- Email: [email protected]
Geometry and electronic structure of rhombohedral C60 polymer
Chemical Physics Letters 380 (2003) 589–594 www.elsevier.com/locate/cplett
Geometry and electronic structure of rhombohedral C60 polymer Takashi Miyake a
a,*
, Susumu Saito
a,b
Department of Physics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan b Department of Physics, University of California, Berkeley, and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Received 31 July 2003; in final form 8 August 2003 Published online: 7 October 2003
Abstract Geometry and electronic structure of the rhombohedral C60 polymer are studied by means of density-functional theory (DFT) within the local-density-approximation (LDA). It is found that stacking sequence proposed by Chen et al. ~ez-Regueiro et al., although the energy difference between the two is very is more stable than the original model by N un small. The material is a semiconductor with the LDA gap of 0.68 eV. Conduction bands show dependence on the way of stacking, and density of states has a sharp peak at the conduction bottom. Bond lengths are also calculated and found to be in good agreement with the results of the X-ray structure analysis. Ó 2003 Published by Elsevier B.V.
1. Introduction Fullerene C60 molecules attract each other and are crystallized via van der Waals interaction. The resultant solid C60 has been classified as Ônew form of carbonÕ [1], which is different from sp3 threedimensional-network diamond as well as from sp2 two-dimensional-network graphite. Under certain conditions, the solid phase is polymerized by forming chemical bonds between C60 Õs. The polymerization was first achieved under illumination of light, though the structure was not clarified [2].
*
Corresponding author. Fax: +81357342739. E-mail address: [email protected] (T. Miyake).
0009-2614/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.cplett.2003.09.067
Later on, chemical doping was found to be another way of polymerization [3], and the polymer turned out to be one-dimensional orthorhombic. Alternatively, pressure also induces polymerization [4], for which three different structures have been known so far; They are one-dimensional orthorhombic phase, two-dimensional tetragonal phase, and two-dimensional rhombohedral phase. Once they are formed, all of these three phases remain stable even at ambient pressure. Therefore, they should be classified as another important new form of carbon. Their dimensionalities are different from the original face-centered cubic (fcc) C60 which consist of finite, zero-dimensional-network spherical units. Moreover, each of three phases possesses both sp2 and sp3 -hybridized C atoms,
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T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
being completely different from fcc C60 consisting only of sp2 C atoms. A structure model for the rhombohedral phase ~ez-Regueiro et al. in 1995 [5]. was proposed by N un In this model, C60 Õs form a layer, on which C60 molecules occupy the lattice points of the two-dimensional triangular lattice. The C60 molecules in the layer are connected via chemical bond by [2+2] cycloaddition of Ô66Õ bond, which is the adjoining edge of two six-membered rings. The layers are piled up so that every three layers are on top of another. This model has been supported by a theoretical calculation using the tight-binding method [6]. The electronic properties, as well as stability, of the model have been studied based on densityfunctional theory (DFT) combined with the localdensity-approximation (LDA) [7]. It was found ~ez-Regthat the rhombohedral phase in the N un ueiro model is a narrow-gap semiconductor. The gap value is only 0.35 eV in LDA, which is smaller than that of another polymer phase with the tetragonal symmetry and much smaller than the fcc solid phase, for which the gap is more than 1 eV [8]. The band dispersion shows three-dimensional character in spite of strong anisotropy in geometry. This suggests importance of interlayer interaction in discussing electronic properties. Difference in the way of stacking may cause new feature in the electronic structure. ~ez-Regueiro model had been Although the N un regarded as a plausible structure for the rhombohedral phase, experimental verification was limited because of difficulty in preparing samples. Recently, Chen et al. have succeeded in synthesizing single crystal of the rhombohedral polymer [9]. They studied the structure by means of the X-ray measurement and clarified that the rhombohedral ~ezphase stacks in a different way from the N un Regueiro model. Motivated by this finding, we reinvestigate the rhombohedral C60 polymer theoretically by using the DFT–LDA. We find that the structure proposed by Chen et al. is indeed more stable than ~ez-Regueiro et al., although the energy that by N un difference is very small. The electronic structure shows a stacking dependence in the unoccupied conduction bands. The Chen model shows a sharp
peak at the bottom of the conduction bands. Detailed analysis on the stable geometry is also presented and is compared with the experiment. This Letter is organized as follows. Computational methods are described in Section 2, and results and discussions are given in Section 3. The Letter is concluded in Section 4.
2. Computational methods The calculation is based on the LDA in the framework of DFT [10,11]. The Perdew–Zunger formula is adopted for the explicit form of the exchange-correlation energy [12,13]. The effects of core electrons are included in the norm-conserving pseudopotential [14] in the Kleinman–Bylander form [15]. The wave functions are expanded by plane waves up to an energy cutoff of 50 Ry. The 23 k-point mesh is used for the Brillouine zone integration. In order to check the reliability of the method, we applied the method to the isolated C60 molecule. The optimized bond lengths for Ô5–6Õ and Ô6– , respectively, 6Õ bonds are 1.433 and 1.378 A whereas the corresponding experimental values are [16]. Thus, the present method 1.458 and 1.401 A is expected to reproduce the C–C bond length . We should note within the error of about 0.02 A that the difference between the 5–6 and 6–6 bond in the calculation. This value is length is 0.055 A , very close to the experimental value of 0.057 A thereby difference between bond lengths is calculated at higher accuracy.
3. Results and discussion In each layer of the rhombohedral C60 polymer, fullerenes occupy the lattice points of the twodimensional triangular lattice and they are connected each other with six neighboring C60 Õs. The sheet contains two kinds of hollow sites which we denote B site and C site, respectively (Fig. 1). The two sites are not equivalent, because rotational symmetry of the sheet is three-folded, not six-folded. If the second layer is arranged so that the C60 Õs are on top of the B sites, adjacent C60 Õs be-
T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
591
Fig. 1. Structure model for the rhombohedral C60 polymer with lattice constants in the hexagonal-cell representation. (a) Top view of a polymerized sheet and (b) side view. Shaded faces in (a) represent pentagons.
tween the two layers contact each other at the hexagonal faces. On the other hand, if C60 Õs on the second layer occupy the C sites, fullerenes contact each other at the pentagons. Therefore, the stacking sequence ÔACBACBÕ is different from the sequence ÔABCABCÕ [9]. This is the difference be~ez-Regueiro model and Chen tween the N un model. The former corresponds to ÔACBACBÕ, and the latter is ÔABCABCÕ. We begin with discussing stability of the structures. The calculation is done with lattice parameters fixed at the experimental values [9], a ¼ 9:175 and c ¼ 24:568 A which should be of high acA curacy, whereas the internal lattice degrees of freedom are fully relaxed. Fig. 2 shows the total energy of the rhombohedral polymers in the Chen ~ez-Regueiro model, compared with model and N un a couple of different structures, where the energy is measured from the isolated C60 molecule. A hypothetical isolated sheet of the rhombohedral polymer (ÔSheetÕ) is less stable than the C60 molecule. When the sheets are stacked, van der Waals interaction stabilizes the material by about 0.8 eV per C60 , consequently, the rhombohedral phase is more stable than the C60 molecule, although it is still less stable than the solid fcc phase. Looking at the energy difference caused by stacking sequence, the Chen model is indeed more stable than the
~ez-Regueiro model. The energy difference beN un tween them is, however, only 0.01 eV/C60 . Optimized geometry of the rhombohedral phase is shown in Fig. 3. The upper panel shows a part of the [2+2] cycloaddition of 66 bonds. Two ÔC1Õ atoms in the figure belong to a C60 fullerene and form a 66 bond between them. The two ÔC1*Õ atoms are in the neighboring C60 and a 66 bond is
0.31
Sheet
0
Molecule
-0.48 -0.49
'~ Nunez-Regueir Chen
-0.78
Fcc
Fig. 2. Total energy diagram for a couple of structures composed of C60 . The energy is measured from the isolated C60 molecule.
