Optical Phonon Interacting with Electrons in Carbon Nanotubes

Effects of interactions with electrons on optical phonons are studied in an effective-mass approximation. The longitudinal mode with displacement in the axis direction is lowered in its frequency, while the transverse mode with displacement in the circumference direction is raised, in metallic nanotubes. The shifts are opposite but their absolute values are smaller in semiconducting nanotubes. Only the longitudinal mode has a considerable broadening in metallic nanotubes. In the presence of an Aharonov-Bohm magnetic flux, the broadening appears for the transverse mode and diverges when the induced gap becomes the same as the frequency of the optical phonon.


§1. Introduction
Carbon nanotubes are quasi-one-dimensional materials made of sp 2 -hybridized carbon networks. 1)The electronic states change from metallic to semiconducting depending on the tubular circumferential vector characterizing a nanotube.−14) Electron-phonon interactions can play an important role in the behavior of optical phonons in carbon nanotubes.−17) The purpose of this paper is to study effects of electron-phonon interactions on long-wavelength optical phonons based on the k•p scheme.
Although electronic properties have been understood by those of the graphite plane using a periodic boundary condition, the phonon modes of nanotubes are not simply given by the zone-folded modes of planes because they fail to give breathing modes. 7)In a previous work, a continuum model suitable for a correct description of long-wavelength acoustic phonons was constructed. 18)In this paper we shall introduce a similar continuum model of optical phonons and derive the Hamiltonian for electron-phonon interactions.Then, we calculate the self-energy of phonon Green's function in the lowest order approximation.The real part of the self-energy gives an energy shift and the imaginary part provides a lifetime.
As is expected, the most important feature comes from the bands in the vicinity of the Fermi level.−24) This gap can be controlled by a magnetic flux passing through the cross section due to the Aharonov-Bohm effect. 14,25,26)Recently, splitting of optical absorption and emission peaks due to the Aharonov-Bohm effect was observed. 27,28)It will be shown that this small gap can manifest itself as considerable broadening of optical phonon.
This paper is organized as follows: In §2, the k•p scheme for the description of energy bands is reviewed very briefly and a continuum model of optical phonons is introduced.The phonon Green's function is calculated and shifts and broadening of phonon modes are discussed in §3.The results are discussed in §4 and a short summary is given in §5.§2.Formulation

Effective-mass description
Figure 1 shows the lattice structure of the twodimensional graphite, the first Brillouin zone, and the coordinate system in nanotubes.In a graphite sheet the conduction and valence bands consisting of π orbitals cross at K and K' points of the Brillouin zone, where the Fermi level is located. 29,30)Electronic states of the π-bands near a K point are described by the k•p equation: 12,14,31,32) with where γ is a band parameter, σ x and σ y are the Pauli spin matrices, and k = ( kx , ky ) = −i∇ is a wave-vector operator.
The structure of a nanotube is specified by a chiral vector L corresponding to the circumference as shown in Fig. 1(a).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 with |a| = |b| = a (= 2.46 Å).In the following we shall choose the x axis in the circumference direction and the y axis in the axis direction, i.e., L = (L, 0), where L is the circumference.The angle η 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 Submitted to Journal of Physical Society of Japan conditions around the circumference direction: 12,14) where ϕ = φ/φ 0 with φ being a magnetic flux passing through the cross section and φ 0 being the flux quantum given by φ 0 = ch/e, and ν is an integer (ν = 0 or ±1) determined by with integer M .Metallic and semiconducting nanotubes correspond to ν = 0 and ±1, respectively.The energy bands are specified by s = ±1 (s = −1 and +1 for the valence and conduction band, respectively), integer n corresponding to the discrete wave vector along the circumference direction, and the wave vector k in the axis direction.The wave function for a band associated with the K point is written as (2.6) where A is the length of the nanotube, and The corresponding energy is given by (2.9) For the K' point, the k•p Hamiltonian is obtained by replacing ky with − ky in eq.(2.2) and the boundary conditions by replacing ν with −ν.Therefore, the energy band is given by eq.(2.9) in which κ νϕ (n, k) is replaced with κ −νϕ (n, k), and the wave function is given by eq.

