Unified Analysis of Fermionic Tunneling, Barrow‐Exponential Entropy Corrections, and Plasma Lensing in Carrollian Reissner‐Nordström Spacetime

ABSTRACT We explore quantum and thermodynamic features of Carrollian Reissner‐Nordström black holes, emphasizing Lorentz‐violating geometry, generalized uncertainty principle corrections, and electromagnetic charge interactions. Using the tunneling formalism for Dirac particles, we derive the Hawking temperature and extend it to include quantum gravity modifications induced by a minimal length scale. The corrected temperature depends on particle charge, mass, and energy, leading to suppressed or enhanced radiation rates depending on the deformation parameter. To capture microscopic corrections to black hole thermodynamics, we introduce a combined Barrow‐Exponential entropy model that unifies fractal and quantum statistical effects. The resulting expressions for Helmholtz free energy and heat capacity reveal nontrivial stability regions and smooth crossover behavior instead of abrupt phase transitions. We also study weak gravitational lensing of light rays in the same Carrollian charged background, both in vacuum and within a non‐magnetized plasma medium. The plasma contribution increases the deflection angle for small impact parameters, indicating a potential observational imprint of Lorentz‐violating and quantum effects. Our results provide a consistent framework connecting fermionic tunneling, entropy‐corrected thermodynamics, and plasma‐modified optical geometry within the Carrollian extension of Reissner‐Nordström spacetime.