Reproducing Gupta and Akhtar 2011: dry and wet oxidation of both faces of 4H-SiC, a second laboratory for Song's constants

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/gupta_akhtar.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

S. K. Gupta and J. Akhtar, “Thermal Oxidation of Silicon Carbide (SiC) – Experimentally Observed Facts”, in M. Mukherjee (ed.), Silicon Carbide – Materials, Processing and Applications in Electronic Devices, InTech (2011), chapter 9, pp. 207–230.

A second laboratory for the SiC growth program. Song 2004's constants (the only SiC growth constants we hold) are run on this chapter's dry-oxygen thickness curves for the Si and C faces at 1000, 1050, 1110 and 1150 °C (48 thicknesses read from Figure 10), and the chapter's own rate constants are set beside Song's. The wet-oxygen curves have no SiC program to run.

How well can this study be reproduced?

Moderate. The furnace cycle is described better than in most papers: load and unload at 800 °C in nitrogen, ramps at 5 °C/min, a dry or bubbler-wet oxygen hold. What is missing is the gas during the ramps, the flow of dry oxygen, and a single statement of the oxidation times: the text, Figure 3 and Figure 10 give three different sets.

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. 4 same, 5 approximated, 1 assumed, 3 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.

Si face (0001), 1150 °C

Simulated time in this history: 8 h 20 min 01 s.

Across the samples

Timing the paper does not give

• The gas during the 5 °C/min ramps (40 to 70 min each way): not stated.
• Whether the oxidation time includes any of the ramp: not stated; Figure 3 draws it as the plateau only.
• The oxidation times: the text says 30 to 180 min in 30 min steps, Figure 3 says 0.5 to 6 h in 0.5 h steps, Figure 10 plots 0.5, 2, 3, 4, 5 and 6 h.
• Time at 800 °C after loading and before unloading, and the cooling from 800 °C: not stated.
• Time from the last HF dip to loading: not stated.
• Dry oxygen flow: not stated (wet: 0.4 LPM through the bubbler).

What the second reading changed

• The project held this chapter as “skimmed”, with the tables transcribed in tables_agent_read_UNVERIFIED.csv and not used “until the units are settled from the chapter's figures”. The units are settled by Figures 15 and 16 (X0 in µm, t/X0 in h/µm): B in µm²/h, B/A in µm/h. The transcription's values agree with the second reading's, entry for entry.
• The transcription keeps the printed “1050” for the fourth column; the text (p. 15, p. 19) and Figure 10 make it 1150 °C.
• No stored recipe exists for this chapter: there was no dataset. The thicknesses scored here are the second reader's readings of Figure 10, not a project dataset.
• Our sample has Song's 0.2 nm native oxide at the start, since the SiC growth program does not read STARTING_OXIDE_NM.

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>.

Si face (0001), 1150 °C

# Si face (0001), 1150 °C
SAMPLE SiC_Si_1150 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 4200 TEMP 800->1150 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1150->1150 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1150->1150 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 4200 TEMP 1150->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

Si face (0001), 1110 °C

# Si face (0001), 1110 °C
SAMPLE SiC_Si_1110 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 3720 TEMP 800->1110 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1110->1110 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1110->1110 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 3720 TEMP 1110->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

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
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 3000 TEMP 800->1050 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1050->1050 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1050->1050 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 3000 TEMP 1050->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

Si face (0001), 1000 °C

# Si face (0001), 1000 °C
SAMPLE SiC_Si_1000 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 2400 TEMP 800->1000 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1000->1000 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1000->1000 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 2400 TEMP 1000->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

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
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 4200 TEMP 800->1150 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1150->1150 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1150->1150 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 4200 TEMP 1150->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

C face (000-1), 1110 °C

# C face (000-1), 1110 °C
SAMPLE SiC_C_1110 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 3720 TEMP 800->1110 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1110->1110 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1110->1110 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 3720 TEMP 1110->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

