ๐ ๐ โ๐ โ๐ 1
(๐ปโฒ)(0)5 2 1 โ1 0 โ2๐ โ๐ โ๐ 0
(๐ปโฒ)(0)6 2 1 โ1 0 2๐ ๐ ๐ 0
๐ข1,๐ป(๐) 1 1 1 1 ๐ ๐ ๐ ๐
๐ข2(๐) 1 โ1 1 ๐ ๐ โ๐ โ๐ โ1๐๐๐๐ป๐(๐) 1 1 1 1 ๐ ๐ ๐ ๐
๐๐๐๐ป๐(๐)๐ข1,๐ป(๐)๐ข2(๐) 1 โ1 1 ๐ โ๐ ๐ ๐ 1
๐ท+1/2
(๐) 2 1 โ1 0 0 โโ3
โ3 0
๐พ0 b ๐ป(1)1 โ๐ท+
1/24 โ1 1 0 0 3๐ 3๐ 0
๐ป(1)3 โ๐ท+
1/24 2 โ2 0 0 0 0 0
๐ปโฒ๐ 8 1 โ1 0 0 3๐ 3๐ 0
into account are given in Table 11, ๐. The bot-tom part of Table 11, ๐ contains the values ofthe coefficients ๐ข1,๐ป(๐) and ๐ข2(๐), which transformthe factor-system ๐1,๐ป(๐2, ๐1) and ๐2(๐2, ๐1), respec-tively, to the standard form, as well as the valuesof the phase factors ๐โ๐k๐ป๐ผ(๐), the coefficient prod-ucts ๐โ๐k๐ป๐ผ(๐)๐ข1,๐ป(๐)๐ข2(๐), and the characters ofthe double-digit spinor representation ๐ท+
1/2(๐) of thepoint symmetry group 6๐2(๐ท3โ). In Table 11, ๐, thecharacters of the products of the projectivev repre-sentations ๐ป
(1)1 โ ๐ท+
1/2 and ๐ป(1)3 โ ๐ท+
1/2 are shown,which make it is easy to find the consistency con-ditions for the irreducible projective representationsfor the spinless orbitals of point ๐ป, the orbitals ๐ป
(1)๐
of the projective class ๐พ1 with irreducible projectiverepresentations of spin orbitals (๐ป โฒ)
(0)๐ of the projec-
tive class ๐พ0, which arise, when the electron spinis taken into account. For instance, if the electronspin is taken into consideration, the sum of spin or-bitals ((๐ป โฒ)
(0)1 +(๐ป โฒ)
(0)3 )+(๐ป โฒ)
(0)6 corresponds to the
spinless orbital ๐ป(1)1 , and the sum of spin orbitals
(๐ป โฒ)(0)5 + (๐ป โฒ)
(0)6 to the spinless orbital ๐ป(1)
3 .
The characters of the projective representation ofall electron ๐-bands taking the electron spin into ac-count (๐โฒ-bands) for point ๐ป in the Brillouin zone ofcrystalline graphite ๐พ-๐ถ, i.e. the representation ๐ป โฒ
๐,are equal to the sums of the corresponding charactervalues of all spin orbitals (๐ป โฒ)
(0)๐ . They are given in
the bottom part of Table 11, ๐. The indicated valuesfor the characters of the projective representation ๐ป โฒ
๐
can be easily obtained using formula (4) (for point ๐ป,it looks like ๐ป โฒ
๐ = ๐ปeqโ๐ป โฒ๐ง, where ๐ป โฒ
๐ง is a representa-tion that characterizes the symmetry of a ๐-electronwith its spin at point ๐ป in the Brillouin zone of crys-talline graphite).
Table 2 demonstrates the characters of the equiva-lence representation for point ๐ป in the Brillouin zoneof crystalline graphite, the representations ๐ปeq; therepresentations ๐ป๐ง โก ฮ๐ง and ๐ป๐ = ๐ป๐๐โ๐ป๐ง describ-ing the spatial symmetry of ๐-electron and electron๐-bands, respectively, without taking the spin intoaccount; the two-valued (spinor) representation ๐ท+
1/2,and the two-valued representations ๐ป โฒ
๐ง = ๐ป๐ง โ๐ท+1/2
and ๐ป โฒ๐ = ๐ปeq โ ๐ป โฒ
๐ง describing the symmetry of ๐-electron and electron ๐-bands making allowance for
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V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
the spin. Table 3 contains the distribution of electron๐โฒ-bands with regard for the spin at point ๐ป over thetwo-valued (spinor) irreducible projective representa-tions of the projective class ๐พ0.
