Physics of Carbon Nanotubes

Transcription

1 Physics of Carbon Nanotubes Tsuneya Ando Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo , Japan Abstract. A brief review is given on electronic and transport properties of carbon nanotubes mainly from a theoretical point of view. The topics include a description of electronic states in a tight-binding model and in an effective-mass or k p scheme. Transport properties are discussed including absence of backward scattering except for scatterers with a potential range smaller than the lattice constant and its extension to multi-bands cases. 1 Introduction Graphite needles called carbon nanotubes (CNs) have been a subject of an extensive study since discovery in 1991 [1,2]. A multi-wall 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 20 and 300 Å and their length can exceed 1 µm. The distance of adjacent sheets or walls is larger than the distance between nearest neighbor atoms in a graphite sheet and therefore electronic properties of CNs are dominated by those of a single layer CN. Single-wall nanotubes were produced in a form of ropes a few years later [3,4]. The purpose of this paper is to give a brief review of recent theoretical study on electronic and 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 [5 14]. These properties can be well reproduced in a k p method or an effectivemass approximation [15]. The theoretical predictions have been confirmed by Raman experiments [16] and direct measurements of local density of states by scanning tunneling spectroscopy [17 19]. The k p scheme has been used successfully in the study of wide varieties of electronic properties. Some of such examples are magnetic properties including the Aharonov-Bohm (AB) effect on the band gap [20], optical absorption spectra [21], exciton effects [22], lattice instabilities in the absence [23,24] and presence of a magnetic field [25], magnetic properties of ensembles of nanotubes [26], effects of spin-orbit interaction [27], junctions [28], vacancies [29], topological defects [30], and properties of nanotube caps [31].

2 2 Tsuneya Ando Transport properties of CNs are interesting because of their unique topological structure. There have been reports on experiments in CN bundles [32] and ropes [33,34]. Transport measurements became possible for a single multi-wall nanotube [35 39] and a single single-wall nanotube [40 44]. Single-wall nanotubes usually exhibit large charging effects presumably due to nonideal contacts [45 49]. Almost ideal contacts were realized also [50]. In this paper we shall mainly discuss electronic states and transport properties of nanotubes obtained theoretically in the k p method combined with a tight-binding model. It is worth mentioning that several papers giving general reviews of electronic properties of nanotubes were published already [51 56]. In section 2 electronic states are discussed in a nearest-neighbor tightbinding model. In section 3, the effective mass equation is introduced. In section 4 the absence of backscattering in the presence of scatterers is discussed. In section 5 the discussion is extended toward the presence of perfectly conducting channel when several bands coexist at the Fermi level. Brief discussions are made on effects of lattice vacancies in section 6 and on Tomonaga- Luttinger liquid behavior in section 7. A short summary is given in section 8. 2 Electronic States The structure of 2D graphite sheet is shown in Fig. 1. We have the primitive translation vectors a=a(1, 0) and b=a( 1/2, 3/2), and the vectors connecting nearest neighbor carbon atoms τ 1 = a(0, 1/ 3), τ 2 = a( 1/2, 1/2 3), and τ 3 = a(1/2, 1/2 3). Note that a b = a 2 /2. The primitive reciprocal lattice vectors a and b are given by a =(2π/a)(1, 1/ 3) and b = (2π/a)(0, 2/ 3), The K and K points are given as K =(2π/a)(1/3, 1/ 3) and K =(2π/a)(2/3, 0), respectively. We have exp(ik τ 1 )=ω, exp(ik τ 2 )= ω 1, exp(ik τ 3 )=1, exp(ik τ 1 )=1, exp(ik τ 2 )=ω 1, and exp(ik τ 3 )=ω, with ω =exp(2πi/3). In a tight-binding model, the wave function is written as ψ(r) = R A ψ A (R A )φ(r R A )+ R B ψ B (R B )φ(r R B ), (1) where φ(r) is the wave function of the p z orbital of a carbon atom located at the origin, R A =n a a+n b b+τ 1, and R B =n a a+n b b with integer n a and n b. Let γ 0 be the transfer integral between nearest-neighbor carbon atoms and choose the energy origin at that of the carbon p z level. Then, we have εψ A (R A )= γ 0 ψ B (R A τ l ), εψ B (R B )= γ 0 ψ A (R B +τ l ). (2) l l

