Nanoscale objects often behave differently than their "normal-sized" counterparts. Sometimes it is enough to be small in just one direction to exhibit unusual features. One example of such a phenomenon is a very specific in-plane magnetic anisotropy observed sometimes in very thin layers of various materials. Here we recall a peculiar form of the free energy functional nicely describing the experimental findings but completely irrelevant and thus never observed in larger objects.
Intriguing Experimental ObservationsIn [1] we find the experimentally observed in-plane magnetic anisotropy energy (MAE) diagrams for multilayer structure Cr(4)/Fe(2)/Cr(d Cr )/Fe(4)/Cr(2), where numbers are thicknesses of components, expressed in nm. The thickness of the middle Cr layer, d Cr , was varied in few nm range, and the complete structures were deposited on Si(100) substrate, covered with natural SiO 2 layer 1.5-2.0 nm thick. The substrate was not perfectly flat-as a result of ion beam erosion it was covered with quite well-ordered ripples (see atomic force microscopy (AFM) images of the substrates, Figure 1 in [1]). The metallic layers were deposited using molecular beam epitaxy (MBE) technique. Their transmission electron microscopy (TEM) cross-sections revealed mostly amorphous structure with small inclusions of polycrystalline character. The values of MAE were derived from hysteresis loop area observed while exciting field was oriented along successive in-plane directions.In principle, samples of this kind should not exhibit any in-plane magnetic anisotropy. This is indeed the case when the substrate is flat (see Figure 4(a) in [1]). On the rippled substrate, however, this is no longer true and the sample exhibits peculiar twofold in-plane anisotropy (coercive field, Figure 4(b) in [1], MAE-in Figure 4(c)). It is peculiar since it is not the uniaxial anisotropy: four maxima are visible instead of just two.
The Surface Magnetic Anisotropy of a CylinderConsider the static configuration of individual spins located on a surface of a long (ideally: infinitely long), hollow ferromagnetic cylinder. In absence of any external field, one may expect that individual spins may adopt one of the two stable configurations:(i) they all may be aligned with C ∞ symmetry axis of a cylinder. This is the lowest exchange energy configuration; (ii) they all may be oriented perpendicularly to the above symmetry axis. Now the exchange energy is no longer at its global minimum. Nevertheless, such a configuration is stable since it realizes a local minimum of exchange energy.In the second case we may again distinguish two cases: either individual spins are aligned with local C 2 symmetry axis (there are infinitely many of them, each perpendicular to C ∞ ) thus pointing inwards or outwards of a cylinder and being perpendicular to the cylinder's surface, or they can be perpendicular to the local C 2 axis (laying on the cylinder's circumference), making a ring-shaped configuration and producing no net magnetization. It is easy to see that both of thos...