CMP Polisher

1. Theory

• Definition: A CMP polisher presses a rotating wafer face down onto a rotating pad wetted with abrasive slurry, removing material by combined chemical softening and mechanical abrasion to planarize the surface (Lai; Boning et al.).
• Preston law: Removal rate is proportional to pressure times relative velocity, with a coefficient that falls roughly inversely with film hardness (Lai).
• Pattern dependence: Raised areas polish faster than recessed ones until the step height falls below the pad contact height; the rate depends on layout density averaged over a planarization length (Boning et al.).
• Metal CMP: Copper polishing ends with dishing of wide lines and erosion of dense arrays (Boning et al.).

2. Working Principle

• Carrier and platen: The carrier holds the wafer at a set pressure, typically tens of kPa, and both rotate at similar speeds; Noh and Lyon used 48 $\mathrm{kPa}$ and 75 $\mathrm{rpm}$ (Noh and Lyon; Hasni et al.).
• Slurry and pad: Alumina abrasives of 50 to 1000 $\mathrm{nm}$ were compared by Lai; uniformity is sensitive to how the pad is roughened by a diamond conditioning disc (Lai; Hasni et al.).
• Endpoint: An in-situ optical reflectance sensor detects when copper clears (Noh and Lyon).
• Post-CMP clean: Cleaning chemistries break the ceria-oxide bond and remove ceria particles down to 10 $\mathrm{nm}$ (Seo).

3. Applications

• Shallow trench isolation: Oxide is polished until the nitride stop layer is exposed, with ceria slurries giving about 45:1 oxide-to-nitride selectivity against 4:1 for silica (ceria patent; Seo).
• Inter-layer dielectric: Planarization of oxide between metal levels (Boning et al.).
• Damascene copper and tungsten: Removal of overburden and barrier to form lines, vias and contacts (Boning et al.; Noh and Lyon).
• Advanced device architectures: FinFET, gate-all-around nanosheet, complementary FET and 3D-stacked processes rely on CMP and post-CMP cleaning (Seo).

Removal Mechanics

Name	Formula	Related Formulae	Variables/Constants
Preston equation	$\frac{dh}{dt}=k_{p}p{v}_R$	$k_p\text{in}\mathrm{P}{\mathrm{a}}^{-1}\left(={\mathrm{m}}^2/N\right)$	$h:removedthickness$ $p:nominalpressure$ $v_R:pad-waferrelativevelocity$ $\begin{array}{r}{k}_p:Prestoncoefficient;Lai:equation3.1,\\ validlocallyandgloballywhenremoval\\ greatlyexceedsroughness,k_{p}plottedfrom\\ about1\times 10^{-9}to1\times 10^{-6}\mathrm{MP}{\mathrm{a}}^{-1};removal330to4\\ nm/minat48\mathrm{kPa}and0.7m/swith300\mathrm{nm}\\ alumina\end{array}$ $\begin{array}{r}(computed:k_p=1.6\times 10^{-7}\mathrm{MP}{\mathrm{a}}^{-1}for330nm/min,\\ 5.6\times 10^{-8}forcopperat112nm/minand\\ 2.0\times 10^{-9}for4nm/min,allinsidethe\\ plottedrange)\end{array}$
Wear-law link	$V=k_w\frac{LS}{H}$	$k_p=\frac{k_w}{H},MRR\propto H^{-1.155}$	$V:removedvolume$ $L:load$ $S:slidingdistance$ $H:hardness$ $\begin{array}{r}{k}_w:wearcoefficient;Lai:equation3.2,k_w\\ about1\times 10^{-4}forallcoatings,log-log\\ slopeofremovalrateagainsthardness\\ -1.155(Buehlerpad-1.147,Rodelpad\\ -1.164)\end{array}$ $\begin{array}{r}(assumed:oxidehardness8GPa;computed:\\ k_w=1\times 10^{-4}predictsabout25nm/minat48\\ \mathrm{kPa}and0.7m/s)\end{array}$
Normalized removal rate	$NMRR=\frac{MRR}{v_R}$	$S=v_{R}t$	$\begin{array}{r}NMRR:thicknessremovedpermetreof\\ sliding;Lai:0.1nm/mfor\mathrm{S}{\mathrm{i}}_3{\mathrm{N}}_{4}upto10\\ nm/mforaluminium;NohandLyon:0.70m/s\\ and1to6mingiving42to252m\end{array}$ $\begin{array}{r}(computed:0.70m/sfor1minis42m,\\ matchingNohandLyon;330nm/minat0.7\\ m/sis7.9nm/m,slightlybelowLa{i}^{\prime }s10\\ nm/mmaximum)\end{array}$

