Family Effects on Excitons in Semiconducting Carbon Nanotubes

Family behavior of excitons are studied in carbon nanotubes based on a k·p scheme. Effects responsible for family pattern are characterized by two parameters β representing higher-order terms in the effective Hamiltonian responsible for trigonal warping and p representing amount of effective flux due to curvature and lattice distortion. These parameters are determined through comparison with empirical formula of the lowest exciton and experiments of two-photon absorption giving excited exciton states. The observed family pattern can be reproduced much better for the first gap than for the second gap.


§1. Introduction
−3) The structure of a nanotube is specified by chiral vector L giving its circumference.−6) The purpose of this paper is to study this family effect based on a k•p scheme.
A carbon nanotube has characteristic optical properties.In fact, the absorption of light polarized perpendicular to the axis is suppressed because of a strong depolarization effect, 7,8) and the optical absorption is dominated by parallel polarization.It was shown that interaction effects significantly enhance the band gap and that the exciton effect is important because of the one-dimensional nature of the nanotube. 9,10)−18) The same scheme was also used for various exciton-related properties such as excitons for cross polarization, 19,20) excitons in metallic nanotubes, 21) twophoton absorption spectra, 22) and exciton fine structure. 23)−28) Quasi-particle spectra of CN were calculated using a first-principles GW method. 29)alculations were performed also for optical absorption spectra with the inclusion of excitonic final state interactions, 30−32) although limited to CN's with very small diameter.−44) In this paper, we calculate the energy levels of excitons taking into account various higher order effects extending the lowest-order k•p scheme in order to clarify the origin of the family behavior.It is shown that these effects are characterized by two parameters specifying effects of trigonal warping and curvature and associated lattice distortion.They are determined by comparison with empirical formula of the exciton energy experimentally determined and with two-photon spectra.
The paper is organized as follows: In §2 various higher-order terms in the k•p scheme are briefly reviewed.In §3 the method of calculation of exciton states is reviewed.In §4 some examples of explicit numerical results are presented.A short summary and conclusion are given in §5.An effective Hamiltonian describing overlap integral and next-nearest neighbor hopping integrals is derived in Appendix A. §2.Effective-Mass Approximation

Chiral Vector
The structure of graphene sheet and the first Brillouin zone are shown in Figs. 1 (a) and (b).A unit cell contains two carbon atoms, which are denoted by A and B and the primitive translation vectors are denoted by a and b, where |a| = |b| = a with lattice constant a = 2.46 Å.The conduction and valence bands consisting of π states cross at K and K' points of the Brillouin zone, where the Fermi level is located. 45,46)he structure of a carbon nanotube is specified by chiral vector L corresponding to the circumference.We have L = n a a+n b b with integers n a and n b .The angle between L and the horizontal axis is called chiral angle denoted by η, which varies between η = 0 corresponding to zigzag nanotubes and η = π/6 corresponding to armchair nanotubes.For tubes with a sufficiently large diameter, the energy bands are obtained by imposing periodic boundary condition ψ(r+L)= ψ(r) or exp(ik•L) = 1.The energy bands are give by straight lines in the k space satisfying this condition.
We have exp(iK •L) = exp(+2πiν/3) and exp(iK • L) = exp(−2πiν/3), where K and K are wave vectors at the K and K' points, respectively, ν is an integer, 0 or ±1, determined by n a + n b = 3N + ν with integer N .For ν = 0, a straight line satisfying exp(ik•L) = 1 passes through K or K' points and the nanotube becomes a metal.For ν = ±1, on the other hand, the nanotube becomes a semiconductor with nonzero gap at the Fermi level.
Let the envelope function associated with the K Submitted to Journal of Physical Society of Japan T. Ando point be Then, periodic boundary conditions in the circumference direction in a nanotube are written as with ϕ K = φ K /φ 0 , where φ K is effective magnetic flux to be introduced below and φ 0 = ch/e is the flux quantum.
For F K (r) at the K' point, ϕ K is replaced with ϕ K and ν with −ν.

Trigonal Warping
The effective Hamiltonian for the K point containing higher order terms is given by 17,47) with where β is a dimensionless parameter of the order of unity, η is the chiral angle, and γ is a band parameter, related to nearest-neighbor hopping integral γ 0 through γ = ( √ 3/2)aγ 0 .In a nearest-neighbor tight-binding model, β = 1. 17,47)or wave function F K (r) at the K' point, we should replace h K ( k) with h K ( k)= −h K ( k) * .This corresponds to the fact that the K and K' points are related to each other via the time-reversal operation T , 48,49) given by F T K = e −iψ σ z F * K and F T K = e −iψ σ z F * K , with ψ being an arbitrary phase.As illustrated in Fig. 2, equi-energy lines in the vicinity of the K and K' points exhibit trigonal warping.

