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Crystalline structure and properties of diamond-like materials
E. A. Belenkov1, M. M. Brzhezinskaya2, V. A. Greshnyakov1 Chelyabinsk State University, Bratiev Kashirinykh, 129, Chelyabinsk, 454001, Russia 2Helmholtz-Zentrum Berlin fur Materialien und Energie, Albert-Einstein-Strasse, 15, Berlin, 12489, Germany belenkov@csu.ru, maria.brzhezinskaya@helmholtz-berlin.de, greshnyakov@csu.ru
PACS 61.50.Ah, 81.05.ug DOI 10.17586/2220-8054-2017-8-1-127-136
Using density functional theory, the structures and properties of carbon diamond-like phases, all of whose atoms occupy equivalent crystallo-graphic positions, were studied. Structures of these phases were obtained by linking or superposing ofcarbon clusters. It was found that thirteen such phases can exist. The calculated structural characteristics and properties (density, cohesive energy, bandgap, bulk modulus, hardness, and x-ray diffraction pattern) of the diamond-like phases under study differ significantly from respective parameters of cubic diamond. Keywords: Diamond crystal, diamond-like carbon, fullerenes, modeling. Received: 18 September 2016 Revised: 21 November 2016
1. Introduction
Fullerene is a nano-allotropic carbon form consisting of 3-coordinated atoms [1-3] whose structure complies with the Euler's rule for convex polyhedrons. The fullerene surface is formed by carbon atoms located in the pentagon and hexagon vertices. Fullerenes can be practically used in medicine and nanophotonics, as well as in producing adsorbers, abrasive agents and construction materials [4,5]. Highly strong construction materials may be obtained by bulk compression of fullerene condensates (fullerites). Pioneering work in this field was performed by several research groups [6-9]. When pressure is about 1 GPa and temperature is 200° C, some atoms of fullerite initially consisting only of tri-coordinated carbon atoms begin transforming to the tetra-coordinated state [10]. This gives rise to inter-fullerene covalent bonds and initiates formation of a hybrid structure consisting of tri- and tetra-coordinated atoms. The increase in pressure results in an increase in the fraction of tetra-coordinated atoms; thus, when pressure exceeds 17 GPa, almost all the atoms transfer to the tetra-coordinated state, and phases of diamondlike structures arise [9,11-13]. Theoretical study of diamond-like phases that can be obtained from fullerenes and carbon clusters was performed in a number of works [14-18]. Theoretical analyses have shown the possibility of existence of six phases of this type: supercubane and tetrahedral diamond [14], "simple cubic fullerite" C24 [15], T-carbon [16], carbon sodalite [17], and face-centered cubic C12 phase (fcc-C12) [18]. Additional investigations should be performed to ascertain if other diamond-like phases can be obtained from carbon clusters. A schematic diagram for deriving diamond-like phases from various precursors was proposed in [19]. Earlier, this scheme was used to study diamond-like phases of the L-, T- and S-groups obtained from graphene layers, carbon nanotubes, and three dimensionally (3D) covalently bonded graphites, respectively [20, 21]. This paper presents calculations for the structures and properties of thirteen carbon diamond-like phases with crystallographically equivalent atomic positions simulated by either linking or superposing the last precursor group, namely, carbon clusters.
2. Calculation methods
The diamond-like carbon phases were obtained from carbon clusters consisting of tri-coordinated atoms via the technique described in [19]. Diamond-like phases arising from cluster nanoprecursors of the [0Dc,3] structural group are referred to as C-phases.
