Rate equation is depending on the discrete energy level position and chemical potential of the quantum dot. A good seminal paper for introduction of graphene quantum dot is found here : Silvestrov et al. Quantum Dots in Graphene PRL 98, 016802 (2007). a microscopic derivation of rate equations for quantum transport is  found in S. A. Gurvitz and Ya. S. Prager, Phys. Rev. B 53, 15932 (1996). For the specific case of graphene, the recent book https://link.springer.com/book/10.1007%2F978-3-662-44611-9 is very helpful. Hope this helps, Vincent 

thank you sir Vincent Bouchiat,

Dear sir :

can you help me about "rate equation of graphene quantum dot laser"

best regard