Scattering amplitudes for dark and bright excitons

TL;DR

This paper derives the effective scattering equations for excitons in GaAs heterostructures, revealing how spin configurations influence brightness and the conditions for optical observation of exciton Bose-Einstein condensation.

Contribution

It introduces a formalism that explicitly accounts for fermion exchanges in exciton scattering and uncovers a key relation between scattering amplitudes affecting condensate observability.

Findings

Most spin configurations lead to brightness-conserving scatterings with equal amplitude.

Dark excitons can scatter into bright states with a different amplitude, depending on carrier exchange parity.

The relation Δ_e + Δ_o = Δ determines when exciton condensates are optically observable.

Abstract

Using the composite boson many-body formalism that takes single-exciton states rather than free carrier states as a basis, we derive the integral equation fulfilled by the exciton-exciton effective scattering from which the role of fermion exchanges can be unraveled. For excitons made of $(\pm1/2)$-spin electrons and $(\pm3/2)$-spin holes, as in GaAs heterostructures, one major result is that most spin configurations lead to brightness-conserving scatterings with equal amplitude $\Delta$, in spite of the fact that they involve different carrier exchanges. A brightness-changing channel also exists when two opposite-spin excitons scatter: dark excitons $(2,-2)$ can end either in the same dark states with an amplitude $\Delta_e$, or in opposite-spin bright states $(1,-1)$, with a different amplitude $\Delta_o$, the number of carrier exchanges being even or odd respectively. Another major result is that these amplitudes are linked by a striking relation, $\Delta_e+\Delta_o=\Delta$, which has decisive consequence for exciton Bose-Einstein condensation. Indeed, this relation leads to the conclusion that the exciton condensate can be optically observed through a bright part only when excitons have a large dipole, that is, when the electrons and holes are well separated in two adjacent layers.