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**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