Reproducing Deal, Sklar, Grove and Snow 1967: fixed charge and the last hot step

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/deal_1967.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

B. E. Deal, M. Sklar, A. S. Grove and E. H. Snow, “Characteristics of the Surface-State Charge (Qss) of Thermally Oxidized Silicon”, J. Electrochem. Soc. 114, 266 (1967).

The origin of the statement that fixed charge is set by the last hot step. Its Table I (charge against pull time) is what our pull model was fitted on, and its Tables II and III give 16 values in the graph. The fixed-charge program the predictor runs was fitted to another laboratory's anneals and uses this paper's steady-state line as a sourced input.

How well can this study be reproduced?

Moderate. The charge values are printed and the pull times are given to the second, but the paper gives a pull as tube travel time, never as wafer temperature, and gives an oxidation time for one table only. The quantity that carries the result, how the slice cools, is the one thing a simulation must invent.

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. 3 same, 4 approximated, 6 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.

(111), 1200 °C, standard fast pull

Simulated time in this history: 1 h 03 min 33 s.

Across the samples

Timing the paper does not give

• The wafer's temperature against time during a pull. The paper gives the time to travel from the centre of the furnace to the end of the tube; where and in what gas the slice then cools is not said.
• The tube length: 50 cm in the text, 54 cm in the Table I caption, 53.3 cm from the tabulated rate times the time.
• Oxidation time for everything except Table I (60 min). Tables II and III give neither time nor thickness.
• How the temperature was changed between the 1200 °C oxidation and the lower-temperature hold in oxygen: the same tube cooling, or a move to a second furnace. “Immediately” is all the paper says.
• The pull after an anneal (speed and gas).
• The 550 °C nitrogen treatment after metallization is “2-5 min”.

What the second reading changed

• The recipe stored with each of these measurements is one step: the pull, or the anneal. The oxidation that came before it (1200 °C, 60 min, dry oxygen, 0.2 µm) is absent, and so is the pull that must follow an anneal.
• The argon anneal at 920 °C is stored as inert_N2. The language has had inert_Ar since the SiC work; the row was never updated.
• Metallized samples were heated at 550 °C for 2 to 5 min in dry nitrogen before measurement. No stored recipe has this step. Samples measured with a gold probe did not get it, and the paper does not say which table values are which.
• Table I's fast-pull value is 1.9 × 10^11 cm^-2 and Tables II to IV give 1.7 for the same recipe; the slow pull of 2 min in Table III (7.4) sits beside 7.3 at 1.0 min and 8.9 at 2.8 min in Table I. The paper's own repeatability is about 0.5 × 10^11.
• The end temperature of the pull (400 °C) is not from the paper. It was chosen by us as the best of six candidates against Table I, so the pull rows are fitted in one more number than their record says.

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>.

(111), 1200 °C, standard fast pull

# (111), 1200 °C, standard fast pull
SAMPLE Slice_111_pull_3s {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 3 TEMP 1200->400 AMBIENT dry_O2 TIMING bound VARY T_end 25..700 VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, pulled in 10 s

# (111), 1200 °C, pulled in 10 s
SAMPLE Slice_111_pull_10s {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 10.02 TEMP 1200->400 AMBIENT dry_O2 TIMING stated VARY T_end 25..700
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, pulled in 30 s

# (111), 1200 °C, pulled in 30 s
SAMPLE Slice_111_pull_30s {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 30 TEMP 1200->400 AMBIENT dry_O2 TIMING stated VARY T_end 25..700
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, pulled in 1 min

# (111), 1200 °C, pulled in 1 min
SAMPLE Slice_111_pull_60s {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 60 TEMP 1200->400 AMBIENT dry_O2 TIMING stated VARY T_end 25..700
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, pulled in 2.8 min

# (111), 1200 °C, pulled in 2.8 min
SAMPLE Slice_111_pull_168s {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 168 TEMP 1200->400 AMBIENT dry_O2 TIMING stated VARY T_end 25..700
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, pulled in 34 min

# (111), 1200 °C, pulled in 34 min
SAMPLE Slice_111_pull_2040s {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 2040 TEMP 1200->400 AMBIENT dry_O2 TIMING stated VARY T_end 25..700
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, then oxygen at 900 °C

