  Using density functional theory (DFT), we calculate the magnetic short-ranged
exchange forces between a magnetic tip and an adatom adsorbed on the
antiferromagnetic Mn monolayer on the W(110) surface [Mn/W(110)]. These
exchange forces can be measured in magnetic exchange force microscopy allowing
atomic-scale imaging of spin structures on insulating and conducting surfaces.
We consider two types of $3d$ transition-metal atoms with intrinsic magnetic
moments: Co and Mn and Ir as an example of a $5d$ transition-metal atom
exhibiting an induced magnetic moment on Mn/W(110). The tips are modeled by Fe
pyramids and terminated either with an Fe or a Mn apex atom. From our total
energy DFT calculations for a parallel and antiparallel alignment between tip
and adatom magnetic moments we obtain the exchange energy $E_{\rm ex}(d)$ as a
function of tip-adatom distance $d$. The exchange forces, $F_{\rm ex}(d)$, are
calculated based on the Hellmann-Feynman theorem. We show that structural
relaxations of tip and sample due to their interaction need to be taken into
account. Due to the exchange interaction the relaxations depend on the
alignment between tip and adatom magnetization -- an effect which will affect
the tunneling magnetoresistance that can be measured by a scanning tunneling
microscope. A maximum in the exchange energy and force curves is obtained for
magnetic adatoms at tip-adatom separations of about 3 to 4~{\AA}. The exchange
forces with an Fe terminated tip reach a maximum value of up to 0.2~nN and
0.6~nN for Co and Mn adatoms, respectively, and prefer an antiferromagnetic
coupling. Surprisingly, we also find an exchange force of up to 0.2~nN for Ir
adatoms. We analyze the exchange interaction between tip and adatom based on
the spin-polarized electronic structure of the coupled system......
