Dynamical Conductivity in Metallic Carbon Nanotubes

In a self-consistent Born approximation the dynamical conductivity is calculated in a metallic carbon nanotube. To the lowest order in scattering strength the Drude conductivity is given by a delta function of the frequency except in the presence of scatterers with range shorter than the lattice constant. When effects of other bands are included, the Drude conductivity is broadened and the static limit of the conductivity becomes finite. In the case of scatterers with range shorter than the lattice constant, the low-frequency conductivity is given by a conventional Drude conductivity with finite relaxation time.


§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 theoretically explore the dynamical conductivity of a metallic carbon nanotube and discuss the Drude conductivity.
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)ransport 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)−29) Theoretical calculations were made on transport properties of CN.In particular, effects of impurity scattering in CNs were studied and the complete absence of back scattering was proved rigorously except for scatterers having a potential range smaller than the lattice constant. 30)This intriguing fact was related to Berry's phase acquired by a rotation in the wave vector space in the system described by a k•p Hamiltonian 5) which is the same as Weyl's equation for a neutrino. 31)ffects of scattering by a short-range and huge potential corresponding to lattice vacancies were studied, 32−38) and the conductance was shown to be quantized into zero, one, and two times of the conductance quantum e 2 /πh depending on the type of the vacancy.−41) In this paper we discuss the Drude conductivity and its relation to the conductance quantization for scatterers with range comparable or larger than the lattice constant.We will consider the electron scattering on impurities in two cases: (i) The range of the scattering potential is shorter than the lattice constant.(ii) The range of scattering potential is larger than the lattice constant but remains much shorter than the typical electron wavelength.Effects of scattering from such impurities are considered self-consistently by standard Green's function technique.
The paper is organized as follows: In §2 energy levels and wavefunctions for a 2D graphite sheet are summarized and two kinds of impurities are introduced.Further, the dynamical conductivity is calculated in a self-consistent Born approximation.In §3 numerical results are presented and in §4 a brief discussion is made on the relation to the absence of backward scattering and the perfect conductance for scatterers with range larger than the lattice constant.In §5 a short summary is given.§2.Dynamical Conductivity

Effective-Mass Description
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 = Submitted to Journal of Physical Society of Japan (2π/a)(1/3, 1/ √ 3) and K = (2π/a)(2/3, 0). 42)Electronic states near a K point of 2D graphite are described by the k•p equation: 43,5) γ( σ • k)F K (r) = εF K (r), (2.1) where γ is the band parameter, k = ( kx , ky ) 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.This is reduced to the periodic boundary condition for the neutrino wave function F (r) in metallic nanotubes.The wave function and the corresponding energy at the K point is written as where s = ±1, n = 0, ±1, . .., L = |L|, A is the length of the nanotube, and (2.7) In the following we shall consider the dynamical conductivity σ yy (ω) which represents the absorption for electromagnetic wave with its electric field along the axis direction.The current operator is given by and its matrix element is given by with Similar expressions can be obtained easily for the K' point.

Self-Consistent Born Approximation
First, we consider scatterers with potential range larger than the lattice constant.In this case, the scattering between K and K' points is negligible.Therefore, it is sufficient to consider only the K point and the total conductivity is obtained by the multiplication of two corresponding to the valley degeneracy.The potential of scatterers is assumed to be In the self-consistent Born approximation, we have where n i is the average number of impurities per unit area, • • • means the average over scatterers, and we have introduced the cutoff function g(ε) given by with the cutoff energy ε c .The cutoff energy is of the order of the band width and therefore n c = Lε c /2πγ is of the order of the total number of the bands present in the nanotube. 37,44)igure 2 shows the diagram representing the equation for the current vertex part J y (n, k; ε , ε) s s in the self-consistent Born approximation.This equation can be solved by putting and where we have introduced the cutoff function g(ε) in the integrand and the dimensionless parameter characterizing the strength of scatterers.This parameter is related to the parameter A introduced in ref. 45  through W = A −1 .The dynamical conductivity becomes Next, we consider scatterers with potential range much smaller than the lattice constant.For scatterers located at A sites, we have and Similarly for scatterers located at B sites, we have and (2.24) with (2.25) In the case that the self-energy is formally given by the same expression as in the case of long-range scatterers, i.e., by eq.(2.12).
It is straightforward to show that vertex corrections vanish identically for the current and therefore the conductivity is given by eq.(2.19) with Ξ = 1.
When both kinds of scatterers coexist, the conductivity is given by eq.(2.19) where δW and (1 − δ)W represent the contribution of long-and short-range scatterers, respectively.

