💠 I-fab ye-semiconductor Imodeli ephilayo

Isisekelo — imodeli engemuva kwezinombolo

Wonke ubudlelwano isilingisi esibusebenzisayo, nomthombo wabo. Izinto ezihlala zingashintshi ezimakwe njengokuthathelwa yizilinganiso zesibonelo.

Izimo zemishini zilandela i-SEMI E10: ukukhiqiza, ukulala, ubunjiniyela, ukuma okuhleliwe nokungahleliwe.

Availability = up tools / 8 · Utilization = productive tool-hours / total[1]

Amathuluzi ahluleka ngokungahleliwe ngesikhathi esiya ekwehluleni se-exponential; ukulungiswa kulinda onjiniyela okhululekile.

P(fail in Δt) = 1 − e^(−Δt/MTBF), MTBF = 250 h; repair ~ Exp(mean 6 h) once a technician is free[17][1]Okuthathelwa: osayizi (MTBF, amazinga okukhiqiza, k, τ) yizilinganiso zesibonelo, hhayi idatha ye-fab yangempela.

Umthetho kaLittle uxhumanisa umsebenzi okuqhubekayo, ukudlula nesikhathi somjikelezo.

CT = WIP / TH = T0 + queue / TH, T0 = 1 h[6][5]

Emugqeni ozinzile umkhiqizo uyalingana nezinga lokukhipha, ngakho ukuqala kwe-wafer okungakhishwanga kuyi-wafer engasoze yaphuma. Ama-wafer alinda isikhathi eside kakhulu phakathi kwezinyathelo aphula iwindi lesikhathi somugqa.

TH = release rate while u < 1 ⇒ wafers out ≈ Σ starts; queue wait = CT − T0 = queue / TH ≤ queue-time window (3.5 h CT in semi-tool-crash)[5][16]Okuthathelwa: umkhawulo wesikhathi somjikelezo wamahora angu-3.5 umele iwindi lesikhathi somugqa (queue-time window); amawindi angempela abekwa ubunjiniyela benqubo isinyathelo ngesinyathelo, futhi ama-wafer awephulayo ngokuvamile alahlwa.

Isilinganiso sika-Kingman: isikhathi somugqa sikhula njengo-u/(1−u) — siqhuma uma ukusetshenziswa kusondela ku-100 %.

CTq ≈ ((ca² + ce²)/2) · (u/(1−u)) · te[5]Kuboniswa ukuze kuqondwe: umugqa wesilingisi uvela ekufikeni okungahleliwe namandla angahleliwe hhayi kule fomula.

Umkhawulo wekilasi le-ISO 14644-1 wezinhlayiya zosayizi D; ku-0.1 µm ikilasi yi-logarithm yokugxila.

Cn = 10^N · (0.1/D)^2.08 ⇒ N = log10(C≥0.1µm)[2]

Igumbi elixutshwe kahle: ukugxila = ukukhiqiza ÷ (ukushintsha komoya × umthamo × ukusebenza kwesihlungi).

C = G / (ACH · V · η)[13][11]Okuthathelwa: osayizi (MTBF, amazinga okukhiqiza, k, τ) yizilinganiso zesibonelo, hhayi idatha ye-fab yangempela.

Umthetho we-fan affinity: amandla akhuphuka ngokuphindaphindwa kathathu kwesivinini sefeni.

P_fan = P_max · (speed)³[10]

Ukukhiqiza kwe-negative-binomial: amaphutha ayaqoqana, ngakho ukukhiqiza kwehla kancane kunalokho okubikezelwa yimodeli ye-Poisson.

Y = (1 + A·D/α)^(−α), α = 2 (α→∞: Y = e^(−A·D))[3][4][15][14]

Ukufunda kokukhiqiza: ukuminyana kwamaphutha kuyehla kuya ezingeni elivuthiwe njengoba isikhathi sobunjiniyela sisetshenziswa.

D(t+Δt) = D∞ + (D − D∞)·e^(−Δt/τ), τ = 120 h · (0.10 / engineering share)[15][18]Okuthathelwa: i-τ icindezelwe isikhathi cishe izikhathi ezingu-100 uma iqhathaniswa namazinga okufunda embonini abikiwe (4–6.5 % ngenyanga, Leachman) ukuze isimo sezinsuku ezingu-14 sibonise umphumela.

Izinhlayiya ezehlela kuma-wafer zengeza amaphutha abulalayo ngokulingana nokugxila komoya.

D_total = D_learn + k · C, k = 4·10⁻⁶ cm⁻² per particle/m³Okuthathelwa: into ehlala ingashintshi yesibonelo — ama-fab angempela alinganisa amazinga amaphutha abulalayo kudatha yawo yokuhlola.

Ukukhiqiza kwe-parametric: ingxenye yenqubo evamile ewela ngaphakathi kwemikhawulo ecacisiwe; i-Cpk ilinganisa isikhala.

