10.1.1.22 subbands in carbon nanotubes under radial deformation

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Journal of the Korean Physical Society, Vol. 37, No. 2, August 2000, pp. 85 88

Subbands in Carbon Nanotubes under Radial Deformation

Yong-Hyun Kim , Chan-Jeong Park and K. J. Chang

Department of Physics, Korea Advanced Institute of Science and Technology, Taejon 305-701

(Received 25 March 2000)

Radial deformations strongly affect the electronic structure of carbon nanotubes. We analyzethe wave functions in the (9, 0) and the (5, 5) nanotubes and nd that the variation of subbandswith radial deformation depends on the wave vector and the helicity. In metallic nanotubes localradial deformations provide internal barriers to electron transport while in small-gap nanotubesthey induce a metallic transition.

Carbon nanotubes, which consist of only carbonatoms, were rst discovered by Iijima [1] and can bethought of as a single layer of graphite that is wrappedinto a cylinder. The cylindrical nanotubes are very sta-ble and are regarded as the strongest bers ever; nan-otubes are extremely rigid to distortions along the tubeaxis whereas they are very exible to those perpendic-ular to the axis [2]. The electronic structure of a per-fect nanotube is known to be either metallic or semicon-ducting [2–6], depending on both the tube diameter andthe wrapping index ( n, m), where n and m describe theprojection of the circumferential vector onto the basisvectors of the graphite lattice. Armchair ( n , n) nan-otubes have a band degeneracy between the so-calledπ and π bands, which are even and odd under mir-ror symmetry operations. Since these two bands crossat the Fermi level, armchair nanotubes exhibit metallicconduction. In ( n, m) nanotubes, where n −m is a multi-ple of 3, a small gap appears due to the curvature effect;in all others, a large gap appears. Recently, a eld-effecttransistor based on an individual nanotube molecule wasdemonstrated [7], and single-electron tunneling and res-onant tunneling effects through single molecular orbitalswas observed [8,9], indicating that nanotubes are promis-ing materials for future molecular devices.

In nanotube-based devices, nanotubes undergo variousdeformations from the perfect cylindrical form. Nan-otubes bent by the tip of an atomic force microscope orby crossing over electrodes undergo mechanical deforma-tions that atten the tubes (see Fig. 1). Many prototypedevices made of carbon nanotubes inevitably contain de-fective regions because metal-nanotube, tip-nanotube,and nanotube-substrate interactions introduce structuraldistortions, such as bends, twists, and kinks [7–13].Thus, understanding the effect of mechanical deforma-tions on the electronic structure and the transport prop-

E-mail: [email protected]

erties is very important for device design; furthermore,the manipulation of such mechanical deformations couldopen a new eld in nanotube applications.

Recently, several theoretical calculations have beendone to investigate the effect of bending distortions invarious nanotubes [14,15]. For attening deformations, asevere modication of the electronic structure was found;radial deformations open the gap in metallic armchairnanotubes if mirror symmetries are broken, while inzigzag nanotubes, the gap closure occurs [16]. This studyprovides a clue for understanding the ballistic conductionof armchair nanotubes and the multiple-quantum-dot be-havior of bent nanotubes [8,10].

In this work, we investigate more precisely the effectsof radial deformations on the electronic structure of (5,5) and (9, 0) nanotubes. Since subbands of nanotubesoriginate from the periodicity along the circumference,radial deformations which perturb the periodicity mod-ify the dispersion of subbands. We nd that radial de-formations induce an overlapping of the wave functions,and the modication of each subband depends on thecorresponding orbital characteristics.

To analyze the characteristics of wave functions in nan-otubes, we start from a two-dimensional graphitic sheet.Figure 1(c) shows the hexagonal unit cell with two basisatoms denoted by A and B in the graphite lattice with

Fig. 1. The atomic structures of (a) perfect and (b) radi-ally deformed (5, 5) nanotubes. In (b), the interlayer distanced is 3.4 A. (c) The graphitic sheet of graphite, and the hexag-onal unit cell with the basis atoms A and B.

