A pulse sized on a blanket wafer is 40× too short for a 3D-NAND wafer.

A chamber model in which every wafer face runs its own hole.

The chamber is an axisymmetric wedge of a 300 mm showerhead reactor: N₂ at 300 Pa and 573 K, 300 sccm, Reynolds number 0.045, a total-flux (Danckwerts) inlet so the TMA entering is exactly what the showerhead delivers. The wafer is a custom OpenFOAM boundary condition: on each of its 150 faces a one-dimensional Knudsen-diffusion, Langmuir-adsorption model of a 100 nm hole of aspect ratio 100 is advanced every time step from the local gas density, and its uptake is returned to the gas as a mixed boundary condition. Flat and hole fluxes blend by open-area fraction. Nothing is fitted to the chamber; the hole model sits within 3 to 11 % of the converged ballistic reference (Argonne's Machball and this repository's own 3D voxel Monte Carlo solver, which agree within 2 %) and was fitted to published lateral-structure thickness profiles.

The chamber and the pattern, explodedshowerhead, 300 mmwafer, 150 facesone hole model per faceoutlet at r = 200 mm

6,000 cells, 5° wedge, 15 mm gap (drawn 3× for legibility). The magnified patch is the pattern every face carries: 100 nm holes at 180 nm hexagonal pitch, half the wafer patterned. Drag to orbit.

Size the pulse on the densest region, not the wafer average.

Real device wafers mix dense arrays, open field and tiers of different depth. The boundary condition takes the hole density and aspect ratio as functions of radius, so two layouts were run: a dense AR-100 array in the centre with bare wafer beyond 100 mm, and a wafer whose hole depth ramps from AR 100 at the centre to AR 60 at the edge. The rule evaluated with the wafer-average wall area under-predicts both (5.1 s against 11.85 s needed; 10.6 s against 12.5 s). Evaluated with the local wall-area ratio of the most demanding region it is conservative by 15 to 21 %, the surplus being precursor the less-demanding neighbourhood feeds back laterally. The showerhead supplies molecules per unit area, so the local ratio is the one that counts.

Saturating pulse across the wafer for three layoutsuniform AR 100dense centre, bare edgeAR 100 → 60 ramp

Hole-bottom time to 90 % coverage against radius. A pattern boundary puts a 26 % saturation gradient inside the array (11.85 s at the centre, 8.75 s at its outer edge) where the uniform wafer varies by 0.4 %. The bare edge is itself 15 to 30 % slower than a blanket wafer because the array drains the gas. Hover.

Layout	wall ratio, wafer avg.	rule, wafer avg.	rule, local	needed
Uniform AR 100	55.9	14.4 s	14.4 s	14.05 s
Dense centre (r < 75 mm), bare beyond 100 mm	19.2	5.1 s	14.4 s	11.85 s
AR 100 at the centre → 60 at the edge	41.0	10.6 s	14.4 s	12.5 s

A 40 % tier-to-tier depth variation gives a 39 % spread in saturation time (12.5 s at AR 100, 7.6 s at AR 60): the recipe is set by the deepest tier and the shallower ones are over-dosed by their depth ratio. The uniform path of the boundary condition was regression-checked to the eighth digit after the change. Files: figures/pattern_layout.csv, reactor/make_layout_cases.py.

Wafers and powders on one curve.

The same normalisation puts a measured fluidized-bed ALD run (Argonne, 10 g of silica gel, 940 m²) at 1.12 supply-limit times, inside the 1.05 to 1.27 band of the chamber runs, and orders the measured precursor utilization of a rotary-drum reactor with its loading. Precursor utilization against total reactive area spans five orders of magnitude on one curve. Separately, the hole model fits the published saturation profiles from 500 nm lateral test channels; the fitted TMA sticking coefficient depends on the slit transport assumed, 8.5 × 10⁻³ with the hydraulic diffusivity and 2.0 × 10⁻² with the ideal-plate correction, both with the same residual, so the profile alone does not pin s.

Precursor utilization against reactive areablanket wafer (model)patterned wafers (model)fluidized bed, measuredrotary drum, measured

Utilization = 1 / (saturation time over supply-limit time). Fluidized bed: Lu et al., JVST A 40 (2022). Rotary drum: Coile et al., JVST A 38, 052403 (2020), Table 2. Hover a point.

The hole model against measured profilesZenodo 3974438, 8 profilesfit, hydraulic slit transport, s = 8.5e-3fit, ideal-plate transport, s = 2.0e-2s = 1e-3 and 1e-2

Al₂O₃ from TMA/H₂O at 300 °C, 500 cycles of 0.1 s, 500 nm lateral channel (Puurunen and Ylivaara, Aalto, CC BY 4.0). Two fits with the same residual (rms 0.038 of the plateau): s = 8.5 × 10⁻³ at 75 Pa with the hydraulic Knudsen diffusivity, s = 2.0 × 10⁻² at 32 Pa with the ideal-plate correction the 3D solver showed is needed for slits (the real cavity, 140 to 200 gaps wide and pillar-supported, sits between the two). Hover.

The chamber runs use s = 5 × 10⁻³; Arts et al. report 0.5 to 2 × 10⁻³ from the same kind of structure. The patterned-wafer conclusions are supply-limited and do not move with s. The blanket-wafer numbers do, and are quoted with that caveat.

Precursor delivery sets the time. Pressure and carrier flow set what it costs.

A face-centred central composite design, 25 OpenFOAM runs over TMA fraction (0.5 to 3 %), carrier flow (150 to 600 sccm), pressure (100 to 1,000 Pa) and gap (10 to 25 mm), fitted in JMP: log₁₀ of the saturating pulse scales as x⁻¹·⁰ Q⁻⁰·⁸ p⁻⁰·¹², R² 0.993, and the gap term is not significant. Utilization of the delivered precursor runs from 51 to 97 %: pressure up and carrier flow down raise it, because a slower gas spends longer over the wafer. A Gaussian-process surrogate on the same 25 runs (MATLAB fitrgp, leave-one-out R² 0.999 on time and 0.988 on utilization) finds the operating point that uses the same precursor supply as the centre and saturates faster.

The response surface100 Pa316 Pa1,000 Pa

log₁₀ t₉₀ from the JMP fit at the centre pressure and gap, with every run as a sphere. Pressure moves a point off the surface by at most ×1.15; gap by less than the fit's residual. Hover a run.

Cost of ownership: precursor per wafer against pulse timethe 25 runssurrogate front, edge lag ≤ 5 %

Precursor use is 1 / utilization: how many times the stoichiometric minimum the showerhead had to deliver before the deepest hole saturated. Hover a run.

The rule has consequences past the TMA pulse.

Water: same pulse, 50× less reactivek_des 00.1 /s1 /s10 /s

On the patterned wafer the H₂O half-cycle (s = 10⁻⁴) saturates in 13.4 s against 14.05 s for TMA: supply, not kinetics, again. With reversible adsorption above about 0.3 /s the surface cannot reach 90 % at 1 % H₂O, and the desorbing water keeps the exhaust wet through the purge.

Plasma: the recombination ceilinglargest AR reached in 5 sAR 100

Radicals lost on every wall collision cannot be bought back with dose: at a recombination probability above about 3 × 10⁻³, aspect ratio 100 is unreachable in any exposure. Measured values for O on SiO₂ and TiO₂ sit near 6 × 10⁻⁵ (Arts et al.).

Temporal against spatial ALDtemporal, 1 % and 3 %spatial, zone 10×30×100×

Cycle time from the rule for a temporal tool (two half-cycles plus two 0.5 s purges) and for a spatial tool whose precursor zone runs at an assumed 10 to 100× the showerhead flux. The zone flux is assumed, not measured; the shape of the comparison is not.

Every number from a script, on a laptop.

git clone https://github.com/Soham-ChemE/ald-conformality.git && cd ald-conformality
python3.12 -m venv .venv && .venv/bin/pip install -e . && PYTHONPATH=src .venv/bin/python -m pytest -q tests
# chamber: OpenFOAM v2606 (gerlero/openfoam-app on Apple silicon), then
cd reactor/lib/aldWallKinetics && wmake && cd ../.. && ./Allrun          # steady flow, blanket pulse, Langmuir wafer
python sweep_pattern_map.py && python doe_patterned.py                        # the 16-run map and the 25-run DOE
python ../analysis/make_doe_jsl.py && ../scripts/run_jsl.sh ../analysis/doe_rsm.jsl   # JMP response surface
matlab -batch "run('../analysis/doe_gp.m')"                                   # MATLAB GP, optimum, Monte Carlo
python ../docs/build_assets.py                                                # this page's data and figures

129 tests (pytest --collect-only -q tests). The same tests, the two C++ libraries with their Python bindings, the OpenFOAM chamber chain in a container and the page assets run in GitHub Actions on every push:

The validation data (Zenodo 3974438) ships in data/; the Argonne numbers are quoted from the papers. CLAIMS.md maps every number on this page and in the paper to the file that produces it. The paper: draft 0.9, LaTeX in paper/.

Sticking coefficient, fab units, chamber-scale checks, a 3D chamber, device geometry, other chemistries.

Seven questions any reviewer or process engineer would ask, each answered with new runs rather than argument. Every scene below is drawn from the repository's own data files; the numbers in the captions trace to CLAIMS.md, and the audit figure behind each scene is linked.

Does the rule depend on the sticking coefficient?blanket waferflat surface between holeshole bottomssupply limits

No. Across s = 5 × 10⁻⁴ to 2 × 10⁻², the published TMA values and both PillarHall fits of section 08, the AR-100 wafer saturates in 14.00 to 14.20 s, 1.17 to 1.18× the supply limit. The blanket wafer (0.28 to 1.01 s) and the flat surface between the holes (1.05 to 6.0 s) carry a factor-4 uncertainty; the rule and the map do not. Eleven chamber runs. Hover a point. Audit figure.

In fab units: step coverage and thicknessAR 20AR 50AR 100

The rule's pulse gives 99 % step coverage, not just 90 %, because the saturation front is 0.3 s wide at every aspect ratio. Fifty cycles of a blanket recipe leave 1.4 nm of a 5.15 nm film on top of an AR-100 array and nothing at the bottom. Audit figure.

The Al₂O₃ cycle, segment by segmentTMA pulsepurge to 1 %H₂O pulsepurge, water desorbing

The purge is not the hole's doing: a 10 µm hole empties in 14 µs by Knudsen diffusion, the chamber reaches 1 % of the pulse-end signal in 1.9 s, and desorbing water can hold the exhaust up for 8 s at a desorption constant of 0.1 /s. An Al₂O₃ cycle is 12 s on a blanket wafer and 38 s on the product wafer: 5.5 against 1.9 wafers per hour for 5 nm. This is why high-aspect-ratio thermal ALD in production runs in batch vertical furnaces (100 or more wafers per load) or multi-station chambers: the long pulse is paid once for many wafers. Hover a segment.

The rule as throughput1 % TMA and H₂O3 %the chamber run

Above a wall-area ratio of about 10, throughput falls as 1/(1 + A_wall/A_wafer) and precursor fraction is the only knob that moves it; tripling the fraction doubles AR-100 throughput and leaves the blanket wafer unchanged. The point is the chamber run, on the curve. Audit figure.

Does the blanket number sit where published data sit?this modelkinetic, from a reported sapplied doseCFD exposure

On the like-for-like basis (the exposure the surface actually sees, 0.146 Pa·s, 1,090 L) the blanket wafer sits within a factor 1.8 of the three 300 °C sources with an explicit sticking coefficient. The feed exposure is 7× larger because the wafer draws the gas down, and published feed exposures are inflated the same way. Hover a point for its source. Audit figure.

A published reactor, reproduced where it can be25 ms, s 3.4e-360 ms, 1.5× amount60 ms, 2.4×25 ms, s 5e-3measured, Gakis 2019

Gakis et al. (2018, 2019) ran 3D CFD of TMA/H₂O in a Fiji F200 and measured thickness maps. In an axisymmetric approximation of their reactor, the TMA exposure delivered per 25 ms pulse agrees with their CFD to 12 to 22 % (0.67 against 0.55 to 0.60 Pa·s) and the saturation state matches the measured growth per cycle. Their 2.4 % measured non-uniformity is not reproduced: it comes from an off-axis inlet, a loading-door purge and a 21 °C wafer temperature gradient that a single-inlet frozen-flow wedge cannot carry. Stated, not hidden. Audit figure.

What a one-sided pump does that the wedge cannot seepump port, 60°

A full 3D chamber (106,000 cells, 60° pump port, 32 minutes for the pulse) with the same patterned wafer. The carrier flow is strongly one-sided (the port-side quadrant drains at 3× the axisymmetric speed) and the gas-side quantities follow at the 10 % level, but the saturation time integrates the supply: worst point 14.17 s on the far side against 14.05 s in the wedge. A one-sided pump costs about 1 % of pulse time; the rule's 1.2 covers it. Drag to orbit, hover a face. Audit figure.

