Phase-Space Ab-Initio Direct and Reverse Ballistic-Electron Emission Spectroscopy: Schottky Barriers Determination for Au/Ge(100)

We develop a phase-space ab-initio formalism to compute Ballistic Electron Emission Spectroscopy current-voltage I(V)'s in a metal-semiconductor interface. We consider injection of electrons into the conduction band for direct bias ($V>0$) and injection of holes into the valence band or injection of secondary Auger electrons into the conduction band for reverse bias ($V<0$). Here, an ab-initio description of the semiconductor inversion layer (spanning hundreds of Angstroms) is needed. Such formalism is helpful to get parameter-free best-fit values for the Schottky barrier, a key technological characteristic for metal-semiconductor rectifying interfaces. We have applied the theory to characterize the Au/Ge(001) interface; a double barrier is found for electrons injected into the conduction band -- either directly or created by the Auger process -- while only a single barrier has been identified for holes injected into the valence band.