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Carbon nanotube elbow connections and tori
PHYSICS LETTERS A
PhysicsLettersA 170 (1992) 37—40 North-Holland
Carbon nanotube elbow connections and tori L.A. Chernozatonskii Institute of ChemicalPhysics, Russian Academy ofSciences, 4 Kosygin Street, 117334 Moscow, Russian Federation Received 15 April 1992; accepted for publication 13 July 1992 Communicated by V.M. Agranovich
Cage carbon structures of an elbow element which connect two tubelenes are modeled as well as torus-like clusters formed by carbon elbows. Electron-guides on the base of such nanostructures are proposed and some oftheir physical properties are discussed.
Nowadays researchers are particularly interested in various cluster compounds consisting of a graphite surface curved by introduced octagons [1], heptagons [21, and pentagons [3—7]in its hexagon net. Sphere-like macromolecules, fullerenes, are the best known of them, including the remarkable footballene C60 [4—7]. Earlier [8] we have analyzed a new family of barrel-like and tube-like clusters t~C2mn. Their main fragment is a cylinder surface formed by n belts of m hexagons attached to each other (the C—C bonds are parallel to the cylinder axis). Around the same time the semi-metallic character ofthe electron spectrum of “infinite” carbon tubes: t-C1 2n [9] and C2om [10] (where the C—C bonds are normal to the axis) was proved theoretically. These calculations as well as the experimental discovery of microtube carbon structures [11,12] inspired our interest in modeling structures, formed by tubelenes: original electron-guides, which can become the base of integral nanoelectronics. First of all we would like to draw attention to constructing cluster fragments connecting two tubes rotated by an angle Here we discuss the elbow connection of two identical tubes. Its surface is part of a torus (fig. 1 a). So it should include a saddle-like fragment. Such a fragment has two principal curvatures of the surface near the inner radius r one positive and one negative. The only way to get it is to introduce polygons 2 bonding) [1].of more than six carbon atoms (for sp The simplest “saddle” is formed by the fragment ~.
S8, consisting of an octagon surrounded by eight hexagons [1] (fig. Ib). We have also found a saddle S7 consisting of two heptagons surrounded by ten hexagons (fig. 1c). The elbow element geometry (fig. la) leads naturally to the estimation formulae connecting the diameter D of the corresponding tubes with the angle ~ and the outer (R) and inner (r) radii of this element’s outer (L) and inner (1) arcs: D=(L—l)~~=R—r,L=~R, 1=9~r.
(1)
The “outer” surface ofan elbow element has two different but both positive curvatures. So it can be formed by a row of hexagons attached to each other with insertion of either a vertical (F~,figs. 1 c, 1 d) or a horizontal (Fh, fig. 1 e) “fullerene” fragment (two pentagons connected and surrounded by hexagons). Thus using fragments depicted in fig. 1 it is possible to model “elbows” connecting tube pairs of various diameter. Elbows connecting th~C2mfltubes, ending by n/2 trapezoidjars, are the simplest to construct with the help of S~and F~fragments. eh-C~ elbows for n =8, 10 are shown in fig. 2. On the base of formulae (1) we estimate the rotation angle ~, and the outer radius of the elbow for the most interesting tubes th~C2om[10] linked with halves of C60 and C70 fifth-order symmetrical fullerenes: çon~70°, R~L (D~5a,L~lOa, 1~4a,where a= 1.4 A is the average bond length in the C60 cluster). So five connections to each a torus outer cageattached radius R 14 Aother and form diameter D with 7 A. Such a torus-like C 440 macromolecule would have the
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers BY. All rights reserved.
37
Volume 170, number 1
PHYSICS LETTERS A
19 October 1992
~ (b)
(a)
(c)
(d)
(e)
Fig. 1. (a) Scheme of an “elbow” connecting identical tubes (diameter D) with an S
8 element, a “Mackay saddle”, having an octagon near the internal radius r; (b) S7 element, an “Anna saddle”, having two heptagons attached to each other; (c) and (d) F. elements having two pentagons of external radius R; (e) Fh element having two pentagons at the opposite sides (from the external radius).
