Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 2 / Section A(ii) / 15 / i

For a uniformly magnetized bilayer that relaxes to one of its easy directions after a saturating in-plane field, the in-plane remanent component isIt is therefore approximately zero when the easy axis is perpendicular at the thin-Co end, and increases towards $M_s$ as the easy axis rotates into the plane. The graph should show an increasing crossover between about $1$ and $4\mathrm{nm}$, but the source does not specify its exact shape. A straight line is not a consequence of the two endpoint orientations.

The trend reflects competition between interface-driven perpendicular magnetic anisotropy and volume or demagnetizing contributions favouring in-plane magnetization. An illustrative effective anisotropy is $K_{\mathrm{eff}}=K_v+K_s^{\mathrm{eff}}/t_{\mathrm{Co}}-{\mu }_0{M}_s^2/2$, with sign convention chosen so positive $K_{\mathrm{eff}}$ favours the normal. A continuous rotation can require higher-order anisotropy terms; the simple sign change alone need not produce a smooth canting interval.

The relation assumes a well-defined remanent direction and no complete cancellation by domains. A vibrating-sample magnetometer measures total moment: if the result is reported without dividing by magnetic volume, the moment per area additionally scales as $t_{\mathrm{Co}}{M}_s\cos \theta$.