Reproducing Hijikata et al. 2013: growth rate against thickness on both faces of 4H-SiC

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/hijikata_2013.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

Y. Hijikata, S. Yagi, H. Yaguchi and S. Yoshida, “Thermal Oxidation Mechanism of Silicon Carbide”, chapter 7 in Physics and Technology of Silicon Carbide Devices (InTech, 2013), doi:10.5772/50748.

The second SiC growth source for challenge system 3, and the only one that measures the thin-oxide regime. Song 2004, whose constants the SiC growth program uses, measured thickness ex situ at 0.25 to 6 h; Hijikata's group measured the growth rate continuously by in-situ ellipsometry and found it higher below about 20 nm than a Deal-Grove law fitted to the thick part. This page runs the SiC growth program (Song's constants, dry, 950 to 1150 °C) on each in-situ run and compares thickness against time, rebuilt from the measured rates, and the rate itself at each thickness read.

How well can this study be reproduced?

Moderate. A review chapter: the process is one sentence (dry O2, 1 atm, a temperature, a face) and every experimental detail is deferred to the authors' earlier papers. What it gives that no other source in our library does is the growth rate measured in situ against thickness, from about 1 nm, on both faces at 4 to 8 temperatures. No times are given: thickness against time has to be rebuilt by integrating the rates, from an unknown time zero.

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, 1 approximated, 2 assumed, 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.

C face (000-1), 890 °C

Simulated time in this history: 17 h 43 min 03 s.

Across the samples

Timing the paper does not give

• No time axis anywhere: every run's duration and time zero are unknown; the durations here are reconstructed by integrating the measured rates.
• Heat-up gas and time before oxygen is admitted (in refs [4, 5, 8-10]).
• The C-face 1 atm runs begin at 5 to 12 nm because the early growth was too fast to follow (p. 16): the time to reach the first point is unknown.
• Cooling and unload: not described (the measurement is in situ, so they do not affect the data).

What the second reading changed

• There is no stored recipe or dataset for this chapter (ontology/library.json: status skimmed, used by nothing). SOURCE.md lists Figures 3, 5, 7 and 9 as the ones with data; Figure 1 (C face at 8 temperatures, linear axes) is the most complete and is used here.
• Figure 3's Si face at 1100 °C and Figure 5's 1100 °C curve agree point for point; they are taken to be the same run and only Figure 5 is used.
• Figure 5's y axis labels only 10⁻⁶ µm/s; the decade was taken from the minor-tick spacing and checked against Figure 3 at 1100 °C (both give 4.0 nm/h at 10 to 40 nm).
• The temperature of Figures 7 to 9 (partial pressures) is not printed in their captions; they are not used.

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>.

C face (000-1), 890 °C

# C face (000-1), 890 °C
SAMPLE SiC_C_890 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 1348.04 TEMP 890->890 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 62435.3 TEMP 890->890 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

890 °C is outside the 950 to 1150 °C of the growth program's constants. Thickness against time reconstructed from Figure 1's growth rates (8 points read; 7 scored).

C face (000-1), 930 °C

# C face (000-1), 930 °C
SAMPLE SiC_C_930 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 265.148 TEMP 930->930 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 44722 TEMP 930->930 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

930 °C is outside the 950 to 1150 °C of the growth program's constants. Thickness against time reconstructed from Figure 1's growth rates (15 points read; 8 scored).

C face (000-1), 970 °C

# C face (000-1), 970 °C
SAMPLE SiC_C_970 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 149.871 TEMP 970->970 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 29214.5 TEMP 970->970 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 1's growth rates (23 points read; 8 scored).

C face (000-1), 1000 °C

# C face (000-1), 1000 °C
SAMPLE SiC_C_1000 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 721.739 TEMP 1000->1000 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 20494.3 TEMP 1000->1000 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 1's growth rates (18 points read; 8 scored).

C face (000-1), 1050 °C

# C face (000-1), 1050 °C
SAMPLE SiC_C_1050 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 885.321 TEMP 1050->1050 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 14806.2 TEMP 1050->1050 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 1's growth rates (16 points read; 8 scored).

C face (000-1), 1090 °C

# C face (000-1), 1090 °C
SAMPLE SiC_C_1090 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 732.69 TEMP 1090->1090 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 17077.4 TEMP 1090->1090 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 1's growth rates (18 points read; 8 scored).