592
T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
Fig. 3. Bond lengths of the rhombohedral C60 phase. Two ÔC1Õ atoms in a C60 form chemical bonds with the ÔC1*Õ atoms in the neighboring C60 . DFT–LDA optimized values in this work, in , are compared with the experimental values [9] in units of A parentheses.
between them as well. In the rhombohedral phase, polymerization is attained in such a way that chemical bonds are formed between C1 and C1*.
The lower panel shows the upper half of the upper panel, looked at from a different angle. Thus all atoms in the lower panel belong to one C60 fullerene. Numbers in the figure are bond lengths in and they are compared to the experiunits of A mental values obtained by the X-ray structure analysis [9] shown in parentheses. We find that the theoretical values are close to the experimental . Espeones. The differences are less than 0.05 A cially, the differences in the rectangle by the cy. These cloaddition of 66 bond are less than 0.01 A excellent agreement between the X-ray structure analysis and the present LDA structure optimization indicates the high reliability of these two independent works. The C1–C1 bond length in the . This value is significantly rectangle is 1.596 A larger than the 66 bond length of the isolated C60 , and is even larger than the molecule, 1.378 A . newly formed C1–C1* length, 1.592 A Finally, we discuss the electronic properties. In Fig. 4, the electronic structure and density of states ~ez-Regueiro model and (b) Chen model of (a) N un are shown. The unit cell size is fixed to the experimental value, while internal lattice degrees of free-
~ez-Regueiro model and (b) Chen model. The energy is measured from the Fig. 4. Electronic structure and density of states of (a) N un top of the valence band.
T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
593
Fig. 5. Geometry and charge density distribution of the isolated C60 molecule. (a) Geometry, (b) charge distribution of the occupied hu states, and (c) charge distribution of the unoccupied t1u states. Charge distributions are plotted on the xy plane which includes the center of mass of the C60 molecule (at the center of the figure), two Ô66Õ bonds (at the upper and lower parts of the figure), centers of four pentagons (next to each 66 bonds), and centers of four hexagons (next to pentagons). Each contour represents twice/half the density of the neighboring line. The maximum contour lines correspond to 0.025 e/(a.u.)3 in (b), and 0.0125 e/(a.u.)3 in (c).
dom are fully relaxed in each case. The former case has been already reported previously [7]. A prominent feature is that the electronic structure shows larger dispersion than that in the fcc C60 solid. This band broadening results in a reduction in the fundamental-gap value. The gap is formed between the Z point (top of the valence band) and the F point (bottom of the conduction bands), and its value is 0.58 eV. The band structure has considerable dispersion not only in the polymerized plane but also along the C–Z line. Therefore, although the geometry is highly two-dimensional, the electronic structure has three-dimensional character [7]. In the Chen model shown in (b), the valence ~ez-Regueiro bands are similar with those of the N un model. Conduction bands, on the other hand, show different structure. The lowest conduction band is flatter in the former model along the F–C and R lines. Consequently, the density of states has a sharp peak at the bottom of the conduction bands. This narrowing raises the band gap to 0.68 eV. The difference in the conduction band between the two models can be understood as follows. The lowest-unoccupied-molecular-orbital (LUMO) of the C60 molecule has large population in the vicinity of the five-membered ring, as shown in Fig. 5c. On the other hand, as described above, the C60 fullerenes in the adjacent layers contact each ~ez-Regueother at five-membered ring in the N un iro model, while they contact each other at sixmembered ring in the Chen model. This means the
conduction electrons hop easier in the former model, thus the conduction band is wider.
4. Summary and concluding remarks We have studied the geometry and electronic structure of the rhombohedral C60 polymer using the LDA of DFT. The total energy analysis supports stacking model by Chen et al., although the energy difference caused by the difference in stacking sequence is very small. Detail of the geometry is analyzed and bond lengths are found to be in good agreement with the experimental reports. This confirms the high reliability of the LDA–DFT analysis of the carbon nanostructures. The electronic structure shows three-dimensional character. The polymer is a semiconductor and the fundamental gap is 0.68 eV which is significantly smaller than the fcc solid phase. The stacking sequence causes change in the width of the lowest conduction band, and Chen model has a sharp density of states there.
