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Physica E 40 (2008) 2053–2055

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Magnetoelectronic properties of finite double-walled carbon nanotubes

C.H. Leea, R.B. Chenb, T.S. Lic, M.F. Lina,�

aDepartment of Physics, National Cheng Kung University,Tainan, Taiwan, ROCbCenter for General Education, National Kaohsiung Marine University, Kaohsiung, Taiwan, ROC

cDepartment of Electrical Engineering, Kun Shan University, Tainan, Taiwan, ROC

Available online 5 November 2007

Abstract

Magnetoelectronic properties of finite double-walled carbon nanotubes, whose structure belongs to D5h, are studied by the tight-

binding model. Energy levels, energy gaps, and density of states strongly depend on intertube hoppings, nanotube length, strength and

direction of the magnetic field, and the Zeeman splitting. Intertube interactions could change level spacings, modify energy gaps, and

destroy state symmetry about the chemical potential. Magnetic field could induce destruction of state degeneracy, increase of low-energy

states, and strong modulation of energy gap. Moreover, the Zeeman splitting plays an important role in the above-mentioned

magnetoelectronic properties.

r 2007 Elsevier B.V. All rights reserved.

PACS: 73.63.Fg; 73.22.�f; 73.61.Wp

Keywords: Carbon nanotubes; Electronic properties

Quasi-one-dimensional carbon nanotubes (CNs) [1] haveattracted a lot of attention, owing to their unique electronicproperties. This work is focused on zero-dimensional finiteCNs produced by cutting a long CN into shorter ones [2].The effects of the reduction in dimensionality have beenobserved [3,4] and theoretically investigated [5–7].

Double-walled CNs (DWCNs) are different from single-walled CNs (SWCNs) in essential physical properties,mainly owing to the intertube interactions. Many studiesare made on theoretical predictions and experimentalmeasurements [8] regarding the electrical transport ofDWCNs, while some on observable changes in the electronicstructures, which are evaluated by the tight-binding model[9,10] and the first-principles calculations [11].

In SWCNs or DWCNs, a magnetic field could alterenergy gap, break degeneracy, and change energy disper-sion [11–13]. Moreover, finite SWCNs exhibit specialmagnetic properties, and their optical spectra are deeplyinfluenced by magnetic fields [6,7]. In this work, electronicproperties of a finite DWCN, their dependence on tubelength, intertube interactions, and strength and direction

e front matter r 2007 Elsevier B.V. All rights reserved.

yse.2007.09.098

ing author.

ess: mflin@mail.ncku.edu.tw (M.F. Lin).

of the magnetic field, are investigated by the tight-bindingmodel.A coaxial ð5; 5Þ–ð10; 10Þ finite double-walled armchair

CN with a symmetric configuration D5h, at which there isan axial displacement between the inner and outer tubes, ischosen for a model study. The inner tube is projected ontothe outer tube and drawn by the thick black lines, as shownin Fig. 1. The tube length is L ¼ ðNl � 1Þ

ffiffiffi3p

b=2, where Nl

is shown in Fig. 1 and b ¼ 1:42 A is the C–C bond length[6]. The lengths of the inner and outer tubes are the same.By using the tight-binding model with 2pz orbitals of

carbon atoms, the Hamiltonian is given by

Hk;l ¼hðyklÞc

y

kcl intratube;

�WhðyklÞeða�dkl Þ=dc

y

kcl intertube:

8<: (1)

cl and cy

k are annihilation and creation operators on site l

and k, respectively. The p-bonding Vppp ð¼ �2:66 eV ¼ g0Þand the s-bonding Vpps ð¼ 6:38 eVÞ are both considered inthe hopping integrals hðyklÞ. For the intratube part, onlythe nearest-neighbor interactions are taken into account inthe tight-binding model. The intertube interactions includean exponential decay function eða�dkl Þ=d, where dkl is the

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Nl=1

2

3

D5h

Fig. 1. The extended D5h geometric structure of coaxial ð5; 5Þ–ð10; 10Þfinite armchair carbon nanotubes. The inner tube is projected onto the

outer one with the thick black lines.

0.2

0.0

−0.2

Ec,

v (� 0)

0.0 0.5 1.5 0.5 1.0

� (�0) � (�0)

Indpt.N1=17

�=0°�=90°

�=0°�=90°

D5hN1=17

Fig. 2. Energy spectra versus field strength at a ¼ 0� and 90� in the

(a) absence and (b) presence of the intertube interactions.

DO

S (S

tate

s/� 0

ato

m)

50

50

50

50

0−0.4 −0.2 0.0 0.2 0.4

�/�0

N1=17 Indpt.

D5h

D5h, �=0.1; �=0°

D5h, �=0.1; �=90°

Fig. 3. Density of states for (b) f ¼ 0, and f ¼ 0:1f0 at (c) a ¼ 0� and

(d) 90�. The effects with the spin-B interactions are considered. Besides,

the independent case (a) is listed for comparison. The arrow represents the

position of Eg.

