Gradient method of mask edge correction
The present invention is directed to a method and apparatus for making mask edge corrections using a gradient method for high density chip designs. The present invention uses a newly defined cost function.
1. A method for making mask edge corrections, comprising steps of:
(a) providing initial parameter values and calibration parameter values;
(b) performing coherent field calculation;
(c) performing process intensity calculation; and
(d) calculating a cost function, said cost function being defined as
Φ
=
∑
i
Φ
i
2
,
wherein
Φ
i
=
{
(
J
in
i
-
C
in
0
)
,
if
J
in
i
<
C
in
0
(
J
out
i
-
C
out
0
)
,
if
J
out
i
>
C
out
0
0
,
in
other
cases
for
dark
background
;
and
Φ
i
=
{
(
J
in
i
-
C
out
0
)
,
if
J
in
i
>
C
out
0
(
J
out
i
-
C
in
0
)
,
if
J
out
i
<
C
in
0
0
,
in
other
cases
for
clear
background
;
where J in i and J out i are process intensity values at points P in i and P out i , respectively, and C out 0 ≦C 0 ≦C in 0 , C 0 being a cutline of a process intensity or aerial image contour, and C out 0 and C in 0 being some cutlines of said process intensity, which characterize an intensity gradient near said aerial image contour, wherein said cost function is used for optical proximity correction (OPC) purposes in making said mask edge corrections.
2. The method of claim 1 , further comprising:
when a current cost function is less than a previous cost function, calculating a gradient component of said current cost function for i-coordinate a i as
V
i
[
i
]
=
∂
∂
a
i
Φ
(
a
⇀
)
|
a
⇀
=
a
⇀
[
i
]
.
3. The method of claim 2 , wherein i-th gradient component for l-stage of an iteration process is represented as follows:
∂
∂
a
i
Φ
(
a
⇀
)
|
a
⇀
=
a
⇀
[
i
]
=
2
∑
j
(
J
j
-
C
j
)
∂
J
j
∂
a
i
|
a
⇀
=
a
⇀
[
i
]
wherein J j ={J in i or J out j } is a process intensity value corresponding to j control point, and
∂
J
j
∂
a
i
is a first derivative of said process intensity J j for said parameter component a i .
4. The method of claim 3 , wherein when a size of a rectangle along a boundary edge is smaller than an optical resolution limit
λ
NA
,
said first derivative of said process intensity J j for said parameter vector component a i is represented as follows:
∂
J
j
∂
a
i
|
a
⇀
=
a
⇀
[
i
]
=
∂
∂
a
i
J
(
x
j
,
y
j
)
|
a
⇀
=
a
⇀
[
i
]
=
L
i
·
K
(
x
j
-
x
0
i
,
y
j
-
y
0
i
)
·
G
(
E
(
x
j
,
y
j
,
a
⇀
[
i
]
)
,
x
j
-
x
0
i
,
y
j
-
y
0
i
)
wherein
K
(
x
,
y
)
=
∫
∫
H
(
ϛ
,
η
)
ⅇ
-
i
·
2
·
π
·
NA
λ
(
ϛ
x
+
η
y
)
d
ϛ
d
η
is a function of a coherent impulse response of an optical system;
G
(
E
(
x
j
,
y
j
,
a
⇀
[
i
]
,
x
j
-
x
0
i
,
y
j
-
y
0
i
)
=
Re
∫
∫
J
(
x
u
,
y
u
)
E
(
x
,
y
,
x
u
,
y
u
,
a
⇀
[
i
]
)
·
ⅇ
-
1
·
2
·
π
·
NA
λ
(
x
u
(
x
-
x
0
)
+
y
u
(
y
-
y
0
)
)
ⅆ
x
u
ⅆ
y
,
is a field weight function for said parameter component a i ;
x 0i and y 0i are coordinates of a middle point of i edge fragment, moved at a distance a i normal to said boundary edge; and
E(x, y, x u , y u , {right arrow over (a)} [l] ) is a process coherent field for a source point (x u , y u ) and said l-stage of said iteration process.
