Validation & Verification (V&V)

Every claim here is enforced by CI.

PlasmaCCP is a trend-level screening reduced-fluid model — it predicts direction and order of magnitude, not absolute values. Every figure below is backed by repository code and CI.

Enforced by CI on every commitEvery benchmark runs automatically against the published literature.

Direction & order-of-magnitudeWe claim the trend, not the absolute value — that honesty is the point.

8CI benchmarks (M1–M8)

10literature anchors (V1–V10)

24presets × geometry test battery

3,200+automated tests

~2 sfull solve

Verification

Are we solving the equations right?

Numerical and regression checks. Directional / ratio criteria — power↑→nₑ, frequency↑→ion energy↓, power-balance closure — are run by CI on every commit.

See benchmarks M1–M8 →

Validation

Are we solving the right equations?

Directional checks anchored to published literature. We confirm agreement with the literature at the order-of-magnitude / direction level, never on absolute values.

See literature anchors V1–V10 →

Benchmark gallery — M1–M8

8 verified · 0 partial. Each case is a directional / ratio criterion, not absolute-value accuracy. Use the "reproduce" link to load the preset straight into the simulator.

Case	Claim tested	Status
M1	GEC Ar reference cell — power sweep	✓ verified
M2	Ar pressure sweep — Te & sheath	✓ verified
M3	Frequency sweep — coupling & ion energy	✓ verified
M4	Asymmetric electrode area ratio — DC self-bias	✓ verified
M5	Dual-frequency (60 MHz + 2 MHz) — role decoupling	✓ verified
M6	O₂ electronegative discharge — density & Te	✓ verified
M7	SiH₄/Ar PECVD — deposition regime	✓ verified
M8	Power-balance closure — absorbed-power consistency	✓ verified

M1verified

GEC Ar reference cell — power sweep

Ar, 0.1 Torr, 13.56 MHz, symmetric parallel-plate GEC cell, power swept 100→500 W

Literature basis

L&L 2005 Ch.10 global model (nₑ ∝ P_abs); Godyak et al. 1991; Olthoff & Greenberg, NIST 1995 (GEC nₑ~1e15–1e16 m⁻³)

Pass criterion

PASS if nₑ increases 3–8x over 100→500 W. FAIL if flat or decreasing. |V_dc| < 10% of the RF amplitude.

Reproduce: open preset ref-gec-ar

M2verified

Ar pressure sweep — Te & sheath

Ar, 13.56 MHz, fixed power, pressure swept 0.05→0.5 Torr

Literature basis

L&L 2005 Ch.10 (Te vs p·L); Godyak, Piejak & Alexandrovich 1991; Child law s ∝ V^0.75/√n

Pass criterion

PASS if Te(0.5 Torr) < Te(0.05 Torr), monotone decreasing. Sheath thins monotonically as pressure (density) rises. FAIL if Te is flat or increases with pressure.

Reproduce: open preset ref-gec-ar

M3verified

Frequency sweep — coupling & ion energy

Ar, fixed power & pressure, frequency 2 / 13.56 / 27 / 60 MHz

Literature basis

VHF CCP literature (density rise); L&L 2005 §11 (sheath voltage ∝ 1/f tendency); Boyle, Ellingboe & Turner 2004

Pass criterion

PASS if ion energy at 60 MHz < ion energy at 2 MHz (direction). nₑ increases with f but bounded by the η_f clamp. FAIL if ion energy rises with f.

Reproduce: open preset sym-research-ar

M4verified

Asymmetric electrode area ratio — DC self-bias

Ar, 13.56 MHz, powered electrode smaller than ground (area ratio A_g/A_p > 1), increasing asymmetry

Literature basis

L&L 2005 §11 capacitive divider V_a/V_b=(C_b/C_a); Godyak & Sternberg 1990; Child law s ∝ V^0.75

Pass criterion

PASS if V_dc is negative (on powered electrode) and |V_dc| grows with asymmetry. Sheath thickens as |V_dc| rises. FAIL if V_dc is positive or insensitive to asymmetry.

Reproduce: open preset baseline-300mm

M5verified

Dual-frequency (60 MHz + 2 MHz) — role decoupling

Ar, HF 60 MHz + LF 2 MHz co-driven, HF power and LF power varied independently

Literature basis

Boyle, Ellingboe & Turner, J. Phys. D 37, 697 (2004); L&L 2005 §11.5 dual-frequency CCP

Pass criterion

PASS if raising HF power moves ion flux more than ion energy, and raising LF power moves ion energy more than ion flux. FAIL if both frequencies move the same output in the same proportion.

Reproduce: open preset etch-ox-60-2

M6verified

O₂ electronegative discharge — density & Te

O₂, 13.56 MHz, compared against Ar at identical power / pressure / geometry

Literature basis

Lafleur et al., PSST 23, 035010 (2014); L&L 2005 Ch.10 electronegative discharges

Pass criterion

PASS if nₑ(O₂) < nₑ(Ar) at matched conditions. FAIL if O₂ gives equal or higher nₑ than Ar.

Reproduce: open preset iso-etch-cf4

M7verified

SiH₄/Ar PECVD — deposition regime

SiH₄/Ar mix, 13.56 MHz, low–moderate power, PECVD conditions (high neutral/ion ratio)

Literature basis

Kushner, J. Appl. Phys. 54, 4958 (1983); L&L 2005 Ch.14–16 (PECVD)

Pass criterion

PASS if nₑ(SiH₄/Ar) < nₑ(pure Ar) at equal power and neutral/ion ratio > 1e3. FAIL if density is indistinguishable from pure Ar.

Reproduce: open preset pecvd-sinx-300

M8verified

Power-balance closure — absorbed-power consistency

Any Ar preset; volume integral of the absorbed-power density P=(1/2)Re(σ)|E|² compared with the requested power

Literature basis

L&L 2005 Ch.11 time-averaged Ohmic heating P=(1/2)Re(σ)|E|²; Boeuf & Pitchford 1995 (2D Ohmic P=J·E)

Pass criterion

PASS if 0.5 ≤ (integrated absorbed power / requested power) ≤ 2. FAIL if the ratio falls outside this band or diverges with power.

Limit of this check The solver rescales the fields by balance = √(P_target/P_integral) before this check runs, so the ratio matches the target by construction — 1.000004–1.000035 across 5/5 shipped presets, and fixed even when balance moves 12x (measured 2026-07-31). M8 therefore checks the plumbing (that the rescale was applied, and that the quadrature agrees), and cannot be cited as evidence that a field map is physically right.

Reproduce: open preset baseline-300mm

Predicted vs observed — CI sweeps against the literature

The solid line is the simulation's CI-measured curve; the dashed line is a guide for the direction/shape the cited literature predicts. The guide is anchored to the sim's own points (its first/last or peak), so it shows only direction/shape and invents no absolute value or slope. Agreement of direction/shape is verified by CI on every commit.

RF frequency → Vpp

✓ direction matches — CI-guarded

Fixed power; drive voltage falls as frequency rises

A direction/shape comparison — not an absolute-value or slope claim

Solid = sim (source: tests/test_rf_frequency_response.py) · Dashed = literature direction (L&L 2005 §11 (sheath voltage ∝ 1/f tendency)), monotone-decrease guide joining the sim's first & last measured points (no slope/shape claim)

RF frequency → ion energy

✓ direction matches — CI-guarded

M3 criterion: ion energy at 60 MHz < at 2 MHz

A direction/shape comparison — not an absolute-value or slope claim

Solid = sim (source: tests/test_rf_frequency_response.py) · Dashed = literature direction (Godyak et al. 1991 / L&L 2005 §11), monotone-decrease guide joining the sim's first & last measured points (no slope/shape claim)

Pressure → density reference

✓ shape matches — CI-guarded

M2 criterion: density_ref rises through the sub-Torr band and saturates near ~1–3 Torr (0D power-balance hump) — so the sheath thins monotonically with pressure

A rise→saturate shape comparison — not an absolute-value claim

Solid = sim (source: B12 parameter sensitivity, Ar 500 W sweep) · Dashed = literature shape (0D power balance (L&L Ch.10) — saturates ~1–3 Torr), sim first point → peak point, then hold the peak level (rise→saturate shape, not absolute values)

Cross-code verification — Zapdos (MOOSE)

A quasi-1D active-gap contract (1 Torr argon, 13.56 MHz, 25.4 mm powered-to-ground gap, zero neutral flow, and single-ended 100 V drive) is run through Zapdos, an independent open-source implementation. The same three axial profiles are compared only for normalized shape; both curves are real computed results, each normalized to its own peak.

What this does and does not show

It shows: where and by how much the two codes diverge in an aligned quasi-1D active-gap comparison. Declared mismatches remain, and the comparison can catch classes of implementation errors.
The active gap is quasi-1D matched. The first revision put a 1-D reference against a 2-D asymmetric GEC counterpart, so it proved nothing. The active gap, coordinate, drive, and zero-flow condition are now matched; a finite grounded sidewall remains, and near-axis radial shape deviation is measured below 1%. Wall closures, chemistry, density scale, and the Zapdos steady-state limit remain declared differences.
It does NOT show: accuracy. Code-to-code agreement is verification, not validation against measurement. Zapdos is not a truth standard either — its own peer-reviewed V&V reports departures from experiment at low pressure.
The sheath is where they part — as expected. Zapdos screens the RF field in the bulk by 33x; we manage 1.35x. That is the same axis as an exclusion already declared under "What we do NOT claim" below (self-consistent space-charge sheath potential profile). This is not a newly found defect — it puts a number on a limit we had already stated.
Symmetry — the sharpest result. In the quasi-1D active-gap contract, Zapdos gives nearly symmetric walls (|E| wall asymmetry 0.001). Ours is 0.357, tilted toward one electrode, and nₑ peaks near the powered face. This is a normalized-shape result, not evidence of identical full geometry.
Sheath width definitions and values differ. Our sheath model returns a lumped thickness scalar; Zapdos reports a spatially resolved density-criterion width. Under the quasi-1D active-gap contract they are 1.90 mm and 3.63 mm respectively, but they are not a like-for-like sheath definition or a promotion metric. The earlier 1.1% claim from the mismatched case is withdrawn.
Where Zapdos is legitimately ahead: chemistry generality (EEDF-table multi-species kinetics), a peer-reviewed V&V record, kinetic wall boundary conditions, and — decisively here — a spatially resolved self-consistent sheath.

|E| electric field

△ deviates even in the bulk (0.781) — sheath-model difference

Bulk 3.8–21.6 mm · max deviation over the whole gap 0.866

A shape comparison. For |E|, the displayed absolute values come from the drive-scale CSV (drive), not the power-balanced product map; absolute peaks are not an agreement claim.

x=0 powered wall, x=25.4 mm ground · mean deviation in the bulk 0.574 / whole gap 0.574

n_e electron density

△ deviates even in the bulk (0.583) — sheath-model difference

Bulk 3.8–21.6 mm · max deviation over the whole gap 0.910

Shape only — our nₑ map is a relative trend field with no absolute scale, so absolute densities cannot be compared.

x=0 powered wall, x=25.4 mm ground · mean deviation in the bulk 0.177 / whole gap 0.268

T_e electron temperature

△ our curve is essentially flat across the gap (varies 0% vs 92%)

Bulk 3.8–21.6 mm · max deviation over the whole gap 0.922

A shape comparison. Tₑ absolute values are shown, but the closures and floor differ, so they are not an agreement claim.

x=0 powered wall, x=25.4 mm ground · mean deviation in the bulk 0.032 / whole gap 0.347

Reproduction — zapdos ab83e0573f77 / moose beb188528893, 120 RF cycles (1 core, 34 min), averaged over the final period. Zapdos resolves the gap with 212 nodes, we use 126 cells. The archived 103→120-cycle normalized-profile movement is at most 6.37%; inputs, checksums, CSVs and scripts live in docs/crosscode-zapdos/ in the repo.

Literature-anchored validation — V1–V10

Direction / order-of-magnitude checks anchoring each model output to published-literature values. Preset links deep-link straight into the simulator. The full table and explicit exclusions are preserved in the model-docs archive below.

Quick-reproduce presets:ref-gec-ar sym-research-ar etch-ox-60-2 iso-etch-cf4

References flagged [verify] have not yet been source-checked against the original and keep that caption in the model-docs archive below.

What we do NOT claim

We keep this as a first-class section, not hidden — being explicit about what it does not do is what builds trust.

Out of validation scope (will not match)

Absolute etch/deposition rate (nm/min)
EEDF shape / non-Maxwellian detail
Azimuthal uniformity
Absolute plasma-density value without calibration
Process-driven change in field-map SHAPE (e.g. with pressure) — levels and geometry dependence do respond, but the shape itself is set by the geometry

What this model does NOT do

Detailed EEDF (Maxwellian assumed)Assumes a Maxwellian distribution; no non-equilibrium tail
Full space-charge self-consistent sheath potentialSheath potential is a Child-law proxy, not a full field solve
Absolute etch / deposition ratesOnly flux/energy proxies — no absolute rate in nm/min
Detailed surface reaction chemistryNo surface coverage/reaction kinetics; only lumped wall loss
Time-resolved RF cycle (phase is reconstructed)Cycle-averaged solve; phase frames are reconstructed after
3D non-axisymmetric effects2D axisymmetric (r,z) only; no azimuthal asymmetry

The full limitations — coordinate conventions, the charge-free (Laplace) electrostatic solve, the deliberate sheath-potential approximation — are documented in the model-docs archive below (Scope & Limitations, mirroring physics-model.md §12–13). Go to model docs →

Confidence system — attached to every result

Every run carries a 3-level confidence chip and up to 37 condition-note codes automatically — these are the actual components the simulator uses, not screenshots.

Trend-level (screening)Trust relative trends (direction, order of magnitude) — not absolute values.

Semi-quantitative (calibrated)Semi-quantitative near the calibration point — still a screening tool.

Low confidence — check geometryThese results are not trustworthy — fix the geometry/inputs first.

Model documentation — full basis, tables, references

Below are the per-component literature basis (1–7), the detailed B1–B16 documentation, result-reading notes, and the full Scope & Limitations. Collapsed by default, with every deep-link anchor preserved.

Expand / collapse full model documentation

Model Validation & Literature Basis

PlasmaCCP is a trend-level reduced-order fluid model. Below each modeling component is justified with the literature that supports the approach.

1. Charge-free electrostatic Laplace solve (2D axisymmetric, dielectric, Dirichlet electrodes)

\nabla \cdot (\varepsilon \nabla \phi) = 0

The simulator discretizes the charge-free Laplace form in cylindrical coordinates with dielectric permittivity, applying Dirichlet conditions on powered and ground electrodes. Full space-charge Poisson is intentionally excluded because the reference mesh does not resolve the Debye length.

2. Complex/phasor RF field and Drude conductivity

\sigma(\omega) = \frac{n_e e^2}{m_e(\nu_m + i\omega)}

The simulator uses a phasor representation of the RF electric field and the Drude conductivity model to approximate the electromagnetic response of the plasma. This closure matches that applied in 13.56 MHz CCP discharges by Gogolides & Sawin (1992).

3. Drift-diffusion transport + Scharfetter-Gummel discretization

Electron and ion transport are described by the drift-diffusion equations, stabilized with Scharfetter-Gummel exponential discretization to prevent numerical blow-up. PIC results of Surendra & Dalvie (1993) support the drift-diffusion closure for capturing qualitative trends.

4. Sheath formation, Bohm criterion, ion energy/flux proxies

s = \frac{\sqrt{2}}{3}\lambda_{\mathrm{De}}\left(\frac{2V_s}{T_e}\right)^{0.75}

Sheath thickness is mean-anchored to the Child-Langmuir law shown below. The sheath voltage is derived from the RF and DC-bias terms; the final thickness is clamped between the Debye length and 0.45·gap so it can never exceed the electrode gap. The radial shape still comes from the φ-drop profile. The mm values are semi-quantitative.

5. Neutral transport (inlet/outlet, residence time, depletion, gas heating)

\tau_{\mathrm{res}} = \frac{pV_p}{Q_{pV}} \frac{T_{\mathrm{STP}}}{T_{\mathrm{wall}}} = \frac{V_p}{Q_{\mathrm{chamber}}}

The reported residence time converts STP-referenced pV throughput to chamber volumetric flow. Gas temperature is estimated by a steady-state lumped energy balance; full reactive chemistry is not included.

6. Multi-RF-source superposition (power-weighted effective frequency and phase)

Multiple RF sources are superimposed using power-weighted effective frequency and phase. Boyle et al. (2004) show in dual-frequency CCP that the high frequency controls ion flux while the low frequency controls ion energy.

7. Volume power-absorption density proxy

P_{\mathrm{abs}} = \frac{1}{2} \operatorname{Re}(\sigma) \lvert E \rvert^2

Power absorption density is computed from the time-averaged Ohmic expression P = (1/2)Re(σ)|E|², using the phasor electric field and Drude conductivity. This form is derived standardly in Lieberman & Lichtenberg (2005) Ch. 11.

B1. Model Classification — Trend-Level Reduced Fluid Plasma Model

This simulator uses a reduced fluid formulation for fast relative-trend analysis of RF/CCP plasma. It estimates qualitative and semi-quantitative trends in plasma density, ion flux, RF impedance, sheath response, and parameter sensitivity. It is not a replacement for fully validated PIC/MCC, commercial multi-fluid simulation, or experiment.

B2. Core Model Choices

Component	Method	Reason	Limitation
Electrostatic field	Poisson solver with dielectric + Dirichlet BC (2D axisymmetric)	Standard approach for GEC-cell CCP fluid models	No electromagnetic (non-electrostatic) effects at very high power/frequency
Electron/ion transport	Drift-diffusion with Scharfetter-Gummel discretization	Numerically stable; PIC-validated for CCP trends	No ion inertia or non-local presheath transport
Electron conductivity	Drude model σ = nₑe²/(mₑ(νₑ+iω))	Standard RF plasma closure at 13.56 MHz CCP	Inaccurate for non-Maxwellian EEDF, especially at low pressure
Sheath	Child-Langmuir mean-anchor s=(√2/3)λ_De(2V_s/Te)^0.75 (clamped [λ_De, 0.45·gap]) + φ-drop radial shape. By default the RF-circuit anchor (area-ratio capacitive divider V_a/V_b=C_b/C_a) is used; Child-Langmuir is the fallback.	Standard trend approximation for ion flux/energy; thickness never exceeds the gap	No stochastic/collisionless sheath heating; mm values are semi-quantitative
Electron rate coefficientsalways-on	Analytical Te-dependent functions (LUT upgrade applied)	Practical for fast parameter sweeps	Inaccurate at high Te or non-Maxwellian distributions
Electron energyalways-on	0D energy balance — electron temperature coupled to particle balance model	Reduces computational cost	No spatial electron energy distribution
Gas heatingalways-on	Kushner (1983) volumetric heating source term	Captures gas rarefaction trends	No full fluid energy equation
Multi-frequency RF	Power-weighted effective frequency superposition	Captures dual-frequency CCP trends	No higher harmonics or cross-modal effects
Chemistryalways-on	Lumped chemistry auto-derived from gas mix (real N2 and Ar/SiH4 sets, electronegative routing)	Sufficient for basic process trends	Real N2 (e, N2⁺, N, N2(A)) and Ar/SiH4 (SiH3/H radicals) sets exist; CF4/CHF3/C4F8/NF3/SF6/Cl2 route to the electronegative set — per-region surface recombination and secondary-electron emission (γ_se) are now included, but full surface reaction chemistry is not modeled
Calibration	Optional experimental calibration factor	Provides simple path to improved absolute accuracy	Without calibration, absolute density uncertainty ~2–5×

B3. Intended Use

Recipe trend comparison (A vs B under the same gas conditions)
Pressure / power / gap / frequency sensitivity analysis
Relative plasma-density trend estimation
Relative ion-flux trend estimation
RF impedance and matching tendency exploration
Educational visualization and physical intuition building
Early-stage process intuition (design-space screening)

B4. Not Intended For (Caution)

Do not rely on this simulator for the following purposes — results may be misleading.

Absolute density values without calibration
Final process qualification or production sign-off
Detailed radical chemistry analysis
Low-pressure nonlocal kinetic analysis
Stochastic sheath-heating analysis
High-frequency electromagnetic wave effects
Hardware design sign-off
Replacement for experiment, PIC-MCC, or validated commercial solvers

B5. Expected Accuracy

Uncalibrated electron density uncertainty is approximately 2–5× of the reference. Absolute ion flux is accurate to roughly one order of magnitude. Calibration with experimental data improves relative trend accuracy near the calibration operating point, but absolute accuracy remains system-dependent. For applications requiring absolute confidence over a wide parameter range, use full PIC/MCC or a validated multi-fluid commercial solver.

B6. Lightweight Accuracy Upgrades

The upgrades below fall into two groups. (a) Always-on — applied to every run with no opt-in. (b) Flag-gated — controlled via environment variables; ION_SOLVER, RF_SHEATH, and TE_PDE default to ON ("0" to disable), while only GUMMEL defaults to OFF (bit-exact when OFF). This remains a trend-level reduced fluid model and does not guarantee absolute accuracy.

(a) Always-on (enabled by default)

Upgrade	Runtime cost	Expected benefit
Per-gas rate-coefficient lookup table (LUT) from BOLSIG+/LXCat	Very low	Improved ionization rate accuracy over a wide Te range
0D electron energy balance (self-consistent Te)	Low	More realistic Te response over power/pressure sweeps
Gas heating source term (Kushner 1983)	Very low	Captures gas rarefaction effects at high power
Lumped chemistry (real N2 and Ar/SiH4 sets, incl. electronegative routing)	Low	Improved density prediction at high power, low pressure, and molecular-gas conditions
Experimental calibration factors (calibration block)	Almost none	Large improvement in absolute density accuracy near the calibration point (neutral defaults = bit-exact)

(b) Flag-gated (ION_SOLVER, RF_SHEATH, TE_PDE default ON; only GUMMEL default OFF)

ION_SOLVER, RF_SHEATH, and TE_PDE are now ON by default and can be disabled by setting the environment variable to "0". Only the GUMMEL outer loop remains OFF by default (opt-in); when OFF, results are bit-exact with the existing pipeline.

Environment flag	Default	What it does
`PLASMACCP_GUMMEL_ITERS`	OFF (1 pass)	Damped ne↔RF Gummel outer loop (self-consistent iteration; falls back to one-shot on divergence)
`PLASMACCP_TE_PDE`	ON ("0"=off)	Screened-diffusion Te(r,z) energy PDE — supplies the spatial electron-temperature shape
`PLASMACCP_RF_SHEATH`	ON ("0"=off)	Analytic two-sheath capacitive divider (area-ratio voltage-division fixed point)
`PLASMACCP_ION_SOLVER`	ON ("0"=off)	Single effective-ion continuity equation → surface ion flux (Bohm edge 0.61·n·u_B)

B7. Validation Workflow

Run a baseline Ar simulation at the documented GEC-cell conditions (Hargis et al. 1994).
Compare computed electron density and ion-flux trends against published PIC/MCC or fluid benchmark results at the same conditions.
Confirm that trend directions (increasing/decreasing) agree across voltage, pressure, and gap sweeps.
If available, derive a calibration factor from experimental data (Langmuir probe, VI probe, matching network).
After calibration, trust absolute numbers only near the closest operating point.
Always review the Scope & Limitations section before drawing conclusions.

B8. Suggested Benchmark Targets

GEC reference cell (Hargis et al. 1994) — the well-documented parallel-plate 13.56 MHz Ar CCP benchmark experiment.
Ar CCP electron density and ion flux — compare directly against literature values (density ~10⁹–10¹¹ cm⁻³, ion flux ~10¹⁵–10¹⁶ m⁻²s⁻¹).
Pressure / power / gap / frequency sweep trends — check directional agreement against Boeuf & Pitchford (1995) and Surendra & Dalvie (1993).
VI-probe and matching-network impedance — compare RF impedance trends against lab measurements.
COMSOL Plasma Module — a validated commercial multi-fluid solver for trend comparison.
Zapdos/MOOSE — open-source MOOSE-based plasma finite-element code for independent cross-check.

B13. Literature-Anchored Validation

The table anchors each model output to published-literature values. Each row lists a preset deep-link (loads directly into the simulator), the model output to read, the literature anchor, and the pass criterion. These are order-of-magnitude and directional checks, not absolute validation.

Check	Preset	Model output	Literature anchor	Pass criterion
V1 Ar GEC cell nₑ order-of-magnitude	ref-gec-ar	`model_upgrades.electron.n_e_0d_m3`	Olthoff & Greenberg, J. Res. NIST 100(4) (1995): nₑ ~1e15–1e16 m⁻³	within one order
V2 Ar GEC Te	ref-gec-ar	`electron.te_eV`	GEC/Godyak: Te≈2–4 eV @0.1 Torr	inside range
V3 Te falls with pressure	ref-gec-ar 0.1→0.5 Torr pressure sweep	Te scalar	Godyak, Piejak & Alexandrovich (1991)	Te(0.5T) < Te(0.1T)
V4 Sheath mm-scale	ref-gec-ar / sym-research-ar	`sheath_metrics.thickness_mean_mm`	Lieberman (1988), L&L Ch.11	0.3–10 mm, never ≥ gap
V5 nₑ linear in power	any Ar preset ×2 power	density reference / 0D nₑ	L&L Ch.10 global model; Godyak	doubling P ≈ doubles nₑ
V6 Sheath thins with power/density, thickens with \|Vdc\|	sweeps	`thickness_mean_mm`	Child law s∝V^0.75/√n	direction
V7 Dual-frequency role split	etch-ox-60-2	ion flux vs ion energy when varying HF vs LF power	Boyle, Ellingboe & Turner, J. Phys. D 37, 697 (2004)	HF→flux, LF→energy
V8 Frequency coupling direction	13.56→60 MHz fixed power	density ref	VHF CCP literature	nₑ↑ bounded by η_f clamp
V9 Electronegative nₑ depression	iso-etch-cf4 vs Ar (ref-gec-ar)	`nₑ`	Lafleur et al., PSST 23, 035010 (2014)	O2 < Ar
V10 PIC benchmark trend cross-check	sym-research-ar	nₑ/Te pressure trends	Turner et al., Phys. Plasmas 20, 013507 (2013)	direction-only (fluid model, no kinetic claims)

Explicit exclusions (out of scope for literature validation)

The following are deliberately out of scope for literature validation (mirrors physics-model.md §12).

Absolute etch/deposition rates
EEDF / non-Maxwellian detail
Space-charge sheath potential profiles (deliberately rejected — unresolved λ_De)
Absolute IEDF eV scale (power absorption / |E| are now in physical units W/m³, V/m, power-balance normalized to the requested power)
IEDF bimodal structure (no RF-cycle resolution)
Stochastic sheath heating (ohmic Drude only)
Absolute impedance (ohms)
Full CFD neutral flow (now a showerhead→pump potential flow — face-selective emission + jet-column directionality — but not Navier–Stokes CFD)

B15. Benchmark validation matrix

The matrix lists the expected trend and pass/fail criterion for each representative benchmark case. These are directional and rough-magnitude (ratio / order-of-magnitude) checks, not absolute-value accuracy. It complements B13 (literature anchors V1–V10). Status chips: verified (green) = the full criterion is asserted by an automated test in tests/test_validation_matrix.py; partial (blue) = the direction is CI-tested but one clause of the criterion is not; expected (neutral) = literature-grounded but not yet a regression check; todo (amber) = no coverage.

ID	Case	Conditions	Expected trend	Basis	Pass criterion	Status
M1	GEC Ar reference cell — power sweep	Ar, 0.1 Torr, 13.56 MHz, symmetric parallel-plate GEC cell, power swept 100→500 W	nₑ rises ~linearly with power (density_ref ∝ P^1.0). Being symmetric, DC self-bias stays small (≈0).	L&L 2005 Ch.10 global model (nₑ ∝ P_abs); Godyak et al. 1991; Olthoff & Greenberg, NIST 1995 (GEC nₑ~1e15–1e16 m⁻³)	PASS if nₑ increases 3–8x over 100→500 W. FAIL if flat or decreasing. \|V_dc\| < 10% of the RF amplitude.	verified
M2	Ar pressure sweep — Te & sheath	Ar, 13.56 MHz, fixed power, pressure swept 0.05→0.5 Torr	Te falls as pressure rises (higher p·L needs lower Te to sustain ionization). Sheath thins with higher density (s ∝ 1/√n).	L&L 2005 Ch.10 (Te vs p·L); Godyak, Piejak & Alexandrovich 1991; Child law s ∝ V^0.75/√n	PASS if Te(0.5 Torr) < Te(0.05 Torr), monotone decreasing. Sheath thins monotonically as pressure (density) rises. FAIL if Te is flat or increases with pressure.	verified
M3	Frequency sweep — coupling & ion energy	Ar, fixed power & pressure, frequency 2 / 13.56 / 27 / 60 MHz	At fixed power, higher f raises nₑ via capacitive coupling efficiency (η_f=(f/13.56)^0.5, clamped [0.45,2.4]). Ion energy falls as sheath voltage drops.	VHF CCP literature (density rise); L&L 2005 §11 (sheath voltage ∝ 1/f tendency); Boyle, Ellingboe & Turner 2004	PASS if ion energy at 60 MHz < ion energy at 2 MHz (direction). nₑ increases with f but bounded by the η_f clamp. FAIL if ion energy rises with f.	verified
M4	Asymmetric electrode area ratio — DC self-bias	Ar, 13.56 MHz, powered electrode smaller than ground (area ratio A_g/A_p > 1), increasing asymmetry	Greater area asymmetry drives a negative DC self-bias on the smaller powered electrode; \|V_dc\| grows. As \|V_dc\| rises, that electrode's sheath thickens.	L&L 2005 §11 capacitive divider V_a/V_b=(C_b/C_a); Godyak & Sternberg 1990; Child law s ∝ V^0.75	PASS if V_dc is negative (on powered electrode) and \|V_dc\| grows with asymmetry. Sheath thickens as \|V_dc\| rises. FAIL if V_dc is positive or insensitive to asymmetry.	verified
M5	Dual-frequency (60 MHz + 2 MHz) — role decoupling	Ar, HF 60 MHz + LF 2 MHz co-driven, HF power and LF power varied independently	HF power mainly sets nₑ / ion flux; LF power mainly sets ion energy (sheath voltage) — the two knobs are approximately decoupled.	Boyle, Ellingboe & Turner, J. Phys. D 37, 697 (2004); L&L 2005 §11.5 dual-frequency CCP	PASS if raising HF power moves ion flux more than ion energy, and raising LF power moves ion energy more than ion flux. FAIL if both frequencies move the same output in the same proportion.	verified
M6	O₂ electronegative discharge — density & Te	O₂, 13.56 MHz, compared against Ar at identical power / pressure / geometry	Attachment loss (negative-ion formation) lowers nₑ vs Ar at the same conditions. Negative ions suppress electron density and shift Te behavior (gas-specific).	Lafleur et al., PSST 23, 035010 (2014); L&L 2005 Ch.10 electronegative discharges	PASS if nₑ(O₂) < nₑ(Ar) at matched conditions. FAIL if O₂ gives equal or higher nₑ than Ar.	verified
M7	SiH₄/Ar PECVD — deposition regime	SiH₄/Ar mix, 13.56 MHz, low–moderate power, PECVD conditions (high neutral/ion ratio)	Lower ionization gives lower nₑ than pure Ar; a high neutral/ion-ratio deposition regime. Radical (SiH₃/H) routing corrects the density estimate.	Kushner, J. Appl. Phys. 54, 4958 (1983); L&L 2005 Ch.14–16 (PECVD)	PASS if nₑ(SiH₄/Ar) < nₑ(pure Ar) at equal power and neutral/ion ratio > 1e3. FAIL if density is indistinguishable from pure Ar.	verified
M8	Power-balance closure — absorbed-power consistency	Any Ar preset; volume integral of the absorbed-power density P=(1/2)Re(σ)\|E\|² compared with the requested power	After the recent power-balance normalization, the absorbed-power volume integral closes to the same order as the requested power. \|E\| and absorbed power are in physical units (V/m, W/m³).	L&L 2005 Ch.11 time-averaged Ohmic heating P=(1/2)Re(σ)\|E\|²; Boeuf & Pitchford 1995 (2D Ohmic P=J·E)	PASS if 0.5 ≤ (integrated absorbed power / requested power) ≤ 2. FAIL if the ratio falls outside this band or diverges with power. Limit of this check The solver rescales the fields by balance = √(P_target/P_integral) before this check runs, so the ratio matches the target by construction — 1.000004–1.000035 across 5/5 shipped presets, and fixed even when balance moves 12x (measured 2026-07-31). M8 therefore checks the plumbing (that the rescale was applied, and that the quadrature agrees), and cannot be cited as evidence that a field map is physically right.	verified

The explicit out-of-scope list (will not match) is kept in one place, → What we don't claim above.

B9. References / Further Reading

References below are grouped by model component. References already cited in the sections above are not duplicated.

Drift-diffusion / fluid model (general)

Lieberman & Lichtenberg, Principles of Plasma Discharges and Materials Processing, 2nd ed., Wiley 2005 — drift-diffusion, global model, sheath (already cited in Section 1)
Boeuf & Pitchford, Phys. Rev. E 51, 1376 (1995) — RF fluid CCP model, Poisson-coupled fluid + power absorption (already cited in Section 1)
Scharfetter & Gummel, IEEE Trans. Electron Dev. 16, 64 (1969) — exponential-fit flux discretization (already cited in Section 3)

Electron rate-coefficient LUT / EEDF (A1)

Hagelaar & Pitchford, Plasma Sources Sci. Technol. 14, 722 (2005) — BOLSIG+ Boltzmann solver for electron transport coefficients and rate coefficients
BOLSIG+ official site (LAPLACE, Toulouse) — free electron Boltzmann solver for generating LUT data
LXCat cross-section database — source data for A1 LUT / BOLSIG+ inputs (electron-neutral cross sections)

0D electron energy / particle balance (A2)

Lee & Lieberman, J. Vac. Sci. Technol. A 13, 368 (1995) — global model 0D energy and particle balance for high-density plasmas

Sheath / Bohm criterion / RF sheath (A3)

Riemann, J. Phys. D 24, 493 (1991) — Bohm criterion and sheath formation (already cited in Section 4)
Lieberman, IEEE Trans. Plasma Sci. 16, 638 (1988) — analytical RF sheath model (already cited in Section 4) [verify IEEE Xplore]
Godyak & Sternberg, Phys. Rev. A 42, 2299 (1990) — dynamic model of electrode sheaths in symmetrically driven RF discharges

GEC benchmark cell (B8)

Hargis et al., Rev. Sci. Instrum. 65, 140 (1994) — GEC RF reference cell: defined parallel-plate 13.56 MHz system for benchmark studies

Comparison / open-source solvers (B8)

COMSOL Plasma Module — validated commercial multi-fluid plasma solver for benchmark comparison
Zapdos (shannon-lab / MOOSE) — free and open-source MOOSE-based finite-element plasma simulation code

B10. Disclaimer

This simulator is a trend-level reduced-order model provided for academic research and educational purposes. Outputs are directional indicators, not absolute references. Do not use the results of this simulator alone for real process decisions, equipment design, safety analysis, or regulatory compliance. All critical applications require specialized software (PIC/MCC, validated multi-fluid solvers, commercial plasma tools) and experimental validation. The developers accept no liability for direct or indirect damages arising from the use of this tool.

