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FLAPW investigation of the stability and equation of state of rectangulated carbon
PERGAMON
Solid State Communications 122 (2002) 473±477
www.elsevier.com/locate/ssc
FLAPW investigation of the stability and equation of state of rectangulated carbon John V. Badding*, Thomas J. Scheidemantel Department of Chemistry, Penn State University, 152 Davey, University Park, PA 16802, USA Received 23 October 2001; received in revised form 12 March 2002; accepted 1 April 2002 by A.K. Sood
Abstract The rectangulated carbon structure can be formed by buckling the graphene layers of graphite and linking them together with four-membered cyclobutane-like rings. Baughman et al. proposed rectangulated carbon as a candidate for the structure of the transparent phase of carbon that forms upon room temperature compression of graphite. Here we present a full-potential linear augmented plane wave (FLAPW) investigation of the stability and equation of state of rectangulated carbon. The total energy and equation of state of graphite and diamond are also calculated for comparison with rectangulated carbon. The local spin-density approximation (LSDA) and the generalized gradient approximation (GGA) give similar results for diamond and rectangulated carbon, but different results for graphite. The pressure estimated for the transition from graphite to rectangulated carbon using the LSDA is slightly higher than is observed for single crystal graphite. The energy differences between diamond and rectangulated carbon are in accord with earlier calculations. The agreement at 25 GPa between the calculated diffraction pattern for rectangulated carbon and the observed diffraction pattern for transparent carbon is not as good as the agreement at 0.1 MPa. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 71.15.Ap; 71.15.Mb; 71.15.Nc; 74.62.Fj Keywords: E. X-ray scattering; B. Crystal structure and symmetry; A. Insulators
1. Introduction The structure of the `transparent phase' of carbon that graphitizes at room temperature remains uncertain. When graphite is compressed to pressures in the range of 14± 18 GPa at ambient temperature, a light transparent phase of carbon is formed [1±11]. In this phase the sp 2 bonded graphene network of graphite is disrupted, presumably as the graphene layers buckle and bond together. Upon release of pressure, the transparent phase reverts back to graphite at 2.5 GPa [8], a behavior that is not found for diamond. The transparent phase can be quenched to ambient pressure at liquid nitrogen temperatures, but reverts to graphite upon warming to ambient temperature, indicating that there is a small activation barrier to reversion [12]. Its optical, electrical, and mechanical properties are consistent with the formation of a sp 3 bonded phase [1±11]. The crystallo* Corresponding author. Tel.: 11-814-777-3054; fax: 11-814865-3314. E-mail address: [email protected] (J.V. Badding).
graphic orientation of the graphite that forms from the transparent phase upon release of pressure is the same as the original graphite from which it was formed upon compression [1]. This suggests that some anisotropy is maintained in an sp 3 bonded form of carbon. Such anisotropy appears to be inconsistent with the presence of a three-dimensionally bonded tetrahedral diamond network [3]. It has been proposed that the transparent phase has the hexagonal diamond (Lonsdaleite) structure [1]. However, diffraction data indicate that the unique c-axis of this structure would be parallel to the graphene layers of graphite. Then one of three equivalent planes in the idealized hexagonal diamond structure must break apart during the reconversion to graphite [1,3,4,13]. It is not clear why only one of these sets of planes would consistently break apart to maintain the observed orientation between graphite and hexagonal diamond [3]. Novel forms of carbon have been of longstanding scienti®c interest [14] and it is important to understand the bonding con®guration of the transparent phase. A nondiamond, anisotropic sp 3 bonded form of carbon could exhibit especially interesting optical, mechanical, and electronic
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(02)00136-9
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Fig. 1. Rectangulated carbon structure optimized at zero pressure using the LSDA. Tetragonal space group I4/mmm. Lattice paraÊ , c 2.4853 A Ê . There is one free atomic posimeters: a 4.3295 A tion x, x, 0 with x 0.31955. The vertical four-membered rings link together the horizontal buckled graphite layers.
