United States Patent (19)
`KolenkoW
`
`USOO583O327A
`Patent Number:
`11
`(45) Date of Patent:
`
`5,830,327
`Nov. 3, 1998
`
`54 METHODS AND APPARATUS FOR
`SPUTTERING WITH ROTATING MAGNET
`SPUTTER SOURCES
`
`75 Inventor: Robert J. Kolenkow, Berkeley, Calif.
`73 Assignee: Intevac, Inc., Santa Clara, Calif.
`
`21 Appl. No.: 724,792
`22 Filed:
`Oct. 2, 1996
`(51) Int. Cl." ..................................................... C23C 14/34
`52 U.S. Cl. ................................ 204/192.12; 204/298.19;
`204/298.2; 204/298.26
`58 Field of Search ......................... 204/192.12, 298.19,
`204/298.2, 298.26, 1922, 298.25
`
`56)
`
`References Cited
`
`U.S. PATENT DOCUMENTS
`
`4,558.388 12/1985 Graves, Jr. ..................... 204/298.25 X
`4,894,133
`1/1990 Hedgcoth. .............
`... 204/398.26 X
`4,995,958 2/1991 Anderson et al.
`... 024/298.2
`5,047,130 9/1991 Akao et al. ............................. 204/192
`5,188,717 2/1993 Broadbent et al. .............. 204/298.2 X
`
`5,248,402 9/1993 Ballentine et al. .................. 204/298.2
`5,252,194 10/1993 Demaray et al. .................... 204/298.2
`5,314,597 5/1994 Harra .............
`... 204/298.2 X
`5,320,728 6/1994 Tepman .............................. 204/298.12
`Primary Examiner Nam Nguyen
`Attorney, Agent, or Firm-Stanley Z. Cole; William R.
`McClellan
`ABSTRACT
`57
`A magnetron Sputtering Source for forming a Sputtered film
`on a Substrate in a magnetron Sputtering apparatus includes
`a target having a Surface from which material is Sputtered
`and a magnet assembly that is rotatable about an axis of
`rotation with respect to the target. The magnet assembly
`produces on the target an erosion profile that is calculated to
`yield a desired depositional thickness distribution and inven
`tory. A method for configuring the rotatable magnet assem
`bly includes the Steps of determining an optimal erosion
`profile that yields the desired depositional thickness distri
`bution and inventory, determining a plasma track on the
`Surface of the target that produces an acceptable approxi
`mation to the optimal erosion profile, and determining a
`magnet Structure that produces the plasma track.
`17 Claims, 15 Drawing Sheets
`
`
`
`CENTER OF
`ROTATION
`
`72O
`
`Page 1 of 26
`
`APPLIED MATERIALS EXHIBIT 1033
`
`

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`U.S. Patent
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`Nov. 3, 1998
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`Sheet 1 of 15
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`5,830,327
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`
`
`ROTATION
`MOTOR
`
`Page 2 of 26
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`

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`U.S. Patent
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`Nov. 3, 1998
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`Sheet 2 of 15
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`5,830,327
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`50 DETERMINE OPTIMAL EROSION
`PROFILE TO PROVIDE DESIRED
`THICKNESS UNIFORMITY
`AND INVENTORY
`
`
`
`52
`
`DETERMINE TRACK SHAPE
`THAT PROVIDES OPTIMA
`EROSION PROFILE
`
`EXPRESS DESIRED TRACK SHAPE
`AS PIECEWISE LINEAR SEGMENTS;
`CALCULATED EROSION PROFILE
`
`ACCEPTABLE FIT TO
`OPTIMAL EROSION PROFILE?
`
`YES
`
`60
`
`LAYOUT MAGNET ARRAY
`
`CALCULATE PLASMA TRACK
`
`64
`
`ACCEPTABLE FTT TO
`DESIRED PLASMA TRACK?
