Electronic structure of collapsed C, BN, and BC3 nanotubes

Yong-Hyun Kim, H.-S. Sim, K.J. Chang *

Department of Physics, Korea Advanced Institute of Science and Technology, 373-1 Kusung-dong, Yusung-ku, Taejon 305-701, South Korea

Received 28 August 2000; accepted 8 September 2000

Abstract

The electrical properties of single-wall C, BN, and BC3 nanotubes in ideally rolled-up forms show a wide spectrum from truly

metals to large band gap semiconductors. In the presence of radial deformations that collapse tubes, the electrical properties are

severely modi®ed such that metals turn into semiconductors and vice versa. Based on ®rst-principles pseudopotential calculations,

we ®nd that metallic C nanotubes have a ®nite band gap if radial deformations break all mirror symmetries of the tubes, and that

original ®nite gaps (�0.5 eV) of semiconducting C and BC3 tubes are closed by collapsing deformations. In BN tubes, band gaps can

be tuned in the range 2±5 eV. On the other hand, the band gaps of armchair BN and zigzag BC3 nanotubes are found to be in-

sensitive to radial deformations. These new ®ndings can be applied to design new types of nanotube-based functional devices using

radial deformations. Ó 2001 Elsevier Science B.V. All rights reserved.

PACS: 71.20.Tx; 71.20.Nr; 71.24.+q

Keywords: Electronic structure; Carbon nanotube; BN nanotube; Collapsed nanotube

1. Introduction

Since the discovery by Iijima in 1991 [1], single-wallcarbon nanotubes have attracted a lot of attention in thephysical and industrial societies [2,3]. From his an-nouncing that a single-wall nanotube is constructed byrolling up a single graphite sheet into a cylinder, it wasimmediately known that the electronic structure of na-notubes can be determined by the band structure ofgraphite and the Bloch wave-vector quantization fol-lowing a periodic condition along the tubular circum-ference [4±6]. So, if nÿ m � 3i (i is an integer), where nand m describe the projection of the circumferentialvector onto the basis vectors of the graphite lattice, the�n;m� C tube is classi®ed as a metal, and if not, the tubeis a semiconductor with a ®nite band gap. In realisticnanotubes, since the surface curvature results in hy-bridizations between p and r states, small modi®cationsarise in the simple three-multiple rule [5,7].

Cohen and his co-workers [8±11] studied bulk layeredstructures and graphene layers of BN, BC3, and BC2Nbefore the discovery of wrapped-up C nanotubes, andtheoretically proposed the existence of BN, BC3, and

BC2N nanotubes, that is, wrapped-up structures ofgraphenes. Due to the ionic origin of the band gap, BNnanotubes with diameters larger than 9.5 �A (whichcorresponds to the diameter of the �12; 0� tube) have avery stable gap of about 5.5 eV, almost independent oftube diameter, helicity, and wall±wall interactions. As Ccomposition increases in BxCyNz nanotubes, the bandgap generally decreases, 2.0 and 0.5 eV for BC2N andBC3 nanotubes, respectively. Thus, the synthesis ofcomposite BxCyNz nanotubes in the designed orderwould make possible applications for electronic andphotonic devices with a variety of electronic properties[12]. In addition, radial deformations o�er rich elec-tronic structure in single-wall nanotubes [13±17].

In this paper, we summarize our recent calculationalresults for the electronic properties of collapsed single-wall C, BN, and BC3 nanotubes and discuss possibleelectronic devices utilizing collapsed nanotubes. Inarmchair C nanotubes, a genuine metal turns into asemiconductor in the presence of symmetry-broken ra-dial deformations, while zigzag nanotubes with bandgaps of 0.5±0.6 eV transform into metals when they arecollapsed. Zigzag BN tubes show a large variation ofband gaps from 5 to 2 eV and armchair BC3 tubes withband gaps of about 0.5 eV lose their semiconductingproperties if the tubes are collapsed until two facinglayers approach to the layer±layer distance of graphite,

Current Applied Physics 1 (2001) 39±44

www.elsevier.com/locate/cap

* Corresponding author. Tel.: +82-42-869-2531; fax: +82-42-869-

2510.

E-mail address: kchang@mail.kaist.ac.kr (K.J. Chang).

1567-1739/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved.

PII: S 1 5 6 7 - 1 7 3 9 ( 0 0 ) 0 0 0 0 8 - 0

3.4 �A. On the other hand, the semiconducting propertiesof armchair BN and zigzag BC3 tubes with gaps of 5 and0.5 eV, respectively, are almost una�ected by radialdeformations. Such modi®cations of the band gaps of C,BN, and BC3 nanotubes can be used to design func-tional electronic and photonic devices in the future.

