Proximity effect correction calculations by the integral equation approximate solution method

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Proximity effect correction calculations by the integral equation approximate solution

method

J. M. Pavkovich

Citation: Journal of Vacuum Science & Technology B 4, 159 (1986); doi: 10.1116/1.583369

View online: http://dx.doi.org/10.1116/1.583369

View Table of Contents: http://scitation.aip.org/content/avs/journal/jvstb/4/1?ver=pdfcov

Published by the AVS: Science & Technology of Materials, Interfaces, and Processing

Articles you may be interested in The representative figure method for the proximity effect correction [III] J. Vac. Sci. Technol. B 9, 3059 (1991); 10.1116/1.585369

Methods for proximity effect correction in electron lithography J. Vac. Sci. Technol. B 8, 1889 (1990); 10.1116/1.585179

Exact solution of the proximity effect equation by a splitting method

J. Vac. Sci. Technol. B 6, 432 (1988); 10.1116/1.583969

Approximate solution of Fredholm integral equations by the maximumentropy method J. Math. Phys. 27, 2903 (1986); 10.1063/1.527267

Solution of Linear Integral Equations Using Padé Approximants

J. Math. Phys. 4, 1506 (1963); 10.1063/1.1703931

Redistribution subject to AVS license or copyright; see http://scitation.aip.org/termsconditions. Download to IP: 132.239.95.91 On: Wed, 09 Apr 2014 06:30:39



Proximity effect correction calculations by the integral equation approximate

solution method

J.

M. Pavkovich

Varian Associates, Palo Alto, California 94303

(Received 17 June 1985; accepted 4 October 1985)

The

task

of successfully deal ing

with

the proximity effect problem involves

many

aspects, all of

which must be

dealt

with in

a reasonably satisfactory

manner. If one

chooses

to

correct

for

the

proximity effect by doing dose compensation one

major

task is

the

solution

of

the i.ntegral

equation whichdescribes

the

resultant exposure

in

terms ofthe incident flux of electrons.

One

well

known method

is

the

self-consistent

method. This paper

describes

a method which

provides

a

relatively accurate approximate solution

to the

integral equation which is easy to calculate and

which

provides

information on where

features

should be fractured to obtain good

dose

compensation.

Although the

relationship between

the incident

flux

and the resultant exposure

is

linear, the development process itself s not. This means that

the

usual integral equation should be

modified slightly so that

the

resultant exposure is defined in a

manner

which

more

closely

matches

the real problem. This paper will attempt to describe how information from the development

process

can be

used

to

define

the

exposure problem in

a manner which

provides

more

desirable

solutions.

I.

INTRODUCTION

In electron beam lithography

the

problem of proximity ef

fect compensation is well

known

and several techniques have

been suggested to solve it. These include shape correction,

dose compensation,z multilayer resist techniques,3 and

the

method

of equalization of

background

dose,4 or GHOST

method.

AU

of these

methods have theiradvantages

and dis

advantages. This paper describes a technique primarily use

ful with

the

dose compensation

method although it can

be

used in conjunction with other methods. This paper

de

scribes a

method

of solving

the

equation for the incident dose

in

terms

of

the

desired energy deposition in

the

resist.

The

solution is an approximate solution, but appears to be accu

rate

to within 2% or 3% of

the

correct solution.

U.

THE INTEGRAL EQUATION AND ITS SOLUTiON

The energy deposition in a resist layer on top of a silicon

substrate subjected to a patterning e-beam

can

be described

by

the

following equation:

E x,y)

= KJ x - x ,y - y )i(x ,y )dx dy , 1)

where

K

is a

constant

relating

the

energy deposition

in the

resist

to the incident current

density,

f(x,y)

is

the incident

current

density, f x,y) is the function which describes

the

distribution of energy which would be deposited

by an

inci

dent delta function

of

current. I t is normalized so

that

f f x,Y)dX dy = 1

2)

t will

be

assumed that

f x,y)

is adequately described by

a

double

Gaussian-type

function

which

will

be written

in

the

following form:

f x,y)

= exp - - --.:..-=--

(1) (

x

2

+ 2 )

1+ 7J 7T{ } f3

}

17(1)

( X 2+ y 2 )

- 1 + 7J rrf3 xp {3

.

