Reproducing Goto and Hijikata 2016: SiC growth rates on two faces, and the Si and C emission model as a candidate structure

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/goto_2016.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

D. Goto and Y. Hijikata, “Unified theory of silicon carbide oxidation based on the Si and C emission model”, J. Phys. D: Appl. Phys. 49, 225103 (2016).

Two things. A third laboratory's dry-oxygen growth on the Si and C faces (eight temperatures across the two faces), run through the SiC growth program with Song 2004's constants. And a candidate structure for that program: the Si and C emission model, whose equations and parameters are recorded below with whether our language can carry them.

How well can this study be reproduced?

Low. A modelling paper whose data are in-situ growth rates against thickness. The oxidation itself is stated (dry oxygen, 1 atm, a lamp heater, argon before oxygen), but not its duration, the clean, or the heat-up, and the model's material parameters are mostly not printed. Thickness against time has to be rebuilt by integrating the rates.

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, 2 approximated, 4 assumed, 1 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), 1200 °C

Simulated time in this history: 5 h 08 min 07 s.

Across the samples

Timing the paper does not give

• Clean and loading: not stated.
• Heat-up rate of the lamp heater and time in argon at temperature: not stated.
• Time from the oxygen admission to the first thickness the ellipsometry resolved: not stated; Figure 3 starts at 3 to 13 nm.
• Total oxidation time of each run: not stated. Times here are integrals of the read rates.
• Cooling: not stated (it does not enter in-situ data).

What the second reading changed

• The project held this paper as “abstract read … nothing digitized”, with no stored recipe and no dataset. The rates and parameters here are the second reader's readings of Figures 3 to 5.
• Text p. 6: activation energies of k0 “0.68, 1.37 and 2.02” for the “Si-, a- and C-faces”; Figure 4(a) labels 2.02 eV on the Si-face line and 0.68 eV on the C-face line, and the paper's own argument (three Si back-bonds on the Si face, one on the C face) requires the figure's order.
• L_S: 1.2 nm at 1100 °C for both faces in the text (for Figure 2), about 2.3 nm at 1100 °C on Figure 5(b) (for Figure 3).
• The Si face here is 8° off axis; Song's, whose constants the program uses, was on axis. The C face is 0.5° off.

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), 1200 °C

# Si face (0001), 1200 °C
SAMPLE SiC_Si_1200 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_Si_face
    CLEAN: not_stated
    DOPING: n_type_1e16
    OFFCUT: 8_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->1200 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 1200->1200 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 437.802 TEMP 1200->1200 AMBIENT dry_O2 TIMING assumed VARY duration 109.451..1751.21
    RUN Oxidation DURATION 17628.8 TEMP 1200->1200 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

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_1e16
    OFFCUT: 8_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->1150 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 1150->1150 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 315.303 TEMP 1150->1150 AMBIENT dry_O2 TIMING assumed VARY duration 78.8258..1261.21
    RUN Oxidation DURATION 15172.2 TEMP 1150->1150 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

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
    DOPING: n_type_1e16
    OFFCUT: 8_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->1100 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 1100->1100 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 1144.77 TEMP 1100->1100 AMBIENT dry_O2 TIMING assumed VARY duration 286.193..4579.09
    RUN Oxidation DURATION 14725.8 TEMP 1100->1100 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

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_1e16
    OFFCUT: 8_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->1050 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 1050->1050 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 1121.91 TEMP 1050->1050 AMBIENT dry_O2 TIMING assumed VARY duration 280.477..4487.63
    RUN Oxidation DURATION 8765.64 TEMP 1050->1050 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

C face (000-1), 1200 °C

# C face (000-1), 1200 °C
SAMPLE SiC_C_1200 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
    DOPING: n_type_5e15
    OFFCUT: 0.5_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->1200 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 1200->1200 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 218.454 TEMP 1200->1200 AMBIENT dry_O2 TIMING assumed VARY duration 54.6134..873.815
    RUN Oxidation DURATION 13917.6 TEMP 1200->1200 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

C face (000-1), 1100 °C

# C face (000-1), 1100 °C
SAMPLE SiC_C_1100 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
    DOPING: n_type_5e15
    OFFCUT: 0.5_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->1100 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 1100->1100 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 334.131 TEMP 1100->1100 AMBIENT dry_O2 TIMING assumed VARY duration 83.5328..1336.53
    RUN Oxidation DURATION 70508.9 TEMP 1100->1100 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

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_5e15
    OFFCUT: 0.5_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->1000 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 1000->1000 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 377.932 TEMP 1000->1000 AMBIENT dry_O2 TIMING assumed VARY duration 94.483..1511.73
    RUN Oxidation DURATION 20071.1 TEMP 1000->1000 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