2. Calculational method

We perform extensive ®rst-principles pseudopotentialcalculations within the local-density-functional approx-imation (LDA) for various squashed single-wall na-notubes. Ab initio norm-conserving pseudopotentialsare generated by the scheme of Troullier and Martins[18], and transformed into the Kleinman±Bylanderseparable form [19]. The Ceperley±Alder exchange andcorrelation potential [20] is used. The wave functions areexpanded in a plane-wave basis set with a kinetic energycuto� of 36 Ry, which ensures the convergence of bandenergy di�erences to within 0.05 eV. The LDA calcula-tions are carried out for tetragonal supercells with tube±tube distances larger than 6 �A. To obtain the collapsedgeometries of single-wall nanotubes, as depicted in Fig.1, we use Terso�'s empirical potentials [21] for C tubesand calculate the Hellmann±Feynman forces based onthe LDA calculations for BN and BC3 tubes. Duringrelaxations, we impose a constraint that atoms are onlyallowed to move between two walls, the separation ofwhich is de®ned as the interlayer distance d of the col-lapsed tube.

The total energy of a deformed tube increases as ddecreases because of the strain energy in the curved re-gions. From the total energy variation, we can make arough estimate of the pressure involved. Assuming thatforces act on the projected area of the tube along theradial direction, the pressure estimated for the �5; 5�tube is less than about 10 GPa for d P 3:35 �A, in goodagreement with experiments that give a critical collaps-ing stress of 100±150 GPa in a multi-wall nanotube with10±20 layers [22]. Similarly, for BN and BC3 nanotubes,a pressure of about 10 GPa is needed to deform thetubes down to d � 3:4 �A. In this case, forces of 1±5 nNare required to ¯atten tubes. Recent atomic force mi-croscopy measurements demonstrated the reversibleradial deformability (up to �40%) of multi-wall tubeswith forces less than 30 nN [23].

3. Results

Defect-free BN and BC3 nanotubes with large diam-eters are semiconductors with band gaps of about 5.5and 0.5 eV, respectively, while C tubes become metallicwith increasing of the diameter. Since the subbands ofnanotubes, which determine the electronic properties of

individual tubes, are originated from the periodicityalong the circumference, radial deformations that per-turb the periodicity modify the location and dispersionof the subbands.

3.1. Collapsed C nanotubes

Fig. 2 shows the LDA band structure for variousdeformed geometries of the �9; 0� and �5; 5� C nanotu-bes. In the perfect form of the �9; 0� tube, whose diam-eter is 7.2 �A, the band gap is 0.09 eV. If the tube is¯attened, then the band gap gradually decreases as ddecreases to 6.0 �A, and then is completely removed atd � 5:3 �A. The analysis of charge densities indicates thatthe lowering state near the band gap is identi®ed as thehybridized singlet state, which is also known to modifythe band gaps of the �6; 0�; �7; 0�, and �8; 0� C tubeswith small radii [7]. For the �13; 0� and �18; 0� C tubeswith diameters of 10.5 and 14.3 �A, respectively, the

Fig. 1. Ball-and-stick models for collapsed �9; 0� C, �9; 0� BN, and

�4; 0� BC3 nanotubes. The opposing layer±layer distances are 5.0 , 4.5,

and 5.0 �A from top to bottom.

40 Y.-H. Kim et al. / Current Applied Physics 1 (2001) 39±44

hybridized singlet states also make the tubes truly me-tallic when they are collapsed to d � 5±6 �A. One inter-esting feature in the band gap variation is that before thehybridized singlet states get into the band gaps of small-gap C tubes �nÿ m � 3i�, the original gaps of a few meVincrease to 0.1±0.2 eV.

For metallic armchair C tubes, the band gap modi-®cations follow the mirror-symmetry selection rules [24].An armchair tube with the rolling index �n; n� has nmirror planes containing the tube axis, with two Diracbands characterized by p and p�, which are even andodd under mirror-symmetry operations. If we imposemirror-symmetry breaking radial deformations (hereaf-ter named as asymmetric deformations) that breakmirror-symmetries transversely on the tube, a mixing ofdegenerate p and p� bands at the Fermi level is expectedbecause the even and odd wave functions cannot be theeigenstates of the symmetry-broken systems. However,when one mirror-symmetry is preserved under defor-mation, the band gap does not appear, as shown in Fig.2(b). We consider two distinct types (asymmetric andsymmetric) of radial deformations for the �5; 5� tubewith d � 2:7 �A. Our LDA calculations show that whenall the mirror symmetries are broken, the originally de-generate Fermi states are clearly removed, but when one

mirror-symmetry remains, no gap is developed despitesimilar deformations. The band gap opening solely re-sults from the mixing of the p and p� states such as1��2p �jpi � jp�i�. The band gap is less than 10 meV ford > 4:0 �A in the LDA calculations. For d below 4.0 �A,the gap starts to increase rapidly because of the layer±layer interactions inside the tube. The LDA gaps arefound to be 0.07 and 0.24 eV at d � 3:35 and 2.71 �A,respectively. These gaps are su�ciently large enough forthe tube to be semiconducting.