Acknowledgements Numerical calculations were performed on Fujitsu VPP5000 and NEC SX-7 at the Research Center for Computational Science, Okazaki National Institute. We appreciate Prof. A. Oshiyama, Prof. T. Nakayama, Dr. M. Saito, and
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Dr. O. Sugino for providing us with the computer code for the electronic structure calculation. This work has been supported by Grant-in-Aid from the Ministry of Education, Science and Culture ofJapan, and by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, U.S. Department of Energy under Contract No. DEAC03-76SF00098. References [1] W. Kr€ atschmer, L.D. Lamb, K. Fostiropoulos, D.R. Huffman, Nature (London) 347 (1990) 354. [2] A.M. Rao, P. Zhou, K. Wang, G.T. Hager, J.M. Holden, Y. Wang, W. Lee, X. Bi, P.C. Eklund, D.S. Cornett, M.A. Duncan, I.J. Amster, Science 259 (1993) 955. [3] O. Chauvet, G. Oszlanyi, L. Forro, P.W. Stephans, M. Tegze, G. Faigel, A. Janossy, Phys. Rev. Lett. 72 (1994) 2721.
[4] Y. Iwasa, T. Arima, R.M. Fleming, T. Siegrist, O. Zhou, R.C. Haddon, L.J. Rothberg, K.B. Lyons, H.L. Carter Jr., A.F. Hebard, R. Tycko, G. Debbagh, J.J. Krajewski, G.A. Thomas, T. Yagi, Science 264 (1995) 1570. ~ez-Regueiro, L. Marques, J.-L. Hodeau, O. [5] M. N un Bethoux, M. Perroux, Phys. Rev. Lett. 74 (1995) 278. [6] C.-H. Xu, G.E. Scuseria, Phys. Rev. Lett. 74 (1995) 274. [7] S. Okada, S. Saito, Phys. Rev. B 55 (1997) 4039; Phys. Rev. B 59 (1999) 1930. [8] S. Saito, A. Oshiyama, Phys. Rev. Lett. 66 (1991) 2637. [9] X. Chen, S. Yamanaka, K. Sako, Y. Inoue, M. Yasukawa, Chem. Phys. Lett. 356 (2002) 291. [10] P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. [11] W. Kohn, L.J. Sham, Phys. Rev. A 140 (1965) 1133. [12] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. 45 (1980) 566. [13] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [14] N. Troullier, J.L. Martins, Phys. Rev. B 43 (1991) 1993. [15] L. Kleinman, D.M. Bylander, Phys. Rev. Lett. 48 (1982) 1425. [16] K. Hedberg, L. Hedberg, D.S. Bethune, C.A. Brown, H.C. Dorn, R.D. Johnson, M. de Vries, Science 254 (1991) 410.
Geometry and electronic structure of rhombohedral C60 polymer Takashi Miyake a
a,*
, Susumu Saito
a,b
Department of Physics, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-8551, Japan b Department of Physics, University of California, Berkeley, and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Received 31 July 2003; in final form 8 August 2003 Published online: 7 October 2003
Abstract Geometry and electronic structure of the rhombohedral C60 polymer are studied by means of density-functional theory (DFT) within the local-density-approximation (LDA). It is found that stacking sequence proposed by Chen et al. ~ez-Regueiro et al., although the energy difference between the two is very is more stable than the original model by N un small. The material is a semiconductor with the LDA gap of 0.68 eV. Conduction bands show dependence on the way of stacking, and density of states has a sharp peak at the conduction bottom. Bond lengths are also calculated and found to be in good agreement with the results of the X-ray structure analysis. Ó 2003 Published by Elsevier B.V.
1. Introduction Fullerene C60 molecules attract each other and are crystallized via van der Waals interaction. The resultant solid C60 has been classified as Ônew form of carbonÕ [1], which is different from sp3 threedimensional-network diamond as well as from sp2 two-dimensional-network graphite. Under certain conditions, the solid phase is polymerized by forming chemical bonds between C60 Õs. The polymerization was first achieved under illumination of light, though the structure was not clarified [2].
