Reproducing Kao et al. 1988: wet oxidation of convex and concave silicon cylinders

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/kao_1988.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.-B. Kao, J. P. McVittie, W. D. Nix and K. C. Saraswat, “Two-Dimensional Thermal Oxidation of Silicon—II. Modeling Stress Effects in Wet Oxides”, IEEE Trans. Electron Devices ED-35, 25–37 (1988). Data from Part I, the same authors, IEEE Trans. Electron Devices ED-34, 1008 (1987), held as the Stanford report DTIC ADA181721.

The challenge's own reference for curved silicon (system 2). Kao Part I etched silicon cylinders and holes 3 µm deep, oxidized them in steam at 800 to 1200 °C for the time that grows about 500 nm on a flat surface, and measured the oxide by SEM against the radius of the silicon after oxidation (0.12 µm to over 10 µm). Part II, this paper, explains the slowdown by the viscous stress of the oxide: the normal stress at the interface slows the reaction (k_s = k_0 exp(−σ_rr ΔΩ/kT)), and the pressure in the bulk of the oxide (tensile on convex, compressive on concave) changes the oxidant solubility and the viscosity. The project's radial program (experiments/muller_nanowire/radial_oxidation.isl.template) is this law for a convex cylinder with one lumped stress time, fitted on Müller's 75 to 110 nm wires; the predictor now runs a curvature-length form instead (exp(−ℓ/a), ℓ = 6.4 nm). Kao's cylinders are 3 to 50 times larger than Müller's wires, and his low temperatures are where the effect is large.

How well can this study be reproduced?

Low. This paper is a model. Its data are Part I's: oxide on silicon cylinders and holes against the radius after oxidation, at five wet temperatures, normalized to the flat oxide of the same run. Part I prints the wet times and the flat (100) oxide (Table 1), so each run's flat-surface rate is known, which is what a curved-silicon program needs. What a simulation must supply: the etched radius (only the radius after oxidation is plotted), the steam source, the furnace cycle, and which flat wall the normalization uses ((100) by the text, (110) by the plots' limit of 1.0). Every curved-surface number is a graph reading; the points are not labelled by temperature, so the least-squares lines are read.

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, 6 approximated, 3 assumed, 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.

Convex cylinder, wet 800 °C, radius 500 nm after oxidation

Simulated time in this history: 24 h 20 min.

Across the samples

Timing the paper does not give

• Furnace cycle: load temperature, push and pull rate, gas during heat-up and cool-down. Neither part states them.
• Steam source (pyrogenic or bubbler) and water pressure.
• The 800 °C time: Part I Table 1 gives 1440 min; this paper's model uses 1050 min “the time required for a flat surface to grow 5000 Å”.
• The 620 °C, 90 min polysilicon cap after oxidation is stated; it is recorded, not simulated.
• Time from the HF dip to the furnace.

What the second reading changed

• The project held no dataset for this paper (ontology/library.json: “Not modelled”). The flat-oxide table was in the history bundle's EXTRACTED_DATA.json (Part I Table 1) and agrees with the second reading; the curved-surface figures had not been digitized. Every curved number here is a new graph reading.
• The radial program reads RUN_FLAT_OXIDE_NM as oxide ÷ steam time, a linear rate, and then divides by its diffusion term 1 + (2a/A) log(b/a) with A = 1.1 µm. For Kao's 460 to 555 nm flat oxides the program's own flat sidewall then grows only 70% of the flat oxide it was given (349 nm for 503 nm at 800 °C). Kao's normalization needs the curved oxide over the flat oxide of the same program.
• The radial program has no temperature. Its A is Deal and Grove's 850 °C value and its curvature term (ℓ = 6.4 nm, or stress_time = 142 s) was fitted at 850 °C; Kao's whole result is that the curvature effect grows as the temperature falls.
• Kao's x-axis is the radius after oxidation; the program needs the etched radius. Taking Kao's radius as the etched one puts the 800 °C, 0.28 µm cylinder at 560 nm instead of 815 nm diameter.
• The text normalizes the (110) oxide to the flat (100) wall, but the curves all tend to 1.0 at large radius while flat (110) grows 1.03 to 1.29 times flat (100) (Part I Table 1). Either the flat reference is (110) or the curved oxide is not (110); it changes the absolute oxide by up to 29% at 800 °C.
• Kao's model includes the oxidant solubility rising under tension (V_s) and the viscosity falling under tension (α); the radial program has neither, so on a convex cylinder it can only slow growth, never speed it. Kao measured slightly thicker oxide than flat on 1 to 2 µm convex cylinders.