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
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 3000 TEMP 800->1050 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1050->1050 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1050->1050 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 3000 TEMP 1050->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

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
    DOPING: n_type
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING stated                         # Degrease, RCA, piranha, HF dips
    EVENT Load KIND load TIMING assumed                          # Loaded at 800 °C in nitrogen
    RUN RampUp DURATION 2400 TEMP 800->1000 AMBIENT inert_N2 TIMING stated
    RUN BeforeClock DURATION 1 TEMP 1000->1000 AMBIENT dry_O2 TIMING assumed VARY duration 1..1800
    RUN Oxidation DURATION 21600 TEMP 1000->1000 AMBIENT dry_O2 TIMING bound
    RUN RampDown DURATION 2400 TEMP 1000->800 AMBIENT inert_N2 TIMING stated
    EVENT Unload KIND other TIMING assumed                       # Unloaded at 800 °C in nitrogen
    EVENT Measure KIND measurement TIMING stated                 # Thickness by ellipsometry, checked by stylus profiler
}

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

17 within, 4 near, 27 off of 48 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 C_1000, C_1150, Si_1000, Si_1150 only; every other number is a test they played no part in.

Together: BeforeClock duration 12.16 s (was 1).

As reproduced: 17 of 48 within tolerance, median error 23.1%. Aligned: 18 of 48 within tolerance, median error 23.0%. On the held-out samples only: 9 of 24 within tolerance, median error 20.7% as reproduced, 9 of 24 within tolerance, median error 20.5% aligned.

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

What was run. The SiC growth program (Song 2004's tabulated constants, Arrhenius-interpolated to 1110 °C and, on the Si face, extrapolated to 1000 °C) on the chapter's dry-oxygen curves: load at 800 °C, a 5 °C/min ramp in nitrogen, the oxidation, a ramp down. As reproduced: 17 of 48 within 10%, median error 23.6%. The two faces part completely. C face: 17 of 24, median signed error −2.4%, and −0.7% from 2 h on; the misses are the 0.5 h points (−10 to −30%, the program grows too little at first) and three single points at 1000 and 1050 °C. Si face: 0 of 24, every point low, median signed −29%: about −28% at 1050 to 1150 °C and −44 to −64% at 1000 °C, where Song has no Si-face constants and the program extrapolates.

What was searched. One unknown, oxygen at temperature before the clock starts (1 s to 30 min), chosen on the 1000 and 1150 °C curves (24 points); the 1050 and 1110 °C curves (24 points) are the test. Two variants: both ramps in dry oxygen, and the text's times (30 to 180 min) in place of Figure 10's (0.5 to 6 h).

The offset cannot fix it. On the chosen points the count stays at 8 or 9 of 24 from 1 s to 8.6 min and falls to 3 of 24 at 30 min, where the C face overshoots while the Si face is still 15 to 20% low. The search kept 12 s (9 of 24, a tie broken on median error). On the held-out 1050 and 1110 °C curves: 9 of 24 before and after, median error 20.7% and 20.5%. Ramps in oxygen add 40 to 70 min of oxide before the clock: 15 of 48 (median 20.7%); the Si face stays low (median signed −26%) and the C face goes high (+5%). The text's times are ruled out: 0 of 48 (median 42.9%). Figure 10's times are the ones that agree with the program on the C face, as they agree with the chapter's own Figure 16.

