1. Internal Conversions in Aromatics
Internal conversions transfer energetically higher lying electronic or excitonic states into energetically lower lying electronic or excitonic states of the same spin multiplicity by a microscopic scattering mechanism. The three electronic structures discussed in the previous section included the extremes of aromaticity as found in, benzene and graphene, and as an intermediate case of SWNT. Knowledge of the relaxation mechanism of the extrema can give insight to processes that may occur in nanotubes. In the case of benzene the electronic excitations are coupled strongly to the vibrational degrees of freedom and relaxation takes place through very efficient internal conversions [Hen71]. Graphene, on the other hand, has not been studied experimentally by optics in its single-layer form until recently [Bla07, Yan07]. Ultrafast optical experiments on high-quality samples of graphite have provided clues to the peculiar
nature of graphene and its linear dispersion relation, but this has yet to be done on single layer graphene. Yet, Raman spectroscopy of optical phonons in metallic SWNT reveal peculiarities in the screening of the ion-ion interaction near the zone boundary and are interpreted as effects from the well-know, Kohn anomaly [Pis07]. This effect explains the difference between Raman spectra of semiconducting and metallic SWNTs, and gives clues that graphene should behave the same way and that excitations to the dipole allowed electronic states relax to electronic states that are coupled strongly to the lattice by “softened” phonon modes.
It can be expected, as in the case in benzene that in isolated excited semiconducting SWNT will undergo molecular like internal conversions by phonon scattering. Conversely, the continuum of states that do not contribute to the formation of excitons may also provide an efficient scattering mechanism for internal conversion.
Until recently the photoluminescence quantum yields have only been reported to be around 0.1%, and most of the energy must be dissipated through nonradiative decay channels as in benzene that has a quantum yield of 0.5% [O’Co00]. There is no evidence for strong vibrational coupling as in benzene in the optical spectra of SWNT, but the peculiar nature of one dimension excitations may hide these effects. As long as energy and momentum is conserved the mechanism of internal conversion may take on both molecular and bulk like characteristics in SWNTs. If both processes compete with each other than the branching ratio between the two is of interest to account for low observed photoluminescence quantum yields.
2. Intersystem Crossing in Aromatics
Intersystem crossing processes transfer energetically higher lying electronic or excitonic states into energetically lower lying or energetically equivalent states of different spin multiplicity. In order to flip a spin that changes the multiplicity of the excitation, the product of the square of the spin orbit interaction and the final density of states must be non-negligible. Intrinsic spin orbit interaction in atomic carbon is on the order of 6 meV or ≈ 70 °K and estimated in isolated benzene to be approximately 0.1 meV or ≈ 1 °K [Min06]. Due to strong electron-phonon coupling, intersystem crossings can overcome internal conversions by mixing π and σ states in benzene [Hen71]. The Franck-Condon factors are extremely important in these processes and determine not only the oscillator strengths of the optical transitions, but also the efficiency of intersystem crossing mechanisms. Despite the small spin-orbit interaction, triplets efficiently quench the fluorescence of benzene and can account for the observed low quantum yield [Daw86]. In SWNT the notion of Franck-Condon factors being important in optical transitions has not been reported, but in transport measurements this interpretation has been used to explain phonon assisted tunneling between electronic states [Sap06].
In graphene it has been estimated that the intrinsic spin orbit coupling is on the order of 1 µeV or ≈ 10 mK and may arise from the local curvature of the topologically imperfect sheet that mixes π and σ orbitals [Min06]. The mixing of π and σ orbitals may be even stronger in small diameter SWNTs and spin orbit interactions of approximately 0.1 meV or ≈ 1 °K have been predicted which is similar to Benzene [Hue06]. Moreover,
the degeneracy of the lowest lying singlet exciton and its triplet counterpart along with the one dimensionality of the Coulomb interaction would lead to a large density of states for the singlets to scatter into even though the spin-orbit interaction is rather small.
Therefore, even though the Condon factors may not play a significant role in nanotubes the extremely large density of states may compensate and allow for an efficient decay channel fro singlet excitons. By determining the non-radiative lifetime of the lowest lying singlet state a good estimate of the spin-orbit interaction should be accessible.
3. Electron and Exciton Energy Transfer
When synthesized, SWNT form large crystallites due to relatively strong van der Waals interactions [Ter94]. The crystallites consist of statistical mixtures of semiconducting and metallic tubes that do not fluorescence. Intertube relaxation pathways may be very important and dominate intratube processes in crystallites. The mechanism involved are not very clear, but theoretical speculations of intertube tunneling and delocalization have been reported with out any clear evidence from experiment [Kwo98, Maa00, Sta00]. In aggregates of molecules and quantum dots, exciton energy transfer can take place depending on the physical distance and spectral overlaps of acceptors and donors [Pow75, Kag96]. Also, electron transfer can lead to ionized states that have a different ground state due to the gain of an extra electron or hole and become dark states or charge transfer excitons.
If energy transfer resulted in emission from a smaller band gap tube it is most likely that the donor singlet state and acceptor singlet state are coupled though a dipolar
field. In the case of non-radiative energy transfer with significant donor emission and acceptor absorption spectral overlap and the interactions between the donor acceptor is relatively week Förster type exciton energy is most likely involved [Vol04]. Typical separations between acceptor and donor exceed 1 nm and the transfer rate follows the well known R−6 dependence. Conversely, if the interaction between acceptor and donor are relatively strong and the interspacing is approximately 1 nm or less wavefunction overlap becomes a critical component of the transfer mechanism and the exchange interaction must be taken into account [Dex53]. Energy transfer in this regime is considered to be Dexter transfer and the rate decreases exponentially as the interspacing is increased. Moreover, since the exchange interaction becomes important in this regime, this is the dominant mechanism for triplet-triplet exciton energy transfer and may or may not lead to emissive states. In essence this is an electron or hole exchange mechanism and can be classified as an ultrafast electron transfer mechanism. [Vol04]. In nanotubes the intertube spacing is on the order of 3 Å and the tubes are bound by approximately 500 meV per micron of tube contact. Therefore, the relative strength of the interaction between tubes can be tested if exciton energy transfer is found to be either Förster or Dexter in nature.