Stylus Profiler

1. Theory

• Definition: A stylus profiler drags a diamond-tipped stylus across a surface at a set contact force and records its vertical deflection, giving step heights, roughness, waviness and wafer curvature along a line (UCSB page; Fraunhofer page).
• Transduction: The stylus arm pivots on a flexure, and a capacitive or inductive sensor converts deflection to height while an electromagnetic actuator holds the force constant (constant-force patent).
• Tip geometry: Tip radius limits lateral resolution and the depth reachable in narrow features (UCSB page; computed in the trench row).
• Stress: Curvature change from film deposition gives film stress through the Stoney equation (Bruker note; Injeti et al.).

2. Working Principle

• Stylus: Diamond tip of 2 or 12.5 $\mu \mathrm{m}$ radius at 0.03 to 15 $\mathrm{mg}$ force (UCSB page; Fraunhofer page).
• Scan: The stage moves the sample under the stylus over 55 $\mu \mathrm{m}$ to 55 mm, or 200 mm with stitching, in 3 to 200 s (Fraunhofer page).
• Sensing: Capacitive vane sensor with temperature drift compensation in the constant-force design (constant-force patent).
• Analysis: Levelling, step averaging, roughness parameters and polynomial curvature fits (Bruker note; Fraunhofer page).

3. Applications

• Film thickness by step: Etched or lifted-off film edges (UCSB page; Fraunhofer page).
• Roughness and waviness: Ra, Rq and related parameters (Fraunhofer page).
• Film stress and bow: Pre- and post-deposition scans for Stoney stress and radial maps (Bruker note).

Stylus Mechanics and Resolution

Name	Formula	Related Formulae	Variables/Constants
Stylus force	$F=m_{set}g$	$F=kx\text{(flexure spring)}$	$F:contactforce$ $m_{set}:forcesettinginmg$ $g:gravitationalacceleration$ $k:pivotstiffness$ $\begin{array}{r}x:deflection;UCSBpage:force0.03to15\\ \mathrm{mg};Fraunhoferpage:1to15\mathrm{mg};Bruker\\ note:atmost15\mathrm{mg}forthe12.5\mu \mathrm{m}stylus\\ onabarewafer,lowerforceforsmaller\\ tipsandsoftfilms;constant-force\\ patent:electromagneticcoilandmagnet\\ setaconstantforce,F=kxcited,\\ capacitivevanesensorwithplatesabout\\ 0.7mmapartandplusorminus0.16mm\\ vanetravel\end{array}$ $\begin{array}{r}(computed:15\mathrm{mg}=147\mu \mathrm{N},1\mathrm{mg}=9.8\mu \mathrm{N},\\ 0.03\mathrm{mg}=0.29\mu \mathrm{N})\end{array}$
Hertzian contact of the tip	$a={\left(\frac{3FR}{4E^{\ast }}\right)}^{1/3},p_0=\frac{3F}{2\pi a^2}$	$\frac{1}{E^{\ast }}=\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}$	$a:contactradius$ $R:tipradius$ $E^{\ast }:reducedmodulus$ $p_0:peakcontactpressure$ $E_1,{\nu }_1:diamond$ $\begin{array}{r}{E}_2,{\nu }_2:sample;VirtualLabs(IITKanpur):\\ theseHertzrelations;UCSBpage:standard\\ tipradius12.5\mu \mathrm{m},optional2\mu \mathrm{m};\\ constant-forcepatent:diamondtipradius\\ 0.01mm\end{array}$ $\begin{array}{r}(assumed:diamond1141\mathrm{GPa}and0.07,\\ resist4\mathrm{GPa}and0.35;Si130\mathrm{GPa}and0.28\\ fromIoffe;computed:12.5\mu \mathrm{m}tipat15\mathrm{mg}\\ givesa=222\mathrm{nm}and1.4\mathrm{GPa}onSi,672\mathrm{nm}\\ and155\mathrm{MPa}onresist)\end{array}$
Vertical digitization	$\Delta z=\frac{Z_{range}}{2^{N_{bit}}}$	$\text{resolution depends on the selected range}$	$\Delta z:heightstep$ $Z_{range}:selectedverticalrange$ $\begin{array}{r}{N}_{bit}:converterbits;UCSBpage:1\mathring{\mathrm{A}}onthe\\ 6.55\mu \mathrm{m}range,0.1\mu \mathrm{m}inthespecification\\ list,rangeto1mm,4\mathring{\mathrm{A}}repeatabilityon\\ a0.1\mu \mathrm{m}step;Fraunhoferpage:betterthan\\ 1\mathring{\mathrm{A}}over6.55\mu \mathrm{m},4\mathring{\mathrm{A}}(1sigma)atstepsup\\ to1\mu \mathrm{m}\end{array}$ $\begin{array}{r}(assumed:N_{bit}=16;computed:1.0\mathring{\mathrm{A}}on6.55\\ \mu \mathrm{m}and15\mathrm{nm}on1mm)\end{array}$

