Effect of oxygen vacancies and strain on the phonon spectrum of HfO$_{\mathrm{2}}$

The effect of strain and oxygen deficiency on the Raman spectrum of monoclinic HfO$_{\mathrm{2}}$ is investigated theoretically using first-principle calculations. In-plane compressive strain is found to blue shift the phonon frequencies, while tensile strain does the opposite. The simulations are compared to and good agreement is found with experimental results of Raman frequencies greater than 110 cm$^{\mathrm{-1}}$. Several Raman modes measured below 110 cm$^{\mathrm{-1}}$ and previously assigned to HfO$_{\mathrm{2}}$ cannot be assigned to HfO$_{\mathrm{2}}$. However, localized vibrational modes introduced by threefold-coordinated oxygen (O$_{\mathrm{3}})$ vacancies are identified at 96.4 cm$^{\mathrm{-1}}$ These results are important for a deeper understanding of vibrational modes in HfO$_{\mathrm{2}}$, which has technological applications in transistors, and particularly in resistive random-access memory (RRAM) whose operation relies on oxygen-deficient HfO$_{\mathrm{x}}$