Long Wavelength Optical Phonon
An equation of motion for optical phonons of the two-dimensional graphite in the long wavelength limit has been derived previously based on a valence-force-field model as 18) M ω(q) 2 u(q) = H op (q)u(q), (2.10) where q is the wave vector and u = (u x , u y ) is the relative displacement of two sub-lattice atoms A and B, defined by The effective Hamiltonian is given by x q x q y q y q x q 2 y − 3 2 where M is the mass of a carbon atom and K 1 and K 2 are the force constants for the bond stretching and bond-angle change, respectively.The resulting phonons are isotropic within the plane of the two-dimensional graphite.
The longitudinal phonon has the frequency ω l (|q|) and the eigen vector e l (q) given by with and (2.15) The transverse phonon has the frequency ω t (|q|) and the eigen vector e t (q) given by with (2.17) The dispersion of the optical phonons is small for |q|a 1 and therefore will be neglected completely in the following.
In the representation of the second quantization, the phonon displacement u(r) is given by u(r) = g,q h 2N M ω 0 (a g,q + a † g,−q )e g (q)e iq•r , (2.18)   where N is the number of unit cells, g denotes the modes (t for transverse and l for longitudinal), and a † g,q and a g,q are the creation and destruction operators, respectively.In carbon nanotubes, the wave vector in the circumference direction becomes discrete, i.e., q x = (2π/L)j with integer j, and that in the axis direction remains continuous, i.e., q y = q.
In the following, we shall confine ourselves to the long wavelength limit, j = 0 and |q|a 1, i.e., q = (0, q).In this case we have which means that the longitudinal mode has lattice displacement along the axis y direction and the transverse mode along the circumference x direction.

Electron-Phonon Interaction
The optical phonon distorts the distance between neighboring carbon atoms and therefore modifies the band structure through the change in the resonance integral between carbon atoms.The corresponding effective Hamiltonian can be obtained easily in a manner similar to the case of acoustic phonons.Details are discussed in Appendix A. When we use a simple nearest-neighbor tight-binding model, the interaction Hamiltonian for the K point is given by and for the K' point where b = a/ √ 3 is the equilibrium bond length, γ 0 is the resonance integral between nearest neighbor carbon atoms appearing in a tight-binding model related to γ through γ = ( √ 3a/2)γ 0 , and This means that the lattice distortion gives rise to a shift in the origin of the wave vector, i.e., u x in the y direction and u y in the x direction.
We have derived the optical phonons starting with the simplest valence-force-field model and the effective Hamiltonian describing their interaction with electrons in the two-dimensional graphite.It should be noticed, however, that the phonon modes are exact in the longwavelength limit if we use an appropriate value of the frequency ω 0 and also that the explicit form of the obtained Hamiltonian is much more general and is valid although the coupling parameter β can be different from that estimated above.

Phonon Green's Function
The phonon Green's function D(q, iω l ) is given by where Π(q, iω l ) is the self energy and ω l = (2πk B T /h)l is the Matsubara frequency with integer l.We shall consider the lowest order self-energy given in Fig. 2. By making an analytic continuation iω l → ω+i0, we have the retarded Green's function The phonon frequency is determined by the pole of D ret (q, ω) as

25)
As will become clear in the following, the phonon selfenergy is small.In this case, the shift of the phonon frequency is given by Δω = Re Π ret (q, ω 0 ), (2.26)   and the broadening is given by Γ = Im Π ret (q, ω 0 ). (2.27) A straightforward calculation gives the contribution from states in the vicinity of the K point, where f (ε) is the Fermi distribution function and the upper and lower sign correspond to g = l and t, respectively.The factor two comes from the electron spin.The contribution from states in the vicinity of the K' point Π gK ret (q, ω) is obtained by appropriate replacement of the wave vectors and the energies and the total self-energy is given by the sum In this paper, we shall calculate the self-energy of optical phonons starting with the known phonon modes in the two-dimensional graphite.Therefore, the direct evaluation of the above self-energy causes a problem of double counting. 33)In fact, if we apply the above formula to the case of infinitely large circumference, we get the frequency shift due to virtual excitations of electrons in the π bands in the two-dimensional graphite.However, this contribution is already included in the definition of ω 0 .In order to avoid such a problem, we have to subtract the contribution in the two-dimensional graphite, which is obtained by the sum over discrete wave vector in the circumference direction κ νϕ (n) replaced with a continuous integration.Because the frequencydependence of the phonon self-energy is negligible, the ω dependence can be ignored completely for this term.
Therefore, the final expression of the self-energy is given by

.30)
This self-energy shows that the renormalization of phonon frequencies in nanotubes from the two-dimension graphite is determined mainly by electronic states in the vicinity of the Fermi level where the discreteness of the bands plays important roles.The contributions from states away from the Fermi level are the same between the nanotube and the two-dimensional graphite and therfore cancel each other in the self-energy.§3.Frequency Shift and Lifetime

Long Wavelength Limit
In the following, we shall consider phonons with q = 0 at zero temperature.Then, the self-energy is simplified considerably where (ν → −ν) represents terms in which ν is replaced with −ν, we have introduced the dimensionless quantities, and α(L) is the coupling parameter defined by with the dimensionless parameter in the two-dimensional graphite, given by For the parameter hω 0 = 0.196 eV corresponding to 1583 cm −1 , γ 0 = 2.63 eV, and β = 2, we have λ = 0.08.
The above shows that the self-energy depends on the circumference L or the diameter d = L/π only through the universal form given by a/L and is independent of the chirality or the individual values of n a and n b .Further, except in extremely thick nanotubes, the phonon energy hω 0 is much smaller than the typical electronic energy 2πγ/L of the order of band gaps, i.e., ω 1.In the following, we shall obtain analytic expressions of the phonon self-energy using this fact explicity.

Metallic Nanotubes
First, we consider a metallic nanotube in the absence of an Aharonov-Bohm magnetic flux, i.e, ν = 0 and ϕ = 0. Expanding the terms with n = 0 in terms of ω in eqs.(3.1) and (3.2), we immediately obtain Because ω 1, the frequency is shifted to the higher energy side and no imaginary part appears in the case of the transverse mode.In the case of the longitudinal mode, on the other hand, the frequency is shifted to the lower energy side and exhibits a logarithmic divergence for small ω.Further, the self-energy has an imaginary part comparable or larger than the real part, meaning that the phonon has a finite life time due to spontaneous emission of an electron-hole pair.This width is qualitatively in agreement with that obtained previously. 17)he effective Hamiltonian describing the electronphonon interaction shows that the lattice displacement u x for the transverse mode causes a shift in the wave vector along the axis direction.This shift does not give rise to any change in the energy of the metallic linear bands n = 0.The frequency lowering due to interactions with electrons with wave vector close to k x = 0 (but not exactly k x = 0) present in the two-dimensional graphite disappears in nanotubes because of quantization of k x into multiples of 2π/L.This quantization is the origin of the frequency increase obtained for the transverse mode.
In the case of the longitudinal mode, the displacement u y causes a shift in the wave vector along the circumference direction, leading to opening of a band gap reducing the energy of electrons in the metallic linear bands.As a result, the phonon frequency is lowered and at the same time the imaginary part appears due to excitations of electrons in the linear bands.−35) This instability is known to be unimportant except in extremely thin nanotubes. 33)ecause of the logarithmic divergence for the longitudinal mode, the phonon Green's function gives a pole in the vicinity of zero frequency.In fact (2.25) has a pole at which is extremely small for conventional single-wall nanotubes L/a > ∼ 10.Its spectral weight, i.e., the integral of the imaginary part of the Green's function around ω * , is calculated as πω * /α(L)ω 0 .This turns out to be negligibly small.Figure 3 shows the frequency shift of the transverse and longitudinal modes as a function of the circumference.The dependence of these shifts on the circumference is determined essentially by the effective coupling parameter α(L) = (a/L)λ.For the longitudinal mode, the broadening is comparable or even larger than the shift.

Semiconducting Nanotubes
In semiconducting nanotubes, to the leading order in ω, we have This shows that in contrast to the case of metallic nanotubes, the transverse mode is shifted to the lower energy side and the longitudinal mode is shifted to the higher energy side.However, the amount of the shifts is very small.Figure 3 contains also the shifts in semiconducting nanotubes.

Aharonov-Bohm Effect
In metallic nanotubes with an Aharonov-Bohm magnetic flux ϕ, we have for ω <2|ϕ|, and for ω > 2|ϕ|.The real part diverges when ω approaches the gap 2|ϕ| due to the Aharonov-Bohm effect from the high frequency side, while it converges a finite value from the low frequency side.Therefore, the phonon Green's function has in principle an extra pole below 4πγ|ϕ|/L, but its intensity is extremely small and can be neglected practically.
The most notable feature is the appearance of the imaginary part for 2|ϕ| < ω giving rise to the broadening of the phonon spectrum.With the increase of the gap 4πγ|ϕ|/L due to the flux, the imaginary part increases in proportion to ϕ 2 , diverges when the gap reaches the optical-phonon energy hω 0 , and vanishes when the gap exceeds the phonon energy.Figure 4 shows this broadening as a function of the Aharonov-Bohm gap.
For the longitudinal mode, we have for ω <2|ϕ| and for ω > 2|ϕ|.The real part remains continuous and does not exhibit divergence at ω = 2|ϕ|.The most notable feature lies in the broadening of the phonon spectrum again.As shown in Fig. 4 the imaginary part gradually decreases with the magnetic flux and vanishes when the gap reaches the phonon energy hω 0 .§4. Discussion In the Raman spectra of single-wall nanotubes the so-called G band is usually fit with two components G + and G − arising from transverse and longitudinal optical phonons at the Brillouin-zone center.Semiconducting tubes have sharp G + and G − , while metallic tubes have a broad down-shifted G − and a sharp G + , 36−44) although some experiments seem to show that the G − peak appears only in nanotube bundles. 45)The G − band shows a strong diameter dependence, being lower in frequency for smaller diameters.This suggested its attribution to a circumferential mode, whose atomic displacements would be most affected by a variation in diameter.−41) It was suggested also that the G band can be understood in terms of double-resonant scattering involving phonon wave vectors throughout the whole Brillouin zone. 46)ptical phonon modes in nanotubes were studied theoretically in various methods such as the zone-folding scheme, 47) force-constant models, 48,49) first-principles methods, 15,16,50−55) and some combinations. 17)In refs.15 and 16, for example, the obtained longitudinal frequency is lowered considerably in metallic tubes, while it is not shifted in semiconducting tubes.Further, the transverse frequency is lowered but its amount is less than the longitudinal mode in metallic nanotubes and is almost independent of whether the tube is metallic or semiconducting.In ref. 17, on the other hand, the calculated longitudinal modes are lowered considerably and the transverse modes remain near the frequency of the zone-center phonon of the two-dimensional graphite independent of whether the nanotube is metallic or semiconducting.
The considerable lowering of the longitudinal mode in metallic nanotubes and the small lowering of the transverse mode in semiconducting nanotubes obtained in this paper are qualitatively in agreement with the results of refs.15 and 16 although being different quantitatively.The present calculations show that the transverse modes also become different between metallic and semiconducting nanotubes and the difference between the longitudinal and transverse frequencies is much smaller in semiconducting cases than in metallic cases.These features are quite different from those of refs.15 and 16 and of ref. 17.
Figure 5 shows the comparison of the present results with the G + and G − peaks experimentally obtained in ref. 40.We have assumed ω 0 = 1570 cm −1 and λ = 0.17, corresponding to β slightly larger than 2. The amount of the splitting of the G + and G − peaks seems to be in good agreement with the experiments for metallic nanotubes, but is smaller than the experiments for semiconducting nanotubes.The bond strength of neighboring carbon atoms due to bonding through σ orbitals is likely to be lowered in nanotubes due to nonzero curvature and as a result the frequency of the optical mode itself can be lowered from that of the two-dimensional graphite depending on the diameter.The actual estimation of such effect is out of the scope of this paper but the renormalization of ω 0 depending on the curvature has a tendency to reduce the disagreement in particular for nanotubes with a small diameter.
Figure 6 compares the width of the phonon spectrum with some of existing experiments for the G − peak.In this comparison the parameters same as in Fig. 5 are used for ω 0 and λ.The broadening predicted theoretically has a right order of magnitude as that observed experimentally.§5.

