I'm currently studying the physics of the PN junction. I went though the derivation of the built-in potential in the PN junction under equilibrium:(adsbygoogle = window.adsbygoogle || []).push({});

Diffusion current density = Drift current density

[tex] D_{p}p\frac{dp}{dx} = EU_ppq [/tex]

where [itex] D_{p} [/itex] : Diffusion coefficient for holes

[itex] p [/itex] = holes density in semiconductor

[itex] E [/itex] = electric field

[itex] U_p [/itex] = hole mobility

[itex] q [/itex] = electron charge

integrating on [itex] x [/itex] we get :

[tex] \int_{junction\ width}^{} D_{p} q \frac{dp}{dx} {d}x = \int_{junction\ width}^{} E U_p p q {d}x \implies [/tex]

[tex] \int_{p_{n0}}^{p_{p0}} D_{p} \frac {1}{p} {d}p = U_p \int_{junction\ width}^{} E {d}x [/tex]

some mathematics and the resulting formula is:

[tex] ln \frac{p_{p0}}{p_{n0}} = \frac{V_{built\text{-}in}}{V_t} \implies [/tex]

[tex] p_{n0} e ^ \frac{V_{built\text{-}in}}{V_t} = p_{p0} [/tex]

Which can be understood as [itex] {V_{built\text{-}in}} [/itex] is the necessary voltage to counter the diffusion from concentrations [itex] p_{p0} [/itex] to [itex] p_{n0} [/itex]

This is called the law of the junction.

Up to this point everything is fine. The problem is:

This law is also used in forward bias. The [itex] V_{built\text{-}in} [/itex] is substituted by the net resultant voltage of the forward bias.

This is wrong, the resultant voltage is NOT countering the diffusion current along the gradient, there is forward current flowing. And the formula we based our derivation on should now be:

[tex] D_{p}p\frac{dp}{dx} = EU_ppq + I_{f} \implies [/tex]

[tex] \int_{junction\ width}^{} D_{p} q \frac{dp}{dx} {d}x = \int_{junction\ width}^{} E U_p p q {d}x \implies + \int_{junction\ width}^{} I_{f}{d}x [/tex]

How is the first formula used as the law of the junction? Am I missing something? Is there some other derivation I'm not aware about ?

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# Law of the PN junction under forward bias. Valid ?

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