Page 49 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
k
q
Equation (2.6) generates the (wave) vectors from the (lattice posi-
tion) vectors, and we see that these are mutually orthogonal. We summa-
rize the classical results of the reciprocal lattice:
Reciprocal • The reciprocal lattice is also a Bravais lattice.
Lattice
• Just as the direct lattice positions are generated with the primitive unit
Properties
vectors q = α a + β b + γ c , the reciprocal lattice can also be so
j j j j
generated using k = δ d + ε e + ζ f . The relation between the two
j j j j
primitive vector sets is
b × c
d = 2π------------------------ (2.7a)
⋅
a ( b × c)
c × a
e = 2π------------------------ (2.7b)
⋅
a ( b × c)
a × b
f = 2π------------------------ (2.7c)
a ( b × c)
⋅
• The reciprocal of the reciprocal lattice is the direct lattice.
• The reciprocal lattice also has a primitive cell. This cell is called the
Brillouin zone after its inventor.
⋅
• The volume of the direct lattice primitive cell is v = a ( b × c) .
• The volume of the reciprocal lattice primitive cell is
⋅
f
V = d ( e × ) . From (2.7c) we see that
2π 2π 2π
d = ------ b ×( c) e = ------ c ×( a) f = ------ a ×( b) (2.8)
v v v
and hence that

( 2π) 3
V = ------------- (2.9)
v
Miller Indices The planes formed by the lattice are identified using Miller indices.
These are defined on the reciprocal lattice, and are defined as the coordi-
nates of the shortest reciprocal lattice vector that is normal to the plane.
Thus, if we consider a plane passing through the crystal, the Miller indi-

46 Semiconductors for Micro and Nanosystem Technology