Our calculation of the Herring criterion for point ๐ปwith the use of the symmetry elements satisfying theequality ๐โฒ๐ = โ๐ โ namely, these are the elements๐โฒ7 = (๐1
2 |๐2) , ๐โฒ8 = (๐1
2
๐56) , ๐โฒ9 = (๐1
2 |๐6) , ๐โฒ10 == (๐1
2 |(๐ขโฒ2)1) , ๐โฒ11 = (๐1
2 |(๐ขโฒ2)2) , ๐โฒ12 = (๐1
2 |(๐ขโฒ2)3) ,
Table 12. (๐) Characters of the one-valued irreducibleprojective representations of the projective class ๐พ0
of the group 3๐ (๐ถ3๐ฃ), which describe the symmetryof vibrational and electron excitations without takingthe spin into account for the spatial symmetry group atpoint ๐ (the wave-vector group of point ๐ ) in the Bril-louin zone of crystalline graphite and are ๐-equivalentto the characters of the one-valued irreducible pro-jective representations corresponding to the standardunity-containing factor-system of this projective class.(b) Characters of the two-valued (spinor) irreducibleprojective representations of point ๐ in the Brillouinzone of crystalline graphite ๐พ-๐ถ (the spatial symme-try group ๐63/๐๐๐ (๐ท4
6โ), the projective class ๐พ0),the two-valued spinor irreducible projective represen-tation ๐ท
+1/2
, the projective representation products
๐(0)1 โ๐ท
+1/2
, ๐ (0)2 โ๐ท
+1/2
, and ๐(0)3 โ๐ท
+1/2
, and the two-valued projective representation of electron ๐โฒ-bandstaking the electron spin into account โ representation๐ โฒ
๐(๐ฟ)
Projec-tiveclass
Notation forirreducibleprojective
representation
3๐(๐ถ3๐)
๐ ๐3 ๐23 3๐๐ขโฒ2
๐พ0 ๐ ๐(0)1 1 1 1 ๐๐๐ง
๐(0)2 1 1 1 โ๐๐๐ง
๐(0)3 2 โ1 โ1 0
๐ข2(๐) 1 โ1 1 ๐
๐พ0 b (๐ โฒ)(0)1 1 โ1 1 ๐๐๐๐ง
(๐ โฒ)(0)2 1 โ1 1 โ๐๐๐๐ง
(๐ โฒ)(0)3 2 1 โ1 0
๐ท+1/2
2 1 โ1 0
๐(0)1 โ๐ท+
1/22 1 โ1 0
๐(0)2 โ๐ท+
1/22 1 โ1 0
๐(0)3 โ๐ท+
1/24 โ1 1 0
๐ โฒ๐ 8 1 โ1 0
* ๐๐๐ง = ๐โ๐๐๐๐1/2 = ๐โ๐๐๐ง๐1/2
๐โฒ13 = (0 |๐) , ๐โฒ14 = (0 |๐๐3) , ๐โฒ15 = (0๐๐23) , ๐โฒ16 =
= (0 |๐(๐ข2)1) , ๐โฒ17 = (0 |๐(๐ข2)2) , ๐โฒ18 = (0 |๐(๐ข2)3) , andtheir squares equal (๐โฒ7)2 = (๐1 |๐) , (๐โฒ8)2 = (๐1
๐๐23) ,
(๐โฒ9)2 = (๐1 |๐3) , (๐โฒ10)
2 = (0 |๐) , (๐โฒ11)2 = (0 |๐) ,
(๐โฒ12)2 = (0 |๐) , (๐โฒ13)
2 = (0 |๐) , (๐โฒ14)2 = (0
๐23) ,
(๐โฒ15)2 = (0 |๐๐3) , (๐โฒ16)
2 = (0 |๐) , (๐โฒ17)2 = (0 |๐)
and (๐โฒ18)2 = (0 |๐) โ testifies that the two-valued
one-dimensional spinor irreducible projective repre-sentations (๐ป โฒ)
(0)1 and (๐ป โฒ)
(0)3 at point ๐ป in the
Brillouin zone of crystalline graphite belong to thecase ๐2 [9] and, owing to their time-reversal in-variance, they must unite, although our case corre-sponds to the union of nonequivalent complex repre-sentations, rather than complex-conjugate ones. Atthe same time, the two-valued two-dimensionalcomplex-conjugate irreducible projective representa-tions (๐ป โฒ)
(0)5 and (๐ป โฒ)
(0)6 belong to the case ๐2 [9] and
do not unite, if the time-reversal symmetry is takeninto account.
3.2.3. Point ๐
The group of equivalent directions of the wave-vectorgroup at point ๐ in the Brillouin zone of crys-talline graphite ๐พ-๐ถ is the group 3๐ (๐ถ3๐ฃ). The group3๐ (๐ถ3๐ฃ) has only one class of projective represen-tations. This is the class ๐พ0. Therefore, all projec-tive representations of this group are ๐-equivalentto vector ones. The wave-vector star at this pointcontains four rays: (๐๐ )1 = โ๐๐ง โ 1
3 (2๐2 โ ๐3),(๐๐ )2 = โ๐๐ง+
13 (2๐2โ๐3), (๐๐ )3 = ๐๐งโ 1
3 (2๐2โ๐3),and (๐๐ )4 = ๐๐ง +
13 (2๐2 โ ๐3).
Table 12 presents (๐) the one-valued irreducibleprojective representations of the projective class ๐พ0
of the group 3๐ (๐ถ3๐ฃ) at point ๐ in the Brillouin zoneof crystalline graphite and (b) the two-valued (spinor)irreducible projective representations, also belongingto the projective class ๐พ0 of this group, of equiva-lent directions; the characters of the spinor represen-tation ๐ท+
1/2; the products of projective representa-
tions ๐(0)1 โ ๐ท+
1/2, ๐(0)2 โ ๐ท+
1/2, and ๐(0)3 โ ๐ท+
1/2;and the characters of the projective representation ofelectron ๐โฒ-bands making allowance for the electronspin, ๐ โฒ
๐. The bottom part of Table 12, ๐ contains thecoefficients ๐ข2(๐) that transform the factor-systemof transformations of the spin variable ๐2(๐2, ๐1) tothe standard form ๐โฒ
2(๐2, ๐1) โก ๐โฒ(0)(๐2, ๐1. The bot-
tom part of the whole Table 12 shows the phase fac-tor value.
