Search

The Henstock integral in ℝn and its relation to the n-dimensional improper Riemann integral are studied. A Hake-type theorem for the Henstock integral in ℝn is proved.

Examining CMP at any scale, one finds coupled contact mechanics and fluid mechanics. Increasingly sophisticated experimental and computational techniques have revealed aspects of solid-solid interaction and slurry flow at the wafer and groove scale and, more recently, at the texture scale. Successful prediction of CMP performance hinges on identifying universal physics that span these scales. In this paper we first review results of novel asperity-scale experiments that characterize the pad texture both as a solid topography subject to contact deformation and as an equivalent porous medium for slurry flow. These measures reveal that much of the texture volume is inactive as flow space, a feature confirmed quantitatively by computational modeling of flow across a conditioned CMP pad surface built from 3-D microscopy images. For hydrodynamics, the findings establish active fluid volume per unit area as the property that bridges from asperity scale to wafer scale. We then derive a fundamental basis for CMP removal rate prediction based on contact and hydrodynamics, using a Sommerfeld number defined across the groove and texture length scales. The resulting equation, containing a single unknown proportionality constant, demonstrates that the often used product of downforce and table speed tracks removal rate only when the hydrodynamic state affords adequate pad-wafer contact. Departures from the Preston equation attributed in other models to chemically-limited regimes of CMP are explained in the present treatment as changes in hydrodynamic film thickness and contact area—a fact confirmed by direct measurement. Removal rate predictions are discussed for ILD, STI, and copper processes using both conventional and non-Prestonian slurries, including variations in downforce, table speed, temperature, pad properties, and groove design. Finally, the influence of regional pad-wafer hydrodynamics is illustrated by applying the contact-hydrodynamics equation to grooves specially configured to vary the slurry film thickness from wafer center to edge. Local removal rates are well predicted using locally defined values of the groove-texture Sommerfeld number, confirming the generality of the contact-hydrodynamic description at least from wafer to asperity scale. Findings are further discussed in the context of next-generation pad architectures—not only to achieve more effective pad-wafer contact and slurry delivery, but also to favorably decouple contact and fluid mechanics in CMP pad design.

Total contact area between a CMP pad and wafer has emerged as a fundamental property that is directly linked to both removal rate and defectivity. Contact area has been shown to depend on surface morphology as it results from pad microstructure and material properties. Pads exhibiting higher contact area, hence imparting lower point stresses to the wafer, are effective in reducing CMP-related defects. The present study quantifies pad-wafer contact characteristics for both hard and soft porous polyurethane pads as a function of the extent of diamond conditioning. An Instron microtester is used to impart controlled quasi-static pad compression against a sapphire cover slip mounted on a Zeiss confocal microscope. Images collected through the microscope at discrete compressive states are analyzed off-line to quantify the total contact area and the size, shape, and distribution of individual contacts. A relationship is then established between these contact measures and the extent of conditioning. Conditioning decreases the mean asperity contact size as expected, but the evolution of contact regions shows non-intuitive features. In particular, total contact area passes through a minimum within the typical conditioning time of commercial CMP processes, then gradually increases. Clustered contact regions observed at short conditioning times spread out to a more uniform lateral spacing at longer conditioning times. These findings are accounted for by considering the texture subdivision achieved by individual diamonds together with bridging by the conditioning disk of higher regions on an unlevel pad surface. Textures produced by different conditioners on the same pad are quantified in terms of total contact area and number and uniformity of contact points. Formation of an ideal texture of abundant, well-spaced and level asperities is expedited by both conditioner type and treatment time. The results illustrate that evolution of pad contact area under compression is an essential measure for understanding polishing metrics such as wafer removal rate, defect count, and pad wear, and indicate clear direction for next-generation pad microstructures to achieve low-stress planarization.

Asperity-scale pad deformation and dynamic pad-wafer contact area are crucial to the fundamental understanding of material removal and defect formation mechanisms in CMP. Pad asperity stress and strain are also central to characterizing pad wear rate during polishing and cut rate during conditioning. While it is very difficult to isolate and measure stress and strain in individual asperities, finite element modeling may be used in conjunction with experimental surface characterization to predict asperity-scale deformation and pad-wafer contact. Asperity sub-domains up to 1270 microns across are reproduced from three-dimensional point cloud data on porous polyurethane CMP pads obtained by confocal microscopy, meshed to high resolution, and analyzed using ABAQUS finite element software. Physical properties are derived from dynamic mechanical experiments. Pad stacks are simulated both with and without sub-pads. Results show that while a sub-pad increases pad-wafer contact area overall, it limits the local spreading of individual contact regions as polishing load increases. This finding identifies a direct mechanical origin of the trade-off in pad design between wafer-scale and die-scale planarity. As expected, the real contact area between a pad and wafer is much smaller than the cross-sectional or “bearing” area, but the difference is notably greater when a sub-pad is present. Values of asperity stress and strain under typical CMP polishing pressures reveal that plastic deformation takes place both on and beneath the contacting surface. Hence upon release of the polishing load the asperities do not fully rebound to their pre-compressed shapes. Each pass under the wafer thus reshapes the pad asperities such that a slightly different texture is presented upon the next pass. These deformation mechanics clarify the impact of top pad and sub-pad properties on real contact area, allowing better optimization of CMP pad performance.