3 η τ τ τ Physics of Carbon Nanotubes 3 η ηπ η η Fig. 1. The lattice structure of a 2D graphite sheet and various quantities, the corresponding Brillouin zone, and the coordinate system on cylinder surface. We shall consider the case that 0 η π/6. The zigzag nanotube corresponds to η =0 and the armchair nanotube to η =π/6. Assuming ψ A (R A ) f A (k) exp(ik R A ) and ψ B (R B ) f B (k) exp(ik R B ), we have γ 0 l 0 γ 0 exp( ik τ l ) l exp(+ik τ l ) 0 ( ) fa (k) = ε f B (k) ( ) fa (k).(3) f B (k) The energy bands are given by ε ± (k) =±γ 0 exp( ik τ l ). (4) l It is clear that ε ± (K) =ε ± (K ) = 0. Near the K and K point, we have ε ± (k+k)=ε ± (k+k )=±γ kx+k 2 y 2 with γ = 3aγ 0 /2. The band structure is shown in Fig. 2. Every structure of single tube CNs can be constructed from a monatomic layer of graphite as shown in Fig. 1 (a). Each hexagon is denoted by the chiral vector ( L = n a a + n b b = n a 1 3 ) 2 n b, 2 n b. (5) In another convention for the choice of primitive translation vectors, L is characterized by two integers (p, q) with p = n a n b and q = n b and the corresponding CN is often called a (p, q) nanotube. We shall construct a nanotube in such a way that the hexagon at L is rolled onto the origin. For convenience, we introduce another unit basis vectors (e x, e y ) as shown in Fig. 1. The direction of e x or x is along the circumference of CN, i.e., e x = L/L with L= L =a n 2 a +n2 b n an b, and e y or y is along the axis of CN. A primitive translation vector in the e y direction is written as T = m a a + m b b, (6)

4 4 Tsuneya Ando Energy (units of γ0) M Γ K E F -2-3 K Γ M K Wave Vector Fig. 2. Calculated band structure of a two-dimensional graphite along K Γ M K shown in the inset with integer m a and m b.now,t is determined by the condition T L=0 or m a (2n a n b ) m b (n a 2n b ) = 0. This can be solved as pm a = n a 2n b, pm b =2n a n b, (7) where p is the greatest common divisor of n a 2n b and 2n a n b. The first Brillouin zone of the nanotube is given by the region π/t k y <π/t with T = a m 2 a +m2 b m am b. The unit cell is formed by the rectangular region determined by L and T. For nanotubes with sufficiently large diameter, effects of mixing between π bands and σ bands and change in the coupling between π orbitals can safely be neglected. Then, the energy bands of a nanotube are obtained simply by imposing periodic boundary conditions along the circumference direction, i.e., ψ(r+l) =ψ(r). This leads to the condition exp(ik L) = 1, which makes the wave vector along the circumference direction discrete, i.e., k x =2πj/L with integer j, but the wave vector perpendicular to L arbitrary except that π/t k<π/t. The number of one-dimensional bands, i.e., j is given by the total number of carbon atoms in a unit cell determined by L and T. The band structure of a nanotube depends critically on whether the K and K points in the Brillouin zone of the 2D graphite are included in the allowed wave vectors when the 2D graphite is rolled into a nanotube. This can be understood by considering exp(ik L) and exp(ik L). We have ( exp(ik L) = exp + i 2πν ), exp(ik L) ( = exp i 2πν 3 3 ), (8) where ν is an integer (0 or ±1) determined by n a + n b =3N + ν, (9) with integer N. This shows that for ν = 0 the nanotube becomes metallic because two bands cross at the wave vector corresponding to K and K points