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The characters of the one-valued projective equiv-alence representations ๐eq, the one-valued projectiverepresentations ๐๐ง and ๐๐, the double-valued projec-tive representations ๐ท+
1/2 and ๐ โฒ๐ง, and the two-valued
(spinor) projective representation ๐ โฒ๐ of electron ๐โฒ-
bands making allowance for the electron spin for point๐ in the Brillouin zone of crystalline graphite aregiven in Table 2. The distributions of projective rep-resentations of electron ๐-bands without taking theelectron spin into account, ๐๐, and taking it into ac-count, ๐ โฒ
๐, over the irreducible one- and two-valued,respectively, projective representations are presentedin Table 3.
The calculation of the Herring criterion for point ๐in the Brillouin zone of crystalline graphite with theuse of the symmetry elements satisfying the require-ment ๐โฒ๐ = โ๐โnamely, these are the elements ๐โฒ10 == (๐1
2 |(๐ขโฒ2)1) , ๐โฒ11 = (๐1
2 |(๐ขโฒ2)2) , ๐โฒ12 = (๐1
2 |(๐ขโฒ2)3) ,
๐โฒ13 = (0 |๐) , ๐โฒ14 = (0 |๐๐3) and ๐โฒ15 = (0๐๐23) , and
their squares equal (๐โฒ10)2 = (0 |๐) , (๐โฒ11)
2 = (0 |๐) ,(๐โฒ12)
2 = (0 |๐) , (๐โฒ13)2 = (0 |๐) , (๐โฒ14)
2 = (0๐23) ,
(๐โฒ15)2 = (0 |๐๐3) โ shows that the two-valued one-
dimensional spinor irreducible projective representa-tions at this point, (๐ โฒ)
(0)1 and (๐ โฒ)
(0)2 , belong to the
case ๐2 [9] and, owing to the time-reversal invari-ance, they unite into the two-dimensional projectiverepresentation ((๐ โฒ)
(0)1 + (๐ โฒ)
(0)2 ). At the same time,
the one-valued irreducible projective representations๐
(0)1 , ๐ (0)
2 , ๐ (0)3 , and the two-value (spinor) projective
representation (๐ โฒ)(0)3 belong to the case ๐2 [9] and do
not unite, if the time-reversal symmetry is taken intoconsideration.
At point ๐ป, which is limiting for points ๐ in theBrillouin zone of crystalline graphite, if the elec-tron spin is neglected, i.e. when the phase factor๐โ๐๐๐ง๐1/2 = ๐ at ๐๐ง = โ๐1/2, the sums of the char-acters of one-valued irreducible projective represen-tations ๐
(0)1 and ๐
(0)2 , as the complex-conjugate rep-
resentations, should transform into the characters ofthe two-dimensional one-valued irreducible projectiverepresentation ๐ป
(1)3 belonging to the projective class
๐พ1 of a higher symmetry group than the symmetrygroup of points ๐ . It is also easy to see from Ta-ble 12, ๐ that, if the electron spin is taken into ac-count, the spin orbital (๐ โฒ)
(0)3 corresponds to the spin-
less orbitals ๐ (0)1 and ๐
(0)2 , and the sum of the united
spin orbitals ((๐ โฒ)(0)1 + (๐ โฒ)
(0)2 ) and the spin orbital
(๐ โฒ)(0)3 corresponds to the spinless orbital ๐ (0)
3 .
3.3. Line MโUโL of crystallinegraphite and point M of single-layer graphene
At points ๐ and ๐ฟ in the Brillouin zone of crystallinegraphite ๐พ-๐ถ and at point ๐ in the Brillouin zone ofsingle-layer graphene ๐ถ๐ฟ,1, the factor-groups of wave-vector groups with respect to the invariant translationsubgroups are isomorphic to the same point symme-try group ๐๐๐ (๐ท2โ), which is the point symmetrygroup of equivalent directions for points ๐ in thosestructures.
3.3.1. Points ๐
Each of the stars of the wave-vector groups at points๐ in the Brillouin zones of the structures con-cerned contains three rays: these are (๐๐ )1 = โ 1
2๐3,(๐๐ )2 = 1
2๐2, and (๐๐ )3 = โ 12 (๐2 โ ๐3) for crys-
talline graphite ๐พ-๐ถ; and (๐๐ )1 = โ 12๐2, (๐๐ )2 =
= 12๐1, and (๐๐ )3 = โ 1
2 (๐1 โ ๐2) for single-layergraphene ๐ถ๐ฟ,1.
First of all, as was done in work [1] for the pointgroup 6/๐๐๐ (๐ท6โ), let us construct a factor system๐2(๐2, ๐1) describing the transformations of spinorsunder the action of symmetry operations for thepoint group ๐๐๐ (๐ท2โ) and determine the coeffi-cients ๐ข2(๐) that transform it to the standard form๐โฒ2(๐2, ๐1).Of three rays of the wave-vector stars at point ๐
in the Brillouin zones of both crystalline graphite andsingle-layer graphene, let us consider the rays (๐๐ )1(points ๐1), for which the elements of symmetry thattransform these rays into the equivalent ones andform the point symmetry group ๐๐๐ are the ele-ments ๐, (๐ข2)1, ๐2, (๐ขโฒ
2)1, ๐, ๐(๐ข2)1, ic2, and ๐(๐ขโฒ2)1. As
the ๐๐๐ group generators, let us choose the ele-ments ๐ = (๐ข2)1, ๐ = ๐2, and ๐ = ๐. This choice makesallowance for the composition principle. According tothe latter, the group ๐๐๐ can be represented asthe direct group product 222 โ 1 (๐๐๐ = 222 โ 1or ๐ท2โ = ๐ท2 โ ๐ถ๐), and the group 222 as the directgroup product 2โฒ โ 2 (222 = 2โฒ โ 2 or ๐ท2 = ๐ถ โฒ
2 โ๐ถ2).By applying the defining relations, let us calculate
all values of ๐2(๐2, ๐1). It is clear that this is the defin-ing relations for the double group (๐๐๐)
โฒ that haveto be taken for this purpose:๐4 = ๐, ๐4 = ๐, ๐2 = ๐,๐๐ = ๐๐๐, ๐๐ = ๐๐, ๐๐ = ๐๐.
