  Discovering new electrodes for sodium-ion battery requires clear
understanding of the material process during battery operation. Using
first-principles calculations, we identify mechanisms of ionic diffusion and
electronic transfer in newly developed cathode material, eldfellite
Na$_x$Fe(SO$_4$)$_2$, reproducing the electrochemical properties in good
agreement with experiment. The inserted sodium atom is suggested to diffuse
along the two-dimensional pathway with preceding movement of the host sodium
atom, and the activation energy is calculated to be reasonable for fast
insertion. We calculate the electronic properties, showing the band insulating
at low composition of inserted sodium, for which the electron polaron formation
and hoping are also suggested. Our results may contribute to opening a new way
of developing innovative cathode materials based on iron and sulfate ion.
