Asymmetry in Raman Resonant Excitation Profiles of Single-Wall Carbon Nanotubes: The Role of Kohn Anomaly, Optical Transition Order, and State-Mixing

Examination of Raman resonant excitation profiles (REPs) of the radial breathing mode (RBM) and G-band phonons has provided new insights into the coupling between excitons and phonons in single-wall carbon nanotubes (SWCNTs). For example, measurements of REPs of the longitudinal optical (LO) and transverse optical (TO) G-band phonons of highly pure, single semiconducting species SWCNT samples displayed a high degree of asymmetry when comparing the Raman intensity at the incident and scattered Raman resonances for excitation via E₂₂^S.  The LO and TO phonon REP asymmetries were explained using a molecular picture as due to a self-interference between Condon and non-Condon terms in the Raman polarizability, due to a vibrational coordinate dependence in transition dipole moment.   This results in the observation that the Raman intensity at the scattered resonance is always lower in intensity than that of the incident resonance, a consequence of the failure of the Franck-Condon approximation.  This is contrary to the symmetrical G-band REP expected by conventional theory and observed for RBM REPs. 

More recently, these Raman REP studies were expanded to include metallic armchair species, (5,5) through (8,8), excited via E₁₁^M.  Similar asymmetries to those of semiconducting SWCNTs were observed in the TO phonons of these armchair species while no asymmetry was observed for their respective RBM REPs, emphasizing the similarity in the excitonic nature of the optical states of these two electronic types of SWCNTs.  In addition, a new model was developed in terms of a condensed matter formalism involving phonon-mediated state-mixing of dark and bright excitonic states to successfully explain the asymmetry in G-band phonon REPs of both semiconductor and armchair SWCNT species.  Here, we extend our understanding of this optical phenomenon by considering the roles that Kohn anomaly, optical transition order and phonon-mediated state-mixing play in the degree of asymmetry observed for SWCNT phonon REPs.

To examine the effect of the Kohn anomaly, we measured G-band phonon REPs of another electronic type of SWCNT, known as narrow-gap semiconductors (NGS), such as the (7,4) and (10,4) species, whose electronic band structure is similar to that of metallic armchair SWCNTs with the exception of a small band gap on the order of 10 meV.  Of special interest in NGS SWCNT structures are the LO phonons, which are strongly frequency-softened and broadened through the Kohn anomaly due to strong coupling of LO phonons with the two Fermi points.  Such a strong interaction should further enhance the observed REP asymmetry due to non-Condon-based self-interference.  Furthermore, NGS-type SWCNT species, such as the (7,4) and (10,4), possess the additional trigonal warping splitting term in their band structure, which causes each optical transition to split into two, allowing for potential quantum interference between upper and lower branch transitions which may produce further asymmetry. 

To examine the role of optical transition order on REP asymmetry, we measured RBM and G-band REPs for multiple semiconducting chiralities such as (8,3), (9,1), (5,4), (7,6), (9,4), and (8,6) across multiple optical transitions ranging from E₁₁^S-E₄₄^S.  Generally, the degree of asymmetry increased with increasing transition order for the same SWCNT species for the LO phonon.  Additionally, we reexamined whether asymmetry exists in RBM REPs of semiconducting species excited via E₁₁^S, where the RBM phonon energy is greater than or equal to the electronic linewidth of the transition, allowing resolution of both incident and scattered resonances of the REP.  This was not possible in previous measurements because the electronic linewidths of the optical transitions involved were greater than the RBM phonon energies.

Finally, to test the validity of the phonon-mediated state-mixing model, we prepared isotopically pure ¹²C- and ¹³C-samples of single chiralities (7,5), (7,6), and (7,7).  Using these samples, excited via E₂₂^S for (7,5) and (7,6) and via E₁₁^M for (7,7), we compared the measured G-band phonon REPs for both isotopes of the same chirality.  While ¹³C-SWCNTs look identical to their ¹²C-counterparts in terms of electronic band structure and optical transition energies, the phonons in their Raman spectra decrease in frequency by a factor of √(12/13) due to the isotope effect.  If G-band phonon REP asymmetry is really due to mixing of dark and bright excitonic states, mediated by G-band or K-point phonons, which in turn causes quantum interference between excitonic states or transfer of oscillator strength from one excitonic state to another, then a change in the phonon energy mediating such an interaction should cause a visible change in the degree of asymmetry of the REP for the same chirality.  The change in the degree of REP asymmetry provides a measure of the magnitude of state-mixing.