properties and could be of technological interest if it could be quenched to ambient conditions. The experimental data available to date are insuf®cient to completely determine the structure of the transparent phase. Thus far, for example, Raman scattering experiments have been hindered by ¯uorescence and interference from sp 2-bonded carbon (both the transparent phase and sp 2-bonded graphite coexist even after the formation of the transparent phase) [12]. There are also considerable experimental challenges in obtaining accurate X-ray diffraction data for the transparent phase. Because of the weak scattering power of carbon and the presence of preferred orientation, it is particularly dif®cult to obtain accurate powder diffraction intensities, which are important for structural re®nement. Yagi et al. have reported the most thorough X-ray diffraction study to date of the transition of graphite to the transparent phase [1]. They proposed a hexagonal diamond (Lonsdaleite) structure for the transparent phase, but noted that the reason why the transparent phase is unquenchable to ambient pressure remains unexplained. A tetragonal `rectangulated carbon' structure that has the intralayer connectivity of graphite, but is composed of buckled sp 3 carbon layers has been proposed for transparent carbon [3]. Like C60 polymers [15], rectangulated carbon contains four member cyclobutane-like rings (Fig. 1). These four member rings link together the buckled graphite layers and could form by [2 1 2] cycloaddition. The tetragonal c-axis is perpendicular to the four-membered rings. It has been suggested that the anisotropy of the rectangulated phase parallel to the original graphene planes could facilitate its reversion back to graphite upon release of pressure [3], although fully rectangulated carbon has two equivalent planes of symmetry parallel and perpendicular to the original graphene layers. Hybrid structures having both rectangulated layers and hexagonal diamond layers in various stacking sequences have also been proposed [3]. These structures are also anisotropic and have stabilities in
between that of hexagonal diamond and rectangulated carbon. For rectangulated carbon to be a viable candidate for the structure of the transparent phase, it must be energetically accessible from graphite in the pressure range between 2 and approximately 18 GPa. Here we investigate the energy and equation of state of rectangulated carbon by means of fullpotential linear augmented plane wave (FLAPW) calculations [16] within the local spin-density approximation (LSDA) [17] and the generalized gradient approximation (GGA) of Perdew et al. [18]. The energies and equations of state of graphite and diamond were also calculated by the same method for comparison with rectangulated carbon. The GGA gave results for the relative energies of graphite and diamond that do not compare well with experiment. Using the LSDA, we ®nd that the estimated transition pressure for the graphite to rectangulated carbon transition is close to the range of transition pressures observed experimentally. Using both the LSDA and GGA, our results for the differences in the cohesive energies between rectangulated carbon and diamond at ambient pressure are consistent with those obtained earlier by means of molecular mechanics calculations [3].
2. Method of calculations We used the WIEN97 code (version 10) running on a 7 CPU Linux Athlon cluster for the FLAPW calculations [16,19]. The RK-max parameter, which controls the number of LAPW basis functions and the level of convergence, was 8. An atomic sphere radius of 1.2 Bohr was used for all three structures. For graphite, p local orbitals [20] located above the Fermi energy were found to slightly improve the level of convergence. The convergence was checked carefully with respect to RKmax and the number of k points and was better than 1 mRyd. We used 72 irreducible k points for diamond, 148 k points for graphite and 159 k points for rectangulated carbon. The graphite c/a ratio was optimized at every volume. Rectangulated carbon has two structural degrees of freedom, a free atomic position (x, x, 0) and the c/a ratio, which were optimized at every volume. The Vinet equation of state was used to ®t the energy± volume curves [21±23]. The Vinet equation has been found to be the most accurate of several common equations of state over a wide range of compression. The correlations among V0, B0, and B 0 0, the zero pressure volume, bulk modulus and bulk modulus pressure derivative, respectively, are also much lower for the Vinet equation [21]. Zero point energies of 17.5 and 16 kJ/mol were added to the energies of diamond and graphite, respectively [24]. Because the zero point energy of rectangulated carbon is unknown, we used the value for diamond, which is structurally more similar to rectangulated carbon than graphite.
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Table 1 Zero pressure volume (V0), bulk modulus (B0) and bulk modulus pressure derivative (B 0 0), cohesive energy (Ec) and difference in the cohesive energy from diamond (D diamond). The diamond reference energy used for D diamond is the energy determined by the same calculational method (e.g. for graphite GGA D diamond is difference in diamond GGA and graphite GGA values). B 0 0 for diamond was ®xed to 4 for comparison to experiment. B 0 0 was also ®xed to 4 for rectangulated carbon. Graphite experimental data is from Refs. [24,32]. Diamond experimental data is from Refs. [24,33]. Cohesive energies are rounded to the nearest kJ/mol
Graphite LSDA a Graphite GGA a Graphite experiment b Diamond LSDA a Diamond GGA a Diamond experiment c Rectangulated carbon LSDA a Rectangulated carbon GGA a a b c
Ê 3/mol) V0 (A
B0 (GPa)
B00
Ec (kJ/mol)
D diamond (kJ/mol)
8.62 9.60 8.78 5.51 5.69 5.67 5.82 6.02
32.8 25.3 33.8 458 433 446 425 397
9.32 7.81 8.9 4 (Fixed) 4 (Fixed) 4 (Fixed) 4 (Fixed) 4 (Fixed)
845 754 711 845 740 709 824 721
0 2 14 22 0 0 0 21 19
Vinet equation of state. Murnaghan equation of state. Birch±Murnaghan equation of state.