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>.

Convex cylinder, wet 800 °C, radius 500 nm after oxidation

# Convex cylinder, wet 800 °C, radius 500 nm after oxidation
SAMPLE Cylinder_800C_500nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 1373
    RADIUS_AFTER_NM: 500
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 502.7
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->800 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 86400 TEMP 800->800 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 800->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.73 read from Kao's least-squares line in Part I Figure 7 at 1/r = 2 per µm; times the flat (100) oxide of Table 1, 502.7 nm. The etched diameter 1373 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 800 °C, radius 280 nm after oxidation

# Convex cylinder, wet 800 °C, radius 280 nm after oxidation
SAMPLE Cylinder_800C_280nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 815
    RADIUS_AFTER_NM: 280
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 502.7
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->800 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 86400 TEMP 800->800 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 800->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.49 read from Kao's least-squares line in Part I Figure 7 at 1/r = 3.57 per µm; times the flat (100) oxide of Table 1, 502.7 nm. The etched diameter 815 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 900 °C, radius 500 nm after oxidation

# Convex cylinder, wet 900 °C, radius 500 nm after oxidation
SAMPLE Cylinder_900C_500nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 1377
    RADIUS_AFTER_NM: 500
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 457.5
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->900 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 18000 TEMP 900->900 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 900->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.81 read from Kao's least-squares line in Part I Figure 7 at 1/r = 2 per µm; times the flat (100) oxide of Table 1, 457.5 nm. The etched diameter 1377 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 900 °C, radius 235 nm after oxidation

# Convex cylinder, wet 900 °C, radius 235 nm after oxidation
SAMPLE Cylinder_900C_235nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 740
    RADIUS_AFTER_NM: 235
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 457.5
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->900 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 18000 TEMP 900->900 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 900->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.56 read from Kao's least-squares line in Part I Figure 7 at 1/r = 4.26 per µm; times the flat (100) oxide of Table 1, 457.5 nm. The etched diameter 740 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 1000 °C, radius 500 nm after oxidation

# Convex cylinder, wet 1000 °C, radius 500 nm after oxidation
SAMPLE Cylinder_1000C_500nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 1458
    RADIUS_AFTER_NM: 500
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 473.5
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1000 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 5760 TEMP 1000->1000 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1000->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.935 read from Kao's least-squares line in Part I Figure 7 at 1/r = 2 per µm; times the flat (100) oxide of Table 1, 473.5 nm. The etched diameter 1458 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 1000 °C, radius 125 nm after oxidation

# Convex cylinder, wet 1000 °C, radius 125 nm after oxidation
SAMPLE Cylinder_1000C_125nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 647
    RADIUS_AFTER_NM: 125
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 473.5
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1000 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 5760 TEMP 1000->1000 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1000->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.72 read from Kao's least-squares line in Part I Figure 7 at 1/r = 7.98 per µm; times the flat (100) oxide of Table 1, 473.5 nm. The etched diameter 647 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 1100 °C, radius 500 nm after oxidation

# Convex cylinder, wet 1100 °C, radius 500 nm after oxidation
SAMPLE Cylinder_1100C_500nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 1504
    RADIUS_AFTER_NM: 500
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 488
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1100 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 2400 TEMP 1100->1100 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1100->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.99 read from Kao's least-squares line in Part I Figure 7 at 1/r = 2 per µm; times the flat (100) oxide of Table 1, 488 nm. The etched diameter 1504 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 1100 °C, radius 132 nm after oxidation

# Convex cylinder, wet 1100 °C, radius 132 nm after oxidation
SAMPLE Cylinder_1100C_132nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 824
    RADIUS_AFTER_NM: 132
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 488
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1100 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 2400 TEMP 1100->1100 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1100->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 0.965 read from Kao's least-squares line in Part I Figure 7 at 1/r = 7.6 per µm; times the flat (100) oxide of Table 1, 488 nm. The etched diameter 824 nm is computed from the radius after oxidation and this oxide.

