Abstract: An integral machine for polishing, cleaning, rinsing and drying workpieces such as semiconductor wafers. A load/unload station has a plurality of platforms for receiving cassettes of wafers to be processed. A dry end-effector of a robot retrieves wafers from the cassettes and transfers them to an index table. A transfer apparatus having wafer carrier elements picks up wafers from the index table, moves the wafers to a polishing table for polishing, and returns the wafers to the index table for further processing. A flipper moves the polished wafers to a cleaning station. The cleaning station includes scrub stations, a rinsing station and a spin dryer station, and a connective system of water tracks. A wet end-effector of the robot transfers rinsed wafers to the spin dryer station. The dry end-effector of the robot moves dried wafers from the spin dryer station back to the cassette of origination.

Abstract: An polishing apparatus consists of a piston which is fixed to the rotation axis, a ceramic plate which is oppositely arranged against the piston via a silicone gel, and a cylinder which houses these components. The wafer is attached on the bottom surface of a backing pad, and will be pressed and rotated by the piston in order to polish the surface thereof.

Abstract: A polishing head of a chemical and mechanical polishing apparatus can uniformly polishing a wafer by sticking the wafer firmly and with uniform pressure to a polishing pad. The polishing head includes a housing through which air is supplied and discharged from the head. A carrier is connected to the housing so as to be movable up and down relative to the housing. A wafer chucking device is mounted to the carrier. The wafer chucking device includes a wafer chuck body movable up and down relative to the carrier. The wafer chuck body defines a vacuum chamber therein for use in adhering a wafer to the wafer chucking device. A retainer is mounted to the periphery of the carrier, guides the wafer chuck body in its up and down movement relative to the carrier, and protects the wafer chucked by the wafer chucking device. A biasing member is used to exert a downward biasing force uniformly on the wafer chuck body.

Abstract: A transparent portion of a polishing pad in the solid phase has an index of refraction nearly matched to the index of refraction of a fluid polishing composition for chemical mechanical polishing, which minimizes scattering of an optical beam at an interface of the transparent portion and the polishing composition.

Abstract: White light from a light source is applied onto a wafer through an observation window which is formed on a polishing pad, and a spectrometric analysis is performed to the light which has been reflected on the wafer, whereby a polishing end point of the wafer is detected. In this case, an amount of the reflected light is measured and brightness of the light source is corrected so that the amount of the reflected light is constant. Thereby, the polishing end point is accurately detected.

Abstract: The present invention provides an improved planarization or polishing apparatus for chemical mechanical planarization and other types of polishing such as metal polishing and optical polishing. In an exemplary embodiment, an apparatus for polishing an object comprises a pad having a polishing surface to be placed on a target surface of the object to be polished. A pad drive member is connected to the pad to move the pad relative to the object to change a position of the polishing surface of the pad on the target surface of the object. A drive support is movably coupled with the pad drive member to support the pad drive member for rotation relative to the drive support around a pivot point which is disposed substantially on the target surface of the object during polishing.

Abstract: A multi-zoned carrier for a chemical-mechanical planarization (CMP) polishing device includes a center cell, a middle cell, and an outer cell. Each of the cells are in fluid communication with each other through multiple conduits equipped with flow restrictors. The combination of cells and conduits allows more uniformity and planarity to be achieved during the chemical-mechanical planarization (CMP) polishing process.

Abstract: A pure water reusing system recovers water discharged from apparatuses in which pure water is used, and regenerates the water to produce pure water. The pure water reusing system has a recovering and regenerating apparatus for recovering water discharged from the apparatuses and regenerating the water by removing impurities from the water to produce pure water, and a regenerated pure water supply line for supplying the regenerated pure water to at least one of the apparatuses. The same process is conducted in the apparatuses in which pure water is used.

Abstract: A polishing apparatus for polishing a surface of a workpiece includes a housing unit, a partition wall partitioning an interior of the housing unit into a first chamber and a second chamber, a polishing section disposed in the first chamber and having a turntable with an abrasive cloth mounted on an upper surface thereof and a top ring positioned above the turntable for supporting the workpiece to be polished and pressing the workpiece against the abrasive cloth, and a cleaning section disposed in the second chamber and cleaning the workpiece which has been polished. The polishing apparatus further includes a transferring device for transferring the workpiece which has been polished from the polishing section to the cleaning section through an opening and an exhaust system for exhausting ambient air from each of the polishing section and cleaning section separately and independently.

