  Electric field effect (EFE) controlled magnetoelectric transport in thin
films of undoped and La-doped Sr$_{2}$IrO$_{4}$ (SIO) were investigated under
the action of ionic liquid gating. Despite large carrier density modulation,
the temperature dependent resistance measurements exhibit insulating behavior
in chemically and EFE doped samples with the band filling up to 10\%. The
ambipolar transport across the Mott gap is demonstrated by EFE tuning of the
activation energy. Further, we observe a crossover from a negative
magnetoresistance (MR) at high temperatures to positive MR at low temperatures.
The crossover temperature was around $\sim$80-90 K, irrespective of the
filling. This temperature and magnetic field dependent crossover is
qualitatively associated with a change in the conduction mechanism from Mott to
Coulomb gap mediated variable range hopping (VRH). This explains the origin of
robust insulating ground state of SIO in electrical transport studies and
highlights the importance of disorder and Coulombic interaction on electrical
properties of SIO.