Convex cylinder, wet 1200 °C, radius 319 nm after oxidation

# Convex cylinder, wet 1200 °C, radius 319 nm after oxidation
SAMPLE Cylinder_1200C_319nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: nanowire
    DIAMETER_NM: 1276
    RADIUS_AFTER_NM: 319
    HEIGHT_NM: 3000
    SIDEWALL_MEASURED: 110
    RUN_FLAT_OXIDE_NM: 555.4
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1200 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 1440 TEMP 1200->1200 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1200->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

Normalized oxide 1.03 read from Kao's least-squares line in Part I Figure 7 at 1/r = 3.13 per µm; times the flat (100) oxide of Table 1, 555.4 nm. The etched diameter 1276 nm is computed from the radius after oxidation and this oxide.

Concave hole, wet 800 °C, radius 735 nm after oxidation

# Concave hole, wet 800 °C, radius 735 nm after oxidation
SAMPLE Hole_800C_735nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: hole
    HOLE_RADIUS_AFTER_NM: 735
    HEIGHT_NM: 3000
    RUN_FLAT_OXIDE_NM: 502.7
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->800 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 86400 TEMP 800->800 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 800->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

A hole: oxide grows on the inside of a cylinder. No program in the predictor has a concave surface, so the history runs and nothing is predicted.

Concave hole, wet 900 °C, radius 1370 nm after oxidation

# Concave hole, wet 900 °C, radius 1370 nm after oxidation
SAMPLE Hole_900C_1370nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: hole
    HOLE_RADIUS_AFTER_NM: 1370
    HEIGHT_NM: 3000
    RUN_FLAT_OXIDE_NM: 457.5
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->900 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 18000 TEMP 900->900 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 900->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

A hole: oxide grows on the inside of a cylinder. No program in the predictor has a concave surface, so the history runs and nothing is predicted.

Concave hole, wet 1000 °C, radius 769 nm after oxidation

# Concave hole, wet 1000 °C, radius 769 nm after oxidation
SAMPLE Hole_1000C_769nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: hole
    HOLE_RADIUS_AFTER_NM: 769
    HEIGHT_NM: 3000
    RUN_FLAT_OXIDE_NM: 473.5
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1000 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 5760 TEMP 1000->1000 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1000->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

A hole: oxide grows on the inside of a cylinder. No program in the predictor has a concave surface, so the history runs and nothing is predicted.

Concave hole, wet 1100 °C, radius 658 nm after oxidation

# Concave hole, wet 1100 °C, radius 658 nm after oxidation
SAMPLE Hole_1100C_658nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: hole
    HOLE_RADIUS_AFTER_NM: 658
    HEIGHT_NM: 3000
    RUN_FLAT_OXIDE_NM: 488
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1100 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 2400 TEMP 1100->1100 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1100->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

A hole: oxide grows on the inside of a cylinder. No program in the predictor has a concave surface, so the history runs and nothing is predicted.

