Reproducing Krzeminski et al. 2007: thin dry oxides, Massoud's kinetics and a furnace cycle at 725 and 750 °C

One of the [studies we reproduced](/writing/interface-studies-reproduced) for [Questions we need Emily to answer](/writing/questions-for-emily-testing-a-theory-of-the-interface), the companion to [the essay](/writing/testing-a-theory-of-the-interface-with-an-ontology). Generated by `scripts/reproduce_studies.py` from `reproductions/krzeminski_2007.py`. Every value from the paper was read by an AI agent and has not been checked by a person; the sample and history were re-read from the PDF alone by a second agent that had not seen our earlier transcription.

The study and why we depend on it

C. Krzeminski, G. Larrieu, J. Penaud, E. Lampin and E. Dubois, “Silicon dry oxidation kinetics at low temperature in the nanometric range: modeling and experiment”, J. Appl. Phys. 101, 064908 (2007); arXiv:1106.3160.

The thin-oxide program the predictor runs on planar silicon in dry oxygen (planar_si_thin_growth in isl_harness/predict.py) is a structure found by scripts/search_growth.py on data/derived/silicon_dry_kinetics.csv: 115 points of Massoud's data as redrawn in this paper's Fig. 4, and 12 of the paper's own at 725 and 750 °C. It was fitted to Massoud at 800 to 1000 °C with 900 °C held out (18 of 25), and the 725 and 750 °C points are its other-laboratory test (6 of 10, 13% low at 725 °C). That test started the law at the measured thickness at time zero and ran only the isothermal stage. The second reading found the whole cycle: loading 50 °C below, a 30 min ramp in nitrogen with 9% oxygen that grows 1.53 or 1.96 nm before time zero, the stage in oxygen, then 10 min down and 30 min of reflow in nitrogen. This page writes that cycle and asks whether it explains the low result. The program printed under “The simulation in our language” is the 1965 program, which also runs on these histories but is not scored here; the thin-oxide program, which the rows compare, exists only as text generated by the search and is printed in full at the end of the page.

How well can this study be reproduced?

Moderate. The paper's own furnace cycle is described step by step, with times, gas flows and the oxide grown before time zero. What a simulation must still supply: the temperatures of loading and reflow (on a figure only), the loading time and gas, whether the 0 min wafers went through the ramp-down and reflow, and the delay in air before ellipsometry. Massoud's data are second-hand, redrawn, with a 1.0 nm starting oxide stated and no history beyond temperature and pressure.

The grade is ours and is about the paper as a source for simulation, not about the quality of the experiment: high means the sample, every step and its timing are stated well enough that a simulation needs no choices of ours; moderate that some choices are ours and are listed; low that the result depends on choices the paper does not constrain.

The sample and its history, played back

The sample is drawn from the simulation: layer heights and the read-outs are what the ISL programs compute at each moment of the history. Steps the language cannot state (a clean, a metallization, a measurement) are on the clock and marked as recorded, not simulated.

Does our sample and history match the paper's?

Left: the paper's own words, with the page of the PDF. Right: what we simulate. 6 same, 4 approximated, 1 assumed, 1 differs, 2 not in ours.

The timing of the history

Time is counted from the start of the first simulated step. A step the language cannot state takes no simulated time; where the paper gives its duration, the duration is in the last column. Stated means the paper gives the duration; partly stated that it gives a bound or a range; not stated; assumed that the duration in our history is ours.

Krzeminski, 725 °C, 60 min main stage

Simulated time in this history: 2 h 10 min.

