Auger recombination of excitons in one-dimensional systems

In tightly confined one-dimensional (1D) systems, the effective Coulomb interaction is greatly enhanced and optical transitions generally lead to the formation of strongly bound excitons. When more than one exciton is present, the Coulomb interaction also leads to rapid exciton-exciton annihilation through an Auger recombination process. This effect, which may be significant even at low exciton densities, can be described by a rate law governing two-body interactions. The Auger recombination rate for excitons in a strongly confined 1D system is analyzed. The rate increases sharply with exciton binding energy, but varies only weakly with temperature. An explicit expression for the Auger recombination rate in terms of the exciton binding energy, optical matrix element and reduced carrier mass is derived for a two-band model in which the Coulomb interaction is approximated by a point-contact potential. Results for the prototypical 1D system of single-walled carbon nanotubes are obtained and compared with experiment.

Figure 1

(Color online) Illustration of scattering processes leading to Auger recombination for two electron-hole pairs. The dotted line describes a pair of electrons that scatter by their Coulomb interaction. The arrows indicate the final state of the electrons after scattering in the Auger process. Processes $A$ and $B$ correspond to scattering of two electrons in the conduction band, while processes $A^{\prime }$ and $B^{\prime }$ involve scattering of an electron in the conduction band with an electron in the valence band.

Figure 2

(Color online) Feynman diagrams for the Auger recombination process. The solid lines with forward- (backward-) going arrows denote electrons (holes), and the wavy lines represent the Coulomb interaction. Time progresses from left to right. The initial state consists of two excitons with momenta of $K$ and $P$; the final state consists of a free electron with momentum $k_e$ and a hole with momentum $k_h$. The exchange of momenta $K$ and $P$ leads to the other four diagrams that are not shown explicitly, but are included in the calculation.