ISSN 10637834, Physics of the Solid State, 2015, Vol. 57, No. 10, pp. 2126–2133. © Pleiades Publishing, Ltd., 2015. Original Russian Text © E.A. Belenkov, A.E. Kochengin, 2015, published in Fizika Tverdogo Tela, 2015, Vol. 57, No. 10, pp. 2071–2078.
GRAPHENES
Structure and Electronic Properties of Crystals Consisting of Graphene Layers L6, L4–8, L3–12, and L4–6–12 E. A. Belenkov* and A. E. Kochengin Chelyabinsk State University, ul. Brat’ev Kashirinykh 129, Chelyabinsk, 454001 Russia *email:
[email protected] Received March 31, 2015; in final form, April 14, 2015
Abstract—The structure and electronic properties of crystals consisting of graphene layers of four main poly morphic modifications, namely, L6, L4–8, L3–12, and L4–6–12, have been calculated in the framework of the density functional theory (DFT) using the generalized gradient approximation (GGA). The structural char acteristics of individual layers and their relative positions that correspond to the minimum energy of interlayer bonds in the crystals have been found from the calculations. The electron densities of states and band struc ture of graphene crystals have been calculated. It has been established that crystals consisting of the main polymorphic modifications of graphene should exhibit metallic properties. DOI: 10.1134/S1063783415100030
1. INTRODUCTION As a rule, graphene is referred to as a separate layer of carbon atoms with a hexagonal structure [1–3]. In conventional hexagonal graphene layers L6, each car bon atom is bound by covalent bonds with three neigh boring atoms (threefoldcoordinated state) so that the atoms form a network of regular hexagons whose ver tices are occupied by carbon atoms and sides are formed by carbon–carbon bonds. However, the exist ence of other polymorphic modifications of graphene is also possible [4–13]. Structural modifications of graphene include lay ered carbon compounds consisting only of atoms in threefoldcoordinated states (the sp2 hybridization of carbon atoms) [6, 14]. Among the main modifications of graphene are those in which all the atoms forming a flat layer are in crystallographically equivalent states. The theoretical analysis performed by Shubnikov and Delone proved that there are only four variants of fill ing the plane with a crystalline network consisting of equivalent threefoldcoordinated sites [15, 16]. Therefore, theoretically, there can exist four main polymorphic modifications of graphene, i.e., conven tional hexagonal graphene layers L6 and three types of graphene layers, namely, L4–8, L3–12, and L4–6–12, which consist of tetragons and octagons, triangles and dodecagons, and tetragons, hexagons, and dodeca gons, respectively (Fig. 1) [6, 14]. In addition to the main structural modifications of graphene, theoreti cally, there can exist an unlimited number of polymor phs in which atomic positions are crystallographically nonequivalent. Experimentally, even the main struc tural modifications of graphene, i.e., L4–8, L3–12, and L4–6–12, have not as yet been obtained. However, their
structure and properties were investigated theoreti cally in sufficient detail in several works, the review of which is given in [4]. The conventional hexagonal graphene monolayers L6 have not been found in nature. They can be revealed only as structural units of graphite crystals. It seems likely that other polymor phic modifications of graphene layers can first be syn thesized in the form of crystals as stacks of graphene layers of the corresponding modifications. The three dimensional structure of such crystals remains unclear. In this work, we calculated the structures of crystals of four main polymorphic modifications of graphene, their band structures, and electron densities of states. 2. METHODICAL PART The structure of the polymorphic modifications of the graphene crystals was calculated in two stages. In the first stage, we calculated the structure of graphene layers. Preliminary calculations of the geometrically optimized structure were performed for fragments of graphene layers of four main structural modifications, namely, L6, L4–8, L3–12, and L4–6–12, using the PM3 semiempirical quantum mechanics method (paramet ric method 3) [17, 18]. Fragments of graphene layers contained from 120 to 180 carbon atoms. Dangling carbon–carbon bonds at the edges of the layers were compensated by hydrogen atoms. As a result of the calculations using the PM3 method, we determined the unit cell parameters of the graphene layers and the coordinates of atoms in the unit cells [19]. These pre liminary results were used for subsequent calculations of the geometrically optimized structure of the graphene layers and the properties of threedimen
2126
STRUCTURE AND ELECTRONIC PROPERTIES
2127
(a)
(b)
R2 R1
R1
R2 b
a
R3
R3
b a
(c)
(d)
b
R1 b
R1
R3
R2
R3 R2 a
a
Fig. 1. Unit cells of graphene layers of four main structural modifications: (a) L6, (b) L4–8, (c) L3–12, and (d) L4–6–12. Designa tions: R1, R2, and R3 are the lengths of interatomic bonds, and a and b are the elementary translation vectors.
