  We investigate the structural, dielectric relaxation, electric modulus and
impedance behavior of Ni-doped NASICON ceramic
Na$_{3+2x}$Zr$_{2-x}$Ni$_{x}$Si$_2$PO$_{\rm 12}$ ($x=$ 0.05--0.2) prepared
using the solid-state reaction method. The increase in dielectric constant with
temperature and decrease with frequency is explained on the basis of space
charge polarization using the two-layer model of Maxwell-Wagner relaxation. The
dielectric loss peak at lower temperatures follows the Arrhenius-type behavior
with frequency having activation energy of 0.27$\pm$0.01~eV of dipolar
relaxation, suggests similar type of defects are responsible for all the doped
samples. The real ($\epsilon$ $^{'}$) and imaginary ($\epsilon$ $^{''}$)
permittivity variation with frequency shows the broad relaxation behavior
indicates the non-Debye type of relaxation in the measured temperature range.
The permittivity values decrease with the amount of doping due to the increased
number of charge carriers upon Ni doping at the Zr site. The grain
contributions are observed at higher frequencies, while grain-boundary
contributions occur at the lower side of frequencies. The imaginary part of the
electric modulus also shows two types of relaxation peaks for all the samples
indicating similar activation energy at low temperatures and variable
activation energy at higher temperatures. The fitting of the imaginary modulus
using KWW function shows the non-Debye type of relaxation. We find that all
modulus curves merge with each other at low temperatures showing a similar type
of relaxation, while curves at high temperatures show the dispersed behavior
above the peak frequency. The {\it a.c.} conductivity data are fitted using the
double power law confirming the grain and grain boundary contributions in total
conductivity.
