| Abstract Scope |
We present a simple thermodynamic model to explain the doping of nanoparticles through minimization of Gibbs free energy. Over the last decade, nanoparticle-related research has become increasingly prominent, and researchers have found that the properties of nanoparticles differ significantly from that of bulk materials. Accordingly, there has been considerable work done on the doping of nanoparticles and nanostructures, with varying levels of success. While it is possible to dope nanoparticles, as their size decreases, the ability to incorporate impurities into the core of the structure becomes increasingly difficult [1]. There are two models commonly used to explain this phenomenon, kinetics and thermodynamics, the former of which requires many assumptions and complex calculations. We propose that thermodynamics on its own is generally sufficient to explain the doping properties of nanoparticles. Gibbs free energy is minimized, with both the enthalpy and entropy terms playing prominent roles on such small scales. We are primarily concerned with electrically-active dopants and therefore differentiate between surface and core atoms, calculating differential Gibbs free energy for each as a function of nanoparticle size. We define the enthalpic term as a difference (generally increase) in bond energy for dopants and consider configurational entropy for both the surface and core. We show that the free energy of surface impurities in very small nanoparticles is lower than that of core impurities; as particle size decreases, competition between the enthalpic and entropic terms becomes more prominent, surface doping therefore occurs preferentially. A critical size for core doping is identified, below which impurity incorporation in the core is thermodynamically limited. In all cases, core impurity concentration is reduced as particle size decreases. However, we confirm that larger-than-bulk impurity concentrations (which have been reported experimentally as increased alloying [2]) are possible, but are confined to surface doping. We also present data which confirms that this simple model is capable of handling real world systems and is in good agreement with experimental results. This model can easily be extended to one- and two-dimensional structures, such as nanowires and quantum wells, and can be applied to material systems with different particle shapes and coordinations. It is also not restricted to bond energy as the dominant enthalpic term, but can be modified to model any other first order effect such as strain. |