Solid materials are classified by the way the atoms are arranged within the solid. Materials in which atoms are placed at random are called amorphous. Materials in
which atoms are placed in a highly ordered structure are called crystalline. Poly-crystalline materials are materials with a high degree of short-range order and no
long-range order. These materials consist of small crystalline regions with random orientation called grains, separated by grain boundaries. Crystals naturally form
as liquid material cools down, since the close proximity of atoms lowers their energy. However, since crystallization typically occurs in multiple locations
simultaneously, one finds that the poly-crystalline structure is quite common except for materials such as glass which tend to be amorphous. Crystalline silicon
dioxide does occur in the form of quartz but only if the temperature and pressure promote crystal formation.
Of primary interest in this text are crystalline semiconductors in which atoms are placed in a highly ordered structure. Crystals are categorized by their crystal
structure and the underlying lattice. While some crystals have a single atom placed at each lattice point, most crystals have a combination of atoms associated with
each lattice point. This combination of atoms is also called the basis.
The classification of lattices, the common semiconductor crystal structures and the growth of single-crystal semiconductors are discussed in the following sections.
Bravais lattices
The Bravais lattices are the distinct lattice types, which when repeated can fill the whole space. The lattice can therefore be generated by three unit vectors,
and a set of integers k, l and m so that each lattice point, identified by a vector , can be obtained from:
In two dimensions, there are five distinct Bravais lattices, while in three dimensions there are fourteen. The lattices in two dimensions are the square lattice, the rectangular lattice, the centered rectangular lattice, the hexagonal lattice and the oblique lattice as shown in Figure 2.2.2.It is customary to organize these lattices in groups, which have the same symmetry. An example is the rectangular and the centered rectangular lattice. As can be seen on the figure, all the lattice points of the rectangular lattice can be obtained by a combination of the lattice vectors 
The centered rectangular lattice can be constructed in two ways. It can be obtained by starting with the same lattice vectors as those of the rectangular lattice and then adding an additional atom at the center of each rectangle in the lattice. This approach is illustrated by Figure 2.2.2 c). The lattice vectors generate the traditional unit cell and the center atom is obtained by attaching two lattice points to every lattice point of the traditional unit cell. The alternate approach is to define a new set of lattice vectors, one identical to and another starting from the same origin and ending on the center atom. These lattice vectors generate the so-called primitive cell and directly define the centered rectangular lattice.
The centered rectangular lattice can be constructed in two ways. It can be obtained by starting with the same lattice vectors as those of the rectangular lattice and then adding an additional atom at the center of each rectangle in the lattice. This approach is illustrated by Figure 2.2.2 c). The lattice vectors generate the traditional unit cell and the center atom is obtained by attaching two lattice points to every lattice point of the traditional unit cell. The alternate approach is to define a new set of lattice vectors, one identical to and another starting from the same origin and ending on the center atom. These lattice vectors generate the so-called primitive cell and directly define the centered rectangular lattice.
Estudiante:
Leonardo Andrés Márquez Fernández
Electrónica del Estado Sólido.
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