Wire-sawing processes: parametrical study and modeling

Solar Energy Materials & Solar Cells 132 (2015) 392–402

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Wire-sawing processes: parametrical study and modeling A. Bidiville a,b,1, K. Wasmer a,n,1, M. Van der Meer c, C. Ballif b a Empa, Swiss Laboratories for Materials Science & Technology, Laboratory for Advanced Materials Processing, Feuerwerkerstrasse 39, CH-3602 Thun, Switzerland b Ecole Polytechnique Fédérale de Lausanne (EPFL), Institute of Microengineering (IMT), Photovoltaics and Thin-Film Electronics Laboratory, Rue A.-L. Breguet 2, CH-2000 Neuchâtel, Switzerland c Applied Materials Switzerland, Precision Wafering Systems, Route de Genève 42, CH-1033 Cheseaux, Switzerland

art ic l e i nf o

a b s t r a c t

Article history: Received 25 July 2013 Received in revised form 8 September 2014 Accepted 15 September 2014

Reducing the wafer breakage rate without changing the wafer thickness and sawing thinner wafers while maintaining constant breakage rate are two possibilities to decrease the costs of solar cells. They are similar in the sense that both require stronger wafers. To achieve this goal, it is important to gain insight into the wire-sawing process, its underlying defect creation mechanisms and the impact of sawing parameters on wafer strength. Consequently, a series of bricks were sawn with different slurry densities, wire tensions and feed rates, and the results were analyzed in terms of the wafer strength measured by bending tests. Roughness and wafer thickness were also measured. It is found that the strongest wafers were obtained by using a low abrasive volume fraction in the slurry, a low wire tension and a slow feed rate. From the analyses, we provide a qualitative interpretation of the effects of the processes at work in slurry-based wafering that explains, for instance, the wafer thickness and roughness variations. Based on physical arguments about the interaction between the wire, the silicon carbide particles and the silicon wafer, a semi-empirical model relating defect creation to the sawing parameters is developed. With this model, the wafer strength distribution can be predicted, thus simplifying optimization of the sawing process. & 2014 Elsevier B.V. All rights reserved.

Keywords: Wire-sawing Silicon Wafer

1. Introduction Wire-sawing is the most common method to produce wafers for the photovoltaic industry. A wire-saw consists of a thin steel wire wound around wire-guides, forming a web of parallel wires (Fig. 1). Even though wires coated with diamond particles have started to be used industrially for sawing monocrystalline wafers, most wafers are cut with an abrasive slurry. The slurry, usually a suspension of silicon carbide (SiC) particles in polyethylene glycol (PEG), is poured on the wire web while the wire is driven through the web. The silicon block to be cut is then pushed through the web. The SiC particles carried by the wire wear the silicon away, hence sawing it into wafers. Due to the pressure for cost reduction over the last decade, wafer thicknesses have decreased from more than 300 to about 180 mm nowadays [1,2], while the wire diameter has also decreased. This has made the sawing process trickier, as thinner wafers and wires are more prone to breaking, and as a silicon brick is sawn into more wafers, the wear of the wire is more important. Therefore, research has been focused on producing

n

Corresponding author. Tel.: þ 41 58 765 62 71; fax: þ 41 33 228 44 90. E-mail address: [email protected] (K. Wasmer). 1 Both authors have contributed equally to this work.

http://dx.doi.org/10.1016/j.solmat.2014.09.019 0927-0248/& 2014 Elsevier B.V. All rights reserved.

stronger wafers, allowing thinner wafers to be sawn with reduced subsequent breakage, thus reducing silicon consumption [3,4]. At the wire-sawing stage, mechanical and geometrical parameters are useful to characterize the process. Whereas the electrical properties and the presence of precipitates are mostly determined by the solidification step, wire-sawing determines the wafer thickness, the wafer thickness variation, the roughness and the breakage stress. Another important aspect of wire-sawing is the breakage rate of wafers during or right after the cut. If, for a given sawing recipe, many wafers are broken before being processed into solar cells, this recipe cannot be deemed efficient as a large amount of high-quality material is lost. Although this aspect cannot be easily evaluated in a research environment as it requires sawing many full-load batches of wafers with the same recipe, it is reasonable to assume that if wafers are mechanically strong, fewer will break in the industrial process. To improve wafer quality, it is important to better understand the sawing process. Some groups rely on finite element simulations for insights into the interactions of the abrasive particle with the wire and the silicon block [5,6]. Such simulations focus on only a very small region (less than 200 mm in length), but still produce interesting results. However, these results depend largely on the assumptions made for the modeling: Wagner and Möller [5] applied a high contact force (5 mN per mm of wire, allegedly reflecting a moderate table speed) on the wire and showed that high contact force could be

A. Bidiville et al. / Solar Energy Materials & Solar Cells 132 (2015) 392–402

reached even if the wire-silicon distance was greater than the abrasive particle diameter, whereas Bierwisch et al. [6] showed that, in the so-called non-contact regime, the pressure of the abrasive particles on the silicon brick was low (in the range of the pressure applied by the wire, i.e. 0.15 N/mm2) and homogeneously distributed around the wire. By contrast, they found that, in the so-called semicontact regime, the force was located mainly at the top of the groove. Other researchers have used instrumented wire-saws to analyze the macroscopic forces present during the sawing process [7–9]. We have approached the problem from another perspective: by analyzing the wafer characteristics from a tribological point of view, we get insight into the material removal mechanisms and into the way the defects present in the wafers were created [10]. A study of the saw groove [11] showed that the sawing conditions are completely different at the top of the saw groove and on the sides (which later form the wafer surface, see inset in Fig. 1). It was also shown that, as the saw groove is thin and long, the sawing conditions change from the wire's entrance to its exit of the silicon brick. Similarly, it was demonstrated that the roughness and thickness variation is dependent of the SiC particle size. The larger the particle size, the larger are the roughness and thickness variation. A study of the effect of silicon debris confirmed these observations, as saw marks (abrupt thickness changes occurring in the form of lines parallel to the wire direction) first appeared at the wire exit side of the wafers when a concentration threshold is reached [10,12]. Furthermore, it was shown that small saw marks do not affect the wafer bending strength whereas large saw marks dramatically decrease it. It was demonstrated that roughness measurement is a good method for detecting saw marks, as roughness increases when saw marks are present. But when this increased roughness was smaller than the roughness at the wire entry side of the wafers, the wafers did not show a bending strength decrease. In this contribution, the effect of the sawing parameters (abrasive volume fraction, wire tension and feed rate) on the wafer bending strength, the surface roughness and the wafer thickness is analyzed. With these results, a microscopic model of the sawing process is developed. It is in agreement with the observations made in our previously published papers [11,12] and allows for a better understanding of the sawing process. From this parametrical study, it is found that the strongest wafers are sawn with a low abrasive volume fraction, a low wire tension and a slow feed rate. Moreover, by looking at the impact of the sawing parameters on wafer characteristics such as the wafer thickness variation, the roughness variation and the breakage stress, insight into the sawing process is gained. This investigation also enables a

