The concept of toroidal moments in condensed-matter physics and their long-range ordering in a so-called ferrotoroidic state is reviewed. We show that ferrotoroidicity as a form of primary ferroic order can be understood both from microscopic (multipole expansion) and macroscopic (symmetry-based expansion of the free energy) points of view. The definition of the local toroidal moment and its transformation properties under the space-inversion and time reversal operations are highlighted and the extension to periodic bulk systems is discussed. Particular attention is paid to the relationship between the toroidal moment and the antisymmetric magnetoelectric effect and to limitations of the magnetoelectric response in ferrotoroidic systems and ferroic materials in general. Experimental access to the ferrotoroidic state by magnetoelectric susceptibility measurements, x-ray diffraction and optical techniques or direct measurement of the bulk toroidization is discussed. We outline the pertinent questions that should be clarified for continued advancement of the field and mention some potential applications of ferrotoroidic materials.

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The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect*

Techniques exist for imaging the magnetization patterns of magnetic thin films and at the surfaces of magnets, but here hard-X-ray tomography is used to image the three-dimensional magnetic structure within a micrometre-sized magnet in the vicinity of Bloch points. Techniques have long existed for imaging the two-dimensional magnetization patterns of thin-film magnets, but the three-dimensional complexities of magnetization structure within the body of a magnet is not so amenable to direct investigation. Claire Donnelly et al. have made substantial progress in lifting this veil by harnessing hard-X-ray tomography to determine the inner magnetic structure of micrometre-sized magnets. The properties of current X-ray sources limit the spatial resolution to about 100 nanometres, but it is anticipated that future instrumental developments could greatly improve on this. In soft ferromagnetic materials, the smoothly varying magnetization leads to the formation of fundamental patterns such as domains, vortices and domain walls1. These have been studied extensively in thin films of thicknesses up to around 200 nanometres, in which the magnetization is accessible with current transmission imaging methods that make use of electrons or soft X-rays. In thicker samples, however, in which the magnetization structure varies throughout the thickness and is intrinsically three dimensional, determining the complex magnetic structure directly still represents a challenge1,3. We have developed hard-X-ray vector nanotomography with which to determine the three-dimensional magnetic configuration at the nanoscale within micrometre-sized samples. We imaged the structure of the magnetization within a soft magnetic pillar of diameter 5 micrometres with a spatial resolution of 100 nanometres and, within the bulk, observed a complex magnetic configuration that consists of vortices and antivortices that form cross-tie walls and vortex walls along intersecting planes. At the intersections of these structures, magnetic singularities—Bloch points—occur. These were predicted more than fifty years ago4 but have so far not been directly observed. Here we image the three-dimensional magnetic structure in the vicinity of the Bloch points, which until now has been accessible only through micromagnetic simulations, and identify two possible magnetization configurations: a circulating magnetization structure5 and a twisted state that appears to correspond to an ‘anti-Bloch point’. Our imaging method enables the nanoscale study of topological magnetic structures6 in systems with sizes of the order of tens of micrometres. Knowledge of internal nanomagnetic textures is critical for understanding macroscopic magnetic properties and for designing bulk magnets for technological applications7.

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Three-dimensional magnetization structures revealed with X-ray vector nanotomography

We reveal the cooperative effect of coherent and dissipative magnon-photon couplings in an open cavity magnonic system, which leads to nonreciprocity with a considerably large isolation ratio and flexible controllability. Furthermore, we discover unidirectional invisibility for microwave propagation, which appears at the zero-damping condition for hybrid magnon-photon modes. A simple model is developed to capture the generic physics of the interference between coherent and dissipative couplings, which accurately reproduces the observations over a broad range of parameters. This general scheme could inspire methods to achieve nonreciprocity in other systems.

Nonreciprocity and Unidirectional Invisibility in Cavity Magnonics.

We report element specific structural and magnetic investigations on Zn(1-x)Co(x)O epitaxial films using synchrotron radiation. Co dopants exclusively occupy Zn sites as revealed by x-ray linear dichroism having an unprecedented degree of structural perfection. Comparative magnetic field dependent measurements by x-ray magnetic circular dichroism and conventional magnetometry consistently show purely paramagnetic behavior for isolated Co dopant atoms with a magnetic moment of 4.8 (mu B). However, the total magnetization is reduced by approximately 30%, demonstrating that Co-O-Co pairs are antiferromagnetically coupled. We find no sign of intrinsic ferromagnetic interactions for isolated or paired Co dopant atoms in Co:ZnO films.