Curvature Effects
In thin nanotubes effects of a nonzero curvature should also be considered.A curvature causes a shift in the origin of kx and ky in the k•p Hamiltonian. 16,50)The shift in the circumferencial x direction can be replaced by an effective magnetic flux φ passing through the cross section, i.e., Δk x = (2π/L)(φ/φ 0 ) with L = |L|, although the effective flux has a different signature between the K and K' points.For the K point, for example, it was estimated as 50) ) , where η is the chiral angle, and V π pp and V σ pp are the conventional tight-binding parameters for neighboring p orbitals. 50)The curvature effect is largest in zigzag nanotubes with η = 0.For usual parameters, we have γ /γ ∼ 8/3 and therefore it is very difficult to make a reliable estimation of p except that |p| < 1. 50) In the presence of curvature, nanotubes metallic in its absence become narrow-gap semiconductors except in armchair nanotubes with η = π/6 and the gap is largest in zigzag nanotubes as has been shown first in ref. 51.In semiconducting nanotubes nonzero p contributes to the so-called family behavior as will be discussed below.

Effects of Lattice Distortion
The curvature always gives rise to some lattice distortion.Effects of lattice distortion can again be included in a from of effective magnetic flux with different signs between the K and K' points.For the K point, we have where u xx , u yy , and u xy are the strain tensors, and the coupling parameter was estimated as g 2 = (3λ 1 λ 2 /4)γ 0 with λ 2 < ∼ 4 and λ 1 ≈ 1/3 within a valence-force-field model. 52)he lattice distortion associated with curvature is expected to satisfy u xx − u yy > 0 and u xy = 0, because the bond strength between carbon atoms lying along the circumference direction becomes weaker due to curvature.Further, to the lowest order in a/L, we have u xx − u yy = λ 3 (a/L) 2 , with λ 3 being a constant of the order of unity.As a result, the effective flux associated with lattice distortion can be given by the expression same as that for the curvature, i.e., eq.(2.5) with Various first-principles calculations have been reported on the lattice distortion in thin nanotubes, 53−56) giving results seemingly different from each other.For examples, reference 54 gives results λ 3 ∼ 0.6, for which p ∼ −0.12 when g 2 ∼ γ 0 .

Asymmetry of Conduction and Valence Bands
When we consider overlap integral in a nearestneighbor tight-binding model, the band structure becomes asymmetric between the conduction and valence bands.Similar asymmetry appears also when we consider next-nearest-neighbor hopping integrals between same sublattice atoms.As discussed in Appendix A, in the absence of a magnetic field, these effects can be incorporated by the diagonal Hamiltonian consisting of a term proportional to k2 , i.e., for both K and K' points, where S is an effective overlapping integral.
As will be shown below, this asymmetry term does not explicitly contribute to exciton energy levels corresponding to light polarized parallel to the axis.The reason lies in the complete cancellation of these terms between the conduction and valence bands.A small effect appears only in the effective dielectric function due to virtual interband transitions, but can safely be neglected.For light polarized perpendicular to the axis, which will not be discussed here, it can have direct influence in absorption spectra because of incomplete cancellation.

Energy Bands and Wave Functions
We have shown that for the K point, in particular, higher order terms in the k•p scheme can fully be incorporated by the Hamiltonian and effects of curvature and associated lattice distortion by the effective magnetic flux ϕ K = ϕ where ϕ is given by eq.(2.5) with appropriate parameter p.For the K' point the Hamiltonian is obtained by replacing h K ( k) with h K ( k) and the flux is replaced with ϕ K = −ϕ.Thus, we have three parameters β, S, and p.
The energy bands at the K point are specified by α = (n, s, k), where n is an integer specifying the wave vector with ν = ±1, s = + for the conduction and − for the valence band, respectively, and k being the wave vector in the axis direction.The corresponding wave function is written as with A being the tube length and where The corresponding energy is given by with and The wave function and the energy for bands associated with the K' point are given in the similar manner.When we neglect short-range part of the Coulomb interaction, 23) the bands associated with the K and K' points are completely separated from each other and degenerate due to the time reversal symmetry in the absence of magnetic field.

