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)Cu4(PO4)4. The present result serves as a useful guide for exploring and designing new ME-active materials based on vortex-like spin arrangements.
In magnetic materials, noncollinear spin arrangements often emerge when spins are placed in a particular lattice geometry such as geometrically frustrated lattices1. Symmetry-breaking properties of an additional multispin degree of freedom inherent in the noncollinearity can generate various anomalous magnetic phenomena. A well-known example is spiral-spin-driven ferroelectricity arising from vector spin chirality with broken space-inversion symmetry2,3,4,5. Another example is the magnetoelectric (ME) effect—magnetic field (B) control of electric polarization (P) and electric field (E) control of magnetization (M)—which originates from magnetic multipole moments that break both space-inversion and time-reversal symmetries6,7,8,9,10,11,12. Three types of symmetrically distinct ME-active multipole moments are known: the toroidal moment (t∝Σnrn × Sn); the monopole moment (a∝Σnrn·Sn); and the magnetic quadrupole moment (qij∝Σn[rniSnj+rnjSni−δijrn·Sn]), where n represents a label of the spin Sn at the position vector rn and i, j denote the x, y or z axis7,10,13,14. These quantities can be finite in a specific vortex-like spin arrangement. For example, the spin arrangements illustrated in Fig. 1a,b have the monopole moment a and the quadrupole moment (=qxx−qyy), respectively, which therefore allow for the linear ME effect determined by the corresponding ME tensor (for more details, see Supplementary Note 1). Previous theoretical studies4,15 predict that a B-induced vortex deformation can generate finite P in toroidal and monopolar vortices (Fig. 1a), and also in the quadrupolar vortex as shown in (Fig. 1b), which is independent of the ME coupling mechanism. Of the three types of multipole moments, the uniform ordering of toroidal moments (ferrotoroidicity) and associated ME activities have been experimentally probed via various techniques9,12,16,17,18,19. In particular, an observation of ferrotoroidic domain structure9 and its hysteretic switching by crossed magnetic and electric fields12,18 have led researchers propose ferrotoroidicity to be a fourth form of primary ferroic orders in the fundamental scheme based on the order-parameter symmetries with respect to the space-inversion and time-reversal operations. The ME activity of the other magnetic multipole moments, however, has never been established experimentally. Here we propose a strategy for the realization of these ME-active multipole moments in a material starting from a simple magnetic cluster.
The magnetic cluster considered here is illustrated in Fig. 1c,d. It consists of four transition metal ions forming a spin plaquette and twelve coplanar ligands such as oxygen. This is a fragment of a very common lattice seen in many inorganic materials including the high-Tc cuprate superconductors20 and infinite-layer iron oxides21. If we introduce a single-ion anisotropy (or exchange anisotropy) normal to the metal-ligand plane and simple ferromagnetic (FM) or antiferromagnetic (AFM) interactions between the spins, the FM and AFM spin-plaquette show an all-up (or all-down) and an up-down-up-down structure, respectively. Both are ME inactive, because they have neither toroidal, monopole nor quadrupole moments. To induce ME activity, we introduce a buckling deformation that transforms the cluster geometry to one of the Johnson solids22 known as square cupola (Fig. 1e,f), while assuming that the anisotropy and exchange interactions remain unchanged. With this buckling, the FM spin plaquette has monopole components in addition to ferromagnetic components, while the AFM spin plaquette has magnetic quadrupole components. Therefore, in principle, both square cupola clusters should exhibit ME activity.
