John A. Tichy

Contact Mechanics and Lubrication Hydrodynamics of Chemical Mechanical Polishing

A preliminary model for the contact mechanics and fluid mechanics of the chemical mechanical polishing process is presented. Only the basic equations of elastic contact surface mechanics and hydrodynamic lubrication are required. Although the model is highly idealized, no ad hoc assumptions or adjustable parameters are required. Some new experimental results are presented, reinforcing the counterintuitive experimental determination of suction fluid pressure below the pad. The model correctly predicts the magnitude of the suction pressure and the effect of load, speed, and roughness.

Interfacial Fluid Mechanics and Pressure Prediction in Chemical Mechanical Polishing

This paper reports on the measurement of fluid (water) pressure distribution at a soft (polyurethane) pad/steel interface. The distribution of the interfacial fluid pressure has been measured with a specially.-designed fixture over the typical range of normal loads and velocities used in the chemical mechanical polishing/planarization of silicon wafers. The results show that, for most cases, the leading two-thirds of the fixture exhibits a subambient pressure, and the trailing third a positive pressure. The average pressure is sub-ambient and may be of the order of 50∼100% of the normal load applied. An analytical model has been developed to predict the magnitude and distribution of the interfacial fluid pressure. The predictions of this model fit the experimental results reasonably well, especially for low sliding velocities.

Multiscale material removal modeling of chemical mechanical polishing

Abstract This paper describes a multiscale model for material removal during conventional chemical mechanical polishing (CMP). Three spatial scales are considered in the integrated model: (i) abrasive particle scale; (ii) asperity scale; (iii) wafer scale. The model is based on the deformation of hyper-elastic asperities attached to a linear-elastic pad. ANSYS is used to perform finite element analyses of a single asperity to obtain the relations between the deformation of the asperity, and the contact stress and area. Those relations are used in an extended Greenwood–Williamson model to compute the local average contact pressures on the pad. The material removal model includes the abrasive wear caused by local contact stress between the abrasive particles and the wafer, the distribution of asperity heights, and the plastic deformation of the wafer. The material removal rate results for unpatterned wafers are used to predict the material removal rates on a feature. The results of a computer simulation of material removal and the time evolution of a feature are shown. Two FEM based codes, ANSYS and EVOLVE are used; the former for the contact stress analysis and the latter to generate a new surface.

Hydrodynamic lubrication theory for the Bingham plastic flow model

When the shear stress magnitude of a Bingham material exceeds the yield shear stress, quasi‐Newtonian flow results, otherwise the material is rigid. The Bingham model has been used in tribology to describe the behavior of greases, but may also be used for electrorheological fluids proposed as ‘‘smart’’ lubricants. For two‐dimensional flow, different modified Reynolds’ equations are obtained, depending on the possible local formation of a rigid core which may be attached to either surface, or float between the surfaces. From the modified Reynolds’ equation it is straightforward to predict bearing behavior, patching together the different core formation cases. Results are presented for two geometries: the squeeze film damper and the journal bearing.

Granular collision lubrication

The flow of powder or granules has been proposed as a mechanism of lubrication suitable for high‐temperature applications where conventional liquid lubrication fails. This study is primarily experimental with a simplified theory presented for interpreting the data. Two key features are present which seem to be additive: (1) a collisional normal stress generated by kinetic energy of the particles, and (2) a lubrication normal stress due to converging surfaces. Experiments are conducted in an annular shear cell with sliding motion between opposing surfaces. The shear surface may be flat or contain three sloping regions with a step. Normal stress (load) and shear stress (friction) are proportional to shear rate squared. In the case of inclined surfaces, stresses are also proportional to the surface slope squared.

Modeling of Thin Film Lubrication

A rheological model is proposed which can be applied to boundary lubrication. The model satisfies the formal continuum mechanics considerations of material objectivity, i.e., it is indifferent to changes in coordinate system and reference frame. The model is applicable to thin films in which the molecular length scale is the same order as the film thickness, due to the use of ensemble, rather than spatial, averaging. The microstructure is described through the use of a director, a unit vector aligned with the molecular orientation. The model contains three material parameters, i.e., the conventional viscosity, a director viscosity which describes varying flow resistance depending on the director orientation, and an elastic modulus relating to moments required to change the director orientation. Typically, the directors are anchored to the boundary surfaces in a favored direction. Calculations are performed for mass velocity, director orientation, and stress in a contact. The lubricant exhibits nearly soli...

