Dzyaloshinskii-Moriya Vector Research Articles

Chiral multiferroicity in two-dimensional hybrid organic-inorganic perovskites.

Chiral multiferroics offer remarkable capabilities for controlling quantum devices at multiple levels. However, these materials are rare due to the competing requirements of long-range orders and strict symmetry constraints. In this study, we present experimental evidence that the coexistence of ferroelectric, magnetic orders, and crystallographic chirality is achievable in hybrid organic-inorganic perovskites [(R/S)-β-methylphenethylamine]2CuCl4. By employing Landau symmetry mode analysis, we investigate the interplay between chirality and ferroic orders and propose a novel mechanism for chirality transfer in hybrid systems. This mechanism involves the coupling of non-chiral distortions, characterized by defining a pseudo-scalar quantity, ( represents the ferroelectric displacement vector and denotes the ferro-rotational vector), which distinguishes between (R)- and (S)-chirality based on its sign. Moreover, the reversal of this descriptor's sign can be associated with coordinated transitions in ferroelectric distortions, Jahn-Teller antiferro-distortions, and Dzyaloshinskii-Moriya vectors, indicating the mediating role of crystallographic chirality in magnetoelectric correlations.

Three-dimensional Heisenberg model with Dzyaloshinskii-Moriya interaction: A Monte Carlo study.

The three-dimensional classical Heisenberg model on a simple cubic lattice with Dzyaloshinskii-Moriya (DM) interactions between nearest-neighbors in all directions has been studied using Monte Carlo simulations. The Metropolis algorithm, combined with single histogram reweighting techniques and finite-size scaling analyses, has been used to obtain the thermodynamic behavior of the system in the thermodynamic limit. Simulations were performed with the same set of interaction parameters for both shifted boundary conditions (SBC) and fluctuating boundary conditions (FBC). Because of an incommensurability caused by the DM interaction, the SBC incorporated a fixed shift angle at the boundary which varies as a function of the DM interaction and lattice size. This SBC method decreases the simulation time significantly, but the distribution of states is somewhat different than that obtained with FBC. The ground state for nonzero DM interaction is a spiral configuration where the spins are restricted to lie in planes perpendicular to the DM vector. We found that this spiral configuration undergoes a conventional second-order phase transition into a disordered, paramagnetic state with the transition temperature being a function of the magnitude of the DM interaction. The limiting case with only DM interaction in the model has also been considered. The critical exponent ν, the critical exponent ratios α/ν, β/ν, γ/ν, as well as the critical temperature T_{c} and fourth-order cumulant of the order parameter U_{4}^{*} at T_{c} have been estimated for different magnitudes of DM interaction. The critical exponents and cumulants at the transition are different from those for the three-dimensional Heisenberg model, but the ratios α/ν, β/ν, γ/ν, U_{4}^{*}/ν are the same, implying that weak universality is valid for all values of DM interaction. Structure factor calculations for particular cases have been performed considering SBC and FBC in the simulations with different lattice sizes at the critical temperatures.

Skyrmions in van der Waals centrosymmetric materials with Dzyaloshinskii–Moriya interactions

Skyrmions can appear in non-centrosymmetric materials because of non-vanishing Dzyaloshinskii–Moriya interactions (DMIs). We investigate the magnetic properties of rhombohedral MX3 (M: V, Cr, Mn, Fe; X: Cl, Br, I) van der Waals materials with centrosymmetric lattices by combining first-principle calculations and Monte Carlo simulations. We determine that the Dzyaloshinskii–Moriya vector acting between the second nearest neighbor sites of the intralayer is non-zero and large in MX3, owing to the breaking of the local inversion symmetry. Large DMIs cause nanoscale magnetic vortices, i.e. that is, skyrmions. We observe both conventional skyrmions in CrCl3 and VCl3 and antiferromagnetic skyrmions in FeCl3 and merons in MnCl3. Furthermore, the skyrmions in CrCl3 and VCl3 have different helicities, indicating the possibility of controlling the helicity by electron/hole doping in MX3 materials. Van der Waals materials have high degrees of freedom in heterostructures and twisted structures, demonstrating promising potential as skyrmion materials.