*
Corresponding author. Fax: +81357342739. E-mail address: [email protected] (T. Miyake).
0009-2614/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.cplett.2003.09.067
Later on, chemical doping was found to be another way of polymerization [3], and the polymer turned out to be one-dimensional orthorhombic. Alternatively, pressure also induces polymerization [4], for which three different structures have been known so far; They are one-dimensional orthorhombic phase, two-dimensional tetragonal phase, and two-dimensional rhombohedral phase. Once they are formed, all of these three phases remain stable even at ambient pressure. Therefore, they should be classified as another important new form of carbon. Their dimensionalities are different from the original face-centered cubic (fcc) C60 which consist of finite, zero-dimensional-network spherical units. Moreover, each of three phases possesses both sp2 and sp3 -hybridized C atoms,
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T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
being completely different from fcc C60 consisting only of sp2 C atoms. A structure model for the rhombohedral phase ~ez-Regueiro et al. in 1995 [5]. was proposed by N un In this model, C60 Õs form a layer, on which C60 molecules occupy the lattice points of the two-dimensional triangular lattice. The C60 molecules in the layer are connected via chemical bond by [2+2] cycloaddition of Ô66Õ bond, which is the adjoining edge of two six-membered rings. The layers are piled up so that every three layers are on top of another. This model has been supported by a theoretical calculation using the tight-binding method [6]. The electronic properties, as well as stability, of the model have been studied based on densityfunctional theory (DFT) combined with the localdensity-approximation (LDA) [7]. It was found ~ez-Regthat the rhombohedral phase in the N un ueiro model is a narrow-gap semiconductor. The gap value is only 0.35 eV in LDA, which is smaller than that of another polymer phase with the tetragonal symmetry and much smaller than the fcc solid phase, for which the gap is more than 1 eV [8]. The band dispersion shows three-dimensional character in spite of strong anisotropy in geometry. This suggests importance of interlayer interaction in discussing electronic properties. Difference in the way of stacking may cause new feature in the electronic structure. ~ez-Regueiro model had been Although the N un regarded as a plausible structure for the rhombohedral phase, experimental verification was limited because of difficulty in preparing samples. Recently, Chen et al. have succeeded in synthesizing single crystal of the rhombohedral polymer [9]. They studied the structure by means of the X-ray measurement and clarified that the rhombohedral ~ezphase stacks in a different way from the N un Regueiro model. Motivated by this finding, we reinvestigate the rhombohedral C60 polymer theoretically by using the DFT–LDA. We find that the structure proposed by Chen et al. is indeed more stable than ~ez-Regueiro et al., although the energy that by N un difference is very small. The electronic structure shows a stacking dependence in the unoccupied conduction bands. The Chen model shows a sharp
peak at the bottom of the conduction bands. Detailed analysis on the stable geometry is also presented and is compared with the experiment. This Letter is organized as follows. Computational methods are described in Section 2, and results and discussions are given in Section 3. The Letter is concluded in Section 4.
2. Computational methods The calculation is based on the LDA in the framework of DFT [10,11]. The Perdew–Zunger formula is adopted for the explicit form of the exchange-correlation energy [12,13]. The effects of core electrons are included in the norm-conserving pseudopotential [14] in the Kleinman–Bylander form [15]. The wave functions are expanded by plane waves up to an energy cutoff of 50 Ry. The 23 k-point mesh is used for the Brillouine zone integration. In order to check the reliability of the method, we applied the method to the isolated C60 molecule. The optimized bond lengths for Ô5–6Õ and Ô6– , respectively, 6Õ bonds are 1.433 and 1.378 A whereas the corresponding experimental values are [16]. Thus, the present method 1.458 and 1.401 A is expected to reproduce the C–C bond length . We should note within the error of about 0.02 A that the difference between the 5–6 and 6–6 bond in the calculation. This value is length is 0.055 A , very close to the experimental value of 0.057 A thereby difference between bond lengths is calculated at higher accuracy.