Across the samples

Timing the paper does not give

• Time zero is the start of the main stage: every thickness at “0 min” already holds 30 min of ramp growth. The language counts time from the first RUN, so the datasets' recipe (one isothermal RUN) and the paper's cycle disagree about what t = 0 means.
• How long the wafers sat at the loading temperature, and in what gas, before the ramp.
• The ramp's shape: 50 °C in 30 min, read from Fig. 9's end points; linear is our assumption.
• Whether the 0 min wafers went through the ramp-down and reflow (Fig. 9 labels the main stage [10–90] min).
• The reflow temperature: drawn in Fig. 9, not in the text.
• Unloading and cooling after the reflow: not in the manuscript.
• The delay in air before ex-situ ellipsometry; an uncaptioned figure shows 0.25–0.37 nm of growth in 90 min of exposure, which is as large as the growth of the first 10 min of the main stage.
• For Massoud's data: ramp, time zero and how the 1.0 nm starting oxide was made are not given in this paper.

What the second reading changed

• The stored recipe for the paper's own wafers is one RUN of 90 min at 725 or 750 °C in dry oxygen. The paper's cycle is a load at 675/700 °C, a 30 min ramp in 9% oxygen, the main stage in oxygen, 10 min of ramp-down and 30 min of reflow in nitrogen. The ramp is the one that matters: it grows 1.53 or 1.96 nm, all of the oxide at “0 min”.
• The other-laboratory score in evaluations/growth_search_report.json (6 of 10, 13% low at 725 °C) started the law at the measured 1.53 or 1.96 nm. The predictor cannot do that: it starts every history at the fitted 1.71 nm. Run through the predictor with the stored recipe, the same points come out between 1% high and 14% low at 725 °C and 8 to 11% low at 750 °C, and the 0 min points (excluded from the search's score) 12% high and 13% low. The two scores of one program on one dataset differ because the starting oxide lives in different places.
• Table II prints 2.71 nm by ellipsometry for the 30 min wafer at 725 °C (TEM 2.7 nm); the dataset, from the figure, has 2.59 nm at 30 min and 2.71 nm at 45 min. One of the two is mislabelled; the dataset follows the figure.
• The main stage is written as pure oxygen. The paper says only “an enhanced oxygen flux of (2 L/min)”; whether the 2 L/min of nitrogen continues is not stated. If it does, the stage is at 0.5 atm of oxygen.
• The source folder's SOURCE.md says “Held, not yet extracted”; the folder holds kinetics_vector_extracted.csv and the dataset is built from it. The note is out of date.
• Massoud's wafers are written as (100). This paper does not state their orientation; the model geometry it uses is (100).

Their terms and ours

The simulation in our language

Each sample is one history file: a SAMPLE and a CONTROL_TRAJECTORY. This is the whole input; run it with PYTHONPATH=. python -m isl_harness.predict <file>.

Krzeminski, 725 °C, 60 min main stage

# Krzeminski, 725 °C, 60 min main stage
SAMPLE Wafer_725C_60min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 675 °C
    RUN Ramp DURATION 1800 TEMP 675->725 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 625..725
    RUN Oxidation DURATION 3600 TEMP 725->725 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 725->675 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 675->675 AMBIENT inert_N2 TIMING bound VARY T_hold 625..725
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 725 °C, 0 min main stage

# Krzeminski, 725 °C, 0 min main stage
SAMPLE Wafer_725C_0min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 675 °C
    RUN Ramp DURATION 1800 TEMP 675->725 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 625..725
    EVENT NoMainStage KIND other TIMING assumed                  # No main stage (the 0 min point)
    RUN RampDown DURATION 600 TEMP 725->675 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 675->675 AMBIENT inert_N2 TIMING bound VARY T_hold 625..725
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 725 °C, 10 min main stage

# Krzeminski, 725 °C, 10 min main stage
SAMPLE Wafer_725C_10min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 675 °C
    RUN Ramp DURATION 1800 TEMP 675->725 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 625..725
    RUN Oxidation DURATION 600 TEMP 725->725 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 725->675 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 675->675 AMBIENT inert_N2 TIMING bound VARY T_hold 625..725
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 725 °C, 25 min main stage