Drude Conductivity
We consider the case that the Fermi level lies in the bands with n = 0, i.e., −2πγ/L < ε < +2πγ/L.First, we neglect terms with nonzero n completely.Then, we have (2.28) The relaxation time is given by with the density of states where the factor 1/2 in eq.(2.29) represents the absence of backward scattering.Therefore, the approximate selfenergy is given by as is expected.
The function φ becomes The dynamical conductivity becomes In the absence of short-range scatterers, we have (2.35)This corresponds well to the absence of backward scattering leading to the perfect conductance of the metallic nanotube except in the presence of scatterers with range shorter than the lattice constant.
In the following we shall confine ourselves to the case δ = 1 and consider effects of the presence of other bands.Define (2.36) To the lowest order in (h/τ )(2πγ/L) −1 , the self energy is given by (2.38) Therefore, we have Further, we have with (2.41) The substitution of the approximate expression of X(ε) obtained above leads to which gives (2.43) with

.44)
This shows that the conductivity now exhibits a Drude behavior with finite τ eff in contrast to the delta function behavior when the presence of other bands is neglected completely.The width of the Drude conductivity τ −1 eff is proportional to W 2 , i.e., the square of the effective scattering strength, because it appears only in higher order.
For large ε c the correction factor f for the selfenergy diverges logarithmically, i.e., f ∼ ln(Lε c /2πγ), while g converges to a small constant g ∼ 0.46.This means that g can be neglected practically, corresponding to the fact that the cutoff function is irrelevant except in the self-energy.It shows that the broadening of the Drude conductivity is larger in thicker nanotubes because of the presence of many bands.§3.Numerical Results Figure 3 shows some examples of calculated dynamical conductivity at zero temperature for the Fermi energy ε F = 0.The parameters are ε c L/2πγ = 10 and W −1 = 50 in Fig. 3(a) and W −1 = 100 in (b).The parameter 1−δ represents the strength of scatterers with range shorter than the lattice constant.
In the presence of scattering, the peak of interband transitions from the valence band ε −,1 (k) to the conduction band ε +,1 (k) is shifted from 4πγ/L to a lower energy.This shift is caused by a band-gap reduction due to level repulsion effects among different bands due to scattering.The broadening of the interband peak for same W is larger for short-range scatterers δ = 0 than for δ = 1, but the difference is not so significant.
Effects of the potential range are much more dramatic for the Drude conductivity because it is reduced to a delta function for δ = 1 when the presence of bands other than those with the linear dispersion in the vicinity of the Fermi level.The frequency dependence of the Drude conductivity deviates from the delta function due to mixing with different bands by scatterers, as has been shown in the previous section.Figure 4 shows a blowup of σ yy (ω) in the frequency range corresponding to the Drude conductivity.
The sharpness of σ yy (ω) can be seen in the static limit σ yy (0) as is clear in eq.(2.43).Figure 5 shows σ yy (0) as a function of the Fermi energy ε F , together with the density of states at ε F .It shows that the exceptionally sharp Drude peak occurs only when the Fermi level lies in the energy range where only metallic bands with linear dispersion exist.The static conductivity exhibits clear oscillation related to the density of states, which is essentially same as that obtained by Boltzmann transport equation for model short-range scatterers except in the energy range −2πγ/L < ∼ ε F < ∼ + 2πγ/L. 46)