The factor-system ๐2(๐2, ๐1) calculated for the ๐๐๐group following this way and using the method of
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Table 13. Factor-system ๐2(๐2, ๐1) for the group ๐๐๐ (๐ท2โ) (a) and the correspondingstandard factor-system ๐โฒ
2(๐2, ๐1) (๐). The bottom part of Table 13, ๐ contains the values of the function๐ข2(๐) that transforms the factor-system ๐2(๐2, ๐1) to the standard form ๐โฒ
2(๐2, ๐1) โก ๐โฒ(1)
(๐2, ๐1)
a b
work [1] is shown in Table 13, ๐. This factor sys-tem belongs to the projective class ๐พ1, because ๐ผ == โ1, ๐ฝ = 1, and ๐พ = 1 for it [1]. In Table 13, ๐,the subscripts near the coefficients of the factor-system ๐2(๐2, ๐1), which contain parenthesized num-bers, compose a multiplication table for the elementsof the ๐๐๐ group (the numbers in parentheses indi-cate the numerical designations of the elements cor-responding to the products ๐2๐1).
With the help of the coefficients ๐ข2(๐) given inthe bottom part of Table 13, ๐, the factor system๐2(๐2, ๐1) is transformed into a ๐-equivalent block-symmetric form, which corresponds to the defini-tion of a standard factor system, i.e. the factor-system ๐โฒ
2(๐2, ๐1). In so doing, the values of thecoefficients ๐ข2(๐) can be calculated using formulas(13.3), (14.18), and (14.19) of work [9]. Alternati-vely, they can be found, when constructing an ex-tended group, the representation group [9], wherethey are determined by its one-valued irreducible rep-resentations, being additional to ordinary vector rep-resentations.
The values of the coefficients ๐ข2(๐), which char-acterize the transformation of spin functions foridentical elements ๐ belonging to different pointgroups โ in our case, to point groups ๐๐๐ and6/๐๐๐, where the group ๐๐๐ is also a subgroupof the group 6/๐๐๐ โ expectedly turned out iden-tical. This means that the factor-system presented inTable 13, ๐ is really a standard factor-system for theprojective class ๐พ1 of the point group ๐๐๐, i.e. thefactor-system ๐โฒ
2(๐2, ๐1) โก ๐โฒ(1)(๐2, ๐1). Solid lines in
Table 13, ๐ distinguish the contours of blocks, inwhich the coefficients have a value of โ1.
Table 14 displays the characters of the irreduciblerepresentations of the double group (๐๐๐)โฒ (๐ทโฒ
2โ),the additional one-valued irreducible representationsof which (additional to the ordinary vector one-valuedirreducible representations of the group ๐๐๐, whichcan be obtained from the representations of the group(๐๐๐)โฒ by simply excluding the element ๐ from allrelations) are either two-valued projective or spinorrepresentations of the ๐๐๐ group. The spinor rep-resentations are denoted by the symbols (๐ธโฒ)+ and(๐ธโฒ)โ in the Mulliken notation or the symbols ฮ+
5
and ฮโ5 in the Koster notation, where the letter ฮ
denotes not only their membership in a certain pointgroup (in the given case, this is the group ๐๐๐),but also in the coinciding group of equivalent direc-tions of the wave-vector group of point ฮ in crys-tals or periodic nanostructures. The symbols K0 andK1 denote the corresponding projective classes, andthe notations ((ฮโฒ)(1))+ and ((ฮโฒ)(1))โ were proposedby us (here, the prime means the two-valued spinorrepresentation, the superscript (the number in theparentheses) indicates the projective class, and thesuperscripts โ+โ and โโโ mean the representationparity).