# Krzeminski, 725 °C, 25 min main stage
SAMPLE Wafer_725C_25min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 675 °C
    RUN Ramp DURATION 1800 TEMP 675->725 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 625..725
    RUN Oxidation DURATION 1500 TEMP 725->725 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 725->675 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 675->675 AMBIENT inert_N2 TIMING bound VARY T_hold 625..725
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 725 °C, 30 min main stage

# Krzeminski, 725 °C, 30 min main stage
SAMPLE Wafer_725C_30min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 675 °C
    RUN Ramp DURATION 1800 TEMP 675->725 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 625..725
    RUN Oxidation DURATION 1800 TEMP 725->725 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 725->675 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 675->675 AMBIENT inert_N2 TIMING bound VARY T_hold 625..725
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 725 °C, 45 min main stage

# Krzeminski, 725 °C, 45 min main stage
SAMPLE Wafer_725C_45min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 675 °C
    RUN Ramp DURATION 1800 TEMP 675->725 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 625..725
    RUN Oxidation DURATION 2700 TEMP 725->725 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 725->675 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 675->675 AMBIENT inert_N2 TIMING bound VARY T_hold 625..725
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 725 °C, 90 min main stage

# Krzeminski, 725 °C, 90 min main stage
SAMPLE Wafer_725C_90min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 675 °C
    RUN Ramp DURATION 1800 TEMP 675->725 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 625..725
    RUN Oxidation DURATION 5400 TEMP 725->725 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 725->675 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 675->675 AMBIENT inert_N2 TIMING bound VARY T_hold 625..725
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 750 °C, 0 min main stage

# Krzeminski, 750 °C, 0 min main stage
SAMPLE Wafer_750C_0min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 700 °C
    RUN Ramp DURATION 1800 TEMP 700->750 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 650..750
    EVENT NoMainStage KIND other TIMING assumed                  # No main stage (the 0 min point)
    RUN RampDown DURATION 600 TEMP 750->700 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 700->700 AMBIENT inert_N2 TIMING bound VARY T_hold 650..750
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 750 °C, 10 min main stage

# Krzeminski, 750 °C, 10 min main stage
SAMPLE Wafer_750C_10min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 700 °C
    RUN Ramp DURATION 1800 TEMP 700->750 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 650..750
    RUN Oxidation DURATION 600 TEMP 750->750 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 750->700 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 700->700 AMBIENT inert_N2 TIMING bound VARY T_hold 650..750
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 750 °C, 30 min main stage

# Krzeminski, 750 °C, 30 min main stage
SAMPLE Wafer_750C_30min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 700 °C
    RUN Ramp DURATION 1800 TEMP 700->750 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 650..750
    RUN Oxidation DURATION 1800 TEMP 750->750 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 750->700 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 700->700 AMBIENT inert_N2 TIMING bound VARY T_hold 650..750
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 750 °C, 60 min main stage

# Krzeminski, 750 °C, 60 min main stage
SAMPLE Wafer_750C_60min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 700 °C
    RUN Ramp DURATION 1800 TEMP 700->750 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 650..750
    RUN Oxidation DURATION 3600 TEMP 750->750 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 750->700 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 700->700 AMBIENT inert_N2 TIMING bound VARY T_hold 650..750
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Krzeminski, 750 °C, 90 min main stage

# Krzeminski, 750 °C, 90 min main stage
SAMPLE Wafer_750C_90min {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    DOPING: p_type_5e15
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # RCA clean, then 1% HF dip
    EVENT Load KIND load TIMING assumed                          # Loaded at 700 °C
    RUN Ramp DURATION 1800 TEMP 700->750 AMBIENT mix:N2=0.909,O2=0.0909 TIMING stated VARY duration 60..1800 VARY T_start 650..750
    RUN Oxidation DURATION 5400 TEMP 750->750 AMBIENT dry_O2 TIMING stated
    RUN RampDown DURATION 600 TEMP 750->700 AMBIENT inert_N2 TIMING stated
    RUN Reflow DURATION 1800 TEMP 700->700 AMBIENT inert_N2 TIMING bound VARY T_hold 650..750
    EVENT Unload KIND other TIMING assumed                       # Unloaded and cooled
    EVENT Measure KIND measurement TIMING assumed                # Ex-situ spectroscopic ellipsometry
}

