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Optical Phonon Interacting with Electrons in Carbon Nanotubes Kohta ISHIKAWA and Tsuneya ANDO Department of Physics, Tokyo Institute of Technology 2–12–1 Ookayama, Meguro-ku, Tokyo 152-8551 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. Keywords: carbon nanotube, graphite, optical phonon, softening, hardening, effective-mass theory, Aharonov-Bohm effect §1. Introduction Carbon nanotubes are quasi-one-dimensional materials made of sp2 -hybridized carbon networks.1) The electronic states change from metallic to semiconducting depending on the tubular circumferential vector characterizing a nanotube. The characteristic properties were predicted by calculations in tight-binding models2−11) and also in a k·p scheme or an effective-mass approximation.12−14) Electron-phonon interactions can play an important role in the behavior of optical phonons in carbon nanotubes. In fact, recent first-principles calculations reported lowering of phonon energy compared to that in the two-dimensional graphite.15−17) 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. In metallic nanotubes a narrow gap can open due to effects of a finite curvature2,14,19,20) and possible lattice strain.14,18,19,21−24) 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 AharonovBohm 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 Submitted to Journal of Physical Society of Japan 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 2.1 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) H0 F (r) = εF (r), with H0 = γ 0 k̂x +ik̂y k̂x −iky 0 (2.1) = γ(σx k̂x +σy k̂y ), (2.2) where γ is a band parameter, σx and σy are the Pauli spin matrices, and k̂ = (k̂x , k̂y ) = −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 L = na a+nbb, (2.3) in terms of two integers na and nb , 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 Page 2 K. Ishikawa and T. Ando conditions around the circumference direction:12,14) ν , (2.4) F (r+L) = F (r) exp 2πi ϕ− 3 where ϕ = φ/φ0 with φ being a magnetic flux passing through the cross section and φ0 being the flux quantum given by φ0 = ch/e, and ν is an integer (ν = 0 or ±1) determined by na +nb = 3M +ν, K2 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 ωl (|q|) and the eigen vector el (q) given by ωl (|q|)2 = ω02 (1−η1 a2 |q|2 ), i qx el (q) = , |q| qy (2.5) 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 1 bνϕ (n, k) Fs,n,k (x, y) = √ exp[iκνϕ (n)x+iky] , s 2AL (2.6) where A is the length of the nanotube, ν 2π κνϕ (n) = n+ϕ− , (2.7) L 3 (2.13) with ω02 = 3(K1 +6K2 ) , M (2.14) η1 = 1 K1 −12K2 . 8 M ω02 (2.15) and The transverse phonon has the frequency ωt (|q|) and the eigen vector et (q) given by ωt (|q|)2 = ω02 [1−(η1 +η2 )a2 |q|2 ], i qy et (q) = , |q| −qx (2.16) with and κνϕ (n)−ik bνϕ (n, k) = . κνϕ (n)2 +k 2 The corresponding energy is given by εsνϕ (n, k) = sγ κνϕ (n)2 +k 2 . (2.8) (2.9) For the K’ point, the k·p Hamiltonian is obtained by replacing k̂y with −k̂y 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 κνϕ (n, k) is replaced with κ−νϕ (n, k), and the wave function is given by eq. (2.6) in which bνϕ (n, k) is replaced with b−νϕ (n, k)∗ .12,14) 2.2 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) = Hop (q)u(q), (2.10) where q is the wave vector and u = (ux , uy ) is the relative displacement of two sub-lattice atoms A and B, defined by 1 u(q) = √ [uA (q)−uB (q)]. (2.11) 2 The effective Hamiltonian is given by 1 0 Hop (q) = 3(K1 +6K2) 0 1 2 K 3K2 2 qx qx qy 1 + − + a (2.12) qy qx qy2 8 2 2 2 3 K1 +3K2 qx +qy 0 − K 2 a2 , 0 qx2 +qy2 2 K1 +6K2 where M is the mass of a carbon atom and K1 and η2 = 3 3K1 +3K2 K2 . 