Profile Analysis

Name	Formula	Related Formulae	Variables/Constants
Step height	$h={\bar{z}}_{top}-{\bar{z}}_{bottom}$	$\text{levelled by fitting the reference plane first}$	$h:stepheight$ $\begin{array}{r}\bar{z}:meanheightoveraregiononeachside\\ ofthestep;averagingrelation(my\\ reading);UCSBandFraunhoferpages:step\\ heightandlayerthicknessasmain\\ applications\end{array}$ $\begin{array}{r}(computed:4\mathring{\mathrm{A}}ona0.1\mu \mathrm{m}stepis0.4per\\ cent)\end{array}$
Tip convolution in trenches	$h_{max}=R-\sqrt{R^2-{\left(w/2\right)}^2}\left(w<2R\right)$	$\text{measured profile = dilation of surface by tip}$	$\begin{array}{r}{h}_{max}:deepestpointasphericaltipreaches\\ inatrenchofwidthw\end{array}$ $\begin{array}{r}R:tipradius;geometry(computedfora\\ sphericaltip);UCSBpage:lateral\\ resolutionlimitedbytipshape,deep\\ high-aspect-ratiotrenchesshowmostlythe\\ tipshape\end{array}$ $\begin{array}{r}(computed:ina2\mu \mathrm{m}trench,40\mathrm{nm}fora\\ 12.5\mu \mathrm{m}tipand268\mathrm{nm}fora2\mu \mathrm{m}tip)\end{array}$
Roughness parameters	$R_a=\frac{1}{L}{\int }_0^L\left|z\right|dx,R_q=\sqrt{\frac{1}{L}{\int }_0^L{z}^{2}dx}$	$R_q/R_a=\pi /\left(2\sqrt{2}\right)\text{(sinusoid)}$	$z:heightfromthemeanline$ $\begin{array}{r}L:evaluationlength;Fraunhoferpage:Ra,\\ Rq,Rz,Rt,RvandRp;Keyence:Raasthe\\ averageabsoluteheightandRqastheroot\\ meansquarealongthesamplinglength,the\\ integralformsandsinusoidratio\\ (computed);scanlength55\mu \mathrm{m}to55mm,200\\ mmwithstitching,scantime3to200s\end{array}$ $\begin{array}{r}(computed:R_q/R_a=1.11forasinusoid;55\\ mmin200sis0.275mm/s)\end{array}$