Family Behavior
Nanotubes with L lying on each of vertical dotted lines in Fig. 1  (2.17  I. Figure 2 shows schematic illustration of equi-energy lines and one-dimensional bands of a semiconducting nanotube.The feature of trigonal warping is shown by a triangle.In the case ν = +1, the first gap determined by n = 0 is lowered and the second gap (n = +1 for the K point and n = −1 for the K' point) is raised by trigonal warping.In the presence of curvature and associated lattice distortion (ϕ> 0 or p < 0), these shifts in the gap are further enhanced.When L is changed along a family line characterized by family number f from η = 0 (zigzag) to η = π/6 (armchair), both effects of the trigonal warping and curvature decrease at the same time.This family behavior was first experimentally clarified by Weisman and Bachilo, 6) who proposed empirical formula giving exciton energies as a function of L. §3.Exciton It has been shown in the previous section that effects of various terms leading to the family behavior are fully included by the change in b K ν (n, k) and ε nk and that the spacial variation of the wave function itself is not affected.As a result, calculations of the exciton spectrum can be carried out by a straightforward extension of the previous works made in the lowest order k•p scheme. 9,10)he exciton wave function for an electron in the K valley and a hole in the K valley is given by where |g is the ground-state wave function corresponding to the filled valence bands and empty conduction bands and c K † s,n,k and c K s,n,k are the creation and annihilation operators for states (s, n, k) of the valley K.The equation of motion for ψ u n (k) becomes and where ε |n−m| (|q|) is the dielectric function describing screening effects due to virtual interband transitions in the vicinity of the K and K' points, and I n (t) and K n (t) are the modified Bessel function of the first and second kind, respectively.The self-energy is calculated as where g 0 (ε) is a cutoff function, defined by The cutoff function (3.6) contains two parameters α c and ε c and should be chosen in such a way that only the contributions from states in the vicinity of the Fermi level, where the k•p scheme is valid, should be included.The difference in the self-energy of the conduction and valence bands depends on the cutoff energy logarithmically, but is essentially independent of the parameter α c as long as the cutoff function decays smoothly but rapidly enough. 57,58)The appropriate value of ε c is the half of the π band width ∼ 3γ 0 .As a result the selfenergy shift has an extra weak logarithmic dependence on the diameter ∝ ln(L/a) in addition to the universal L −1 scaling in the lowest order k•p scheme.
Using the velocity operator vK y = ∂H K /h∂ ky , the matrix element for optical transition is calculated as The diagonal term proportional to S does not contribute to above interband matrix elements because of the orthogonality of the wave functions.So far, parameter S giving the asymmetry between the conduction and valence bands does not explicitly appear in the equations determining the exciton energy and wave function.It appears only in the dielectric function, given by where Here, terms associated with parameter S appear only in the denominator in the form of the difference ξ n+m (k + q) − ξ n (k) in contrast to the form of the sum ε n+m (k+q)+ε m (k) for p and β, and therefore its effects are quite weak.In fact, explicit numerical calculations show that the correction due to S gives negligible contribution to the dielectric function and to the exciton states for S < ∼ 0.15.This shows that we can safely neglect the presence of the S terms.
The strength of the Coulomb interaction is specified by dimensionless parameter (e 2 /κL)(2πγ/L) −1 , which is estimated as (e 2 /κL)(2πγ/L) −1 ≈ 0.4/κ for γ 0 ≈ 2.7 eV and a = 2.46 Å.In bulk graphite we have κ ≈ 2.4. 59)In CN, this simple constant screening is valid only approximately because of the cylindrical form with hollow vacuum inside and surrounding material outside.Therefore, κ should be treated as a parameter.A comparison with two-photon absorption experiments places (e 2 /κL)(2πγ/L) −1 between 0.1 and 0.2. 22)In the following, therefore, we shall present explicit results only for (e 2 /κL)(2πγ/L) −1 = 0.15.§4.Numerical Results and Discussion Numerical calculations have been performed for various values of parameters 1 ≤ β ≤ 2.5, −0.5 ≤ p ≤ +0.5, and 0 ≤ S ≤ 0.15.The results show essentially no dependence on S and therefore we shall confine ourselves to the case S = 0.It has been shown that positive values of p corresponding to negative ϕ tend to give family behavior opposite to that of experiments or that described by the empirical formula.It turns out further that the observed family effect can be reproduced much better for the first gap than for the second gap.
Figure 3 shows some examples of calculated exciton energies as a function of the circumference or the diameter for β = 1 and 1.5 and p = −0.2 and −0.5.The dashed lines show band gaps 4πγ/3L and 8πγ/3L obtained in the lowest-order k•p scheme neglecting curvature effects.The open and closed symbols denote excitons for ν = −1 and +1, respectively.The numbers denoting each line represent family number f and the dotted lines denote the empirical expressions given in ref. 6.We have assumed γ 0 = 2.7 eV used for comparison with experiments in previous works. 10,22)t is certainly not possible to exactly reproduce the empirical result by using two parameters β and p, and therfore impossible to assign definite values to β and p.The appropriate values are likely to be 1 < ∼ β < ∼ 1.5 and −0.5 < ∼ p < ∼ − 0.2 for the exciton associated with the first gap.For the second gap, the reduction of the exciton energy for ν = −1 in particular cannot be reproduced by using the parameter values used in calculations.It seems that further higher order terms such as ∝ (ka) 3 etc. in the k•p scheme giving rise to stronger trigonal warping are needed for the second gap.
Figures 4 and 5 show calculated ground and firstexcited exciton energies as a function of the circumference or the diameter together with band edges for (a) β = 1.5 and p = −0.2 and for (b) β = 1.0 and p = −0.5.We notice that within each family the ground state, first excited states, and the band gaps are almost parallel.This shows that effects on the band gap without Coulomb interaction dominate the family behavior and effects on the band-gap shift and exciton binding energy are less important although being not negligible.−62) The experimental results are well reproduced by the calculated results for the parameter range discussed above.
The dimensionless oscillator strength of the lowest exciton absorption peak has been defined as 10) with ε u being the exciton energy.Figure 6 shows this oscillator strength as a function of the circumference or the diameter.The intensity also exhibits family behavior in such a way that the intensity of the lower-energy excitons is smaller and that of the higher-energy is larger.This is mainly due to the fact that a band with higher energy tends to have a larger effective mass and therefore a larger exciton binding energy or stronger electron-hole binding.This feature should be experimentally observed.§5.