5. The method of claim 2 , further comprising:
when a length L i of a rectangle along an i-th edge fragment is greater than a optical resolution limit
λ
NA
,
dividing said i-th edge fragment into
P
i
=
[
L
i
·
NA
λ
]
+
1
parts, each part with a length less than
λ
NA
and with a height a i , wherein said first derivative of said process intensity J j for said parameter vector component a i is represented as follows:
∂
J
j
∂
a
i
|
a
⇀
=
a
⇀
[
i
]
=
∑
p
=
1
P
i
L
ip
·
K
(
x
j
-
x
0
ip
,
y
j
-
y
0
ip
)
·
G
(
E
(
x
j
,
y
j
,
a
⇀
[
i
]
)
,
x
j
-
x
0
ip
,
y
j
-
y
0
ip
)
wherein x 0ip and y 0ip are coordinates of a middle point of p-th part of said i-th edge fragment, moved at a distance a i along a direction normal to said i-th edge fragment; and L ip is a length of said p-th part of said i-th edge fragment.
6. The method of claim 2 , further comprising:
calculating a step of iteration, calculating new parameter values and returning to said step (b).
7. The method of claim 1 , further comprising:
stopping an iteration process when a current cost function is equal to a previous cost function.
8. The method of claim 1 , further comprising:
decreasing a step of iteration when a current cost function is greater than a preceding cost function.
9. An apparatus for making mask edge corrections, comprising:
(a) means for providing initial parameter values and calibration parameter values;
(b) means for performing coherent field calculation;
(c) means for performing process intensity calculation; and
(d) means for calculating a cost function, said cost function being defined as
Φ
=
∑
i
Φ
i
2
,
wherein
Φ
i
=
{
(
J
in
i
-
C
in
0
)
,
if
J
in
i
<
C
in
0
(
J
out
i
-
C
out
0
)
,
if
J
out
i
>
C
out
0
0
,
in
other
cases
for
dark
background
;
and
Φ
i
=
{
(
J
in
i
-
C
out
0
)
,
if
J
in
i
>
C
out
0
(
J
out
i
-
C
in
0
)
,
if
J
out
i
<
C
in
0
0
,
in
other
cases
for
clear
background
;
where J in i and J out i are process intensity values at points P in i and P out i , respectively, and C out 0 ≦C 0 ≦C in 0 , C 0 being a cutline of a process intensity or aerial image contour, and C out 0 and C in 0 being some cutlines of said process intensity, which characterize an intensity gradient near said aerial image contour, wherein said cost function is used for optical proximity correction (OPC) purposes in making said mask edge corrections.
10. The apparatus of claim 9 , further comprising:
when a current cost function is less than a previous cost function, means for calculating a gradient component of said current cost function for i-coordinate a i as
V
i
[
i
]
=
∂
∂
a
i
Φ
(
a
⇀
)
|
a
⇀
=
a
⇀
[
i
]
.
11. The apparatus of claim 10 , wherein i-th gradient component for t-stage of an iteration process is represented as follows:
∂
∂
a
i
Φ
(
a
⇀
)
|
a
⇀
=
a
⇀
[
i
]
=
2
∑
j
(
J
j
-
C
j
)
∂
J
j
∂
a
i
|
a
⇀
=
a
⇀
[
i
]
wherein J j ={J in i or J out j } is a process intensity value corresponding to j control point, and
∂
J
j
∂
a
i
is a first derivative of said process intensity J j for said parameter component a i .
