Wafer Curvature Stress Gauge

1. Theory

• Definition: A wafer curvature stress gauge measures the curvature of a wafer before and after film deposition, usually by reflecting a laser from the surface, and converts the change into film stress with the Stoney equation (TI patent; scanning patent).
• Optical lever: A change in surface slope deflects the reflected beam by twice the angle, which a position detector at a long path length converts to a large displacement (TI patent; reflection geometry, computed).
• Stress-thickness: The measurement gives the product of film stress and thickness, independent of film elastic constants, since none appear in the Stoney form (Injeti et al.; computed).
• In situ: Multi-beam sensors follow curvature during growth, separating incremental stress from relaxation (Gilardi et al.).

2. Working Principle

• Point and scan: Two-point deflection with a quadrant detector, or scanned slope with split photodiodes across rotated diameters (TI patent; scanning patent).
• Multi-beam: An array of parallel beams on a camera, with curvature from spot spacing, insensitive to vibration (Gilardi et al.; vibration insensitivity my reading).
• Subtraction: Pre-deposition curvature is removed by subtracting the blank scan (scanning patent).
• Heating: Stress-temperature cycling on a hot stage gives thermal expansion mismatch and plastic deformation (Bigl et al.).

3. Applications

• Dielectric and metal films: Nitride, oxide and metal stress control after deposition (TI patent; scanning patent).
• Epitaxy and oxide growth: Real-time strain relaxation during film growth (Gilardi et al.).
• Metal fatigue: Thermo-mechanical cycling of Cu metallizations (Bigl et al.).

Optical Curvature Measurement

Name	Formula	Related Formulae	Variables/Constants
Two-point beam deflection	$\frac{1}{R}=\frac{D}{2LZ}$	$\Delta {\theta }_{beam}=2\Delta {\theta }_{surface}$	$R:radiusofcurvature$ $\begin{array}{r}D:changeindetectorpositionbetweentwo\\ points\end{array}$ $L:opticalpathfromwafertodetector$ $\begin{array}{r}Z:stagedisplacementbetweenthepoints;\\ TIpatent:0.4mWHeNelaser,quadrant\\ detectornulledwithmicrometersattwo\\ pointsabout60mmapart,pathLviaa\\ flatreflector,concavemirrorofknown\\ radiusforverification;relationderived\\ fromreflectiongeometry(computed)\end{array}$ $\begin{array}{r}(assumed:L=0.5m;computed:a92m\\ radiusgivesD=0.65mmoverZ=60mm)\end{array}$
Scanned slope and curvature	$\theta \left(x\right)=\frac{dz}{dx},\kappa =\frac{d\theta }{dx}$	$Q=\frac{A-B}{A+B}$	$\theta :localsurfaceslope$ $\kappa :curvature$ $Q:normalizedsplit-photodiodesignal,A$ $\begin{array}{r}B:photodiodesignals;scanningpatent:\\ 780\mathrm{nm}diodelaserbelow2mWwitha3mm\\ spot,about300mmreflectedpath,steps\\ ofabout0.025mm,rotation360/N_{s}degrees\\ with1<N_s<100,detectablechangeabout\\ 0.005\mu \mathrm{m}orroughly1\times 10^{-8}radwitha20\mu \mathrm{m}\\ detectorspot,blankscansubtractedfrom\\ post-depositionscan\end{array}$ $\begin{array}{r}(computed:0.005\mu \mathrm{m}over300mmis1.7\times 10^{-8}\\ rad;witha100mmscan,curvature\\ resolution8.3\times 10^{-8}{\mathrm{m}}^{-1},a12,000kmradius)\end{array}$
Multi-beam spot spacing	$\frac{1}{\rho }=-\frac{\cos \alpha }{2L}\frac{\delta d}{D_0}$	$\text{tensile strain: positive curvature}$	$\rho :radiusofcurvature$ $\alpha :incidenceangle$ $L:opticalpath$ $D_0:initialmeandifferentialspotspacing$ $\begin{array}{r}\delta d:itschange;Gilardietal.:3by3\\ beamarrayonaCCD,30degreeincidence,\\ about1mmspotspacing,compressive\\ straingivingnegativeandtensile\\ positivecurvature,equationaswritten\end{array}$ $\begin{array}{r}(assumed:L=0.5m,\delta d=0.1\mu \mathrm{m};\\ computed:8.7\times 10^{-5}{\mathrm{m}}^{-1})\end{array}$