Since in this work we studied diamond-like phases with crystallographically equivalent atom positions, clusters with equivalent atom positions were considered as possible precursors for their formation. Such clusters are carbon molecules with carbon skeletons shaped as regular and semi-regular polyhedrons and prisms [19]. The number of regular and semi-regular clusters is limited to three and seven, respectively, while the number of prismatic clusters is infinite. Further decrease in the number of clusters is because any three dimensionally ordered structure of the diamond-like phase can have only 2-, 3-, 4- and 6-fold symmetry axes. Due to this restriction, the following clusters cannot be used as precursors: all 53m semiregular clusters (C20, two polymorphs of C6o, and C120) and prismatic clusters except for C6, C12, C16, and C24. In the two last prismatic clusters (C16 and C24), 8- and
12-fold symmetry axes (respectively) may degenerate into the 4- and 6-fold axes in case of changes in lengths of a half of the prism base bonds. Fig. 1 presents carbon clusters used to construct diamond-like phases.
Fig. 1. Fullerene-like clusters
The primary mechanism for modeling C-phase structures is cluster linking that leads to formation of bonds between pairs or groups of atoms of neighboring clusters. Phases formed as a result of cluster linking are referred to as CA-phases [22]. The other method for obtaining diamond-like C-phases consists of superposing equivalent faces of neighboring clusters with subsequent removal of a half of combined atoms. The neighboring clusters may be combined by either vertices, or edges, or faces. In cases where the edges or vertices are combined, the atom coordination number will be five or six, respectively. The atom coordination number in diamond-like phases should be four; this number occurs only when cluster faces are combined. To designate C-phases obtained by superposing carbon clusters, symbols "CB" are used.
To create 3D crystalline structures of diamond-like phases, each cluster should be linked or superposed with minimum three nearest clusters so as to form a structural motif similar to uninodal 3D lattices of the [3Dc, 3] structural group [19]. The cluster coordination number corresponding to close packing may be a maximum of twelve. However, not every type of 3D cluster packing can be used to form C-phases with equivalent atom positions. The possibility of constructing diamond-like phases is restricted by the number and shape of carbon cluster faces. Table 1 presents the results of analyzing all possible coordination numbers of clusters from which diamond-like phases may be derived.
Geometrical optimization of diamond-like phase crystalline structures was performed by using the density functional theory (DFT) method [23] realized in program code Quantum Espresso [24]. Energetic characteristics and densities of the C-phase electronic states were calculated in the Generalized Gradient Approximation (GGA) and Local Density Approximation (LDA) using the B3LYP [25] and Perdew-Zunger [26] exchange-correlation energy functionals respectively. The ion core effect was accounted for via the norm-conserving pseudopotentials. Calculation was performed by using Brillouin-zone k-point grids 12x12x12 calculated by the Monkhorst-Pack method [27]; the cut-off energy was 60 Rydberg.
Bulk moduli of the diamond-like phases were determined by the method presented in [28]. Atomic volumes and total energies necessary for calculating the bulk moduli were evaluated by the DFT-GGA method at the relative phase volume variation of up to 3.5 %. The phase hardness was calculated by the Li's method [29]. C-phase powder x-ray diffraction (XRD) patterns were simulated via the standard technique described in [30] assuming that Acu-a = 1.5405 A and using the unit cell parameters and atom coordinates calculated by the DFT-LDA method.
3. Results and Discussion
Theoretical analysis performed in the scope of this work showed the possibility of creating fifteen different diamond-like structures from carbon clusters with equivalent atom positions. Fourteen of these structures can be formed by cluster linking, while the remaining structure can be created by cluster superposition. Geometric
Table 1. Structural characteristics and coordinations of fullerene-like clusters necessary for obtaining the diamond-like phases
Cluster Ring parameter Point group Coordination number
Linking Superposition
C4 33 43m 4 -
Ce 31 4 to 6m2 3, 6 -
C8 43 m3m 4, 8 -
C12 4261 6/mmm 3, 6, 12 3
C12 3162 43m 4, 6, 12 4
Ci6 4281 8/mmm 4 4
C24 4162 m3m 4, 6, 8, 12 4, 6
C24 3182 m3m 3, 6, 8, 12 3, 8
C24 42121 12/mmm 6, 12 6
C48 416181 m3m 6, 8, 12 6, 8, 12
optimization of those structures by the DFT method proved the possible stable existence of thirteen diamond-like C-phases whose structures are shown in Fig. 2. Twelve CA-phases (Fig. 2(a-l)) were obtained by linking together carbon clusters depicted in Fig. 1. Clusters corresponding to each phase are presented in Table 2. Only one phase, i.e., CB, was formed by cluster superposition (Table 2). The symmetry of the final phases' crystal lattices did not change in the course of geometrical optimization of initial structures obtained from linked clusters. In the diamond-like phases, the shape of cluster-like structural units remained similar to that of initial clusters. Two C-phases appeared to be unstable: a highly symmetrical (I432) structure formed by linked C24 clusters in the bcc packing and a chiral (P6222) structure consisting of polymerized tetrahedral C4 clusters located in the phase SA2 lattice sites [26]. In the process of DFT optimization, lattice destruction of these phases was observed.
