Conductivity in Carbon Nanotubes with Aharonov-Bohm Flux

The Boltzmann conductivity is calculated for carbon nanotubes in the presence of an Aharonov-Bohm magnetic flux. Effects of impurity scattering are considered at low temperatures and those of electron-phonon scattering are considered at room temperature. Effects of strains and curvature manifest themselves as a prominent conductivity peak as a function of the flux. The appearance of the peak corresponds to the absence of backscattering in metallic linear bands.


Introduction
Carbon nanotubes are novel quantum wires consisting of rolled graphite sheets. 1)][4] Their cylindrical shape leads to a strong Aharonov-Bohm (AB) effect in the band structure due to a magnetic field parallel to the axis. 5,6)The purpose of this paper is to study conductivity in the presence of AB flux.
The electronic states change from metallic to semiconducting depending on the tubular circumferential vector characterizing a nanotube.,18) The AB effect on the band gap in nanotubes was theoretically predicted in the effective-mass approximation.5,6,19,20) Recently, splitting of optical absorption and emission peaks due to the AB effect was observed.21) A shift in the gate voltage corresponding to the conductance peak was also observed in Fabry-Perot interference regime and attributed to the AB effect.][25] When several bands are occupied, a perfectly conducting channel transmitting through the system without being scattered back is present.26) These intriguing features are due to the existence of a special symmetry present in the Schro ¨dinger equation in the lowest-order kÁp equation.The absence of backscattering is robust, but the perfect channel is fragile against various symmetry breaking effects such as a magnetic field and flux, 27) short-range scatterers, 28) and trigonal warping of the bands appearing in higher-order kÁp terms.29) Metallic nanotubes are almost ballistic with a mean free path exceeding 1 mm even at room temperature. 30,31) A lttice vacancy with strong and short-range potential causes an interesting conductance quantization.[32][33][34][35][36][37] Mechanical deformation is known to modify the band gap.38,39) In fact, the presence of a lattice distortion can be incorporated in the kÁp scheme as an effective flux.0,43) Recent experiments on the conductance in metallic nanotubes are understood by taking a small gap induced by an effective flux into account.22,44) Nanotubes millimeters long synthesized recently by a water-assisted technique is likely to show a diffusive behavior.4) In this paper, we shall calculate the Boltzmann conductivity in the kÁp scheme and explore effects of AB flux on transport properties of nanotubes in the diffusive region.In §2, the kÁp scheme is reviewed very briefly.Some examples of explicit results of the conductivity in the presence of flux are presented in §3 for impurity scatterings and in §4 for the electron-phonon scattering, demonstrating a possibility to directly determine a small gap present in metallic nanotubes experimentally. Theesults are discussed in §5 and summarized in §6.