To experimentally verify this ME design strategy, we have searched for a real material that comprises magnetic square cupola clusters. The candidate compound found is Ba(TiO)Cu4(PO4)4, a recently synthesized magnetic insulator crystallizing in a chiral tetragonal structure with space group P4212 (ref. 23). The crystal structure is illustrated in Fig. 2a,b. The magnetic properties are dominated by Cu2+ ions (S=1/2) with square planar coordination of oxygen ions. Notably, the crystal structure comprises an irregular Cu4O12 square cupola cluster formed by four corner-sharing CuO4 planes—that is, an experimental realization of Fig. 1e,f. Two types of square cupola clusters, A (upward) and B (downward), distinguished by their direction with respect to the c axis, align alternatingly in the ab-plane to form a layered structure, which we call a square cupola layer. Importantly, the nature of the square cupola is expected to be preserved in this compound because neighbouring square cupola clusters do not share any oxygen, suggesting weak inter-cluster couplings (Fig. 2b). These features make this material suitable for testing our ME design strategy. Therefore, we have studied magnetic and ME properties of Ba(TiO)Cu4(PO4)4, demonstrating that the magnetic structure is describable by an antiferroic magnetic quadrupole order and, moreover, that an associated ME activity is manifested in the magnetodielectric properties. These results successfully verify our ME design strategy.
The temperature (T) dependence of magnetization (M) divided by B (χ≡M/B) applied along the  and  axes is shown in Fig. 2c. Fits to the high-temperature data (T>100 K) of χ and χ using the Curie–Weiss law yield effective moments of 1.92(1) μB/Cu and 1.96(1) μB/Cu, respectively, which are typical for Cu2+ ions, and AFM Weiss temperatures (θCW) of −33.2(6) and −30.1(2) K. On cooling, a broad maximum appears at around 17 K, followed by a clear anomaly at TN=9.5 K, below which χ shows an anisotropy. Because of the two-dimensional (2D) nature of the square cupola layers due to the separation by nonmagnetic Ba and Ti layers, the broad maximum suggests a development of short-range correlation within each cluster and/or 2D layer. The anomaly at TN=9.5 K indicates the onset of AFM long-range order due to weak inter-layer couplings. Notably, both χ and χ remain finite at the lowest temperature measured, which indicates that the magnetic structure is not a simple collinear antiferromagnet. No metamagnetic transition is observed up to B=7 T, as demonstrated by approximately linear magnetization curves at T=1.8 K (Fig. 2d). The AFM transition is also evidenced by a peak in specific heat CP, as shown in Fig. 2e.
For microscopic characterization of the magnetic properties, we use neutron diffraction. As depicted in Fig. 3a, below 9.5 K we observe new Bragg peaks emerging corresponding to magnetic ordering of moments. The magnetic reflections can be indexed using a single propagation wave vector k=(0, 0, 0.5), which corresponds to a doubling of the unit cell along the  direction. Symmetry analysis indicates that the magnetic representation can be decomposed into Γmag=3Γ1+3Γ2+3Γ3+3Γ4+6, where the irreducible representation is 2D and the others are one-dimensional. The best-fit magnetic structure obtained by the Rietveld refinement with the magnetic R-factor of 11.5% (Γ3 irreducible representation) at 1.5 K is illustrated in Fig. 3b for a single square cupola layer. The magnetic structure is noncollinear, with the moments tilted away from the c axis in such a way that they are approximately perpendicular to the CuO4 plane. Notably, in each Cu4O12 square cupola (Fig. 3c), the c axis components of magnetic moments align in the up-down-up-down manner while the ab-plane components rotate by 90°. This demonstrates an experimental realization of the AFM square cupola considered in Fig. 1b, which carries a magnetic quadrupole moment composed of an almost pure component. Moreover, the components in every Cu4O12 square cupola align uniformly, suggesting that the magnetic structure can be considered as a uniform order of mangnetic quadrupole moments within the ab-plane. Further details on this quadrupole order can be found in Supplementary Fig. 1 and Supplementary Note 2. An ordered moment of 0.80(3)μB is found on each Cu site. The details of refinement are provided in Supplementary Figs 2–4, Supplementary Tables 1 and 2, and Supplementary Note 3.
