  One prominent structural feature of ionic liquids near surfaces is formation
of alternating layers of anions and cations. However, how this layering
responds to applied potential is poorly understood. We focus on the structure
of 1-butyl-1-methylpyrrolidinium tris(pentafluoroethyl) trifluorophosphate
(BMPY-FAP) near the surface of a strontium titanate (SrTiO$_3$) electric
double-layer transistor. Using x-ray reflectivity, we show that at positive
bias, the individual layers in the ionic liquid double layer thicken and the
layering persists further away from the interface. We model the reflectivity
using a modified distorted crystal model with alternating cation and anion
layers, which allows us to extract the charge density and the potential near
the surface. We find that the charge density is strongly oscillatory with and
without applied potential, and that with applied gate bias of 4.5 V the first
two layers become significantly more cation rich than at zero bias,
accumulating about $2.5 \times 10^{13}$ cm$^{-2}$ excess charge density.
