A simple thermodynamic model for the doping and alloying of nanoparticles

Impurity incorporation into nanoparticles is modeled using thermodynamics. For small particles, entropically driven impurity incorporation is reduced, rendering doping difficult. We show that the free energy of surface impurities in small nanoparticles is lower than core impurities, surface doping therefore occurs preferentially. A critical size for core doping is identified, below which it is energetically unfavorable. In all cases, core impurity concentration is reduced as particle size decreases. We show larger than bulk impurity concentrations are possible, corresponding to increased alloying.