Presence of Perfectly Conducting Channel in Metallic Carbon Nanotubes

It is proved that a conducting channel transmitting perfectly without backscattering is present independent of energy in metallic carbon nanotubes for scatterers with potential range larger than the lattice constant. In the case that several traveling channels are present, the conductance decreases from the ideal value determined by the number of traveling modes to the single-channel value when the length exceeds the mean free path determined by a Boltzmann transport equation. Further, inelastic scattering makes the conductance decrease in proportion to the inverse of the length in qualitative agreement with the Boltzmann result.


§1. Introduction
Carbon nanotubes (CN's) are quasi-one-dimensional materials made of sp 2 -hybridized carbon networks 1) and have been a subject of an extensive study. In particular, the electronic structure of a single CN has been studied theoretically, which predicted that CN becomes either metallic or semiconducting depending on its chiral vector, i.e., boundary conditions in the circumference direction. 2−11) These predictions have been confirmed by Raman experiments 12) and direct measurements of local density of states by scanning tunneling spectroscopy. 13−15) In this paper we shall investigate effects of potential scattering in metallic nanotubes in the case of the presence of several current carrying channels.
Transport properties are particularly interesting because of their unique topological structures. 16) For impurity scattering, it was shown theoretically that there is no backscattering for potentials with a range larger than the lattice constant in metallic CN's. 17) The absence of backscattering was related to Berry's phase acquired by a rotation in the wave-vector space in the system described by a k·p Hamiltonian. 18) It has been confirmed by numerical calculations in a tight-binding model. 19) There have been some reports on experiments which support this theoretical prediction. 20,21) In understanding transport properties of nanotubes, a k·p method or an effective-mass approximation 11) has been quite useful. It has been used successfully in the study of wide varieties of electronic properties of CN's. Some of such examples are magnetic properties 22) including the Aharonov-Bohm effect on the band gap, optical absorption spectra, 23) exciton effects, 24) lattice instabilities in the absence 25,26) and presence of a magnetic field, 27) magnetic properties of ensembles of nanotubes, 28) effects of spin-orbit interaction, 29) and electronic properties of nanotube caps. 30) Effects of scattering by a lattice vacancy in armchair nanotubes have been studied within a tight-binding model. 31,32) 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 /πh for ∆N AB ≥ 2, ∆N AB = 1, and ∆N AB = 0, respectively, where ∆N AB is the difference between the number of removed atoms at A and B sites. 32,33) This rule was analytically derived in a k·p scheme. 34,35) Calculations for realistic defects were performed also. 36) Long wavelength phonons and electron-phonon scattering have been studied. 37) In this paper, we consider scattering in metallic carbon nanotubes containing scatterers with potential range larger than or comparable to the lattice constant in the case that several traveling modes are present at the Fermi energy. We shall prove analytically the presence of a channel transmitted through the system without backscattering. Further, numerical calculations are performed to clarify how the conductance decreases from the ideal value determined by the number of traveling modes to that of a single mode with the increase of the length. Effects of inelastic scattering are studied also within a model that an electron looses its phase memory after traveling over a distance determined by a phase coherence length.
The paper is organized as follows: In §2 the treatment of electronic states in a k·p scheme is reviewed briefly. Some exact properties of reflection coefficients in metallic nanotubes are derived in §3. Numerical results on the length dependence of the conductance are presented in §4. In §5, a Boltzmann transport equation is shown to give a finite conductivity when there are several current carrying bands at the Fermi level, quite in contrast to the exact result showing the presence of a perfectly transmitting channel. In §6 a model calculation is performed to clarify effects of inelastic scattering. §2. Effective-Mass Equation The lattice structure of a two-dimensional (2D) graphite and corresponding first Brillouin zone are shown in Fig. 1. A unit cell contains two carbon atoms denoted by A and B. The coordinate system (x , y ) is fixed onto the graphite sheet (x is in the direction of a with a = |a| being the lattice constant) and (x, y) is chosen in such a way that the x axis is in the chiral direction of CN, i.e., the direction along the circumference, and the y axis in the direction of the axis. The angle between the x and x direction is called a chiral angle and denoted as η. We have η = 0 for a zig-zag nanotube and η = π/6 for an armchair nanotube. The conduction and valence bands of 2D graphite cross each other at K and K' points where Submitted to Journal of Physical Society of Japan the bands have approximately a linear dispersion as a function of the wave vector.
