# ξ η ξ ξ ξ η ξ ξ η ξ ξ η ξ ξ η ξ η η ξ η ξ ξ η ξ ξ η ξ ξ η ξ ξ 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