sional graphene crystals, which were performed with the Quantum ESPRESSO software package [20] using the density functional theory (DFT) method [21] in the generalized gradient approximation (GGA) [22]. In the calculations, we used a 16 × 16 × 16 kpoint mesh in the Brillouin zone and the cutoff energy of 70 Ry for the planewave basis set. The interatomic bond lengths (Ri, i = 1, 2, 3), bond angles (β12, β13, β23), and elementary translation vec tor lengths a and b were calculated as the parameters characterizing the structure of the graphene layers. The notation of the interatomic bond lengths in differ ent layers is given in the caption of Fig. 1. The obtained bond lengths and bond angles were used for the calcu lation of the deformation parameter Def and the stress parameter Str, which characterize the degree of defor mation of the structure of polymorphic modifications as compared to the structure of the hexagonal graphene layers. The parameter Def was determined as the sum of absolute values of the deviations of the angles βij in a particular graphene layer from 120°. The stress parameter Str was calculated as the sum of abso lute values of the difference between the interatomic bond lengths Li in the graphene layers L4–8, L3–12, PHYSICS OF THE SOLID STATE
Vol. 57
No. 10
and L4–6–12 and the C–C bond length in the graphene layer L6. In the second stage, we calculated the three dimensional structure of graphene crystals in the four main modifications. In these calculations, it was assumed that the crystals consist of graphene layers ordered in stacks so that the structure of an individual layer remains unchanged upon interlayer interactions and corresponds to the structure found in the DFT– GGA calculations of the first stage. The interaction between the layers in the graphene crystals is provided by van der Waals forces, which are not quite accurately calculated using the semiempirical and ab initio quan tum mechanics methods. Therefore, in the second stage of the calculations, we used the atom–atom potential method [23] approved in the simulation of the crystal structure of different carbon materials, such as graphite crystallites [24], carbon fibers [25], multi walled carbon nanotubes [26], carbyne crystals [27], and graphyne crystals [28]. The calculation of the threedimensional structure of the crystals consisted in finding the relative positions of graphene layers, which correspond to the minimum energy of the van der Waals bonds. The threedimensional structure of 2015
2128
BELENKOV, KOCHENGIN
the graphene crystals is determined by the relative positions of the adjacent layers, because the interac tion of layers at larger distances Ef can be ignored due to the small contribution from this interaction to the total bond energy E (Ef < E × 10–3). Therefore, we cal culated the relative positions of the adjacent pairs of the graphene layers. The energy of van der Waals bonds E was calculated as the sum of the pair interaction energies of all the atoms forming one layer with all atoms of the other layer: N
E =
N
∑ ∑ [ –AR
–6 ij
+ Be
–αR ij
],
i = 1j = 1
where Rij is the distance between the ith atom of the first layer and the jth atom of the second layer; N is the number of atoms in each layer; and A, B, and α are the coefficients determined from the experimentally mea sured energies of van der Waals bonds in different car bon compounds [23]. A comparison of different poly morphic carbon structures is possible in the calcula tions of the specific energy Eε per carbon atom: Eε = E/N. The threedimensional structure was calculated for four polymorphic modifications of graphene: L6, L4–8, L3–12, and L4–6–12. In these calculations, the graphene layers were considered as a set of rectangular unit cells. The rectangular unit cells with the elementary transla tion vectors a' and b' for the graphene layers were cho sen for reasons of convenience and simplicity of the calculations. The numbers of atoms in these unit cells of the graphene layers L6, L4–8, L3–12, and L4–6–12 are equal to 4, 8, 12, and 24, respectively. For the calcula tions of the specific energy of bonds, we calculated the energy of bonds of the unit cells in the graphene layers. The replacement of the calculations of the total bond energy of single crystals by the calculation of specific energies per unit cell or per atom is