B11. Magnetic Field Model — Trend-Level Magnetized-Electron Confinement Approximation

When the magnetic field (B) input is active, the simulator applies a trend-level magnetized-electron confinement model. Core physics: the static B field magnetizes electrons, forming the Hall parameter β = ω_ce/ν_m (ω_ce = eB/m_e: electron cyclotron frequency, ν_m: electron-neutral collision frequency). Large β reduces cross-field electron transport by the factor 1/(1+β²), applied directionally as aniso_r and aniso_z. The result is enhanced electron confinement and a density increase trend captured by mag_gain.

This magnetic field model does NOT include: full Hall-tensor anisotropic transport / ExB-drift rotation / electron cyclotron resonance (ECR) heating / magnetized-sheath physics. It is a trend-oriented approximation based on directional Hall-parameter anisotropic transport + a lightweight ExB net-trend gain. Do NOT use it as an absolute magnetized-plasma result.

B12. Parameter Sensitivity — How Each Input Trends Each Output

The table below summarises the relative trend directions this reduced-fluid model reproduces, verified by parameter sweeps. Arrows (↑ ↓ ~) indicate which way outputs move — they are not calibrated absolute magnitudes. For absolute use, calibration against experimental data is required.

Input change	Density nₑ	Electron temp Tₑ	Ion energy	Sheath thickness	RF impedance
Frequency ↑	↑ (capacitive coupling efficiency η_f=(f/13.56MHz)^0.5, clamp [0.45,2.4]; the 0D balance itself is f-independent)	~ (set by p·L, not f)	↓ (lower sheath V)	↓	↓ (∝ 1/f, capacitive)
Pressure ↑	∩ (hump: rises sub-Torr, saturates 1–3 T; Ar 500 W sweep: 0.03T 0.79→0.1T 0.97→0.5T 1.25→1T 1.32→5T 1.28, anchored 1.0 at 0.12 T; 0D power-balance shape, M2 fix 2026-07-09)	↓ (strong)	↓	↓ (monotone thinning, CI-asserted)	~
RF power ↑	↑ (≈ linear; density reference ∝ P^1.0 anchored at 500 W; formerly P^0.74)	~ (slight ↑)	↑	↓ (thinner at higher density)	↓
Gap ↑	↓ (more wall loss)	↓ (larger diffusion length)	↑	↑	~
Electronegative gas (O₂/SiF₄/SiH₄)	↓ (attachment loss)	gas-specific	~	~	↑
Magnetic field \|B\| ↑	↑ (cross-field confinement + ExB gain)	~	~	~	~

↑ = increases, ↓ = decreases, ~ = approximately flat. Magnitudes are trend-level; calibration against measurements is needed for absolute use.

B14. Model Internals — Te(r,z) map, nₑ observable, discharge gap, calibration knobs

Te(r,z) map (fields.te_eV)

The fields.te_eV heatmap is, by default, a heuristic energy map mean-matched to the calibrated scalar Te (a geometric blend of the heuristic and the 0D particle balance, clamped to [1,9] eV). With PLASMACCP_TE_PDE, the spatial shape instead comes from a screened-diffusion PDE heated by the RF absorption map.

nₑ observable (fields.ne)

The displayed fields.ne = v/(v+n_sat) ∈ [0,1] is a saturating transfer (a per-request fixed curve so A/B legends stay comparable). The absolute density scale lives in density_ref / the 0D block.

Electrode gap

Gap-derived quantities (the Te gap factor, the Child-Langmuir clamp, and Λ=gap/π) use the grid-derived discharge gap — the median plasma-cell span between the powered and ground blocks. z_max is used only as a fallback.

Calibration knobs (calibration block)

Knob	Applies to
`eta_abs`	RF drive power (all power trends)
`density_scale`	density_ref (direct multiply)
`nu_m_scale`, `tg_gain`	RF collision frequency
`k_wall`	wall-loss map
`gamma_sec`, `a_eff_scale`, `l_loss_scale`, `ion_flux_scale`, `impedance_scale`	0D report block only

Neutral defaults (1.0 / 0.0) = bit-exact with prior results.

B16. Computational Methods — Grid, Solvers, Numerical Honesty

B1–B14 above describe what is modeled; this section documents how it is actually computed — the grid, the discretization, the linear-algebra solvers underneath, and the verification gates layered on top.

Component	Numerical method
Grid & coordinates	Axisymmetric r-z finite volume, reference mesh 144(r)×176(z). Only the radial face coefficients of the electrostatic Poisson and RF complex-admittivity assemblers (4 scalar+vectorized twins) get cylindrical weighting rw=r_face/max(r_cell,0.25·dr) (BC2, 2026-07-09). z-faces are coordinate-free. The ne SG assembler, the helper PDEs, and the native C path remain planar.
Electrostatics	Variable-ε 5-point finite volume. Dirichlet applies only to powered electrodes (per-tag matched amplitude + DC offset) and to ground electrodes/solid_wall (=0); every other cell (dielectric/floating-conductor interior, the axis, unassigned outer edges) is natural no-flux (the axis preserves symmetry via a doubled radial weight). Sparse direct solve via scipy splu (SuperLU LU factorization) / spsolve, with an LRU cache (≤16 entries) keyed on (ε, dr, dz, nz, nr, dirichlet_mask) that skips refactorization for repeated shapes.
RF complex field / sheath	The same cylindrically-weighted complex-admittivity system is solved once as a steady-state (cycle-averaged) phasor, then renormalized to absolute power via the cylindrical volume integral ∫0.5σ\|E\|²dV=P_target. The sheath is a Lieberman-style two-sheath capacitive-divider fixed point (area-ratio V_a/V_b=C_b/C_a), blended continuously — not with a hard cutover — against the Child-Langmuir fallback by relative exit step delta_rel: w=clamp(1−log10(delta_rel/tol),0,1), exposed as rf_sheath_convergence in [0,1].
Electron transport	Scharfetter-Gummel (Bernoulli) 5-point discretization. The default path is assemble_python_system_vectorized (numpy CSR), verified to parity ≤1e-9 against the scalar reference implementation. Boundaries are absorbing walls carrying a surface-loss sink k_s at every plasma→non-plasma face. One-shot by default; a damped Gummel outer loop (α=0.4, tol 1e-2, falls back to one-shot on divergence) only engages when PLASMACCP_GUMMEL_ITERS>1 (opt-in). PLASMACCP_TG_FEEDBACK (on by default) is a one-shot correction pass injecting the local neutral-density modulation.
Auxiliary PDEs	Te(r,z) is a screened-diffusion PDE (1−λ_ε²∇²)Te=q (λ_ε=λ_m·√(M/6m_e)·0.35 for molecular gases), mean-matched to the scalar Te. Tg(r,z) uses the same form with a different screening length λ_g=gap/3 (clamped). Ion continuity is a single effective-species ambipolar drift-diffusion solve, reported as surface flux Γ_i(r) only. Neutral transport is potential flow ∇·(κ∇p)=0. All four share the same sparse direct solve.

Numerical honesty: the goal is trend direction, not absolute value. CI pins on every commit — parity of every vectorized path against its scalar reference, off=bit-exact regression for every new flag, directional assertions (power↑→density↑, etc.), an r-alternation (checkerboard) guard, and the M1–M8 benchmark suite (must pass unmodified).

Bit-exact escape hatches: PLASMACCP_PLANAR_RADIAL_OPERATOR=1 (reproduces the old planar operator), PLASMACCP_TG_FEEDBACK=0, calibration neutral values (1.0/0.0), and RF_SHEATH="0" (Child-law fallback only). The native C assembler stays disabled by default in prod — it carries a checkerboard M-matrix bug at high cell aspect ratio and, since BC2, is also physically stale (still planar); a lone ENABLE=1 is ignored (with a warning) by a consistency guard unless paired with PLANAR_RADIAL_OPERATOR=1.

Reading the results

The Process summary tab reports wafer-level trend indicators. The subsections below explain what each metric means, why a process engineer cares, and how the simulator computes it.

Radial ion-flux nonuniformity

Measures the radial flatness of the ion-flux profile — a primary indicator of how uniformly a process proceeds across the wafer surface. Larger values can translate to greater etch-depth, film-thickness, and critical-dimension (CD) variation between center and edge. The simulator computes nonuniformity (%) = (max − min) / (2 × mean) × 100 over the radial profile; lower is better. If the offered wafer interval contains a zero or non-finite sample, the scalar is hidden and used/total column coverage is shown because missing surface support cannot be distinguished from a physical zero. Real-tool uniformity also depends on gas-flow patterns, chuck temperature, and focus-ring condition — factors not fully captured here.

Ion energy / IEDF

The energy of ions impinging on the substrate determines etch anisotropy (directional selectivity), sputter yield, and substrate damage. The shape and mean of the ion energy distribution function (IEDF) are central to process-window definition. The simulator computes a time-averaged ion energy proxy based on Child-Langmuir scaling at each radial position — this is a relative trend indicator, not an absolute eV value. In dual-frequency configurations, the high frequency (HF) primarily controls ion flux while the low frequency (LF) primarily controls ion energy (Boyle et al. 2004). Real IEDFs show a characteristic double-peak structure arising from RF sheath phase-averaging (Godyak et al. 1991); this simulator does not resolve that structure.

Sheath thickness

The sheath is the positive space-charge region near electrodes where ions are accelerated toward the substrate. Sheath thickness is governed by the applied voltage, plasma density, and frequency (Child-Langmuir law), and directly influences ion bombardment energy and angular distribution. A thicker sheath generally corresponds to a longer ion acceleration path and higher ion energy. The simulator estimates sheath thickness at each radial position using the Bohm velocity criterion and Child-Langmuir scaling — interpret as a trend signal, not an absolute measurement.

Etch/deposition rate proxy

This is NOT an absolute etch rate (nm/min). It is a relative comparison proxy grounded in the physics of ion-assisted etching, where the rate depends on both ion flux and ion energy (Coburn & Winters 1979). The simulator computes ion flux × √(ion energy) at each radial position and takes the radial mean. Surface chemistry, sputter yield, reaction by-products, and mask selectivity are not modeled at all — use this value only for relative trend comparison within the same simulation set.

Scope & Limitations

Trend-level reduced-order fluid model. Each modeling choice (Poisson-coupled drift-diffusion, Drude conductivity, Scharfetter-Gummel, Bohm-criterion sheath, power-weighted multi-frequency) is grounded in the literature above. Deliberately omitted or approximated: non-Maxwellian EEDF (especially at low pressure), stochastic/collisionless sheath heating, ion inertia/non-local presheath transport, detailed plasma chemistry (ionization/dissociation/radical kinetics), and electromagnetic (non-electrostatic) effects at very high frequency/power. Realistic accuracy: density within ~2–5x, ion flux within ~order of magnitude. For higher accuracy use full PIC/MCC (Birdsall 1991; Vahedi & Surendra 1995) or validated multi-fluid solvers benchmarked to the GEC cell (Boeuf & Pitchford 1995; Lymberopoulos & Economou 1995).

Additional notes: the electrostatic solve is charge-free ∇·(ε∇φ)=0 (Laplace). A full space-charge Poisson was evaluated and REJECTED — λ_De (0.05–0.5 mm) is unresolved on the 144×176 mesh, producing mesh artifacts rather than physics. The RF phasor boundaries also exclude the DC offset, but dc_bias_regions is reflected in the sheath/ion-energy path in addition to the electrostatic solve.