  The 2D electron gas (2DEG) formed at the surface of SrTiO$_3$(001) has
attracted great interest because of its fascinating physical properties and
potential as a novel electronic platform, but up to now has eluded a
comprehensible way to tune its properties. Using angle-resolved photoemission
spectroscopy with and without spin detection we here show that the band filling
can be controlled by growing thin SrTiO$_3$ films on Nb doped SrTiO$_3$(001)
substrates. This results in a single spin-polarised 2D Fermi surface, which
bears potential as platform for Majorana physics. Based on our results it can
furthermore be concluded that the 2DEG does not extend more than 2 unit cells
into the film and that its properties depend on the amount of SrO$_x$ at the
surface and possibly the dielectric response of the system.
