fracture analysis and distribution of surface cracks in...

S. SaffarDepartment of Structural Engineering,

Norwegian University of Science

and Technology,

Trondheim NO-7491, Norway

S. GouttebrozeSINTEF Materials and Chemistry,

Oslo NO-0314, Norway

Z. L. Zhang1

Department of Structural Engineering,

Norwegian University of Science

and Technology,

Trondheim NO-7491, Norway

e-mail: [email protected]

Fracture Analysis andDistribution of SurfaceCracks in MulticrystallineSilicon WafersSolar silicon wafers are mainly produced through multiwire sawing. The sawing processinduces micro cracks on the wafer surface, which are responsible for brittle fracture.Hence, it is important to scrutinize the crack geometries most commonly generated in sili-con wafer sawing or handling process and link the surface crack to the fracture of wafers.The fracture of a large number of multicrystalline silicon wafers has been investigated bymeans of 4-point bending and twisting tests and a failure probability function is presented.By neglecting the material property variation and assuming that one surface crack is dom-inating the wafer breakage, 3D finite element models with various crack sizes (depth,length, and orientation) have been analyzed to identify the distribution of surface crackgeometries by fitting the failure probability from the experiments. With respect to the 63%probability, the existing surface cracks in the wafers studied appear to have depth andlength ratios less than 0.042 and 0.19, respectively. Furthermore, it has been shown thatthe surface cracks with depth in the range from 10 to 20 lm, length up to 10 mm andangles in the range of 30 deg–60 deg, can be considered as the most common crackgeometries in wafers we tested. Finally, it has been found that the mechanical strength ofthe wafers tested parallel to the sawing direction is approximately 15 MPa smaller thanthose tested perpendicular to the sawing direction. [DOI: 10.1115/1.4025972]

Keywords: brittle fracture, surface crack, silicon wafer, sawing direction, probability

1 Introduction

Cracking of solar cells is one of the major sources of solar mod-ule failure and rejection. Therefore, it is not only important toinvestigate the electrical properties of silicon solar cells but alsothe mechanical properties, especially the strength. The fracturestrength and the mechanism of breakage have to be understood inorder to minimize the fracture rate and to optimize the processsteps. Although the surface roughness is less important for thicksections, it becomes crucial in thin wafers, and sometimes it canbe considered as initial surface crack. Some studies have focusedon the relation between the silicon wafer strength and surfaceroughness. They showed that the wafer strength increase signifi-cantly by polishing and different types of etching treatments[1–6]. The fracture stress of individual solar silicon wafer is gov-erned by the surface damage depth [7] and the deepest microcrackon the wafer surface or edges dominate the failure. In particular,microcracks at the edges induced by sawing are frequently deeperand can deteriorate the strength dramatically [8–10]. Dependingon the purpose, different modes of fracture tests such as 4-pointbending and twisting can be carried out. The 4-point bending testexerts a homogeneous stress between the inner bars, both on thesurface and specimen edges [11–13]. In contrast, the twisting testis more sensitive to the surface defects [14]. The two tests investi-gate the fracture behavior of different populations of cracks. Bothtests lead to large deflections of specimens and therefore analyti-cal solutions for fracture strength from measured forces cannot beutilized [15].

In this study, both 4-point bending and twisting have been car-ried out to identify the critical surface crack geometries of solarsilicon wafers by comparing the 3D finite element analysis resultswith the experimental ones. The paper is organized as follows: thetheoretical aspects of the study including the failure probability,material properties, and models for thin silicon wafers are pre-sented first. The experimental setup, parameter study, results anddiscussion are presented in Sec. 4. Some concluding remarks aregiven at the end of paper.