`
`
`
`
`
`
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`58
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`ALTER
`TRACK
`SHAPE
`
`
`
`
`
`
`
`
`
`
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`66
`
`ALTER
`MAGNET
`ARRAY
`
`YES
`
`DONE
`
`FIG, 2
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`Page 3 of 26
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`U.S. Patent
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`Nov. 3, 1998
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`Sheet 3 of 15
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`5,830,327
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`SUBSTRATE
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`FIG, 3
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`Page 4 of 26
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`U.S. Patent
`US. Patent
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`Nov. 3, 1998
`Nov. 3, 1998
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`Sheet 4 of 15
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`FIG, 4
`FIG. 4
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`Page 5 of 26
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`Nov. 3, 1998
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`Nov. 3, 1998
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`U.S. Patent
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`Nov. 3, 1998
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`Sheet 14 of 15
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`U.S. Patent
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`Nov. 3, 1998
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`Sheet 15 of 15
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`5,830,327
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`
`
`SPUTTERING
`SOURCE 1
`
`SPUTTERING
`SOURCE 2
`
`FIG, .5
`
`Page 16 of 26
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`

`

`1
`METHODS AND APPARATUS FOR
`SPUTTERING WITH ROTATING MAGNET
`SPUTTER SOURCES
`
`FIELD OF THE INVENTION
`This invention relates to deposition of sputtered films on
`Substrates and, more particularly, to rotating magnet Sput
`tering methods and apparatus which provide long target life,
`broad erosion patterns and depositional thickness unifor
`mity.
`
`BACKGROUND OF THE INVENTION
`Sputter deposition, also known as Sputter coating, is a
`technique for depositing thin films of desired materials on a
`Substrate Such as, for example, a magnetic disk for a hard
`disk drive or a Semiconductor wafer. In general, inert gas
`ions from a gas plasma are accelerated toward a target of the
`material to be deposited. Free atoms of the target material
`are expelled when the ions collide with the target. A portion
`of the free atoms form a thin film on the Surface of the
`Substrate.
`One well known Sputtering technique is magnetron Sput
`tering. Magnetron Sputtering uses a magnetic field to con
`centrate the Sputtering action. Magnets are positioned
`behind the target, and magnetic field lines penetrate the
`target and form arcs over its Surface. The magnetic field
`helps to confine free electrons in an area near the Surface of
`the target. The resulting increased concentration of free
`electrons produces a high density of inert gas ions and
`enhances the efficiency of the Sputtering process.
`Both fixed and movable magnet structures have been
`utilized in magnetron Sputtering. In a typical structure uti
`lizing a moving magnet, the target is circular and the magnet
`Structure rotates with respect to the center of the target. In
`either configuration, the Sputtering process produces an
`erosion pattern on the target that is nonuniform. To avoid
`contaminating the Substrate, Sputtering must be stopped
`before the erosion pattern has consumed the full thickness of
`the target material at any point. The target must be replaced
`when the erosion at any point approaches a Substantial
`fraction of the targets initial thickness. Thus in a given
`production process, only a certain number of Substrates can
`be coated from one target. The Sputtering apparatus must be
`shut down during a target change and is nonproductive
`during this period, with a consequent undesirable and costly
`decrease in average throughput.
`Three basic approaches may be used to increase the
`number of Substrates a target can coat before the target must
`be replaced. The thickness of the target can be increased to
`increase the volume of material to be removed from the
`target before it is spent. Second, the shape of the erosion
`profile can be altered by design to make greater use of the
`target Volume. Finally, the target-to-Substrate distance can
`be decreased So as to capture a larger percentage of the
`material Sputtered from the target. However, performance
`may be degraded as the thickness of the target is increased.
`In particular, the field Strength at the target Surface may be
`decreased, decreasing the efficiency of Sputtering. Also,
`deposition uniformity may show greater variation over the
`target life because of the variation in target-to-SubStrate
`distance.
`The design and physical realization of a Suitable erosion
`profile has remained a problem in magnetron Source design.
`Sources have been designed having uniform erosion over
`much of the target, which maximizes use of the target
`Volume, but the corresponding deposition uniformity under
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`typical proceSS conditions has not been acceptable. Decreas
`ing the target-to-Substrate distance can degrade deposition
`uniformity, unless the erosion profile is redesigned to com
`pensate.
`Another problem in current Sputter coating Systems is that
`certain areas of the target, especially the center region,
`experience no Sputtering. Redeposition from Sputtered
`atoms turned back to the target by gas Scattering can
`accumulate in the nonsputtered regions. The accumulated
`redeposition may be a poor conductor and may promote a
`low Voltage arc breakdown, with consequent undesirable
`generation of particles that can contaminate the Substrate
`being coated. In the prior art, Sputtering of the center region
`has been achieved by complex mechanical motion of the
`magnet Structure relative to the target.