Abstract: An apparatus for chemically and mechanically planarizing a workpiece surface employs a polishing slurry and a microreplicated pad having a surface for planarizing the workpiece surface in the presence of the slurry. The surface of the microreplicated pad has a regular array of structures having a sharp distal apexes which contact the workpiece surface during planarization and which are subject to ablating during planarizing thereby becoming substantially blunt. The workpiece moves in a rotational, orbital or translational motion relative to the microreplicated pad.

Abstract: A wafer polishing apparatus according to this invention has a polishing unit that includes three platens and two polishing heads. Injection ports are disposed between the platens. The two polishing heads can linearly move between the platens. In a two-step polishing process, while the polishing head holds a wafer after polishing by a platen, the wafer is transferred to a center platen which provides different polishing characteristics. In this instance, washing water from the injection ports washes away slurry, etc, adhering to the wafer. The wafer is then carried to the center platen beyond a waiting unit and is subjected to further polishing.

Abstract: An apparatus for controlling a polishing process, in particular for detecting an end point of the polishing process, comprising a rotating or orbiting platen with a pad, a rotating head that supports an object to be treated, e.g., a semiconductor wafer, and performs radial movements with respect to the platen, and a polishing process control system comprising a plurality of groups of various sensing devices for detecting an end point of the process. In the illustrated embodiment one group of the sensing devices is a group of high-frequency acoustic emission sensors built on various levels into components of the rotating head. Another group of sensing devices is represented by force/torque sensors connected with various elements of the rotating head and the platen, respectively, and intended for direct measurement of compression force and friction response (force or torque) between the head and the platen and a coefficient of friction between the wafer and the polishing pad.

Abstract: A polishing apparatus that employs a polishing media retention arrangement to prevent slippage or wrinkles in the polishing media during polishing. The polishing media is drawn against a support surface by a vacuum applied between the polishing media and the support surface. Also, a porous layer may be placed between the polishing media and the support surface to form dimples in the polishing media upon the application of vacuum. An alternative arrangement draws the polishing media against a carrier and the substrate to be polished.

Abstract: An improved system and method for automated fiber polishing overcomes the limitations of conventional systems and methods. In particular, the present invention provides for continuous determination of the quality of the polish during a polishing procedure. An optical signal is transmitted to a fiber tip, and any back-reflected signal is detected and monitored to determine polish quality. In this manner, automated polishing quality assurance is provided in real time.

Abstract: An apparatus for planarizing semiconductor wafers in a chemical-mechanical planarization process comprises a polishing pad, a wafer carrier, and at least one acoustic sensor for receiving acoustic emissions produced during the chemical-mechanical planarization process. The wafer carrier is positioned adjacent the polishing pad and is adapted for carrying a wafer in a manner so that the wafer engages the polishing pad. The wafer carrier and the polishing pad are moveable relative to one another in a manner to planarize the wafer. The acoustic sensor is mounted to the wafer carrier in a manner so that the sensor is in contact with the wafer. The acoustic sensor receives acoustic emissions produced during a chemical-mechanical planarization process. The received acoustic emissions are then analyzed to identify and determine surface characteristics of the wafer.

Abstract: A method and apparatus for planarizing a microelectronic substrate. In one embodiment, the apparatus can include a fixed abrasive polishing pad having metal abrasive elements selected to be a compound of metal in the substrate. Alternatively, the metal abrasive elements can include a refractory metal where the substrate includes a refractory metal. Where the substrate includes two metals, the abrasive elements can be selected to planarize the first metal at a rate that is less than approximately twice the rate at which it planarizes the second metal. A single fixed abrasive polishing pad and a single planarizing liquid can be used to planarize both metals.

Abstract: A chemical mechanical polishing machine is provided with a platen that has an integral sub-pad. A fixed abrasive polishing layer is mounted, without adhesive between the polishing layer and the sub-pad, to the top surface of the platen and the sub-pad. The polishing layer is vacuum mounted, for example, to the integral sub-pad of the platen. A cooling arrangement is provided in the platen that cools the polishing layer and improves product quality.

