Ionic Diffusion and Electronic Transport in Eldfellite Na$_x$Fe(SO$_4$)$_2$

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.