Massoud (as redrawn), 800 °C (fitted)

# Massoud (as redrawn), 800 °C (fitted)
SAMPLE Massoud_800C {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    STARTING_OXIDE: 1.0_nm
}
CONTROL_TRAJECTORY History {
    EVENT Prepare KIND other TIMING assumed                      # Starting oxide of 1.0 nm
    RUN Oxidation DURATION 4455 TEMP 800->800 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # In-situ ellipsometry
}

Massoud (as redrawn), 850 °C (fitted)

# Massoud (as redrawn), 850 °C (fitted)
SAMPLE Massoud_850C {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    STARTING_OXIDE: 1.0_nm
}
CONTROL_TRAJECTORY History {
    EVENT Prepare KIND other TIMING assumed                      # Starting oxide of 1.0 nm
    RUN Oxidation DURATION 4132.2 TEMP 850->850 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # In-situ ellipsometry
}

Massoud (as redrawn), 900 °C (held out)

# Massoud (as redrawn), 900 °C (held out)
SAMPLE Massoud_900C {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    STARTING_OXIDE: 1.0_nm
}
CONTROL_TRAJECTORY History {
    EVENT Prepare KIND other TIMING assumed                      # Starting oxide of 1.0 nm
    RUN Oxidation DURATION 4598.4 TEMP 900->900 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # In-situ ellipsometry
}

Massoud (as redrawn), 950 °C (fitted)

# Massoud (as redrawn), 950 °C (fitted)
SAMPLE Massoud_950C {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    STARTING_OXIDE: 1.0_nm
}
CONTROL_TRAJECTORY History {
    EVENT Prepare KIND other TIMING assumed                      # Starting oxide of 1.0 nm
    RUN Oxidation DURATION 3162 TEMP 950->950 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # In-situ ellipsometry
}

Massoud (as redrawn), 1000 °C (fitted)

# Massoud (as redrawn), 1000 °C (fitted)
SAMPLE Massoud_1000C {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    STARTING_OXIDE: 1.0_nm
}
CONTROL_TRAJECTORY History {
    EVENT Prepare KIND other TIMING assumed                      # Starting oxide of 1.0 nm
    RUN Oxidation DURATION 1355.4 TEMP 1000->1000 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # In-situ ellipsometry
}

The programs the predictor ran on these histories: experiments/deal_qss/two_equilibria.isl.template, experiments/vitkavage_dit/oxide_growth.isl.template, scripts/search_growth.py (structure generated from the grammar), scripts/search_trap_density.py (structure generated from the grammar).

The model itself, experiments/vitkavage_dit/oxide_growth.isl.template, as it is written. Every equation the simulation uses is in this text; the runtime holds no physics.