Dominant exchange interactions
The quadrupole-based description of the magnetic structure is valid if intraplaquette exchange interactions dominate over interplaquette ones. To examine this point, we have performed inelastic neutron-scattering measurements. Colour plot of inelastic neutron spectra at T=2 K is shown in Fig. 3d. We observe two strong, flat bands of intensity close to 3.2 and 4.2 meV (Fig. 3e). This is consistent with the energy scale estimated from θCW≈−30 K. In addition, we find branches dispersing from magnetic zone centres with a gap energy of around 1 meV. On warming to 30 K, shown in Fig. 3e, we find that both the gapped branches and the flat bands of intensity disappear as would be expected for excitations, which are magnetic in origin. Further measurements using Ei=12.1 meV did not reveal any additional excitations. The observed inelastic spectrum is consistent with strongly coupled plaquettes with weak inter-plaquette interactions and an anisotropy, such as from Dzyaloshinskii–Moriya interaction, which is symmetrically allowed in this material. The former would result in dispersive collective excitations and the latter create a spin-gap. Therefore, Ba(TiO)Cu4(PO4)4 could potentially be an exciting system to examine the cross-over from local quantum levels to dispersive spin-waves of coupled spin clusters such as in Cu2Te2O5(Cl,Br)2 (ref. 24).
Further insights to exchange interactions are provided by density functional theory (DFT) calculations. The electronic structure of Ba(TiO)Cu4(PO4)4 was calculated by using GGA+U method25,26, where the on-site Coulomb repulsion Ueff was set to 4 eV. We have calculated the magnetic exchange coupling constants Jk (k=1–6) between the spins (Fig. 2b). Here positive (negative) J represents FM (AFM) interactions. We find that the strongest interaction is the nearest-neighbour intraplaquette J1=−3.0 meV, followed in order by orthogonal interplaquette interaction J5=−0.7 meV and orthogonal intraplaquette J4=−0.5 meV. The other interactions are one order of magnitude smaller than J1 (J2=−0.2 meV, J3=0.2 meV and J6=−0.1 meV). From these coupling constants the Weiss temperature is calculated to be −31 K, which is comparable to the experimental value of θCW≈−30 K. We have also calculated Jk with different values of Ueff (0 and 7 eV) and confirmed that the relative strengths of Jk is almost insensitive to Ueff. Thus, our experimental and theoretical studies establish that the present material is a weakly coupled plaquette antiferromagnet, which means that the quadrupole-based description of the magnetic structure is valid. Furthermore, a preliminary spin-wave calculation27 using the exchange parameters (J1–J6) obtained from DFT calculations gives a qualitative agreement to the measured inelastic neutron spectrum, which confirms the consistency between the experiments and the DFT results (Supplementary Fig. 5). However, because of a relatively large number of exchange paths which must be taken into account, single-crystal study is necessary for a detailed comparison of inelastic neutron spectrum and a model based on DFT parameters.
The task is now to experimentally confirm the ME activity of these quadrupole moments. Usually, it can be easily probed through the measurements of B-induced P or E-induced M. In the present case, however, it is not straightforward because, as indicated by the 00 magnetic wave vector, the quadrupole moments are antiferroically coupled along the c axis, which results in the cancellation of the associated linear ME response. Indeed, our high-precision pyroelectric current measurement (<0.1 pA) at a high-B (<9 T) does not show any signal indicative of a macroscopic B-induced P near TN. Nonetheless, we observe a clear signature for the ME-activity in the dielectric constant (ɛ), as shown in the following.
Figure 4a shows the T-dependence of ɛ along the  direction (ɛ) measured at selected B applied along the  direction (B). While ɛ shows only a slight anomaly at TN in the absence of B, the application of B induces a divergent peak towards TN. Together with the absence of P, this result suggests the onset of the B-induced antiferroelectric (AFE) order, which is similar to the B-induced ferroelectric order observed in linear ME materials such as Cr2O3 (ref. 28). To quantitatively analyse the data, we define the B-induced component in ɛ as Δɛ(B)≡ɛ(B)−ɛ(0). The inset of Fig. 4a shows Δɛ(B) divided by the square of B, Δɛ/, as a function of a reduced temperature, t≡(T−TN)/TN. Strikingly, all the data approximately collapse onto a single curve, meaning that Δɛ(B) is proportional to the square of B. This scaling behaviour of the B-induced divergent peak is characteristic in AFM linear ME materials such as MnTiO3, which is accounted for on the basis of Landau free energy expansion involving the linear ME coupling term29. No divergent behaviour is observed for ɛ in B and B, and for ɛ in all B directions. (Supplementary Fig. 7).
Because different ME-active multipole moments (t, a and qij) lead to different forms of the ME tensor (Supplementary Fig. 6 and Supplementary Note 1), identifying the ME tensor in Ba(TiO)Cu4(PO4)4 is crucial to verify that the observed ME response originates from the ME activity of the quadrupole moments. To this end, we first discuss the observed ME response in terms of the linear ME effects in each square cupola layer. According to the transformation properties of the present Γ3 magnetic structure (Supplementary Table 2), the magnetic point group symmetry of each square cupola layer is 4′22′, which breaks both the space-inversion and time-reversal symmetries and therefore allows for a linear ME effect given by the ME tensor30
This ME tensor predicts that the application of B induces P along the  direction (P). In addition, the 00 magnetic wave vector indicates that the sign of the ME tensor of the neighbouring layers is opposite to each other. This means that the system exhibits a B-induced AFE behaviour along the  direction, consistent with the observed ME response. Note that this ME tensor allows, in principle, a macroscopic -type quadrupole moment within the ab-plane (Supplementary Fig. 6), which is consistent with the uniform alignment of quadrupole moments on the individual square cupolas. Consistently, our group theory analysis in the framework of Landau theory of phase transitions31,32,33 shows that the quadrupole order can lead to the same ME effect as expected from the ME tensor in equation (1). Details of the analysis are provided in Supplementary Fig. 8, Supplementary Table 3 (refs 34, 35, 36, 37) and Supplementary Note 4.
Next, we associate the observed result with the ME activity of individual magnetic quadrupole moments. As schematically illustrated in Fig. 1b, while no net P emerges from the magnetic quadrupole moment at B=0, the application of B along the x or y axis induces P parallel or antiparallel to the applied B, where the local principal axes parallel to the inward and outward spins are taken as the x and y axis, respectively. By applying this scenario to the single square cupola layer of the present system, we find that under B all the quadrupole moments generate P approximately parallel to the B direction, resulting in macroscopic P in the single square cupola layer (Fig. 3b). Although a small diagonal component of local P may be generated in each quadrupole moment due to the slight deviation of the inward spin direction from the  direction, this component cancels out in neighbouring quadrupole moments because of the symmetry. The direction of B-induced P is again consistent with that of the observed ME response and the ME tensor in equation (1). Note that there must be some contributions to the ME activity from the interplaquette interactions. However, the leading exchange interaction is the intraplaquette J1, which provides a reasonable basis for the interpretation of the observed ME activity in terms of the ME activity of individual square cupolas. Owing to an AFM stacking of quadrupole moments along the  direction, the induced P aligns antiferroelectrically along the  direction.
Furthermore, on the basis of the ME tensor given by equation (1) as well as the ME activity of individual magnetic quadrupole moments (Fig. 1b), one can expect that P is induced by the application of B[1–10] but not by B, and the magnitude of B[1–10]-induced P is the same as that of B-induced P. This expectation is experimentally demonstrated in the effects of B[1–10] and B on ɛ (Fig. 4b), together with the scaling behaviour of ɛ with the magnitude of Δɛ/ comparable to that of Δɛ/ (Fig. 4b, inset). Thus, the observed AFE behaviour can be nicely explained in terms of the ME activity of quadrupole moments and their antiferroic alignment.
Finally, we discuss a possible microscopic origin for the observed ME activity. The previous theoretical work demonstrated that a magnetostriction mechanism via non-relativistic superexchange interactions can generate large ME effects in monopolar and toroidal vortices15. Applying the similar discussion to the present quadrupolar vortex on a square cupola, we have found that the magnetostriction mechanism can also lead to the B-induced P, and the direction of B and P is consistent with the experimental observation (Supplementary Fig. 9). However, because other known mechanisms associated with a relativistic spin orbit interaction, for example, the d–p hybridization mechanism38 and the spin-current3 (or inverse DM5) mechanism, can also be possible. Therefore, more detailed understanding of the microscopic mechanism for the ME activity is left for future study.
