Optoelectronics with Carbon Nanotubes

optical absorption peaks for different samples and their corresponding diameter measurements
by transmission electron microscopy (TEM) 14 . The so-called Kataura plot and its subsequent
enhancements (especially Ref. 15) have been widely used to identify the diameter and even
particular species of SWNTs, although it was developed within the van-Hove transition scheme.
While Kataura mentions the effect of excitons briefly in his original paper, it is treated as a slight
deviation from the calculated values.
The discrepancy between the single-particle picture and the nature of excitation and
emission observations first became apparent in the so-called “ratio problem” 16, 17 , in which the
ratio of the energy of the second-excited state to the first one deviates significantly from the
expected value of 2 expected from the single-particle picture (see Figure I-5). While there are
additional complicating factors because of the chirality and the warping that comes from the
rolled geometry, the average ratio should approach 2 as the diameter increases and the system
goes from one dimension to two. The actual picture turns out to be an interplay of electron self-
energy Eee and the exciton binding energy Eb: the effect of the former interaction remains the
same as the diameter increases because it is an intrinsic property of graphene, while the latter
scales roughly as d -1 (Ref. 18). Since Eee blueshifts the first and the second transitions by nearly
the same amount, the ratio approaches a value smaller than 2 (Ref. 19). Recently, a successful
direct measurement of electronic bandgap (≈Eg + Eee) under different dielectric environments
was conducted using scanning tunneling spectroscopy (STS) in which the self-energy correction
was deduced to be in the order of 500 meV 20 . It should also be noted from this experiment that
the environmental medium also plays a significant role, as it screens the electron-electron
interaction. The effect is thought to scale as the inverse of the dielectric constant, ε 21, 22 .
Now we discuss the excitonic effect, which also modifies the optical transition energies.
Beginning with Ando’s theoretical work 23 , it became increasingly apparent that the binding
energy Eb of excitons in nanotubes is a significant fraction of the bandgap. The Coulomb
interaction between an electron and a hole is greatly enhanced in one dimension, and the charge
screening is also reduced, all contributing to a large Eb. Collaborating theoretical 24, 25 and
experimental 26-28 works have provided definitive evidence for the excitonic nature of observed
optical transitions in SWNTs, especially the two-photon excitation experiments 18, 26, 28 and the
observation of sidebands from an exciton-phonon complexes 27, 29, 30 . Perebeinos, et al. 24 gives a
general expression for Eb that agrees well with the results of the two-photon experiments if one
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