C face (000-1), 900 °C

# C face (000-1), 900 °C
SAMPLE SiC_C_900 {
    FAMILY: SiC_SiO2
    ORIENTATION: 4H_SiC_C_face
    CLEAN: not_stated
    DOPING: n_type_5e15
    OFFCUT: 0.5_degrees
}
CONTROL_TRAJECTORY History {
    EVENT Clean KIND clean TIMING assumed                        # Surface preparation
    RUN HeatInArgon DURATION 120 TEMP 25->900 AMBIENT inert_Ar TIMING assumed
    RUN ArgonAtTemperature DURATION 300 TEMP 900->900 AMBIENT inert_Ar TIMING assumed
    RUN ToFirstPoint DURATION 956.195 TEMP 900->900 AMBIENT dry_O2 TIMING assumed VARY duration 239.049..3824.78
    RUN Oxidation DURATION 12631.7 TEMP 900->900 AMBIENT dry_O2 TIMING assumed
    EVENT Measure KIND measurement TIMING assumed                # In-situ spectroscopic ellipsometry
}

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

29 within, 15 near, 29 off of 73 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_1100, Si_1100 only; every other number is a test they played no part in.

Together: ToFirstPoint duration 1× (was 1).

As reproduced: 29 of 73 within tolerance, median error 13.5%. Aligned: 29 of 73 within tolerance, median error 13.5%. On the held-out samples only: 21 of 53 within tolerance, median error 12.8% as reproduced, 21 of 53 within tolerance, median error 12.8% aligned.

How the comparison is made. Figure 3 gives growth rate against thickness at 1 atm. The second reader read 7 to 12 points per curve; the time between points is the integral of 1/rate over thickness (trapezoid rule), which turns each curve into thickness against time measured from its first point. The history anchors that first point where the program's own constants put it (the time the program needs to grow the first read thickness, 3 to 13 nm), so the scored points test the growth from there on, not the start. Points just after the anchor agree almost by construction; the last point of each curve is the real test. Every number is labelled outside the tested scope: 1200 °C and 900 °C are outside Song's 950 to 1150 °C, and the argon heat-up trips the program's scope check (see the gaps below).

As reproduced: 29 of 73 within 10%, median error 13.5%. The two faces err in opposite directions. Si face: the program is too slow, and more so as the oxide thickens; last points −36% (1200 °C, 98 nm), −33% (1150 °C, 56 nm), −17% (1100 °C, 29 nm), −17% (1050 °C, 14 nm). At 1150 °C and 43 nm Goto and Hijikata measure 11.0 nm/h; the program's rate there is B/(2X + A) = 4.3 nm/h. C face: the program is too fast at 15 to 100 nm, by up to +74% at 1200 °C and +73% at 1100 °C; at 1100 °C and 32 nm they measure 26 nm/h, the program gives 65 nm/h. The gap closes as the oxide thickens (+20% at 330 nm, where the rates are 10.8 and 13.1 nm/h) and at low temperature (900 °C: −13% at the last point).

What was searched. One unknown: the anchor, a factor from 0.25 to 4 on the time to the first read point, shared by every curve; chosen on the 1100 °C curves (20 points), tested on the other six (53 points). It is not flat: 8 of 20 at the program's own anchor, 5 at 0.63, 4 at 1.59, 0 at 0.25. The search keeps the program's own anchor. Held-out: 21 of 53 before and after. No time offset rescues a curve whose slope is wrong in the middle, and on these two faces the slope is wrong in opposite directions.

What it says about Song's constants. On the C face, Song's B and B/A transfer to Gupta and Akhtar's furnace (their page) and not to Goto and Hijikata's in-situ data. The C-face difference is largest at 20 to 100 nm, the region Goto and Hijikata call the interface-oxidation stage with deceleration; Song's linear-parabolic fit has no deceleration and Gupta and Akhtar's first point is already 21 to 78 nm. On the Si face, both other laboratories grow faster than Song's constants allow (Gupta and Akhtar by about 1.4 times in thickness, Goto and Hijikata's 8° off-axis wafers by up to 2.5 times in rate at 40 nm). The Si-face constants are the weak part of the program.

Heating and cooling transitions (above): no number moves. The argon heat-up and the assumed nitrogen transitions grow nothing in this program, and in-situ data have no cool-down.

21 within, 12 near, 20 off of 53 numbers with the aligned settings (the samples used to choose the unknowns are left out).