To conclude, we propose a magnetic square cupola cluster as a promising structural unit that carries the ME-active multipole moments, and confirm this idea by magnetodielectrically detecting the ME activity of the magnetic quadrupole moments in a real material. This result indicates that ME activity arising from magnetic multipoles can also be found in other square cupola based materials (for example, Na5ACu4(AsO4)4Cl2 (A=Rb, Cs)39 and [NH4]Cu4Cl(PO3F)4 (ref. 40)). In particular, the monopolar vortex, which can appear in the FM square cupola (Fig. 1a), is interesting because it possesses a ferromagnetic component controllable by an electric field through the ME coupling. Another aspect that deserves attention in the present study is the discovery of the ME response due to the antiferroic ME-active multipole order. For example, this ME response allows for an E- or B-induced finite-q magnetization or electric polarization, which might be useful for future nanoscale spintronics. Moreover, the present discovery demonstrates that dielectric constant measurements can provide a rather straightforward, macroscopic signature for antiferroic order of any types of ME-active multipole moments. This is an important finding to explore a new state of matter such as a recently discussed antiferromonopolar state and antiferrotoroidic state14.
Sample preparation and characterization
Single crystals of Ba(TiO)Cu4(PO4)4 were grown by the flux method23. Powder X-ray diffraction measurements on crushed single crystals confirmed a single phase. The crystals were oriented using Laue X-ray diffraction. The crystal structures displayed in this article were drawn using VESTA software41. Magnetization (M) measurements down to temperature (T) of 1.8 K and magnetic field (B) up to 7 T were performed using a commercial superconducting quantum interference device magnetometer (Quantum Design MPMS3). The specific heat (CP) was measured down to 2 K by a thermal relaxation method using a commercial calorimeter (Quantum Design PPMS). For dielectric measurements, single crystals were cut into thin plates and subsequently silver electrodes were vacuum deposited on the pair of widest surfaces. The dielectric constant ɛ was measured using an LCR meter at an excitation frequency of 100 kHz. Pyroelectric current was measured by an electrometer (Keithley 6517) to monitor electric polarization.
Neutron diffraction measurements were performed on a powder sample using the time-of-flight neutron diffractometer WISH at ISIS42 and the DMC diffractometer at the SINQ spallation source43. Magnetic and nuclear structure refinements were performed using FullProf44. The inelastic neutron-scattering measurements were carried out on a 17 g powder sample using the time-of-flight spectrometer FOCUS at the SINQ spallation source45. Using incident neutron energies of Ei=6 and 12.1 meV, the energy resolution in the energy transfer range of interest was ∼0.2 and 0.7 meV, respectively.
DFT calculations were performed to estimate the magnitude of dominant magnetic interactions. The VASP (Vienna ab initio simulation package)46,47,48,49 was used with a projector-augmented wave basis set50,51. The electronic exchange and correlation were described by the Perdew–Burke–Ernzerhof generalized gradient approximation (PBE-GGA)25. The DFT+U method26 was used for correction for strongly correlated Cu-3d states. We first calculated the electronic structure of Ba(TiO)Cu4(PO4)4 with the experimental crystal structure. Then we obtained the total energy differences among several magnetic phases with different spin structures by performing spin-constrained DFT calculations. The magnetic exchange coupling constants Jk were estimated as the best fit for the energy differences within an effective classical Heisenberg model. Our model Hamiltonian is defined as follows:
Here is a unit vector that point to the direction of the spin at site l and Jlm the effective coupling constant between sites l and m. The factor of 1/2 removes double counting. Jk is the average of Jlm taken over the k-th nearest-neighbour spin pairs. Note that the magnitude of spin (ideally S=1/2 for Cu2+ ions) is renormalized in Jk and that the summation over k terminates at 6 as we consider up to J6 (Fig. 2b).
The data that support the findings of this study are available from the corresponding author on request.
How to cite this article: Kimura, K. et al. Magnetodielectric detection of magnetic quadrupole order in Ba(TiO)Cu4(PO4)4 with Cu4O12 square cupolas. Nat. Commun. 7, 13039 doi: 10.1038/ncomms13039 (2016).
Greedan, J. E. Geometrically frustrated magnetic materials. J. Mater. Chem. 11, 37–53 (2001).