`An apparently related phenomenon is the growth of
`poorly conducting nodules on the target Surface when Sput
`tering from a carbon target, as is used in magnetic disk
`coating. In production runs, it may be necessary to halt the
`machine from time to time to clean the carbon targets.
`U.S. Pat. No. 4,995,958 issued Feb. 26, 1991 to Anderson
`et al. discloses a Sputtering apparatus with a rotating magnet
`array having a geometry which produces a Selected erosion
`profile on the target. The Selected erosion profile is typically
`nearly constant with radius over a Selected region. The
`centerline of the magnet Structure is described by an equa
`tion. The disclosed track equation is not fully general and
`does not describe all possible plasma tracks. In particular,
`the track equation cannot be used to describe an erosion
`profile that extends to the center of the target.
`U.S. Pat. No. 5,252,194 issued Oct. 12, 1993 to Demaray
`et al. discloses a magnetron Sputter Source which includes a
`rotating magnet assembly that is Stated to produce uniform
`target erosion acroSS the full target Surface, including the
`center. The target Surface may be planar or dished.
`U.S. Pat. No. 5,314,597 issued May 24, 1994 to Harra
`discloses a Sputtering apparatus including a rotatable magnet
`configuration for obtaining a desired Sputter target erosion
`profile and a desired film characteristic. In developing the
`magnet configuration, a heart-shaped plasma track having an
`erosion profile near the desired profile is utilized. A Static
`erosion test, with the magnet Structure not rotating, is run to
`develop a measurable Static erosion groove in the target. The
`depth of erosion is measured at various points on the target
`in Such a way as to quantify the erosion along radial spokes
`at fixed values of polar angle. The magnet configuration is
`then adjusted to provide an erosion profile that is closer to
`the desired profile. The process is repeated until the desired
`profile is achieved. The 597 patent discloses a relationship
`for finding thickneSS uniformity given an erosion profile, but
`does not disclose how to find an erosion profile given a
`desired thickness variation.
`U.S. Pat. No. 5,120,417 issued Jun. 9, 1992 to Takahashi
`et al. discloses a magnetron Sputtering apparatus including a
`rotating magnet Structure which is Stated to erode the central
`region of the target and thereby reduce the number of
`particulates deposited on the Substrate.
`U.S. Pat. No. 5,130,005 issued Jul. 14, 1992 to Hurwitt et
`al. discloses a magnetron Sputter coating apparatus including
`a rotating magnet Structure comprising a Stack of flexible
`plasticized ferrite and Several auxiliary magnets which pro
`vide a desired plasma track. The magnet Structure is rotated
`in a cavity filled with water. The surface of the target is
`machined to a cylindrically Symmetric shape which is
`thicker near the outer rim.
`U.S. Pat. No. 5,188,717 issued Feb. 23, 1993 to Broad
`bent et al. discloses a magnetron Sputtering apparatus
`
`Page 17 of 26
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`

`

`3
`including a rotating magnet assembly which produces a
`plasma track comprising a closed curve that is Stated to have
`the shape of a kidney bean. The closed curve is generated in
`part by a spiral curve generated by the same equation
`disclosed by Anderson et al. in U.S. Pat. No. 4,995,958. The
`magnet assembly is simultaneously rotated about a center of
`rotation and is caused to oscillate radially with respect to the
`center of rotation So as to produce erosion over the entire
`target Surface.
`U.S. Pat. No. 5,248,402 issued Sep. 28, 1993 to Ballentine
`et al. discloses a magnetron Sputtering System including a
`rotating magnet assembly that is characterized as apple
`shaped. The disclosed magnet assembly is Stated to produce
`uniform coatings and erosion over the entire Surface of the
`target.
`U.S. Pat. No. 5,417,833 issued May 23, 1995 to Harra et
`al. discloses a magnetron Sputtering apparatus including a
`rotating magnet assembly and a pair of Separately driven
`Stationary electromagnets. The electromagnets are used to
`increase target utilization at its center and to compensate for
`the change in shape of the target and distance from the target
`to the substrate with depletion.
