Hi there,

I'm new here so I'm not sure if this is in the best section, but here goes...

I'm trying to establish how a formula (relating the carrier lifetime in a semiconductor active region with the injected current) from a paper I have read has been derived. The formula is:

$\displaystyle n=\frac{1}{qV} \int_0^I{\tau} dI$ (1)

wherenis the carrier density,qis the elementary charge,Vis the volume of the semiconductor active area.

From another source (textbook) I have:

$\displaystyle \frac{1}{\tau} = \frac{\partial R}{\partial n}$ (2)

where

$\displaystyle R(n) = An + Bn^2 + Cn^3$ (3)

and also the injected currentIis related tonas follows:

$\displaystyle I = qVR(n)$ (4)

I have a complete mental block on how whether I can derive the first equation from the following three - any help would be appreciated!

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Also, in the textbook, it uses equations 2,3 and 4 above to define the relationship between $\displaystyle \tau$ andIas being:

$\displaystyle \frac {1}{\tau^2} = A^2 + \frac{4B}{qV}I}$ (5)

I just keep going round in circles when I try to derive this from equations 2,3 and 4

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Any help in how to derive (1) or (5) would be appreciated! I realise that this is a 'physicsy" type question being asked on a math forum, but I hope someone can give me some pointers. Cheers!