# ==============================================================================
# ISL MODEL: DRY THERMAL OXIDATION OF SILICON (growth sector)
#
# Linear-parabolic (Deal-Grove) growth written as one ISL operator. Nothing is
# fitted to the test data: every parameter keeps the value declared here.
#
# Provenance of the constants: the rate constants are an Arrhenius fit to Table II of
# Deal and Grove (1965), which is for (111) silicon; see scripts/derive_deal_grove_constants.py.
# The orientation factor for other faces is not in that paper and remains a hypothesis.
# Fields in {{braces}} are filled by scripts/run_vitkavage_growth.py.
# ==============================================================================
SAMPLE {sample_id} {{
    FAMILY: Si_Planar
    ORIENTATION: Si{orientation}
}}
# Oxide thickness. The initial value is Deal and Grove's effective starting thickness
# for dry oxygen, which stands in for the fast initial regime their model does not describe.
STATE x_ox {{
    UNIT: cm
    INITIAL: x_i
    MIN: 0
    MAX: 1.0e-3
}}
CONTROL T {{ UNIT: K }}
CONTROL p_O2 {{ UNIT: atm }}
# Parabolic rate constant B = B0 * exp(-E_B / kT).
PARAMETER B0 {{ VALUE: 3.6053e-09; UNIT: cm^2 s^-1; STATUS: sourced; SOURCE: Deal and Grove, J. Appl. Phys. 36, 3770 (1965), Table II, dry oxygen on (111), Arrhenius least-squares fit, table agent-read }}
PARAMETER E_B {{ VALUE: 1.2860; UNIT: eV; STATUS: sourced; SOURCE: Deal and Grove, J. Appl. Phys. 36, 3770 (1965), Table II, dry oxygen on (111), Arrhenius least-squares fit, table agent-read }}
# Linear rate constant B/A on (111) = BA0 * exp(-E_BA / kT).
PARAMETER BA0 {{ VALUE: 2.4265e-01; UNIT: cm s^-1; STATUS: sourced; SOURCE: Deal and Grove, J. Appl. Phys. 36, 3770 (1965), Table II, dry oxygen on (111), Arrhenius least-squares fit, table agent-read }}
PARAMETER E_BA {{ VALUE: 2.0303; UNIT: eV; STATUS: sourced; SOURCE: Deal and Grove, J. Appl. Phys. 36, 3770 (1965), Table II, dry oxygen on (111), Arrhenius least-squares fit, table agent-read }}
# Orientation factor on the linear rate, relative to (111).
PARAMETER r_or {{ VALUE: {r_or}; UNIT: 1; STATUS: hypothesis }}
# Effective initial thickness for dry oxygen.
PARAMETER x_i {{ VALUE: 2.3e-6; UNIT: cm; STATUS: sourced; SOURCE: Deal and Grove, J. Appl. Phys. 36, 3770 (1965), stated intercept 230 +/- 30 angstrom for dry oxygen, agent-read }}
# Both rate constants are proportional to oxidant pressure; the constants above are for 1 atm.
PARAMETER p_ref {{ VALUE: 1.0; UNIT: atm; STATUS: constant }}
PARAMETER k_B {{ VALUE: 8.617333262e-5; UNIT: eV K^-1; STATUS: constant }}
CONTROL_TRAJECTORY {trajectory_id} {{
    RUN Oxidation DURATION {duration_s} TEMP {T_C}->{T_C} AMBIENT dry_O2
}}
# dx/dt = B / (2x + A), with A = B / (B/A).
OPERATOR F_LinearParabolicGrowth : OxideGrowth {{
    EQUATION: d(x_ox)/dt = (p_O2 / p_ref) * B0 * exp(-E_B / (k_B * T)) / (2 * x_ox + B0 * exp(-E_B / (k_B * T)) / (r_or * BA0 * exp(-E_BA / (k_B * T))))
}}
OBSERVATION_MODEL Obs_Oxide_Thickness {{
    TARGET: Thickness
    UNIT: cm
    EQUATION: Thickness = x_ox
}}
# Silicon consumed by the oxide grown in this recipe: 0.44 of its thickness, by the ratio of molar volumes.
OBSERVATION_MODEL Obs_Consumed_Silicon {{
    TARGET: Consumed_substrate
    UNIT: cm
    EQUATION: Consumed_substrate = 0.44 * (x_ox - x_i)
}}
EVALUATION_CONTRACT {contract_id} {{
    FROZEN: true
    DATASET: data/derived/vitkavage_1990_thickness.csv
    DATASET_DIGEST: sha256:{dataset_sha256}
    SPLIT: test
    TARGET: Thickness = {target} +/- {tolerance} rel
}}

Our simulation against the paper

107 within, 19 near, 1 off of 127 numbers. Tolerance is stated per row; nothing was fitted for this page.

