High Pressure Structural Stability, and Optical Properties of Scheelite type ZrGeO$_4$ and HfGeO$_4$ X-ray Phosphor Hosts
\emph{Ab-initio} calculations were performed on the scheelite type MGeO$_4$ (M = Hf, and Zr) compounds which find wide range of applications such as in x-ray imaging. We have studied the high pressure structural stability, elastic constants, electronic structure and optical properties of these compounds through density functional theory calculations. Two different density functional approaches namely plane wave pseudopotential method (PW-PP) and full potential linearized augmented plane wave method (FP-LAPW) were used for the present study. The ground state structural and vibrational properties are calculated and found to be in good agreement with experimental data. The compressibility of Zr and Hf germanates is found to be anisotropic as the a-axis is less compressible over c-axis due to the presence of Ge-O bonds along a-axis which is further confirmed from the ordering of the elastic constants that follows C$_{11}$ $>$ C$_{33}$. The electronic structure of the compounds has been calculated through recently developed Tran Blaha-modified Becke Johnson potential. The calculated electronic structure shows that the compounds are insulators with a gap of 5.39 eV for ZrGeO$_4$ and 6.25 eV for HfGeO$_4$ respectively. Optical anisotropy of these compounds are revealed from the computed optical properties such as complex dielectric function, refractive index, and absorption coefficient. In addition, it is observed that Ti doped ZrGeO$_4$ and HfGeO$_4$ turns out to be a good phosphor as the pristine compounds have the energy gap greater than the visible range upon Ti doping bandgap reduces as a result emission spectra occurs in the visible region and is well explained in the present study.