Wafer Geometry Gauge

1. Theory

• Definition: A wafer geometry gauge maps the thickness and shape of a whole wafer and reports standard parameters such as thickness, TTV, bow, warp, sori and site flatness (Vitrek; MTI; Corning).
• Median surface: Shape parameters use the surface midway between front and back, removing thickness variation from bow and warp (Vitrek).
• Reference planes: Bow and warp use a free wafer referenced to a plane through edge points, while flatness uses least-squares or ideal planes over the wafer or a site (Vitrek; Corning).
• Lithography link: Site flatness must stay well inside the scanner’s depth of focus, $k_2\lambda /N{A}^2$ (Semiconductor History Museum of Japan; my reading).

2. Working Principle

• Capacitance gauges: Opposed capacitance probes measure the gaps to both surfaces, giving thickness from their sum and shape from their difference (Vitrek; MTI).
• Interferometric gauges: Frequency-stepping or Fizeau interferometry maps surface height and, with the refractive index, thickness of transparent wafers (Corning).
• Scanning: Full-surface or standard scan patterns are recorded and parameters computed by standard algorithms (MTI).

3. Applications

• Incoming wafer qualification: SEMI and ASTM geometry specifications for prime silicon and glass carriers (MTI; Corning).
• Lithography readiness: Site flatness for focus control (Corning).

Thickness and Shape Parameters

Name	Formula	Related Formulae	Variables/Constants
Thickness and TTV	$TTV=T_{max}-T_{min}$	$T_w=G_{total}-\left(A+B\right)$	$TTV:totalthicknessvariation$ $T:localthickness$ $T_w:waferthickness$ $\begin{array}{r}{G}_{total}:calibratedgapbetweenopposedprobes,\\ A\end{array}$ $\begin{array}{r}B:upperandlowerprobe-to-surfacegaps;\\ Vitrek:definitionsafterASTMF657,\\ dual-proberelation;MTI:thickness(ASTM\\ F533)andTTV(F657)over\pm 500\mu \mathrm{m}with\pm \\ 0.25\mu \mathrm{m}accuracy;Corning:TTVreportedas\\ GBIR,0.05\mu \mathrm{m}accuracyand0.01\mu \mathrm{m}\\ repeatability\end{array}$ $\begin{array}{r}(assumed:G=2000\mu \mathrm{m},A=600\mu \mathrm{m},B=624.5\\ \mu \mathrm{m};computed:T_w=775.5\mu \mathrm{m})\end{array}$
Bow	$\text{Bow}=z_{median}\left(0,0\right)-z_{ref}\left(0,0\right)$	$z_{median}=\frac{z_{front}+z_{back}}{2}$	$z_{median}:median-surfaceheight$ $\begin{array}{r}{z}_{ref}:three-pointreferenceplanenearthe\\ edge;Vitrek:bowascentredeviationof\\ themediansurfaceofafree,unclamped\\ wafer(ASTMF534),positiveabovethe\\ plane,localthicknesseffectsremovedby\\ themediansurface;MTI:\pm 250\mu \mathrm{m}range\\ with\pm 2.0\mu \mathrm{m}accuracy;Corning:\\ front-surfacebow,0.5\mu \mathrm{m}accuracy(bow\\ below10\mu \mathrm{m})and0.2\mu \mathrm{m}repeatability\end{array}$ $\begin{array}{r}(myreading:bowisasingle-pointvalue,\\ soasaddle-shapedwafercanhavezerobow\\ butlargewarp)\end{array}$
Warp and sori	$\text{Warp}=RP{D}_{max}-RP{D}_{min}$	$\text{Warp}\ge 0$	$\begin{array}{r}RPD:median-surfacedeviationfromthe\\ referenceplane;Vitrek:warpafterASTM\\ F1390,example1.5-(-1.5)=3,always\\ positive;MTI:warpandsori(ASTMF1451)\\ over\pm 250\mu \mathrm{m}with\pm 2.0\mu \mathrm{m}accuracy;\\ Corning:bow,warpandSORIonthefront\\ surface\end{array}$ $\begin{array}{r}(computed:Vitre{k}^{\prime }sexamplegives3\mu \mathrm{m}\\ warp)\end{array}$

