Structural Chirality of Polar Skyrmions Probed by Resonant Elastic X-Ray Scattering.

An escalating challenge in condensed-matter research is the characterization of emergent order-parameter nanostructures such as ferroelectric and ferromagnetic skyrmions. Their small length scales coupled with complex, three-dimensional polarization or spin structures makes them demanding to trace out fully. Resonant elastic x-ray scattering (REXS) has emerged as a technique to study chirality in spin textures such as skyrmions and domain walls. It has, however, been used to a considerably lesser extent to study analogous features in ferroelectrics. Here, we present a framework for modeling REXS from an arbitrary arrangement of charge quadrupole moments, which can be applied to nanostructures in materials such as ferroelectrics. With this, we demonstrate how extended reciprocal space scans using REXS with circularly polarized x rays can probe the three-dimensional structure and chirality of polar skyrmions. Measurements, bolstered by quantitative scattering calculations, show that polar skyrmions of mixed chirality coexist, and that REXS allows valuation of relative fractions of right- and left-handed skyrmions. Our quantitative analysis of the structure and chirality of polar skyrmions highlights the capability of REXS for establishing complex topological structures toward future application exploits.

Chiral structures of electric polarization vectors quantified by X-ray resonant scattering.

Resonant elastic X-ray scattering (REXS) offers a unique tool to investigate solid-state systems providing spatial knowledge from diffraction combined with electronic information through the enhanced absorption process, allowing the probing of magnetic, charge, spin, and orbital degrees of spatial order together with electronic structure. A new promising application of REXS is to elucidate the chiral structure of electrical polarization emergent in a ferroelectric oxide superlattice in which the polarization vectors in the REXS amplitude are implicitly described through an anisotropic tensor corresponding to the quadrupole moment. Here, we present a detailed theoretical framework and analysis to quantitatively analyze the experimental results of Ti L-edge REXS of a polar vortex array formed in a PbTiO3/SrTiO3 superlattice. Based on this theoretical framework, REXS for polar chiral structures can become a useful tool similar to x-ray resonant magnetic scattering (XRMS), enabling a comprehensive study of both electric and magnetic REXS on the chiral structures.

Electric field control of chirality.

Polar textures have attracted substantial attention in recent years as a promising analog to spin-based textures in ferromagnets. Here, using optical second-harmonic generation­based circular dichroism, we demonstrate deterministic and reversible control of chirality over mesoscale regions in ferroelectric vortices using an applied electric field. The microscopic origins of the chirality, the pathway during the switching, and the mechanism for electric field control are described theoretically via phase-field modeling and second-principles simulations, and experimentally by examination of the microscopic response of the vortices under an applied field. The emergence of chirality from the combination of nonchiral materials and subsequent control of the handedness with an electric field has far-reaching implications for new electronics based on chirality as a field-controllable order parameter.

A unified formulation of dichroic signals using the Borrmann effect and twisted photon beams.

Dichroic X-ray signals derived from the Borrmann effect and a twisted photon beam with topological charge l = 1 are formulated with an effective wavevector. The unification applies for non-magnetic and magnetic materials. Electronic degrees of freedom associated with an ion are encapsulated in multipoles previously used to interpret conventional dichroism and Bragg diffraction enhanced by an atomic resonance. A dichroic signal exploiting the Borrmann effect with a linearly polarized beam presents charge-like multipoles that include a hexadecapole. A difference between dichroic signals obtained with a twisted beam carrying spin polarization (circular polarization) and opposite winding numbers presents charge-like atomic multipoles, whereas a twisted beam carrying linear polarization alone presents magnetic (time-odd) multipoles. Charge-like multipoles include a quadrupole, and magnetic multipoles include a dipole and an octupole. We discuss the practicalities and relative merits of spectroscopy exploiting the two remarkably closely-related processes. Signals using beams with topological charges l ≥ 2 present additional atomic multipoles.

Antiferro-quadrupolar structures in UPd3 inferred from x-ray resonant Bragg diffraction.

A systematic analysis of resonant x-ray Bragg diffraction data for UPd(3), with signal enhancement at the U M(IV) edge, including possible structural phase transitions leads to a new determination of the space groups of the material in the phases between T(0)=7.8 K and T(+1)=6.9 K, as P 222(1), and between T(-1)=6.7 K and T(2)=4.4 K, as P2(1). In addition, the quadrupolar order parameters, {Q(ab)}, inferred from diffraction data for the phase between T(-1) and T(2), are {Q(xz)} and {Q(yz)} at the (103) Bragg reflection and {Q(xy)} at the (104) reflection.

Calculated chiral and magneto-electric dichroic signals for copper metaborate (CuB(2)O(4)) in an applied magnetic field.

Expressions for dichroic signals in terms of electron multipoles have been used to analyse optical data gathered on a crystal of copper metaborate in the presence of a magnetic field. Calculated signals comply with the established crystal and magnetic structures of CuB(2)O(4), and respect the global symmetries of parity-even and parity-odd dichroic signals in full. We have success in describing five different experiments in total. The claim by Saito et al (2008 Phys. Rev. Lett. 101 117402) that they observe magnetic control of crystal chirality in one of their five experiments is challenged.

Chirality, magnetic charge and other strange entities in resonant x-ray Bragg diffraction.

Subtleties in the electronic structure of complex materials can be directly observed, in great detail, by means of the Bragg diffraction of x-rays whose energy matches an atomic resonance. Strange atomic multipoles can be encountered in the interpretation of measured Bragg intensities, e.g., chirality and magnetic charge. Additionally, the x-ray technique allows the direct observation of the enantiomorphic screw-axis in chiral crystals, such as tellurium, low quartz and berlinite.

Reply to comment on 'Calculated chiral and magneto-electric dichroic signals for copper metaborate (CuB(2)O(4)) in an applied magnetic field'.

From the dawn of modern electromagnetism it has been known that a magnetic field is not handed (chiral). Arima and Saito (2009 J. Phys.: Condens. Matter 21 498001) persist with unwisdom in their repeated claim to have observed control of chirality using a magnetic field by and in itself. In our reply to their claim, we demonstrate damning errors in all challenges in the comment levelled at our analysis of the observation reported by Saito et al (2008 Phys. Rev. Lett. 101 117402) and made on a crystal of copper metaborate.

Right handed or left handed? Forbidden x-ray diffraction reveals chirality.

Enantiomers, or stereoisomers, have crystal structures that are mirror images of each other and are thus handed, like our right and left hands. The physical properties of enantiomers are identical except for optical activity, which rotates linearly polarized light by equal amounts but in opposite directions. While conventional x-ray Bragg diffraction can determine crystal structures, it does not distinguish between right- and left-handed crystals. We show resonant Bragg diffraction using circularly polarized x rays reveals the handedness of crystals by coupling x-ray helicity to a crystal screw axis. The intensity of resonantly allowed reflection of alpha-quartz is well described by an admixture of a parity-even and a parity-odd process. Our results are of general importance and demonstrate a new method to directly study chiral motifs in structures that include biomaterials, liquid crystals, magnets, multiferroics, etc.

Resonant diffraction of circularly polarized x-rays by a chiral crystal (low quartz).

A correlation in x-ray resonant scattering between crystal chirality and circular polarization (helicity) is explored in the context of an analysis of Bragg diffraction from low quartz (α-SiO(2)). There is a one-to-one correlation between chirality and helicity when one resonant event is present in diffraction and thus, in this simple case, resonant Bragg diffraction of circularly polarized x-rays is a direct probe of crystal chirality. The presence of more than one resonant event is shown to add phase relations to the scattering amplitude and then coupling of helicity and chirality is no longer transparent.