💠 Fab tas-semikondutturi Mudell ħaj

Bażi — il-mudell wara n-numri

Kull relazzjoni li juża s-simulatur, bis-sors tagħha. Il-kostanti mmarkati bħala assunzjonijiet huma kalibrazzjonijiet illustrattivi.

L-istati tat-tagħmir isegwu SEMI E10: produttiv, standby, inġinerija, waqfa skedata u mhux skedata.

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

L-għodod jfallu b'mod każwali b'ħin sal-falliment esponenzjali; ir-riparazzjonijiet jistennew tekniku ħieles.

P(fail in Δt) = 1 − e^(−Δt/MTBF), MTBF = 250 h; repair ~ Exp(mean 6 h) once a technician is free[17][1]Assunzjoni: il-kobor (MTBF, rati ta' ġenerazzjoni, k, τ) huma kalibrazzjonijiet illustrattivi, mhux dejta minn fab reali.

Il-liġi ta' Little torbot ix-xogħol fil-proċess, il-ħruġ u ċ-ċiklu tal-ħin.

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

F'linja stabbli l-output huwa ugwali għar-rata tar-rilaxx, għalhekk bidu ta' wafer li ma jiġix rilaxxat huwa wafer li qatt ma joħroġ. Wafers li jistennew wisq bejn il-passi jaqbżu t-tieqa tal-queue-time.

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]Assunzjoni: il-limitu ta' 3.5 h taċ-ċiklu tal-ħin jirrappreżenta tieqa ta' queue-time; it-twieqi reali jiġu stabbiliti pass pass mill-inġinerija tal-proċess, u l-wafers li jaqbżuhom tipikament jintremew.

Approssimazzjoni ta' Kingman: il-ħin fil-kju jikber bħal u/(1−u) — jisplodi hekk kif l-użu joqrob lejn 100 %.

CTq ≈ ((ca² + ce²)/2) · (u/(1−u)) · te[5]Murija għall-intuwizzjoni: il-kju tas-simulatur joħroġ minn wasliet każwali u kapaċità każwali aktar milli minn din il-formula.

Limitu tal-klassi ISO 14644-1 għal partikuli ta' daqs D; f'0.1 µm il-klassi hija l-logaritmu tal-konċentrazzjoni.

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

Kamra mħallta tajjeb: konċentrazzjoni = ġenerazzjoni ÷ (bidliet tal-arja × volum × effiċjenza tal-filtru).

C = G / (ACH · V · η)[13][11]Assunzjoni: il-kobor (MTBF, rati ta' ġenerazzjoni, k, τ) huma kalibrazzjonijiet illustrattivi, mhux dejta minn fab reali.

Liġi tal-affinità tal-fans: il-qawwa togħla bil-kubu tal-veloċità tal-fan.

P_fan = P_max · (speed)³[10]

Rendiment binomjali negattiv: id-difetti jinġabru, għalhekk ir-rendiment jinżel aktar bil-mod milli jbassar il-mudell ta' Poisson.

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

Tagħlim tar-rendiment: id-densità tad-difetti tonqos lejn livell matur hekk kif jintefaq ħin tal-inġinerija.

D(t+Δt) = D∞ + (D − D∞)·e^(−Δt/τ), τ = 120 h · (0.10 / engineering share)[15][18]Assunzjoni: τ huwa kkompressat fil-ħin madwar 100× meta mqabbel mar-rati ta' tagħlim irrappurtati tal-industrija (4–6.5 % fix-xahar, Leachman) biex xenarju ta' 14-il jum juri l-effett.

Il-partikuli li jaqgħu fuq il-wafers iżidu difetti qattiela proporzjonalment mal-konċentrazzjoni tal-arja.

D_total = D_learn + k · C, k = 4·10⁻⁶ cm⁻² per particle/m³Assunzjoni: kostanti illustrattiva — fabs reali jikkalibraw ir-rati tad-difetti qattiela mid-dejta tal-ispezzjoni tagħhom stess.

Rendiment parametriku: is-sehem ta' proċess normali li jaqa' ġewwa l-limiti tal-ispeċifikazzjoni; il-Cpk ikejjel il-marġni.

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

SPC: medji ta' n kampjuni, b'limiti ta' kontroll ±3σ/√n u regoli taż-żoni.

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]

Kostanti operazzjonali oħra użati mill-mudell.

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 %Assunzjoni: il-kobor (MTBF, rati ta' ġenerazzjoni, k, τ) huma kalibrazzjonijiet illustrattivi, mhux dejta minn fab reali.

Dies grossi għal kull wafer ta' 300 mm għal die b'erja A.

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

Każwalità: ġeneratur mulberry32 b'seed; distribuzzjonijiet użati — uniformi, esponenzjali (CDF invers), normali (Box–Muller), Poisson (Knuth). Is-seed jintwera u jista' jinqasam.

Sorsi

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