# (111), 1200 °C, then oxygen at 900 °C
SAMPLE Slice_111_O2_900 {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN OxygenHold DURATION 600 TEMP 900->900 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 3 TEMP 900->400 AMBIENT dry_O2 TIMING bound VARY T_end 25..700 VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, then oxygen at 550 °C

# (111), 1200 °C, then oxygen at 550 °C
SAMPLE Slice_111_O2_550 {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN OxygenHold DURATION 5400 TEMP 550->550 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 3 TEMP 550->400 AMBIENT dry_O2 TIMING bound VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 1200 °C, then nitrogen at 1200 °C

# (111), 1200 °C, then nitrogen at 1200 °C
SAMPLE Slice_111_N2_1200 {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN NitrogenAnneal DURATION 600 TEMP 1200->1200 AMBIENT inert_N2 TIMING stated
    RUN Pull DURATION 3 TEMP 1200->400 AMBIENT inert_N2 TIMING bound VARY T_end 25..700 VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(111), 920 °C, then argon at 920 °C

# (111), 920 °C, then argon at 920 °C
SAMPLE Slice_111_Ar_920 {
    FAMILY: Si_Planar
    ORIENTATION: Si111
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation920 DURATION 3600 TEMP 920->920 AMBIENT dry_O2 TIMING assumed VARY duration 1800..86400
    RUN ArgonAnneal DURATION 1800 TEMP 920->920 AMBIENT inert_Ar TIMING stated
    RUN Pull DURATION 3 TEMP 920->400 AMBIENT inert_Ar TIMING bound VARY T_end 25..700 VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(100), 1200 °C, standard fast pull

# (100), 1200 °C, standard fast pull
SAMPLE Slice_100_pull_3s {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 3 TEMP 1200->400 AMBIENT dry_O2 TIMING bound VARY T_end 25..700 VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(100), 1200 °C, then oxygen at 550 °C

# (100), 1200 °C, then oxygen at 550 °C
SAMPLE Slice_100_O2_550 {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation DURATION 3600 TEMP 1200->1200 AMBIENT dry_O2 TIMING stated
    RUN OxygenHold DURATION 5400 TEMP 550->550 AMBIENT dry_O2 TIMING stated
    RUN Pull DURATION 3 TEMP 550->400 AMBIENT dry_O2 TIMING bound VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

(100), 920 °C, then argon at 920 °C

# (100), 920 °C, then argon at 920 °C
SAMPLE Slice_100_Ar_920 {
    FAMILY: Si_Planar
    ORIENTATION: Si100
    CLEAN: not_stated
    DOPING: p_type_boron_1p4e16
}
CONTROL_TRAJECTORY History {
    EVENT Polish KIND clean TIMING assumed                       # Solvent clean and chemical polish
    EVENT Load KIND load TIMING assumed                          # Slices flat on a quartz boat, into the tube
    RUN Oxidation920 DURATION 3600 TEMP 920->920 AMBIENT dry_O2 TIMING assumed VARY duration 1800..86400
    RUN ArgonAnneal DURATION 1800 TEMP 920->920 AMBIENT inert_Ar TIMING stated
    RUN Pull DURATION 3 TEMP 920->400 AMBIENT inert_Ar TIMING bound VARY T_end 25..700 VARY duration 1..3
    EVENT Strip KIND etch TIMING assumed                         # 50–100 Å of oxide removed in dilute HF, back oxide removed
    EVENT Metal KIND metallization TIMING assumed                # Aluminium field plates, 375 µm
    RUN MetalAnneal DURATION 210 TEMP 550->550 AMBIENT inert_N2 TIMING range VARY duration 120..300
    EVENT CV KIND measurement TIMING assumed                     # C-V at about 100 kc, 25 °C
}

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/deal_qss/two_equilibria.isl.template, as it is written. Every equation the simulation uses is in this text; the runtime holds no physics.

