When you are looking at a device model like this, it's a good idea to always repeat it 3x3 in A and B just to check how it looks. If you do that, you will find that the gate is repeated to, since there is always also a geometric periodic boundary condition in both A and B.
So the only chance this geometry is going to work is if you make a double-gate. In that case you can use periodic boundary conditions also for the Poisson solver.