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Introduction&lt;br/&gt;Multiferroic materials have been attracting much attention because they have multiple ordered (ferroelastic, ferroelectric, and ferromagnetic) states. In particular, magnetic and ferroelectric material has a property that magnetization &lt;i&gt;M&lt;/i&gt; (electric polarization &lt;i&gt;P&lt;/i&gt;) can be controlled by electric &lt;i&gt;E&lt;/i&gt; (magnetic &lt;i&gt;H&lt;/i&gt;) field. Such property is called “magnetoelectric (ME) effect”. In magnetic and ferroelectric material, there is also a possibility of observing an optical response that the rate of absorption is different between the cases that light enters from one direction and the opposite direction. Such a large nonreciprocal response in the visible light region was discovered in CuB&lt;sub&gt;2&lt;/sub&gt;O&lt;sub&gt;4 &lt;/sub&gt;[1]. The observed nonreciprocal response in light is thought to be originated from dynamical AC ME response.&lt;br/&gt;The present compounds, CaBa&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; (&lt;i&gt;M&lt;/i&gt;=Co, Fe), have been attracting interest for their nontrivial magnetism because of the geometrically frustrated crystal structure consisting of triangular and Kagome layers. From the viewpoints of the studies of multiferroics, we have expected that the magnetic ordering couples with the spontaneous electric polarization &lt;i&gt;P&lt;/i&gt; due to the crystal structure with a polar point-group of &lt;i&gt;mm&lt;/i&gt;2. Indeed, we have observed the change of &lt;i&gt;P&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt; induced by the weak ferromagnetic ordering in CaBaCo&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; [2]. Then the magnetic-field &lt;i&gt;H&lt;sub&gt;ab&lt;/sub&gt;&lt;/i&gt;-induced giant variation in &lt;i&gt;P&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt; was reported and the magnetostriction mechanism was proposed to explain the observed giant ME effect [3]. Also in the case of CaBaFe&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt;, the ME coefficient &lt;i&gt;&lt;sup&gt;H&lt;/sup&gt;α&lt;sub&gt;cc&lt;/sub&gt;&lt;/i&gt; defined as &lt;i&gt;△P&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;=&lt;i&gt;&lt;sup&gt;H&lt;/sup&gt;α&lt;sub&gt;cc&lt;/sub&gt;&lt;/i&gt;&lt;i&gt;H&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt; with a maximum value of 39 ps/m was reported for the measurements of &lt;i&gt;H&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;-induced change of &lt;i&gt;P&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt; at 80 K below the ferrimagnetic transition temperature of 275 K [4].&lt;br/&gt;In this work, we have systematically investigated nonreciprocal DC ME response, i.e., crystalline (polar) direction dependence of ME coefficients for CaBa&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; (&lt;i&gt;M&lt;/i&gt;=Co, Fe) single crystals. Especially, we focused on the electric-field (&lt;i&gt;E&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;)-induced magnetization (&lt;i&gt;M&lt;sub&gt;ab&lt;/sub&gt;&lt;/i&gt; or&lt;i&gt; M&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;) in CaBa&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; (&lt;i&gt;M&lt;/i&gt;=Co, Fe), which had almost not yet been performed.&lt;br/&gt;&lt;br/&gt;Results and Discussion&lt;br/&gt;Figure 1 shows the Schematic crystal structure of CaBa&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; (&lt;i&gt;M&lt;/i&gt; = Co, Fe). The space group of CaBaCo&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; at room temperature is orthorhombic &lt;i&gt;Pbn&lt;/i&gt;2&lt;sub&gt;1&lt;/sub&gt;, which is distorted and lowered symmetry from the higher one of hexagonal &lt;i&gt;P&lt;/i&gt;6&lt;sub&gt;3&lt;/sub&gt;&lt;i&gt;mc&lt;/i&gt;. Its (magnetic) point group is (&lt;i&gt;m&lt;/i&gt;’&lt;i&gt;m&lt;/i&gt;2’) &lt;i&gt;mm&lt;/i&gt;2. Those of CaBaFe&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; are &lt;i&gt;Pbn&lt;/i&gt;2&lt;sub&gt;1&lt;/sub&gt; and &lt;i&gt;mm&lt;/i&gt;2, which are the same as CaBaCo&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; [2,4]. Its magnetic point group is unknown yet. Both materials have quasi-triangular lattices and quasi-Kagome ones stacked along the &lt;i&gt;c&lt;/i&gt;-axis direction. There exists multidomain structure within the orthorhombic &lt;i&gt;ab&lt;/i&gt; plane in spite of no such one in the &lt;i&gt;c&lt;/i&gt; axis. A spontaneous electric polarization along the &lt;i&gt;c&lt;/i&gt; axis (&lt;i&gt;P&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;) is expected from the polar point group of &lt;i&gt;mm&lt;/i&gt;2.&lt;br/&gt;Figure 2(i) shows the electric field dependence of the magnetization of CaBaCo&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; in constant magnetic fields. The magnetic field (&lt;i&gt;H&lt;sub&gt;ab&lt;/sub&gt;&lt;/i&gt;) was applied within the magnetic easy &lt;i&gt;ab&lt;/i&gt; plane perpendicular to the &lt;i&gt;c&lt;/i&gt; axis, and the electric field (&lt;i&gt;E&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;) was applied along the &lt;i&gt;c&lt;/i&gt; axis which is the direction of spontaneous electric polarization (&lt;i&gt;P&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;). The configuration among the external magnetic field, electric field, and crystalline direction (spontaneous electric polarizaion) is presented in the inset of Fig. 2(ii). When the electric field is applied in the parallel direction (p) to the crystalline direction, the magnetization increases as the electric field increases. Such trends can be clearly seen as solid symbols in Fig. 2(i). Conversely, in the antiparallel (a) case, the magnetization decreases with electric field (See open symbols).&lt;br/&gt;Figure 2(ii) shows the magnetic field dependence of &lt;i&gt;&lt;sup&gt;E&lt;/sup&gt;α&lt;/i&gt;&lt;sub&gt;32&lt;/sub&gt; which is a component of linear ME tensor and defined as &lt;i&gt;M&lt;sub&gt;ab&lt;/sub&gt;&lt;/i&gt;≡&lt;i&gt;&lt;sup&gt;E&lt;/sup&gt;α&lt;/i&gt;&lt;sub&gt;32&lt;/sub&gt;&lt;i&gt;E&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;. Depending on the configurations between the electric field and the crystal direction, &lt;i&gt;&lt;sup&gt;E&lt;/sup&gt;α&lt;/i&gt;&lt;sub&gt;32&lt;/sub&gt; shows opposite signs. The observed crystalline-direction (polarity) dependence of change in magnetization can be regarded as a DC nonreciprocal response. As is well known for the diode, nonreciprocal response is indispensable in device applications. Therefore, we explore the novel nonreciprocal phenomena as mentioned above in the present polar magnetic compounds, CaBa&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; (&lt;i&gt;M&lt;/i&gt; = Co, Fe), for the future application such as “thermal diode”.