admissible, because these specific energies are equal to each other for any unit cell and for any atom of infinite graphene layers if the atoms in them are located in identical crystallographic positions. The total interaction energy E can be found as the product of the specific interaction energy of the unit cell by the number of unit cells. The total energy E will have a minimum value if the specific interaction energy for a single unit cell will also have a minimum value. Therefore, the calculation of the energy can be performed only for the interaction of N' atoms of the unit cell of the first layer with the second layer of infinite size. This calculation can be further simplified, because the van der Waals interaction between the atoms rapidly weakens with an increase in the interatomic distance. The estimated sizes of the second layer, for which the limited layer adequately simulates an infinite layer, demonstrate that, if the sizes of the second layer exceed 8.0 nm, the specific energy of bonds almost reaches the limiting value, so that a further increase in the layer size leads
to a general change in the interaction energy of less than 0.01%. In the calculations of the threedimensional struc ture of graphene crystals, first of all, we set the values of the displacement vector S, which determines the relative displacement of the layers: S = xi + yj, where i and j are the vectors of elementary translations along the x and y axes, respectively. For the layers L6, L4–8, and L3–12, the projection of the displacement vector along the x axis varied in the range from 0 to a' with a step a'/10, while the step of variations for the L4–6–12 layer was equal to a'/20. In this case, the projection of the displacement vector along the y axis varied from 0 to b' with a step b'/10. Thus, the interaction energies were calculated using 121 different values of the vector S for the layers L6, L4–8, and L3–12 and 231 values of the vector S for the layer L4–6–12. Then, for a given value of the displacement vector, we varied the distance d between the layers and found the interlayer distance d0, which corresponded to the minimum energy of the van der Waals bonds. Using the obtained energies, we constructed the dependences of the specific energy of bonds per atom on the displacement vector. From the constructed dependences, we found the values of the vectors for which the adjacent layers were located so that the energy of their interaction had a minimum value. For the obtained values of the vector S, we cal culated the refined interlayer distance dS, which corre sponded to the absolute minimum of the bond energy. 3. RESULTS Using the DFT–GGA calculations of the geomet rically optimized structure of the graphene layers, we found that carbon atoms in the studied modifications of graphene are located in crystallographically equiva lent positions. The numerical values of the bond lengths and bond angles measured in the geometrically optimized graphene layers are presented in Table 1. In the L6 layer, all the calculated bond lengths and bond angles are respectively equal to each other (R1 = R2 = R3; β12 = β13 = β23 = 120°). The numerical values of the calculated bond lengths in the L6 layer are equal to 1.436 Å, which is close enough to the experimental value of this parameter (1.42 Å) in hexagonal graphene layers. In the L4–8 layer, the bond lengths R1 and R3 are equal to 1.477 Å and exceed the bond length R2 = 1.388 Å. The bond angles are as follows: β12 = β23 = 135° and β13 = 90° (Table 1). In the L3–12 layer, the bond lengths R1 and R3 are equal to 1.448 Å and exceed the bond length R2 = 1.364 Å. The bond angles are as follows: β12 = β23 = 150° and β13 = 60° (Table 1). In the L4–6–12 layer, all bonds have different lengths: R1 = 1.484 Å, R2 = 1.376 Å, and R3 = 1.481 Å. In this case, the numerical values of the bond angles β13, β23, and β12 are different and equal to 90°, 120°, and 150°, respectively (Table 1). Different lengths of carbon
PHYSICS OF THE SOLID STATE
Vol. 57
No. 10
2015
STRUCTURE AND ELECTRONIC PROPERTIES Table 1. Structural parameters of the crystals consisting of graphene layers of four main polymorphic modifications Structural parameter
2129
Table 2. Coordinates of atoms in unit cells of the crystals consisting of graphene layers of the main structural modifi cations
Graphene layer L6
L4–8
L3–12
