Simulation of the Deformation Behaviour of Large Thin Silicon Wafers and Comparison with Experimental Findings J. Schicker1 , T. Arnold1 , C. Hirschl1 , A. Iravani, and M. Kraft1 1 CTR
Carinthian Tech Research AG, Europastraße 4/1, 9524 Villach, Austria Email:
[email protected], Phone: +43 (0)4242 56300–217
Abstract
The deformation of large thin uncoated silicon wafers without remaining intrinsic misfit stresses resting on a ring is investigated. We use both, Finite Element simulations and THz tomography mapping. Specific attention is given the scaling of the warping for increasing slenderness of those wafers. We follow the approach of starting with a known solution for a compact wafer and increase the slenderness, i.e. increase the radius and decrease the thickness, using simulation models. Then, we measure the warping by THz mapping for some slender wafers and compare the data to simulation results. We compare the maximum warpage for given loadings and we compare the deflected shapes. Due to the geometric ratio radius/thickness of over 1000:1 and the anisotropic material behaviour, simulations can only be done effectively using shell element modelling of a spatial plate. And due to large warpages in the order of 10 times of the thickness, only incremental update Lagrange nonlinear calculations give reliable results. Simulations using the available shell elements overestimate slightly the values measured by tomography, but still yield acceptable values with errors less than 10% for very slender wafers and below for more compact ones. For invariable loading conditions, a logarithmic scaling function gives an acceptable estimate for the maximum warpage for increasing slenderness. An additional important observation was that the warpage of thin wafers is heavily affected by the size of the contact radius of a weight.
rotations, contact boundary conditions and huge poor conditioned matrices complicate solutions. When we began our research on various effects concerning the behaviour of thin large wafers we encountered the problem that we did not know how to rate the accuracy of our numerical results. The reasons were that the obtained solutions for different solution approaches seemed not to converge to each other for rough estimates, i.e. reasonable mesh sizes, whereas fine enough meshes needed days of CPU time to a solution or often failed. On the other hand, reference solutions or experimentally gained values for large thin wafers were not to be found in the literature. Our successional efforts to generate some reference values are presented here. We started with an approach to rate the different available models in Finite Element Analysis (FEA) with regard to accuracy and suitability for large anisotropic, highly deforming plates based on a well documented solution for a more compact wafer [1]. From this, solutions for thinner and/or larger wafers could successively be found and rated by plausibility [2]. Finally, we measured the deflections of warped wafers by THz tomography and compared the measurements to simulation results. 2. Ball on ring test and calibration of the model Usually, a load-deflection curve for the centre of a wafer is experimentally determined by a ball or ring on ring test, see [1, 3]. A principle drawing of the ball-on-ring is shown in Figure 1. This test takes advantage of the brittle
1. Introduction In silicon wafer fabrication the trend is to produce larger wafers to reduce costs per unit. Actually, wafer sizes of 300 mm diameter are already state of the art and manufacturers are on the verge of initiating a 450 mm diameter wafer. On the second side, many applications require thinned wafers, where the thickness of a standard 700 µm-wafer is ground down to a thickness of 100 µm and even below. The handling of these large and thin wafers poses a challenge. Optimizations of the handling or elsewhere in the process chain often involve numerical simulations as a key feature. The possibility to predict reliably the deformation behaviour under various loading conditions and thus the stresses is a crucial step in this task. The simulation of slender, i.e. large thin, wafers is numerically challenging since material anisotropy, large
Figure 1.
Principle drawing of the ball-on-ring test layout.
material behaviour and the resultant scatter of the fracture strength: For a large number of specimens the load and the corresponding warpage are measured in the instant of fracture. Due to the scatter of the fracture strengths of the tested specimens a load vs. deflection curve of the wafer centre is obtained. The shape and deflection values of the wafer between centre and rim stay unknown in this test.
