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Physica E 41 (2009) 1226–1231

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physe

Low-energy electronic properties of finite double-walledcarbon nanotubes under external fields

C.H. Lee a, W.S. Su a,c, R.B. Chen b, M.F. Lin a,�

a Department of Physics, National Cheng Kung University, Tainan 701, Taiwanb Center for General Education, National Kaohsiung Marine University, Kaohsiung 811, Taiwanc Center for General Education, Tainan University of Technology, Tainan 710, Taiwan

a r t i c l e i n f o

Article history:

Received 14 August 2008

Accepted 6 February 2009Available online 20 February 2009

PACS:

73.63.Fg

73.22.�f

73.61.Wp

Keywords:

Carbon nanotubes

Electronic properties

77/$ - see front matter & 2009 Elsevier B.V. A

016/j.physe.2009.02.005

esponding author. Tel.: +886 6 2757575 6527

ail address: mflin@mail.ncku.edu.tw (M.F. Lin

a b s t r a c t

Using the 2pz tight-binding model, we investigated the electronic properties of finite double-walled

carbon nanotubes (DWCNTs) under static electric and magnetic fields. It was found that for low-energy

electronic structures, their energy levels, density of states and energy gap are obviously affected by

symmetric configurations as well as electric and magnetic fields. The intertube atomic interactions

remarkably change the state energies, modulate the energy gap, and break the state symmetry about

the Fermi level. Moreover, both electric and magnetic fields can destroy the degeneracy and modulate

the above-mentioned electronic properties. The magnetic fields with various directions also induce

variation of electronic structure. As the magnitude of electric fields perpendicular to the tube axis

increases, more energy states gather around the Fermi energy, which, as a result, causes a complicated

variation of energy gap.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Carbon nanotubes (CNTs) [1], which are in the form of cylinderand can be viewed as a rolled-up two-dimensional graphite sheet,belong to quasi-one-dimensional systems and have attractedmuch attention due to their remarkable properties. Their physicalproperties are under influence of the radius and chirality of CNTs.A quasi-one-dimensional CNT can be shortened by the scanningtunneling microscope (STM) and, thus, produce zero-dimensionalfinite CNTs [2]. The effects of the reduction in dimensionality canbe observed in STM and transport experiments [3–7]. Through thefirst-principles and the tight-binding calculations, the theoreticalinvestigations on finite single-walled CNTs (SWCNTs) discover theinteresting physical properties [8–12]. Besides, double-walledCNTs (DWCNTs) are the simplest type of the multiwalled CNTs andexhibit many interesting physical properties. DWCNTs provide agood platform for research on the effect of intertube atomicinteractions on the electronic properties, optical properties, andtransport properties [13–17].

External fields can modulate the physical characteristics ofCNTs and have been extensively used in many experiments[5–7,13,18]. For instance, magnetic fields can vary the stateenergies, destroy the state degeneracy, and change the energygap [11,12,19–21]. Similarly, many physical properties can also be

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).

affected by electric fields. For example, the electronic properties ofCNTs, such as the energy gap modulation and semiconductor-metal transition, are subjected to external electric fields [22–25].In this work, the electronic properties of finite DWCNTs withelectric and magnetic fields are calculated by the tight-bindingmethod. Their dependence on the symmetric configurations, thestrength of the transverse electric field ðFÞ, the magnitude ðBÞ anddirection ðaÞ of the magnetic field as well as the Zeeman splittingare also investigated.

2. Theory

A SWCNT can be regarded as a graphite sheet rolled from theorigin to the vector Rx ¼ ma1 þ na1, where a1 and a2 arethe primitive vectors of a graphite sheet. The radius andthe chiral angle of the ðm;nÞ CNT are, respectively,

r ¼ bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ðm2 þmnþ n2Þ

p=2p and y ¼ tan�1½�

ffiffiffi3p

n=ð2mþ nÞ�, where

b ¼ 1:42 A is the C–C bond length. The coaxial (5,5)–(10,10) finitedouble-walled armchair CNTs with three symmetric configura-tions, C5, D5h, and S5, in the present paper, will be studied anddiscussed. In the three targeted cases, their tube lengths (L) areequal to ðNl � 1Þ

ffiffiffi3p

b=2, where Nl is the number of the dimer lines[12] and both inner and outer tubes have the same length alongthe tube axis. Fig. 1 presents our three models with variousconfigurations. Fig. 1(a) shows the plot of the C5 system and thecarbon atoms on the inner tube (thick black lines) are directly