3. Results and discussion There are two different types of carbon±carbon bonds within the rectangulated carbon structure (Fig. 1): those Ê long within the four-membered rings, which are 1.514 A and the zigzag bonds connecting the four-membered rings, Ê when optimized at ambient pressure with which are 1.46 A the LSDA. These zigzag bonds and the horizontal bonds of the four-membered rings in Fig. 1 form the buckled graphite Ê bond layers. These bond lengths are shorter than the 1.541 A Ê distance found in diamond, but longer than the 1.418 A distance found in graphite. In addition to the 908 bond angles in the four member rings, there are bond angles of 111.1 and 113.68. The strain associated with these angles destabilizes
Fig. 2. Cohesive energies of graphite, diamond, and rectangulated carbon using the LSDA and the GGA. The curved lines are ®ts to the Vinet equation of state.
the rectangulated carbon structure relative to diamond and graphite. The nature of the bonding in layered graphite and three dimensionally bonded structures such as cubic diamond or rectangulated carbon is very different and comparison of the energies and equations of state of these structures provides a challenging test for theory. The equations of state calculated within the LSDA for graphite and diamond compare well with experiment (Table 1, Fig. 2). The LSDA value for the ambient pressure volume is slightly less than experiment. An earlier FLAPW investigation of graphite overestimated the value of the bulk modulus by approximately 47%, but gave a value for the ambient pressure volume that was slightly too large [25]. Other calculations, such as a more recent all-electron, full potential linear combinations of Gaussian-type orbitals-®tting function technique [26], have given values for the equation of state parameters that are closer to experiment. The cohesive energies obtained for both graphite and diamond are too large by more than 100 kJ/mol, which is typical for LSDA calculations of the energies of carbon structures [25,27,28]. The difference in the energies of graphite and diamond, for which experiment gives a value of 2 kJ/mol [24], is close to 0 when the LSDA is used (Table 1). This agreement with experiment may be fortuitous in view of the tendency for the LSDA to overestimate the binding in the covalent graphite planes and underestimate the weak van der Waals interaction between the layers [26]. When the GGA of Perdew et al. [18] is used (Fig. 2), the equation of state obtained for graphite is quite different from the experimental equation of state (Table 1). Although the absolute values of the cohesive energies for diamond (740 kJ/mol) and graphite (754 kJ/mol) are closer to experiment (Table 1), they differ by 14 kJ/mol, much more than the difference in the experimental values. The GGA appears to be less accurate for predicting equations of state and
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J.V. Badding, T.J. Scheidemantel / Solid State Communications 122 (2002) 473±477
Table 2 Calculated diffraction pattern for rectangulated carbon at 25 GPa. Ê , c 2.4454 A Ê . Miller indices, Lattice parameters: a 4.2502 A calculated intensities and calculated interplanar spacings for rectangulated carbon and observed interplanar spacings for the transparent phase at 25 GPa [1] are shown hkl
I
dcalc
dobs
110 020 011 220 121 310 301 002 112 040 231
21 72 62 7 12 9 34 9 1 0.5 19
3.005 2.125 2.120 1.503 1.501 1.344 1.226 1.227 1.133 1.063 1.062
, 2.75 2.139 2.055
1.234
1.057
energy differences for some layered structures that have van der Waals bonding, such as graphite and MoS2 [29]. The GGA tends to bind less than the LSDA and the V0 value for graphite is much too large (Table 1). The differences between the energies of diamond and rectangulated carbon calculated within the LSDA and the GGA are 21 and 19 kJ/mol, respectively. Thus, in contrast to the results for the graphite±diamond energy difference, both methods give almost the same value for the stability of rectangulated carbon relative to diamond. Unlike graphite, neither of these structures has van der Waals bonding, which is probably why both the LSDA and the GGA give similar results. Molecular mechanics and ab initio molecular dynamics calculations [3] give values for the energetic differences between diamond and rectangulated carbon of 24 and 23 kJ/mol, respectively, in good agreement with the LSDA and GGA results. Rectangulated carbon is a low energy form of carbon, thermodynamically more stable than some known allotropes, such as C60 [30]. The bulk modulus calculated for rectangulated carbon using the LSDA is 425 GPa (Table 1), which compares with a value of 403 GPa determined for rectangulated carbon by means of molecular mechanics [3]. The LSDA value for the bulk modulus of rectangulated carbon is slightly less than the LSDA value for diamond, as expected from the structural differences between these two phases. By assuming the hexagonal diamond structure for the transparent phase, a value for the bulk modulus of 425 ^ 25 GPa was determined from X-ray diffraction data [1]. Thus the calculated bulk modulus for rectangulated carbon is close to the experimental bulk modulus. The transition pressure for the conversion of graphite to rectangulated carbon can be estimated from the common tangent between the equations of state for graphite and rectangulated carbon. The transition from graphite to rectangulated carbon becomes possible once the enthalpy for