For thirteen C-phases, ring parameters (Wells' parameters), space groups of symmetry, unit cell parameters (a, c, and Z), and deformation parameters (Def and Str) were determined (Table 2). Analyzing the ring parameters of the diamond-like phases, we have found out that C-phase structures inevitably inherit three covalent bond rings that are characteristic of the clusters-precursors (see also Table 1). Three-membered rings were observed in four phases, four-membered rings - in eleven phases, 6-membered rings - in eight phases, 8-membered rings - in nine phases, and twelve-membered rings were observed in the structures of two C-phases. The C-phase unit cells belong to the following syngonies: cubic (CA1, CA3, CA4, CA6-CA9, CA12, CB), hexagonal (CA2, CA11), tetragonal (CA5), and trigonal (CA10) (see Table 2). Calculated unit-cell parameters agree rather well with the experimental parameters for cubic diamond [2], hexagonal diamond (lonsdaleite) [31], C8 [32], and simple cubic diamond polymorph [33].
Crystalline structure distortion of the diamond-like phases relative to the cubic diamond structure (LA1) is characterized by parameters Def [20] and Str [21]. The C-phase deformation parameter Def accounting for the deviation of the interatomic bond angles of a certain structure from the diamond angle of 109.47° varies from 81.1 (CA6) to 254° (CA1) (Table 2). The C-phase stress parameter Str characterizing deviations of interatomic bond lengths from the diamond bond length ranges from 0.02 (CA6) to 0.25 A (CA1) for the DFT-GGA method and from 0.01 (CA6) to 0.23 A (CA1) for the DFT-LDA method (Table 2).
The diamond-like phase densities were determined by using calculated equilibrium parameters of unit cells at zero pressure (Table 3). All the C-phase densities were shown to be lower than the diamond one from 18 to 65% for the CA6 and CA12 phases, respectively.
In addition, cohesive energies (Ecoh) were determined for the C-phases (their values are listed in Table 3). The cubic diamond (LA1) cohesive energies calculated by the GGA and LDA methods proved to exceed experimental values [34] by 0.49 and 1.59 eV/atom, respectively. Cohesive energies of the diamond-like C-phases proved to be lower than that of diamond by a value ranging from 5.3 (CA6) to 19.2 % (CA12). Phases CA3, CA6, and CB are the most stable among all the C-phases since their cohesive energies are maximal. A correlation between the deformation parameters and the cohesive energy is observed: Ecoh decreases linearly with increasing a combination of Str and Def (Fig. 3). The dependence presented in Fig. 3 is common for the diamond-like C-phases and diamond-like phases of the L-, T- and S-types previously studied [20-22]. As another evidence of stability of not-yet synthesized carbon C-phases, one can consider the fact that they contain structural units existing in