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Fig. 2. (a) The real part of f( k )| f( k ) | . The left (right) inset

shows the smallest building block in armchair (zigzag) nan-otubes. The folded zones (heavy lines) of the (b) armchairand (c) zigzag nanotubes in the graphitic BZ. Dotted linesdenote allowed k -point lines in the nanotubes.

the lattice constant a (= √ 3dC −C ), where dC −C is thedistance between two neighboring carbon atoms. Usingtwo Bloch functions in the π-electron approximation [17],which are constructed from the π orbitals for A and B,we can express the Hamiltonian for the graphite sheet as

H = 2 p tf(k )tf(k ) 2 p

, (1)

where 2 p is the energy of the 2 p orbital of the C atom,t is a positive parameter for the overlap integral repre-

senting the hopping rate between two neighboring car-bon atoms, and f( k ) = eik x a/ √ 3 + 2 eik x a/ 2√ 3 cos k y a

2 .For the Hamiltonian, we easily obtain the eigenstatesψ± = 1

N (1, f( k )

|f( k ) | ), where N is a normalization factor.In this case, ψ+ corresponds to the higher energy state,the so-called π state, while ψ− stands for the lower en-ergy π state. The real part of f( k )

|f( k )| is +1 in the rstBrillouin zone (BZ) (see Fig. 2(a)) and −1 in the darkregions outside the rst BZ. Thus, ψ± are reduced to

1√ 2 (1, 1), which are simply represented as (+, −) and(+, +), respectively.

For armchair and zigzag nanotubes, a rectangular cell

containing four consecutive carbon atoms (see the insetsin Fig. 2(a)) is the smallest building block in construct-ing the nanotube lattice. Since the rectangular cell istwice as large as the unit cell of a graphene, the corre-sponding unit cell in reciprocal-lattice space is half of thehexagonal BZ, as shown in Figs. 2(b)-(c). In this case,the folding direction in the BZ is determined by the di-mension of the rectangular cell, and the band structurein the folded region turns into subbands of nanotubesnear the Fermi level. From the circumferential periodic-ity, we obtain the quantization rule for the wave vectoras kx L = q ·2π, where kx is the wave vector perpendic-ular to the tube axis ( y direction) in the hexagonal BZ,L is the circumferential length of the tube, and q is aninteger corresponding to the quantized momentum for

each subband. Then, we can select allowed k -point lineslying parallel to the tube axis, as shown in Figs. 2(b)-(c).Each k -point line forms a subband with a specic valueof q . For instance, the line of q = 0 passes through the

Γ point in the hexagonal BZ and gives rise to two linearbands in armchair nanotubes and to a singlet state [6]in zigzag nanotubes with small diameters. In this case,however, the folded band has a phase factor of ei 2k B ·r ,where k B is the wave vector at the zone boundary in thefolding direction. In the (5, 5) tube, when zone-foldingoccurs along the y direction, two atoms in the rectangu-lar cell are located at y = 0 with the phase factor of 1,and the other two at y = D y

2 have the phase factor of −1,where Dy is the cell length along the y direction. In the(9, 0) tube, since the zone-folding takes place along the xdirection, the phase factor is 1 for the two atoms at x = 0and −1 for the others at x = D x

2 , where D x is the cell

length along the x direction. Thus, the wave functionsof the subbands in nanotubes can be characterized bythose of the graphite sheet on the allowed k -point linesand the phase factors arising due to the folding effect.

Based on an analysis of the phase factors, we representthe two linear bands for q = 0 in armchair nanotubes,which are called the ascending π and the descending πbands, by the (+ , −, −, +) and the (+ , + , −, −) states,respectively. Since the allowed k -point lines for q = 1and 2 pass through similar regions to that for q = 0, asshown in Fig. 2(b), the characteristics of their subbandsare quite similar to the two linear bands for q = 0. In the(9, 0) nanotube, only the states of q = 3 and 4 intersect

the zone boundaries; thus, their wave functions are verydifferent from those for q = 0 −2; the unoccupied statesfor q = 0 − 2 are characterized by (+ , + , −, −) whereasthose for q = 3 and 4 are characterized by (+ , −, −, +).On the other hand, the occupied states have the char-acteristics of (+ , −, −, +) and (+ , + , −, −) for q = 0 − 2and q = 3 −4, respectively. We also nd the same featurein both rst-principles and tight-binding calculations.

With information for the wave functions of the sub-bands, we investigate the electronic structures of radiallydeformed (9, 0) and (5, 5) nanotubes. We employ Ter-soff’s empirical potentials [18] to obtain the optimized ge-ometries of deformed nanotubes and calculate the band

structure using a rst-principles pseudopotential method[19]. We also perform tight-binding calculations with aSlater-Koster-type non-orthogonal basis [20].

