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Pn junction characteristics: How were the lengths and minority carrier concentrations derived from the above information?

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An example says,
Na=10^18cm^-3
Nd=10^16cm^-3
Dp=10cm^2/s
Dn=25cm^2/s
tpo=10^-7s
tno=10^-7s

"We can then calculate the following parameters:"

Lp=1.0x10^-3cm
Ln=1.58X10^-3cm
pno=2.25x10^4cm^-3
npo=2.25x10^2cm^-3

How were the lengths and minority carrier concentrations derived from the above information? And mathematically what is Shottky barrier lowering?

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Please see the attached file for a full derivation

The Diffusion length (Ln or Lp) is related to the minority carrier lifetime (tn or tp) and the diffusion coefficient ...

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