Direct Determination of Exciton Wave Function Amplitudes by the Momentum-Resolved Photo-Electron Emission Experiment

We study conceptional problems of a photo-electron emission (PEE) process from a free exciton in insulating crystals. In this PEE process, only the electron constituting the exciton is suddenly emitted out of the crystal, while the hole constituting the exciton is still left inside and forced to be recoiled back to its original valence band. This recoil on the hole is surely reflected in the spectrum of the PEE with a statistical distribution along the momentum-energy curve of the valence band. This distribution is nothing but the square of the exciton wave function amplitude, since it shows how the electron and the hole are originally bound together. Thus, the momentum-resolved PEE can directly determine the exciton wave function. These problems are clarified, taking the $\Gamma$ and the saddle point excitons in GaAs, as typical examples. New PEE experiments are also suggested.