Page 58 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Elastic Properties: The Stressed Uniform Lattice
Linearized
Potential To illustrate the derivation of the elastic energy in terms of the strain and
the elastic constants of the crystal, we will use an interatomic potential
Energy
that only involves two-atom interactions. For a more general derivation,
i
see e.g. [2.1]. Denote the relative vector position of atom by u , and its
i
absolute position by r = R + u , where R localizes the regular lattice
i i i i
site. For a potential binding energy E between a pair of atoms as
depicted in Figure 2.17 (II), we form the potential energy of the whole
crystal with
N N N N
1 1
U = --- ∑ ∑ E r –( i r ) = --- ∑ ∑ E R –( i R + u – u )
j
j
i
j
2 2
i = 1 j = 1 i = 1 j = 1
. (2.14)
N N
1
= --- ∑ ∑ E R +( ij u )
ij
2
i = 1 j = 1
where R = R – R j and u = u – u j . The factor 12⁄ arises because
i
ij
i
ij
the double sum counts each atom pair twice. We expand the energy about
the lattice site R ij using the Taylor expansion for vectors
1 2
(
(
(
(
E R + u ) = E R ) + ( u ⋅ ∇)E R ) + --- u ⋅( ∇) E R ) + … (2.15)
ij ij ij ij ij ij ij
2
because of the assumption that the atom displacements u i and hence
n
(
their differences u ij are small. The terms u ⋅( ij ∇) E R ) must be read
ij
as u ⋅( ij ∇) n operating n times on the position-dependent energy E
evaluated at the atom position R ij . Applying (2.15) to (2.14) we obtain
N N N N
1 1
(
(
U = --- ∑ ∑ E R ) + --- ∑ ∑ u • ∇ E R )
ij
ij
ij
2 2
i = 1 j = 1 i = 1 j = 1
(2.16)
N N
1 2
(
(
+ --- ∑ ∑ ( u • ∇) E R )) + …
ij
ij
4
i = 1 j = 1
The first term in (2.16) is a constant for the lattice, and is denoted by U .
o
The second term is identically zero, because the energy gradient is evalu-
ated at the rest position of each atom where by definition is must be zero
Semiconductors for Micro and Nanosystem Technology 55