Absence of intrinsic ferromagnetic interactions of isolated and paired Co dopant atoms in Zn1-xCoxO with high structural perfection.

Electronic properties of crystalline materials observed in X-ray diffraction

A 220 diamond phase plate was combined with an energy-dispersive absorption spectrometer to convert linearly polarized X-rays into circularly polarized photons and to detect circular magnetic X-ray dichroism (CMXD) from ferromagnetic samples. In these experiments, carried out at LURE, the energy-dispersive spectrometer was equipped with a bent Si (311) polychromator and vertically collimating slits accepting essentially a linearly polarized incident beam. The quarter-wave plate was operated in the Bragg geometry but well outside the range of quasi total reflection, with the consequence that the forward-diffracted beam was circularly polarized with a polarization rate approaching 80% over the whole energy bandpass of the polychromator. CMXD spectra of GdFe2 and GdCo2 intermetallic compounds were recorded at ca 7.2 keV near the Gd LIII absorption edge: they are essentially identical to the spectra commonly recorded with elliptically polarized X-ray photons collected out of the orbit plane of the storage ring. It is suggested that the energy-dispersive phase plate will be very useful to detect CMXD spectra with energy-dispersive spectrometers exploiting the well collimated linearly polarized emission of standard undulators installed on the storage rings of the third generation.

Energy-Dispersive Phase Plate for Magnetic Circular Dichroism Experiments in the X-ray Range

We report the first observation of a nonreciprocal x-ray linear dichroism caused by the time-reversal odd, real part zeta of the complex gyrotropy tensor zeta(*) which is dominated by electric dipole-electric quadrupole E1E2 interference terms. A nonreciprocal transverse anisotropy was observed in the low temperature insulating phase of a Cr doped V2O3 Mott crystal when a single antiferromagnetic domain was grown by magnetoelectric annealing along the hexagonal c axis. This new element (edge) specific spectroscopy could nicely complement x-ray magnetic circular dichroism which is silent for antiferromagnetic materials.

First observation of nonreciprocal X-Ray gyrotropy

In a large Czochralski-grown Si1−xGex (0.02 \leq x \leq 0.07) gradient crystal, diffraction patterns have been measured in symmetrical Laue geometry using synchrotron radiation in the energy range 100–200 keV. The experimental data are in very good agreement with the results from geometrical optics theory for distorted crystals, if the creation of new wavefields for strain gradients larger than a critical value is taken into account. In this sense, the crystal behaves like an ideal gradient crystal. If the normal absorption is disregarded, for reflection 111 and 100 keV energy, the full width at half-maximum values and the peak reflectivities of the diffraction patterns range from 14.6′′ and 97%, respectively, to 70.9′′ and 74%, respectively, for a variation in the Ge concentration from 3.5 to 5.3 at.%.

Diffraction of 100 to 200 keV X-rays from an Si1-xGex gradient crystal: comparison with results from dynamical theory.

This paper reports on advanced instrumentation developed for the ESRF beam line ID12A which is dedicated to polarization dependent x-ray absorption, excitation and emission spectroscopies at photon energies hν ≥ 2.0 keV. Emphasis is laid on problems associated with either the control or the conversion of the polarization state of the x-ray photons. Several examples have been selected to illustrate the performances of the beam line and show what type of novel spectroscopic information can be obtained.

Instrumentation Developments for Polarization Dependent X-ray Spectroscopies

Diamond crystals were used as quarter‐wave plates (QWPs) and half‐wave plates (HWPs) to produce and completely analyze circular and vertical linear polarization of monochromatic x rays from an undulator source (ID10) at the European Synchrotron Radiation Facility. The horizontal linear polarization state of the forward diffracted beam in the Bragg or Laue geometry was efficiently converted into circular (99%) and vertical polarization (97%) by tuning the crystal conveniently away from the Bragg peak. Also, the combination of two QWPs was used to obtain linear polarization at any desired azimuth (e.g., 95% at 45°). Further, the second QWP and the linear polarimeter were used to completely characterize the polarization state of the circularly polarized beam after the first QWP. High efficiencies at QWPs and HWPs and high transmitted intensity (40%) were achieved with the diamond crystal in the asymmetric Laue geometry.

C. Malgrange

Papers

Energy-Dispersive Phase Plate for Magnetic Circular Dichroism Experiments in the X-ray Range

First observation of nonreciprocal X-Ray gyrotropy

Diffraction of 100 to 200 keV X-rays from an Si1-xGex gradient crystal: comparison with results from dynamical theory.

Instrumentation Developments for Polarization Dependent X-ray Spectroscopies

X‐ray polarimetry with phase plates