Stress from Curvature

Name	Formula	Related Formulae	Variables/Constants
Stoney stress-thickness product	${\sigma }_f{t}_f=\frac{M_s{t}_s^2}{6}\left({\kappa }_{post}-{\kappa }_{pre}\right)$	$M_s=\frac{E_s}{1-{\nu }_s}$	${\sigma }_f:filmstress$ $t_f,t_s:filmandsubstratethickness$ $M_s:substratebiaxialmodulus$ $\begin{array}{r}\kappa :curvaturebeforeandafterdeposition;\\ Gilardietal.:Stoneyequationin\\ curvatureformwithsubstratePoisson\\ ratio,Youn{g}^{\prime }smodulusandthickness;\\ scanningpatent:blankanddepositeddata\\ subtracted;Stoneyform(Injetietal.;\\ Bigletal.)\end{array}$ $\begin{array}{r}(assumed:t_s=725\mu \mathrm{m},t_f=100\mathrm{nm},M_s=\\ 180.6\mathrm{GPa}computedas{C}_{11}+C_{12}-2C_{12}^2/C_{11}from\\ Ioffe,thecubicformmyreading;\\ computed:theoreticalresolution0.0013\\ N/mor0.013\mathrm{MPa}at8.3\times 10^{-8}{\mathrm{m}}^{-1})\end{array}$
Two-point stress formula	${\sigma }_f=\frac{E{T}^{2}D}{12\left(1-\nu \right)LZF}$	$K=\frac{M_s}{6}\text{(stress constant)}$	$T:waferthickness$ $F:filmthickness$ $\begin{array}{r}E,\nu :substrateconstants;TIpatent:E=\\ 1.66\times 10^{12}dyn/c{\mathrm{m}}^2,\nu =0.217,Z=60mm,F\\ about15,000\mathring{\mathrm{A}}ofcompressivenitride,\\ passbetween1.0\times 10^{-9}and4.0\times 10^{-9}(units\\ writtenasdynes);scanningpatent:\\ defaultstressconstant3.02\times 10^{18}dyn/c{\mathrm{m}}^2\\ timesangstrompermmforSi(100)\end{array}$ $\begin{array}{r}(computed:TIconstantsgive{M}_s=212\mathrm{GPa};\\ thescanningconstantimplies181\mathrm{GPa};\\ ratio1.17)\end{array}$
In situ incremental stress	$\frac{d\left(\sigma t\right)}{d{t}_f}={\sigma }_{inc}$	$\sigma t=\frac{Y{\tau }_s^2}{6\left(1-\nu \right)}\frac{1}{\rho }$	${\sigma }_{inc}:incrementalstressofthelayerbeingadded$ $\begin{array}{r}{\tau }_s,Y,\nu :substratethickness,modulusand\\ Poissonratio;Gilardietal.:10by10by\\ 0.5mmsubstratesofAl2O3,LaAlO3and\\ SrTiO3at750{}^{\circ }C,detectablechangeof\\ -2.5\mathrm{GPa}\mathrm{nm}at1nm,changesat8to10\mathring{\mathrm{A}}\\ forabout0.4percentstrainand350\mathrm{GPa},\\ curvatureslopeconstantuntilstrain\\ release;sloperelation(computedfromthe\\ Stoneyform)\end{array}$ $\begin{array}{r}(assumed:Y=345\mathrm{GPa},\nu =0.29,{\tau }_s=0.5\\ mm;computed:2.5N/mneeds1.2\times 10^{-4}{\mathrm{m}}^{-1},and\\ 8.7\times 10^{-5}{\mathrm{m}}^{-1}gives1.8N/m)\end{array}$