C face (000-1), 1150 °C

# C face (000-1), 1150 °C
SAMPLE SiC_C_1150 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 580.05 TEMP 1150->1150 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 17304.6 TEMP 1150->1150 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 1's growth rates (19 points read; 8 scored).

C face (000-1), 1220 °C

# C face (000-1), 1220 °C
SAMPLE SiC_C_1220 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 165.675 TEMP 1220->1220 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 5078.68 TEMP 1220->1220 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

1220 °C is outside the 950 to 1150 °C of the growth program's constants. Thickness against time reconstructed from Figure 1's growth rates (20 points read; 8 scored).

C face (000-1), 1090 °C (Figure 3)

# C face (000-1), 1090 °C (Figure 3)
SAMPLE SiC_C_1090_fig3 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 604.544 TEMP 1090->1090 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 73492.1 TEMP 1090->1090 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 3's growth rates (26 points read; 8 scored).

Si face (0001), 1050 °C

# Si face (0001), 1050 °C
SAMPLE SiC_Si_1050 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 927.924 TEMP 1050->1050 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 27691.2 TEMP 1050->1050 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 5's growth rates (22 points read; 8 scored).

Si face (0001), 1080 °C

# Si face (0001), 1080 °C
SAMPLE SiC_Si_1080 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 340.479 TEMP 1080->1080 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 31744 TEMP 1080->1080 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 5's growth rates (26 points read; 8 scored).

Si face (0001), 1100 °C

# Si face (0001), 1100 °C
SAMPLE SiC_Si_1100 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 278.856 TEMP 1100->1100 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 31527 TEMP 1100->1100 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

Thickness against time reconstructed from Figure 5's growth rates (26 points read; 8 scored).

Si face (0001), 1200 °C

# Si face (0001), 1200 °C
SAMPLE SiC_Si_1200 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
}
CONTROL_TRAJECTORY History {
    EVENT Load KIND load TIMING assumed                          # Loaded into the in-situ ellipsometry furnace and heated
    RUN ToFirstPoint DURATION 255.687 TEMP 1200->1200 AMBIENT dry_O2 TIMING assumed VARY duration 1..7200
    RUN Oxidation DURATION 33710 TEMP 1200->1200 AMBIENT dry_O2 TIMING bound
    EVENT Measure KIND measurement TIMING assumed                # Growth rate measured in situ by spectroscopic ellipsometry
}

1200 °C is outside the 950 to 1150 °C of the growth program's constants. Thickness against time reconstructed from Figure 5's growth rates (29 points read; 8 scored).

The programs the predictor ran on these histories: experiments/sic_interface_states/one_way_anneal.isl.template, experiments/song_sic/oxide_growth_sic.isl.template.

The model itself, experiments/song_sic/oxide_growth_sic.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 4H-SiC (growth sector on a second material)
#
# The OPERATOR and OBSERVATION_MODEL blocks below are copied character for character
# from experiments/vitkavage_dit/oxide_growth.isl.template, the silicon program; a test
# checks that they are identical. Only the declared parameters differ.
#
# Song et al. (2004) fit their SiC data with the same linear-parabolic law, reading its
# two constants as including out-diffusion of the carbon product. The constants here are
# Arrhenius fits to that paper's Tables I and II, per crystal face
# (scripts/build_song_dataset.py). Nothing is fitted by us.
# Fields in {{braces}} are filled by scripts/run_song_sic.py.
# ==============================================================================
SAMPLE {sample_id} {{
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_{face}_face
}}
STATE x_ox {{
    UNIT: cm
    INITIAL: x_i
    MIN: 0
    MAX: 1.0e-3
}}
CONTROL T {{ UNIT: K }}
CONTROL p_O2 {{ UNIT: atm }}
PARAMETER B0 {{ VALUE: {B0}; UNIT: cm^2 s^-1; STATUS: sourced; SOURCE: {source} }}
PARAMETER E_B {{ VALUE: {E_B}; UNIT: eV; STATUS: sourced; SOURCE: {source} }}
PARAMETER BA0 {{ VALUE: {BA0}; UNIT: cm s^-1; STATUS: sourced; SOURCE: {source} }}
PARAMETER E_BA {{ VALUE: {E_BA}; UNIT: eV; STATUS: sourced; SOURCE: {source} }}
# The constants above are already specific to the crystal face, so no further factor is applied.
PARAMETER r_or {{ VALUE: 1.0; UNIT: 1; STATUS: constant }}
# The paper sets the time offset to zero and reports a native oxide of about 2 angstrom.
PARAMETER x_i {{ VALUE: 2.0e-8; UNIT: cm; STATUS: sourced; SOURCE: Song et al., J. Appl. Phys. 95, 4953 (2004), native oxide about 2 angstrom, agent-read }}
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
}}
EVALUATION_CONTRACT {contract_id} {{
    FROZEN: true
    DATASET: data/derived/song_2004_sic_thickness.csv
    DATASET_DIGEST: sha256:{dataset_sha256}
    SPLIT: {split}
{targets}
}}