**References:**

[1] M.Saito &lt;i&gt;et al.&lt;/i&gt;, J. Phys. Soc. Jpn. 77, 013705 (2008).&lt;br/&gt;[2] H. Iwamoto &lt;i&gt;et al.&lt;/i&gt;, J. Phys: Conf. Ser. 400, 032031 (2012).&lt;br/&gt;[3] V. Caignaert &lt;i&gt;et al.&lt;/i&gt;, Phys. Rev. B 88, 174403 (2013).&lt;br/&gt;[4] V. Kocsis &lt;i&gt;et al.&lt;/i&gt;, Phys. Rev. B 93, 014444 (2016).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ae2bea3298433e5c1ccf14561452cfc9.jpg)
Schematic crystal structure of CaBa&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt; (&lt;i&gt;M&lt;/i&gt; = Co, Fe) projected onto the &lt;i&gt;bc&lt;/i&gt; plane (a). Its quasi-triangular (b) and quasi-Kagome layers (c) are projected onto the &lt;i&gt;ab&lt;/i&gt; plane. The rectangular boxes represent the boundaries of unit cell [2].

![](https://assets.underline.io/uploads/markdown_image/1/image/0c8e519eab7cbb2fc00b2b326156b7ec.jpg)
(i) Electric field dependence of magnetization in CaBaCo&lt;sub&gt;4&lt;/sub&gt;O&lt;sub&gt;7&lt;/sub&gt;. Solid symbols correspond to the parallel configuration between the electric field and the crystalline (polar) direction, and open ones to the antiparallel configuration. (ii) Magnetic field dependence of &lt;i&gt;&lt;sup&gt;E&lt;/sup&gt;α&lt;/i&gt;&lt;sub&gt;32&lt;/sub&gt; (≡&lt;i&gt;M&lt;sub&gt;ab&lt;/sub&gt;&lt;/i&gt;/&lt;i&gt;E&lt;sub&gt;c&lt;/sub&gt;&lt;/i&gt;) deduced from the data shown in (i). The inset is a schematic diagram of the configuration among the external magnetic/electric fields and the crystal direction.

INTERMAG 2021

 Polarity dependence of change in electric polarization (magnetization) induced by magnetic (electric) fields in CaBaM4O7(M=Co, Fe) single crystals

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Introduction Multiferroic materials have been attracting much attention because they have multiple ordered (ferroelastic, ferroelectric, and ferromagnetic) states. In particular, magnetic and ferroelectric material has a property that magnetization M (electric polarization P) can be controlled by electric E (magnetic H) field. Such property is called “magnetoelectric (ME) effect”. In magnetic and ferroelectric material, there is also a possibility of observing an optical response that the rate of absorption is different between the cases that light enters from one direction and the opposite direction. Such a large nonreciprocal response in the visible light region was discovered in CuB2O4 [1]. The observed nonreciprocal response in light is thought to be originated from dynamical AC ME response. The present compounds, CaBaM4O7 (M=Co, Fe), have been attracting interest for their nontrivial magnetism because of the geometrically frustrated crystal structure consisting of triangular and Kagome layers. From the viewpoints of the studies of multiferroics, we have expected that the magnetic ordering couples with the spontaneous electric polarization P due to the crystal structure with a polar point-group of mm2. Indeed, we have observed the change of Pc induced by the weak ferromagnetic ordering in CaBaCo4O7 [2]. Then the magnetic-field Hab-induced giant variation in Pc was reported and the magnetostriction mechanism was proposed to explain the observed giant ME effect [3]. Also in the case of CaBaFe4O7, the ME coefficient Hαcc defined as △Pc=HαccHc with a maximum value of 39 ps/m was reported for the measurements of Hc-induced change of Pc at 80 K below the ferrimagnetic transition temperature of 275 K [4]. In this work, we have systematically investigated nonreciprocal DC ME response, i.e., crystalline (polar) direction dependence of ME coefficients for CaBaM4O7 (M=Co, Fe) single crystals. Especially, we focused on the electric-field (Ec)-induced magnetization (Mab or Mc) in CaBaM4O7 (M=Co, Fe), which had almost not yet been performed. Results and Discussion Figure 1 shows the Schematic crystal structure of CaBaM4O7 (M = Co, Fe). The space group of CaBaCo4O7 at room temperature is orthorhombic Pbn21, which is distorted and lowered symmetry from the higher one of hexagonal P63mc. Its (magnetic) point group is (m’m2’) mm2. Those of CaBaFe4O7 are Pbn21 and mm2, which are the same as CaBaCo4O7 [2,4]. Its magnetic point group is unknown yet. Both materials have quasi-triangular lattices and quasi-Kagome ones stacked along the c-axis direction. There exists multidomain structure within the orthorhombic ab plane in spite of no such one in the c axis. A spontaneous electric polarization along the c axis (Pc) is expected from the polar point group of mm2. Figure 2(i) shows the electric field dependence of the magnetization of CaBaCo4O7 in constant magnetic fields. The magnetic field (Hab) was applied within the magnetic easy ab plane perpendicular to the c axis, and the electric field (Ec) was applied along the c axis which is the direction of spontaneous electric polarization (Pc). The configuration among the external magnetic field, electric field, and crystalline direction (spontaneous electric polarizaion) is presented in the inset of Fig. 2(ii). When the electric field is applied in the parallel direction (p) to the crystalline direction, the magnetization increases as the electric field increases. Such trends can be clearly seen as solid symbols in Fig. 2(i). Conversely, in the antiparallel (a) case, the magnetization decreases with electric field (See open symbols). Figure 2(ii) shows the magnetic field dependence of Eα32 which is a component of linear ME tensor and defined as Mab≡Eα32Ec. Depending on the configurations between the electric field and the crystal direction, Eα32 shows opposite signs. The observed crystalline-direction (polarity) dependence of change in magnetization can be regarded as a DC nonreciprocal response. As is well known for the diode, nonreciprocal response is indispensable in device applications. Therefore, we explore the novel nonreciprocal phenomena as mentioned above in the present polar magnetic compounds, CaBaM4O7 (M = Co, Fe), for the future application such as “thermal diode”.

