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Theory of ballistic transport in carbon nanotubes Tsuneya Andoa,*, Hajime Matsumuraa,1, Takeshi Nakanishib a

Institute for Solid State Physics, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan b Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands

Abstract A brief review of electronic and transport properties of carbon nanotubes is given mainly from a theoretical point of view. The topics include an effective-mass description of electronic states, the absence of backward scattering except for scatterers with a potential range smaller than the lattice constant, a conductance quantization in the presence of lattice vacancies, junction systems, and effects of Stone–Wales defects. r 2002 Elsevier Science B.V. All rights reserved. PACS: 72.10d; 73.50h; 73.40h Keywords: Carbon nanotubes; Effective-mass approximation; Aharonov–Bohm effect; Conductance quantization; Impurity scattering; Lattice vacancy; Topological defect; Junction

1. Introduction Graphite needles called carbon nanotubes (CNs) were discovered recently [1,2] and have been a subject of an extensive study. A CN is a few concentric tubes of two-dimensional (2D) graphite consisting of carbon-atom hexagons arranged in a helical fashion about the axis. The diameter of ( and their CNs is usually between 4 and 300 A length can exceed 1 mm: Single-wall nanotubes are produced in the form of ropes [3,4]. The purpose of this paper is to give a brief review of recent theoretical study on transport properties of carbon nanotubes. Carbon nanotubes can be either a metal or semiconductor, depending on their diameters and *Corresponding author. Fax: +81-3-5734-2739. E-mail address: [email protected] (T. Ando). 1 Present address: Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

helical arrangement. The condition whether a CN is metallic or semiconducting can be obtained based on the band structure of a 2D graphite sheet and periodic boundary conditions along the circumference direction. This result was ﬁrst predicted by means of a tight-binding model. These properties can be well reproduced in a k p method or an effective-mass approximation [5]. In fact, the method has been used successfully in the study of wide varieties of electronic properties of CN. Some of such examples are magnetic properties [6] including the Aharonov–Bohm effect on the band gap, optical absorption spectra [7,8], exciton effects [9], lattice instabilities in the absence [10,11] and presence of a magnetic ﬁeld [12,13], magnetic properties of ensembles of nanotubes [14], and effects of spin–orbit interaction [15]. Transport properties of CNs are interesting because of their unique topological structure. There have been some reports on experimental study of transport in CN bundles [16] and ropes

0921-4526/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 0 9 6 4 - X

T. Ando et al. / Physica B 323 (2002) 44–50

[17,18]. Transport measurements became possible for a single multi-wall nanotube [19–24] and a single single-wall nanotube [25–29]. In this paper, we shall mainly discuss the transport properties obtained theoretically.

2. Energy bands Fig. 1 shows the lattice structure and the ﬁrst Brillouin zone of a 2D graphite together with the coordinate systems. The unit cell contains two carbon atoms denoted as A and B. A nanotube is speciﬁed by a chiral vector L ¼ na a þ nb b with integer na and nb and basis vectors a and b ðjaj ¼ ( In the coordinate system ﬁxed jbj ¼ a ¼ 2:46 AÞ: onto a p graphite sheet, we have a ¼ ða; 0Þ and b ¼ ﬃﬃﬃ ða=2; 3a=2Þ: For convenience, we introduce another coordinate system where the x direction is along the circumference L and the y direction is along the axis of CN. The direction of L is denoted by the chiral angle Z: A graphite sheet is a zero-gap semiconductor in the sense that the conduction and valence bands consisting of p states cross at K and K0 points of the Brillouin zone, whose pﬃﬃﬃ wave vectors are given by K ¼ ð2p=aÞð1=3; 1= 3Þ and K 0 ¼ ð2p=aÞð2=3; 0Þ [30]. Electronic states near a K point of 2D graphite are described by the k p equation [5]: ! K F ðrÞ A # K ðrÞ ¼ eF K ðrÞ; F K ðrÞ ¼ gð~ s kÞF ; ð2:1Þ FBK ðrÞ where g is the band parameter, k# ¼ ðk#x ; k#y Þ is a wave-vector operator, e is the energy, and sx and

