The magnetic anisotropy of a free-standing transition-metal monolayer versus the band filling is systematically studied. By using a tight-binding Hamiltonian we show that it is possible to obtain an accurate description of the anisotropy energy. We show that the magnetic anisotropy energy versus d-band filling must exhibit a minimum number of nodes, with irregular oscillations imposed on this curve. Their origin, which is likely to be related to energy-band crossings, is discussed in detail. Our results are consistent with the available data. For Fe and Co (001) monolayers, a very small magnetic anisotropy is found, as well as for a Ni (001) monolayer. Moreover, the shape of the anisotropy energy curve might explain the controversial situation for iron.

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