V
/
/,~~/N
/~‘Th~
N ~
N
hi h
/ /
Fig. 2. Scheme ofelbows eh-C70 (left) and eh-C98 (right) connecting h-tubes th-C16,,, and th-C20m; the h axis lies in the horizontal plane of the tube profile.
following real sizes: larger dimension DT = 2R +1w 31 A and “thickness” dT = D +1w 10 A, where lw~3A is the van der Waals diameter. The elbows ev-C1~2mn, connecting v-tubes ending by m triangular jars are constructed of S7 and F~(or Fh) fragments. Such models for m = 7, 8, 9 are presented in fig. 3. It shows also a torus cage model T~38
C340 consisting of five elbows for v-tubes t-C1 8n• The dimensions of its macromolecule are as follows: DT=2R+1w~25 A, dT~10 A, less than those ofthe C440 torus. It is convenient to use molecule “orange-peel” graphs for building the Kekulé structure and finding out the symmetry type of the elbows. Such graphs for
Volume 170, number I
PHYSICS LETTERS A
19 October 1992
Fig. 3. Models of~Ibowsev-C
46, ev-C62 and ev-C68 connecting v-tubes C,4,,, C16,, and C18,,, the torus model is situated in the center of the
figure.
e~- c62
e~- C46
~
ev-C&6
~
Il~I
elI- C70
elI - Cg~
Fig. 4. Molecule graphs of elbow carbon cluster elements: the upper row: ~‘v”-elbows, where two (0) atoms belong to two adjacent heptagons: the lower row: “h”-elbows. (The mean coordination number of the whole elbow net is 6. which is the same as for a plane net.) 39
Volume 170, number I
PHYSICS LETFERS A
the elbows demonstrated in figs. 2 and 3 are presented in fig. 4. Construction of ev and eh elbows for tubes of greater diameters, helical microtubeles including elbows [11] follows the same scheme, based on S7, S8, F~,and Fh fragments. We are going to demonstrate them in later works. It is natural to assume that the elbow net as well as fullerenes can include other “three-bond” atoms besides carbon, such as B, Si etc. The existence of an inner “channel” allows one to create a continuous chain structureof a metal “vein” at tube bends by doping metal atoms inside the tubes. This provides the possibility of a quasi-single-dimension electric charge transfer in such an original nanocable with a turning of the current spatial direction. Tori (ring tube structures) are of great interest due to their potential magnetic and superconductivity properties. Most probably they inherit the remarkable peculiarity of tubes, metal type conductivity [9,10]. The existence of strong electron—phonon interaction at barrel-like (running round the torus) vibrations of the torus cage due to the presence of a large tangent component of the electric polarization (the torus surface is not mirror symmetrical regarding the inner and outer spaces of the torus) can lead
40
19 October 1992
to its superconductivity. The ring torus structure is also the origin for the appearance of a strong magnetic moment, created by ring currents. I am sincerely thankful to E.G. Galpern, D.A. Kirjnitz, I.V. Stankevitch for useful discussions and especially to A.G. Rogovina for remarkable transformations of speculations into real models.
References [1] A.L. Mackay and H. Terrones, Nature 352
(1991) 762. [2] P.W. Fowler, J. Chem. Soc. Faraday Trans. 87 (1991) 1945. [31H.W. Kroto, Science 243 (1988) 1139. [4] D.E.H. Jones, New Sci. 32 (1966) 245. [5] E. Osawa, Kasaku (Kyoto) 25 (1970) 854. [6] D.S. Bochvar and E.G. Halpern, Dokl. Akad. Nauk SSSR 209 (1973) 619. [7] H.W. Kroto et al., Nature 318 (1985) 162.
[81L.A. Chernozatonskii, Phys. Lett. A 166 (1992)55. [9] E.G. Galpern et al., Pis’ma Zh. Eksp. Teor. Fiz. 55(1992) 469. [10] J.W. Mintmire, B.I. Dunlap and CT. White, Phys. Rev. Lett. 68 (1992) 631. [11] S. lijima, Nature 354 (1991) 56. [12] L.A. et al., in: June Proc.1992, Adriatico Conf.Chernozatonskii on Clusters and fullerenes, Trieste,Research J. Mod. Phys., tobe published.