# ==============================================================================
# ISL MODEL: FIXED CHARGE WITH AN EQUILIBRIUM IN EACH GAS
#
# In oxygen the charge moves toward a steady value that falls with temperature (the
# line of Deal et al. 1967, unchanged, with the oxygen kinetics fitted there).
# In a gas without oxygen it moves toward a lower equilibrium that also falls with
# temperature, Qi_ref * exp(E_q / k_B * (1/T - 1/T_iref)), at a rate of its own.
#
# The inert clock and the (100) inert equilibrium are fitted to the argon time series
# of Akinwande, Ho and Plummer (1982). The (111) inert equilibrium is fitted to their
# cycling experiment, with the clock carried over from (100). The model is then
# tested, unchanged, on the Fairchild wafers of 1967 and 1980.
# Fields in {{braces}} are filled by scripts/run_fixed_charge_two_lines.py.
# ==============================================================================
SAMPLE Wafer_{tag} {{
    FAMILY: Si_Planar
    ORIENTATION: Si{orientation}
}}
# The wafer starts at the steady value for its oxidation temperature.
STATE Q_f {{ UNIT: cm^-2; INITIAL: Q_hot + slope * (T_hot - T_ox); MIN: 0; MAX: 1.0e14 }}
CONTROL T {{ UNIT: K }}
CONTROL p_O2 {{ UNIT: atm }}
PARAMETER Q_hot {{ VALUE: {Q_hot}; UNIT: cm^-2; STATUS: sourced; SOURCE: Deal et al. 1967 Table II, steady value after dry oxidation at 1200 C }}
PARAMETER slope {{ VALUE: {slope}; UNIT: cm^-2 K^-1; STATUS: sourced; SOURCE: Deal et al. 1967 Table II, line through the 1200 C and 920 C steady values }}
PARAMETER T_hot {{ VALUE: 1473.15; UNIT: K; STATUS: constant }}
PARAMETER T_ox {{ VALUE: {T_ox_K}; UNIT: K; STATUS: constant }}
# ---- fitted to the pull-rate series ------------------------------------------------
PARAMETER k0 {{ VALUE: 1.0e3; UNIT: s^-1; STATUS: fitted }}
PARAMETER E_a {{ VALUE: 1.0; UNIT: eV; STATUS: fitted }}
# ---- fitted to the argon anneals of Akinwande, Ho and Plummer (1982) -----------------
PARAMETER k_i0 {{ VALUE: 1.0e1; UNIT: s^-1; STATUS: fitted }}
PARAMETER E_i {{ VALUE: 0.9; UNIT: eV; STATUS: fitted }}
PARAMETER Qi_ref {{ VALUE: 2.5e10; UNIT: cm^-2; STATUS: fitted }}
PARAMETER E_q {{ VALUE: 0.4; UNIT: eV; STATUS: fitted }}
PARAMETER T_iref {{ VALUE: 1273.15; UNIT: K; STATUS: constant }}
PARAMETER p_ref {{ VALUE: 1.0; UNIT: atm; STATUS: constant }}
PARAMETER k_B {{ VALUE: 8.617333262e-5; UNIT: eV K^-1; STATUS: constant }}
# ---- where the exchange with oxygen stops as the wafer cools (hypothesis, added 10 November 2026) ----------------------
# The oxygen rate above was fitted with an activation energy at its lower bound (0.05 eV), so on its own it keeps moving
# charge at room temperature. Every history it was fitted and tested on ended its pull at 400 C, a cut-off that stood in for
# this. Reproducing Stanford 1982, Deal 1967 and Deal and Razouk 1980 with a wafer cooling to room temperature made the charge
# 3 to 5 times too high. The cut-off is now stated: the exchange is switched off over about 20 K around T_freeze. At 400 C it
# is 99% on, so every fitted and tested history is unchanged; how high the true freezing point is, is an open question
# (the Deal and Razouk alignment prefers pulls ending near 530 C).
PARAMETER T_freeze {{ VALUE: 623.15; UNIT: K; STATUS: hypothesis }}
PARAMETER w_freeze {{ VALUE: 10.0; UNIT: K; STATUS: hypothesis }}
QUANTITY rate {{ UNIT: s^-1; EQUATION: rate = k0 * exp(-E_a / (k_B * T)) / (1 + exp(-(T - T_freeze) / w_freeze)) }}
QUANTITY rate_inert {{ UNIT: s^-1; EQUATION: rate_inert = k_i0 * exp(-E_i / (k_B * T)) }}
QUANTITY Q_inert {{ UNIT: cm^-2; EQUATION: Q_inert = Qi_ref * exp((E_q / k_B) * (1 / T - 1 / T_iref)) }}
QUANTITY oxidizing {{ UNIT: 1; EQUATION: oxidizing = p_O2 / p_ref }}
QUANTITY Q_steady {{ UNIT: cm^-2; EQUATION: Q_steady = Q_hot + slope * (T_hot - T) }}
CONTROL_TRAJECTORY History_{tag} {{
{runs}
}}
# In oxygen the charge follows the temperature-dependent steady value.
OPERATOR F_OxidizingEquilibration : Relaxation {{
    EQUATION: d(Q_f)/dt = -rate * oxidizing * (Q_f - Q_steady)
}}
# Without oxygen it relaxes, at its own rate, toward an equilibrium of its own that falls as temperature rises.
# Akinwande, Ho and Plummer (1982) show the annealed charge cycling reversibly with temperature in argon.
OPERATOR F_InertRelaxation : Relaxation {{
    EQUATION: d(Q_f)/dt = -rate_inert * (1 - oxidizing) * (Q_f - Q_inert)
}}
OBSERVATION_MODEL Obs_Fixed_Charge {{ TARGET: Qss; UNIT: cm^-2; EQUATION: Qss = Q_f }}
EVALUATION_CONTRACT Contract_{tag} {{
    FROZEN: true
    DATASET: data/derived/deal_1967_qss.csv
    DATASET_DIGEST: sha256:{dataset_sha256}
    SPLIT: {split}
    TARGET: Qss = {target} +/- {tolerance} abs
}}

