`
`(cid:44)(cid:49)(cid:55)(cid:40)(cid:47) EXHIBIT 10(cid:22)(cid:22)
`
`
`
`US. Patent
`
`Nov. 3, 1998
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`Sheet 1 of 15
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`5,830,327
`
` ROTATION
`
`MOTOR
`
`I____J___._........_..........J
`
`Page 2 of 26
`Page 2 of 26
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`US. 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
`
`DErERMINE TRACK SHAPE
`THAT PROVIDES OPTIMAL
`EROSION PROFILE
`
`
`
`54
`
`EXPRESS DESIRED TRACK SHAPE
`AS PIECEWISE LINEAR SEGMENTS:
`CALCULATED EROSION PROFILE
`
`58
`
`ALTER
`TRACK
`SHAPE
`
`NO
`
`50
`
`56
`
`
`
`
`ACCEPTABLE FIT TO
`OPTIMAL EROSION PROFILE?
`
`
`
`
`YES
`
`LAYOUT MAGNET ARRAY
`
`CALCULATE PLASMA TRACK
`
`
`
`
`ACCEPTABLE FIT TO
`DESIRED PLASMA TRACK?
`
`
`
`YES
`
`DONE
`
`FIG. 2
`
`ALTER
`MAGNET
`ARRAY
`
`65
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`Page 3 of 26
`Page 3 of 26
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`US. 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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`AXISOF ROTATION
`
`FIG. 3
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`Page 4 of 26
`Page 4 of 26
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`US. Patent
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`Nov. 3, 1998
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`Sheet 4 of 15
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`5,830,327
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`FIG. 4
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`Page 5 of 26
`Page 5 of 26
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`US. Patent
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`Nov. 3, 1998
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`Sheet 5 of 15
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`5,830,327
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`FIG.5
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`Page 6 of 26
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`US. Patent
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`5,830,327
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`Page 7 of 26
`Page 7 of 26
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`US. Patent
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`Nov. 3, 1998
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`Sheet 7 of 15
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`5,830,327
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`Page 8 of 26
`Page 8 of 26
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`US. Patent
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`NM. 3, 1998
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`Sheet 8 of 15
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`5,830,327
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`US. Patent
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`Nov. 3, 1998
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`Sheet 9 of 15
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`5,830,327
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`US. Patent
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`Nov. 3, 1998
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`Sheet 10 0f 15
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`5,830,327
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`RADIUS(mm)
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`Page 11 of 26
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`US. Patent
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`Nov. 3, 1993
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`Sheet 11 of 15
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`5,830,327
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`OFROTATION
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`FIG.I|
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`Page 12 of 26
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`US. Patent
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`Nov. 3, 1998
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`Sheet 12 0f 15
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`5,830,327
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`RADIUS(mm)
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`Page 13 of 26
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`US. Patent
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`Nov. 3, 1998
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`Sheet 13 0f 15
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`5,830,327
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`Nov. 3, 1998
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`US. Patent
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`Nov. 3, 1998
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`Sheet 15 0f 15
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`5,830,327
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`SOURCE 2
`
`SPU'ITERING
`SOURCE 1
`
`SPUTTERING
`
`FIG.
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`I 5
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`Page 16 of 26
`Page 16 of 26
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`5 330,3 2?
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`1
`METHODS AND APPARATUS FOR
`SPU'ITERING WITH ROTATING MAGNET
`SPUT'I‘ER SOURCES
`
`FIELD OF THE INVEN'I'ION
`
`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 iLs 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
`whett
`the erosion at any point approaches a substantial
`fraction of the target’s 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 nonproduetive
`during this period, with a consequent undesirable and costly
`decrease in average throughput.
`Three basic approaches may be used to increase the
`number ol~ substrates a target can coat belhre the target must
`be replaced. The thickness of the target can be increased to
`increase the volume ol~ 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—suhstrate
`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
`
`10
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`typical process conditions has not been acceptable. Decreas-
`ing the targelvto—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 he a poor conductor and may promote a
`low voltage are breakdown, with consequent undesirable
`generation 01. particles that can contaminate the substrate
`being coated. 1n the prior art, sputtering of the center region
`has been achieved by complex mechanical motion of the
`magnet structure relative to the target.
`Art 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.
`US. Pat. No. 4,995,958 issued Feb. 26, 1991 to Anderson
`et a]. discloses a sputtering apparatus with a rotating magnet
`array having a geometry which produces a selected erosion
`profile on the target. The selected erosion prolile 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.
`US. 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.
`US. Pat. No. 5,314,597 issued May 24, 1994 to Ilarra
`discloses a sputtering apparattts 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 '59? 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.
`US. 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 01. flexible
`plasticieed ferrite and several auxiliary magnets which pro—
`vide a desired plasma track. The magnet structure is rotated
`in a cavity tilled with water. The surface of the target is
`machined to a cylindrically symmetric shape which is
`thicker near the outer rim.
`US. Pat. No. 5,188,717 issued Feb. 23, 1993 to Broad-
`benl el al. discloses a nragnetron sputtering apparatus
`
`Page 17 of 26
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`5 330,3 27
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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 slated 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.; US. 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
`
`31]
`
`35
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`40
`
`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
`
`I?
