CARBON NANOTUBES AS A PERFECTLY CONDUCTING CYLINDER

Transcription

1 International Journal of High Speed Electronics and Systems Vol. 0, No. 0 (0000) c World Scientific Publishing Company CARBON NANOTUBES AS A PERFECTLY CONDUCTING CYLINDER Tsuneya ANDO Department of Physics, Tokyo Institute of Technology Ookayama, Meguro-ku, Tokyo , Japan ando@stat.phys.titech.ac.jp Received (received date) Revised (revised date) Accepted (accepted date) 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, its extension to multi-bands cases, and longwavelength phonons and electron-phonon scattering. Keywords: graphite: carbon nanotubes; impurity scattering; phonon scattering. 1. Introduction Graphite needles called carbon nanotubes (CNs) have been a subject of an extensive study since discovery in ,2 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 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 Figure 1 shows a computer graphic image of a single-wall carbon nanotube. 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 ignoring the effect of the tube curvature. 5,6,7,8,9,10,11,12,13,14 These properties can be well reproduced in a 1

2 2 T. Ando Fig. 1. A computer graphic image of a single-wall armchair nanotube. k p method or an effective-mass approximation. 15 The theoretical predictions have been confirmed by Raman experiments 16 and direct measurements of local density of states by scanning tunneling spectroscopy. 17,18,19 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 Transport properties of CNs are interesting because of their unique topological structure. There have been reports on experimental study of transport in CN bundles 32 and ropes. 33,34 Transport measurements became possible for a single multi-wall nanotube 35,36,37,38,39 and a single single-wall nanotube. 40,41,42,43,44 Single-wall nanotubes usually exhibit large charging effects presumably due to nonideal contacts. 45,46,47,48,49 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. 51,52,53,54,55,56 In section 2 electronic states are discussed in a nearest-neighbor tight-binding model. In section 3, the effective mass equation is introduced. In section 3 the absence of backscattering in the presence of scatterers is discussed. In section 4 the discussion is extended toward the presence of perfectly conducting channel when several bands coexist at the Fermi level. In section 5 a continuum model for phonons in the long-wavelength limit is introduced and effective Hamiltonian describing electron-phonon interaction is derived. A short summary is given in section Electronic States The structure of 2D graphite sheet is shown in Fig. 2. We have the primitive

3 Carbon Nanotubes as a Perfectly Conducting Cylinder 3 (m a,m b ) A τ 1 τ 2 τ 3 A B A y x T (0,0) η y x L (n a,n b ) b a (a) Fig. 2. (a) The lattice structure of a 2D graphite sheet and various quantities. We shall consider the case that 0 η η. The zigzag nanotube corresponds to η = 0 and the armchair nanotube to η =π/6. (b) The coordinate system on cylinder surface. (b) translation vectors a = a(1, 0) and b = a( (1/2), 3/2), and the vectors connecting between 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 the relations 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 ), l (2) εψ B (R B )= γ 0 ψ A (R B + τ l ). l 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

4 4 T. Ando Energy (units of γ 0 ) M Γ K E F -2-3 K Γ M K Wave Vector Fig. 3. Calculated band structure of a two-dimensional graphite along K Γ M K shown in the inset. have 0 γ 0 exp( ik τ l ) l γ 0 exp(+ik τ l ) 0 l ( ) fa (k) = ε f B (k) ( ) fa (k). (3) f B (k) The energy bands are given by ε ± (k) =±γ cos ( akx ) cos 2 ( 3ak ) y +4 cos 2 ( akx ) 2. (4) 2 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 2 +k2 y with γ = 3aγ 0 /2. The band structure is shown in Fig. 3. Every structure of single tube CNs can be constructed from a monatomic layer of graphite as shown in Fig. 2 (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 sometimes 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. 2. 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. Further, the origin x=0 is chosen always at a point corresponding to the top side when the sheet is rolled and the point x = L/2 at point corresponding to the bottom side as is shown in Fig. 2 (b).