3)

In this equation /3 = width parameter of forward scattered

energy,

{3

b

= width parameter

of

backward

scattered ener

gy, and

J = ratio

of total energy

in backward portion to the

total

energy in

the forward

portion. t

should be noted that

K,

3

f,/3

b and 1] are also all functions ofz the vertical coordi

nate

in

the

resist. For

the moment

however, we will assume

that

a

z coordinate can be chosen which adequately charac

terizes

the

average

energy

deposition vertically.

The

backward

scattering

parameter

{J b

can be

of

the

order

of 1 or 2

J1

and is

the

source of

the proximity

problem.

The

backward scattered energy is quite broad and slowly varying

spatially

and thus

generates

a background

level of energy

on

which the more sharply

defined energy deposited

by the

for

ward scattering rests.

The forward scattering paramet er f f is small compared to

the minimum

feature size.

This must

be so, otherwise

there

would

be no

hope

ofeven

doing lithography

since

one

feature

edge would directly affect another. However, {3 does affect

the

sharpness of

the

energy profile

at the

beam edge. More

over,

the

steep

portion of the

energy deposition profile is

determined

not

only by

the

broadening due

to

the

forward

scattering, but by the profile of the incident

beam

itself. Since

both

are

usually

represented

by

Gaussian-type

functions,

and the convolution

of

Gaussian functions is still Gaussian

it

is appropriate to approximate the forward scattering por

tion of

the

distribution function by a

delta

function. The

forward scattering,

although

still

important can

be consid

ered to contribute

to simply a less

sharply

defined

incident

beam. Also it is more appropriate to consider the beam edge

broadening problemseparately from

the

proximity problem

and

define

the proximity problem

as

the problem caused by

only

the broad

background scattering.

Thus

the distribution

function for

the

proximity problem

can

be written as

159 J.

Vac. ScI.

Technol

B .. (1),

Jan/Fab

1986

0134-211)(/86/010159-05 01.00 @

1986

American

Vacuum SOCiety

159




160

J. M. Pavkovich: Proximity effect correction calculations

f(x,y) = _ 1 _

8

(x,y) + _ YJ-S(x,y),

1

+ YJ

1 + YJ

(4)

where

S(x,y) = exp - ,

X2

+

y2

17 3 f

(5)

and will be referred to simply as the spreading function. By

writing things in this manner, we can separate the proximity

problem, which is really global in nature, from the precise

shape

ofthe

beam edge, which is really a

rather

local prob

lem. With this simplification, Eq. 1) can be written in the

form

E

(x,y)

=

K [_1_ i(x,y) +

_TJ_

1+TJ 1+TJ

X

II

(x

-x ,y - y )i(x ,y )dx d y l (6)

I f we further consider only one dimension, the above equa

tion becomes simply

E (x)

=

K

[_1_

i(x)

+

_TJ_

f

(x

- X )z (X )dX ],

(7)

1+1] l+TJ

This equation describes

the

resulting exposure from an inci

dent beam,

For the moment consider the three limiting cases depicted

in

Fig.

1.

Case I is a very narrow isolated line. If the line is

very narrow, there will

be

almost no contribution to the

background and thus the resultant exposure will consist only

of the first term in Eq. 7). Case II is a narrow gap between

two infinitely large regions. In this case, the background is at

its maximum value over the whole region. Thus even in the

gap there is essentially a full contribution from the second

term in Eq.

7).

Case

III

is the edge of an infinitely large

region.

At

the edge, the value

of

the integral in the second

term of Eq. 7) is 0.5.

Thus

the resultant exposure has the

step caused by the first term sitting

on

top of he contribution

of he second term. Far outside the region the contribution of

the second term will drop

to

zero.

Far

inside, the contribu

tion will approach T I

(1 +TJ)·

The proximity problem arises because of the significant

change in the profile and level

of

the energy deposition when

the feature size approaches the value of b When the feature

size is large, all the edges have an energy profile similar to

that depicted in case III. Even though the background expo

sure level is present, it has a relatively constant value at all

edges.

I t

should also be noted that the method of equaliza

tion

of

background dose attempts to make the edge profiles

all look like those for case II.

I t

s important to realize that it

is not whether the edge profiles are good or bad in some

sense,

but

whether they are all similar

that

is important.