PhysicsLettersA 170 (1992) 37—40 North-Holland
Carbon nanotube elbow connections and tori L.A. Chernozatonskii Institute of ChemicalPhysics, Russian Academy ofSciences, 4 Kosygin Street, 117334 Moscow, Russian Federation Received 15 April 1992; accepted for publication 13 July 1992 Communicated by V.M. Agranovich
Cage carbon structures of an elbow element which connect two tubelenes are modeled as well as torus-like clusters formed by carbon elbows. Electron-guides on the base of such nanostructures are proposed and some oftheir physical properties are discussed.
Nowadays researchers are particularly interested in various cluster compounds consisting of a graphite surface curved by introduced octagons [1], heptagons [21, and pentagons [3—7]in its hexagon net. Sphere-like macromolecules, fullerenes, are the best known of them, including the remarkable footballene C60 [4—7]. Earlier [8] we have analyzed a new family of barrel-like and tube-like clusters t~C2mn. Their main fragment is a cylinder surface formed by n belts of m hexagons attached to each other (the C—C bonds are parallel to the cylinder axis). Around the same time the semi-metallic character ofthe electron spectrum of “infinite” carbon tubes: t-C1 2n [9] and C2om [10] (where the C—C bonds are normal to the axis) was proved theoretically. These calculations as well as the experimental discovery of microtube carbon structures [11,12] inspired our interest in modeling structures, formed by tubelenes: original electron-guides, which can become the base of integral nanoelectronics. First of all we would like to draw attention to constructing cluster fragments connecting two tubes rotated by an angle Here we discuss the elbow connection of two identical tubes. Its surface is part of a torus (fig. 1 a). So it should include a saddle-like fragment. Such a fragment has two principal curvatures of the surface near the inner radius r one positive and one negative. The only way to get it is to introduce polygons 2 bonding) [1].of more than six carbon atoms (for sp The simplest “saddle” is formed by the fragment ~.
S8, consisting of an octagon surrounded by eight hexagons [1] (fig. Ib). We have also found a saddle S7 consisting of two heptagons surrounded by ten hexagons (fig. 1c). The elbow element geometry (fig. la) leads naturally to the estimation formulae connecting the diameter D of the corresponding tubes with the angle ~ and the outer (R) and inner (r) radii of this element’s outer (L) and inner (1) arcs: D=(L—l)~~=R—r,L=~R, 1=9~r.
(1)
The “outer” surface ofan elbow element has two different but both positive curvatures. So it can be formed by a row of hexagons attached to each other with insertion of either a vertical (F~,figs. 1 c, 1 d) or a horizontal (Fh, fig. 1 e) “fullerene” fragment (two pentagons connected and surrounded by hexagons). Thus using fragments depicted in fig. 1 it is possible to model “elbows” connecting tube pairs of various diameter. Elbows connecting th~C2mfltubes, ending by n/2 trapezoidjars, are the simplest to construct with the help of S~and F~fragments. eh-C~ elbows for n =8, 10 are shown in fig. 2. On the base of formulae (1) we estimate the rotation angle ~, and the outer radius of the elbow for the most interesting tubes th~C2om[10] linked with halves of C60 and C70 fifth-order symmetrical fullerenes: çon~70°, R~L (D~5a,L~lOa, 1~4a,where a= 1.4 A is the average bond length in the C60 cluster). So five connections to each a torus outer cageattached radius R 14 Aother and form diameter D with 7 A. Such a torus-like C 440 macromolecule would have the
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers BY. All rights reserved.
37
Volume 170, number 1
PHYSICS LETTERS A
19 October 1992
~ (b)
(a)
(c)
(d)
(e)
Fig. 1. (a) Scheme of an “elbow” connecting identical tubes (diameter D) with an S
8 element, a “Mackay saddle”, having an octagon near the internal radius r; (b) S7 element, an “Anna saddle”, having two heptagons attached to each other; (c) and (d) F. elements having two pentagons of external radius R; (e) Fh element having two pentagons at the opposite sides (from the external radius).