Song's constants against Gupta and Akhtar's, same faces and temperatures (a test of transfer between laboratories). Parabolic B, C face: 1.40, 0.96 and 0.95 times Song's printed value at 1000, 1050 and 1150 °C (1.14 times the Arrhenius value at 1110 °C). Si face: 1.85 and 1.61 times Song's printed value at 1050 and 1150 °C (1.53 times at 1110 °C; 3.6 times the extrapolation at 1000 °C). The activation energies of B agree: 2.03 eV against Song's 1.98 eV (C face), 2.86 against 3.12 eV (Si face). The linear constants B/A do not transfer: 0.76 and 1.31 times Song's at 1000 and 1050 °C on the C face but 5.8 times at 1150 °C, and 1.5 times (1050 °C) rising to 13.8 times (1150 °C) on the Si face; activation energies 2.27 and 2.81 eV against Song's 1.29 and 1.34 eV. The chapter fits with τ = 0 and no point below 7 nm, so B/A, which is set by the thin end, is poorly determined in both papers. The thickness curves say the same thing as the B comparison: Song's C face transfers to this laboratory within 10% after 2 h, Song's Si face is 1.5 to 1.9 times too slow in B.

Heating and cooling transitions (the section above): none of the four scenarios moves any number by more than 2%, because every assumed heat-up and cool-down is in nitrogen, which grows nothing in this program. The chapter's own ramps are also in nitrogen in our history, so the same holds for them.

9 within, 3 near, 12 off of 24 numbers with the aligned settings (the samples used to choose the unknowns are left out).

The Si-face deficit is in the constants, not the history: no offset or ramp gas reaches 10% on the Si face without putting the C face 10% high. The program has one set of Si-face constants, from Song's three temperatures on on-axis wafers. This chapter does not state its off-cut (Figure 1(a) has no text layer); Goto and Hijikata's 8° off-axis Si face is also faster than Song's (see their page). OFFCUT read by the program, with constants per off-cut, is what this would need.

The 0.5 h points are 10 to 30% low on the C face and 42 to 64% low on the Si face: the linear-parabolic law with Song's B/A has no fast initial regime. A thin-oxide term (Massoud's, or Goto and Hijikata's R_S exp(−X/L_S)) is the structural change this asks for; no history change reaches it.

Wet oxidation has no SiC program. The chapter gives what one needs: B and B/A for both faces at four temperatures in 85 °C-bubbler wet oxygen (Tables 1 and 2). Its Si-face wet constants at 1150 °C (B = 1.2×10⁻³ µm²/h, B/A = 0.885 µm/h) give 48.3 nm after 2 h; Yano et al. 2002 measured 45 nm after 120 min of wet oxidation at 1150 °C on (0001), another laboratory. A wet SiC program would be the growth operator with these constants, scaled by water pressure the way the silicon wet operator is (WET_GROWTH in isl_harness/predict.py), with the bubbler temperature written as the water partial pressure. The (03-38) and a faces would still have none.

What our language and ontology cannot yet say

• OFFCUT read by the SiC growth program (constants per face and off-cut). Song's Si face is on axis, Goto and Hijikata's is 8° off, this chapter's is unrecorded, and the Si-face constants do not transfer between them (28 to 64% low here). The field exists on samples and no program reads it.
• A ramp rate as a statement (RAMP RATE 5 C/min FROM 800 TO 1150), so the chapter's stated rate is written as the paper gives it, not as a duration we derive.
• The sic_growth program's scope check labels a history outside its tested scope whenever any step is not dry oxygen, so a load and ramp in nitrogen, which grows nothing, marks every number here “outside the tested scope”. The check should look only at the steps that grow oxide.
• Bubbler temperature as an ambient statement (H2O FROM bubbler AT 85 C) that the runtime turns into a water partial pressure, so a wet recipe is written as the paper states it rather than as our computed mixture.
• A cumulative or replicate series: REPLICATES 4 on a measurement, so the four samples per point are part of the record (and their scatter, if a paper gives it, becomes the tolerance).

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. In a furnace that loads at 800 °C in nitrogen and ramps at 5 °C/min, would the oxygen normally be switched on at the start of the ramp or at the plateau?
2. Figure 10 and the text disagree on the times (6 h against 3 h for the longest). Which would you trust?
3. The chapter's Si-face B/A at 1150 °C is 14 times Song's. Is a linear constant from a fit with τ = 0 and no thin-oxide points worth comparing at all?