the most successfull modelling approaches the same wafer was calculated again, now using the material properties of silicon, Figure 2. This was meant as a reference solution for a compact, i.e. a small and thick, silicon wafer. Due to find a credible solution for a large thin wafer in the ball on ring test we successively increased the wafer diameters and reduced the thickness. The calculation strategy that was successful for the last wafer size was used as a starting point for the next. Thus it was possible to obtain convergence by modifying the model only slightly, i.e. modifying tolerances for the convergence criteria and the contact formulations or decreasing the mesh and the time step size. Then, the resulting warpages were plotted and it was decided whether they show a plausible development. Only for the ultimate aimed wafer size of 2R = 300 mm and h = 0.06 mm convergence could not be achieved. As a final depiction we chose the warpage vs. the slenderness. It shows an approximate logarithmic trend, see Figure 3. The idea was to compare the trend to the
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As starting point of our investigation, reported values of a ball-on-ring test for a compact GaAs wafer were considered. The material as reported in [1], c11 = 119, c12 = 53.8, c44 = 59.5 GPa and ρ = 5.3167 g/cm3 , is only slightly softer than that of silicon, has a similar cubic anisotropy, wheras it has a higher density (ρSi = 2.336 g/cm3 ). Elastic constants for silicon can be found e.g. in [4]: c11 = 166, c12 = 64, c44 = 80 GPa. The reported test was done for 150 mm diameter wafers with a thickness of h = 0.675 mm. The used ring had an inner radius Ri = 71 mm and the steel ball had a radius of Rball = 1.5875 mm (2R = 3.175 mm = 1/8”). The considered wafers are cut in the (100) plane. Between ball and wafer Hertz’ian pressure is applied. Between wafer and ring simple frictionless contact is assumed wheras the ring is considered as infinitely stiff. Different models were tested on this setup and accuracy of the results were compared to the reported values and set in relation to the calculation effort. Both solid modelling and shell modelling (the shell elements base on a Mindlin plate model) can be used on this (comparable small) system, but in either way large deflection formulation (NLGEOM), invoking an update Lagrangian algorithm, must be used. An isotropic approximation and a rotational symmetric approach, however, lead to large errors. Similarly, a simplified boundary condition that avoids the contact problem does not yield appropriate results. Figure 2 depicts the resulting force-deflection curve from two simulations of this problem in comparison to the reported values. Obviously, solid element modelling yields better
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Figure 3. Calculated values for the warping of different sized Si wafers in a ball-on-ring test at a uniform load. The respective wafer sizes (radius-thickness) are marked at each point. The unit thickness h0 is introduced to make the slenderness parameter R2 h0 /h3 dimensionless.
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Figure 2. Comparison of the stiffness of identical sized compact (R=75 mm, h=0.675 mm) GaAs and Silicon wafers in a ball-on-ring test. The reported GaAs data are taken from [1].
results than shell elements modelling. The shell modelling slightly overestimates the deflection for about 4%. But on the other hand the calculation time for appropriate results using solid elements is between 3 and 10 times higher than when using shell elements, depending on the respective element type, used mesh size and needed time steps. With
scaling of an analytic small deformation solution for a circular isotropic plate loaded by a center point load, where the deflection depends only on the applied force, the material properties and the slenderness R2 /h3 . To make this parameter dimensionless, we multiplied it with a unit thickness h0 . For this study we did not take the strength into account, it means we did not pay attention to the question, whether the deflection can realistically be reached before the wafer fractures. 3. THz measurements Because we had neither a possibility to measure wafer deflection by means of a ball or ring on ring test nor enough specimens to destroy, we decided to measure wafer deflections in a nondestructive way by Terahertz (THz) tomography. Thus, we gained no information about wafer strengths, but in exchange we could measure the
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Figure 5. Warping of a 200 µm thick wafer from dead load in a ring: the points are measurements and the line is from simulation. Different point colours indicate the two measured sides.
eter hole to measure the deepest warping point through the weight. Because of the need to store a certain amount of mass and a limited height for the THz piloting device, the weight above the contacting bulk was larger than the contact area. This lower bulk guaranteed that no other parts than the contact area came into contact with the wafer. Unfortunately, the weight blocked the sight for the THz beam to the central part of the wafer outside of the hole. In Figure 8 this is the part where no measured points can bee seen. Figure 6 depicts the maximum warpage values from 3 0 0
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deflected shape of the whole wafer surface. The method is based on a pulsed THz beam that is reflected at the wafer surface. The measurement uses the time delay of the beam to return. We ordered a retaining ring with an inner radius Ri = 97 mm and an outer radius Ra = 100 mm to accommodate a 200 mm diameter wafer. The ring was manufactured as axact as possible. Along its circumference the highest measured point deviated from the lowest less than 20 µm. It was mounted perpendicular to gravity and the light beam, which is piloted by a precise two axial movement device. We used pure uncoated silicon wafers cut in (100) direction of various thicknesses, nominally 700, 500, 300, 200, and 100 µm wafers. The 700 µm wafers are only cut from the ingot, whereas the thinner wafers are made from 700 µm wafers by backside grinding, i.e. only one face is ground down. First the exact thickness of each wafer was determined by precise indenter measurement at different spots of the wafers. Then, the wafers’ sagging caused by pure dead load was measured twice for each wafer, one time the ground side bottom-up, one time it was turned to show the undisturbed face on top. The THz measurements were done in a grid of 2x2 mm, and the following drawings of the wafer warpings are done by rotating the measured points about the vertical axis at the centre of the wafer. It must also be said, that the scatter of the measured values is too big (about 3%) to observe an influence of the anisotropy. Some wafers exhibited different warping depending on what side faced upwards, Figure 4. It is known, that grind-
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Figure 4. Warping of a 300 µm thick wafer from dead load in a ring with Ri = 97 mm: one THz mapping is done with the ground side sagging and one mapping is done after turning the wafer’s ground side bottomup. As reason for the different warpings residual stresses from grinding come into consideration.
