Methods, Results & Figures

Abstract

This document presents a methods‑ and results‑focused v2 of the BSM-SG/QFG–InSe / In₂Se₃ study. It explicitly documents the numerical simulations, parameter sweeps, metrics, score functions, and Pareto analyses performed in this session, with direct references to the generated figures and datasets. The work bridges BSM‑SG atomic geometry with QFG reduced‑order field simulations, extending from atoms and molecules to grain structure and wafer‑scale process windows.

1. Atomic‑Scale Foundations (Context)

This work builds on the BSM‑SG Atlas view of atoms as extended geometric objects composed of:

• φ‑cores (nuclear geometry),
• ψ‑resonant structures (coherent field loops),
• coherence boundaries defined by a critical density ρ₀.

Indium (In) acts as a soft, donor‑like bridge, while Selenium (Se) acts as a directional, phase‑locking clamp. This atomic complementarity motivates both InSe and In₂Se₃ stability and informs the mesoscopic modeling choices that follow.

2. QFG Reduced‑Order Model (Methods)

2.1 Order Parameters

Crystalline structure is represented by multiple competing order parameters ηₖ(x,y,t), each corresponding to a grain orientation k.

The total QFG density is:

ρ(x,y,t) = Σₖ ηₖ(x,y,t)

Crystallinity is operationally defined as regions where ρ > 0.7.

2.2 Free‑Energy‑Like Functional (Conceptual)

Each ηₖ evolves under:

• a local double‑well potential (amorphous vs crystalline),
• mutual competition between orientations,
• gradient energy with anisotropy,
• external driving and locking terms.

This is a QFG‑consistent ROM, not a full microscopic Hamiltonian, but it preserves geometric and coherence principles.

2.3 Moving Front Activation

Crystallization is activated by a moving Gaussian front:

W(x,y,t) = exp(−(x − v_f t)² / 2σ_f²)

where v_f is the front_speed.

Behind the front, a locking term b_lock increases monotonically, suppressing further evolution and freezing grain geometry.

2.4 Anisotropy and Pinning

Each grain k is assigned a random crystallographic angle θₖ, yielding anisotropic gradient penalties:

κₓᵏ = κ₀(1 + α cos 2θₖ)
κᵧᵏ = κ₀(1 − α cos 2θₖ)

A quenched pinning field hₖ(x,y), with amplitude h_amp, represents substrate disorder, defects, and local ψ‑misalignment.

3. Simulation Outputs (Results)

3.1 Density Evolution and Front Locking

Figures:

• InSe_SLS_QFG_rho_evolution.png

• InSe_SLS_QFG_polycrystal_evolution.png

Observations:

• The moving front produces area‑wide crystallization rather than a narrow stripe.
• Post‑front locking arrests late‑stage coarsening.
• Final states are stable over long simulated times.

This behavior is interpreted in QFG terms as phase‑coherence locking behind the growth front.

3.2 Grain Identity and Boundary Formation

Figures:

• InSe_SLS_QFG_grain_boundaries.png

• InSe_SLS_QFG_defect_map.png

Observations:

• Without pinning, grain coarsening leads to single‑grain dominance.
• With pinning, stable polycrystalline mosaics form.
• Grain boundaries are tortuous and spatially heterogeneous, consistent with real wafers.

3.3 A2b, A2c, A2d Regimes

Figures:

• InSe_SLS_QFG_A2b_front_lock_*
• InSe_SLS_QFG_A2c_*
• InSe_SLS_QFG_A2d_*

These regimes demonstrate progressive control over:

• coverage,
• grain multiplicity,
• boundary density,

by tuning front_speed, b_lock, and h_amp.

4. Parameter Sweeps

4.1 Sweep A2d: front_speed × b_lock

Figures:

• InSe_SLS_QFG_A2d_sweep_heatmap_fast_coverage.png

• InSe_SLS_QFG_A2d_sweep_heatmap_fast_graincount.png

• InSe_SLS_QFG_A2d_sweep_heatmap_fast_boundarydensity.png

Key results:

• b_lock strongly controls final crystalline coverage.
• front_speed mainly affects transient dynamics.
• Final topology is weakly sensitive to front_speed alone.

4.2 Sweep‑2: front_speed × h_amp

Data:

• InSe_SLS_QFG_A2d_sweep2_frontspeed_x_hamp.csv

Key results:

• h_amp governs grain multiplicity and mosaic character.
• front_speed governs fill rate.
• The two parameters are largely orthogonal in effect.

5. Optimization Analyses

5.1 Single‑Objective Score Maps

Score function:

S = coverage − λ · boundary_density

Figures:

• InSe_SLS_QFG_A2d_scoremap_lambda_0p05.png

• InSe_SLS_QFG_A2d_scoremap_lambda_0p1.png

• InSe_SLS_QFG_A2d_scoremap_lambda_0p2.png

These maps identify regimes favoring uniform, low‑defect wafers.

5.2 Multi‑Objective Score Maps

Extended score:

S = coverage − λ · boundary_density + μ · grain_count

Figures:

• InSe_SLS_QFG_A2d_multi_score_l0p1_m0p02.png

• InSe_SLS_QFG_A2d_multi_score_l0p1_m0p05.png

• InSe_SLS_QFG_A2d_multi_score_l0p1_m0p1.png

These maps expose trade‑offs between uniformity and polycrystallinity.

