Page 136 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Periodic Potentials in Crystal
Equation (3.92) together with (3.93) is called the Bloch theorem. Thus
electronic wavefunctions in a perfect crystal lattice are plane waves mod-
ulated by a lattice periodic function. We immediately recognize that the
problem left is to determine the form of u x() . Therefore, we insert
k
(3.92) into the stationary Schrödinger equation, which gives
2
—
------- k –( 2 2ik∇∇ ) + V x() u x() = E , u x() (3.94)
2
–
2m k k j k
We see that u x() and E k j depend on the plane wave-vector k. In addi-
,
k
tion, (3.94) allows for given k a series of eigen-values accounted for by
the index j.

3.3.2 Formation of Band Structure

To construct a simple model semiconductor we take a one-dimensional
potential composed of barriers repeated periodically with the periodicity
length L = a + b , e.g.

 V , x ∈ – [ b 0]
,
Vx() =  0 (3.95)
 0, x ∈ [ 0 a]
,
as sketched in Figure 3.10. We insert (3.95) in (3.94) and solve the one-

Vx()

V 0

– b 0 a x
Figure 3.10. Bloch waves in a
periodic crystal lattice potential.

dimensional eigenvalue problem with the ansatz

Semiconductors for Micro and Nanosystem Technology 133