The characters of the irreducible representations ofthe point group ๐๐๐ (๐ท2โ) of the projective classes๐พ0 (ordinary one-valued or vector) and ๐พ1 (two-valued projective or spinor) for the standard factor-systems ๐โฒ
(0)(๐2, ๐1) and ๐โฒ(1)(๐2, ๐1) of the point group
๐๐๐ are presented in Table 15. The irreducible pro-
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Table 14. Characters of the irreducible representations of the double group (๐๐๐)โฒ (๐ทโฒ2โ)
(๐๐๐)โฒ(๐ทโฒ2โ) ๐ ๐
(๐ข2)1,๐(๐ข2)1
๐2,๐๐2
(๐ขโฒ2)1,
๐(๐ขโฒ2)1
๐ ๐๐๐(๐ข2)1,๐๐(๐ข2)1
๐๐2,๐๐๐2
๐(๐ขโฒ2)1
๐๐(๐ขโฒ)1
๐พ0 ฮ+1 ๐ด+
1 1 1 1 1 1 1 1 1 1 1ฮโ1 ๐ดโ
1 1 1 1 1 1 โ1 โ1 โ1 โ1 โ1ฮ+2 ๐ด+
2 1 1 1 โ1 โ1 1 1 1 โ1 โ1ฮโ2 ๐ดโ
2 1 1 1 โ1 โ1 โ1 โ1 โ1 1 1
ฮ+3 ๐ต+
1 1 1 โ1 1 โ1 1 1 โ1 1 โ1ฮโ3 ๐ตโ
1 1 1 โ1 1 โ1 โ1 โ1 1 โ1 1ฮ+4 ๐ต+
2 1 1 โ1 โ1 1 1 1 โ1 โ1 1ฮโ4 ๐ตโ
2 1 1 โ1 โ1 1 โ1 โ1 1 1 โ1
๐พ1 ((ฮโฒ)(1))+ ฮ+5 (๐ธโฒ)+ 2 โ2 0 0 0 2 โ2 0 0 0
((ฮโฒ)(1))โ ฮโ5 (๐ธโฒ)โ 2 โ2 0 0 0 โ2 2 0 0 0
Table 15. Characters of the one-valued(vector) and two-valued (spinor) irreducibleprojective representations of the group ๐๐๐ (๐ท2โ)
corresponding to the standard factor systems๐โฒ
(0)(๐2, ๐1) (the projective class ๐พ0) and ๐โฒ
(1)(๐2, ๐1)
(the projective class ๐พ1), respectively
๐๐๐ (๐ท2โ) ๐ (๐ข2)1 ๐2 (๐ขโฒ2)1 ๐ ๐(๐ข2)1 ๐๐2 ๐(๐ขโฒ
2)1
๐พ0 ๐ด+1 1 1 1 1 1 1 1 1
๐ดโ1 1 1 1 1 โ1 โ1 โ1 โ1
๐ด+2 1 1 โ1 โ1 1 1 โ1 โ1
๐ดโ2 1 1 โ1 โ1 โ1 โ1 1 1
๐ต+1 1 โ1 1 โ1 1 โ1 1 โ1
๐ตโ1 1 โ1 1 โ1 โ1 1 โ1 1
๐ต+2 1 โ1 โ1 1 1 โ1 โ1 1
๐ตโ2 1 โ1 โ1 1 โ1 1 1 โ1
๐พ1 (๐ (1))+ 2 0 0 0 2 0 0 0(๐ (1))โ 2 0 0 0 โ2 0 0 0
jective representations for ๐ points in the Brillouinzones of crystalline graphite and single-layer grapheneare identical to them and are shown in Table 16. Inthe group ๐๐๐ (๐ท2โ), the second-order axis (๐ข2)1is a senior axis in the structurally distinguished se-nior subgroup [the axis ๐2 is involved into the ex-tension of the group consisting of the elements ๐ and(๐ข2)1 (the group ๐ถ โฒ
2) to the group ๐ท2]. In other words,in the formation of the direct product of the groups๐ถ โฒ
2 [(๐2)1] and ๐ถ2, the axis (๐ข2)1 plays the role of theprincipal axis, according to which the symbols (num-bers) of irreducible representations are determined. Itis easy to see that the characters of the irreducibleprojective representations of the class ๐พ1 of the group๐๐๐ for the standard factor-system of this class,
Table 16. Characters of the one-valued (vector)(the projective class ๐พ0) and two-valued (spinor)(the projective class ๐พ1) irreducible projectiverepresentations of points ๐ in the Brillouin zonesof crystalline graphite and single-layer graphene
๐๐๐ (๐ท2โ) ๐ (๐ข2)1 ๐2 (๐ขโฒ2)1 ๐ ๐(๐ข2)1 ๐๐2 ๐(๐ขโฒ
2)1
๐พ0 ๐+1 1 1 1 1 1 1 1 1
๐โ1 1 1 1 1 โ1 โ1 โ1 โ1
๐+2 1 1 โ1 โ1 1 1 โ1 โ1
๐โ2 1 1 โ1 โ1 โ1 โ1 1 1
๐+3 1 โ1 1 โ1 1 โ1 1 โ1
๐โ3 1 โ1 1 โ1 โ1 1 โ1 1
๐+4 1 โ1 โ1 1 1 โ1 โ1 1
๐โ4 1 โ1 โ1 1 โ1 1 1 โ1
๐พ1 ((๐ โฒ)(1))+ ๐+5 2 0 0 0 2 0 0 0
((๐ โฒ)(1))โ ๐โ5 2 0 0 0 โ2 0 0 0
where they are denoted as (๐ (1))+ and (๐ (1))โ, onthe one hand, and the characters of the irreducibleprojective representations of the class ๐พ1 of the group๐๐๐ for points ๐ , which are denoted by the sym-bols ๐+
5 and ๐โ5 [or ((๐ โฒ)(1))+ and ((๐ โฒ)(1))โ],
on the other hand, coincide with the character ofthe spinor irreducible representations of the doublegroup (๐๐๐)
โฒ.Table 2 exhibits the characters of the projective
equivalence representation at point ๐ (the repre-sentation ๐eq), the characters of the representation๐๐ง โก ฮ๐ง, which determines the spatial symmetry ofthe ๐๐ง orbital, the characters of the representationof the electron ๐-bands without taking the spin intoaccount (the representation ๐๐), and the charactersof the two-valued representations ๐ท+
1/2, ๐ โฒ๐ง โก ฮโฒ
๐ง,
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Table 17. Characters of the irreducible projective representationsof the projective class ๐พ4 of the group ๐๐๐ (๐ท2โ) corresponding to the standardfactor-system of this class, ๐โฒ
