We present a detailed first principles study on the magnetic structure of anFe monolayer on different surfaces of 5d transition metals. We use thespin-cluster expansion technique to obtain parameters of a spin model, andpredict the possible magnetic ground state of the studied systems by employingthe mean field approach and in certain cases by spin dynamics calculations. Wepoint out that the number of shells considered for the isotropic exchangeinteractions plays a crucial role in the determination of the magnetic groundstate. In the case of Ta substrate we demonstrate that the out-of-planerelaxation of the Fe monolayer causes a transition from ferromagnetic toantiferromagnetic ground state. We examine the relative magnitude of nearestneighbour Dzyaloshinskii-Moriya (D) and isotropic (J) exchange interactions inorder to get insight into the nature of magnetic pattern formations. For theFe/Os(0001) system we calculate a very large D/J ratio, correspondingly, a spinspiral ground state. We find that, mainly through the leading isotropicexchange and Dzyaloshinskii-Moriya interactions, the inward layer relaxationsubstantially influences the magnetic ordering of the Fe monolayer. For theFe/Re(0001) system characterized by large antiferromagnetic interactions wealso determine the chirality of the $120^{\circ}$ N\'eel-type ground state.
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