Page 51 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
r p)
r n +
fixed
Since
kr =
.
1
3
2
and ka⋅
⋅
=
ka = by , ka⋅ ⋅ = Constant = 2π r m +( 2πp , the space axis inter-
2πn
2πm
1 2 3
⁄
cepts for the plane are therefore r = Constant 2πm ,
1
⁄
⁄
r = Constant 2πn and r = Constant 2πp .
2 3
Silicon We have seen that the silicon crystal can be represented by a face-cen-
Reciprocal tered cubic lattice with a basis. This means that its reciprocal lattice is a
Lattice Shape
body-centered lattice with a basis. This has the following implications.
The Wigner-Seitz cell for the silicon direct lattice is a rhombic dodecahe-
dron, whereas the first Brillouin zone (the Wigner-Seitz cell of the recip-
rocal lattice) is a truncated dodecahedron.
2.3 Elastic Properties: The Stressed Uniform
Lattice
In a broad sense the geometry of a crystal’s interatomic bonds represent
the “structural girders” of the crystal lattice along which the forces act
that keep the crystal intact. The strength of these directional interatomic
forces, and the way in which they respond to small geometrical perturba-
tions; these are the keys that give a crystal lattice its tensorial elastic
properties and that enable us to numerically relate applied stress to a
strain response. In this section we derive the Hooke law for crystalline,
amorphous and poly-crystalline materials based on lattice considerations.

2.3.1 Statics

Atomic Bond It is well known that atoms form different types of bonds with each other.
Model The classification is conveniently viewed as the interaction between a
pair of atoms:
• Ionic—a “saturated” bond type that is characterized by the fact that
one atom ties up the electrons participating in the bond in its outer-
most shell. This leaves the two atoms oppositely charged. The cou-

48 Semiconductors for Micro and Nanosystem Technology