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Nl = 1

2

3

Fig. 1. The extended geometric structures of coaxial (5,5)–(10,10) finite armchair DWCNs are shown in (a) C5, (b) D5h , and (c) S5. The inner tube is projected onto the outer

one with the thick black lines. It is noted that Nl is the number of dimer lines along the axial direction.

C.H. Lee et al. / Physica E 41 (2009) 1226–1231 1227

projected on the outer tube. Then, the configuration switchesfrom C5 to D5h with the inner tube displaced axially by

ffiffiffi3p

b=4, asillustrated in Fig. 1(b). Finally, the S5 system can be attained whenthe inner tube is rotated with an angle of p=30 in the D5h system,as plotted in Fig. 1(c).

Under the tight-binding model with 2pz-orbitals of carbonatoms, the Hamiltonian matrix can be divided into two parts andbe expressed as

Hk;l ¼hðyklÞc

y

kcl intratube part;

�WhðyklÞeða�dklÞ=dcykcl intertube part:

8<: (1)

cl and cyk are annihilation and creation operators on sites l and k.Since the 2pz orbitals perpendicular to the cylindrical surface arenot parallel to each other, both the p-bonding Vppp ¼ �2:66 eVð¼ �g0Þ and the s-bonding Vpps ¼ 6:38 eV must be taken intoaccount in the intratube (intertube) hopping integrals hðyklÞ,where ykl is the angle between two radial vectors of the kth atomand the lth atom. For the intratube part, only the nearest-neighborinteractions are considered in our tight-binding model, while forthe intertube part, an additional exponential function eða�dklÞ=d

term is involved, where dkl is the distance between two 2pz

orbitals, a is that between inner and outer tubes, and d equals0:45 A. Besides, the intertube atomic interactions are cut off with

dkl greater than 3:9 A [26] and the strength of the intertube atomicinteractions are modified by W term so as to fit the ab initio

calculation and experimental data.Under external fields, the Hamiltonian elements are expected

to change, and, as a result, affect the electronic properties of thesystem. When a finite DWCNT exists in a transverse electric fieldF, the diagonal Hamiltonian matrix elements can be expressed asHii ¼ �eVicosðFiÞ, where Vi ¼ RiF is the electric potential from theeffective electric field, and the position of the 2pz orbital i isdescribed by the cylindrical coordinate. For a finite DWCNT with amagnetic field, the vector potential A ¼ rB cosa=2Fþ rB sina sinFz

is chosen, which corresponds to the magnetic field B ¼ B cosazþ

B sinaF, where a is the angle between the direction of themagnetic field and the tube axis. Thus, the Peierls phase differenceDGk;l would be obtained, which is related to the vector potential AbyR k

l A � dr and l. As a consequence, the Hamiltonian matrix of thesystem can be expressed as

H ¼X

kl

Hk;l � eiðe=_ÞDGk;l , (2)

where

DGk;l ¼12RkRl sinðFk �FlÞB cosaþ 1

2ðzk � zlÞðRk sinFk þ Rl sinFlÞB sina. (3)

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C.H. Lee et al. / Physica E 41 (2009) 1226–12311228

It is noted that the magnetic flux in our calculations is f ¼ pR2oB,

where Ro is the radius of the outer (10,10) CNT and the unit ofmagnetic flux is a fundamental magnetic flux f0 ¼ hc=e.

3. Results

In Fig. 2(a), the energy states without the intertube atomicinteractions are symmetric about the Fermi level EF ¼ 0. The statenearest to EF ðE ¼ �0:02g0Þ is contributed by the inner (5,5) tube,while the other state ðE ¼ �0:075g0Þ comes from the outer (10,10)one. Though both states vary with an external electric fields, moreintense variations are caused by the states from the outer tubethan by those from the inner one. The reason is that the statesfrom the outer (10,10) tube would be influenced by the largerelectric potential due to its larger radius. Moreover, when themagnitude of the electric field rises, three consequences happen:the oscillation of the state energies, the gather of many states nearthe Fermi level, and the occurrence of the state crossings morefrequently.

