Why does a higher gate voltage increase electron density + mobility in a 2DEG?

Question

I have a question concerning a 2DEG created in a n-AlGaAs / GaAs heterostructure: If the n-doped AlGaAs is on the left side and the GaAs on the right side, the following band structure will appear: Applying a positive gate voltate now results in a higher electron density and increased mobility inside the 2DEG. Unfortunately, I am not sure whether I understood well why this is happening. My explanation for these phenomena is:

A positive gate voltage causes the electron levels to drop, but as the Fermi level stays constant, there are more electrons inside the well --> charge density is increased. As scattering with the dopants at the left side limits the mobility, applying a positive gate voltate results in an increased mobility, because the "new" electrons inside the 2DEG are shifted to the right side (see band structure), thus further away from the dopants.

Are these explanations correct?

Answer 1

The expected effects at elementary level would be that the positive gate potential would attract the electrons into the well, and also that it would counteract the built -in field orthogonal to the surface. Orthogonal field reduces mobility, so reducing it increases mobility.
This elementary view may not, of course, be correct.