In the traditional method of dose compensation, the ob

jective is to choose the incident current so

that

the resultant

exposure is uniform in all exposed areas. No constraints are

placed on the exposure in unexposed areas. Equation

7)

can

not be solved for i(x) since one does not know how to choose

(x) outside

of

features

to

generate

an

acceptable non-nega

tive) solution for i(x). Thus to obtain the equation to be solved

for

i(x),

one must multiply Eq.

7)

by

P (x),

where

P

(x) repre

sents the patterns to

be

exposed and is equal to 1 inside a

J. Vac. Sci. Techno . S, Vol. 4, No.1, Jan/Feb 1986

CASE I

NARROW

LINE

w4 i

i3b

l

CASE II

NARROW GAP

Hli

n-,l, .

I I

w __

w4 i

ilb) I H I I l IHIIIIIIHII

UK

I --;-+:;) 0

I I

I I

I I

, I

I

l

, I

I ,

_w_

CASE III EDGE OF LARGE FEATURE

IHIIIIIIIIIIlIl

K

1 ( 7) .

2 1 +

r;

0

160

FIG.

1. Three

limiting cases illustrating

the

proximity effect. Case I is a very

narrow line. Case II is a very narrow gap between two large features. Case

III is

the

edge of

an

isolated large feature.

feature and 0 outside. The result is

EoP(x) = K [_1_ i(x)P(x) +

_TJ_P(x)

l+TJ 1+1]

X

I (x - X )i(X )dX ] '

(8)

where Eo is the desired value of exposure within a feature.

Since

P

(x) is 0 where i(x) is 0, and 1 where i(x) is not zero, Eq.

8)

can be written in the form

aP (x) = K [_1_ i(x) + _ 1 ] _

I+TJ

I+TJ

X

I (x)S (x - x )P (x )i(x )dx ].

(9)

This is a linear integral equation of the second kind with a

symmetric kernel. Techniques are available to solve such

equations,

but

they are not very useful for the problem

at

hand. Moreover, the kernel is a function ofthe pattern and

thus there is no general characteristic type transfer function

relating

output

to input as one sees with linear filters.

The situation, however, is not hopeless. The functions




161

J. M. Pavkovich: Proximi ty

effect

correction calculations

within the integral are all well behaved and never negative.

The form

of

the integral is like a convolution which for this

case tends to make the result more slowly varying than the

input functions. The spreading function S

(x

-

x')

is a local-

ized function and thus tends to smooth out the variations in

i(x).

Moreover,

it

is really the changes in

Pix)

which contri-

bute most to the variation

of

the value

of

the integraL Thus

one is led to consider moving

i(x )

outside of he integral; thus

I

(x)S (x - x')P (x')i{x')dx' --+P (x)i(x)

J

(x - x')P

(x')dx'.

10)

With this approximation, the integral can be evaluated and

the resulting equation solved for

fIx).

The resulting approxi-

mate solution is then given

by

i(x)

=

(EoIK)P(x) _

(1/1 +

11)

+ ( 11/

1

+

I])(p*S)

11)

where we have written (P *S) for the integral since it now

truly represents the convolution of the pat tern function

P

(xl

with the spreading function S (x).

This approximate solution has the correct limiting behav-

Ior. In the case

of

an isolated line, (P S is essentially zero

and the dose should be increased by a factor

of

(1

+

1J).

In

the case

of

a small gap between two large features,

(P

>I<

S

is

close to 1 and thus there is no need to increase the incident

dose. Since this approximate solution was not derived ana-

lytically and since it is impossible to solve the integral equa-

tion analytically, it is somewhat difficult to assess its accura-

cy.

In

order to get some idea of the quality of this

approximate solution, a program was written which calcu-

lated both the approximate solution as well as determined

the correct solution numerically for arbitrary cases of lines

and spaces. Also

it

was possible to consider slight modifica-

tions to the approximate solution.

The kinds of modifications considered were adding terms

of the form [V (P >I<

S

F or

[ V

(P

*S 1where

' . a . a k a

v = l - J - .

ax Jy az

Such terms

will

be zero when

(P S)

is constant and are coor-

dinate system independent. This effort produced two results.

First, the edge of a large region really represents one of the

most difficult cases since it represents the fastest transition

from a region where (P S

is

1 to a region where it is O

Secondly, the results of various test problems indicated that

the above approximate solution could be improved slightly

by adding a term

of

the form

_ I1_)2[V(P*SlF

1+1]

12)

to Eq. 11). Thus the recommended approximate solution is

given by

r 1

i(x) = EoIK p x l--------

. [(l/1 +

1])

+

(1]/1

+ 1J)(P*S)

+

c: 1JY[V(P>I<

S

W} .

