Page 75 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Crystal Lattice System
stiffnesses are taken to be f =
neighbor bonds. 0.6 times the stiffnesses of the nearest
a
8 4 7

a
Figure 2.21. Model for a 2D mon- 5 1 3
atomic square lattice. The atom
9 2 6
numbers are markers for the deri-
vation of the equations of motion.

Following the discussion for the linear lattices, we formulate the equa-
tions of motion as

2 n 2
∂U
,
mu˙˙ = – ∑ ---------- = – ∑ ∑ D u , α = 12 ,
αi ∂u αβij βj
β = 1 βi j = 1 β = 1
,,
,
i = 12 … n (2.63)
where the dynamical matrix D αβij represents the stiffness of the bond
between atom sites and and between the directions α and . The
β
i
j
current model has a nine-atom 2D interaction, for which we can use an
18 × 18 stiffness matrix. The nonzero unique elements of the matrix are
E
d = D = D = D = D = – --- (2.64a)
1 1115 2214 1113 2212
a
fE
d = D = D = D = D = – ------ (2.64b)
2 1119 2218 1117 2216
a
Equation (2.63) thus becomes

E fE
mu ˙˙ = --- 4u –( 1 u – u – u – u ) + ------ 4u –( 1 u – u – u – u ) (2.65)
1
8
6
7
3
5
2
9
4
a a
We make an harmonic ansatz for the atom displacement of the form
72 Semiconductors for Micro and Nanosystem Technology