Real contact area between a CMP pad and wafer is a key factor in local contact pressure, friction and pad wear, all of which impact material removal and defect formation. A new optical method is introduced that quantifies the real contact area during polishing. Confocal reflectance interference contrast microscopy (C-RICM) uses a single focal plane to image the pad-wafer contact interface. A sapphire cover slip is used to provide optical transparency and to match the refractive index of the pad. Imaging the pad surface through the cover slip reveals areas of no reflection (pad-cover slip contact), areas of reflection (non-contact), and interference fringes (near contact). The C-RICM method was validated using micro-fabricated pads having uniform arrays of cylindrical surface structures of known contact area. Experiments conducted on porous polyurethane pads revealed that the real contact area is less than 10% of the total presented area. However Greenwood-Williamson (G-W) theory, widely used in CMP material removal models, predicts a contact area at least a factor of ten smaller. The discrepancy was found to result at least in part because the individual contact zones are not elliptical as assumed in G-W theory. In fact many contacting structures are crescents located at the perimeter of individual pores. These findings underscore the need for accurate control of pore density and morphology in polishing pads, in both initial manufacture and surface conditioning during CMP. The C-RICM method allows non-destructive benchmarking of polishing media in terms of pad-wafer contact, essential for developing improved pad architectures that achieve lower CMP defect levels.

Material removal in CMP occurs during intervals of pad-wafer contact separated by intervals of non-contact. One predictable sequence of non-contact intervals for a fixed point on the wafer is the traverse of the pad grooves, during which the wafer surface is renewed with fresh chemistry and heat is conveyed away. It is well understood that good uniformity requires machine kinematics that expose all points on the wafer to the same total contact time, mean slurry concentration, and temperature. Less widely known is that coherent structures tens to hundreds of nanometers high and matching the pitch of the pad groove pattern may be formed on an otherwise planar wafer despite multiple rotary motions. This unexpected phenomena is of interest not only because it manifests the impact of grooves and transport at scales not easily studied, but also because shrinking device architectures will ultimately disqualify even nanoscale departures from planarity. Wafer polish experiments are conducted alternately using circular, Cartesian grid, and spiral groove patterns using specialized pad conditioning and CMP recipes to amplify groove-induced nanotopography. Polish results illustrate sharp patterns in finished wafers (visible to the naked eye) that should not survive dual-axis tool kinematics. Computational 3-D model results are then presented for transient slurry mixing in the pad-wafer gap of a 200-mm polisher using the same groove patterns. A direct correspondence is found between observed wafer nanotopography and predicted groove-scale slurry mixing dynamics. In particular, the surface structures are underpolished areas traceable to intervals of non-contact protracted by depleted polish chemistry that prevails in groove segments when oriented relative to the local pad and wafer motion in a way that suspends transverse mixing in the groove crosssection. The study conclusively defines the features required in a groove pattern and polish recipe to form coherent structures matching the groove pitch. As validation of the theory, a groove pattern expected to form no surface topography is defined, experimentally tested, and shown to perform as predicted. Findings are discussed in the context of next-generation pad grooving and texturing as required for progressively more demanding applications of CMP.

A 3-D fluid flow and kinetics model of the full pad-wafer gap of a commercial CMP machine, including both grooves and land area flow properties, was developed to research the influence of these essential pad features at a scale not previously studied. Concentric circular grooves of two different dimensions were examined. Heat transfer and chemical reaction rates at the wafer surface were studied by including a combined frictional/chemical heat release model and firstorder kinetics for slurry activity. Results revealed sharp variations in velocity between the grooves and land areas, and a strong impact of groove depth and width on transient slurry mixing in the pad-wafer gap. These effects were more pronounced at higher polish pressures. Flow patterns in individual grooves beneath the rotating wafer were highly variable from one location to another such that fluid pathlines were dissimilar point to point even within the same groove. The effects of pad and wafer rotation, amplified by the unequal flow resistances of grooves and land areas, led to features in the steady-state temperature and concentration profiles at the same spacing as the groove pitch. This unusual finding identified a direct mechanism by which polish irregularities may be formed at the scale of the grooves.