Following above case without considering the intertube atomicinteractions, Figs. 2(b)–(d) further illustrate the cases with theinteractions. In Fig. 2(b), the effects of the intertube atomicinteractions strongly alter the state energies and break the

Fig. 2. The energy spectra versus a electric field with Nl ¼ 18 in (a) independent case, (b)

splitting and additional magnetic fields f ¼ 0:1f0 at a ¼ 0� and 90� , respectively.

symmetry of the energy levels in the C5 system without theelectric field. It is clearly seen that the stronger hybridizationinduces drastic energy shifts in the range from 0:075g0 and 0:02g0

to 0:164g0 and �0:032g0, respectively. For such two statesðE ¼ 0:164g0 and � 0:032g0Þ, the contributions of their tight-binding functions from the inner and the outer tubes are roughlyequivalent under this stronger hybridization. However, theweaker one occurs between the two states below the Fermilevel, and the state energies are modified mildly ð�0:02g0 !

0:032g0 and � 0:075g0 !�0:07g0Þ. In this case, the contri-bution of the tight-binding functions for the two states ðE ¼0:032g0 and � 0:07g0Þ almost comes from the original self tube.Hence, the hybridization causes the interchange between theunoccupied and occupied states, of which both are nearest toEF ð�0:02g0 ! 0:032g0 and 0:02g0 !�0:032g0Þ. Besides the mod-ulation of energy levels, the hybridization of the inner and outertight-binding functions is also drastically affected by the electricfield. For example, the state E ¼ �0:032g0 in the absence of theelectric field is dominated by both the (5,5) and (10,10) tubes,while for a large enough F, it is dominated by the (10,10) tube.

As shown in Fig. 2(c), the changes of energy levels withoutF near EF in the D5h system ð0:02g0 ! 0:012g0 and � 0:02g0 !

0:007g0Þ are weaker than that in the C5 system and there is asmaller Eg formed. On the other hand, the dissimilar boundary

C5, (c) D5h, and (d) S5. (e) and (f) are the same plots as (b) with considering Zeeman

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C.H. Lee et al. / Physica E 41 (2009) 1226–1231 1229

structures on the two extremities of the finite DWCNT lead to theresult that the tight-binding functions are not symmetricallydisposed about the middle of the finite DWCNT. With F, the statecrossing can take place at Fo0:05g0=eA due to the smaller Eg.In Fig. 2(d), the variation of the energy levels in the S5 systemwithout and with the transverse electric field, respectively, aresimilar to the results of the D5h system. It means that the mainelectronic properties would depend on the boundary structureson the two extremities of the finite DWCNT.

Now, two types of additional magnetic fields (f ¼ 0:1f0 anda ¼ 0� as well as a ¼ 90�) are considered in the C5 system, asshown in Figs. 2(e) and (f), respectively. The state energies areexpressed as Eðf;sÞ ¼ Eðf;sÞ þ Ez, where Ez ¼ gsf=m�r2f0 and iscalled the Zeeman energy. The g factor is assumed to be the sameas that ð2Þ of graphite, s ¼ �1=2 is the electron spin, and m� isthe bare electron mass. For a parallel magnetic field ða ¼ 0�Þ, itaffects the state energies and induces the Zeeman splitting, whichwill lead to the rigid shift between the spin-up and spin-downstates, as plotted in Fig. 2(e). Moreover, it not only results in the

Fig. 3. Density of states of C5 with Nl ¼ 18 with different conditions (F ¼ 0; F ¼ 0:02g0=

D5h in (b) and S5 in (c). The inset in (a) indicates the independent system at F ¼ 0 and

energy shift, but also makes the state crossing near EF caused bythe electric field disappear. For a magnetic field perpendicular tothe nanotube axis ða ¼ 90�Þ, the trend of the change in the stateenergies can be viewed to be similar to the results of Fig. 2(b) withthe Zeeman splitting, as shown in Fig. 2(f).

