💠 Halfgeleier-fab Lewendige model

Grondslag — die model agter die syfers

Elke verband wat die simuleerder gebruik, met sy bron. Konstantes gemerk as aannames is illustratiewe kalibrasies.

Toerustingstoestande volg SEMI E10: produktief, bystand, ingenieurswese, geskeduleerde en ongeskeduleerde stilstand.

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

Toestelle faal lukraak met 'n eksponensiële tyd-tot-faling; herstelwerk wag vir 'n vrye tegnikus.

P(fail in Δt) = 1 − e^(−Δt/MTBF), MTBF = 250 h; repair ~ Exp(mean 6 h) once a technician is free[17][1]Aanname: groottes (MTBF, opwekkingskoerse, k, τ) is illustratiewe kalibrasies, nie data van 'n werklike fab nie.

Little se wet verbind werk-in-proses, deurset en siklustyd.

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

In 'n stabiele lyn is deurset gelyk aan die vrystellingskoers, so 'n wafelbegin wat nie vrygestel word nie, is 'n wafel wat nooit uitkom nie. Wafels wat te lank tussen stappe wag, oorskry die wagtydvenster.

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]Aanname: die siklustydlimiet van 3,5 h staan in vir 'n wagtydvenster; werklike vensters word stap vir stap deur prosesingenieurswese gestel, en wafels wat dit oorskry, word gewoonlik geskrap.

Kingman se benadering: wagtyd in die tou groei soos u/(1−u) — dit ontplof namate benutting 100 % nader.

CTq ≈ ((ca² + ce²)/2) · (u/(1−u)) · te[5]Vir intuïsie gewys: die simuleerder se tou ontstaan uit lukrake aankomste en lukrake kapasiteit eerder as uit hierdie formule.

ISO 14644-1-klaslimiet vir deeltjies van grootte D; by 0.1 µm is die klas die logaritme van die konsentrasie.

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

Goed gemengde kamer: konsentrasie = opwekking ÷ (lugwisselings × volume × filterdoeltreffendheid).

C = G / (ACH · V · η)[13][11]Aanname: groottes (MTBF, opwekkingskoerse, k, τ) is illustratiewe kalibrasies, nie data van 'n werklike fab nie.

Waaier-affiniteitswet: krag styg met die derde mag van waaierspoed.

P_fan = P_max · (speed)³[10]

Negatief-binomiale opbrengs: defekte groepeer, so opbrengs daal stadiger as wat die Poisson-model voorspel.

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

Opbrengsleer: defekdigtheid neem af na 'n volwasse vlak namate ingenieurstyd bestee word.

D(t+Δt) = D∞ + (D − D∞)·e^(−Δt/τ), τ = 120 h · (0.10 / engineering share)[15][18]Aanname: τ is ongeveer 100× tydgekomprimeer in vergelyking met gerapporteerde bedryfsleerkoerse (4–6.5 % per maand, Leachman) sodat 'n 14-dae-scenario die effek wys.

Deeltjies wat op wafels land voeg moorddefekte by in verhouding tot lugkonsentrasie.

D_total = D_learn + k · C, k = 4·10⁻⁶ cm⁻² per particle/m³Aanname: 'n illustratiewe konstante — werklike fabs kalibreer moorddefekkoerse uit hul eie inspeksiedata.

Parametriese opbrengs: die deel van 'n normale proses wat binne die spesifikasielimiete val; Cpk meet die marge.

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

SPC: gemiddeldes van n monsters, met ±3σ/√n-beheerlimiete en sone-reëls.

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]

Ander bedryfskonstantes wat deur die model gebruik word.

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 %Aanname: groottes (MTBF, opwekkingskoerse, k, τ) is illustratiewe kalibrasies, nie data van 'n werklike fab nie.

Bruto skyfies per 300 mm-wafel vir 'n skyfie met oppervlakte A.

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

Lukraakheid: 'n gesaaide mulberry32-genereerder; verdelings gebruik — uniform, eksponensieel (inverse CDF), normaal (Box–Muller), Poisson (Knuth). Die saad word gewys en kan gedeel word.

Bronne

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