R1, Å R2, Å 1.436 1.477 1.448 R3, Å 1.388 1.364 β13, deg 90 60 β12, deg 120 135 150 β23, deg Es, eV/atom –0.046 –0.042 –0.033 S, a (2/3, 1/3) (1/2, 1/2) (2/3, 1/3) N, atom 4 8 12 a(b), Å 2.471 3.429 5.130 c, Å 6.713 6.679 6.603 γ, deg 120 90 120 3 1 2 Rng 6 48 31122 Etot, eV/atom –157.34 –156.78 –156.22 ΔEtot, 0 0.56 1.12 eV/atom Esub, eV/atom 7.78 7.22 6.66 ρ, kg/m3 2245.4 2029.9 1588.9 Def, deg 0 60 120 Str, Å 0 0.130 0.096
L4–6–12 1.484 1.481 1.376 90 150 120 –0.038 (1/2, 1/2) 24 6.713 6.687 120 1 4 61121 –156.65 0.69
Vol. 57
L6 L4–8
L3–12
L4–6–12
7.09 1832.5 60 0.153
bonds in the graphene layers L4–8, L3–12, and L4–6–12 are apparently caused by different orders of covalent bonds. In the hexagonal graphene layers, covalent bonds have the order of 1.33, and their calculated length is 1.436 Å. In the other polymorphic modifications of graphene, the second bond is significantly shorter (1.364–1.388 Å) most likely due to the higher bond order (~ 1.6). The ring parameters (Wells parameters Rng) of the polymorphic modifications of the graphene layers L4⎯8 and L3–12 do not account for hexagonal rings (Table 1). In the L4–6–12 layer, apart from the hexagonal ring, there are fourmembered and twelvemembered rings. In the hexagonal graphene, there are only sixmem bered rings. The smallest difference between the ring parameter and the value of Rng for the hexagonal graphene is observed for the L4–8 layer (six units). For the layers L4–6–12 and L3–12, the corresponding differ ences are larger and equal to 8 and 15 units, respec tively. The threedimensional structure of the graphene crystals was determined from the calculations per formed using the atom–atom potential method. The dependences of the energy of interlayer bonds on the relative displacement for pairs of adjacent graphene PHYSICS OF THE SOLID STATE
Layer No.
No. 10
1 2 1 2 3 4 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12
X
Y
Z
No.
X
Y
Z
0.333 0.667 0.5 0.807 0.5 0.193 0.147 0.427 0.427 0.573 0.573 0.853 0.126 0.126 0.324 0.550 0.550 0.324 0.450 0.676 0.874 0.874 0.676 0.450
0.667 0.333 0.193 0.5 0.807 0.5 0.573 0.853 0.573 0.427 0.147 0.427 0.450 0.676 0.874 0.874 0.676 0.450 0.324 0.550 0.550 0.324 0.126 0.126
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
3 4 5 6 7 8 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
0 0.333 0 0.307 0 0.693 0.480 0.760 0.760 0.907 0.907 0.187 0.626 0.626 0.824 0.050 0.050 0.824 0.950 0.176 0.374 0.374 0.176 0.950
0 0.667 0.693 0 0.307 0 0.240 0.520 0.240 0.093 0.814 0.093 0.950 0.176 0.374 0.374 0.176 0.950 0.824 0.050 0.050 0.824 0.626 0.626
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
layers are shown in Fig. 2. From these dependences, we obtained the vectors of relative displacements of the adjacent layers, which correspond to the minimum specific energy of interlayer bonds. For different graphene layers, the relative displacement vectors S expressed in fractions of elementary translation vec tors a are equal to (2/3, 1/3) for the L6 layer, (1/2, 1/2) for the L4–8 layer, (2/3, 1/3) for the L3–12 layer, and (1/2, 1/2) for the L4–6–12 layer (Table 1). The calcu lated relative displacement of the L6 layers is identical to that experimentally observed for graphite crystals. This suggests that the obtained values of the displace ment vectors for the other polymorphic modifications are also calculated accurately. The relative positions of pairs of the adjacent graphene layers, which corre spond to the minimal energy of interlayer bonds, are graphically shown in Fig. 3. The vectors of relative dis placements of the adjacent layers in the crystals are such that they can be packed in the form of the layer polytype 2H, for which the length of the elementary translation vector along the crystallographic c axis is equal to twice the interlayer distance (Table 1). The 2015
2130
BELENKOV, KOCHENGIN (a)
(b) E, J/mol −3800 −3850 −3900
E, J/mol −4350 −4375 −4400 0.2
0.4
0.4 0.2
0.2 0.1 0 y, n −0.1 m −0.2 −0.4
−0.2
0 m x, n
y, n
0.2
0 m
−0.2
−0.4
−0.4
0.4
0 m x, n
−0.2
(c)
(d)
E, J/mol −2800 −2850 −2900
E, J/mol −3500 −3525
0.8 0.4 y, n 0 m
−0.4 −0.8
0.4 0 0.2 −0.2 nm −0.4 x,
1.0 0.4 y, n 0 m −0.4
−1.0
−0.6
−0.2
x
0.2 , nm
0.6
Fig. 2. Change in the specific energy of bonds as a function of the relative displacement for pairs of adjacent graphene layers: (a) L6, (b) L4–8, (c) L3–12, and (d) L4–6–12.
