| Recent research found that in epitaxy experiment, Pb atoms can grow into highly self-organized islands. Such islands have a set of stable heights which are called "magic thickness". Similar phenomena were also found in some other metal/semiconductor systems. To study the driving force of the formation for Pb island grown on Si surface is an important research subject in this area.When the de Broglie wavelength of electrons inside a nanoscale metal particle is comparable to its size, the quantum size effects (QSE) appear. It is thought that the growing mechanism for Pb island must be related to QSE. However, how can QSE dominate the growing process is still unclear.In the first chapter, the stable heights of islands grown on Pb/Si(111)-( 7×7 ) and surfaces were studied. Following the previouselectronic growth model, a new effect - degeneracy pressure of electrons is taken into account. By considering this new effect, the theoretical results successfully predict the most stable heights for the Pb islands grown on these two different surfaces. Combining the experimental information on the Fermi energy of electrons inside Pb island on Pb/Si(111)-(7x7) and the contraction of its height, the new effect can illustrate all the "magic thickness" of Pb island of this phase, which indicates its significance in determining the preferred heights of metal islandIn second chapter, the size of Pb island was studied. Experiments found that on different Pb/Si phases, the Pb islands have different sizes. In the island growing process, its sizes remain nearly unchanged. Because island's size is about ten times as large as that of its height. Previous theoris treat island as a 2D film, which is not reasonable. In fact, because the island's size is important to determine the energy levels of electrons inside island, it should affect the stable heights indirectly according to electronic growth model. Using elasticity mechanics, the elastic energy of the interface between island and substratecan be calculated, which is positive and makes the system energy increase. While another term - capacitance effect, decreases the system's energy. These two terms compete and decide the size of island. Due to the lack of values for some parameters in our formula, we cannot calculate the island size directly. But this formula can be used to understand other characteristics of Pb island and the simple estimation of island's size is consistent with the experimental observation. At the end of this chapter, an important interaction between islands was proposed, which would change the opinion that the distribution of island is completely stochastic.In epitaxy experiments, the substrate is always thought to be smooth and uniform. However, the real crystal surface is far from being like that. The vicinal surfaces of a crystal consist of step structure, which directly affects the chemical and physical process on the surface. Theoretically, the step structures can be viewed as the realization of the special statistic model at low dimension. In third chapter, a model of step structure was introduced, and the fixed points of its Hamiltonian renormalization equations can be used to classify the different Si surfaces. Then the formation energy of step was calculated using its spontaneous formation mechanism. At last, due to the similarity of the alignment of atoms on a surface and that of a polymer, SSH model can be applied. Not only the step formation energy can be worked out, but two interactions can be obtained.
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