Figure 1

Left: the excitation pathway leading to the ratio problem in SWNT fluorescence spectroscopy. Electron hole pairs excited into the second azimuthal subband (a) relax to the first azimuthal subband where they radiatively recombine (b). Right: the electron hole pair in the second subband can decay into a single electron hole pair in the first subband (c), or to a two electron–two hole state (d),(e)Reuse & Permissions

Figure 2

The Coulomb interaction between the electron and hole is described by two fundamental scattering processes: (a) intraband scattering, conserving the azimuthal components of crystal momenta and (b) interband scattering with finite momentum transfer in the azimuthal direction. Interactions denoted by the dashed lines preserve the azimuthal crystal momentum; those denoted by double dashed lines carry one unit of the azimuthal crystal momentum $\hslash /R$. (c) is the multiple intraband scattering that leads to exciton formation and (d) describes the decay of the second subband exciton into electron hole pairs in the first subband.Reuse & Permissions

Figure 3

An electron hole pair in the second azimuthal subband can relax into a final state containing two electrons and two holes: (a) A typical amplitude for this relaxation process; (b) the general vertex for the relaxation; (c) the exciton self-energy produced by this process.Reuse & Permissions

Figure 4

Absorption line shapes for the second subband exciton resonating with first subband excitations plotted as a function of the dimensionless energy $\epsilon =ER/\hslash v_F$. The inset shows the spectral densities for the electron-hole and two electron–two hole continua. The bold dot denotes the location of the unrenormalized second subband exciton. The curves show the dimensionless line shapes computed from Eq. (4) for three values of the screened dimensionless coupling constant of the theory [$\tilde{\alpha }=0.15$ (dashed), 0.20 (solid), 0.25 (dot-dashed)]. The lower panel gives the ratio of the energy of the renormalized exciton to the energy of the fluorescence. The unrenormalized theory gives the dashed line and the shaded region is obtained from the experiments of Ref. 6.Reuse & Permissions