13)

J.

Vac. ScI. Techno .

e,

Vol. 4, No.1, Jan/Feb 1986

161

TABI.E 1. Comparison of

the approximate

solution

and

the numerical solu-

tion for the case

of an

isolated edge for increasing values

of

11.

Maximum

Maximum

Edge

dose

exposure

exposure

T

error

error

error

(%)

(%) (%)

0.75

2.6

2.1

0.75

1.00

3.5

2.8

0.37

1.25

4.5

3.6 0.20

LSO

6.2

4.2

0.86

III. COMPARISON WITH NUMERICAL SOLUTIONS

For

the proximity problems in which we are interested,

values

of 11

range up to around 1.0

or

perhaps a little more.

At a value of 1.0, halfof he deposited energy

is

going into the

background. It should be expected that as the value

of

'I]

increases, the accuracy

of

the approximate solution should

decrease.

In

this work, three measures were used to compare

the approximate soluti.on with the numerical solution. They

were, first, the greatest error between the correct and ap-

proximate solution, second, the greatest error between the

resulting exposures, and third, the difference between the

resultant exposure at the edge. The results are tabulated in

Table I for various values

of

1]. As can be seen, the results are

reasonably good, even for values

of 11

greater than

1.

Also

plotted in Fig. 2 is a plot of the correct numerical solution at

the edge of a large region for 1] =

1

IV.

CALCULATION OF THE APPROXIMATE

SOLUTION

The calculation

of

the approximate solution is quite

straightforward. One first overlays the pattern with a square

grid

of

mesh points. The size

of

the mesh is not related to the

feature size, but rather is related to

f

b the backward scatter-

ing width parameter. We wish to calculate the valueof P *

S )

at each mesh point. t is important to think

of

the mesh

points as sample points at which the value of (P

*

S is calcu-

lated rather than an average over some area. The value

of

(P

*S) at any arbitrary point can then be found by interpola-

tion. Since the spreading function S (x) is extremely wen be-

haved, (P

*S)

is also extremely wen behaved, This means one

can use a very coarse mesh and a high order interpolation

INCiDENT

DOSE

DEPOSITED

ENERGY

1.414

'I =

1

0.293

FIG. 2. Numerical solution of he integral equation illustrating the incident

dose and the resulting deposited energy for the edge of a large isolated

feature.




162

J. M.

Pavkovich: Proximity

effect

correction calculations

formula to find the value of(P

*S

at an arbitrary point,

Thus

how one chooses the mesh size is a compromise between the

number of mesh points at which one calculates

(P ..

S and the

type of interpolation one wishes to use. t is likely that the

optimum approach is to use a very coarse mesh initially

and

then generate a finer grid

of

points by interpolation so that

simple interpolation can be used to evaluate

(P

*S at arbi

trary

points.

The calculation of

(P

*S) at the grid points is very simple

for rectangles. The formula for the value

of

P

*S at an arbi

trary point is given by the following equation:

P *S) = [erf X 2 p ~ X _

rf

X l p ~ X

]

X ~ [ e r f Y ~ ~ Y ) - e r f Y ~ ~ Y ) l

14)

where erf(x) is the

error

function and is defined by

erf(x) = - e u

2

du

LX

[iTo

15)

and the coordinates

of

the rectangle are defined by

x

X

•

Y

and

Y2

Notice that the calculation is the product of two

terms where each term is dependent on only one coordinate.

This means

that

one can calculate a factor to be associated

with a row of grid points which depends only on they coordi

nates of a rectangle. A factor can also be calculated for a

column which depends only on the

x

coordinates of a rectan

gle. The contribution at each grid point is then simply the

row factor times the column factor.

After calculating (P

*S

at all the grid points, the values at

each grid point can be converted to values

of

the approxi

mate solution and thus the desired values of incident current

intensity.

The

more difficult

part of

applying this method

is

then to

determine how to actually do the exposure since even vector

scan machines cannot vary the exposure continuously. Typi

cally, each rectangle which is exposed can have its own level

of ntensity, but the intensity is constant across the rectangle.