ing induces stresses in the ground surface, see e.g. [5]. We assumed residual stresses from grinding for these wafers. The conclusion is, that these stresses increase the warping of the wafer when the stress acts concertedly with gravity and diminishs the warping when acting opposite, and the warping of an unstressed wafer is anywhere between both measurements. Wafers without residual stresses exhibited equal warping from either side, see Figure 5. A second series of tests is done with a weight as a defined force. A 100 g weight with an contact diameter 2Rw = 21.93 mm was centrical positioned on the wafer. The weight was manufactured with a central 5 mm diam-
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Figure 6. Warping of R=100 mm Si wafers of different thicknesses in a Ring with Ri = 97 mm from dead load.Thicknesses are 100, 200, 300 500 and 700 µm, resp. The measurements of the thicker (compacter) wafers scatter due to residual misfit stresses from grinding.
dead load obtained from measurements for the various wafers sizes and Figure 7 depicts these values from loading by the additional weight. The measurements for the wafers with intrinsic stresses are retained as hints for upper and lower bounds. We chose here to depict the wafer thickness as logarithmic slenderness, ln(R2 h0 /h3 ) with R = 100 mm and h0 = 1 mm, only the thicknesses vary between h = 0.12 mm (nominal 100 µm, log slenderness= 15.57) and h = 0.7 mm (log slenderness= 10.28), to demonstrate, that they obey a logarithmic function, too. 4. Comparison of measurements and Simulations We used the same simulation models and strategies as already shown in chapter 2 and did not try to alter or
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slender wafers took our attention. The usual warpage of more compact wafers has a parabolic shape. In the slender wafers, however, we observed a change of sign of curvature in about a one third distance from the centre to the rim, Figure 8. Only on seeing the same effect in the
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Figure 7. Warping of the same wafers as in Figure 6 but loaded by a 100g-weight (plus dead load). Here, the weight dominates the residual misfit stresses from grinding.
modify them to gain better agreement with the measurements, but we did some additional tests on modelling. The thickness values that were determined for each wafer scattered about 1–5% of the wafer thickness at different spots. One wafer was thinner at the rim than in the centre, at others it scattered randomly over the wafer surface. Also, we measured at some wafers different thicknesses at one spot from both sides. But most of the scatter in the indendation measurements where within a range of 1–3 µm around a mean. Only the unground wafer had deviations in its thickness of more than 20 µm over its area. For the simulations we used a mean of the thickness values determined for each wafer. Only the nominal 700 µm thick wafer was twice calculated with two differnt constant thickness assumptions. Also the THz signal yielded in scattering values. When fitting the measured data of a warped wafer to a spatial smooth function this became obvious and the scatter can be quantified to be in the range of about 5 µm around this function. The impression of higher scatter nearer to the rim in Figure 4 comes from the larger amount of data that are rotated into the drawing plane. It can be seen from the drawings in Figures 5–7 that the simulations overestimate the deflections for the silicon wafers as it was already be seen in Figure 2 for the ball on ring test for the GaAs wafer. The overestimation is the bigger the thinner the wafers are. That excludes the imprecision of the measurements as reason. Even with largely decreased time steps or even using a very fine mesh of solid Hex20-elements (3.5 million nodes for a quarter wafer and 33 time steps for pure dead load loading, i.e. 80 hours of cpu time) the result did not converge nearer towards the measured values. But nevertheless, the calculated warpage values represent the measurements not too badly and can be used – with some caution – for further studies. Two observations concerning the shape of the deflected
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Figure 8. Warping of a 100 µm thick wafer from dead load plus a 100 g weight (points) in comparison to the simulated wafer warp (line). The central points are measured though the hole in the weight, whereas the shadowed area from the weight is without points.
measured data, we were convinced not to see a numerical artefact, and could find an explication, see chapter 5. The simulations of the compact wafers with the weight were first done using the contacting area with a sharp angle, since we were not interested in the local stresses. But, comparing the warpage of the slender wafers, the error to the measured data became too big. Then we realized that there is a tiny fillet at the edge and took it into account for the simulation. It became obvious in the simulations, that the maximum warpage of very slender wafers varies extremely with the size of the contact area of the weight. (The contact area is better described as
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Figure 9. Warpage of the 100 µm thick wafer in a ring from dead load plus a 100 g weight with different contact sizes: The weight has a diameter of 2Rw = 21.95 mm with 3 different fillets of R f = 0.5, 1.5, and 10 mm, resp. Thus the contact sizes are 2Rc = 21.85, 18.95, and 1.95 mm, resp.
