As we already know that for an intrinsic semiconductor, the relation between recombination rate (r) and intrinsic carrier concentration (n) is given as the following:
$$r = a \cdot n^2$$
where a: recombination coefficient.
My question is will recombination rate (r) increase or decrease or remain same with doping.
Firstly think about intrinsic semiconductor. When thermal equilibrium satisfied excess holes and electrons are equal. $$n=p$$
Think about electrons at the conduction band. They want to recombine with holes to decreases their energy states.
When you add donors to semiconductor now there are more electrons want to recombine with fixed holes. Therefore recombination rate must be larger.
All figures are taken Middle East Technical University EE212 Lecture Notes of Prof Dr Cengiz Beşikçi