393

better understanding of the particles' trajectories through the saw groove.

2. Material and experimental procedures A modified HCT wire-saw, dedicated for research and development, was used to saw 125  125 mm2 monocrystalline {100} silicon ingots with their edges along the o 1004 directions. The wire had a diameter of 140 mm and its pitch was 380 mm. The wire velocity was 11.5 m/s giving a total wire length of 230–250 km. Three size distributions of SiC abrasive were used: F600 (median size of 9.3 mm), F800 (median size of 6.5 mm) and a 1:1 mixture of F600 and F800, resulting in wafers with thicknesses of 200– 220 mm. To keep the experiments at a reasonable price, the bricks had only a length of 50 mm so that about 120 wafers were produced per cut. However, no influence on the results is expected. The slurry flow was 112 kg/min. After ungluing the wafers from the sawing support, the wafers were cleaned with water and dried. For each cut, the volume fraction of abrasive in the slurry, the wire tension and the sawing speed were investigated. Table 1 presents the sawing parameters used for each cut made with F800. The bricks cut with F600 and the mixture of F600 and F800 were cut according to parameters A–C in Table 1. For each sawing condition, the thickness of ten wafers chosen randomly across the whole brick was measured as illustrated in Fig. 2. Table 1 Sawing parameters for the cuts made with F800 abrasive. The volume fraction of abrasive in the slurry is VSiC, the wire tension is T and the feed rate is f. Cut

VSiC

T [N]

f [mm/min]

A B C D E F G H I J K L M

0.261 0.236 0.211 0.211 0.211 0.211 0.211 0.211 0.236 0.236 0.236 0.236 0.236

30.0 30.0 30.0 25.2 30.0 25.2 21.2 21.2 30.0 25.2 25.2 21.2 21.2

450 450 450 378 378 450 450 378 378 378 450 450 378

Fig. 1. Simplified view of a wire-saw. Silicon bricks are glued onto holders (not shown) and pushed progressively through the wire web as the wire is running and slurry is poured onto the web. The inset on the right shows a schematic view of the wire entering the silicon ingot. The created grooves separate the wafers at the end of the cut.

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particles entering and exiting the silicon ingot. The breakage forces measured were then converted into breakage stresses [15] and Weibull statistics [16] were carried out to compare the wafers sawn with different sets of parameters.

3. Results 3.1. Roughness and thickness measurements

Fig. 2. Schematic view of a wafer showing the lines where the roughness was measured and the points where the thickness was measured. For each sawing condition, the thickness was measured on ten wafers and the roughness on five different wafers.

The thickness was measured every centimeter on two lines parallel to the wire direction, one 40 mm away from the glued side of the wafer and one 40 mm away from the beginning side of the cut. These measurements were made with a mechanical sensor having a circular area with a diameter of 1 mm and a precision of 1 mm. For roughness measurements, five wafers, not used for thickness measurements, were measured for each sawing condition with a white light profilometer (COTEC Altisurf 500) as illustrated in Fig. 2. Roughness was measured on a length of 5.6 mm, as described in norm ISO 4287 [13]. The measurements were made every 5 mm perpendicularly to a line 40 mm under the glued edge of the brick. The roughness values of the five wafers were averaged for each position. The particle size distribution of the slurry was measured with a Malvern Mastersizer S. This instrument uses the scattering of laser light passing through a suspension of particles to determine the particle size distribution. Wafer strength was measured by bending tests. In order to have a statistically relevant wafer bending strength measurement, 50 wafers were bent until breakage for each sawing condition on a four-line bending rig, with the supports parallel to the wire direction during sawing. This test type has been considered in this study, as compared to tests such as biaxial and twist tests, since it has the largest area under constant tensile stress (that is between the inner lines). The two outer lines were spaced 100 mm apart, and the two inner lines were spaced 50 mm apart, giving a constant tensile area of 6'250 mm2. The tests were carried out on a UTS TESTSYSTEME apparatus. During the test, the displacement of the top rig and the applied force were measured until breakage with measurement precisions of the displacement and force on the range of 1 μm and 5 mN, respectively. However, great care must be taken prior to measuring wafer strength with a four-line test as wafer edges play an important role in the overall wafer strength [14]. Actually, Wasmer et al. [14] found that the wafer edges are more fragile than the wafer surface. As the aim of this work was to study the impact of the sawing parameters on the sawing mechanism and not simply the overall wafer strength, it was necessary to measure the breakage stress due to cracks on the wafer surface and not at the wafer edges. Thus, the wire entrance and exit edges of the wafers were polished before the bending test for two main reasons. First, the squaring of the ingot is a much coarser process than the sawing and so creating larger defects. Second, it is to remove any damage caused by the wire and

