  In Bi$_2$Te$_2$Se, the period of quantum oscillations arising from surface
Dirac fermions can be increased 6-fold using ionic liquid gating. At large gate
voltages, the Fermi energy reaches the $N$ = 1 Landau level in a 14-Tesla
field. This enables the $\frac12$-shift predicted for the Dirac spectrum to be
measured accurately. A surprising result is that liquid gating strongly
enhances the surface mobility. By analyzing the Hall conductivity, we show that
the enhancement occurs on only one surface. We present evidence that the gating
process is fully reversible (hence consistent with band-bending by the
$E$-field from the anion layer accumulated). In addition to the surface
carriers, the experiment yields the mobility and density of the bulk carriers
in the impurity band. By analyzing the charge accumulation vs. gate voltage, we
also obtain estimates of the depletion width and the areal depletion
capacitance $C_d/A$. The value of $C_d/A$ implies an enhanced electronic
polarizability in the depletion region.