**References:**

[1] M.Saito et al., J. Phys. Soc. Jpn. 77, 013705 (2008). [2] H. Iwamoto et al., J. Phys: Conf. Ser. 400, 032031 (2012). [3] V. Caignaert et al., Phys. Rev. B 88, 174403 (2013). [4] V. Kocsis et al., Phys. Rev. B 93, 014444 (2016).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ae2bea3298433e5c1ccf14561452cfc9.jpg)
Schematic crystal structure of CaBaM4O7 (M = Co, Fe) projected onto the bc plane (a). Its quasi-triangular (b) and quasi-Kagome layers (c) are projected onto the ab plane. The rectangular boxes represent the boundaries of unit cell [2].

![](https://assets.underline.io/uploads/markdown_image/1/image/0c8e519eab7cbb2fc00b2b326156b7ec.jpg)
(i) Electric field dependence of magnetization in CaBaCo4O7. Solid symbols correspond to the parallel configuration between the electric field and the crystalline (polar) direction, and open ones to the antiparallel configuration. (ii) Magnetic field dependence of Eα32 (≡Mab/Ec) deduced from the data shown in (i). The inset is a schematic diagram of the configuration among the external magnetic/electric fields and the crystal direction.

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Artificial spin ice (ASI) is a magnetic metamaterial designed to exhibit frustration [1]. It consists of an array of magnetostatically interacting nanomagnets arranged in a geometry that prevents these interactions to be all satisfied simultaneously. They were originally conceived to replicate, in an artificial manner, interesting frustration induced phenomena found in natural frustrated magnets known as spin ices [2]. One of those most remarkable phenomena is the emergence of quasiparticles similar to magnetic monopoles when such materials are excited above the ground state [3]. In ASI, this behavior is experimentally reproduced by subjecting the system to a magnetization cycle [4]. The magnetization reversal proceeds by creation of one-dimensional strings of flipped nanomagnets, referred as Dirac strings, that host the magnetic monopoles at their ends. This reversal mechanism has been already reproduced by Monte Carlo-based simulations, where the nanomagnets are treated as Ising variables and a phenomenological random switching field distribution is employed [5]. In this work, we reproduced the same behavior using a more fundamental approach, that was lacking in literature, based on micromagnetic simulations, applied particularly to a large scale (more than a thousand magnets) kagome lattice arrangement of ASI [6]. We regarded a more realistic description of the shape of each nanomagnet, including its finite size and roughness at the edges, in such a way that no a priori switching field distribution was required but that came out naturally. Furthermore, our simulations predict a critical angle between the applied field and the direction of a sublattice that separates different reversal regimes: the process is mostly 2D for angles above the critical value, and it is 1D (i.e., via Dirac strings and monopoles) otherwise.

**References:**

[1] S. H. Skjærvø, C. H. Marrows and R. L. Stamps, Nature Review Physics, Vol. 2, p. 13–28 (2020) [2] R. F. Wang, C. Nisoli and R. S. Freitas, Nature, Vol. 439, pp. 303–306 (2006) [3] C. Castelnovo, R. Moessner and S. L. Sondhi, Nature, Vol. 451, p. 42–45 (2008) [4] E. Mengotti, L. J. Heyderman and A. F. Rodríguez, Nature Physics, Vol. 7, p. 68–74 (2011) [5] R. V. Hügli G., G. Duff and B. O'Conchuir, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci., Vol. 370, p. 5767–5782 (2012 [6] M. F. Velo, B. M. Cecchi, and K. R. Pirota, Physical Review B, Vol. 102, p. 224420 (2020)

**Images:**

 Micromagnetic simulations of magnetization reversal in kagome artificial spin ice

