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Understanding simultaneous double-disk grinding: operation principle and material removal kinematics in silicon wafer planarization
Precision Engineering 29 (2005) 189–196
Understanding simultaneous double-disk grinding: operation principle and material removal kinematics in silicon wafer planarization Georg J. Pietsch∗ , Michael Kerstan Research and Development, Siltronic AG, D-84489 Burghausen, Germany Received 30 December 2002; received in revised form 28 January 2004; accepted 7 July 2004 Available online 25 September 2004
Abstract Simultaneous double-disk grinding (DDG) is a novel and powerful technology for precision-machining mono-crystalline silicon slices (“wafers”). With DDG the extreme degrees of planarity can be achieved, which the fabrication of micro-electronic devices with minimum lateral feature dimensions of 90 nm and below demands. In DDG, both sides of the wafer are ground simultaneously between two opposite grinding wheels on collinear spindle axes. It is a chuck-less process, in which the workpiece is machined in “free-floating” fashion. Machining kinematics, removal mechanism, and resulting wafer shape differ from those known from (chuck-mounted) single-side grinding or doublesided batch lapping, which are conventionally used in mechanical wafer shaping. This article explains the kinematics of DDG and derives the basic, method-inherent features always observed on DDG-ground wafers from simple kinematic considerations without further model assumptions: global wafer shape, center dip (“navel”), edge thickness roll-off, and symmetries. The expected results are compared with experimental data. © 2004 Elsevier Inc. All rights reserved. Keywords: Double-disk grinding; Simultaneous grinding; Flatness; Wafer preparation; Kinematics; Grinding wheel; Nanotopography
1. Introduction Minimum feature dimensions of microelectronic components continue to shrink by about 30–40% each design-rule generation as defined by the industry’s overall roadmap for semiconductor devices and materials [1]. The key processes used to pattern devices make extreme demands on the flatness of the raw wafer substrate, as the properties of the starting surface ultimately limit the achievable process results. Photolithography, for instance, due to its sub-100 nm depth of focus during pattern exposure, and CMP (chemomechanical planarization for interlayer dielectric planarization [2,3]) due to the elasticity of the polishing pad. A very critical flatness parameter is the wafer “nanotopology” (NT) [4], which is the front-side-referenced flatness within any e.g. 2 mm × 2 mm or 10 mm × 10 mm measurement field throughout the entire wafer surface. For current 65 nm (volume) and 45 nm (sampling) design-rules NT within 10 mm × ∗
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10 mm and 2 mm × 2 mm must never exceed 28 and 20 nm, respectively. To achieve this flatness, a sequence of mechanical and wet-chemical shaping processes are applied after a single-crystalline silicon ingot has been grown. Such a process chain conventionally consists of: 1. ingot slicing [mostly multi-wire slicing (MWS) or innerdiameter slicing (ID); ID is practical for small wafers only]; 2. mechanical planarization [double-sided batch lapping; or sequential single-side grinding (SSG)]; 3. wet-chemical etching (caustic, alkaline, or a combination of both); 4. stock-removal polishing [chemo-mechanical/shape polishing; batch or single-wafer single-side polishing (SSP); or simultaneous batch double-side polishing (DSP)]; 5. final polishing (batch or single-wafer; single-side). Since the same tools as for CMP are used, final wafer substrate polishing is often termed “CMP” also. Among all the theoretically possible combinations of these individual steps into complete process sequences, only com-
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binations which contain at least one “free-floating” process (FFP) are of practical importance [5–9]. In an FFP, such as lapping or DSP, the wafer is processed without forcemounting it onto a chuck or clamping device during machining. Chuck processes alone cannot remove wafer waviness (parallel corrugations of front and back at basically constant thickness) because of the elasticity of the thin wafer. When the wafer is forced flat on the mounting chuck by vacuum, corrugations are elastically copied back to its front and are thus preserved during machining. Lapping [10–12] is an old, versatile, and powerful machining technique. However, it produces a poor surface with deep crystal damage and has bad run-to-run end thickness control. This makes a costly thickness sorting step necessary before subsequent polishing. Furthermore, it is difficult to automate for positive single-wafer traceability. DSP [13], on the other hand, can eliminate geometric defects, but only at the expense of large material removal, long processing time and high cost. A novel FFP is simultaneous double-disk grinding (DDG). Several DDG tools have independently been developed [14–20]. Their differences in principle of operation, workpiece guidance, set-up, design, and process integration have already been discussed elsewhere [21], along with their specific suitability for precision wafer preparation [22]. Fig. 1 highlights the superiority of DDG over conventional sequential SSG, especially with respect to critical MWS waviness removal.
2. DDG: principle In simultaneous double-disk grinding, both sides of the wafer are ground, simultaneously in a single process step and one wafer at a time (single-wafer process), between a pair of plano-parallel grinding cup wheels on opposite
collinear axes. Among the different DDG principles developed [14–20], the one with fully free-floating hydrostatic workpiece suspension [15] produces the best results with respect to wafer flatness and nanotopology [21]. Fig. 2 shows its principle (explosion view). The wafer is loosely guided between water cushions formed by a pair of “hydro-pad” fixtures. The spindles are axially fed through openings in the hydro-pads. The wafer rotation is driven by a “notch-finger”, which meshes with the wafer’s mechanical fiducial. The notch-finger is part of a thin plastic frame mounted within a driven carrier-ring that loosely guides the wafer along its axis. The spindles typically rotate in opposite directions for a nearly vanishing net torque on the wafer and, thus, vanishing force on the sensitive wafer notch during grinding. Since the free-floating wafer is accessible from both sides, a built-in in-process thickness-gauge can measure the momentary wafer thickness in situ. Thus, the end thickness of the wafer can typically be controlled within <±0.8 m deviations from the target (wafer-to-wafer; 2σ). The kinematics of the DDG process is explained in Fig. 3(a). Unlike in SSG, where a large, intentionally tilted grinding wheel for an edge-to-center radial cut is preferred for reasons detailed elsewhere [23], the grinding wheel in DDG is smaller and in full peripheral contact with the wafer. This yields an optimum balance of forces, precision of component adjustment and minimum net load on the wafer [21,22]. The wafer W is simultaneously being ground on both faces between two typically counter-rotating cup wheels (disks) D1, D2, which are mounted on collinear axes DA1, DA2. The grinding wheels are typically, but not necessarily (see below), in full peripheral contact with the wafer. Thus, each wheel exhibits a “leading edge” with high momentary material removal and a “trailing edge” with low momentary material removal. The full peripheral removal of leading and trailing edges together form a characteristic crosscut pattern (Fig. 3(b)). This pattern is in contrast to SSG where the edge-
Fig. 1. Nanotopography (NT) map of 300 mm wafers from a “wire breakage” lot after DDG (+ polish; (a)) and SSG (+ polish; (b)). The accidental breakage and replacement of the slicing wire during MWS produced an abrupt step in the cutting plane due to thermal drift. (a) DDG, since free-floating, completely eliminates this step. (b) In SSG, since fixed-referenced mounting (chuck), the step survives (marked area in (b)). (DDG and SSG wafers were taken from neighboring positions of the same ingot. All prior and subsequent processes were identical for both wafers. Method of NT measurement: interferometry.)