Aligning the simulation: what the unstated things would have to be

Every step whose timing or temperature the paper leaves open carries a range in the history above (VARY). scripts/align_study.py searched those ranges against the paper's numbers through the same predictor, first one unknown at a time, then all together. No parameter a program declares fitted was touched. The unknowns were chosen on k725_0, k725_10, k725_25, k725_30, k725_45, k725_60, k725_90, massoud_1000, massoud_800, massoud_850, massoud_900, massoud_950 only; every other number is a test they played no part in.

Together: Ramp duration 1800 s (was 1800); Ramp T start 33.33 °C shift (was 0); Reflow T hold 0 °C shift (was 0).

As reproduced: 107 of 127 within tolerance, median error 5.1%. Aligned: 108 of 127 within tolerance, median error 5.1%. On the held-out samples only: 5 of 5 within tolerance, median error 9.3% as reproduced, 5 of 5 within tolerance, median error 8.6% aligned.

Choices of program or material property, each scored on its own and on top of the aligned history:

What was searched. Three unknowns on the paper's own wafers: the ramp's duration (60 s to 30 min, standing in for how much oxygen the ramp delivers, since no program reads the 9% fraction), the ramp's start temperature and the reflow temperature (each ±50 °C about Fig. 9's 50 °C below). They were chosen on the 725 °C wafers (with Massoud's 115 points, which no unknown touches) and tested on the five 750 °C wafers. Massoud's curves are unchanged throughout: 97 of 115 within tolerance, the same as the search's own 79 of 90 fitted and 18 of 25 held out at 900 °C.

Written as stated, the cycle makes the program run high, not low. With the ramp, ramp-down and reflow, 4 of 7 at 725 °C (median +9.9%) and 2 of 5 at 750 °C (median +20.8%). Three things add oxide that the paper's wafers did not get: the program starts at its fitted 1.71 nm rather than an HF-dipped surface; it grows the ramp as if in pure oxygen (0.32 nm at 725 °C, 0.57 nm at 750 °C); and it grows 0.2 to 0.4 nm in the 40 min of nitrogen, where the paper expects nothing. The 0 min wafers show it plainly: 2.28 against 1.53 nm and 2.71 against 1.96 nm.

What the unknowns would have to be. The search shortens the ramp to 580 s (about a third of the stated 30 min) and lowers its start and the reflow by 33 °C. The 725 °C wafers barely choose: every ramp from 60 s to 580 s gives 102 or 103 of 122, and the temperatures move one point. On the held-out 750 °C wafers the aligned history gives 5 of 5 (median +4.6%), against 2 of 5 as stated. This is not a measurement of the ramp. A 580 s ramp in pure oxygen stands in for a 1800 s ramp at 9% oxygen, a program start that is too thick, and growth in nitrogen that should not happen; the three cannot be told apart with a program that reads no gas.

The variants say the same. The stored recipe (main stage only): 4 of 6 at 725 °C (median −7.5%), 2 of 4 at 750 °C (−10.2%). Leaving out the nitrogen steps, as the paper's “no growth is expected” would: 5 of 7 (+2.1%) and 4 of 5 (+8.0%). Leaving out the ramp and keeping the nitrogen steps: 6 of 6 and 4 of 4, the best of all, because the program's growth in nitrogen (an artefact) happens to fill the main stage's shortfall. None of these is chosen: the best-scoring one is the least physical.

5 within of 5 numbers with the aligned settings (the samples used to choose the unknowns are left out).

What alignment cannot fix. The program's main-stage growth is too slow below its fitted range: from the measured thickness at time zero (side calculation, below) the law adds 70% of the oxide the 725 °C wafers gained in 90 min and 88% at 750 °C. No unknown in the history touches that, because the shortfall comes after time zero. The ramp's own 1.53 and 1.96 nm are beyond any program we hold: from an HF-dipped surface (0.3 to 0.5 nm) the fitted law grows to at most 1.0 nm in 30 min of pure oxygen over the 725 °C ramp, and 1.4 nm over the 750 °C ramp.

