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Capillary Self-Alignment of Polygonal Chips: a Generalization for the Shift-Restoring Force J

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Capillary Self-Alignment of Polygonal Chips: a Generalization for the Shift-Restoring Force J Capillary self-alignment of polygonal chips: A generalization for the shift-restoring force J. Berthier,1 S. Mermoz,2 K. Brakke,3 L. Sanchez,4 C. Fr´etigny,5 and L. Di Cioccio4 1)Biotechnology Department, CEA-Leti, 17 avenue des Martyrs, 38054 Grenoble, France 2)STMicroelectronics, 850 rue Jean Monnet, 38926 Crolles Cedex, France 3)Mathematics Department, Susquehana University, 514 University avenue, Selingrove, PA 17870, USA 4)Microelectronic Department, CEA-Leti, 17 avenue des Martyrs, 38054 Grenoble, France 5)ESPCI, 10 rue Vauquelin, 75005, Paris, France (Dated: 6 May 2012) Capillary-driven self-alignment using droplets is currently extensively investigated for self-assembly and mi- croassembly technology. In this technique, surface tension forces associated to capillary pinning create restor- ing forces and torques that tend to bring the moving part into alignment. So far, most studies have addressed the problem of square chip alignment on a dedicated patch of a wafer, aiming to achieve 3D microelectronics. In this work, we investigate the shift-restoring forces for more complex moving parts such as regular { convex and non-convex { polygons and regular polygons with regular polygonal cavities. A closed-form approximate expression is derived for each of these polygonal geometries; this expression agrees with the numerical results obtained with the Surface Evolver software. For small shifts, it is found that the restoring force does not depend on the shift direction or on the polygonal shape. In order to tackle the problem of microsystem pack- aging, an extension of the theory is done for polygonal shapes pierced with connection vias (channels) and a closed form of the shift-restoring force is derived for these geometries and again checked against the numerical model. In this case, the restoring force depends on the shift direction. Finally, a non-dimensional number, the shift number, is proposed that indicates the isotropic or anisotropic behavior of the chip according to the shift direction. I. INTRODUCTION man have developed CSA methods for fluidic assembly of optical components19, Zhang and colleagues have investi- gated the CSA of drosophila embryos on 2-D arrays of hy- Capillary-driven self-alignment using droplets, or drophobic sites on a silicon substrate in water20, Stauth capillary self-assembly (CSA), is currently exten- and Parviz have developed the self-assembly of single- sively investigated for self-assembly and microassembly crystal silicon circuits on plastic21, and Fukushima and 1{8 technology . In this technique, surface tension forces colleagues have studied the use of CSA for microsystem associated to capillary pinning create restoring forces and packaging22. These investigations are mainly experimen- torques that tend to bring the moving part into align- tal (except for the work of Zhang et al.). ment. In the field of 3D microelectronics, the method In this work, we present a novel theoretical and numer- aims to be an alternate approach to the \pick and place" ical approach to the shift restoring forces for polygonal approach. In the capillary technique, the chip is de- chips. For many different polygonal shapes, the shift posited on top of a water droplet and is brought into restoring force is calculated analytically for an approxi- alignment by the action of capillary forces, because the mation, and numerically in more detail. In the partic- liquid interface tends to minimize its free area9. Af- ular case of regular polygons { convex or not { of the ter alignment, which is fast, evaporation brings the chip same perimeter, it is demonstrated that the magnitudes into contact with the pad on the wafer, and direct of the restoring forces at short range { small initial shift bonding is possible if the two surfaces are sufficiently or at the end of the alignment process { are equal for hydrophilic10{13. It is expected that self-alignment meth- all polygonal shapes. Also, the restoring force at small ods could be faster and more precise than the conven- shift does not depend on the direction of the shift for tional robotic method. Developments have been fast, and these regular polygonal shapes. The theoretical results it has been found that a square chip can be aligned by the are in agreement with numerical results obtained with the action of restoring forces and torques if a certain number Surface Evolver software23. Moreover, it is shown that of precautions are taken14,15. A review of the different the approach can be extended to