Toroidal Order in Magnetic Metamaterials

In the formation of ice flowers after breathing against a cold window pane, nature impressively displays its tendency to spontaneously order in well-defined symmetric configurations. The example depicts an elementary feature of phase transitions

The following Chapter introduces three main topics of my work: ferroic order, nanomagnetism and metamaterials. First, the concept of ferroic order is presented. A symmetry-based classification is given together with a brief discussion of phase transitions, the emergence of a ferroic order parameter, spontaneous domain formation and the manipulation of an order parameter with a conjugate field. This part closes by unravelling ferrotoroidicity. Second, magnetic properties of sub-micrometre-sized objects made from a ferromagnetic material are discussed. Here, the formation of different kinds of spin structures is explained that serve as building blocks of nanomagnetic arrays. The suppression or—more important here—the support of long-range order in extended magnetostatic-coupled arrays is explained. The third part of this Chapter introduces metamaterials, a class of matter that is assembled on length scales comparable with the wavelength of radiation that interacts with it and that provides design-determined novel material properties and functionalities.

This chapter provides an overview of techniques that I have used for this work. After introducing the fabrication of artificial crystals, the major part discusses magnetic force microscopy as one of the two main techniques that have been applied for studying and manipulating the magnetic state of nanomagnetic arrays. The chapter continues by providing background for the second main part—optical methods, mainly the magneto-optical Kerr effect, that are able to detect changes in the magnetisation, but eventually of the toroidisation as well, exploiting more sophisticated experimental settings. Since the as-grown magnetic state of the fabricated crystals cannot simply be changed by annealing procedures, a concept for a non-thermal relaxation of the samples is presented to disprove a possible pinning of the magnetic configuration. A section of this chapter discusses a statistical analysis scheme to relate the local energy landscape to the formation of particular micromagnetic states. Moreover, a self-written two-dimensional micromagnetic calculation script is presented that gives access to magnetic fields that cause correlation between the building blocks in magneto-toroidal arrays. Last, the MFM-revealed microscopic magnetisation pattern do not show the corresponding toroidal order directly. Therefore, the chapter closes with ideas for uncovering contrast between different toroidal domain states using suitable image post-processing.

For modelling and implementing toroidal order in artificial spin systems on mesoscopic length scales, a number of requirements have to be fulfilled. First, the distribution of magnetic moments within a single unit cell has to favour a compensated, vortex-like configuration. Second, this local magnetic arrangement has to couple to neighbouring unit cells to allow for a collective behaviour. This chapter introduces my model systems and explains the challenges in finding suitable parameters for their successful experimental realisation.

One of the two key requirements of n-dimensional ferroic systems is the spontaneous formation of homogeneously ordered areas, called domains, separated one from another by distinct (\(n-1\))-dimensional entities, called domain walls, see Sect. 2.​1.​4 on page 16. These walls constitute natural interfaces within the system between two or more energetically degenerate realisations of a particular order. For the artificial nanomagnetic arrays investigated throughout this work, the spontaneous formation of long-range order connected to an order parameter is a hallmark for their classification as primary ferroic order [1]. Suitable symmetry groups allowing for toroidal order have already been identified a few decades ago and are listed in Sect. 2.​1.​6 on page 19. The nanomagnetic arrays that have been investigated here, see Fig. 4.​1 on page 83, fall in one of these groups, setting the basis for their further scrutiny not only on the macroscopic but also on the microscopic scale.

The basis for key applications of ferroic materials is the ability to reverse the orientation of the order parameter and, accompanied by that, the sign or direction of a physical vectorial or tensorial quantity [1, 2]. This characteristic allows one to distinguish otherwise symmetry-equivalent material classes such as pyroelectric materials (polar but not switchable) from ferroelectric materials (polar and switchable). The feature to experimentally manipulate—to “write”—a material property that is connected to the order parameter extends the applicability of ferroic materials from merely passive devices like sensors to new fields such as actuators, memory cells or logic gates. While some of the established ferroic materials, in particular ferromagnets and ferroelectrics, are widely used in scientific and technological applications, ferrotoroidic materials are far from being technologically applicable. This is especially because the generation of an actual conjugate field seems elusive so far.

Both, linear magneto-optical effects [1‐3], as introduced in Section 3.​2.​2, and the anomalous Hall effect [4] are based on the violation of time-reversal symmetry that manifests in a coupling to electromagnetic fields by energy-state shifts due to spin-orbit interaction.

The goal of this work was to establish experimental pathways for investigating ferrotoroidicity in appropriately engineered two-dimensional arrays composed of nanomagnetic building blocks.