Parametric instabilities in semiconducting quantum plasma with effect of Fermi pressure and Bohm potential

In this theoretical study, we investigate non linearity in semiconductor quantum plasma which is the Couse of parametric interaction between pump wave and acoustic fluctuation. The non-linear interaction result into amplitude modulation in acoustic wave. The quantum plasma is subjected to static magnetic field applied in angle $\theta$ and an inhomogeneous density. quantum correction through Fermi temperature and Bohm potential terms modifies the threshold characteristic and growth rate. We apply Quantum hydrodynamic model to investigate how these waves interact, considering how the magnetic field and inhomogeneous density affect the process. These results can better explain wave patterns in quantum plasmas and may enhance technologies such as semiconductors and plasma devices. The plasma inhomogeneity amplifies the instability in regions of high-density gradient, and the magnetic field has a limited role in balanced configurations, consistent with classical findings.