Weak trimer distortion and planar spin configuration in hexagonal Lu0.6In0.4FeO3

Improper ferroelectricity and canted ferromagnetism in antiferromagnetically ordered hexagonal ferrites with A2-type spin configuration have been intensely studied due to their potential for room-temperature multiferroicity with strong magnetoelectric couplings. However, the subtle interplay between the magnetic structure and trimer structural distortion, which is a critical ingredient for ferroelectricity, has not been clearly verified in experiments due to the lack of control over trimer distortion. In this study, we report on Lu0.6In0.4FeO3, which exhibits weaker trimer distortion primarily related to the smaller tilting of the FeO5 bipyramids. The reduced vertical displacement of the equatorial oxygen of FeO5 located at the center of the trimer lowers the magnitude of the in-plane Dzyaloshinskii–Moriya vector component, resulting in the absence of canted ferromagnetism with the planar A1-type spin configuration, rather than the canted A2-type observed in other hexagonal ferrites. Our findings demonstrate that the degree of trimer distortion plays an important role in determining the spin configuration in related hexagonal systems.

Controlling Dzyaloshinskii-Moriya interaction in a centrosymmetric nonsymmorphic crystal

Presence of the Dzyaloshinskii-Moriya (DM) interaction in limited noncentrosymmetric materials leads to novel spin textures and exotic chiral physics. The emergence of DM interaction in centrosymmetric crystals could greatly enrich material realization. Here we show that an itinerant centrosymmetric crystal respecting a nonsymmorphic space group is a new platform for the DM interaction. Taking the P4/nmm space group as an example, we demonstrate that the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction induces the DM interactions, in addition to the Heisenberg exchange and the Kaplan-Shekhtman-Entin-wohlman-Aharony (KSEA) interaction. The direction of DM vector depends on the positions of magnetic atoms in the real space, and the amplitude depends on the location of the Fermi surface in the reciprocal space. The diversity stems from the position-dependent site groups and the momentum-dependent electronic structures guaranteed by the nonsymmorphic symmetries. Our study unveils the role of the nonsymmorphic symmetries in affecting magnetism, and suggests that the nonsymmorphic crystals can be promising platforms to design magnetic interactions.

Control of magnetic states and spin interactions in bilayer CrCl3 with strain and electric fields: an ab initio study

Using ab initio density functional theory, we demonstrated the possibility of controlling the magnetic ground-state properties of bilayer CrCl_{3} by means of mechanical strains and electric fields. In principle, we investigated the influence of these two fields on parameters describing the spin Hamiltonian of the system. The obtained results show that biaxial strains change the magnetic ground state between ferromagnetic and antiferromagnetic phases. The mechanical strain also affects the direction and amplitude of the magnetic anisotropy energy (MAE). Importantly, the direction and amplitude of the Dzyaloshinskii–Moriya vectors are also highly tunable under external strain and electric fields. The competition between nearest-neighbor exchange interactions, MAE, and Dzyaloshinskii–Moriya interactions can lead to the stabilization of various exotic spin textures and novel magnetic excitations. The high tunability of magnetic properties by external fields makes bilayer CrCl_{3} a promising candidate for application in the emerging field of two-dimensional quantum spintronics and magnonics.

Linear response based theories for Dzyaloshinskii-Moriya interactions

We investigate abilities of various linear response based techniques for extracting parameters of antisymmetric Dzyaloshinskii-Moriya (DM) interactions from the first-principles electronic structure calculations. For these purposes, we further elaborate the idea of Sandratskii [Phys. Rev. B 96, 024450 (2017)], which states that $z$ component of the DM vector can be computed by retaining only spin-diagonal part of the spin-orbit (SO) coupling. We start our analysis with the magnetic force theorem (MFT), which relies on additional approximations resulting in the linear dependence of the exchange interactions on the response tensor, and compare it with the exact approach formulated in terms of the inverse response. We propose the downfolding procedure transferring the effect of these ligand spins into parameters of effective interactions between the localized spins. These techniques are applied for the series of CrCl$_3$ and CrI$_3$ based materials, including bulk, monolayer, bilayer, and three-layer systems. Particularly, we discuss how the DM interactions are induced by the inversion symmetry breaking at the surface or by the electric field. As long as the SO interaction of the heavy ligand atoms is taken into account in the calculations of the response tensor between the Cr $3d$ states, the MFT appears to be a good approximation for the DM interactions, being in contrast with the isotropic exchange, for which MFT and the exact method provide quite a different description. Finally, we discuss the relevance of our approach to other techniques ever proposed for calculations of the DM interactions. Particularly, we argue that the spin-current model for the DM interactions can be derived from the MFT based expression and is the relativistic counterpart of the double exchange, occurring in metallic systems in the limit of infinite exchange splitting.

