Establishing the discipline of physics-based CMP modeling
03/01/2002
Scott R. Runnels, Scott Runnels Consulting, Austin, Texas
Thomas Laursen, SpeedFam-IPEC, Chandler, Arizona
overview
For the past decade, a physically based comprehensive process model for chemical mechanical polishing has eluded the semiconductor industry. However, a long-term collaborative effort has now resulted in a workable version of that approach. The highly fundamental model is based on advanced finite element analysis and is beginning to show promise in CMP process development.
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When chemical mechanical polishing (CMP) modeling first emerged in the early 1990s, the computational cost of "solving the CMP problem" with brute force simulations was considered beyond reach. Thus, the term "CMP process model" became associated with the many phenomenological models created by investigators in the field. These models are actually mathematical encapsulations of investigators' understanding of the CMP process. As such, the models reproduce existing knowledge but cannot be used to explore new physical situations. Instead of replacing expensive experimental work, these models require more tests for validation, and then are limited to exploration in process segments close to where they have been validated. For this reason, the popularity and range of application of phenomenological models have been limited. Although to some extent, belief that the process can be modeled using physics has suffered, many of the key aspects of CMP are actually well understood in classical physics.
The challenge has not been to develop those individual models from scratch, but rather, to undertake the substantial challenge of integrating them. Meeting that challenge requires confidence and support that must be gained through a well-managed collaborative effort.
CMP software development
In 1996, SpeedFam-IPEC and Southwest Research Institute (SwRI) began a relationship aimed at building and establishing confidence in physics-based modeling. Their shared vision was to start with simple models, gradually add physics as needed, and always integrate end users. The incremental approach was designed to maintain accountability of the highly theoretical work by ensuring timely results, while also establishing a firm base of support. The first major accomplishment occurred when SwRI developed the pattern-scale simulator Mesa, which models the process by solving the pad-wafer elastic contact problem, a classical challenge in computational mechanics [1]. Mesa represents the motion of multiple asperity tips as one continuous flat pad, modeled as a matrix of springs, so that the pad's internal stresses and the contact forces are in overall equilibrium with the process down-force. The contact stresses from the pad are then used to determine the varying rates of material removal, point by point. Mesa demonstrated relatively accurate predictions over a wide range of pattern densities for oxide, copper, and STI processes [2-5], providing the first proof that robustness and independence from experimental validation can be achieved when the appropriate physics are correctly included. At the same time, a user-friendly version of the software was developed and deployed, which helped increase the base of support for Mesa's more physical and computationally demanding approach to process modeling.
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Figure 1. Overview of the integrated fundamental mechanics CMP process model embodied in Compass. a) Composite carrier with moment of inertia, mass, and carrier ring that can interact with the pad and wafer; input: pc, shear, and force from wafer on retaining ring; output: tilt; b) pressure distribution pc obtained from carrier film model; c) carrier film model; input: ps; output: compressed shape; d) wafer model; input: ps, shear on both sides and position of retaining ring; output: distorted shape force on carrier ring; e) pressure distribution ps obtained from slurry transport model; f) slurry transport model; input: shape of interface, velocities, pad roughness; output: fluid pressure, percent contact, shear; and g) pad stack compression model; input: ps; output: compressed shape.
Mesa's effectiveness, combined with its acceptance by end users, helped renew confidence in physics-based modeling to the point that work on the comprehensive computational mechanics model shown in Fig. 1 could be undertaken. The comprehensive model was assembled from several fundamental mechanics models that incorporate gradual (viscoelastic) compression and rebound of the pad; elastic-plastic asperity wafer-pad contact (permanent asperity deformation); intra-asperity slurry flow; wafer deformation; carrier-film contact; and carrier dynamics.
The theoretical backbone
The backbone of the integrated model is the application of advanced finite element analysis (FEA) methods, which seek the approximate solution to partial differential equations describing specific classical laws of physics, such as the conservation of mass and energy, and the balance of forces within continuous media. FEA rests on a discretization (breaking up) of geometries into small, nicely shaped pieces called "elements," the collection of which is called a grid or mesh. Grid points, or nodes, are typically placed at the corners of these elements and an approximate solution is comprised of discrete solution values at those points.
