Two-phonon resonant Raman scattering in single-walled carbon nanotubes

Fig. 1. (a) Electronic band structure close to the Fermi energy, chosen as zero. The optical transition E₂₂ is denoted by vertical arrows. (b) Phonon dispersion in the extended zone representation. The pairs of close vertical lines bracket the phonons for electron scattering between the two extrema of the bands for the E₂₂ transition. (c) Electronic broadening (solid line) in comparison with that for graphene (dotted line).

The Raman scattering of light in carbon nanotubes is usually observed under resonant conditions, i.e., the Raman signal is enhanced for laser photon energies, close to an optical transition of a nanotube [1,2]. The quantum-mechanical description of the Raman scattering process can be done considering the system of electrons, photons and phonons, and their interactions [3,4]. The resonant Raman intensity for Stokes processes is derived in fourth-order quantum-mechanical perturbation theory

Here, , ; is the energy of the initial state, is the incident photon energy (laser excitation); , , are the energies of the intermediate (a, b, c) and final (f) states of the system;are the matrix elements between initial, intermediate, and final states; and are momentum matrix elements; and are electron-phonon matrix elements [5]; γ is the broadening parameter, equal to the sum of the halfwidths of conduction and valence states [6]. The two-phonon Raman process can be represented by a sequence of virtual processes of electron-hole creation and annihilation, and electron-phonon scattering.

Calculations of the two-phonon Raman bands of a number of SWNTs, using the expression above, have been reported recently [7]. As an example for calculation of the two-phonon Raman bands of SWNTs we consider nanotube (6,5) [8]. The translational unit cell of this nanotube contains N = 182 two-atom unit cells. The CPU for direct calculations of the intensity increases rapidly with the increase of the number of atoms in the unit cell and the calculations are prohibitively expensive even for the studied nanotube. In order to make the calculations feasible, we use the chiral symmetry of the nanotubes and reformulate the electronic and dynamical problems for two-atom unit cells [1 – 8].

The electronic states of a nanotube are labeled by the one-dimensional wavevector k and the integer quantum number l. The phonons are labeled by the one-dimensional wavevector q and the integer quantum number λ. The electron-phonon scattering processes obey selection rules.

Figure 1 shows the electronic band structure, phonon dispersion, and electronic level broadening γ for nanotube (6,5). Close to the optical transition E₂₂, the main contribution to the resonant Raman intensity comes from scattering of electrons, occupying the bands in bold in Fig. 1(a), by phonons between the close vertical lines in Fig. 1(b). These phonons give rise to various overtone and combination bands, the most intense one being the 2D band (or 2TO) due to scattering by in-plane transverse optical (TO) phonons.

The two-phonon Raman spectrum of nanotube (6,5) for laser excitation E_L = 2.35 eV is given in Fig. 2. Along with the 2TO, 2LO, and TOLA bands, which are also observed in graphene, the spectrum shows TOZA and TOZO bands. The latter are not observed in graphene, because the coupling of electrons with out-of-plane acoustic (ZA) phonons and out-of-plane optical (ZO) phonons is zero. However, the non-zero curvature of the nanotubes makes this coupling non-zero and the corresponding combination bands – observable [7].

Figure 3 shows the evolution of the 2D band with laser excitation in the vicinity of transition E₂₂. For E_L ≈ E₂₂, the 2D band has a symmetric shape. With the increase of E_L, the 2D band widens and, at E_L ≈ E₂₂ + E_ph, where E_ph is the energy of the scattering phonon, the onset of a splitting of the band into two subbands can be observed. With further increasing E₂₂, the splitting of the 2D band increases resulting in a “Y”-shaped form of the peaks of the subbands. The origin of the splitting will be discussed below. The resonant Raman profile of the 2D band shows three peaks at E₂₂, E₂₂ + E_ph, and E₂₂ + 2E_ph, the third one being most intense (Fig. 3, right).

The behavior of the 2D band with EL can be explained with the scattering processes that have largest contribution. The Raman intensity, given by the expression above, is resonantly enhanced for E_L, for which one, two, or all three are close to zero. These three cases are referred to as single, double and triple resonance.

In order to identify the contribution of these resonances to the peaks of the resonant Raman profile (Fig. 3, right), we consider the scattering processes, which contribute to these resonances. We note that the Raman processes can be denoted as ee, hh, and eh, depending on the scattered particles – electrons (e) or holes (h) (Venezuela). In Fig. 4, the scattering processes, giving rise to (a) single, (b) double, and (c) triple resonance are given.

The single resonances give negligible contribution to the 2D band of graphene. In nanotubes, if the electron scattering takes place between the extrema of the conduction or valence bands, due to the high electronic density of states there (Fig. 4(a)), the single resonance has visible contribution to all three peaks at E₂₂, E₂₂ + E_ph, and E₂₂ + 2E_ph of the resonant Raman profile (Fig. 3, right). All ee, hh, and eh scattering processes contribute to the Raman intensity.

The double resonance processes, initially proposed by Thomsen, are due to ee and hh scattering processes (Fig. 4(b)). In nanotubes, these processes are most effective if the final electron/hole state is close to the conduction or valence extrema. These processes then contribute to the peak at E₂₂ + 2E_ph of the resonant Raman profile (Fig. 3, right).

The triple resonance processes, identified as giving largest contribution the two-phonon Raman intensity of graphene by Venezuela, are due to eh scattering processes (Fig 4(c)). In nanotubes, these processes are most effective if the final electron/hole state is close to the conduction or valence extrema. Similarly to the double resonances, these processes contribute to the peak at E₂₂ + 2E_ph of the resonant Raman profile (Fig. 3, right).

Finally, we discuss the “Y”-shaped behavior of the Raman intensity (Fig. 3, left). The single resonances arise from scattering of electrons and holes close to the extrema of the conduction/valence bands. Therefore, the wavevector q₀ of the scattering phonon will be close to the separation between these extrema and the contribution of the single resonances to the 2D band will be centered at 2ω(q₀) (Fig. 4(d)).

With the increase of the E_L, double and triple resonances are “switched-on”. The largest contribution of these resonances comes from scattering of electrons/holes with final states close to the extrema of the conduction or valence bands. However, the initial electron/hole states will be away from these extrema. Consequently, the wavevectors of the scattering phonons, q₁ and q₂, will be less or larger than q₀. The two resulting subbands of the 2D band will be centered at 2ω₁ and 2ω₂ (Fig. 4(d)), wherefrom the “Y” shape of the Raman intensity appears (Fig. 3, left).

References:

1. V. N. Popov, New J. Phys. 6 (2004) 17.

2. V. N. Popov and L. Henrard, Phys. Rev. B 70 (2004) 115407.

3. V. N. Popov, L. Henrard, and Ph. Lambin, Nano Letters 4 (2004) 1795-1799.

4. V. N. Popov, L. Henrard, and Ph. Lambin, Phys. Rev. B 72 (2005) 035436.

5. V. N. Popov and Ph. Lambin, Phys. Rev. B 73 (2006) 165425.

6. V. N. Popov and Ph. Lambin, Phys. Rev. B 74 (2006) 075415.

7. Ch. Tyborski et al., Phys. Rev. B 97 (2018) 214306.

8. V. N. Popov, Phys. Rev. B 98 (2018) 085413.

Fig. 2. Resonant two-phonon Raman spectrum at E_L = 2.35 eV.

Fig. 3. Left panel: Evolution of the 2D band with laser excitation. Right panel: resonant Raman profile of the 2D band.

Fig. 4.

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Valentin Popov

June 28, 2018