Table 2. Precursors and structural characteristics of diamond-like C-phases based on the GGA and LDA calculations (LDA calculations are given in brackets)
Phase Precursor Ring parameter Space group a, A c, A Z, at. Def° Str, A
c-diam. (LA1) Le p6mm 6e Fd3m 3.597 (3.557) 3.597 (3.557) 8 0.0 0.00
experiment 3.567° 3.567 - -
h-diam. (LA2) Le p6mm 6e P63/mmc 2.528 (2.499) 4.203 (4.157) 4 1.8 (2.6) 0.04 (0.03)
experiment 2.52 ° 4.14 - -
Cg (LA4) Le p6mm 6e Ia3 4.504 (4.448) 4.504 (4.448) 16 44.99 (46.03) 0.34 (0.313)
experiment 4.279 ° 4.279 - -
CA1 C4(33) 33123 Fd3m 7.588 (7.516) 7.588 (7.516) 32 254 (254) 0.25 (0.23)
CA2 Ce (3M2) 314283 P63/mmc 5.015 (4.949) 5.062 (5.031) 12 153 (152) 0.13 (0.12)
CA3 Cg(43) 4383 Im3m 4.895 (4.851) 4.895 (4.851) 16 106 (106) 0.19 (0.21)
CA4 C24 (4162) 436281 Pm3m 5.946 (5.891) 5.946 (5.891) 24 105 (105) 0.20 (0.21)
experiment 5.545 ° 5.545 - -
CA5 C16 (4281) 4383 I4/mmm 7.185 (7.124) 4.802 (4.730) 32 114 (114) 0.13 (0.13)
CA6 C24 (4162) 4264 Im3m 4.393 (4.352) 4.393 (4.352) 12 81.1 (81.1) 0.02 (0.01)
CA7 C48 (416181) 436182 Im3m 7.552 (7.477) 7.552 (7.477) 48 120 (120) 0.09 (0.09)
CA8 C48 (416181) 436182 Im3m 9.319 (9.228) 9.319 (9.228) 96 109 (109) 0.12 (0.13)
CA9 C24 (4162) 4362121 Fd3m 12.44 (12.32) 12.44 (12.32) 192 114 (114) 0.11 (0.12)
CA10 C12 (4261) 436182 R3m 6.843 (6.783) 7.371 (7.262) 36 109 (109) 0.18 (0.18)
CA11 C12 (4261) 436182 P63/mmc 6.831 (6.779) 4.966 (4.890) 24 110 (110) 0.15 (0.15)
CA12 C4(33) 3381122 Im3m 9.249 (9.157) 9.249 (9.157) 48 252 (252) 0.23 (0.22)
CB C48 (416181) 31416282 Fm3m 9.513 (9.420) 9.513 (9.420) 96 141 (141) 0.13 (0.12)
°Ref. [2]; bRef. [31]; cRef. [32]; dRef. [33]
Fig. 2. The diamond-like phase structure projection onto the (001)-plane: (a) CA1; (b) CA2; (c) CA3; (d) CA4; (e) CA5; (f) CA6; (g) CA7; (h) CA8; (i) CA9; (j) CA10; (k) CA11; (l) CA12; (m) CB
Table 3. Density, cohesive energy (Ecoh), bandgap (Eg), bulk modulus (B0) and hardness (H) of diamond-like C-phases calculated by the GGA and LDA methods (LDA calculations are given in brackets)
Phase Density, g/cm3 Ecoh, eV/atom Eg ,eV Bo, GPa H, GPa
c-diam. (exp.) 3.517° 7.37b 5.48b 443b 90c
LA1 3.428 (3.546) 7.86 (8.96) 5.44 (4.61) 445 86.6 (90.8)
LA2 3.429 (3.549) 7.83 (8.93) 5.03 (3.64) 445 86.5 (90.7)
LA4 3.492 (3.627) 7.11 (8.22) 2.70 (2.12) 410 86.0 (90.5)
CA1 1.461 (1.503) 6.43 (7.32) 2.41 (1.98) 177 36.5 (38.1)
CA2 2.171 (2.242) 6.92 (7.87) 1.90 (1.55) 250 54.4 (56.9)
CA3 2.720 (2.795) 7.15 (8.15) 3.95 (3.06) 346 67.6 (70.2)
CA4 2.276 (2.342) 7.06 (8.04) 2.91 (2.30) 285 55.9 (58.2)
CA5 2.575 (2.658) 6.97 (7.97) 2.66 (1.92) 316 63.8 (66.7)