FEA is a proven and effective method for solving all the classical conservation laws present within the major components of a CMP process. However, commercial FEA software does not accommodate the complexities of the partial wafer-pad contact, moving pad material, and slurry flow. Therefore, custom FEA software, equipped with advanced methods, was required.
The first advanced method incorporated in the custom FEA software allowed simultaneous handling of two types of formulations known as Eulerian and Lagrangian. The Eulerian approach uses a stationary finite element mesh that allows material to move through it, as is used in computational fluid dynamics. The Lagrangian approach uses a moving mesh that remains fixed to the material even if the material moves.
The second advanced method, known as the "overset grid method" [6], was implemented to enable proper resolution of the wafer geometry. Since the wafer may move laterally, and its round shape demands a properly fitted mesh, two meshes in the pad are useful. One pad mesh, Eulerian, is fixed under the wafer and is circular to provide good spatial resolution. The other pad mesh, Lagrangian, follows the shape and movement of the pad. These two meshes communicate so that their solutions are mathematically consistent. The overset grid method, in combination with the simultaneous Eulerian-Lagrangian formulation, enables the advanced FEA software to include all of the physics traditionally handled by FEA, but enables it to do so on the pad, wafer, and carrier assembly at the same time, despite their large-scale relative motions.
 Figure 2. Overview of the elastohydrodynamic wafer-pad-slurry contact model in Compass. |
A third advanced method added to the FEA software was implemented for handling the various modes of contact present in CMP, particularly the elastohydrodynamic (EHD) contact between the pad and wafer, collectively referred to as the "slurry transport model." In Fig. 2, a schematic of the overall EHD contact model shows the total pressure, imposed on the wafer and pad surfaces. In the region of contact, the total pressure is the sum of the slurry pressure and the asperity contact pressure. In the regions of noncontact, it is the slurry pressure only. The slurry pressure is determined by using the finite element method to solve the "average flow model" [7, 8], a form of the Reynolds equation, reduced for thin film flow. Asperities interfere with pressure-driven flow and enhance sliding boundary-driven flow. The average flow model results when parameters, called "flow factors," are introduced into the Reynolds equation to represent the bounding surfaces' roughness, and orientation of the surface asperities [7, 8].
The method for handling the second component of the slurry transport model, the asperity-wafer contact pressure, is based on the classical conceptual technique of geometrical decomposition. As illustrated at the top of Fig. 2, the ultra-thin asperity layer is sliced away, leaving a perfectly smooth pad surface. This asperity layer is so thin that it has no bending strength, and therefore conforms perfectly to the shape of the bulk pad's smooth surface. Conceptually, this thin layer could be subjected to various compressions from a rigid surface, and could be characterized in a compact model describing the percent contact and strength of the material as a function of compression. A wide variety of statistically identical 3-D asperity surfaces could be simulated to produce plots for those compression-strength and compression-contact relationships. Once obtained, they may be used to relate the amount of protrusion from the wafer into the conceptual asperity layer to the effective strength of the layer and the contact stress.
The characterization of the asperity layer and the computation of the slurry flow factors are the means for integrating pad topography into the integrated CMP process model in a scientifically disciplined way. For example, to obtain closed-form characterization equations, the asperity layer is modeled as a group of individual pillars, whose heights are randomly and uniformly distributed. In this simplified case, deriving a relationship analytically between the amount of protrusion of the wafer into the asperity layer, the effective modulus of that layer, and the contact pressure, is a straightforward process.
More complex surface topographies and nonelastic material behavior can be considered. For instance, the EHD contact model may be implemented through the exchange of stress boundary conditions between the pad and wafer or through specialized elements that join the wafer and pad mesh. When the element approach is used, the strength of the elements is varied according to the effective modulus of the compressed asperity layer and pressure terms are added to represent the slurry pressure.