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Fig. 2. Basic machine elements of a simultaneous double-disk grinding tool (DDG; explosion view).
to-center cut yields a radial cut (inset Fig. 3(b); for more details see [21–23]). Due to the typical counter-rotation in DDG the role of the leading edges of the opposite wheels will also be opposed: if one wheel’s leading edge performs its high removal on its way from wafer edge to center (“inward cut”), then the leading edge of the opposite wheel will, consequently, perform its high removal on its way from wafer
center to edge (“outward cut”). The material removal per angular increment in wafer rotation, and thus per unit time, is highest at the very edge of the wafer and decreases to zero towards the wafer center. Thus, with regard to material removal rate, an “inward cut” can be considered a “downhill cut” and an “outward cut” can be considered an “uphill cut”, respectively. Thus, there will always be a slight asymmetry in wheel
Fig. 3. (a) DDG kinematics. Crosscut with intersecting grooves (1) . . . (7) and (1 ) . . . (7 ), respectively. DA1, DA2: axes of opposite grinding disks; D1, D2: grinding disks; Ca: carrier ring assembly consisting of R: (plastic) retainer ring and F: (metal) frame; N: wafer notch (mechanical fiducial). (b) Micrograph of crosscut pattern on DDG wafer (inset: radial cut on SSG wafer for comparison).
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is described by z= ei ωt + e±i Ωt , r0
Fig. 4. Schematic of the DDG grinding geometry. Ω: Angular velocity of the rotating workpiece (silicon wafer); ω: angular velocity of the grinding wheel; C: center of wafer; c: center of grinding wheel; R0 : wafer radius, r0 : wheel radius (R0 ≥ 2r); N: wafer notch; P: a reference point on the wheel rim.
where r0 is the wheel radius, and ω and Ω are the angular velocities of wheel and wafer, respectively. z- is complex, with absolute value z- = z. The upper and lower signs in the exponent of the second term are for inward (“downhill”) and outward (“uphill”) cut of the two counter-rotating DDG grinding wheels, respectively, and describe known epitrochoidal and hypotrochoidal motions. In polar coordinates, the path length s(t) swept by P within a time interval t = t1 − t0 is t1 s(t) = ϕ˙ 2 r 2 + r˙ 2 dt, (2) t0
where r ω±Ω = z- = 2 cos t and r0 2 r˙ d r 2 = z- = −(ω ∓ Ω) 1 − . r0 dt 2r0
wear (self-dress or clog) and wafer material removal. Over many wafers the wafer position will drift to the side with the higher wheel wear and will thus become off center with the hydro-pads. This yields a gradual loss of balance. Restoring forces will build up that deform the wafer and deteriorate its geometry and nanotopology. To compensate this drift and periodically reset the spindle home positions according to the asymmetric wheel wear, there is another in-process gauge built into the grinding chamber that measures the absolute wafer position within the hydro-pads [24].
Thus,
3. Calculations
s˙ = r0
The purpose of this article is to derive a simple model that explains the basic features characteristic to any wafer machined by DDG. The goal is a simple, self-explanatory model based on clear initial boundary constraints without any toolspecific assumptions and without any further model-fitting parameters. (The more parameters, the less self-evident explanatory power the model will have.) Consequently, the scope of this description of DDG kinematics will be limited to the inherent key properties of the DDG kinematics and will fall short of reproducing secondary properties in all detail. Predictions will be compared to practical results, and deviations, which result from the simplicity of the model and the absence of additional model-fitting assumptions, will be explained.
(1)
(3)
The velocity v = s˙ along path s = s(t) of point P over the wafer is 2 (d/dt)z s˙ r 2 = = (ω ∓ Ω) ± ωΩ . (4) r0 r0 r0 (ω ∓ Ω)2 ± 4ωΩ cos2
=
ω∓Ω t 2
ω2 + Ω2 ± 2ωΩ cos(ω ∓ Ω)t.
(5)
The amount of material removed by the wheel at any point r = r(ϕ) = r(t) on the wafer surface during a time dt is removal =
(path swept by wheel) (wheel rim width) dt. wafer area swept by wheel
Thus,
=
ds dt (ds/dt)dt 2 s˙ = , = 2 r dr dϕ r˙rϕ˙ r(ds/dt)(dϕ/dt)dt
(6)
where denotes the local material removal. Using Eqs. (3) and (4) and sinωt ± sinΩt cosωt + cosΩt (ω ± Ω) d 1 ⇒ ϕ˙ = , since arctan x = 2 dx 1 + x2 ϕ(t) = arctan
3.1. Paths and removal In a frame of reference that rotates with the wafer (Fig. 4), the motion of a point P on the rim of the grinding cup-wheel
(7)
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˙ respectively, yields for r and r˙ , s˙ , and φ, (ω ∓ Ω)2 ± ωΩ(r/r0 )2
=− [(ω2 − Ω2 )/2]r 1 − (r/2r0 )2 =−
ω2 + Ω2 ± 2ωΩ cos (ω ∓ Ω) t . [(ω2 − Ω2 )/2] sin(ω ∓ Ω)t
(8)
Here, the approximation that the rim width w of the cupwheel is small compared to the wheel radius r0 has been made, i.e. w was set unity. The expression for implies that the shape of the wafer surface is determined by the DDG wheel/wafer kinematics solely, i.e. the wheel is assumed to follow any wafer surface contour completely. In reality, all wheels, depending on bond type (vitrified, metal, resin), have a finite stiffness that will dampen the singularities in Eq. (8) to finite values and additional (and debatable) assumptions about elastic properties of the wheel and abrasive would have to be made. This is beyond the intended scope of the model whose goal is it to see, how far the DDG wafer characteristics can be understood from purely kinematical considerations alone without any additional material-specific assumptions and model-fitting parameters. 3.2. Wafer shape What characteristic features can now be concluded from these kinematic considerations alone, that are inherent to all wafers ground with DDG? 1. The wafer shape, within the model (Eq. (8)), is radially symmetric∗ . 2. There always is a distinct dip in the center of the wafer (pole of Eq. (8) at r = 0). (This is also found and known in SSG). 3. For r0 ∼ R0 /2, as in all practical DDG tools [14–20] and for the reasons given in Section 2 and in [21,22], there will always be a tapering off towards the edge of the wafer (“edge roll-off”). This is the pole at r = 2r0 in Eq. (8). In SSG, since typically r0 ≥ R0 , the pole is safely outside the wafer area and an edge roll-off thus not observed [23]. 4. Due to the typical counter-rotation of the opposite DDG grinding wheels, the wafer front and back surface will always be dissimilar (different signs in Eq. (8); conjugate epitrochoids and hypotrochoids). This is a serious limitation, which is characteristic to DDG. In contrast to DDG, SSG can—in principle and irrespective of its dominating chuck-related process limitations—form identical front and back surfaces. Rotation of the wheels in the ∗ In practical DDG experiments, the actual residual wafer rotational asymmetry is, indeed, much smaller than the radial flatness variation. The rotational asymmetry is solely determined by the lift-off mark (cross-cut left incomplete under the wheel at the end of the grinding cycle), which is very weak. (The step height usually is 100 nm.) This is not discussed here since the lift-off feature is beyond the scope of the model used.
