$K$ dependent exchange interaction of the $1S$ ortho exciton in Cu$_2$O

TL;DR

This paper derives the correct higher order $K$ dependent exchange interaction terms for the 1S exciton in Cu$_2$O, showing they are too small to explain observed splittings, which are instead due to exciton dispersion.

Contribution

It provides a rigorous derivation of $K$ dependent exchange interactions using $oldsymbol{k}oldsymbol{ullet}oldsymbol{p}$ theory and clarifies their limited role in exciton splitting in Cu$_2$O.

Findings

Higher order $K$ dependent exchange terms are negligible in Cu$_2$O.

Observed exciton splitting is primarily due to dispersion effects.

Coupling between longitudinal and transverse excitons arises from $K$ dependent nonanalytic exchange.

Abstract

When treating the exchange interaction of Wannier excitons, usually only the leading terms of the analytic and the nonanalytic exchange interaction are considered. However, higher order terms can lead to a splitting of exciton states, for which reason a splitting of the $1S$~exciton in cuprous oxide $\left(\mathrm{Cu_{2}O}\right)$ depending on its total momentum $\hbar K$ has been attributed to a $K$ dependent analytic exchange interaction by Dasbach et al [Phys. Rev. Lett. 91, 107401 (2003)]. Going beyond the common treatment of the exchange interaction, we derive the correct expressions for these $K$ dependent higher order terms using $\boldsymbol{k}\cdot\boldsymbol{p}$ perturbation theory. We prove that the appearance of a $K$ dependent exchange interaction is inseparably connected with a $K$ independent exchange interaction of $P$ and $D$ excitons. We estimate the magnitude of these terms for $\mathrm{Cu_{2}O}$ from microscopic calculations and show that they are far too small to explain the observed $K$ dependent splitting. Instead, this splitting has to be treated in terms of the dispersion of the excitons. Furthermore, we prove the occurence of a coupling between longitudinal and transverse excitons in $\mathrm{Cu_{2}O}$ due to the $K$ dependent nonanalytic exchange interaction.