5 Carbon Nanotubes as a Perfectly Conducting Cylinder 5 A primitive translation vector in the e y direction is written as T = m a a + m b b, (6) with integer m a and m b. Now, T is determined by the condition T L = 0, which can be written as m a (2n a n b ) m b (n a 2n b )=0, (7) where use has been made of a b= a 2 /2. This can be solved as pm a = n a 2n b, pm b =2n a n b, (8) 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 onedimensional 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 + 2πi ) ( 3 (n a+n b ) = exp exp(ik L) ( = exp 2πi ) ( 3 (n a+n b ) = exp where ν is an integer (0 or ±1) determined by + i 2πν ), 3 i 2πν 3 ), (9) n a + n b =3N + ν, (10) 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 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 ( exp(ik T) = exp + i 2πµ ), 3 i 2πµ 3 ), (11)

6 6 T. Ando ( 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. 4. Some examples of the band structure obtained in a tight-binding model for zigzag nanotubes. where µ=0 or ±1 is determined by m a + m b =3M + µ, (12) 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 = n a 2n b = m and pm b =2n a n 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 = n a 2n b = 0 and pm b =2n a n 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. In a tight-binding model, the periodic boundary condition ψ(r + L) = ψ(r) is converted into ψ A (R A +L) =ψ A (R A ), ψ B (R B +L) =ψ B (R B ). (13) Then, it is straightforward to calculate the band structure of nanotubes when a tight-binding model is used. Figures 4 and 5 show some examples for zigzag and armchair nanotubes. Figure 6 shows the band gap of zigzag nanotubes as a function of the circumference length. 3. Neutrino on Cylinder Surface

7 Carbon Nanotubes as a Perfectly Conducting Cylinder 7 Energy (units of γ0) K Armchair L/a=14 3 K Wave Vector (units of 2π/a) Fig. 5. Some examples of the band structure obtained in a tight-binding model for armchair nanotubes. Energy (ev) (n a,n b )=(L/a,0) 4πγ/3L Circumference L/a Fig. 6. The band gap of zigzag nanotubes with (n a,n b )=(m, 0) as a function of m. The dotted line shows the band gap obtained in a k p approximation and the black dots those of a tight-binding model. 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 K points in the 2D graphite. The effective-mass equation for the K point is given by γ( σ ˆk)F(r) =εf(r), F(r) = ( FA (r) F B (r) ), (14) 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 7 shows the energy dispersion and the density of states. An important feature is the presence of a topological singularity at k =0. A

8 8 T. Ando Fig. 7. The energy dispersion and density of states in the vicinity of K and K points obtained in a k p scheme. 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 ). (15) 3 This corresponds to the presence of a fictitious Aharonov-Bohm (AB) magnetic flux along the axis determined by L. 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, (16) where + and stand for conduction and valence bands, respectively. Figure 8 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. (14) and the boundary conditions are obtained by the replacement ν by ν.

9 Carbon Nanotubes as a Perfectly Conducting Cylinder 9 Fig. 8. Energy bands of a nanotube obtained in the effective-mass approximation for ν = 0 (left) and ν =+1 (right). 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 9 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. 10 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. 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

10 10 T. 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. 9. 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 Fig. 10. Schematic illustration of a back scattering process (solid arrows) and corresponding time-reversal process (dashed arrows). 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. 45,46,47,48,49 Recently, almost ideal contacts were realized and a Fabry-Perot type oscillation due to reflection by contacts was claimed to be observed 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

11 Carbon Nanotubes as a Perfectly Conducting Cylinder 11 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. (17) This leads to the absence of backward scattering rᾱα = rᾱα = 0 in the singlechannel 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 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. (18) 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), i.e., n c Ψ α (r) =ψ α (r)+ ψ β(r) r βα, (19) where Ψ α (r) is the wave function in the left hand side corresponding to mode α. The vanishing determinant of r, i.e., Eq. (18), shows that there exists at least one nontrivial solution for the equation n c α=1 β=1 r βα a α =0. (20) Then, there is no reflected wave for the incident wave function α a αψ α (r). As a matter of fact, Eq. (19) gives Ψ(r) = n a α Ψ α (r) = α=1 n a α ψ α (r), (21) demonstrating the presence of a mode which is transmitted through the system with probability one without being scattered back. Figure 11 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. α=1

12 12 T. Ando Conductance (units of 2e 2 /πh) Mean Free Path εl/2πγ W -1 = u/2γl = Length (units of L) Fig. 11. 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) Conductivity (units of 2e 2 L/πhW) Conductivity Density of States Energy (units of 2πγ/L) Fig. 12. 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,...) Density of States (units of 1/2πγ)