How to read the aligned numbers: the 725 °C rows chose the unknowns and are not in the chart; the five 750 °C rows are the test. Massoud's rows are scored the same way before and after, and 90 of them were used to fit the program, so they are not a test either.

What our language and ontology cannot yet say

• A starting oxide a program reads. The single most useful addition. The thin-oxide program starts every history at its fitted 1.71 nm, so Massoud's 1.0 nm wafers, Krzeminski's HF-dipped wafers and a 1.53 nm post-ramp oxide all start the same. With STARTING_OXIDE read by the growth programs (default: the program's fitted value), the search's other-laboratory test (start at the measured 1.53 nm) and the predictor would give one answer instead of two.
• Growth programs that read the oxygen pressure. The thin-oxide program has no p_O2 in its rate. It grows in nitrogen (0.2 nm in a 30 min reflow at 675 °C) and grows a 9% oxygen ramp as pure oxygen. Multiplying B (or the power law's constant) by p_O2/p_ref, as the 1965 program does, would make the stated cycle meaningful; the alignment's ramp of 580 s is a symptom.
• Time zero as a statement. The datasets count time from the start of the main stage; the language counts from the first RUN. A marker such as EVENT TimeZero KIND other that results can refer to, or a t0 on the dataset rows, would let the stored recipe hold the whole cycle without moving every time.
• A curve of separate wafers. Each of the paper's points is a wafer that went through the whole cycle; Massoud's are one wafer watched in situ. The study format can sample in situ (curves within a RUN) but needs one case per wafer for the ex-situ points. The language should say which kind of series a dataset is.
• A fast first regime. Neither the 1965 law nor the searched power law grows the first 1.5 nm of a bare surface; both fold it into a starting thickness. An operator for it (or a second state for the first monolayers) is what Krzeminski's ramp asks for, and what its own reaction-rate model provides.
• Exposure to air before an ex-situ measurement. An uncaptioned figure shows 0.25–0.37 nm of growth in 90 min of air. A storage EVENT with a duration and an ambient (room air) that a program can read would carry it.

With heating and cooling transitions

No paper we hold states how its wafers heated up or cooled down. The project owner specified scenarios for one Vitkavage wafer (isl_harness/vitkavage_transitions.py); isl_harness/thermal_transitions.py applies the same numbers to every sample here: a heat-up in nitrogen before the first furnace step (a hot push of 1 to 5 min, or a cold-furnace ramp of 60 to 120 min), the owner's radiative cooling knots after the last furnace step or after the paper's own pull (fast, or every time multiplied by 5), and ramps around an anneal at 300 to 500 °C. The paper's own steps are kept as written. All transitions are assumptions, written in the history with TIMING assumed.

The paper's own conclusions

What the reproduction shows

Does writing the ramp explain the low result? No. The 13% shortfall at 725 °C is in the growth after time zero, not in where the curve starts. Run from the measured 1.53 nm at the start of the main stage (a side calculation with the fitted law, integrated outside the predictor, which reproduces the search's numbers), the law adds 0.20 nm in the first 10 min and 1.51 nm in 90 min; the paper's wafers added 0.35 and 2.17 nm. At 750 °C it adds 2.31 against 2.61 nm. The ramp happened before time zero and is already in the 1.53 nm; whatever the ramp does, the main-stage increments are 30% short at 725 °C and 12% short at 750 °C. What would explain it is something that adds oxide after time zero to every wafer but the 0 min one, or a law that is too slow below the 800 °C it was fitted at. The second is what the paper itself says of Deal–Grove and Massoud's correction.

What the ramp would take. From an HF-dipped surface (0.3 to 0.5 nm) the fitted law grows 0.37 to 0.56 nm in the ramp at the stated 9% oxygen (if the rate scales with oxygen pressure), and 0.90 to 1.03 nm even in pure oxygen at 725 °C; the paper measured 1.53 nm. The law has no fast first regime: on Massoud's data that regime is folded into its fitted 1.71 nm start. The ramp's oxide is that regime, grown in a stated history, and no program we hold can grow it.

