electronic structure of chiral graphene tubules

(a) (n,,rz,)=(IO,O) T 1.0 E ma bl ‘i5 = $ .= 0.5 5 3 3 &!?I 2 0.0 -4.0 -3.0 -2.0 -1 .o 0.0 EneWy, 1.0 2.0 3.0 4.0 04 h,7n,)=P70) 8 1.0 II! % 6 E = 8 .= 0.5 g 2 0.0 ,,ig,,,d .I ,,,,,,, 2 I 5 FIG. 2. (a) Graphene tubules are made by rolling a graphene sheet into a cylinder. The tubules are uniquely determined by their lattice vectors ck The chiral angle is denoted by 8, while a, and a, denote the unit vectors of graphite. (b) Possible vectors for chiral fibers. The circled dots and dots, respectively, denote metallic and semiconducting behavior for each fiber. -4.0 -3.0 -2.0 -1.0 0.0 Enera% 1.0 2.0 3.0 FIG. 3. Electronic density of states for two (n,,n,) zigzag fibers: (a) (10,O) and (b) (9,0). there is no mirror symmetry for a fiber, both right- (6 > 00) and left- (8 < 0”) handed optical isomers are possible, as are also chiral fibers with 161 > 30”. The chiral angle 0 should determine the optical activity of the fiber and the speed (or stability) of fiber growth. From a theoretical standpoint, graphite tubules are in- teresting as the embodiment of a one-dimensional ( 1D) periodic structure along the fiber axis. Confinement in the radial direction is provided by the monolayer thickness of the fiber. In the circumferential direction, periodic bound- ary conditions apply to the enlarged unit cell that is formed in real space and the subsequent zone folding that occurs in reciprocal space. For the fiber geometry, there is some mix- ing of the 4 2p,) and a( 2s and 2p,,) carbon orbitals due to the fiber curvature, but this mixing is small and can be neglected near the Fermi level.6 Thus, we consider only 7r orbitals. The two-dimensional (2D) energy dispersion re- lations for n- bands of graphite, E,,,, are given by9 E,,=hyO[ ,.,COS(+)COS(y) ch-k = 25-m, (2) where m is an integer, we get 1D energy bands for general chiral structures. In other words, 1D energy bands can be obtained by sIicing the 2D energy dispersion relations of JZq. ( 1) in the directions expressed by Eq. (2). In Figs. 3 (a) and 3 (b), the density of states for two zigzag fibers with (yl& = ( 10,O) and (9,0), respectively, are plotted in units of states per unit cell of 2D graphite. We also plot the corresponding density of states of 2D graphite (dotted lines) in both figures for comparison. The l/ @ singularities characteristic of 1D energy bands ap- pear at the band edges of each energy band. In Fig. 3 (a), there is an energy gap at the Fermi level (E-O), while we have a finite density of states for Fig. 3 (b). Thus, we can have both semiconducting [Fig. 3(a)] and metallic [Fig. 3(b)] fibers by merely changing the fiber diameter. The energy gap for the semiconducting fibers decreases with increasing fiber diameter d and in the limit of d- 00, we obtain the 2D case of a zero-gap semiconductor. The con- dition for a fiber to be metallic is 2n1+n2=3q, (3) where q in an integer. This condition is easily obtained by substituting the k vector of the degenerate point of 2D graphite (corner of the hexagonal Brillouin zone) into Eq. (2). Since two optical isomers for - 8 and 8 give the same kyz -j-4 cos2 2 ( )I “’ , (1) where ‘yo is the nearest-neighbor overlap integral.” Elimi- nating k, or kv by using the periodic boundary condition, results for the energy gap (optical selection rules are dif- 2205 Appl. Phys. Lett., Vol. 60, No. 18, 4 May 1992 Saito et al. 2205 Downloaded 31 Aug 2011 to 222.195.80.102. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions

ferent), we show chiral vectors only for - 30”<8<0” in Fig. 2(b). Here, the metallic and semiconducting fibers are de- noted by circled dots and simple dots, respectively. In par- ticular, all armchair fibers are metallic, and zigzag fibers are metallic when nl is a multiple of three. A finite density of states results from the crossing of two 1D energy bands at degenerate points of the 2D graph- ite energy-band structure. Metallic 1D energy bands are generally unstable under a Peierls distortion. However, the Peierls energy gap obtained for the metallic cases is found to be greatly suppressed by increasing the fiber diameter and the Peierls gap quickly approaches the zero-energy gap of 2D graphite.67*’ Thus, if we consider finite temperatures or fluctuation effects, such a small Peierls gap can be ne- glected. It is surprising that the calculated electronic struc- ture can be either metallic or semiconducting depending on the fiber diameter and on the chiral angle 8, though there is no difference in the local chemical bonding between the carbon atoms, and no doping impurities are present. If the distribution of ch vectors shown in Fig. 2 is uniform, l/3 of the fibers will be metallic and 2/3 semi- conducting. However, we may obtain a larger fraction of metallic fibers if the initial seed of the fiber caps is centered about a pentagon, which yields an armchair fiber. If the initial seed is not a pentagon but a hexagon, growth of the planar graphite structure seems more likely. In this sense, nature may prefer armchair-type fibers which are metallic for all (p,p) . From the results of this letter, one could imagine de- signing a minimum-size conductive wire consisting of two concentric graphene tubules with a metallic inner tubule covered by a semiconducting (or insulating) outer tubule. These concepts could further lead to the design of meso- scopic metal-semiconductor devices with cylindrical geom- etry which are optically active, without introducing any doping impurities. There are, of course, many other possi- bilities for arranging graphene tubules with interesting po- tential applications which could be stimulated by the re- sults presented here. Two of the authors (R.S. and M.F.) have carried out this work while they are visiting scientists at MIT as Over- seas Research Scholars of the Ministry of Education, Sci- ence and Culture of Japan. We gratefully acknowledge Na- tional Science Foundation Grant No. DMR88-196 for support for this research. ’ M. S. Dresselhaus, G. Dresselhaus, and R. Saito, Phys. Rev. B 45,6234 (1992). ‘S. Iijima, Nature 354, 56 (1991). ’ M. Endo, H. Fujiwara, and E. Fukunaga, Abstract of Second C,, Sym- posium (Japan Chemical Society, Tokyo, 1992), pp. 101-104. 4G. Dresselhaus, M. S. Dresselhaus, and P. C. Eklund, Phys. Rev. B 45, 6923 (1992). ‘F. Diederich and R. L. Whetten, Act. Chem. Res. 25, 119 (1992). ‘R. Saito. M. Fuiita. G. Dresselhaus, and M. S. Dresselhaus, MRS Symp. Proc. 247,” 333 (1992). . ‘M. Fujita, R. Saito, G. Dresselhaus, and M. S. Dresselhaus (unpub- lished) . ‘P. W. Fowler, Chem. Phys. Lett. 131, 444 (1986). ‘P. R. Wallace, Phys. Rev. 71, 622 (1947). “M. S. Dresselhaus and G. Dresselhaus, Adv. Phys. 30, 139 (1981). “J. W. Mintmire, B. I. Dunlap, and C T. White, Phys. Rev. Lett. 68, 631 (1992). 2206 Appl. Phys. Lett., Vol. 60, No. 18, 4 May 1992 Saito et a/. 2208 Downloaded 31 Aug 2011 to 222.195.80.102. Redistribution subject to AIP license or copyright; see http://apl.aip.org/about/rights_and_permissions