# Rate Equations for a Three-Level Model in Low-Dimensional Exciton Systems

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This Demonstrations shows a three-level scheme useful for describing the luminescence decays of low-dimensional quantum systems, such as single-walled carbon nanotubes and quantum dots. The excited-state photophysics of these systems can be modeled with an optically active exciton state, namely bright exciton , with an optically forbidden level below, known as dark exciton , separated by a splitting energy (of the order of a few meV). Recombination from both levels to the electronic ground state is recovered by the related rates (mainly nonradiative) and (purely nonradiative), respectively. The inverse quantities give the lifetimes of these excitons.

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Contributed by: Jessica Alfonsi (March 2011)

(University of Padova, Italy)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

For more information, see [1], [2], and [3]. In [3] is a treatment of spin-flip processes using the Fermi-Dirac distribution function, but with similar rate equations as in [1] and [2].

References

[1] T. Gokus, L. Cognet, J. G. Duque, M. Pasquali, A. Hartschuh, and B. Lounis, "Mono- and Biexponential Luminescence Decays of Individual Single-Walled Carbon Nanotubes," *Journal of Physical Chemistry C* 114(33), 2010 pp. 14025–14028.

[2] R. Matsunaga, Y. Miyauchi, K. Matsuda, and Y. Kanemitsu, "Symmetry-Induced Nonequilibrium Distributions of Bright and Dark Exciton States in Single Carbon Nanotubes," *Physical Review B,* 80, 2009 p. 115436.

[3] P. A. Dalgarno, J. M. Smith, B. D. Gerardot, A. O. Govorov, K. Karrai, P. M. Petroff, and R. J. Warburton, "Dark Exciton Decay Dynamics of a Semiconductor Quantum Dot," *Physica Status Solidi A* 202(14), 2005 pp. 2591–2597.

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