`Magnetron Sputtering Systems which utilize a rotating
`magnet assembly are also disclosed in U.S. Pat. No. 4,444,
`643 issued Apr. 24, 1984 to Garrett; U.S. Pat. No. 4,714,536
`issued Dec. 22, 1987 to Freeman et al., U.S. Pat. No.
`4,746,417 issued May 24, 1988 to Ferenbach et al.; U.S. Pat.
`No. 5,047,130 issued Sep. 10, 1991 to Akao et al.; U.S. Pat.
`No. 5,194,131 issued Mar. 16, 1993 to Anderson; and U.S.
`Pat. No. 5,320,728 issued Jun. 14, 1994 to Tepman.
`All of the known prior art magnetron Sputtering Systems
`utilizing rotating magnet assemblies have had one or more
`disadvantages, including but not limited to Short target life,
`nonuniform depositional thickness, variations in perfor
`mance over the life of the target, contamination of the
`Substrate, complex mechanical drive Structures and a
`requirement for nonplanar target Surfaces.
`SUMMARY OF THE INVENTION
`According to the present invention, a magnetron Sputter
`ing Source is provided for forming a Sputtered film on a
`Substrate in a magnetron Sputtering apparatus. The magne
`tron Sputtering Source comprises a target having a Surface
`from which material is Sputtered and a magnet assembly that
`is rotatable about an axis of rotation with respect to the
`target. The magnet assembly produces on the target an
`erosion profile that approximates a Solution to an equation of
`the form
`
`where e(r) is the erosion profile, t(r) is a desired radial
`thickness distribution of the sputtered film, K(r,r) is a
`function depending on the Sputter geometry and proceSS
`conditions, r is the radial position on the Substrate, r" is the
`radial position on the target, and a and b are the radial limits
`of erosion on the target. Although for many applications it
`may be desirable to choose t0r)=constant to specify uniform
`thickness, the invention also includes cases where t0r) varies
`in a specified non-uniform fashion across the Substrate.
`According to another aspect of the invention, a method is
`provided for configuring a rotatable magnet assembly for
`use in the magnetron Sputtering apparatus. The magnetron
`Sputtering apparatus includes a target having a Surface from
`
`5,830,327
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`4
`which material is Sputtered to form a Sputtered film on a
`Substrate. The method comprises the Steps of determining an
`erosion profile on the target that approximates a Solution to
`an equation of the form
`
`where e(r) is the erosion profile, t(r) is a desired radial
`thickness distribution of the sputtered film, K(r,r) is a
`function depending on the Sputter geometry and process
`conditions, r is the radial position on the Substrate, r" is the
`radial position on the target, and a and b are the radial limits
`of erosion on the target, and determining a magnet Structure
`for the rotatable magnet assembly that produces an accept
`able approximation to the erosion profile e(r). Although for
`many applications it may be desirable to choose tr)=
`constant to Specify uniform thickness, the invention also
`includes cases where tOr) varies in a specified non-uniform
`fashion across the Substrate.
`According to another aspect of the invention, a magnetron
`Sputtering Source is provided for forming a Sputtered film on
`a Substrate in a magnetron Sputtering apparatus. The mag
`netron Sputtering Source comprises a target having a Surface
`from which material is Sputtered and a magnet assembly that
`is rotatable about an axis of rotation with respect to the
`target. The magnet assembly produces on the Surface of the
`target a plasma track having a shape characterized by a pair
`of Symmetrical lobes, a deeply indented first inward cusp
`located near the axis of rotation and a moderately indented
`Second inward cusp. The first and Second cusps are located
`on opposite Sides of the plasma track. Each of the lobes has
`a relatively long Section of Substantially constant radius with
`respect to the axis of rotation.
`According to another aspect of the invention, a magnetron
`Sputtering Source is provided for forming a Sputtered film on
`a Substrate in a magnetron Sputtering apparatus. The mag
`netron Sputtering Source comprises a target having a Surface
`from which material is Sputtered and a magnet assembly that
`is rotatable about an axis of rotation with respect to the
`target. The magnet assembly produces on the target an
`erosion profile characterized by a relatively deep first cir
`cular groove near an outer periphery of the target, a rela
`tively shallow Second circular groove near the center of the
`target and an intermediate region between the first and
`Second grooves. The intermediate region has a shallower
`erosion depth than the first and Second grooves.