Next, a set of auxiliary models was placed on the backbone of the finite element models. Of primary importance is the rigid body dynamics model for the carrier. The finite element mesh for the carrier is used to compute the mass, center of mass, and moments of inertia, which are then used in the equations of rigid body motion for a symmetrical object forced to rotate about its axis and allowed to move in the vertical direction. The three components of the force and moment vectors applied to the bottom of the wafer are used in the carrier's rigid body motion equations. These vectors are computed by integrating the pressure and shear stress at the wafer-pad interface.
Finally, the more phenomenological Preston model is used to compute the spatially and temporally varying removal rate across the surface of the wafer. The use of the Preston model is still justified, based on its accuracy in bulk material. By arming it with accurate pressure, temperature, and species concentration distributions, the Preston model is expected to yield a substantially improved prediction of material removal on the scale of the wafer. However, since the overall integrated model also provides values for temperature, chemical species concentration, and the full stress state, more sophisticated removal models can also be implemented.
New validation paradigm
The benefits of physics-based modeling emerge in the validation process because it is more practical and scientifically disciplined. Specifically, each component of the integrated model can be validated separately in a laboratory removed from the CMP process. For example, it is not legitimate to question whether or not forces and inertia balance each other inside the pad, but one could question the viscoelastic constitutive model. That model can be validated, however, using thorough material testing at various temperatures in a laboratory. Likewise, the use of a Newtonian model for the slurry can also be addressed by considering small, isolated samples of slurry. The same is true of all the integrated components, with the exception of the material removal model, which is more phenomenological. However, the global model is not rendered invalid if any of these individual constitutive models are determined invalid. Rather, the results of the global model are interpreted with those deficiencies in mind. Furthermore, when comparing the model results with experimentally derived trends, differences in the model predictions can be attributed to certain specific groups of physical phenomena not included in the model.
The challenges in validating an integrated physics-based model are twofold. First, it must be verified that the intended computations are, in fact, occurring. To that end, specific test cases of the classical models are solved by the model software and the results compared to either analytical solutions or known-good solutions obtained by other means. Second, and probably more critical, the implicit assumptions must be continually addressed. The CMP process is sufficiently complex that it is well within human oversight to inadvertently assume away a key detail of the process. Surprisingly, it is that aspect which has been the biggest challenge and has proven the most educational throughout development of the model. To guard against this, it is necessary to check continually to ensure that the model correctly and reliably predicts all well-known trends. If it does not, then foundational assumptions of the model must be revisited.
 Figure 3. The two test cases used to explore the integrated CMP process model. |
Lessons learned
The integrated model is capable of simulating most aspects of polishing. However, because of the model's complexity, it is more productive to activate aspects gradually, thereby understanding their ramifications individually. Two simple examples have been considered, the first for a rotational platform with a single-pad layer (Fig. 3a). The model included a rigid carrier, elastic carrier film and wafer, slurry, viscoelastic pad, and a rigid platen. The simulation was isothermal, and since it was for oxide polishing, chemical species concentration was not considered. The vertical position and angle of attack for the carrier were primary unknowns during the simulation and determined through the rigid body dynamics model. Considering only the dynamics of the carrier, however, was found to be inappropriate because doing so allowed the possibility of high-frequency carrier vibrations. The addition of appropriate mass and frictional dampening terms resolved that problem.
 Figure 4. Pad deformation at 2 sec, predicted using the overset grid finite element method. |
In the first few milliseconds of the polish simulation, the pad responded elastically, descending nearly uniformly under the wafer, and exhibiting a substantial edge effect. New pad material entering under the wafer gradually compressed during its traversal, rebounding gradually after emerging. Steady state was reached at about two seconds. The deformation of the pad at steady state, highlighting the use of the overset grids, is shown in Fig. 4. The initial edge effect due to the elastic deformation softened at steady state because of the pad's viscoelasticity. When the simulation was rerun with a 5¥ storage modulus, the pad deformation and pressure distributions on the wafer exhibited steep edge pressure gradients.