2 Failure Probability Theory

The fracture strength of brittle solids is ultimately controlled bythe presence of defects that act as stress concentrators. Defects areusually distributed randomly and, as a consequence, the strengthshows scatter and requires statistical treatment. Weibull theory[16] of brittle fracture is based on a weakest link hypothesis,according to which the strength of a body involves the product ofsurvival probabilities of the individual volume elements, so thatfailure of the whole body will take place as soon as the materialstrength is surpassed at one element containing the critical defect.Weibull distribution assumed a power-law probability functionfor the survival of the elements, which is given by Ref. [17] asfollowing well-known equation:

Pf ¼ 1� exp � rF

r0

� �m� �(1)

where Pf is the probability of failure2 at the applied fracture stress,rF, and is described by the material parameter m and the parame-ter r0, which are depends on the material and also on the size

1Corresponding author.Contributed by the Solar Energy Division of ASME for publication in the

JOURNAL OF SOLAR ENERGY ENGINEERING. Manuscript received July 2, 2013; finalmanuscript received October 31, 2013; published online December 19, 2013. Assoc.Editor: Santiago Silvestre.

2Usually, for the ith strength value Pf,i¼ (i� 0.5)/N, where N is the total numberof measurements.

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effect of the component and the stress distribution in the compo-nent (for detail see Sec. 8.3 of Ref. [17]).

The parameters r0 and m can be determined from experimentalstrength data using a Weibull plot [18]. Briefly, a Weibull plot isconstructed by assigning a probability of failure, Pf, to each exper-imental strength value, rF, and then plotting ln (ln (1/(1�Pf)))versus ln (rF). According to Eq. (1), a linear fit allows the charac-teristic strength and Weibull modulus to be determined. Equation(1) gives the failure probability with only one active defectpopulation (unimodal strength distribution).

In twisting test due to the complexity of deformation, the forceat the moment of fracture (FF) and the correspond displacement(DispF) are measured instead of stress. Therefore, the F0 is usedas strength parameter instead of r0 for the twisting test.

3 Finite Element Models

The dimensions of the silicon wafers studied in the presentstudy are 156� 156� 0.18 mm3. 20-node quadratic brick ele-ments (C3D20R) in ABAQUS-standard have been used to model thewafer. Surface cracks with different geometries are placed in thecenter of silicon wafer with orientation (b) varying from 0 deg to90 deg (see Fig. 1). The following assumptions have been used:

• Crack plane is perpendicular to the wafer surface.• The entire wafer fractures once a single crack starts to

propagate.• No interaction between the cracks with the wafer edge and

with each other.• Although silicon is an anisotropic material, it is considered in

this study as a linear elastic isotropic material. The values ofYoung’s modulus and Poisson ratio are taken asE¼ 162.5 GPa and t¼ 0.223, respectively [19].

The parameters to be varied are shown in Table 1. It can beseen from the table, in total 9� 5� 5 (numbers in crack depth,crack length, and crack orientation) simulations are used to coverall the parameters for every loading case (4-point bending andtwisting). The J-integral is computed from the well-designed finite

Fig. 1 Finite element model for (a) four-point bending (b) twisting

Table 1 Crack parameters

Crackdepth-a (lm)

Cracklength-c (mm)

Crackangle-b (deg)

5 0.5 0 (parallel the sawing direction)10 1 3012 10 4516 50 6020 100 90 (perpendicular to

the sawing direction)40 — —80 — —120 — —160 — —

021024-2 / Vol. 136, MAY 2014 Transactions of the ASME


element mesh around the crack tip. Figure 1 shows a typical crackmodel for a crack with 20 lm depth, 1 mm length and zero angle(b¼ 0) used for both 4-point bending and twisting loading. In the4-point bending case, due to symmetry only one quarter of theoverall model is applied. However, a whole model is necessaryfor the twisting loading (Fig. 1). The wafer was restrained in thevertical direction at the points of support and loading is applied bydisplacement control. The analysis accounts for geometric nonli-nearity associated with large deformation of thin wafers. Prelimi-nary analysis showed that the differences in the results obtainedfrom linear and nonlinear simulations are sufficiently large toindicate that a linear simulation is not adequate for these wafers.Similar conclusion was also drawn in Ref. [20]. Although

geometrical nonlinearity is considered, the simulation times forthese models are reasonable (less than 1 h on a single W3670,3.2 GHz processor machine).