rectangulated carbon becomes less than or equal to that of graphite. The enthalpy is the Legendre transform of the energy with respect to pressure [31]. Therefore, the slope and intercept of the tangent lines to the energy vs. volume curve give the pressure and enthalpy, respectively. The transition pressure estimated from the tangent to the E±V curves calculated with the LSDA is 20 GPa (Fig. 2). The experimental values for the transition from graphite to the transparent phase upon compression range from 14 to 23 GPa, depending on how hydrostatic the stress conditions are and the type of graphite used as a starting material. When single crystal graphite is used, the transition pressure ranges from 14 to 18 GPa [1]. The degree to which kinetic factors control the transition remains uncertain, but the calculated transition pressure is close to the range of pressures observed experimentally. The diffraction pattern calculated for rectangulated carbon using the structural parameters optimized at a volume corresponding to 25 GPa is shown in Table 2, together with the experimental diffraction data at 25 GPa of Yagi et al. [1]. Previously, the comparison between the calculated and observed (or extrapolated) diffraction patterns was made at ambient pressure. In contrast to the ambient pressure diffraction patterns [3], the interplanar spacing at 25 GPa for the rectangulated carbon (110) Ê ) is considerably different from diffraction line (3.005 A Ê ). This experimental that observed experimentally (,2.75 A diffraction line has been attributed to graphite [1,4]. However, the absence of a diffraction line cannot be used to rule out a crystal structure. The largest remaining discrepancy is in the position of the (011) line (Table 2). This line shifts to smaller interplanar spacings when hexagonal diamond-like layers are incorporated into the rectangulated carbon structure [3]. Such mixed structures are more stable than fully rectangulated carbon and could be expected to form at lower pressures. At 25 GPa the predicted diffraction pattern for fully rectangulated carbon does not match as well as at ambient pressure [3]. Experimental diffraction data collected at yet higher pressures might reveal enough difference between calculated and experimental diffraction patterns to con®rm or rule out rectangulated carbon or a rectangulated carbon/hexagonal diamond polytype as the structure for the transparent phase of carbon. The c/a ratio for rectangulated carbon shifts according to the equation c=a 0:60325 2 0:0050481V; where the volume V is in cubic angstroms (Table 3). Table 3 Volume, atomic position x, and c/a ratio for rectangulated carbon at several pressures Pressure (GPa)
Ê 3) Volume (A
x
c/a
8.4 31.2 47.8 73.3
5.710 5.456 5.300 5.096
0.31947 0.31934 0.31926 0.31916
0.5744 0.5757 0.5765 0.5775
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Acknowledgements We thank the NSF Division of Materials Research for ®nancial report and J.O. Sofo for valuable discussions. References [1] T. Yagi, W. Utsumi, M. Yamakata, T. Kikegawa, O. Shimomura, Phys. Rev. B 46 (1992) 6031. [2] R.B. Aust, H.G. Drickamer, Science 140 (1963) 817. [3] R.H. Baughman, A.Y. Liu, C. Cui, P.J. Schields, Synth. Met. 86 (1997) 2371. [4] F.P. Bundy, J.S. Kasper, J. Chem. Phys., 46, 3437. [5] C. Frondel, U.B. Marvin, Nature 214 (1967) 587. [6] A.F. Goncharov, I.N. Makarenko, S.M. Stishov, Sov. Phys. JETP 69 (1989) 380. [7] R.E. Hanneman, H.M. Strong, F.P. Bundy, Science 155 (1967) 995. [8] W. Utsumi, T. Yagi, Science 252 (1991) 1542. [9] H.K. Mao, J.F. Shu, J.Z. Hu, R.J. Hemley, EOS Trans. Am. Geophys. Union 71 (1990) 1667. [10] K.J. Takano, H. Harashima, M. Wakatsuki, Jpn. J. Appl. Phys. 30 (1991) L860. [11] M. Han¯and, K. Syassen, R. Sonnenschein, Phys. Rev. B 40 (1989) 1951. [12] E.D. Miller, D.C. Nesting, J.V. Badding, Chem. Mater. 9 (1997) 18. [13] K. Lonsdale, Am. Mineral. 56 (1971) 333. [14] K.M. Merz, R. Hoffmann, A. Balaban, J. Am. Chem. Soc. 109 (1987) 6742. [15] P.W. Stephens, G. Bortel, G. Faigel, M. Tegze, A. Janossy, S. Pekker, G. Oszlanyi, L. Forro, Nature 370 (1994) 636.