Y_param = Φ((USL−μ)/σ) − Φ((LSL−μ)/σ), Cpk = min(USL−μ, μ−LSL)/(3σ)[7][8][19]

I-SPC: izilinganiso zamasampula angu-n, nemikhawulo yokulawula ±3σ/√n nemithetho yezindawo.

x̄ ~ N(μ, σ/√n), control limits ±3σ/√n; zone rules (2 of 3 beyond 2σ, 4 of 5 beyond 1σ, 8 on one side)[8][7][12]

Ezinye izinto ezihlala zingashintshi zokusebenza ezisetshenziswa imodeli.

8 scanners × 6 wafer-starts/h · metrology −0.5 % capacity per sampled wafer/h · rollback: 2 h at half capacity + 4 wafer-equivalents reworked · drift 0.25 nm/h · crash repair 18 technician-hours · leak crew +2 people for 3 h · tools 150 kW busy / 60 kW idle, fans 150 kW at 100 %Okuthathelwa: osayizi (MTBF, amazinga okukhiqiza, k, τ) yizilinganiso zesibonelo, hhayi idatha ye-fab yangempela.

Ama-die aphelele ngewafa elingu-300 mm lie-die elinendawo u-A.

DPW = π·d²/(4A) − π·d/√(2A), d = 300 mm[9]

Ukungahleliwe: i-generator ye-mulberry32 enembewu; ukusabalalisa okusetshenzisiwe — uniform, exponential (inverse CDF), normal (Box–Muller), Poisson (Knuth). Imbewu iyabonisa futhi ingabelwana.

Imithombo

1. SEMI E10 — Specification for Definition and Measurement of Equipment Reliability, Availability, and Maintainability (RAM) and Utilization — SEMI
2. ISO 14644-1:2015 Cleanrooms and associated controlled environments — Part 1: Classification of air cleanliness by particle concentration — ISO, 2015
3. C. H. Stapper, F. M. Armstrong, K. Saji — Integrated circuit yield statistics — Proceedings of the IEEE 71(4), 1983
4. J. A. Cunningham — The use and evaluation of yield models in integrated circuit manufacturing — IEEE Trans. Semiconductor Manufacturing 3(2), 1990
5. W. J. Hopp, M. L. Spearman — Factory Physics (3rd ed.), ch. 7–8: Little’s law, Kingman (VUT) equation — Waveland Press, 2008
6. J. D. C. Little — A Proof for the Queuing Formula L = λW — Operations Research 9(3), 1961
7. D. C. Montgomery — Introduction to Statistical Quality Control (x̄ charts, process capability Cpk) — Wiley, 2019
8. NIST/SEMATECH e-Handbook of Statistical Methods — 6.3 Univariate and Multivariate Control Charts; 6.1.6 Process capability — NIST
9. Dies-per-wafer estimate DPW = πd²/(4S) − πd/√(2S) (de Vries, “Investigation of gross die per wafer formulas”, IEEE TSM 18(1)) — IEEE, 2005
10. Fan affinity laws: flow ∝ speed, pressure ∝ speed², power ∝ speed³ — U.S. DOE — Improving Fan System Performance: A Sourcebook for Industry
11. EN 1822-1:2019 High efficiency air filters (EPA, HEPA and ULPA) — classification (U15 ≥ 99.9995 % at MPPS) — CEN, 2019
12. Western Electric Statistical Quality Control Handbook (1956) — zone rules for control charts — Western Electric / NIST e-Handbook 6.3.2
13. W. Whyte — Cleanroom Technology: Fundamentals of Design, Testing and Operation (2nd ed.), ch. on dispersion of particles from people; well-mixed room dilution equation — Wiley, 2010
14. Yu. I. Bogdanov, N. A. Bogdanova, V. L. Dshkhunyan — Statistical Yield Modeling for IC Manufacture: Hierarchical Fault Distributions (§2 Compound Poisson distribution, after Eq. (17), p. 4 of the arXiv PDF — large-area clustering negative binomial model: the cluster parameter’s “typical values approximately range from 0.3 to 7”) — arXiv physics/0303039, 2003
15. R. C. Leachman — Yield Modeling and Analysis (Poisson, Murphy, Seeds, negative-binomial models; §8: SMLY survey with C. N. Berglund for International SEMATECH, 2002–03 — yield loss fitted as YL(t) = YL(0)·e^(−λt), Table 2 averages 4.4 / 4.0 / 6.5 %/month at 350 / 250 / 180 nm) — UC Berkeley, IEOR 130 course notes (unpublished), 2014
16. A. Klemmt, L. Mönch — Scheduling jobs with time constraints between consecutive process steps in semiconductor manufacturing (time windows set by process engineering against native oxidation and contamination; jobs that violate them are scrapped) — Proceedings of the 2012 Winter Simulation Conference, 2012
17. NIST/SEMATECH e-Handbook of Statistical Methods — 8.1.6.1 Exponential distribution (constant failure rate) — NIST
18. C. Weber — Yield learning and the sources of profitability in semiconductor manufacturing and process development — IEEE Trans. Semiconductor Manufacturing 17(4), 2004
19. M. Abramowitz, I. Stegun — Handbook of Mathematical Functions, 26.2.17 — NBS, 1964