Surface texture of CMP polishing pads varies considerably with the intrinsic microstructure of the pad, the addition of perforations or grooves, and the means of surface conditioning. The state of pad-wafer contact is determined by both large and small-scale texture in the top pad asperity layer and by the interaction of the top pad and sub-pad.A flow-based texture characterization test was applied to several types of CMP polishing pads to describe the asperity layer as a porous media having a void fraction and characteristic length. Fluid pressure loss profiles were measured in a radial flow geometry across pad samples pressed against a flat instrumented plate. Symmetry of the profiles revealed the extent of contact between the pad and the plate, and curvature of the profiles showed the relative contributions of viscous and inertial flow among the asperities. Sub-pads, grooving, and conditioning all increased padwafer contact and the effective resistance of the surface texture, improving flow uniformity. Soft pads showed higher inertial flow influence than fixed abrasives with regularly spaced asperities. The results demonstrate that pad surface texture has a strong influence on heat and mass transfer at the wafer surface in CMP. Implications are discussed for both pad design and CMP modeling.

Parathyroid hormone levels in primary hyperparathyroid patients and patients with malignant disease and hypercalcaemia are reported. The value of PTH measurement in conjunction with serum calcium and bicarbonate values in differential diagnosis is discussed. Theories for the production of hypercalcaemia in malignant disease are reviewed.

Amethod is described for determination of serum ionised calcium, using an ion specific electrode. Values recorded in 24 normal subjects, and in various disease states are presented and the clinical value of this measurement discussed.

A Retrospective review of 46 osteomalacic patients is reported. There was a marked female preponderance (80%). The mean age at time of diagnosis was 50 years and the majority of patients (80%) were symptomatic. Chronic renal failure (30%) and adult coeliac disease (24%) were the two most common underlying conditions. The cause of osteomalacia was identified in all but four cases (9%). The biochemical parameters among the series are described. Serum 25 (OH) vitamin D was estimated in 35 cases, and was reduced in 25 of 28 not receiving vitamin D supplements. We suggest that a) coeliac disease should be excluded in any patients with osteomalacia of undetermined origin, b) that a normal serum 25-OH vitamin D level is a useful negative index for significant osteomalacia.

Conflicting opinions are held on the efficacy and safety of low-dose oral vitamin D supplementation. In this study, the value of short-term and long-term low-dose vitamin D supplementation (20 μg D3 daily) is demonstrated in elderly subjects. Short-term therapy readily restored serum 25-hydroxy-vitamin D (25OHD) values to normal, and ameliorated biochemical abnormalities of calcium, phosphate, and alkaline phosphatase, consistent with healing of mild osteomalacia. Prolonged therapy (16 months) maintained serum 25OHD levels within the young adult range, and was not associated with hypercalcemia. Withdrawal of therapy was associated with a steady decline in serum 25OHD levels. Low-dose vitamin D supplementation is recommended for elderly subjects to prevent the development of hypovitaminosis D osteopathy.

Renal function was seen to deteriorate as total serum calcium rose in 40 patients with primary hyperparathyroidism. There is no significant difference between creatinine clearance values measured pre- and post-parathyroidectomy even after an interval of 8 years. This indicates that lowering of serum calcium by parathyroidectomy stabilises renal function, but creatinine clearance does not return to normal values once renal damage has occurred.

A review is presented of 35 cases of primary hyperparathyroidism, studied before and after parathyroidectomy. Presenting clinical features varied widely but were mostly related to renal stone disease. The critical value of serum ionised calcium and parathyroid hormone measurements are emphasized. Metabolic acidosis due to renal bicarbonate wastage was present in 68% and improved significantly following operation.

Five subjects presenting with renal stone, hypercalciuria and normal total serum calcium showed persistent elevation of ionised plasma calcium and marginal elevation of plasma parathyroid hormone concentration. Neck exploration revealed a parathyroid adenoma in each, and parathyroidectomy has been followed by reduction in both plasma ionised calcium and urine calcium. Ionised calcium measurement is an important aid in the differential diagnosis of idiopathic hypercalciuria from ‘normo-calcaemic’ hyperparathyroidism.

1. A clinical case of osteomalacia secondary to idiopathic steatorrhoea was found to exhibit the biochemical features of renal tubular acidosis. 2. Treatment with vitamin D resulted in reversal of systemic acidosis with restoration to normal of renal tubular acidifying power. 3. It is suggested that vitamin D is necessary for renal tubular (and perhaps gastro-intestinal) transport of hydrogen and phosphate ions. Evidence from the literature is quoted in favour of the further concept that transport of phosphate, hydrogen ion, glucose and amino-acids may be vitamin D dependent.