Using ﬁrst-principles calculations, we investigate the application of ferromagnetic and antiferromagnetic tips in magnetic exchange force microscopy, a novel technique which allows imaging the magnetic structure of insulating and conducting surfaces on the atomic scale by detecting the exchange forces between magnetic tip and sample. We apply density functional theory using the full-potential linearized augmented plane-wave method to calculate the short-ranged exchange forces between Fe and Cr tips with the antiferromagnetic Fe monolayer on W(001). We include structural relaxations of the tip and sample due to their interaction and consider pure Fe and Cr cluster tips consisting of 14 atoms as well as alloyed Fe-Cr tips. For single atom tips we observe enhanced exchange energies and forces for Cr atoms with respect to Fe due to the larger extent of the 3d orbitals of Cr. Surprisingly, the exchange forces for unrelaxed cluster tips of Fe and Cr are of very similar magnitude and in the noncontact regime the Fe tip exhibits a slightly larger exchange force than the Cr tip. This effect stems from the competition of Cr apex atom and base atoms which contribute with opposite sign to the exchange force due to the antiferromagnetic coupling within the tip. For the ferromagnetic Fe cluster tips, the base atoms contribute less to the total exchange force and with the same sign. The Cr cluster tips display a characteristic sign change of the exchange interaction from ferromagnetic at large tip-sample distances to antiferromagnetic at short separations due to the transition from an indirect double exchange mediated by conduction electrons to a direct exchange mechanism due to overlap of d orbitals. Structural relaxations turn out to be more important for Cr-terminated tips as a consequence of larger exchange forces on the apex atom. The magnitude of the exchange energy rises more quickly due to the relaxations and the exchanges forces are signiﬁcantly enhanced. At intermediate tip-sample separations, relaxed Fe and Cr cluster tips result in similar exchange forces, while at short distances, Fe tips exhibit larger forces. Fe-based tips can be further improved by adsorbing single Cr atoms at their apex. The analysis of spin-resolved charge density difference plots conﬁrms the competing exchange mechanisms and reveals a signiﬁcant contribution from base atoms to the exchange interaction at small tip-sample separations.
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