Strength of Si Wafers with Microcracks: A Theoretical Model

Przemyslaw (Peter) Rupnowski and Bhushan Sopori

National Renewable Energy Laboratory • Golden, CO 80401

 Production yield losses resulting from wafer 

breakage can be as high as 5%–10% in a typical 

manufacturing facility. 

 Fracture strength becomes even more important

when the new thinner and large-area wafers are 

manufactured. 

 To successfully reduce silicon usage and maintain 

high production yield, one needs to understand the 

fracture behavior of silicon wafers. 

Introduction Objective

Classical Weibull equation describes the probability 

FV(σ) that the brittle specimen survives load σ:

where γV, αV, and ωV are the three parameters of the 

Weibull distribution.

In the case of the multiaxial stress state, σ represents 

position-dependent effective stress. 

Classical strength modeling

Weibull distribution

Decomposition into three

modes of failure

The weakest-link yields the effective probability of 

survival for the entire brittle specimen with bulk, 

surface and edge imperfections as:

or:

How to predict the Weibull parameters?

How to calculate the effective stress?

General strength model

Silicon wafers are manufactured by casting of a 

multicrystalline silicon (mc-Si) ingot followed block cutting, 

wafer sawing and etching. 

The material removal during sawing can be regarded as a 

series of micro-indentations that lead to subsurface 

microcracking. 

The surface of a post-sawn wafer contains deep subsurface 

damage (micro-cracks 10–70 μm deep).

Etching does not remove the entire layer with the sawing 

damage.

Due to the large surface area and relatively small volume, 

the volume failure mode can be neglected. 

If blocks are polished before slicing then the edge failure 

mode can be also neglected. 

Cast unpolished PV wafer

specific conditions

Calculation procedure:

Monte Carlo simulation

for strength of PV wafer

An example of 5" x 5” wafer with surface micro-cracks is shown. For the 

sake of legibility, the cracks are magnified 20 times in the figure. In our 

example the wafers were subjected to uni-axial tension.

Monte Carlo simulation

assumptions

Monte Carlo simulation

results

Comparison with experiment

 In our virtual experiment, the strength of wafers 

varied from 100 to 125 MPa

 The most probable strength value was 106 MPa. 

 The obtained distribution can be accurately fitted by 

the Weibull equation. 

 The predicted distribution compares well with the 

available experimental results from the literature. 

 The strength distribution predicted in this study fits 

between experimental distributions for the as-sawn and 

5-μm etched wafers. 

Discussion

1) A new analytical expression that takes into account 

the surface, edge, and bulk properties of a wafer has 

been proposed to describe the strength of the brittle 

materials. 

2) A new proposed fracture-mechanics numerical 

simulation successfully predicted the strength of the cast 

silicon wafers. 

3) It has been shown that the predicted wafer strength 

distribution agrees well with the available experimental 

results. 

ACKNOWLEDGMENTS

This work was supported by the U.S. Department of Energy under 

Contract No. DE-AC36-99GO10337 with the National Renewable Energy 

Laboratory.

Conclusions and Acknowledgments

The information contained in this poster is subject to a government license.

Presented at the 33rd IEEE Photovoltaic Specialists Conference • San Diego, CA • May 11–16, 2008 • NREL/PO-520-43306

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 − = − ∫

V V

V FV dV

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σ γ σ

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 − = − ∫

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L F dL

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σ γ (σ ) exp

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σ γ (σ ) exp

• Analytical description

• Numerical simulation

Develop model for the strength 

of photovoltaic (PV) wafers.

• Effect of micro-cracks generated during wafer sawing

Identify the features which limit 

the wafer strength

• How to increase wafer strength?

• How to handle fragile wafers?

Optimize handling and processing 

to improve production yield

Weibull fitting results:

αA=0.205 MPa m

ωA 2/

ωA=2.49

γA =109 MPa

Assumptions: 

a) γ0 for Si equals 2.47 J/m2

b) density of cracks is 0.32/cm2

c) semi-elliptical shape of cracks 

d) crack plane is perpendicular to the 

wafer surface

e) cracks length varies from 0 to 

20 μm and the depth up to 200 μm

f) wafer fractures once a single crack 

starts to propagate

g) cracks do not interact neither with 

the edge nor with other cracks

F

F

We generate a set of 100 virtual wafers. Each 

wafer contains 100 randomly distributed and 

oriented surface cracks. 

The maximum load for each wafer is 

calculated using the fracture-mechanics 

methods and the weakest-link principle. 

Statistical analyses of the results for all 100 

wafers are performed to obtain the strength 

distribution. 

We determine if the Weibull distribution can 

accurately fit the obtained results

Experimental data 

were take from:

[1] C. Funke, E. Kullig, 

M. Kuna, and H. J. 

Möller, "Biaxial fracture 

Test of Silicon Wafers," 

Advanced Engineering 

Materials, vol. 6, pp. 

594-598, 2004.

[2] H. J. Möller, C. 

Funke, M. Rinio, and S. 

Scholz, "Multicrystalline

silicon for solar cells," 

Thin Solid Films, vol. 

487, pp. 179-187, 2005.

strength models: