Fragmentation of magnetism in artificial dipolar spin ice

The difficulty of depicting such a system in real space is alleviated in reciprocal space. Indeed, the diffraction pattern from such an exotic state, if experimentally accessible, would display the two features simultaneously, corresponding to order and disorder. The pattern would feature very intense points in specific regions of the reciprocal space, called Bragg peaks, reflecting the periodic arrangement and the symmetry of the crystal. Further, it would also show a diffuse background associated to the disorder present in the system. It is important to stress once more that these two characteristics would describe a system in thermodynamic equilibrium.

To investigate the properties of such an exotic state of matter, researchers at the Néel and Jean Lamour Institutes have fabricated a specially designed magnetic metamaterial called spin ice. This is an array of elongated nanomagnets, artificially arranged as a lattice of equilateral triangles connected by their corners (kagome structure). By carefully tuning the nanomagnet size, their magnetization can be set in two and only two states: an ''up'' state or a ''down'' state''. We can then shake the system (using a magnetic field or the temperature) to bring it into a magnetic state minimizing its configurational energy (a nanomagnet ''up'' prefers to be surrounded by a nanomagnet ''down'', and vice versa).

Figure 1. Experimental kagome array of nanomagnets. The nanomagnets are composed of a ferrimagnetic alloy and have dimensions of the order of 500x100x10 nm³. Left inset: topographic image obtained with an atomic force microscope. Right: magnetic image of the same array obtained with a photoemission electron microscope. The black and white contrast gives the local direction of the magnetization, within each individual nanomagnet. The field of view is 10x11 µm².

The magnetic properties of the spin ice system were characterized at the Nanospectroscopy beamline of Elettra using x-ray circular dichroism photoemission electron microscopy. When imaged in real space, the magnetic configuration of such a system seems, essentially, disordered. On the other hand, looking at it more carefully, a well-trained eye may recognize periodic patterns, see Figure 1. These features appear local and cannot be spotted everywhere. This is exactly why the resulting magnetic configuration is better visualized in reciprocal space, see Figure 2. When doing so, the corresponding diffraction pattern shows a coexistence of Bragg peaks and a diffuse background, thus indicating that the system is both ordered and disordered. The challenge is then to demonstrate that this is not a coexistence of two out-of-equilibrium phases, but a state of matter that is both liquid and solid, everywhere in the lattice, at thermodynamic equilibrium.

A detailed analysis of the magnetic configuration reveals that the system acts as if each individual nanomagnet of the lattice was split into two separate components. A fraction (1/3 to be exact) of each nanomagnet contributes to the formation of a magnetic solid, i.e. an ordered, periodic magnetic structure, while the remaining fraction (2/3) fluctuates magnetically and leads to a strongly disordered phase. What is particularly surprising is that each nanomagnet is in a well-defined magnetic state, either ''up'' or ''down''. But collectively, the fractional parts of nanomagnets behave as two separate entities with different properties. This is why we say that magnetism is fragmented: the same internal degree of freedom (the magnetization of the nanomagnets) splits into two parts to generate, collectively, a phase which is at the same time ordered and disordered, solid and liquid.

Figure 2. Representation in reciprocal space of the magnetic phase deduced from the experimental configuration. The diffuse background, associated with the magnetic disorder, appears in yellow. Bragg peaks are also visible in certain regions of the reciprocal space (indicated by a red circle). The coexistence of this diffuse background and the Bragg peaks illustrates here the fragmentation of magnetism.