Saturation time around the waferr = 50 mm100 mm140 mm

Azimuthal spread 0.09 % at r = 50 mm, 0.37 % at 100 mm, 0.96 % at 140 mm: a single lobe, earliest at the port, latest opposite it. The flat surface, saturating at 2 s under a strongly drawn-down gas, is the more sensitive indicator (4.8 % spread at 140 mm).

Other chemistries, by the same ruleAl₂O₃ / TMAHfO₂ / TDMAHTiN / TiCl₄W / WF₆

The rule is chemistry-agnostic through s, N₀ and c₀: HfO₂ from TDMAH 9.96 s (the chamber run gives 11.6 s, 1.17×, the excess mostly the 2.3× lower gas diffusivity), TiN from TiCl₄ 4.4 s, W from WF₆ 57 s. Per nanometre of film the pulse scales with the metal-atom density. Literature inputs and their caveats in the hover.

Tapered holescoatedbare

Real channel holes narrow toward the bottom. With the hole model extended to a depth-varying opening, a bottom at 90 / 80 / 60 % of the mouth costs 1 to 4 % of pulse time on the supply-limited wafer (14.25 / 14.40 / 14.65 s against 14.05 s straight), because the lost conductance outweighs the lost area; at constant mouth density the cost is 4 to 20 %. The rule with the frustum area moves the other way; the two bracket a tapered-hole wafer. Hover a hole. Audit figure.

Word-line recesses filled through a slitcoatedbare

The hardest ALD step in 3D-NAND: lateral recesses (29 nm gap, 365 nm deep, 51 nm pitch) fed through a 100 nm slit 5 µm deep, modelled as nested one-dimensional problems. The last recess bottom lags the slit bottom by 0.5 to 9 % across the four geometries run, and the structure behaves as an area-equivalent slit within 4 %, so the rule's area-only treatment holds to recess aspect ratio 35. A fully patterned wafer of this kind carries 150 to 190× its area and would need 38 to 48 s at 1 % TMA. Hover a recess. Audit figure.

Gate-all-around nanosheets, with the 3D solvercoatedbare

IBM 2017 cell, TMAafter 3 nm of oxide, TiCl₄time —

A three-sheet cell from the 2017 IBM/Samsung/GlobalFoundries stack (5 nm sheets, 10 nm slots, 40 nm wide, 12 nm gate length) and a scaled four-sheet cell, run with the voxel solver for TMA, TDMAH and TiCl₄. The last point to saturate is always the centre of the bottom slot; TMA reaches 90 % everywhere in 0.56 s at 0.3 Pa against 0.47 s on a flat surface, and with the 3 nm interfacial oxide and high-k already in place TiCl₄ in the 4 nm slot lags 28 % centre to edge. Per wafer such a stack is only 4 to 6× the flat area. No per-sheet thickness measurement is public, so this is an application of the validated solver, with the testable prediction stated: a 3 to 8 % slot-centre deficit after a one-third pulse. The scene is every exposed voxel face of the cell (8,400 faces at 1.25 nm; the narrowed cell's 36,864 faces of 0.5 nm merged to 1 nm tiles) at a third, two thirds and all of the time to 90 % everywhere; slots are drawn 3× their height and the trench above the stack at half height, so the slot interiors can be seen. Hover a face; drag to orbit. Audit figure · Geometries and sources.

A current NAND word-line cellcoatedbareslowest recess

time —

A 253-gate cell at a 48 nm word-line pitch, 12.7 µm slit depth, 1 µm recesses and four rows of support pillars (TechInsights, YMTC 232L), 5.7 million exposed voxel faces. TMA saturates the slit floor in 14.6 s but the slowest recess in 33 s, and that recess sits at 87 to 90 % of the depth, not at the floor: the saturated floor reflects the stream into the lowest recesses. Per wafer the stack carries 448× its area; the rule gives 113 s at 1 % TMA, 3.4× what the 0.3 Pa cell needs, and for TiN the margin is only 17 %. The scene is the 5 µm cell with pillars (1.19 million faces) cut through its pillar row: the faces in a two-voxel slab, merged to 6,205 tiles, at 3, 6, 9 s and the end of the TMA run at 0.3 Pa, with the pillar wall coloured by its coverage per recess; the slit lies horizontally with its mouth at the left, drawn 4× wider across the slit than deep, each face 100 nm long along the slit. Hover a tile or a pillar segment; drag to orbit. figures/device_geometries.csv · Audit figure.