Semiconductor Statistics

Abstract

The periodic potential distribution of an electron in a crystal shown in Fig. 2.4 involves N discrete levels if the crystal contains N atoms, as we have seen in Fig. 2.8. A discussion of these levels can be confined to the first Brillouin zone. We saw in the last chapter that due to the crystal periodicity, the electron wave functions, which in one dimension are ψ (x) = u (x) exp (i k x), also have to be periodic (Bloch functions). Hence, from

$$ u\left( {x + Na} \right) = u(x) $$

(3.1)

and

$$ \exp \left( {ikx + ikNa} \right)u\left( {x + Na} \right) = \exp \left( {ikx} \right)u(x), $$

(3.2)

we obtain

$$ \exp \left( {{\text{i}}kNa} \right) = 1 $$

(3.3)

or

$$ k = n2\pi /Na;\,n = 0, \pm 1, \pm 2, \ldots \pm N/2, $$

(3.4)

where a is the lattice constant. We notice that (3.1) is actually valid for a ring-shaped chain which means that we neglect surface states (Sect. 14.1). 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.