Bow and Film Stress

Name	Formula	Related Formulae	Variables/Constants
Bow from curvature	$\delta =\frac{L^2}{8R_c}$	$\kappa =\frac{y^{\prime \prime }}{{\left(1+y^{\prime 2}\right)}^{3/2}}\approx y^{\prime \prime }$	$\delta :sagittaoverscanlengthL$ $R_c:radiusofcurvature$ $\kappa :curvature$ $\begin{array}{r}y:fittedprofile;Brukernote:eachscan\\ fittedwitha5th-orderpolynomialby\\ leastsquaresanddifferentiatedto{y}^{\prime }and\\ y^{\prime \prime },Zrange1mmusuallysufficient\\ unlessthewaferisheavilybowed;\\ relations(computedfromthecircle\\ geometryandthecurvaturedefinition)\end{array}$ $\begin{array}{r}(assumed:L=200mm,R_c=50m;computed:\\ 100\mu \mathrm{m}bow;1mmbowat{R}_c=5m)\end{array}$
Stoney film stress	${\sigma }_f=\frac{E_s}{1-{\nu }_s}\frac{t_s^2}{6t_f}\left(\frac{1}{R_{post}}-\frac{1}{R_{pre}}\right)$	${\sigma }_f<0\text{compressive (convex)},{\sigma }_f>0\text{tensile (concave)}$	$E_s/(1-{\nu }_s):substratebiaxialmodulus$ $t_s:substratethickness$ $t_f:filmthickness$ $\begin{array}{r}{R}_{pre},R_{post}:radiusbeforeandafter\\ deposition;Brukernote:variablesand\\ signconventionasstated,equationshown\\ asanimage;Injetietal.:Stoneyform\\ {\sigma }_f=E_s{h}_s^2\kappa /[6(1-{\nu }_s)h_f],writtenherewith\\ thecurvaturechange\end{array}$ $\begin{array}{r}(biaxialmodulusofSi(100)180.6\mathrm{GPa}\\ computedas{C}_{11}+C_{12}-2C_{12}^2/C_{11}fromIoff{e}^{\prime }s\\ constants,thecubicformmyreading;t_s=\\ 725\mu \mathrm{m},t_f=1\mu \mathrm{m},flatto50m;computed:\\ 317\mathrm{MPa})\end{array}$
Two-dimensional stress map	${\sigma }_f\left(\theta \right)\text{from diameter traces at angle}\theta$	$N_{traces}\le 180,\Delta \theta \ge 1^{\circ }$	$\begin{array}{r}\theta :scanangle;Brukernote:radialmap\\ withrotationbetweenscans,upto180\\ diametertraces,minimumthetastep1\\ degree,360degreeextenttypical,\\ re-leveluntiltworawscansagreewithin\\ plusorminus20\mu \mathrm{m}inslope,measure\\ lengthequaltothewaferdiameterwith\\ automaticstitching\end{array}$ $\begin{array}{r}(myreading:rotatingtracesturnaline\\ measurementintoanareamap,revealing\\ anisotropicstressthatonediameter\\ misses)\end{array}$

Important Questions

Ques: How does a stylus profiler measure a step height?

By recording stylus deflection as the sample moves underneath it. In the constant-force patent the stylus arm pivots on a flexure, a capacitive vane between two plates about 0.7 mm apart senses deflection over plus or minus 0.16 mm, and an electromagnetic coil and magnet hold the force constant under DSP feedback. The profile is levelled on a reference region and the step taken as the difference of mean heights on each side (my reading). The UCSB page gives 4 $\mathring{\mathrm{A}}$ repeatability on a 0.1 $\mu \mathrm{m}$ step, 0.4 per cent (computed), and the Fraunhofer page 4 $\mathring{\mathrm{A}}$ (1 sigma) for steps up to 1 $\mu \mathrm{m}$. Scans run 55 $\mu \mathrm{m}$ to 55 mm in 3 to 200 s. The same profiles give roughness parameters such as Ra, the average absolute height, and Rq, the root mean square height (Fraunhofer page; Keyence). Reference: ‘Constant-force profilometer with stylus-stabilizing sensor assembly, dual-view optics, and temperature drift compensation’, US patent 5,705,741 (fetch-tool summary; number from the URL, assignee and date not in the text); UCSB Nanofab wiki, ‘Step Profilometer (DektakXT)’ (fetch-tool summary); Fraunhofer ENAS, ‘Tactile Profilometry’ (DEKTAK 150 and DEKTAK XT specifications) (fetch-tool summary); Keyence, surface roughness parameter pages: Ra as the average of the absolute value and Rq as the root mean square along the sampling length, after ISO 4287 (fetch-tool summary).

Ques: Can the stylus damage the film it measures?

