Maxwell's/Gauss equations define the electric fields and current density at the interface of metal/dielectric. Since semiconductor is a dielectric material, I would expect that these boundaries together with the continuity of the potential will satisfy the boundaries at metal/semiconductor interface. However, I've never seen these boundaries being applied to the semiconductor equations.
Instead, the boundaries are defined as ohmic or schottky contacts, which has no connection to the metal parameters (such as conductivity or dielectric constant).
Moreover, if we do use the ohmic/schottky conditions, it follows that there is a discontinuity of the displasment or the normal component of the current density. It means, surface charge or surface current density is presented, respectively, and as I understand, both shouldn't exist in an interface between non-perfect conductor and dielectric.
Furthermore, the boundary condition for the potential is the built-in potential minus the applied voltage, whereas the potential at the conductor is approximately the applied voltage (approximately is due to a finite conductivity). This causes also discontinuity of the potential at the interface.
Can someone explain the physical justification using ohmic or schottky contacts instead of Maxwell's boundary conditions?
Where is the gap?
There is zelion books that describe contact phenomena. Basically, charge carriers in different materials have different energy. Therefore,, at the contact, some of electrons from one material move to the other and create a double layer that balances the electrochemical potentials aka Fermi levels in both materials.. That layer creates a discontinuity in the electric potential. if the region of the double layer is thin, that is atomic thick, carriers can tunnel through it and the contact is ohmic. But in semiconductors , the region of the space charge is thick. That creates the Schottky barrier or p-n-junction if both materials are semiconductors. That's how all electronic works.
This is only a sketch. Books will be more useful.
In addition, there are no perfect semiconductor surfaces outside ultra-high vacuum chambers. They are either oxidised or covered by hydro-cabons (=dirty). If you clean them chemically and then heat them up, they will stay clean only for a very brief time.
If I understand correctly, the electric potential is continuous but changes on a very small scale. The potential of the double layer at the metal side is Va (applied voltage) and at the semiconductor side it is Vbi-Va. Can't this double layer together with the rest of the semiconductor be described dy the continuity equation, using the continuity of the potential as the boundary condition?
In my opinion, the main reason it is that only the semiconductor layer is simulated. Thus, you fix the boundary conditions (BCs) in the own semiconductor. For the BCs, it is easier to work with the magnitudes of physical parameters as charge concentrations, which you can control, than with the magnitude of the electric field. Only by means of the difference of the Fermi Level of the semiconductor and the work function of the metal, you can fix the type of contact (ohmic or Schottky).
Sure, off course it is prefferrble to simulate the device with accounting only the semiconductor layer, but in the end, the fundemental maxwell's boundary conditions must be satisfied (potential continuity, and continuity and discontinuity of electric fields). But then, when using both schottky or ohmic contact alongside with the Maxwell's B.C.,.it seems that there are to many conditions
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