12. The apparatus of claim 11 , wherein when a size of a rectangle along a boundary edge is smaller than an optical resolution limit
λ
NA
,
said first derivative of said process intensity J j for said parameter vector component a i is represented as follows:
∂
J
j
∂
a
i
a
→
=
a
→
[
i
]
=
∂
∂
a
i
J
(
x
j
,
y
j
)
a
→
=
a
→
[
i
]
=
L
i
·
K
(
x
j
-
x
0
i
,
y
j
-
y
0
i
)
·
G
(
E
(
x
j
,
y
j
,
a
→
[
i
]
)
,
x
j
-
x
0
i
,
y
j
-
y
0
i
)
wherein
K
(
x
,
y
)
=
∫
∫
H
(
ϛ
,
η
)
ⅇ
-
ⅈ
·
2
·
π
·
NA
λ
(
ϛ
x
+
η
y
)
ⅆ
ϛ
ⅆ
η
is a function of a coherent impulse response of an optical system;
G
(
E
(
x
j
,
y
j
,
a
→
[
i
]
,
x
j
-
x
0
i
,
y
j
-
y
0
i
)
=
Re
∫
∫
J
(
x
u
,
y
u
)
E
(
x
,
y
,
x
u
,
y
u
,
a
→
[
i
]
)
·
ⅇ
-
1
·
2
·
π
·
NA
λ
(
x
u
(
x
-
x
0
)
+
y
u
(
y
-
y
0
)
)
ⅆ
x
u
ⅆ
y
i
is a field weight function for said parameter component a i ;
x 0i and y 0i are coordinates of a middle point of i edge fragment, moved at a distance a i normal to said boundary edge; and
E(x, y, x u , y u , {right arrow over (a)} [l] ) is a process coherent field for a source point (x u , y u ) and said l-stage of said iteration process.
13. The apparatus of claim 10 , further comprising:
when a length L i of a rectangle along an i-th edge fragment is greater than a optical resolution limit
λ
NA
,
means for dividing said i-th edge fragment into
P
i
=
[
L
i
·
NA
λ
]
+
1
parts, each part with a length less than
λ
NA
and with a height a i , wherein said first derivative of said process intensity J j for said parameter vector component a i is represented as follows:
∂
J
j
∂
a
i
a
→
=
a
→
[
i
]
=
∑
p
=
1
P
i
L
ip
·
K
(
x
j
-
x
0
ip
,
y
j
-
y
0
ip
)
·
G
(
E
(
x
j
,
y
j
,
a
→
[
i
]
)
,
x
j
-
x
0
ip
,
y
j
-
y
0
ip
)
wherein x 0ip and y 0ip are coordinates of a middle point of p-th part of said i-th edge fragment, moved at a distance a i along a direction normal to said i-th edge fragment; and L ip is a length of said p-th part of said i-th edge fragment.
14. The apparatus of claim 10 , further comprising:
means for calculating a step of iteration, calculating new parameter values and returning to said step (b).
15. The apparatus of claim 9 , further comprising:
means for stopping an iteration process when a current cost function is equal to a previous cost function.
16. The apparatus of claim 9 , further comprising:
means for decreasing a step of iteration when a current cost function is greater than a preceding cost function.
17. A computer-readable medium having computer-executable instructions for performing a method for making mask edge corrections, said method comprising steps of:
(a) providing initial parameter values and calibration parameter values;
(b) performing coherent field calculation;
(c) performing process intensity calculation; and
(d) calculating a cost function, said cost function being defined as
Φ
=
∑
i
Φ
i
2
,
wherein
Φ
i
=
{
(
J
in
i
-
C
in
0
)
,
if
J
in
i
<
C
in
0
(
J
out
i
-
C
out
0
)
,
if
J
out
i
>
C
in
0
0
,
in
other
cases
for
dark
background
;
and
Φ
i
=
{
(
J
in
i
-
C
out
0
)
,
if
J
in
i
>
C
out
0
(
J
out
i
-
C
in
0
)
,
if
J
out
i
<
C
in
0
0
,
in
other
cases
for
clear
background
;
where J in i and J out i are process intensity values at points P in i and P out i , respectively, and C out 0 ≦C 0 ≦C in 0 , C 0 being a cutline of a process intensity or aerial image contour, and C out 0 and C in 0 being some cutlines of said process intensity, which characterize an intensity gradient near said aerial image contour, wherein said cost function is used for optical proximity correction (OPC) purposes in making said mask edge corrections.
