Replication Data for the manuscript: Adjoint-State Identifiability of Piezo-Tunable Valley Splitting in 2D Magnetic Heterostructures

Title: Replication Data for: Adjoint-State Identifiability of Piezo-Tunable Valley Splitting in 2D Magnetic Heterostructures Description: This repository contains the complete multiscale computational framework and dataset required to reproduce the findings in the manuscript "Adjoint-State Identifiability of Piezo-Tunable Valley Splitting in 2D Magnetic Heterostructures". The workflow demonstrates how to bridge microscopic first-principles quantum predictions into a macroscopic, partial differential equation-constrained adjoint-state inverse solver. It utilizes a three-part multi-physics approach: a density functional theory (DFT) pipeline to establish the chiral C$_3$ symmetry and extract the exchange-strain coupling ($\eta$), a continuum elastodynamic simulation to generate the true-trace piezoelectric strain field of an AlN actuator, and a finite-difference drift-diffusion-Poisson solver to compute the macroscopic transverse voltage observable. To ensure strict reproducibility, the raw DFT output files containing the residual self-consistent-field noise are provided in full. The adjoint-state gradient validations and Cramér-Rao lower bound (CRLB) resolution limits (bounding the 262.0 V/cm to 24,420.5 V/cm operational window) are mathematically hardcoded with fixed random seeds (rng(42)) in the provided MATLAB scripts. Repository Structure The compressed archive is organized into three primary directories. To comply with VASP licensing restrictions, proprietary POTCAR files have been omitted (a POTCAR-INFO.txt file is provided detailing the exact PAW-PBE metadata required for reconstruction). WAVECAR and CHGCAR files have been omitted due to file size constraints, as they are fully regenerable from the provided inputs. 1. DFT/ (Density Functional Theory - VASP Data) Stacking-Comparison/: Contains the structural relaxations for Hollow, Cr-aligned, and Br-aligned geometries. The OUTCAR files contain the final energetic penalties (3.16 eV and 0.99 eV) and confirm the C$_3$ symmetry. Band-Structure/: The EIGENVAL and PROCAR files in this directory provide the K-valley conduction band minimum data used for the t = 1.1601 eV parabolic fit. This also represents the 0% strain proximity splitting baseline ($M_0 = 0.081$ meV). Convergence-Sweep/: Contains the OUTCAR and VASP-SOC.log files for the plane-wave cutoff (400 eV to 650 eV) and K-mesh (3x3 to 9x9) evaluations. Strain-Sweep/: Contains the contiguous 0%, 1%, 2%, 3%, and 4% biaxial strain runs. (Note: The 0pct directory is an exact duplicate of the Band-Structure baseline, provided here so the MATLAB regression script can parse all five data points seamlessly from a single path). The OUTCAR and log files disclose the AMIX/AMIX_MAG pinning and the SCF convergence behavior responsible for the sub-meV scatter discussed in the Supplementary Information. 2. COMSOL/ (Elastodynamic Actuator) Contains the .mph files modeling the two-dimensional aluminum nitride membrane. The extracted true-trace strain field utilized by the MATLAB transport solver is included as a CSV export. 3. MATLAB/ (Inverse Problem & Transport Solver) Bootstrap-Regression.m: Parses the DFT strain sweep data to reproduce the -0.0014 meV/% OLS slope and the statistical confidence intervals reported in the manuscript. VH-Final-Trust-Boundary.m: The master transport script. Assembles the forward state, computes the continuous adjoint sensitivity kernel, executes the gradient convergence check, and extracts the operational window bounded by the noise floor and the Caughey-Thomas velocity saturation limit. D-Optimal-Comparison.m: Evaluates the Fisher information matrix traces to compare the uniform vs. split-electrode geometries. Note: All scripts use a strict $O(\delta\eta^2)$ centered-difference scheme for validation and incorporate a random seed (rng(42)) to ensure exact reproducibility of the stated limits. Step-by-Step Reproduction Guide Phase 1: First-Principles Parameterization Navigate to the DFT/Strain-Sweep/ directory. Run the Bootstrap-Regression.m MATLAB script. This will parse the discrete splitting values from the OUTCAR and EIGENVAL files to reproduce the exact regression statistics, p-values, and bootstrap histograms shown in Table S3 and Figure 3. Phase 2: Continuum Mechanical Generation Open the .mph project file in the COMSOL/ directory using COMSOL Multiphysics. Compute the structural mechanics study to generate the uniform active-channel strain profile. Export the resulting 2D spatial grid along the longitudinal transport axis to a CSV file to be loaded by the macroscopic transport solver. Phase 3: Macroscopic Transport & Identifiability Inversion Launch MATLAB and navigate to the MATLAB/ directory. Execute the primary trust boundary and D-Optimal-Comparison scripts. The solvers will output the exact baseline gradient match (-97.8490), the D-optimal Fisher information traces, the 262.0 V/cm statistical resolution limit, and the corresponding velocity saturation trust boundaries. Software Requirements Density Functional Theory: VASP (Vienna Ab initio Simulation Package) version 6.x (PAW PBE). Mechanical Modeling: COMSOL Multiphysics (v6.0 or higher recommended). Mathematical/Transport Solver: MATLAB (R2023a or higher, no external PDE toolboxes required). License The computational datasets (DFT inputs/outputs, COMSOL exports) are distributed under the Creative Commons Attribution 4.0 International (CC-BY-4.0) license. The custom MATLAB scripts are distributed under the MIT License.