Stephen M. Heinrich

Thermal Excitation and Piezoresistive Detection of Cantilever In-Plane Resonance Modes for Sensing Applications

Thermally excited and piezoresistively detected bulk-micromachined cantilevers vibrating in their in-plane flexural resonance mode are presented. By shearing the surrounding fluid rather than exerting normal stress on it, the in-plane mode cantilevers exhibit reduced added fluid mass effects and improved quality factors in a fluid environment. In this letter, different cantilever geometries with in-plane resonance frequencies from 50 kHz to 2.2 MHz have been tested, with quality factors as high as 4200 in air and 67 in water.

Prediction of Solder Joint Geometries in Array-Type Interconnects

An approximate mathematical model is developed for predicting the shapes of solder joints in an array-type interconnect (e.g., a ball-grid array or flip-chip interconnect). The model is based on the assumption that the geometry of each joint may be represented by a surface of revolution whose generating meridian is a circular arc. This leads to simple, closed-form expressions relating stand-off height, solder volume, contact pad radii, molten joint reaction force (exerted on the component), meridian curvature, and solder surface tension. The qualitative joint shapes predicted by the model include concave (hourglass-shaped), convex (barrel-shaped, with a truncated sphere as a special case), and truncated-cone geometries. Theoretical results include formulas for determining the maximum and minimum solder volumes that can be supported by a particular pair of contact pads. The model is used to create dimensionless plots which summarize the general solution in the case of a uniform array (i.e., one comprising geometrically identical joints) for which the contact pads on the component and substrate are of the same size. These results relate the values of joint height and width (after reflow) to the solder joint volume and the molten-joint force for arbitrary values of the pad radius and solder surface tension. The graphs may be applied to both upright and inverted reflow, and can be used to control stand-off for higher reliability or to reduce bridging and necking problems causing low yields. A major advantage of the model is that it is numerically efficient (involving only simple, closed-form expressions), yet generates results that are in excellent agreement with experimental data and more complex models. Thus, the model is ideally suited to performing parametric studies, the results of which may be cast in a convenient form for use by practicing engineers. Although in the present paper the array is assumed to be doubly-symmetric, i.e., possess two orthogonal planes of symmetry, the model may be extended to analyze arrays of arbitrary layout. The motivation for predicting joint geometries in array-type interconnects is two-fold: (1) to achieve optimal joint geometries from the standpoint of improved yield and better reliability under thermal cycling and (2) to take full advantage of the flexibility of new methods of dispensing solder, such as solder-jet and solder-injection technologies, which enable the volume of each individual joint to be controlled in a precise manner. Use of dispensing methods of these types permits the solder volumes in the array to be distributed in a non-uniform manner. Results such as those presented here (in combination with appropriate fatigue studies) can be used to determine the optimal arrangement of solder volumes.

Characteristics of laterally vibrating resonant microcantilevers in viscous liquid media

The characteristics of microcantilevers vibrating laterally in viscous liquid media are investigated and compared to those of similar microcantilevers vibrating in the out-of-plane direction. The hydrodynamic loading on the vibrating beam is first determined using a numerical model. A semi-analytical expression for the hydrodynamic forces in terms of the Reynolds number and the aspect ratio (beam thickness over beam width) is obtained by introducing a correction factor to Stokes’ solution for a vibrating plate of infinite area to account for the effects of the thickness. The results enable the effects of fluid damping and effective fluid mass on the resonant frequency and the quality factor (Q) to be investigated as a function of both the beam’s geometry and liquid medium’s properties and compared to experimentally determined values given in the literature. The resonant frequency and Q are found to be higher for laterally vibrating microcantilevers compared to those of similar geometry experiencing transv...

Liquid-Phase Chemical Sensing Using Lateral Mode Resonant Cantilevers

Liquid-phase operation of resonant cantilevers vibrating in an out-of-plane flexural mode has to date been limited by the considerable fluid damping and the resulting low quality factors (Q factors). To reduce fluid damping in liquids and to improve the detection limit for liquid-phase sensing applications, resonant cantilever transducers vibrating in their in-plane rather than their out-of-plane flexural resonant mode have been fabricated and shown to have Q factors up to 67 in water (up to 4300 in air). In the present work, resonant cantilevers, thermally excited in an in-plane flexural mode, are investigated and applied as sensors for volatile organic compounds in water. The cantilevers are fabricated using a complementary metal oxide semiconductor (CMOS) compatible fabrication process based on bulk micromachining. The devices were coated with chemically sensitive polymers allowing for analyte sorption into the polymer. Poly(isobutylene) (PIB) and poly(ethylene-co-propylene) (EPCO) were investigated as sensitive layers with seven different analytes screened with PIB and 12 analytes tested with EPCO. Analyte concentrations in the range of 1-100 ppm have been measured in the present experiments, and detection limits in the parts per billion concentration range have been estimated for the polymer-coated cantilevers exposed to volatile organics in water. These results demonstrate significantly improved sensing properties in liquids and indicate the potential of cantilever-type mass-sensitive chemical sensors operating in their in-plane rather than out-of-plane flexural modes.