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same direction to overcome this limitation in DDG is impractical, as for any practical wheel with acceptably low sub-surface damage generation (abrasive mesh number #325, i.e. diamond grain size 50 m) the net torque resulting from the wheels will break the the wafer. In SSG, the chuck allows much higher forces, and breakage very rarely occurs. 5. For ω/Ω 1, the epitrochoids and hypotrochoids gradually become alike and a more symmetric wafer front and back can be achieved (Eq. (8)): lim ( ) = −
ω/Ω1
1
[(ωr)/2] 1 − (r/2r0 )2
.
(9)
For grinding conditions with ω ∼ Ω the solutions become degenerate, e.g., the radicand in Eq. (4) becomes negative for one of the conjugate epitrochoid and hypotrochoid. Eq. (8), for instance, then no longer holds. As in practice always ω >··· Ω (high wheel RPM, slow wafer RPM), this can be neglected here. Example front and back surfaces for different values of ω and Ω, calculated from Eq. (8), confirm these findings (Fig. 5(a–c)). The total thickness (front + back profiles) and corrugation (“warp”; front–back profiles) for case Fig. 5(c), ω = 8500 RPM and Ω = 7.5 RPM, is given in Fig. 5(d). Although the front and back surface can be made nearly identical for ω Ω, it is obviously never possible to achieve an inherently flat surface. The radial position of maximum wafer thickness (lowest material removal) between the center “navel” and the edge roll-off is obtained by d ! rmin Ω
=0 ⇔ = 2 1∓ dr r0 ω 2ω (10) r0 (ω2 − Ω2 ) √ For ω Ω, rmin approaches 2 r0 . For small ω/Ω, rmin becomes smaller for the epitrochoids and larger for the hypotrochoids (see e.g., Fig. 5(a)). The full width at half maximum (FWHM) of this ring is obtained by finding radial positions r2 min where (r2 min ) = 2 min . Solving 2 min for r2 min , with min taken from Eq. (10), yields
r2 min 3 Ω 2 Ω = 2± 3− (11a) − r0 4 ω ω ⇒ min = −
and r2 min r0
3 Ω 2 Ω = 2± 3− + . 4 ω ω
(11b)
Eq. (11a) is for the epitrochoids (inward/downhill cut) and Eq. (11) for the hypotrochoids (outward/uphill cut), respectively.
194 G.J. Pietsch, M. Kerstan / Precision Engineering 29 (2005) 189–196 Fig. 5. Calculated shape (3D view) of wafer front and back as resulting from DDG with different wheel and wafer rotational speeds. (a–c) With increasing ratio ω/Ω of wafer, ω, to wheel, Ω, RPM the width and depth of the “navel” (center hole) and the edge roll-off decreases, and front and back wafer surface shape become similar. (d) Total thickness and overall wafer warpage derived from the sum and the difference of front (here inward/downhill) and back (outwards/uphill) surface height profiles, respectively.
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The width of the ring (FWHM) is the difference of the larger and the smaller value in each Eq. (11a and b), given by the positive and the negative sign under the root, respectively, i.e. r2 min |“+ and r2 min |“− . The area of these annuli enclosed by these r2 min |“+ and r2 min |“− be defined as the “flatness area” Aflat of the wafer, 2 2 Aflat r2 min r2 min :=π − r0 “+” r0 “−” (r0 )2
2
3 Ω ω 2π − . (12) = 2 3− Ω 4 ω r0 Due to the squaring and subtraction in Eq. (12), the size of the flatness areas of the front and back of the wafer are equal, in contrast to their radial positions alone as given by Eq. (11). For ω/Ω 1, e.g., ω/Ω ≥ 10, rmin from Eq. (11) approaches √ 1 lim (r2 min ) = 2 ± 3 ≈ 1.93 ∨ 0.52, compare r0 (ω/Ω)1 r2 min ω = 10 ≈ 1.83 ∨ 0.62. (13) r0 Ω For the practical case of r0 = 80 mm (160 mm diameter grinding wheel) and R0 = 150 mm (300 mm wafer) the flatness area, relative to the entire wafer area, thus approaches √ 2π 3(80 mm)2 Aflat = ≈ 98.5% (14) πR0 2 π(150 mm)2 (∼92.7% for ω/Ω = 10, and ∼86.7% for ω/Ω = 5 etc.). In the example given, the center “navel” and the edge roll-off thus account for at least ∼1.5% of the wafer surface.
