This article was downloaded by: [New York University] On: 24 April 2013, At: 11:03 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Fullerene Science and Technology Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lfnn19 Polymorphism of Extended Fullerene Networks: Geometrical Parameters and Electronic Structures Mitsutaka Fujita a , Mitsuho Yoshida b & Kyoko Nakada c a University of Tsukuba, Institute of Materials Science, Tsukuba, 305, Japan b Department of Knowledge-Based Information Engineering, Toyohashi University of Technology, Toyohashi, 441, Japan c Doctoral Research Course in Human Culture, Ochanomizu University, Otsuka, Tokyo, 112, Japan Version of record first published: 19 Aug 2006. To cite this article: Mitsutaka Fujita , Mitsuho Yoshida & Kyoko Nakada (1996): Polymorphism of Extended Fullerene Networks: Geometrical Parameters and Electronic Structures, Fullerene Science and Technology, 4:3, 565-582 To link to this article: http://dx.doi.org/10.1080/10641229608001571 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms- and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan,
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This article was downloaded by: [New York University]On: 24 April 2013, At: 11:03Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK
Fullerene Science andTechnologyPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/lfnn19
Polymorphism of ExtendedFullerene Networks:Geometrical Parameters andElectronic StructuresMitsutaka Fujita a , Mitsuho Yoshida b & KyokoNakada ca University of Tsukuba, Institute of MaterialsScience, Tsukuba, 305, Japanb Department of Knowledge-Based InformationEngineering, Toyohashi University of Technology,Toyohashi, 441, Japanc Doctoral Research Course in Human Culture,Ochanomizu University, Otsuka, Tokyo, 112, JapanVersion of record first published: 19 Aug 2006.
To cite this article: Mitsutaka Fujita , Mitsuho Yoshida & Kyoko Nakada (1996):Polymorphism of Extended Fullerene Networks: Geometrical Parameters andElectronic Structures, Fullerene Science and Technology, 4:3, 565-582
To link to this article: http://dx.doi.org/10.1080/10641229608001571
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,
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The publisher does not give any warranty express or implied or make anyrepresentation that the contents will be complete or accurate or up todate. The accuracy of any instructions, formulae, and drug doses should beindependently verified with primary sources. The publisher shall not be liablefor any loss, actions, claims, proceedings, demand, or costs or damageswhatsoever or howsoever caused arising directly or indirectly in connectionwith or arising out of the use of this material.
Polymorphism of Extended F'ullerene Networks: Geometrical Parameters and Electronic Structures
Mitsutaka Fujita' Institute of Materials Science, University of Tsukuba,
Tsukuba 305, Japan
Mitsuho Yoshida Department of Knowledge-Based Information Engineering, Toyohashi University of Technology, Toyohashi 441, Japan.
Kyoko Nakada Doctoral Research Course in Human Culture, Ochanomizu University,
Otsuka, Tokyo 112, Japan.
The morphology of fullerene networks can be widely extended by in- troducing heptagonal or octagonal rings, which produce a Gaussian nega- tive curvature. Their presence makes it possible to form donut-, coil- and sponge-shaped networks of carbon atoms. We discuss the geometry of the polymorphous forms based on the net diagram method relative to a hon- eycomb lattice, and further study the electronic structures constructed by the network of a electrons system. Special emphasis is put on how the ge- ometrical parameters, which specify the relative arrangement of polygonal rings, control the electronic structures in the various extended-fullerene net- works. In addition, we mention that the presence of a certain type of edge in fullerene network derives critical localized edge states at the Fermi level.
'To whom correspondence should be addressed.
This paper was presented before the Symposium 3 on 'Expanded Horizons of Fullerene Science and Technology' organized by L. Y. Chiang, E. asawa, H. Temnes and M. Saunders at the 4th International Conference on Advanced Materials (ICAM-IV), Aug. 27 to Sept. 1, 1995, Cancun, Mexico.
565
Copyright 0 1996 by Marcel Dekker, Inc.
