Large electron-hole exchange interaction of carbon nanotubes

Effect of electron-hole asymmetry on cross-polarized excitons in carbon nanotubes

“…[74][75][76] It was applied to cross-polarized excitons in carbon nanotubes under a rotatingwave approximation. 41,77 The Bethe-Salpeter-type equation is often solved with a so-called exchange term, approximately describing the depolarization effect, under the condition that only a single electronhole pair is excited, 18,21,26,78,79 i.e., in a single configuration interaction approximation equivalent to the Tamm-Dancoff approximation. 80 Both rotating-wave and Tamm-Dancofftype approximations are certainly valid when the spectrum is dominated by a single resonance peak.…”

“…7,22,[30][31][32][33][34] In this scheme, higher-order corrections describe trigonal warping of energy bands, 7,30,35 curvature, 32,36 lattice distortion, 32,[37][38][39][40] and electron-hole asymmetry. 7,41 A recent study on optical properties for parallel polarization revealed that these corrections can well reproduce experimental family patterns. 7 Higher-order terms play different roles for perpendicular polarization.…”

“…42 The degeneracy is actually lifted by electron-hole asymmetry present in the band structure and, therefore, such effects were already included in various calculations based on first principles and a tight-binding model. 18,19,21,26 Quite recently, experimental observation of brightened dark excitons in the perpendicular polarization was reported 43 and some attempts of analyzing the experiments based on an effective-mass scheme were made, 41,43 although they were unsatisfactory, as will be discussed in the following.…”

Environment effect on cross-polarized excitons in carbon nanotubes

“…41,42,44 Further recent studies for cross-polarized excitons revealed that dark excitons, located below bright excitons in energy, become optically allowed due to weak electron-hole asymmetry. [45][46][47] Optical spectra of carbon nanotubes have been measured in various environments such as for self-assembled tubes, 3,5,[48][49][50] tubes suspended in air, 4,7,8,10,12,51 and tubes in the presence of surrounding materials such as surfactants and solvents. 1,3,6,7,9,11,[13][14][15][16]34 It has often been shown that exciton energies in the above environments are reduced in comparison with those in tubes in air or vacuum.…”