4. Experiment versus calculations The preceding calculations were done for two dimensions, i.e. wafer and wheel in same plane (no spindle tilt). In prac-
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tical DDG grinding, however, an intentional spindle tilt is one of the key measures to fine-tune the wafer shape (radial thickness profile) and geometry (TTV, nanotopography, etc.). A vertical tilt of the spindle axes downwards (see Fig. 2), for instance, would yield a stronger tapering-off of the wafer profile towards its edges. This slightly decreases the center “navel”, since the height of the apex of the “flatness area” according to Eq. (10) is slightly reduced. (The navel clearly dominates the thickness profile of a wafer ground with wellaligned spindles and thus determines the achievable minimum TTV.) Vice versa, an upward tilt reduces the edge rolloff but yields a deeper navel. A horizontal spindle tilt (refer to Fig. 2) increases the material in a region nearly half-way between wafer center and wafer edge. This can help to reduce the height of the apex of the flatness area. It does so, however, at the expense of a balanced cross-cut; the material removal performed by each wheel’s leading edge is increased at the expense of the removal by its respective trailing edge, or vice versa. This change in material removal bears the risk of unbalancing the wafer’s perfectly parallel and centered alignment between the hydro-pads or stalling the self-dressing operation of the wheels. The former deteriorates the nanotopology due to additional bending forces. The latter disturbs the initial balance of the front and back wheel wear, since the inwardcutting wheel wears faster than the opposite outward-cutting. (Any departure from a balanced leading-edge to trailing-edge removal will cause the inward-cutting wheel to wear even faster and the opposite outward-cutting wheel to wear slower. The latter wheel then has a higher risk to clog and leave the region of continuous self-dressing operation.) The equations previously derived were implemented into a numerical program. Additional vertical spindle tilt was taken into account by subtracting straight lines from the calculated geometry (single-side) or thickness (both sides) profiles, with slopes according to the chosen vertical tilt angle. For horizontal spindle tilts elliptical segments were subtracted according to the respective tilt angle. Also, a finite rim width of the cup wheel, so far neglected in the derivation of Eq. (8)
Fig. 6. Wafer thickness radial profile for various grinding spindle misalignments: calculation vs. experimental data. (a) Dominant vertical “+” tilt misalignment. √ (b) Dominant horizontal mistilt (with slight vertical “−” tilt; compare to (a)). (c) Oblique: (weaker) vertical and (stronger) horizontal tilt. The near - 2r0 -apex, (a), is converted into a pit by the strong horizontal spindle tilt such, that the total thickness variation throughout the profile is reduced to <1 m (as shown; <0.3–0.4 m in optimum cases). Strong horizontal spindle tilt means to partially abandon crosscut (→ radial cut). Calculation (circles) and experiment (solid line) agree well, including overall shape, apex and edge roll-off; but fail to reproduce the center navel correctly (see text).
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etc., was included (typically 3–5 mm), and an adjustable lateral offset between the axis of wafer rotation and the spindle axes was introduced to allow for overcut and undercut of the wafer center. The latter is used in practice as a key measure for the reduction of the depth of the center grinding navel. A fitting routine was used to fit the parameters to the measured geometry profiles from actual DDG runs. Results are shown for three wafers ground with characteristic spindle tilts in Fig. 6(a) (vertical spindle tilt), Fig. 6(b) (predominantly horizontal tilt), and Fig. 6(c) (oblique tilt). The profile in Fig. 6(c) is close to the best TTV currently practically achievable by a DDG tool with the listed technical features (TTV ∼ 0.5 m). There is a good agreement between calculation and measurement—within the expected limits and according to the simplifications made in the model assumption (e.g., neglecting finite wheel elasticity). All features characteristic for DDG kinematics are reproduced despite the simplicity of the model: the predicted edge roll-off, center navel, general slope (convex or concave, depending on spindle tilt) etc. As expected, significant deviations occur in the wafer center due to the neglected wheel stiffness properties, where the model simply displays a singularity. This is in agreement with the purpose of the model, which is aimed at predicting DDGinherent geometry features without sacrificing simplicity or making additional ad hoc assumptions. 5. Summary Simultaneous double-disk grinding is a novel and powerful technology for silicon wafer planarization as required for the latest and upcoming future microelectronic device design rules. DDG uses two opposite grinding wheels, which simultaneously flatten both sides of the free-floating wafer between them in a single process step. The basic characteristics inherent to the specific DDG kinematics can be understood with simple analytical calculations and the resulting wafer geometry predicted within a simple model. This model shows; 1) an inevitable center navel and edge thickness roll-off limit the TTV to around 0.5 m (which is far better than what subsequent wafer preparation steps such as polishing require as minimum starting quality), (2) radial symmetry, (3) asymmetric front and back removals (for standard counter-rotating grinding wheels), and (4) the dependence of the thickness profile on specific kinematics and dimensions (ω, Ω, r0 , R0 ; relative sense of rotations). References [1] Semiconductor Industry Association (SIA); Internet: www.semichips. org. International Technology Roadmap for Semiconductors; internet; public.itrs.net: 2003 edition. [2] Patrick WJ, Guthrie WL, Standley CL, Schiable PM. Application of chemical mechanical polishing to the fabrication of VLSI circuit interconnections. J Electrochem Soc 1991;138:1778– 84.
[3] Beyer KD, Guthrie WL, Makarewicz SR, Mendel E, Patrick WJ, Perry KA, Pliskin WA, Riseman J, Schiable PM, Standley CL. Chem-mech polishing method for producing coplanar metal/insulator films on a substrate. US Pat 4,944,836 (1990). [4] SEMI standard M43-0301. Guide for reporting wafer nantopography. Semiconductor Equipment and Materials International (SEMI), Milpitas, CA, USA, 2001. Internet: www.semi.org. [5] Kishimoto J. Semiconductor wafer and method for fabrication thereof. Europ Pat EP 1 005 069 A2 (2000). [6] Kato T, Shima S, Nakano M, Aza Haranaka, Kudo, H. Method for production of wafer. Europ Pat Appl 0 617 457 A2. [7] Masumua H, Suzuki K, Kudo H. Method of manufacturing semiconductor mirror wafers. US Pat 5,827,779 (1998). [8] Davis E, Trentman BM, Dangel M, Smith WL, Smith JD. Method of improving the flatness of polished semiconductor wafers. Internat’l Pat Appl WO 99/31723 (1999). [9] Kato T, Masumura H, Okuni S, Kudo H. Method of manufacturing semiconductor wafers. US Pat 5,942,445 (1999). [10] Blum G. L¨appen. In: Spur G, St¨oferle T, editors. Handbuch der Ferti-gungs-tech-nik, 3/2. M¨unchen, Germany: Hanser; 1980. p. 366–83 [in German]. [11] Feinstbearbeitung. In: Polieren, L¨appen, Honen, Preßpolieren, Polierdrehen, editors. Ausschuß f¨ur wirtschaftliche Fertigung beim Reichskuratorium f¨ur