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In vortex-like spin arrangements, multiple spins can combine into emergent multipole moments. Such multipole moments have broken space-inversion and time-reversal symmetries, and can therefore exhibit linear magnetoelectric (ME) activity. Three types of such multipole moments are known: toroidal; monopole; and quadrupole moments. So far, however, the ME activity of these multipole moments has only been established experimentally for the toroidal moment. Here we propose a magnetic square cupola cluster, in which four corner-sharing square-coordinated metal-ligand fragments form a noncoplanar buckled structure, as a promising structural unit that carries an ME-active multipole moment. We substantiate this idea by observing clear magnetodielectric signals associated with an antiferroic ME-active magnetic quadrupole order in the real material Ba(TiO)Cu 4 (PO 4 ) 4 . The present result serves as a useful guide for exploring and designing new ME-active materials based on vortex-like spin arrangements.

Magnetoelectric properties are studied by a combined experimental and theoretical study of a quasi-two-dimensional material composed of square cupolas, Ba(TiO)Cu4(PO4)4. The magnetization is measured up to above the saturation field, and several anomalies are observed depending on the field directions. We propose a S=1/2 spin model with Dzyaloshinskii-Moriya interactions, which well reproduces the full magnetization curves. Elaborating the phase diagram of the model, we show that the anomalies are explained by magnetoelectric phase transitions. Our theory also accounts for the scaling of the dielectric anomaly observed in experiments. The results elucidate the crucial role of the in-plane component of Dzyaloshinskii-Moriya interactions, which is induced by the noncoplanar buckling of square cupola. We also predict a 'hidden' phase and another magnetoelectric response both of which appear in nonzero magnetic field.PACS numbers: 77.80. Fm,75.85.+t,75.30.Kz Spatial asymmetry is a source of interesting phenomena in a broad range of condensed matter physics. A well-known example is the molecular asymmetry of water H 2 O, which leads to an electric polarization in each molecule. The asymmetry is at play also in magnets: the loss of inversion symmetry activates the asymmetric interactions through the relativistic spin-orbit coupling, such as the Dzyaloshinskii-Moriya (DM) interaction [1,2]. The asymmetric interactions lead to intriguing magnetism, e.g., weak ferromagnetism in antiferromagnets and spin-spiral ordering in helimagnets. They have also attracted growing interest as an origin of the magnetoelectric (ME) effect, that is, cross correlations between dielectricity and magnetism [3,4].Recently, an interesting series of chiral antiferromagnets, A(TiO)Cu 4 (PO 4 ) 4 (A = Ba, Sr) with space group P 42 1 2, was newly synthesized [5]. The materials have a quasi-two-dimensional structure, composed of an alternating array of Cu 4 O 12 clusters, as shown in Fig. 1(a). Each Cu 4 O 12 cluster consists of four corner-sharing CuO 4 plaquettes, forming a noncoplanar buckled structure termed (irregular) square cupola. The asymmetric unit can carry ME-active magnetic multipoles [6] associated with Cu spins. Indeed, a divergent anomaly of the dielectric constant was observed at the Néel temperature (T N =9.5 K) in magnetic fields applied along the [100] and [110] directions for A = Ba [7]. Although the ME response was argued by the magnetic quadrupole associated with noncoplanar antiferromagnetic ordering, the microscopic understanding is not fully obtained. It is highly desired to clarify how the unique asymmetry arising from the square cupolas affects the magnetic and dielectric properties in this series of compounds. [010][001] In this Letter, combining experimental and theoretical studies, we clarify the microscopic mechanism of ME behavior in A(TiO)Cu 4 (PO 4 ) 4 . First, from the magnetization measurement for the compound with A = Ba up to above the saturation field, we find several anomalies depending on the field dire...

Single crystals of two novel tetragonal chiral materials, A­(TiO)­Cu4(PO4)4 (A = Ba, Sr), were grown from Na2Mo2O7 flux, and their crystal and chiral domain structures were characterized. Polarized-light microscopy studies of the chiral domain structures in the crystals show that Ba­(TiO)­Cu4(PO4)4 mostly hosts a multidomain state, while a monodomain state predominantly appears in Sr­(TiO)­Cu4(PO4)4. To explain this striking difference, we quantified the chirality strength of these materials by comparing atomic positions in the chiral and nearest-achiral crystal structures, revealing larger chirality strength in Sr­(TiO)­Cu4(PO4)4 than in Ba­(TiO)­Cu4(PO4)4. Our proposed mechanisms linking the chirality strength and domain formation can account for the different occurrence frequency of chiral domains in this system.

Ferroic magnetic quadrupole order exhibiting macroscopic magnetoelectric activity is discovered in the novel compound A(TiO)Cu4(PO4)4 with A = Pb, which is in contrast with antiferroic quadrupole order observed in the isostructural compounds with A = Ba and Sr. Unlike the famous lone-pair stereochemical activity which often triggers ferroelectricity as in PbTiO3, the Pb 2+ cation in Pb(TiO)Cu4(PO4)4 is stereochemically inactive but dramatically alters specific magnetic interactions and consequently switches the quadrupole order from antiferroic to ferroic. Our firstprinciples calculations uncover a positive correlation between the degree of A-O bond covalency and a stability of the ferroic quadrupole order. I. INTORODUCTIONEarlier works demonstrated that the usage of specific elements with characteristic chemical properties is effective to realize desired ferroic order. For example, lonepair stereochemical activity of a heavier post-transition metal cation with an s 2 electron configuration such as Pb 2+ and Bi 3+ , which we call an s 2 -cation, is known as a driving force for ferroelectric order [1,2], as discussed in perovskite oxides PbTiO 3 [3], BiMnO 3 [4], and BiFeO 3 [5]. The stereochemically active s 2 -cations are surrounded by "hemidirected" local coordination, in which there is a void in the distribution of bonds to the ligands [1]. The origin for this directional bonding is explained by the hybridization between nominally empty metal p states with anti-bonding states formed by filled metal s states and ligand p states [2]. Such a hybridization is possible only when the inversion-symmetry at the cation site is broken. This is a driving force for off-center distortion and thus ferroelectric order.Another potential role of s 2 -cations is an impact on magnetism in insulating magnetic oxides. There, dominant magnetic superexchange interactions are usually mediated by O 2p orbitals near Fermi energy (E F ) [6]. As exemplified by the comparison between PbTiO 3 and BaTiO 3 [3], s 2 -cations tend to exhibit strong orbital hybridization with O ions, which should significantly affects superexchange interactions. Note that s 2 -cations are not necessarily stereochemically active; there are comparable number of compounds containing such cations located in "holodirected" local environment without a void in the * kentakimura@edu.k.u-tokyo.ac.jp ligand bond distribution [1]. In this case, substituting s 2 -cations can be a promising way of fine-tuning magnetic interactions without large distortion of the original structure.Among various ferroic orders, a particular class of magnetic order with broken space-inversion and time-reversal symmetries has recently attracted considerable interest because it can exhibit symmetry-dependent unique phenomena, such as magnetoelectric (ME) effects [7][8][9][10][11][12][13][14][15] and unconventional nonreciprocal electromagnetic responses [16][17][18]. From a symmetry point of view, it is known that ferroic order of magnetic multipole moments (toroidal, monopole, and quadrupole m...