Frustrated magnetic structures hold configurations wherein localized magnetic moments or spins interact through competing exchange interactions, rendering macroscopic degeneracy of exotic ground states like spin ice and providing an excellent platform for discovering novel states and properties. One example is the pyrochlore spin ice, in which four nearest-neighbor Ising-like spins on the vertices of the corner-sharing tetrahedra are forced to form the local “2-in-2-out” configuration for each tetrahedron, which forms spin ice in analogy to the water ice[1]. In two dimensions (2D), ice rules can be similarly defined for in-plane Ising like classical spins residing on the kagome lattice, which require “2-in-1-out” or “1-in-2-out” local arrangements of the spins on its triangles[2]. A √3×√3 ground state can be selected by further neighbor exchange couplings or the long-range dipolar interaction. As a result, kagome spin ices show a characteristic multi-stage ordering behavior under changing temperature. Experimentally, kagome spin ices have only been realized in artificial spin ice systems formed by nanorods of ferromagnets organized into honeycomb networks[3-5]. However, the large magnetic energy scales and system sizes make it challenging to explore the rich phase diagram of spin ices in the thermodynamic limit. Neutron scattering, though measuring the energy and momentum of in-coming and out-going neutrons, offers one of the most powerful tools in the study of magnetic structures and dynamics of magnets. Through neutron scattering and other experimental and theoretical approaches, we discover that the intermetallic compound HoAgGe is a naturally existing kagome spin ice that exhibits a fully ordered ground state[6]. The multi-stage ordering behavior was confirmed by Single-crystal neutron diffraction: below 11.6 K the Ho spins enter a partially ordered state with one-third of them still fluctuating, while below ~7 K, all spins order into the √3×√3 ground state of the kagome ice with crystalized magnetic monopoles. All these observations are in good agreement with Monte Carlo simulations of a classical spin model including both dipolar interactions and up to 4th nearest-neighbor exchange couplings on the noncentrosymmetric kagome lattice of HoAgGe.

**References:**

[1] Bramwell, S. T., and Gingras, M. J. P. Science 294, 1495–1501 (2001) [2] Chern, G.-W. et al. Phys. Rev. Lett. 106, 207202 (2011) [3] A. S. Wills, R. Ballou and C. Lacroix, Phys. Rev. B 66, 144407 (2002) [4]G. Möller, R. Moessner, Phys. Rev. B 80, 140409 (2009) [5]G.-W. Chern, P. Mellado and O. Tchernyshyov, Phys. Rev. Lett. 106, 207202 (2011) [6] Zhao, K., Deng, H., Chen, H. et al. Science 367, 1218–1223 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d4ab6a1eaf76a141378b86c81da328f8.jpg)
The refined magnetic structures of HoAgGe at 10K and 4K

 Neutron Scattering Revealing Kagome Spin Ice in a Frustrated Intermetallic Compound

A particular interest to the behavior of magnetic frustrated spin systems is explained that the problem of finding a ground state, i.e., the state with the lowest energy is a very difficult task due to the macroscopic degeneracy and frustrations that are present in most realistic spin glass models. This problem is fundamentally difficult to solve since it often requires the search for potential solutions over huge spaces of possible states. Among conventional methods which are using in statistical physics, one of the most powerful approaches is the Monte Carlo simulation, consisting of stochastic sampling over a state space and an evaluation of estimators for physical quantities. However, the ability of modern machine learning techniques to classify, identify, or interpret massive data sets provides a complementary paradigm to the above approach in statistical physics to analyze the exponentially large number of states. The main research object is the two-dimensional Edwards-Anderson model, with the exchange integral J as a random function and the average value of J is zero in a system. In such the system, one half of the spins' interaction is ferromagnetic, and the other is antiferromagnetic. It is well known that critical phenomena occur in condensed matter under certain conditions, when an abrupt change in its properties occurs. Near the critical points, various phenomena may arise. The critical region can be described by a set of state parameters (order parameters), which allow one to obtain information about the anomalous behavior of thermodynamic averages, internal processes, and the nature of the objects of study. The first research direction related to the fact, that the temperature behavior of the average magnetization modulus equals to zero for the Edwards-Anderson model. Because of this, we calculated an average size of the percolation cluster, which is defined as the relative size of maximum cluster, which include spins in the ground state, to the total number of spins. The average size of the percolation was chosen as an order parameter. This order parameter, in contrast to other thermodynamic characteristics, makes it possible to describe the behavior of the system with a change in temperature. Using this parameter, we calculated main critical exponents of these two models: correlation length, magnetization, specific heat and susceptibility. The second research direction was about the problem of finding a ground state for the Edwards-Anderson model. In present work, for simulation we used a combination of our Hybrid Multispin Method (HMM) and the Restricted Boltzmann Machine (RBM) to predict the ground states for the Edwards–Anderson spin glass model. The problem of spin glass simulation is determined by the high complexity of obtaining configurations with the lowest possible energy level (ground state). At the moment, no universal algorithm exists to directly generate the ground state. To predict the GS, we used the data of the HMM to train our neural network and to predict spin glass state with a lower energy level than in the training data sets. We used small spin systems (4 x 4 and 6 x 6) to verify our results and compare them with those of the exact solution calculated by a brute-force method (BFM). Our research has shown that the ground states of spin glass systems can be predicted using a neural network. However, given the absence of an exact solution to determine the number of degenerations at the ground energy level for big systems, it is impossible to check whether our algorithms reach the global energy minimum. Nonetheless, machine-learning algorithms can make a significant contribution to solving NP-complete problems in the future. This work was supported by the state task of the MSHE of Russia No. 075-00400-19-01.

**References:**

S. Edwards, P. Anderson, Phys. F: Metal Phys., Vol. 5 (5), p. 965 (1975) A. Patashinski, V. Pokrovskii, Pergamon Press (1979) A.G. Makarov et al., JETP Letters., Vol. 110 (10), pp. 702-706 (2019) R. Salakhutdinov, A. Mnih, G. Hinton, Restricted Boltzmann machines for collaborative filtering //Proceedings of the 24th international conference on Machine learning, pp. 791-798 (2007)