Abstract: The invention provides structure and method for achieving a uniformly polished or planarized substrate such as a semiconductor wafer including achieving substantially uniform polishing between the center of the semiconductor wafer and the edge of the wafer. In one aspect the invention provides a polishing apparatus including a housing, a carrier for mounting a substrate to be polished, a retaining ring circumscribing the carrier for retaining the substrate, a first coupling attaching the retaining ring to the carrier such that the retaining ring may move relative to the carrier, a second coupling attaching the carrier to the housing such that the carrier may move relative to the housing, the housing and the first coupling defining a first pressure chamber to exert a pressure force against the retaining ring, and the housing and the second coupling defining a second pressure chamber to exert a pressure force against the subcarrier. In one embodiment, the couplings are diaphragms.

Abstract: A chemical mechanical head includes a retaining ring to maintain a substrate beneath the mounting surface during polishing. The retaining ring has a lower portion with a bottom surface for contacting a polishing pad during polishing and made of a first material, such as plastic, and an upper portion made of a second, different material, such as metal.

Abstract: In the polishing machine, bad influence caused by a surface condition of a retainer ring can be reduced with a simple structure. The polishing machine comprises: a rotatable polishing plate; a top ring including a holding plate for holding and pressing a wafer onto polishing cloth of the polishing plate; a retainer ring in which the holding plate is freely inserted, the retainer ring independently rotating, the retainer ring including a pressing member, which presses the polishing cloth along an outer edge of the wafer so as to make level of the polishing cloth pressed by the pressing member substantially equal to that of the polishing cloth pressed by the wafer; and a positioning member for correctly positioning the retainer ring on the polishing cloth while the retainer ring is rotated.

Abstract: A polishing tool comprising a support member, and polishing means fixed to the support member. The polishing means is composed of felt having a density of 0.20 g/cm3 or more and a hardness of 30 or more, and abrasive grains dispersed in the felt. A polishing method and apparatus involving pressing the polishing means against a surface of a workpiece to be polished, while rotating the workpiece and also rotating the polishing tool.

Abstract: A chemical mechanical polishing apparatus comprises a platen having a polishing pad thereon, a wafer carrier holding a wafer on the polishing pad, a washer having a cleaning device; and a dresser. The dresser comprises a bottom disk contacting the polishing pad. The dresser can move between the polishing pad and the washer. When the dresser moves into the washer, a diamond zone on the bottom disk cleans the polishing pad, and the cleaning device cleans the diamond zone.

Abstract: A compressible film that is detachably securable to a surface of a rigid structure in the carrier head. The compressible film has a plurality of apertures positioned to create a non-uniform pressure distribution on a substrate during polishing so as to improve polishing uniformity. The apertures may be spaced and positioned to provide a pressure distribution on the substrate that is locally uniform but globally non-uniform.

Abstract: A retaining ring structure of a carrier head designed for use in a chemical mechanical polishing system (CMP) is provided. The retaining ring includes a retaining ring support and a sacrificial retaining ring, which is designed to confine a substrate to be polished. The included sacrificial retaining ring has an upper surface and a contact surface. The upper surface of the sacrificial retaining ring is configured to be attached to the retaining ring support, such that the retaining ring support holds the sacrificial retaining ring. Preferably, the contact surface of the sacrificial retaining ring is configured to be substantially planer with a top surface of the substrate being polished. In a preferred example, the sacrificial retaining ring can include a plurality of capillary tubes and is constructed from a material having substantially the same characteristics as the surface of the substrate to be polished.

Abstract: A carrier head assembly of a substrate polishing apparatus and a substrate supporting carrier pad is disclosed. A down force is uniformly distributed over the backside of the substrate by the carrier pad adapted to be internally pressurized by the down force.

Abstract: The present invention is a method and apparatus for CMP processing that reduces scratching of the insulating film and conductor lines of a wafer. More specifically, the method and apparatus introduce an aqueous solution to the polishing pad and wafer during various intervals of the polishing procedure.

Abstract: A polishing apparatus is used for polishing a surface of a workpiece such as a semiconductor wafer or a glass substrate. The polishing apparatus comprises a polishing table having a polishing surface thereon, a plurality of workpiece holders each for holding a workpiece and pressing the workpiece against the polishing surface, and a controller for controlling the workpiece holders individually so that polishing operations of the workpiece holders are controlled independently of each other.