Nonreciprocal Spin-Wave Propagation in a Ferromagnet With Stepwise Interfacial Dzyaloshinskii–Moriya Interaction

In this article, boundary conditions for Landau–Lifshitz equations at the interface between two ferromagnets with different interfacial Dzyaloshinskii–Moriya interactions (iDMIs) are derived. We calculated and verified the analytical expression for the energy flux density continuity for the spin-wave (SW) propagation through the ferromagnet with stepwise iDMI considering the boundary conditions mentioned. Analytical expressions for reflection, transmission coefficients, and the nonreciprocity factor are calculated in the case of SW propagation through a ferromagnet with the stepwise iDMI. Two different types of nonreciprocity effects for SW propagation through a ferromagnet with the stepwise interface Dzyaloshinskii–Moriya interaction are revealed for opposite orientations of Dzyaloshinskii–Moriya vector. The material parameters of a ferromagnet and the stepwise iDMIs are found for which the extremely high nonreciprocity factor (>10) is expected according to the results of calculations. The results of this article deepen the knowledge about the SW propagation control in magnonic devices.

Frustrated antiferromagnetic triangular lattice with Dzyaloshinskii–Moriya interaction: Ground states, spin waves, skyrmion crystal, phase transition

We study in this article a triangular lattice with Heisenberg spins interacting with each other via an antiferromagnetic exchange interaction J and a Dzyaloshinskii–Moriya (DM) interaction D, between nearest-neighbors (NN). We consider two cases: the first case in which the DM vector D is perpendicular to the lattice plane and the second case where it lies in the plane. A magnetic field H is applied perpendicular to the spin plane in both cases. The ground state (GS) of this system is calculated by minimizing the energy using the very fast steepest-descent method in the two cases. In the case of perpendicular D with H=0, the GS is periodic. We analytically determine the GS configuration which is characterized by two well-defined angles. We calculate the spin-wave spectrum in this case which shows that for small wave vectors the spin waves are forbidden in the system. When H≠0, the GS in the perpendicular D case shows no skyrmions. However, in the case of in-plane D with H≠0, we find a crystal of skyrmions at T=0 composed of three interpenetrating skyrmion sublattice crystals, in agreement with low-T spin textures found in earlier works. We show by Monte Carlo simulations that this skyrmion crystal is stable at finite temperatures below a critical temperature.

Skyrmion Creation and Annihilation by Electric Current Vorticity

Motivated by the recent experiment done by Yu <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> [X. Z. Yu, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et. al.</i> , Sci. Adv., 6, 25, eaaz9744 (2020)] on magnetic skyrmion creation and annihilation by electric currents in FeGe devices with a notch structure, we examine the interaction between the electric current vorticity near the notch and localized spins, and obtain three effects of the current vorticity on the localized spins. The first one is an effective Zeeman field induced by the current vorticity. This effective Zeeman field is responsible for skyrmion creation. The second effect is a vorticity-induced Dzyaloshinskii–Moriya (DM) interaction with the direction of the DM vector being parallel to the vorticity direction and is different from the vector of the intrinsic DM interaction. This vorticity-induced DM interaction makes the skyrmions distorted. The third effect is a spin current due to the gradient of the vorticity and the spin current drives the magnetic skyrmions far from the notch. Our theory opens a new path to design the magnetic textures by using structural settings.

