Derivation of relationship between carrier density and current
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:
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:
and also the injected current I is related to n as follows:
I have a complete mental block on how whether I can derive the first equation from the following three - any help would be appreciated!
Also, in the textbook, it uses equations 2,3 and 4 above to define the relationship between and I as being:
I just keep going round in circles when I try to derive this from equations 2,3 and 4 :(
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!