The thin-oxide program as the predictor runs it (generated by scripts/search_growth.py with the fitted values the predictor applies; the sample and recipe are replaced by the history's). It reads no gas and no sample field: every history starts at 10^log_xi cm = 1.71 nm and grows in any RUN, nitrogen included.

# GENERATED by scripts/search_growth.py. Structure: power, activation energy own.
# A candidate from the grammar, not from a paper. Every parameter is fitted.
SAMPLE History { FAMILY: Si_Planar }
STATE x_ox { UNIT: cm; INITIAL: x_unit * exp(ln10 * log_xi); MIN: 0; MAX: 1.0e-3 }
CONTROL T { UNIT: K }

PARAMETER log_k { VALUE: -8.04985; UNIT: 1; STATUS: fitted } PARAMETER E_k { VALUE: 2.06976; UNIT: eV; STATUS: fitted } PARAMETER n { VALUE: 0.545154; UNIT: 1; STATUS: fitted } PARAMETER log_xi { VALUE: -6.76621; UNIT: 1; STATUS: fitted } PARAMETER x_i { VALUE: 2e-08; UNIT: cm; STATUS: constant } PARAMETER x_unit { VALUE: 1.0; UNIT: cm; STATUS: constant } PARAMETER T_start { VALUE: 998.15; UNIT: K; STATUS: constant } PARAMETER x_ref { VALUE: 1.0e-6; UNIT: cm; STATUS: constant } PARAMETER T_ref { VALUE: 1373.15; UNIT: K; STATUS: constant } PARAMETER v_unit { VALUE: 1.0; UNIT: cm s^-1; STATUS: constant } PARAMETER B_unit { VALUE: 1.0; UNIT: cm^2 s^-1; STATUS: constant } PARAMETER ln10 { VALUE: 2.302585092994046; UNIT: 1; STATUS: constant } PARAMETER k_B { VALUE: 8.617333262e-5; UNIT: eV K^-1; STATUS: constant } CONTROL_TRAJECTORY History_History { RUN Oxidation DURATION 60.0 TEMP 725.0->725.0 AMBIENT dry_O2 } OPERATOR F_Growth : OxideGrowth { EQUATION: d(x_ox)/dt = v_unit exp(ln10 log_k) exp(-(E_k / k_B) (1 / T - 1 / T_ref)) exp(-n log(x_ox / x_ref)) } OBSERVATION_MODEL Obs { TARGET: Thickness; UNIT: cm; EQUATION: Thickness = x_ox } EVALUATION_CONTRACT Contract_History { FROZEN: true DATASET: data/derived/silicon_dry_kinetics.csv DATASET_DIGEST: sha256:07f03482625934e8719e93e520d6be7807d5032b36c6a964873ddbb2d1e3abcb SPLIT: search TARGET: Thickness @ 60.0 = 1e-06 +/- 1e-07 abs } ```

What we would like Emily to check

1. The ramp in 9% oxygen grows 1.53 nm in 30 min at 675–725 °C from an HF-dipped surface, while the main stage in pure oxygen at 725 °C adds only 0.35 nm in the next 10 min. Is that the fast first regime of a bare surface, and should a program treat the ramp's oxide as a starting oxide rather than grow it?
2. Does the main stage keep the nitrogen flow (0.5 atm oxygen) or switch to pure oxygen?
3. Does a 30 min reflow at 675 °C in nitrogen grow anything on a 2–4 nm oxide? The paper expects not; our thin-oxide program grows 0.2 nm in it because it does not read the gas.
4. The paper's thickness at 30 min is 2.71 nm in Table II and 2.59 nm in the figure. Which is right?
5. Is the ex-situ growth in air (0.25–0.37 nm over 90 min in an uncaptioned figure) part of every point here?