LT 21 Proceedings of the 21st International Conference on Low Temperature Physics Prague, August 8-14, 1996

Part $5 - LT Properties of Solids 2: Lattice structure and dynamics

Effect of Topological Defects in Graphite.

M a s a t s u r a Igami , = K y o k o Nakada , = M i t s u t a k a Fuj i ta , ~ and Koichi K u s a k a b e b

alnstitute of MateriMs Science, University of Tsukuba, Tsukuba 305, Japan

blnstitute for Solid State Physics, University of Tokyo, Tokyo 106, Japan

The simplest topological defect in a graphite sheet can be introduced by switching one bond with the movement of two carbon atoms, which transforms four hexagons to two pentagons and two heptagons. Putting the topological defects periodically in a graphite sheet, we study how the electronic state changes based on the tight binding model. Depending on the combination of geometrical parameters which specify the arrangement of the defects, we find that the electronic states can be classified into the zero-gap semiconducting, semi- metallic and metallic cases. The localized charge density states at the defect sites emerge just above the Fermi level. Curious states having a flat band at the Fermi level appear in some particular geometrical arrangements of the defects. In addition, we propose to design a series of graphitic polymers with a flat lowest-unoccupied band.

1. I n t r o d u c t i o n

Carbon nanotubes have been recognized as per- fect graphite sheets of cylindrical form without de- fects. However, some recent experiments reveal the presence of defects in nanotubes which gives rise to the change of morphology of tubes more or less[l]. Perfect nanotubes should be defined as the hexago- nal networks of sp 2 carbon forming seamless cylin- ders. One of possible types of defect in the networks is the topological defect, which can be recognized as polygonal defects other than hexagon. Their intro- duction keeps the three coordinated connection of the network. In this paper, we study how the topo- logical defects affect the electronic states and change the basic band structure of graphite.

The simplest topological defect in a planar hexag- onal network without changing the Euler character- istic of the structure is double pairs of pentagonal and heptagonal defects, which corresponds to the azupyrene structure as shown in Fig.1. We can pro- duce it by switching one bond with the movement of two carbon atoms in a graphite sheet, and therefore four hexagons are transformed into two pentagons and two heptagons. This kind of topological defects might need less energy, in particular, in curved struc- tures like tubes.

2. Band s tructure o f defect ive graphite

Putting the azupyrene defects periodically in a graphite sheet, we study how the electronic state

changes from that of graphite based on the tight binding band model. The electronic states of the defective tubes can be derived by imposing the peri- odic boundary condition on it.

Depending on the combination of geometrical pa- rameters which specify the arrangement of azupyrene defects, we find that the electronic states of the de- fective system drastically vary. The criterion for the classification of the electronic state is given for the geometrical parameters on the basis of the three sub- lattices structure relative to the hexagonal network as displayed in Fig.2, where one third of hexagons shaded in dark forms a sublattice. Hereafter we call if the vector spanned by the primitive vectors ff and

connects the same sublattice sites, the vector sat- isfies the "Kekul6 rule".

Fig.1 Azupyrene defect m a graphite sheet.

Fig.2 The vector which satisfies the kekul~ rule.

Czechoslovak Journal of Physics, Vol. 46 (1996), Suppl. $5 2715

Tile superlattice of tile defects in a graphite can be specified by the two vectors i 'and rfi. If one or the other vector does not satisfy the Kekuld rule or the both vectors do not satisfy the Kekul~ rule, the system kcops a zero-gap semiconductor as same as graphite. Tile valence and conduction bands touch at two points in the first Brillouin zone and have linear dispersion around them like around the K point of graphite. We exhibit the band structure and the den- sity of states(DOS) near the Fermi level for the case that / '=(-3,6)(- -3E+6b') and rf i=(7,0)in Figs.3(a) and (b). We should mention that the critical change emerges a little above the Fermi level showing a sharp peak of DOS, the states of which are localized in the defect sitcs(Fig.3(c)).

,:0 --- -f:- ----:- . . . .

r u M X r - I

(a) o

(b) m

(c) Fig.3 Band sturucture(a), DOS(b) and unit cell with

charge density(c) when i'=(-3,6) and rfi=(7,0).

(a) (b)

v

m

(c) Fig.4 Band sturucture(a), DOS(b) and unit cell with

charge density(c) when !'=(-3,6) and r~=(6,0).

(a) (b)

v

m

(c) Fig.3 Band sturucture(a), DOS(b) and unit cell(c)

when 1"=(-3,6) and ~=(3,0).

While when the both of two geometrical vec- tors satisfy the Kekul~ rule the electronic state varies drastically to be metallic. In the circum- stance the parabolic valence and conduction bands

touch each other giving tile finite DOS at Fermi level(Figs.4(a)and(b)). Tile Fermi point always ap- pears at F point where the state of K point in graphite is mapped by introducing the supercell structure. Same as tile case above, tile localized states at the defect sites situate over the Fermi level(Fig.4(c)).

Fig.6 the wave function of flat band state at F point.

(b)

(c)

(d)

Fig.7 Graphite ribbon(a) and a series of polymer with flat band.

If we put one of the vectors as (3,0) in the case the system becomes metallic, a partly flat band comes out at the conduction band along one of the sym- metric lines from F point(Figs.5(a)-(c)) . The state in the flat band has the nodal lines sandwiched be- tween the cis-polyacetylene textures. As an exam- ple the wave function at F point in the flat band is depicted in Fig.6. This lead us to design the new one-dimensional materials in which the lowest unoc- cupied band is flat. Based on the graphite ribbon in Fig.7(a)[2], we can obtain a series of polymer by in- troduce the topological defect periodically(Figs.7(b)- (d)). Although they are insulating except (b), we expect the electron doping to the flat band may in- duce some peculiar properties[3]. Optical properties relating to the excited state should be interesting.

Re fe rences

[1] T. W. Ebbesen and T. Takada, Carbon, 33, 973(199 )

[2] M. Fujita, M. Yoshida and K. Nakada. Fnllerene Sci. Technol., 1996, Vol.4, (in press).

[3] N. Shima and H. Aoki, Phys. Rev. Left, 71, 4389(1993)

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