Dirac Quasinormal Modes, Quality Factor and Gravitational Lensing in Nonlinear Electrodynamics Black Holes with Barrow Entropy

We investigate a static and spherically symmetric black hole solution of Einstein gravity sourced by a nonlinear electromagnetic field in the presence of a phantom global monopole. The monopole induces a constant solid-angle deficit, rendering the spacetime asymptotically non-flat, while the nonlinear electromagnetic sector introduces subleading corrections that become relevant in the strong-field regime. The thermodynamics is analyzed using Barrow entropy, which modifies the standard area law through a single deformation parameter. We derive the corresponding internal energy, Helmholtz free energy, and heat capacity, and show that the Barrow deformation parameter significantly affects the phase structure and stability properties, particularly for small horizon radii. The dynamical response of the spacetime is further analyzed through massless Dirac field perturbations. The effective fermionic potential exhibits a single-barrier structure outside the event horizon, allowing for a reliable WKB treatment of the quasinormal modes. The resulting spectra indicate that the imaginary parts of the frequencies remain negative throughout the parameter space considered, confirming linear stability against fermionic perturbations. To further quantify the damping properties of the fermionic ringdown, we analyze the quality factor of Dirac quasinormal modes, which provides a compact measure of the balance between oscillation frequency and decay rate and highlights the influence of the nonlinear electromagnetic sector on the persistence of fermionic perturbations. In addition, we study gravitational lensing using the Gauss–Bonnet theorem, providing a global description of light deflection in both vacuum and plasma environments. The combined effects of nonlinear electrodynamics, the phantom monopole, and plasma dispersion lead to measurable deviations from the Schwarzschild case in the small impact-parameter regime, while the weak-field limit is recovered at large distances.