Wirtschaftlichkeit. RKW-Ver¨offentlichungen, 74. Berlin, Germany: Beuth; 1932. p. 48–9 [in German]. [12] L¨atzig W L¨appen. Grundlagen und praktische Anwendungen. M¨unchen, Germany: Hanser; 1950 [in German]. [13] Haisma J, van der Kruis FJHM, Spierings BACM, Baalbergen JJ, Bijsterveld BH, Brehm R, et al. Improved geometry of double-sided polished parallel wafers prepared for direct wafer bonding. Appl Opt OT 1994;33(34):7945. [14] Nishi K, Nukui M, Murai S, Nakajima K, Wada T. Grinding method, surface grinder, work piece support mechanism, and work rest. Europ Pat Appl EP 0 868 974 A2 (1998). [15] Ikeda J, Ishii T, Sasakura S, Yoshimura Y, Ueda K. Double side grinding apparatur for flat disklike work. US Patent 5,989,108 (1999). [16] Abe K, Isobe S, Tomita Y, Hara K, Masaki R, Iwase A, Nagata H. Double side grinding apparatur and double side polishing apparatus. Europ Pat Appl EP 0 940 220 A1 (1999). [17] Yukio O, Masatoshi T. Vertical shaft double-head surface grinding machine. Jap Pat JP 9-168951 (1997) [in Japanese]. [18] Nobuto S, Akihide T. Double grinding device for high brittle material. Jap Pat JP 9-262747 (1997) [in Japanese]. [19] Yamato S. Method and device for grinding both side. Jap Pat JP 11-048107 (1999) [in Japanese]. [20] Hara K, Isobe S, Nagata H, Iwase A, Tomita Y. Proceedings of the 14th annual meeting of The American Society for Precision Enginieering (ASPE). NC, USA: Raleigh; 1999. p. 559– 62. [21] Kerstan M, Pietsch GJ. Silicon wafer substrate planarization using simultaneous double-disk grindng: impact on wafer surface, geometry, morphology and subsurface crystal damage. In: Yasunaga N, Tamaki J, Wong WCK, Chen BC, Choi HZ, Gao YS, Salmon SS, editors. Proceedings of the ISAAT conference on Advances in abrasive technology III, Hawaii, 2000. p. 211–8. [22] Pietsch GJ, Kerstan M. In: Balsamo A, Evans CJ, Frank A, Knapp W, Mana G, Mortarino M, Sartori S, Thwaite EG, editors. Proceedings of the 2nd international conference of the european society for precision engineering and nanotoechnology. Bedford, UK: Euspen; 2002. p. 644–8. [23] Huber A, Drexler R. Wafer holder and method of producing a semiconductor wafer. US Pat 6,117,776 (2000). [24] Okura K. Both face grinding method for thin pate disclike workpiece and device thereof. Jap Pat 2002307303 A (2002).
Understanding simultaneous double-disk grinding: operation principle and material removal kinematics in silicon wafer planarization Georg J. Pietsch∗ , Michael Kerstan Research and Development, Siltronic AG, D-84489 Burghausen, Germany Received 30 December 2002; received in revised form 28 January 2004; accepted 7 July 2004 Available online 25 September 2004
Abstract Simultaneous double-disk grinding (DDG) is a novel and powerful technology for precision-machining mono-crystalline silicon slices (“wafers”). With DDG the extreme degrees of planarity can be achieved, which the fabrication of micro-electronic devices with minimum lateral feature dimensions of 90 nm and below demands. In DDG, both sides of the wafer are ground simultaneously between two opposite grinding wheels on collinear spindle axes. It is a chuck-less process, in which the workpiece is machined in “free-floating” fashion. Machining kinematics, removal mechanism, and resulting wafer shape differ from those known from (chuck-mounted) single-side grinding or doublesided batch lapping, which are conventionally used in mechanical wafer shaping. This article explains the kinematics of DDG and derives the basic, method-inherent features always observed on DDG-ground wafers from simple kinematic considerations without further model assumptions: global wafer shape, center dip (“navel”), edge thickness roll-off, and symmetries. The expected results are compared with experimental data. © 2004 Elsevier Inc. All rights reserved. Keywords: Double-disk grinding; Simultaneous grinding; Flatness; Wafer preparation; Kinematics; Grinding wheel; Nanotopography
1. Introduction Minimum feature dimensions of microelectronic components continue to shrink by about 30–40% each design-rule generation as defined by the industry’s overall roadmap for semiconductor devices and materials [1]. The key processes used to pattern devices make extreme demands on the flatness of the raw wafer substrate, as the properties of the starting surface ultimately limit the achievable process results. Photolithography, for instance, due to its sub-100 nm depth of focus during pattern exposure, and CMP (chemomechanical planarization for interlayer dielectric planarization [2,3]) due to the elasticity of the polishing pad. A very critical flatness parameter is the wafer “nanotopology” (NT) [4], which is the front-side-referenced flatness within any e.g. 2 mm × 2 mm or 10 mm × 10 mm measurement field throughout the entire wafer surface. For current 65 nm (volume) and 45 nm (sampling) design-rules NT within 10 mm × ∗
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10 mm and 2 mm × 2 mm must never exceed 28 and 20 nm, respectively. To achieve this flatness, a sequence of mechanical and wet-chemical shaping processes are applied after a single-crystalline silicon ingot has been grown. Such a process chain conventionally consists of: 1. ingot slicing [mostly multi-wire slicing (MWS) or innerdiameter slicing (ID); ID is practical for small wafers only]; 2. mechanical planarization [double-sided batch lapping; or sequential single-side grinding (SSG)]; 3. wet-chemical etching (caustic, alkaline, or a combination of both); 4. stock-removal polishing [chemo-mechanical/shape polishing; batch or single-wafer single-side polishing (SSP); or simultaneous batch double-side polishing (DSP)]; 5. final polishing (batch or single-wafer; single-side). Since the same tools as for CMP are used, final wafer substrate polishing is often termed “CMP” also. Among all the theoretically possible combinations of these individual steps into complete process sequences, only com-
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binations which contain at least one “free-floating” process (FFP) are of practical importance [5–9]. In an FFP, such as lapping or DSP, the wafer is processed without forcemounting it onto a chuck or clamping device during machining. Chuck processes alone cannot remove wafer waviness (parallel corrugations of front and back at basically constant thickness) because of the elasticity of the thin wafer. When the wafer is forced flat on the mounting chuck by vacuum, corrugations are elastically copied back to its front and are thus preserved during machining. Lapping [10–12] is an old, versatile, and powerful machining technique. However, it produces a poor surface with deep crystal damage and has bad run-to-run end thickness control. This makes a costly thickness sorting step necessary before subsequent polishing. Furthermore, it is difficult to automate for positive single-wafer traceability. DSP [13], on the other hand, can eliminate geometric defects, but only at the expense of large material removal, long processing time and high cost. A novel FFP is simultaneous double-disk grinding (DDG). Several DDG tools have independently been developed [14–20]. Their differences in principle of operation, workpiece guidance, set-up, design, and process integration have already been discussed elsewhere [21], along with their specific suitability for precision wafer preparation [22]. Fig. 1 highlights the superiority of DDG over conventional sequential SSG, especially with respect to critical MWS waviness removal.
