Magneto-electric Multipoles Research Articles

Neutron scattering from local magnetoelectric multipoles: A combined theoretical, computational, and experimental perspective

Neutron scattering from local magnetoelectric multipoles: A combined theoretical, computational, and experimental perspective

Magnetic octupole tensor decomposition and second-order magnetoelectric effect

We discuss the second-order magnetoelectric effect, in which a quadratic or bilinear electric field induces a linear magnetization, in terms of the ferroic ordering of magnetic octupoles. We present the decomposition of a general rank-3 tensor into its irreducible spherical tensors, then reduce the decomposition to the specific case of the magnetic octupole tensor, Mijk=∫μi(r)rjrkd3r. We use first-principles density functional theory to compute the size of the local magnetic multipoles on the chromium ions in the prototypical magnetoelectric Cr2O3, and show that, in addition to the well established local magnetic dipoles and magnetoelectric multipoles, the magnetic octupoles are non-zero. The magnetic octupoles in Cr2O3 have an anti-ferroic arrangement, so the net second-order magnetoelectric response is zero. Therefore they form a kind of hidden order, which could be revealed as a linear magnetic (antiferromagnetic) response to a non-zone-center (uniform) quadratic electric field.

Magnetoelectric Classification of Skyrmions.

We develop a general theory to classify magnetic skyrmions and related spin textures in terms of their magnetoelectric multipoles. Since magnetic skyrmions are now established in insulating materials, where the magnetoelectric multipoles govern the linear magnetoelectric response, our classification provides a recipe for manipulating the magnetic properties of skyrmions using applied electric fields. We apply our formalism to skyrmions and antiskyrmions of different helicities, as well as to magnetic bimerons, which are topologically, but not geometrically, equivalent to skyrmions. We show that the nonzero components of the magnetoelectric multipole and magnetoelectric response tensors are uniquely determined by the topology, helicity, and geometry of the spin texture. Therefore, we propose straightforward linear magnetoelectric response measurements as an alternative to Lorentz microscopy for characterizing insulating skyrmionic textures.

Anti-symmetric Compton scattering in LiNiPO 4: Towards a direct probe of the magneto-electric multipole moment.

Magnetoelectric multipoles, which break both space-inversion and time-reversal symmetries, play an important role in the magnetoelectric response of a material. Motivated by uncovering the underlying fundamental physics of the magnetoelectric multipoles and the possible technological applications of magnetoelectric materials, understanding as well as detecting such magnetoelectric multipoles has become an active area of research in condensed matter physics. Here we employ the well-established Compton scattering effect as a possible probe for the magnetoelectric toroidal moments in LiNiPO 4. We employ combined theoretical and experimental techniques to compute as well as detect the antisymmetric Compton profile in LiNiPO 4. For the theoretical investigation we use density functional theory to compute the anti-symmetric part of the Compton profile for the magnetic and structural ground state of LiNiPO 4. For the experimental verification, we measure the Compton signals for a single magnetoelectric domain sample of LiNiPO 4, and then again for the same sample with its magnetoelectric domain reversed. We then take the difference between these two measured signals to extract the antisymmetric Compton profile in LiNiPO 4. Our theoretical calculations indicate an antisymmetric Compton profile in the direction of the t y toroidal moment in momentum space, with the computed antisymmetric profile around four orders of magnitude smaller than the total profile. The difference signal that we measure is consistent with the computed profile, but of the same order of magnitude as the statistical errors and systematic uncertainties of the experiment. While the weak difference signal in the measurements prevents an unambiguous determination of the antisymmetric Compton profile in LiNiPO 4, our results motivate further theoretical work to understand the factors that influence the size of the antisymmetric Compton profile, and to identify materials exhibiting larger effects.

Hidden k-Space Magnetoelectric Multipoles in Nonmagnetic Ferroelectrics.

In condensed matter systems, the electronic degrees of freedom are often entangled to form complex composites, known as hidden orders, which give rise to unusual properties, while escaping detection in conventional experiments. Here we demonstrate the existence of hidden k-space magnetoelectric multipoles in nonmagnetic systems with broken space-inversion symmetry. These k-space magnetoelectric multipoles are reciprocal to the real-space charge dipoles associated with the broken inversion symmetry. Using the prototypical ferroelectric PbTiO_{3} as an example, we show that their origin is a spin asymmetry in momentum space resulting from the broken space inversion symmetry associated with the ferroelectric polarization. In PbTiO_{3}, the k-space spin asymmetry corresponds to a pure k-space magnetoelectric toroidal moment, which can be detected using magnetic Compton scattering, an established tool for probing magnetism in ferromagnets or ferrimagnets with a net spin polarization, which has not been exploited to date for nonmagnetic systems. In particular, the k-space magnetoelectric toroidal moment combined with the spin-orbit interaction manifest in an antisymmetric magnetic Compton profile that can be reversed using an electric field. Our work suggests an experimental route to directly measuring and tuning hidden k-space magnetoelectric multipoles via specially designed magnetic Compton scattering measurements.

