  Modulation doped GaAs-AlGaAs quantum well based structures are usually used
to achieve very high mobility 2-dimensional electron (or hole) gases. Usually
high mobilities ($>10^{7}{\rm{cm}^{2}\rm{V}^{-1}\rm{s}^{-1}}$) are achieved at
high densities. A loss of linear gateability is often associated with the
highest mobilites, on account of a some residual hopping or parallel conduction
in the doped regions. We have developed a method of using fully undoped
GaAs-AlGaAs quantum wells, where densities
$\approx{6\times10^{11}\rm{cm}^{-2}}$ can be achieved while maintaining fully
linear and non-hysteretic gateability. We use these devices to understand the
possible mobility limiting mechanisms at very high densities.