Energy bands and wave functions
The structure of a two-dimensional graphite is shown in Fig. 1 together with the first Brillouin zone and coordinate systems to be used in the following.In a two-dimensional graphite, two bands having approximately a linear dispersion cross the Fermi level (chosen at " ¼ 0) at K and K 0 points of the first Brillouin zone.Electronic states of the -bands near a K point are described by the kÁp equation: 17,20,45) where is a band parameter, x and y are the Pauli spin matrices, and Àir is a wave-vector operator.Two components of the wave function FðrÞ correspond to the amplitude at A and B sites in a unit cell.
The structure of a nanotube is specified by a chiral vector L corresponding to the circumference as shown in Fig. 1.It is written as in terms of two integers n a and n b , where a and b are the primitive translation vectors of a graphite sheet.In the following we shall choose the x axis in the circumference direction and the y axis in the axis direction, i.e., L ¼ ðL; 0Þ, where L is the circumference.The angle between L and the horizontal axis is called the chiral angle.Electronic states of a nanotube with a sufficiently large diameter are obtained by imposing the boundary conditions around the circumference direction: Fðr þ LÞ ¼ FðrÞ expð2iÈÞ; ð2:3Þ where È ¼ À=3 with integer ¼ 0 or AE1 determined by with integer M. Metallic and semiconducting nanotubes correspond to ¼ 0 and AE1, respectively.A nonzero curvature causes a shift in the origin of k k x and k k y in the kÁp Hamiltonian.The shift in the y direction is irrelevant and that in the x direction can be replaced by an effective flux ' s .The flux was estimated as 20,43) ' s ¼ 2 4 ffiffi ffi 3 p a L p cos 3; ð2:5Þ with a being the lattice constant of the two-dimensional graphite, p ¼ 1 À ð3=8Þ 0 =, ¼ Àð ffiffi ffi 3 p =2ÞV p p a, and 0 ¼ Àð ffiffi ffi 3 p =2ÞðV p p À V p p Þa, where V p p (¼ À 0 ) and V p p are the conventional tight-binding parameters for neighboring p orbitals. 43)The curvature effect is largest in zigzag nanotubes with ¼ 0 and absent in armchair nanotubes with ¼ =6.
The presence of a lattice distortion u ¼ ðu x ; u y ; u z Þ causes also an effective flux.It is estimated as 31) where u (, ¼ x, y) denotes the lattice strain given by ð2:7Þ and g 2 is the electron-phonon interaction energy given by g 2 ¼ ð=2Þ 0 with $ 1, where ¼ ffiffi ffi 3 p a 0 =2. 31,42)This shows that twist and stretch deformation give rise to nonzero flux in armchair ¼ =6 and zigzag ¼ 0 nanotubes, respectively.
In the presence of such flux ' s due to finite curvature and strain, we have È ¼ ' e with In the presence of flux due to a magnetic field parallel to the axis, further, we have where ' ¼ = 0 with the magnetic flux quantum 0 ¼ ch=e.
An effective flux, defined by e ¼ ' e 0 ; ð2:10Þ is used also in the following.The energy bands are specified by s ¼ AE1, integer n corresponding to the discrete wave vector along the circumference direction, and the wave vector k in the axis direction, where s ¼ þ1 and À1 denote the conduction and valence bands, respectively.The wave function for a band associated with the K point is written as where A is the tube length, The corresponding energy is given by The energy bands and wave functions for the K 0 point are obtained by replacing k k y by À k k y and in the boundary conditions ' e by À' e .Correspondingly, we have 0 ¼ 'À' e ðnÞ and b 0 ¼ b 'À' e ðn; kÞ Ã .Figure 2 shows an example of the energy dispersion of the band n ¼ 0 near the Fermi energy for small ' e .In the absence of magnetic flux ', as shown by dotted lines, the energy dispersions are the same between the K and K 0 points with gap 2j' e jð2=LÞ.When ' e > 0, the gap diminishes at the K 0 point by applying ', while it increases at the K point, as shown by solid lines.The Fermi wave-number in the axis direction is denoted by k 0 for the K point and k 0 0 for the K 0 point, with ð2:15Þ for the Fermi energy ", where 'AE' e 'AE' e ð0Þ; ð2:16Þ for simplicity.When the electron concentration is sufficiently small, the K point or the K 0 point is depopulated by electrons for sufficiently large magnetic flux '.At ' ¼ ' e , the dispersion becomes completely linear at the K 0 point.The same thing happens for the K point at ' ¼ À' e .The conductivity is evaluated under two different conditions, fixed energy and electron density.The latter is expected to be more appropriate for long nanotubes exhibiting diffusive conduction.The electron density is characterized by the average Fermi wave number k þ defined by

Effective potential of impurity
Most of scatterers including charged centers are expected to be characterized by a potential with range larger than the lattice constant a.For such scatterers, the Hamiltonian is given by a diagonal matrix for both K and K 0 points, and matrix elements between K and K 0 points are safely neglected. 23)These scatterers will be called long-range although their potential range can be comparable to the lattice constant and therefore can actually be much shorter than the electron wavelength typically of the order of the nanotube circumference.In the following, we shall confine ourselves to the case that the potential range of long-range scatterers is sufficiently smaller than the circumference.In this case we can replace the potential of each scatterer by a delta function with strength u L .
When the potential range becomes shorter than the lattice constant, the potential for A and B sub-lattice points in a unit cell can be different and matrix elements between K and K 0 points can no longer be neglected.As such scatterers, we shall consider those with potential range much smaller than the lattice constant and strength u S , giving rise to the same amplitude for scattering within the K and K 0 points and for scattering between the K and K 0 points.These scatterers are called short-range in the following.
In the lowest Born approximation the scattering strength for long-and short-range scatterers is characterized by the dimensionless quantity W L and W S , respectively, with hju S j 2 i 4 2 ; ð2:18Þ where n L and n S are the concentration of long-and shortrange scatterers in a unit area, respectively. 28)In the following, we shall fix the total strength W with and vary the parameter defined by in order to see effects of short-range scatterers. 28)

