Theory of Ballistic Transport in Carbon Nanotubes

Theory of Ballistic Transport in Carbon Nanotubes Tsuneya Ando a, Hajime Matsumura a *, and Takeshi Nakanishi b 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 A brief review is given of electronic and transport properties of carbon nanotubes 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. 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 CNs is usually between 4 and 300 Å and their length can exceed 1 µm. Single-wall nanotubes are produced in a 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 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 [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 transport properties obtained theoretically. 2. Energy Bands Figure 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 = n a a+n b b with integer n a and n b and basis vectors a and b ( a = b =a=2.46 Å). In the coordinate system fixed onto a graphite sheet, we have a=(a, 0) and b=( a/2, 3a/2). 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 η. A graphite sheet is a zero-gap semiconductor in the sense that the conduction and valence bands consisting of π states cross at K and K points of the Brillouin zone, whose wave vectors are given by K =(2π/a)(1/3, 1/ 3) and K =(2π/a)(2/3, 0) [30]. Electronic states near a K point of 2D graphite are described by the k p equation [5]: ( ) γ( σ ˆk)F K (r)=εf K (r), F K F K (r)= A (r) FB K(r), (2.1) where γ is the band parameter, ˆk =(ˆk x, ˆk y ) is a wavevector operator, ε is the energy, and σ x and σ y are the Pauli spin matrices. Equation (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 Ψ(r +L)=Ψ(r) except for extremely thin CNs. The Bloch functions at a K point change their phase by exp(ik L) = exp(2πiν/3), where ν is an integer defined by n a +n b =3M +ν with integer M and can take 0 and ±1. Because Ψ(r) is written as a product of the Bloch function and the envelope function, this phase change should be canceled by that of the envelope functions and the boundary conditions for the envelope functions are given by F K (r+l)=f K (r) exp( 2πiν/3). Energy levels in CN for the K point are obtained by putting k x = κ ν (n) with κ ν (n) =(2π/L)[n (ν/3)] and k y = k in the above k p equation as ε ν (±) (n, k) = ±γ κ ν (n) 2 +k 2 [5], where L = L, n is an integer, and the upper (+) and lower ( ) signs represent the conduction and valence bands, respectively. The Hamiltonian for F K for the K point is obtained by replacing ˆk y by ˆk y and therefore the corresponding energy levels are obtained by replacing ν by ν. This shows that CN becomes metallic for ν = 0 and semiconducting with gap E g =4πγ/3L for ν =±1. 3. Absence of Backward Scattering In the presence of impurities, electronic states in the Submitted to Physica B * Present address: Frontier Science Laboratories, Sony Corporation, Yokohama 240-0005, Japan

Page 2 vicinity of K and K points can be mixed with each other. Therefore, we should use a 4 4 Schrödinger equation ( ) F K HF =εf, H=H 0 +V, F =. (3.1) F K The effective potential of an impurity is written as [31] V = u A (r) 0 e iη u A (r) 0 0 u B (r) 0 ω 1 e iη u B (r) e iη u, A (r) 0 u A (r) 0 0 ωe iη u B (r) 0 u B (r) (3.2) where ω = exp(2πi/3). 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 u A (r) and u B (r) 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 tight binding 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 tight-binding model [37,38]. It has been shown that the conductance at ε=0 in the absence of a magnetic field is quantized into zero, one, or two times of the conductance quantum e 2 /π h 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 10 5 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 pentagonheptagon pair has been observed in the transmission electron microscope [2]. Some theoretical calculations on CN junctions within a tight-binding 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. Figure 2 shows the development of a junction system onto a 2D graphite sheet [43]. We have a pair of a pentagon (R 5 ) and heptagon (R 7 ) ring, and L 5 and L 7 are the chiral vector of the thick and thin nanotube, respectively. Therefore, R 5 L 5 and R 7 L 7 are rolled on to R 5 and R 7, respectively. An equilateral triangle with a base connecting R 5 and R 5 L 7 and another with a base connecting R 7 and R 7 L 7 have a common vertex point at R. The angle between L 5 and L 7 is denoted as θ. 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 π/3 around R as shown in Fig. 2. Then, we have F [R π/3 r]=t π/3 F (r), T π/3 = 0 0 0 e iψ(r) 0 0 ω 1 e iψ(r) 0 0 e iψ(r) 0 0 ωe iψ(r) 0 0 0, (5.1) with e iψ(r) = exp[i(k K) R], where ω = exp(2πi/3) and R π/3 describes a π/3 rotation around R. Because of the boundary conditions, states near the K and K point mix together in the junction region. Figure 3 shows the topological structure of the junction system. Under these boundary conditions, the Schrödinger 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 ε = 0 gives T =4L 3 5 L3 7 /(L3 5 +L3 7 )2, R =(L 3 5 L 3 7) 2 /(L 3 5 +L 3 7) 2. We have T 4(L 7 /L 5 ) 3 in the long junction (L 7 /L 5 1). When they are separated into different components, T KK = T cos 2 (3θ/2), T KK = T sin 2 (3θ/2), R KK = 0, and R KK = R, where the subscript KK means intravalley scattering within K or K point and KK stands for intervalley scattering between K and K points. As for the reflection, no intravalley scattering is allowed. Explicit calculations can

Page 3 be performed also for ε 0 [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 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 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 effectivemass 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 R A1 R B3 and R B1 R A2 and connect R A1 R A2 and R B1 R B3. (c) Cut R A1 R B2 and R B1 R A3 and connect R A1 R A3 and R B1 R B2. (d) Add R C1 and R C2 and connect R C1 R A2, R C1 R B2, R C2 R A3, R C2 R B3, and R C1 R C2. Set the amplitude of wave functions zero at R A1 and R B1 by introducing an on-site potential u 0 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. Figure 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 2e 2 /π h. With the increase of the circumference, the dip energies approach the band 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 tight-binding 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 ε=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. 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Page 5 Fig. 1 Fig. 2 Fig. 3 2.0 Conductance (units of e 2 /πh) 1.5 1.0 Tight-binding (d) (b),(c) AC nanotube with a SW defect L=18 3a k c a/2π=0.43 0.5-1.0 0.0 1.0 Energy (unit of 2πγ/L 5 ) Fig. 4 Fig. 5