In the effective-mass approximation, the electronic states of a 2D graphite near a K point are described by a k·p equation as 11,38) γ (σ ·k) F (r) = ε F (r), 2) where γ is the band parameter, σ = (σ x , σ y ) is the Pauli spin matrix,k = (k x ,k y ) is a wave vector operator defined byk = −i∇, and F A and F B represent the amplitude at A and B carbon atoms, respectively. The wave functions and energies of this Hamiltonian are written as where L is the circumference of CN, A is the length, and s = +1 and −1 denote conduction and valence bands, respectively. In metallic CN's the wave vector in the k x direction is quantized into k x = κ(n) with κ(n) = 2πn/L where n is an integer. In general, we can write eigenvector F sk as where φ s (k) is an arbitrary phase factor, θ(k) is the angle between wave vector k and the k y axis, i.e., k x + ik y = i|k|e iθ(k) , R(θ) is a spin-rotation operator, given by (2.6) with σ z being a Pauli matrix, and |s) is the eigenvector for the state with k in the positive k y direction, given by Obviously, we have (2.8) Further, because R(θ) describes the rotation of a spin, it has the property which gives R(−π) = −R(+π). In order to define states and corresponding wave functions uniquely, we shall assume in the following that −π < θ(k) ≤ +π. (2.10) By choosing the phase φ s (k) in an appropriate manner, the wave function can be chosen as either continuous or discontinuous across the point corresponding to θ = +π and −π in the k space. The results are certainly independent of such choices. §3. Reflection Coefficient In the following, we shall consider the reflection coefficient rβ α from a state with wave vector k α to a state with kβ ≡ −k β due to an arbitrary external potential having a range larger than or comparable to the lattice constant 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 in the k x direction as mentioned above. We shall confine ourselves to states in the vicinity of the K point, but the extension to states near a K' point is straightforward.
Introduce a T matrix defined by where V is the impurity potential given by a diagonal matrix, i.e., ε is the energy, and H 0 = γσ·k is the Hamiltonian in the absence of the potential. The (p+1)th order term of the T matrix corresponding to scattering α →β is written as For each term in eq. (3.3), there is a term obtained through the replacement Both matrix elements of the impurity potential and energy denominators are same as those for the corresponding quantities of (s,ᾱ|T (p+1) |s, β). Figure  2 shows a schematic illustration of processes (s,β|T (p+1) |s, α) and (s,ᾱ|T (p+1) |s, β). Define This quantity describes matrix elements of a rotation in the spin space corresponding to (s,β|T (p+1) |s, α). Under the replacement (3.4), it changes into S According to the present definition of the angle given by eq. (2.10), we have where sgn(t) is the signature of t defined by Note the definition for t = 0. Note also that the choice θ(−k j ) = θ(k j ) − π or θ(−k j ) = θ(k j ) + π causes only the difference in the signature of the wave function. For intermediate states, therefore, the results are independent of these choices because θ(−k j ) always appears in a pair, In the following we shall put θ(−k j ) = θ(k j )−π for j = 1, . . . , p and have Now, with the use of eq. (3.7) we have where S p+1 αβ is obtained from eq. (3.5) just by the exchange α ↔ β.