Anti-symmetric Compton scattering in LiNiPO4: Towards a direct probe of the magneto-electric multipole moment

Background: Magnetoelectric multipoles, which break both space-inversion and time-reversal symmetries, play an important role in the magnetoelectric response of a material. Motivated by uncovering the underlying fundamental physics of the magnetoelectric multipoles and the possible technological applications of magnetoelectric materials, understanding as well as detecting such magnetoelectric multipoles has become an active area of research in condensed matter physics. Here we employ the well-established Compton scattering effect as a possible probe for the magnetoelectric toroidal moments in LiNiPO4. Methods: We employ combined theoretical and experimental techniques to compute as well as detect the antisymmetric Compton profile in LiNiPO4. For the theoretical investigation we use density functional theory to compute the anti-symmetric part of the Compton profile for the magnetic and structural ground state of LiNiPO4. For the experimental verification, we measure the Compton signals for a single magnetoelectric domain sample of LiNiPO4, and then again for the same sample with its magnetoelectric domain reversed. We then take the difference between these two measured signals to extract the antisymmetric Compton profile in LiNiPO4. Results: Our theoretical calculations indicate an antisymmetric Compton profile in the direction of the ty toroidal moment in momentum space, with the computed antisymmetric profile around four orders of magnitude smaller than the total profile. The difference signal that we measure is consistent with the computed profile, but of the same order of magnitude as the statistical errors and systematic uncertainties of the experiment. Conclusions: While the weak difference signal in the measurements prevents an unambiguous determination of the antisymmetric Compton profile in LiNiPO4, our results motivate further theoretical work to understand the factors that influence the size of the antisymmetric Compton profile, and to identify materials exhibiting larger effects.

Revealing hidden magnetoelectric multipoles using Compton scattering

Magneto-electric multipoles, which are odd under both space-inversion $\cal I$ and time-reversal $\cal T$ symmetries, are fundamental in understanding and characterizing magneto-electric materials. However, the detection of these magneto-electric multipoles is often not straightforward as they remain "hidden" in conventional experiments in part since many magneto-electrics exhibit combined $\cal IT$ symmetry. In the present work, we show that the anti-symmetric Compton profile is a unique signature for all the magneto-electric multipoles, since the asymmetric magnetization density of the magneto-electric multipoles couples to space via spin-orbit coupling, resulting in an anti-symmetric Compton profile. We develop the key physics of the anti-symmetric Compton scattering using symmetry analysis and demonstrate it using explicit first-principles calculations for two well-known representative materials with magneto-electric multipoles, insulating LiNiPO$_4$ and metallic Mn$_2$Au. Our work emphasizes the crucial roles of the orientation of the spin moments, the spin-orbit coupling, and the band structure in generating the anti-symmetric Compton profile in magneto-electric materials.

Imaging switchable magnetoelectric quadrupole domains via nonreciprocal linear dichroism

Parity-odd magnetoelectric multipoles such as magnetic quadrupoles and toroidal dipoles contribute to various symmetry-dependent magnetic phenomena and formation of exotic ordered phases. However, the observation of domain structures emerging due to symmetry breaking caused by these multipoles is a severe challenge because of their antiferromagnetic nature without net magnetization. Here, we report the discovery of nonreciprocal linear dichroism for visible light (~4% at 1.8 eV) in a magnetic quadrupole ordered phase of antiferromagnetic Pb(TiO)Cu4(PO4)4, which enables the identification of magnetic quadrupole domains of opposite signs. Symmetry considerations indicate that nonreciprocal linear dichroism is induced by the optical magnetoelectric effect, i.e., the linear magnetoelectric effect for electromagnetic waves. Using the nonreciprocal linear dichroism, we successfully visualize spatial distributions of quadrupole domains and their isothermal electric-field switching by means of a transmission-type polarized light microscope. The present work exemplifies that the optical magnetoelectric effect efficiently visualizes magnetoelectric multipole domains responding to external perturbations.

