Many of the materials employed to create devices used for electronics, optoelectronics, and sensoring are given category names, such as metals, insulators, and semiconductors.
Material Ba Cr Cs Fe Nb Rb Ta W
Lattice constant (a) 5.02 2.886.05 2.873.30 5.59 3.31 3.16 Some materials which crystallize
in monoatomic bcc structures
F i g u r e 1.3: The body-centered cubic lattice along with a choice of primitive vectors. Also shown are lattice constants of some materials that crystallize in the monoatomic bcc structure.
Material Ag Al Au Ca Ce Cu La Ni Pb Pd Pt Th
Lattice constant (a) 4.09
4.05 4.08 5.58 5.16 3.61 5.30 3.52 4.95 3.89 3.92 5.08 Materials with monoatomic fee
structures
Figure 1.4: Primitive basis vectors for the face-centered cubic lattice. Also shown are some materials that crystallize in the monoatomic fee structure.
1.2. Crystalline materials
\ /
\ /
\ /
/ \
\ a /
\ /
/ \
\ /
\
/ \
I ai\ H ôJ = a
/ \
Figure 1.5: The simple hexagonal Bravais lattice. Two-dimensional triangular nets (shown in inset) are stacked directly above one another, a distance c apart. Also shown are the three unit vectors.
Depending upon applications, they are also categorized as ceramics, polar materials, ferroelectrics, ferromagnetics, etc. These materials have a crystal structure, ranging from the very simple with one atom per basis to complex ones with several atoms on a basis. Also in many materials the positions of atoms in the structure are not ideal, due to "spontaneous" effects arising from charges on the ions.
Monoatomic body-centered cubic
There are many metals which have the bcc lattice with one atom per basis. In Fig. 1.3 we show some of these materials.
Monoatomic face-centered cubic
Many metals crystallize in the fee lattice and have just one atom per basis. In Fig. 1.4 we show some of the important metals that fall into this category.
Sodium chloride structure
The sodium chloride (NaCl) structure is based on the fee lattice and a basis of one Na atom and one Cl atom separated by half of the body diagonal of the cube. The basis atoms are at 0 and a/2(x 4- y + z). The structure is shown in Fig. 1.6, along with some materials which crystallize in this structure.
Cesium chloride structure
The cesium chloride structure is shown in Fig. 1.7. The cesium and chloride atoms are placed on the points of a bcc lattice so that each atom has eight neighbors. The underlying lattice is simple cubic with two atoms per basis. The atoms are at 0 and a/2(x + y + z). In Fig. 1.7 we show some important materials which have the CsCl structure.
Diamond and zinc blende structures
Most semiconductors of interest for electronics and optoelectronics have an underlying fee lattice. However, they have two atoms per basis. The coordinates of the two basis
Material AgBr
KC1 LiH MgO MnO NaCl PbS
Lattice Constant (a) A 5.77 6.294.08 4.20 4.435.63 5.92
Figure 1.6: The sodium chloride crystal structure. The space lattice is fee, and the basis has one Na+ ion at 0 0 0 and one Cl~ ion at \\\- The table shows some materials with NaCl structure.
Material Lattice constant (a)
A
AIM BeCu CsCl LiHg
2.88 2.7 4.11 3.29
Some materials that have the cesium chloride structure.
Figure 1.7: The cesium chloride crystal structure. The space lattice is simple cubic, and the basis has one Cs+ ion and one Cl~ ion at \\\- The table shows some materials with the cesium chloride structure.
1.2. Crystalline materials
Figure 1.8: The zinc blende crystal structure. The structure consists of the interpenetrating fee lattices, one displaced from the other by a distance (f f f) along the body diagonal. The underlying Bravais lattice is fee with a two atom basis. The positions of the two atoms is (000)
and (fff).
atoms are
(O00)-d(|,!,f)
Since each atom lies on its own fee lattice, such a two atom basis structure may be thought of as two interpenetrating fee lattices, one displaced from the other by a trans- lation along a body diagonal direction (f f f ) .