Our simulation against the paper

21 within, 16 near, 66 off of 103 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. There are no held-out samples here: the unknowns were chosen on the same numbers they are scored on, so the aligned score is an upper bound, not a test.

Together: ToFirstPoint duration 1 s (was 1348.04); ToFirstPoint duration 1 s (was 265.148); ToFirstPoint duration 1 s (was 149.871); ToFirstPoint duration 1 s (was 721.739); ToFirstPoint duration 372.9 s (was 885.321); ToFirstPoint duration 1 s (was 732.69); ToFirstPoint duration 1 s (was 580.05); ToFirstPoint duration 1 s (was 165.675); ToFirstPoint duration 1 s (was 604.544); ToFirstPoint duration 19.31 s (was 927.924); ToFirstPoint duration 1 s (was 340.479); ToFirstPoint duration 84.85 s (was 278.856); ToFirstPoint duration 1 s (was 255.687).

As reproduced: 21 of 103 within tolerance, median error 29.0%. Aligned: 37 of 103 within tolerance, median error 17.7%.

What was searched. The chapter gives no times. Each run's thickness against time was rebuilt by integrating 1/rate over the points read, so the only unknown is when the first point was reached: the time in dry oxygen before it, one unknown per run (13), from 1 s to 2 h, starting from the first thickness over the first rate. Each unknown moves only its own run. There are no held-out runs: the unknowns are chosen on the numbers they are scored on, so the aligned score is an upper bound.

The data want no time before the first point, and that is not enough. Ten of 13 unknowns go to the 1 s minimum; the C face at 1050 °C keeps 6 min (assumed 15), the Si face at 1050 °C 19 s and at 1100 °C 85 s. With Song's constants the program reaches each run's first measured thickness sooner than any time zero allows: it grows thin oxide too fast. As reproduced: 21 of 103 within 10%, median error 29.0%; aligned: 37 of 103, median 17.7%. Inside the program's 950 to 1150 °C: 16 of 72 (median 22.4%) as reproduced, 30 of 72 (12.6%) aligned; outside it (890, 930, 1220 °C C face, 1200 °C Si face): 5 of 31, then 7 of 31.

Where it stays wrong. Inside the range, points of 50 nm and above: 15 of 27 within, median error 9.8%, median signed +1.7%. Points below 20 nm: 5 of 29 within, median 34%, signed +24%. The C face at 1050 °C is the best run (7 of 8) and 1150 °C the worst (0 of 8, +10 to +24% from 20 nm on). On the Si face the error changes sign along each run: at 1100 °C +17 to +34% in the first 15 nm, −14% at 41 nm. Song's Si-face constants put the change from linear to parabolic growth at A = 5.8 nm (1100 °C), so the program slows to 1.9 nm/h at 40 nm while the measured rate stays at 4.0 nm/h from 10 to 40 nm (Figure 5; Hijikata: reaction-limited “up to several μm”). On the C face A is 80 nm at 1090 °C.

The rate itself (table under “What the reproduction shows”): below 10 nm the program's rate is 1.5 to 2.7 times the measured one on both faces inside the range, and 2.3 to 3.1 times at 1200 to 1220 °C. Above 100 nm on the C face the ratio rises with temperature, 0.55 at 930 °C to 1.14 at 1150 °C and 1.67 at 1220 °C: Song's parabolic activation energy (1.98 eV) is steeper than these data.