Kimura, T. et al. Magnetic control of ferroelectric polarization. Nature 426, 55–58 (2003).
Katsura, H., Nagaosa, N. & Balatsky, A. V. Spin current and magnetoelectric effect in noncollinear magnets. Phys. Rev. Lett. 95, 057205 (2005).
Mostovoy, M. Ferroelectricity in spiral magnets. Phys. Rev. Lett. 96, 067601 (2006).
Sergienko, I. A. & Dagotto, E. Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites. Phys. Rev. B 73, 094434 (2006).
Astrov, D. N. The magnetoelectric effect in antiferromagnetics. Sov. Phys. JETP 11, 708–709 (1960).
Schmid, H. On ferrotoroidics and electrotoroidic, magnetotoroidic and piezotoroidic effects. Ferroelectrics 252, 41–50 (2001).
Fiebig, M. Revival of the magnetoelectric effect. J. Phys. D 38, R123–R152 (2005).
Aken, B. V., Rivera, J.-P., Schmid, H. & Fiebig, M. Observation of ferrotoroidic domains. Nature 449, 702–705 (2007).
Spaldin, N. A., Fiebig, M. & Mostovoy, M. The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect. J. Phys.: Condens. Matter 20, 434203 (2008).
Yamaguchi, Y. & Kimura, T. Magnetoelectric control of frozen state in a toroidal glass. Nat. Commun. 4, 2063 (2013).
Zimmermann, A. S., Meier, D. & Fiebig, M. Ferroic nature of magnetic toroidal order. Nat. Commun. 5, 4796 (2014).
Dubovik, V. M. & Tugushev, V. V. Toroid moments in electrodynamics and solid-state physics. Phys. Rep. 187, 145–202 (1990).
Spaldin, N. A., Fechner, M., Bousquet, E., Balatsky, A. & Nordström, L. Monopole-based formalism for the diagonal magnetoelectric response. Phys. Rev. B 88, 094429 (2013).
Delaney, K. T., Mostovoy, M. & Spaldin, N. A. Superexchange-driven magnetoelectricity in magnetic vortices. Phys. Rev. Lett. 102, 157203 (2009).
Popov, Y. F. et al. Magnetoelectric effect and toroidal ordering in Ga2−xFexO3 . JETP 87, 146–151 (1998).
Ressouche, E. et al. Magnetoelectric MnPS3 as a candidate for ferrotoroidicity. Phys. Rev. B 82, 100408 (2010).
Baum, M. et al. Controlling toroidal moments by crossed electric and magnetic fields. Phys. Rev. B 88, 024414 (2013).
Tolédano, P. et al. Primary ferrotoroidicity in antiferromagnets. Phys. Rev. B 92, 094431 (2015).
Tokura, Y. & Arima, T. New classification method for layered copper oxide compounds and its application to design of new high T c superconductors. Jpn J. Appl. Phys. 29, 2388–2402 (1990).
Tsujimoto, Y. et al. Infinite-layer iron oxide with a square-planar coordination. Nature 450, 1062–1065 (2007).
Johnson, N. W. Convex polyhedra with regular faces. Can. J. Math. 18, 169–200 (1966).
Kimura, K., Sera, M. & Kimura, T. A2+ cation control of chiral domain formation in A(TiO)Cu4(PO4)4 (A=Ba, Sr). Inorg. Chem. 55, 1002–1004 (2016).
Prša, K. et al. Anomalous magnetic excitations of cooperative tetrahedral spin clusters. Phys. Rev. Lett. 102, 177202 (2009).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Liechtenstein, A. I., Anisimov, V. I. & Zaanen, J. Density-functional theory and strong interactions: orbital ordering in Mott-Hubbard insulators. Phys. Rev. B 52, R5467–R5470 (1995).
Toth, S. & Lake, B. Linear spin wave theory for single-Q incommensurate magnetic structures. J. Phys.: Condens. Matter 27, 166002 (2015).
Iyama, A. & Kimura, T. Magnetoelectric hysteresis loops in Cr2O3 at room temperature. Phys. Rev. B 87, 180408 (2013).
