Page 59 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
N
N
1
(
--- ∑ ∑ u • ∇ E R ) = 0 (2.17)
ij
ij
2
i = 1 j = 1
This leaves us with the zeroth and second and higher order terms
N N
o ∑
2
(
U = U + 1 ∑ ( u • ∇) E R ) + … (2.18)
---
ij
ij
4
i = 1 j = 1
The first term in (2.18) is the energy associated with the atoms at the lat-
tice positions, i.e., at rest, and represents the datum of energy for the
crystal. The second term is the harmonic potential energy, or the small-
displacement potential energy. When it is expanded, we obtain
N N
h 1
(
•
U = --- ∑ ∑ u • ∇ ( ∇ E R )) u ij (2.19)
ij
ij
4
i = 1 j = 1
Elasticity Equation (2.19) illustrates two terms, the deformation felt between two
Tensor sites u , and the second derivative of the interatomic potential at zero
ij
(
deformation (∇ ∇ E R )) . The second derivative is the “spring constant”
ij
of the lattice, and the deformation is related to the “spring extension”. We
can now go a step further and write the harmonic potential energy in
terms of the elastic constants and the strain.
Moving towards a continuum view, we will write the quantities in (2.19)
in terms of the position R alone. Consider the term
i
(
∇ [ ∇ E R )] = ∇ [ ∇ E R – R )] . We expect E R( ) to fall off rapidly
(
ij i j ij
away from R when R is far removed, so that for the site R , most of
i j i
the terms in (2.19) in the sum over j are effectively zero. To tidy up the
notation, we denote DR( ) as the components of the dynamical tensor
i αβ
(
DR ) = [∇ ∇ E R )] . The remainder of the sites R are close to R ,
(
i ij j i
making R small, so that we are justified in making an expansion of the
ij
components of DR( ) about R
ij i
•
{
(
(
DR ) = DR ) + ∇ DR ) } R + … (2.20)
(
ij αβ i αβ i αβ ij
56 Semiconductors for Micro and Nanosystem Technology