(a)
(b)
(c)
(d)
Fig. 3. Relative displacements of adjacent layers, which correspond to the minimum energy of interatomic bonds, in crystals con sisting of graphene layers: (a) L6, (b) L4–8, (c) L3–12, and (d) L4–6–12. PHYSICS OF THE SOLID STATE
Vol. 57
No. 10
2015
STRUCTURE AND ELECTRONIC PROPERTIES 6
EF
(b)
5 DOS, eV −1
DOS, eV −1
EF
(a)
3
2131
2
1
4 3 2 1
0 −20 12
−10
0
10
EF
0 −20 20
20
−10
0
10
EF
(c)
20 (d)
DOS, eV −1
DOS, eV −1
15 8
4
10
5 0 −20
−10
0 E, eV
10
0 −20
20
−10
0 E, eV
10
20
Fig. 4. Electron density of states for crystals consisting of graphene layers: (a) L6, (b) L4–8, (c) L3–12, and (d) L4–6–12.
calculated specific energies of interlayer bonds ES for crystals of the graphene layers L6, L4–8, L3–12, and L4⎯6–12 are equal to –0.046, –0.042, –0.033, and ⎯0.038 eV/atom, respectively. The strongest interlayer bonds are observed in hexagonal graphene crystals, weaker bonds exist in the crystals consisting of the lay ers L4–8 and L4–6–12, and the L3–12 layers are bound by the weakest bonds (the bond energy of these layers are almost 30% less than the energy of bonds in graphite crystals consisting of the L6 layers). The calculated value of ES for the crystals consisting of the L6 layers is in good agreement with the experimental value of this parameter (–0.043 eV/atom) for the graphite powder [29]. The unit cells of the graphene layers L6, L4–8, L3–12, and L4–6–12 contain 2, 4, 6, and 12 atoms, respectively, whereas the number of atoms in the unit cells of the graphene crystals is two times larger (Table 1). The ele mentary translation vector lengths a and b for primi tive cells of the graphene layers are pairwise equal to each other (a = b). Their numerical values are pre sented in Table 1. The lengths of the elementary trans lation vectors of the graphene crystals along the crys tallographic c axis are equal to twice the distance between the adjacent graphene layers (from 6.603 to PHYSICS OF THE SOLID STATE
Vol. 57
No. 10
6.713 Å). The calculated parameter c = 6.713 Å for the hexagonal graphene crystals is in good agreement with the experimental value of 6.7079 Å for graphite [30]. The crystal lattices of the graphene layers L6, L3–12, and L4–6–12 belong to the hexagonal system, whereas the crystal lattice of the L4–8 layer corresponds to the tetragonal system. The numerical values of the atomic coordinates expressed in fractions of elementary translation vectors are listed in Table 2. The maximum deformation of the structure, as compared to the structure of the hexagonal graphene layer, is observed for the structural modification of the L3–12 layer: the deformation parameter Def for this layer is two times greater than that for the layers L4–8 and L4–6–12 (Table 1). The maximum stress parameter Str = 0.153 Å is observed for the L4–6–12 layer. Thus, among the polymorphic modifications of the layers L4–8, L3–12, and L4–6–12, the least deformed structure is observed for the L4–8 layer. The calculations of the total energy of the graphene crystals demonstrated that the L6 layer has the mini mum energy (Etot = –157.34 eV/atom). The other structural modifications are characterized by higher energies (Table 1). The smallest difference ΔEtot = 2015
2132
BELENKOV, KOCHENGIN (a)
4
2 EF
0
E, eV
E, eV
2
0
EF
−2
−2
−4
−4
Γ
M
L
A
Γ
K
H
Γ
A
(c)
4
R
Γ
X
M
X
(d)
2 E, eV
EF
0
EF 0
−2
−2
−4
−4
Γ
Z
4
2 E, eV
(b)
4
M
L
A
Γ
K
H
A
Γ
M
L
A
Γ
K
H
A
Fig. 5. Band structure of crystals consisting of graphene layers: (a) L6, (b) L4–8, (c) L3–12, and (d) L4–6–12.