Tool Kinematics and Consumables

Name	Formula	Related Formulae	Variables/Constants
Rotary kinematics	${\vec{v}}_R=\left({\omega }_p-{\omega }_c\right)\times \vec{r}+{\omega }_p\times {\vec{R}}_{cc}$	${\omega }_p={\omega }_c\Rightarrow v_R={\omega }_p{R}_{cc}\text{everywhere}$	${\omega }_p,{\omega }_c:platenandcarrierangularvelocities$ $\vec{r}:positiononthewaferfromitscentre$ $\begin{array}{r}{R}_{cc}:carrier-to-platencentredistance;\\ Hasnietal.:relativevelocityasthe\\ vectordifferenceofpadandwafer\\ velocities,magnitude{\omega }_p\alpha \sqrt{(r\mu {)}^2+2r\mu \cos \theta +1};\\ equal-speedreduction(computedfromthat\\ difference);NohandLyon:equalspeeds\\ makeanin-situsensortracethesame\\ locus,sopracticeoffsetsthewaferspeed\\ slightly;75\mathrm{rpm}and0.70m/s\end{array}$ $\begin{array}{r}(computed:0.70m/sat75\mathrm{rpm}implies{R}_{cc}=\\ 89mm)\end{array}$
Abrasive size dependence	$MR{R}_{Cu}=51,112,133nm/min$	$d_{abr}=50,300,1000\mathrm{nm}$	$\begin{array}{r}{d}_{abr}:abrasivediameter;Lai:copperremoval\\ withaluminaabrasives;particleload\\ about6\mu \mathrm{N},about7\times 10^{7}particlescarrying\\ theloadagainstabout1\times 10^{11}neededto\\ covera100mmwafer\end{array}$ $\begin{array}{r}(computed:removalrisesas{d}^{0.44}from50to\\ 300\mathrm{nm}andonly{d}^{0.14}from300to1000\mathrm{nm})\end{array}$
Friction and contact	${\mu }_f=\frac{F_t}{F_n}$	${\mu }_f\approx 0.1\Rightarrow \text{contact mode}$	$\begin{array}{r}{F}_t,F_n:tangentialandnormalforces;Lai:\\ steady-statefriction0.15to0.28\\ (Buehler)and0.09to0.17(Rodel),\\ wafer-paddirect-contactfriction0.5to\\ 0.7,padmoduliabout500\mathrm{MPa}(top)and60\\ \mathrm{MPa}(bottom),stifferpadsgivelower\\ WIWNU\end{array}$ $\begin{array}{r}(myreading:measuredfrictionisa\\ load-weightedmixofparticle-mediatedand\\ directpadcontact)\end{array}$