**Images:**

 Numerical Studies of Spin Glasses in the Frame of Edwards-Anderson Model

In our information society, the importance of energy conservation is increasing year by year. To tackle this issue, the transition from volatile memory to non-volatile memory can be a solution. One of the candidates for non-volatile memories is a racetrack memory[1], driven by current-induced domain wall motion (CIDWM). For practical use, faster CIDWM and lower threshold current density are key factors. In order to reach this goal, we have been focusing on and investigating Mn4N as a promising material. Mn4N film is an antiperovskite ferrimagnet without rare-earth elements, which is advantageous in fast magnetization reversal due to its perpendicular magnetic anisotropy (Ku ~ 1.1×105 J/m3) and a small saturation magnetization (MS ~ 80 kA/m)[2]. Previous study on 1-2μm-wide Mn4N strips showed the fastest spin-transfer-torque-driven domain wall motion (vDW ~ 900 m/s at 1.3×1012 A/m2) at room temperature (RT)[2], comparable to those reported in the system including rare-earth 4f magnets or heavy metals. To achieve faster vDW, Mn4N based mixed crystals have been studied in pursuit of the use of magnetic and/or angular momentum compensation for more efficient CIDWM thanks to diverged damping constants. Recently, Mn4-xNixN films have been suggested to have a magnetic compensation (MC) point between x = 0.1 and 0.25 at RT[3]. We found that Mn4-xCoxN films have a compensation point between x = 0 and 0.8 from x-ray magnetic circular dichroism (XMCD) measurements[4]. However, there is a lack of information about the magnetic behavior of Mn4-xCoxN at values x much smaller or larger than 0.8. Considering that the compensation in Mn4-xNixN takes place in a small range of composition x, further compensation points can be found when x is far from 0.8. In this work, we performed XMCD measurements on Mn4-xCoxN epitaxial films at x = 0.2 and 1.3 and investigated the change in magnetic structures by composition ratio to verify MC at RT. 20-30 nm-thick Mn4-xCoxN films with x = 0-1.3 were epitaxially grown on SrTiO3(001) substrates by molecular beam epitaxy. SiO2 or Ta capping layers were sputtered in-situ on the surface to prevent oxidation. X-ray absorption spectroscopy (XAS) and XMCD measurements were performed at BL-16A of KEK-PF for x = 0.2 and 1.3. In these measurements, we applied an external magnetic field of 5 T perpendicular to sample surfaces. The incidence angle of the circularly polarized x-ray was 54.7°(magic angle) to the plane in order to simplify the sum rule calculation. For x = 0.8, the XMCD measurements were carried out at BL23SU of SPring-8[4]. Figures 1(a)-1(c) show the XAS and XMCD spectra of Mn in Mn4-xCoxN at x = 0.2, 0.8[4], and 1.3, respectively. In these figures, the sharp peak near 640 eV comes from Mn atoms at corner sites (I sites), and the broad peak near 643 eV originates from those at face-centered sites (II sites)[4]. We observed the sign reversals of XMCD signals between x = 0.2 and 0.8, and also between x = 0.8 and 1.3. Similar sign reversals were also observed in the XMCD signals of Co atoms. These results indicate that MC occurs twice in the range of x = 0-1.3 in Mn4-xCoxN. Besides, the XAS spectra of Co indicate that Co preferentially occupied the corner sites. We also calculated the mean magnetic moment of Co by using the sum rule analysis. Figure 2 shows the expected magnetic structures derived from the XMCD measurements. Around x = 0.2, Co atoms preferentially replaced Mn(I) and thus the total magnetic moment of I site and that of face-centered site (II site) became closer. With further increasing x, the total magnetic moment became zero. In this manner, the first MC occurred between x = 0.2 and 0.8. After the first MC, the total magnetic moment of II sites became larger and this led to the reversal of magnetic moments of all atoms in order to minimize the Zeeman energy. Above x = 0.8, Co atoms gradually got to occupy II sites and thus the second MC occurs. As far as we investigated, Mn4-xCoxN is the only material which has two MC points at RT. This property can be useful for spintronic devices, for example, the use of compensation in a wide range of composition. The XMCD experiment for x = 0.2 and 1.3 was performed with the approval of the Photon Factory Program Advisory Committee (Proposal No. 2019G574).

**References:**

[1] S. Parkin and S. Yang, Nat. Nanotechnol., Vol.10, p.195 (2015) [2] T. Gushi et al., Nano Lett., Vol.19, p.8716 (2019) [3] T. Komori et al., JAP, Vol.127, p.043903 (2020) [4] K. Ito et al., Phys. Rev. B, Vol.101, p104401 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/afcad013ae3441a446c71d093ae63afb.jpg)
Fig. 1 XAS (blue) and XMCD (black) spectra in (a) Mn3.8Co0.2N, (b) Mn3.2Co0.8N[3], and (c) Mn2.7Co1.3N films at the Mn L2,3 absorption edges.

![](https://assets.underline.io/uploads/markdown_image/1/image/3f59e17fa54814000208a461221f56c6.jpg)
Fig. 2 Schematic of the expected magnetic structures of Mn4-xCoxN.

 Two magnetic compensation compositions in Mn4-xCoxN epitaxial films at room temperature proved by X-ray magnetic circular dichroism