0.56 eV/atom (as compared to the hexagonal graphene) is observed for the L4–8 layer. The calculated values of the total energy of the crystals and the energy of an isolated carbon atom were used to calculate the sublimation energy (Table 1). The maximum sublima tion energy Esub = 7.78 eV/atom is observed for the graphene crystals consisting of the L6 layers. The cal culated sublimation energy Esub is close to the experi mentally measured value (7.43 eV/atom) for graphite [30]. The sublimation energies of the graphene layers, which were calculated by other authors using the DFT method in different approximations, lie in the range from 6.95 to 8.96 eV/atom [31]. Our value of Esub also lies within this range. The maximum density of 2245.4 kg/m3 is observed for graphene crystals consist ing of the L6 layers (the experimental value for graphite is 2260 kg/m3 [32]). For the graphene crystals consist ing of the layers L4–8, L3–12, and L4–6–12, the densities are lower by 10, 29, and 19%, respectively (Table 1). The results of the DFT–GGA calculations of the electronic structure of the polymorphic modifications of the graphene crystals are presented in Figs. 4 and 5. For the hexagonal graphene layers L6, there is a touch ing of the valence band and the conduction bands at the characteristic point K of the Brillouin zone (Fig. 5a). This indicates metallic conductivity, the
correctness of the performed calculations, and the agreement with the experimental data on the conduct ing properties of graphite crystals. The overlap of the valence band and the conduction bands in the crystals consisting of the L4–8 layers occurs in the vicinity of the M point (Fig. 5b). For the crystals consisting of the layers L3–12, and L4–6–12, the overlap of the bands is observed in the vicinity of the points K and H (Figs. 5c, 5d). The calculations of the electron density of states indicate that, for all polymorphic modifications of graphene, there is no band gap near the Fermi energy (Fig. 4); i.e., the crystals consisting of the layers L6, L4–8, L3–12, and L4–6–12 should have metallic conduc tivity. 4. CONCLUSIONS Thus, in this work, we calculated the threedimen sional structure of crystals consisting of graphene lay ers of four main polymorphic modifications, namely, L6, L4–8, L3–12, and L4–6–12. The unit cell parameters of graphene layers and threedimensional crystals formed from these layers, as well as the coordinates of atoms in the unit cells, were found from the performed calculations. Also, we calculated the specific energies of interlayer bonds, sublimation energies, electron densities of states, and band structures of three
PHYSICS OF THE SOLID STATE
Vol. 57
No. 10
2015
STRUCTURE AND ELECTRONIC PROPERTIES
dimensional crystals of four main structural modifica tions of graphene. All polymorphs of graphene should possess metallic conductivity. The structures of the layers L4–8, L3–12, and L4–6–12 can be considered to be deformed as compared to the structure of the hexago nal graphene layer L6. It was found that there is a dependence of the sublimation energy of graphene crystals on the degree of deformation of their struc ture, which is characterized by the numerical values of the parameters Def and Str: the more severe is the deformation of the structure, the lower is the sublima tion energy. Thus, the least deformed structure and the maximum sublimation energy were revealed for the crystals consisting of the layers L4–8 (without regard for the polymorph L6). This polymorphic modification should apparently be the most stable under normal conditions. It is this modification that, first of all, should be obtained experimentally, although the sta bility of the L4–8 layer can be broken during avalanche formation of defects [33]. New polymorphs of graphene can be synthesized according to the mecha nism proposed in [34, 35], i.e., through the polymer ization of molecular compounds with the structure of a carbon skeleton similar to the structure of fragments of the layers L4–8, L3–12, and L4–6–12. REFERENCES 1. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science (Washington) 306 (5696), 666 (2004). 2. A. V. Eletskii, I. M. Iskandarova, A. A. Knizhnik, and D. N. Krasikov, Phys.—Usp. 54 (3), 227 (2011). 3. S. V. Morozov, K. S. Novoselov, and A. K. Geim, Phys.—Usp. 51 (7), 744 (2008). 4. A. L. Ivanovskii, Usp. Khim. 81 (7), 571 (2012). 5. A. N. Enyashin and A. L. Ivanovskii, Phys. Status Solidi B 248 (8), 1879 (2011). 6. E. A. Belenkov and V. A. Greshnyakov, Phys. Solid State 55 (8), 1754 (2013). 7. V. H. Crespi, L. X. Benedict, M. L. Cohen, and S. G. Louie, Phys. Rev. B: Condens. Matter 53 (20), R13303 (1996). 8. H. Terrones, M. Terrones, E. Hernández, N. Grobert, J. C. Charlier, and P. M. Ajayan, Phys. Rev. Lett. 84 (8), 1716 (2000). 9. X. Rocquefelte, G. M. Rignanese, V. Meunier, H. Ter rones, M. Terrones, and J. C. Charlier, Nano Lett. 4 (5), 805 (2004). 10. D. J. Appelhans, Z. Lin, and M. T. Lusk, Phys. Rev. B: Condens. Matter 82 (7), 073410 (2010). 11. S. Zhang, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe, and P. Jena, Proc. Natl. Acad. Sci. USA 112 (8), 2372 (2015).