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[16] D. Singh, Plane Waves, Psuedopotentials, and the LAPW Method, Kluwer, New York, 1994. [17] J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [18] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [19] P. Blaha, K. Schwarz, J. Luitz, in: K. Schwarz (Ed.), WIEN97, A Full Potential Linearized Augmented Plane Wave Package for Calculating Crystal Properties, Techn. Univ. Wien, Vienna, 1999 ISBN 3-9501031-0-4. Updated version of P. Blaha, K. Schwarz, P. Sorantin, S.B. Trickey, Comp. Phys. Commun. 59 (1990) 399. [20] D. Singh, Phys. Rev. B 43 (1991) 6388. [21] R.E. Cohen, O. Gulseren, R.J. Hemley, Am. Mineral. 85 (2000) 338. [22] P. Vinet, J.H. Rose, J. Ferrante, J.R. Smith, J. Phys.: Condens. Matter 1 (1989) 1941. [23] P. Vinet, J. Ferrante, J.H. Rose, J.R. Smith, J. Geophys. Res.ÐSolid Earth Planets 92 (1987) 9319. [24] M.T. Yin, M.L. Cohen, Phys. Rev. B 29 (1984) 6996. [25] H.J.F. Jansen, A.J. Freeman, Phys. Rev. B 35 (1987) 8207. [26] J.C. Boettger, Phys. Rev. B 55 (1997) 11202. [27] M. Weinert, E. Wimmer, A.J. Freeman, Phys. Rev. B 26 (1982) 4571. [28] J.R. Chelikowsky, S.G. Louie, Phys. Rev. B 29 (1984) 3470. [29] J.O. Sofo, personal communication. [30] J. Cioslowski, N. Rao, D. Moncrieff, J. Am. Chem. Soc. 122 (2000) 8265. [31] H.B. Callen, Thermodynamics and an Introduction to Thermostatistics, Wiley, New York, 1985. [32] M. Han¯and, H. Beister, K. Syassen, Phys. Rev. B 39 (1989) 12598. [33] P. Gillet, G. Fiquet, I. Daniel, B. Reynard, M. Han¯and, Phys. Rev. B 60 (1999) 14660.
Solid State Communications 122 (2002) 473±477
www.elsevier.com/locate/ssc
FLAPW investigation of the stability and equation of state of rectangulated carbon John V. Badding*, Thomas J. Scheidemantel Department of Chemistry, Penn State University, 152 Davey, University Park, PA 16802, USA Received 23 October 2001; received in revised form 12 March 2002; accepted 1 April 2002 by A.K. Sood
Abstract The rectangulated carbon structure can be formed by buckling the graphene layers of graphite and linking them together with four-membered cyclobutane-like rings. Baughman et al. proposed rectangulated carbon as a candidate for the structure of the transparent phase of carbon that forms upon room temperature compression of graphite. Here we present a full-potential linear augmented plane wave (FLAPW) investigation of the stability and equation of state of rectangulated carbon. The total energy and equation of state of graphite and diamond are also calculated for comparison with rectangulated carbon. The local spin-density approximation (LSDA) and the generalized gradient approximation (GGA) give similar results for diamond and rectangulated carbon, but different results for graphite. The pressure estimated for the transition from graphite to rectangulated carbon using the LSDA is slightly higher than is observed for single crystal graphite. The energy differences between diamond and rectangulated carbon are in accord with earlier calculations. The agreement at 25 GPa between the calculated diffraction pattern for rectangulated carbon and the observed diffraction pattern for transparent carbon is not as good as the agreement at 0.1 MPa. q 2002 Elsevier Science Ltd. All rights reserved. PACS: 71.15.Ap; 71.15.Mb; 71.15.Nc; 74.62.Fj Keywords: E. X-ray scattering; B. Crystal structure and symmetry; A. Insulators
1. Introduction The structure of the `transparent phase' of carbon that graphitizes at room temperature remains uncertain. When graphite is compressed to pressures in the range of 14± 18 GPa at ambient temperature, a light transparent phase of carbon is formed [1±11]. In this phase the sp 2 bonded graphene network of graphite is disrupted, presumably as the graphene layers buckle and bond together. Upon release of pressure, the transparent phase reverts back to graphite at 2.5 GPa [8], a behavior that is not found for diamond. The transparent phase can be quenched to ambient pressure at liquid nitrogen temperatures, but reverts to graphite upon warming to ambient temperature, indicating that there is a small activation barrier to reversion [12]. Its optical, electrical, and mechanical properties are consistent with the formation of a sp 3 bonded phase [1±11]. The crystallo* Corresponding author. Tel.: 11-814-777-3054; fax: 11-814865-3314. E-mail address: [email protected] (J.V. Badding).