Since the seminal papers of Jones and Varma [1,2], the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction Y between two Kondo impurities is often modeled by a Heisenberg coupling term JH. It gives rise to a quantum phase transition between the Kondo and the RKKY phases in the two-impurity Kondo model for arbitrarily large Kondo couplings, even in the presence of charge fluctuations (the relevant phase diagram, obtained with a numerical renormalization group technique is shown in Figure 1 for a comparison with further results). However, the significance of this result is still controversial. Firstly, the transition is fragile to particle-hole asymmetry, smearing the critical point into a crossover in its presence [3,4]. This has lead to the common belief that it cannot be realized in a realistic 2-impurity system and has made its relevance for lattice models debatable. Moreover, in this model the Y and the Kondo exchange JK are considered independent, although Y is genuinely generated from JK, and the Kondo temperature depends on Y, as has been shown experimentally [5] and theoretically [6]. Recently, it has been shown that the quantum phase transition can be restored for weaker particle-hole symmetry by parameter fine-tuning [7]. We revisit the problem by considering a geometry of two impurities, where each one is coupled to a different host as in [5], and the RKKY interaction is induced solely by JK and an inter-host exchange coupling JY. We show by numerical renormalization group (NRG) calculations that, for a properly symmetric case, this causes a transition, if the Kondo coupling is not too strong. Moreover, another phase transition occurs then at very strong JY (of the order of the bandwidth D), which is not present in the Jones-Varma model [1,2], and drives the system to yet another phase, with non-universal impurity spectral density. The two phase transition lines meet for increasing JK and, hence, are replaced by a single crossover above a critical value of JK. The corresponding phase diagram is presented in Figure 2. The phase diagram can be understood by analyzing the relevant quasiparticle pictures. In the Kondo regime, the relevant quasi-particles are spatially somewhat extended objects, and their effective bandwidth is given by the width of the Abrikosov-Suhl resonance, which is of the order of the Kondo temperature TK. Therefore, the first transition (as recognized by Jones and Varma [2]) happens at the RKKY interaction strength of the order of TK. In the RKKY phase the Kondo quasiparticles are destroyed and the conduction electron states alone comprise the relevant quasiparticles. The second phase transition then happens when the inter-host coupling becomes of the order of the bandwidth D. For inter-host couplings exceeding the second critical value, the band electrons become irrelevant again and the effective quasiparticles (which we call Heisenberg quasi-particles) are somewhat analogous to the Kondo quasi-particles, in that the two hosts mutually spin-screen each other. The latter observation also explains why a continuous crossover is possible between the Kondo phase and the Heisenberg one. The two transitions present for genuine RKKY interaction can be recognized in the NRG results as abrupt changes of the local density of states of electrons localized at impurities and of conduction band electrons at the impurity sites, as well as from the divergence of corresponding staggered magnetic susceptibilities. This may make them detectable despite the fact that all static spin-spin correlation functions are continuous through both transitions (like in the Jones-Varma model). As the inter-host exchange must be of the order of the width of the conduction band for the second transition to be observed, the phenomenon seems relevant to for narrow-bandwidth materials, such as twisted bilayer graphene or other flat-band Moirés [8].

**References:**

[1] B. A. Jones and C. M. Varma, Phys. Rev. Lett., vol. 58, p. 843 (1987). [2] B. A. Jones, C. M. Varma, and J. W. Wilkins, Phys. Rev. Lett., vol. 61, p. 125 (1988). [3] R. M. Fye, Phys. Rev. Lett., vol. 72, p. 916 (1994). [4] I. Affleck, A. W. W. Ludwig, and B. A. Jones, Phys. Rev. B, vol. 52, p. 9528 (1995). [5] J. Bork et al., Nat. Phys., vol. 7, p. 901 (2011). [6] A. Nejati, K. Ballmann, J. Kroha, Phys. Rev. Lett., vol. 118, p. 117204 (2017). [7] F. Eickhoff, B. Lechtenberg, F. Anders, Phys. Rev. B, vol. 98, 115103 (2018). [8] L. Balents et al., Nat. Phys., vol. 16, p. 725 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5e803c783776de126295c07071d80a03.jpg)

![](https://assets.underline.io/uploads/markdown_image/1/image/db9cda654a3d809d5608b02b8e9a837e.jpg)

 Surprises in the phase diagram of two impurities on flat-band hosts coupled by genuine RKKY interaction

Single molecule magnetism (SMM) presents a novel avenue for the pursuit of a bottom up approach to the design of next generation magnets, their molecular origin of magnetism and sensitivity to their chemical environment allows SMMs to be designed and exploited at the molecular level1. Applications include proposals for SMMs to be used as qubits and components in spintronic devices2,3. Here we present the synthesis and magnetic properties of mononuclear Ln(III) complexes, where two Er(III) complexes exhibit SMM behavior and blocking temperatures of around 40K.4 We used a synthetic approach and ligand fine-tuning where the ligand field is favorable for the Er(III) compounds to exhibit SMM properties. Single crystal structure analysis was carried out to obtain structural information on all the presented complexes. The field dependence of the magnetization for both Er(III) compounds were measured at 2, 3 and 5 K; the magnetization curves did not saturate at high fields indicating the presence of significant magnetic anisotropy and/or low-lying excited states. In order to investigate the potential presence of slow relaxation of the magnetization caused by SMM behavior, ac magnetic susceptibility measurements were performed, in applied dc-fields from 0 to 3000 Oe, in attempts to suppress any quantum tunneling of the magnetization (QTM). For both compounds the in-phase and out-of-phase ac susceptibility show field-dependent signals; the optimum field was found to be 500 Oe and this was used for further ac measurements. In both cases, the ac measurements reveal temperature dependent in- and out-of-phase signals with clear maxima up to 3.5 K. The maxima of the frequency dependent out-of-phase susceptibility curves were used extract the relaxation time as a function of temperature.

**References:**

[1] B. M. Day, F.-S. Guo, R. A. Layfield, Acc. Chem. Res. 51,1880–1889 (2018). [2] M. Shiddiq et al., Nature 531, 348–351 (2016). [3] S. Thiele et al., Science 344, 1135–1138 (2014). [4] I.A. Kühne, et al., Dalton Trans 48, 15679 (2019)

**Images:**

 The Influence of Electron-Withdrawing Groups on the Relaxation Time of Field-Induced Er(III) Single Molecule Magnets.

Two-dimensional arrays of interacting magnetic nanostructures allow the exploration of the many body physics associated with highly frustrated magnets in a convenient manner. Based on a lab-on-chip approach, artificially made structures can be viewed as experimental simulators of frustrated magnetism, complementing what can be done with chemically synthesized compounds. For example, collective magnetic phenomena such as emergent charge crystallization, magnetic moment fragmentation, and Coulomb phase physics have been recently evidenced in artificial systems. If the initial motivation was to realize an artificial ice magnet, a wide range of systems have been now investigated, including structures that have no counterparts in bulk materials. Whether they are chemically synthesized or nanofabricated, the spin dynamics of frustrated magnets generally freezes before magnetic ordering eventually sets in at very low temperature. Here, we will show how dynamical freezing can be bypassed in an artificial kagome ice to access low-energy states [1]. More specifically, we will demonstrate that the a priori dynamically inaccessible ordered ground state and fragmented spin liquid configurations can be obtained reproducibly, imaged in real space at room temperature, and studied conveniently (Figs.1 and 2). We will then identify the mechanism by which dynamical freezing occurs in the dipolar kagome ice. Besides, although artificially made structures were initially designed to address the physics of frustrated spin models, they can be used to investigate vertex models as well. Experimental strategy to fabricate artificial vertex models will be also briefly discussed [2].