PHYSICS OF THE SOLID STATE
Vol. 57
No. 10
2133
12. M. Maruyama and S. Okada, Appl. Phys. Express 6 (9), 095101 (2013). 13. H. Lu and S. D. Li, J. Mater. Chem. C 1, 3677 (2013). 14. E. A. Belenkov and V. A. Greshnyakov, New Carbon Mater. 28 (4), 273 (2013). 15. A. V. Shubnikov, Izv. Imp. Akad. Nauk, Ser. VI 10 (9), 755 (1916). 16. B. N. Delone, Izv. Akad. Nauk SSSR, Ser. Mat. 23 (3), 365 (1959). 17. J. J. P. Stewart, J. Comput. Chem. 10 (2), 209 (1989). 18. J. J. P. Stewart, J. Comput. Chem. 10 (2), 221 (1989). 19. A. E. Kochengin, T. E. Belenkova, V. M. Chernov, and E. A. Belenkov, Vestn. Chelyab. Gos. Univ., Fiz. 316 (25), 40 (2013). 20. X. P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi, R. Gebauer, et al., J. Phys.: Con dens. Matter 21 (39), 395502 (2009). 21. W. Koch and M. C. Holthausen, A Chemist’s Guide to Density Functional Theory (Wiley, Weinheim, 2002). 22. J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D. J. Singh, and C. Fiolhais, Phys. Rev. B: Condens. Matter 46 (11), 6671 (1992). 23. A. I. Kitaigorodsky, Molecular Crystals and Molecules (Nauka, Moscow, 1971; Academic, New York, 1973). 24. E. A. Belenkov, Inorg. Mater. 37 (9), 928 (2001). 25. E. A. Belenkov, Crystallogr. Rep. 44 (5), 749 (1999). 26. E. A. Belenkov, Izv. Chelyab. Nauchn. Tsentra Ural. Otd. Ross. Akad. Nauk, No. 1, 25 (2001). 27. E. A. Belenkov and V. V. Mavrinsky, Crystallogr. Rep. 53 (1), 83 (2008). 28. E. A. Belenkov and V. V. Mavrinskii, Izv. Chelyab. Nauchn. Tsentra Ural. Otd. Ross. Akad. Nauk, No. 2, 13 (2006). 29. L. A. Girifalco and R. A. Lad, J. Chem. Phys. 25 (4), 693 (1956). 30. D. R. Lide, CRC Handbook of Chemistry and Physics, 86th ed. (CRC Press, London, 2005). 31. G. Graziano, J. Klimes, F. FernandezAlonso, and A. Michaelides, J. Phys.: Condens. Matter 24 (42), 424216 (2012). 32. H. O. Pierson, Handbook of Carbon, Graphite, Dia mond, and Fullerenes: Properties, Processing, and Appli cations (Noyes, Park Ridge, Illinois, United States, 1993). 33. A. I. Podlivaev and L. A. Openov, Phys. Solid State 55 (12), 2592 (2013). 34. F. Diederich and Y. Rubin, Angew. Chem., Int. Ed. Engl. 31 (9), 1101 (1992). 35. F. Diederich, Nature (London) 369 (3), 199 (1994).
Translated by O. BorovikRomanova
2015