From the numerical results, the J-integral, as a key fracture pa-rameter is used to compare with critical J-integral (JIC). In thisstudy, JIC¼ 0.00446 N/mm value is calculated from the fracturetoughness, KIC, (0:7560:06 MPa�

ffiffiffiffimp

for poly crystalline sili-con material [21]) by using the J ¼ G ¼ K2

I =E� (where, KI is thestress intensity factor and E*is the elastic modulus of the siliconwafer). This value is used in all simulations as a failure criterion.

4 Experimental Setup

Since silicon material is brittle and it is difficult to prepare testspecimens of required geometry and grip the samples withoutfracturing, the tensile properties cannot be easily determined by atensile test. Therefore, bending or twisting test is commonlyapplied to test the silicon wafer [22–24]. A set of 160 multicrystal-line silicon wafers of dimensions 156� 156 mm2 with averagethickness of 180 lm, was manufactured by and tested with twodifferent set-ups at REC (wafer and solar module manufacturer inPorsgrun, Norway) in the frame of a collaborative project.

In our study, wafer breakage force and displacement are meas-ured using a customized fixture attached to a 10 N load cell, in auniversal testing machine (Zwick/Roell, Zwick Material Testing).Tests were performed with a cross head speed of 25 mm/min. Thedetails about the two different loading geometries are:

(i) 4-point bending test: cylinders of 6 mm diameter made ofstainless-steel were used as supports. The relatively largediameter was chosen in order to avoid high contact stressesduring loading. The outer span length selected was140 mm, and the inner span length was 80 mm. The detailof the experimental setup and wafer positions can be seenin Fig. 2(a). The specimens were separated into 2 groups,one tested parallel and another perpendicular to the sawingdirection.

(ii) Twisting test: hemispheres of 10 mm diameter were used.The load and support points were located 35.6 mm awayfrom the edges. Figure 2(b) shows the detail of the twistingtest fixture and the different sides of wafer. The same as 4-point bending test, the specimens were tested in 2 groups(A and B directions). Two pins which are located on theback of the wafer keep fixed and two pins which arelocated on the top of the wafer move vertically toward thesurface of the wafer (the position of the pins is shown inFig. 1(b)).

5 Results and Discussion

The multicrystalline silicon wafer is modeled with different ma-terial orientations to investigate the effect of crystalline on staticloading. Figure 3 shows the built model from the realistic multi

Fig. 2 Experimental setups for (a) four point bending and (b)twisting test

Fig. 3 Built model from the realistic MC silicon wafer

Journal of Solar Energy Engineering MAY 2014, Vol. 136 / 021024-3


crystalline silicon wafer used for a static 4-point bending test. Theeffect of anisotropy on the stress ratio (average stress distributedin anisotropic wafer divided over the average stress distributed inisotropic wafer) has been studied. An approximate 4% discrep-ancy is observed between different cases (50 different anisotropiccases) with different material orientation. Therefore, the isotropicsilicon model works as well as the anisotropic multicrystalline sil-icon wafer.

Figure 4(a) shows the stress distribution (von Mises) in thewafers under 4-point bending when upper rods moved 15 mmdown (normal to the wafer). The highest tensile stresses can befound along the edges and on the lower surface between the innerspan (space between two upper rods). This loading configurationtests both the surface and edge. Figure 4(b) depicts that the higheststresses (shear stress) in a twisting test occur within the middle ofthe wafers, far from the edges. Hence, this loading configurationaims to test the defects located on the surface of the wafers. In

particular, Fig. 4, displays a uniform tensile stress field in middleof the wafer (1 quarter was modeled) in the 4-point bending and auniform shear stress in the twisting test. The J-integral versustime (displacement) is presented in Fig. 5 for the same crack ge-ometry under bending. With Fig. 5, the breakage displacement12.05 mm can be calculated by using JIC¼ 0.00446. The sameapproach has been used to calculate the breakage displacement forall the crack sizes considered in Table 1 under both the 4-pointbending and twisting. In Sec. 5.1, the breakage displacementsobtained from numerical analyses will be compared with thosefrom experiments to determine the distribution of the crackgeometry.

Because of statistical nature, failure probability, and frequencydistribution of breakage displacements are also obtained by com-paring the experimental data and to compare with the finite ele-ment method (FEM) results.