Our simulation against the paper

9 within, 4 near of 13 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 111_pull_10s, 111_pull_168s, 111_pull_2040s, 111_pull_30s, 111_pull_3s, 111_pull_60s only; every other number is a test they played no part in.

Together: Pull T end 400 °C (was 400); Pull duration 1 s (was 3); MetalAnneal duration 180 s (was 210).

As reproduced: 9 of 13 within tolerance, median error 12.5%. Aligned: 10 of 13 within tolerance, median error 9.3%. On the held-out samples only: 4 of 7 within tolerance, median error 13.6% as reproduced, 4 of 7 within tolerance, median error 13.1% aligned.

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

What was searched. Four things the paper leaves open: the temperature at which a pull ends (25 to 700 °C, one value for every pull that starts hot), the length of the standard fast pull (“less than 3 sec”: 1 to 3 s), the 550 °C nitrogen treatment after metal (“2-5 min”) and the 920 °C oxidation time (not given; 30 min to 24 h). They were chosen on Table I's six pulls only; the seven values of Tables II and III are the test.

The pull end temperature is pinned, and 400 °C stands. Scanned alone on Table I it passes 2 of 6 at 25 °C, 2 at 137 °C, 3 at 250 °C, 5 at 362 and 400 °C, 6 at 475 °C, 2 at 587 °C and 2 at 700 °C. The window that works is about 360 to 480 °C; outside it the six pulls cannot all be matched by any one end temperature. Combined with the other unknowns the search returned to 400 °C. This number is a fit: it was first chosen against Table I when the pull model was built, and it is chosen against Table I again here.

The other three unknowns barely matter. The fast-pull length moves the median error on Table I from 3.5% (3 s) to 3.1% (1 s) and changes no verdict. The 550 °C nitrogen treatment moves one Table I value across its tolerance (6 of 6 at 2 to 2.5 min, 5 of 6 at 3 to 5 min): the program's charge relaxes slightly without oxygen even at 550 °C. The 920 °C oxidation time changes nothing, because the charge program starts from its steady state and does not simulate the oxidation. Together: pull 1 s, metal treatment 2.5 min, pull end 400 °C, the 920 °C oxidation left at 60 min (its scan is flat).

