  For spintronics applications, it is highly desirable to realize
highly-spin-polarized two-dimensional (2D) electron systems in
electrically-controllable epitaxial ultrathin films on semiconductor
substrates. Through systematic first-principles investigation, we propose the
TcO$_2$ uni-cell layer (one-unit-cell thickness) on rutile TiO$_2$ (001)
substrate as a semiconductor heterostructure and use electric field to
manipulate its electronic and magnetic properties. Our study shows that the
heterostructure is a narrow-gap semiconductor with an antiferromagnet-like
ordering when the applied electric field is less than 0.026 V/\AA{}, and then
it transits to a half-metallic ferrimagnet with 100\% spin polarization. Our
further analysis indicates that the magnetization density and the electronic
states near the Fermi level originate mainly from the TcO$_2$ uni-cell layer,
with the remaining minor part from the interfacial Ti-O$_2$ monolayers, and the
bonds and bond angles quickly converge to the corresponding values of bulk
TiO$_2$ when crossing the interface and entering the TiO$_2$ layer. Therefore,
the heterostructure is actually a 2D electron system determined by the TcO$_2$
uni-cell layer and the TiO$_2$ substrate. Because the half-metallic phase with
100\% spin polarization can be achieved at 0.026 V/\AA{}, this epitaxial 2D
electron system should be usable in spintronics applications.