5.1 Failure and Crack Existence Probabilities. In order tocalculate the failure probability, Pf, the material parameters, mand r0 must be determined first. The material parameters obtainedfrom the Weibull plot, as shown in Fig. 6, are presented in Table 2for the bending and twisting data sets (mixed with sawing direc-tion). By substituting these parameters into Eq. (1), the failureprobability versus displacement curves are shown in Fig. 7 forboth 4-point bending and twisting. In this study, by fitting the fail-ure probabilities 3D finite element analyses of surface crackedwafers are employed to find the distribution of the most commoncrack geometries. From one hand, each breakage displacementhas an associated failure probability with respect to the experi-ment results. On the other hand in a deterministic way, each crack

Fig. 5 J-integral versus time (displacement) for a crack with1 mm length, 20 lm depth, and zero direction (in parallel direc-tion to the sawing)

Fig. 6 Material parameters fitted for (a) four point bending and(b) twisting

Fig. 4 Stress distribution in specimens under (a) von Misesstress for four-point bending and (b) shear stress for twisting

Table 2 Material parameter identification for both 4-pointbending and twisting tests

Test type r0 m

Bending 12.504 15.951Twisting 7.5037 14.425



geometry breaks at a characteristic displacement. However, onecharacteristic displacement will correspond to a group of crackgeometries. Here, we assign the group of crack geometries whichbreaks at a specific displacement calculated from FEM models to

Fig. 8 Probability of existing crack at b 5 0 deg for (a) fourpoint bending and (b) twisting loading

Fig. 7 Failure probability versus displacement for (a) fourpoint bending and (b) twisting





the failure probability determined from the experiments on thatspecific displacement. In this regards, the breakage displacementscalculated from FEM are linked to the associated value fromexperiments and a corresponding failure probability is obtainedfor each crack geometry. For instance, the failure probability ofthe wafer broken at 12 mm in a 4-point bending test is 0.39. Oneof the crack geometries obtained from the simulation whichbreaks at 12 mm is a crack with depth, length and angle of 20 lm,10 mm, and b¼ 0 deg. Therefore, if this crack geometry exists inthe wafer, its failure probability would be 0.39. In other words,the probability of the finding this crack geometry is 0.39 in thewafer. Hence, the probability of existing crack can be used insteadof the failure probability. A large set of calculations have beencarried out and Figs. 8–12 show the probability of existing crackversus crack depth and length ratios at different crack angles forboth the 4-point bending and twisting tests. By comparing the 4-point bending results shown in Figs. 8–12, it can be seen that theprobability of finding larger crack geometries increases with theincrease of crack angles. 63% failure probability is a well-knownparameter to evaluate the mechanical strength of the siliconwafers. In this regard, as shown in Fig. 13, the maximum crackdepth ratio (crack depth over the thickness of wafer) at b¼ 0 degis 0.042 for 63% failure probability under 4-point bending, whilethe same value is 0.146 at b¼ 90 deg. The twisting results of Figs.8–12 indicate that the probability of finding deeper but shortercracks is higher at both b¼ 0 deg and b¼ 90 deg, and the proba-bility of existing longer but shallower cracks is higher atb¼ 30 deg and 60 deg. Also, the minimum crack geometry occursat b¼ 45 deg. For the 63% probability of existing crack, the maxi-mum crack depth and length ratios (crack length over the width ofwafer) are 0.15 and 0.2 for b¼ 0 deg or 90 deg, respectively. But,they are 0.05 and 0.641 for b¼ 45 deg (Fig. 13). Indeed, it can be



Fig. 13 Crack geometry for the 63% probability of existingcrack in wafers



understood from the Figs. 8–13 that the breakage of the wafer ismore sensitive to the crack depth than crack length. It is observedthat the crack length up to 10 mm has no significant effect on thebreakage displacement. This phenomenon is due to the muchlarger width/length than thickness of the wafer. The stress concen-tration varies from b¼ 0 deg to b¼ 90 deg and catching its mini-mum value at b¼ 45 deg. In order to be able to find thecorrelation between the 4-point bending and twisting results, it isassumed that all wafers have the same systematic group of distri-bution of flaws. With this assumption, it can be expected that theintersection area between the 4-point bending and twisting resultsindicates the common crack geometries. Therefore, the hatchedarea in Fig. 13, represents that the 63% failure probability of thestudied wafers are able to have the cracks with depth and lengthratios less than 0.042 and 0.19, respectively.

