Sub-resolution assist features

1. A method of semiconductor device fabrication, comprising:

receiving a first mask design comprising a first mask function;

determining a transmission cross coefficient (TCC) of an exposure tool;

decomposing the TCC into a plurality orders of eigenvalues and a plurality orders of eigenfunctions;

calculating a kernel based on the plurality orders of eigenvalues and the plurality orders of eigenfunctions;

determining a first sub-resolution assist feature (SRAF) seed map by convoluting the first mask function and the kernel;

storing the kernel in a memory medium;

receiving a second mask design comprising a second mask function different from the first mask function;

retrieving the kernel stored in the memory medium;

determining a second SRAF seed map by convoluting the second mask function and the stored kernel;

processing the second SRAF seed map to obtain a second SRAF map;

modifying the second mask design according to the second SRAF map to obtain a second modified mask design; and

performing photolithography using the exposure tool and the second modified mask design.

2. The method of claim 1 , wherein the first SRAF seed map comprises coordinates of peak positions for placement of a plurality of SRAFs.

3. The method of claim 1 , further comprising:

processing the first SRAF seed map to obtain a first SRAF map;

modifying the first mask design according to the first SRAF map to obtain a first modified mask design; and

performing photolithography using the exposure tool and the first modified mask design.

4. The method of claim 3 , wherein the first SRAF map comprises:

a plurality of SRAFs; and

a polygonal shape of each of the plurality of SRAFs.

5. The method of claim 1 , wherein the exposure tool comprises an extreme ultraviolet (EUV) exposure tool or a deep ultraviolet (DUV) exposure tool.

6. The method of claim 1 , wherein the TCC comprises information about an illumination intensity of the exposure tool, a numerical aperture of the exposure tool, a thickness of a resist stack to be patterned, or a range of an aberration.

7. The method of claim 1 ,

wherein the plurality orders of eigenvalues are denoted as λ i ,

wherein the plurality orders of eigenfunctions are denoted as ϕ i ,

wherein the kernel is denoted as Ω(x,y),

wherein the calculating of the kernel comprises use of the following mathematical formula:

Ω

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

*

(

0

,

0

)

⁢

ϕ

i

⁡

(

-

x

,

-

y

)

]

.

8. A method of semiconductor device fabrication, comprising:

receiving a first mask design comprising a first mask function (a (x,y));

providing an exposure tool that includes a set of exposure conditions;

determining a transmission cross coefficient (TCC) of the exposure tool based on the set of exposure conditions;

decomposing the TCC into a plurality orders of eigenvalues (Xi) and a plurality orders of eigenfunctions (ϕ i (x,y));

calculating a kernel (Ω(x,y)) based on the following mathematical formulae:

Ω

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

*

(

0

,

0

)

⁢

ϕ

i

⁡

(

-

x

,

-

y

)

]

and

determining a first sub-resolution assist feature (SRAF) seed map (Γ(x,y)) by convoluting the kernel and the first mask function using the following mathematical formula:

Γ( x,y )= a ( x,y )⊗Ω( x,y ).

9. The method of claim 8 , further comprising:

determining a first sub-resolution assist feature (SRAF) seed map (Γ(x,y)) by convoluting the kernel and the first mask function using the following mathematical formula:

Γ( x,y )= a ( x,y )⊗Ω( x,y ),

wherein the first mask design is assumed to be implemented as an ideal mask.

10. The method of claim 8 ,

wherein the first mask function includes an X-X component (a xx (x,y)), an X-Y component (a xy (x,y)), a Y-X component (a yx (x,y)), and a Y-Y component (a yy (x,y)),

wherein the plurality orders of eigenvalues include a first plurality orders of X-X interaction eigenfunctions (ϕ i xx (x,y)), a second plurality orders of X-Y interaction eigenfunctions (ϕ i xy (x,y)), a third plurality orders of Y-X interaction eigenfunctions (ϕ i yx (x,y)), and a fourth plurality orders of Y-Y interaction eigenfunctions (ϕ i yy (x,y)),

wherein the kernel includes an X-Y kernel component (Ω xy (x.y)), a Y-X kernel component (Ω yx (x.y)), and a Y-Y kernel component (Ω yy (x.y)) respectively expressed by the following mathematical formulae:

Ω

XX

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

XX

*

(

0

,

0

)

⁢

ϕ

i

XX

⁡

(

-

x

,

-

y

)

]

,

⁢

Ω

X

⁢

Y

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

XY

*

(

0

,

0

)

⁢

ϕ

i

X

⁢

Y

⁡

(

-

x

,

-

y

)

]

,

⁢

Ω

YX

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

YX

*

(

0

,

0

)

⁢

ϕ

i

YX

⁡

(

-

x

,

-

y

)

]

,

and

Ω

YY

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

YY

*

(

0

,

0

)

⁢

ϕ

i

YY

⁡

(

-

x

,

-

y

)

]

.

