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Using first-principles calculations based on density functional theory (DFT),we investigate the exchange interaction between a magnetic tip and a magneticsample which is detected in magnetic exchange force microscopy (MExFM) and alsooccurs in spin-polarized scanning tunneling microscopy (SP-STM) experiments. Asa model tip-sample system, we choose Fe tips and one monolayer Fe on W(001)which exhibits a checkerboard antiferromagnetic structure and has beenpreviously studied with both SP-STM and MExFM. We calculate the exchange forcesand energies as a function of tip-sample distance using different tip modelsranging from single Fe atoms to Fe pyramids consisting of up to 14 atoms. Wefind that modelling the tip by a single Fe atom leads to qualitativelydifferent tip-sample interactions than using clusters consisting of severalatoms. Increasing the cluster size changes the calculated forces quantitativelyenhancing the detectable exchange forces. Rotating the tip with respect to thesurface unit cell has only a small influence on the tip-sample forces.Interestingly, the exchange forces on the tip atoms in the nearest andnext-nearest layers from the apex atom are non-negligible and can be oppositeto that on the apex atom for a small tip. In addition, the apex atom interactsnot only with the surface atoms underneath but also with nearest-neighbors inthe surface. We find that structural relaxations of tip and sample due to theirinteraction depend sensitively on the magnetic alignment of the two systems. Asa result the onset of significant exchange forces is shifted towards largertip-sample separations which facilitates their measurement in MExFM. At smalltip-sample separations, structural relaxations of tip apex and surface atomscan either enhance or reduce the magnetic contrast measured in SP-STM

physical review. b, condensed matter and materials physics

Role of tip size, orientation, and structural relaxations in  first-principles studies of magnetic exchange force microscopy and  spin-polarized scanning tunneling microscopy