§4. Discussion
As has been discussed in §2, the Drude conductivity is proportional to δ(ω) and its static limit σ yy (0) becomes infinite in the case of long-range scatterers without inter-valley scattering when only the contribution of the metallic linear bands is considered.This corresponds well to the absence of backward scattering leading to the ideal conductance in the presence of scatterers. 30,31)hen effects of other conduction and valence bands are included, however, the conductivity is broadened and σ(0) becomes finite.This is quite in contrast to the fact that the absence of backward scattering is valid even in the presence of other bands.In fact, it can be understood as a direct consequence of Berry's phase due to the presence of a topological anomaly at k = 0 in the k•p equation. 31,47)It can be understood also by a peculiar signature change of the wave function under quasi-time-reversal symmetry of the Hamiltonian. 48)n the presence of scatterers, a state with energy ε is no longer an eigenstate of the momentum and its wave function is written as a linear combination such as with an expansion coefficient C ε snkj , where j = K and K denote the K and K' points, respectively.In the case of long-range scatterers, states associated with different valleys are independent and therefore we can suppress the summation over j.The matrix element of the current j y is given by where (j nk y ) ss is defined in eq.(2.10).For the linear bands with n = 0, we have where sgn(k) represents the signature of k.When the linear bands are classified into right and left movers with positive (v = +γ/h) and negative velocity (v = −γ/h) instead of s = ±1, we have obviously In the case of long-range scatterers the matrix element of the impurity potential between right and left movers vanishes identically and the wave function for a state with velocity v is written as as long as effects of other bands are not included.Therefore, the matrix element of the current operator is simply proportional to the overlapping of the wave functions and vanishes because of the orthogonality between states with different energies, i.e., This leads to the conclusion that the dynamical conductivity vanishes identically for nonzero ω.
In the case of short-range scatterers causing backward scattering, the wave function contains components of states with opposite velocity while there is no current matrix-element between states with different velocities.Therefore, eq.(4.6) is no longer valid and the dynamical conductivity becomes nonzero for nonzero ω, leading to the broadening of the Drude conductivity.
When the presence of other bands is considered, the current matrix element (4.2) is not proportional to the overlap integral even in the case of long-range scatterers giving vanishing backward scattering because the current matrix element depends on the wave vector and the band index.Therefore, the matrix element remains nonzero in general for states with different energies.The broadening of the Drude conductivity can be understood also from Fig. 3.There is a long low-frequency tail of the dynamical conductivity corresponding to interband transition between states with n = ±1, ±2 . ... The Drude peak interacts with such interband transitions and therefore likely to be broadened due to mixing.
The Landauer formula shows that the conductance is given by the transmission probability and becomes ideal in the absence of backward scattering.In deriving Landauer's formula for the conductance, we assume that the system is attached to infinitely long ideal leads each of which is connected eventually to a reservoir.As a result the conductance is given by the transmission probability between eigenstates of the Hamiltonian in the absence of scatterers with exactly same energy.The absence of backward scattering and the perfect conductance occur only between these states.
In the dynamical conductivity, however, the optical transition occurs between eigen states consisting of mixtures of different bands in the presence of scatterers.It is reasonable, therefore, that the Drude conductivity is broadened although the conductance remains ideal due to the special feature of the absence of backward scattering.The relation between the Drude conductivity calculated using the Kubo formula and the perfect conductance obtained by the Landauer formula constitutes an intriguing and fundamental theoretical problem.§5.Summary We have calculated the dynamical conductivity in metallic carbon nanotubes in a self-consistent Born approximation.It has been shown that to the lowest order in scattering strength the Drude conductivity is given by a delta function of the frequency except in the presence of scatterers with range shorter than the lattice constant.This corresponds to the absence of backward scattering and the perfect conductance.When effects of other bands are included, however, the Drude conductivity is broadened and the static limit of the conductivity becomes finite.In the case of scatterers with range shorter than the lattice constant, the low-frequency conductivity is given by a conventional Drude conductivity with finite relaxation time.

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

Fig. 2
Fig. 2 The self-consistent Born approximation.(a) The self-energy.(b) The current vertex part.

Fig. 5
Fig. 5 The static limit of the dynamical conductivity as a function of the Fermi energy.The solid lines represent σ(0) and the dashed line the density of states.(a) W −1 = 50.(b) W −1 = 100.