1.1. band structure of pristine SWCNTsband structure of pristine SWCNTs

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Effect of Helical Perturbation on Exciton Binding Energy in Semiconducting Carbon NanotubesBenjamin Tayo and Slava V. Rotkin

Department of Physics, Lehigh University, Bethlehem, PA

We study the effect of an externally applied helical potential induced by DNA wrapping on electron-hole excitations in single-walled carbon nanotubes. These coupled electron-hole states referred to as excitons play a significant role in the photophysics of semiconductors and by extension in device application. One important quantity which can easily be measured is the binding energy of an exciton. So its imperative that we study how exciton binding energies in pristine carbon nanotubes (CNT) differ from exciton binding energies in DNA-CNT hybrids. We present below the theory (which we are currently implementing numerically) that is used to study excitonic effects in DNA-functionalized CNTs.

Electronic energy bands of single-walled carbon nanotubes are obtained from those of graphene by means of the zone folding approximation. The band structure of graphene is calculated using the tight binding approximation (TBA).

In the TBA (band limit), conduction is described as a hoping process from sites to sites. This leads to the Hamiltonian:

Basis

change

yields

In matrix form, the TBA Hamiltonian, its eigenvalues and eigenfunctions are

Band structure of a (3,0) tube. This tube is ungapped, meaning its metallic in character.

Band structure of a (5,0) nanotube. This tube is gapped, hence it’s a semiconductor.

2. band structure of SWCNTs in the presence of an 2. band structure of SWCNTs in the presence of an external helical potential induced by DNA wrapexternal helical potential induced by DNA wrap

By definition, an exciton is a bound state of a photo-excited electron (in the conduction band) and the hole (in the valence band) which it leaves behind. In order to formulate the theory of excitons in DNA-CNT hybrids, we must first determine band structure modulations induced by the presence of the single-stranded (ss) DNA polymer wrapping the tube. The perturbed energy bands are easily calculated in first order perturbation theory. The total Hamiltonian including DNA-CNT interaction (V) is now

The Interacting Hamiltonian above due to its one-body nature, can be diagonalized easily by introducing a unitary transformation which we denote U. Under the unitary transformation U, H assumes the following diagonal form:

We conclude that the electrons in a CNT are “dressed” by the external potential. The dressed (quasiparticle) wave functions which enter the Bethe-Salpeter (BS) equation are given by:

The exciton creation operator is defined as

where “a” and “d” are the electron and hole creation operators, respectively.

The wave function amplitudes

are obtained by solving the BS equation

In these expressions, w is the screened Coulomb interaction and v, the unscreened Coulomb interaction

3. exciton states in DNA-CNT hybrids 3. exciton states in DNA-CNT hybrids

Scheme of electron-hole transitions corresponding to exciton levels. Feynman diagram corresponds to exciton state. The interaction line is screened.

Exciton binding energies for the first four excitonic states for a (5,0) SWCNT are shown below:

Our goal is to compute exciton binding energies for a DNA-CNT hybrid and compare its binding energy with those obtained above.

index

Index Energy (eV)

1 0.34

2 0.38

3 1.10

4 1.15

References

References

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