5 Physics of Carbon Nanotubes ( 8,0) ( 9,0) (10,0) Energy (ev) 5 0 E F 5 0 E F 5 0 E F Wave Vector (π/ 3a) Fig. 3. Some examples of the band structure obtained in a tight-binding model for zigzag nanotubes without a gap. When ν = ±1, on the other hand, there is a nonzero gap between valence and conduction bands and the nanotube is semiconducting. For translation r r + T the Bloch function at the K and K points acquires the phase ( exp(ik T) = exp + i 2πµ ), exp(ik T) ( = exp i 2πµ ), (10) 3 3 where µ=0 or ±1 is determined by m a + m b =3M + µ, (11) with integer M. When ν = 0, therefore, the K and K points are mapped onto k 0 =+2πµ/3T and k 0 = 2πµ/3T, respectively, in the one-dimensional Brillouin zone of the nanotube. At these points two one-dimensional bands cross each other without a gap. A nanotube has a helical structure for general L. There are two kinds of nonhelical nanotubes, zigzag with (n a,n b )=(m, 0) and armchair with (n a,n b )=(2m, m). A zigzag nanotube is metallic when m is divided by three and semiconducting otherwise. We have pm a = m and pm b =2m, which give m a = 1 and m b = 2, and µ = 0 and T = 3a. When a zigzag nanotube is metallic, two conduction and valence bands having a linear dispersion cross at the Γ point of the one-dimensional Brillouin zone. On the other hand, an armchair nanotube is always metallic. We have pm a =0 and pm b =3m, which gives m a = 0 and m b = 1, and µ = 1 and T = a. Thus, the conduction and valence bands cross each other always at k 0 = ±2π/3a. It is straightforward to calculate the band structure of nanotubes when a tight-binding model is used. Figures 3 and 4 show some examples for zigzag and armchair nanotubes. 3 Neutrino on Cylinder Surface Essential and important features of electronic states become transparent when we use a k p scheme in describing states in the vicinity of K and

6 6 Tsuneya Ando Energy (units of γ0) K Armchair L/a=14 3 K Wave Vector (units of 2π/a) Fig. 4. Some examples of the band structure obtained in a tight-binding model for armchair nanotubes K points in the 2D graphite. The effective-mass equation for the K point is given by ( ) γ(σ ˆk)F(r) FA (r) =εf(r), F(r) =, (12) F B (r) where the origin of energy ε is chosen at K or K points, σ =(σ x,σ y )is the Pauli spin matrix, γ is a band parameter, ˆk = i, and F A and F B represent the amplitude at two carbon sites A and B, respectively [15]. The above equation is same as Weyl s equation for a neutrino with vanishing rest mass and constant velocity independent of the wave vector. The energy becomes ε s (k)=±γ k and the velocity is given by v = γ/ h independent of k and ε. The density of states becomes D(ε)= ε /2πγ 2. Figure 5 shows the energy dispersion and the density of states schematically. An important feature is the presence of a topological singularity at k=0. A neutrino has a helicity and its spin is quantized into the direction of its motion. The spin eigen function changes its signature due to Berry s phase under a 2π rotation. Therefore the wave function acquires phase π when the wave vector k is rotated around the origin along a closed contour [57,58]. The signature change occurs only when the closed contour encircles the origin k = 0 but not when the contour does not contain k = 0. This topological singularity at k = 0 causes a zero-mode anomaly in the conductivity [59,60]. It is also the origin of the absence of backscattering and perfect conductance in metallic carbon nanotubes as discussed below. The wave function is written as a product of the neutrino wave function and the Bloch function at the K point in the k p scheme. The Bloch function ψ K acquires a phase under translation, i.e., ψ K (r+l)=ψ K (r) exp(ik L). Therefore, the boundary conditions for the neutrino wave function becomes [15] ( F(r+L) =F(r) exp 2πiν ). (13) 3

