From the bulk to surfaces and clusters

We finally arrive at a point where we can relate what we have learned to the principal theme of the text. Recall that small clusters essentially consist of surface whereas band theory applies to situations where the number of surface atoms is so small with respect to atoms in the bulk they can be ignored. In the Al77 cluster we have a shell-like structure of All A1i2 A144 (A1R)2o or a ratio of 20/57 = 0.35 surface to bulk. In addition, each of the 20 surface atoms is coordinated to 

In fact, the surface of a transition metal is highly delocalized. Let s take a look at it. Focus on the d band where the Fermi level is situated regardless of the metal. For the sake of simplicity, we do not consider the overlap and mixing with the s 

Now that the qualitative aspects of surfaces are understood, we may use these ideas to address our problem of the connection between cluster MOs and bulk COs. Consider a crystalline him of a metal deposited on a planar support. Now the bulk, which constitutes the middle part of the him, can no longer be considered as inhnite. Surface states will dominate the total density of states if the thickness of the him is not large. In fact, in a thin him consisting of a few layers of atoms the bulk no longer exists and the total DOS is composed of surface states. The DOS should be narrower 

In this chapter the electronic-structure model developed for extended main-group systems describes properties that do not depend on the surface states. It is simply 

Work out the band structure for the px, py and pz functions of a two-dimensional square array of B atoms in the xy plane separated by a distance d. This might be a model for a material made up of weakly interacting sheets or of a very thin film or surface layer of a solid. Hint carry out the operations used to generate the band structure in Exercise 6.9 with p functions rather than s functions. 

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