Numerical Study of the Dynamical Conductivity in Carbon Nanotubes

The dynamical conductivity in carbon nanotubes in the presence of impurity scattering is studied within the effective mass approximation based on an exact numerical diagonalization of the k · p Hamiltonian. By controlling the Fermi energy and the magnetic flux, the number of conducting channels at the Fermi energy Nc is varied between one, two, and three without breaking symplectic symmetry. The exact dynamical conductivity deviates significantly from that in the self-consistent Born approximation in a small frequency region corresponding to the absence of backscattering for Nc =1, the Anderson localization for Nc =2, and the presence of a perfect conducting channel for Nc =3.


Introduction
Since the discovery of carbon nanotubes, 1) the electronic structure and transport in carbon nanotubes have been a subject of an extensive experimental and theoretical study. 2,3) t has been clarified that carbon nanotubes show many interesting electronic properties because of their unique topological structure.The main purpose of this paper is to investigate effects of impurity scattering on the dynamical conductivity in metallic carbon nanotubes.
Band structure of a single wall carbon nanotube can be either metallic or semiconducting depending on their diameters and helical arrangement.The k • p scheme based on the effective mass approximation can well reproduce such characteristic band structure. 4)Furthermore, because of its simplicity many important electronic properties of carbon nanotubes are clarified in the k • p scheme, including the Aharonov-Bohm (AB) effect on the band gap, 5) optical absorption spectra, 6) exciton effects, 7) and interaction effects. 8)(See also the review article Ref. 3 and references therein.)Recently, the AB effect on the optical absorption peaks was observed experimentally. 9,10) e k • p scheme is also useful for the study of transport properties.It was shown that backward scattering is absent in metallic carbon nanotubes when the potential range of impurities is larger than the lattice constant. 11,12) his can be understood by Berry's phase acquired by a rotation in the wave vector space. 12)Further, the existence of a perfectly conducting channel in metallic nanotubes without short-range scatterers has been proved by using the fact that the k • p Hamiltonian has symplectic symmetry and odd number of propagating channels. 13)[22][23] In addition to the dc-transport property, the actransport property is also important.The dynamical conductivity in metallic carbon nanotubes with longand short-range impurities was calculated in the selfconsistent Born approximation. 24)The weak localization correction was calculated also. 25)More detailed study is desirable to clarify quantum interference effects due to impurity scattering on the dynamical conductivity in carbon nanotubes and to understand the relation between the perfectly conducting channel obtained by the Landauer formula and the conductivity obtained by the Kubo formula.
In this paper, we study the dynamical conductivity in carbon nanotubes with long-range impurities based on a numerical diagonalization of the k • p Hamiltonian.The results are compared with those in the self-consistent Born approximation.Numerical results of the density of states are also reported.This paper is organized as follows: In §2, the effective mass approximation is described.Further, we explain our method to calculate the density of states and the dynamical conductivity.In §3 numerical results are presented and in §4 a discussion is made.The last section is devoted to a short summary.