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51. Introduction
We know that carbon is a unique element to bond with itself through
sp3, sp2 and even sp hybridization, which is quite different from silicon with
the same number of valence electrons. The resultant structures of carbon
crystalline materials are diamond and graphite, and recently discovered C a
crystal, while silicon forms only the crystal structure of diamond. This va-
riety of carbon bonding leads the polymorphous forms in disordered carbon
materials, some of which are playing an important role in our modern life
as activated carbons, carbon blacks, carbon fibers etc. However, the micro-
scopic understanding for the structures of these amorphous carbon(a-C) has
not been well established yet.
The intrinsic features of the local structure in disordered carbon materi-
als can be recognized as the following two: the co-existence of two-, three-
and four-fold coordinated carbon atoms, and the presence of 5-, 7- and 8- membered rings in a graphitic network. The latter comes from the fact that
graphite which is the most thermodynamically stable at ambient tempera-
tures and pressures may contain some polygonal defects in the hexagonal
network. We could say that fullerene is a kind of art made by pentago-
nal carbon rings in soot[l], as well known that the name is derived from the
architect R. Buckminster Fuller who designed a lot of geodesic domes. There-
fore, fullerenes are considered as the latter structural disorder in a graphite
network, although they are molecules.
As a curious extension of fullerene networks, introduction of heptagonal
and octagonal rings gives more fascinating forms of new fullerene families
by producing Gaussian negative curvature on a honeycomb network, where
the three-coordinated carbon networks span periodically not in plane but
in space. Their presence has already been suggested by the observation
of morphologies of fullerene tubules via transmission electron microscopy[2,
31. In this paper, we propose the polyhedral construction method of new
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS 567
carbon forms with negative curvature, by demonstrating various forms of
donut fullerenes, coiled fullerene tubules and spongy graphites with 7- or
8-membered rings, and show how their electronic states are controlled by
the geometrical parameters in the carbon network. In addition, we briefly
note that if a graphite sheet is peripheral, a localized edge state can exist
depending on the shape of the edge.
$2. Polygonal Ring in a Hexagonal Network
Let us begin by briefly reviewing the geometry of c60 and c70 molecules.
The former is called a truncated-icosahedron, which means that if we put
an icosahedron and cut all vertices off, twelve pentagons appear. In other
words, if we decorate each face of the icosahedron with a honeycomb patch as in Fig.l(b), you can easily get the atomic network of Ceo. While if we try
to obtain the atomic network of C70 in the same way, we decorate honeycomb
patches based on the polyhedral frame which is given by combining a pentag-
onal prism at the body with pentagonal pyramids at the both edges via their
pentagonal faces as shown in Fig.Z(b). We should note that twelve pentagons
appear at each vertex of the polyhedral frame because totally 30O"of angles
meet.
Cutting along the edges of the basis polyhedral frames of Fig.l(b) for C a
and Fig.l(d) for C,o, we can open them on a plane as exhibited in Fig.l(d)
and (b) for c60 and C70, respectively, where we see that the connectivity
of twelve pentagons for each fullerene molecule can be represented by their
distribution relative to a honeycomb network[$, 51. An arbitrary fullerene
can be completely specified by the arrangement of twelve pentagons in a
honeycomb lattice in the same way. In other words, the structure of any
fullerene can be regarded as a polyhedron with twelve vertices whose faces
are decorated with honeycomb patches. On each face the polyhedral vertices
are placed at the center of the honeycomb hexagons, and at each vertex the
Fig.2. Net diagrams on a honeycomb lattice for (a) C60 and (b) C70.
angular deficiency is 60°, thereby a pentagonal ring is produced at the site.
Briefly speaking, the network of fullerenes can be produced by the patchwork
of honeycomb pieces on the polyhedral faces. Moreover, it can be expressed
as a net diagram on a honeycomb lattice.
The role of a pentagonal ring in a single graphite sheet is displayed in
Fig.S(a) where we see around an apical pentagonal ring a conical surface
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS 569
Fig.3 (a) Conical surface around a pentagonal ring. (b) Saddle-shaped surface around a heptagonal ring.
consisting of hexagons is formed. While if you put a heptagonal ring in a
sheet as well, a saddle-shaped surface appears around it (Fig.3(b), which
means a surface with negative curvature is introduced by a heptagon. A
polygonal ring except a hexagon can be regarded as a defect on a graphitic
sheet [4].
