We use first-principles calculations based on density functional theory to study how the magnetic properties of an Fe monolayer on a W(001) surface---exhibiting a $c(2\ifmmode\times\else\texttimes\fi{}2)$ antiferromagnetic ground state---can be modified by an additional $4d$ transition-metal overlayer. To obtain an overview of how the $4d$-band filling influences the exchange interactions in the Fe layer we have calculated the energy dispersion of spin spirals for $4d/\mathrm{Fe}/2\mathrm{W}$ unsupported quadlayers, in which the W(001) substrate is represented by only two atomic layers. Hybridization with the overlayer leads to a reduced ferromagnetic nearest-neighbor exchange interaction and the next-nearest neighbor exchange gains in strength. Surprisingly, we find that the $c(2\ifmmode\times\else\texttimes\fi{}2)$ antiferromagnetic state is unfavorable for all systems with a $4d$ overlayer. For $4d$ overlayers from the beginning (Nb) or end (Pd) of the series we find a ferromagnetic ground state. As one moves to the center of the series there is a transition via a spin spiral (Mo, Rh) to a $p(2\ifmmode\times\else\texttimes\fi{}1)$ antiferromagnetic ground state (Tc, Ru). We have studied the Mo, Ru, and Pd overlayer on Fe/W(001) representing the surface by a sufficiently large number of W layers to obtain bulklike properties in its center. The energy dispersions of spin spirals show qualitatively the same results as those from the $4d/\mathrm{Fe}/2\mathrm{W}$ quadlayers. The Dzyaloshinskii-Moriya interaction calculated upon including spin-orbit coupling shows significant strength and considerable frustration effects. The calculated magnetocrystalline anisotropy energy is large as well. All $4d/\mathrm{Fe}/\mathrm{W}(001)$ films are potential candidates for complex noncollinear spin structures.
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