Effective-mass approximation
We consider a metallic carbon nanotube with circumference L and length A (A L).We take the x direction as the circumference direction and the y direction as the axis direction.For convenience, we impose periodic boundary conditions in the axis direction, i.e., we consider a carbon nanotube with a ring geometry.
A nanotube with a ring geometry can be constructed from a rectangular graphite sheet by imposing periodic boundary conditions in both directions.A graphite sheet has conduction and valence bands consisting of π states which cross at K and K points of the Brillouin zone.In the effective mass approximation, electronic states near the K point are described by the k • p equation: 3,4,26,27) Here γ is a band parameter, k = ( kx , ky ) the wave-vector operator defined by k = −i∇ + eA/c with A being a vector potential, σ = (σ x , σ y ) the Pauli spin matrices, the energy, and the subscripts A and B indicate two carbon atoms in a unit cell.
The eigenvalues of the k • p Hamiltonian for a carbon nanotube are written as where s = ±1 (s = +1 for the conduction band and s = −1 for the valence band).Because of the periodic boundary conditions, the wave number takes discrete values, with n x and n y being integers.Here we have supposed that a magnetic flux φ is passing through the cross section, and φ 0 (= ch/e) is the magnetic flux quantum.The corresponding eigen functions are written as where ψ(k) is the angle of the wave vector k Figure 1 shows the band structure of the k • p Hamiltonian for φ/φ 0 = 0 and φ/φ 0 = 1/2 for an infinitely long nanotube.The energy bands shown in the figure are two-fold degenerate except for the linear band which appears in the absence of magnetic flux.When φ/φ 0 = 0, therefore, the number of bands N c at a Fermi energy F is odd irrespective of the value of the Fermi energy, while N c becomes even for φ/φ 0 = 1/2.We call N c number of channels hereafter in analogy with Landauer's transport theory.In the present work, the dynamical conductivity is calculated for three cases: In the case (i) there is no backward scattering for the conducting channel and the conductance is given by an ideal value independent of length. 11)In the case (ii) the direct backward scattering is forbidden but electronic states are all localized as in usual quantum wires in an infinitely long nanotube.In the case (iii) there is a single perfectly conducting channel which transmits through the system without being scattered back in spite of the presence of interband scattering. 13)e consider impurities whose range of scattering potential is larger than the lattice constant.For such scatters, we can neglect the scattering between K and K points and these two points can be treated independently. 11)In the present work, therefore, we consider only the K point.At the same time, the potential range is supposed to be shorter than the electron wave length, typically 2π/L, and therefore we consider delta scatterers, Here N i is the number of impurities, u i the strength of ith impurity, and r i its position.The positions r i are distributed randomly and independently in the system and we assume equal amount of attractive and repulsive scatters, u i = ±u.
According to the theory in the self-consistent Born approximation, 24) the strength of disorder is characterized by the dimensionless parameter Here n i (= N i /LA) is the average impurity density.This parameter W is equal to the ratio between n i u 2 /2Lγ and 2πγ/L, which are the level broadening /2τ obtained in the Born approximation when only the linear band is considered 24) and the distance of two consecutive bands at k y = 0, respectively.The half magnetic-flux quantum does not break time reversal symmetry and therefore all systems considered have symplectic symmetry. 25,28) sually systems with symplectic symmetry are realized in the presence of strong spin-orbit coupling. 29)In the k • p scheme two sites A and B in a unit cell play a role of a pseudospin, and symplectic systems are realized without spin-orbit coupling.Note that symplectic symmetry can be broken in practical nanotubes by the trigonal warping when the energy is far away from = 0. 30) For the purpose of numerical calculations, we restrict the basis by a cutoff energy c .We use |α, k (α = A, B and γ|k| < c ) as a basis.

Density of states
We calculate eigenvalues of the k • p Hamiltonian and the average density of states per unit area, which is defined by where n represents nth eigenvalue and • • • means the sample average.In practice, we can use the symmetry of the average density of states, ρ( ) = ρ(− ), to reduce the numerical effort.
We also calculate the density of states in a selfconsistent Born approximation for the same finite size systems.The retarded and advanced Green's functions in the self-consistent Born approximation are written as where the self-energy Σ( ± i0) is given by ).We solve the above equations selfconsistently.We then calculate the average density of states by with the use of eq. ( 12).

Dynamical conductivity
According to the Kubo formula, the dynamical conductivity in the tube axis direction is given by Here f ( ) is the Fermi distribution function, n and |ψ n represent nth eigenvalue and eigenvector, respectively, and ĵy is the current operator of the k • p Hamiltonian, which is given by In practical calculation based on the numerical diagonalization, we calculate the dynamical conductivity averaged over ω in a small interval δ ω as The integral over ω was taken to avoid the delta function in (14).
We have also calculated the dynamical conductivity in the self-consistent Born approximation with taking account of the vertex correction.Using the Green's functions obtained from ( 11) and ( 12), the dynamical conductivity is written as 24) σ yy (ω Here with In practice, the energy integral in ( 17) is performed numerically.