7 Physics of Carbon Nanotubes 7 Fig. 5. The energy dispersion and density of states in the vicinity of K and K points obtained in a k p scheme Fig. 6. Energy bands of a nanotube obtained in the effective-mass approximation for ν =0 (left) and ν =+1 (right) This corresponds to the presence of a fictitious AB magnetic flux along the axis. In the k p scheme, therefore, electrons in a nanotube can be regarded as neutrinos on a cylinder surface with an AB flux. The neutrino wave function is written as F(r) exp[iκ ν (n)x+iky] with κ ν (n) =(2π/L)(n ν/3), where the x axis and y axis are chosen in the circumference and axis direction, respectively, n is an integer, and k is the wave vector in the axis direction. The corresponding energy levels are ε (±) ν (n, k) =±γ κ ν (n) 2 +k 2, (14) where + and stand for conduction and valence bands, respectively. Figure 6 shows a schematic illustration of the bands for ν = 0 and +1. When ν =0, there are bands with a linear dispersion without a gap and CN becomes a metal, while when ν =±1, on the other hand, there is a nonzero gap and CN becomes a semiconductor. There is a one-to-one correspondence between the AB flux and the band gap. The effective-mass equation for the K point is obtained by replacing σ y by σ y in Eq. (12) and the boundary conditions are obtained by the replacement ν by ν.

8 8 Tsuneya Ando Conductance (units of 2e 2 /πh) Length (units of L) ν = 0 φ/φ 0 = 0.00 εl/2πγ = 0.0 Λ/L = 10.0 u/2lγ = Magnetic Field: (L/2πl) 2 Fig. 7. Calculated conductance of finite-length nanotubes at ε = 0 as a function of the effective strength of a magnetic field (L/2πl) 2 in the case that the effective mean free path Λ is much larger than the circumference L. The conductance is always given by the value in the absence of impurities at H = 0 [61] 4 Absence of Backscattering The nontrivial Berry s phase leads to the unique property of a metallic nanotube that there exists no backscattering and the tube is a perfect conductor even in the presence of scatterers [58,61]. In fact, it has been proved that the Born series for backscattering vanish identically [61]. Further, the conductance has been calculated exactly for finite-length nanotubes containing many impurities with the use of Landauer s formula [62]. Figure 7 shows an example of such a conductance as a function of a magnetic field applied perpendicular to the axis. The conductance is given by 2e 2 /π h independent of length in the absence of a magnetic field, while it decreases with length in magnetic fields in consistent with Ohm s law. The absence of backward scattering has been confirmed by numerical calculations in a tight binding model [63]. As shown in Fig. 8 backscattering corresponds to a rotation of the k direction by ±π (in general (2n +1)π with integer n). In the absence of a magnetic field, there exists a time reversal process corresponding to each backscattering process. The time reversal process corresponds to a rotation of the k direction by ±π in the opposite direction. The scattering amplitudes of these two processes are same in the absolute value but have an opposite signature because of Berry s phase. As a result, the backscattering amplitude cancels out completely. In semiconducting nanotubes, on the other hand, backscattering appears because the symmetry is destroyed by a nonzero AB magnetic flux.

9 Physics of Carbon Nanotubes 9 Fig. 8. Schematic illustration of a back scattering process (solid arrows) and corresponding time-reversal process (dashed arrows) An important information has been obtained on the mean free path in nanotubes by single-electron tunneling experiments [40,44]. 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 [64]. This behavior is consistent with the presence of backscattering leading to a localization of the wave function. In metallic nanotubes, the wave function is extended in the whole nanotube because of the absence of backscattering. With the use of electrostatic force microscopy the voltage drop in a metallic nanotube has been shown to be negligible [65]. Single-wall nanotubes usually exhibit large charging effects due to nonideal contacts. Contacts problems were investigated theoretically in various models and an ideal contact was suggested to be possible under certain conditions [45 49]. Recently, almost ideal contacts were realized and a Fabry-Perot type oscillation due to reflection by contacts was claimed to be observed [50]. 5 Presence of Perfectly Transmitting Channel When the Fermi level moves away from the energy range where only linear bands are present, interband scattering appears because of the presence of several bands at the Fermi level. Let r βα be the reflection coefficient from a state with wave vector k α to a state with k β k β in a 2D graphite sheet. Here, β stands for the state with wave vector opposite to β. Only difference arising in nanotubes is discretization of the wave vector. Apart from a trivial phase factor arising from the choice in the phase of the wave function, the reflection coefficients satisfy the symmetry relation [66]: r βα = rᾱβ. (15) This leads to the absence of backward scattering rᾱα = rᾱα = 0 in the single-channel case as discussed in the previous section. Define the reflection matrix r by [r] αβ = rᾱβ. Then, we have r = t r, where t r is the transpose of r. In general we have det t P = det P for any matrix P, where detp is the

