System and method to emulate finite element model based prediction of in-plane distortions due to semiconductor wafer chucking

Patent metadata

Field	Value
Publication number	US-9430593-B2
Application number	US-201313735737-A
Country	US
Kind code	B2
Filing date	Jan 7, 2013
Priority date	Oct 11, 2012
Publication date	Aug 30, 2016
Grant date	Aug 30, 2016

Abstract

Systems and methods for prediction of in-plane distortions (IPD) due to wafer shape in semiconductor wafer chucking process is disclosed. A process to emulate the non-linear finite element (FE) contact mechanics model based IPD prediction is utilized in accordance with one embodiment of the present disclosure. The emulated FE model based prediction process is substantially more efficient and provides accuracy comparable to the FE model based IPD prediction that utilizes full-scale 3-D wafer and chuck geometry information and requires computation intensive simulations. Furthermore, an enhanced HOS IPD/OPD prediction process based on a series of Zernike basis wafer shape images is also disclosed.

First claim

What is claimed is:

1. A computer implemented method for providing in-plane distortion (IPD) prediction, the method comprising:
(a) generating a plurality of sample wafer shapes, the plurality of sample wafer shapes generated by varying a set of coefficients B in a predefined shape equation;
(b) performing finite element (FE) model based IPD prediction for the plurality of sample wafer shapes, wherein IPD predictions with respect to x-axis (X-IPD) and IPD predictions with respect to y-axis (Y-IPD) are obtained for the plurality of sample wafer shapes;
(c) fitting equations to the FE model based X-IPD predictions and the FE model based Y-IPD predictions for the plurality of sample wafer shapes to obtain a set of coefficients A, wherein the set of coefficients A corresponds to the set of coefficients B of the shape equation used in step (a) to generate the plurality of sample wafer shapes;
(d) analyzing the set of coefficients B and the set of coefficients A to determine a mapping function ƒ suitable for calculating coefficients A based on coefficients B of the shape equation;
(e) calculating an interim shape for a given wafer utilizing the mapping function ƒ and a shape equation representing the initial shape of the given wafer; and
(f) performing IPD prediction for the given wafer based on the interim shape calculated for the given wafer.

2. The method of claim 1 , wherein the predefined shape equation for describing a wafer shape Z 0 is Z 0 (x,y)=b 1 +b 2 x 2 +b 3 xy+b 4 y 2 ,b i εB.

3. The method of claim 2 , wherein the equations to be fitted to the X-IPD and Y-IPD predictions include a pair of Taylor polynomials or a pair of Zernike polynomials.

4. The method of claim 2 , wherein the X-IPD and Y-IPD predictions are modified by dividing by a neutral surface factor before fitting the equations.

5. The method of claim 4 , wherein the equations to be fitted to the modified X-IPD and Y-IPD predictions include:

∂
z

∂
x

=

a
1

⁢
2
⁢
x

+

a
2

⁢
y

+

a
4

⁢
4
⁢

x
3

+

a
5

⁢
3
⁢

x
2

⁢
y

+

a
6

⁢
2
⁢
x
⁢

⁢

y
2

+

a
7

⁢

y
3

for fitting to the FE model based X-IPD prediction; and

∂
z

∂
y

=

a
2

⁢
x

+

a
3

⁢
2
⁢
y

+

a
5

⁢

x
3

+

a
6

⁢
2
⁢

x
2

⁢
y

+

a
7

⁢
3
⁢
x
⁢

⁢

y
2

+

a
8

⁢
4
⁢

y
3

for fitting to the FE model based Y-IPD prediction.

6. The method of claim 5 , wherein the mapping function ƒ is determined by solving equations:
a i =c 1 b 3 +c 2 b 2 2 +c 3 b 3 2 +c 4 b 4 2 +c 5 b 2 b 4 +c 6 b 2 b 3 +c 7 b 4 b 3 +c 8 b 2 3 +c 9 b 4 3 +c 10 b 2 2 b 3 +c 11 b 2 2 b 4 +c 12 b 3 2 b 2 +c 13 b 3 2 b 4 +c 14 b 4 2 b 3 +c 15 b 4 2 b 2

wherein the values of b j εB are known for each sample wafer shape, and the values of a i εA are determined by fitting the pair of polynomial equations to the modified FE model based X-IPD and Y-IPD predictions for each sample wafer shape.

7. The method of claim 6 , wherein calculating the interim shape for the given wafer further comprises:
calculating the values of coefficients B by fitting the predefined shape equation Z 0 (x,y)=b 1 +b 2 x 2 +b 3 xy+b 4 y 2 to the initial shape of the given wafer;
calculating the values of coefficients A based on the mapping function ƒ and the values of coefficients B; and
obtaining the interim shape Z as X(x,y)=a 1 x 2 +a 2 xy+a 3 y 2 +a 4 x 4 +a 5 x 3 y+a 6 x 2 y 2 +a 7 xy 3 +a 8 y 4 , a i εA.

8. The method of claim 7 , wherein predicting the X-IPD and Y-IPD for the given wafer based on the interim shape further comprises:
calculating a x-slope of the interim shape,

∂
z

∂
x

Assignees

• Kla Tencor Corp

Inventors

• Vukkadala Pradeep
• Veeraraghavan Sathish
• Sinha Jaydeep
• Chen Haiguang
• Kirk Michael

Classifications

• H10P72/0616

  Monitoring of warpages, curvatures, damages, defects or the like · CPC title

• G06F30/23 Primary

  using finite element methods [FEM] or finite difference methods [FDM] · CPC title

• G06F17/5018 Primary

  Physics · mapped topic

• H01L21/67288

  Electricity · mapped topic