Effective potential for phonon
The electron-phonon interaction is considered in a continuum model for long-wavelength acoustic phonons.An effective Hamiltonian for the electron-phonon interaction for the K point is given by 31) where the strain tensor is defined in eq.(2.7).The Hamiltonian for the K 0 point is obtained by replacing V 2 with ÀV Ã 2 and V Ã 2 with ÀV 2 .The diagonal term represents coupling through the deformation potential g 1 and the offdiagonal coupling g 2 through the bond-length change.It has been shown that g 1 ) g 2 . 31)Note that eq.(2.6) is derived easily from eq. (2.21).
Phonons contributing to the electron scattering are described well by the potential-energy functional where B and are the bulk and shear modulus, respectively.An elastic scattering approximation is employed, because the phonon velocity is much slower than the electron.Further, a high-temperature approximation is used for the phonon distribution function.Then, the relaxation time due to electron-phonon scattering becomes independent of the chirality . 31)2. A schematic view of the bands near the Fermi energy.In the absence of a magnetic flux shown by dotted lines, the energy gaps of 4j' e j=L are present at the K and K 0 point.As shown by solid lines, the band gap diminishes at the K 0 point and increases at the K point in the presence of magnetic flux for ' e > 0. k 0 and k 0 0 are the Fermi wavenumber measured from K and K 0 points, respectively.

Boltzmann conductivity
In this study, we calculate the Boltzmann conductivity where Ã m ð"Þ is the mean free path. 27,46)It satisfies the transport equation where m and m 0 denote the bands crossing the energy ", and þ (À) the wave vector corresponding to the positive (negative) velocity in the y direction.The kernel for the transport equation is given by for 6 ¼ , where v is the velocity of mode ðmAEÞ and hÁ Á Ái denotes an average over impurities or a thermal average.The diagonal elements are defined by 3. Impurity Scattering

Conductivity
When a magnetic flux is present, the occupation of the bands varies and some bands can be depopulated by electrons completely.First, we shall consider the case that such depopulation does not occur and the lowest conduction band is occupied by electrons for both K and K 0 points.In this case the Fermi energy " is related to k þ given by eq.(2.17) through This shows that " increases with the magnetic flux '.
The conductivity is given by where Ã K and Ã K 0 are the mean free path for the K and K 0 points.For the K point, we have ð3:3Þ where Ã K L and Ã K S are contributions of long-and short-range scatterers, respectively.The results for the K 0 point, Ã K 0 , Ã K 0 L , and Ã K 0 S , are given by exchanging k 0 and k 0 0 and replacing ' e with À' e .
The absence of backward scattering 23) corresponds to the divergence of Ã K L at ' ¼ À' e and Ã K 0 L at ' ¼ þ' e , giving rise to infinitely large conductivity in the absence of shortrange scatterers, i.e., ¼ 0. At these values of the flux, therefore, the mean free path and the conductivity of the corresponding point are determined by scattering by shortrange scatterers.

Metallic nanotubes
In metallic nanotubes, the effective flux is very small, i.e., j' e j ( 1, and therefore the condition ' ¼ þ' e or ' ¼ À' e can be reached easily using a conventional magnet.In the vicinity of ' ¼ À' e , for example, the mean free path for the K point is written as Therefore, the mean free path exhibits a Lorentzian-like peak at ' ¼ À' e with width Á'.The condition that ð'Þ has actually a peak can be obtained approximately by the condition ð' e Þ > ð0Þ as Á' < ffiffi ffi 2 p j' e j.For Lk þ =2 ) j' e j, for example, we have k 0 % k 0 0 % k þ and therefore the condition for the peak appearance becomes ) j' e j, the contribution of long-range scatterers can be neglected safely and the conductivity is given by This does not exhibit any peak structure at ' $ AE' e , but rather decreases with the increase of ' and approaches the value independent of ' and ' e , Figure 3 shows some examples of the conductivity for ¼ 0:01 as a function of '=' e with the fixed electron density.A prominent peak appears for the small electron density k þ .Note that, results for a fixed Fermi energy are similar to those for the fixed density in this region of the electron density.
Figure 4 shows the conductivity for several values of at ðk þ L=2Þ=' e ¼ 5.For ¼ 0 the conductivity diverges at ' ¼ ' e .For nonzero , the peak appears and its height decreases with the increase of .The peak disappears around ¼ 0:04, in agreement with the condition obtained above.