Behavior of a bingham-like viscous fluid in lubrication flows

Abstract When the shear stress magnitude of a Bingham fluid exceeds the yield shear stress, quasi-Newtonian flow results, otherwise the material behaves as a rigid body. Yield surfaces may occur in the flow, and discontinuities due to changes from fluid to solid behavior must be respected. The behavior of Bingham fluids is mimicked by a purely viscous fluid with high viscosity at low rate of shear. The results of predictions of one such model are presented in this paper. The goals of this investigation are (1) to present a continuous formulation (i.e. applicable throughout the flow field) to characterize the behavior of such a material, (2) to calculate the behavior in several lubrication flows, and (3) to compare these results to predictions of the Bingham theory where applicable. The rheological equation is a function which can be reduced to a Newtonian fluid and the Bingham-like purely viscous fluid by changing the value of a parameter. A generalized Reynolds equation is presented in order to use the continuous constitutive law. Several applications are studied: a plane slider bearing in two-dimensional flow as well as two parallel plates with an imposed pressure in three-dimensional flow. Several curious results occur which are not anticipated from the behavior of either Newtonian or Bingham fluids.

Granular Flow Lubrication: Continuum Modeling of Shear Behavior

Because at extreme temperatures, conventional liquid lubrication breaks down, researchers have proposed using flows of solid particles as a lubricating mechanism. The particles may be powders, which tend to coalesce and slide over one another in sustained contact, or granules, which collide with one another in fluctuating motion. Distinction between these two regimes is elucidated. The behavior of various granular flows is studied using a granular kinetic lubrication (GKL) model. Our GKL model is a continuum approach that applies proper rheological constitutive equations for stress, conduction and dissipation to thin shearing flows of granular particles, as well as the most rigorous boundary conditions for momentum and energy transport. A robust numerical code, utilizing Newtons finite differencing method, is developed to apply GKL theory to the problem of simple shearing flow. The code solves two second-order, coupled nonlinear ordinary differential equations with coupled boundary conditions of the first-order. As a result, new parametric curves for the local flow properties of the large-particle granular flows are constructed. Results from the GKL model agree qualitatively with past experiments using glass granules in an annular shear cell.

Review of solid mechanics in tribology

Abstract The study of solid mechanics is essential to the field of tribology, (friction, lubrication and wear). Tribology is of immense economic importance. The potential savings, were tribological principles better understood and applied to friction and wear reduction) may be several percent of the gross national product. Solutions to tribology problems often enable current technologies in a broad spectrum of applications from friction contact in the turbine shrouds of aircraft engines, to bearing contact in motor vehicle gear assemblies, to the sliding contact of magnetic storage disk drives. Conversely, tribology issues, e.g., the coefficient of friction, may impact solid mechanics problems and tangential tractions are essentially free parameters in many cases. Active issues of research in tribology where solid mechanics is applied include: friction and wear in dynamic loading of bearings to extend bearing life; models for contact and thermal stresses of sliding surface asperities; design criteria for magnetic recording heads, and behavior of human artificial joints to extend service life. Countless other applications exist, requiring the development of essential theories of conforming and non-conforming surface behavior. Information such as the frictional response of surfaces in relative motion, and modes of stress and deformation emerges from the fusion of solid mechanics and tribology.

A Multiscale Elastohydrodynamic Contact Model for CMP

We present a three-dimensional multiscale elastohydrodynamic lubrication (EHL) contact model for chemical mechanical planarization (CMP) based on satisfying the fundamental mechanical force and moment balance equations appropriate for a rotational CMP tool. We describe surface-surface and fluid-surface interactions in conditions where pad asperities are in direct contact with the wafer and the effective film thickness is comparable in size to the roughness of the bounding surfaces. A hyperelastic material is used for our asperity scale model to account for large deformations of the soft polymer asperities. We iteratively solve the soft elastohydrodynamic contact problem. i.e., the solid-solid contact problem coupled with fluid pressure, until the global force and moment balances are satisfied for given operating conditions. The multiscale model computes contact stress across the wafer surface due to asperity deformation that is, in turn, caused by the externally applied load and interfacial slurry pressures. Finally, a relative measure of material removal rate across the wafer is represented by the computed local asperity contact stress. Flat, concave, and convex rigid wafers are considered. The work of this paper focuses on understanding fundamental mechanical aspects of CMP and presents a methodology to simulate mechanical aspects of the CMP process.