(4)(๐2, ๐1) (๐), and the characters of two-valued (spinor)
irreducible projective representations of point ๐ฟ in the Brillouin zone of crystalline graphite ๐พ-๐ถ (๐)
Projectiveclass
Notation for irreducibleprojective representation
๐๐๐ (๐ท2โ)
๐ (๐ข2)1 ๐2 (๐ขโฒ2)1 ๐ ๐(๐ข2)1 ๐๐2 ๐(๐ขโฒ
2)1
๐พ4 a ๐(4)1 2 2 0 0 0 0 0 0
๐(4)2 2 โ2 0 0 0 0 0 0
๐ข1,๐ฟ(๐) 1 1 1 1 1 1 โ1 โ1
๐ข2(๐) 1 ๐ ๐ โ1 1 ๐ ๐ โ1
๐โ๐๐๐ฟ๐(๐) 1 1 ๐ ๐ 1 1 ๐ ๐
๐๐๐๐ฟ๐(๐)๐ข1,๐ฟ(๐)๐ข2(๐) 1 ๐ โ1 โ๐ 1 ๐ 1 ๐
๐ท+1/2
2 0 0 0 2 0 0 0
๐พ4 ๐ฟ(4)1 2 2๐ 0 0 0 0 0 0
b ((๐ฟโฒ)(4)1 + (๐ฟโฒ)
(4)2 )<
๐ฟ(4)2 2 โ2๐ 0 0 0 0 0 0
๐ฟโฒ๐ 8 0 0 0 0 0 0 0
and ๐ โฒ๐. The distributions of electron excitations at
points ๐ in the Brillouin zones of crystalline graphite๐พ-๐ถ and single-layer graphene ๐ถ๐ฟ,1 for ๐-bands with-out taking and taking the electron spin into accountover the one- and two-valued (spinor), respectively, ir-reducible projective representations of points ๐ aregiven in Table 3.
3.3.2. Point ๐ฟ
As was already mentioned above, the factor-group ofthe wave-vector group with respect to the invarianttranslation subgroup at points ๐ฟ in the Brillouin zoneof crystalline graphite is also isomorphic to the pointgroup ๐๐๐ (๐ท2โ). The wave-vector star of point ๐ฟfor the graphite ๐พ-๐ถ structure also contains threevectors: these are (๐๐ฟ)1 = โ 1
2 (๐1 + ๐3), (๐๐ฟ)2 == โ 1
2 (๐1 โ ๐2), and (๐๐ฟ)3 = โ 12 (๐1 + ๐2 โ ๐3).
Similarly to what was done earlier for point ๐ , ofthree rays of the wave-vector stars at point ๐ฟ in theBrillouin zone of crystalline graphite, let us considerthe ray (๐๐ฟ)1, for which the symmetry elementsthattransform the rays of the wave-vector star at point ๐ฟinto the equivalent ones and form the point symme-try group ๐๐๐ (๐ท2โ) are the elements ๐, (๐ข2)1, ๐2,(๐ขโฒ
2)1, ๐, ๐(๐ข2)1, ic2, and ๐(๐ขโฒ2)1. Again, as was done
for the group of point ๐ , let us choose the elements๐ = (๐ข2)1, ๐ = ๐2, and ๐ = ๐ to be the ๐๐๐ groupgenerators.
In work [1], it was demonstrated that the factor-system ๐1,๐ฟ(๐2, ๐1) belongs to the projective class ๐พ5
of the group ๐๐๐ (๐ท2โ) and can be reduced to thestandard form ๐โฒ
1,๐ฟ(๐2, ๐1) โก ๐โฒ(5)(๐2, ๐1) with the
help of the coefficients ๐ข1,๐ฟ(๐) given in the bottompart of Table 17, ๐ in work [1]. It was also shownthat the factor-system ๐2(๐2, ๐1) of spinor transfor-mations at the symmetry operations of the group๐๐๐ (๐ท2โ), as it occurs for point ๐ , belongs to theprojective class ๐พ1 and is reduced to the standardform ๐โฒ
2(๐2, ๐1) โก ๐โฒ(1)(๐2, ๐1) with the help of the co-
efficients ๐ข2(๐) given in the bottom part of Table 13, ๐of this work. This means that the factor-system atpoint ๐ฟ making allowance for the spin, ๐2,๐ฟ(๐2, ๐1),is the product of the factor-system ๐1,๐ฟ(๐2, ๐1) (theprojective class ๐พ5), which is determined by thestructure of the spatial group of crystalline graphiteat point ๐ฟ making no allowance for the spin, andthe factor-system ๐2(๐2, ๐1) (the projective class ๐พ1),which describes the transformations of spinors atpoint ๐ฟ (in the point symmetry group ๐๐๐ (๐ท2โ)),i.e. ๐2,๐ฟ(๐2, ๐1) = ๐1,๐ฟ(๐2, ๐1)๐2(๐2, ๐1).
The standard factor-system taking the spin intoaccount for point ๐ฟ, ๐โฒ
2,๐ฟ(๐2, ๐1) = ๐โฒ(5)(๐2, ๐1)ร
ร๐โฒ2(๐2, ๐1) = ๐โฒ
(5)(๐2, ๐1)๐โฒ(1)(๐2, ๐1), belongs to the
projective class ๐พ4 of the group ๐๐๐ (๐ท2โ) (be-cause K5 ยท K1 = K4) and coincides with the stan-dard factor-system of the projective class ๐พ4 of the
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Fig. 4. Dispersion of the electron energy ๐โฒ-bands calculated taking the electron spin intoaccount
group ๐๐๐ (๐ท2โ); i.e., ๐โฒ2,๐ฟ(๐2, ๐1) โก ๐โฒ
(4)(๐2, ๐1)with the coefficients of transformation to the stan-dard form, ๐ข2,๐ฟ(๐), being equal to the product of thecoefficients of transformation of the factor-systems๐1,๐ฟ(๐2, ๐1) and ๐2(๐2, ๐1) to the standard form,๐ข1,๐ฟ(๐) and ๐ข2(๐).