Density of states
We set the nanotube length to be 100 times as long as the circumference, A/L = 100, and the cutoff energy to be c L/2πγ = 3.0.The matrix size of the Hamiltonian is more than 5000.The strength of disorder is set to be W −1 = 100, 50, 20, and 10.In the self-consistent Born approximation, W is the single parameter characterizing the strength of disorder.In the calculation with exact diagonalization, we set u/2γL = 0.01 √ π and n i L 2 = 100, 200, 500, and 1000 for the above values of W −1 .We have accumulated 100 samples to calculate the average density of states.
Figures 2 and 3 show the numerical results of the density of states for φ/φ 0 = 0 and φ/φ 0 = 1/2.In these figures we have multiplied the factor four which comes from the presence of K and K points and from the spin degeneracy.The results in the self-consistent Born approximation are nearly the same as those of numerical diagonalization, indicating that the self-consistent Born approximation is sufficient for the broadening of the density of states.Only in the gap between the conduction and valence bands for φ/φ 0 = 1/2 in Fig. 3 there is a difference between the results in the self-consistent Born approximation and the exact results.Similar disappearance of the density of states in the self-consistent Born approximation was obtained for the Landau bands in quantum Hall systems. 31,32) igher order effects are needed to describe the behavior near the band edge.

Dynamical conductivity
We set A/L = 100, c L/2πγ = 3, and W −1 = 100, 50, 20, and 10.The zero temperature is assumed.In the calculation with exact diagonalization, we set u/2γL = 0.01 √ π and n i L 2 = 100, 200, 500, and 1000 for these values of W .Only for (ii) φ/φ 0 = 1/2, F L/2πγ = 1, and N c = 2 we perform an additional calculation for n i L 2 = 3000 to see localization effects.The corresponding strength of disorder is W −1 = 3.3.We accumulate 30 samples for the sample average.We set δ ω L /2πγ =  0.03, which is a few times larger than the mean level spacing, which is of order ∆ L/2πγ ∼ 0.01.In the figures presented here, we have multiplied the factor four which comes from the presence of K and K points and from the spin degeneracy.

SCBA exact
Figure 4 shows the dynamical conductivity in the case (i) φ/φ 0 = 0, F L/2πγ = 0, and N c = 1.When ω is comparable or larger than 2πγ/L , there is no significant difference between the results in the self-consistent Born approximation and the exact results.The large structures in the dynamical conductivity, which are present in regions ωL /2πγ ∼ 2 and ωL /2πγ ∼ 4 in the absence of impurities, are broadened due to impurities.Further, when W −1 = 100 or W −1 = 50, we can see that there are two small structures in regions with ωL /2πγ ∼ 1 and ωL /2πγ ∼ 3.They correspond to transitions between n x = 0 and n x = ±1 bands and between n x = ±1 and n x = ±2 bands, respectively.These transitions, forbidden in the absence of impurities, occur because of the inter-band mixing.
In Fig. 4 there is a qualitative difference between the results in the self-consistent Born approximation and the exact results near ω = 0.The Drude behavior with finite broadening is indicated in the self-consistent Born approximation.On the other hand, such Drude behavior does not appear in the exact calculation.Figure 4(c) shows the expansion of the results near ω = 0.Although large statistical fluctuations make it difficult to deduce definite conclusions, the present result is likely to show that the Drude conductivity is totally absent.
Figure 5 shows the dynamical conductivity in the case (ii) φ/φ 0 = 1/2, F L/2πγ = 1, and N c = 2. Again, the exact results and those in the self-consistent Born approximation are in quantitative agreements in the large frequency region with ωL /2πγ > 1.In this case states are all localized in an infinitely long nanotube.The localization length ξ was estimated to be ξ ≈ 20L when W −1 = 100. 28)The length of the nanotube A = 100L considered here is larger than the localization length even for W −1 = 100.However, an indication of such localization effects can be seen only in the case of the strongest disorder, i.e., W −1 = 3.3.In fact, a local peak appears in the exact dynamical conductivity at about ω c L /2πγ ≈ 0.1 and below ω c the conductivity decreases as the frequency decreases.An effective transport relaxation time τ tr can be estimated by fitting σ(ω) = σ 0 /(ω 2 τ 2 tr + 1) to the numerical data in the self-consistent Born approximation.We find τ tr 2πγ/L ≈ 0.73W −1 in this case and Figure 6 shows the dynamical conductivity in the case (iii) φ/φ 0 = 0, F L/2πγ = 1.5, and N c = 3.We do not find quantitative difference between the exact results and those in the self-consistent Born approximation in a region with sufficiently large ω (ωL /2πγ > 0.5).When W −1 = 20 and W −1 = 10, we find a significant difference in a small frequency region.The exact dynamical conductivity exhibits a peak at a certain critical frequency ωc .The peak frequency is about ωc L /2πγ ≈ 0.06 when W −1 = 20 and about ωc L /2πγ ≈ 0.1 when W −1 = 10.A transport relaxation time can be estimated with the use of the results in the self-consistent Born approximation to be τ tr 2πγ/L ≈ 0.38W −1 and gives ωc τ tr ≈ 0.4, (21)   which is twice as large as ω c in the case (ii).The origin of this intriguing behavior will be discussed in the next section.

