Bending of energy bands To me energy band diagram is a one dimensional plot which shows the differences between the energy bands of a material.  In case of an insulator  an energy band diagram shows a huge gap between the valence band and the conduction band , for a semiconductor a small gap between the conduction and valence bands and for a conductor the minimum permissible energy for existence of an electron in conduction band is less than the maximum amount of energy that a valence band electron can attain. However, as I was going through a CMOS technology text ,  it mentioned a case of bending of energy bands when the silicon dioxide layer , the aluminium of gate and the substrate of a CMOS comes in contact. Ii wonder what exactly this means in regard to the difference between the minima of conduction band and the maxima of valence band . Can anyone explain this query of mine ? That book also mentions the Fermi level of the bulk of substrate being larger than the fermi level at the interface of substrate and the silicon dioxide layer. I will be glad someone provides a reason for this fermi level difference.
 A: Bending of the band diagram is caused by the volume charge of the depletion layer at the interface. In this region, the difference between conduction and valence band edges remains the same. However, if the electric filed of this charge and, consequently, band bending are strong enough to provide quantum confinement of electrons or holes (forming 2D electron gas), the gap between electron and holes energies can be changed due to quantization of the energy band which leads to its splitting into sub-bands.
The Fermi level should be constant for the structure in thermodynamical equilibrium. However, it can be positional-dependent if the electrical current does not equal zero but the system is still in quasi-equilibrium state. In this case positional-dependence of the Fermi level follows the spatial distribution of non-equilibrium charge carriers.
Useful trick is considering the conduction band edge as potential energy of a single electron (in sense of quasi-particle), as well as the valence band edge as potential energy of a single hole.