Current-driven spiral domain wall in a ferrimagnet near the magnetization compensation point

A spiral domain wall emerges when the Dzyaloshinskii-Moriya vector is along the easy axis. While the ferromagnetic spiral wall has been well studied, its ferrimagnetic counterpart has nevertheless not been explored. Here, using the collective variable approach, we derive the spiral domain-wall solutions in a ferrimagnetic nanostrip with a longitudinal easy axis and a bulk Dzyaloshinskii-Moriya interaction. At the magnetization compensation point, the wall rotates and propagates steadily in a wide current range. The (angular) velocity of this wall increases almost linearly with the experimentally feasible currents. Also, the wall exhibits a relativistic-like contraction with increasing velocity. Near this point, the wall moves smoothly with its velocity linearly depending on the current below the Walker breakdown. Above this critical current, all the internal parameters of the wall, such as the azimuthal angle, width, and spiral pitch, as well as their rates of change, oscillate periodically. The wall moves forward with its velocity varying periodically. These different behaviors at or near the compensation point are ascribed to the tunable demagnetization effect. The adjustability of FiM parameters, combining with the twist induced by the Dzyaloshinskii-Moriya interaction, can effectively change the DW's propagation and rotation.

Relief of spin frustration through magnetic anisotropy in the quasi-one-dimensional S=12 antiferromagnet Na2CuSO4Cl2

We report the magnetic structure and anisotropy of the quasi-one-dimensional $S=\frac{1}{2}$ antiferromagnet ${\mathrm{Na}}_{2}{\mathrm{CuSO}}_{4}{\mathrm{Cl}}_{2}$ obtained by single-crystal neutron scattering, electron spin resonance (ESR), and magnetization measurements, following an earlier study of its dynamics [M. Fujihala et al., Phys. Rev. B 101, 024410 (2020)]. A N\'eel-type spin structure is formed within the chain of this compound, where the spins point along the $b$ axis, and ESR data indicate an antisymmetric exchange with a uniform Dzyaloshinskii-Moriya (DM) vector pointing along the $b$ axis. The anisotropy $g$ factor and magnetic structure are strong indicators of magnetic anisotropy originating from a symmetric anisotropic exchange interaction and/or a magnetic dipole interaction. These results suggest that these terms of the anisotropic spin Hamiltonian counteract the effect of the DM interaction and stabilize the N\'eel-type structure in ${\mathrm{Na}}_{2}{\mathrm{CuSO}}_{4}{\mathrm{Cl}}_{2}$.

Magnetic Skyrmions in a Thickness Tunable 2D Ferromagnet from a Defect Driven Dzyaloshinskii-Moriya Interaction.

There is considerable interest in van der Waals (vdW) materials as potential hosts for chiral skyrmionic spin textures. Of particular interest is the ferromagnetic, metallic compound Fe3 GeTe2 (FGT), which has a comparatively high Curie temperature (150-220 K). Several recent studies have reported the observation of chiral Néel skyrmions in this compound, which is inconsistent with its presumed centrosymmetric structure. Here the observation of Néel type skyrmions in single crystals of FGT via Lorentz transmission electron microscopy (LTEM) is reported. It is shown from detailed X-ray diffraction structure analysis that FGT lacks an inversion symmetry as a result of an asymmetric distribution of Fe vacancies. This vacancy-induced breaking of the inversion symmetry of this compound is a surprising and novel observation and is a prerequisite for a Dzyaloshinskii-Moriya vector exchange interaction which accounts for the chiral Néel skyrmion phase. This phenomenon is likely to be common to many 2D vdW materials and suggests a path to the preparation of many such acentric compounds. Furthermore, it is found that the skyrmion size in FGT is strongly dependent on its thickness: the skyrmion size increases from ≈100 to ≈750nm as the thickness of the lamella is increased from ≈90nm to ≈2µm. This extreme size tunability is a feature common to many low symmetry ferro- and ferri-magnetic compounds.

