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Self-localized spin structures can substantially enrich the scope of the field-free antiferromagnetic spintronics [1]. One of them - the domain wall (DW) is a topological soliton that bridge a connection between two ground states, similar to a link in a Josephson junction (JJ) between two superconductors. JJs are intensively investigated for decades as they promise important advantages for electronics. In particular, JJ-based qubits are already successfully realized and commercialized.&lt;br/&gt;&lt;br/&gt;We demonstrate that the DW in bi-axial AFM with the easy-axis type of primary anisotropy driven by a spin current is a close analogy of a long Josephson junction (LJJ). Thus, the dynamics of the Neel vector angle inside the DW (Josephson phase Φ) is described by a sine-Gordon type of equation, where applied spin transfer torque (STT) represents a bias current through the junction, anisotropy in a basal plane defines a plasma frequency, and characteristic speed of magnons, defined by nonhomogeneous exchange energy, corresponds to Swihart velocity.&lt;br/&gt;&lt;br/&gt;In contrast with a conventional LJJ, a DW has additional degrees of freedom, namely the position of its center X and distribution of the current along a DW. We show by micromagnetic simulations, that when the polarization of the STT is parallel to the easy axis of an AFM, the above allows to nucleate and propel a single kink-type Φ soliton [2], which is Bloch line (Fig.1) that corresponds to the Josephson vortex in LJJ, as well as their series. Withal, a weak alternate current excites Φ-type intrawall spin waves, an analog of plasma waves in LJJ, which spectrum has a bandgap, defined by the anisotropy (Fig.2). STT with a perpendicular polarization allows manipulating the X position, particularly to excite the flexural vibrations of the DW with a gapless spectrum [3] (Fig.2).&lt;br/&gt;&lt;br/&gt;The variety of possible spin transport regimes together with the possibility of DW movement or bending under the action of applied spin current, offers considerable scope for developing novel spintronics devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/ec00686c8da48a4e0f62c64d5949d0c0.jpg)

Generation of the Bloch line, - an anlog of the Josephson vortex.

![](https://assets.underline.io/uploads/markdown_image/1/image/b8a610941c6971f9af5c49d7f985cadd.jpg)

Dispersion law for Φ (red) and X (blue) intrawall modes in easy-axis Cr2O3 antiferromagnet with the presence of in-plane anisotropy.

2022 Joint MMM-INTERMAG

Spin transport in a long Josephson junction based on an antiferromagnetic domain wall.

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Self-localized spin structures can substantially enrich the scope of the field-free antiferromagnetic spintronics [1]. One of them - the domain wall (DW) is a topological soliton that bridge a connection between two ground states, similar to a link in a Josephson junction (JJ) between two superconductors. JJs are intensively investigated for decades as they promise important advantages for electronics. In particular, JJ-based qubits are already successfully realized and commercialized. We demonstrate that the DW in bi-axial AFM with the easy-axis type of primary anisotropy driven by a spin current is a close analogy of a long Josephson junction (LJJ). Thus, the dynamics of the Neel vector angle inside the DW (Josephson phase Φ) is described by a sine-Gordon type of equation, where applied spin transfer torque (STT) represents a bias current through the junction, anisotropy in a basal plane defines a plasma frequency, and characteristic speed of magnons, defined by nonhomogeneous exchange energy, corresponds to Swihart velocity. In contrast with a conventional LJJ, a DW has additional degrees of freedom, namely the position of its center X and distribution of the current along a DW. We show by micromagnetic simulations, that when the polarization of the STT is parallel to the easy axis of an AFM, the above allows to nucleate and propel a single kink-type Φ soliton [2], which is Bloch line (Fig.1) that corresponds to the Josephson vortex in LJJ, as well as their series. Withal, a weak alternate current excites Φ-type intrawall spin waves, an analog of plasma waves in LJJ, which spectrum has a bandgap, defined by the anisotropy (Fig.2). STT with a perpendicular polarization allows manipulating the X position, particularly to excite the flexural vibrations of the DW with a gapless spectrum [3] (Fig.2). The variety of possible spin transport regimes together with the possibility of DW movement or bending under the action of applied spin current, offers considerable scope for developing novel spintronics devices.