2 2K1 +6K2 M ω02 (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 h̄ u(r) = (ag,q + a†g,−q )eg (q)eiq·r , (2.18) 2N M ω 0 g,q where N is the number of unit cells, g denotes the modes (t for transverse and l for longitudinal), and a†g,q and ag,q are the creation and destruction operators, respectively. In carbon nanotubes, the wave vector in the circumference direction becomes discrete, i.e., qx = (2π/L)j with integer j, and that in the axis direction remains continuous, i.e., qy = 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 0 i , et (q) = , (2.19) el (q) = i 0 which means that the longitudinal mode has lattice displacement along the axis y direction and the transverse mode along the circumference x direction. 2.3 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 Optical Phonon in Carbon Nanotubes in Appendix A. When we use a simple nearest-neighbor tight-binding model, the interaction Hamiltonian for the K point is given by √ βγ 0 uy (r)+iux(r) K Hint = − 2 2 , (2.20) uy (r)−iux(r) 0 b and for the K’ point √ βγ 0 uy (r)−iux(r) K =− 2 2 , (2.21) Hint uy (r)+iux(r) 0 b √ where b = a/ 3 is the equilibrium bond length, γ0 is the resonance integral between nearest neighbor carbon atoms appearing √ in a tight-binding model related to γ through γ = ( 3a/2)γ0 , and β=− d ln γ0 . d ln b (2.22) This means that the lattice distortion gives rise to a shift in the origin of the wave vector, i.e., ux in the y direction and uy 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 ω0 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. 2.4 Phonon Green’s Function The phonon Green’s function D(q, iωl ) is given by D(q, iωl ) = 2ω0 , 2 2 (iωl ) −ω0 −2ω0 Π(q, iωl ) (2.23) where Π(q, iωl ) is the self energy and ωl = (2πkB T /h̄)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 iωl → ω+i0, we have the retarded Green’s function Dret (q, ω) = 2ω0 . (ω+i0)2 −ω02 −2ω0 Πret (q, ω) (2.24) Page 3 from states in the vicinity of the K point, dk βγ 2 h̄ (q, ω) = −2 ΠgK ret 2π b2 N M ω0 n s,s × ss [κνϕ (n)2 −k(k−q)] 1± 2 κνϕ (n)2 +k 2 κνϕ (n)2 +(k−q)2 × f [εsνϕ (n, k)]−f [εsνϕ(n, k−q)] , h̄ω −εsνϕ(n, k)+εsνϕ (n, k−q)+i0 1 (2.28) 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 ΠgK 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 gK Πgret (q, ω) = ΠgK ret (q, ω) + Πret (q, ω). (2.29) 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 ω0 . 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 κνϕ (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 ΠgK ret (q, ω) = −2 s,s n 1/2 dt −1/2 dk βγ 2 h̄ 2π b2 N M ω0 1 ss [κνϕ (n)2 −k(k−q)] 1± × 2 κνϕ (n)2 +k 2 κνϕ (n)2 +(k−q)2 The phonon frequency is determined by the pole of Dret (q, ω) as ω 2 2 −1 = ReΠret (q, ω) (2.25) ω0 ω0 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 Πret (q, ω0 ), (2.26) and the broadening is given by Γ = Im Πret (q, ω0 ). (2.27) A straightforward calculation gives the contribution f [εsνϕ (n, k)]−f [εsνϕ(n, k−q)] h̄ω −εsνϕ(n, k)+εsνϕ (n, k−q)+i0 ss [κνϕ (n+t)2 −k(k−q)] 1 1± − 2 κνϕ (n+t)2 +k 2 κνϕ (n+t)2 +(k−q)2 f [εsνϕ (n+t, k)]−f [εsνϕ(n+t, k−q)] × . (2.30) −εsνϕ (n+t, k)+εsνϕ (n+t, k−q)+i0 × 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 Page 4 K. Ishikawa and T. Ando cancel each other in the self-energy. §3. Frequency Shift and Lifetime 3.1 Long Wavelength Limit In the following, we shall consider phonons with q = 0 at zero temperature. Then, the self-energy is simplified considerably 1/2 1 t dt dk̃ Πret (ω) = − α(L)ω0 2 −1/2 n (n+ϕ−ν/3)2 4 × [(n+ϕ−ν/3)2 + k̃ 2 ]1/2 4[(n+ϕ−ν/3)2 + k̃ 2 ]− ω̃ 2 (n+t)2 − + (ν → −ν) , (3.1) [(n+t)2 + k̃ 2 ]3/2 1/2 1 Πlret (ω) = − α(L)ω0 dt dk̃ 2 −1/2 n 4 k̃ 2 × 2 2 1/2 [(n+ϕ−ν/3) + k̃ ] 4[(n+ϕ−ν/3)2 + k̃ 2 ]− ω̃ 2 k̃ 2 − + (ν → −ν) , (3.2) [(n+t)2 + k̃ 2 ]3/2 where (ν → −ν) represents terms in which ν is replaced with −ν, we have introduced the dimensionless quantities, 2πγ −1 2π −1 ω̃ = h̄ω × , k̃ = k × , (3.3) L L and α(L) is the coupling parameter defined by a α(L) = λ , L (3.4) with the dimensionless parameter in the two-dimensional graphite, given by λ= h̄2 1 2 27 2 β γ0 . π 2M a2 h̄ω0 (3.5) For the parameter h̄ω0 = 0.196 eV corresponding to 1583 cm−1 , γ0 = 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 na and nb . Further, except in extremely thick nanotubes, the phonon energy h̄ω0 is much smaller than the typical electronic energy 2πγ/L of the order of band gaps, i.e., ω̃ 1. In the following, we shall obtain analytic expressions of the phonon self-energy using this fact explicity. 