The model, as printed (for a future program). Interface reaction rate (eq. 5): k = k0 (1 − C_Si^I/C_Si^0)(1 − C_C^I/C_C^0). Growth (eq. 2 with eq. 8): N0 dX/dt = k C_O^I (1 − ν_Si) + ∫ (κ1 C_Si C_O + κ2 C_Si C_O²) dx + N0 R_S exp(−X/L_S), N0 = 2.25×10²² cm⁻³. Diffusion in the oxide (eq. 6) for Si and C interstitials and O2, with absorption R1 = κ C C_O and R2 = κ C C_O², surface exchange through η and the gas-phase coefficient h. Interface fluxes (eq. 7): D_Si ∂C_Si/∂x = −ν_Si k C_O^I, D_C ∂C_C/∂x = −ν_C k C_O^I, D_O ∂C_O/∂x = (3/2 − ν_Si − ν_C/2) k C_O^I (reconstructed from a damaged text layer; check against the page). Substrate side (eqs. 9 to 12): C interstitials diffuse into SiC and recombine with C vacancies (γ C_C^b V_C), entering with flux ν_C^b N_s dX/dt, ν^b = 1.2×10⁻⁹ ν, V_C0 = 10¹³ cm⁻³, N_s = 4.8×10²² cm⁻³; Si interstitials likewise with D_Si^b = 5.8×10⁻¹⁰ exp(−1.53 eV/kT) cm²/s. Oxygen solubility C_O^0 scales with pressure.

Parameters. Printed or readable: k0 (Figure 4a; activation energy 2.02 eV on the Si face, 1.37 on the a face, 0.68 on the C face by the figure's labels; the text gives the reverse order), ν_Si and ν_C (Figure 4b; ν_Si 0.20 to 0.38 on the Si face at 1050 to 1200 °C, 0.002 to 0.30 on the C face at 900 to 1200 °C; ν_C about 0.04 to 0.10, the same on every face), R_S (Figure 5a; 18 to 100 nm/h on the Si face, 52 to 270 nm/h on the C face; 2.02 and 1.01 eV), L_S (Figure 5b; about 1.5 nm at 900 °C to 5 nm at 1200 °C, common to the faces; the text says 1.2 nm at 1100 °C). Not printed: D_Si, D_O, η, κ1, κ2, C_Si^0 (taken from Uematsu, Kageshima and Shiraishi's silicon work, refs 32 and 38), and every C-related parameter (D_C, η′, κ1′, κ2′, C_C^0) and C_O^0 (fitted, values not given). The model cannot be run from this paper alone.

Can the project's growth law carry it? Partly. (1) The steady-state limit (the paper says it is Massoud's relation) can be written today: one more term R_S exp(−X/L_S) in d(x_ox)/dt beside B/(2x + A). With Song's Si-face constants and Goto and Hijikata's R_S = 32 nm/h, L_S = 2.3 nm at 1100 °C, the rate rises from 10.3 to 13.9 nm/h at 5 nm (measured 12.5) and is unchanged above 10 nm, where the Si-face deficit is (3.6 against 5.3 nm/h at 20 nm). So the term fixes the first nanometres, not the Si face. (2) The substrate side (eqs. 9 to 12) can be written in the language as it is: a grid in SiC that travels with the front, a boundary flux ν^b N_s d(x_ox)/dt at the interface, a vacancy field and a recombination source. (3) The oxide side cannot: its fields live on a layer whose thickness is the growing state x_ox, and the language's grids have fixed length (“a layer whose own thickness changes, such as the growing oxide itself, has no grid of varying length yet”). The interface concentrations would be left(C_Si), which exists; the absorption integral ∫ κ C_Si C_O dx needs the integral of a product of two fields, which does not.

What our language and ontology cannot yet say

• A grid whose length is a state (GRID oxide { FROM: 0; TO: x_ox; CELLS: 50 }), or a stretched coordinate with position-dependent velocity, so interstitial and oxidant profiles can live in the growing oxide. This is the one change that would let the Si and C emission model be written as a candidate structure beside the linear-parabolic law.
• integral(expr) over a product of fields (the paper's ∫ κ C_Si C_O dx), not only integral(F) of one field.
• A growth rate against thickness as an observable (OBSERVE rate OF x_ox AT Thickness), so in-situ rate data can be scored directly instead of integrated into times that carry every reading error.
• A time origin at gas admission and an observation that starts later (MEASURE … FROM first_resolved_thickness): the paper's clock starts when argon is replaced by oxygen, its data at 3 to 13 nm; we had to anchor the clock with the program's own constants.
• OFFCUT read by the growth program: this Si face is 8° off axis, Song's on axis, and Song's Si-face constants are up to 36% slow here. The C face is 0.5° off.
• The sic_growth scope check marks a history outside its tested scope when any step is not dry oxygen, including an argon heat-up that grows nothing. It should check only the steps where oxide grows (and the temperatures of those).

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. Is a time integral of rates read from a log-log figure an acceptable way to compare with a thickness-against-time model, or should the comparison stay in rates?
2. Goto and Hijikata's Si face is 8° off axis and Song's is on axis. How much faster would you expect the off-axis Si face to oxidize?
3. For a program that has to predict 20 to 100 nm gate oxides, is the Massoud-type surface term (R_S, L_S) enough, or are the interstitial profiles needed?