Pressure- and field-dependent Raman studies of magnetodielectric behavior in Mn$_{3}$O$_{4}$*

We present simultaneous pressure- and field-dependent Raman scattering studies of the magnetodielectric spinel, Mn$_{3}$O$_{4}$, which allow us to investigate both the microscopic origins and pressure dependence of magnetodielectric phenomena in this material. We identify the specific phonon modes responsible for the dramatic magnetodielectric behavior observed in this material, and show that these modes provide quantitative information regarding magnetodielectric behavior in Mn$_{3}$O$_{4}$ via the Lyddane-Sachs-Teller (LST) relationship. We also show that pressure can induce monoclinic distortions accompanied with magnetic ordering in Mn$_{3}$O$_{4}$. Finally, by exploring the field-dependent phonon spectrum at different pressures, we are able to map out the pressure-field structural phase diagram of Mn$_{3}$O$_{4}$ and explore the pressure-dependence of magnetodielectric behavior in this material.