Abstract: A method for polishing wafers includes providing a wafer having a grating structure including a trench and a process layer formed in the trench; illuminating at least a portion of the process layer and the grating structure with a light source; measuring light reflected from the illuminated portion of the process layer and the grating structure to generate a reflection profile; and identifying dishing in the process layer based on the reflection profile. A metrology tool adapted to receive a wafer having a grating structure including a trench and a process layer formed in the trench, includes a light source, a detector, and a data processing unit. The light source is adapted to illuminate at least a portion of the process layer overlying the grating structure. The detector is adapted to measure light reflected from the illuminated portion of the grating structure and the process layer to generate a reflection profile.

Abstract: In a chemical-mechanical planarization apparatus, a pad spindle includes a pad chuck operative for selective attachment and detachment of a polishing pad. The pad chuck includes a plurality of pivoting links which cooperate to provide a clamping action to retain the polishing pad. A detachment station engages the pivoting links to detach the polishing pad.

Abstract: There is provided a planarizing and polishing apparatus and a planarizing and polishing method for measuring the polishing conditions of a polishing work during the polishing process in view of obtaining a fault-free polishing work. The planarizing and polishing apparatus is provided with a detecting unit for detecting a change in surface reflectivity of the polishing work and a control unit for recognizing the additional polishing part of the polishing work based on the detected value from the detecting unit and then automatically generating, for the feedback purpose, the polishing conditions of the additional polishing part and the portion other than the additional polishing part.

Abstract: After a hole is formed in a polishing pad, a transparent window plate is inserted into the hole. Here, a gap is left between the upper surface of the transparent window plate and the outermost surface constituting the working surface of the polishing pad. During polishing, the polishing head holding the wafer applies a load to the polishing pad by means of a load-applying mechanism, so that the polishing pad and transparent window plate are compressed. In this case, the system is arranged so that the gap remains constant, and so that a dimension equal to or greater than a standard value is maintained. Since the upper surface of the transparent window plate is recessed from the upper surface of the polishing pad, there is no scratching of the surface of the transparent window plate during dressing. Accordingly, the polishing pad has a long useful life.

Abstract: A polishing apparatus is used for polishing a surface of a workpiece such as a semiconductor wafer or a glass substrate. The polishing apparatus comprises a polishing table having a polishing surface thereon, a plurality of workpiece holders each for holding a workpiece and pressing the workpiece against the polishing surface, and a controller for controlling the workpiece holders individually so that polishing operations of the workpiece holders are controlled independently of each other.

Abstract: An arrangement for polishing a workpiece in a chemical mechanical polishing apparatus has a rotatable platen with an aperture that allows a laser interferometric measuring device to measure the surface condition of a workpiece being polished. A transparent block is flexibly attached to the top surface of the platen over the platen aperture to rotate with the platen. A polishing pad is disposed on the top surface of the platen, and has a hole extending through the pad configured to fit over the transparent block attached to the platen.

Abstract: The present invention provides device and system for suppressing edge instability during a chemical mechanical planarization (CMP) process for planarizing a surface topography on a wafer. A wafer carrier holds and rotates a wafer on a polishing surface of a polishing pad that is arranged to move in a first direction. The edge instability suppressing device includes a front process unit and a second process unit. The front process unit is disposed around a first portion of the wafer facing the first direction and arranged to align to the polishing surface of the polishing pad independent of the wafer during CMP processing. The back process unit is disposed around a second portion of the wafer opposite the first portion and is arranged to align to the polishing surface of the polishing pad independent of the wafer during the CMP processing. In so doing, the aligned front and back process units substantially reduce edge effects on the wafer during the CMP processing.

Abstract: A method and apparatus for planarizing a microelectronic substrate. The apparatus can include a planarizing medium having a relatively hard polishing pad and a planarizing liquid disposed on a generally non-porous planarizing surface of the polishing pad. The planarizing liquid can include a colloidal suspension of colloidal particles having generally smooth external surfaces. The colloidal particles can have a variety of shapes, including a spherical shape, a cylindrical shape, a cubic shape, and a hexagonal shape, among others. The colloidal particles can be formed from a variety of materials, including silicon dioxide, manganese oxide, and cerium oxide.

