formula for the electric charge stored accros a Schottky junction

Most formulae give incremental capacitance C(V) as a function of voltage V accross the junction, but that a capacitance in general is a derived quantity, whereas the physical phenomenon involves V and charge Q... Some textbokks give a formula for Q, however:

(1) Q(V) = Cjo Vj ( 1 - V / Vj )^(1-M) / (M-1) , where V>0 in forward bias, Cjo is the zero-bias incremental capacitance, and Vj is the diffusion potential (see for example the books of Stephen Maas 1. on microwave mixers 2. and on nonlinear microwave and RF circuits)

Formula (1) which states that Q(0)<0 is wrong, and MathWorks got it right*:

(2) Q(V) = Cjo Vj ( 1 - V / Vj )^(1-M) / (M-1) - Cjo Vj / (M-1)

I am looking for a published reference (a paper or a textbook) which quotes (2)

Please help (-:

* SPICE Diode - SPICE-compatible diode - MATLAB (mathworks.com)

Answer 1

[Umar]

Hi @Jaisson, I've gone through the SPICE Diode documentation and looked into the junction charge formula issue you raised. You're absolutely right about the normalization problem. The formula that appears in some textbooks, including Maas, gives Q(V) = Cjo Vj (1 - V/Vj)^(1-M) / (M-1), which creates a real problem at zero bias. Since M is typically between 0.3 and 0.5, the term (M-1) is negative, so at V=0 this gives Q(0) = Cjo Vj / (M-1), which is negative. That's physically nonsensical because there should be zero stored charge at zero bias. MathWorks corrected this by adding the normalization constant, giving Q(V) = Cjo Vj [(1 - V/Vj)^(1-M) - 1] / (M-1). This ensures Q(0) = 0, which makes physical sense. The charge formulation has to satisfy the boundary condition that at zero applied voltage, there's no excess charge stored. As for finding a published reference that explicitly shows equation (2), that's proving tricky. Most textbooks tend to present the incremental capacitance C(V) rather than the charge Q(V), and when they do give Q(V), they often omit or gloss over the normalization constant. I searched through references on SPICE modeling, semiconductor device physics, and nonlinear circuit analysis, but haven't found a textbook that explicitly writes out the normalized form. Here are a few suggestions on where you might find it: The original Berkeley SPICE source code and documentation might have it, particularly the SPICE2 and SPICE3 technical memos from Vladimirescu. These often include the derivations and references to the original papers. The source code comments sometimes cite the foundational work. Since you mentioned Maas gives the unnormalized version, it might be worth checking if there's an erratum or if later editions corrected this. He's still active in the field, so reaching out to him directly could clarify whether the normalization was intentionally omitted or if it's an oversight. You could also look at papers on SPICE diode model parameter extraction from IEEE Transactions on Electron Devices. Papers dealing with extracting CJO, VJ, and M from measurements sometimes show the complete charge equation because they need the proper formulation for fitting. The other option is to treat this as a straightforward derivation from first principles. Starting from C(V) = dQ/dV, integrating the standard depletion capacitance formula C(V) = Cjo / (1 - V/Vj)^M gives you the integral plus an arbitrary constant. The boundary condition Q(0) = 0 then determines that constant to be exactly -Cjo Vj / (M-1). If you write this up with that justification, it stands on its own even without finding it explicitly stated in a textbook. The reason this normalization matters isn't just mathematical correctness. In circuit simulation, having the wrong charge formulation can cause convergence issues and give incorrect transient behavior. SPICE implementations use the normalized form specifically to maintain numerical stability and physical consistency. Hope this helps!

Answer 2

[jaisson]

Thank you for looking into my query, Umar. I could find any information in the literature, apart from MathWorks' good tip...

I overlooked the potential convergence issue that you rightly pointed out, because I have been looking at an analytic solution to the network problem at hand. From the point of you of an iterative solver such as Spice, it makes no sense indeed to formulate such a problem in terms of incremental capacitances rather than charges. My suspicion is that the developers of the SPICE engine actually implemented charge equations, not capacitance equations, but:

1. They wrote their user manual with industrial data in mind — namely capacitances, not charges.

2. Because analytical solutions are rarely pursued nowadays, many obvious issues are overlooked. As a result, most academics take the easier route (not the most didactic one!) and use in their lectures and publications what diode manufacturers can measure more easily, namely the incremental capacitance.

Regardless, the obvious way to circumvent the illiteracy of the literates is to integrate the expression for the said capacitance while enforcing zero charge at zero potential, as you suggested,

Dr. D. Jaisson