The main features of the electronic states can be observed indensity of states (DOS), defined as Dðo;fÞ ¼

PG=p½ðo�

Eðf;sÞÞ2 �G2�, where Gð¼ 10�4g0Þ is the phenomenological

broadening parameter. Low-energy DOS of the C5 system DWCNTwith Nl ¼ 18 under external fields are shown in Fig. 3(a). Fordiscrete states, DOS is delta-function-like, and the heights ofpeaks are related to the state degeneracy. In the independentsystem, DOS is symmetrical about the Fermi level EF ¼ 0 (the insetof Fig. 3(a)). If the spin degeneracy is considered, the two fourfolddegenerate peaks near 0:35g0 come from the outer (10,10) tubeand the others are doubly degenerate. When the intertube atomicinteractions are taken into account, the low-energy peak frequen-cies vary more distinctly, while their height is not reduced.However, the electric field both lifts the fourfold degeneracy and

eA; F ¼ 0:02g0=eA with f ¼ 0:1f0 at a ¼ 0� or 90� is shown in (a). Similar plots for

f ¼ 0.

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Fig. 4. The magnetic-flux-dependent energy gaps with Nl ¼ 18 for (a) C5, (b) D5h , and (c) S5 are calculated at different a’s, where a means the angle between the direction of

the magnetic field and the tube axis. The result at f ¼ 0 is calculated for comparison.

C.H. Lee et al. / Physica E 41 (2009) 1226–12311230

changes the frequencies of the peaks. Moreover, the doubledegeneracy is further broken by the spin-B interaction after theparallel magnetic field is applied. The same behavior happensagain at a ¼ 90�. The effects caused by the intertube atomicinteractions and external fields could also be found in othersystems with different symmetric configurations (Figs. 3(b) and(c)). The different symmetric configurations significantly alter thepeak frequency.

Fig. 4(a) presents the energy gap as a function of the electricfield for the C5 system. As F increases, Eg slightly increases firstly,and decreases sharply later because the higher state gets closer toEF . The complete energy-gap modulation (CEGM; Eg ¼ 02Ega0)occurs at F40:05g0=eA in the absence of magnetic fields (the solidline with circles). At large F, the sharp variation of the energy gapor the CEGM is derived from more low-energy states caused bythe electric field. As a magnetic field is turned on, the energy gapdiminished. The differences between a’s are less discrepantat small F excluding a ¼ 90�. The variation of the energy gap ata ¼ 90� is similar to that at f ¼ 0. In Fig. 4(b), the finite DWCNTwith D5h system and Nl ¼ 18 owns a smaller energy gap, but Eg

becomes larger by applying a magnetic field with a ¼ 0�. The

effects of different a’s on the energy gap in the transverse electricfield are strong because of the smaller Eg. For the S5 system, thechanges of Eg would be analogous to that of the D5h system, exceptfor CEGM at Fo0:05g0=eA in the absence of B (Fig. 4(c)).

4. Conclusion

The electronic properties of finite DWCNTs depend strongly onthe symmetric configurations, the transverse electric field, thedirection and magnitude of the magnetic field, and Zeeman effect.In finite SWCNTs, the electronic structures would be determinedby the length and boundary structure. However, finite DWCNTsare much more different from finite SWCNTs in terms of electronicproperties, mainly owing to the intertube atomic interactions. Theenergy shifts due to the intertube atomic interactions correspondto the symmetric configurations. The new boundary structures onthe two extremities, which are caused by the displacement alongthe z-axis, dominate the electronic properties in finite DWCNTs.The electric and magnetic fields will destroy the state degeneracy,change the energy spacing, and induce many low-energy states.

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Furthermore, the Zeeman effect plays an important role in themodification of the electronic properties. The symmetric config-urations influence the modulation of the electronic properties inelectric fields and additional magnetic fields. The directions of themagnetic field can affect the occurrence of the CEGMs.

Acknowledgments

This work was supported in part by the National ScienceCouncil of Taiwan under the Grant nos. NSC 95-2112-M-006-028-MY3 and NSC 95-2112-M-022-001-MY2.

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