Figure 1.8 gives details of this important structure. If the two atoms of the basis are identical, the structure is called diamond. Semiconductors such as Si, Ge, C, etc. fall into this category. If the two atoms are different, the structure is called the zinc blende structure. Semiconductors such as GaAs, AlAs, CdS, etc. fall into this category. Semi- conductors with the diamond structure are often called elemental semiconductors, while the zinc blende semiconductors are called compound semiconductors. The compound semiconductors are also denoted by the position of the atoms in the periodic chart, e.g., GaAs, AlAs, InP are called III-V (three-five) semiconductors, while CdS, HgTe, CdTe, etc., are called II-VI (two-six) semiconductors.
Hexagonal close-pack structure
The hexagonal close-pack (hep) structure is an important lattice structure and many metals have this underlying lattice. Some semiconductors, such as BN, A1N, GaN, SiC, etc., also have this underlying lattice (with a two-atom basis). The hep structure is formed as shown in Fig. 1.9a. Imagine that a close-packed layer of spheres is formed.
Each sphere touches six other spheres, leaving cavities, as shown. A second close-packed layer of spheres is placed on top of the first one so that the second-layer sphere centers are in the cavities formed by the first layer. The third layer of close-packed spheres can now be placed so that centers of the spheres do not fall on the centers of the starting spheres (left side of Fig. 1.9a) or coincide with the centers of the starting spheres (right side of Fig. 1.9b). These two sequences, when repeated, produce the fee and hep lattices.
Spheres on the starting layer
Centers of spheres on the second layer Centers of spheres on the third layer
fee * * hep
a a a
• •
(a)
(c)
Figure 1.9: (a) A schematic of how the fee and hep lattices are formed by close packing of spheres, (b) The hep structure is produced by two interpenetrating simple hexagonal lattices with a displacement, as discussed in the text. Arrangement of lattice points on an hep lattice.
1.2. Crystalline materials 11
Threefold axis [111]
Sixfold axis
Cubic Hexagonal
Figure 1.10: The stacking of tetrahedral layers in cubic and hexagonal ZnS. The large atoms are S; the small atoms are Zn. The vertical axis of hexagonal ZnS is a six-fold screw axis involving a translation of one-half c for each 60 degrees of rotation.
Underlying the hep structure is a simple hexagonal lattice (discussed earlier).
The hep structure consists of two interpenetrating simple hexagonal lattices as shown in Fig. 1.9b. The two lattices are displaced from each other by ai/3 + &2/3 + a3/2 as shown. The magnitude of a3 is denoted by c and in an ideal hep structure
(1.9) Wurtzite structures
A number of important semiconductors crystallize in the hep structure with two atoms per lattice site. The coordination of the atoms is the same as in the diamond or zinc blende structures. The nearest neighbor bonds are tetrahedral and are similar in both zinc blende and wurtzite structures. The symmetry of rotation is, however, different as shown in Fig. 1.10.
In Tables 1.2 and 1.3 we show the structural properties of some important materials that crystallize in the diamond, zinc blende, and wurtzite structures.
Perovskite structure
Materials like CaTiOa, BaTiO3, SrTiOs, etc., have the perovskite structure using BaTiO3 as an example. The structure is cubic with Ba2+ ions at the cube corners and O2~ ions at the face centers. The Ti4 + ion is at the body center.
Perovskites show a ferroelectric effect below a temperature called Curie tem- perature and have spontaneous polarization due to relative movements of the cations and anions. As shown in Fig. 1.11 the Ba2+ ions and Ti4 + ions are displaced relative
T i4 +
o-
(a)
Cube corners: Ba++
Cube face centers: O - Cube center: Ti4+
T i4 +
Displacement of positive charges with respect to negative charges = £ >
ferroelectric effect (b)
Figure 1.11: (a) The structure of a typical perovskite crystal illustrated by examining barium titanante. (b) The ferroelectric effect is produced by a net displacement of the positive ions with respect to the negative ions.
to the 02~ ions creating a dipole moment. As will be discussed later in this book the polarization can be controlled by an external electric field.
For many applications, one uses alloys made from two or more different mater- ials. The lattice constant of the alloy is given by Vegard's law, according to which the alloy lattice constant is the weighted mean of the lattice constants of the individual components.
aalloy = xaA + (1 ~~ x)aB C1-10) where aajio v is the lattice constant of the alloy AXB\-XJ and a A and a& are the lattice constants of materials A and B, respectively.