It is evident from Fig. 3 that wafer roughness decreases between the wire entrance and the wire exit side of the wafers; this observation is consistent with previous work [11]. Note that wafers sawn with a mixture of F600 and F800, and wafers sawn with F600 only, have comparable roughnesses. Furthermore, along the first 30 mm, the roughness of the wafers cut with F600 is higher than the one cut with F800. Coming close to the wire exit side of the wafers, the roughness of the wafers cut with F600 becomes similar to that of wafers cut with F800 abrasive. A comparison between the thickness of wafers sawn with a mixture of F600 and F800 and of those sawn only with F600 is presented in Fig. 4. In this figure, each line represents the mean thickness of ten wafers chosen randomly across the whole brick. This figure demonstrates that the fine portion of the abrasive size distribution (F800) has a marginal impact on wafer thickness. As a reference, the histogram of the measured particle size of F600 and F800 abrasive is shown in Fig. 5. 3.2. Wafer strength The Weibull plots of the wafers broken for the cuts presented in Table 1 are shown in Fig. 6. The cuts have been combined into different plots to show the effect of the sawing parameters tested on the wafer strength. As one cut can be compared to several different cuts to illustrate the effect of the different sawing parameters, all the cuts appear several times in the graphs. Some sawing parameters have a large effect on the Weibull modulus but a small effect on characteristic stress (e.g. changing the abrasive volume fraction, Fig. 6a), whereas other parameters do not have much impact on the Weibull modulus, but a sizeable impact on the characteristic stress (e.g. the wire tension with the low feed rate and low abrasive volume fraction, Fig. 6g). However, the amplitude of a given parameter impact also depends on the other parameters (e.g. changing the wire tension with a low abrasive volume fraction and high feed rate, Fig. 6e, has less effect than when the feed rate is low, Fig. 6g).

4. Discussion 4.1. Roughness and thickness measurements The thickness and roughness measurements suggest that only the coarsest particles are responsible for the damage at the wafer surface. The wafers sawn with a mixture of F600 and F800 demonstrate that, as they have a roughness and thickness comparable to the wafers sawn with just F600, only a small amount of particles is required to create these defects. The fact that the roughness and thickness change from the wire entrance to the wire exit side of the wafers proves that the active particle size decreases. Thus, it can be deduced that particles are either ejected from the cutting region, breaking down during the cutting process or a combination of both processes. However, there is an important difference between the roughness and the thickness measurements: whereas the thickness monotonically increases from the wire entrance side to the exit side, the roughness first

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Fig. 3. Roughness measurements for cuts with F600, F800 and a 1:1 mixture of F600 and F800 with constant tension and feed rate (T¼ 30.0 N – f¼ 450 μm/min) for abrasive volume fractions of (A) VSiC ¼ 0.261, (B) VSiC ¼ 0.236, and (C) VSiC ¼0.211.

Fig. 4. Mean wafer thicknesses at different positions from the wire entry for cuts with F600 and a 1:1 mixture of F600 and F800. These measurements were made according to Fig. 2. On wafers sawn at constant tension and feed rate (T ¼ 30.0 N – f¼ 450 μm/min) for various abrasive volume fractions of VSiC ¼ 0.261, VSiC ¼ 0.236, and VSiC ¼ 0.211. The thinnest region is found at the wire entrance side.

decreases close to the wire entrance side, but reaches a plateau around half of the wafer width. Analyzing the particle size distribution (Fig. 5), it is seen that the coarsest particles are very few, but finer particles are found in much larger quantities. Hence, as large particles are continuously ejected from the cutting area, there is a shift in the particle size distribution to the left of the distribution shown in Fig. 5. Obviously, to have a further shift of the particle size distribution to the left, the number of particles to be either ejected or broken down has to increase. Therefore, the shift of the distribution to smaller particles is not linear and this explains the sharp decrease in roughness close to the wire entrance side of the wafer and why the roughness reaches a plateau near the wire exit side of the wafer.

Fig. 5. Particle size histograms of the F600 and F800 abrasives before cutting.

The continuous increase in wafer thickness can be explained by also taking into account the volume fraction of particles in the slurry. Given that silicon is worn out when direct wire – abrasive particle – silicon contact occurs as illustrated in Fig. 7, the wire has to be held in place laterally by two or more particles on one side of the groove to apply a sufficiently large force on another particle on the other side of the groove to damage the silicon and remove some material. It is evident from Fig. 7 that by increasing the distance d between the two particles, the required bending of the wire will decrease so that the particles are able to pass around the wire in a narrower saw groove without removing some silicon. Consequently, as particles are ejected from the cutting area, their concentration decreases so that the wire bends less sharply around them, resulting in a smaller kerf leading to increased

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Fig. 6. Weibull plots of cuts made with F800 slurry using different parameters. (a) and (b) The effect of the abrasive volume fraction; (c) and (d) the effect of the feed rate; (e), (f), (g) and (h) the effect of the wire tension. The characteristic stress σ0 is the stress at which 63.2% of the wafers have broken and the Weibull modulus m represents the slope of the fitted lines. Both values are given between parentheses for every cut.

wafer thickness. In contrast, the roughness depends mainly on the particle size and not on particle concentration [10,11,17], and this explains why the roughness measurements reach a plateau (Fig. 4) whereas the thickness measurements show a continuous increase (Fig. 3).

The situation at the top of the sawing groove is different than on the groove sides [11] (the different positions are indicated on the schematic view of a silicon brick being sawn in Fig. 1). Indeed, at the top of the groove, a given particle indents the silicon with a smaller force as compared to the groove side. The main reason is

A. Bidiville et al. / Solar Energy Materials & Solar Cells 132 (2015) 392–402

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Fig. 8. Schematic cross-section of the wire pushing the particles from the top of the groove to the sides. Fig. 7. Schematic view from the bottom of a saw groove when a particle is indenting the silicon. The wire (with tension T) bends around particles, making an angle ϕ with the groove axis and pushing the particle in the middle with a force P. The distance between two particles on one side of the wire is defined as d.