polygonal chips with concepts for 3D integration has recently been published polygonal cavities, such as those used by Fukushima and by Lee and coworkers16. coworkers to seal microsystems22. If connection vias or In parallel, following the pioneering work of the White- microchannels pierce the sides of the chip, the isotropic sides group17, new investigations have started: Knuesel behavior is lost, and the former analytical expression has and colleagues have used the method for the assembly of to be corrected by an anisotropic factor. Finally, it is segmented monocrystalline solar cells18, Avital and Zuss- shown that anisotropic chip geometries have restoring 2 forces different than regular shapes and the isotropicity Approximating the surface energy of the whole interfacial is characterized by a non-dimensional number which we area by the sum of the surface energy of all \facets" { i.e. call the shift number. not considering the inward curvature of the interface at the junction of two facets or any curvature in the middle of the face { the total surface energy after the shift is II. THEORY X X q 2 2 2 E2 = Ei = γA2 = γ si h + x sin θi: (6) Consider a chip in shape of a polygon and a shift x in i i the direction as sketched in figure 1. Edge number k has The restoring force is then length sk and angle from horizontal θk. X X 2 1 F = − Fi = −γx si sin θi p :(7) 2 2 2 i i h + x sin θi FIG. 1. Sketch of a polygon with shift direction. FIG. 2. sketch of a shifted interface. Now consider the deformation of one edge of length s and angle θ, as shown in figure 2. The initial interface { An interesting observation is that, when x is very small assuming it is flat { has an area of in comparison with h, relation (7) collapses to x X F = −γ s sin2 θ : (8) A1 = sh; (1) h i i i where h is the height of the liquid layer. The surface Relation (8) shows that when x is small, the restoring energy is then force is a linear function of x. On the other hand, if the shift is large compared to the vertical dimension of the E1 = γA1 = γsh; (2) fluid layer h, then (7) collapses to where γ is the surface tension. The shifted interface { X − j j still assuming it is flat { has an area of F = sign(x)γ si sin θi : (9) p i 2 2 2 A2 = s h + x sin θ; (3) In this last case, the restoring force is constant. If we apply relation (8) and (9) to the case of the square, we where x is the shift. The corresponding surface energy is retrieve the relations given in previous work9. A geomet- p rical interpretation of (9) is given in Appendix A. E = γA = γs h2 + x2 sin2 θ: (4) 2 2 Consider now polygons, convex or not, with equal Note that an interface parallel to the direction of the edges. Then, for all i, we have si = s, and relation (8) becomes shift keeps the same surface area (θ = 0), and an interface X perpendicular to the shift has the energy E = γA = x 2 p 2 2 F = −γs sin θi: (10) 2 2 h γs h + x , which is what was found in our previous i work in the case of a square chip and a shift perpendicular to an edge9. The restoring force corresponding to this Relation (10) can be simplified by introducing the poly- single interface is gon perimeter p and using the fact that the average of sin 2θ over equally spaced angles is: −δE2 − 2 p 1 x p F = = γsx sin θ : (5) F = −γ ; (11) δx h2 + x2 sin2 θ h 2 3 where ϵ is the aspect ratio 2h=p. An important remark is angle or any regular polygonal shape. For small shifts that the restoring force F at small range does not depend it is predicted that they are equal, and similar for large on the shift direction or on the shape of the polygon. shifts, because the shift number is similar for all shapes On the other hand, (9) becomes an asymptotic value and all orientations. for the shift restoring force for large shifts: X F = −sign(x)γs j sin θij: (12) i Note that the restoring force F in this latter expression depends on the polygonal shape and on the direction of the shift, but not on the magnitude of the shift. For a convex polygon, (12) is just the surface tension times twice the transverse width of the polygon, which is to be expected, since a large shift just stretches two horizon- tal surfaces, top and bottom, each of the width of the polygon. III. REGULAR POLYGONS (CONVEX) FIG. 3. Shift number as a function of the shift direction for 5 regular polygons. We investigate first the case of regular convex poly- gons. In Appendix B, it is shown that relation (12) for large shifts can be expressed by h ( ) ( )] γs X π π F = − cos 2(i + 1) − cos 2i sin α IV. STAR POLYGONS (REGULAR AND NOT CONVEX) 2 sin π n n hn i( ) ( )] π π + sin 2(i + 1) − sin 2i cos α ; (13) Consider now the case of star polygons, a class of poly- n n gons defined in24; and let us examine the case of three where α is the shift angle.
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