On the held-out tables nothing improves. Tables II and III: 4 of 7 within tolerance as reproduced and 4 of 7 aligned, median error 13.6% and 13.1%. The (111) hold in oxygen at 900 °C moves inside its tolerance (5.02 against 4.6 × 10^11 cm^-2) and the (100) hold at 550 °C moves just outside (3.70 against 4.2). The nitrogen anneal at 1200 °C stays at about half the paper's value (0.97 against 1.8) and the (100) argon anneal at 920 °C stays at half (0.36 against 0.7): those are the program's equilibria without oxygen, which no timing unknown reaches.

The shape of the pull. An exponential fall toward 25 °C with a time constant equal to the pull time does as well as the straight fall to 400 °C (9 of 13, median 12.5%; 9.7% on the aligned history); it ends each pull at about 460 °C. So does an exponential fall toward 400 °C with a time constant of half the pull time (9 of 13). A faster exponential, time constant 0.3 × the pull time toward 25 °C, ends below 100 °C and fails (5 of 13, median 28.8%, simulation high). The data cannot tell a linear from an exponential pull; what they fix is that the charge must stop moving once the slice is somewhere around 400 to 480 °C.

Cooling to room temperature: a finding about the program, not the paper. The program's oxygen rate was fitted with an activation energy of 0.05 eV (k0 = 0.081 s^-1), so in oxygen its time constant is about 18 s at 1200 °C and still about 90 s at 25 °C, and the value it relaxes toward is the steady-state line extrapolated down in temperature: 14.3 × 10^11 cm^-2 on (111) and 6.5 × 10^11 on (100) at 25 °C. Twenty more seconds of cooling in oxygen after the pull takes the score from 9 of 13 to 2 of 13 (median 72%, every value high); ten minutes of cooling to 25 °C in oxygen gives 1 of 13 (median 210%), and in air 1 of 13 (median 172%). The same ten minutes in nitrogen changes nothing (9 of 13). A slice in 1967 did cool to room temperature, in oxygen or air, and kept 1.9 × 10^11. So the program reproduces Table I only because its histories stop at 400 °C: the end temperature of the pull is standing in for an activation energy the oxygen kinetics do not have.

4 within, 3 near of 7 numbers with the aligned settings (the samples used to choose the unknowns are left out).

Alignment cannot fix the oxygen kinetics. With a 0.05 eV activation energy the charge keeps following the extrapolated steady-state line at any temperature, which no measurement supports below about 550 °C (the lowest temperature in Table III). A freeze-out is needed: a rate with a real activation energy (refitted on Table I with histories that cool to room temperature), or a steady-state line that stops falling below some temperature. That is a change to a fitted program, which this page does not make.

Alignment cannot fix the anneals without oxygen either: nitrogen at 1200 °C and argon at 920 °C on (100) come out at about half the paper's values whatever the timing, because the inert equilibrium was fitted to another laboratory's wafers.

What our language and ontology cannot yet say

• A pull as a motion through a furnace profile: PULL DISTANCE 54 cm DURATION t PROFILE <furnace axial temperature profile>, so that the wafer's temperature against time follows from the paper's own words (time and distance) instead of an end temperature and a shape we invent. The paper gives distance and time; we have to write a temperature ramp.
• A history that ends at the measurement temperature by default (a final cooling to 25 °C written explicitly, with its gas). Our histories stop at the end of the last hot step, which is what hid the low-temperature behaviour of the charge program.
• A way to search a PROFILE TAU as an unknown (VARY tau <lo>..<hi>, or VARY tau_factor relative to the duration). The exponential pull had to be tried as a handful of declared variants.
• A tag for which samples were measured with a gold probe on bare oxide and which were metallized and given the 550 °C treatment: an optional step (EVENT … OPTIONAL, or two histories under one measured value), since the paper does not say which tabulated value is which.
• An ambient for “the end of the tube”: where the slice sits after the pull (open tube, flowing oxygen, room air) is part of the history and has no word.

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 we would like Emily to check

1. What does a slice's temperature do during and after a pull of this kind (a thin quartz boat, about 50 cm of tube)? A linear fall to 400 °C over the pull time is the weakest assumption on this page.
2. Between the 1200 °C oxidation and a hold at 900 or 550 °C, was the practice of the time to move the boat to a second furnace?
3. Does the 550 °C nitrogen treatment after aluminium change the fixed charge, in your experience, or only the interface traps?