13 Carbon Nanotubes as a Perfectly Conducting Cylinder 13 Conductance (units of 2e 2 /πh) L φ /L εl/2πγ = 1.5 W -1 =100.0 u/2γl = 0.10 Boltzmann Length (units of L) Fig. 13. 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 Figure 12 shows an example of the conductivity obtained by solving the Boltzmann equation. The Boltzmann transport equation can be converted into the equation in terms of mean free path for each band in quasi-one-dimensional 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 13 shows some examples of calculated 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

14 14 T. Ando 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 6in the present case), the conductance becomes close to the Boltzmann result given by the dotted line. 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 pseudo-spin 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, 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 This subject will not be discussed further. 6. Phonons and Electron-Phonon Scattering At nonzero temperatures, lattice vibrations usually constitute the major source of electron scattering and limit the resistivity. Among phonons, long-wavelength acoustic phonons consisting of twisting, stretching, and breathing modes as shown in Fig. 14 are most important. The twisting mode is made of pure circumferencedirectional deformation and its velocity is equal to that of the TA mode of a graphite sheet. The breathing mode corresponds to a radial deformation with a frequency inversely proportional to the radius R of CN. The stretching mode corresponds to the deformation in the nanotube-axis direction and is coupled with the breathing mode when their energies cross each other. These modes can be treated by a continuum model. 73 The potential-energy functional for displacement u=(u x,u y,u z ) is written as U[u] = dxdy 1 ( B(u xx +u yy ) 2 + µ [ (u xx u yy ) 2 +4u 2 2 xy] ), (22) with u xx = u x x + u z R, u yy = u y y, 2u xy = u x y + u y x, (23) where the term u z /R is due to the finite radius R = L/2π of the nanotube. The parameters B = λ+µ and µ denote the bulk modulus and the shear modulus for a

15 Carbon Nanotubes as a Perfectly Conducting Cylinder 15 Fig. 14. Schematic illustration of long-wavelength phonons consisting of transverse twist modes and longitudinal stretching and breathing modes. graphite sheet (λ and µ are Lame s constants). The corresponding kinetic energy is written as K[u] = dxdy M [ ( ux ) 2 +( u y ) 2 +( u z ) 2], (24) 2 where M is the mass density given by the carbon mass per unit area, M = kg/m 2. The corresponding equations of motion are given by Mü x =(B+µ) 2 u x x 2 + µ 2 u x y 2 + B 2 u y x y + B+µ R Mü y = B 2 u x u y x y +(B+µ) 2 y 2 + u y µ 2 x 2 Mü z = B+µ u x R x B µ R u y y B+µ R 2 u z. u z x, + B µ u z R y, (25) The phonon modes are specified by the wave vector along the circumference χ(n)=2πn/l and that along the axis q, u(r)=uexp[iχ(n)x+iqy]. When n=0 and χ=0, the eigen equation becomes Mω 2 u x u y = µq (B+µ)q 2 i(b µ)qr 1 u x u y, (26) u z 0 i(b µ)qr 1 (B+µ)R 2 u z which has three eigenmodes called twisting, stretching, and breathing modes. The twisting mode is made of pure circumference-directional deformation and its velocity v t is equal to that of the TA mode of a graphite sheet, µ ω T (q) =v T q, v T = vt G = M. (27) In the long wavelength limit q = 0, the radial deformation generates a breathing mode with a frequency B+µ 1 ω B = M R, (28) which is inversely proportional to the radius R of the CN. In the case qr 1, the deformation in the nanotube-axis direction generates stretching modes. When