![](https://assets.underline.io/uploads/markdown_image/1/image/ec00686c8da48a4e0f62c64d5949d0c0.jpg)

Generation of the Bloch line, - an anlog of the Josephson vortex.

![](https://assets.underline.io/uploads/markdown_image/1/image/b8a610941c6971f9af5c49d7f985cadd.jpg)

Dispersion law for Φ (red) and X (blue) intrawall modes in easy-axis Cr2O3 antiferromagnet with the presence of in-plane anisotropy.

technical paper

Modern magnetic-memory technology requires all-electric control of perpendicular magnetization with low energy consumption. While spin-orbit torque (SOT) in heavy metal/ferromagnet (HM/FM) heterostructures holds promise for applications in magnetic random access memory, till today, it is limited to the in-plane direction. Such in-plane torque can switch perpendicular magnetization only deterministically with the help of additional symmetry breaking, e.g., through the application of an external magnetic field, an interlayer coupling or an asymmetric design. Instead, an out-of-plane spin-orbit torque could directly switch perpendicular magnetization. In current presentation, we report that we observe an out-of-plane spin-orbit torque in an HM/FM bilayer of L11-ordered CuPt/CoPt and demonstrate field-free switching of the perpendicular magnetization of the CoPt layer.1 The low symmetry point group (3m1) at the CuPt/CoPt interface gives rise to this spin torque, herein after referred as 3m torque, which strongly depends on the relative orientation of current flow and crystal symmetry. We observe a 3-fold angular dependence in both the field-free switching and the current-induced out-of-plane effective field as shown in Figure 1. Because of the intrinsic nature of the 3m torque, the field-free switching in CuPt/CoPt shows good endurance in cycling experiments. Experiments with the wide variety of SOT bilayers with low-symmetry point groups at the interface may uncover further unconventional spin-torques in future.

![](https://assets.underline.io/uploads/markdown_image/1/image/e032f3aa438f0424fd4d0cd491e3db92.jpg)

Figure 1| Symmetry-dependent magnetic field-free magnetization switching. a, Schematic of the CuPt/CoPt Hall bar for electrical transport measurement. The red arrow represents the magnetization (M). The grey arrow represents the current (I) flowing. b, The definition of current flowing direction (θI). The current is applied along the Hall bar, which has an azimuth angle of θI with respect to the [1-10] direction. c, Anomalous Hall effect of the bilayer for θI = 0°. d, Current angle dependence of the SOT induced magnetization switching. The solid line is a cosine fit to the data. e, Current-induced magnetization switching with different θI.

Symmetry Dependent Field-free Switching of Perpendicular Magnetization

A magnetic tunnel junction (MTJ) is one of the most important device structures in spintronic applications. However, significant improvement in the tunnel magnetoresistance (TMR) ratio at room temperature (RT) is necessary to establish novel spintronic applications. Even in a simple Fe/MgO/Fe(001) MTJ, the experimental TMR ratios (180-220% at RT [1]) are much smaller than the theoretical values &gt;1,000% [2, 3]. In addition, the origin of TMR oscillation with the barrier thickness in experiments is still an open question [1]. In this study, we revisited Fe/MgO/Fe(001) based structures to bridge the discrepancies between experiments and theories. We deposited Fe/MgO/Fe(001) MTJs on a Cr-buffered MgO(001) single crystal substrate. By combining sputtering and electron-beam evaporation, we improved the barrier interface crystallinity [4]. We further examined an MgAl2O4 (MAO) based barrier to reduce the lattice mismatch with Fe. An MAO composition of Mg:Al = 4:1 (Mg rich) was selected. Fig. 1 shows TMR ratios of Fe/MgO/Fe and Fe/MAO/Fe MTJs as a function of the barrier thickness at RT. The maximum TMR ratio of Fe/MgO/Fe is 417% at RT (917% at 5 K) [4], which is ~twice (~3 times) of the previous report [1]. The TMR ratio shows significant oscillatory behavior with the MgO thickness. The peak-to-peak of 80% is also much larger (~7 times) than the previous report [1]. A TMR ratio of 496% at RT was observed by CoFe insertion at the bottom Fe/MgO interface [4]. The use of an MAO barrier improved the TMR ratio and oscillation peak-to-peak of Fe/MgO/Fe; the maximum ratio (peak-to-peak) reaches 429% (120%) in Fe/MAO/Fe at RT, which may be due to reduced defects at barrier interfaces. This demonstration will be an important step to achieve much larger TMR ratios for widening the range of spintronic applications. This study was partly supported by the ImPACT Program of the Council for Science, Technology and innovation (Cabinet Office, Government of Japan), JSPS KAKENHI 16H06332, 21H01750, 21H01397, and project JPNP16007, commissioned by the New Energy and Industrial Technology Development Organization (NEDO).

