Suppose that I am doing photoconductivity experiment with some intrinsic semiconductor/insulator. When light generates e-h pairs, under basic assumption, they are pulled opposite way by applied electric field. Neglecting diffusion, the continuity equations for electrons is
dn/dt = mun d(nE)/dx + Gn - Rn
and similarly for holes,
dp/dt = -mup d(pE)/dx +Gp -Rp
Where G and R are generation and recombination rates.
For band to band transition, I think the generation rates are equal. However, the recombination rates which depend on the present concentration (n and p which depend on boundary conditions/contacts) seem to be difficult for me. Most people use minority carrier lifetime to determine this as
Rp = (n-n0)/tau for P type
Rn = (p-p0)/\tau for N type.
But in case of intrinsic SC/insulator, none of the carrier is minority. Though they are generated by equal rate, their concentration varies spatially because of applied field.
How can I define the recombination rate in this situation to deal individually with electrons and holes. Are these two recombination rates coupled?
The recombination is $\gamma n p$ in absence of traps or sinks.
$\gamma = 4 \pi D R $ where R is the radius of recombination, D is diffusion coefficient. The model supposes that the carriers recombine when they come together closer than R.
The formula for Recombination which does not assume any minority carrier is
R = B*n*p + Rsrh
where B is a constant for direct recombination, and Rsrh is the Shockley-Read-Hall formula for trap recombination. See equation 2.91 here,
http://www.iue.tuwien.ac.at/phd/entner/node11.html
It is valid for any carrier concentrations n and p in steady state, but it uses the trap concentration, energy, and capture cross section.
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