11. The method of claim 10 , further comprising:

determining a first sub-resolution assist feature (SRAF) seed map (Γ(x,y)) by convoluting the kernel and the first mask function using the following mathematical formula:

Γ( x,y )= a xx ( x,y )⊗Ω xx ( x,y )+ a xy ( x,y )⊗Ω xy ( x,y )+ a yx ( x,y )⊗Ω yx ( x,y )+ a yy ( x,y )⊗Ω yy ( x,y ),

wherein the first mask design is assumed to be implemented as a real-world mask.

12. The method of claim 8 , wherein the set of exposure conditions comprises an illumination intensity of the exposure tool, a numerical aperture of the exposure tool, a thickness of a resist stack to be patterned, or a range of an aberration.

13. The method of claim 8 , further comprising storing the kernel in a memory medium.

14. The method of claim 8 , further comprising:

processing the first SRAF seed map to obtain a first SRAF map;

modifying the first mask design according to the first SRAF map to obtain a first modified mask design; and

performing photolithography using the exposure tool and the first modified mask design.

15. The method of claim 14 , wherein the processing of the first SRAF seed map comprises:

filtering the first SRAF seed map to remove noise, resulting in a filtered first SRAF seed map; and

fitting polygonal shapes onto the filtered first SRAF seed map.

16. A method of semiconductor device fabrication, comprising:

receiving a first mask design comprising a first mask function (a(x,y)), the first mask function (a(x,y)) including an X-X component (a xx (x,y)) and a Y-Y component (a yy (x.y));

providing an exposure tool that includes a set of exposure conditions;

determining a transmission cross coefficient (TCC) of the exposure tool based on the set of exposure conditions;

decomposing the TCC into a plurality orders of eigenvalues (Xi), a first plurality orders of X-X interaction eigenfunctions (ϕ i xx (x,y)), a second plurality orders of Y-Y interaction eigenfunctions (ϕ i yy (x,y));

calculating an X-X kernel component (Ω xx (x.y)) and a Y-Y kernel component (Ω yy (x.y)) based on the following mathematical formulae:

Ω

XX

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

XX

*

(

0

,

0

)

⁢

ϕ

i

XX

⁡

(

-

x

,

-

y

)

]

,

and

Ω

YY

⁡

(

x

,

y

)

=

Re

⁡

[

∑

i

=

1

N

⁢

λ

i

⁢

ϕ

i

YY

*

(

0

,

0

)

⁢

ϕ

i

YY

⁡

(

-

x

,

-

y

)

]

;

and

determining a first sub-resolution assist feature (SRAF) seed map (Γ(x,y)) by convoluting the kernel and the first mask function using the following mathematical formula:

Γ( x,y )= a xx ( x,y )⊗Ω xx ( x,y )+ a yy ( x,y )⊗Ω yy ( x,y ).

17. The method of claim 16 ,

wherein the exposure tool comprises a pupil,

wherein the X-X kernel component comprises a simulated interaction between an X-polarized radiation on the first mask design and an X-polarized radiation on the pupil,

wherein the Y-Y kernel component comprises a simulated interaction between a Y-polarized radiation on the first mask design and a Y-polarized radiation on the pupil.

18. The method of claim 16 , further comprising:

processing the first SRAF seed map to obtain a first SRAF map;

modifying the first mask design according to the first SRAF map to obtain a first modified mask design; and

performing photolithography using the exposure tool and the first modified mask design.

19. The method of claim 18 , wherein the processing of the first SRAF seed map comprises:

filtering the first SRAF seed map to remove noise, resulting in a filtered first SRAF seed map; and

fitting polygonal shapes onto the filtered first SRAF seed map.

20. The method of claim 16 , wherein the exposure tool comprises an extreme ultraviolet (EUV) exposure tool or a deep ultraviolet (DUV) exposure tool.