**References:**

[1] V. Schanilec et al., Phys. Rev. Lett. 125, 057203 (2020). [2] Y. Perrin et al., Phys. Rev. B 99, 224434 (2019).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/06cf2e370a40d127454b84c97e8d5755.jpg)
Real space magnetic images with their associated spin-charge representations in an artificial kagome ice trapped within a fragmented spin state. Scale bar is 6 μm.

![](https://assets.underline.io/uploads/markdown_image/1/image/c53af84486a627fed47bc581cd73f8c2.jpg)
Analysis of the pairwise spin correlations.

Dynamical freezing in an artificial kagome ice magnet

The realization of physical systems whose electronic and magnetic properties can be easily manipulated by playing with valence electrons are very promising from both the fundamental and applied perspectives. Half-metallic ferromagnets are ideal candidates for spintronic applications because of their high spin polarization of charge carriers at the Fermi level [1,2]. High Curie temperature is one of the important precondition of the half-metallic ferromagnets from the application point of view. Co2- based intermetallic Heusler alloys (L21 structure, space group Fm-3m, # 225) are among the most attractive half-metallic systems due to their high Curie temperatures, high spin polarization and the structural similarity to binary semiconductors [1,2]. The Co2FeSi (CFS) system is one such candidate of interest for spintronic applications due to its high Curie temperature crystallizing in the L21 structure but Fermi level falls on the edge of the minority conduction band (in both generalized gradient approximation (GGA) and GGA+U approach) making the system shy from being robust half metallic. High Tc = 1100K and large magnetic moment of 6.00 μB/f.u. is reported in Ref [3]. If one of the Co atom in CFS is replaced by low valence transition metal atom Y = Sc, Ti, V, Cr, Mn, and Fe, alloys with rich and useful properties can be tailored. The quaternary Heusler alloy CoFeMnSi [4] is observed theoretically and experimentally to be spin gapless semiconductor, CoFeCrSi [5] is reported to be half-metallic with some structural disorder, CoFeVSi and CoFeTiSi are reported theoretically to be nearly half-metallic and half-metallic respectively, but both are observed experimentally to show multi-phase behavior in bulk form. The Fermi level lies on the edge of the minority valence band in all Y substituted quaternary Heusler alloys, leading to an unstable half-metallicity. So, as we go from CFS to CoFeYSi, Fermi level is shifted from lower edge of conduction band to upper edge of valence band in minority spin channel which is in accordance to Ref. [2,6]. They have suggested that an expansion of the lattice should shift the Fermi level deeper in energy and the contraction should shift it higher in energy. Therefore, one can expect robust half metallicity with Fermi level exactly at the middle of the band gap for some intermediate Y concentrations in CFS due to the expansion of the lattice when Y with larger atomic radius substitutes for Co. The substitution of Y to Co may be also seen as d-electron deficiency. In this work, we have introduced the substitution of all low valence transition metal atoms Y = Sc, Ti, V, Cr, Mn, and Fe to Co atoms in the parent CFS system, and examined the structural, electronic, magnetic, and mechanical properties of quaternary Heusler compounds Co2-xYxFeSi (0≤x≤1) to get a global overview of the electronic, magnetic and mechanical properties, promising candidates for spintronics applications. All single phase alloys exhibit face centered cubic crystal structure with a strong tendency towards L21 ordering, as corroborated by X-ray diffraction. The low-temperature saturation magnetic moments agree fairly well with our theoretical results and also obey the Slater-Pauling rule, a prerequisite for half metallicity. Fig.1 shows the linear decrease of the spontaneous magnetizations measured at 2K with the valence electron counts per formula units with element Y = Ti, V, Cr, Mn, Fe, and Co in Co2-xYxFeSi (x = 0.50). All alloys are soft ferromagnets with high Curie temperatures, ranging from 700K to 1100K, allowing for applications at room temperature and above. The Curie temperatures are observed to vary almost linearly with the saturation magnetic moment which is also expected in half-metallic systems. First-principles calculations also predicted finite band gap in the minority spin channel for most of the alloys in Co2-xYxFeSi series. Fig. 2 shows the calculated density of states (DOS) plots using GGA approach for majority and minority spin channels of Co2-xMnxFeSi (0≤x≤1) alloys, where the Fermi level is represented by the zero energy. It can be seen clearly that the system exhibits half-metallic behavior after Mn substitution. In the case of parent compound CFS, there is significant amount of DOS at the Fermi level on both the spin channels. After Mn substitution, the Fermi level is observed to shift deeper in energy-levels and falls at the middle of the energy gap on spin down channel for x = 0.50. Further increase of Mn content shifts the Fermi level towards the lower edge of energy gap due to lattice expansion. But, still Fermi level lies within energy gap making the system Half-metallic. In addition, conduction band and valence band in majority spin channel touch each other at Fermi level in x = 1, making the system spin gapless semiconductor. The energy gap is observed to increase with the increase of Mn content. Although, the calculated minority spin gap is the highest for x = 1, the half-metallic gap is greater for x = 0.50. Relatively high mechanical hardness values among the reported Heusler alloys are also observed, approaching 16 GPa for the most Ti-rich material [6-8].