![](https://assets.underline.io/uploads/markdown_image/1/image/6f165c2448768fb7457ce2e234352b60.jpg)

Fig.1 TMR ratio as a function of barrier thickness of Fe/MgO/Fe and Fe/MgAlO/Fe at RT.

Giant tunnel magnetoresistance ratio and oscillation in Fe/MgO/Fe(001) and Fe/MgAlO/Fe(001) magnetic tunnel junctions

In a recent work [1], we investigated the spin transfer toque (STT) in epitaxial magnetic tunnel junctions (MTJs) based on ferrimagnetic tetragonal Mn3Ga [2] electrodes and MgO barrier (Fig.1a,b) from first principles ballistic non-collinear spin transport (NEGF + SDFT[LSDA]), using the Smeagol code [3]. We found a long-range spatial oscillation of the in-plane STT (defined as in Ref. [4]) decaying on a length scale of a few nm. We discussed how this STT oscillation can be anticipated from the bulk electronic structure of Mn3Ga and the spin-filtering properties of the MgO barrier, and it is robust against variations in the stack geometry. This is expected to result in a net in-plane torque which oscillates as a function of the Mn3Ga thickness, but stabilizes for sufficiently thick layers (&gt;40 MLs) at larger values than the net STT in conventional Fe/MgO/Fe MTJs [1]. We also discussed a novel all-ferrimagnetic MTJ in which the Fe electrode is substituted with Mn3Ga and found similar STT properties, particularly enhanced in the case of narrow MgO barrier (of 3 MLs), due to resonant interface states. Here we focus on the TMR effect in these MTJ stacks as a function of the applied bias voltage and find a maximum value of over 300%, asymmetry and multiple sign changes in the range ± 2V (Fig.1 c) for the Fe-based junction, in a noteworthy correspondence with experimental observations for similar ferrimagnetic junctions [5]. In the case of the fully symmetric novel ferrimagnetic MTJ, we find a symmetric TMR, peaking at 0 V. This is particularly enhanced for the thin barrier (3 ML) case due to the aforementioned interface resonances. We will discuss effects of geometry optimisation and possible Hubbard U corrections to the LSDA xc-functional on the transport properties, at computed bulk parameters of Mn3Ga closer to the experimental lattice constant and magnetic moments, as well as the role of different interface terminations. In view of the stronger TMR at finite bias for the Fe-based MTJs, our calculations suggest that further efforts are justified on the Fermi-level and interface engineering of ferrimagnetic electrodes, by means of both composition and substrate-induced strain, for demanding spintronic applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/68696b51c4439eecb133e13652ee671a.jpg)

High bias TMR in ferrimagnetic MTJs based on Mn3Ga

Emerging phenomena, such as the spin-Hall effect (SHE), spin pumping, and spin-transfer torque (STT), allow for interconversion between charge and spin currents and the generation of magnetization dynamics that could potentially lead to faster, denser, and more energy efficient, non-volatile memory and logic devices. Present STT-based devices rely on ferromagnetic materials as their active constituents. However, the flexibility offered by the intrinsic net magnetization and anisotropy for detecting and manipulating the magnetic state of ferromagnets also translates into limitations in terms of density (neighboring elements can couple through stray fields), speed (frequencies are limited to the GHz range), and frequency tunability (external magnetic fields needed). A new direction in the field of spintronics is to employ antiferromagnetic materials. In contrast to ferromagnets, where magnetic anisotropy dominates spin dynamics, in antiferromagnets spin dynamics are governed by the interatomic exchange interaction energies, which are orders of magnitude larger than the magnetic anisotropy energy, leading to the potential for ultrafast information processing and communication in the THz frequency range, with broadband frequency tunability without the need of external magnetic fields. I will present the first evidence of sub-terahertz coherent spin pumping at the interface of a uniaxial insulating antiferromagnet MnF2 and a platinum thin film, measured by the ISHE voltage signal arising from spin-charge conversion in the platinum layer. The ISHE signal depends on the chirality of the dynamical modes of the antiferromagnet, which is selectively excited and modulated by the handedness of the circularly polarized sub-THz irradiation. Contrary to the case of ferromagnets, antiferromagnetic spin pumping exhibits a sign dependence on the chirality of dynamical modes, allowing for the unambiguous distinction between coherent spin pumping and the thermally-driven spin Seebeck effect. Our results open the door to the controlled generation of coherent pure spin currents with antiferromagnets at unprecedented high frequencies.