that when the force on the particle becomes too large, it pushes the wire away from the cutting zone, thus limiting the maximal force applied on the particles. In contrast – and maybe counterintuitively – on the groove side, particles are ejected from the cutting zone on both sides of the wire (see Fig. 7), so that when the wire is blocked on one side by particles, there is nothing limiting the maximum force applied to a particle on the other side of the wire as the wire has no room to move away. However, the wear rate is far greater at the top of the groove than at the sides. Actually, based on Fig. 8, it becomes obvious that the wear rate decreases from a maximum value at the top of the groove to zero at the groove sides which becomes the wafer surface. This can be easily explained by considering the amount of silicon cut when the wire moves into the ingot by a given distance. Obviously, the amount of silicon cut at the top of the groove is far larger than at the groove sides. Therefore, there are few impacts made with a large force on the groove sides, in contrast to the top of the groove where there are many indentations made with a relatively small force [11]. However, it is important to keep in mind that the situation at the sides depends also on the situation at the top of the groove, since particles ejected from the top of the groove proceed to the sides, where they may indent the silicon before exiting the sawing region. The wafer thickness measurements in Fig. 4 show that, in some cases, the point closest to the wire exit edge (at 12 cm) is thinner than the one at 11 cm. This behavior cannot be explained by a change in abrasive size or concentration. However, it is explained by the wire vibration outside the silicon ingot. It was previously shown by Wei and Kao [18] that the wire vibrates between the wire-guide and the silicon ingot. Once the wire enters the silicon, the vibrations are quickly damped by the silicon groove, so that they have an impact only on the first and last few millimeters at the wafer edges. Hence, at the wafer edge, the vibration is not yet damped, allowing the wire to move laterally, thus taking more space than further in the ingot. The lateral movement of the wire might also push the abrasive particles harder into the silicon, creating larger defects. This phenomenon would then depend mainly on the wire tension and is one additional reason for polishing the wafer edges prior to the breakage tests. From all previous observations, it is possible to identify three fundamental aspects that influence the wafer strength: – The wire bends around particles. The less the wire bends, either due to a larger distance between particles or lower wire

tension, the lower the indentation force of the particles into the silicon, resulting in stronger wafers. – Particles are ejected from the sawing region. The higher the wire pressure at the top of the groove, the more particles are ejected. This depends on the feed rate and on the volume fraction of particles. – The wire vibrates outside the silicon ingot. The larger the vibrations, the more damage are made at the wire entrance and exit sides of the silicon. The wire tension mostly determines the vibration amplitude [18].

4.2. Characteristic stress Based on fracture mechanics principles, it is possible to relate the breakage stress with the crack depths of the wafers. Assuming a semi-circular crack shape, the crack size varies with the square of the breakage stress [19] (given that the crack orientation remains the same and as the wafers are monocrystalline, this is a reasonable hypothesis [10]). In this study, it is possible to illustrate this by considering the wafer series sawn with F800 which breaks typically at stresses ranging from 120 to 170 MPa as evident from Fig. 6. Translated into crack size, assuming the case of a semicircular crack, this gives typical crack sizes from 18 to 39 μm. Obviously, the strongest wafers have a breakage-inducing crack that is half the size of that in the weakest wafers. This shows that the size of the largest crack present inside a wafer can vary considerably from one wafer to the next. Fig. 6 presents Weibull plots of wafer strength for several batches sawn with F800 while varying the wire tension, feed rate and abrasive volume fraction. Wafer strength cannot be characterized by a single value, as wafers break within a range of values, usually described by a Weibull distribution [16]. Two parameters are needed to describe such a distribution: the characteristic stress σ0, which is the stress at which 63.2% of the wafers are broken (Fig. 6), and the Weibull modulus m, which determines the spread of the distribution (represented by the slope of the fit, with a large m-value indicating a narrow distribution). In terms of wafer defects, the characteristic stress represents the critical crack size present in the wafers sawn with a given set of parameters at which 63.2% of the wafers are broken. The Weibull modulus, from a fracture mechanics point of view, can be seen as the variation of this critical crack size between wafers sawn with the same parameters. Evidently, the characteristic stress depends on the typical sawing conditions and can be described by the parameters that globally affect the sawing conditions. In Fig. 6, it can be seen that, in general, one sawing parameter has always the same effect on the wafer strength, independently of the other fixed sawing

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reasons that will become clear later when the model is refined to be more accurate. Based on Table 2, the ranking of the relative effect in decreasing order is as follows:

parameters. However, the amplitude of the effect caused by varying this parameter is interdependent on other parameters. For instance, in Fig. 6b, increasing the abrasive volume fraction decreases the Weibull characteristic stress (indicated by a horizontal line on the graphs) as well as the Weibull modulus (the slope of the fitted line), but the decrease in characteristic stress is much more important with a wire tension of 21.2 N than with a tension of 25.2 N. The sawing parameters were varied according to a factorial design with two levels for each parameter except the abrasive volume fraction (with only one extra cut at a higher abrasive volume fraction) and the tension, which was tested at three different values. A more extensive experimental plan would have been too expensive to carry out. In point of fact, this set of experiments was chosen in order to investigate parameters expected to have a significant impact from earlier studies and their respective interactions. With such an experiment plan, it is possible to model the effect of the sawing parameters on the characteristic stress by assuming a model of the form:

σ 0 ¼ C 0 þ C 1 V SiC þ C 2 T þ C 3 f þ C 12 V SiC T þ C 13 V SiC f þ C 23 Tf þ C 123 V SiC Tf þ ε

jRE1 j 4 jRE12 j 4 jRE3 j 4 jRE2 j4 jRE23 j 4 jRE123 j 4jRE13 j However, by analyzing the impact of the different terms have on the characteristic stress, it appears that the contribution of the interaction of the abrasive volume fraction with the feed rate (C13) as well as the interaction of all parameters together (C123) are not significant. For this model, the coefficient of determination R2 is 0.87. Removing the terms C13 and C123 and performing a new regression, the coefficient of determination does not change. The interaction that has the largest influence on the model is the wire tension T with the abrasive volume fraction VSiC. By getting a fundamental understanding of this interaction, it is possible to improve the model precision [10]. Subsequently, a closer look at the way abrasive particles are pushed into the silicon and what is determining the crack size caused by such indentation is insightful. Indeed, the thickness measurements imply that the wire is bending around particles (see Fig. 7). From this observation, it is reasonable to assume that the amounts of wire bending as well as the wire tension are important parameters determining the wafer strength. Funk and Dinger [20] showed that the distance between two neighboring particles in a suspension is inversely proportional to the volume fraction of particles in the slurry according to:

ð1Þ

where V SiC is the abrasive volume fraction, T is the wire tension and f is the feed rate. Cx represents the constants used to fit the model and ε is the error term. A least square regression of the experimental results gives the coefficients shown in Table 2. Also given in Table 2 are the relative effects of each constant. The relative effect of the fraction of abrasive in the slurry (RE1) is given by: ΔV RE1 ¼ ðC 1 þ C 12 T Mid þ C 13 f Mid þ C 123 T Mid f Mid Þ SiC

d 1 p 2 V SiC

ð2Þ

sMid

Furthermore, based on the thickness measurements presented in the previous section, it can be assumed that the groove width varies (to a first approximation) with the volume fraction of abrasive in the slurry (the higher the volume fraction of abrasive in the slurry, the thinner the wafers). Thus, the lateral displacement of the wire (h in Fig. 7) is proportional to the abrasive volume fraction so that:

where T Mid ¼ ððMaxðTÞ þ MinðTÞÞ=2Þ is the wire tension in the middle of the tested range. Similar definitions stand for fMid and VSiC, Mid. The amplitude of the abrasive fraction range is defined by ΔV SiC ¼ ððMaxðV SiC Þ  MinðV SiC ÞÞ=2Þ and ΔT and Δf are defined similarly. σMid is the characteristic stress obtained from Eq. (1) using the parameters VSiC, Mid, TMid and fMid. For our set of experiments, we get σMid ¼ 153.79 MPa. The relative effect of the wire tension (RE2) and feed rate (RE3) are calculated using the same approach as for RE1 in Eq. (2). The relative effect of the interaction between the fraction of abrasive and wire tension were calculated as follows: RE12 ¼ ðC 12 þ C 123 f Mid Þ

ΔV SiC ΔT

hp V SiC ;


 h V p SiC p V 2SiC tan ϕ ¼ d=2 1=V SiC

ΔV SiC ΔTΔf

P ¼ 2T sin ðϕÞ:

ð8Þ

For small values of ϕ as it is the case here, then tan(ϕ)Esin(ϕ) so that the indentation force P becomes:

ð4Þ

sMid

ð7Þ

The indentation force P depends on the wire tension T and wire bending angle ϕ according to:

Similar equations were used for the interaction of the fraction of abrasive and feed rate as well as the interaction of the wire tension and feed rate. For the interaction of all three parameters together, the relative effect was calculated with: RE123 ¼ C 123

ð6Þ

where h and d/2 are defined as in Fig. 7. Consequently, based on Eqs. (5) and (6), the angle ϕ at which the wire is bent is given by:

ð3Þ

sMid

ð5Þ

P p 2TV 2SiC

These equations give the same relative effect of the parameter as when a standard normalization and centering of the variables is carried out. However, such a procedure was not performed for

ð9Þ

Lawn et al. [21] showed that the crack depth depends on the indenter geometry (i.e. the abrasive shape) and on the applied force to the power 2/3, so with the help of Eqs. (8) and (9), the

Table 2 Fitted constants and relative effect for the characteristic stress model presented in Eq. (1). For this model, the characteristic stress when all the parameters are in the middle of their range is σMid ¼ 153.79 MPa. Variable

Constant

VSiC

T

Unit

MPa

MPa

MPa N

Coefficient Size Relative effect [%]

C0 1110

C1  3530  3.340

C2  31.4  0.914

VSiC  T

f MPa

ðμm= minÞ C3  1.73  1.891

MPa N

C12 118 2.180

VSiC  f MPa

ðμm= minÞ C13 6.15 0.434

fT

VSiC  T  f

MPa Nðμm= minÞ

MPa Nðμm= minÞ

C23 0.0578 0.808

C123  0.211  0.546

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presented. The relative effect of the abrasive fraction is given by:

crack depth c depends on: 

c p P 2=3 p 2TV 2SiC

2=3

ð10Þ

Based on the fracture mechanics principle, the breakage stress (σc) depends on the square root of the crack depth [21,22] according to: K

σ c ¼ pIcffiffiffiffiffiffi; Y πc

ð11Þ

where KIc is the fracture toughness and Y is a non-dimensional

' ¼ RE12

399

RE1' ¼

' ð1=ðV 2 1=3 C 1 DV SiC þ ð1=2ÞC 12  1=ðV 2SiC;Min T Mid Þ1=3 Þ Sic;Max T Mid Þ

sMid

ð15Þ Similar equation is used for the relative effect of the feed rate. VSiC, Max and VSiC, Min are the highest and lowest abrasive fraction tested. The same definitions apply to TMax and TMin. The equation giving the relative effect of the interaction of the abrasive fraction and the wire tension is given by:

C '12 ðð1=ðV 2SiC;Max T Max Þ1=3 Þ  ð1=ðV 2SiC;Max T Min Þ1=3 Þ  ð1=ðV 2SiC;Min T Max Þ1=3 Þ þ ð1=ðV 2SiC;Min T Min Þ1=3 ÞÞ 4sMid

function of the crack size and component dimensions. So when the wire bends around particles and pushes them into the wafer surface, cracks are created as described by Eq. (10). These cracks have an impact on the wafer strength described by Eq. (11). By combining these two equations together, a new equation taking into consideration the interaction of the wire tension and the abrasive volume fraction is obtained, that has the form:

σ ρ; T ¼ C 012 

1

TV 2SiC

1=3 ;

ð12Þ

where C 012 is a fitting constant. Substituting the term C 12 V SiC T in Eq. (1) by the term presented in Eq. (12), we get a new model equation:

σ 0 ¼ C 0 þ C 1 V SiC þ C 2 T þ C 3 f þ C 012 

1 TV 2SiC

1=3 þ C 23 Tf þ ε

ð13Þ

As compared to Eq. (1), the coefficient of determination R2 ¼0.95 is significantly enhanced. However, the impact of the term C 2 T is extremely small and so can be removed from the model, so that the characteristic stress is finally given by:

σ 0 ¼ C 0 þ C 1 V SiC þ C 3 f þ C 012 

1

TV 2SiC

1=3 þ C 23 Tf þ ε

ð14Þ

ð16Þ

The terms with the most important effect are the fraction of particle in the slurry ðC 1 Þ, as well as the feed rate ðC 3 Þ (both terms change the characteristic stress by about 2.2%, see Table 3). The interaction of the wire tension and the feed rate ðC 23 Þ has a less important impact of 1.2%. Finally, the interaction of the wire tension and the fraction of particles in the slurry ðC 012 Þ has the least important effect: 1.1%. By changing all the parameters, the largest measured characteristic stress difference is 24.9 MPa, while the model predicts it to be 21.3 MPa. 4.3. Weibull modulus The same approach can be followed for modeling the Weibull modulus as for modeling the characteristic stress. In this case, the model is: m ¼ D0 þ D1 V SiC þ D2 T þ D3 f þ D12 V SiC T þ D13 V SiC f þD23 Tf þ D123 V SiC Tf þ ε

ð17Þ

The results of fitting with this model, including the relative effect for each main parameter and interactions calculated in the same fashion as for the characteristic stress (Eqs. (2)–(4)), are given in Table 4. The coefficient of determination R2 is 0.85. Based on Table 4, the ranking of the relative effect in decreasing order is as follows: jRE13 j 4 jRE1 j 4 jRE123 j4 jRE12 j4 jRE23 j 4 jRE3 j4 jRE2 j

Table 3 summarizes the coefficients as well as the relative effects of σ0 for each main parameter and interaction in Eq. (14). The coefficient of determination is R2 ¼0.97. To evaluate the relative effects induced by each parameter, slightly different equations than Eqs. (2)–(4) had to  be used  1=3 1=3to take the term 1= TV 2SiC into account. Actually, as TV 2SiC is in the denominator, the equation resulting from a centering and normalization of the variables would be complicated and impossible to simplify, thus making the centering and normalization procedure impracticable. This is the reason why, for the sake of coherence, no centering and normalization was carried out on the models Table 3 Fitted constants and relative for the characteristic stress model presented in Eq. (14). For this model, the characteristic stress when all the parameters are in the middle of their range is σMid ¼ 151.50 MPa. Variable

Constant

VSiC

f

Unit

MPa

MPa

MPa ðμm= minÞ

Coefficient Size Relative effect [%]

C0  428.7

C1 973.8  2.23

C3  0.3764  2.22

1

ðTV 2SiC Þ

1=3

MPa U N1=3 C0 12 437.9 1.07

Tf

The most obvious outlying point of the model is the cut made with the densest slurry. To agree with the model, the Weibull modulus should be much lower than it actually is. Adding a new term D11 V 2SiC to the model allows taking this observation into account and a new regression achieves an R2 of 0.98. However, looking at the effect of wire tension on the Weibull modulus, it appears that this effect is negligible. Indeed, the model can be simplified by removing all the terms depending on the tension, so that Eq. (17) becomes: m ¼ D0 þ D1 V SiC þ D3 f þ D13 V SiC f þD11 V 2SiC þ ε

ð18Þ

The regression coefficients obtained with this model and their respective relative effects are given in Table 5 and the R2 becomes 0.97, which is excellent considering that the model takes finally into consideration only half of the initial terms. Based on this table, the ranking of the relative effect in decreasing order is as follows: jRE11 j 4 jRE13 j 4jRE3 j 4 jRE1 j

MPa ðμm= minÞ

C23 0.01105 1.16

The power two dependence of the abrasive volume fraction on the Weibull coefficient can be thought as follows: when many particles are active, direct particle–particle interactions become important, thus increasing the probability of having comparable

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Table 4 Fitted constants and relative effect for the Weibull modulus model presented in Eq. (17). For this model, the Weibull modulus when all the parameters are in the middle of their range is mMid ¼17.40. Variable

Constant

VSiC

T

Unit

MPa

MPa

MPa N

Coefficient Size Relative effect [%]

D0  674.7

D1 3281  27.60

D2 49.49 4.30

VSiC  T

f MPa

ðμm= minÞ D3 2.026 8.24

MPa N

D12  226.0 11.20

VSiC  f MPa

ðμm= minÞ D13  9.473 29.10

fT

VSiC  Tf

MPa Nðμm= minÞ

MPa Nðμm= minÞ

D23  0.1291 8.86

D123 0.5885 13.50

Table 5 Fitted constants and relative effect for the Weibull modulus model presented in Eq. (18). For this model, the Weibull modulus when all the parameters are in the middle of their range is mMid ¼ 16.59. Variable

VSiC

f

D0 995.4

D1  6642 3.14

ðμm= minÞ D3  0.9711 3.52

Unit Coefficient Size Relative effect [%]

VSiC  f

Constant

1

V 2SiC

1

ðμm= minÞ D13 4.183 22.8

D11 10450 39.5

critical defects on the wafers resulting in a high Weibull modulus. Similar results are obtained at the other extremity that is at a much lower volume fraction of abrasive. In such situation, there are much fewer direct particle–particle interactions and so the probability of having comparable critical defects is rather high, leading again to a situation with a high Weibull modulus but at a different characteristic stress. At an intermediate volume fraction of abrasive, the particle–particle interactions are more random as compared to both high and low volume faction, leading to a wide crack size distribution giving a lower Weibull modulus. Interestingly – and in contrast to the characteristic stress model – there is no term depending on the wire tension in Eq. (18). Indeed, as far as our measurements show, the tension has no impact on the Weibull modulus, or its impact is too low to be detected within our experimental plan. Similarly to the characteristic stress model, the Weibull modulus model has limitations. The square dependency on the abrasive fraction, especially, makes the model vary quickly when it is used with parameters outside the range tested for this work, and extrapolating any data is risky at best.