It can on soft films, which is why force is reduced for them. A 15 $\mathrm{mg}$ setting is 147 $\mu \mathrm{N}$ (computed). Hertz contact of an assumed 12.5 $\mu \mathrm{m}$ diamond tip gives a 222 $\mathrm{nm}$ contact radius and 1.4 $\mathrm{GPa}$ peak pressure on silicon, well below silicon’s hardness, but 155 $\mathrm{MPa}$ on a 4 $\mathrm{GPa}$ resist, which may exceed what soft polymers tolerate (assumed values; Virtual Labs Hertz relations; computed; judgement my reading). A 2 $\mu \mathrm{m}$ tip at the same force raises the resist pressure to 527 $\mathrm{MPa}$ (computed). The Bruker note recommends at most 15 $\mathrm{mg}$ for the 12.5 $\mu \mathrm{m}$ stylus on a bare wafer and lower force for smaller tips and soft films, and the UCSB page notes adjustable force for hard and soft surfaces. Reference: ‘Constant-force profilometer with stylus-stabilizing sensor assembly, dual-view optics, and temperature drift compensation’, US patent 5,705,741 (fetch-tool summary; number from the URL, assignee and date not in the text); Virtual Labs (IIT Kanpur), ‘Estimation of Hertzian Contact Pressure during Nanoindentation’: a = (3PR/4E*)^(1/3), p0 = 3P/(2 pi a^2), 1/E* = (1 - nu_s^2)/E_s + (1 - nu_i^2)/E_i (fetch-tool summary); Ioffe Institute, NSM archive, ‘Si - Silicon: Mechanical properties’: C11 = 16.60e11 and C12 = 6.40e11 dyn/cm2, Young’s modulus [100] 13.0e11 dyn/cm2, Poisson ratio 0.28 (fetch-tool summary); Bruker, ‘TN-504: Thin Film Stress Measurement Using Dektak Stylus Profilers’ (fetch-tool summary; Stoney equation shown as an image); UCSB Nanofab wiki, ‘Step Profilometer (DektakXT)’ (fetch-tool summary).

Ques: Why can’t a stylus measure deep narrow trenches?

Because a spherical tip cannot fit into a trench narrower than its own width. The deepest point a tip of radius $R$ reaches in a trench of width $w$ is $R-\sqrt{R^2-(w/2{)}^2}$ (computed from the tip geometry). In a 2 $\mu \mathrm{m}$ trench a 12.5 $\mu \mathrm{m}$ tip reaches only 40 $\mathrm{nm}$ and a 2 $\mu \mathrm{m}$ tip 268 $\mathrm{nm}$ (computed), and the recorded profile is the surface dilated by the tip shape. The UCSB page notes that lateral resolution is limited by the tip shape and that deep, high-aspect-ratio trenches mostly show the tip. Sidewall angle of the conical shank adds a further limit (my reading). For narrow etch features, AFM, cross-section SEM or scatterometry are the usual alternatives (my reading). Reference: UCSB Nanofab wiki, ‘Step Profilometer (DektakXT)’ (fetch-tool summary).

Ques: Is the vertical resolution 1 angstrom or 0.1 micrometre?

The UCSB page states both: 1 $\mathring{\mathrm{A}}$ on the 6.55 $\mu \mathrm{m}$ range in its overview and 0.1 $\mu \mathrm{m}$ in its specification list. The Fraunhofer page gives better than 1 $\mathring{\mathrm{A}}$ over a 6.55 $\mu \mathrm{m}$ scan. Resolution depends on the selected range: if the signal is digitized with an assumed 16 bits, 6.55 $\mu \mathrm{m}$ divided by 65,536 gives 1.0 $\mathring{\mathrm{A}}$, while the full 1 mm range gives 15 $\mathrm{nm}$ (computed). The 1 $\mathring{\mathrm{A}}$ figure is therefore more likely correct for the small range, and the 0.1 $\mu \mathrm{m}$ entry probably refers to the large range or is a rounded specification (my reading). The 4 $\mathring{\mathrm{A}}$ repeatability on a 0.1 $\mu \mathrm{m}$ step is consistent with sub-nanometre resolution on the small range. Reference: UCSB Nanofab wiki, ‘Step Profilometer (DektakXT)’ (fetch-tool summary); Fraunhofer ENAS, ‘Tactile Profilometry’ (DEKTAK 150 and DEKTAK XT specifications) (fetch-tool summary).

Ques: Is the lowest stylus force 0.03 mg or 1 mg?

