💠 半导体晶圆厂 实时模型

依据：数字背后的模型

模拟器使用的每一个关系式及其来源。标注为假设的常数是示意性校准值。

设备状态遵循 SEMI E10：生产、待机、工程、计划停机和非计划停机。

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

设备随机发生故障，故障前时间服从指数分布；维修要等有空闲技术员。

P(fail in Δt) = 1 − e^(−Δt/MTBF), MTBF = 250 h; repair ~ Exp(mean 6 h) once a technician is free[17][1]假设：各量级（MTBF、产生率、k、τ）均为示意性校准值，并非真实晶圆厂的数据。

利特尔定律把在制品、吞吐量和周期时间联系起来。

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

在稳定的产线中，产出等于投片速率，所以没投出去的一片就是永远出不来的一片。工序之间等待过久的晶圆会超出排队时间窗口。

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]假设：3.5 h 的周期时间上限代替排队时间窗口（Q-time）；实际窗口由工艺工程逐道工序设定，超出窗口的晶圆通常会报废。

金曼近似：排队时间按 u/(1−u) 增长，利用率接近 100 % 时会急剧飙升。

CTq ≈ ((ca² + ce²)/2) · (u/(1−u)) · te[5]仅为帮助直观理解而显示：模拟器中的队列源于随机到达和随机产能，而不是这个公式。

粒径为 D 的颗粒的 ISO 14644-1 等级限值；在 0.1 µm 时，等级是浓度的对数。

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

充分混合的房间：浓度 = 产生量 ÷（换气次数 × 体积 × 过滤效率）。

C = G / (ACH · V · η)[13][11]假设：各量级（MTBF、产生率、k、τ）均为示意性校准值，并非真实晶圆厂的数据。

风机相似定律：功率随风机转速的立方上升。

P_fan = P_max · (speed)³[10]

负二项良率：缺陷会成簇出现，所以良率的下降比泊松模型预测的更慢。

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

良率学习：随着投入工程时间，缺陷密度向成熟水平衰减。

D(t+Δt) = D∞ + (D − D∞)·e^(−Δt/τ), τ = 120 h · (0.10 / engineering share)[15][18]假设：τ 相比业界报告的学习速率（每月 4–6.5 %，Leachman）压缩了约 100 倍，以便 14 天的场景能展现出这一效果。

落在晶圆上的颗粒会按空气浓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假设：这是示意性常数，真实晶圆厂会根据自己的检测数据校准致命缺陷率。

参数良率：正常工艺落在规格限内的比例；Cpk 衡量裕量。

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

SPC：n 个样本的平均值，带 ±3σ/√n 控制限和区域规则。

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]

模型使用的其他运行常数。

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 %假设：各量级（MTBF、产生率、k、τ）均为示意性校准值，并非真实晶圆厂的数据。

面积为 A 的芯片在每片 300 mm 晶圆上的毛芯片数。

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

随机性：带种子的 mulberry32 生成器；所用分布包括均匀分布、指数分布（逆 CDF）、正态分布（Box–Muller）和泊松分布（Knuth）。种子会显示出来，并且可以分享。

来源

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