Coherent Sub-Terahertz Spin Pumping from an Insulating Antiferromagnet

The spin current with out-of-plane spin polarization (sigma-z) is favorable for the efficient switching of perpendicular magnetized devices, which is required for high-density memories [1,2]. Recently we observed sigma-z both in non-collinear antiferromagnets Mn3Pt [3] and Mn3SnN [4], as well as in collinear antiferromagnets Mn2Au [5,6] and RuO2 [7] thin films. Strong out-of-plane polarized spin current is generated when the charge current is applied along the magnetic mirror plane of non-collinear antiferromagnet, but tiny when perpendicular. With the aid of strong sigma-z, current induced spin-orbit torque (SOT) switching of adjacent perpendicular ferromagnet is realized without external magnetic field. In Mn2Au films, sigma-z is generated at two spin sublattices with broken spatial symmetry, and the antiparallel antiferromagnetic moments play an important role. The out-of-plane polarized spin current can be controlled by the current- and electric field-induced Néel vector switching. Thus this Néel vector-dependent spin Hal effect is termed as “antiferromagnetic spin Hall effect”, which adds another twist to the atomic-level control of spin currents via the antiferromagnetic spin tructure. RuO2 is a tetragonal itinerant collinear antiferromagnet with nonmagnetic atoms at noncentrosymmetric positions [8]. This particular crystal structure produces anisotropic band structure，enabling RuO2 a efficient spin source due to the spin splitting effect. Spin current generated by spin splitting effect do not meet conventional “cross product relationship”. The spin current flow direction is related to the crystal structure，while the spin polarization direction is related to the Néel vector. Spin splitting effect has more efficient spin-charge convention and more controllable spin polarization. The role of sigma-z for the field-free SOT switching is theoretically analyzed, and different roles of sigma-z and conventional in-plane spin polarization (sigma-y) are presented with different external magnetic field. Our findings would help to use out-of-plane polarization for efficient SOT-MRAM to combine the advantage of SOT and spin-transfer torque.

![](https://assets.underline.io/uploads/markdown_image/1/image/2cd5af1ecc73195bd6fc92b4627ec76c.jpg)

Fig. 1. Schematic of out-of-plane polarized spin current and field-free spin-orbit torque switching.

Out-of-plane spin polarization and antiferromagnetic spin Hall effect

The interplay between electronic transport and the antiferromagnetic order has attracted a surge of interest. Recent studies have shown that a moderate change in the spin orientation of a collinear antiferromagnet may significantly affect the electronic band structure and subsequently manifest itself in transport properties [1]. Among numerous electrical probes to read out such magnetic order, unidirectional magnetoresistance (UMR), where the resistance changes under the reversal of the current direction, can provide rich insights into the transport properties of spin-orbit coupled systems. However, UMR has never been observed in antiferromagnets before, given the absence of intrinsic spin-dependent scattering. Here, we report that a sizable UMR can emerge in an antiferromagnetic system, specifically in a collinear-antiferromagnet/heavy-metal bilayer. The observed UMR in the antiferromagnetic phase of a FeRh/Pt bilayer evolves nonlinearly with increasing magnetic field, persists to large values at high magnetic field and, most notably, exhibits a sign change at a critical field, which is in stark contrast with the UMRs observed in ferromagnetic and non-magnetic systems [2-3]. We attribute this UMR to the strong effective magnetic field due to spin canting in FeRh. Our results can motivate the growing fields of non-centrosymmetric and topological systems, and suggest a route to the development of tunable antiferromagnet-based spintronics devices. This research was primarily supported by the NSF through the University of Illinois at Urbana-Champaign Materials Research Science and Engineering Center DMR-1720633 and was carried out in part in the Materials Research Laboratory Central Research Facilities, University of Illinois. Thin-film growth at Argonne National Laboratory was supported by the U.S. Department of Energy, Office of Science, Materials Science and Engineering Division.