Perfect (3 i, 0) nanotubes ( i is an integer) have ideallyzero gap; however, because of the curvature effect [3],they are small-gap semiconductors. Figure 3 shows theband structure of the perfect and deformed (9, 0) tubes,as well as the energy variations of the subbands with theinterlayer distance d. As d decreases, the curvature effectis enhanced, shifting the Fermi point away from the highsymmetric K point; thus, the gap initially increases. Forfurther deformations, the small-gap nanotube becomesmetallic due to a lowering of the singlet state with q = 0;

in fact, gap closure occurs for d = 5.3 A. The lowering of the q = 0 state is in part due to an increase in the σ -π



Subbands in Carbon Nanotubes under Radial Deformation – Yong-Hyun Kim et al. -87-

Fig. 3. (a) The band structure for the perfect and de-formed (9, 0) nanotubes. (b) The variations of the subbandsnear the Fermi level under radial deformation. Here, E F rep-resents the Fermi level in the perfect tube, and the numbersdenote the value of quantized wave vector q .

hybridization effect in the curved regions [6].In the (9, 0) nanotube, since the q = 0 state is repre-

sented by the wave function (+ , + , −, −) for four consec-utive atoms in the rectangular cell, half of the bonds areformed by orbitals with different signs. If the rectangu-lar cells are repeated along the circumferential directionuntil the primitive cell is made, the bonds formed byorbitals with equal signs are aligned along the circum-ference. When the nanotube is attened by radial de-

Fig. 4. Contour plots of the charge densities for the sub-bands with q = 0 and 1 in (a) perfect and (b) deformed (9, 0)tubes. The q = 1 states are doubly degenerate, and d = 6.0A is chosen in (b). Filled circles represent the carbon atomsin the cross section.

Fig. 5. Contour plots of the charge densities for the (a) πand (b) π states at the Γ point in the deformed (5, 5) tubewith d = 3.4 A.

formations, the distance between two neighboring bondsalong the circumference is reduced in the curved regions.

Then, the overlap of the charge densities at the hexagoncenter increases, as shown in Fig. 4, and the energy of the q = 0 state decreases, closing the gap. We also ndthe same origin in the lowering behavior of the unoccu-pied states with q = 1 and 2. The q = 1 states are doublydegenerate in the perfect tube while they are split withdifferent variations under deformation. One of the de-generate states, which has more overlap of the chargedensities at the hexagon center in the curved regions,decreases under deformation while the other with chargedensities accumulated in the attened regions increases.A large splitting is also found for the degenerate stateswith q = 2; however, the high-momentum states with q

= 3 and 4 show minor changes under deformation.The perfect (5, 5) armchair nanotube has ve mir-ror symmetries, with two linear bands degenerate at theFermi point. If radial deformations break all the mirrorsymmetries, the band gap opens due to mixing of the πand the π states. The mixed orbitals reside alternatelyon the atomic sites along the circumference, but theiratomic positions differ by a C −C bond [16]. For smalldistortions with d > 4.0 A, there is almost no overlapbetween the charge densities in the two attened layers;thus, the gap is small, less than 10 meV. For d below4.0 A, the gap starts to increase rapidly because of thelayer-layer interactions [16]. The band gaps are found

to be 0.07 and 0.24 eV at d = 3 .35 and 2.71 A, respec-tively. These gaps are sufficiently large for the tube tobe semiconducting.

In the perfect tube, the Fermi point is positioned at2π/ (3a). As d runs from 6.88 to 2.71 A, the location of the highest occupied orbital state in the BZ is shiftedfrom 0.648 to 0.576 π/a . This shift can be understoodby analyzing the wave functions in the building block.Since the π state is characterized by (+ , −, −, +) in thebuilding block, two neighboring orbitals on a cross sec-tion have opposite signs, resulting in the anti-bondingstate (see Fig. 5(a)). Radial deformations enhance theanti-bonding characteristics so that the linear π bandis lifted. On the other hand, for the π band, two neigh-boring orbitals on the circumference have equal signs be-



-88- Journal of the Korean Physical Society, Vol. 37, No. 2, August 2000

cause the building block is represented by (+ , + , −, −).Then, a bonding state is formed between two neighbor-ing C atoms, as shown in Fig. 5(b). Thus, deformationsincrease the linear π band. Because of the opposite be-

havior of the π and the π bands under deformation, theFermi point moves to the Γ point.In conclusion, we have investigated the variations of

subbands in nanotubes under radial deformation. Froman analysis of the wave functions for the subbands, wend that the gap closure in the (9, 0) zigzag tube resultsfrom an overlap of the charge densities in the curvedregions, which lowers the zero-momentum state. In the(5, 5) armchair tube, symmetry-breaking deformationsinduce a mixing of the π and the π states, opening thegap. Because the π and the π states are characterizedby bonding and anti-bonding states, respectively, theirvariations with deformation are found to be opposite.

ACKNOWLEDGMENTS

We thank Dr. I-H. Lee at Korea Institute for Ad-vanced Study (KIAS) for useful discussions. One of theauthors (Y-H. Kim) thanks Dr. M. Mehl for his distri-bution of the tight-binding static freeware code throughthe internet. This work is supported by the Ministryof Science and Technology (MOST) and the QuantumFunctional Semiconductor Research Center (QSRC) atDongguk University.

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10.1.1.22 subbands in carbon nanotubes under radial deformation

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