More explicitly, we have (3.12) which is either real or pure imaginary. Because the number of matrix elements for different bands is always even in the expression of S (p+1) , we see immediately that S (p+1) is given by a real number. Therefore, we have (3.13) This immediately gives the relation with and which satisfies |u α | = 1. The above relation for the T matrix immediately gives This leads to the absence of backward scattering shown previously. 17,18) Define the reflection matrix r by and a unitary matrix U by Then, we have where t r is the transpose of r. In general we have det t P = det P for any matrix 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 By definition, rβ α represents the amplitude of an out-going modeβ with wave function ψβ(r) for the reflected wave corresponding to an in-coming mode α with wave function ψ α (r), i.e., Then, there is no reflected wave for the incident wave function α a α ψ α (r). As a matter of fact, eq. (3.23) gives demonstrating the presence of a mode which is transmitted through the system with probability one without being scattered back. This conclusion is a natural extension of the absence of backward scattering when the Fermi level lies in the range corresponding to the presence of a single traveling channel proven previously. 17,18) It is interesting to note that a perfect channel can exist when the number n c of the traveling channels is odd and cannot when n c is an even integer because det(r) = 0 in this case. In fact, in a quantum wire with a strong spin-orbit interaction similar symmetry relations between reflection coefficients exist but the mode number is always even. 39) The relation given by eq. (3.17) can be derived also by the property of a time reversal operator. Corresponding to the 2×2 equation containing Pauli's spin matrices, a time reversal operation T is defined as (3.26) where the wave function is denoted by ψ for convenience, ψ * represents the complex conjugate of ψ, and K is an antisymmetric unitary matrix The corresponding operation for an operator P is given by Note that For this definition, we have the identity where (ψ, ψ ) stands for the inner product of ψ and ψ . It is easy to show that the Hamiltonian H 0 in the absence of scatterers is invariant under the time reversal operation. Further, the potential is given by a real and diagonal matrix and therefore invariant also. Consequently the T matrix is also invariant under time reversal. Under the time reversal operation, the wave functions are transformed as which show that ψ sα and ψ sᾱ are indeed connected to each other through the time reversal. With the use of eq. (3.31), we have which agrees with the result obtained directly using eq. (3.29), Because of the time-reversal invariance of T matrix, we have which gives the relation showing that transmission coefficient t βα is same as that of the time reversal processes tᾱβ apart from a trivial phase factor arising from the phase of the wave functions, i.e., t βα = u α tᾱβu * β and |t βα | 2 = |tᾱβ| 2 . For reflection processes, on the other hand, we have which gives This is equivalent to eq. (3.17) or eq. (3.21). The negative sign in the right hand side appears because apart from a trivial phase factor ψ T α can be replaced by ψᾱ while ψ T β should be replaced by −ψ β . The time reversal symmetry does not give any relation between the reflection coefficients for waves incident from left and those from right.
Because the phase of the wave function can be chosen arbitrarily as long as it is uniquely defined for each state, it is possible to choose phases in such a way that the transmission coefficients satisfy tᾱβ = −t βα and rᾱ β = rβ α for some α and β. However, it is impossible to choose phases so as to make all transmission coefficients antisymmetric and reflection coefficients symmetric. In fact, the symmetry for diagonal terms is independent of phases, i.e., t αα = tᾱᾱ and rᾱ α = −rᾱ α . Note, further, that the determinant of the reflection matrix vanishes independent of such different choices of the phase.

§4. Numerical Results
In order to see the dependence of the conductance on the length of CN explicitly, we shall consider a model system described by scatterers with potential where r i denotes the position of the ith scatterer and u i its strength. In actual calculations, r i distributes randomly and equal amount of attractive and repulsive scatterers are assumed, i.e., u i = ±u. In the lowest Born approximation the scattering strength is characterized by the dimensionless quantity where n i is the concentration of impurities in a unit area and · · · means the average over scatterers. 40) It is straightforward to calculate a scattering matrix for an impurity given by eq. (4.1) and a conductance of a finite-length nanotube containing many impurities, combining scattering matrices as discussed previously. 17,41) Figure 3 shows some examples of the length dependence of the calculated conductance for long-range scatterers 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 the next section). We have chosen W −1 = 1000 in Fig. 3 (a) and W −1 = 100 in (b). Although detailed behavior varies as a function of energy relative to the band edge and of W , we can safely conclude that the relevant length scale over which the conductance decreases down to the single-channel result is given by the mean free path. §5. Transport Equation The transport equation in a CN is the same as that in quantum wires 42) and written as where E is the electric field in the y direction and g mk is the distribution function for states specified by the wave vector k in the y direction and the band index m. The scattering probability is given in the Born approximation by where V m k mk is the scattering amplitude. The transport equation can be solved to the lowest order in the electric field as where f (ε) is the Fermi distribution function, τ mk is the relaxation time, and v mk is the group velocity. Substituting eq. (5.3) into eq. (5.1), we obtain the following equation for the mean free path Λ m (ε) = |v mk |τ m for band m: where m and m denote the bands crossing the energy ε, and + (−) the wave vector corresponding to the positive (negative) velocity in the y direction. The kernel for the transport equation is given by The diagonal elements are defined by The conductivity becomes with f being the Fermi distribution function and The actual conductivity should be obtained by multiplication of factor four corresponding to the electron spin and the presence of K and K' points. Note that eq. (2.16) of ref. 43 should be multiplied by two. Figure 4 shows an example of the conductivity obtained by solving the Boltzmann equation. 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.
