Types of surface wavefunctions

As we have shown in Chapters 2 and 3, under the normal operating conditions of STM, the tunneling current can be calculated from the wavefunctions a few A from the outermost nuclei of the tip and the sample. The wavefunctions at the surfaces of solids, rather than the wavefunctions in the bulk, contribute to the tunneling current. In this chapter, we will discuss the general properties of the wavefunctions at surfaces. This is to fill the gap between standard solid-state physics textbooks such as Kittel (1986) and Ashcroft and Mermin (1985), which have too little information, and monographs as well as journal articles, which are too much to read. For more details, the book of Zangwill (1988) is helpful. 

At crystalline surfaces, there are three types of wavefunctions as shown in Fig. 4.1. (1) The Bloch states are terminated by the surface, which become evanescent into the vacuum but remain periodic inside the bulk. (2) New states created at the surfaces in the energy gaps of bulk states, which decay both into the vacuum and into the bulk, the so-called surface states. (3) Bloch states in the bulk can combine with surface states to form surface re.sonances, which have a large amplitude near the surface and a small amplitude in the bulk as a Bloch wave. 

In the bulk of a perfect crystal, the electronic states follow the Bloch theorem, which have the form 

The vacuum tails of Bloch waves are relatively straightforward to understand. Comparing with other experimental methods, STM is more sensitive to the surface states, both the sample and the tip. Therefore, we will spend more time to explain the concept of surface states, from a theoretical point of view and an experimental point of view. 

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