  We develop an efficient back gate for silicon-on-insulator (SOI) devices
operating at cryogenic temperatures, and measure the quadratic hyperfine Stark
shift parameter of arsenic donors in isotopically purified $^{28}$Si-SOI layers
using such structures. The back gate is implemented using MeV ion implantation
through the SOI layer forming a metallic electrode in the handle wafer,
enabling large and uniform electric fields up to $\sim$ 2 V/$\mu$m to be
applied across the SOI layer. Utilizing this structure we measure the Stark
shift parameters of arsenic donors embedded in the $^{28}$Si SOI layer and find
a contact hyperfine Stark parameter of $\eta_a=-1.9\pm0.2\times10^{-3}
\mu$m$^2$/V$^2$. We also demonstrate electric-field driven dopant ionization in
the SOI device layer, measured by electron spin resonance.