C.H. Lee et al. / Physica E 40 (2008) 2053–20552054

distance between two 2pz orbitals, a is that between theinner and outer tubes, and d ¼ 0:45 A. Moreover, theintertube interactions are cut off when dkl is greater than3:9 A [14]. At last, the strength of the intertube interactionsare modified by W to fit the ab initio calculation andexperimental data [15].

When DWCNs exist in a magnetic field B, a change inthe Hamiltonian element is induced by multiplying acomplex factor. H ¼

PHk;l expðiðe=_ÞDGk;lÞ, where

DGk;l ¼RkRl

2sinðFk � FlÞB cos a

þðzk � zlÞ

2ðRk sinFk þ Rl sinFlÞB sin a. ð2Þ

The positions of two 2pz orbitals k and l are chosen byusing the cylindrical coordinate. DGk;l is the phase

differenceR k

lA � dr caused by the vector potential

A ¼ rB cos a=2Fþ rB sin a sinFz. a is the angle betweenB and the nanotube axis. In this work, the magnetic flux is

f ¼ pR2oB, where f0 ¼ hc=e is one fundamental magnetic

flux, and Ro is the radius of the outer nanotube. The

Zeeman energy is Ez ¼ gsf=m�r2f0. The g factor is

assumed to be the same as that ð�2Þ of graphite. s ¼ � 12

is the electron spin and m� is bare electron mass.Following the theoretical description, electronic struc-

tures of finite DWCN for Nl ¼ 17 are investigated indetail. In Fig. 2(a), electronic states without the intertubeinteractions are symmetric about the Fermi level EF ¼ 0.Besides, their tight-binding functions with Nl ¼ 17 are inthe standing-wave form along the tube axis. Therefore, thestate nearest to EF is contributed by the outer ð10; 10Þ tubeand intermittent with a period f0 in the parallel magneticfield ða ¼ 0�Þ; the next state from the inner ð5; 5Þ tube has alarger period because of its smaller radius. When amagnetic field is perpendicular to the tube axis, morelow-energy states will occur as the field strength rises.

After the intertube interactions are considered, the stateenergies of D5h will be changed drastically, as shown inFig. 2(b). The energy levels near EF would be changed bythe strong hybridization of the inner and outer tight-binding functions ð�0:018g0! 0:007g0 and 0:018g0!0:012g0Þ. The contributions of the inner and outer tubesfor larger transformation are almost the same, but the

contributions for smaller one comes mainly from theoriginal nanotube. When a parallel magnetic field isapplied, energy gap will alter and there is an unusual turnat f�0:3f0. It is depicted by Fig. 2(a) and attributed by thedifferent periods of the inner and outer tubes in the parallelmagnetic field. There are more low-energy states as thetransverse magnetic field continues to rise, but no statecrossing is found at fof0.Finite DWCNs own the discrete states and the delta-

functional-like peaks in density of states (DOS). Theheights of prominent peaks represent state degeneracy. InFig. 3(a), DOS is symmetrical about the Fermi levelEF ¼ 0. The only fourfold degenerate peak near 0:3g0originates from the outer ð10; 10Þ tube, whereas all theother peaks are doubly degenerate if the spin degeneracy istaken into account. The peaks of the two states near EF

would approach to each other due to the intertubeinteractions, and cause energy gap to decline, as shown inFig. 3(b). The frequencies of four-degeneracy peaks would

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0.2

0.1

0.00.0 0.5 1.0

Eg

(�0)

� (�0)

D5hN1=17

�=0°; no spin−B�=0°; spin−B�=30°�=60°�=90°

Fig. 4. Magnetic-flux-dependent energy gaps for different a’s. Both effects

with and without the spin-B interactions are considered.

C.H. Lee et al. / Physica E 40 (2008) 2053–2055 2055

not be altered drastically and their heights hardly dependon the intertube interactions. However, the degeneracy andfrequencies would be altered with the magnetic field. Thefrequencies of the low-energy peaks induce the apparentshift as the magnetic field is added; the difference betweena ¼ 0� and 90� is relatively obvious for degenerate statesand low-energy states, as presented in Figs. 3(c) and (d).

Energy gap as a function of an applying magnetic fieldwith different a’s is presented in Fig. 3. Eg without the spin-B interactions or without the Zeeman splitting, in theparallel magnetic field has a bend at f�0:3f0 and decreasesover 0:5f0 because of the above-mentioned differentperiods between the inner and outer tubes. Furthermore,the complete energy-gap modulations ðEg ¼ 02Ega0Þwould take place by means of the spin-B interactions atfpf0 because Eg of the finite ð10; 10Þ SWCN for Nl ¼ 17

is smaller. When the magnetic field gradually deviates fromthe tube axis, Eg would be reduced and there exist thecomplete energy-gap modulations at larger a mainly due tothe Zeeman splitting. (Fig. 4)In conclusion, electronic properties of finite DWCNs

strongly depend on intertube interactions, tube length,direction and magnitude of the magnetic field, and theZeeman effect. A magnetic field would destroy statedegeneracy, change energy spacings, and induce morelow-energy states, which could also be enhanced by theZeeman splitting.

This work was supported in part by the National ScienceCouncil of Taiwan under Grant no. NSC 95-2112-M-006-028-MY3.

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