**References:**

1. C. Felser, G. H. Fecher, and B. Balke, Angewandte Chemie International Edition, 46, 668 (2007). 2. I. Galanakis, P. Dederichs and N. Papanikolao, Physical Review B, 66, 174429 (2002). 3. E. Fischer, J. Karel and S. Fabbrici, Physical Review B, 94, 024418 (2016). 4. L. Bainsla, A.I. Mallick and M.M. Raja, Physical Review B, 91, 104408 (2015). 5. Y. Jin, P. Khareland P. Lukashev, Journal of Applied Physics, 120, 053903 (2016). 6. R. Mahat, S. KC and U. Karki, Physical Review B, Submitted (2021). 7. R. Mahat, S. KC and D. Wines, Journal of Alloys and Compounds, 830, 154403 (2020). 8. R. Mahat, S. KC and D. Wines, Journal of Magnetism and Magnetic Materials, 521, 167398 (2021).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7c947ceb5f0d173267903a665062a366.jpg)
Fig.1. The saturation magnetic moment versus valence electron counts per formula units with element Y = Ti, V, Cr, Mn, Fe, and Co in Co2-xYxFeSi (x = 0.50), both experimental and expected from Slater Pauling rule for half metals. The black data points represent reported literature values [3].

![](https://assets.underline.io/uploads/markdown_image/1/image/42867fe7839357c6b915951bb21bee55.jpg)
Fig.2. Spin polarized total DOS for Co2-xYxFeSi (Y = Mn, 0≤x≤1). Number of states is scaled with respect to one formula unit.

 Effect of mixing the low-valent transition metal atoms Y = Sc, Ti, V, Cr, Mn and Fe on the properties of quaternary Heusler compounds Co2-xYxFeSi (0≤x≤1)

Strongly correlated materials with multiple order parameters provide unique insights into the fundamental interactions in condensed matter systems and present opportunities for innovative technological applications. A class of antiferromagnetic honeycomb lattices compounds, A4B2O9 (A = Co, Fe, Mn; B = Nb, Ta), have been explored owing to the occurrence of linear magnetoelectricity. From our investigation of magnetoelectricity on single crystalline Co4Ta2O9, we discovered strongly nonlinear and antisymmetric magnetoelectric behavior above the spin-flop transition for magnetic fields applied along two orthogonal in-plane directions. This observation suggests that two types of inequivalent Co2+ sublattices generate magnetic-field-dependent ferroelectric polarization with opposite signs. The results motivate fundamental and applied research on the intriguing magnetoelectric characteristics of these buckled-honeycomb lattice materials.

**References:**

[1] N. Lee, D. G. Oh and Y. J. Choi, Scientific Reports, vol. 10, 12362 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/14735df7b8d80e6faf7c4ad0add6d601.jpg)
Fig. 1 Temperature dependence of the anisotropic ferroelectric polarization. (a–c) T dependence of Pa obtained by integrating the pyroelectric current after poling from 100 to 2 K in at Ha, Hb*, and Hc, respectively. Pa was measured at H = 1, 3, 5, 7 and 9 T. (d–f) T dependence of Pb* measured at Ha = 9 T, Hb*, and Hc, respectively. (g–i) T dependence of Pc measured at Ha = 9 T, Hb*, and Hc, respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/1dddc0b497f4c43b1b9e466269999736.jpg)
Fig. 2 Comparison of electric and magnetic properties. (a) Hb* dependence of Pa at T = 2 K. (b) Hb* dependence of the magnetodielectric effect along the a axis, MDa (%) = {ε′(H) − ε′(0T)} / ε′(0T) × 100 , measured with AC excitation of Ea = 1 V at f = 100 kHz and T = 2 K. (c) Hb* derivative of Mb* at 2 K.

 Highly nonlinear magnetoelectric effect in buckled honeycomb antiferromagnetic Co4Ta2O9

In rare-earth orthoferrites, strongly correlated order parameters have been thoroughly investigated, which aims to find multiple fuctionalities such as multiferroic or magnetoelectric properties. We have discovered highly anisotropic and nonlinear magnetodielectric effect from detailed measurements of magnetoelectric properties in single-crystalline orthoferrite, ErFeO3. Isothermal dielectric constant varies in shapes and signs depending on the relative orientations between the external electric and magnetic fields, which may be ascribed to the spin-phonon couplings. In addition, a dielectric constant with both electric and magnetic fields along the c axis exhibits two symmetric sharp anomalies, which are closely relevant to the spin-flop transition, below the ordering temperature of Er3+ spins, TEr = 3.4 K. We speculate that the magnetostriction form the exchange couplings between Er3+ and Fe3+ magnetic moments would be responsible for this relationship between electric and magnetic properties. Our results present significant characteristics of the orthoferrite compounds and offer a crucial guide for exploring suitable materials for magnetoelectric functional applications.

**References:**

[1] H. J. Shin, Y.J. Choi and N. Lee, Scientific Reports., 10:11825 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/5564792e58f2025c698fb955501095ae.jpg)
Magnetodielectric effect of ErFeO3. Magnetodielectric (MD) effect defined as 
MD (%) = (ε′(H)−ε′(0T))/ε′(0T) × 100 for (a)–(c) a (MDa), (d)–(f) b (MDb), and (g)–(i) c (MDc) axes, respectively, at Ha, Hb , and Hc up to ± 9 T and T = 2 K.

![](https://assets.underline.io/uploads/markdown_image/1/image/7b3b7a8f4622c4c168fd19b029722ed0.jpg)
Temperature evolution of magnetodielectric effect along the c axis. (a)–(f) Hc dependence of MDc at T = 2, 2.5, 3, 3.5, 5 and 10 K. (g)–(l) Hc derivative of Mc and dMc/dHc measured at T = 2, 2.5, 3, 3.5, 5 and 10 K.

Polarity dependence of change in electric polarization (magnetization) induced by magnetic (electric) fields in CaBaM4O7(M=Co, Fe) single crystals

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