Statistical Thermodynamics and Kinetics of Long-Range Order in Metal-Doped Graphene

Abstract

The statistical-thermodynamics and kinetics models of atomic ordering in a metal-doped graphene (binary two-dimensional planar graphene-type crystal lattice) at 1/8, 1/4, and 1/2 stoichiometries are proposed. Impossibility of (completely) atomic-ordered distribution at 1/6 and 1/3 stoichiometries is ascertained in a graphene-type crystal lattice (in case of a short-range interatomic interactions at least). If a graphene is doped by the short-range interacting metal atoms, the superstructures described only by a one LRO parameter are possible; and if it is doped by the long-range interacting metal atoms, the new superstructures with the two or three LRO parameters may appear as well. If stoichiometry is 1/4, the structure has a one long-range order (LRO) parameter is more thermodynamically favorable than those have one or two LRO parameters. It is established that kinetics curves of LRO parameters can be non-monotonic for structures where there are two or three LRO parameters (because graphene-type lattice contains two sublattices, and mixing energy is different for each of them). It is shown that the most ordered is structure with equal atomic fractions of carbon and metal atoms, while the least one is structure with a maximal difference of carbon and metal atoms. Kinetics results confirm statistical-thermodynamic ones: firstly, equilibrium values of LRO parameter coincide within the framework of both models, secondly, equilibrium (and instantaneous) value of LRO parameter in a nonstoichiometric binary graphene-type structure (where atomic fraction of a doping component deviates from the stoichiometry to the side of the higher concentrations) may be higher than it is in a stoichiometric one. The dominance of the same physical mechanisms of atomic ordering in both mixed nanosystems and macrosystems is assumed.