10 10 Tsuneya Ando determinant of P. In metallic nanotubes, the number of traveling modes n c is always given by an odd integer and therefore det( r)= det(r), leading to det(r) = 0. By definition, r βα represents the amplitude of an out-going mode β with wave function ψ β(r) for the reflected wave corresponding to an in-coming mode α with wave function ψ α (r). The vanishing determinant of r shows that there exists at least one nontrivial solution for the equation n c r βα a α =0. (16) α=1 Then, there is no reflected wave for the incident wave function α a αψ α (r), demonstrating the presence of a mode which is transmitted through the system with probability one without being scattered back. Figure 9 shows some examples of the length dependence of the calculated conductance for different values of the energy. There are three and five traveling modes for 1 < εl/2πγ <2 and 2 < εl/2πγ <3, respectively. The arrows show the mean free path of traveling modes obtained by solving the Boltzmann transport equation (see below). The relevant length scale over which the conductance decreases down to the single-channel result is given by the mean free path. Figure 10 shows an example of the conductivity obtained by solving the Boltzmann equation. The Boltzmann transport equation can be converted Conductance (units of 2e 2 /πh) Mean Free Path εl/2πγ W -1 = u/2γl = Length (units of L) Fig. 9. An example of the length dependence of the calculated conductance. The arrows show the mean free path of traveling modes obtained by solving the Boltzmann transport equation. For lower three curves there are three bands with n =0 and ±1 and for upper three curves there are five bands with n=0, ±1, and ±2. The mean free path is same for bands with same n and decreases with the increase of n [66]. (b)

11 Physics of Carbon Nanotubes 11 Conductivity (units of 2e 2 L/πhW) Conductivity Density of States Energy (units of 2πγ/L) Fig. 10. An example of the energy dependence of the conductivity calculated using Boltzmann transport equation. The conductivity is infinite in the energy range 1 < εl/2πγ < +1 but becomes finite in the other region where there are several bands coexist at the Fermi level. The dashed line shows the density of states and thin solid lines show the contribution of each band to the conductivity (the lowest curve represents the contribution of m =0, the next two m =±1,...) [66] Density of States (units of 1/2πγ) into the equation in terms of mean free path for each band in quasi-onedimensional systems [67,68] and the conductivity becomes the sum of the mean free path of each band. The Boltzmann equation gives an infinite conductivity as long as the Fermi level lies in the energy range 1<εL/2πγ <+1 where only the linear metallic bands are present. However, the conductivity becomes finite when the Fermi energy moves away from this energy range into the range where other bands are present. Further, the increase of the number of conducting modes gives no enhancement of electronic conduction, because the inter-band scattering becomes increasingly important than the number of conducting modes. This conclusion is quite in contrast to the exact prediction that there is at least a channel which transmits with probability unity, leading to the conclusion that the conductance is given by 2e 2 /π h independent of the energy for sufficiently long nanotubes. The difference originates from the absence of phase coherence in the approach based on a transport equation. In fact, in the transport equation scattering from each impurity is treated as a completely independent event after which an electron looses its phase memory, while in the transmission approach the phase coherence is maintained throughout the system. Effects of inelastic scattering can be considered in a model in which the nanotube is separated into segments with length of the order of the phase coherence length and the electron looses the phase information after the transmission through each segment. Figure 11 shows some examples of calculated

