💠 半導體晶圓廠 即時模型

依據：數字背後的模型

模擬器使用的每一個關係式及其來源。標示為假設的常數是示意校準值。

設備狀態遵循 SEMI E10：生產、待機、工程、計劃停機及非計劃停機。

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

機台隨機發生故障，故障前時間服從指數分布；維修要等有空閒的技術員。

P(fail in Δt) = 1 − e^(−Δt/MTBF), MTBF = 250 h; repair ~ Exp(mean 6 h) once a technician is free[17][1]假設：各數量級（MTBF、產生率、k、τ）均為示意校準值，並非真實晶圓廠的數據。

利特爾定律把在製品、產出率及週期時間連繫起來。

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

在穩定的生產線中，產出等於投片速率，所以沒有投出的一片，就是永遠出不來的一片。在工序之間輪候太久的晶圓，會超出輪候時間窗口。

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]假設：3.5 h 的週期時間上限代表輪候時間窗口（Q-time）；實際窗口由製程工程逐個工序設定，超出窗口的晶圓通常會報廢。

金曼近似：輪候時間按 u/(1−u) 增長，使用率接近 100 % 時會急升。

CTq ≈ ((ca² + ce²)/2) · (u/(1−u)) · te[5]只為幫助直觀理解而顯示：模擬器中的隊列來自隨機到達及隨機產能，而不是這條公式。

粒徑為 D 的微粒之 ISO 14644-1 級別限值；在 0.1 µm 時，級別是濃度的對數。

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

充分混合的房間：濃度 = 產生量 ÷（換氣次數 × 體積 × 過濾效率）。

C = G / (ACH · V · η)[13][11]假設：各數量級（MTBF、產生率、k、τ）均為示意校準值，並非真實晶圓廠的數據。

風扇相似定律：功率隨風扇轉速的立方上升。

P_fan = P_max · (speed)³[10]

負二項良率：缺陷會成羣出現，所以良率的下降比泊松模型預測的更慢。

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

良率學習：隨着投入工程時間，缺陷密度朝成熟水平衰減。

D(t+Δt) = D∞ + (D − D∞)·e^(−Δt/τ), τ = 120 h · (0.10 / engineering share)[15][18]假設：τ 相比業界匯報的學習速率（每月 4–6.5 %，Leachman）壓縮了約 100 倍，令 14 日的情景能顯示出這個效果。

落在晶圓上的微粒，會按空氣濃度的比例增加致命缺陷。

D_total = D_learn + k · C, k = 4·10⁻⁶ cm⁻² per particle/m³假設：這是示意常數，真實晶圓廠會按自己的檢測數據校準致命缺陷率。

參數良率：正常製程落在規格界限內的比例；Cpk 量度裕度。

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

SPC：n 個樣本的平均值，附 ±3σ/√n 管制界限及區域規則。

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]

模型使用的其他運作常數。

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 %假設：各數量級（MTBF、產生率、k、τ）均為示意校準值，並非真實晶圓廠的數據。

面積為 A 的晶粒在每片 300 mm 晶圓上的毛晶粒數。

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

隨機性：帶種子的 mulberry32 產生器；所用分布包括均勻分布、指數分布（反 CDF）、常態分布（Box–Muller）及泊松分布（Knuth）。種子會顯示出來，並可分享。

來源

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