![](https://assets.underline.io/uploads/markdown_image/1/image/756eaf87bcd78eaae92bc5c59ee591d2.jpg)

Fig. 1: UMR in the antiferromagnetic phase of FeRh [110]/Pt (T = 10 K), as a function of field B at various current densities J.

![](https://assets.underline.io/uploads/markdown_image/1/image/71626e5a0aa513b3c7c5af64e05003bc.jpg)

Fig 2: Generation of a nonlinear charge current transverse to the applied B field (top left), which increases considerably in the presence of canting (bottom right).

Observation of unidirectional magnetoresistance in collinear-antiferromagnet/heavy-metal bilayers

Antiferromagnetic (AF) spintronics received an enormous impulse after the first demonstrations of electrical switching of the AF order [1]. However, a lot of these observations can equally well be explained by non-magnetic, parasitic effects caused by the same electrical current pulses that are intended to manipulate the magnetic state of the AF [2]. While it is possible to distinguish between magnetic and non-magnetic effects using imaging techniques that can resolve the AF order [3], these techniques are usually very complex. Moreover, since they require access to the AF layer, they are not suitable for systems where the AF is buried beneath other layers. Our approach is to distinguish between the magnetic and non-magnetic effects by suppressing the magnetic effects using high magnetic fields. In this contribution, we study the high magnetic field behavior of the electrical switching of thin-film AFs NiO and CoO, using the devices shown in Fig. 1. We perform switching experiments in magnetic fields up to 16 Tesla (Fig. 2). The measurements show that these high magnetic fields indeed suppress the current-induced switching of the AF order, whereas the thermal effects remain unaffected. Hence, this technique allows to irrefutably separate the magnetic and non-magnetic effects. These results are corroborated by a model accounting for the multi-domain character of the AF. Finally, the experiments with CoO yield an unexpected dependence of the magnetic configuration on the direction of the high magnetic fields. This indicates an unexpectedly complex magnetic anisotropy in CoO that depends on the magnetic field strength and orientation, most likely due to unquenched orbital momentum. These results both demonstrate that combining electrical methods with strong magnetic fields can be a valuable tool for AF spintronics, and give invaluable insight into the interplay of spin-Hall magnetoresistance and thermal effects as well as the domain structure and complex anisotropy of thin-film antiferromagnets.

![](https://assets.underline.io/uploads/markdown_image/1/image/1fb212d0af0688a1f81826c7b8bf2f9a.jpg)

Fig. 1: Micrograph of the device used for the experiments.

![](https://assets.underline.io/uploads/markdown_image/1/image/e89ac11416c899724e28149f7ee39cf4.jpg)

Fig. 2: Switching amplitude as a function of magnetic field. The lines are calculated with the model; the pie charts represent the modeled domain distribution.

Suppressing Electrical Switching of Antiferromagnets with High Magnetic Fields

The proximity-coupling of a chiral non-collinear antiferromagnet (AFM)1,2 with a singlet superconductor allows spin-unpolarized singlet Cooper pairs to be converted into spin-polarized triplet pairs3,4, thereby enabling non-dissipative, long-range spin correlations3,4. The mechanism of this conversion derives from fictitious magnetic fields that are created by a non-zero Berry phase5 in AFMs with non-collinear atomic-scale spin arrangements1,2. In this talk, I would like to describe our recent achievement of long-ranged lateral Josephson supercurrents through an epitaxial thin film of the triangular chiral AFM Mn3Ge6. The Josephson supercurrents in this chiral AFM decay by approximately 1–2 orders of magnitude slower than would be expected for singlet pair correlations3,4 and their response to an external magnetic field reflects a clear spatial quantum interference. Given the long-range supercurrents present in both single- and mixed-phase Mn3Ge, but absent in a collinear AFM IrMn, our results pave a way for the topological generation of spin-polarized triplet pairs3,4 via Berry phase engineering5 of the chiral AFMs.