graphic orientation of the graphite that forms from the transparent phase upon release of pressure is the same as the original graphite from which it was formed upon compression [1]. This suggests that some anisotropy is maintained in an sp 3 bonded form of carbon. Such anisotropy appears to be inconsistent with the presence of a three-dimensionally bonded tetrahedral diamond network [3]. It has been proposed that the transparent phase has the hexagonal diamond (Lonsdaleite) structure [1]. However, diffraction data indicate that the unique c-axis of this structure would be parallel to the graphene layers of graphite. Then one of three equivalent planes in the idealized hexagonal diamond structure must break apart during the reconversion to graphite [1,3,4,13]. It is not clear why only one of these sets of planes would consistently break apart to maintain the observed orientation between graphite and hexagonal diamond [3]. Novel forms of carbon have been of longstanding scienti®c interest [14] and it is important to understand the bonding con®guration of the transparent phase. A nondiamond, anisotropic sp 3 bonded form of carbon could exhibit especially interesting optical, mechanical, and electronic
0038-1098/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(02)00136-9
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Fig. 1. Rectangulated carbon structure optimized at zero pressure using the LSDA. Tetragonal space group I4/mmm. Lattice paraÊ , c 2.4853 A Ê . There is one free atomic posimeters: a 4.3295 A tion x, x, 0 with x 0.31955. The vertical four-membered rings link together the horizontal buckled graphite layers.
properties and could be of technological interest if it could be quenched to ambient conditions. The experimental data available to date are insuf®cient to completely determine the structure of the transparent phase. Thus far, for example, Raman scattering experiments have been hindered by ¯uorescence and interference from sp 2-bonded carbon (both the transparent phase and sp 2-bonded graphite coexist even after the formation of the transparent phase) [12]. There are also considerable experimental challenges in obtaining accurate X-ray diffraction data for the transparent phase. Because of the weak scattering power of carbon and the presence of preferred orientation, it is particularly dif®cult to obtain accurate powder diffraction intensities, which are important for structural re®nement. Yagi et al. have reported the most thorough X-ray diffraction study to date of the transition of graphite to the transparent phase [1]. They proposed a hexagonal diamond (Lonsdaleite) structure for the transparent phase, but noted that the reason why the transparent phase is unquenchable to ambient pressure remains unexplained. A tetragonal `rectangulated carbon' structure that has the intralayer connectivity of graphite, but is composed of buckled sp 3 carbon layers has been proposed for transparent carbon [3]. Like C60 polymers [15], rectangulated carbon contains four member cyclobutane-like rings (Fig. 1). These four member rings link together the buckled graphite layers and could form by [2 1 2] cycloaddition. The tetragonal c-axis is perpendicular to the four-membered rings. It has been suggested that the anisotropy of the rectangulated phase parallel to the original graphene planes could facilitate its reversion back to graphite upon release of pressure [3], although fully rectangulated carbon has two equivalent planes of symmetry parallel and perpendicular to the original graphene layers. Hybrid structures having both rectangulated layers and hexagonal diamond layers in various stacking sequences have also been proposed [3]. These structures are also anisotropic and have stabilities in
between that of hexagonal diamond and rectangulated carbon. For rectangulated carbon to be a viable candidate for the structure of the transparent phase, it must be energetically accessible from graphite in the pressure range between 2 and approximately 18 GPa. Here we investigate the energy and equation of state of rectangulated carbon by means of fullpotential linear augmented plane wave (FLAPW) calculations [16] within the local spin-density approximation (LSDA) [17] and the generalized gradient approximation (GGA) of Perdew et al. [18]. The energies and equations of state of graphite and diamond were also calculated by the same method for comparison with rectangulated carbon. The GGA gave results for the relative energies of graphite and diamond that do not compare well with experiment. Using the LSDA, we ®nd that the estimated transition pressure for the graphite to rectangulated carbon transition is close to the range of transition pressures observed experimentally. Using both the LSDA and GGA, our results for the differences in the cohesive energies between rectangulated carbon and diamond at ambient pressure are consistent with those obtained earlier by means of molecular mechanics calculations [3].
2. Method of calculations We used the WIEN97 code (version 10) running on a 7 CPU Linux Athlon cluster for the FLAPW calculations [16,19]. The RK-max parameter, which controls the number of LAPW basis functions and the level of convergence, was 8. An atomic sphere radius of 1.2 Bohr was used for all three structures. For graphite, p local orbitals [20] located above the Fermi energy were found to slightly improve the level of convergence. The convergence was checked carefully with respect to RKmax and the number of k points and was better than 1 mRyd. We used 72 irreducible k points for diamond, 148 k points for graphite and 159 k points for rectangulated carbon. The graphite c/a ratio was optimized at every volume. Rectangulated carbon has two structural degrees of freedom, a free atomic position (x, x, 0) and the c/a ratio, which were optimized at every volume. The Vinet equation of state was used to ®t the energy± volume curves [21±23]. The Vinet equation has been found to be the most accurate of several common equations of state over a wide range of compression. The correlations among V0, B0, and B 0 0, the zero pressure volume, bulk modulus and bulk modulus pressure derivative, respectively, are also much lower for the Vinet equation [21]. Zero point energies of 17.5 and 16 kJ/mol were added to the energies of diamond and graphite, respectively [24]. Because the zero point energy of rectangulated carbon is unknown, we used the value for diamond, which is structurally more similar to rectangulated carbon than graphite.
