Imagine there is a cubic 3D domain of GaAs / AlGaAs heterostructure with 2DEG underneath of surface. Clearly for x and y (surface dimensions) that goes to infinite using plane wave leads to 1D solution of Schrödinger Poisson equation only in z direction. However, it is almost impossible to implement 3D numerical solution for a real cm or even micro meter domain. Secondly if a negative biased gate with width of 'd' from x = 0 to x = d (infinite on y direction) implant on the surface then it will push electrons away on x direction. The real question appears here. How far the electrons pushed back? Due to small width of gates the wave function in x direction still can be express with wave plan. The accessible position for wave function in x direction is limited depend on bias and more likely it is possible to still keep the plan wave interpretation for wave function in x direction. Can any body suggest me how to calculate the the depletion distance via shelf- consistent solutions in this case?

Maybe some sort of perturbation method or 2D solution on x and z direction. Finally what would be the solution if the gate is not rectangular shape or it has different widths for different y?