3. Results and discussion In each layer of the rhombohedral C60 polymer, fullerenes occupy the lattice points of the twodimensional triangular lattice and they are connected each other with six neighboring C60 Õs. The sheet contains two kinds of hollow sites which we denote B site and C site, respectively (Fig. 1). The two sites are not equivalent, because rotational symmetry of the sheet is three-folded, not six-folded. If the second layer is arranged so that the C60 Õs are on top of the B sites, adjacent C60 Õs be-
T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
591
Fig. 1. Structure model for the rhombohedral C60 polymer with lattice constants in the hexagonal-cell representation. (a) Top view of a polymerized sheet and (b) side view. Shaded faces in (a) represent pentagons.
tween the two layers contact each other at the hexagonal faces. On the other hand, if C60 Õs on the second layer occupy the C sites, fullerenes contact each other at the pentagons. Therefore, the stacking sequence ÔACBACBÕ is different from the sequence ÔABCABCÕ [9]. This is the difference be~ez-Regueiro model and Chen tween the N un model. The former corresponds to ÔACBACBÕ, and the latter is ÔABCABCÕ. We begin with discussing stability of the structures. The calculation is done with lattice parameters fixed at the experimental values [9], a ¼ 9:175 and c ¼ 24:568 A which should be of high acA curacy, whereas the internal lattice degrees of freedom are fully relaxed. Fig. 2 shows the total energy of the rhombohedral polymers in the Chen ~ez-Regueiro model, compared with model and N un a couple of different structures, where the energy is measured from the isolated C60 molecule. A hypothetical isolated sheet of the rhombohedral polymer (ÔSheetÕ) is less stable than the C60 molecule. When the sheets are stacked, van der Waals interaction stabilizes the material by about 0.8 eV per C60 , consequently, the rhombohedral phase is more stable than the C60 molecule, although it is still less stable than the solid fcc phase. Looking at the energy difference caused by stacking sequence, the Chen model is indeed more stable than the
~ez-Regueiro model. The energy difference beN un tween them is, however, only 0.01 eV/C60 . Optimized geometry of the rhombohedral phase is shown in Fig. 3. The upper panel shows a part of the [2+2] cycloaddition of 66 bonds. Two ÔC1Õ atoms in the figure belong to a C60 fullerene and form a 66 bond between them. The two ÔC1*Õ atoms are in the neighboring C60 and a 66 bond is
0.31
Sheet
0
Molecule
-0.48 -0.49
'~ Nunez-Regueir Chen
-0.78
Fcc
Fig. 2. Total energy diagram for a couple of structures composed of C60 . The energy is measured from the isolated C60 molecule.
592
T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
Fig. 3. Bond lengths of the rhombohedral C60 phase. Two ÔC1Õ atoms in a C60 form chemical bonds with the ÔC1*Õ atoms in the neighboring C60 . DFT–LDA optimized values in this work, in , are compared with the experimental values [9] in units of A parentheses.
between them as well. In the rhombohedral phase, polymerization is attained in such a way that chemical bonds are formed between C1 and C1*.
The lower panel shows the upper half of the upper panel, looked at from a different angle. Thus all atoms in the lower panel belong to one C60 fullerene. Numbers in the figure are bond lengths in and they are compared to the experiunits of A mental values obtained by the X-ray structure analysis [9] shown in parentheses. We find that the theoretical values are close to the experimental . Espeones. The differences are less than 0.05 A cially, the differences in the rectangle by the cy. These cloaddition of 66 bond are less than 0.01 A excellent agreement between the X-ray structure analysis and the present LDA structure optimization indicates the high reliability of these two independent works. The C1–C1 bond length in the . This value is significantly rectangle is 1.596 A larger than the 66 bond length of the isolated C60 , and is even larger than the molecule, 1.378 A . newly formed C1–C1* length, 1.592 A Finally, we discuss the electronic properties. In Fig. 4, the electronic structure and density of states ~ez-Regueiro model and (b) Chen model of (a) N un are shown. The unit cell size is fixed to the experimental value, while internal lattice degrees of free-
~ez-Regueiro model and (b) Chen model. The energy is measured from the Fig. 4. Electronic structure and density of states of (a) N un top of the valence band.