CA6 2.824 (2.904) 7.44 (8.46) 3.96 (3.18) 351 70.9 (73.8)
CA7 2.223 (2.290) 6.86 (7.84) 3.32 (2.68) 270 54.6 (56.9)
CA8 2.366 (2.436) 6.96 (7.95) 3.11 (2.50) 275 58.3 (60.7)
CA9 1.987 (2.048) 6.82 (7.79) 3.69 (3.03) 239 48.3 (50.4)
CA10 2.402 (2.482) 7.02 (8.00) 4.37 (3.56) 273 59.3 (62.0)
CA11 2.386 (2.459) 6.98 (7.97) 4.18 (3.35) 280 58.8 (61.4)
CA12 1.210 (1.247) 6.35 (7.24) 2.14 (1.72) 141 29.6 (30.9)
CB 2.224 (2.290) 7.16 (8.14) 3.34 (2.70) 278 56.3 (58.6)
°Ref. [2]; bRef. [34]; cRef. [35]
synthesized saturated hydrocarbons. These phases are: (i) CA1 whose structure is based on fragments like the carbon tetrahedral skeleton of the tetrahedrane molecule [36]; (ii) CA2 containing carbon skeleton fragments similar to the carbon skeleton of experimentally obtained prismane [37]; (iii) CA3 and CA4 whose structural elements look like the cubane molecule carbon skeleton [38]; (iv) CA6 consisting of the four-membered carbon rings.
Figure 4 presents calculated electronic densities of states (DOS) of the cubic diamond and diamond-like phases derived from carbon clusters. From the DOS patterns, the bandgaps (Eg) of the phases under study were obtained (Table 3). The diamond bandgap calculated by the GGA and LDA methods differ from the experimental value [34] by 0.04 and 0.87 eV, respectively. Calculations have shown that the diamond-like phase bandgaps are narrower than that of LA1 by values varying from 20 (CA10) to 66 % (CA2), i.e., C-phases are either semiconductors or insulators. Semiconductor properties may be exhibited by carbon materials based on the CA2 and CA12 phases whose structures are formed by great numbers of covalently bonded prismatic or tetrahedral clusters, respectively. Other C-phases have wider bandgaps typical for insulators. We have determined that the bandgap decreases with increasing deformation parameter Str.
To assess the mechanical characteristics of the diamond-like phases, bulk moduli (at high compression: 0.965 < V/V0 < 1) and Knoop hardness (see Table 3) were calculated. Calculations of bulk modulus B0 and hardness H of the cubic diamond fit well the measurements [34,35]. The bulk moduli of all the C-phases are below the LA1 B0 by 21-68 %. Phases CA3, CA5, and CA6 are expected to be the least compressible among the C-phases since their bulk moduli are comparable with that of titanium nitride [39]. A linear decrease in B0 is observed when parameter Def increases. Theoretical hardness of the C-phases is also lower than the diamond hardness by 18 to 66 %. The densest phases CA3 and CA6 should exhibit high strength characteristics comparable with those of boron nitride [40]. Theoretical hardness of other C-phases is higher than that of silicon carbide or almost the same [40].