Semiconducting nanotubes
In semiconducting nanotubes, the effective flux is given by ' e % AE1=3 (AE depending on the structure).It is almost impossible to reach this amount of the flux using conventional magnets for which ' ( 1.We can discuss only whether the flux tends to increase or decrease the conductivity.The first derivative d=d' at ' ¼ 0 vanishes identically because dÃ K =d' and dÃ K 0 =d' cancel each other, and therefore we have When the electron concentration is not extremely large we have Lk þ =2 ( 1=3.In this case, we have It is concluded, therefore, that a usual semiconducting nanotube exhibits a positive magnetoconductivity, its amount being strongly dependent on the electron concentration.Only dirty semiconducting nanotubes containing large amount of short-range scatterers exhibit a negative magnetoconductivity.The tendency that dominant longrange scatterers cause a positive magnetoconductivity, while short-range scatterers cause a negative magnetoconductivity, corresponds to the behavior of the peak structure in metallic nanotubes, discussed in the previous section.

Band depopulation
When the electron concentration is small and the band edge of K and K 0 points cross the Fermi level as a function of the flux, the conductivity exhibits a singular behavior depending strongly on whether the electron density or the Fermi level is fixed.Several examples of calculated Fermi energy are shown with the fixed electron densities as a function of the magnetic flux for ' e > 0 in Fig. 5.The band edges j 'À' e j and j 'þ' e j are also shown by thin solid and dotted lines, respectively.Only the K 0 point is populated in the region between the solid and dotted thin lines, and both K and K 0 points are populated above the dotted thin lines.Near the band edges, the corresponding energy strongly depends on the magnetic flux.
The conductivity under the condition of a fixed energy and a fixed electron density are shown in Figs. 6 and 7, respectively.At a critical flux the conductivity vanishes due   Flux (units of φ e ) Energy (units of (2πγ/L)(φ e /φ 0 )) (k + L/2π)/(φ e /φ 0 ) 10.00 5.00 2.50 2.00 1.25 1.00 .50.25 Fig. 5.The Fermi energy as a function of the magnetic flux ' ¼ = 0 for several values of the electron density k þ for ' e > 0. The band edges are shown by thin lines, where solid and dotted lines are for K 0 and K point, respectively.Only the K 0 point is populated in the region between the solid and dotted thin lines and both K and K 0 points are populated above the dotted thin lines.
to the divergence in the scattering probability caused by the infinite density of states characteristic of the one-dimensional system.Beyond this flux the K point is depopulated completely and electrons are only in the band associated with the K 0 point.Apart from this singular behavior, the conductivity has a general tendency to exhibit a peak at ' ¼ ' e (also at ' ¼ À' e ).

Electron-Phonon Scattering
In the presence of the electron-phonon interaction eq.(2.21), where the scattering is only within the K point or K 0 point, we have the conductivity and Ã K 0 obtained by replacing 'þ' e with 'À' e and k 0 with k 0 0 .When only a single valley is occupied, the conductivity is given by its contribution alone.For ' ¼ ' e ¼ 0, in particular, this expression is reduced to ; ð4:3Þ obtained previously. 31)igure 8 shows the conductivity of a metallic nanotube as a function of the magnetic flux under the condition of a fixed electron density for g 1 =g 2 ¼ 10 and B= ¼ 3= ffiffi ffi 2 p . 31) A peak with height $ A =2 appears at ' ¼ ' e when the electron density is low.With the increase of the electron density, the peak becomes less prominent and disappears around ðk þ L=2Þ=ð e = 0 Þ $ 10.This behavior is very similar to the case of the impurity scattering in Fig. 3 for ¼ 0:01.The conductivity approaches A for sufficiently large electron density.The figure contains also some results for a small electron density [ðk þ L=2Þ=' e ¼ 1, 0.5, and 0.25] for which the K or K 0 valley is depopulated with the increase of the flux.No singular behavior appears even in such a case.
In semiconducting nanotubes with ' e % AE1=3 and k þ L=2 ( 1, we have eq.(3.8) with ð4:4Þ Therefore, a semiconducting nanotube almost always exhibits a positive magnetoconductivity and its amount is strongly dependent on the electron concentration.