Long-range supercurrents through a chiral non-collinear antiferromagnet in lateral Josephson junctions

Antiferromagnets are at the forefront of research in spintronics. Recent results demonstrate their high potential for revolutionizing memory technologies. However, many of the underlying phenomena remain to be explored. Here, we study the domain structure and magnetic reversal of the canted antiferromagnet α-Fe2O3. Using x-ray magnetic linear dichroism (XMLD) measurements, we observe a strongly field-dependent domain structure of α-Fe2O3 (see Fig. 1a). To study the magnetic field control of the Néel order and demagnetization process in more detail, we employ spin Hall magnetoresistance (SMR) measurements in α-Fe2O3/Pt bilayer devices. The angle-dependent SMR shows the characteristic saturated negative SMR signal of antiferromagnets (light red curve in Fig. 1b) in agreement with previous findings1,2. The dark red curve shows the remanent resistance as a function of the angle of the preceding saturating magnetic field. The data show a simple scaling between the saturated SMR signal and the remanent resistance by a factor of 5. This implies that the internal fields driving the demagnetization do not follow the crystal symmetry of α-Fe2O3. Furthermore, the Dzyaloshinskii-Moriya interaction (DMI) induces canting of the two magnetic sublattices giving rise to a ferromagnetic moment at room temperature. The ferromagnetic moment lifts the degeneracy of the sublattices and consequently the orientation of the Néel vector with respect to the applied magnetic field resulting in hysteresis in the SMR signal. The irreversibility upon switching the direction of the magnetic field elucidates the nature and formation of antiferromagnetic as well as ferromagnetic domain walls at low magnetic fields. The insights gained from our work serve as a foundation for further studies of electrical and optical manipulation of the domain structure of canted antiferromagnets.

![](https://assets.underline.io/uploads/markdown_image/1/image/d3101936d30b393ec81080a0faba3817.jpg)

Fig. 1: (a) XMLD image of the domain structure of α-Fe2O3 in zero magnetic field (left) and in 250 mT in-plane magnetic field (right). (b) Transverse remanent SMR signal Rrem (dark red) of α-Fe2O3/Pt Hall cross after H has been reduced from a saturated state Rsat (light red) to zero field as a function of angle of the applied magnetic field.

Roles of Destressing and Dzyaloshinskii-Moriya Interaction for Magnetic Reversal in α-Fe2O3

A magnetization m is the well-known hallmark of magnetic order in ferromagnets. A counterpart to m describing the magnetic order in antiferromagnets has conspicuously remained absent in the world of magnetism. We derive a Néel operator τ that describes collinear antiferromagnetic order, in the same way a magnetization m signals ferromagnetic order [1]. While m is even under space inversion and odd under time reversal, τ describes a toroidal moment that is odd under both symmetries. Thus m and τ quantify complementary aspects of magnetic order in solids. Orbital ferromagnetic order can be attributed to dipolar equilibrium currents that give rise to a magnetization. We show that, in the same way, antiferromagnetic order arises from quadrupolar equilibrium currents that generate the toroidal moment τ. Considering an antiferromagnetic diamond structure as an example (Fig. 1), we derive explicit analytical expressions for the Néel operator τ and the effects of AFM order on the itinerant electrons in this system [1]. As an example for the efficacy of our approach, we develop a unified theory for magnetoelectricity in quasi-2D paramagnets, ferromagnets and antiferromagnets. In magnetoelectric media, an electric field can induce a magnetization and a magnetic field can induce a polarization, while the system remains in thermal equilibrium [2,3]. This effect appeals to recent efforts to manipulate magnetic order electrically [4]. It requires that both space-inversion and time-reversal symmetry are broken. We present a comprehensive theory [1] for magnetoelectricity in quasi-2D systems, where antiferromagnetic order and toroidal moments play a central role. We obtain quantitative expressions for the magnetoelectric responses due to electric and magnetic fields that reveal explicitly, for the first time, the inherent duality of these responses required by thermodynamics.

![](https://assets.underline.io/uploads/markdown_image/1/image/fa70f074e078d7f874157bfe833bf03c.jpg)

Figure 1: The antiferromagnetic diamond structure breaks time-reversal symmetry θ and inversion symmetry I (though the joint operation θI remains a good symmetry).

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