Page 60 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Elastic Properties: The Stressed Uniform Lattice
for which we only keep the leading term. We now apply a similar series
expansion to the term u , because we expect it to vary little in the vicin-
j
ity of R , and from now on we do not consider atom sites that are far
i
away, to obtain
u = u + R • ∇ u R() + … (2.21)
j i ij i i
Taking the first right-hand-side term of (2.20) and the first two terms of
(2.21), and inserting these into (2.19), we obtain
N N
h 1 T
(
•
U = – --- ∑ ∑ ( R • ∇ u ) • DR ) ( R • ∇ u ) (2.22)
i
ij
ij
i
i
4
i = 1 j = 1
The result of (2.22) can now be rewritten again in terms of the original
quantities, to give
N N
i ∑
h 1 T
(
U = – --- ∑ ( ∇ u ) : { R DR )R }: u∇ i
i
ij
ij
4
i = 1 j = 1
(2.23)
N
1 T
= – --- ∑ ( ∇ u ) :FR(): u∇ i
i
i
2
i = 1
The rank four elastic material property tensor FR in the vicinity of the
()
i
lattice site R is defined as
i
N
1
FR() = --- ∑ ( R DR( )R ) (2.24)
i ij i ij
2
j = 1
in terms of the crystal constituent positions and the resulting net inter-
atom binding energy. We expect this tensor to be translationally invariant
with respect to the lattice. Therefore, we can now move from a discrete
crystal description to the continuum, by considering the crystal as a col-
V
lection of primitive cells of volume with an average “density” of elas-
⁄
tic material property in any particular unit cell to be E = F V , so that
we can replace the sum in (2.23) by an integral to obtain
Semiconductors for Micro and Nanosystem Technology 57