Introduction of negative curvature makes it possible to construct topo-
logically more complicated structures than those of normal fullerenes which
are topologically equivalent to a sphere. The Euler characteristic plays an
important role in the network on such complex objects. The extended Euler
law is given by w-e+f=K, where K is the Euler characteristic, w, e and f are the numbers of vertices, edges and faces of the polyhedral object, respec-
tively. If we describe the number of a n-membered ring as f n , we additionally
obtain 2e=Cn n f,, and f =En fn. Further, since we are now focusing on the
fullerene-type network where each carbon atom is connected with three neigh-
boring atoms forming sp2 hybrid orbitals, we obtain another relation 2e=3v.
These four equations lead to Cn(6-n)fn=6K. We note that K is defined
by the genus g which is the number of holes as K=2(1-g), hence K=2 for a
sphere and K=O for a donut.
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570 FUJITA, YOSHDA, AND NAKADA
$3. F’ullerene Donuts and Coiled F’ullerene Tubules
The net diagram method is useful even for extended fullerene families
with negative curvature[6]. As the first example, we apply it to fullerene
donuts[7, 81. The net diagram of a fullerene donut C504 having chirality
(Fig.4(a)) is shown in Fig.4(b). Since in donuts f5=f7 because K=O, we find
twelve pairs of pentagonal and heptagonal rings in it. The light- and dark-
shaded hexagons indicate the places of 5- and ?-membered rings, respectively.
Each heptagonal site is to be surrounded by seven hexagonal rings. ---f
The geometry of fullerene donuts can be specified by four vectors; A , B , 2 and D shown in Fig.4(b). Hereafter we express the components of
a vector V as (K,&)=K%+ & b , where 2 and b are the primitive
vectors of the hexagonal lattice shown in Fig.4(b). Here the four vectors
are assigned to (5,1), (3,0), (-4,2) and (0,2), respectively. The total number
of carbon atoms for the M-fold donut is given by N=2M(IA1&-A2&[ +
-+ ---f
+ + +
101 C2-& C1 I + (A:+Al A2+A; - C,Z - Ci C2 4;)). If we joint the units of an unclosed donut one after another, we obtain
a coiled fullerene tubule[9]. In Fig.5(a) we exhibit one of examples whose
unit is same as the donut in Fig.4. Certainly it is composed of pentagonal,
hexagonal, and heptagonal rings as well. While if we set C and D to be 0
in the net diagram of Fig.4(b), a coiled tubule with square, and octagonal
rings instead of pentagonal and heptagonal ones can be formed.
+ +
One of our interests in the polymorphous forms of the extended fullerene
network is to reveal the variety of the electronic structures depending on the
geometrical parameter. In order to examine the electronic states for coiled
tubules, we utilize the simple tight binding model with a constant transfer
integral t o , under the knowledge that in the sp2 network materials, the T
electrons mainly contribute to the electronic states near the Fermi level. The
curvature, which must be one of the important structural factors, may give
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS
Fig.4 (a) Fullerene donut CEO4 and (b) its net diagram.
57 1
Fig.5 (a) Coiled tubule with the same unit as Fig.4(b) and (b) Coiled tubule
where C and D are set to be 0. ---f ---f
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512 FUJITA, YOSHIDA, AND NAKADA
the effect of reduction for t o . In a qualitative stage of discussion, however,
the simplified calculation is very useful to reveal the characteristic electronic
states. Previous to showing the results we should note that for cylindrical
fullerene tubules the simple model for the 7~ electron network explains the ge-
ometrical classification between the metallic and semiconducting tubules[lO].