Abstract: The present invention delineates a carrier for an apparatus (10) which polishes a surface of a semiconductor wafer (56, 124). In a preferred embodiment, the carrier includes a rigid plate (34) connected to one or more diaphragms (40, 42) of soft, flexible material that provide pressurizable cavities (50, 52) having respective surfaces for contacting the back surface of the wafer. A plurality of conduits (28a, 28c) are used to selectively pressurize the diaphragm cavities. The carrier head may also include an inter-diaphragm cavity (54) formed between a portion of one diaphragm, a portion of another diaphragm, and the semiconductor wafer. The inter-diaphragm cavity is provided with its own conduit (28b) by which a source of pressurized fluid and a source of vacuum are selectively connected to the inter-diaphragm cavity. During operation, pressure and/or vacuum may be applied through one or more cavities to chuck (90) a wafer, and to pressurize (96) the cavities during polishing.

Abstract: The invention is a chemical-mechanical polishing wafer carrier that is able to apply a plurality of different pressures, with minimal discontinuities at the interfaces between different pressures, through a diaphragm to a back surface of a wafer. A plurality of concentric balloons, that may be individually pressurized, is used to support and press on the back surface of the diaphragm. The walls of the balloons are preferably thin and elastic and preferably do not attach to the diaphragm. This helps to minimize any pressure discontinuities on the diaphragm along the interfaces between the balloons. A wafer may be placed against the front surface of the diaphragm allowing the front surface of the diaphragm to retain and press against the back surface of the wafer during a planarization process.

Abstract: A polishing apparatus applicable to a dry-in/dry-out polishing system is capable of increasing the capacity of processing substrates to be polished, e.g. semiconductor wafers, per unit of time and per unit area of installation. A polishing unit includes a turntable having a polishing surface, and at least two wafer carriers each adapted to hold a wafer and to press the wafer against the polishing surface. Each wafer carrier is supported by a pivot shaft and movable between a polishing position where the wafer is polished on the turntable and a transfer position for transferring the wafer. The at least two wafer carriers can be situated at the polishing position simultaneously and also alternately.

Abstract: A polishing method uses a dry particulate solids composition that is reconstituted into an aqueous composition for delivery to a polishing apparatus. In one aspect of the invention, the dry particulate solids composition is provided in a package size that is just substantially sufficient to make a reconstituted slurry for completing the polishing of a predetermined number of work pieces. The quantity of dry particulate solids delivered to a reconstitution apparatus can be an amount appropriate for polishing one work piece or a small number of work pieces, or the system can operate in larger batches or a continuous flow mode. The reconstituted aqueous polishing solution can be monitored for physical or chemical properties, filtered, blended with other chemical mixtures, or modified in other ways prior to being used in the polishing apparatus.

Abstract: In a polishing apparatus, a wafer is sustained by a wafer carrier and the wafer is ground with a polishing pad which is rotationally driven. A slurry is fed on to the polishing pad through a slurry feed nozzle. A filter is disposed upstream from the wafer in the direction of rotation of the polishing pad and traps the swarf generated on the polishing pad. A defense barrier is provided around the polishing pad to prevent the slurry on the polishing pad from spilling out therefrom during polishing and thus retains the slurry thereon, thereby recycling the slurry on the polishing pad.

Abstract: A polishing apparatus includes an arrangement of a plurality of units to deal with various operations and a robot having at least one arm. The plurality of units are disposed around the robot and include a loading unit for receiving thereon a, e.g. dry, workpiece to be polished, a polishing system including at least one polishing unit for polishing the workpiece, a washing system and a drying system at least including one washing unit for washing and drying the polished workpiece, and an unloading unit for receiving thereon a resultant clean and dry polished workpiece.

Abstract: A polishing pad platen that is equipped with a guard ring, or a slurry retaining collar, used in chemical mechanical polishing (CMP) for conserving usage of polishing slurry is described. A method for conserving slurry solution during a CMP process is further described. In the novel polishing pad platen, a guard ring is mounted to the platen by sealiningly engaging an outer periphery of the platen for preventing spilling out of slurry solution during a polishing operation. The guard ring is mounted to slidingly engage the platen in such a way that the ring may be lowered to be completely out of the way during a pad condition process in which the spinning out of a pad conditioning solution from a top surface of the polishing pad is necessary.