Concave hole, wet 1200 °C, radius 833 nm after oxidation

# Concave hole, wet 1200 °C, radius 833 nm after oxidation
SAMPLE Hole_1200C_833nm {
    FAMILY: Si_Nanowire
    ORIENTATION: Si100
    CLEAN: peroxide_HF
    SHAPE: hole
    HOLE_RADIUS_AFTER_NM: 833
    HEIGHT_NM: 3000
    RUN_FLAT_OXIDE_NM: 555.4
    DOPING: not_stated_undoped
}
CONTROL_TRAJECTORY History {
    EVENT Lithography KIND other TIMING assumed                  # Circles and holes on the mask (smallest 1 um), AZ1470 resist, 4:1 projection aligner
    EVENT Etch KIND etch TIMING bound                            # Anisotropic plasma etch to 3 um, SF6 + chlorofluorocarbon, about 6.5 min per um
    EVENT Clean KIND clean TIMING stated                         # RCA clean ending in 50:1 HF, 30 s, and a rinse
    EVENT Load KIND load TIMING assumed                          # Into the furnace
    RUN PushIn DURATION 600 TEMP 700->1200 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    RUN WetOxidation DURATION 1440 TEMP 1200->1200 AMBIENT steam TIMING stated
    RUN PullOut DURATION 600 TEMP 1200->700 AMBIENT inert_N2 TIMING assumed VARY duration 60..1800
    EVENT PolySiCap KIND deposition TIMING stated                # 1 um LPCVD polysilicon, 620 °C, 500 mTorr, 90 min
    EVENT Lap KIND other TIMING assumed                          # Lapped to mid-height
    EVENT EtchOut KIND etch TIMING bound                         # Oxide etched out in 6:1 buffered HF, 100 Å gold sputtered
    EVENT MeasureSEM KIND measurement TIMING assumed             # SEM, x20,000 to x50,000: oxide on the (110) part of the wall, and the silicon radius after oxidation
}

A hole: oxide grows on the inside of a cylinder. No program in the predictor has a concave surface, so the history runs and nothing is predicted.

The programs the predictor ran on these histories: experiments/muller_nanowire/radial_oxidation.isl.template with the length exponential curvature term.

