  At the LaAlO$_3$-SrTiO$_3$ interface, electronic phase transitions can be
triggered by modulation of the charge carrier density, making this system an
excellent prospect for the realization of versatile electronic devices. Here,
we report repeatable transistor operation in locally gated LaAlO$_3$-SrTiO$_3$
field-effect devices of which the LaAlO$_3$ dielectric is only four unit cells
thin, the critical thickness for conduction at this interface. This extremely
thin dielectric allows a very efficient charge modulation of
${\sim}3.2\times10^{13}$ cm$^{-2}$ within a gate-voltage window of $\pm1$ V, as
extracted from capacitance-voltage measurements. These also reveal a large
stray capacitance between gate and source, presenting a complication for
nanoscale device operation. Despite the small LaAlO$_3$ thickness, we observe a
negligible gate leakage current, which we ascribe to the extension of the
conducting states into the SrTiO$_3$ substrate.