5.2 Effect of Sawing Direction on Failure. In order to inves-tigate the effect of sawing direction on the mechanical strength ofthe silicon wafers, the probability of fracture Pf is calculated sepa-rately for parallel and perpendicular to the sawing direction forboth 4-point bending and twisting tests. The correspondingmaterial parameters, m and r0 obtained in both parallel andperpendicular to the sawing directions for the 4-point bending andA and B directions (see Fig. 2(b)) for the twisting test are shownin Table 3. By employing these parameters, the failure probabilityversus stress/force and displacement are displayed in Fig. 14. Itcan be observed from Fig. 14(a) that fracture stress and displace-ment corresponding to the parallel with the sawing direction areapproximately 15 MPa and 1.5 mm smaller than corresponding

ones for perpendicular direction. Therefore, the wafers are stron-ger in perpendicular than parallel direction under 4-point bendingtest. It is interesting to note that no significant differences in fail-ure probability between the parallel and perpendicular directionscan be found for the twisting test, Fig. 14(b). Because the appliedstress has the same configuration for parallel and perpendicular tothe sawing directions in twisting.

5.3 Frequency Distribution of Displacement. Analysis offrequency distribution of breakage displacement is more helpfulto understand the distribution of the generated cracks during theproduction and handling process. In order to calculate the fre-quency distribution of the breakage displacement, each 1 mm dis-placement is treated as one group and the number of brokenwafers is the corresponding frequency distribution for that group.For instance, 0–1 mm displacement is considered as group 1, dis-placements which are in 1–2 mm range are considered as group 2,and so on. By using this definition, the value of each group isdetermined by the number of breakages in that group. For exam-ple, group 14 is the relevant group for those wafers which havebeen broken at displacement between 13 and 14 mm. This defini-tion has been applied to all broken wafers. Table 4 displays thefrequency distribution of displacement group for both 4-pointbending and twisting tests. It is observed from Table 4 that themaximum frequency distribution takes places at groups 13 and 8(displacement between 12–13 mm and 7–8 mm) for the 4-pointbending and twisting tests, respectively. Therefore, we may con-clude that the corresponding crack geometries to these groups arethe most common crack geometries in silicon wafers tested.

Table 3 Material parameter (based on displacement data) identification for both 4-point bending and twisting tests

Test type Stress data Displacement data

r0 for bending and F0 for twisting m r0 m

Bending Perpendicular to the sawing direction 155.786 47.854 13.02 29.182Parallel the sawing direction 146.924 20.824 11.73559 19.385

Twisting A direction 3.457247 15.744 7.547654 17.093B direction 3.380341 12.483 7.456113 12.508

Fig. 14 Failure probability of stress/force and displacement for (a) four point bending and (b)twisting



Although there is difference between breakage displacementsdetermined by 4-point bending and twisting tests, some similarflaws exist in silicon wafers due to the same production and han-dling condition. Consequently, the common crack geometries intested silicon wafers are covered by the intersection of crack geo-metries between 4-point bending and twisting results. The com-parison between frequency distribution of breakage displacements

in Table 4 shows that the crack geometries with depth in range of10–20 lm, length shorter than 10 mm and direction in range of0 deg–90 deg is located in the intersection area between the 4-point bending and twisting. It is interesting to note that the fre-quency distribution of those displacements which are in range of30 deg–60 deg is higher for twisting (see Table 4). Finally, thecrack geometries with the depth in range of 10–20 lm, length up

Table 4 Crack geometries corresponding to the breakage displacement for both 4-point bending and twisting tests

Crack depth (lm)Range (lm)

Crack length (mm)Range (mm)

Angle(deg)

Displacement (mm)Range (mm)