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

# ==============================================================================
# ISL MODEL: OXIDATION OF A SILICON NANOWIRE (challenge system 2)
#
# Data: Mueller et al., Nanotechnology 34, 105303 (2023), Figure 5: vertical silicon
# nanowires oxidized wet at 850 C, oxide thickness and remaining core against the
# as-etched diameter.
#
# The wire is a cylinder of silicon of radius a inside an oxide shell of outer radius b.
#   - Oxide forms at the silicon surface at the planar linear rate, times a sidewall
#     factor, reduced by diffusion through the shell in cylindrical geometry
#     (the linear-parabolic law written for a cylinder, after Kao et al.).
#   - New oxide takes more room than the silicon it replaces, so the shell must flow
#     outward. In a viscous shell that puts the silicon surface under a radial
#     compression proportional to the growth rate and to (1/a^2 - 1/b^2). The reaction
#     slows exponentially with that compression. One parameter, stress_time, lumps the
#     oxide viscosity and the activation volume of the reaction.
#   - The stress is carried as a state that follows its viscous value with a short lag,
#     because the viscous value depends on the rate it reduces.
# The outer radius is not a state: it follows from the silicon consumed.
#
# Not included: the effect of pressure on oxidant diffusion and solubility, stress-
# dependent viscosity, and faceting. Fields in {{braces}} are filled by
# scripts/run_muller_nanowire.py.
# ==============================================================================
SAMPLE Wire_{tag} {{
    FAMILY: Si_Nanowire
    CLEAN: not_stated
    ORIENTATION: Si100
    SHAPE: nanowire
    DIAMETER_NM: {diameter_nm}
}}
STATE a {{ UNIT: cm; INITIAL: a_0; MIN: 1.0e-9; MAX: 1.0e-4 }}
STATE stress {{ UNIT: 1; INITIAL: 0; MIN: 0; MAX: 50 }}
CONTROL T {{ UNIT: K }}
CONTROL p_H2O {{ UNIT: atm }}
PARAMETER a_0 {{ VALUE: {a_0_cm}; UNIT: cm; STATUS: sourced; SOURCE: Mueller et al. 2023 Figure 5, half the as-etched diameter }}
# ---- taken from the paper's own planar and flat-sidewall measurements --------------
PARAMETER rate_planar {{ VALUE: {rate_planar}; UNIT: cm s^-1; STATUS: sourced; SOURCE: Mueller et al. 2023 Figure A1, slope of bulk (100) oxide thickness against time at 850 C wet }}
PARAMETER sidewall {{ VALUE: {sidewall}; UNIT: 1; STATUS: sourced; SOURCE: Mueller et al. 2023 p. 8, nanosheet oxide 44.1 nm over bulk oxide 19 nm in the same run }}
PARAMETER A_dg {{ VALUE: 1.1e-4; UNIT: cm; STATUS: sourced; SOURCE: Deal and Grove 1965 wet oxidation, A extrapolated from 920 and 1000 C to 850 C }}
# ---- fitted to the larger wires -----------------------------------------------------
PARAMETER stress_time {{ VALUE: 1.0; UNIT: s; STATUS: fitted }}
# ---- assumptions --------------------------------------------------------------------
PARAMETER lag {{ VALUE: 2.0; UNIT: s; STATUS: hypothesis }}
# Growth is switched off smoothly as the core vanishes, so a fully consumed wire stops instead of going negative.
PARAMETER a_end {{ VALUE: 2.0e-8; UNIT: cm; STATUS: hypothesis }}
PARAMETER expansion {{ VALUE: 2.27; UNIT: 1; STATUS: constant }}
PARAMETER consumed {{ VALUE: 0.44; UNIT: 1; STATUS: constant }}
PARAMETER p_ref {{ VALUE: 1.0; UNIT: atm; STATUS: constant }}
QUANTITY b {{ UNIT: cm; EQUATION: b = sqrt(a * a + expansion * (a_0 * a_0 - a * a)) }}
QUANTITY growth {{ UNIT: cm s^-1; EQUATION: growth = (p_H2O / p_ref) * sidewall * rate_planar * exp(-stress) * (a / (a + a_end)) / (1 + (2 * a / A_dg) * log(b / a)) }}
QUANTITY viscous_stress {{ UNIT: 1; EQUATION: viscous_stress = stress_time * (1 - consumed) * a * growth * (1 / (a * a) - 1 / (b * b)) }}
CONTROL_TRAJECTORY History_{tag} {{
{runs}
}}
# Silicon is consumed at 0.44 of the oxide thickness formed at its surface.
OPERATOR F_RadialOxidation : OxideGrowth {{
    EQUATION: d(a)/dt = -consumed * growth
}}
# The compression at the silicon surface follows the value a viscous shell would carry.
OPERATOR F_ShellFlowStress : Relaxation {{
    EQUATION: d(stress)/dt = (viscous_stress - stress) / lag
}}
OBSERVATION_MODEL Obs_Oxide {{ TARGET: Thickness; UNIT: cm; EQUATION: Thickness = sqrt(a * a + expansion * (a_0 * a_0 - a * a)) - a }}
OBSERVATION_MODEL Obs_Core {{ TARGET: Core_diameter; UNIT: cm; EQUATION: Core_diameter = 2 * a }}
OBSERVATION_MODEL Obs_Consumed {{ TARGET: Consumed_substrate; UNIT: cm; EQUATION: Consumed_substrate = a_0 - a }}
EVALUATION_CONTRACT Contract_{tag} {{
    FROZEN: true
    DATASET: data/derived/muller_2023_nanowires.csv
    DATASET_DIGEST: sha256:{dataset_sha256}
    SPLIT: {split}
    TARGET: Thickness = {oxide_cm} +/- {oxide_tol_cm} abs
    TARGET: Core_diameter = {core_cm} +/- {core_tol_cm} abs
}}

Our simulation against the paper

1 within, 3 near, 5 off, 5 not modelled of 14 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 convex_1000C_r125nm, convex_1000C_r500nm, convex_1100C_r132nm, convex_1100C_r500nm, convex_1200C_r319nm only; every other number is a test they played no part in.

Together: PushIn duration 600 s (was 600); PullOut duration 600 s (was 600); PushIn duration 600 s (was 600); PullOut duration 600 s (was 600); PushIn duration 600 s (was 600); PullOut duration 600 s (was 600).

As reproduced: 1 of 14 within tolerance, median error 25.9%. Aligned: 2 of 14 within tolerance, median error 13.4%. On the held-out samples only: 1 of 9 within tolerance, median error 15.2% as reproduced, 0 of 9 within tolerance, median error 36.6% aligned.

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

The aligned numbers below use the variant flat oxide stated for the program's diffusion term.

The unknowns were chosen on the five convex cylinders at 1000, 1100 and 1200 °C; the four convex cylinders at 800 and 900 °C, where Kao's effect is large, and the five holes are the test. Each temperature is its own furnace run, so each has its own push and pull.