In this case, the slurry pressure was only 2% of the total. Its contribution was small because the asperity height was set at 100μm, which prevented the wafer surface from getting very close to the pad's mean (average) surface. If the table and carrier speeds were doubled to 40rpm and the mean asperity height set to 50μm, the slurry pressure jumped to 25% of the total pressure. The anticipated trend of higher speeds and a smoother pad leading to decreased pad-wafer contact and the onset of hydroplaning was correctly predicted.
Despite these interesting results, this first example did not predict the widely observed trend that increasing the down-force improves uniformity. Instead, the opposite trend was predicted. This seemingly unfortunate result actually became a guide that eventually pointed to another important implicit assumption, which was that the wafer and pad were modeled as being flat.
Those two assumptions were directly addressed in the second example: a single-head orbital polish geometry, hard platen CMP system (Fig. 3b), including a rigid carrier without a wear ring, elastic carrier, and wafer (with initial internal stress); a single-layer (IC1000) pad with 24μm thickness variations at a wavelength of about 20 cm; and a rigid platen. The models for this example were like those listed for the first example, except that the carrier was held in a fixed position, as is often done in practice. The use of a pad with the thickness nonuniformity (as described above) revealed some important behaviors. During the imposition of the carrier's downward displacement, the pad's uneven surface was flattened somewhat due to the relatively slow viscoelastic response of the polyurethane. After about 2 sec, the bulk compression of the pad was completed and the pad's uneven surface peaks began to reappear. An important finding was that the pad's overall bulk downward displacement oscillates throughout the simulation because different combinations of its peaks and valleys are drawn underneath the wafer as the pad orbits. This same behavior can be expected in a rotational tool, and to some extent, a carrier with a fixed down-force. And because of the pad's uneven thickness, the most appreciable direct pad-wafer contact is limited to the thick areas of the pad, which is in constant motion relative to the wafer.
 Figure 5. Results from the integrated model demonstrating how key trends are predicted without the use of adjustable parameters. |
As shown in Fig. 5a, the model predicted that the material removal rate increases and nonuniformity improves with increasing downward carrier displacement. Figure 5b shows how the material removal rate, with units of length/sec, decays with time for two different carrier displacements. At the beginning of the polish, the rate increases with time because the carrier is displaced gradually at a rate of 30μm/sec, and, in addition, the pad compresses viscoelastically over a period of a few seconds. Due to the high stresses at the pad peaks, plastic deformation occurs, reducing the initial point-wise level of pad-wafer contact. The plastic deformation appears to occur in the first 10 sec, a time scale that is likely associated with the viscoelastic compression time scale of the bulk polyurethane.
Moving forward
Deployment of these results and capabilities to end users and developers will continue to be a focus as the modeling program moves forward. While Mesa addresses feature-scale planarization on a local scale, the integrated model predicts polish performance on the wafer scale, which is sensitive to all of the mechanical components in a polish tool. Working together, the pressure, contact, and polish profiles calculated by the integrated model, Compass, can be used as input for Mesa, which can then be used to calculate, for example, within-wafer variation of dishing and erosion.
The degree to which any model is based on physics determines the extent that it can be used for exploration. One of the chief benefits of such a physically based model is that it directly challenges implicit assumptions. This benefit is realized best when a close collaboration between theoreticians and practitioners is established. To the extent that the integrated model challenges those assumptions through such collaboration and its scientifically disciplined approach, it can contribute to CMP understanding.
Acknowledgments
Mesa is a trademark of SpeedFam-IPEC and Southwest Research Institute [2, 9]. Compass is a trademark of SpeedFam-IPEC. The calculations reported here were performed with Compass 1.0 and 2.0, a comprehensive computational mechanics model for CMP guidance.
References
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Scott Runnels is the author of Compass and owner of Scott Runnels Consulting, 6406 Back Bay Lane, Austin, TX 78739; ph 512/394-0064, fax 512/394-9465, e-mail [email protected], www.srconsult.com.
Thomas Laursen is technical manager of SpeedFam-IPEC's Copper CMP Group. He has more than 80 published papers, 2 patents, and 6 patents pending in CMP, ion beams, and material science.