T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
593
Fig. 5. Geometry and charge density distribution of the isolated C60 molecule. (a) Geometry, (b) charge distribution of the occupied hu states, and (c) charge distribution of the unoccupied t1u states. Charge distributions are plotted on the xy plane which includes the center of mass of the C60 molecule (at the center of the figure), two Ô66Õ bonds (at the upper and lower parts of the figure), centers of four pentagons (next to each 66 bonds), and centers of four hexagons (next to pentagons). Each contour represents twice/half the density of the neighboring line. The maximum contour lines correspond to 0.025 e/(a.u.)3 in (b), and 0.0125 e/(a.u.)3 in (c).
dom are fully relaxed in each case. The former case has been already reported previously [7]. A prominent feature is that the electronic structure shows larger dispersion than that in the fcc C60 solid. This band broadening results in a reduction in the fundamental-gap value. The gap is formed between the Z point (top of the valence band) and the F point (bottom of the conduction bands), and its value is 0.58 eV. The band structure has considerable dispersion not only in the polymerized plane but also along the C–Z line. Therefore, although the geometry is highly two-dimensional, the electronic structure has three-dimensional character [7]. In the Chen model shown in (b), the valence ~ez-Regueiro bands are similar with those of the N un model. Conduction bands, on the other hand, show different structure. The lowest conduction band is flatter in the former model along the F–C and R lines. Consequently, the density of states has a sharp peak at the bottom of the conduction bands. This narrowing raises the band gap to 0.68 eV. The difference in the conduction band between the two models can be understood as follows. The lowest-unoccupied-molecular-orbital (LUMO) of the C60 molecule has large population in the vicinity of the five-membered ring, as shown in Fig. 5c. On the other hand, as described above, the C60 fullerenes in the adjacent layers contact each ~ez-Regueother at five-membered ring in the N un iro model, while they contact each other at sixmembered ring in the Chen model. This means the
conduction electrons hop easier in the former model, thus the conduction band is wider.
4. Summary and concluding remarks We have studied the geometry and electronic structure of the rhombohedral C60 polymer using the LDA of DFT. The total energy analysis supports stacking model by Chen et al., although the energy difference caused by the difference in stacking sequence is very small. Detail of the geometry is analyzed and bond lengths are found to be in good agreement with the experimental reports. This confirms the high reliability of the LDA–DFT analysis of the carbon nanostructures. The electronic structure shows three-dimensional character. The polymer is a semiconductor and the fundamental gap is 0.68 eV which is significantly smaller than the fcc solid phase. The stacking sequence causes change in the width of the lowest conduction band, and Chen model has a sharp density of states there.
Acknowledgements Numerical calculations were performed on Fujitsu VPP5000 and NEC SX-7 at the Research Center for Computational Science, Okazaki National Institute. We appreciate Prof. A. Oshiyama, Prof. T. Nakayama, Dr. M. Saito, and
594
T. Miyake, S. Saito / Chemical Physics Letters 380 (2003) 589–594
Dr. O. Sugino for providing us with the computer code for the electronic structure calculation. This work has been supported by Grant-in-Aid from the Ministry of Education, Science and Culture ofJapan, and by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, U.S. Department of Energy under Contract No. DEAC03-76SF00098. References [1] W. Kr€ atschmer, L.D. Lamb, K. Fostiropoulos, D.R. Huffman, Nature (London) 347 (1990) 354. [2] A.M. Rao, P. Zhou, K. Wang, G.T. Hager, J.M. Holden, Y. Wang, W. Lee, X. Bi, P.C. Eklund, D.S. Cornett, M.A. Duncan, I.J. Amster, Science 259 (1993) 955. [3] O. Chauvet, G. Oszlanyi, L. Forro, P.W. Stephans, M. Tegze, G. Faigel, A. Janossy, Phys. Rev. Lett. 72 (1994) 2721.
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