Complex magnetic textures in Ni/Irn/Pt(111) ultrathin films

A combined approach using first-principles calculations and spin dynamics simulations is applied to study $\mathrm{Ni}/{\mathrm{Ir}}_{n}/\mathrm{Pt}(111)$ ($n=0,1,2$) films. The lowest-energy states are predicted to be spin spirals but with a minute (of the order of a few $\ensuremath{\mu}\mathrm{eV}/\mathrm{atom}$) energy difference with skyrmionic states. The spontaneous low-temperature skyrmions, with $\ensuremath{\sim}15$ nm to $\ensuremath{\sim}35$ nm size, arise from a large Dzyaloshinskii-Moriya (DM) and Heisenberg exchange interactions ratio and, in particular, from a large in-plane DM vector component for nearest neighbors. The skyrmions become larger and more dispersed with the enhancement of the Ir buffer thickness. Also, with increasing $n$, the skyrmions' stability decreases when an external magnetic field is applied or the temperature is raised. For $n=0$ and $n=1$, we find that metastable skyrmioniums can occur, which are characterized by a slightly lower stability with respect to the external fields and larger critical currents, compared to skyrmions.

Thermal Properties and Instability of a U(1) Spin Liquid on the Triangular Lattice.

We study the effect of Dzyaloshinskii-Moriya (DM) interaction on the triangular lattice U(1) quantum spin liquid (QSL) which is stabilized by ring-exchange interactions. A weak DM interaction introduces a staggered flux to the U(1) QSL state and changes the density of states at the spinon Fermi surface. If the DM vector contains in-plane components, then the spinons gain nonzero Berry phase. The resultant thermal conductances κ_{xx} and κ_{xy} qualitatively agree with the experimental results on the material EtMe_{3}Sb[Pd(dmit)_{2}]_{2}. Furthermore, owing to perfect nesting of the Fermi surface, a spin density wave state is triggered by larger DM interactions. On the other hand, when the ring-exchange interaction decreases, another antiferromagnetic (AFM) phase with 120° order shows up which is proximate to a U(1) Dirac QSL. We discuss the difference of the two AFM phases from their static structure factors and excitation spectra.

Hybrid improper ferroelectricity and magnetoelectric coupling in a two-dimensional perovskite oxide

Unlike van der Waals materials, perovskite oxide may undergo structural reconstruction when reduced to two dimensions. Therefore, it is still an open question whether some of the ferroelectric mechanisms and magnetoelectric coupling effects found in perovskite bulks can be maintained at the two-dimensional (2D) limit. Here we demonstrate that the ferroelectricity induced by octahedral rotation and the magnetoelectric coupling mechanism dependent on Dzyaloshinskii-Moriya (DM) interaction exist in a proposed perovskite bilayer. Although the octahedral rotation distortion of the ${\mathrm{Ca}}_{3}{\mathrm{Mn}}_{2}{\mathrm{O}}_{7}$ bilayer is reconstructed with respect to its layered bulk phase, tensile strain can induce the same octahedral rotation type as its bulk phase, resulting in the emergence of ferroelectric phase. We show that the modulation of the DM vector by octahedral rotation distortion provides a coupling between polarization and the induced magnetization. In ferroelectric switching, the polarization is reversed by a ${180}^{\ensuremath{\circ}}$ rotation of the tilt axis within the $ab$ plane, which also results in a reversal of magnetization. This work is expected to provide a guidance for realizing electric-field control of magnetization direction in 2D perovskite oxides.

Dynamics of local magnetic moments induced by itinerant Weyl electrons

We derive the effective interactions between local magnetic moments which are mediated by Weyl electrons in magnetic topological semimetals. The resulting spin dynamics is governed by the induced Heisenberg, Kitaev and Dzyaloshinskii-Moriya (DM) interactions with extended range and oscillatory dependence on the distance between the spins. These interactions are realized in multiple competing channels shaped by the multitude of Weyl nodes in the electron spectrum. Microscopic spins need to be spatially modulated with a channel-dependent wavevector in order to take advantage of the interactions. The DM vector is parallel to the displacement between the two interacting spins, and requires the presence of Weyl electron Fermi surfaces. We also derive the Weyl-induced chiral three-spin interaction in the presence of an external magnetic field. This interaction has an extended range as well, and acts upon the spatially modulated spins in various channels. Its tendency is to produce a skyrmion lattice or a chiral spin liquid which exhibits topological Hall effect. Ultimately, the theory developed here addresses magnetic dynamics in relativistic metals even when chiral magnetism is microscopically precluded. We discuss insights into the ordered state of the magnetic Weyl semimetal NdAlSi.

