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Primitive Cell
A substance is considered to be in a crystalline state (known as a crystal) if its constituent atoms are arranged in a regular and repetitive pattern.
Crystals can be described by an object that, when repeated, generates the entire crystal. Such an object can be considered as a “building block” called a unit cell. The unit cell is spanned by three independent vectors a 1 , a 2 , and a 3 , which may or may not lie along the Cartesian coordinate axes. It contains only one lattice point, and its volume is
V 5 ða 1 3 a 2 Þ · a 3
Its vertices define the lattice points. Each unit cell has the same shape, volume, and atomic arrangements. The choice of unit cell is not unique, and there is an infinite number of possible ways to choose a unit cell. The smallest possible unit cell is called the primitive cell. The vectors that define a primitive cell are called primitive lattice vectors.
The symmetry of a crystal is determined by the arrange- ment of atoms in its unit cell. A simple cubic unit cell is the simplest 3D case. Each of its eight corners contains identical atoms. More-complicated cases can have atoms in the center of the faces or at the center of the cube, known as a face-centered or body-centered cubic unit cell, respectively. They are important structures because many metals, ionic solids, and intermetallic compounds crystal- lize in such structures. Experimental techniques for studying crystal structures include
- x-ray diffraction structural analysis,
- neutron diffraction, and
- electron diffraction.
a (^2)
a (^3)
a (^1)
Bravais Lattice
An ideal crystal, i.e., without defects or imperfections, consists of a regular arrangement of atoms. A Bravais lattice (BL) is used to characterize such systems. It is a mathematical set of points that are equivalent to each other. The various symmetry operations (see “Elements of Symmetry”) possible for a 3D set of points result in 14 different BLs. Each lattice represents a set of points in a space that forms a periodic structure. Each point has exactly the same environment. The lattice points are specified by a lattice vector R as R 5 n 1 a 1 þ n 2 a 2 þ n 3 a 3
where n 1 , n 2 , and n 3 are integers, and a 1 , a 2 , and a 3 are three independent vectors.
Representative unit cells of the 14 BLs are shown on page 4. An important role is played by a face-centered cubic lattice, which is a structure that appears in many semiconductors.
It is often important to know the arrangement of neighboring atoms, especially the nearest neighbors (NNs). The number of NNs of an atom is known as the coordination number. A BL can also be introduced in two dimensions; a 2D lattice can be specified by vectors a and b and the angle g (of specific value) between them. Other 2D lattices are square, rectangular, centred rectangular, and hexagonal. Symmetry operations, like those for 3D strustures, can be used to describe 2D structures.
For crystals with many atoms, a building block of atoms, called the basis, is associated with each lattice point. Together, they create the crystal structure.
a
γ
b
Summary of Bravais Lattices
Wigner–Seitz Cell
A Wigner–Seitz (WS) cell is a special primitive cell that contains one lattice point. The WS cell is defined for a general lattice as the smallest polyhedron bounded by planes that are the perpendicular bisectors joining one lattice point to the others. The WS cell has the smallest possible volume (3D) and area (2D).
The WS cell in the reciprocal lattice defines the first BZ and has the full symmetry of the BL. It is constructed around an arbitrary lattice point, and when translated over all lattice points, it will fill the space of the entire crystal without overlapping.
It is difficult to illustrate the construction of a WS cell in 3D, so the image here depicts the construction of a WS cell in 2D real space.
- Select an arbitrary atom 0.
- Determine the NN atoms of the same species as the chosen atom.
- Draw lines from the atom 0 to all NNs, e.g., lines from atom 0 to all atoms 1, 2, 3, 4.
- Draw lines (planes, in 3D) passing through the middle of the lines. The cell enclosed by those lines (planes) is the WS primitive cell. The 3D WS cell for the body- centered cubic lattice shown here is a truncated octahedron with volume ½a^3. The cube around it is a conventional body-centered cubic cell with a lattice point on each vertex and at its center.
0
1
2
3
4
Diamond Structure
In a diamond structure in the conventional fcc lattice (shown here with the locations of atoms and the bonds (lines) between them), all atomic sites are occupied by carbon atoms. Semiconductors such as diamond (C), silicon (Si), germanium
The structure is a combination of two identical interpene- trating fcc lattices. One of the sublattices is shifted along the body diagonal of the cubic cell by one quarter of the length of the diagonal. The diamond structure is thus fcc with a basis containing two identical atoms. A simple method for constructing a diamond lattice considers it as a fcc structure with an extra atom placed at ¼a 1 þ ¼a 2 þ ¼a 3 from each of the fcc atoms. The basic element of the struc- ture is a tetrahedron where a C atom is at the center, and its four NNs are at the corners of the cube (or vice versa). Each atom forms four bonds with its NNs. Atoms in diamond-type crystals form covalent bonding. The bonding energy is associated with the shared valence electrons between atoms and depends on the relative orientation of atoms.
The atomic arrangement in the diamond structure helps explain its mechanical, chemical, and metallurgical properties. These semiconductor crystals can be cleaved along certain atomic planes to produce excellent planar surfaces, e.g., diamonds used in jewelry. Such surfaces are used as Fabry–Pérot reflectors in semiconductor lasers. Chemical reactions performed with such crystals, such as etching, often occur preferentially in certain directions.
[111]
109 o^ [111]
__
[111]_ _
[111]__
a
(Ge), and grey tin (α-Sn) crystallize in this structure.
Zincblende and Hexagonal Structures
A zincblende structure has the form of a diamond except that the two sublattices are occupied by different types of atoms, e.g., As and Ga, as shown here. It is a fcc structure with a two-atom basis. The structure is typical for semiconductors such as GaAs, ZnS, ZnSe, SiC, and GaP. For example, in ZnS, the Zn atoms occupy positions ð 0 ; 0 ; 0 Þ; ð 0 ; 12 ; 12 Þ and ð^12 ; 0 ; 12 Þ, ð^12 ; 12 ; 0 Þ, and S 1 4
1 4
3 4
3 4
1 4
3 4
3 4
1 atoms occupy positions ð 4 1 4 ;^ ;^ Þ;^ ð
1 4 ;^ ;^ Þ^ and^ ð
3 4 ;^ ;^ Þ;^ ð
3 4 ;^ ;^ Þ. Each atom has four equally distant neighbors of the opposite type located at the corners of a regular tetrahedron. There are four molecules of ZnS per conventional cell. In this compound, the presence of two different kinds of atoms means that the system does not have inversion symmetry.
As atom Ga atom
Another important structure is the hexagonal close-packed (^) A (hcp) structure. All atoms in the A-plane have an identical environment and can be taken as B lattice points. Atoms that form the B-plane “feel” different A environmentally and do not lie on lattice points. The unit cell contains a basis of an A atom 1 3
1 at (0, 0, 0) and a B atom at ð 2 2 3 ;^ ;^ Þ. There are two atoms in each primitive unit cell. The close-packed A-planes form two horizontal planes, and the B-planes are halfway between. The fcc structure has a hexagonal packing in the direction along [111], the cube diagonal. However, it is a close packing with three layers ABC instead of two, as in the hcp (e.g., Ti, Mg, Cd, Zn, He).
