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I have a problem i'm trying to work out and I would like some advice please.

The intrinsic carrier equation for a semiconductor is the following: $$n_i = BT^{3/2}e^{\bigl(\frac{-E_g}{2KT}\bigr)}$$

my question is, is there a way to solve for $T$? $$\frac{n_i}{B} = T^{3/2}e^{\bigl(\frac{-E_g}{2KT}\bigr)}$$

I would like to take the log natural at some point, but I think you can see what I perceive as my dilemma as T appears twice in this equation.

Bucephalus
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  • Only by numerics.For example:https://brilliant.org/wiki/newton-raphson-method/ – Mariusz Iwaniuk Aug 22 '20 at 13:32
  • That's good thanks @MariuszIwaniuk I have since found another formula that I can use to get around trying to use this formula to obtain the temperature $T$, thanks for your help. – Bucephalus Aug 22 '20 at 13:41

1 Answers1

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Make this manipulation $$ \left(\frac{n_i}{B}\right)^{2/3} = Te^{\bigl(\frac{-E_g}{3KT}\bigr)}\\ \left(\frac{B}{n_i}\right)^{2/3} = \frac{1}{T}e^{\bigl(\frac{E_g}{3KT}\bigr)}\\ \frac{E_g}{3K}\left(\frac{B}{n_i}\right)^{2/3} = \frac{E_g}{3KT}e^{\bigl(\frac{E_g}{3KT}\bigr)} $$ then use Lambert W function $$ W\left[\frac{E_g}{3K}\left(\frac{B}{n_i}\right)^{2/3}\right] = \frac{E_g}{3KT}\\ T=\frac{E_g}{3K}W\left[\frac{E_g}{3K}\left(\frac{B}{n_i}\right)^{2/3}\right]^{-1} $$

Vincenzo Tibullo
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