45

sy are the Pauli spin matrices. Eq. (2.1) has the form of Weyl’s equation for neutrinos. The electronic states can be obtained by imposing the periodic boundary condition in the circumference direction Cðr þ LÞ ¼ CðrÞ except for extremely thin CNs. The Bloch functions at a K point change their phase by expðiK LÞ ¼ expð2pin=3Þ; where n is an integer deﬁned by na þ nb ¼ 3M þ n with integer M and can take 0 and 71: Because CðrÞ is written as a product of the Bloch function and the envelope function, this phase change should be cancelled by that of the envelope functions and the boundary conditions for the envelope functions are given by F K ðr þ LÞ ¼ F K ðrÞexpð2pin=3Þ: Energy levels in CN for the K point are obtained by putting kx ¼ kn ðnÞ with kn ðnÞ ¼ ð2p=LÞ½n ðn=3Þ and ky ¼ k in the above k p equation as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ eð7Þ ðn; kÞ ¼ 7g kn ðnÞ2 þ k2 [5], where L ¼ jLj; n n is an integer, and the upper ðþÞ and lower ðÞ signs represent the conduction and valence bands, 0 respectively. The Hamiltonian for F K for the K0 point is obtained by replacing k#y by k#y and therefore the corresponding energy levels are obtained by replacing n by n: This shows that CN becomes metallic for n ¼ 0 and semiconducting with gap Eg ¼ 4pg=3L for n ¼ 71:

3. Absence of backward scattering In the presence of impurities, electronic states in the vicinity of K and K0 points can be mixed with

Fig. 1. (a) Lattice structure of two-dimensional graphite sheet. Z is the chiral angle. The coordinates are chosen in such a way that x is along the circumference of a nanotube and y is along the axis. (b) The ﬁrst Brillouin zone and K and K0 points. (c) The coordinates for a nanotube.

T. Ando et al. / Physica B 323 (2002) 44–50

46

each other. Therefore, we should use a 4 4 . Schrodinger equation ! FK : ð3:1Þ HF ¼ eF; H ¼ H0 þ V ; F ¼ 0 FK The effective potential of an impurity is written as [31] 0

uA ðrÞ

B B 0 V ¼B B eiZ u0 ðrÞn @ A 0

0

eiZ u0A ðrÞ

uB ðrÞ

0

0

uA ðrÞ

oeiZ u0B ðrÞn

0

0

1

C o1 eiZ u0B ðrÞ C C; C 0 A uB ðrÞ

ð3:2Þ where o ¼ expð2pi=3Þ: If we use a tight-binding model, we can obtain the explicit expressions for the potentials. With the increase of the range, the off-diagonal intervalley term decreases and vanishes when the range exceeds the lattice constant, and the diagonal terms uA ðrÞ and uB ðrÞ become equal to each other also. This leads to the absence of the backward scattering and the realization of a perfect conductor in the presence of scatterers. It has been proved that the Born series for backscattering vanish identically [31]. This has been ascribed to a spinor-type property of the wave function under a rotation in the wave vector space [32]. The absence of backward scattering has been conﬁrmed by numerical calculations in a tightbinding model [33]. More detailed discussion can be found in a review [34]. Because of the presence of large contact resistance between a nanotube and metallic electrode, the conductance usually exhibits a prominent effect of a single electron tunneling due to charging effects. An important information can be obtained on the effective mean free path and the amount of backward scattering in nanotubes [25–29]. In fact, the Coulomb oscillation in semiconducting nanotubes is quite irregular and can be explained only if nanotubes are divided into many separate spatial regions in contrast to that in metallic nanotubes [35]. This behavior is consistent with the presence of considerable amount of backward scattering leading to a strong localization of the wave function in semiconducting tubes. In metallic nanotubes, the wave function is extended in the whole region of a

nanotube because of the absence of backward scattering. With the use of electrostatic force microscopy the voltage drop in a metallic nanotube has been shown to be negligible in comparison with an applied voltage [36].

