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not the full 0.2e11 band" }, "radiative_check": { "formula": "dT/dt = -2 eps sigma (T^4 - T_env^4) / (rho cp d), T in kelvin", "parameters": { "eps": 0.7, "sigma_W_m2_K4": 5.670374419e-08, "rho_kg_m3": 2330, "cp_J_kg_K": 900, "d_m": 0.000525, "T_env_K": 298.15 }, "comparison": [ { "seconds": 0, "prompt_C": 1000, "equation_C": 1000.0000000000001 }, { "seconds": 1.5, "prompt_C": 800, "equation_C": 800.668216225628 }, { "seconds": 4.7, "prompt_C": 600, "equation_C": 601.7516060438472 }, { "seconds": 13, "prompt_C": 400, "equation_C": 402.41669782527015 }, { "seconds": 41, "prompt_C": 200, "equation_C": 211.881998901249 }, { "seconds": 90, "prompt_C": 100, "equation_C": 117.32386127775521 }, { "seconds": 300, "prompt_C": 25, "equation_C": 38.121989518242856 } ], "implementation": "The four histories use the prompt's knots exactly. Their final 25 C endpoint is an assumed tail, not the exact radiative solution (about 38.1 C at 300 s). The equation approaches 25 C asymptotically. Slow cooling is the prompt's phenomenological 5x time scaling." }, "provenance": [ { "number": "(100); 1000 C for 8 h", "source": "paper", "where": "Vitkavage 1990, Table I, PDF p. 37", "timing": "stated" }, { "number": "400 C for 30 min in forming gas", "source": "paper", "where": "PDF p. 7", "timing": "stated" }, { "number": "forming gas: 10% H2 / 90% N2", "source": "paper plus transfer assumption", "where": "PDF p. 7, explicitly specified for non-cleanroom set; adopted for cleanroom", "timing": "assumed" }, { "number": "pull shorter than 5 s in oxidizing gas", "source": "paper", "where": "PDF p. 6, non-cleanroom set only", "timing": "bound" }, { "number": "5 s travel at 1000 C, dry O2, applied to this cleanroom case", "source": "prompt plus modelling assumption", "timing": "assumed" }, { "number": "25 C at clean/load/store/metal/CV; N2 throughout preheat; dry O2 begins with the full 8 h clock", "source": "prompt", "timing": "assumed" }, { "number": "A: preheat 1\u20135 min; PMA heatup 1\u20135 min; PMA cooldown 5\u201315 min", "source": "prompt", "timing": "assumed" }, { "number": "B: preheat 60\u2013120 min; PMA heatup 20\u201340 min; PMA cooldown 30\u201390 min", "source": "prompt", "timing": "assumed" }, { "number": "fast cooling knots 0,1.5,4.7,13,41,90,300 s at 1000,800,600,400,200,100,25 C; slow times x5", "source": "prompt", "timing": "assumed" }, { "number": "emissivity 0.7; wafer thickness 525 um; density 2330 kg/m3; heat capacity 900 J/kg/K; surroundings 25 C", "source": "prompt", "timing": "assumed" }, { "number": "representative durations are range midpoints; hot-push curves use a normalized exponential, tau = duration/5, approximated by 12 linear segments; A PMA cooldown uses the same shape", "source": "implementation choices within prompt ranges", "timing": "assumed" }, { "number": "Qf comparison threshold +/-0.1e11 cm^-2", "source": "prompt, referencing the paper's 1000 C spread", "where": "Table III prints a spread for the 1 h temperature-series row; Table VI's 8 h row has no individual spread" } ], "limits": [ "No transition temperatures or durations were fitted to measurements.", "Qf starts at the oxidation equilibrium: preheat and forming-gas steps are omitted by its existing model, so their insensitivity is built in.", "Dit uses the current fitted hydrogen rate without temperature dependence. Its changes during long forming-gas ramps are model sensitivity, not validated physical kinetics.", "The growth model retains its existing 23 nm Deal\u2013Grove intercept; this is not a measured starting oxide. No material parameter was changed.", "The paper's 8 h Table VI/VII rows do not give individual spreads; the Qf threshold comes from the prompt and the 1000 C temperature-series spread.", "The 32 endpoint combinations sample the specified duration ranges; they are not a proof of extrema over every possible trajectory." ] }