Effect of Coating Viscoelasticity on Quality Factor and Limit of Detection of Microcantilever Chemical Sensors

Microcantilevers with polymer coatings hold great promise as resonant chemical sensors. It is known that the sensitivity of the coated cantilever increases with coating thickness; however, increasing this thickness also results in an increase of the frequency noise due to a decrease of the quality factor. By taking into account only the losses associated with the silicon beam and the surrounding medium, the decrease of the quality factor cannot be explained. In this paper, an analytical expression is obtained for the quality factor, which accounts for viscoelastic losses in the coating. This expression explains the observed decrease of the quality factor with increasing polymer thickness. This result is then used to demonstrate that an optimum coating thickness exists that will maximize the signal-to-noise ratio and, thus, minimize the sensor limit of detection

Effect of Chip and Pad Geometry on Solder Joint Formation in SMT

An analytical model of solder joint formation during a surface mount reflow process is developed for two-dimensional fillets whose flow may be restricted due to “finite” metallizations on a leadless component and the printed circuit board. Although these height and length constraints on the fillet geometry may result in obtuse contact angles, the solution is obtained in the form of an explicit integral, similar to that previously derived by the authors for the case of acute contact angles. This solution may also be recast into the form of elliptic integrals of the first and second kinds, thereby permitting one to evaluate the fillet geometry using mathematical tables or special function software, if desired, rather than resorting to a computer-based numerical quadrature. In addition an approximate zero-gravity solution is given by means of simple closed-form expressions relating the height, length, contact angles, and cross-sectional area of the fillet. Numerical results generated by implementing the “exact” integral solution for the joint profile are given in the form of dimensionless plots, relating fillet geometry to the solder properties (surface tension and density), amount of solder, chip height, and pad length. Also presented in dimensionless form are the approximate results from the zero-gravity model, which are independent of solder properties, yet are of sufficient accuracy for “small” joints. Because of their dimensionless nature, the results of the present paper may be of maximum utility to process engineers aiming to achieve desired joint geometries (e.g., to maximize fatigue life or to eliminate bridging problems), or to board designers responsible for selecting efficient footprint patterns to maximize board density. Models of solder joint formation, such as the one presented here, may be of most value when used in conjunction with stress analysis packages (e.g., finite element programs) and appropriate fatigue models. In this way an integrated approach to the design of solder joints and circuit boards may be taken, resulting in improved product reliability and performance.

An analytical model of a thermally excited microcantilever vibrating laterally in a viscous fluid

To achieve higher quality factors (Q) for microcantilevers used in liquid-phase sensing applications, recent studies have explored the use of the lateral (in-plane) flexural mode. In particular, we have recently shown that this mode may be excited electrothermally using integrated heating resistors near the micro-cantilever support, and that the resulting increase in Q helps to make low-ppb limits of detection a possibility in liquids. However, because the use of electrothermally excited, liquid-phase, microcantilever-based sensors in lateral flexure is relatively new, theoretical models are lacking. Therefore, we present here a new analytical model for predicting the vibratory response of these devices. The model is also used to successfully confirm the validity of our previously derived Q formula, which was based on a single-degree-of-freedom (SDOF) model and a harmonic tip force. Comparisons with experimental data show that the present model and, thus, the analytical formula provide excellent Q estimates for sufficiently thin beams vibrating laterally in water and reasonable upper-bound estimates for thicker beams.

Sorption-induced Static Bending of Microcantilevers Coated with Viscoelastic Material

Absorption of a chemical analyte into a polymer coating results in an expansion governed by the concentration and type of analyte that has diffused into the bulk of the coating. When the coating is attached to a microcantilever, this expansion results in bending of the device. Assuming that absorption (i.e., diffusion across the surface barrier into the bulk of the coating) is Fickian, with a rate of absorption that is proportional to the difference between the absorbed concentration and the equilibrium concentration, and the coating is elastic, the bending response of the coated device should exhibit a first-order behavior. However, for polymer coatings, complex behaviors exhibiting an overshoot that slowly decays to the steady-state value have been observed. A theoretical model of absorption-induced static bending of a microcantilever coated with a viscoelastic material is presented, starting from the general stress/strain relationship for a viscoelastic material. The model accounts for viscoelastic str...

Generalized model of resonant polymer-coated microcantilevers in viscous liquid media.

Expressions describing the resonant frequency and quality factor of a dynamically driven, polymer-coated microcantilever in a viscous liquid medium have been obtained. These generalized formulas are used to describe the effects the operational medium and the viscoelastic coating have on the device sensitivity when used in liquid-phase chemical sensing applications. Shifts in the resonant frequency are normally assumed proportional to the mass of sorbed analyte in the sensing layer. However, the expression for the frequency shift derived in this work indicates that the frequency shift is also dependent on changes in the sensing layers loss and storage moduli, changes in the moment of inertia, and changes in the medium of operations viscosity and density. Not accounting for these factors will lead to incorrect analyte concentration predictions. The derived expressions are shown to reduce to well-known formulas found in the literature for the case of an uncoated cantilever in a viscous liquid medium and the case of a coated cantilever in air or in a vacuum. The theoretical results presented are then compared to available chemical sensor data in aqueous and viscous solutions.

Improved yield and performance of ball-grid array packages: design and processing guidelines for uniform and non-uniform arrays

Two models are presented for analyzing ball-grid array (BGA) solder interconnects: (a) a model for predicting solder joint geometries after an upright or inverted reflow process and (b) a reliability model for estimating maximum shear strain and fatigue life under a global thermal mismatch load condition. The reliability model includes a newly derived closed-form expression relating the maximum shear strain in a solder joint to the load, material, and joint shape parameters. The models are used to generate design/processing guidelines that may be used to improve the yield of a BGA soldering operation and the in-service reliability of the interconnect. Potential benefits of using a nonuniform array (i.e. one including joints of different size and shape) are explored.