Concepts from the linear magnetoelectric effect that might be useful for antiferromagnetic spintronics

We discuss the correspondence between the current-induced spin polarization in non-centrosymmetric magnetic metals and the linear magnetoelectric effect in non-centrosymmetric magnetic insulators using a linear-response theory and the concept of magnetoelectric multipoles. We show that the magnetoelectric toroidal moment is a particularly useful quantity since it determines the ground-state antiferromagnetic domain of a non-centrosymmetric antiferromagnet in the presence of a steady-state electric current. We analyze two prototypical antiferromagnetic spintronic materials—Mn2Au and CuMnAs—and show that the experimentally reported domain reorientations are consistent with the alignment of their toroidal moments parallel to the applied electric current. Finally, we determine whether similar behavior should be expected in the prototypical insulating magnetoelectric materials, Cr2O3 and LiMPO4, if they could be doped into a semiconducting or metallic regime.

Magnetoelectric multipoles in metals

In this paper, we demonstrate computationally the existence of magnetoelectric multipoles, arising from the second-order term in the multipole expansion of a magnetization density in a magnetic field, in non-centrosymmetric magnetic metals. While magnetoelectric multipoles have long been discussed in the context of the magnetoelectric effect in non-centrosymmetric magnetic insulators, they have not previously been identified in metallic systems, in which the mobile carriers screen any electrical polarization. Using first-principles density functional calculations, we explore three specific systems: first, a conventional centrosymmetric magnetic metal, Fe, in which we break inversion symmetry by introducing a surface, which both generates magnetoelectric monopoles and allows a perpendicular magnetoelectric response. Next, the hypothetical cation-ordered perovskite, SrCaRu2O6, in which we study the interplay between the magnitude of the polar symmetry breaking and that of the magnetic dipoles and multipoles, finding that both scale proportionally to the structural distortion. Finally, we identify a hidden antiferromultipolar order in the non-centrosymmetric, antiferromagnetic metal Ca3Ru2O7, and show that, while its competing magnetic phases have similar magnetic dipolar structures, their magnetoelectric multipolar structures are distinctly different, reflecting the strong differences in transport properties.This article is part of the theme issue 'Celebrating 125 years of Oliver Heaviside's 'Electromagnetic Theory''.

Dirac multipoles in diffraction by the layered room-temperature antiferromagnets BaMn2P2 and BaMn2As2

Properties of two $\mathrm{ThC}{\mathrm{r}}_{2}\mathrm{S}{\mathrm{i}}_{2}$-type materials are discussed within the context of their established structural and magnetic symmetries---the structure type accounts for more than 400 compounds of the $\mathrm{A}{\mathrm{B}}_{2}{\mathrm{X}}_{2}$ composition. Both materials develop collinear, G-type antiferromagnetic order above room temperature, and magnetic ions occupy acentric sites in centrosymmetric structures (magnetic crystal class ${4}^{\ensuremath{'}}/{m}^{\ensuremath{'}}{m}^{\ensuremath{'}}m$). We refute a previous conjecture that $\mathrm{BaM}{\mathrm{n}}_{2}\mathrm{A}{\mathrm{s}}_{2}$ is an example of a magnetoelectric material with hexadecapole order using Dirac (magnetoelectric) multipoles by exposing flaws in supporting arguments: principally, an omission of discrete symmetries enforced by the symmetry ($\overline{4}{m}^{\ensuremath{'}}{2}^{\ensuremath{'}}$) of sites used by Mn ions and, also, improper classifications of the primary and secondary order parameters. G-type antiferromagnetism using conventional magnetic dipoles provides the primary order parameter. Dirac quadrupoles are secondary, and possibly visible in Bragg diffraction patterns. Patterns caused by conventional magnetic dipoles and Dirac multipoles are predicted to be fundamentally different, which raises the intriguing possibility of a unique and comprehensive examination of the magnetoelectric state by diffraction; Dirac multipoles create Bragg spots with Miller index $l$ even (and $h+k$ even) while magnetic dipoles appear for $l$ odd (and $h+k$ odd). A rotoinversion operation ($\overline{4}$) in Mn site symmetry is the ultimate source of distinguishing features in multipole types.

Quasistatic magnetoelectric multipoles as order parameter for pseudogap phase in cuprate superconductors

We introduce a mechanism in which coupling between fluctuating spin magnetic dipole moments and polar optical phonons leads to a non-zero ferroic ordering of quasi-static magnetoelectric multipoles. Using first-principles calculations within the LSDA$+U$ method of density functional theory, we calculate the magnitude of the effect for the prototypical cuprate superconductor, HgBa$_2$CuO$_4$. We show that our proposed mechanism is consistent, to our knowledge, with all experimental data for the onset of the pseudo-gap phase and therefore propose the quasi-static magnetoelectric multipole as a possible pseudo-gap order parameter. Finally, we show that our mechanism embraces some key aspects of previous theoretical models, in particular the description of the pseudo-gap phase in terms of orbital currents.

