  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 $c(2 \times 2)$ 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/Fe/2W 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 $c(2 \times 2)$
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 $p (2 \times 1)$ 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 bulk like properties in its center. The energy dispersion of spin
spirals show qualitatively the same results as those from the 4\textit{d}/Fe/2W
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/Fe/W(001) films are potential candidates for complex non-collinear spin
structures.
