Semiconductor Statistics

Abstract

The periodic potential distribution of an electron in a crystal shown in Fig.2.5 involves N discrete levels if the crystal contains N atoms as we have seen in Fig. 2.9. A discussion of these levels can be confined to the first Brillouin zone. We have seen in the last chapter that due to the crystal periodicity the electron wave functions, which in one dimension are ψ(x) = u(x) exp(ikx), have also to be periodic (“Bloch functions”). Hence from

$$$$

(3.1)

and

$$$$

(3.2)

we obtain

$$$$

(3.3)

or

$$$$

(3.4)

where a is the lattice constant. We notice that Eq.(3.1) is actually valid for a ring-shaped chain which means that we neglect surface states (see Chap. 14a). Since for the first Brillouin zone k has values between −π/a and +π/a, we find that the integer n is limited to the range between −N/2 and +N/2. In Fig.3.1 the discrete levels are given for a “crystal” consisting of N = 8 atoms.