^{1}and K. L. Sebastian

^{1,a)}

### Abstract

We study the process of electronic excitation energy transfer from a fluorophore to the electronic energy levels of a single-walled carbon nanotube. The matrix element for the energy transfer involves the Coulombic interaction between the transition densities on the donor and the acceptor. In the Förster approach, this is approximated as the interaction between the corresponding transition dipoles. For energy transfer from a dye to a nanotube, one can use the dipole approximation for the dye, but not for the nanotube. We have therefore calculated the rate using an approach that avoids the dipole approximation for the nanotube. We find that for the metallic nanotubes, the rate has an exponential dependence if the energy that is to be transferred, is less than a threshold and a dependence otherwise. The threshold is the minimum energy required for a transition other than the and transition. Our numerical evaluation of the rate of energy transfer from the dye pyrene to a (5,5) carbon nanotube, which is metallic leads to a distance of up to which energy transfer is appreciable. For the case of transfer to semiconducting carbon nanotubes, apart from the process of transfer to the electronic energy levels within the one electron picture, we also consider the possibility of energy transfer to the lowest possible excitonic state. Transfer to semiconducting carbon nanotubes is possible only if . The long range behavior of the rate of transfer has been found to have a dependence if . But, when the emission energy of the fluorophore is in the range , the rate has an exponential dependence on the distance. For the case of transfer from pyrene to the semiconducting (6,4) carbon nanotube,energy transfer is found to be appreciable up to a distance of .

R.S.S. acknowledges Council of Scientific and Industrial Research (CSIR), India and Bristol-Myers-Squibb fellowship for financial support. The work of K.L.S. was supported by the J.C. Bose Fellowship of the DST (India).

I. INTRODUCTION

II. MODEL FOR THE RATE OF ENERGY TRANSFER

A. Case I ( and )

B. Case II (at least one of and is not zero)

1. Case IIa

2. Case IIb

III. METALLIC CARBON NANOTUBES

IV. SEMICONDUCTING CARBON NANOTUBES

V. ENERGY TRANSFER TO THE EXCITONS

VI. RESULTS

VII. DISCUSSION

VIII. CONCLUSIONS

### Key Topics

- Carbon nanotubes
- 110.0
- Energy transfer
- 73.0
- Semiconductor nanotubes
- 24.0
- Nanotubes
- 21.0
- Excitons
- 20.0

## Figures

A schematic representation of a single-walled carbon nanotube formed by rolling up a sheet of graphene. The figure is appropriate for a zig-zag nanotube. On the right, we show the transition dipole of the fluorophore along with the angles that it makes with the coordinate axes.

A schematic representation of a single-walled carbon nanotube formed by rolling up a sheet of graphene. The figure is appropriate for a zig-zag nanotube. On the right, we show the transition dipole of the fluorophore along with the angles that it makes with the coordinate axes.

Band structure of the (5,5) carbon nanotube. Note that the nanotube is metallic and that electron (hole) bands are symmetric about the zero of energy.

Band structure of the (5,5) carbon nanotube. Note that the nanotube is metallic and that electron (hole) bands are symmetric about the zero of energy.

Band structure of the (6,4) carbon nanotube. Note that the nanotube is semiconducting and bands are farther away from compared to the bands.

Band structure of the (6,4) carbon nanotube. Note that the nanotube is semiconducting and bands are farther away from compared to the bands.

The total rate of energy transfer as a function of the distance between the fluorophore and the (5,5) carbon nanotube. The rates are evaluated for the transfer of a fixed amount of energy given by , the emission maximum of pyrene.

The total rate of energy transfer as a function of the distance between the fluorophore and the (5,5) carbon nanotube. The rates are evaluated for the transfer of a fixed amount of energy given by , the emission maximum of pyrene.

A comparison of the total rate of energy transfer as a function of the distance between the fluorophore and the (5,5) carbon nanotube with a single emission frequency for the fluorophore and with the experimental emission spectrum of pyrene incorporated into the analysis.

A comparison of the total rate of energy transfer as a function of the distance between the fluorophore and the (5,5) carbon nanotube with a single emission frequency for the fluorophore and with the experimental emission spectrum of pyrene incorporated into the analysis.

The total rate of energy transfer to the band gap states of the (6,4) carbon nanotube as a function of the distance between the fluorophore and the tube. The rates are evaluated for the transfer of a fixed amount of energy given by , the emission maximum of pyrene.

The total rate of energy transfer to the band gap states of the (6,4) carbon nanotube as a function of the distance between the fluorophore and the tube. The rates are evaluated for the transfer of a fixed amount of energy given by , the emission maximum of pyrene.

The rate of energy transfer to the lowest excitonic state of the (6,4) carbon nanotube as a function of the distance. The linear fit in the long distance limit shows that the rate follows an exponential dependence for large .

The rate of energy transfer to the lowest excitonic state of the (6,4) carbon nanotube as a function of the distance. The linear fit in the long distance limit shows that the rate follows an exponential dependence for large .

A plot showing the total rate of energy transfer to the band gap states and the rate to the lowest excitonic state of the (6,4) carbon nanotube as a function of the distance.

A plot showing the total rate of energy transfer to the band gap states and the rate to the lowest excitonic state of the (6,4) carbon nanotube as a function of the distance.

The total rate of energy transfer from pyrene to the (6,4) carbon nanotube as a function of the distance.

The total rate of energy transfer from pyrene to the (6,4) carbon nanotube as a function of the distance.

## Tables

A summary of all the three cases analyzed above. Whether a particular case arises or not for the metallic and the semiconducting tubes is also shown.

A summary of all the three cases analyzed above. Whether a particular case arises or not for the metallic and the semiconducting tubes is also shown.

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