Mufti, N. et al. Magnetoelectric coupling in MnTiO3 . Phys. Rev. B 83, 104416 (2011).
Birss, R. R. Symmetry and Magnetism North-Holland (1966).
Landau, L. D. & Lifshitz, E. M. Statistical Physics Part IPergamon (1980).
Yamauchi, K., Barone, P. & Picozzi, S. Theoretical investigation of magnetoelectric effects in Ba2CoGe2O7 . Phys. Rev. B 84, 165137 (2011).
Yamauchi, K., Oguchi, T. & Picozzi, S. Ab-initio prediction of magnetoelectricity in infinite-layer CaFeO2 and MgFeO2 . J. Phys. Soc. Jpn 83, 094712 (2014).
Campbell, B. J., Stokes, H. T., Tanner, D. E. & Hatch, D. M. ISODISPLACE: a web-based tool for exploring structural distortions. J. Appl. Crystallogr. 39, 607–614 (2006).
Aroyo, M. I. et al. Crystallography online: Bilbao crystallographic server. Bulg. Chem. Commun. 43, 183–197 (2011).
Aroyo, M. I. et al. Bilbao Crystallographic Server: I. Databases and crystallographic computing programs. Z. Kristallogr. 221, 15–27 (2006).
Aroyo, M. I., Kirov, A., Capillas, C., Perez-Mato, J. M. & Wondratschek, H. Bilbao Crystallographic Server. II. Representations of crystallographic point groups and space groups. Acta Crystallogr. A 62, 115–128 (2006).
Arima, T. Ferroelectricity induced by proper-screw type magnetic order. J. Phys. Soc. Jpn 76, 073702 (2007).
Hwu, S.-J. et al. A new class of hybrid materials via salt inclusion: novel copper(II) arsenates Na5ACu4(AsO4)4Cl2 (A=Rb, Cs) composed of alternating covalent and ionic lattices. J. Am. Chem. Soc. 124, 12404–12405 (2002).
Williams, E. R., Marshall, K. & Weller, M. T. Copper (II) chlorofluorophosphate: a new layered square-net for intercalating amines. CrystEngCommn. 17, 160–164 (2015).
Momma, K. & Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011).
Chapon, L. C. et al. Wish: the new powder and single crystal magnetic diffractometer on the second target station. Neutron News 22, 22–25 (2011).
Fischer, P., Keller, L., Schefer, J. & Kohlbrecher, J. Neutron diffraction at SINQ. Neutron News 11, 19–21 (2000).
Rodriguez-Carvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Physica B 192, 55–69 (1993).
Janßen, S., Mesot, J., Holitzner, L., Furrer, A. & Hempelmann, R. Focus: a hybrid TOF-spectrometer at SINQ. Physica B 234–236, 1174–1176 (1997).
Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558–561 (1993).
Kresse, G. & Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium. Phys. Rev. B 49, 14251–14269 (1994).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169–11186 (1996).
Kresse, G. & Furthmüller, J. Efficiency of ab initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mat. Sci. 6, 15–50 (1996).
Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953–17979 (1994).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758–1775 (1999).
We thank Y. Kato and Y. Motome for helpful discussions and H. Tada for specific heat measurements. We are grateful for the initial neutron diffraction measurements by L. Keller. Experiments at the ISIS Pulsed Neutron and Muon Source were supported by a beamtime allocation from the Science and Technology Facilities Council. Neutron-scattering experiments were carried out at the continuous spallation neutron source SINQ at the Paul Scherrer Institut at Villigen PSI (Switzerland). This work was partially supported by JSPS KAKENHI Grants 26610103, 16K05449, 26800186 and 24244058, by European Research Council grant CONQUEST, and by the Swiss National Science Foundation and its Sinergia network MPBH.
The authors declare no competing financial interests.
About this article
Cite this article
Kimura, K., Babkevich, P., Sera, M. et al. Magnetodielectric detection of magnetic quadrupole order in Ba(TiO)Cu4(PO4)4 with Cu4O12 square cupolas. Nat Commun 7, 13039 (2016). https://doi.org/10.1038/ncomms13039
npj Quantum Materials (2021)