16 16 T. Ando Fig. 15. Some examples of deformation of the cross section of CN with n=0, ±1, and ±2. ω ω B, we have from the last equation of the above u z i B µ B+µ qr u y. (29) Upon substitution of this into the second equation, we have ω S = v S q, v S = 4Bµ (B+µ)M. (30) The velocity v S is usually smaller than that of the LA mode of the graphite v G L = (B+µ)/M. We set v G L =21.1 km/s and v G T =15.0 km/s, and we obtain v s =21.1 km/s, v t =15.0 km/s, and hω B = ev, or 237 K for an armchair CN with L=(20, 10)a and R=6.785 Å. These values show good agreement with recent results by Saito et al. 74 and can never be reproduced by a zone-folding method. The above model is too simple when dealing with modes with n 0. In order to see this fact explicitly, we shall consider the case with q = 0. In this case there is a displacement given by sin 2inx L u x = u n u z = u cos 2inx L = u n sin nx R, (31) nx = u cos R, with arbitrary u. This displacement gives u xx =0 identically and also u yy =u xy =0, giving rise to the vanishing frequency. For n = ±1, this vanishing frequency is absolutely necessary because the displacement corresponds to a uniform shift of a nanotube in a direction perpendicular to the axis. For n>1, on the other hand, the displacement corresponds to a deformation of the cross section of the nanotube as shown in Fig. 15. Such deformations should have nonzero frequency in actual graphite because otherwise CN cannot maintain a cylindrical form. Actually, we have to consider the potential energy due to nonzero curvature of the 2D graphite plane. It is written as U c [u] = 1 [( 2 2 a2 Ξ dxdy x ) ] 2, R y 2 u z (32) where Ξ is a force constant for curving of the plane. The presence of the term 1/R 2 guarantees that the deformation with n = ±1 given in Eq. (31) has a vanishing

17 Carbon Nanotubes as a Perfectly Conducting Cylinder n ω/ω B ω/ωb 0.5 No Curvature Effect qr (a) n With Curvature Effect qr (b) Fig. 16. Frequencies of phonons obtained in the continuum model. (a) Without out-of-plane curvature effect and (b) with out-of-plane curvature effect. 73 frequency. This curvature energy is of the order of the fourth power of the wave vector and therefore is much smaller than U[u] as long as qr 1 for n = 0 and 1 but becomes appreciable for n>1. Figure 16 shows phonon dispersions calculated in this continuum model. An acoustic phonon gives rise to an effective electron-phonon coupling called the deformation potential V 1 = g 1 (u xx + u yy ), (33) proportional to a local dilation, with u xx =( u x / x)+(u z /R) and y yy = u y / y, where u x, u y, and u z represent lattice displacements in the x, y, and z directions, respectively. This term appears as a diagonal term in the matrix Hamiltonian in the effective-mass approximation. Consider a square a a with area S =a 2. In the presence of a lattice deformation, the area S changes into S +δs(r) with δs(r) =a 2 (r), where (r) =u xx + y yy. Therefore, the ion density changes locally by n 0 n 0 [1 (r)]. The electron density should change in the same manner due to the charge neutrality condition. Consider a 2D electron gas. The potential energy δε(r) corresponding to the density change should satisfy δε(r)d(ε F )=n 0 (r), where D(ε F ) is the density of states at the Fermi level D(ε) =m/π h 2 independent of energy. Therefore, n 0 = D(ε F )ε F, leading to δε(r) =ε F (r). This shows g 1 = ε F. In the 2D graphite, the electron gas model may not be so appropriate but can be used for a very rough estimation of g 1 as the Fermi energy measured from the bottom of the valence bands (σ bands), i.e., ev.