This rule can be illustrated as follows: First we paint one of three sublat-
tice sites in a graphite sheet as shown in Fig.6 (a) and (b). The cylindrical
fullerene tubule can be made by cutting out a strip whose width is specified
by the vector R and by rolling it into a cylinder where I R I corresponds to --+ t
the circumference. If we can roll it without misfit of the decoration (Fig.G(c)),
the tubule becomes metallic, otherwise insulating (Fig.G(d)). Mathematically
it is represented as when R = (m,n) the tubule is metallic in the case that
m--71 is a multiple of 3, ot,herwise insulating. Hereafter we call the condition
for the circumferential vector R to give a metallic tubule as "the KekulC
rule".
+
--t
To discuss the electronic structure of coiled tubules[ll], it is easier to
examine the coiled tubules with all even-membered rings, because they have
less geometrical parameters and show more symmetric energy bands, which
means they are determined only by .A and B and are bipartite under the
double unit cells. The rule for the electronic states is simple as follows: the
tubule becomes metallic when either A or B satisfies the KekulC rule. In
other words, it is insulating when the total atomic number in the unit cell
is a multiple of 3, and otherwise metallic. The extension to the case of non-
zero C and D is stra.ightforward. In addition to the metallic condition for
A and B , the coiled tubule with pentagonal and heptagonal rings becomes
metallic when both C and D satisfy the KekuM rule. In Fig.7(a) and (b),
we display the tight-binding energy band for the tubules of Fig.G(a) and (b),
t --t
+ t
4 -+
t t
-+ t
respectively.
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS 573
t Fig.6 If the circumferential vector R of a cylindrical tubule satisfies(a) and does not satisfy(b) the Kekulh rule, the resultant tubule becomes metallic(c) and insulating(d), respectively.
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(4 1 .o
0.5
E 0.0 l-----l -0.5
-1.0
0 X I 2 1c
Q
FUJITA, YOSHIDA, AND NAKADA
1.0
0.5
E 0.0
-0.5
-1.0
0 d 2 K (0
Fig.7 Tight-binding energy bands of the coiled tubules in Fig.G(a) and (b), respectively, where is a phase shift between the neighboring unit cells.
54. Graphitic Sponges (Pearcene)
Let us now discuss periodic graphite surfaces with negative curvature in-
cluding heptagonal or octagonal rings in three dimensional (3D) space. Such
“graphitic sponges” have been intensively examined by many authors[l3, 14,
151. Most of them are created mathematically based on the theory of periodic
minimal surfaces[l6]. A surface is minimal because, e.g., when a soap-film
spans a non-planar loop of wire, it minimizes its energy by having a minimum
of area. Here we propose a new method of constructing graphitic sponges by
means of packing regular or semi-regular polyhedra in space[l7].
The basic idea of our construction method is that if we have the 3D networks of equilateral triangles or hexagons, we can easily produce the
3D graphitic surfaces by decorating each face with triangular or hexagonal
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS 575
patches in a honeycomb lattice as shown in Fig.8. Each patch is specified by
vectors (m,n) along one of the edges. Since Peter Pearce, a modern archi-
tectural designer, has elegantly designed his unique architectures[l8] having
negative surfaces based on the polyhedral fillings and packings, hereafter we
call the resultant structures as “Pearcene”.
We begin by considering a space filling with truncated octahedra shown
in Fig.S(a) which gives the connections of hexagonal faces in space. Since at
each vertex four hexagons meet, octagonal rings are created as displayed in
one example of the Pearcenes with (2,O) in Fig.S(b), where they are darkly
shaded. We examine the electronic structures using the simple tight binding
model. The Pearcenes based on the structure of Fig.S(a) are found to be
metallic for ( 1 , O ) and insulating for (l,l), (2,1), (3,O) and (3,1), and semi-
conducting with zero gap for (2,O) where the DOS behaves as near
EF. In these Pearcenes, the number of carbon atoms per unit cell is given
by 48Sm,n(-m2 + mn + n’). We should note that the network of (2,O) is
topologically equivalent to the Mackay and Terrones’s graphitic sponge de-
rived from the P surface[l3], and the network of (1 ,O) is one of the O’Keeffe’s
polybenzenes[lS].