The characters of the irreducible projective rep-resentations of the point group ๐๐๐ (๐ท2โ) of theprojective class ๐พ4 for the standard factor-system๐โฒ(4)(๐2, ๐1) of the point group ๐๐๐ are presented
in Table 17, ๐. In the bottom part of this table,the values of the coefficients ๐ข1,๐ฟ(๐), ๐ข2(๐), thephase factors ๐โ๐๐๐ฟ๐ผ(๐), the products of coefficients๐โ๐๐๐ฟ๐ผ(๐)๐ข1,๐ฟ(๐)๐ข2(๐) โก ๐โ๐๐๐ฟ๐ผ(๐)๐ข2,๐ฟ(๐), and thecharacters of the irreducible spinor projective repre-sentation ๐ท+
1/2 are given. The characters of the two-valued (spinor) irreducible projective representationsat point ๐ฟ in the Brillouin zone of crystalline graphite,which were calculated using the formula
(๐ฟโฒ)(4)๐ = ๐โ๐๐๐ฟ๐ผ(๐)(๐)๐ข1,๐ฟ(๐)๐ข2(๐)๐
(4)๐ =
= ๐โ๐๐๐ฟ๐ผ(๐)๐ข2,๐ฟ(๐)๐(4)๐ , (9)
are shown in Table 17, ๐. The bottom part of thistable contains the characters of the two-valued spinor
projective representation of electron ๐โฒ-bands makingallowance for the electron spin.
The characters of the projective equivalence rep-resentation at point ๐ฟ (the representation ๐ฟeq), therepresentation ๐ฟ๐ง โก ฮ๐ง determining the spatial sym-metry of the ๐๐ง orbital, the representation ๐ฟ๐ of theelectron ๐-bands making no allowance for the spin,and the two-valued representations ๐ท+
1/2, ๐ฟโฒ๐ง โก ฮโฒ
๐ง,and ๐ฟโฒ
๐ are given in Table 2. The distributions of elec-tron excitations at point ๐ฟ in the Brillouin zone ofcrystalline graphite for ๐-bands without and with tak-ing the electron spin into account over the one-valuedand, accordingly, two-valued (spinor) irreducible pro-jective representations of point ๐ฟ are presented inTable 3.
Our calculation of the Herring criterion using thesymmetry elements for point ๐ฟ in the Brillouin zoneof crystalline graphite, which satisfy the equality๐โฒ๐ = โ๐ โ in particular, these are the elements๐โฒ1 = (0 |๐ ), ๐โฒ4 = (0 |(๐ข2)1 ), ๐โฒ7 = (๐1
2 |๐2 ), ๐โฒ10 == (๐1
2 |(๐ขโฒ2) 1), ๐
โฒ13 = (0 |๐ ), ๐โฒ16 = (0 |๐(๐ข2)1 ), ๐โฒ19 =
= (๐1
2 |๐๐2 ) and ๐โฒ22 = (๐1
2 |๐(๐ขโฒ2) 1), whose squares
equal (๐โฒ1)2 = (0 |๐ ), (๐โฒ4)2 = (0 |๐ ), (๐โฒ7)2 = (๐1 |๐ ),(๐โฒ10)
2 = (0 |๐ ), (๐โฒ13)2 = (0 |๐ ), (๐โฒ16)
2 = (0 |๐ ),
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V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
Fig. 5. Dispersion of the electron energy (๐) ๐-bands (not taking the electron spin into account) and (๐) ๐โฒ-bands (taking theelectron spin into account) in the K โ P โH direction of the Brillouin zone in graphite ๐พ-๐ถ crystals. The spin-dependent finestructure of ๐โฒ-bands in panel ๐ is shown schematically, with the energy scale for the splitting of electron ๐โฒ-bands being enlargedby a factor of about 103
(๐โฒ19)2 = (0 |๐ ) and (๐โฒ22)
2 = (๐1 |๐ ), โ testifiesthat the two-valued (spinor) irreducible projectiverepresentations of point ๐ฟ in the Brillouin zone ofcrystalline graphite that belong to the projectiveclass ๐พ4 โ these are the representations (๐ฟโฒ)
(4)1 and
(๐ฟโฒ)(4)2 โ belong to the case ๐2 [9]. Being the rep-
resentations with complex-conjugate characters, iftheir time-reversal invariance is taken into considera-tion, they unite into four-dimensional representations((๐ฟโฒ)
(4)1 +(๐ฟโฒ)
(4)2 )1 and ((๐ฟโฒ)
(4)1 +(๐ฟโฒ)
(4)2 )2, so that the
degree of degeneration for each of the electron statesincreases to four. In Table 17, ๐, just this union ofrepresentations with complex-conjugate characters isindicated.
In Fig. 4, the dispersion of the electron energy ๐-bands making allowance for the electron spin (๐โฒ-bands) in graphite crystals is shown schematically.The letters are used to mark points in the Brillouinzone, and the letters with indices to mark the two-valued spinor irreducible projective representationsof the corresponding projective classes (the latterare indicated by the parenthesized superscripts). Thedispersion of electron ๐-bands is schematically illus-trated for all high-symmetry points in the Brillouinzone of crystalline graphite. The curves agree wellat the qualitative level with the results of numer-ical calculations carried out in works [11, 12] tak-
ing no electron spin into account, i.e. in the case ofweak spin-orbit interaction. Nevertheless, the curvesdemonstrate the qualitative behavior of the disper-sion of the electron ๐-bands along the line ฮโฮโ๐ด.
