Electrons at Crystal Surfaces

Abstract

Abstract Since crystals in their natural state are finite, i.e., limited by surfaces, the question arises as to how the energy spectrum of an electron propagating in a finite crystal differs from that in an infinite one. The partial answer is provided by using the Born-von K árm án (1912, 1913) boundary condition (2.138), which shows that, if a finite crystal of N atoms is cut from an infinite one, then the previous continuous allowed band becomes a sequence of N discrete levels (2.141). Clearly, this is not a big difference, if the crystal (i.e., N) is very large. However, there is another important consequence of taking the crystal size into account, namely, that the presence of the surface produces discrete energy levels, which lie in the FEG regions that are inaccessible to electrons in infinite crystals. Tamm (1932) was the first to demonstrate the existence of these surface states by terminating the KP potential as a means of representing the surface. Because of the importance of Tamm’s work in establishing the new idea of surface states, we shall now briefly discuss his findings.