Discussion
The dynamical conductivity in infinitely long metallic nanotubes in the absence of magnetic flux was studied previously in the self-consistent Born approximation. 24)The results show that in the case (i) φ/φ 0 = 0, F L/2πγ = 0, and N c = 1, the dynamical conductivity is given by a delta function, corresponding to infinite transport time, when only the linear band of a metallic nanotube is considered.The divergence of the dynamical conductivity at ω = 0 is consistent with the absence of backward scattering.When small effects of other bands are included, however, the Drude conductivity is broadened and the static limit of the conductivity becomes finite in contrast to the absence of backward scattering.
Our result shown in Fig. 4 indicates that the Drude behavior obtained in the self-consistent Born approxi-mation is not valid.Figure 4 seems to suggest that the dynamical conductivity in the vicinity of ω = 0 is not broadened and given by a delta function.This singular behavior is consistent with the absence of backward scattering.However, we cannot exclude the possibility that there is a very narrow Drude behavior around ω = 0 because σ(ω) for very small ω cannot be studied in the present frame work.In conventional systems with a freeelectron like dispersion, the so-called frequency sum-rule can give information on the conductivity for small ω.In the present system such a sum-rule does not provide meaningful conclusion because of the presence of infinite numbers of states below the Fermi level. 6)n the case (ii) φ/φ 0 = 1/2, F L/2πγ = 1, and N c = 2, the Anderson localization takes place as in usual quantum wires.In the limit A → ∞ the dynamical conductiv-ity for sufficiently small ω is expected to behave as [33][34][35][36][37] σ(ω) ∝ ω 2 (ln ω) 2 . ( The crossover frequency ω c below which the localization effect becomes relevant can be given by the condition where ω (ω) = √ Dω −1 with D being the diffusion constant, C a factor of order unity, and ξ the localization length.
The length scale ω (ω) is the distance that an electron travels in the interval ω −1 .The condition (23) is obtained based on an argument that electrons behave as if they are extended as long as motion of electrons induced by an applied oscillating electric field is in a scale which is shorter than the localization length ξ.Above ω c , the standard Drude conductivity can be expected.Assuming that the diffusion constant is written as D = Λ 2 /τ tr with Λ being the mean free path and τ tr being the transport time, the crossover frequency multiplied by the transport time is written as This indicates that the ratio of the localization length ξ to the mean free path Λ is relevant to the crossover between the Drude regime and the Anderson localization regime.
In the case (iii) φ/φ 0 = 0, F L/2πγ = 1.5, and N c = 3, electronic states are not localized because there is a perfectly conducting channel.However, Fig. 6 indicates that there can be a crossover from the Drude regime in the large frequency region to a different regime in the small frequency region.In this case the conductance of a finitelength nanotube decreases from the ideal value corresponding to N c to that in the single-channel case with the length, 13,[21][22][23] and the deviation from the single channel value decays exponentially, giving a length scale ξ. Wen the motion of electrons due to an applied oscillating electric field is in a scale which is shorter than ξ they can move diffusively by using all three channels, but they cannot when the scale of the oscillating motion is longer than ξ.This suggests that there can be a crossover between the Drude regime to the "single-channel transport regime." The crossover frequency ωc is determined by where C is a constant of order unity.The diffusion constant is written as D = Λ 2 /τ tr with Λ being an appropriate average of the mean free path.At the Fermi energy there are three channels and the mean free path depends on the channel.We use the simple arithmetic average in the following argument among several kinds of average.The choice of the average is not important since the channel dependence of the mean free path is weak enough in the case (ii).Then, Eq. ( 25) is written as This indicates that the ratio of the decay length ξ to the average mean free path Λ is relevant to this crossover.According to the numerical calculation of the Landauer conductance in the former case for W −1 = 1000, 38) the ratio of the localization length to the mean free path is ξ/Λ ≈ 5. Hence the crossover frequency multiplied by the transport time becomes According to the numerical calculation of the Landauer conductance in the latter case for W −1 = 1000, 17) the ratio of the decay length for the excess conductance to the average mean free path is ξ/Λ ≈ 3, giving Therefore, the numerical results ( 20) and ( 21) are explained quite well for C ≈ C ≈ 1/2.Because ( 23) and ( 25) are derived based on similar arguments, two constants C and C should be close to each other.Therefore, the fact that almost the same values are obtained for C and C shows the validity of the present analysis to a certain extent.Note that ξ/Λ and ξ/Λ actually depend on the strength of disorder W and can also depend on the cutoff energy although weakly. 17)Therefore the actual values of C and C may not be so meaningful.The so-called DMPK equation gives ξ = 2(N c − 1)Λ and ξ = (2/3)(N c − 1)Λ. 21)Although these approximate ξ and ξ are different from the numerical results, they give also the result C ≈ C.
In the case (iii) there is a perfectly conducting channel which transmits through the system without being scattered back and gives an ideal conductance in infinitely long nanotubes.The weak localization correction was calculated for systems with N c ≥ 2. 25) In the calculation the presence of the perfectly conducting channel was not obtained, but the correction to the dynamical conductivity ∝ 1/ √ ω as in conventional one-dimensional systems was obtained.The present result does not show a clear indication of this weak-localization conductivity but suggests that the dynamical conductivity of this perfect channel is given by a delta function without broadening.
][41][42] Scaling of the dynamical property in the vicinity of zero frequency was found to be governed by the generalized fractal dimension D 2 and the dynamical exponent z at the critical point of the quantum Hall transition. 43,44) n the carbon nanotubes studied in the present work, wave functions are not critical but extended (in the cases of odd N c ) or localized (even N c ).It seems that neither of the generalized fractal dimension nor the dynamical exponent are needed to describe the dynamical conductivity in the carbon nanotubes.