4. Lattice vacancies Effects of scattering by a vacancy in armchair nanotubes have been studied within a tightbinding model [37,38]. It has been shown that the conductance at e ¼ 0 in the absence of a magnetic ﬁeld is quantized into zero, one, or two times of the conductance quantum e2 =p_ for a vacancy consisting of three B carbon atoms around an A atom, of a single A atom, and of a pair of A and B atoms, respectively [38]. Numerical calculations were performed for about 1:5 105 different types of vacancies and demonstrated that such quantization is quite general [39]. This rule was analytically derived in a k p scheme later [40,41].

5. Junction systems A junction which connects CNs with different diameters through a region sandwiched by a pentagon–heptagon pair has been observed in the transmission electron microscope [2]. Some theoretical calculations on CN junctions within a tightbinding model were reported for junctions between metallic and semiconducting nanotubes and those between semiconducting nanotubes [42,43]. In particular, tight-binding calculations for junctions consisting of two metallic tubes with different chirality or diameter demonstrated that the conductance exhibits a universal power-law dependence on the ratio of the circumference of two nanotubes [44]. The k p scheme is ideal to clarify electronic states and their topological characteristics in such junction systems. Fig. 2 shows the development of a junction system onto a 2D graphite sheet [43]. We have a pair of a pentagon ðR5 Þ and heptagon ðR7 Þ ring, and L5 and L7 are the chiral vector of the thick and thin nanotube, respectively. Therefore, R5 –L5

T. Ando et al. / Physica B 323 (2002) 44–50

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Fig. 3. Schematic illustration of the topological structure of a junction. In the tube regions, two cylinders corresponding spaces for the K and K0 points are independent of each other. In the junction region, they are interconnected to each other. Fig. 2. The structure of a junction consisting of two nanotubes having an axis not parallel to each other (y is their angle).

and R7 –L7 are rolled on to R5 and R7 ; respectively. An equilateral triangle with a base connecting R5 and R5 –L7 and another with a base connecting R7 and R7 –L7 have a common vertex point at R: The angle between L5 and L7 is denoted as y: Boundary conditions can be derived by considering such a structure of the junction [45]. In the junction region, where any point on the development moves onto the corresponding point after making a rotation by p=3 around R as shown in Fig. 2. Then, we have F½Rp=3 r ¼ Tp=3 FðrÞ; 0 0 0 B B 0 0 Tp=3 ¼ B B icðRÞ 0 e @ oe

icðRÞ

0

0 o1 eicðRÞ 0 0

eicðRÞ

1

C 0 C C 0 C A 0 ð5:1Þ

icðRÞ

0

with e ¼ exp½iðK KÞ R ; where o ¼ expð2pi=3Þ and Rp=3 describes a p=3 rotation around R: Because of the boundary conditions, states near the K and K0 point mix together in the junction region. Fig. 3 shows the topological structure of the junction system.

. Under these boundary conditions, the Schrodinger equation can be solved analytically. An approximate expression for the transmission T and reﬂection probabilities R can be obtained by neglecting evanescent modes decaying exponentially into the thick and thin nanotubes [45]. The solution at e ¼ 0 gives T ¼ 4L35 L37 =ðL35 þ L37 Þ2 ; R ¼ ðL35 L37 Þ2 =ðL35 þ L37 Þ2 : We have TB4ðL7 =L5 Þ3 in the long junction ðL7 =L5 51Þ: When they are separated into different components, TKK ¼ T cos2 ð3y=2Þ; TKK0 ¼ T sin2 ð3y=2Þ; RKK ¼ 0; and RKK0 ¼ R; where the subscript KK means intravalley scattering within K or K0 point and KK0 stands for intervalley scattering between K and K0 points. As for the reﬂection, no intravalley scattering is allowed. Explicit calculations can be performed also for ea0 [46]. Effects of a magnetic ﬁeld perpendicular to the axis were also studied [47]. The results show a universal dependence on the ﬁeld component in the direction of the pentagonal and heptagonal rings, similar to that in the case of vacancies [38,41]. On the other hand, more recent k p calculation shows that the junction conductance is independent of the magnetic ﬁeld [48]. The origin of such disagreement is not clear. A bend junction was observed experimentally and the conductance

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T. Ando et al. / Physica B 323 (2002) 44–50

across such a junction between a (6,6) armchair CN and a (10,0) zigzag CN was discussed [49]. The bend junction is a special case of the general junction shown in Fig. 2. Junctions can contain many pairs of topological defects. Effects of three pairs present between metallic (6,3) and (9,0) nanotubes were studied [37], which shows that the conductance vanishes for junctions having a three-fold rotational symmetry, but remains nonzero for those without the symmetry.