18. The computer-readable medium of claim 17 , wherein said method further comprising:
when a current cost function is less than a previous cost function, calculating a gradient component of said current cost function for i-coordinate ai as
V
i
[
i
]
=
∂
∂
a
i
Φ
(
a
→
)
a
→
=
a
→
[
i
]
.
19. The computer-readable medium of claim 18 , wherein i-th gradient component for l-stage of an iteration process is represented as follows:
∂
∂
a
i
Φ
(
a
→
)
a
→
=
a
→
[
i
]
=
2
∑
j
(
J
j
-
C
j
)
∂
J
j
∂
a
i
a
→
=
a
→
[
i
]
wherein J j ={J in i or J out j } is a process intensity value corresponding to j control point, and
∂
J
j
∂
a
i
is a first derivative of said process intensity J j for said parameter component a i .
20. The computer-readable medium of claim 19 , wherein when a size of a rectangle along a boundary edge is smaller than an optical resolution limit
λ
NA
,
said first derivative of said process intensity J j for said parameter vector component a i is represented as follows:
∂
J
j
∂
a
i
a
->
=
a
->
[
i
]
=
∂
∂
a
i
J
(
x
j
,
y
j
)
a
->
=
a
->
[
i
]
=
L
i
·
K
(
x
j
-
x
0
i
,
y
j
-
y
0
i
)
·
G
(
E
(
x
j
,
y
j
,
a
->
[
i
]
)
,
x
j
-
x
0
i
,
y
j
-
y
0
i
)
wherein
K
(
x
,
y
)
=
∫
∫
H
(
ϛ
,
η
)
ⅇ
-
i
·
2
·
π
·
NA
λ
(
ϛ
x
+
ηy
)
ⅆ
ϛ
ⅆ
η
is a function of a coherent impulse response of an optical system;
G
(
E
(
x
j
,
y
j
,
a
->
[
i
]
,
x
j
-
x
0
i
,
y
j
-
y
0
i
)
=
Re
∫
∫
J
(
x
u
,
y
u
)
E
(
x
,
y
,
x
u
,
y
u
,
a
->
[
i
]
)
·
ⅇ
-
1
·
2
·
π
·
NA
λ
(
x
u
(
x
-
x
0
)
+
y
u
(
y
-
y
0
)
)
ⅆ
x
u
ⅆ
y
i
is a field weight function for said parameter component a i ;
x 0i and y 0i are coordinates of a middle point of i edge fragment, moved at a distance a i normal to said boundary edge; and
E(x, y, x u , y u , {right arrow over (a)} [l] ) is a process coherent field for a source point (x u , y u ) and said l-stage of said iteration process.
21. The computer-readable medium of claim 18 , wherein said method further comprising:
when a length L i of a rectangle along an i-th edge fragment is greater than a optical resolution limit
λ
NA
,
dividing said i-th edge fragment into
P
i
=
[
L
i
·
NA
λ
]
+
1
parts, each part with a length less than
λ
NA
and with a height a i , wherein said first derivative of said process intensity J j for said parameter vector component a i is represented as follows:
∂
J
j
∂
a
i
❘
a
->
=
a
->
[
i
]
=
∑
p
=
1
P
i
L
ip
·
K
(
x
j
-
x
0
ip
,
y
j
-
y
0
ip
)
·
G
(
E
(
x
j
,
y
j
,
a
->
[
i
]
)
,
x
j
-
x
0
ip
,
y
j
-
y
0
ip
)
wherein x 0ip and Y 0ip are coordinates of a middle point of p-th part of said i-th edge fragment, moved at a distance a i along a direction normal to said i-th edge fragment; and L ip is a length of said p-th part of said i-th edge fragment.
22. The computer-readable medium of claim 18 , wherein said method further comprising:
calculating a step of iteration, calculating new parameter values and returning to said step (b).
23. The computer-readable medium of claim 17 , wherein said method further comprising:
stopping an iteration process when a current cost function is equal to a previous cost function.
24. The computer-readable medium of claim 17 , wherein said method further comprising:
decreasing a step of iteration when a current cost function is greater than a preceding cost function.