Hello everyone. I am having difficulty in making up a differential equation. Please help me to correctly draw up a differential equation and display graphs, as in this document (http://www.dspmuranchi.ac.in/pdf/Blog/Lec10.pdf) on page 6.

The first difficulty: I don’t know how to correctly write the boundary conditions for pnet on page 7.

I will be very grateful if you can help me.
Built-in potential_3.sm (31 КиБ) скачан 33 раз(а).

Hi. You have "true" boundary conditions in 2 or 3D, but in the one dimensional case you can integrate rho between -inf and +infinity. Given that the field is zero outside the interval (-Wp,Wn) you can set the bounds as (-Wp,0) and (0,Wn) and integrate rho for each interval, which is the integral of a constant.

Notice that you can use units in SMath, which is a very powerful tool detecting issues in practical calculus. For instance, here I don't know how to handle ni, which have units of cm^-3, and I need to introduce that factor as a correction for the bias potential, but using meters, not centimeters. Check that, please.

Built-in potential_3.sm (48 КиБ) скачан 39 раз(а).
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Best regards.
Alvaro.

Wrote

Hi. You have "true" boundary conditions in 2 or 3D, but in the one dimensional case you can integrate rho between -inf and +infinity. Given that the field is zero outside the interval (-Wp,Wn) you can set the bounds as (-Wp,0) and (0,Wn) and integrate rho for each interval, which is the integral of a constant.

Notice that you can use units in SMath, which is a very powerful tool detecting issues in practical calculus. For instance, here I don't know how to handle ni, which have units of cm^-3, and I need to introduce that factor as a correction for the bias potential, but using meters, not centimeters. Check that, please.

Built-in potential_3.sm (48 КиБ) скачан 39 раз(а).
Built-in potential_3.pdf (170 КиБ) скачан 35 раз(а).

Best regards.
Alvaro.

Thank you very much, now I almost all understand. Also thanks for the unit hint. Instead of centimeters, there should be nanometers. But SmutnStudio gives Ф/m, but we need nm. Interestingly, in the program you can somehow change the meters to nanometers. In the last formula (potential), I removed the minus sign and everything became as it should.

The data I took from this document. (https://lampx.tugraz.at/~hadley/psd/problems/pn/S.pdf)

Thank you very much for your help, I am very grateful to you.

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Wrote

...

The data I took from this document. (https://lampx.tugraz.at/~hadley/psd/problems/pn/S.pdf)
...
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Ah, yes: Na, Nd have units, are number of charges by cm^3, I forgot that. Now there are not any mysterious unit factor in the formulation, and have the same answers than in the manual calculus from the pdf.

Built-in potential_3_with_Na_Nd_units.sm (51 КиБ) скачан 32 раз(а).
Built-in potential_3_with_Na_Nd_units.pdf (183 КиБ) скачан 39 раз(а).

Best regards.
Alvaro.