`According to another aspect of the invention, a magnetron
`Sputtering Source is provided for forming a Sputtered film on
`a Substrate in a magnetron Sputtering apparatus. The mag
`netron Sputtering Source comprises a target having a Surface
`from which material is Sputtered and a magnet assembly that
`is rotatable about an axis of rotation with respect to the
`target. The magnet assembly and the target produce a radial
`thickness distribution of the sputtered film on the substrate
`that is uniform to better than about +5% for a source-to
`Substrate distance of less than about 35 millimeters.
`According to another aspect of the invention, a method is
`provided for configuring a rotatable magnet assembly for
`use in a magnetron Sputtering apparatus including a target
`having a Surface from which material is Sputtered to form a
`Sputtered film on a Substrate. The method comprises the
`Steps of determining an erosion profile on the target that
`produces a desired radial thickness distribution of the Sput
`tered film on the substrate and a desired inventory of the
`Sputtered material on one or more Substrates, determining a
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`S
`plasma track on the Surface of the target that produces an
`acceptable approximation to the erosion profile and deter
`mining a magnet Structure for the rotatable magnet assembly
`that produces an acceptable approximation to the plasma
`track.
`According to another aspect of the invention, a magnetron
`Sputtering apparatus is provided. The magnetron Sputtering
`apparatus comprises a first magnetron Sputtering Source for
`forming a Sputtered film on a first Surface of a Substrate and
`a Second magnetron Sputtering Source for forming a Sput
`tered film on a second Surface of the Substrate. The first
`magnetron Sputtering Source includes a first target having a
`Surface from which material is Sputtered and a first magnet
`assembly that is rotatable about an axis of rotation with
`respect to the first target. The Second magnetron Sputtering
`Source includes a Second target having a Surface from which
`material is Sputtered and a Second magnet assembly that is
`rotatable about an axis of rotation with respect to the Second
`target. The first and Second magnet assemblies produce on
`the Surfaces of the first and Second targets plasma tracks,
`each having a shape characterized by a pair of Symmetrical
`lobes, a deeply indented first inward cusp located near the
`axis of rotation and a moderately indented Second inward
`cusp. The first and Second cusps are located on opposite
`Sides of the plasma track. Each of the lobes has a relatively
`long Section of Substantially constant radius with respect to
`the axis of rotation. The magnetron Sputtering apparatus
`further includes a vacuum system for producing a vacuum in
`regions between each target Surface and the Substrate.
`BRIEF DESCRIPTION OF THE DRAWINGS
`For a better understanding of the present invention, ref
`erence is made to the accompanying drawings, which are
`incorporated herein by reference and in which:
`FIG. 1 is a simplified Schematic diagram of a rotating
`magnet Sputtering System;
`FIG. 2 is a flow diagram that illustrates the process for
`configuring a rotating magnet Structure in accordance with
`the invention;
`FIG. 3 is a Schematic diagram showing the geometrical
`parameters involved in calculating the erosion profile on the
`target,
`FIG. 4 is a plan view of a first embodiment of a rotating
`magnet assembly in accordance with the invention;
`FIG. 5 is a schematic plan view of one half of a plasma
`track in accordance with the first embodiment of the inven
`tion;
`FIG. 6 is a graph of erosion depth as a function of radial
`position for a copper target, showing optimal, measured and
`predicted erosion profiles,
`FIG. 7 is a graph of deposited thickness as a function of
`radius for a chromium target;
`FIG. 8 is a graph of deposited thickness as a function of
`radius for a magnetic alloy target;
`FIG. 9 is a graph of deposited thickness as a function of
`radius for a carbon target;
`FIG. 10 is a graph of erosion depth as a function of radial
`position, showing a comparison of the optimized ideal
`erosion profile and the predicted erosion profile for a Second
`embodiment in accordance with the invention;
`FIG. 11 is a schematic plan view of one half of the center
`line of a plasma track for the Second embodiment in accor
`dance with the invention;
`FIG. 12 is a graph of erosion depth as a function of radial
`position, showing a comparison of the optimized ideal
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`6
`erosion profile and the predicted erosion profile for a third
`embodiment in accordance with the invention;
`FIG. 13 is a schematic plan view of one half of the center
`line of a plasma track for the third embodiment in accor
`dance with the invention;
`FIG. 14 is a graph of relative thickness as a function of
`radius, showing a comparison of the ideal radial thickness
`variation and the predicted radial thickness variation for the
`third embodiment in accordance with the invention; and
`FIG. 15 is a block diagram of a Sputter coating System
`having two opposed Sputtering Sources.
