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The spin-one-half Falicov-Kimball model with spin-dependent on-siteinteraction between localized ($f$) and itinerant ($d$) electrons is studied bysmall-cluster exact-diagonalization calculations and a well-controlledapproximative method in two dimensions. The results obtained are used tocategorize the ground-state configurations according to common features (chargeand spin ordering) for all $f$ and $d$ electron concentrations ($n_f$ and$n_d$) on finite square lattices. It is shown that only a few configurationtypes form the basic structure of the charge phase diagram in the $n_f-n_d$plane. In particular, the largest regions of stability correspond to the phasesegregated configurations, the axial striped configurations and configurationsthat can be considered as mixtures of chessboard configurations and the full(empty) lattice. Since the magnetic phase diagram is much richer than thecharge phase diagram, the magnetic superstructures are examined only atselected values of $f$ and $d$ electron concentrations.

Charge and magnetic order in the spin-one-half Falicov-Kimball model with Hund coupling in two dimensions