Summary and Conclusion
Effects of trigonal warping and curvature and associated lattice distortion on excitons have been studied in semiconducting carbon nanotubes based on a k•p scheme and the family behavior has been demonstrated.Higher order terms beyond the lowest-order k•p scheme are characterized by three parameters β, p, and S. Here, β represents the term in the Hamiltonian second order in (ka) 2 with k the wave vector and a the lattice constant, p represents effects of curvature and associated lattice distortion, and S represents effects of overlap integral or next-nearest-neighbor hopping.It has been shown that terms proportional to S give negligible contributions and therefore can be neglected.Comparison with empirical formula experimentally determined has tentatively placed the values of the parameters in the range 1 < ∼ β < ∼ 1.5 and −0.5 < ∼ p < ∼ − 0.2.The energy of the first excited exciton state observed in two-photon absorption is also well understood by these parameter values. where Because we are confining ourselves to the small energy region, we can expand as follows: and εF B (r) = γ kx +i ky 0 0 kx −i ky Then, we have (A7) The above shows that the overlapping integral can be incorporated in the form of a simple effective mass in the diagonal element of the Hamiltonian in the absence of a magnetic field.In a magnetic field, however, we have an additional term similar to the spin-Zeeman energy, which guarantees the fact that the Landau level at ε = 0 is not shifted in energy.
When we consider next-nearest-neighbor interactions between same sublattice points, we have a term proportional to k2 in the diagonal elements.In the absence of a magnetic field, therefore, we have a diagonal Hamiltonian (a) belong to a family characterized by integer f called family index, f = 2n a −n b .

Fig. 1
Fig. 1 (a) Lattice structure of a two-dimensional graphite sheet.Two primitive translation vectors are denoted by a and b and vectors τ l (l = 1, 2, 3) connect neighboring A and B sites.Nanotubes with chiral vector L lying on a vertical dotted line belong to a family characterized by family number f .η is the chiral angle.(b) The first Brillouin zone and K and K' points.(c) Some examples of family number f and corresponding values of ν.

Fig. 2 AFig. 3 (
Fig. 2 A schematic illustration of equi-energy lines and one-dimensional bands of a semiconducting nanotube.(a) ν = +1.(b) ν = −1.The thin dotted lines show trigonal warping of equi-energy lines and the solid lines perpendicular to chiral vector L with chiral angle η denote allowed wave vectors or one-dimensional bands determined by boundary conditions along the circumference.In the presence

5 Fig. 4 (
Fig. 4 (Color online) Calculated ground and firstexcited exciton energies as a function of the circumference or the diameter together with band edges for ν = +1.Experimental results of one-photon and two-photon absorption energies 60−62) are shown by squares and triangles.The numbers denoting each line represent family number f .γ 0 = 2.7 eV. (a) β = 1.5 and p = −0.2.(b) β = 1.0 and p = −0.5.

Fig. 5 (
Fig. 5 (Color online) Calculated ground and firstexcited exciton energies as a function of the circumference or the diameter together with band edges for ν = −1.

Table I
The family index f and the corresponding ν and (n a , n b ).Semiconducting and metallic nanotubes are denoted by S and M, respectively.