K dependent exchange interaction of the 1S ortho exciton in Cu₂O

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\(mathrm{Cu₂O}right\) 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}cdotboldsymbol{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{Cu₂O} 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{Cu₂O} due to the K dependent nonanalytic exchange interaction.