XRD patterns of the diamond-like phases were calculated based on structural data obtained by the LDA method; this approximation was chosen since it provides the best fit between the calculated and measured parameters of the cubic diamond and lonsdaleite unit cells (see Table 2). The obtained C-phase XRD patterns were compared with the experimental XRD patterns of cubic diamond [41], lonsdaleite [31], cubic diamond polymorph [33], hexagonal (2H) graphite [42], and also with those of fcc C60 [43] and hexagonal close-packed (hcp) C70 [44];
Fig. 3. Dependences of the cohesive energy (Ecoh) on the deformation parameters (Str and Def), obtained by the DFT-GGA (a) and DFT-LDA (b) methods
three main diffraction peaks are given for each in Table 4. The comparison has shown that XRD patterns of all the C-phases differ significantly from those of diamond, lonsdaleite, and cubic diamond polymorph. XRD patterns of theoretically studied diamond-like phases are comparable with those of carbon compounds consisting of 3-coordinated atoms. For instance, positions of the CA3-phase 100 % maximum and CA10-phase 9.7 % maximum have been shown to be close to the main 002-maximum of graphite (Ad — 0.05 A). The most intense maxima of phases CA1 and CA2 are located close to the 70 %-maximum 311 of fullerite C6o (Ad — 0.05 A). In the case of the CA7 diamond-like phase, the 100 % maximum almost coincides with the main 110 peak of fullerite C70 (Ad = 0.01 A). In addition, the 79 % maximum 112 of fullerite C70 may be overlapped by the 100 % maxima of phases CA8 and CA10 (Ad - 0.07 A). The main diffraction maxima of phases CA1, CA2, CA4, CA7-CA12, and CB are within the diffraction angle range (26) of 12.4 to 20.6 which makes much easier to identify these phases. The most intense peaks of diamond-like phases CA3, CA5, and CA6 are rather close (25.0-29.0 °) to the 100 % maximum of hexagonal graphite; however, it may be possible to identify them by secondary diffraction peaks.
Fig. 4. Electronic density of states of diamond-like phases (the maximum energy of valence band electrons is assumed to be zero): (a) LA1; (b) CA1; (c) CA2; (d) CA3; (e) CA4; (f) CA5; (g) CA6; (h) CA7; (i) CA8; (j) CA9; (k) CA10; (l) CA11; (m) CA12; (n) CB
Table 4. The most intense x-ray diffraction maxima of synthesized carbon compounds and theoretically studied diamond-like phases (d and I are the interplanar distance and relative intensity of the diffraction peak, respectively)
Phase di, A 11, % d2, A 12, % d3, A I3, %
diamond a 2.060 100 1.261 27 1.0754 16
lonsdaleite b 2.19 100 2.06 100 1.26 75
cubic phase c 2.770 100 2.467 100 3.208 50
2H graphite d 3.376 100 2.039 6 1.6811 4
fcc Ceo e 8.160 100 5.007 92 4.272 70
hcp C70 f 5.294 100 4.510 79 8.630 63
CA1 4.339 100 2.657 8.9 2.170 3.5
CA2 4.304 100 3.270 33.2 2.152 8.0
CA3 3.430 100 2.426 33.8 1.534 23.3
CA4 5.891 100 4.165 28.9 3.401 14.1
CA5 3.562 100 3.941 63.3 2.643 31.0
CA6 3.077 100 1.777 19.1 1.256 12.4
CA7 5.287 100 3.052 19.4 3.739 7.6
CA8 4.614 100 6.526 54.5 2.918 9.7
CA9 7.113 100 4.356 5.1 3.715 3.5
CA10 4.548 100 3.393 9.7 2.122 3.6
CA11 5.871 100 3.758 27.4 3.390 13.9
CA12 6.475 100 3.738 13.9 4.579 7.5
CB 4.710 100 5.439 27.9 2.840 14.4
aRef. [41]; bRef. [31]; cRef. [33]; dRef. [42]; eRef. [43]; fRef. [44]
4. Conclusions
In this paper, we have theoretically calculated structures and properties of carbon diamond-like phases derived from carbon clusters. Crystalline structures of twelve C-phases were modeled by cluster linking. The structure of one of the phases (CB) was formed by cluster superposition. Based on the DFT-GGA and DFT-LDA calculations, various structural characteristics, densities, cohesive energies, bandgaps, bulk moduli, and hardness have been determined for the C-phases. To make possible experimental identification of diamond-like phases, powder XRD patterns were simulated. We also revealed a correlation between the structural parameters and properties of the diamond-like phases derived from carbon clusters, namely: density, cohesive energy, bulk modulus and bandgap decrease with increasing deformation parameters characterizing the phase structure stress with respect to the cubic diamond.
Acknowledgements
The reported study was funded by RFBR according to the research project No. 16-33-00030 mol_a. References
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