Fig. 9. (a) Characteristic breakage stress σ0 for different sawing conditions. The points are the measured values and the lines represent the fitted model. (b) Measured and fit Weibull modulus for different sawing conditions.

4.4. Findings from the model The models of the characteristic stress (Eq. (14)) and the Weibull modulus (Eq. (18)) are fitted to the experimental measurements, with the results given in Tables 3 and 5. Fig. 9 presents the measured and predicted values from both models for different sawing conditions. These models bring important insights into the wire-sawing process. First, the interdependence of the sawing parameters is highlighted, as most of the terms in Eqs. (14) and (18) contain two parameters. Within the parameters tested, the strongest wafers are obtained by sawing with a low abrasive volume fraction (0.211), a low wire tension (21.2 N) and a low feed rate (378 mm/min), as seen in Fig. 9. It is interesting to note that this combination of parameters also gives a high Weibull modulus (Fig. 9b). However, the Weibull modulus model is less accurate than the characteristic stress model. This is easily explained by the fact that tests at the tails of the pdf distribution (low and high percentile of the distribution in Fig. 6) have a large impact on the value of the Weibull modulus, whereas the characteristic stress is not so sensitive. It is important to keep in mind that wafer sawing is a very slow process. With a feed rate of 450 mm/min and not taking into account the bow of the wire, it takes 4 h 47 min to cut a

125  125 mm2 wafer. Using the same feed rate, it takes 1 h 51 min to saw 50 mm of silicon, which corresponds to the distance between the two inner lines of the four-line test where the maximal tensile stress during the bending tests is more or less constant. This corresponds to a wire length of 77 km. This is to emphasize that a tremendous amount of abrasive particles hit the silicon block during typical sawing. Although one might think that so many defect-inducing events should induce comparable defects in each wafer, there are still large differences in breakage stress between wafers (see Fig. 6). This illustrates that the event determining the wafer strength is extremely rare. The model describing the Weibull modulus in Fig. 9b shows that there is a local minimum at intermediate volume fractions of abrasive. The position of this minimum depends on the feed rate: a higher feed rate moves its position to a lower volume fraction of abrasive. To explain the presence of this local minimum, we can imagine a sawing process with two types of defect-creation processes. One mechanism involves just a few particles and is mostly active at a low concentration of abrasive, whereas the other mechanism involves many particles and is mostly active at a high concentration of abrasive. The defects created by these types of interactions

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Fig. 10. Schematic bottom view of the wire-sawing process and the different mechanisms at play on the wafer scale.

are likely to be different. When only one type of interaction takes place, the defects are comparable in all wafers, leading to a high Weibull modulus. However, at an intermediate abrasive concentration, both types of interactions are present, leading to a wider range of defect sizes in the wafers and to a low Weibull modulus. The position of this minimal Weibull modulus depends on the amount of particles present at the side of the wire which, in turn, depends not only on the mean volume fraction of abrasive in the slurry, but also on the rate at which particles are ejected from the top of the wire. Consequently, it is also dependent on the feed rate: at a high feed rate, more particles are ejected from the top of the wire, so that the amount of particles present at the side of the wire is higher than at a low feed rate. Hence, at a high feed rate, the position of the minimal Weibull modulus is at a lower volume fraction of abrasive than at a low feed rate, as can be seen in Fig. 9b. The interest of the presented model aims at describing the strength-limiting cracks, while focused on the wafer scale. In that aspect, it is a bridge between the microscopic particle–silicon interactions that are creating cracks and the macroscopic sawing parameters that can be directly controlled by the wire-saw. Nevertheless, it is not incompatible with models focused on different scales, e.g. the model presented by Bierwisch et al. [6] as well as larger scale sawing models like the one developed by Möller [1]. Furthermore, these models focus on the hydro-dynamical aspects of wire-sawing, whereas ours is centered on wafer strength and surface structure.

5. Conclusions The effect of the abrasive volume fraction in the slurry, the feed rate and the wire tension on the wafer strength distribution has been investigated for wafers cut with F800 slurry. These results were compared, in specific cases, with the results of cuts made with F600 only or a 1:1 mixture of F600 and F800. For F800, a model has been developed, based on the measured breakage stress as well as on the wafer roughness and thickness evolution. Fig. 10 shows a schematic view of the sawing process as it is described in this article. The results from modeling the impact of the sawing parameters on the wafer strength distribution have been

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interpreted with the help of this model, so that the impact of the different sawing parameters could be connected to meaningful interactions between the wire, the abrasive particles and the silicon block. Due to the fact that the interactions described are very complex, the model is valid only in the parameter range tested, and an extension towards more extreme processing ranges would require additional experiments and modeling. It was found that the strongest wafers were obtained by using a low abrasive volume fraction (0.211), a low wire tension (21.2 N) and a low feed rate (378 mm/min). This set of parameters might not be optimal in terms of productivity, but this has to be put into perspective, as stronger wafers may be sawn thinner, thus saving on silicon feedstock. Indeed, by using the optimal sawing parameters, it is possible to increase the breakage stress by more than 20% as compared to the reference wafers or assuming the same breakage force, to reduce the wafer thickness by 19 μm. Furthermore, assuming that the cell process line alone does not impose a given force on the wafer, but that the wafer bending and/or its weight are also involved in the breakage, it is possible to cut even thinner wafers. Actually, thinner wafers are lighter so that any acceleration imposed requires less force and, in addition, they can bend more before reaching a critical surface stress than thicker wafers. Using the best set of parameters to saw 140-mm-thick wafers with a wire diameter of 100 mm and F800 abrasive, it is possible to decrease silicon consumption by 28% as compared to standard 180-mm-thick wafers sawn with F600 and a wire diameter of 140 mm.

Acknowledgments Part of this work was supported by the Swiss Commission for Technology and Innovation (CTI), Project no. 7730.2 NMPP-NM. The authors would like to thank Dr Philippe Nasch, Dr Laurent Québatte and Applied Materials Precision Wafering Systems for allowing us to saw the wafers in their facilities. The authors would like also to thank Mrs Katarzyna Grinberg and Cvetomir Dimov for fruitful discussion about statistics and Dr Lara Sands for improving the manuscript.