2. DDG: principle In simultaneous double-disk grinding, both sides of the wafer are ground, simultaneously in a single process step and one wafer at a time (single-wafer process), between a pair of plano-parallel grinding cup wheels on opposite
collinear axes. Among the different DDG principles developed [14–20], the one with fully free-floating hydrostatic workpiece suspension [15] produces the best results with respect to wafer flatness and nanotopology [21]. Fig. 2 shows its principle (explosion view). The wafer is loosely guided between water cushions formed by a pair of “hydro-pad” fixtures. The spindles are axially fed through openings in the hydro-pads. The wafer rotation is driven by a “notch-finger”, which meshes with the wafer’s mechanical fiducial. The notch-finger is part of a thin plastic frame mounted within a driven carrier-ring that loosely guides the wafer along its axis. The spindles typically rotate in opposite directions for a nearly vanishing net torque on the wafer and, thus, vanishing force on the sensitive wafer notch during grinding. Since the free-floating wafer is accessible from both sides, a built-in in-process thickness-gauge can measure the momentary wafer thickness in situ. Thus, the end thickness of the wafer can typically be controlled within <±0.8 m deviations from the target (wafer-to-wafer; 2σ). The kinematics of the DDG process is explained in Fig. 3(a). Unlike in SSG, where a large, intentionally tilted grinding wheel for an edge-to-center radial cut is preferred for reasons detailed elsewhere [23], the grinding wheel in DDG is smaller and in full peripheral contact with the wafer. This yields an optimum balance of forces, precision of component adjustment and minimum net load on the wafer [21,22]. The wafer W is simultaneously being ground on both faces between two typically counter-rotating cup wheels (disks) D1, D2, which are mounted on collinear axes DA1, DA2. The grinding wheels are typically, but not necessarily (see below), in full peripheral contact with the wafer. Thus, each wheel exhibits a “leading edge” with high momentary material removal and a “trailing edge” with low momentary material removal. The full peripheral removal of leading and trailing edges together form a characteristic crosscut pattern (Fig. 3(b)). This pattern is in contrast to SSG where the edge-
Fig. 1. Nanotopography (NT) map of 300 mm wafers from a “wire breakage” lot after DDG (+ polish; (a)) and SSG (+ polish; (b)). The accidental breakage and replacement of the slicing wire during MWS produced an abrupt step in the cutting plane due to thermal drift. (a) DDG, since free-floating, completely eliminates this step. (b) In SSG, since fixed-referenced mounting (chuck), the step survives (marked area in (b)). (DDG and SSG wafers were taken from neighboring positions of the same ingot. All prior and subsequent processes were identical for both wafers. Method of NT measurement: interferometry.)
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Fig. 2. Basic machine elements of a simultaneous double-disk grinding tool (DDG; explosion view).
to-center cut yields a radial cut (inset Fig. 3(b); for more details see [21–23]). Due to the typical counter-rotation in DDG the role of the leading edges of the opposite wheels will also be opposed: if one wheel’s leading edge performs its high removal on its way from wafer edge to center (“inward cut”), then the leading edge of the opposite wheel will, consequently, perform its high removal on its way from wafer
center to edge (“outward cut”). The material removal per angular increment in wafer rotation, and thus per unit time, is highest at the very edge of the wafer and decreases to zero towards the wafer center. Thus, with regard to material removal rate, an “inward cut” can be considered a “downhill cut” and an “outward cut” can be considered an “uphill cut”, respectively. Thus, there will always be a slight asymmetry in wheel
Fig. 3. (a) DDG kinematics. Crosscut with intersecting grooves (1) . . . (7) and (1 ) . . . (7 ), respectively. DA1, DA2: axes of opposite grinding disks; D1, D2: grinding disks; Ca: carrier ring assembly consisting of R: (plastic) retainer ring and F: (metal) frame; N: wafer notch (mechanical fiducial). (b) Micrograph of crosscut pattern on DDG wafer (inset: radial cut on SSG wafer for comparison).
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is described by z= ei ωt + e±i Ωt , r0
Fig. 4. Schematic of the DDG grinding geometry. Ω: Angular velocity of the rotating workpiece (silicon wafer); ω: angular velocity of the grinding wheel; C: center of wafer; c: center of grinding wheel; R0 : wafer radius, r0 : wheel radius (R0 ≥ 2r); N: wafer notch; P: a reference point on the wheel rim.
where r0 is the wheel radius, and ω and Ω are the angular velocities of wheel and wafer, respectively. z- is complex, with absolute value z- = z. The upper and lower signs in the exponent of the second term are for inward (“downhill”) and outward (“uphill”) cut of the two counter-rotating DDG grinding wheels, respectively, and describe known epitrochoidal and hypotrochoidal motions. In polar coordinates, the path length s(t) swept by P within a time interval t = t1 − t0 is t1 s(t) = ϕ˙ 2 r 2 + r˙ 2 dt, (2) t0
where r ω±Ω = z- = 2 cos t and r0 2 r˙ d r 2 = z- = −(ω ∓ Ω) 1 − . r0 dt 2r0
wear (self-dress or clog) and wafer material removal. Over many wafers the wafer position will drift to the side with the higher wheel wear and will thus become off center with the hydro-pads. This yields a gradual loss of balance. Restoring forces will build up that deform the wafer and deteriorate its geometry and nanotopology. To compensate this drift and periodically reset the spindle home positions according to the asymmetric wheel wear, there is another in-process gauge built into the grinding chamber that measures the absolute wafer position within the hydro-pads [24].
Thus,
3. Calculations
s˙ = r0
The purpose of this article is to derive a simple model that explains the basic features characteristic to any wafer machined by DDG. The goal is a simple, self-explanatory model based on clear initial boundary constraints without any toolspecific assumptions and without any further model-fitting parameters. (The more parameters, the less self-evident explanatory power the model will have.) Consequently, the scope of this description of DDG kinematics will be limited to the inherent key properties of the DDG kinematics and will fall short of reproducing secondary properties in all detail. Predictions will be compared to practical results, and deviations, which result from the simplicity of the model and the absence of additional model-fitting assumptions, will be explained.
(1)
(3)
The velocity v = s˙ along path s = s(t) of point P over the wafer is 2 (d/dt)z s˙ r 2 = = (ω ∓ Ω) ± ωΩ . (4) r0 r0 r0 (ω ∓ Ω)2 ± 4ωΩ cos2
=
ω∓Ω t 2
ω2 + Ω2 ± 2ωΩ cos(ω ∓ Ω)t.
