36
3 Crystals
(a)
(b)
Fig. 3.1 a Two-dimensional lattice. It can be generated by various pairs of translation vectors. b Elementary cells of the
lattice. Primitive elementary cells are shaded
Fig. 3.2 Crystal structure,
consisting of a lattice and a
base
Base
The crystal structure is made up by the lattice and the building block that is attached to each lattice
point. This building block is called the base (Fig. 3.2). In the simplest case, e.g. for crystals like Cu,
Fe or Al, this is just a single atom (monoatomic base). In the case of C (diamond), Si or Ge, it is a
diatomic base with two identical atoms (e.g. Si–Si or Ge–Ge), in the case of zincblende compound
semiconductors, such as GaAs or InP, it is a diatomic base with nonidentical atoms such as Ga–As or
In–P. For wurtzite crystals such as GaN or ZnO the base has four atoms like Ga–N–Ga–N. There exist
far more involved structures, e.g. NaCd 2 where the smallest cubic cell contains 1192 atoms. In protein
crystals, the base of the lattice can contain 10,000 atoms.
In summary: Crystal structure = Lattice × Base.
3.3 Lattice
As described in Sect. 3.2 the lattice is spanned by three vectors a i . The lattice symmetry is decisive for
the physical properties of the semiconductor. It is described by the appropriate groups of the symmetry
operations.
3.3.1 2D Bravais Lattices
There are five two-dimensional (2D) Bravais lattices (Fig. 3.3) which are distinct and fill all (2D) space.
These are very important for the description of symmetries at surfaces. The 2D Bravais lattices are the
square, hexagonal, rectangular and centered-rectangular lattice.
3 Crystals
(a)
(b)
Fig. 3.1 a Two-dimensional lattice. It can be generated by various pairs of translation vectors. b Elementary cells of the
lattice. Primitive elementary cells are shaded
Fig. 3.2 Crystal structure,
consisting of a lattice and a
base
Base
The crystal structure is made up by the lattice and the building block that is attached to each lattice
point. This building block is called the base (Fig. 3.2). In the simplest case, e.g. for crystals like Cu,
Fe or Al, this is just a single atom (monoatomic base). In the case of C (diamond), Si or Ge, it is a
diatomic base with two identical atoms (e.g. Si–Si or Ge–Ge), in the case of zincblende compound
semiconductors, such as GaAs or InP, it is a diatomic base with nonidentical atoms such as Ga–As or
In–P. For wurtzite crystals such as GaN or ZnO the base has four atoms like Ga–N–Ga–N. There exist
far more involved structures, e.g. NaCd 2 where the smallest cubic cell contains 1192 atoms. In protein
crystals, the base of the lattice can contain 10,000 atoms.
In summary: Crystal structure = Lattice × Base.
3.3 Lattice
As described in Sect. 3.2 the lattice is spanned by three vectors a i . The lattice symmetry is decisive for
the physical properties of the semiconductor. It is described by the appropriate groups of the symmetry
operations.
3.3.1 2D Bravais Lattices
There are five two-dimensional (2D) Bravais lattices (Fig. 3.3) which are distinct and fill all (2D) space.
These are very important for the description of symmetries at surfaces. The 2D Bravais lattices are the
square, hexagonal, rectangular and centered-rectangular lattice.