Summary and Conclusion
We have calculated the density of states and the dynamical conductivity in carbon nanotubes in the selfconsistent Born approximation and by exact numerical diagonalization.Impurities whose range of potential is larger than the lattice constant are considered.Our numerical results indicate that the self-consistent Born ap-proximation is sufficient for the broadening of the density of states except in the tail regions.It is also a good approximation for the dynamical conductivity when the frequency ω is comparable or larger than 2πγ/L .Some significant differences between the results in the self-consistent Born approximation and the exact results appear in the dynamical conductivity when ω is small.In our exact calculation, the Drude behavior cannot be seen in the case of N c = 1.Possibly there might be a delta function at zero frequency.In the case of N c = 2, Anderson localization takes place as in usual quantum wires.The behavior of the dynamical conductivity in this case can be understood with the standard theory of localization.When N c = 3, we have obtained a nonmonotonic behavior of the dynamical conductivity near ω = 0 when disorder is sufficiently strong although electronic states are not localized.We have argued that there is a crossover frequency ωc which is related with the length scale ξ ( ξ is the scale over which the conductance decreases down to the single channel value).
Our results indicate that the dynamical conductivity in systems with odd number of channels is, in particular, interesting.More detailed study of the dynamical conductivity in a small ω region is highly expected.

Fig. 2 .Fig. 3 .
Fig. 2. Density of states for φ/φ 0 = 0, A/L = 100, and cL/2πγ = 3 obtained in the self-consistent Born approximation (a) and by exact numerical diagonalization (b).The dashed line indicates the density of states in the absence of impurities for an infinitely long nanotube.

Fig. 4 .
Fig. 4. Dynamical conductivity for (i) φ/φ 0 = 0, F L/2πγ = 0, and Nc = 1 obtained in the self-consistent Born approximation (a) and by exact numerical diagonalization (b).A/L = 100 and cL/2πγ = 3.The dashed line indicates the result in the absence of impurities for an infinitely long nanotube.The blowup for small ω is shown in (c), in which the ordinate is shifted by an appropriate constant depending on W −1 .In (c), the exact results are shown like a histogram reflecting the fact that the conductivity is averaged over ω in a finite interval δω.