  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.