The furnace cycle is invisible. Push and pull in nitrogen score the same at every length from 1 to 30 min (0 of 5 on the fit cylinders at any value). The radial program grows only in steam, and with RUN_FLAT_OXIDE_NM its rate is the run's flat oxide over the steam time, so even the steam time cancels out. The four heating-and-cooling scenarios of the section below move no number by more than 2% for the same reason.

The flat-oxide input is the largest error, and it is the program's, not the paper's. As stated (the flat (100) oxide of Part I Table 1) the program's own flat sidewall grows 70% of it, because the program treats the flat oxide as a linear rate and then applies its diffusion term (A = 1.1 µm). Every convex cylinder comes out 0.63 to 0.70 of the flat oxide: 1 of 14 within tolerance, median error 26%, and the one pass (800 °C, 0.5 µm) is two errors cancelling. Stating the flat oxide as X(1 + X/A), so that the program's flat sidewall grows the measured flat oxide, is the variant used. It is not fitted (A is the program's own constant). It gives 2 of 14, median 13%: the cylinders at 1000 and 1100 °C and 0.5 µm within tolerance (−3% and −8%).

Temperature is what the program lacks. With the flat oxide converted, the program gives 0.82 to 0.91 of the flat oxide at every temperature: its curvature length (ℓ = 6.4 nm) changes the rate by 1 to 5% at radii of 125 to 500 nm, so what remains is the geometry of diffusion and volume conservation. Kao measured 0.49 and 0.56 at 800 and 900 °C near 0.25 µm, and 0.97 to 1.03 at 1100 and 1200 °C. The held-out 800 and 900 °C cylinders are 12 to 73% too thick (0 of 4 within tolerance after alignment; 1 of 4 as reproduced, by cancellation). At 1100 and 1200 °C the small cylinders are 12 to 14% too thin: Kao's convex oxide is at or above flat, and the program has no mechanism that makes a convex shell grow faster than the geometry allows.

The other readings of the paper do not help. Reading the normalization as to flat (110) gives 2 of 14 (median 12%); reading Kao's radius as the etched radius gives 2 of 14 (median 22%), with the small cylinders thinner but still far from 800 and 900 °C. The predictor's previous form, the viscous shell with Müller's stress time (142 s), gives 0.82 to 0.91 as well (scratch run, flat converted): at Kao's growth rates (0.3 to 20 nm/min) and radii a stress time of 142 s builds almost no stress.

Kao's own viscosity in the project's stress form (a scratch run, not a variant: it sets a fitted parameter). Kao's normal stress σ_rr = 2ηξ(1/b² − 1/a²) with ξ = a × growth × 25/45 and k_s ∝ exp(−σ_rr ΔΩ/kT) is the program's stress with stress_time = 2η ΔΩ/kT. Table I's η_0 gives 3.4 × 10⁵ s at 800 °C, 9.3 × 10⁴ s at 900, 8.5 × 10³ s at 1000, 1.6 × 10³ s at 1100 and 148 s at 1200 °C. With these, relative to the program's own no-stress cylinder, the ratios become 0.70 and 0.57 at 800 °C (Kao 0.73, 0.49), 0.69 and 0.53 at 900 (0.81, 0.56), 0.86 and 0.67 at 1000 (0.94, 0.72), 0.93 and 0.82 at 1100 (0.99, 0.97), 0.98 at 1200 (1.03). The temperature order is right and the low-temperature size is close; at 1000 and 1100 °C it slows too much, as expected without Kao's α (tension lowers the viscosity) and V_s (tension raises the solubility). The project's fitted stress time, 142 s at 850 °C, is Kao's 1200 °C viscosity; Kao's interpolated 850 °C value would be about 1.8 × 10⁵ s, 1200 times larger.

Aligned: 2 of 14 within tolerance (median error 13%) from 1 of 14 (26%). On the held-out cylinders and holes: 0 of 9 from 1 of 9, and the held-out median error rises from 15% to 37%: the conversion that fixes 1000 and 1100 °C makes the 800 and 900 °C cylinders, which the program does not slow enough, thicker still. The five holes have no program and count against the score.

1 near, 3 off, 5 not modelled of 9 numbers with the aligned settings (the samples used to choose the unknowns are left out).

What alignment cannot fix: the temperature dependence of the curvature effect, which is Kao's result. The radial program has no temperature anywhere (its A, its curvature length and its stress time are 850 °C numbers), so 800 and 1200 °C cylinders of the same size get the same normalized oxide. A viscosity with an activation energy (Kao: 4.25 eV above about 950 °C, 1.25 eV below) in the stress form, with stress_time = 2η(T)ΔΩ/kT, is what the data ask for; the selected curvature-length form cannot carry it.

Convex oxide thicker than the geometry allows (1100 and 1200 °C, and Kao's 1 to 2 µm cylinders above 1.0) needs Kao's tension terms: oxidant solubility rising with tension (V_s = 7.5 ų) and viscosity falling with it (α). The program has neither.

Concave holes: no program. A concave program needs the opposite sign in the cylindrical diffusion term (Kao's Eq. 8: a hole supplies less oxidant), the oxide forced inward with b < a, and the bulk pressure compressive (Eqs. 37–40), which lowers solubility and raises viscosity. Kao's own model is off by more than 10% at 800 and 900 °C on holes.

The curved-surface numbers are graph readings of least-squares lines, ±0.02 in the ratio, and the starting radius is back-computed from the measured oxide. A table of Kao's points would replace both.

What our language and ontology cannot yet say

• A reference sample whose measured value is an input, read as a thickness. RUN_FLAT_OXIDE_NM says “the oxide on a flat sidewall in the same run”, but the radial program reads it as oxide ÷ steam time and then applies its own diffusion term, so a 500 nm flat oxide becomes 349 nm on the program's flat sidewall. The field should mean what it says: the program should choose its flat rate so that its own flat sidewall grows the stated oxide (for this program, F = X(1 + X/A)). This one change moves the high-temperature cylinders from 0 of 5 to 2 of 5.
• Temperature in the curved-silicon program. Kao's five temperatures are the point of the paper. A stress_time = 2η(T)ΔΩ/kT with an Arrhenius viscosity, and an A that follows Deal and Grove's temperature law, would let the program be scored at 800 to 1200 °C; today every temperature is 850 °C to it.
• SHAPE: hole (concave). The language can write SHAPE hole; no program reads it. The radial program with the concave sign (Kao's Eqs. 8, 22, 37–40) would cover holes and trenches, which are half of Kao's data and all of Krzeminski's rings.
• The radius after oxidation as a sample field and as an observable. Kao, like most curved-surface papers, reports the radius after oxidation. A field RADIUS_AFTER_NM (or an observable Core_radius scored against it) would let the program start from an unknown etched radius and be scored on the measured final one, instead of back-computing the start from the measured oxide.
• A normalized observable. Kao reports curved oxide over flat oxide of the same run. An observable defined as a ratio of two samples' thicknesses in one BATCH would score exactly what was measured, without the (100)/(110) choice of the flat reference.
• Sidewall orientation. Kao reads the (110) part of a wall whose orientation varies around the circumference; flat (110) grows up to 1.29 times flat (100) in steam. A field for the measured orientation that a curved program reads (the planar program already has r_or for 110) would remove one of the paper's ambiguities from the comparison.

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 the oxide in Part I Figures 7 and 8 normalized to flat (100) or flat (110)? The text says (100); the curves tend to 1.0, while flat (110) grows 1.29 times (100) at 800 °C.
2. Why does Part II model 800 °C for 1050 min when Part I's Table 1 gives 1440 min for 5027 Å on (100)? Is 1050 min the time for 5000 Å on (110)?
3. Can Part I's convex points be assigned to temperatures (the symbols are not legible in the report scan)? A table of radius, temperature and thickness would replace these line readings.
4. Were the etched radii recorded? The program needs the radius before oxidation; we back-compute it from the oxide.
5. Was the steam pyrogenic or from a bubbler, and was the wafer pushed in steam or in nitrogen?