12 12 Tsuneya Ando Conductance (units of 2e 2 /πh) εl/2πγ = 1.5 W -1 =100.0 u/2γl = 0.10 Boltzmann σ = (4e 2 /πh) 2L φ Length (units of L) L φ /L Samples Fig. 11. An example of the conductance in the presence of inelastic scattering as a function of the length for different values of the phase coherence length L φ [66]. conductance for ε(2πγ/l) 1 =1.5 with n c =3. As long as the length is smaller than or comparable to the inelastic scattering length L φ, the conductance is close to the ideal value 2e 2 /π h corresponding to the presence of a perfect channel. When the length becomes much larger than L φ, the conductance decreases in proportion to the inverse of the length. When L φ becomes comparable to the mean free path (Λ 0 /L 6 in the present case), the conductance becomes close to the Boltzmann result given by the dotted line. These results correspond to the fact that a conductivity can be defined for finite L φ. The figure shows that this conductivity is roughly proportional to L φ. 6 Lattice Vacancies Short-Range Scatterers So far, we have exclusively considered the case that scattering potential has a range larger than the lattice constant. When the range becomes smaller than the lattice constant, the effective potential for A and B sites in a unit cell can be different. In this case the scatterer becomes dependent on pseudospin and causes backscattering due to pseudo-spin-flip scattering. Further, it causes intervalley scattering between K and K points, which causes also backscattering and leads to nonzero resistance [61]. One typical example of such short-range and strong scatterers is a lattice vacancy. Effects of scattering by a lattice vacancy in armchair nanotubes have been studied within a tight-binding model [69,70]. It has been shown that the conductance at zero energy 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,

13 Physics of Carbon Nanotubes 13 and of a pair of A and B atoms, respectively [70]. Numerical calculations were performed for about different kinds of vacancies and demonstrated that such quantization is quite general [71]. This rule was analytically derived in a k p scheme later [29,72]. 7 Tomonaga Luttinger Liquid A metallic CN has linear bands in the vicinity of the Fermi level. In such a onedimensional metal, the Coulomb interactions induce a breakdown of the Fermi liquid theory by causing strong perturbations. The resulting system is known as a Tomonaga-Luttinger liquid, which is predicted to show non-fermi-liquid behavior such as the absence of Landau quasi-particles, spin-charge separation, suppression of the tunneling density of states, and interaction-dependent power laws for transport quantities. For armchair CN s, in particular, there have been many theoretical works in which an effective low-energy theory was formulated and explicit predictions were made on various quantities like the energy gap at the Fermi level, tunneling conductance between the CN and a metallic contact, etc. [75 79]. Some experiments suggested the presence of such many-body effects [42,43,80]. In ref. [80], for example, the conductance of bundles (ropes) of singlewall CN s are measured as a function of temperature and voltage. Electrical connections to nanotubes can be achieved by either depositing electrode metal over the top of the tubes (end contacted) or by placing the tubes on the top of predefined metal leads (bulk contacted). The measured differential conductance displays a power-law dependence on temperature and applied bias. The obtained different values of the power between two samples of bulk and end contacted roughly agree with the theoretical predictions on the exponent of the electron tunneling into the bulk and the end of the Tomonaga-Luttinger liquid. 8 Summary and Conclusion In summary, a brief review has been given on electronic and transport properties of carbon nanotubes mainly from a theoretical point of view. The topics include a description of energy bands in a tight-binding model and in an effective-mass or k p scheme. In the latter scheme electrons in nanotubes are regarded as neutrinos on cylinder surface with a fictitious Aharonov-Bohm flux passing through the cross section. The k p description is particularly useful for revealing extraordinary properties of metallic nanotubes. In fact, in metallic carbon nanotubes, there is at least a single channel transmitting through the system without backscattering independent of energy for scatterers with potential range comparable to or larger than the lattice constant. This channel is sensitive to inelastic scattering when several bands coexist at the Fermi level, however. This has been demonstrated in a model that an

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