Optical Phonon Tuned by Fermi Level in Carbon Nanotubes

The dependence of the frequency and broadening of optical phonons on carrier concentration is studied in carbon nanotubes within an effective-mass approximation. In metallic nanotubes, the frequency shift exhibits a logarithmic divergence and the broadening vanishes discontinuously when the Fermi level reaches the half of the optical-phonon frequency for the longitudinal mode with displacement in the axis direction, while the transverse mode is not affected. In semiconducting nanotubes, the frequency is raised for both transverse and longitudinal modes, exhibiting a behavior similar to level crossing.


§1. Introduction
−3) In a previous work, 4) the interaction effect was shown to cause characteristic diameter dependence of the frequency and broadening of phonons in metallic and semiconducting nanotubes within the continuum model for both electrons and phonons.Quite recently, large change in the phonon spectra was experimentally observed for varying electron concentration by the application of a gate voltage. 5,6)The purpose of this paper is to study the Fermi-level dependence of optical phonons.
The electronic states of a nanotube change from metallic to semiconducting depending on the tubular circumferential vector.The characteristic properties, first predicted in a tight-binding model, can also be described well in a k•p scheme or an effectivemass approximation. 7)The long-wavelength phonons and their interaction with electrons are also described quite well in a continuum model. 8)In this paper, we calculate the frequency shift and broadening of phonons in the lowest order approximation as a function of the Fermi level, using the continuous phonon model within the k•p scheme.
This paper is organized as follows: In §2, a very short review is given on the k•p scheme for the description of energy bands and the continuum model of optical phonons.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 graphene, the first Brillouin zone, and the coordinate system in nanotubes.In a graphene sheet the conduction and valence bands consisting of π orbitals cross at K and K' points at the corners of the Brillouin zone, where the Fermi level is located. 9,10)Electronic states of the π bands near a K point are described by the k•p equation: 7,11,12) γ(σ• k)F (r) = εF (r), (2.1)   where γ is a band parameter, σ = (σ x , σ y ) is the Pauli spin matrix, 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.Electronic states of a nanotube with a sufficiently large diameter are obtained by imposing the boundary conditions around the circumference direction: 7) 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).
Metallic and semiconducting nanotubes correspond to ν = 0 and ±1, respectively.The energy bands are specified by s (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 The corresponding energy is given by with  7) The wave functions F ϕ,K and F ϕ,K with flux ϕ for the K and K' points, respectively, are related to each other through F ϕ,K = σ z F * −ϕ,K e iψ and F ϕ,K = σ z F * −ϕ,K e iψ by the time reversal, where σ z is the Pauli matrix and ψ is an arbitrary phase. 13,14)

Long Wavelength Optical Phonon
The phonon modes are specified by the wave vector q = (q x , q y ).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, q y = q.In the long-wavelength limit j = 0 and |qa| 1 with a being the lattice constant, both longitudinal and transverse phonons have the frequency ω 0 .The eigen vectors for the longitudinal e l (q) and transverse e t (q) modes are given by e l (q) = 0, i , e t (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.The phonon displacement u(r) is given by where N is the number of unit cells, μ denotes t or l, M is the mass of a carbon atom, and b † μq and b μq are the creation and destruction operators, respectively.
The interaction between optical phonons and an electron near the K point is given by 4) where γ 0 is the hopping integral between nearest neighbor carbon atoms related to γ through γ = ( √ 3a/2)γ 0 .The corresponding Hamiltonian for the K' point can be obtained by replacement of σ with σ * .

Phonon Green's Function
The phonon Green's function is written as (2.12) In the following, we shall consider retarded Green's function.The phonon frequency is determined by the pole of D μ (q, ω) as The phonon self-energy is usually small and therefore, the shift of the phonon frequency Δω μ and the broadening Γ μ are given by As will be shown in explicit results presented below, the self-energy shift sometimes exhibit a logarithmic divergence.In such a case, the phonon Green's function gives a pole in the vicinity of zero frequency.As has been discussed previously, 4) its spectral weight is extremely small and gives a negligible contribution to the phonon spectral function.−21) This instability is not important except in extremely thin nanotubes. 18,21)