`rt
`f K(r.r"}e{r’jr‘a'r' a tfrj
`
`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 Spuller 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 t(r)=constanl to specify uniform
`thickness, the invention also includes cases where t(r) varies
`in a specilied 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
`
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`Page 18 of 26
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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
`
`b
`a
`I K(r'.r’Je(r"Jr"ri'r" = t(r]
`
`where e(r‘) is the erosion profile, ttr) 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 t(r)=
`constant to specify uniform thickness,
`the invention also
`includes cases where t( r) 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 prolile 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 25% 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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`5 330,3 2?
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`5
`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 lirst 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
`
`IO
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`15
`
`31]
`
`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 hall‘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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`Page 19 of 26
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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 ofone 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
`lltird 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. (iood depo—
`sitional thickness uniformity is typically less than 25% and
`preferably less than 23%.
`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 invento
`can be more than 15 M A and preferably
`more than 18 M .
`
`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. Aplasma track
`shape is then generated that
`is predicted to produce the
`optimised target erosion profile to sufiicient accuracy. A
`magnet structure is then designed to produce the desired
`track shape. The plasma track design utilizes magnetostatie
`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
`
`
`
`5 330,3 2?
`
`7
`it
`an optimal erosion prolile subject to the condition that
`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" ot'Com—
`prtrnriortnl Mcrhodr for Integral Equations, Cambridge Uni-
`versily Press (1985). An example of this type of equation is
`as follows:
`
`IO
`
`15
`
`8
`mity and inventory have both reached acceptable values.
`Calculation of inventory is discussed below.
`During iterationai optimization, it may happen that the
`expansion for the erosion shape has a d ilferent algebraic sign
`for some radius or range of radii, which is not allowable
`physically. This is a global efiect, depending on the behavior
`of the whole series expansion, and cannot be Prevented by
`conditions on the coelficients 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 olfending regions is set to zero and the new
`modified profile. nowofone sign, is fit to a (Thebyshev series
`using the inverse of equation (2) according to methods
`familiar to those skilled in the art. This procedure generates
`a new modified set ot'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 (It). 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.
`is predicted to
`A plasma track is then designed that
`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 are
`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.
`the tentative
`in track development,
`As an analysis tool
`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 1th 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 (PWHM). The best value of lin-IM 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 FWlIM
`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.
`
`(I J
`
`31]
`
`2
`
`if:
`a
`I K(r.t’k{r’lr'rl‘r' .. tfri
`This equation has the same form as the equation for express-
`ing the depositional radial thickness distribution [(r) 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 t(r) and the limits {a -
`and b) are specified. The function K(r,r‘) can be modeled to
`sufiicient accuracy, as diseussed 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
`J
`(
`(11’) -= Ell uniirfa)
`The variables r' and s are linearly related so as to bring the
`argument of TH into the required range of definition. The
`expansion has N+l
`terms, coefficients a" are to be
`determined, and functions T" are the first N+l members of
`a complete set of functions (typically orthogonal
`polynomials]. The functions TN were selected to be Cheby-
`shcv polynomials, because these functions are robust curve
`filters. The argument of Chehyshev polynomials is defined
`to he —1§s§1, so that the linear relation between s and r' is
`
`35
`
`40
`
`45
`
`21'" — b + n
`
`(3)
`
`The value of N must be sufficiently large to give good
`accuracy in representing the solution. A value of N=20 is
`usually sufiicient. Software for solving equation (1) incor-
`porates a bounding method for the coefficients a” described
`by Delves and Mohamed as an aid to stability. Delves and
`Mohamed also point out
`that for stability the algebraic
`equations should be solved not directly, but by minimization
`(equivalently, optimization}. The work ot‘ the optimisaer is to
`choose the ooelficients a“ so as to make left and right hand
`sides of the integral equation as equal as possible. The
`software preferably uses a version of the downhill simplex
`optimizing method (AMOEBA) discussed by Press, et al. in
`Nurttericrti Recipes, Cambridge University Press, 1986,
`1992. The AMOEBAoptimizer proceeds by iteration, gradu-
`ally making the left hand side of the integral equation as
`close to the specified right hand side as possible.
`Because inventory tends to decrease as thickness unifor-
`mity improves, the software stops at every 50 iterations and
`calculates the uniformity and inventory based on the current
`erosion shape. The program is terminated when the unifor—
`
`55
`
`till
`
`65
`
`Page 20 of 26
`Page 20 of 26
`
`
`
`5 330,3 2?
`
`9
`The final task is to design a magnet array that gives the
`desired plasma track shape at the target surface. The magnet
`array preferably comprises a series of magnet bars posi-
`tioned along the plasma track so as to produce the desired
`plasma track shape. As a zero order approximation,
`the
`plasma track at the target surface is assumed to lie imme-
`diately above the center line of the magnet bars. Three-
`dimensional magnetic modeling has shown that the plasma
`track tends to expand outward,
`increasing with the gap
`between the top of the magnet array and the target surface.
`Thus, the first approximation is to move the magnet bars
`inwardly so that their centerlines lie within the cenlcrline of
`the desired plasma track. In typical applications, this dis—
`lance may be 2 to 8 millimeters. The magnet bars are then
`placed on the trial magnet center line produced by contrac-
`tion in such manner, typically so that each magnet bar is
`locally perpendicular or nearly perpendicular to the trial
`magnet structure center line, and typically so that one
`magnet bar nearly touches a corner of its neighbor (step 60).
`The trial magnet array geometry is then entered into the 3-D
`modeling program, and the field at the target surface is
`calculated (step 62}. The predicted plasma track on the