Now let us try to obtain other types of Pearcenes based on infinite net-
works of triangles in space. The most elegant way to get such a network
is to consider the open packing of three regular polyhedra, i.e., tetrahedron,
octahedron and icosahedron whose faces are equilateral triangles[l8]. One ex-
ample is exhibited in Fig.S(c). If octahedra are placed on four of eight (111)
faces of the nodal icosahedron, while the octahedra of branch share two par-
allel, oppositely disposed faces with nodal icosahedra, a diamond structure is
obtained as an open packing of icosahedra and octahedra. In this structure
seven triangular faces meet at the vertex, hereby heptagonal rings are formed.
Depicting a honeycomb network with triangular (m,n) patches in Fig.8, we
can get a series of Pearcenes again having the labyrinth of diamond structure.
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576 FUJITA, YOSHUIA, AND NAKADA
Fig.8 Triangular or hexagonal patches in a honeycomb lattice.
One of examples with (2,O) is shown in Fig.S(d). They have 56S,,, carbon
atoms per unit cell. The electronic structures are metallic for (l,O), (2,O) and
(2,1), and insulating for (l,l), (2,2), (3,O). The network with the ( 1 , O ) patch
is topologically equivalent to the Vanderbilt and Tersoff’s graphitic sponge
from the D surface[l4].
Placing octahedra at nodal sites of the diamond network with four addi-
tional octahedra which serve as a branch, we obtain another diamond struc-
ture as shown in Fig.S(e). An octagonal ring is placed at each vertex where
eight triangles join. For each (m,n) of triangular patches, we find 325,,,
atoms in a unit cell. None of the networks in this series of Pearcenes have
been reported so far in our knowledge. Figure 9(f) shows the Pearcene with
(2,O) which is found to be metallic and further to have a flat band all over
the Brillouin zone. Other electronic structures in this series of Pearcenes are
insulating for (lJ), (3,O) and (2,2), and zero-gap semiconducting for ( 1 , O )
and (2,l) where the DOS behaves as E2 near EF because they have linear
dispersions around some Fermi points.
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS 577
Fig.9 Polyhedral frame for graphitic sponges (a)(c)(e) and the resultant network of graphitic sponges by using the patch (2,O).
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FUJITA, YOSHIDA, AND NAKADA
55. Localized Electronic States at Graphite Edges
Fullerene molecules form closed cages without an edge. As for fullerene
tubules, when Iijima first observed multi-layered tubules[20], they were sug-
gested to be composed of concentric and cylindrical graphite sheets, which
leads us to believe perfectibility of the crystallization. It is more confirmed by
the successful synthesis of single layered tubules[21]. Recently another type of
structural model for tubules is proposed, however, to explain the observation
of the presence of singular fringes in multiple layers, where it is proposed that
a multiple layered tubule is partly constructed by a scroll-shaped graphite
sheet[22]. Now we are microscopically faced with the presence of a graphite
edge.
There are some theoretical works for the electronic states of peripheral
graphite systems, mainly based on semi-empirical molecular orbital calcu-
lations. Stein and Brown[23] reported that overall electronic structures for
hexagonal graphite patches strongly depend on the shape of peripheral struc-
tures. Hosoya et al.[24] examined the distribution of the highest occupied
molecular orbital for the one-dimensional system of graphite strips, and
demonstrated that it localizes at edge sites or widely spreads over the sys-
tem depending on the peripheral shape. In this last chapter, we show the
electronic band structures of one-dimensional strip-shaped graphites with
two kinds of typical peripheries, and discuss the existence of a peculiar edge
state at the Fermi level.
When we cut a graphite sheet along a straight line, two typical kinds
of peripheral shapes appear depending on the axial directions, i.e., (1,l) or
(1,O). Usually they are named as the armchair edge for the former and the
zigzag one for the latter. Further if we cut it out along another parallel line,
we obtain a one-dimensional strip-shaped graphite as displayed in Fig.10.
The width N is define by the number of the dimer lines for a strip with
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS 579
Fig.10. Two typical edge structures in graphite. (a) armchair. (b) zigzag.
the armchair edge and by the number of the zigzag lines for a strip with
the zigzag edge, respectively. We calculate the energy band structure for
both systems based on the tight binding model for the 7r electron system.
