This research paper investigates the Kerr effect and third harmonic generation in a gapped graphene layer determined by the Rashba spin–orbit coupling and the bandgap, aiming to develop graphene-based spintronic instruments. The eigenstates and energy dispersion of the system are obtained through computation of the time-independent Schrödinger equation. Main outcomes proved that incrementing the Rashba coupling metric (λ) considerably reduces the \({\sigma }_{{\text{inter}}_{\text{min}}}^{(1)}\) , whereas when the temperature jumps from 4 to 27 ℃, the linear conductivity decreases. Moreover, when λ sweeps from 60 to 80 meV, the \({\sigma }_{{\text{inter}}_{\text{max}}}^{(1)}\) is rapidly enhanced, especially when photon frequencies exeeds 6 × 1013 Hz. Our calculations demonstrated that when chemical potential increments, more states around the Dirac points become accessible for electronic transitions, enhancing the interband optical conductivity. Finally, for \({\sigma }_{\text{inter}}^{(1)}\) (Kerr effect), a quasi-oscillatory behavior is witnessed which rapidly diminishes and is minimized as the mass metric increased from 1 to 2 meV. Our detailed exploration provides fruitful insights into the design of advanced graphene-optoelectronics and spintronic devices.