J.V. Badding, T.J. Scheidemantel / Solid State Communications 122 (2002) 473±477
475
Table 1 Zero pressure volume (V0), bulk modulus (B0) and bulk modulus pressure derivative (B 0 0), cohesive energy (Ec) and difference in the cohesive energy from diamond (D diamond). The diamond reference energy used for D diamond is the energy determined by the same calculational method (e.g. for graphite GGA D diamond is difference in diamond GGA and graphite GGA values). B 0 0 for diamond was ®xed to 4 for comparison to experiment. B 0 0 was also ®xed to 4 for rectangulated carbon. Graphite experimental data is from Refs. [24,32]. Diamond experimental data is from Refs. [24,33]. Cohesive energies are rounded to the nearest kJ/mol
Graphite LSDA a Graphite GGA a Graphite experiment b Diamond LSDA a Diamond GGA a Diamond experiment c Rectangulated carbon LSDA a Rectangulated carbon GGA a a b c
Ê 3/mol) V0 (A
B0 (GPa)
B00
Ec (kJ/mol)
D diamond (kJ/mol)
8.62 9.60 8.78 5.51 5.69 5.67 5.82 6.02
32.8 25.3 33.8 458 433 446 425 397
9.32 7.81 8.9 4 (Fixed) 4 (Fixed) 4 (Fixed) 4 (Fixed) 4 (Fixed)
845 754 711 845 740 709 824 721
0 2 14 22 0 0 0 21 19
Vinet equation of state. Murnaghan equation of state. Birch±Murnaghan equation of state.
3. Results and discussion There are two different types of carbon±carbon bonds within the rectangulated carbon structure (Fig. 1): those Ê long within the four-membered rings, which are 1.514 A and the zigzag bonds connecting the four-membered rings, Ê when optimized at ambient pressure with which are 1.46 A the LSDA. These zigzag bonds and the horizontal bonds of the four-membered rings in Fig. 1 form the buckled graphite Ê bond layers. These bond lengths are shorter than the 1.541 A Ê distance found in diamond, but longer than the 1.418 A distance found in graphite. In addition to the 908 bond angles in the four member rings, there are bond angles of 111.1 and 113.68. The strain associated with these angles destabilizes
Fig. 2. Cohesive energies of graphite, diamond, and rectangulated carbon using the LSDA and the GGA. The curved lines are ®ts to the Vinet equation of state.
the rectangulated carbon structure relative to diamond and graphite. The nature of the bonding in layered graphite and three dimensionally bonded structures such as cubic diamond or rectangulated carbon is very different and comparison of the energies and equations of state of these structures provides a challenging test for theory. The equations of state calculated within the LSDA for graphite and diamond compare well with experiment (Table 1, Fig. 2). The LSDA value for the ambient pressure volume is slightly less than experiment. An earlier FLAPW investigation of graphite overestimated the value of the bulk modulus by approximately 47%, but gave a value for the ambient pressure volume that was slightly too large [25]. Other calculations, such as a more recent all-electron, full potential linear combinations of Gaussian-type orbitals-®tting function technique [26], have given values for the equation of state parameters that are closer to experiment. The cohesive energies obtained for both graphite and diamond are too large by more than 100 kJ/mol, which is typical for LSDA calculations of the energies of carbon structures [25,27,28]. The difference in the energies of graphite and diamond, for which experiment gives a value of 2 kJ/mol [24], is close to 0 when the LSDA is used (Table 1). This agreement with experiment may be fortuitous in view of the tendency for the LSDA to overestimate the binding in the covalent graphite planes and underestimate the weak van der Waals interaction between the layers [26]. When the GGA of Perdew et al. [18] is used (Fig. 2), the equation of state obtained for graphite is quite different from the experimental equation of state (Table 1). Although the absolute values of the cohesive energies for diamond (740 kJ/mol) and graphite (754 kJ/mol) are closer to experiment (Table 1), they differ by 14 kJ/mol, much more than the difference in the experimental values. The GGA appears to be less accurate for predicting equations of state and
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Table 2 Calculated diffraction pattern for rectangulated carbon at 25 GPa. Ê , c 2.4454 A Ê . Miller indices, Lattice parameters: a 4.2502 A calculated intensities and calculated interplanar spacings for rectangulated carbon and observed interplanar spacings for the transparent phase at 25 GPa [1] are shown hkl
I
dcalc
dobs
110 020 011 220 121 310 301 002 112 040 231
21 72 62 7 12 9 34 9 1 0.5 19
3.005 2.125 2.120 1.503 1.501 1.344 1.226 1.227 1.133 1.063 1.062
, 2.75 2.139 2.055
1.234
1.057