3.2 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 ω̃ in eqs. (3.1) and (3.2), we immediately obtain π Πt (ω) = α(L)ω0 2− ω̃ 2 , 9 2π ω̃ π 2 2 Πl (ω) = α(L)ω0 2 ln − ω̃ +iπ . e 18 2 (3.6) (3.7) Because ω̃ 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 ω̃. 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) The effective Hamiltonian describing the electronphonon interaction shows that the lattice displacement ux 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 kx = 0 (but not exactly kx = 0) present in the two-dimensional graphite disappears in nanotubes because of quantization of kx 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 uy 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. The logarithmic divergence for ω̃ = 0 corresponds to an instability toward a spontaneous lattice deformation accompanying opening of a small gap obtained in the adiabatic approximation.3,33−35) This instability is known to be unimportant except in extremely thin nanotubes.33) Because 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 1 eγ exp − , (3.8) ω∗ = L 4α(L) 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)ω0 . 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. 3.3 Semiconducting Nanotubes In semiconducting nanotubes, to the leading order in ω̃, we have 4π 2 ω̃ 2 , Πtret (ω) = α(L)ω0 − 9 1 2π 2 2 Πlret (ω) = α(L)ω0 ln 3 − ω̃ . 2 9 (3.9) (3.10) This shows that in contrast to the case of metallic nanotubes, the transverse mode is shifted to the lower Optical Phonon in Carbon Nanotubes 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. 3.4 Aharonov-Bohm Effect In metallic nanotubes with an Aharonov-Bohm magnetic flux ϕ, we have 8ϕ2 ω̃ +2 Πtret (ω) = α(L)ω0 − sin−1 2 2 2|ϕ| ω̃ 4ϕ − ω̃ 1 π2 1 2 − ω̃ , (3.11) − 3 sin2 πϕ ϕ2 for ω̃ < 2|ϕ|, and Πtret (ω) ω̃+ ω̃ 2 −4ϕ2 = α(L)ω0 ln +2 2 2 2 2 ω̃ ω̃ −4ϕ ω̃− ω̃ −4ϕ 1 π2 4πϕ2 1 − , (3.12) − 2 ω̃ 2 + i 2 3 sin πϕ ϕ ω̃ ω̃ 2 −4ϕ2 4ϕ2 for ω̃ > 2|ϕ|. The real part diverges when ω̃ 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| < ω̃ 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 ϕ2 , diverges when the gap reaches the optical-phonon energy h̄ω0 , 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 2 2 sin πϕ 2 4ϕ − ω̃ 2 ω̃ sin−1 + 2 ln Πlret (ω) = α(L)ω0 ω̃ 2|ϕ| e 1 π2 1 (3.13) − − 2 ω̃ 2 , 2 6 sin πϕ ϕ for ω̃ < 2|ϕ| and Πlret (ω) 2 ω̃ −4ϕ2 ω̃ + ω̃ 2 −4ϕ2 ln = α(L)ω0 ω̃ ω̃ − ω̃ 2 −4ϕ2 2 sin πϕ 1 π 2 1 + 2 ln − 2 ω̃ 2 − 2 e 6 sin πϕ ϕ 2ϕ 2 + iπ 1− , (3.14) ω̃ for ω̃ > 2|ϕ|. The real part remains continuous and does not exhibit divergence at ω̃ = 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 h̄ω0 . §4. Discussion In the Raman spectra of single-wall nanotubes the Page 5 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. Thus, the G+ and G− peaks were often assigned to longitudinal (axial) and transverse (circumferential) modes, respectively.36,37,39−41) 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) Optical 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 ω0 = 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 ω0 depending on the curvature has Page 6 K. Ishikawa and T. Ando 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 ω0 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 that of the carbon pz orbital. Then, we have εψA (RA ) = −γ0 The authors thank Dr. Mikito Koshino for valuable discussion. This work was supported in part by a 21st Century COE Program at Tokyo Tech “Nanometer-Scale Quantum Physics” and by Grant-in-Aid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology Japan. Appendix A: Electron-Phonon Interaction The method to obtain the effective Hamiltonian for electron-phonon interaction is essentially the same as that for acoustic phonons discussed in refs. 14 and 18. We shall start with a nearest-neighbor tight-binding model and then