This conclusion is quite in contrast to the exact prediction discussed in the previous section that there is at least a channel which transmits with probability unity, leading to the conclusion that the conductance is given by 2e 2 /πh independent of the energy for sufficiently long nanotubes. The difference is highly likely to originate 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. §6.

Effects of Inelastic Scattering
In order to take into account effects of inelastic scattering, we shall assume that 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 as illustrated in Fig. 5. The transmission probability of the nth segment is described by T n for modes incident from the left hand side and Tn for modes incident from the right hand side. Similarly, the reflection probability of the nth segment is described by R n for modes incident from the left hand side and Rn for modes incident from the right hand side.
Explicitly, they are defined by where t n and r n are transmission and reflection coefficients of the nth segment. When there are n c right-going modes and n c left-going modes, T and R are n c × n c matrices. In the present case with the time reversal symmetry we have Tn = T n as has been discussed in §3.
The right-i n and left-going current in satisfy where i n and in are n c -component vectors consisting of the current associated with traveling modes. At the left reservoir, we have where i ext is the current determined by the left source reservoir and therefore same for all existing channels. At the right reservoir, we have because there is no reflection at the reservoir. This equation can be solved recursively. Definẽ T n ,Tn,R n , andRn to be the total transmission and reflection matrices for the segments 1, 2, . . ., n combined. Then, when a segment n+1 is added, we have i n =T n i 0 +Rnin, and (6.6) Equation (6.5) is solved as (6.7) Therefore, eq. (6.6) gives In terms of the transmissionT N of the whole segments combined, the conductance is given by 10) The actual conductance should be obtained by multiplication of factor four corresponding to the electron spin and the presence of K and K' points.
In numerical calculations, the tube is divided into different segments with random length. The length of each segment is distributed according to the Poisson distribution (6.11) where L φ is the inelastic scattering length. We have (6.12) Figure 6 shows some examples of calculated conductance for W −1 = 100 and 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 /πh 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.
In the present model, we have assumed that the phase coherence is maintained perfectly in each segment and is lost suddenly after traversal over a segment. This is certainly a quite simplification of inelastic scattering effects and can become unrealistic when the phase coherence length becomes comparable to the mean free path. However, it is expected to be less unrealistic with the increase of the coherence length because phase-breaking scattering events occur less and less frequently. One possible way to take into account inelastic-scattering effects more properly is to attach many wires with current channels at random and connect them to a reservoir with a certain voltage 44) as has been made in quantum Hall systems. 45,46) Such calculations are very tedious and time consuming, and are left for a future study.

§7. Summary and Conclusion
In summary, the presence of a single channel transmitting through the system without backscattering independent of energy has been proved in metallic carbon nanotubes for scatterers with potential range comparable to or larger than the lattice constant. Numerical calculations have shown that the conductance decreases from the ideal value determined by the number of traveling modes to that of a single mode with the increase of the length. The characteristic length is of the order of the mean free path obtained from a Boltzmann transport equation. Effects of inelastic scattering have been studied within a model that an electron looses its coherence for a distance determined by a phase coherence length. For a finite coherence length, the conductance decreases almost in proportion to the inverse of the length when the nanotube is sufficiently long. The conductance approaches that obtained by the transport equation with the decrease of the coherence length.

Fig. 5
A schematic illustration of separation of the nanotube into finite-length tubes determined by the phase coherence length L φ . The length is chosen randomly so as to obey a Poisson distribution. The nanotube separated into N tubes is connected to the left and right reservoirs. Each tube is characterized by matrices T and R giving transmission and reflection probabilities among different in-coming and out-going channels.