Thermal Stress and Calibration

Name	Formula	Related Formulae	Variables/Constants
Thermal mismatch stress	${\sigma }_{th}=M_f\left({\alpha }_s-{\alpha }_f\right)\Delta T$	$\Delta T=T-T_{dep}$	$M_f:filmbiaxialmodulus$ $\begin{array}{r}{\alpha }_s,{\alpha }_f:substrateandfilmexpansion\\ coefficients\end{array}$ $\begin{array}{r}{T}_{dep}:depositiontemperature;Bigletal.:\\ mismatchstrainfromtheCTEdifference\\ andtemperaturechange,multipliedhereby\\ thefilmbiaxialmodulus(computed)\end{array}$ $\begin{array}{r}(assumed:Al70\mathrm{GPa},0.35,23.1ppm/K;\\ SiO272\mathrm{GPa},0.17,0.5ppm/K;Si2.6\\ ppm/K;computed:+883\mathrm{MPa}forAlcooled\\ 400Kand-164\mathrm{MPa}forSiO2cooled900K)\end{array}$
Stress-temperature cycling	$\frac{d{\sigma }_f}{dT}=-M_f\left({\alpha }_f-{\alpha }_s\right)\text{(elastic)}$	$\left|{\sigma }_f\right|\le {\sigma }_y\left(T\right)\text{(plastic limit)}$	$\begin{array}{r}{\sigma }_y:temperature-dependentyieldstress;\\ relationsandhysteresisinterpretation\\ (myreading);Bigletal.:plastic\\ deformationinCumetallizationsunder\\ thermo-mechanicalloading\end{array}$ $\begin{array}{r}(computed:elasticslope-2.2\mathrm{MPa}perK\\ forAlonSi;myreading:aclosedloop\\ withaflattopmarksyielding)\end{array}$
Bow and calibration	$\delta =\frac{L_w^2}{8R}$	$R_{ref}\text{from a concave mirror of known radius}$	$\begin{array}{r}\delta :bowoverwaferdiameter{L}_w;TIpatent:\\ concavemirrorwithknownradiusfor\\ verificationwithoutastressedreference\\ wafer;sagittarelation(computed);Stylus\\ Profilerpage:profiler-basedcurvature\\ forcomparison\end{array}$ $\begin{array}{r}(computed:a92mradiusgives30.6\mu \mathrm{m}bow\\ over150mm)\end{array}$

Important Questions

Ques: How does a laser curvature gauge measure film stress?

By measuring how the wafer bends when a stressed film is added. A curved wafer tilts the reflected beam by twice the change in surface slope, so a detector a distance $L$ away sees a displacement $D=2LZ/R$ over a stage move $Z$ (computed from the reflection geometry). The TI patent nulls a quadrant detector at two points about 60 mm apart, using a 0.4 mW HeNe laser and a flat reflector to lengthen the path, and verifies with a concave mirror of known radius. With an assumed 0.5 m path, a 92 m radius gives 0.65 $\mathrm{mm}$ of detector travel (computed). The Stoney equation then gives stress-thickness from substrate modulus and thickness alone, independent of the film’s own elastic constants. Reference: M. K. Forgey and S. Garza, ‘Laser stress measurement apparatus and method’, US patent 4,991,964, Texas Instruments (filed August 1988) (fetch-tool summary).

Ques: What are the units of the TI patent’s pass and fail limits?

The patent writes them as $1.0\times 10^{-9}$ to $4.0\times 10^{-9}$ dynes, which cannot be a stress. Stress in its formula has units of $dyn/c{\mathrm{m}}^2$, and a positive exponent makes physical sense: $1.0\times 10^9$ to $4.0\times 10^9$ $dyn/c{\mathrm{m}}^2$ is 100 to 400 $\mathrm{MPa}$ (computed), a plausible film-stress range (my reading). The positive-exponent reading is therefore far more likely, and the negative exponent is probably a transcription error (my reading). For 100 $\mathrm{MPa}$ in 1.5 $\mu \mathrm{m}$ of nitride on an assumed 625 $\mu \mathrm{m}$ wafer with the patent’s constants, the radius is 92 m, giving 30.6 $\mu \mathrm{m}$ of bow over 150 mm (computed), well within the resolution of the method. Reference: M. K. Forgey and S. Garza, ‘Laser stress measurement apparatus and method’, US patent 4,991,964, Texas Instruments (filed August 1988) (fetch-tool summary).