Pattern Dependence and Endpoint

Name	Formula	Related Formulae	Variables/Constants
Effective pattern density	$MRR\left(x,y\right)=\frac{K}{{\rho }_{eff}\left(x,y,PL\right)}$	${\rho }_{eff}={\rho }_{local}\ast w_{PL}$	$K:blanketremovalrate$ $\begin{array}{r}{\rho }_{eff}:layoutdensityaveragedoverthe\\ planarizationlengthPLbyconvolution\\ withapad-responseweight{w}_{PL};Boninget\\ al.:removalinverselyproportionalto\\ effectivedensity,elasticpadresponse\\ function,sixfittedparametersPL,K,\tau ,\\ a_1,a_2,a_3\end{array}$ $\begin{array}{r}(assumed:K=250nm/min,{\rho }_{eff}=0.5;\\ computed:500nm/minontheraisedarea)\end{array}$
Step-height reduction	$h\left(t\right)=h_0-\frac{K}{\rho }t\left(h>h_c\right)$	$h\left(t\right)=h_c{e}^{-\left(t-t_c\right)/\tau }\left(h\le h_c\right)$	$h:stepheight$ $\begin{array}{r}{h}_c:contactheightabovewhichthepad\\ touchesonlyraisedareas\end{array}$ $\begin{array}{r}\tau :decaytime;Boningetal.:linear\\ step-heightdecreaseforanincompressible\\ pad,exponentialbelowthecontactheight\\ (Grillaerttransitionmodel),contact\\ heightfittedasafunctionofdensity;\\ RMSerror273and253\mathring{A}(densitymodel)\\ against98and83\mathring{A}(integratedmodel),\\ run-to-runcontrolwithin100\mathring{A}\end{array}$ $\begin{array}{r}(myreading:theextractedequationswere\\ garbled,sothesearethestandardforms\\ thetextdescribes)\end{array}$
Copper dishing and endpoint	${\frac{d{h}_{Cu}}{dt}=\frac{d{h}_{ox}}{dt}|}_{d=d_{ss}}$	$\frac{{\sigma }_R}{{\mu }_R}\text{(reflectance non-uniformity)}$	$\begin{array}{r}{d}_{ss}:steady-statedishing,{\sigma }_R/{\mu }_R:standard\\ deviationovermeanofreflectance;Boning\\ etal.:overpolishcopperratefallsand\\ oxiderateriseswithdishinguntilequal\\ at{d}_{ss};NohandLyon:1\mu \mathrm{m}PVDcopper,20\mathrm{nm}\\ Ta,1\mu \mathrm{m}trenches,780to990\mathrm{nm}sensor,\\ wafer-levelendpointatabout7min\end{array}$ $\begin{array}{r}(myreading:endpointiscalledwhen\\ reflectancenon-uniformitydropsascopper\\ clearseverywhere)\end{array}$

Important Questions

Ques: What does the Preston equation say about CMP removal rate?

It says the removal rate equals $k_{p}p{v}_R$, the Preston coefficient times nominal pressure times relative velocity (Lai). It holds when the removed thickness is much larger than surface roughness. In Lai’s tests at 48 $\mathrm{kPa}$ and 0.7 $m/s$ with 300 $\mathrm{nm}$ alumina, removal ranged from 330 $nm/min$ down to 4 $nm/min$ across coatings, so $k_p$ spans $1.6\times 10^{-7}$ to $2\times 10^{-9}$ $\mathrm{MP}{\mathrm{a}}^{-1}$, and copper at 112 $nm/min$ gives $5.6\times 10^{-8}$ $\mathrm{MP}{\mathrm{a}}^{-1}$ (computed). These lie inside Lai’s plotted range. Dividing removal rate by velocity gives removal per metre of sliding, 7.9 $nm/m$ for the fastest film (computed), close to Lai’s quoted maximum of 10 $nm/m$ for aluminium. Hasni et al. note that Preston’s 1927 glass-polishing law does not capture velocity well, and that removal saturates at high velocity, which they attribute to particle aggregation and lower abrasion efficiency. Reference: MIT CMP group, PhD thesis chapter 3, ‘Mechanisms of Material Removal in the CMP Process’ (J.-Y. Lai, hosted at web.mit.edu/cmp; author and year not printed in the fetched chapter) (fetch-tool summary); A. Hasni, S. M. Raza, H. A. Siddiqui and M. H. Qureshi, ‘Computational and Experimental Study of Velocity Anisotropy and Slurry Hydrodynamics in Wafer-Pad CMP’, Jordan Journal of Mechanical and Industrial Engineering 20, 17 (2026) (fetch-tool summary).

Ques: Why does removal rate fall slightly faster than inversely with hardness?