V
/
/,~~/N
/~‘Th~
N ~
N
hi h
/ /
Fig. 2. Scheme ofelbows eh-C70 (left) and eh-C98 (right) connecting h-tubes th-C16,,, and th-C20m; the h axis lies in the horizontal plane of the tube profile.
following real sizes: larger dimension DT = 2R +1w 31 A and “thickness” dT = D +1w 10 A, where lw~3A is the van der Waals diameter. The elbows ev-C1~2mn, connecting v-tubes ending by m triangular jars are constructed of S7 and F~(or Fh) fragments. Such models for m = 7, 8, 9 are presented in fig. 3. It shows also a torus cage model T~38
C340 consisting of five elbows for v-tubes t-C1 8n• The dimensions of its macromolecule are as follows: DT=2R+1w~25 A, dT~10 A, less than those ofthe C440 torus. It is convenient to use molecule “orange-peel” graphs for building the Kekulé structure and finding out the symmetry type of the elbows. Such graphs for
Volume 170, number I
PHYSICS LETTERS A
19 October 1992
Fig. 3. Models of~Ibowsev-C
46, ev-C62 and ev-C68 connecting v-tubes C,4,,, C16,, and C18,,, the torus model is situated in the center of the
figure.
e~- c62
e~- C46
~
ev-C&6
~
Il~I
elI- C70
elI - Cg~
Fig. 4. Molecule graphs of elbow carbon cluster elements: the upper row: ~‘v”-elbows, where two (0) atoms belong to two adjacent heptagons: the lower row: “h”-elbows. (The mean coordination number of the whole elbow net is 6. which is the same as for a plane net.) 39
Volume 170, number I
PHYSICS LETFERS A
the elbows demonstrated in figs. 2 and 3 are presented in fig. 4. Construction of ev and eh elbows for tubes of greater diameters, helical microtubeles including elbows [11] follows the same scheme, based on S7, S8, F~,and Fh fragments. We are going to demonstrate them in later works. It is natural to assume that the elbow net as well as fullerenes can include other “three-bond” atoms besides carbon, such as B, Si etc. The existence of an inner “channel” allows one to create a continuous chain structureof a metal “vein” at tube bends by doping metal atoms inside the tubes. This provides the possibility of a quasi-single-dimension electric charge transfer in such an original nanocable with a turning of the current spatial direction. Tori (ring tube structures) are of great interest due to their potential magnetic and superconductivity properties. Most probably they inherit the remarkable peculiarity of tubes, metal type conductivity [9,10]. The existence of strong electron—phonon interaction at barrel-like (running round the torus) vibrations of the torus cage due to the presence of a large tangent component of the electric polarization (the torus surface is not mirror symmetrical regarding the inner and outer spaces of the torus) can lead
40
19 October 1992
to its superconductivity. The ring torus structure is also the origin for the appearance of a strong magnetic moment, created by ring currents. I am sincerely thankful to E.G. Galpern, D.A. Kirjnitz, I.V. Stankevitch for useful discussions and especially to A.G. Rogovina for remarkable transformations of speculations into real models.
References [1] A.L. Mackay and H. Terrones, Nature 352
(1991) 762. [2] P.W. Fowler, J. Chem. Soc. Faraday Trans. 87 (1991) 1945. [31H.W. Kroto, Science 243 (1988) 1139. [4] D.E.H. Jones, New Sci. 32 (1966) 245. [5] E. Osawa, Kasaku (Kyoto) 25 (1970) 854. [6] D.S. Bochvar and E.G. Halpern, Dokl. Akad. Nauk SSSR 209 (1973) 619. [7] H.W. Kroto et al., Nature 318 (1985) 162.
[81L.A. Chernozatonskii, Phys. Lett. A 166 (1992)55. [9] E.G. Galpern et al., Pis’ma Zh. Eksp. Teor. Fiz. 55(1992) 469. [10] J.W. Mintmire, B.I. Dunlap and CT. White, Phys. Rev. Lett. 68 (1992) 631. [11] S. lijima, Nature 354 (1991) 56. [12] L.A. et al., in: June Proc.1992, Adriatico Conf.Chernozatonskii on Clusters and fullerenes, Trieste,Research J. Mod. Phys., tobe published.