With heating and cooling transitions no number moves by more than 2% in any of the four scenarios: the added heat-up is in nitrogen, where the growth program grows nothing, and the cooling comes after the last point, which in situ data do not see. There is no hydrogen step, so no Quantum ESPRESSO comparison.

37 within, 22 near, 44 off of 103 numbers with the aligned settings.

No time zero fixes the thin regime: the program's linear rate B/A (Song's, from ex-situ points at 0.25 h and later) is the program's rate at zero thickness, and it is 1.5 to 2.7 times what was measured in situ at 1 to 10 nm. This is the opposite of the chapter's own finding against its own Deal-Grove fit (rates above D-G below 20 nm): the two laboratories' B/A differ by about a factor of 2 on the C face (Song 108 nm/h at 1090 °C; Hijikata's solid line in Figure 1 starts near 50). Alignment cannot touch B/A or B: they are sourced constants.

The Si face needs a different B, not a different history: with Song's B the program cannot grow at a constant 4 nm/h from 10 to 40 nm at 1100 °C. Song's Si-face activation energy for B (3.12 eV) rests on three temperatures.

Two-stage deceleration (break at 7 nm, initial rate independent of oxygen pressure, Figures 7 and 8) needs a program with a second growth channel; the Si and C emission model the chapter describes is one, and the language can write it (below).

What our language and ontology cannot yet say

• A rate observable, measured at a thickness, not at a time. The chapter's data are dX/dt against X. The language observes states at the end of a history, so the rate had to be integrated to thickness against a reconstructed time with an unknown time zero. An observation of a state's time derivative (EQUATION: Growth_rate = d(x_ox)/dt) and a result indexed by a state (MEASURE Growth_rate AT Thickness = 1.0e-6) would compare the rates directly and need no time zero; it would also have answered “is the initial rate enhanced?” in one row.
• A measurement during a RUN (MEASURE <observable> DURING <run> EVERY <s> or in situ), so an in-situ ellipsometry run is one history with many measured points instead of a curve sampled by re-running the history to each time.
• Clock reference (as for Song 2004): TIME_ZERO at_gas_on | at_first_point, so an in-situ run's time axis can be stated rather than searched.
• Predictor: a thin-oxide channel in the SiC growth program, with its own pressure law. The Si and C emission model is writable today: FIELDs for Si and C interstitials and oxygen in the oxide on a grid travelling with the interface, FLUX with D_Si, D_C, D_O, BOUNDARY fluxes ν_Si k C_O, ν_C k C_O and (2 − ν_Si − ν_C − α/2) k C_O at the interface (Eq. 10), an OxideGrowth rate k0 (1 − left(C_Si)/C_Si⁰)(1 − left(C_C)/C_C⁰) C_O / N0 (Eq. 8). Table 1 gives the oxide-side constants; ν_Si, ν_C and k0 per face and temperature exist only as points on Figure 6, so they would be graph reads. Its grid would need to follow the growing oxide (no grid of varying length today).
• Predictor: the program scales its whole rate by p_O2. Hijikata's initial rate R_r is nearly independent of pressure while R_g is proportional (Figure 8). Any SiC program used for low-pressure or ramp-in-diluted-oxygen histories needs the two separated.

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

Growth rate against thickness, measured (in situ, read from Figures 1 and 5) and the SiC growth program's (Song's constants, dX/dt = B/(A + 2X) at the measured thickness; checked against finite differences of the predictor's thickness). Each cell: median measured rate / program rate in nm/h, and their ratio, over the points read in that thickness band.

The initial fast regime the chapter reports is fast relative to its own Deal-Grove fit. Relative to the program it is slow: on both faces, at every temperature, the program's rate below 10 nm is above the measured one. The program is closest between 20 and 100 nm on the C face at 970 to 1090 °C (ratios 0.77 to 1.32), the regime Song's ex-situ points covered.

What we would like Emily to check

1. Is a growth rate read from in-situ ellipsometry at 1 to 5 nm on SiC trustworthy, given the 1.5 nm interface layer the same group reports?
2. Song 2004 and Hijikata disagree on the Si face: Song's B makes growth parabolic at a few nm, Hijikata's rate is flat to 80 nm. Which would you trust for a 50 nm gate oxide at 1100 to 1400 °C?
3. Should the growth program carry a thin-oxide term (Massoud-like, or the emission model) on SiC, or is Song's linear-parabolic law the reference for the challenge?