§3. Frequency Shift and Broadening
Figure 2 shows the lowest order diagram for the phonon self-energy.The contribution of the states in the vicinity of the K point in the long-wavelength limit q → 0 is written as 4) where f (ε) is the Fermi distribution function, g s = 2 the spin degeneracy, δ represents a phenomenological broadening due to disorder, and the upper and lower sign correspond to μ = l and t, respectively, and f 0 (ε)= θ(−ε) with the step function: Because we start with the phonon mode with frequency ω 0 in graphene, we have to subtract the corresponding contribution in the self-energy.The second term in the square bracket containing n+t in the right hand side of eq.(3.1) represents this subtraction.The contribution of states in the vicinity of the K' point, Π K μ (ω), is obtained by appropriate replacement of the wave vectors and the energies, and the total self-energy is given by the sum Within the present scheme, the phonon self-energy is symmetric between the case of the Fermi level lying in the conduction s = +1 and valence s = −1 bands.Therefore, we shall consider the former case explicitly and restrict ourselves to the case that the Fermi level lies in the lowest conduction band with n = 0. Further, the phonon frequency is smaller than the gap for the bands n = 0 and therefore is neglected in the denominator for n = 0.In a previous work, 4) corrections to this approximation were also calculated, but turned out to be small.After a little manipulation as given in Appendix A, we then have and where we have introduced the dimensionless coupling parameter For hω 0 = 0.196 eV corresponding to 1583 cm −1 , γ 0 = 2.63 eV, and 2 < β < 3, we have 0.08 × (a/L) < α(L) < 0.17×(a/L).At zero temperature, the integral over k in the above equations can be performed analytically in the clean limit δ → 0. Explicit expressions for the real part are complicated and therefore presented in Appendix B. The imaginary part is simple and given by where k F is the Fermi wave vector.
In semiconducting nanotubes, we take the limit b |a| and have The imaginary part vanishes identically.The corresponding results for the K' point are obtained by replacement of ν with −ν in a and therefore the same as those for the K point in the absence of flux, ϕ = 0.The frequency of the longitudinal and transverse modes is both shifted to higher frequency side and the shift is smaller for the longitudinal mode for small k F . Figure 3 shows the shifts.It is interesting to note that the behavior of two modes as a function of k F is similar to that of a "level crossing." In metallic nanotubes in the absence of flux, we take the limit a → 0 and have The results for the K' point are the same as those for the K point.The transverse mode is not affected by the doping at all.This is to be expected because the band n = 0 does not give any contribution to the phonon self-energy due to the vanishing matrix element (see eq. (3.1) for ϕ = 0).For the longitudinal mode, the term remaining for k F = 0 represents the downward frequency shift in the undoped case obtained previously. 4)For nonzero k F , the self-energy has a logarithmic divergence at γk F = hω 0 /2 and increases logarithmically with k F for γk F > hω 0 /2.Further, the self-energy has an imaginary part comparable to or larger than the real part for γk F < hω 0 /2, meaning that the phonon has a finite life time due to spontaneous emission of an electron-hole pair.
Figure 4 shows the frequency shift relative to that for k F = 0 and the broadening as a function of the Fermi energy E F ≡ γk F .The logarithmic singularity at γk F = hω 0 /2 is smoothed out with the increase of the broadening δ.This behavior of the logarithmic singularity at γk F = hω 0 /2 and the reduction of the broadening is essentially the same as that predicted theoretically 22−24) and observed experimentally in the monolayer graphene. 25,26)igure 5 shows some examples of the spectral function (−1/π)ImD(q, ω) of the longitudinal mode in metallic nanotubes.The broad peak for γk F /hω 0 < 1/2 becomes much sharper for γk F /hω 0 > 1/2.The peak position exhibits a dip at γk F /hω 0 = 1/2 corresponding to the logarithmic singularity in the ideal case.The peak shift at γk F /hω 0 = 1/2 is quite sensitive to the disorder as has been shown in Fig. 4.
The transverse mode is not affected by the doping of the band n = 0 because the band has no contribution as mentioned above.In the presence of the Aharonov-Bohm flux, however, the transverse mode becomes affected by the doping as shown in Fig. 6 in the case E φ /hω 0 = 0.1, where This flux roughly corresponds to a magnetic field ∼ 40 T for typical single-wall nanotubes with diameter ∼ 1.4 nm.A certain amount of effective flux is present due to the curvature and lattice distortion in actual nanotubes, 7) and therefore a slight peak shift and change in the broadening can be observable experimentally.
Figure 7 shows the frequency shift and broadening as a function of the gap 2E φ induced by the Aharonov-Bohm flux.The singular behavior in the broadening of the transverse mode at 2E φ /hω 0 ≈ 1, predicted previously, 4) becomes weaker with the increase of the doping, but the large broadening for 2E φ /hω 0 ≈ 1 does not disappear completely as long as the doping is small.
Figure 8 shows the spectral function of the longitudinal mode for various values of the gap 2E φ induced by the Aharonov-Bohm flux for two cases of (a) negligible doping γk F /hω 0 = 0.01 and (b) larger doping γk F /hω 0 = 0.2.No appreciable dependence on the flux gap is observed.
Figure 9 shows the spectral function for the transverse mode.In this case the singularity in the vicinity of 2E φ = hω 0 is strongly influenced by the doping.Thus, the Aharonov-Bohm effect for the transverse mode, predicted in ref. 4, is quite sensitive to carrier doping.§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. 27,28)Semiconducting tubes have sharp G + and G − , while metallic tubes have a broad down-shifted G − and a sharp G + , 29−37) although some experiments seem to show that the G − peak appears only in nanotube bundles. 38)The G − band shows a strong diameter dependence, being lower in frequency for smaller diameters.
Optical phonons in nanotubes were calculated in various methods such as a zone-folding scheme, 39) forceconstant models, 40,41) first-principles methods, 1,2,42−47) and some combinations. 3)Although there remains quantitative difference among the results of these theories and the present result, the qualitative features seem to be the same.For example, the considerable lowering of the longitudinal mode with small shift of the transverse mode in metallic nanotubes seems to be the common feature of the results.Further, the shift is not appreciable for the longitudinal and transverse modes in semiconducting nanotubes.
Quite recently, the Raman spectra of individual nanotubes were measured as a function of a gate voltage controlling the Fermi level, although the measurement seems to be in a preliminary stage yet.In ref. 5, for example, the frequency of the G − peak was observed to be lowered considerably at a voltage presumably corresponding to γk F = hω 0 /2 in metallic nanotubes.In ref. 6, further, it was reported that the broadening of the G − peak is substantially reduced with the increase of k F .The present results are qualitatively in agreement with such experimental observations and expected to be useful for quantitative analysis of such experiments.§5.