When chiral C tubes are radially deformed, the hy-bridization of the non-modulated state, which corre-sponds to the singlet state in zigzag tubes, causes asemiconductor±metal transition, as in zigzag tubes.Compared to zigzag tubes, more severe radial defor-mations are needed to close the primary band gaps ofchiral tubes. For example, the band gap closures occurat d � 4:5 �A and about 3.0 �A for the �8; 2� and �6; 3�tubes, respectively.

3.2. Collapsed BN nanotubes

The band structures of the zigzag �9; 0� and armchair�5; 5� BN tubes are plotted and compared for variousradial deformations in Fig. 3. For the �9; 0� nanotube,

Fig. 2. (a) The LDA band structure of the collapsed �9; 0� C nanotubes with d � 7:2 (perfect), 6.0, and 5.3 �A. Shaded regions and dotted line denote

the band gaps and the Fermi level, respectively. (b) The band structure of the perfect �5; 5� C tubes and collapsed tubes with asymmetric and

symmetric deformations of d � 2:7 �A.

Y.-H. Kim et al. / Current Applied Physics 1 (2001) 39±44 41

the band gap is reduced from 3.5 to about 2 eV as thetube is ¯attened to d � 3:0 �A, varying almost linearlywith decreasing of the interlayer distance d. When wecollapse the tube down to d � 2:0 �A, the gap decreasesfurther to 1.0 eV. However, a gap closure never occursin our calculations. The lowering state is turned out tobe the hybridized singlet state. When the tube is ¯at-tened, the charge densities of the singlet state are mostlylocalized in the curved regions, inducing an overlap ofthe charge densities. Then, because of the formation ofweak bonds between the B atoms in the curved regions,the singlet state is lowered under ¯attening deformation.Similar behavior was also found in zigzag C tubes,where the singly degenerate state decreases due to theenhanced accumulation of charge densities in the curvedregions [13,14]. Since zigzag BN tubes with large diam-eters have the gap of �5 eV, radial deformation allowstuning of energy gap in the range 2±5 eV using trans-verse pressures of about 10 GPa.

In contrast, the band gaps of armchair BN tubes arelittle a�ected by radial deformations, as illustrated inFig. 3(b). The band gap of 4.2 eV in the ¯attened �5; 5�BN tube with d � 5:0 �A is quite similar to that of theperfect tube. A charge transfer from the ¯attened layersto the curved regions is not found in contrast to zigzag

tubes. Due to the ionic B±N bonds, the pz orbitalsstrongly reside at one type of atoms for both the valenceand conduction bands. Because of the large distancebetween the same species along the circumference inarmchair BN tubes, an overlap of charge densities doesnot e�ectively occur in the curved regions. For d lessthan the graphite interlayer spacing of 3.35 �A, the gapsuddenly decreases by 0.5 eV due to interactions be-tween the two facing layers of the collapsed tube. Thissituation is similar to the case of collapsed armchair Cnanotubes, where very small gaps less than 10 meV areinduced by symmetry-broken deformations of d P 4:0�A, while the gap increases to about 0.1 eV as the layer±layer interactions become signi®cant at d � 3:35 �A.

3.3. Collapsed BC3 nanotubes

In previous tight-binding calculations for BC3 na-notubes, the band gap of perfect zigzag tubes was foundto decrease as tube diameter decreases, while armchair�n; n� (n � 3; 4; 5) tubes have an almost constant gapindependent of their diameters [10]. Thus, a gap closureis more likely to occur in zigzag BC3 tubules under¯attening deformation. In our LDA calculations forcollapsed �4; 0� and �3; 3� BC3 nanotubes, we only ®nd agap closure in the armchair tube, as shown in Fig. 4.This attributes to the di�erent characteristics of thestates near the band gaps from those of C and BN tubes.Because the B atom lacks one valence electron comparedto the C atom, one p band is completely unoccupied.Moreover, the top of the valence bands is known to haver characters [25,26]. The folded bands of the unoccu-pied p band of a single BC3 sheet have a band width ofabout 1.5 eV and are located just above the band gap.As the zigzag BC3 tube is collapsed, the p band widthincreases slightly with a small upward movement at theC point, but the fundamental gap is stable around 0.6eV. On the other hand, in the armchair BC3 tube, theband gap of 0.4 eV is gradually reduced and eventuallyclosed. We ®nd that the overlap of charge densities inthe curved regions occurs only in the armchair tubes.