Sensing and Gravity

Name	Formula	Related Formulae	Variables/Constants
Capacitance probe	$C=\frac{{\epsilon }_0{A}_p}{g}$	$\frac{dC}{dg}=-\frac{{\epsilon }_0{A}_p}{g^2}$	$C:probecapacitance$ $A_p:sensorarea$ $\begin{array}{r}g:probe-to-wafergap;VirtualLabs\\ (COEP):capacitanceinverselyproportional\\ toplatedistance,parallel-plateform;\\ derivative(computed);MTI:non-contact\\ push-pullcapacitancesensors,150,200\\ and300\mathrm{mm}wafers,user-definedor\\ ASTM/SEMIscanpatterns,1000\mu \mathrm{m}thickness\\ rangewithoutrecalibration\end{array}$ $\begin{array}{r}(assumed:5\mathrm{mm}diameterprobeat0.5\mathrm{mm}\\ gap;computed:C=0.35\mathrm{pF}and0.70\mathrm{fF}per\\ micrometreofgapchange)\end{array}$
Interferometric thickness and shape	$OPD=2n_{g}t$	$\Delta z=\frac{\lambda \Delta \varphi }{4\pi }$	$\begin{array}{r}OPD:opticalpathdifferencethroughthe\\ wafer\end{array}$ $n_g:groupindex$ $t:thickness$ $\begin{array}{r}\Delta \varphi :phasedifference;Corning:\\ frequency-steppinginterferometry,\\ thicknessrequiringrefractive-index\\ knowledge,thicknessaccuracy0.075\mu \mathrm{m}\\ (below2\mathrm{mm})and0.25\mu \mathrm{m}(above2\mathrm{mm}),\\ repeatability0.01and0.02\mu \mathrm{m},300\mathrm{mm}\\ glasswafers,under60s;relations(my\\ reading)\end{array}$ $\begin{array}{r}(computed:a1percentindexerrorgives\\ a7.75\mu \mathrm{m}erroron775\mu \mathrm{m},soindex\\ calibrationdominates)\end{array}$
Gravity sag of a free wafer	$w_{max}=\frac{\left(5+\nu \right)q{a}^4}{64\left(1+\nu \right)D},D=\frac{E{t}^3}{12\left(1-{\nu }^2\right)}$	$q=\rho gt$	$\begin{array}{r}{w}_{max}:centredeflectionofanedge-supported\\ plate\end{array}$ $q:loadperarea$ $a:radius$ $D:flexuralrigidity$ $E:Youn{g}^{\prime }smodulus$ $\nu :Poissonratio$ $t:thickness$ $\begin{array}{r}\rho :density;Structville:platerelations\\ forsimplysupportedandclampededges;\\ Ioffe:E,\nu and\rho ofSi;Vitrek:bowand\\ warpdefinedforafree,unclampedwafer\end{array}$ $\begin{array}{r}(assumed:SiE=130\mathrm{GPa},\nu =0.28,t=\\ 775\mu \mathrm{m},a=150\mathrm{mm};computed:D=5.47Nm,\\ q=17.7Pa,w_{m}ax=106\mu \mathrm{m}edge-supported,26\\ \mu \mathrm{m}clamped)\end{array}$

Flatness and Standards

Name	Formula	Related Formulae	Variables/Constants
Site flatness	$SFQR={\max }_{site}\left(z-z_{LSQ}\right)-{\min }_{site}\left(z-z_{LSQ}\right)$	$GBIR=TTV,SBIR=LTV$	$\begin{array}{r}SFQR:sitefront-surfacerangeabouta\\ least-squaresplane\end{array}$ $\begin{array}{r}GBIR:globalback-surfaceideal-plane\\ range\end{array}$ $\begin{array}{r}SBIR:localthicknessvariation;Corning:\\ GBIR(TTV),SBIR(LTV),SFQR(LTIR),SFQD,\\ GF3R,GF3D,GFLR,SF3R;Vitrek:GFLRand\\ GBIRnamed;MTI:siteandglobalflatness\\ (ASTMF1530)over8\mathrm{mm}with\pm 0.15\mu \mathrm{m}\\ accuracy;epitaxial-waferpatent:GBIRand\\ SBIRasglobalandsite\\ backsurface-referencedidealplanerange,\\ attributedtoSEMI\end{array}$ $\begin{array}{r}(myreading:thefourlettersencode\\ globalorsite,frontorbacksurface,\\ referenceplanetypeandrangeor\\ deviation)\end{array}$
Flatness against depth of focus	$DOF\approx k_2\frac{\lambda }{N{A}^2}$	$SFQR\ll DOF$	$DOF:depthoffocus$ $k_2:processfactor$ $\lambda :exposurewavelength$ $\begin{array}{r}NA:numericalaperture;Semiconductor\\ HistoryMuseumofJapan:DOF=k_2\lambda /N{A}^2;\\ CorningandMTI:siteflatnessreporting\end{array}$ $\begin{array}{r}(assumed:k_2=0.5,193\mathrm{nm},NA=1.35;\\ computed:DOF=53\mathrm{nm},sositeflatness\\ mustbeafewtensofnanometres)\end{array}$
Standards and scan patterns	$\text{ASTM F533, F534, F657, F1390, F1451, F1530}$	$\text{SEMI M1 wafer specifications}$	$\begin{array}{r}VitrekandMTI:ASTMstandardsfor\\ thickness,bow,TTV,warp,soriand\\ flatness,ASTM/SEMI-compliantscan\\ patterns,referenceplanefromthreeedge\\ points;SEMIM1parameterset(Corningand\\ theepitaxial-waferpatent)\end{array}$ $\begin{array}{r}(computed:MT{I}^{\prime }s\pm 250\mu \mathrm{m}bowrangeagainst\\ \pm 2.0\mu \mathrm{m}accuracyis0.8percentofrange)\end{array}$

Important Questions

Ques: What is the difference between bow, warp and TTV?