(5)
The amount of material removed by the wheel at any point r = r(ϕ) = r(t) on the wafer surface during a time dt is removal =
(path swept by wheel) (wheel rim width) dt. wafer area swept by wheel
Thus,
=
ds dt (ds/dt)dt 2 s˙ = , = 2 r dr dϕ r˙rϕ˙ r(ds/dt)(dϕ/dt)dt
(6)
where denotes the local material removal. Using Eqs. (3) and (4) and sinωt ± sinΩt cosωt + cosΩt (ω ± Ω) d 1 ⇒ ϕ˙ = , since arctan x = 2 dx 1 + x2 ϕ(t) = arctan
3.1. Paths and removal In a frame of reference that rotates with the wafer (Fig. 4), the motion of a point P on the rim of the grinding cup-wheel
(7)
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˙ respectively, yields for r and r˙ , s˙ , and φ, (ω ∓ Ω)2 ± ωΩ(r/r0 )2
=− [(ω2 − Ω2 )/2]r 1 − (r/2r0 )2 =−
ω2 + Ω2 ± 2ωΩ cos (ω ∓ Ω) t . [(ω2 − Ω2 )/2] sin(ω ∓ Ω)t
(8)
Here, the approximation that the rim width w of the cupwheel is small compared to the wheel radius r0 has been made, i.e. w was set unity. The expression for implies that the shape of the wafer surface is determined by the DDG wheel/wafer kinematics solely, i.e. the wheel is assumed to follow any wafer surface contour completely. In reality, all wheels, depending on bond type (vitrified, metal, resin), have a finite stiffness that will dampen the singularities in Eq. (8) to finite values and additional (and debatable) assumptions about elastic properties of the wheel and abrasive would have to be made. This is beyond the intended scope of the model whose goal is it to see, how far the DDG wafer characteristics can be understood from purely kinematical considerations alone without any additional material-specific assumptions and model-fitting parameters. 3.2. Wafer shape What characteristic features can now be concluded from these kinematic considerations alone, that are inherent to all wafers ground with DDG? 1. The wafer shape, within the model (Eq. (8)), is radially symmetric∗ . 2. There always is a distinct dip in the center of the wafer (pole of Eq. (8) at r = 0). (This is also found and known in SSG). 3. For r0 ∼ R0 /2, as in all practical DDG tools [14–20] and for the reasons given in Section 2 and in [21,22], there will always be a tapering off towards the edge of the wafer (“edge roll-off”). This is the pole at r = 2r0 in Eq. (8). In SSG, since typically r0 ≥ R0 , the pole is safely outside the wafer area and an edge roll-off thus not observed [23]. 4. Due to the typical counter-rotation of the opposite DDG grinding wheels, the wafer front and back surface will always be dissimilar (different signs in Eq. (8); conjugate epitrochoids and hypotrochoids). This is a serious limitation, which is characteristic to DDG. In contrast to DDG, SSG can—in principle and irrespective of its dominating chuck-related process limitations—form identical front and back surfaces. Rotation of the wheels in the ∗ In practical DDG experiments, the actual residual wafer rotational asymmetry is, indeed, much smaller than the radial flatness variation. The rotational asymmetry is solely determined by the lift-off mark (cross-cut left incomplete under the wheel at the end of the grinding cycle), which is very weak. (The step height usually is 100 nm.) This is not discussed here since the lift-off feature is beyond the scope of the model used.
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same direction to overcome this limitation in DDG is impractical, as for any practical wheel with acceptably low sub-surface damage generation (abrasive mesh number #325, i.e. diamond grain size 50 m) the net torque resulting from the wheels will break the the wafer. In SSG, the chuck allows much higher forces, and breakage very rarely occurs. 5. For ω/Ω 1, the epitrochoids and hypotrochoids gradually become alike and a more symmetric wafer front and back can be achieved (Eq. (8)): lim ( ) = −
ω/Ω1
1
[(ωr)/2] 1 − (r/2r0 )2
.
(9)
For grinding conditions with ω ∼ Ω the solutions become degenerate, e.g., the radicand in Eq. (4) becomes negative for one of the conjugate epitrochoid and hypotrochoid. Eq. (8), for instance, then no longer holds. As in practice always ω >··· Ω (high wheel RPM, slow wafer RPM), this can be neglected here. Example front and back surfaces for different values of ω and Ω, calculated from Eq. (8), confirm these findings (Fig. 5(a–c)). The total thickness (front + back profiles) and corrugation (“warp”; front–back profiles) for case Fig. 5(c), ω = 8500 RPM and Ω = 7.5 RPM, is given in Fig. 5(d). Although the front and back surface can be made nearly identical for ω Ω, it is obviously never possible to achieve an inherently flat surface. The radial position of maximum wafer thickness (lowest material removal) between the center “navel” and the edge roll-off is obtained by d ! rmin Ω
=0 ⇔ = 2 1∓ dr r0 ω 2ω (10) r0 (ω2 − Ω2 ) √ For ω Ω, rmin approaches 2 r0 . For small ω/Ω, rmin becomes smaller for the epitrochoids and larger for the hypotrochoids (see e.g., Fig. 5(a)). The full width at half maximum (FWHM) of this ring is obtained by finding radial positions r2 min where (r2 min ) = 2 min . Solving 2 min for r2 min , with min taken from Eq. (10), yields
r2 min 3 Ω 2 Ω = 2± 3− (11a) − r0 4 ω ω ⇒ min = −
and r2 min r0
3 Ω 2 Ω = 2± 3− + . 4 ω ω
(11b)
Eq. (11a) is for the epitrochoids (inward/downhill cut) and Eq. (11) for the hypotrochoids (outward/uphill cut), respectively.
194 G.J. Pietsch, M. Kerstan / Precision Engineering 29 (2005) 189–196 Fig. 5. Calculated shape (3D view) of wafer front and back as resulting from DDG with different wheel and wafer rotational speeds. (a–c) With increasing ratio ω/Ω of wafer, ω, to wheel, Ω, RPM the width and depth of the “navel” (center hole) and the edge roll-off decreases, and front and back wafer surface shape become similar. (d) Total thickness and overall wafer warpage derived from the sum and the difference of front (here inward/downhill) and back (outwards/uphill) surface height profiles, respectively.