Zigzag BC3 tubes have a gap between the p and p�

bands, which is analogous to the real gaps in C na-notubes, at the C point, and this gap decreases under¯attening deformation, similar to collapsed C nanotubes(see Figs. 2 and 4). This gap is in fact closed for an in-terlayer spacing between 5 and 6 �A. On the other hand,in armchair BC3 tubes, the p and p� conduction bandscross at a 2/3 point on the C±X line. Since we preserveone mirror-symmetry during deformation, the degener-acy at the crossing point is maintained. The crossingpoint of the p and p� bands becomes closer to the Xpoint as ¯attening deformation is enhanced, indicatingthat the movements of the p and p� bands are opposite.In collapsed armchair C nanotubes, the crossing point

Fig. 3. The variation of the band gaps with radial deformation in the

(a) �9; 0� and (b) �5; 5� BN nanotubes. The shaded regions denote the

band gaps.

42 Y.-H. Kim et al. / Current Applied Physics 1 (2001) 39±44

of the p and p� bands was found to move to the C point[13,14].

4. Device applications and conclusions

We suggest the quantum dot behavior of armchair Cnanotubes with locally collapsed regions. When anarmchair C nanotube is locally collapsed with breakingthe mirror-symmetries, the collapsed region can act as alocal transmission barrier because of the gap opening; ifincident channels from an undeformed region have anenergy inside the gap, these channels are mostly re¯ectedby the collapsed region and only evanescent modes aretransmitted. This feature indicates that if such collapsedregions occur serially in armchair C nanotubes, thesandwiched undeformed region can behave as a quan-tum dot. We study the �5; 5� nanotube with the twocollapsed regions where the mirror-symmetries arebroken (see Fig. 5(a)). Each deformed region that con-sists of Nf unit cells is collapsed down to 2.7 �A andgradually recovers the perfect nanotube through adja-cent bu�er regions, each of which has Nb (� 8.5) unitcells, while the sandwiched undeformed region consistsof Nd unit cells. We optimize the geometries of locally

collapsed tubes employing Terso�'s potentials [21], andcalculate the transmission probability T �E� as a functionof energy E of incident channels from the undeformedregion, using Green's function approach [27] and atight-binding Hamiltonian [28]. We note that two-ter-minal conductance G can be obtained as G�E� ��2e2=h�T �E�.

For a dot with �Nd;Nf�� �30; 65� in the �5; 5� arm-chair nanotube, we ®nd a transmission dip near E �ÿ0:1eV (see Fig. 5). This dip is caused by the gap opening dueto the broken mirror-symmetries, and the collapsed re-gion behaves as a barrier. In the dip region, T �E� and theLDOS in the middle of the dot show sharp resonantpeaks, indicating that bound states are formed within thedot. It is interesting to note that the resonant peaks arenearly degenerate as shown in Fig. 5(b). This pairedpeaks occur because the p and p� channels inside the dotare completely mixed at the barrier when the energy of

Fig. 5. (a) A schematic diagram for a nanotube dot formed between

two collapsed regions in the �5; 5� C nanotube. For �Nd; Nf � ��30; 65�, (b) T �E� (solid) is drawn and compared with that (dotted) for

a tube with one ¯attened region; the inset shows a peak pair near

E � ÿ0:1 eV. (c) The local densities-of-states (LDOS) in the middle of

the nanotube dot (solid) and in the left lead (dotted).

Fig. 4. The variation of the band gaps with radial deformation in the

(a) �4; 0� and (b) �3; 3� BC3 nanotubes. The shaded regions denote the

band gaps.

Y.-H. Kim et al. / Current Applied Physics 1 (2001) 39±44 43

the channels is inside the dip. Such degenerate boundstates are a general feature in armchair nanotubes withlocal symmetry-breaking deformations. Because theelectron transport in single-wall C nanotubes is known tobe ballistic and coherent [29], the resonant degeneratebound states in the nanotube quantum dot is observableexperimentally and applicable in future electronic de-vices such as molecular single electron transistors.

In conclusion, we have studied the e�ects of radialdeformations on the electronic properties of single-wallC, BN, and BC3 nanotubes through the ®rst-principlespseudopotential calculations. The modi®cation of theband gaps of single-wall nanotubes by radial deforma-tions has important implications in fabricating newelectronic devices.

Acknowledgements

This work is supported by the Ministry of Scienceand Technology (MOST) and the Quantum FunctionalSemiconductor Research Center (QSRC) at DonggukUniversity.

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