energy differences for some layered structures that have van der Waals bonding, such as graphite and MoS2 [29]. The GGA tends to bind less than the LSDA and the V0 value for graphite is much too large (Table 1). The differences between the energies of diamond and rectangulated carbon calculated within the LSDA and the GGA are 21 and 19 kJ/mol, respectively. Thus, in contrast to the results for the graphite±diamond energy difference, both methods give almost the same value for the stability of rectangulated carbon relative to diamond. Unlike graphite, neither of these structures has van der Waals bonding, which is probably why both the LSDA and the GGA give similar results. Molecular mechanics and ab initio molecular dynamics calculations [3] give values for the energetic differences between diamond and rectangulated carbon of 24 and 23 kJ/mol, respectively, in good agreement with the LSDA and GGA results. Rectangulated carbon is a low energy form of carbon, thermodynamically more stable than some known allotropes, such as C60 [30]. The bulk modulus calculated for rectangulated carbon using the LSDA is 425 GPa (Table 1), which compares with a value of 403 GPa determined for rectangulated carbon by means of molecular mechanics [3]. The LSDA value for the bulk modulus of rectangulated carbon is slightly less than the LSDA value for diamond, as expected from the structural differences between these two phases. By assuming the hexagonal diamond structure for the transparent phase, a value for the bulk modulus of 425 ^ 25 GPa was determined from X-ray diffraction data [1]. Thus the calculated bulk modulus for rectangulated carbon is close to the experimental bulk modulus. The transition pressure for the conversion of graphite to rectangulated carbon can be estimated from the common tangent between the equations of state for graphite and rectangulated carbon. The transition from graphite to rectangulated carbon becomes possible once the enthalpy for
rectangulated carbon becomes less than or equal to that of graphite. The enthalpy is the Legendre transform of the energy with respect to pressure [31]. Therefore, the slope and intercept of the tangent lines to the energy vs. volume curve give the pressure and enthalpy, respectively. The transition pressure estimated from the tangent to the E±V curves calculated with the LSDA is 20 GPa (Fig. 2). The experimental values for the transition from graphite to the transparent phase upon compression range from 14 to 23 GPa, depending on how hydrostatic the stress conditions are and the type of graphite used as a starting material. When single crystal graphite is used, the transition pressure ranges from 14 to 18 GPa [1]. The degree to which kinetic factors control the transition remains uncertain, but the calculated transition pressure is close to the range of pressures observed experimentally. The diffraction pattern calculated for rectangulated carbon using the structural parameters optimized at a volume corresponding to 25 GPa is shown in Table 2, together with the experimental diffraction data at 25 GPa of Yagi et al. [1]. Previously, the comparison between the calculated and observed (or extrapolated) diffraction patterns was made at ambient pressure. In contrast to the ambient pressure diffraction patterns [3], the interplanar spacing at 25 GPa for the rectangulated carbon (110) Ê ) is considerably different from diffraction line (3.005 A Ê ). This experimental that observed experimentally (,2.75 A diffraction line has been attributed to graphite [1,4]. However, the absence of a diffraction line cannot be used to rule out a crystal structure. The largest remaining discrepancy is in the position of the (011) line (Table 2). This line shifts to smaller interplanar spacings when hexagonal diamond-like layers are incorporated into the rectangulated carbon structure [3]. Such mixed structures are more stable than fully rectangulated carbon and could be expected to form at lower pressures. At 25 GPa the predicted diffraction pattern for fully rectangulated carbon does not match as well as at ambient pressure [3]. Experimental diffraction data collected at yet higher pressures might reveal enough difference between calculated and experimental diffraction patterns to con®rm or rule out rectangulated carbon or a rectangulated carbon/hexagonal diamond polytype as the structure for the transparent phase of carbon. The c/a ratio for rectangulated carbon shifts according to the equation c=a 0:60325 2 0:0050481V; where the volume V is in cubic angstroms (Table 3). Table 3 Volume, atomic position x, and c/a ratio for rectangulated carbon at several pressures Pressure (GPa)
Ê 3) Volume (A
x
c/a
8.4 31.2 47.8 73.3
5.710 5.456 5.300 5.096
0.31947 0.31934 0.31926 0.31916
0.5744 0.5757 0.5765 0.5775
J.V. Badding, T.J. Scheidemantel / Solid State Communications 122 (2002) 473±477
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