obtain the effective-mass equation. The resulting Hamiltonian itself is much more general, however. Let −γ0 be the resonance integral between nearestneighbor carbon atoms and choose the energy origin at ψB (RA −τl ), l=1 εψB (RB ) = −γ0 3 (A1) ψA (RB +τl ), l=1 where RA and RB denote the positions of the A and B site, respectively, and ψA (R) and ψB (R) are the amplitude of the pz orbital, and τl (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 ψA (RA ) = eiK·RA FAK (RA ) + eiη eiK ·RB ψB (RB ) = −ωeiη eiK·RB FBK (RB ) + eiK FAK (RA ), ·RB FAK (RB ), (A2) in terms of the slowly-varying envelope functions FAK , FBK , FAK , and FBK . This can be written as ψA (RA ) = a(RA )† FA (RA ), (A3) ψB (RB ) = b(RB )† FB (RB ), with a(RA )† = ( eiK·RA † eiη eiK iη iK·RB b(RB ) = ( −ωe e and FA = FAK FAK e , ·RA ), iK ·RB ), FB = FBK FBK . (A4) (A5) Introduce a smoothing function g(r) varying smoothly in the range |r| < ∼ a and decaying rapidly for |r| a. It should satisfy the condition g(r−RA ) = g(r−RB ) = 1, (A6) RA and Acknowledgments 3 RB g(r)dr = Ω0 , (A7) where √ 2 Ω0 is the area of a unit cell given by Ω0 = 3a /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) ≈ Ω0 δ(r). In order to obtain equations for F , we first substitute eq. (A3) into eq. (A1). Multiply the first equation by g(r−RA )a(RA ) and then sum it over RA . With the full use of the slowly-varying nature of the envelope function, the equation is obtained as14) k̂x −ik̂y 0 εFA (r) = γ (A8) FB (r). 0 k̂x +ik̂y The lattice displacement corresponding to optical phonons causes a change in the distance between nearest neighbor carbon atoms and in the resonance integral. Let uA (RA ) and uB (RB ) be a lattice displacement at A and B site, respectively. Then the resonance integral between an atom at RA and RA −τl changes from −γ0 Optical Phonon in Carbon Nanotubes by the amount ∂γ0 (b) τl +uA (RA )−uB (RA −τl ) − b ∂b ∂γ0 (b) 1 =− τl · uA (RA )−uB (RA −τl ) , (A9) ∂b b √ where b = |τl | = a/ 3. Therefore, the extra term appearing in the right hand side of eq. (A8) is calculated as ∂γ (b) 0 g(r−RA )a(RA )b(RA −τl )† − ∂b l RA 1 × τl · uA (RA )−uB (RA −τl ) FB (r) b −ωeiη e−iK·τl ∂γ0 (b) 0 − = 0 e−iη e−iK ·τl ∂b l 1 × τl · uA (r)−uB (r−τl ) FB (r). 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The longitudinal mode is lowered in the frequency and has large broadening due to interactions with electrons in metallic nanotubes. 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 h̄ω0 , the broadening disappears for the longitudinal mode and becomes infinitely large for the transverse mode. Fig. 5 Comparison of the calculated frequency shifts and experimental results of the Raman G+ and G− peaks (ref. 40). Fig. 6 Comparison of the calculated broadening of the longitudinal mode and the half-width-half-maximum for experimentally observed G− peaks in metallic nanotubes. The experiments are those of Brown et al.,39) Jorio et al.,40) Maultzsch et al.,42) Oron-Carl et al.,43) and Doorn et al.44) Optical Phonon in Carbon Nanotubes Axis y (a) q Page 9 y x φ LO (Longitudinal) B τ2 b (b) K A τ1 TO (Transverse) τ3 s ,n,k−q,εm−ωl L K Γ x η Circumference a (c) ωl, q ωl, q s,n,k,εm Fig. 1 Fig. 2 10 Transverse Longitudinal LO Broadening (units of (a/L)λω0) Frequency Shift (units of λω0) TO 0.0 TO LO -0.5 5 Semiconducting Metallic ( Width) -1.0 10 15 20 25 Circumference (units of a) Fig. 3 30 0 0.0 0.5 1.0 Energy Gap (units of hω0) Fig. 4 Page 10 K. Ishikawa and T. Ando Diameter (nm) 1.0 0.5 1.5 2.0 TO 1550 LO 1500 Metallic Semiconducting Experiments (Jorio et al) ω0 = 1570 cm-1 λ = 0.17 -0.5 Broadening (units of λω0) LO Frequency (cm-1) Frequency Shift (units of λω0) TO 0.0 1.5 2.0 Experiments Brown et al Jorio et al Maultzsch et al Oron-Carl et al Doorn et al ω0 = 1570 cm-1 λ = 0.17 0.6 1650 1600 1.0 0.7 1700 0.5 0.4 150 100 0.3 0.2 50 1450 0.1 1400 0.0 10 15 20 25 Circumference (units of a) Fig. 5 30 0 10 15 20 25 Circumference (units of a) Fig. 6 30 Broadening (cm-1) Diameter (nm)