1.2. CRYSTAL STRUCTURE 7
(a)(b)+S phe res on the star ti ng l ay er
C enter s of sphe res on the second l ay er
C enter s of sphe res on the th i rd l ay er+ + ++ + + + + + + + + ++ + + ++fcc hc pa bcaba(c)Figure 1.5: (a) A schematic of how the fcc and hcp lattices are formed by close packing of
spheres. (b) The hcp structure is produced by two interpenetrating hexagonal lattices with a
displacement discussed in the text. (c) Arrangement of lattice points on an hcp lattice.
In figure 1.5b and figure 1.5c we show the detailed positions of the lattice points in the hcp
lattice. The three lattice vectorsa 1 ,a 2 ,a 3 are shown asa, b, c. The vectora 3 is denoted byc
and the termc-axis refers to the orientation ofa 3. The hexagonal planes are displaced from each
other bya 1 /3+a 2 /3+a 3 / 2. In an ideal structure, if|a|=|a 1 |=|a 2 |,
c
a=
√
8
3
(1.2.8)
In table 1.1 we show the structural and some important electronic properties of some important
semiconductors. Note that two or more semiconductors are randomly mixed to produce an alloy,
AxB 1 −x, 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 (1.2.9)1.2.3 Notation to Denote Planes and Points in a Lattice: Miller Indices
To represent the directions and planes in a crystalline structure an agreed upon scheme is
used. The planes or directions are denoted by a series of integers called the Miller indices.