This means that one must determine the appropriate average

intensity to use in a rectangle and whether or not to fracture

any particula r rectangle into two or more pieces to achieve

the best appropriate exposure.

How

such decisions are made

depend on defining rules of thumb to build into a program

which define how great the deviations from the continuous

solution can be. This, in

tum,

depends on understanding the

development characteristics ofthe resists

5

and

how great the

latitudes are in the development process.

V. WHAT IS THE CORRECT PROBLEM?

Thus far it has been assumed that the goal

of

the exposure

process was to achieve uniform deposited energy within each

feature. This in fact may

not

be the best goal.

In

a region

of

high background, the edges will continue to develop

at

a

faster rate than where the background is low. t may there

fore be worthwhile, in fact, to require that the total resultant

exposure decrease as the background level increases. Thus

one would replace the left-hand side of Eq, 9) by a term of

the form

J. Vae. Sci. Techno . e, Vol.

4,

No,

1,

Jan/Feb 1986

162

16)

where

k

=

a factor which relates how the deposited energy

should be reduced as the background level increases.

The

effect of introducing this factor in the equation is simply to

increase the effective value of J and to modify the definition

ofEo. The same approximate solution techniques can still be

used although the constants change slightly.

Another

choice might be to define the problem to be

solved in a manner which tried to keep the center

of

the

forward scattering portion

of

the energy deposited at a con

stant level. This approach would be useful with a resist

which

had

a very high contrast. The important point is that

defining the problem to be solved depends to some degree on

the resist characteristics.

VI. CONCLUSIONS

An approximate method of solving the integral equation

relating the incident current density to the deposited energy

has been described. This solution technique appears to give

quite good results for the values of parameters typically en

countered in practice. This technique should be useful

if

one

wishes to do proximity correction using dose compensation.

However, the dose compensation technique requires good

understanding

of

the resist development characteristics in

order to determine a set of development parameters which

provide sufficient latitude. Also i t should be clear from this

work that one should use parameters which characterize the

deposited energy and background level when studying resist

development. Incident current density is

not

an appropriate

parameter.

The

GHOST

technique of proximity correction is clearly

the method

of

choice in a raster scan machine. Although

approximations are made in that method in terms

of

achiev

ing background dose equalization, it is likely

that

the lati

tudes required in the resist development are smaller than

with dose compensation. However, in a vector scan machine,

dose compensation may provide faster throughput than the

GHOST method. Whether

or

not this is true depends largely

on

the characteristics of the resist since it is the resist charac

teristics which determine how

much

fracturing is necessary.

The technique described ignored the effect

of

forward

scattering. Forward scattering can dearly be a problem,

even the dominant problem for very narrow lines and thick

resists. Moreover, ifbias

6

is used, the lines which

are

exposed

become even narrower, which compounds the seriousness

of

forward scattering. The problem of forward scattering

should be treated separately from the backward scattering

problem since it is a local problem and occurs because the

region related to the development of a line edge directly

overlaps the region associated with the development of the

edge of a neighboring feature.

Thus

one should compensate

for forward scattering by performing some form

of

shape

adjustment. This in

turn

could depend somewhat

on

the lev

el of background present. Again, it is true that any shape

adjustment requires understanding the resist development

process and how the energy density and background level

affect

that

process.




163

J.

M.

Pavkovlch: Proximity

effect correction

calculations

IN.

D. Wittels

and C.

I.

Youngman in

Proceedings

of

he 8th International

Conference on Electron

and

Ion eam Science and Technology

edited by

Robert

Bakish (The

Electrochemical

Society,

New York

1978),

p.

361.

2Mihir

Parikh J.

Appl.

Phys.

SO 6,

4:>.71

(1979).

3J

B. Kruger

P.

Rissman

and

M. S. Chang J.

Vac. Sci. TechnoL 19, 1320

(1981).

J. Vac Sci Technol S Vol

4

No.1 Jan/Feb 1986

163

4G. Owen

and

P. Rissman J.

Appl. Phys.

54, 6, 3573 (1983).

5A.

S. Chen

A.

R. Neureuther andJ.M.

Pavkovich

J.

Vac. Sci. Techno . B

3, 148 (1985).

6M.

G. Rosenfeld, A. R. Neureuther,

and

C. H. Ting, J. Vac. Sci. Technol.

19, 1242 (1981).

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Proximity effect correction calculations by the integral equation approximate solution method

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