`DETAILED DESCRIPTION
`A simplified Schematic diagram of a rotating magnet
`sputter coating system is shown in FIG. 1. A substrate 10,
`Such as for example a magnetic disk, is positioned in a
`Vacuum chamber 12. A rotating magnet Sputter Source 20
`includes a Sputtering target 22 of a material to be deposited
`on Substrate 10, a rotating magnet assembly 24, and a
`rotation motor 30 which causes the rotating magnet assem
`bly 24 to rotate about an axis of rotation 32 with respect to
`target 22. A magnet array in rotating magnet assembly 24
`produces magnetic fields which penetrate target 22 and form
`arcs over a surface 26 of target 22 facing substrate 10. The
`target 22 is cooled by a target cooling System 28.
`The magnetic field helps to confine free electrons in an
`area near the Surface 26 of the target. The increased con
`centration of free electrons produces high densities of inert
`gas ions, typically argon, and enhances the efficiency of the
`Sputtering process. In particular, the region of most intense
`ionization forms a closed loop plasma track on the Surface
`26 of target 22. The configuration of the plasma track is
`discussed in detail below. AS the rotating magnet assembly
`24 rotates, the plasma track follows the instantaneous posi
`tion of the rotating magnet assembly and Sputters areas of
`the target. Important characteristics of the Source
`performance, including the Volume of erosion through target
`life and depositional thickneSS uniformity on the Substrate
`10, depend on the detailed shape of the plasma track.
`The present invention provides rotating magnet Sputter
`Sources that have a broad erosion pattern and relatively
`Small Source-to-Substrate distances for extended target life.
`The Sputter Sources of the invention also have good depo
`Sitional thickness uniformity on the Substrate. Good depo
`sitional thickness uniformity is typically less than +5% and
`preferably less than +3%.
`It is useful to define the term “inventory” as used in the
`disk coater business to quantify the total thickness that can
`be deposited on Substrates during target life. The units are
`usually millions of angstroms (M A). For chromium, where
`the film thickness on each Substrate is relatively large,
`desirable inventory can be more than 15 M A and preferably
`more than 18 M A.
`According to one aspect of the invention, a method for
`configuring rotating magnet Sputter Sources that have the
`above desirable characteristics is provided. The basis of the
`method is to define, typically by iterative optimization
`techniques, a target erosion profile that is calculated to yield
`a desired depositional thickness distribution. A plasma track
`shape is then generated that is predicted to produce the
`optimized target erosion profile to Sufficient accuracy. A
`magnet Structure is then designed to produce the desired
`track shape. The plasma track design utilizes magnetostatic
`modeling Software Such as “Amperes' (Integrated Engineer
`ing Software, Winnipeg, Canada).
`The design method is illustrated in the flow diagram of
`FIG. 2. The foundation of the design method is to determine
`
`Page 19 of 26
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`5,830,327
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`7
`an optimal erosion profile Subject to the condition that it
`produce approximately a desired depositional thickness uni
`formity on a Substrate located a known distance from the
`target surface (step 50). A prescribed inner diameter of the
`erosion profile, a prescribed outer diameter, and a fixed
`target-to-Substrate distance also constrain the erosion shape.
`The proper plasma track to achieve the optimal erosion
`profile is inferred from the erosion profile.
`Stable methods of solving Fredholm integral equations of
`the first kind are discussed by Delves and Mohamed in
`Chapter 12, “Integral Equations of the First Kind” of Com
`putational Methods for Integral Equations, Cambridge Uni
`versity Press (1985). An example of this type of equation is
`as follows:
`
`8
`mity and inventory have both reached acceptable values.
`Calculation of inventory is discussed below.
