By Paul Roman
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Additional resources for Introduction to Quantum Field Theory
As the nucleation continues, the remaining concentration of precursors decreases, and the temperature drops. Fairly rapidly, a threshold is crossed beyond which nucleation of new nanocrystals stops. The temperature is then held at some lower value (around 200 ı C), which allows the nuclei to continue to grow. Once the nanocrystals have reached the desired size, the temperature is lowered further, and the growth is arrested. As long as the time during which nucleation occurs is short compared to the growth time, the ensemble of nanocrystals will be fairly uniform in size.
Neither of these approaches is ideal. It is unclear whether the assumptions of a bulk material still apply to a nanocrystal, which may just contain a few hundred atoms. On the other hand even with just a few hundred atoms, accurate first-principles calculations become difficult. Nevertheless, both approaches have yielded results that are consistent with experimental evidence. Even in structures containing layers just a few atoms thick, the effective mass approximation has proved fairly successful in predicting the energy levels.
There are two approaches: to treat the nanocrystal as a small piece of bulk material using the effective mass approximation, essentially solving the Schrödinger equation with the appropriate boundary conditions, or to treat it as a collection of individual atoms, finding the energy level spectrum using, for example, density functional theory. Neither of these approaches is ideal. It is unclear whether the assumptions of a bulk material still apply to a nanocrystal, which may just contain a few hundred atoms.
Introduction to Quantum Field Theory by Paul Roman