Ques: Which silicon modulus should the Stoney equation use: 181 or 212 GPa?

The two patents imply different values. The TI patent uses E = $1.66\times 10^{12}$ $dyn/c{\mathrm{m}}^2$ and a Poisson ratio of 0.217, giving a biaxial modulus of 212 $\mathrm{GPa}$ (computed). The scanning patent’s default constant of $3.02\times 10^{18}$ $dyn/c{\mathrm{m}}^2$ times angstrom per mm for Si(100) equals the biaxial modulus divided by six with thicknesses in mm and angstrom, implying 181 $\mathrm{GPa}$ (computed). Ioffe’s elastic constants give a (100) biaxial modulus of 180.6 $\mathrm{GPa}$ (computed), so 181 $\mathrm{GPa}$ is more likely correct for (100) wafers. Using 212 $\mathrm{GPa}$ overestimates stress by 17 per cent (computed). Isotropic E and Poisson values do not describe the anisotropic (100) plane well (my reading). Reference: M. K. Forgey and S. Garza, ‘Laser stress measurement apparatus and method’, US patent 4,991,964, Texas Instruments (filed August 1988) (fetch-tool summary); ‘Method and apparatus for measuring stress in a film applied to surface of a workpiece’, US patent 5,369,286 (continuation of 08/144,141, filed October 1993; parents include US 5,233,201 and US 5,118,955) (fetch-tool summary; assignee not in the text); Ioffe Institute, NSM archive, ‘Si - Silicon: Mechanical properties’: C11 = 16.60e11 and C12 = 6.40e11 dyn/cm2 (fetch-tool summary).

Ques: How sensitive is a scanning curvature gauge?

Very sensitive, because the optical lever is long. The scanning patent detects about 0.005 $\mu \mathrm{m}$ of beam displacement, roughly $1\times 10^{-8}$ rad, with a 20 $\mu \mathrm{m}$ detector spot and a 300 mm path; 0.005 $\mu \mathrm{m}$ over 300 mm is $1.7\times 10^{-8}$ rad (computed). Over an assumed 100 mm scan, this corresponds to a curvature of $8.3\times 10^{-8}$ ${\mathrm{m}}^{-1}$, a 12,000 km radius (computed). For an assumed 725 $\mu \mathrm{m}$ wafer and 100 $\mathrm{nm}$ film, the theoretical stress resolution is 0.013 $\mathrm{MPa}$ (computed). In practice, repeatability is set by wafer placement, initial bow and thermal drift rather than detector noise, which is why the patent subtracts the blank scan and averages rotated diameters (my reading). Reference: ‘Method and apparatus for measuring stress in a film applied to surface of a workpiece’, US patent 5,369,286 (continuation of 08/144,141, filed October 1993; parents include US 5,233,201 and US 5,118,955) (fetch-tool summary; assignee not in the text); S. S. Injeti, N. Vyas, S. Natarajan and R. K. Annabattula, ‘Modified Stoney’s equation with anisotropic substrates undergoing large deformations’, arXiv:1712.07974, 2017: sigma_f = E_s h_s^2 kappa/[6(1 - nu_s) h_f] (fetch-tool summary).

Ques: How does a multi-beam sensor follow stress during growth?

By watching the spacing of an array of reflected spots change. Gilardi et al. reflect a 3 by 3 array of parallel beams, about 1 mm apart, from the substrate at 30 degrees onto a camera, and curvature is $1/\rho =-(\cos \alpha /2L)(\delta d/D_0)$. Parallel beams move together under vibration, so only the spacing carries curvature (my reading). For 0.5 mm substrates they detect -2.5 $\mathrm{GPa}\mathrm{nm}$ at 1 nm, changes at 8 to 10 $\mathring{\mathrm{A}}$. With an assumed 0.5 m path and 0.1 $\mu \mathrm{m}$ spacing change, curvature is $8.7\times 10^{-5}$ ${\mathrm{m}}^{-1}$, giving 1.8 $N/m$ on Al2O3 (assumed; computed). A constant slope of stress-thickness against thickness means coherent strain; a falling slope marks relaxation, as their films on LaAlO3 showed near 25 nm. Reference: E. Gilardi, A. Fluri, T. Lippert and D. Pergolesi, ‘Real-time monitoring of stress evolution during thin film growth by in situ substrate curvature measurement’, J. Appl. Phys. 125, 082513 (2019), arXiv:1905.05060 (fetch-tool summary).