They describe thickness, centre deflection and overall shape. Vitrek defines TTV as maximum minus minimum thickness over the scan (ASTM F657), bow as the deviation of the centre of the median surface of a free wafer from a three-point edge reference plane (ASTM F534), positive if above, and warp as the range of median-surface deviations (ASTM F1390), always positive; its example of +1.5 and -1.5 gives 3 $\mu \mathrm{m}$ warp (computed). Because the median surface lies midway between front and back, thickness variations do not enter bow or warp. A saddle-shaped wafer can therefore have near-zero bow but large warp, so both are specified (my reading). Corning reports TTV as GBIR in SEMI notation. Reference: Vitrek (MTI Instruments), ‘About Wafer Bow and Warp Measurement Systems’ application page (fetch-tool summary); Corning, ‘Tropel FlatMaster MSP-300 Glass Wafer Analysis System’ product sheet (fetch-tool summary).

Ques: How do capacitance gauges measure thickness and shape at once?

With opposed probes on either side of the wafer. Vitrek gives $T_w=G_{total}-(A+B)$, where $G_{total}$ is the calibrated probe separation and A and B the gaps to the top and bottom surfaces: with a 2000 $\mu \mathrm{m}$ gap and gaps of 600 and 624.5 $\mu \mathrm{m}$, the wafer is 775.5 $\mu \mathrm{m}$ thick (assumed values; computed). The difference of the gaps locates the median surface, which gives bow and warp. Each probe’s capacitance follows the parallel-plate form $C={\epsilon }_0{A}_p/g$, inversely proportional to the gap (Virtual Labs, COEP): a 5 $\mathrm{mm}$ probe at 0.5 $\mathrm{mm}$ gap is 0.35 $\mathrm{pF}$ and changes by 0.70 $\mathrm{fF}$ per micrometre (computed), easily resolved. MTI’s push-pull capacitance tool covers 150 to 300 $\mathrm{mm}$ wafers with a 1000 $\mu \mathrm{m}$ range and quotes plus or minus 0.25 $\mu \mathrm{m}$ thickness accuracy. Reference: Vitrek (MTI Instruments), ‘About Wafer Bow and Warp Measurement Systems’ application page (fetch-tool summary); Virtual Labs (COEP), ‘Design of capacitive sensors and its applications’: plate area A, separation d, permittivity epsilon, capacitance inversely proportional to the distance (fetch-tool summary; equation as image); Vitrek (MTI Instruments), ‘Semi-Automated Metrology System’ (Proforma 300iSA) product page (fetch-tool summary).

Ques: Is bow accurate to 0.5 or to 2 micrometres?

Corning quotes 0.5 $\mu \mathrm{m}$ accuracy and 0.2 $\mu \mathrm{m}$ repeatability for bow, warp and sori on its interferometric MSP-300, but only for bow below 10 $\mu \mathrm{m}$, whereas MTI’s capacitance Proforma 300iSA lists plus or minus 2.0 $\mu \mathrm{m}$ over a plus or minus 250 $\mu \mathrm{m}$ range, 0.8 per cent of range (computed). Both are plausible because they cover different regimes: interferometry excels on flat, smooth and transparent wafers, while capacitance gauges handle large warp on opaque or patterned wafers (my reading). For prime silicon with small bow the 0.5 $\mu \mathrm{m}$ figure is more likely achievable; for strongly bowed processed wafers 2 $\mu \mathrm{m}$ is the realistic accuracy. The same pattern holds for TTV: 0.05 $\mu \mathrm{m}$ (Corning) against 0.25 $\mu \mathrm{m}$ (MTI). Reference: Corning, ‘Tropel FlatMaster MSP-300 Glass Wafer Analysis System’ product sheet (fetch-tool summary); Vitrek (MTI Instruments), ‘Semi-Automated Metrology System’ (Proforma 300iSA) product page (fetch-tool summary).

Ques: How large is gravity sag compared with bow specifications?

Larger than many specifications, so it must be removed. A 300 $\mathrm{mm}$, 775 $\mu \mathrm{m}$ silicon wafer has a flexural rigidity $D=E{t}^3/[12(1-{\nu }^2)]$ of 5.47 N m and a self-weight load of 17.7 Pa, so supported at its edge it sags about 106 $\mu \mathrm{m}$ at the centre, and about 26 $\mu \mathrm{m}$ if clamped (E = 130 $\mathrm{GPa}$ and $\nu$ = 0.28 from Ioffe; computed from Structville’s plate formulas). Since Vitrek defines bow and warp for a free, unclamped wafer, gauges measure in a defined support state and subtract gravity, often by measuring both sides up or by modelling. Thinner wafers sag much more, since D scales as $t^3$: halving thickness raises sag fourfold for the same load per unit thickness (computed). Reference: U. Obinna, ‘Elastic Analysis of Circular Plates’, Structville: w_max = (q a^4/64D)(5 + v)/(1 + v) simply supported, q a^4/64D clamped, D = E h^3/[12(1 - v^2)] (fetch-tool summary); Ioffe Institute, NSM archive, ‘Si - Silicon: Mechanical properties’: density 2.329 g/cm3, Young’s modulus [100] 13.0e11 dyn/cm2, Poisson ratio 0.28 (fetch-tool summary); Vitrek (MTI Instruments), ‘About Wafer Bow and Warp Measurement Systems’ application page (fetch-tool summary).

Ques: Why does site flatness matter for lithography?

Because the scanner focuses each exposure field on a local plane. Site flatness such as SFQR is the front-surface range about a least-squares plane within a site, and Corning reports SFQR, SFQD, SBIR and global parameters in SEMI notation; MTI quotes plus or minus 0.15 $\mu \mathrm{m}$ accuracy for site and global flatness. The Rayleigh depth of focus $k_2\lambda /N{A}^2$ is only about 53 $\mathrm{nm}$ for 193 $\mathrm{nm}$ immersion at NA 1.35 with $k_2$ = 0.5 (Semiconductor History Museum of Japan for the form; $k_2$ assumed; computed), so the residual height within a field must be a few tens of nanometres. Global bow is corrected by chucking and focus control, but local topography within a site cannot be, which is why site parameters, not bow, govern yield at advanced nodes (my reading). Reference: Corning, ‘Tropel FlatMaster MSP-300 Glass Wafer Analysis System’ product sheet (fetch-tool summary); Vitrek (MTI Instruments), ‘Semi-Automated Metrology System’ (Proforma 300iSA) product page (fetch-tool summary); Semiconductor History Museum of Japan, lithography exhibit page (version 2019/1/13): R = k1 lambda/NA, DOF = k2 lambda/NA^2 (fetch-tool summary); US patent family, ‘Epitaxially coated silicon wafer and method for producing epitaxially coated silicon wafers’ (priority DE 10 2009 004 557.0, 2009): GBIR global backsurface-referenced ideal plane/range, SBIR site backsurface-referenced ideal plane/range, attributed to SEMI (fetch-tool summary).

References

1. Vitrek (MTI Instruments), “About Wafer Bow and Warp Measurement Systems.” Available: https://vitrek.com/applications/wafer-bow-and-warp/
2. Corning, “Tropel FlatMaster MSP-300 Glass Wafer Analysis System,” product information sheet. Available: https://www.corning.com/media/worldwide/csm/documents/FlatMaster_MSP_300_Glass_Wafer_PI_Sheet.pdf
3. U. Obinna, “Elastic Analysis of Circular Plates,” Structville, 2020. Available: https://structville.com/2020/11/elastic-analysis-of-circular-plates.html
4. Ioffe Physico-Technical Institute, “Si - Silicon: Mechanical properties,” New Semiconductor Materials. Available: https://www.ioffe.ru/SVA/NSM/Semicond/Si/mechanic.html
5. Semiconductor History Museum of Japan, lithography exhibit page (version 2019/1/13). Available: https://www.shmj.or.jp/english/pdf/process/exhibi414E.pdf
6. Virtual Labs, COEP, “Design of capacitive sensors and its applications,” theory page. Available: https://sil-coep.vlabs.ac.in/exp/capacitive-sensors-applications/theory.html
7. “Epitaxially coated silicon wafer and method for producing epitaxially coated silicon wafers,” US patent family, priority DE 10 2009 004 557.0, 2009. Available: https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/8268708
8. Vitrek (MTI Instruments), “Semi-Automated Metrology System” (Proforma 300iSA). Available: https://vitrek.com/mti-instruments/metrology-systems/semi-automated-metrology-system/

Related pages in this vault: Laser-Scattering Particle Inspection, Overlay Metrology Tool, White-Light Interferometer, Thermal Stress, Patterned-Wafer Optical Inspection, Optical CD Scatterometer (OCD), Mechanical and Package Stress, Stylus Profiler, Wafer Curvature Stress Gauge, CMP Polisher, Wafer Backgrinding and Thinning Tool.