`During iterational optimization, it may happen that the
`expansion for the erosion shape has a different algebraic sign
`for Some radius or range of radii, which is not allowable
`physically. This is a global effect, depending on the behavior
`of the whole Series expansion, and cannot be prevented by
`conditions on the coefficients a, An algorithm incorporated
`into the Software as a Subroutine prevents this from happen
`ing. After a chosen predetermined number of iterations,
`typically 10, the current erosion profile is calculated from
`equation (2) and examined by the Software to determine if a
`change of Sign has occurred anywhere. If So, the erosion
`profile in the offending regions is Set to Zero and the new
`modified profile, now of one sign, is fit to a Chebyshev Series
`using the inverse of equation (2) according to methods
`familiar to those skilled in the art. This procedure generates
`a new modified Set of the coefficients an, which are then used
`to reinitialize the optimizer for the Succeeding iterations. In
`practice, this algorithm keeps the erosion profile of one sign
`to typically within 1% of the maximum erosion.
`The iterative optimization proceSS provides an optimal
`erosion profile Suitable for the given Source-to-Substrate
`distance and Specified inside diameter and outside diameter
`of the erosion region (the inside diameter can be 0). The
`erosion profile is also determined during the iterative opti
`mization process to have a Satisfactory thickneSS uniformity
`and a Satisfactory inventory, insofar as this is possible under
`the Specified conditions.
`A plasma track is then designed that is predicted to
`provide the optimal erosion pattern, to Satisfactory accuracy
`(step 52). The design of the plasma track uses the funda
`mental principle that the rate of erosion at a given radius is
`proportional to the angle Subtended by the plasma track arc
`at that radius. For example, to obtain Substantial erosion at
`a large radius, as much plasma track length as possible is
`placed at or near that radius. AS another example, to extend
`the plasma track from a larger radius to near the center
`without having excessive erosion at Small radii, the plasma
`track is made radial or nearly radial to minimize the Sub
`tended angle.
`AS an analysis tool in track development, the tentative
`track shape is expressed as piecewise linear segments (Step
`54). Typically, the segment length is 1.5 to 2.0 millimeters,
`so that there may be 100 or more segments describing half
`the plasma track. The plasma track is preferably Symmetric,
`and only half of the track is analyzed. A computer program
`calculates the predicted erosion, assuming for the plasma
`track a constant Gaussian width of chosen full width at half
`maximum (FWHM). The best value of FWHM is chosen
`with reference to measurements of the groove shape from
`Static erosion tests using plasma tracks of Similar shape.
`Although the FWHM is observed experimentally to vary
`somewhat along the plasma track, the constant FWHM
`approximation has proved to be acceptable in practice.
`If the predicted erosion pattern is determined in step 56
`not to be Sufficiently close to the optimal erosion profile over
`certain radii, the shape of the tentative plasma track is
`altered over those radii in step 58 by manipulating the
`location and orientation of the appropriate Segments. The
`result may not fit the optimal erosion profile perfectly, So it
`is checked by calculating the predicted thickness uniformity
`and inventory. In practice, a reasonably good fit to the
`optimal erosion profile usually yields good predicted uni
`formity and inventory. Small discrepancies from the optimal
`erosion profile do not appear to be critical. As a Secondary
`condition, the plasma track is manipulated So that it predicts
`Some erosion in the central region of the target.
`
`This equation has the same form as the equation for express
`ing the depositional radial thickness distribution tCr) in terms
`of the target erosion profile e(r) and a known modeled
`function K(r,r). The symmetry is purely azimuthal, with
`only radial dependence. The lower limit a and upper limit b
`are fixed values equal to the extent of erosion. In Solving
`equation (1), the thickness distribution tCr) and the limits (a
`and b) are specified. The function K(rr) can be modeled to
`Sufficient accuracy, as discussed below, and e(r) is to be
`determined.
`The unknown erosion profile e(r) is expressed as a finite
`linear expansion to convert the integral equation into a Set of
`linear algebraic equations
`
`N
`e(r') - an.T.(s)
`
`(2)
`
`The variables r" and S are linearly related So as to bring the
`argument of T into the required range of definition. The
`expansion has N+1 terms, coefficients a, are to be
`determined, and functions T are the first N+1 members of
`a complete set of functions (typically orthogonal
`polynomials). The functions T were selected to be Cheby
`shev polynomials, because these functions are robust curve
`fitters. The argument of Chebyshev polynomials is defined
`to be -1s SS1, So that the linear relation between S and r" is
`
`(3)
`
`S =
`
`(b - a)
`The value of N must be suffici