I What should be continuous at the interface of two materials?

[Dor]

At the interface between:
1) conductor/conductor
2) conductor/semiconductor (or dielectric)
3) semiconductor/semiconductor (or dielectric/dielectric)

What quantity should be continuous?
Is it the electrochemical potential, only the chemical potential or is it the electric potential?

Since they are all related by V_ec=V_c+V_e, if two of them are continuous then all of them are continuous, but I think this situations cannot be true.
However, I know that the Fermi energy is continuous (which I think is the chemical potential but I'm not sure since there's a lot of misleading information). Also the electric potential should be continuous, otherwise the electric field (defined by E=-grad(V_e)) would be infinity.

The question refers both for equilibrium and for an applied voltage (steady currents).

I couldn't find any information on this matter. All books on semiconductors or electrostatic/dynamic does not mention it and does not get to much deep into such physics.

 

[.Scott]

What is the context of this question?
The word "should" implies that there is an objective.

Of out context, many answers are possible. For example, the temperature should be continuous.
If they are talking about flashlight components, then voltage should be continuous - because all junctions would be ohmic.

 

[Dor]

l'm trying to understand the physics behind the interfaces of to materials, especially between semiconductors and metal (or poor conductor) electrodes. Both at equilibrium and at applied voltage

 

[marcusl]

I haven't studied such things for many decades, and I'm not near my books at work (and was never much of an expert in any case) so I hope my answers are correct--you should check them. The chemical potential μ is continuous across metal-metal, semi-semi and metal-semi junctions. The Fermi energy is related and is often confused for μ as you have noticed, but it is the chemical potential that you want to look at. Am less sure about electric potential. One can build up surface charge on a dielectric-metal interface, for example, which introduces discontinuities in E field (see any discussion of boundary conditions for Maxwell's equations), but that should still lead to a continuous potential.

You might search out the book by Henisch, I think it's called Semiconductor Contacts. It specializes on the topic of semiconductor interfaces under various biases and with and without current injection. There is also a quite old but very clear book on metal/insulator systems by Lampert, something like Current Injection in Solids.

 

[Dor]

Thanks for the answer. I will look for the books you suggested.
However, there is a problem with Maxwell's boundaries. If the electric potential was continuous, then the voltage drop on a diode for example was that of the source without any consideration of the built-in potential. Than obviously it is not continuous. So how do I describ these interfacess from fundamental prinsiples (like Maxwell's equation for example)?
And how does the electrochemical potential renters this story?

 

[marcusl]

The potential for even abrupt pn junctions is smooth and continuous. This is covered in all semiconductor texts. Do you have Sze, for example?

 

[Dor]

Sure, but is it true also for metal/semiconductor interface?

 

[Useful nucleus]

I did not study metal/semiconductor carefully, but for semiconductor/semiconductor:

1- The electric potential is continuous
2- The electric field is discontinuous
3- The charge density is discontinuous
4- The electric displacement field is continuous
5- The electron chemical potential (also known is Fermi level) is continous
6- The species chemical potentials (e.g. Silicon, oxygen, ...) is continuous and in fact constant under equilbrium
7- Band edges are discontinuous.

 

[DrDu]

At the interface between:
1) conductor/conductor
2) conductor/semiconductor (or dielectric)
3) semiconductor/semiconductor (or dielectric/dielectric)

What quantity should be continuous?
Is it the electrochemical potential, only the chemical potential or is it the electric potential?

Well, in equilibrium, the electrochemical potential should be constant over the sample. On the surface the usual boundary conditions hold for D and E. However, the polarisation typically leads to a generation of a charge layer at the surface, which may make it appear as if the electric fields are discontinuous on a macroscopic scale.