References [1] H.J. Moller, Wafering of silicon crystals, Phys. Status Solidi A 203 (4) (2006) 659–669. http://dx.doi.org/10.1002/pssa.200564508. [2] G. Del Coso, C. del Cañizo, W.C. Sinke, The impact of silicon feedstock on the PV module cost, Sol. Energy Mater. Sol. C 94 (2) (2010) 345–349. http://dx.doi.org/ 10.1016/j.solmat.2009.10.011. [3] P. Rupnowski, B. Sopori, Strength of silicon wafers: fracture mechanics approach, Int. J. Fract. 155 (1) (2009) 67–74. http://dx.doi.org/10.1007/ s10704-009-9324-9. [4] C. Funke, S. Wolf, D. Stoyan, Modeling the tensile strength and crack length of wire-sawn silicon wafers (paper ID 011012), J. Sol. Energy – T. ASME (2009), http://dx.doi.org/10.1115/1.3028048. [5] T. Wagner, H.J. Möller, A 3D wire saw model of the slurry flow to predict forces exerted upon silicon ingots during cutting, in: Proceedings of the 23rd European Photovoltaic Solar Energy Conference, Valencia, Spain (2008) pp. 1315–1320, DOI: 10.4229/23rdEUPVSEC2008-2CV.3.46. [6] C. Bierwisch, B. Weber, R. Kübler, M. Moseler, G. Kleer, Contact regimes in the wire sawing process: explicit 3D modeling of PEG/SiC slurry, in: Proceedings of the 23rd European Photovoltaic Solar Energy Conference, Valencia, Spain (2008) pp. 1104–1108, DOI: 10.4229/23rdEUPVSEC2008-2BO.2.6. [7] Rietzschel R, Wagner T, Funke C, Möller HJ. Optimization of the wire sawing process using force- and temperature- measurements, in: Proceedings of the 23rd European Photovoltaic Solar Energy Conference, Valencia, Spain (2008) pp. 1301–1304, DOI: 10.4229/23rdEUPVSEC2008-2CV.3.42. [8] B. Weber, C. Bierwisch, R. Kübler, G. Kleer, Investigation on the sawing of solar silicon by application of wires of 100 μm diameter, in: Proceedings of the 23rd European Photovoltaic Solar Energy Conference, Valencia, Spain (2008) pp. 1285–1288, DOI: 10.4229/23rdEUPVSEC2008-2CV.3.33. [9] H.J. Moller, Basic mechanisms and models of multi-wire sawing, Adv. Eng. Mater. 6 (7) (2004) 501–513. http://dx.doi.org/10.1002/adem.200400578.

402

A. Bidiville et al. / Solar Energy Materials & Solar Cells 132 (2015) 392–402

[10] A. Bidiville, Wafer sawing processes: from microscopic phenomena to macroscopic properties (Ph.D. thesis), University of Neuchâtel, Neuchâtel, Switzerland, 2010. [11] A. Bidiville, K. Wasmer, J. Michler, P.M. Nasch, M. Van der Meer, C. Ballif, Mechanisms of wafer sawing and impact on wafer properties, Prog. Photovolt.: Res. Appl. 18 (8) (2010) 563–572. http://dx.doi.org/10.1002/pip.972. [12] A. Bidiville, I. Neulist, K. Wasmer, C. Ballif, Effect of debris on the silicon wafering for solar cells, Sol. Energy Mater. Sol. C 95 (8) (2011) 2490–2496. http://dx.doi.org/10.1016/j.solmat.2011.04.038. [13] Norm ISO 4287, Geometrical Product specifications (GPS) – surface texture: profile method – terms, definitions and surface texture parameters, International Organization for Standardization, 1997. [14] K. Wasmer, A. Bidiville, F. Janneret, J. Michler, C. Ballif, M. van der Meer, P.M. Nasch, Effects of edges defects due to multi-wire sawing on the wafer strength, in: Proceedings of the 23rd European Photovoltaic Solar Energy Conference, Valencia, Spain (2008) pp. 1305–1310, DOI: 10.4229/23rdEUPVSEC2008-2CV.3.43. [15] K. Wasmer, A. Bidiville, J. Michler, C. Ballif, M. Van der Meer, P.M. Nasch, Effect of strength test methods on silicon wafer strength measurements, in: Proceedings of the 22nd European Photovoltaic Solar Energy Conference, Milan, Italy, 2007.

[16] W. Weibull, A Statistical Theory Of The Strength Of Materials, Ingeniörsvetenskapsakademiens Handlingar Nr 151, 1939, Generalstabens Litografiska Anstalts Förlag, Stockholm. (this publication can be downloaded at: 〈http:// www.barringer1.com/wa.htm〉). [17] M. Buijs, K. Korpel-van Houven, 3-body abrasion of brittle materials as studied by lapping, Wear 66 (2) (1993) 237–245. http://dx.doi.org/10.1016/0043-1648 (93)90267-P. [18] S.B. Wei, I. Kao, Vibration analysis of wire and frequency response in the modern wiresaw manufacturing process, J. Sound Vib. 231 (5) (2000) 1383–1395. http://dx.doi.org/10.1006/jsvi.1999.2471. [19] I.S. Raju, J.C. Newman Jr., Stress-intensity factors for a wide range of semielliptical surface cracks in finite-thickness plate, Eng. Fract. Mech. 11 (4) (1979) 817–829. http://dx.doi.org/10.1016/0013-7944(79)90139-5. [20] J.E. Funk, D.R. Dinger, Slip control using particle-size analysis and specific surface area, Am. Ceram. Soc. Bull. 67 (5) (1988) 890–894. [21] B.R. Lawn, A.G. Evans, D.B. Marshall, Elastic/plastic indentation damage in ceramics: the median/radial crack system, J. Am. Ceram. Soc. 63 (9–10) (1980) 574–581. http://dx.doi.org/10.1111/j.1151-2916.1980.tb10768.x. [22] A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond. A 221 (1921) 163–198.