How is this transformation then used to calculate charge excess delta N at band bending with sc of the semiconductor?
Ref:
Semiconductor Electrochemistry Wiley-VCH, 2001-Rüdiger Memming P.88-91
@Artur Braun Sorry, yes, its the poison differential equation. I am very used to the former name. I edited the question already. Thank you.
Dear Sim,
well, there is a simple answer to your question, although it will not help you at this stage at all. The answer namely is that you can solve the problem numerically, if analytic solution is not available.
So, what else can I suggest to you ? Two things :
1. One of the most simple, beautiful and for most purposes fully sufficient is the way Richard Feynman deals with this problem in his lectures "Feynman Lectures on Physics", volume II, chapter 7, page 7-8. Do not get distracted by the fact that it is the colloidal particle in a electrolyte, the same Maxwell equation is valid (one of the Maxwell equations can be rewritten in form of Poisson equation). The main argument for the form of solution comes from thermodynamics. Once that is introduced , the exponential form of the solution for the potential (and the electrical charge density), together with the characteristic screening length (Debye screening length), result almost automatically. Check that.
2. A word of caution. In semiconductors, the reason for the existence of Debye "screening" length is not the charge on the surface of colloidal particle, it is the difference in the electrochemical potential across the interface (for example metal electrode-semiconductor interface). But be aware, there are in fact two "screening lengths", one which is the Debye screening length Ldebye (high density of mobile charges) and the second one (parabolic approximation), where the density of mobile charges is relatively low and one encounters a full depletion of charges near surfaces. This characteristic length is called the depletion length Ldepletion . Ldepl > Ldebye, but both these characteristic lengths have the same origin, namely the difference in the electrochemical potential across the interface.
I hope this might help.
with best regards
Petr
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