The Archard wear law $V=k_{w}LS/H$ predicts removal inversely proportional to hardness, which corresponds to a log-log slope of -1 (Lai). Lai measures slopes of -1.155 overall, -1.147 on the Buehler pad and -1.164 on the Rodel pad, with a wear coefficient of about $10^{-4}$ for all coatings. The two values differ by about 15 per cent in exponent. The measured slope is more likely the right one for CMP, because harder films such as nitride and oxide also react more slowly with the slurry chemistry, so the chemical contribution falls with hardness as well (my reading). With $k_w=10^{-4}$ and an assumed oxide hardness of 8 $\mathrm{GPa}$, the wear law predicts about 25 $nm/min$ at 48 $\mathrm{kPa}$ and 0.7 $m/s$ (computed). Pad choice changes metal removal (copper 1.9 times higher on Buehler) but hardly changes oxide or nitride. Reference: MIT CMP group, PhD thesis chapter 3, ‘Mechanisms of Material Removal in the CMP Process’ (J.-Y. Lai, hosted at web.mit.edu/cmp; author and year not printed in the fetched chapter) (fetch-tool summary).

Ques: Why do platen and carrier rotate at almost the same speed?

Because equal angular velocities give the same relative velocity everywhere on the wafer. The relative velocity is $({\omega }_p-{\omega }_c)\times r+{\omega }_p\times R_{cc}$, so with ${\omega }_p={\omega }_c$ it reduces to ${\omega }_p{R}_{cc}$, independent of position (computed from the velocity difference that Hasni et al. use). Noh and Lyon ran 75 $\mathrm{rpm}$ with a linear velocity of 0.70 $m/s$, which implies a centre offset of 89 $\mathrm{mm}$ (computed). They note that equal speeds make an in-situ sensor trace the same path every turn, so in practice the wafer speed is offset slightly. Their table lists 391 N on a 200 $\mathrm{mm}$ wafer and 48 $\mathrm{kPa}$, but 391 N over 200 $\mathrm{mm}$ is only 12.4 $\mathrm{kPa}$; 48 $\mathrm{kPa}$ matches a 102 $\mathrm{mm}$ sample (computed), so the 48 $\mathrm{kPa}$ figure is more likely correct and the load belongs to a 100 $\mathrm{mm}$ test coupon (my reading). Reference: K. Noh and E. Lyon, ‘Statistical Analysis of In-Situ End-Point Detection in Copper Chemical-Mechanical Polishing’, MIT 6.780 term project (Spring 2003), MIT OpenCourseWare (fetch-tool summary); A. Hasni, S. M. Raza, H. A. Siddiqui and M. H. Qureshi, ‘Computational and Experimental Study of Velocity Anisotropy and Slurry Hydrodynamics in Wafer-Pad CMP’, Jordan Journal of Mechanical and Industrial Engineering 20, 17 (2026) (fetch-tool summary).

Ques: How do pattern-density models predict planarization?

They make the local removal rate the blanket rate divided by effective density, $K/{\rho }_{eff}$, where effective density averages layout density over the planarization length using an elastic pad response (Boning et al.). With an assumed blanket rate of 250 $nm/min$, an area of 50 per cent effective density polishes at 500 $nm/min$ (computed). The step height first falls linearly while the pad touches only raised areas, then decays exponentially once it drops below the contact height. Adding the step-height stage cut RMS error from 273 and 253 angstrom to 98 and 83 angstrom on raised and recessed areas, and run-to-run control held thickness within 100 angstrom. For STI, ceria slurries with about 45:1 oxide-to-nitride selectivity, against 4:1 for silica, stop on the nitride (ceria patent), and Seo calls their removal self-stopping. Reference: D. Boning, B. Lee, C. Oji, D. Ouma, T. Park, T. Smith and T. Tugbawa, ‘Pattern dependent modeling for CMP optimization and control’, MRS Spring Meeting, Symposium P (April 1999), reprint (fetch-tool summary); US patent 6,498,102, ‘Method for planarizing a semiconductor device using ceria-based slurry’ (USPTO full text; assignee not shown in the fetched text) (fetch-tool summary); J. Seo, ‘Shallow Trench Isolation Chemical Mechanical Planarization: Slurry chemistry, Cleaning chemistry, and Mechanisms’, 17th US-Korea Forum on Nanotechnology abstract (2023) (fetch-tool summary).

