  We investigate the controllability of quantum electrons trapped in a
two-dimensional device, typically a MOS field-effect transistor. The problem is
modeled by the Schr\"odinger equation in a bounded domain coupled to the
Poisson equation for the electrical potential. The controller acts on the
system through the boundary condition on the potential, on a part of the
boundary modeling the gate. We prove that, generically with respect to the
shape of the domain and boundary conditions on the gate, the device is
controllable. We also consider control properties of a more realistic nonlinear
version of the device, taking into account the self-consistent electrostatic
Poisson potential.
