Derivation of carrier density formula

[McKendrigo]

Hi there,

Not sure if this is in the best section, but here goes...

I'm trying to establish how a formula from a paper I have read has been derived. The formula is:

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

where n is the carrier density, q is the elementary charge, V is the volume of the semiconductor active area.

From another source (textbook) I have:

$\frac{1}{\tau} = \frac{\partial R}{\partial n}$

where

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

and also the injected current I is related to n as follows:

$I = qVR(n)$

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

 

[McKendrigo]

I am also confused about something from the textbook: it uses equations 2,3 and 4 above to define the relationship between $$\tau$$ and I as being:

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

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