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Lachlan Black

Schred V2: Simulating n-type substrates and Vfb determination

Using the compiled Schred V2 code and supplied sample input files I am able to successfully model structures with a uniformly doped p-type Si substrate. I assume that to model an n-type substrate I need to change the sign of the parameter “Na”, which specifies the doping concentration. However, when I do this, I receive NaN values in the output “cv.dat” file for the capacitance in strong accumulation. The rest of the C-V curve looks alright. Am I doing something wrong here, or is this a bug in the code? Also, is there an easy way to determine the value of the flatband voltage from the simulation output? At the moment the best option I can see is to interpolate the values of fi_sc vs Vg to zero in the “av_dist.dat” file, but this is not precise.

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    Gokula Kannan

    The SCHRED V2.0 solves the schrodinger wave equation(SWE) only for the electrons, so the code supports only p-type substrate. The previous version of SCHRED I guess supports the hole SWE as well. The boundary conditions are defined at the start of the gate and the end of the substrate, where we force the potential (voltage) – Dirichlet boundary conditions. So, unfortunately there is no way of forcing fi_sc to zero to find the exact value of Vfb. So i suggest you take smaller steps of Vg once you determine the range at which fi_sc approaches zero, and you should get a much accurate plot of fi_Sc Vs Vg. That should be give you a reasonably accurate value of fi_sc. Hope that helps!

    gokul

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      Lachlan Black

      OK, I understand the limitation with Vfb. I thought it might be that way but just wanted to make sure I wasn’t missing something. Regarding the other issue I’m a little confused. If the software only solves the SWE for the electrons, shouldn’t that mean that it supports n-type substrates in accumulation, and p-type in inversion, when electrons are the majority carriers? And wouldn’t it mean that it could not provide a full C-V curve (inversion to accumulation), at least in quantum mechanical mode, for either substrate? Also, if that is the case, it contradicts what is written in the Schred_201_Tutorial.pdf document, which specifically states that V2.0 can solve quantum mechanically for inversion or accumulation in either p- or n-type substrates, with the qualification that it only treats the majority carriers in each case quantum mechanically (p7). Does this in fact apply only to version 1?

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        Gokula Kannan

        Sorry for the delayed response. Yes you are right, I am sorry I missed something. Let me explain. The QM mode begins in a p-type substrate under inversion only, because that is when the potential well is formed, and the SWE is solved for electrons for the potential. Now when solving for the corresponding accumulation on p-type it switches to the regular semi-classical method and solves for the potential for the semi-classical electron/hole density. Similarly when you have n-type substrate, under inversion you would have to solve SWE for holes- which the tool does not support- so it would solve for it semi-classically. And when n-type is under accumulation, it may not form a deep potential well, so the Quantum part may not be kicked in and the solver continues to be on the semi-classical mode to solve for the same. But this again depends on the doping of the substrate, if it is lightly doped, you can have a well deep enough in the n-type accumulation to have some subbands so that SWE can be solved for the wavefunctions, if not (doping is high and it does not form a well) it will do the semi-classical computation for the same. So in a strict sense, it is capable of solving in the QM mode for electrons for both p-type and n-type. Hope that helps!

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