Abstract: A wafer processing apparatus and method of processing a wafer utilizing a processing slurry are provided. The wafer processing disk comprises a processing disk body and a plurality of processing teeth secured to the processing disk body. The plurality of processing teeth project from the disk body to define respective processing surfaces. The plurality of processing teeth include at least one pair of spaced adjacent teeth defining a processing channel there between. The processing channel is shaped such that the cross sectional area of the processing channel decreases as a function of its distance from the processing disk body.

Abstract: A CMP system and methods make repeatable measurements of eccentric forces applied to carriers for wafer or polishing pad conditioning pucks. Force applied to the carrier may be accurately measured even though such force is eccentrically applied to such carrier. The CMP system and method provide the repeatable measurement features while supplying fluids within the carrier to the wafer and to a wafer support without interfering with the polishing operations. Similarly, the CMP system and methods remove fluids from the wafer or puck carrier without interfering with the CMP operations. An initial coaxial relationship between an axis of rotation and a carrier axis is maintained during application of the eccentric force, such that a sensor is enabled to make repeatable measurements, as defined above, of the eccentric forces, and the carrier may be a wafer or a puck carrier.

Abstract: An aqueous based ceria slurry system and method for chemical mechanical polishing of semiconductor wafers, the slurry comprising less than 5 wt % abrasive cerium oxide particles and up to about the critical micelle concentration of a cationic surfactant, absent other abrasives, in a neutral to alkaline pH solution is disclosed. Also disclosed is slurry comprising a blend of surfactants including a pre-existing amount of anionic surfactant and an added amount of cationic and/or non-ionic surfactant.

Abstract: In the wafer abrasive machine of the present invention, a gravity center and a rotational axis of a wafer can be corresponded while abrading the wafer and a holding plate can be smoothly moved in a head member. The abrasive machine comprises: the head member including a concave section, in which the holding plate is accommodated; an elastic sheet member suspending the holding plate and being reinforced by a cloth-formed reinforcing member; a space for storing pressure fluid which pushes the holding plate toward the abrasive plate, the space being formed between the elastic sheet member and the concave section; and a plurality of spherical bodies being provided between an outer circumferential face of the holding plate and an inner circumferential face of the concave section, the spherical bodies simultaneously point-contact the both circumferential faces.

Abstract: A method for fabricating a semiconductor device includes grindstone surface activation treatment by means of a brush or ultrasonic wave carried out when a concave/convex pattern of a semiconductor wafer is planarized by polishing a semiconductor wafer held by a wafer holder by using a grindstone constituted of abrasive grains and material for holding the abrasive grains onto which the semiconductor wafer is pressed with relative motion. The semiconductor wafer is processed with high removal rate and the polishing thickness is controlled accurately.

Abstract: A workpiece carrier has a top ring body for holding a workpiece, a drive shaft for rotating the top ring body and moving the top ring body toward a turntable to press the workpiece against a polishing surface, and a universal joint for transmitting a pressing force from the drive shaft to the top ring body while allowing the drive shaft and the top ring body to be tilted relative to each other. The universal joint includes two members having curved surfaces formed along arcs having a predetermined radius of curvature from a center positioned on a surface of the workpiece which is held in contact with the polishing surface on the turntable, and four rollers held in rolling contact with the curved surfaces. Two of the rollers held in rolling contact with each respective two of the curved surfaces to allow the top ring body to be tilted relative to the drive shaft about a point positioned on the surface which is held in contact with the polishing surface on the turntable.

Abstract: A method and an apparatus for releasably attaching a polishing pad to a support surface under the polishing pad. In one embodiment of the invention, a polishing pad has a first surface for planarizing a substrate assembly, a second surface contacting the support surface, and an interlocking element. The support surface has a retaining member configured to engage the interlocking element on the polishing pad. The interlocking element and retaining member can be any one of several configurations, including: tongue and groove, protuberance and depression, reciprocal elongated ridges, or teeth.

Abstract: A polishing pad is provided, having its face shaped to produce controlled nonuniform removal of material from a workpiece. Non-uniformity is produced as a function of distance from the pad's rotational axis (the working radius). The pad face is configured with both raised, contact regions and voided, non-contact regions such that arcuate abrasive contact varies nonuniformly as a function of distance from the pad's rotational axis. Void density at any distance may be produced by several techniques such as varying void size as a function of working radius or varying the number of voids per unit area as a function of working radius. Either technique produces variation in voided area per total unit area for rings of pad surface concentric with the rotational axis having infintesimally small width.