Nonlinear Buckling Analysis of Vertical Wafer Probe Technology by W.H. Müller1), T. Hauck2) STAMM 2010 Berlin August 30 – September 2, 2010 1) Technische Universität Berlin Institut für Mechanik - LKM Einsteinufer 5 D-10587 Berlin 2) Freescale Halbleiter Deutschland GmbH Schatzbogen 7 81829 München

Outline Introduction and motivation: Buckling beams and vertical probe card technology Theoretical approach to non-linear buckling Finite element approach to non-linear buckling Comparison with an experiment using a macroscopic beam structure Prediction and comparison of vertical buckling beam technology experiments Conclusions

Outline Introduction and motivation: Buckling beams and vertical probe card technology Theoretical approach to non-linear buckling Finite element approach to non-linear buckling Comparison with an experiment using a macroscopic beam structure Prediction and comparison of vertical buckling beam technology experiments Conclusions

Buckling beams and vertical probe card technology I Objective: Testing the electrical connectivity of dies with buckling needles

Buckling beams and vertical probe card technology II Pads at the periphery of the chip, which will later be wire-bonded. Each pad is probed by a needle in parallel and at the same time. It is possible to probe several chips simultaneously or even all chips on a wafer. probe needles: pads:

Buckling beams and vertical probe card technology III Principle: Dies with pads are pressed against needles in tool Needles start buckling. Buckling beam technology guarantees consistent contact pressure on every point tested, independent of travel. Optimal tolerance even under changing planarity conditions.

Buckling beams and vertical probe card technology IV With applied force: After release of applied force: Conclusion: This is a completely reversible process, buckling eqns. apply

Outline Introduction and motivation: Buckling beams and vertical probe card technology Theoretical approach to non-linear buckling Finite element approach to non-linear buckling Comparison with an experiment using a macroscopic beam structure Prediction and comparison of vertical buckling beam technology experiments Conclusions

Theoretical approach to non-linear buckling I Recall the 4 Euler Cases Objective: Exact calculation of the displacements w(s) and x(s) → non-linear problem

Theoretical approach to non-linear buckling II Procedure following Timoshenko and Gere “Theory of elastic stability”: Free-body-diagram Equilibrium of moments Geometry non-lin. DE for bending Differentiation and rearrangement Note: only for Euler case 3

Theoretical approach to non-linear buckling III Integration for all Euler cases Constant of integration, , from boundary conditions Case 1: (free end) Case 2: (pin joints, symmetry) Case 3: (pin joint) Case 4: or (two points of inflection)

Theoretical approach to non-linear buckling IV Results for Euler case 1 (cf., Timoshenko & Gere; Theory of Elastic Stability) This implicit relation allows to compute the load F required to achieve an angle a0 for a given length l: Critical load

Theoretical approach to non-linear buckling V Results for Euler case 1 cont. (hold also for cases 2 and 3) Critical load: with: and:

Theoretical approach to non-linear buckling VI Results for Euler case 1 cont. Vertical displacement Horizontal displacement Analogous results for Euler cases 2 and 4 due to self-similarity with case 1

Theoretical approach to non-linear buckling VII Euler case 1 Euler case 2 Euler case 4

Theoretical approach to non-linear buckling VIII These three relations allow us to compute (numerically) for a given horizontal push F (i) the vertical force and (ii) the angle of deflection at the hinge, a0, and (iii) the angle of deflection at the inflection point, a1, for a given beam length l. In addition the position s1 of the point of inflection can be obtained. Euler case 3; (i) relation for total length of the beam (ii) Vertical displacement at pinned end vanishes (iii) Moment vanishes at (unknown position of) point of inflection with

Theoretical approach to non-linear buckling IX Euler case 3 Deformation pattern: Current position of point of inflection:

Outline Introduction and motivation: Buckling beams and vertical probe card technology Theoretical approach to non-linear buckling Finite element approach to non-linear buckling Comparison with an experiment using a macroscopic beam structure Prediction and comparison of vertical buckling beam technology experiments Conclusions

Finite element approach to non-linear buckling (Example) Buckling Beam Column build-in pinned F initial displacement F´ u x Dimensions length = 520 mm thickness = 0.7 mm width = 20 mm Young’s modulus E = 210 GPa ANSYS finite element code, uniaxial beam elements with consistent tangent stiffness matrix, large deformation option NLGEOM,ON Initial displacement with a small perturbation  : 100 elements "3-D elastic beam" = 2 nodal elements für bending, tension-compression, and torsion, each node with 6DOFs A total of 101 nodes and 606 DOFs Application of displacement in x-direction in 2000 loading steps

Outline Introduction and motivation: Buckling beams and vertical probe card technology Theoretical approach to non-linear buckling Finite element approach to non-linear buckling Comparison with an experiment using a macroscopic beam structure Prediction and comparison of vertical buckling beam technology experiments Conclusions

Comparison with macroscopic experiment I

Comparison with macroscopic experiment II

Outline Introduction and motivation: Buckling beams and vertical probe card technology Theoretical approach to non-linear buckling Finite element approach to non-linear buckling Comparison with an experiment using a macroscopic beam structure Prediction and comparison of vertical buckling beam technology experiments Conclusions

Comparison with beam technology experiments I Note: These are measurements averaged w.r.t. many needles; co-planarity issues, steady increase of deflection, not an abrupt one beam dimensions: d = 3 and d=2.5 mil, l = 7.8 mm , d = 2 mil, l = 5.33 mm

Comparison with beam technology experiments II Probe force prediction (treated as Euler case 4) E = 110000 N/mm² 118mN 57 mN 50 mN

Conclusions The buckling of micrometer size beams used in VI-probe card technology can be treated in closed-form using Timoshenko’s non-linear deflection approach. All four Euler cases have been analyzed based on non-linear buckling theory. The results for the Euler cases 1, 2, and 4 are essentially the same due to their self-similarity. Euler case 3 (one pinned and one clamped end) needs a special treatment. The Euler length (position of the point of inflection) in case 3 changes from 0.699 l to slightly higher values as the horizontal push increases. The closed-form solution agrees well with FE results that take large deformation and compressibility of the beam into account. FE and closed form solutions can both be used to predíct the experimentally observed deformation pattern both for macroscopic as well as microscopic bucking beams