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The width of the ring (FWHM) is the difference of the larger and the smaller value in each Eq. (11a and b), given by the positive and the negative sign under the root, respectively, i.e. r2 min |“+ and r2 min |“− . The area of these annuli enclosed by these r2 min |“+ and r2 min |“− be defined as the “flatness area” Aflat of the wafer, 2 2 Aflat r2 min r2 min :=π − r0 “+” r0 “−” (r0 )2
2
3 Ω ω 2π − . (12) = 2 3− Ω 4 ω r0 Due to the squaring and subtraction in Eq. (12), the size of the flatness areas of the front and back of the wafer are equal, in contrast to their radial positions alone as given by Eq. (11). For ω/Ω 1, e.g., ω/Ω ≥ 10, rmin from Eq. (11) approaches √ 1 lim (r2 min ) = 2 ± 3 ≈ 1.93 ∨ 0.52, compare r0 (ω/Ω)1 r2 min ω = 10 ≈ 1.83 ∨ 0.62. (13) r0 Ω For the practical case of r0 = 80 mm (160 mm diameter grinding wheel) and R0 = 150 mm (300 mm wafer) the flatness area, relative to the entire wafer area, thus approaches √ 2π 3(80 mm)2 Aflat = ≈ 98.5% (14) πR0 2 π(150 mm)2 (∼92.7% for ω/Ω = 10, and ∼86.7% for ω/Ω = 5 etc.). In the example given, the center “navel” and the edge roll-off thus account for at least ∼1.5% of the wafer surface.
4. Experiment versus calculations The preceding calculations were done for two dimensions, i.e. wafer and wheel in same plane (no spindle tilt). In prac-
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tical DDG grinding, however, an intentional spindle tilt is one of the key measures to fine-tune the wafer shape (radial thickness profile) and geometry (TTV, nanotopography, etc.). A vertical tilt of the spindle axes downwards (see Fig. 2), for instance, would yield a stronger tapering-off of the wafer profile towards its edges. This slightly decreases the center “navel”, since the height of the apex of the “flatness area” according to Eq. (10) is slightly reduced. (The navel clearly dominates the thickness profile of a wafer ground with wellaligned spindles and thus determines the achievable minimum TTV.) Vice versa, an upward tilt reduces the edge rolloff but yields a deeper navel. A horizontal spindle tilt (refer to Fig. 2) increases the material in a region nearly half-way between wafer center and wafer edge. This can help to reduce the height of the apex of the flatness area. It does so, however, at the expense of a balanced cross-cut; the material removal performed by each wheel’s leading edge is increased at the expense of the removal by its respective trailing edge, or vice versa. This change in material removal bears the risk of unbalancing the wafer’s perfectly parallel and centered alignment between the hydro-pads or stalling the self-dressing operation of the wheels. The former deteriorates the nanotopology due to additional bending forces. The latter disturbs the initial balance of the front and back wheel wear, since the inwardcutting wheel wears faster than the opposite outward-cutting. (Any departure from a balanced leading-edge to trailing-edge removal will cause the inward-cutting wheel to wear even faster and the opposite outward-cutting wheel to wear slower. The latter wheel then has a higher risk to clog and leave the region of continuous self-dressing operation.) The equations previously derived were implemented into a numerical program. Additional vertical spindle tilt was taken into account by subtracting straight lines from the calculated geometry (single-side) or thickness (both sides) profiles, with slopes according to the chosen vertical tilt angle. For horizontal spindle tilts elliptical segments were subtracted according to the respective tilt angle. Also, a finite rim width of the cup wheel, so far neglected in the derivation of Eq. (8)
Fig. 6. Wafer thickness radial profile for various grinding spindle misalignments: calculation vs. experimental data. (a) Dominant vertical “+” tilt misalignment. √ (b) Dominant horizontal mistilt (with slight vertical “−” tilt; compare to (a)). (c) Oblique: (weaker) vertical and (stronger) horizontal tilt. The near - 2r0 -apex, (a), is converted into a pit by the strong horizontal spindle tilt such, that the total thickness variation throughout the profile is reduced to <1 m (as shown; <0.3–0.4 m in optimum cases). Strong horizontal spindle tilt means to partially abandon crosscut (→ radial cut). Calculation (circles) and experiment (solid line) agree well, including overall shape, apex and edge roll-off; but fail to reproduce the center navel correctly (see text).
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etc., was included (typically 3–5 mm), and an adjustable lateral offset between the axis of wafer rotation and the spindle axes was introduced to allow for overcut and undercut of the wafer center. The latter is used in practice as a key measure for the reduction of the depth of the center grinding navel. A fitting routine was used to fit the parameters to the measured geometry profiles from actual DDG runs. Results are shown for three wafers ground with characteristic spindle tilts in Fig. 6(a) (vertical spindle tilt), Fig. 6(b) (predominantly horizontal tilt), and Fig. 6(c) (oblique tilt). The profile in Fig. 6(c) is close to the best TTV currently practically achievable by a DDG tool with the listed technical features (TTV ∼ 0.5 m). There is a good agreement between calculation and measurement—within the expected limits and according to the simplifications made in the model assumption (e.g., neglecting finite wheel elasticity). All features characteristic for DDG kinematics are reproduced despite the simplicity of the model: the predicted edge roll-off, center navel, general slope (convex or concave, depending on spindle tilt) etc. As expected, significant deviations occur in the wafer center due to the neglected wheel stiffness properties, where the model simply displays a singularity. This is in agreement with the purpose of the model, which is aimed at predicting DDGinherent geometry features without sacrificing simplicity or making additional ad hoc assumptions. 5. Summary Simultaneous double-disk grinding is a novel and powerful technology for silicon wafer planarization as required for the latest and upcoming future microelectronic device design rules. DDG uses two opposite grinding wheels, which simultaneously flatten both sides of the free-floating wafer between them in a single process step. The basic characteristics inherent to the specific DDG kinematics can be understood with simple analytical calculations and the resulting wafer geometry predicted within a simple model. This model shows; 1) an inevitable center navel and edge thickness roll-off limit the TTV to around 0.5 m (which is far better than what subsequent wafer preparation steps such as polishing require as minimum starting quality), (2) radial symmetry, (3) asymmetric front and back removals (for standard counter-rotating grinding wheels), and (4) the dependence of the thickness profile on specific kinematics and dimensions (ω, Ω, r0 , R0 ; relative sense of rotations). References [1] Semiconductor Industry Association (SIA); Internet: www.semichips. org. International Technology Roadmap for Semiconductors; internet; public.itrs.net: 2003 edition. [2] Patrick WJ, Guthrie WL, Standley CL, Schiable PM. Application of chemical mechanical polishing to the fabrication of VLSI circuit interconnections. J Electrochem Soc 1991;138:1778– 84.
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