Discussion
An effective flux due to the curvature of a nanotube [eq.(2.5)] is estimated as ' s $ 0:05 Â p cos 3 for a typical nanotube with L $ 5 nm with a parameter jpj < 1. 43) This flux is the greatest in the zigzag tubes with ¼ 0, followed by chiral tubes, and is nonexistent in the armchair tubes with ¼ =6, as confirmed by an experiment with numerical simulations. 22)An effective flux appears in the presence of uniaxial stress as well as a hydrostatic pressure because of eq.(2.6).The total effective flux can vary among different nanotubes although their structure is the same.The present calculation shows that the effective flux can be measured directly by a sharp peak as a function of the applied magnetic flux in metallic nanotubes.In fact, the flux can be as large as = 0 ¼ 5 Â 10 À3 at 10 T for a typical nanotube with L $ 5 nm.
The sharpness of the conductivity peak as a function of the flux is sensitive to the electron density and the amount of short-range scatterers.The electron density may be controlled by a gate voltage and therefore a careful analysis of the gate-voltage dependence in the presence of a flux can give important information on dominant scatterers in metallic nanotubes.
At a room temperature, electron-phonon scattering constitutes the main origin of the resistivity, and the conductivity is likely to exhibit a peak structure as a function of an applied flux in metallic nanotubes.A careful analysis of possible gate-voltage dependence can reveal the relative strength of the electron-phonon interaction through a deformation potential and a bond-length change.
In metallic nanotubes with linear dispersion, effects of electron-electron interaction of a Tomonaga-Luttinger type [47][48][49][50][51][52][53] can be important and may modify the flux dependence. Te fact that the conductivity exhibits a sharp peak as a function of a magnetic flux is expected to be valid even if such interaction effects are considered.
Whether we have positive or negative magnetoconductivity in semiconducting nanotubes in the presence of a flux can depend on the change in the strength of screening effects.The static screening constant in the long-wavelength limit is determined by the density of states under the condition of a fixed electron density.The second derivative is positive also under the condition of a fixed energy.Thus, the screening has a tendency to be enhanced due to flux.For the conductivity, however, screening for the wave number 2k þ is important.The logarithmic divergence of the screening function at 2k þ is known to be responsible to a power-law temperature dependence of the conductivity. 54,55)This screening problem in the presence of flux is left for a future study.

Conclusion
We have calculated the Boltzmann conductivity in carbon nanotubes in the presence of a magnetic flux in the kÁp scheme.A gap induced by strain or curvature effects can be identified as a peak in the conductivity by applying an Aharonov-Bohm flux in metallic nanotubes.The peak corresponds to the absence of backscattering in metallic nanotubes with no effective flux.Various information on curvature and strain effects and the relative amount of shortrange scatterers can be obtained by careful measurement of the conductivity in the presence of the magnetic flux and by changing the electron density through a gate voltage.Semiconducting nanotubes exhibit a positive magnetoconductivity and its amount increases with the decrease of the electron concentration.Experiments along these lines are highly desired.

Fig. 1 .
Fig. 1.(a) Lattice structure of a two-dimensional graphite sheet.The coordinates ðx 0 ; y 0 Þ are fixed on the graphite sheet and ðx; yÞ are chosen in such a way that x is along the circumference and y is along the axis. is the chiral angle.(b) The first Brillouin zone and K and K 0 points.(c) The coordinates for the nanotube.An Aharonov-Bohm flux is applied in the axis direction of the nanotube.

Fig. 3 .
Fig.3.Calculated Boltzmann conductivity for several values of the electron density k þ , when electrons occupy the lowest conduction band of the K and K 0 points.The ratio of the short-range scatterers is chosen as ¼ 0:01.