Fracture Distribution

Simulation Experiment

Four point bending 1–2 0 0Twisting 160 100 45 1 0Four point bending 2–3 0 0Twisting 100 50–100 45 10 0

160 50–100 30 and 60160 0.5–50 45

Four point bending 3–4 0 0Twisting 100 0.5–10 45 11 0

160 100 0 and 90160 0.5–10 30 and 60

Four point bending 160 0.5–100 0 4–5 6 0160 100 30

Twisting 40–80 0.5–100 45 12 0100 100 30

Four point bending 160 0.5–50 30 5–6 4 0Twisting 80 50–100 30 and 60 20 4

100 0.5–50 30 and 60160 0.5–50 0 and 90

Four point bending 100 100 0 6–7 6 0160 0.5–100 45

Twisting 5–16 50–100 0 and 90 66 2820–100 0.5–100 0 and 90

40 0.5–100 30 and 6080 0.5–10 30 and 60

Four point bending 100 0.5–50 0 7–8 10 0100 1–100 30160 100 60–90

Twisting 10–20 0.5–100 30–60 84 3910–20 0.5–10 0 and 90

Four point bending 160 0.5–50 60–90 8–9 8 05 0.5–100 30–60

Twisting 5 0.5–10 0 and 90 21 95 0.5–100 30–60

Four point bending 40–80 100 0–30 9–10 20 0100 0.5 30100 0.5–100 45–90

Four point bending 20 100 0 10–11 29 1340 0.5–50 080 0.5–50 0–3040 50 3040 100 4580 0.5–100 45–6080 1–100 90

Four point bending 10–16 50–100 60 11–12 49 2120 10–50 020 50–100 3040 0.5–10 3020 50–100 45–6040 0.5–50 45–9040 100 60–9080 0.5 90

Four point bending 5 100 0 12–13 69 3210–20 0.5–10 0–9010–20 50–100 90

Four point bending 5 0.5–50 0–60 13–14 20 135 100 30–90

Four point bending 5 50 90 14–15 1 1Four point bending 5 0.5–10 90 15–16 3 0



to 10 mm and direction in range of 30 deg–60 deg can beconsidered as the most common cracks found in the silicon wafersstudied.

The same results as shown in Fig. 14 can be obtained by calcu-lating the frequency distribution for parallel and perpendicular tothe sawing direction for both 4-point bending and twisting tests(Fig. 15). As shown in Fig. 15(a), approximately, 1 mm differenceobserves between breakage displacement of parallel and perpen-dicular directions in 4-point bending test. In contrast, Fig. 15(b)shows small differences between parallel and perpendicular to thesawing directions for twisting.

6 Conclusions

This study focused on the correlation between surface crackgeometries and wafer fracture. The fracture behavior of multicrys-talline silicon wafers have been investigated by means of 4-pointbending and twisting tests. Finite element modeling has beenapplied to identify the corresponding surface crack geometriesand their failure probability by fitting the experimental results.The finding of the paper can be summarized as follows:

• With respect to the 63% probability of existing surfacecracks, the existing surface cracks in studied wafers usuallyhave depth and length ratios less than 0.042 and 0.19,respectively.

• The highest frequency distribution of breakage displacementtakes place between 12–13 mm and 7–8 mm for 4-point bend-ing and twisting tests, respectively.

• The surface cracks with depth in range of 10–20 lm, lengthup to 10 mm and angles in span of 30 deg–60 deg, can be con-sidered as the most common crack geometries occurred ininvestigated silicon wafers

• Finally, the mechanical strength of the wafers bent parallelthe sawing direction is approximately 15 MPa smaller thanthose bent perpendicular to the sawing direction.

Acknowledgment

The authors would like thank the NEXTGEN_SI projectfinanced by the REC group and Norwegian Research Council.Also the authors would like to express their sincere appreciationto Dr. Katrin Nord-Varhaug and Jon Aasrum of REC group fortheir help and carrying the experimental work.

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Fig. 15 Frequency distribution versus displacement (a) fourpoint bending for both parallel the sawing and perpendicular tothe sawing directions and (b) twisting test for both A and Bdirections



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