Exact Solutions for Bimodal Distributions under Stochastic Plasma Irradiation in Thin Films

A persistent paradox complicates the study of plasma-irradiated thin films, where bimodal grain distributions and ambiguous scaling laws, roughly shifting between $\Phi^{-1/2}$ and $\Phi^{-1}$, or a general inverse dependence on plasma flux, are empirical yet remain theoretically unreconciled. Existing models fail to unify noise-driven evolution, defect saturation kinetics, and nucleation-loss balance within a single, self-consistent formalism. This work resolves these discrepancies by developing the first exact analytical theory for this system. We derive the closed-form steady-state grain area distribution, $P_{ss}(A)$, establish the precise dimensionless threshold for bimodality onset at $\Pi_c = 4/(3\sqrt{3})$, and demonstrate that defect saturation physics mandate a universal $\langle A \rangle \propto \kappa^2 \Phi^{-1} e^{+2E_b/k_B T_s}$ scaling law. The framework reveals how competition between stochastic impingement and deterministic growth triggers microstructure fragmentation, resolving long-standing ambiguities in irradiation-induced surface evolution and providing a predictive foundation for materials processing.