Page 139 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electronic System
repulsion and quantum mechanical attraction has a minimum for a finite
. b
3.3.3 Types of Band Structures

Valence and The graphical solution of (3.97) as given in Figure 3.11 provides us with
Conduction the possible electronic states that can be occupied by electrons. In a semi-
Bands
conductor the Fermi energy lies in the gap between two bands, i.e., there
is an uppermost completely filled band called the valence band. The fol-
lowing unoccupied band is called the conduction band. As we shall see in
Section 5.4, this band contains a certain number of electrons depending
on the temperature of the electronic system.

k
Si, Ge, GaAs The wave-vector entering the r.h.s. of (3.97) so far was taken in its lim-
⁄
iting cases k = 0 and k = π L , resulting in the upper and lower
bounds of the bands by the fact that cos 0 = 1 and cos π = – 1 . We
know ask for the structure of the bands in between those two limits for
,
⁄
arbitrary k ∈ [ 0 π L] . From Section 2.2 we know that this corresponds
to the center and the edge of the Brillouin zone. The crystal structure is
not isotropic and therefore we shall have different band structures in the
respective crystal directions. This leaves us with a function E k() , where
i
i is the band index in Figure 3.11 and is now a real three-dimensional
k
vector. In Figure 3.13 the typical band structure for cubic crystals is
shown in different crystallographic directions. There are several things to
observe in Figure 3.13:

• after passing the edge of the Brillouin zone they periodically repeat;
• the valence band, i.e., the uppermost occupied band is separated by an
energy gap E from the conduction band (the valence bands are high-
g
lighted);
• in general, the minimum energy of the conduction band is not neces-
sarily found at the same k-value as the maximum of the valence band
(points B or D are lower in energy than point C). This is called an
indirect band gap as is the case for Si and Ge. GaAs shows a direct

136 Semiconductors for Micro and Nanosystem Technology