Spatial localization of magnetic-branch force contrasts in substitutional Mn-doped Ge: A first-principles study

Magnetic energy gaps do not reveal the spatial distribution of magnetic-state-dependent atomic forces. We compare ferromagnetic and antiferromagnetic forces at identical nuclear positions in 64-site substitutional Mn-doped Ge using collinear PBE calculations. For nearest-neighbour Mn pairs, the radial force projection reverses sign between magnetic branches. In the second-neighbour geometry, the shared Ge neighbour has the largest force contrast, about 2.2 times that of equally near Ge sites coordinated to one Mn. Bridge-displacement gap slopes agree with direct force derivatives across four numerical settings. Tightening the electronic threshold by 100-fold preserves magnetic ordering and the leading site, while the force-contrast vector changes by 1.08% despite a nearly unchanged energy gap. The results identify coordination-associated localization of magnetic-branch force contrasts within the frozen model, without establishing relaxed distortions, full numerical convergence or a unique exchange mechanism.