Figure 5, ๐ exhibits the dispersion of the electron๐-bands along the line ๐พโ๐โ๐ป in the Brillouin zoneof crystalline graphite calculated making no allowancefor the electron spin. Figure 5, ๐ qualitatively demon-strates the spin-dependent fine structure of the energy๐-bands for the splitting of electron states shown inFig. 5, ๐ (the energy scale of the spin-dependent finestructure is enlarged by a factor of about 103). Thisfine structure is obtained, if the methods of theo-retical symmetry-group analysis are consistently ap-plied to determine the dispersion of electron ๐-bandsin crystalline graphite taking the electron spin intoaccount. The spin-dependent splitting can be sub-stantial, e.g., for dichalcogenides of transition met-als with the same spatial symmetry group. However,it is small for crystalline graphite and single-layergraphene, because it is caused by a low spin-orbitinteraction energy for carbon atoms and, as a conse-quence, carbon structures.
4. Conclusions
1. For the first time, a theoretical symmetry-groupdescription of the dispersion of electron ๐-bands
364 ISSN 2071-0194. Ukr. J. Phys. 2020. Vol. 65, No. 4
Energy Spectra of Electron Excitations
Ta
ble
18
(A
pp
en
dix
) .
Calc
ula
tion
of
the H
errin
g c
rit
erio
n
ISSN 2071-0194. Ukr. J. Phys. 2020. Vol. 65, No. 4 365
V.O. Gubanov, A.P. Naumenko, M.M. Bilyi et al.
in crystalline graphite (the space symmetry group๐63/๐๐๐ (๐ท4
6โ)) and single-layer graphene (thediperiodic space group ๐6/๐๐๐ (๐ท๐บ80)) is made.The consistency conditions for the irreducible pro-jective representations making allowance for the elec-tron spin and the changes of the projective classes aredetermined for various high-symmetry points in theBrillouin zones of those materials.
2. A correlation between the electron excitationsin crystalline graphite making allowance for the elec-tron spin and the spinor excitations in single-layergraphene is shown.
3. With the help of theoretical symmetry-groupmethods, the existence of a fine structure for elec-tron ๐-bands has been predicted for the first time. Itarises, when the electron spin is taken into account,even in the case of weak spin-orbit interaction. Thecorresponding results include the appearance of asmall (about 1.0โ1.5 meV according to the estima-tions of work [2]) band gap between the valence andconduction bands at the Dirac points and in theirvery small vicinities in crystalline graphite and single-layer graphene. A new interpretation is also given toa small splitting (the spin-dependent fine structure)of electron ๐-bands, if the electron spin is taken intoaccount, at point ๐ป of crystalline graphite on the ba-sis of the established change in the projective classesof irreducible projective representations of the wave-vector groups, which excludes the intersection of thedispersion curves of electron bands near point ๐ป.
APPENDIX:Calculation of the Herring criterion
Table 18 illustrates the stages and the results of cal-culations of the characters ๐k,๐ท๐
[(๐โฒ)2], ๐k,๐ทโฒ๐[(๐โฒ)2],
and the corresponding values of the Herring criterionfor irreducible representations at points ฮ and ๐ด.
1. V.O. Gubanov, A.P. Naumenko, M.M. Bilyi, I.S. Dotsenko,O.M. Navozenko, M.M. Sabov, L.A. Bulavin. Energy spec-tra correlation of vibrational and electronic excitations andtheir dispersion in graphite and graphene. Ukr. J. Phys.63, 431 (2018).
2. M.I. Katsnelson. Graphene: Carbon in Two Dimensions(Cambridge Univ. Press, 2012).
3. J.D. Bernal. The structure of graphite. Proc. Roy. Soc.London A 106, 749 (1924).
4. T. Hahn. International Tables for Crystallography. Vol. A.Space Group Symmetry (D. Reidel, 1983).
5. C. Herring. Effect on time-reversal symmetry on energybands of crystals. Rhys. Rev. 52, 361 (1937).
6. C. Herring. Accidental degeneracy in the energy bands ofcrystals. Rhys. Rev. 52, 365 (1937).
7. E.A. Wood. The 80 diperiodic groups in three dimensions.Bell System Techn. J. 43, 541 (1964).
8. M.S. Dresselhaus, G. Dresselhaus, A. Jorio. Group Theory.Application to the Physics of Condensed Matter (Springer,2008).
9. G.L. Bir, G.E. Pikus. Symmetry and Strain-Induced Ef-fects in Semiconductors (Wiley, 1974).
10. D.S. Balchuk, M.M. Bilyi, V.P. Gryschuk, V.O. Gubanov,V.K. Kononov. Symmetry of vibrational modes, invarianceof energy states to time inversion, and Raman scatter-ing in 4H- and 6H-SiC crystals. 1. Classification of energystates in Brillouin zones. Ukr. Fiz. Zh. 41, 146 (1996) (inUkrainian).
11. E. Doni, G. Pastori Parravicini. Energy bands and opticalproperties of hexagonal boron nitride and graphite. NuovoCimento B 64, 117 (1969).
12. F. Bassani, G. Pastori Parravicini. Electronic States andOptical Transitions in Solids (Pergamon Press, 1975).
Received 22.08.19.Translated from Ukrainian by O.I. Voitenko
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