How to create giant Dzyaloshinskii-Moriya interactions? Analytical derivation and ab initio calculations on model dicopper(II) complexes.

This paper is a theoretical "proof of concept" on how the on-site first-order spin-orbit coupling (SOC) can generate giant Dzyaloshinskii-Moriya interactions in binuclear transition metal complexes. This effective interaction plays a key role in strongly correlated materials, skyrmions, multiferroics, and molecular magnets of promising use in quantum information science and computing. Despite this, its determination from both theory and experiment is still in its infancy and existing systems usually exhibit very tiny magnitudes. We derive analytical formulas that perfectly reproduce both the nature and the magnitude of the Dzyaloshinskii-Moriya interaction calculated using state-of-the-art ab initio calculations performed on model bicopper(II) complexes. We also study which geometrical structures/ligand-field forces would enable one to control the magnitude and the orientation of the Dzyaloshinskii-Moriya vector in order to guide future synthesis of molecules or materials. This article provides an understanding of its microscopic origin and proposes recipes to increase its magnitude. We show that (i) the on-site mixings of 3d orbitals rule the orientation and magnitude of this interaction, (ii) increased values can be obtained by choosing more covalent complexes, and (iii) huge values (∼1000 cm-1) and controlled orientations could be reached by approaching structures exhibiting on-site first-order SOC, i.e., displaying an "unquenched orbital momentum."

Chiral magnetism: a geometric perspective

We discuss a geometric perspective on chiral ferromagnetism. Much like gravity becomes the effect of spacetime curvature in theory of relativity, the Dzyaloshinski-Moriya interaction arises in a Heisenberg model with nontrivial spin parallel transport. The Dzyaloshinskii-Moriya vectors serve as a background SO(3) gauge field. In 2 spatial dimensions, the model is partly solvable when an applied magnetic field matches the gauge curvature. At this special point, solutions to the Bogomolny equation are exact excited states of the model. We construct a variational ground state in the form of a skyrmion crystal and confirm its viability by Monte Carlo simulations. The geometric perspective offers insights into important problems in magnetism, e.g., conservation of spin current in the presence of chiral interactions.

Thermal Hall effect from a two-dimensional Schwinger boson gas with Rashba spin-orbit interaction: Application to ferromagnets with in-plane Dzyaloshinskii-Moriya interaction

Recently, uncovering the sources of thermal Hall effect in insulators has become an important issue. In the case of ferromagnetic insulators, it is well known that Dzyaloshinskii-Moriya (DM) interaction can induce magnon thermal Hall effect. Specifically, the DM vector parallel to the magnetization direction induces complex magnon hopping amplitudes, so that magnons act as if they feel Lorentz force. However, the DM vector which is orthogonal to the magnetization direction has hitherto been neglected as a possible source of magnon thermal Hall effect. This is because they play no role in the linear spin wave theory, an often invoked approximation when computing the magnon thermal Hall effect. Here, we challenge this expectation by presenting the self-consistent Schwinger-boson mean field study of two-dimensional magnets with ferromagnetic Heisenberg interaction and in-plane DM interaction. We find that the relevant Schwinger-boson mean field Hamiltonian takes the form of two-dimensional electron gas with Rashba spin-orbit interaction, which is known to show anomalous Hall effect, spin Hall effect, and Rashba-Edelstein effect, whose thermal counterparts also appear in our system. Importantly, the thermal Hall effect can be induced when out-of-plane magnetic field is applied, and persists even when the magnetic field is large, so that the spins are significantly polarized, and the linear spin wave theory is expected to be a reasonable approximation. Since the linear spin wave theory predicts vanishing thermal Hall effect, our result implies that linear spin wave is not a sufficient approximation, and that magnon-magnon interaction must be taken into account to predict the correct thermal Hall conductivity.