Probabilities for dopant pair-state formation in a nanocrystal: Simulations and theory

For certain dopants, luminescence measurements allow one to distinguish between single-ion and pair-state dopant emission in a (semiconductor) host. In a bulk crystal the concentration of each of these dopant states can be calculated from the dopant fraction present in the material and is found to correlate with luminescence measurements. However, for a nanocrystalline host lattice, these concentrations cannot be calculated due to the difference in coordination numbers for ions at the surface (a substantial fraction in nanocrystals) and in the bulk. Here simulations of dopant pair-state distributions are presented for a zinc-blende nanocrystal. The probability of finding at least one pair state in the nanocrystal and the percentage of dopants forming part of a pair state were calculated on the basis of a statistical average of $1\times {10}^5$ simulations for the same crystal size and dopant concentration. Furthermore, the distribution of nanocrystal lattice positions over the surface and the bulk of the crystal are computed from the simulations and found to agree well with a first-order theory. Finally, a closed-form approximation of the probabilities (valid in any crystal lattice) and a rigorous upper bound for the error in the approximation are discussed.