  In this paper, we present the spin-phonon coupling and dielectric response of
nano-crystalline Pr$_2$CoMnO$_6$ employing Raman and dielectric spectroscopic
study. Pr$_2$CoMnO$_6$ is a manganite compound , it undergoes a paramagnetic to
ferromagnetic (PM-FM) phase transition around $T_c$ $\sim$172 K.
Temperature-dependent Raman scattering experiment is carried out across $T_c$
to study the spin-phonon behavior in this material. The results from Raman
study reveal an obvious softening of the phonon mode involving stretching
vibrations of the (Co/Mn)O$_6$ octahedra in ferromagnetic temperature regions,
indicating a close correlation between magnetism and lattice in Pr$_2$CoMnO$_6$
and conform the spin phonon coupling. Further, we have carried out detailed
study on dielectric response, impedance spectroscopy, electric modulus and AC
conductivity of Pr$_2$CoMnO$_6$ ceramics in the temperature range of 20 K - 300
K and frequency range of 1kHz - 5.5MHz. We found Pr$_2$CoMnO$_6$ shows strong
frequency dependence with large dispersion and large dielectric constant. It is
found that thermally activated relaxation mechanism is involve and material is
deviated from Debye's model which is confirmed by Nyquist plot and complex
modules behavior.