The UCSB page and the Bruker note give a force range of 0.03 to 15 $\mathrm{mg}$ for the DektakXT, while the Fraunhofer page lists 1 to 15 $\mathrm{mg}$ for both its DEKTAK 150 and DEKTAK XT. The lower limit corresponds to 0.29 $\mu \mathrm{N}$ against 9.8 $\mu \mathrm{N}$ (computed), a factor of 33. The difference most likely reflects configuration: the lowest forces need a low-force sensor option, and Fraunhofer reports the setup installed in its line (my reading). The 0.03 $\mathrm{mg}$ value is more likely the instrument capability, and 1 $\mathrm{mg}$ the practical lower limit of a standard head. For soft resists, the Hertz estimate shows that peak pressure scales as force to the one-third power, so lower force reduces damage only gradually (computed). Reference: UCSB Nanofab wiki, ‘Step Profilometer (DektakXT)’ (fetch-tool summary); Bruker, ‘TN-504: Thin Film Stress Measurement Using Dektak Stylus Profilers’ (fetch-tool summary; Stoney equation shown as an image); Fraunhofer ENAS, ‘Tactile Profilometry’ (DEKTAK 150 and DEKTAK XT specifications) (fetch-tool summary).

Ques: How is film stress measured with a stylus profiler?

By scanning wafer curvature before and after deposition. The Bruker note fits each scan with a 5th-order polynomial, differentiates it to get curvature, and applies the Stoney equation with substrate modulus, Poisson ratio, substrate and film thicknesses, and radii before and after; negative stress is compressive (convex) and positive tensile (concave). With a biaxial modulus of 180.6 $\mathrm{GPa}$ for Si(100), computed from Ioffe’s elastic constants, a 725 $\mu \mathrm{m}$ wafer, a 1 $\mu \mathrm{m}$ film and a change from flat to 50 m radius, stress is 317 $\mathrm{MPa}$ (computed). The bow over 200 mm is then 100 $\mu \mathrm{m}$, within the 1 mm Z range. Rotated diameter traces, up to 180, give a radial stress map. Reference: Bruker, ‘TN-504: Thin Film Stress Measurement Using Dektak Stylus Profilers’ (fetch-tool summary; Stoney equation shown as an image); 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); Ioffe Institute, NSM archive, ‘Si - Silicon: Mechanical properties’: C11 = 16.60e11 and C12 = 6.40e11 dyn/cm2, Young’s modulus [100] 13.0e11 dyn/cm2, Poisson ratio 0.28 (fetch-tool summary).

References

1. “Constant-force profilometer with stylus-stabilizing sensor assembly, dual-view optics, and temperature drift compensation,” US patent 5,705,741. Available: https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/5705741
2. UCSB Nanofabrication Facility, “Step Profilometer (DektakXT),” wiki page. Available: https://wiki.nanofab.ucsb.edu/wiki/Step_Profilometer_(DektakXT)
3. Fraunhofer ENAS, “Tactile Profilometry.” Available: https://www.enas.fraunhofer.de/en/Business_Units/Cross-Functional_Topics/Fab-Management/InlineMetrology_and_Data/profilometry/tactile-profilometry.html
4. Virtual Labs, IIT Kanpur, “Estimation of Hertzian Contact Pressure during Nanoindentation,” theory page. Available: https://wtm-iitk.vlabs.ac.in/exp/hertzian-contact-pressure/theory.html
5. 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
6. Ioffe Physico-Technical Institute, “Si - Silicon: Mechanical properties,” New Semiconductor Materials. Available: https://www.ioffe.ru/SVA/NSM/Semicond/Si/mechanic.html
7. Keyence, “Arithmetical Mean Height (Ra)” and “Root Mean Square Deviation (Rq),” surface roughness parameter pages. Available: https://www.keyence.com/ss/products/microscope/roughness/line/parameters.jsp
8. Bruker, “TN-504: Thin Film Stress Measurement Using Dektak Stylus Profilers,” technical note. Available: https://www.bruker.com/pt/products-and-solutions/test-and-measurement/stylus-profilometers/resource-library/tn-504-thin-film-stress-measurement-using-dektak-stylus-profilers.html

Related pages in this vault: Wafer Geometry Gauge, White-Light Interferometer, Atomic Force Microscope (AFM), Nanoindenter, Mechanical and Package Stress, Thermal Stress, Confocal Laser Scanning Microscope, Spectral Reflectometer, Optical CD Scatterometer (OCD), Wafer Curvature Stress Gauge, Quartz Crystal Microbalance (QCM), CMP Polisher, Wafer Backgrinding and Thinning Tool, Wet Etch Bench, Deep Reactive Ion Etcher (Bosch DRIE).