Ques: What does a stress-temperature cycle reveal about a metal film?

Thermal mismatch and plasticity. On heating or cooling, the film stress changes as $M_f({\alpha }_s-{\alpha }_f)\Delta T$, the biaxial modulus times Bigl et al.’s mismatch strain (computed). For Al on Si with assumed constants, the elastic slope is -2.2 $\mathrm{MPa}$ per K, so cooling 400 K would add 883 $\mathrm{MPa}$ of tension (computed), far above the yield stress of thin Al, so the film yields and the loop shows flat plastic segments and hysteresis (my reading). A thermal oxide cooled 900 K becomes 164 $\mathrm{MPa}$ compressive because silica expands less than silicon (computed). The scanning patent’s blank-subtraction method supports such curvature measurements at each temperature. Bigl et al. use such thermo-mechanical cycling to study fatigue of Cu metallizations, assuming a Cu-Si mismatch of 14 ppm/K. Reference: S. Bigl, S. Wurster, M. J. Cordill and D. Kiener, ‘Substrate-Influenced Thermo-Mechanical Fatigue of Copper Metallizations: Limits of Stoney’s Equation’, Materials 10(11), 1287, 2017: Stoney equation, thermal mismatch strain from the CTE difference and temperature change, Cu-Si mismatch 14 ppm/K, twin-boundary migration as plastic deformation (fetch-tool summary); ‘Method and apparatus for measuring stress in a film applied to surface of a workpiece’, US patent 5,369,286 (continuation of 08/144,141, filed October 1993; parents include US 5,233,201 and US 5,118,955) (fetch-tool summary; assignee not in the text).

References

1. M. K. Forgey and S. Garza, “Laser stress measurement apparatus and method,” US patent 4,991,964 A, Texas Instruments, filed August 26, 1988. Available: https://patents.google.com/patent/US4991964
2. S. S. Injeti, N. Vyas, S. Natarajan and R. K. Annabattula, “Modified Stoney’s equation with anisotropic substrates undergoing large deformations,” arXiv:1712.07974, 2017. Available: https://arxiv.org/abs/1712.07974
3. S. Bigl, S. Wurster, M. J. Cordill and D. Kiener, “Substrate-influenced thermo-mechanical fatigue of copper metallizations: limits of Stoney’s equation,” Materials, vol. 10, no. 11, 1287, 2017. Available: https://www.mdpi.com/1996-1944/10/11/1287
4. Ioffe Physico-Technical Institute, “Si - Silicon: Mechanical properties,” New Semiconductor Materials. Available: https://www.ioffe.ru/SVA/NSM/Semicond/Si/mechanic.html
5. “Method and apparatus for measuring stress in a film applied to surface of a workpiece,” US patent 5,369,286 (continuation of US application 08/144,141, filed October 27, 1993). Available: https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/5369286
6. E. Gilardi, A. Fluri, T. Lippert and D. Pergolesi, “Real-time monitoring of stress evolution during thin film growth by in situ substrate curvature measurement,” J. Appl. Phys. 125, 082513 (2019). Available: https://arxiv.org/pdf/1905.05060

Related pages in this vault: Stylus Profiler, Wafer Geometry Gauge, Thermal Stress, Mechanical and Package Stress, Metal and Interconnect, X-Ray Diffractometer (XRD and HRXRD), Raman Spectrometer, Nanoindenter, White-Light Interferometer, Pyrometers and Wafer Temperature Sensors, Wafer Backgrinding and Thinning Tool, LPCVD Furnace, PECVD Reactor.