Ques: How does copper CMP end, and what sets dishing?

Copper CMP removes overburden, then barrier, then overpolishes. In Boning’s removal-rate picture, during overpolish the copper rate in a line falls and the oxide rate between lines rises as dishing grows, until both are equal at the steady-state dishing $d_{ss}$, the maximum expected (Boning et al.). Endpoint detection prevents unnecessary overpolish: Noh and Lyon used a 780 to 990 $\mathrm{nm}$ reflectance sensor on 1 $\mu \mathrm{m}$ PVD copper over 20 $\mathrm{nm}$ Ta and 1 $\mu \mathrm{m}$ trenches, and wafer-level endpoint began after about 7 $\min$. They tracked reflectance non-uniformity as standard deviation over mean, which falls as copper clears across the wafer (my reading). Larger alumina abrasives raise copper removal from 51 to 133 $nm/min$ between 50 and 1000 $\mathrm{nm}$, but most of the gain comes below 300 $\mathrm{nm}$ (Lai; computed). Reference: D. Boning, B. Lee, C. Oji, D. Ouma, T. Park, T. Smith and T. Tugbawa, ‘Pattern dependent modeling for CMP optimization and control’, MRS Spring Meeting, Symposium P (April 1999), reprint (fetch-tool summary); K. Noh and E. Lyon, ‘Statistical Analysis of In-Situ End-Point Detection in Copper Chemical-Mechanical Polishing’, MIT 6.780 term project (Spring 2003), MIT OpenCourseWare (fetch-tool summary); MIT CMP group, PhD thesis chapter 3, ‘Mechanisms of Material Removal in the CMP Process’ (J.-Y. Lai, hosted at web.mit.edu/cmp; author and year not printed in the fetched chapter) (fetch-tool summary).

References

1. J.-Y. Lai, “Mechanisms of Material Removal in the CMP Process,” chapter 3 of PhD thesis, Massachusetts Institute of Technology. Available: https://web.mit.edu/cmp/publications/thesis/jiunyulai/ch3.pdf
2. A. Hasni, S. M. Raza, H. A. Siddiqui and M. H. Qureshi, “Computational and Experimental Study of Velocity Anisotropy and Slurry Hydrodynamics in Wafer-Pad CMP,” Jordan Journal of Mechanical and Industrial Engineering, vol. 20, no. 1, pp. 17-27, 2026. Available: https://jjmie.hu.edu.jo/vol20/vol20-1/02-JJMIE-324-25.pdf
3. K. Noh and E. Lyon, “Statistical Analysis of In-Situ End-Point Detection in Copper Chemical-Mechanical Polishing,” MIT 6.780 term project, Spring 2003. Available: https://ocw.mit.edu/courses/6-780-semiconductor-manufacturing-spring-2003/01962e9a0397bce85da79472401e8c0e_lyons_noh_rep.pdf
4. “Method for planarizing a semiconductor device using ceria-based slurry,” US Patent 6,498,102. Available: https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/6498102
5. J. Seo, “Shallow Trench Isolation Chemical Mechanical Planarization: Slurry chemistry, Cleaning chemistry, and Mechanisms,” 17th US-Korea Forum on Nanotechnology, 2023. Available: https://www.cmu.edu/nanotechnology-forum/Forum_17/USA_Abstract/Jihoon%20Seo_Abstract.pdf
6. D. Boning, B. Lee, C. Oji, D. Ouma, T. Park, T. Smith and T. Tugbawa, “Pattern Dependent Modeling for CMP Optimization and Control,” MRS Spring Meeting, Symposium P, 1999. Available: https://boning.mit.edu/wp-content/uploads/2022/11/MRS99-reprint.pdf

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