System and method for film stress and curvature gradient mapping for screening problematic wafers

1. A method of testing a wafer during semiconductor manufacturing, comprising:

forming a current top layer over device structures for integrated circuit chips being formed on the wafer;

collecting stress data for the wafer after the forming of the current top layer, the stress data being derived from changes in wafer curvature;

calculating a stress gradient vector for each area of the set of finite areas;

calculating a norm for each stress gradient vector for each area of the set of finite areas; and

identifying the finite areas having its stress gradient vector norm greater than a predetermined high-stress-gradient-vector-norm threshold.

2. The method of claim 1 , further comprising rejecting the wafer as defective if a number of the finite areas having its stress gradient vector norm greater than a predetermined high-stress-gradient-vector-norm threshold exceeds a predetermined threshold.

3. The method of claim 1 , wherein the predetermined high-stress-gradient-vector-norm threshold is based on an average stress gradient vector norm for the finite areas of the wafer.

4. The method of claim 1 , wherein the predetermined high-stress-gradient-vector-norm threshold is based on a moving average stress gradient vector norm across a certain set of wafers tested.

5. The method of claim 1 , wherein the stress data comprises:

stress-xx in an x direction and stress-yy in a y direction for each area of a set of finite areas on the wafer, the stress-xx and stress-yy both being derived from wafer-curvature-change-xx in the x direction for each area of the set of finite areas and from wafer-curvature-change-yy in the y direction for each area of the set of finite areas, wherein an x-y plane is along a top surface of the wafer and wherein the x direction is orthogonal to the y direction; and

stress-xy being a shear stress in the x-y plane for each area of the set of finite areas, the stress-xy being derived from wafer-curvature-change-xy, wherein wafer-curvature-change-xy is a change in wafer twist in the x-y plane for each area of the set of finite areas.

6. The method of claim 5 , wherein the stress data is calculated using the following equations:

σ

xx

=

E

s

·

t

s

2

6

·

t

f

⁡

(

1

-

v

s

2

)

⁢

(

Δ

⁢

⁢

κ

xx

+

υ

s

⁢

Δ

⁢

⁢

κ

yy

)

σ

yy

=

E

s

·

t

s

2

6

·

t

f

⁡

(

1

-

v

s

2

)

⁢

(

Δ

⁢

⁢

κ

yy

+

υ

s

⁢

Δ

⁢

⁢

κ

xx

)

σ

xy

=

E

s

·

t

s

2

12

·

t

f

⁡

(

1

+

υ

s

)

⁢

(

Δ

⁢

⁢

κ

xy

)

where

σ xx is the stress-xx in the x direction;

σ yy is the stress-yy in the y direction;

σ xy is the stress-xy in the x-y plane;

E s is a Young's Modulus of a substrate of the wafer;

v s is a Poisson ratio of the wafer substrate;

t s is a thickness of the wafer substrate;

t f is a thickness of the accumulated investigating layer (single or multiple layers);

Δκ xx is the wafer-curvature-change-xx in the x direction;

Δκ yy is the wafer-curvature-change-yy in the y direction; and

Δκ xy is the wafer-curvature-change-xy in the x-y plane.

7. The method of claim 6 , wherein the calculating of the stress gradient vectors is performed using the following equation:

∇

σ

=

(

∂

σ

xx

∂

x

+

∂

σ

xy

∂

y

)

⁢

e

→

i

+

(

∂

σ

xy

∂

x

+

∂

σ

yy

∂

y

)

⁢

e

→

j

where

∇σ is the stress gradient vector for each area of the set of finite areas;

{right arrow over (e)} i is a unit vector in the x direction; and

{right arrow over (e)} j is a unit vector in the y direction.

8. The method of claim 7 , wherein the calculating of the norm for each stress gradient vector is performed using the following equation:



∇

σ



=

(

∂

σ

xx

∂

x

+

∂

σ

xy

∂

y

)

2

+

(

∂

σ

xy

∂

x

+

∂

σ

yy

∂

y

)

2

where

|∇σ| is the norm for each stress gradient vector.

9. A method of testing a wafer, comprising:

forming a current top layer over the wafer;

collecting stress data for the wafer after the forming of the current top layer, the stress data being derived from changes in wafer curvature;

calculating a stress gradient vector for each area of the set of finite areas; and

evaluating the forming of the current top layer based on the stress gradient vectors.

10. The method of claim 9 , wherein the stress data comprises:

stress-xx in an x direction and stress-yy in a y direction for each area of a set of finite areas on the wafer, the stress-xx and stress-yy both being derived from wafer-curvature-change-xx in the x direction for each area of the set of finite areas and from wafer-curvature-change-yy in the y direction for each area of the set of finite areas, wherein an x-y plane is along a top surface of the wafer and wherein the x direction is orthogonal to the y direction; and

stress-xy being a shear stress in the x-y plane for each area of the set of finite areas, the stress-xy being derived from wafer-curvature-change-xy, wherein wafer-curvature-change-xy is a change in wafer twist in the x-y plane for each area of the set of finite areas.

11. The method of claim 9 , further comprising:

calculating a norm for each stress gradient vector for each area of the set of finite areas; and

evaluating the forming of the current top layer based on stress gradient vector norms.

12. The method of claim 11 , further comprising determining a high-stress-gradient-vector-norm threshold for use in testing during manufacturing based on the stress gradient vector norms.

13. The method of claim 11 , wherein the predetermined high-stress-gradient-vector-norm threshold is based on an average stress gradient vector norm for the finite areas of the wafer.

14. The method of claim 11 , wherein the predetermined high-stress-gradient-vector-norm threshold is based on a moving average stress gradient vector norm across a certain set of wafers tested.

15. A method of troubleshooting to find a cause of defects in semiconductor manufacturing, comprising:

testing a wafer after various suspected defect-causing steps in a semiconductor manufacturing process, each testing comprising:

collecting stress data for the wafer after the forming of a current layer being tested, the stress data being derived from changes in wafer curvature;

calculating a stress gradient vector for each area of the set of finite areas;

calculating a norm for each stress gradient vector for each area of the set of finite areas; and

identifying the finite areas having its stress gradient vector norm greater than a predetermined high-stress-gradient-vector-norm threshold; and

determining which of the suspected defect-causing steps is causing stress gradient vector norms for the wafer to be higher than the predetermined high-stress-gradient-vector-norm threshold for a given layer.

16. The method of claim 15 , wherein the stress data comprises:

stress-xx in an x direction and stress-yy in a y direction for each area of a set of finite areas on the wafer, the stress-xx and stress-yy both being derived from wafer-curvature-change-xx in the x direction for each area of the set of finite areas and from wafer-curvature-change-yy in the y direction for each area of the set of finite areas, wherein an x-y plane is along a top surface of the wafer and wherein the x direction is orthogonal to the y direction; and

stress-xy being a shear stress in the x-y plane for each area of the set of finite areas, the stress-xy being derived from wafer-curvature-change-xy, wherein wafer-curvature-change-xy is a change in wafer twist in the x-y plane for each area of the set of finite areas.

17. The method of claim 15 , wherein the predetermined high-stress-gradient-vector-norm threshold is based on an average stress gradient vector norm for the finite areas of the wafer.

18. The method of claim 15 , wherein the predetermined high-stress-gradient-vector-norm threshold is based on a moving average stress gradient vector norm across a certain set of wafers tested.