Summary and Conclusion
An effective Hamiltonian has been derived for describing the interaction between long-wavelength optical phonons and electrons in carbon nanotubes within the continuum approximation for phonons and the effectivemass approximation for electrons.It has been used for the evaluation of the frequency shift and broadening of phonons due to interactions with electrons.The results are summarized as follows: The longitudinal mode with displacement in the axis direction is lowered in its frequency, while the transverse mode with displacement in the circumference direction is raised, in metallic nanotubes.The shifts are opposite but their absolute values are smaller in semiconducting nanotubes.These shifts increase almost in proportion to the inverse of the diameter with the decrease of the diameter.
The longitudinal mode has a considerable broadening or a spectral width due to interactions with electronic excitations.For the transverse mode, the broadening appears in the presence of an Aharonov-Bohm magnetic flux and diverges when the induced gap becomes the same as the frequency of the optical phonon for the transverse mode.For the longitudinal mode, on the other hand, the broadening decreases gradually and disappears as a function of the flux.Therefore, the Aharonov-Bohm effect manifests itself also in optical phonons in metallic carbon nanotubes where R A and R B denote the positions of the A and B site, respectively, and ψ A (R) and ψ B (R) are the amplitude of the p z orbital, and τ l (l = 1, 2, 3) is the vector connecting neighboring carbon atoms as shown in Fig. 1.
In the following we shall consider the coordinates rotated by η.In the vicinity of the Fermi level ε = 0, we can write ) in terms of the slowly-varying envelope functions F K A , F K B , F K A , and F K B .This can be written as and where Ω 0 is the area of a unit cell given by Ω 0 = √ 3a 2 /2.This function g(r) can be replaced by a delta function when it is multiplied by a smooth function such as envelopes, i.e., g(r) ≈ Ω 0 δ(r).In order to obtain equations for F , we first substitute eq.(A3) into eq.(A1).Multiply the first equation by g(r−R A )a(R A ) and then sum it over R A .With the full use of the slowly-varying nature of the envelope function, the equation is obtained as 14) εF A (r) = γ kx −i ky 0 0 kx +i ky F B (r). (A8) The lattice displacement corresponding to optical phonons causes a change in the distance between nearest neighbor carbon atoms and in the resonance integral.Let u A (R A ) and u B (R B ) be a lattice displacement at A and B site, respectively.Then the resonance integral between an atom at R A and R A −τ l changes from −γ 0 The first Brillouin zone and K and K' points.
Fig. 2 A Feynman diagram for the self-energy for optical phonons with a wave vector q along the axis (y) direction.Fig. 4 The broadening of optical phonons in metallic nanotubes as a function of the band gap induced by an Aharonov-Bohm flux.When the gap reaches the optical-phonon energy hω 0 , the broadening disappears for the longitudinal mode and becomes infinitely large for the transverse mode.

)
Introduce a smoothing function g(r) varying smoothly in the range |r| < ∼ a and decaying rapidly for |r| a.It should satisfy the condition RA g(r−R A ) = RB g(r−R B ) = 1, (A6) and g(r)dr = Ω 0 , (A7)

Fig. 1
Fig. 1 (a) A schematic illustration of the lattice structure of the two-dimensional graphite and the lattice displacement for transverse and longitudinal optical phonons.(b) The coordinate system in the nan-

Fig. 3
Fig. 3 Calculated frequency shifts of the optical phonons.The transverse and longitudinal modes are denoted by TO and LO, respectively.The shadowed region shows the broadening Γ.The longitudinal mode is lowered in the frequency and has large broadening due to interactions with electrons in metallic nanotubes.

Fig. 5
Fig. 5 Comparison of the calculated frequency shifts and experimental results of the Raman G + and G − peaks (ref.40).