18 18 T. Ando A long-wavelength acoustic phonon also causes a change in the distance between nearest neighbor carbon atoms and in the transfer integral. Adding this contribution, the Hamiltonian describing electron-phonon interaction is given by with H K = ( V1 V 2 V 2 V 1 ), (34) V 2 = g 2 e 3iη (u xx u yy +2iu xy ), (35) and g 2 =(3αβ/4)γ 0, where β = (b/γ 0 )( γ 0 / b) = d ln γ 0 /d ln b, γ 0 is the transfer integral between nearest-neighbor atoms, b is the bond length, and α is a quantity smaller than unity dependent on microscopic models of phonons. We have γ = 3aγ 0 /2. Usually, we have β 2 75 and α 1/3, which give g 2 γ 0 /2or g ev. This coupling constant is much smaller than the deformation potential constant g 1 30 ev. Off-diagonal parts correspond to the previous results 76 without the reduction factor α. Apart from the spatial part of the wave function, the pseudo-spin part of the wave function is given by s+) = 1 ( ) is, s ) = 1 ( ) +is, (36) where s = +1 for the conduction band and 1 for the valence band, and + and for the right- and left-going wave, respectively. Then, we have ( ( ) s ) V1 V s 2 + V2 = irev V 2. (37) 1 This means that the diagonal deformation-potential term does not contribute to the backward scattering as in the case of impurities and only the real part of the much smaller off-diagonal term contributes to the backward scattering. We have ReV 2 = g 2 [cos 3η(u xx u yy ) 2 sin 3ηu xy ]. (38) In armchair nanotubes with η = π/6, we have ReV 2 = 2g 2 u xy and only shear or twist modes contribute to the scattering. In zigzag nanotubes with η = 0,on the other hand, ReV 2 = g 2 (u xx u yy ) and only stretching and breathing modes contribute to the scattering. When a high-temperature approximation is adopted for phonon distribution function, the resistivity for an armchair nanotube is calculated as ρ A (T )= h 1 g2k 2 B T e 2 R 2γ 2 µ, (39) where µ is the shear modulus. At temperatures much higher than the frequency of the breathing mode ω B, the resistivity of a zigzag nanotube is same as ρ A (T ), i.e.,

19 Carbon Nanotubes as a Perfectly Conducting Cylinder 19 Resistivity (units of ρ B ) η π/6 (Armchair) π/12 (Chiral) 0 (Zigzag) Temperature (units of T B ) Fig. 17. The resistivity of armchair (solid line) and zigzag (dotted line) nanotubes in units of ρ A (T B ) which is the resistivity of the armchair nanotube at T = T B, and T B denotes the temperature of the breathing mode, T B = hω B /k B. 73 ρ Z (T )=ρ A (T ). At temperatures lower than ω B, on the other hand, the breathing mode does not contribute to the scattering and therefore the resistivity becomes smaller than that of an armchair nanotube with same radius. Figure 17 shows calculated temperature dependence of the resistivity. Because of the small coupling constant g 2 the absolute value of the resistivity is much smaller than that in bulk 2D graphite dominated by much larger deformation-potential scattering. The resistivity of an armchair CN is same as that obtained previously 77 except for a difference in g 2. The mean free path Λ at high temperature is given by Λ= µa 2 3k B Tκ 2 L. (40) β2 Using the parameters we obtain Λ 600 L at room temperature. The mean free path is as large as 1 µm for thin armchair nanotubes with diameter 1.5 nm and increases in proportion to L with the increase of L. This means that a metallic CN becomes a one-dimensional ballistic conductor even at room temperature. When the energy is outside of the region 1 <ε(2πγ/l) 1 < +1 where only metallic linear bands are present, it is necessary to solve a Boltzmann transport equation taking scattering between bands into account. 68,78 Figure 18 shows obtained Fermi energy dependence of conductivity for metallic and semiconducting CN s for g 1 /g 2 =10. For metallic nanotubes, around ε 0, the diagonal potential g 1 causes no backward scattering between two bands with linear dispersion, and smaller off-diagonal potential g 2 determines resistivity. However, when the Fermi energy becomes higher and the number of bands increases, g 1 dominates the conductivity due to inter-band scattering and the conductivity drops drastically. On the other hand, such a drastic

20 20 T. Ando σ(ε)/σ A (0) g 1 /g 2 =10 Metal Semiconductor ε(2πγ/l) -1 Fig. 18. Fermi-energy dependence of conductivity for metallic (solid line) and semiconducting (broken line) CN s with g 1 /g 2 = 10 in units of σ A (0) denoting the conductivity of an armchair CN with ε =0. 73 change disappears for semiconducting CN s and smaller conductivity compared to that of a metallic CN shows dominance of the diagonal potential independent of the Fermi energy. 7. Summary 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 electron looses its coherence for a distance determined by a phase coherence length. Acknowledgments This work has been supported in part by Grants-in-Aid for COE (12CE2004 Control of Electrons by Quantum Dot Structures and Its Application to Advanced Electronics ) and Scientific Research from the Ministry of Education, Science and Culture, Japan. The author would like to thank Hidekatsu Suzuura, Tatsuya Yaguchi, Hideaki Sakai, Hajime Matsumura, Takeshi Nakanishi, and Hiroshi Ajiki for collab-

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