Field-Theoretic Criterion for DRAM High-k Dielectrics: A Two-Dimensional Criterion Space (Soft-Mode × Bandgap) and Quantitative Doping-Pinning

DRAM capacitor dielectrics face a hard leakage×EOT scaling constraint below the 1z-nm node (IMW 2020); for 30 years industry has responded by doping-pinning the tetragonal phase of high-k dielectrics (ZrO₂/HfO₂-based) to stabilize it—yet this long-standing practice lacks a unified criterion language. Using a field-theoretic framework (materials as density-cluster fields ρ=m·p; phase-transition candidates as failure-line R'=1 equilibrium regions; shear soft modes as signatures of transition activity), this paper provides a computable criterion coordinate. Core results: (1) industry hit validation of the soft-mode criterion—ZrO₂, the only material with antiferroelectric (AFE) activity, is exactly the only Hf/Zr oxide in the soft-mode region (C44=20.3 GPa≤22.7), with all other stable dielectrics not soft (positive 1/1, negative 4/4); (2) two-dimensional criterion space—soft-mode axis × bandgap axis (leakage wall), tetragonal ZrO₂ P4₂/nmc the unique Pareto-optimal phase (C44=30.7+Eg=4.03), explaining 30 years of industrial pinning of the tetragonal phase; (3) quantitative doping-pinning—C44=83.7·r−30.1 GPa (R²=0.9998), dopant ionic radius as a quantitative tuning knob; Y³⁺ extrapolation 55.3 GPa within the measured 8YSZ range 47-66 GPa; (4) martensitic mirror—the same t→m transformation is exploited in ceramics but is a failure mode in DRAM dielectrics; (5) criterion-D data engineering—matching-rule audit found 44.3% multi-entry contamination, energy-window criterion makes semantics explicit as '0K instability signal of a real metastable phase', with a 546-compound list released. All values machine-verified; references individually verified via CrossRef. v2.0: unified storage-medium criterion extension section (four dimensions). v2.1: criterion extended from four to five dimensions (+Dim 5 Retention quality A=1/τ) — from a 'transition spectrum' to a 'retention spectrum', with charge-trap memory (3D NAND CTF, exponential end) and phase-change memory (PCM, power-law end) as two realizations of the same A=1/τ coordinate; CTF retention is experimentally a distribution of A=1/τ (widely distributed time constants). Chinese version is the companion translation.