According to Wikipedia, the saturation current in a diode is given by the following formula:
$$I_S = qAn_i^2\left(\frac{1}{N_D}\sqrt{\frac{D_p}{\tau_p}} + \frac{1}{N_A}\sqrt{\frac{D_n}{\tau_n}}\right)$$ If my understanding is correct, saturation current is caused by minority carriers reaching the depletion region, getting accelerated by the electric field inside this region, and crossing over to the side where they are majority carriers.
My question is then: how do the doping level, the diffusion coefficient, and the carrier lifetime influence the saturation current?
I would expect that a large doping level will increase the probability that minority carriers recombine before reaching the depletion region, so I understand the inverse proportionality. A large diffusion coefficient means that the minority carriers can diffuse fast and might have a large chance of reaching the depletion region before recombination occurs. However, I specifically don't understand why a short carrier lifetime implies a larger saturation current.