An example is depicted in Fig.11 for each system. We note that the Fermi
level is at E = 0. For the armchair system (Fig.ll(a)), both the valence
band top and the conduction band bottom are located at the r point. The
direct energy gap tends to vanish with increasing width. We can ascribe this
degeneracy to the one of the energy bands for an infinite graphite sheet at
the I< point in the hexagonal Brillouin zone. We emphasize here that in the
armchair system the global band structure is mapped to the one of graphite
in the limit of infinite width without exception.
While in the zigzag system (Fig.ll(b)), a striking difference appears in the
energy band structure. The degeneracy between the valence and conduction
bands in the graphite sheet should be mapped only to the points at ,k = h / 3 .
Nevertheless we find almost flat bands sit steady at the Fermi level. The
electronic state in these flat bands can be characterized as the edge states
where the electron is localized near the zigzag edge. We can express the edge
states in a mathematically analytic form[25]. We should note that the edge
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580 FUJITA, YOSHIDA, AND NAKADA
Fig.11. armchair and (b) zigzag edges.
Energy bands for one-dimensional band-shaped graphites with (a)
state at k = T /2 which consist of only the edge site states will be analytically
connected to the state at k = f7r/3 where the electronic state spreads over
the lattice in the infinite width limit.
In a long history of carbon materials, one of our interests lies on how
to control the macro-, meso- and micro-scopic structure. For example, con-
trolling the degree and type of porosity of activated carbons we gain specific
adsorption capacity. In spite of the remarkable usefulness of carbon materi-
als, the functionality has not been investigated from a microscopical point of
view. The appearance of the fullerene family tells us the importance of pen-
tagonal and heptagonal rings in a basic hexagonal network of sp2 carbons.
Moreover, e.g., if we control a size of the graphite sheet, we have to pay atten-
tion to the existence of edges, which is found to affect the overall electronic
state of a graphite patch. Finally we would like to introduce a noteworthy
magnetic measurement for disordered carbon materials. Nakayama et aZ.[26]
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POLYMORPHISM OF EXTENDED FULLERENE NETWORKS 58 1
found that a certain type of activated carbon fibers(ACF), which has huge
specific surface areas ranging to 3000m2/g and is believed to consist of an
assembly of micro-graphite with a dimension of ca. 20Ax20A, shows param-
agnetic behavior near room temperature, although graphite and other ACF are diamagnetic. It may suggest that an amount of mysterious density of
states exists near the Fermi level, which might be related to the edge states
above.
The authors is grateful to E. Osawa, H. Hosoya, A. Nakayama, K. Kusak-
abe and K. Miyazaki for helpful discussions. They also thank K. Wakabayashi
for the preliminary energy-band calculations of peripheral graphites shown in
Fig.10 and 11. This work has been supported by a Grant-in-Aid for Scientific
Research from the Ministry of Education, Science and Culture, Japan and
partly by Mikitani Science Foundation(M.F.).
References [l] Kroto H,, Heath J. R., Curl S.C.R.F. and Smalley R. E., Nature, (1985) 318,
[2] lijima S., Ichihashi T. and Ando Y., Nature, (1992) 356, 776. [3] Iijima S., Ajayan P. M. and Ichibashi T., Phys. Rev. Lett., (1992) a, 3100. [4] Fujita M., Umeda T. and Yoshida M., Phys. Rev. B, (1992) 45, 13834. Fujita
[5] Yoshida M. and Osawa E., Bull. Chem. SOC. Jpn., (1995) 68, 2073 and ibid.
[6] Fujita M., Yoshida M. and Osawa E., Fullerene Sci. Tech., (1995) 3, 93. [7] Dunlap B. I., Phys. Rev. B, (1992) 46, 1933. [8] Itoh S., Ihara S. and Kitakami J., Phys. Rev. B, (1993) 47, 1703. [9] Ihara S., Itoh S. and Kitakami J., Phys. Rev. B, (1993) a, 5643.
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