Summary and Conclusion
The self-energy of long-wavelength optical phonons has been calculated in nanotubes for nonzero carrier concentrations.In metallic nanotubes, the frequency shift of the longitudinal mode exhibits a logarithmic divergence and the broadening vanishes discontinuously when the Fermi level reaches the half of the opticalphonon frequency, while the transverse mode is not affected.In semiconducting nanotubes, the frequency is raised for both transverse and longitudinal modes, although the details of the dependence on the Fermi level is different.The results are in qualitative agreement with recent preliminary experiments, 5,6) and can be quite useful for quantitative analysis of Raman experiments, in particular, the G + and G − peaks.

Acknowledgments
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: Phonon Self-Energy
The phonon self-energies (3.1) are explicitly written as and By neglecting ω in the denominator except in the term with n = 0 and making a rearrangement of the terms, we have For the transverse mode, the integration over k can be performed with the use of the formula giving eq.(3.5).In the case of the longitudinal mode, with the use of the integration over k can be performed for some of the terms.After the integration over k, the term I containing the summation over n inside the square brackets of eq.(A3) becomes Appendix B: Ideal Limit at Zero Temperature In the limit δ → 0 and at zero temperature, we can obtain analytic results for the real part of the phonon self-energy.For the longitudinal mode, we have for b > a and for for b < a.
For the transverse mode, on the other hand, we have for b > a and  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. 3 The frequency shift in a semiconducting nanotube as a function of the Fermi wave vector k F .The solid and dashed lines represent the longitudinal and transverse modes, respectively.
Fig. 4 The frequency shift (solid lines) and the broadening (dashed lines) of the longitudinal mode in metallic nanotubes as a function of the Fermi energy γk F measured in units of the phonon energy hω 0 .δ is phenomenological broadening due to disorder.
Fig. 5 The spectral function of the longitudinal mode for varying Fermi energy γk F .The downward shift at γk F = hω 0 and the sharpening for γk F > hω 0 /2 are quite noteworthy.
Fig. 6 The frequency shift (solid lines) and the broadening (dashed lines) of the transverse mode in metallic nanotubes in the presence of flux E φ /hω 0 = 0.1 as a function of the Fermi energy γk F measured in units of the phonon energy hω 0 .

. 10 )
where the vector product for a = (a x , a y ) and b = (b x , b y ) in two dimension is defined by a × b = a x b y − a y b x and b = a/ √ 3 is the equilibrium bond length.The dimensionless parameter β is given by a+t) sin πa = − ln |2 sin πa|, (A7) where use has been made of Euler's integral π 0 ln | sin x|dx = −π ln 2, (A8) in the final equality.This gives eq.(3.4).
) for b < a.

Figure Captions Fig. 1
Figure CaptionsFig.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 nanotube with an Aharonov-Bohm magnetic flux.(c) The first Brillouin zone and K and K' points.

Fig. 7 Fig. 8 Fig. 9
Fig. 7 The frequency shift (solid lines) and the broadening (dashed lines) of the (a) longitudinal and (b) transverse mode in metallic nanotubes as a function of the gap 2E φ induced by flux.E F ≡ γk F .Fig. 8 The spectral function of the longitudinal mode for varying the gap 2E φ due to flux.(a) E F ≡ γk F = 0.01.(a) E F = 0.2.Fig. 9 The spectral function of the transverse mode for varying the gap 2E φ due to flux.(a) E F ≡ γk F = 0.01.(a) E F = 0.2.