Environmental Engineering Reference
In-Depth Information
Thesetwounitsaredeterminedinsuchawaythat
ε
ischosentogive
f
2
depth-1, and
σ
is chosen to make
f
2
(2
1
/
6
) vanish.
The reduced pair potential
f
2
reads
A
(
Br
−
p
−
r
−
q
)exp[(
r
−
a
)
−
1
],
r
<
a
=
f
2
(
r
)
,
(1.8)
0,
r
≥
a
where
a
is the cut-off distance beyond which interaction vanishes.
The three-body interaction
f
3
is defined as
f
3
(
r
i
,
r
j
,
r
k
)
=
h
(
r
ij
,
r
ik
,
θ
jik
)
+
h
(
r
ji
,
r
jk
,
θ
ijk
)
+
h
(
r
ki
,
r
kj
,
θ
ikj
),
(1.9)
+
γ
(
r
ik
−
a
)
−
1
]
cos
θ
jik
+
2
1
3
h
(
r
ij
,
r
ik
,
θ
jik
)
=
λ
exp[
γ
(
r
ij
−
a
)
−
1
,
(1.10)
where
θ
jik
is the angle between
r
j
and
r
k
subtended at vertex
i
.The
ideal tetrahedral geometry of the diamond structure is favored by
the angle
θ
t
=
cos
−
1
(
−
1
/
3) in Eq. 1.10.
Although SW potential can model the properties of diamond-
structured silicon quite well, it fails to describe the behavior of non-
tetrahedral polytypes of silicon. To overcome this limitation, Tersoff
proposed a new approach [21] to model covalent interaction based
on the concept of bond order: the strength of a bond between two
atoms depends on the local environment. It is a summation of two-
body interactions expressed as[22]
V
ij
,
V
ij
=
f
C
(
r
ij
)
f
R
(
r
ij
)
+
b
ij
f
A
(
r
ij
)
,
1
2
E
=
(1.11)
=
i
j
where
f
R
and
f
A
describe the repulsive and attractive forces,
respectively, and
f
C
is the cut-off function. The bond order term
b
ij
is defined as [22]
n
i
i
n
i
ij
)
−
1
/
2
n
i
,
b
ij
=
χ
ij
(1
+
β
ζ
ζ
ij
=
ω
ik
g
(
θ
ijk
),
f
C
(
r
ik
)
k
=
i
,
j
c
i
/
d
i
+
θ
ijk
)
2
, (1.12)
c
i
/
d
i
−
θ
ijk
)
g
(
=
1
+
(
h
i
−
cos
where the local information aboutanother atom
k
isinvolved.
BasedonTersoff'sformalism,Brennerincludedadditionalterms
to account for nonlocal effect and extended the application to the
