Coordinates for Tip-Semiconductor System in SEMITIP VERSION 3
In SEMITIP VERSION 3 and higher, a new set of coordinates in the vacuum is used that are a generalization of the usual prolate spheroidal coordinates. The new coordinates
ξ and
η
in the vacuum are related to the cylindrical coordinates r and z by where c
≡ z
0
ξ 2 r
2
−
1
+
( z
−
ξ ac
η
)
2
2
= a
2 , (1a)
/ ( a
η
T
) with
η
T
− r
2
− η 2
+
( z
− ac
η
)
2
η 2
= a
2 (1b)
1
being the
η
value defining the hyperboloid that corresponds to the boundary of the probe tip and z
0
being the center point of this hyperboloid. The values of
η
thus run from 0 on the surface to
η
T
at the tip. The value of
ξ
is 1 on the central axis and increase with distance away from that axis. For z
0
=
0 these equations reduce to the standard definition of prolate spheroidal coordinates. With specified values for R , b , and s we have
η
T
=
1 / 1
+ b
−
2 , a
=
R b
2
/
η
T
, and z
0
= s
− a
η
T
. The inverse equations to (1a) and (1b) are z
= a
ξη + ac
η = a (
ξ + c )
η
, (2a) r
= a [(
ξ 2 −
1 ) ( 1
− η 2
)]
1 / 2
Laplace's equation for the electrostatic potential energy
φ
in the vacuum is found to be f
1
(
ξ
,
η
)
+ f
4
∂ 2
φ
∂ ξ
2
+
(
ξ
,
η
) f
2
(
ξ
,
η
)
∂ 2
φ
∂ ξ ∂ η
+
∂ 2
φ
∂ η
2
+ f
5
(
ξ
,
η
) f
3
(
ξ
,
η
)
∂
∂
φ
ξ
+
∂ 2
φ
∂ θ
2
+ f
6
(
ξ
,
η
)
∂ φ
∂ η
=
0
(3a) where f
1
(
ξ f
2
(
ξ
,
η
)
= f
3
,
η
)
(
ξ
,
η
)
=
=
(
ξ
2 −
1 ) [(
ξ
ξ
(
ξ
+ c )
2
+ c )
− η
− η
2
2
( 2 c
ξ
( c
ξ
+ c
2
+
1 )
ξ
(
( 1
ξ
− η
+ c )
2
) (
ξ
− η
2 − η
2
( c
ξ
2
)
+
1 )
ξ
(
ξ
(
ξ
+
2 c )
− η
2
−
1 ) ( 1
−
( c
ξ
η
2
)
+
1 )
, f
4
(
ξ
,
η
) f
5
(
ξ
,
η
)
=
=
−
ξ
(
2 c
ξ
[
ξ
(
ξ
η
(
ξ
2 −
1 ) ( 1
− η
+
+ c )
− η
2
( c
ξ g c )
5
−
(
ξ
η
,
η
)
2
( c
ξ
2
)
+
1 )
,
+
1 )]
2
,
+
1 )]
, (3b)
(3c)
(3e)
(3f) and f
6
(
ξ
,
η
)
=
[
ξ
(
ξ + g
6 c )
(
ξ
− η
,
η
)
2
( c
ξ +
1 )]
2 with
1

g
5
(
ξ
,
η
)
= c
3 +
3 c
2 ξ + c ( 2
+ c
2
)
ξ 2 +
3 c
2 ξ 3 +
4 c
ξ 4 +
2
ξ 5
η 4
[ c
3
2
η 2
[ c ( c
2
+
( 2
+
3 c
2
−
1 )
+
3 c
)
ξ
2 ξ
+ c ( 6
+
+ c ( 6
+ c
2
)
ξ 2 c
2
)
ξ 2
+
3 c
+
( 2
2 ξ 3
]
−
+
3 c
2
)
ξ 3
+
+ c
ξ 4
]
(3h) and g
6
(
ξ
,
η
)
= − η
{ c
2 +
4 c
ξ + c
2 ξ 2
2
η 2
[ c
2
+
2
ξ 4
+
4 c
ξ
+ η 4
[ 2
+
+
( 2
+ c
2 c
2
)
ξ
+
4 c
ξ
2
] } .
+ c
2 ξ 2
]
−
(3i)
In VERSIONS 3, 4, and 5 there is not
θ
dependence, so the second derivative with respect to
θ
[third term of Eq. (3a)] is zero.
Grid points are labeled by ( i,j ) with i referring to the radial direction and j referring to the direction perpendicular to the surface. In the new coordinate system for the vacuum the grid points are given by (
ξ i
,
η j
). The spacing between the
η j
values is chosen to be simply
Δ η = η
T
/ NV , so that
η j
= j (
η
T
/ NV ) . The
ξ i
values can be obtained by noting that the radial grid values at the surface are matched between the vacuum and the grid.
Thus, using the R i
values output by the program and with j
=
0 referring to the surface, we have r i 0
=
R i
ξ i
=
{ 1
+
[ R i
/ a ]
2
}
1 / 2
. On the surface,
η =
0 , so that
. The particular values of r ij
and z ij r i 0
=
R i
= a (
ξ i
2 −
1 )
1 / 2 so that
corresponding to the grid points can therefore be obtained by:
(i) for z ij
we have z ij
= a (
ξ i
+ c )
η j
=
([ a
2 +
R i
2
]
1 / 2 + a c ) j
η
T
DELV i
as ([ a
2 +
R(I)
2
]
1 / 2 + a c )
η
T
/ NV , so that z ij
=
/ NV . The program defines j
×
DELV i
.
(ii) for the r ij
values, we again note that the radial values at the surface are
R i
= a (
ξ i
2 −
1 )
1 / 2 . Then, away from the surface we have r ij
= a [(
ξ i
2 −
1 ) ( 1
− η 2 j
)]
1 / 2 =
R i
{ 1
−
[ j (
η
T slope.
/ NV )]
2
}
1 / 2 with
η
T
=
1 / 1
+ b
−
2 and where b is the user specified shank
2