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Figure 10.
Detail of the pinnacle of wafer and fillets in Figure 9.
a ring as only the outer rim stays in contact to the wafer after deflection.) A slighty larger fillet of the contacting bulk of the weight, hence a slightly smaller ring, results
in an unproportional larger maximum warpage, Figure 9 and detail in Figure 10. It can be seen that this does not affect the rest of the wafer deformation.
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Although shell elements are known for poor stress results, we did not examine the stresses in more detail, since the stresses in the warped wafers are rather low. The maximum principal stresses for dead load deformed wafers stay well in the order at or below 1 MPa, and for the wafers loaded by the constant 100 g weight we obtained a maximum in the thinnest wafer of about 20 MPa near the contact edge of the weight. A thorough stress examination was only done for the change of sign of curvature in the slender wafers with applied weight. We took the radial normal in-plane stresses at the bottom and top of the wafer and split them into a membrane part (constant value over the thickness) and a bending part (pure bending). The results for a 500 µm thick wafer can be seen in Figure 11 and for a 100 µm thick wafer in Figure 12. (Observe the scalings). It becomes clear, 5 0 0 µ m w a fe r + 1 0 0 g : σr r a l o n g [ 1 0 0 ] 1
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5. Some remarks on stresses
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Figure 12. Distribution of the axial stress into its membrane and bending ratio of a 100 µm thick wafer under pure dead weight.
test environment similar to a ring on ring for both dead load and a constant weight. From these measurements we obtained the magnitude of warping and the resulting shape of the deflected wafers that both could be compared to results of our simulations. From the simulations we found for a constant transverse force the maximum deflection w to be approximately a logarithmic function of the increasing value of slenderness, i.e. w ∝ ln(R2 h0 /h3 ). The shapes of the deformed wafers are nearly parabolic for thick and moderately thin wafers, whereas thin wafers show a more complex deformed shape resulting from a high fraction of membrane stresses in the centre of the wafer but predominant bending forces near the rim. This shape could be confirmed by the measurements. When applying a weight to a slender wafer, a small ring bulges the wafer locally for about 10% more than a larger ring, whereas the rest of the wafer stays nearly unaffected by the size of the weight ring with the equal weight.
Figure 11. Distribution of the axial stress into its membrane and bending ratio of a 500 µm thick wafer under pure dead weight.
7. Acknowledgements
that bending dominates everywhere in the thicker wafer, whereas the thin wafer must be stiff enough to carry its load by bending forces at the rim, but to the centre the membrane forces dominate and it resembles a strained rope or film.
This work is performed in the project EPT300, cofunded by grants from Austria, Germany, Italy, and the Netherlands and the ENIAC Joint Undertaking (ENIAC JU Grant Agreement n. 304668). It is co-funded within the programme "Forschung, Innovation und Technologie für Informationstechnologie" by the Austrian Ministry for Transport, Innovation and Technology.
6. Summary We used shell elements based on Mindlin plates and partially solid elements and large deflection formulation for Finite Element simulations of wafers in a ring. From the calibration with a well documented solution on the deflections of a wafer with moderate radius-to-thickness ratio, R/h=111, in a ball on ring test the radius-to-thickness ratio was increased up to that of a large thin wafer with R/h=2500. Moreover, we measured the deflected shapes of wafers with radius 100 mm by THz tomography in a
References [1] Frank Duderstadt. Anwendung der von Kárman’schen Plattentheorie und der Hertz’schen Pressung für die Spannungsanalyse zur Biegung von GaAs-Wafern im modifizierten Doppelringtest. PhD thesis, Technische Universität Berlin, 2003. [2] Armin Iravani. Suitability of different models in ansys to calculate wafer deformation, specifically in a ballon-ring test with emphasis on slender wafers. Mas-
ter’s thesis, Carinthia University of Applied Sciences, Villach, Austria, Jan. 2015. [3] J. Barredo, L. Hermanns, I. del Rey, A. Fraile, and E. Alarcón. Comparison of different finite element models for the simulation of the ring-ball on ring test. In B.H.V. Topping, J.M. Adam, F.J. Pallarés, R. Bru, and M.L. Romero, editors, Proceedings of the Tenth International Conference on Computational Structures Technology. Civil-Comp Press, Stirling-
shire, Scotland, 2010. Paper 252. [4] Matthew A. Hopcroft, William D. Nix, and Thomas W. Kenny. What is the young’s modulus of silicon? Journal of Microelectromechanical Systems, 19:229–235, April 2010. [5] Nathan R Draney, Jun Jun Liu, and Tom Jiang. Experimental investigation of bare silicon wafer warp. In IEEE Workshop on Microelectronics and Electron Devices, pages 120–123, 2004.