Neutron diffraction and the electronic properties of BaFe2Se3

It is argued on the basis of previously published experimental data that, the magnetic space-group Cac (#9.41) is the correct description of magnetically ordered BaFe2Se3. The corresponding crystal class m1′ allows axial and polar dipoles and forbids bulk ferromagnetism. Magneto–electric multipoles that are both time-odd and parity-odd are allowed, e.g., a magnetic charge (monopole) and an anapole (magnetic toroidal dipole). The experimental observation of magneto–electric multipoles must shed light on valence electrons involved in bonding, including charge transfer using 3d(Fe) and p-states of ligand ions. We provide the appropriate structure factors for the Bragg diffraction neutrons, together with estimates of atomic form factors. Structure factors for resonant x-ray Bragg diffraction are also considered, because the analysis of successful experiments will yield complementary information about electronic properties. Magneto–electric multipoles, over and above those that contribute to magnetic neutron diffraction, include the magnetic monopole. A time-odd, parity-even monopole created from the magnetic dipole and an electric toroidal dipole, which is a manifestation of a structural rotation, is allowed in BaFe2Se3 but it is not visible in diffraction, nor is the corresponding dipole.

Ordered state of magnetic charge in the pseudo-gap phase of a cuprate superconductor (HgBa2CuO4+δ)

A symmetry-based interpretation of published experimental results demonstrates that the pseudo-gap phase of underdoped HgBa2CuO4+δ (Hg1201) possesses an ordered state of magnetic charge epitomized by Cu magnetic monopoles. Magnetic properties of one-layer Hg1201 and two-layer YBa2Cu3O6+x (YBCO) cuprates have much in common, because their pseudo-gap phases possess the same magnetic space-group, e.g. both underdoped cuprates allow the magneto-electric (Kerr) effect. Differences in their properties stem from different Cu site symmetries, leaving Cu magnetic monopoles forbidden in YBCO. Resonant x-ray Bragg diffraction experiments can complement the wealth of information available from neutron diffraction experiments on five Hg1201 samples on which our findings are based. In the case of Hg1201 emergence of the pseudo-gap phase, with time-reversal violation, is accompanied by a reduction of Cu site symmetry that includes loss of a centre of inversion symmetry. In consequence, parity-odd x-ray absorption events herald the onset of the enigmatic phase, and we predict dependence of corresponding Bragg spots on magneto-electric multipoles, including the monopole, and the azimuthal angle (crystal rotation about the Bragg wavevector).

Ferro-type order of magneto-electric quadrupoles as an order-parameter for the pseudo-gap phase of a cuprate superconductor

There is general agreement within the community of researchers that investigate high-Tc materials that it is most important to understand the pseudo-gap phase. To this end, many experiments on various cuprates have been reported. Two prominent investigations—Kerr effect and neutron Bragg diffraction—imply that underdoped YBCO samples possess long-range magnetic order of an unusual kind. However, other measurements do not support the existence of magnetic order. Here we show that the Kerr effect and magnetic Bragg diffraction data are individual manifestations of ordered magneto-electric quadrupoles at Cu sites. While the use of magneto-electric multipoles is new in studies of the electronic properties of cuprates, they are not unknown in other materials, including an investigation with x-rays of the parent compound CuO. We exploit the recent prediction that neutrons are deflected by magneto-electric multipoles. The outcome of our study is a theory for the order-parameter of the pseudo-gap phase without the aforementioned conflict with other measurements, and the first experimental evidence that neutrons interact with multipoles belonging to a state of magnetic charge.

A model of magneto-electric multipoles

A long-known Hamiltonian of electrons with entangled spin and orbital degrees of freedom is re-examined as a model of magneto-electric multipoles (MEs). In the model, a magnetic charge and simple quantum rotator are tightly locked in action, some might say they are enslaved entities. It is shown that MEs almost perfectly accord with those inferred from an analysis of magnetic neutron diffraction data on a ceramic superconductor (YBCO) in the pseudo-gap phase. Nigh on perfection between Stone's model and inferred MEs is achieved by addition to the original model of a crystal-field potential appropriate for the magnetic space group used in the published data analysis. An impression of thermal properties of multipoles is sought from a molecular-field model.