6. Stone–Wales defect One typical example of topological defects present in two-dimensional (2D) graphite is a Stone–Wales defect, composed of two heptagon– pentagon pairs lying next to each other [50] as shown in Fig. 4(a). It is realized by bond alternation within four nearest honeycombs, and is known to be energetically quasi-stable. An effec-

tive-mass Hamiltonian for a nonlocal Stone–Wales defect can be derived by following a procedure similar to that of impurities or short-range defects [51]. To express this system we have various choices, as shown in Fig. 4(b)–(d): (b) Cut RA1 –RB3 and RB1 –RA2 and connect RA1 – RA2 and RB1 –RB3 : (c) Cut RA1 –RB2 and RB1 –RA3 and connect RA1 – RA3 and RB1 –RB2 : (d) Add RC1 and RC2 and connect RC1 –RA2 ; RC1 – RB2 ; RC2 –RA3 ; RC2 –RB3 ; and RC1 –RC2 : Set the amplitude of wave functions zero at RA1 and RB1 by introducing an on-site potential u0 and letting it sufﬁciently large. These bond constructions are classiﬁed into two types, asymmetrical way (b) and (c) and symmetrical way (d). The effective Hamiltonian depends on these choices. Fig. 5 shows results of numerical calculations for each model. The conductance exhibits two dips at a positive energy and a negative energy. Except in the vicinity of the dips, the conductance is close to the ideal value 2e2 =p_: With the increase of the circumference, the dip energies approach the band

Conductance (units of e2/πh)

2.0

1.5

Tight-binding (d) (b),(c) 1.0 AC nanotube with a SW defect L=18 3a kca/2π=0.43

Fig. 4. (a) Stone–Wales defect in 2D graphite sheet. Bond alternation makes this type of defect, consisting of heptagons and pentagons. (b) and (c) Two ways to make non-local potential of the Stone–Wales defect in the k p scheme. In this case, bond alternation is asymmetric contrary to the original defect. (d) Another way of making a Stone–Wales defect. The bond potential becomes symmetric.

0.5 -1.0

0.0

1.0

Energy (unit of 2πγ/L5) pﬃﬃﬃ Fig. 5. Conductance of armchair nanotubes L ¼ 3ma (m ¼ 18) with a Stone–Wales defect. Results of tight-binding calculation are also plotted in a dot–dashed line.

T. Ando et al. / Physica B 323 (2002) 44–50

edges and the deviation of the conductance from the ideal value becomes smaller. The results in the k p scheme are in good agreement with tightbinding results. The two-dip structure corresponds to the case of a pair of vacancies at A and B sites although two dips lie at positions asymmetric around e ¼ 0: A Stone–Wales defect can be obtained by ﬁrst removing a pair of neighboring A and B sites and then adding a pair in the perpendicular direction. The above result seems to show that the removal of a pair is likely to be more dominant than the addition of nonlocal transfer between neighboring sites. Effects of a magnetic ﬁeld can also be studied [51]. The conductance is scaled completely by the ﬁeld component in the direction of a defect and the ﬁeld-dependence is similar to that of an A and B pair defect. The bond alternation manifests itself as a small conductance plateau in high magnetic ﬁelds. The conductance of CN with L=a ¼ 10 containing a Stone–Wales defect has been calculated by pseudopotential method [52].

Acknowledgements The author acknowledges the collaboration with R. Saito, H. Suzuura, M. Igami, and T. Yaguchi. This work was supported in part by Grants-in-Aid for Scientiﬁc Research and for COE (12CE2004 ‘‘Control of Electrons by Quantum Dot Structures and Its Application to Advanced Electronics’’) from Ministry of Education, Science and Culture in Japan.

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