Fig. 4: In situ magnetic flux guiding effect to boost low-field sensitivity.

a Normalized MR curves and b sensitivity curves comparison of printed Fe/Fe₃O₄ core-shell particles (cyan curve), with which removed 10 gain factor (purple dashed line curve) and printed Fe₃O₄ particles (red curve). c Magnetic state and magnetic gain factor at the junction of two spheres with the core radius 120 nm and shell thickness 10 nm (simulations, cubic mesh of 2.5 nm size). A magnetic field of 100 mT is applied along the symmetry axis of the system. The top part shows the equilibrium magnetic state with the color-code indicating M_x component of magnetization normalized by the saturation magnetization of the Iron core and arrows indicating magnetization direction. The bottom part shows the magnetic gain factor calculated based on the B_x component of the local value of the stray field related to the external field. d Gain factor as a function of the shell thicknesses of the core-shell spheres with the fixed core diameter of 240 nm and magnetic field 100 mT applied along the x-axis. The gain factor is calculated for the average B-field at the junction between spheres. e Same, as a function of the core radius of the core-shell spheres with the fixed shell thickness of 10 nm. f Normalized MR curves and g sensitivity curves of printed Fe/Fe₃O₄ core shell particles with different thicknesses that were thermally oxidized for different times (30 min, 60 min, 90 min, and 120 min along the arrow) at 235 °C. h SEM images of Fe/Fe₃O₄ core-shell particles with different sizes (90 nm, 3 μm, and 15 μm), Scale bars: 500 nm, 10 μm, 100 μm from top to bottom. For the distribution of particle diameters, see Supplementary Figs. 19 and 20. All three size classes were oxidized under the same conditions (235 °C, medium vacuum 0.5 mbar, 30 min) to keep the oxide-shell formation comparable across samples. i Normalized MR curves and j sensitivity curves of printed Fe/Fe₃O₄ core-shell particles with different sizes.