Page 135 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
3.3 Periodic Potentials in Crystal
Lattice periodic structures create a very special situation for the elec-
trons. Thus the properties of electrons in such kinds of potentials are dif-
ferent from those of a free electron [3.3]–[3.5].

3.3.1 The Bloch Functions

Symmetry A perfect crystal has a given periodicity. The special property of the elec-
tronic potential V is that it has exactly the same periodicity, i.e.

(
V x + l) = V x() (3.90)
where l is a arbitrary lattice vector as discussed in Chapter 2. We write
(
(3.90) as T l()V x() = V x + l) = V x() , where we used the translation
operator T l() , that transforms a function to its value at the place shifted
by the lattice vector l from the input. Operating on the product
ˆ
H x()Ψ x() it is shifted by l. Since the Hamiltonian is invariant under
translation, we have

ˆ
ˆ
ˆ
(
T l()H x()Ψ x() = H x()Ψ x + l) = H x()T l()Ψ x() (3.91)
ˆ
ˆ
ˆ
which means T l()H x() – H x()T l() = 0 , i.e., and H commute. This
T
implies that Ψ x() is also an eigenfunction of T , which means
TΨ x() = λΨ x() . Applying the translation operator n times yields
n
T Ψ x() = λ Ψ x() . Since the wavefunction must be bound λ ≤ . 1
n
Suppose λ < 1 and replace l by –l, than again the wavefunction would
not be bound in the inverse direction, thus only λ = 1 remains as a
solution. Let us write λ = exp ( ikl) and therefore
(
Ψ x + l) = exp ( ikl)Ψ x() . To fulfil this we write the wavefunction as
Ψ x() = exp ( ikx)u x() (3.92)
k
where u x() is a lattice periodic function
k
(
u x + l) = u x() (3.93)
k
k

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