Chiral properties of hematiteα-Fe2O3inferred from resonant Bragg diffraction using circularly polarized x rays

Chiral properties of the two phases - collinear motif (below Morin transition temperature, TM=250 K) and canted motif (above TM) - of magnetically ordered hematite ({\alpha}-Fe2O3) have been identified in single crystal resonant x-ray Bragg diffraction, using circular polarized incident x-rays tuned near the iron K-edge. Magneto-electric multipoles, including an anapole, fully characterize the high-temperature canted phase, whereas the low-temperature collinear phase supports both parity-odd and parity-even multipoles that are time-odd. Orbital angular momentum accompanies the collinear motif, while it is conspicuously absent with the canted motif. Intensities have been successfully confronted with analytic expressions derived from an atomic model fully compliant with chemical and magnetic structures. Values of Fe atomic multipoles previously derived from independent experimental data, are shown to be completely trustworthy.

Unabridged Resonant Contribution to the X-ray Scattering Length

Resonance-enhanced Bragg scattering of x-rays and concomitant dichroic signals are valuable tools for investigation of material properties. In this context parity-odd events E1–E2 and E1–M1 merit special attention. Theoretical analysis of the product of matrix-elements defining the scattering length for the resonant photon absorption and emission process leads to sums of electronic multipoles calculated for equilibrium electron states. The unabridged result retains explicit dependence on the labels of the intermediate state and recovers a common unit tensor present in the formulae describing parity-odd events. In order to add physical insight, the various forms of the unit tensor are correlated with magneto-electric and polar multipoles, which benefit from representations using familiar operators. The description of parity-odd events requires the use of appropriate combinations of photonic tensors and electronic multipoles to achieve correct symmetry properties. After discussing some special cases the new...

Parity-odd multipoles, magnetic charges, and chirality in hematiteα-Fe2O3

Collinear and canted magnetic motifs in haematite were investigated by Kokubun et al. (2008) using x-ray Bragg diffraction magnified at the iron K-edge, and analyses of observations led to various potentially interesting conclusions. We demonstrate that the reported analyses for both non-resonant and resonant magnetic diffraction at low energies near the absorption K-edge are not appropriate. In its place, we apply a radically different formulation, thoroughly tried and tested, that incorporates all magnetic contributions to resonant x-ray diffraction allowed by the established chemical and magnetic structures. Essential to a correct formulation of diffraction by a magnetic crystal with resonant ions at sites that are not centres of inversion symmetry are parity-odd atomic multipoles, time-even (polar) and time-odd (magneto-electric), that arise from enhancement by the electric-dipole (E1) - electric-quadrupole (E2) event. Analyses of azimuthal-angle scans on two space-group forbidden reflections, hexagonal (0, 0, 3)h and (0, 0, 9)h, collected by Kokubun et al. above and below the Morin temperature (TM = 250K), allow us to obtain good estimates of contributing polar and magneto-electric multipoles, including the iron anapole. We show, beyond reasonable doubt, that available data are inconsistent with parity-even events only (E1-E1 and E2- E2). For future experiments, we show that chiral states of haematite couple to circular polarization and differentiate E1-E2 and E2-E2 events, while the collinear motif supports magnetic charges.

Experimental evidence of anapolar moments in the antiferromagnetic insulating phase ofV2O3obtained from x-ray resonant Bragg diffraction

We have investigated the antiferromagnetic insulating phase of the Mott-Hubbard insulator vanadium sesquioxide $({\text{V}}_{2}{\text{O}}_{3})$ by resonant x-ray Bragg diffraction at the vanadium $K$ edge. Combining the information obtained from azimuthal angle scans, linear incoming polarization scans and by fitting collected data to the scattering amplitude derived from the established chemical $I2/a$ and magnetic space groups we provide direct experimental evidence of the ordering motif of vanadium magnetoelectric multipoles in ${\text{V}}_{2}{\text{O}}_{3}$. Experimental data (azimuthal dependence and polarization analysis) collected at space-group forbidden Bragg reflections are successfully accounted within our model in terms of vanadium magnetoelectric multipoles. We demonstrate that resonant x-ray diffraction intensities in all space-group forbidden Bragg reflections of the kind ${(hkl)}_{m}$ with odd $h$ are produced by an $E1\ensuremath{-}E2$ event. The determined multipolar parameters offer a test for ab initio calculations in this material, which can lead to a deeper and more quantitative understanding of the physical properties of ${\text{V}}_{2}{\text{O}}_{3}$.