Strong electron correlation on the Fe$_3$O$_4$(0 0 1) surfaces

Magnetite Fe$_3$O$_4$ is a fascinating material that still is not well understood and presents challenges for the state-of-the-art computational methods. This transition metal oxide undergoes a first-order metal-insulator transition at $T_V$=120 K. The ferrimagnetic properties of Fe$_3$O$_4$ makes it a promising material for spintronic applications. We use a plane wave density functional theory in the generalized gradient approximation adding a Hubbard-$U$ parameter to describe properly the strongly correlated Fe--3$d$ electrons. Based on previous results \footnote{H. Pinto, S. Elliott, J.Phys.: Condens. Matter \textbf{18}, 10427 (2006)}, we compute the surface structure, magnetic properties and electronic structure of several Fe$_3$O$_4$(0 0 1) surfaces with $(\sqrt{2}\times\sqrt{2})R45^{\circ}$ reconstruction. The simulated scanning tunneling microscopy images of these surfaces are compared and discussed in the light of available experimental data. Finally, we analyze the possible existence of charge ordering on the Fe$_3$O$_4$(0 0 1) surface and the effect on the surface electronic structure with changing the value of the Hubbard-$U$ parameter on the superficial Fe sites.