5.3 Pareto Front

Objectives:

• maximize coverage,
• maximize grain_count,
• minimize boundary_density.

Figures:

• InSe_SLS_QFG_A2d_pareto_plot_coverage_vs_boundary.png

• InSe_SLS_QFG_A2d_pareto_plot_coverage_vs_grains.png

The Pareto front isolates a small number of physically meaningful and industrially relevant operating regimes.

6. Discussion: Linking Scales

The simulation results reflect atomic‑scale geometry:

• Indium softness → easy bridge formation → rapid crystallization.
• Selenium directionality → phase locking → sharp grain boundaries.
• Pinning ↔ local ψ‑misalignment and substrate disorder.

Thus, BSM‑SG atomic geometry → QFG fields → grain physics → wafer statistics forms a continuous causal chain.

7. Significance and Outlook

Scientific

• Demonstrates scale‑bridging from atomic geometry to wafer‑scale structure.
• Provides a concrete, testable extension of BSM‑SG concepts.

Engineering

• Acts as a ROM‑based digital twin for InSe / In₂Se₃ growth.
• Enables process‑window identification via Pareto analysis.

Next steps

• Explicit tiling of In₂Se₃ clusters into 2D layers.
• Coupling QFG order parameters to electronic and ferroelectric observables.
• Extension to other layered chalcogenides.

Conclusion

This v2 document formalizes the numerical and graphical results of the QFG–InSe / In₂Se₃ study. It demonstrates how atomic‑scale geometric principles propagate upward to mesoscopic and wafer‑scale behavior, providing both scientific insight and practical engineering leverage.

Problem

Wafer-scale, phase-pure, and uniform growth of InSe / In₂Se₃ is challenging.
Key industrial pain points include:

• lack of predictable wafer-scale growth windows,
• competing phases and micro-variability,
• defects and grain boundaries degrading device reliability,
• reliance on trial-and-error rather than predictive process control.

Solution

We present a QFG-based reduced-order simulation framework that:

• models wafer-scale growth with a minimal, physically interpretable parameter set,
• bridges atomic geometry (BSM-SG) → grain physics → wafer statistics,
• identifies optimal process windows using numerical optimization (Pareto fronts),
• runs efficiently (ROM), enabling rapid process iteration.

Key Results

From our simulations we demonstrate that:

• wafer-scale uniform coverage can be achieved reproducibly,
• grain multiplicity and defect density are orthogonally controllable,
• phase/topology selection is governed by a small set of process parameters, not stochasticity.

Key extracted metrics:

• Crystalline area fraction (coverage)
• Grain count
• Boundary length density (defect proxy)

Pareto Front & Wafer Windows

Using Pareto optimization (maximize coverage and grain count, minimize boundary density):

• we isolate a small set of optimal operating regimes,
• clearly separating:
  • high-uniformity / low-defect wafers,
  • high-coverage / polycrystalline wafers.

➡️ These constitute actionable wafer-scale process windows, not empirical guesses.

Technical Highlights

Simulation Framework

• 2D wafer-scale reduced-order model (QFG-consistent),
• moving crystallization front with post-front locking,
• anisotropy and quenched pinning (substrate / defects),
• multiple competing order parameters (grain orientations).

Key Control Parameters

• front_speed — growth kinetics,
• b_lock — post-growth phase locking,
• h_amp — defect / substrate pinning strength.

Graphic Figures (provided as PNGs)

Density & Growth

• ρ-evolution maps → wafer-scale filling behavior,
• polycrystal evolution → grain arrest after front passage.

Grain & Defects

• grain ID maps,
• boundary density / defect maps.

Parameter Sweeps

• A2d sweep heatmaps:
  • coverage,
  • grain count,
  • boundary density.

Optimization

• single-objective score maps,
• multi-objective score maps,
• Pareto front plots (coverage vs defects / grains).

Simulation Artifacts (Delivered)

Data

• CSV files containing per-run metrics:
  • crystalline_area_frac,
  • grain_count,
  • boundary_length_density.

Visual Outputs

• PNG figures:
  • density evolution,
  • grain boundaries,
  • defect maps,
  • sweep heatmaps,
  • score maps,
  • Pareto front plots.

Documentation

• concise explanations for:
  • how each figure is generated,
  • how to interpret it,
  • which parameters were used.

➡️ All results are reproducible and traceable.

Potential Impact

Phase Control

• The framework enables biasing toward stable configurations,
• provides a direct path toward explicit α/β phase selection in In₂Se₃.

Defect Reduction

• Boundary density is directly optimizable,
• defects emerge as process outcomes, not random noise.

Uniformity Predictors

• replaces trial-and-error with predictive wafer windows,
• enables:
  • higher yield,
  • thinner layers,
  • reduced material usage.

Industrial Value

• ROM enables fast iteration,
• deployable as:
  • a process-planning tool,
  • a materials digital twin,
  • a pre-TCAD material layer.

Bottom Line

We demonstrate that wafer-scale control of InSe / In₂Se₃ is a numerically tractable problem, not an empirical mystery.
The QFG framework provides a direct bridge from atomic geometry to industrial process control, delivering measurable and optimizable results.

In2Se3 Figure: