Magnetoelectric behavior from cluster multipoles in square cupolas: Study of Sr(TiO)Cu 4 (PO 4 ) 4 in comparison with Ba and Pb isostructurals
Yasuyuki Kato, Kenta Kimura, Atsushi Miyake, Masashi Tokunaga, Akira Matsuo, Koichi Kindo, Mitsuru Akaki, Masayuki Hagiwara, Shojiro Kimura, Tsuyoshi Kimura, Yukitoshi Motome
MMagnetoelectric behavior from cluster multipoles in square cupolas:Study of Sr(TiO)Cu (PO ) in comparison with Ba and Pb isostructurals Yasuyuki Kato , Kenta Kimura , Atsushi Miyake , Masashi Tokunaga , Akira Matsuo , Koichi Kindo ,Mitsuru Akaki , Masayuki Hagiwara , Shojiro Kimura , Tsuyoshi Kimura , and Yukitoshi Motome Department of Applied Physics, University of Tokyo, Bunkyo, Tokyo 113-8656, Japan Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8561, Japan Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan Center for Advanced High Magnetic Field Science, Graduate School of Science,Osaka University, Toyonaka, Osaka 560-0043, Japan Institute for Materials Research, Tohoku University, Sendai, Miyagi 980-8577, Japan (Dated: October 30, 2018)We report our combined experimental and theoretical study of magnetoelectric properties of anantiferromagnet Sr(TiO)Cu (PO ) , in comparison with the isostructurals Ba(TiO)Cu (PO ) andPb(TiO)Cu (PO ) . The family of compounds commonly possesses a low-symmetric magnetic unitcalled the square cupola, which is a source of magnetoelectric responses associated with the magneticmultipoles activated under simultaneous breaking of spatial inversion and time reversal symmetries.Measuring the full magnetization curves and the magnetic-field profiles of dielectric constant forSr(TiO)Cu (PO ) and comparing them with the theoretical analyses by the cluster mean-fieldtheory, we find that the effective S = 1 / (PO ) and Pb(TiO)Cu (PO ) , well explains the experimental results by tuningthe model parameters. Furthermore, elaborating the phase diagram of the model, we find that thesquare cupolas could host a variety of magnetic multipoles, i.e., monopole, toroidal moment, andquadrupole tensor, depending on the parameters that could be modulated by deformations of themagnetic square cupolas. Our results not only provide a microscopic understanding of the series ofthe square cupola compounds, but also stimulate further exploration of the magnetoelectric behaviorarising from cluster multipoles harboring in low-symmetric magnetic units. I. INTRODUCTION
The magnetoelectric (ME) effect is a cross correlationbetween magnetic and electric properties of matters, andenables us to control the electric (magnetic) polarizationby the magnetic (electric) field. The ME effect in a solidwas firstly conjectured for Cr O by Dzyaloshinskii in1959 [1], and indeed observed by Astrov in 1960 [2]. Ithas attracted renewed interest since the discovery of ahuge ME effect in TbMnO in 2003 [3]. Materials host-ing such a huge ME response have been extensively stud-ied as they are potentially useful for future power-savingdevices functioning without electric currents [4].The necessary condition for linear ME effects (ME re-sponses proportional to the applied magnetic and electricfields) is the absence of both spatial inversion and time re-versal symmetries. This condition is satisfied in magnet-ically ordered states on noncentrosymmetric structures.Among them particularly interesting are the systems in-volving noncentrosymmetric clusters made of magneticions, such as magnetic trimers. In such systems, the lin-ear ME effect is explained by magnetic multipoles definedon each cluster [5–8]. In the cluster multipole description,a spin texture on a cluster is decomposed into the mag-netic monopole, toroidal moment, and quadrupole ten-sor, all of which are odd under the operations of spatialinversion and time reversal. Each multipole is associatedwith a particular ME tensor, and hence, the decompo-sition provides systematic understanding of the ME re-sponses in these cluster systems. Recently, single crystals of a series of ME active in-sulating antiferromagnets, A (TiO)Cu (PO ) ( A TCPO, A = Ba, Sr, and Pb), have been synthesized [9, 10].These compounds are composed of magnetic clustersCu O resembling the square cupola that is the fourthJohnson solid [11]. Each square cupola accommodatesfour S = 1 / cations.The family of compounds has a quasi-two-dimensionallattice structure composed of a periodic array of thesquare cupolas. More precisely, as schematically shownin Fig. 1, upward ( α ) and downward ( β ) square cupolasare alternately arranged in each layer. In the absenceof an external magnetic field, these compounds exhibita finite-temperature ( T ) phase transition to an antifer-romagnetically ordered phase where each square cupolahosts a q x − y quadrupole type spin texture [12] (theN´eel temperature is T N (cid:39) A = Ba, Pb, and Sr, respectively). This leads to ME re-sponses, such as a dielectric anomaly at T N in BaTCPOand SrTCPO [10, 13] and a magnetic-field-induced netelectric polarization in PbTCPO [10]. The differenceoriginates from the way of layer stacking: the magneticlayers are stacked in a staggered manner in the Ba and Srcases (layered antiferroic order of the q x − y quadrupole),while in a uniform manner in the Pb case (ferroic or-der of the q x − y quadrupole). These ME behaviorsare understood in terms of the cluster multipoles of thequadrupole type. The theoretical analyses based on a mi-croscopic model were also reported for BaTCPO [14] andPbTCPO [15]. For both compounds, the cluster mean- a r X i v : . [ c ond - m a t . s t r- e l ] O c t (a) Cu O
011 1098 76 5 43 2 1 151413 12 ↵
In this section, we describe the experimental meth-ods for the measurements of magnetization and dielec-tric constant. We also introduce the theoretical modeland method for analyzing the microscopic property ofthe antiferromagnetic square cupola systems, A TCPO.
A. Experimental method
Single crystals of SrTCPO were grown by the fluxmethod as described previously [9]. Powder X-raydiffraction (XRD) measurements on crushed single crys-tals confirmed a single phase. The crystal orientation wasdetermined by the Laue X-ray method. A superconduct-ing magnet system up to 18 T and down to 1.6 K at theTohoku University was used for measurements of dielec-tric properties. For dielectric measurements, single crys-tals were cut into thin plates and subsequently electrodeswere formed by painting silver pastes on a pair of thewidest surfaces. The dielectric constant ε was measuredusing an LCR meter (Agilent E4980) at an excitation fre-quency of 100 kHz. Pyroelectric current was measured byan electrometer (Keithley 6517) to monitor electric polar-ization. High-field magnetization in magnetic fields up to45 T was measured at 1.4 K using an induction methodwith a multilayer pulsed magnet installed at the Inter-national MegaGauss Science Laboratory of Institute forSolid State Physics at The University of Tokyo. Multi-frequency electron spin resonance (ESR) measurements(600–1400 GHz) in pulsed magnetic fields were performedat the Center for Advanced High Magnetic Field Sciencein Osaka University to obtain the g -values for the fielddirections along [100], [110] and [001]. The g -values werefound to be isotropic within the experimental accuracy: g = 2 . B. Model and theoretical method
We consider an effective model for the S = 1 / cations, which wasfirst introduced for BaTCPO [14] and later applied toPbTCPO [15]. The model includes four dominant anti-ferromagnetic exchange interactions, J , J , J (cid:48) , and J (cid:48)(cid:48) ,where J and J are intracupola exchange interactions,and J (cid:48) and J (cid:48)(cid:48) are intralayer and interlayer interactionsbetween the cupolas, respectively (Fig. 1). In addition,we take into account the Dzyaloshinskii-Moriya (DM) in-teraction originating from the relativistic spin-orbit cou-pling on the J bonds as well as the Zeeman coupling toan external magnetic field. The Hamiltonian reads H = (cid:88) (cid:104) i,j (cid:105) [ J S i · S j − D ij · S i × S j ] + J (cid:88) (cid:104)(cid:104) i,j (cid:105)(cid:105) S i · S j + J (cid:48) (cid:88) ( i,j ) S i · S j + J (cid:48)(cid:48) (cid:88) (( i,j )) S i · S j − gµ B (cid:88) i B · S i , (1)where S i = ( S xi , S yi , S zi ) represents the S = 1 / i , and the sums for (cid:104) i, j (cid:105) , (cid:104)(cid:104) i, j (cid:105)(cid:105) , ( i, j ), and(( i, j )) run over the J , J , J (cid:48) , and J (cid:48)(cid:48) bonds, respec-tively. The last term represents the Zeeman couplingwith the isotropic g -factor g and the Bohr magneton µ B .The DM interaction is characterized by the DM vector D ij . For simplicity, we assume that the Cu O magneticunits have the same symmetry with the perfect square cupola, C v . Then, referring the Moriya rules [17], weset D ij in the plain perpendicular to the corresponding J bond with a common angle θ ij = θ from the [001]axis, and a common strength D = | D ij | [the yellow ar-rows in Fig. 1(a)]. Note that some features are omitted inthe present model, such as the chiral twist of the squarecupolas and anisotropic exchange interactions other thanthe DM.In the previous analysis for BaTCPO, the effectivemodel in Eq. (1) successfully reproduces the entire mag-netization curves up to above the saturation field and thedielectric anomaly observed at the N´eel temperature inthe low magnetic field regime with the parameter set [14]: J = 1 , J = 1 / , J (cid:48) = 1 / , J (cid:48)(cid:48) = 1 / ,D = 0 . , and θ = 80 ◦ , (2)on the basis of an estimate of J = 3 .
03 meV by first-principles calculations [13]. Furthermore, by switch-ing the sign of J (cid:48)(cid:48) from antiferromagnetic to ferromag-netic with slight changes of other parameters, this modelis capable of reproducing the uniform manner of layerstacking with the net electric polarization appearing inPbTCPO when B (cid:107) [110] [15]. In particular, the unusualsign change of the polarization observed in the high fieldregime is explained by the model analysis. Through theanalyses of BaTCPO [14] and PbTCPO [10, 15], the mainorigin of the ME effects is identified as the nonrelativisticexchange striction mechanism [18].In the present analysis, we optimize the model parame-ters to reproduce the experimental magnetization curvesmeasured for SrTCPO, as discussed in Sec. III A. In thecalculations, following the previous analyses [14, 15], weemploy the CMF method, which is suitable for cluster-based magnetic insulators. In the CMF method, theweak intercupola interactions ( J (cid:48) and J (cid:48)(cid:48) terms) aredealt with by the conventional mean-field approximation,namely, S i · S j is decoupled as S i · S j (cid:39) (cid:104) S i (cid:105) · S j + S i ·(cid:104) S j (cid:105) − (cid:104) S i (cid:105) · (cid:104) S j (cid:105) , where (cid:104) S i (cid:105) is the expectation value ofthe spin operator S i . On the other hand, the intracupolainteractions are dealt with by the exact diagonalization,and therefore, quantum fluctuations in each cupola arefully taken into account. In this paper, we consider foursquare cupolas allocated as shown in Fig. 1(a) in theCMF method, namely we consider 16 sublattices. III. RESULTS
In this section, the results of experiments and theoret-ical calculations are shown. In Sec. III A, we show theexperimental data of the full magnetization curves forthree different directions of magnetic fields for SrTCPO,and determine the optimal parameter set of the theoreti-cal model (1) to reproduce the experimental results. Wedemonstrate the validity of the model for the dielectricconstant and the phase diagram in Secs. III B and III C,respectively. In Sec. III D, we show the detailed analy-sis of the antiferromagnetic order parameters and electricpolarizations in each phase. In Sec. III E, we show theground-state phase diagrams of the theoretical model inan extended parameter space: an interpolation betweenSrTCPO and BaTCPO, and a change of the DM angle θ for the Sr parameter set, for the latter of which we findadditional phases. Finally, in Sec. III F, we investigatethe ME responses for all the phases appearing in this pa-per by the cluster multipole decomposition of the spinconfiguration of each phase. A. Magnetization curves and model setup d m / d ( g µ B B ) g µ B B m g µ B B B ||[001] B ||[100] B ||[110] d M / d B ( µ B / C u + / T ) B (T) M ( µ B / C u + ) B (T) B ||[001] B ||[100] B ||[110] (a)(b) (c)(d) B [100] c
40 T, the magnetization for allthe B directions shows a saturation at ∼ µ B / Cu .The saturation-magnetization values are corrected by the g -values determined by the ESR. We note that dM/dB shows a hump at B (cid:39)
35 T only for B (cid:107) [110] as shownin Fig. 2(b).A significant difference between the magnetizationcurves of SrTCPO and those of BaTCPO and PbTCPOis found in the relative magnitude of B [001] c and B [100] c , namely, B [001] c > B [100] c for SrTCPO while B [001] c < B [100] c for BaTCPO and PbTCPO. Furthermore, the ratio of thecritical field to the saturation field, b [001] c ≡ B [001] c /B [001]sat ,is much larger: b [001] c ∼ .
75 for SrTCPO while b [001] c ∼ . b [001] c ∼ . J as J (cid:46) .
6, while keeping the otherparameters as those for BaTCPO in Eq. (2). We notethat the smaller J is also reasonable to reproduce thesmaller saturation fields ∼
40 T compared to ∼
60 Tin BaTCPO [14]. At the same time, however, we findthat the parameter change leads to an additional phasetransition not observed in experiments in the higher fieldregime for B (cid:107) [110]. This is remedied by a slight increaseof θ . Consequently, we obtain the optimal parameter setfor SrTCPO by adjusting only J and θ as J = 0 . θ = 90 ◦ , (3)from Eq. (2) for BaTCPO.The main difference of the model parameters be-tween SrTCPO and BaTCPO is in the magnitude of thenearest-neighbor exchange interaction J ; J for SrTCPOis taken as 60% of that for BaTCPO. The parameterchange is consistent with the fact that both the satura-tion field and the Curie Weiss temperature of SrTCPOare approximately 2/3 of those of BaTCPO in experi-ments [10]. We note that J was estimated to be ∼ T , respectively. The entire magnetization curvesare well reproduced by the optimal parameter set, inthe following aspects: (i) B [001]sat < B [110]sat < B [100]sat , (ii) B [100] c < B [110] c < B [001] c , (iii) b [001] c ∼ .
75, (iv) the fieldderivative for out-of-plane field ( B (cid:107) [001]) lower thanthat for in-plane field ( B (cid:107) [100] or [110]) in the low fieldregime, and (v) a hump near the saturation in the fieldderivative for B (cid:107) [110]. B. Dielectric anomaly
Since the maximum field of 18 T available in thepresent dielectric measurements cannot access the criticalfield for B (cid:107) [001], we performed the dielectric measure-ments only in B (cid:107) [100] and B (cid:107) [110]. Figures 3(a) and3(b) show the experimental data of the dielectric constantat low T measured for SrTCPO in B (cid:107) E (cid:107) [100], and B (cid:107) [110] and E (cid:107) [001] up to B = 18 T, respectively ( E isthe electric field). The dielectric constant exhibits sharpanomalies at the magnetic fields where the magnetizationchanges discontinuously. Note that the pyroelectric cur-rent measurement does not detect any signal indicative ofan onset of a macroscopic electric polarization associatedwith these dielectric anomalies. ε [ ] B (T) T = 4.2 K4.0 K3.5 K3.0 K2.5 K2.0 K ε [ ] B (T) T = 2 K ε [ ] g µ B BT = 0.30.20.1 (a) B k [100]
15 T at low T , and the critical fields slightly in-crease while raising T . The critical temperatures of thehigh field phase to the paramagnetic state show a smallincrease in the narrow field region of the measurement.We show the finite- T phase diagram obtained by theCMF method in Fig. 6. Figures 6(b) and 6(c) correspondto the experimental results in Fig. 5. We find that thephase diagrams for B (cid:107) [100] and B (cid:107) [110] are simi-lar to each other; we call the low-field ordered phase Z(Z’) and the high field one Y (Y’) for B (cid:107) [100] ([110]).The results indicate that our theory well reproduces theexperimental results in Fig. 5, except for the small en-hancement of the critical temperature in the high fieldphase. This discrepancy might be reconciled by takinginto account the fluctuation effect beyond the CMF ap-proximation which may play an important role in thephase competing region. Based on the good agreementbetween the experiment and theory, we identify the lowfield phases in experiments as Z and Z’ and the high fieldphases as Y and Y’. We will discuss the order parametersand electric polarizations in these phases in Sec. III D.In addition, we also show the phase diagram for B (cid:107) [001] in Fig. 6(a), in which the high-field phases (IIand III) are not accessible in the present dielectric ex-periments. The phase diagram is similar to that forBaTCPO [14]: the stabilized phases are common, in-cluding the hidden phase III. We note that the phasediagrams in Figs. 6(b) and 6(c) are also similar to thosefor BaTCPO (see Ref. [14] for B (cid:107) [100] and Appendix Bfor B (cid:107) [110]). D. Order parameters and electric polarizations
Based on the similarity of the phase diagrams, herewe analyze the theoretical results for SrTCPO by theantiferromagnetic order parameters used in the study ofBaTCPO [14]: m AF ≡ N spin (cid:88) (cid:96) ( − (cid:96) p (cid:96) (cid:104) S (cid:96) (cid:105) , (6)where p (cid:96) = +1( −
1) for the upper (lower) layer inFig. 1(a), and N spin is the number of spins. Figures 7(a-c) show the magnetic field dependence of m AF at zero T for the three different field directions [19]. While only the z component of the order parameter is nonzero ( m z AF (cid:54) = 0and m x,y AF = 0) for the low field phase including B = 0, theorientation of m AF changes to the perpendicular direc-tion to the z axis through a first-order phase transitionwith the magnetization jump: | m x AF | = | m y AF | (cid:54) = 0 for B (cid:107) [001], m y AF (cid:54) = 0 for B (cid:107) [100], and m [1¯10]AF (cid:54) = 0 for B (cid:107) [110]. m AF vanishes continuously at the saturationfield for all the directions.Figures 7(d-f) show the field dependence of the stag-gered component of the electric polarization, P AF , com-puted based on the exchange striction mechanism [18].In the present system, a ferroelectric polarization canappear in each layer, but the direction is antiparallel be-tween the neighboring layers, resulting in the vanishingnet polarization. Thus, we define the interlayer-staggeredcomponent as [20] P AF = (cid:88) (cid:104) i,j (cid:105) p i n ij (cid:104) S i · S j (cid:105) . (7) P AF behaves differently for three field directions: | P x AF | = | P y AF | (cid:54) = 0 in the phase II for B (cid:107) [001], P x AF (cid:54) = 0in Z for B (cid:107) [100], and P [1¯10]AF (cid:54) = 0 in Z’ and P [001]AF (cid:54) = 0in Y’ for B (cid:107) [110]. Note that P [001]AF changes its sign inthe phase Y’. Similar behavior was found in PbTCPO asa sign change of the net electric polarization parallel to[001] [15].The results for m AF and P AF are summarized in Ta-ble I. The table includes other phases found in Sec. III Eby changing the model parameters. E. Ground-state phase diagram in an extendedparameter space
Thus far, we have discussed the model in Eq. (1) withthe parameter set for SrTCPO. In this section, we extendthe parameter space and try to find other interesting MEbehaviors for future material investigation.First, considering a solid solution of the Sr and Bacompounds, we study the interpolation between the pa-rameter sets for SrTCPO and BaTCPO. Figure 8 showsthe ground-state phase diagrams computed by chang-ing the parameters continuously between SrTCPO and (c)(b)(a) T g µ B B I IIIII B k [001]
0] - P x AF (cid:54) = P y AF Z’ [001] [1¯10] - m AF ⊥ P AF B (cid:107) [110] Y’ [1¯10] [001] - m AF ⊥ P AF M’ - - [110] P (cid:107) B B = 0 ( θ > θ c ) [001] - - B = 0 ( θ < θ c ) - - -TABLE I. Direction of the antiferromagnetic order parame-ter m AF [Eq. (6)], the interlayer-staggered component of theelectric polarization, P AF [Eq. (7)], and the net electric po-larization P [Eq. (5)] in each phase. The symbol “-” indicatesthat the order parameter vanishes. θ c is the critical angle at B = 0: θ c = 12 . ± . ◦ (see the text for details). imply no qualitatively new ME phase for a solid solution(Sr,Ba)TCPO.Next, we study the ground-state phase diagram bychanging only the DM angle θ for the parameter set forSrTCPO. Such a change may be possible by a defor-mation of square cupolas, e.g., by an external pressureand chemical substitutions. Figure 9 shows the resultsas functions of θ and the magnetic field B . In additionto the phases appearing in the previous sections (I, II,III, Z, Y, Z’, and Y’), we find additional phases IV, V,M, T, S, and M’ in the small θ region.When B = 0, the system exhibits a phase transition atthe critical angle θ c = 12 . ± . ◦ between the spin con-figuration of monopole type for θ < θ c [Fig. 10(a)] and of a t x t y t z q xy q xx q yy remarksI * - - - - (cid:88) (cid:88) II * (cid:88) (cid:88) - * * * | t x | = | t y | in a layer B (cid:107) [001] III * (cid:88) /- -/ (cid:88) - - * * Either t x or t y is nonzero.IV ◦ - - - - ◦ ◦ q xx = q yy V * - - (cid:88) / ◦ - * * t z depends on the layer stacking; q xx = q yy Z (cid:88) - * - - (cid:88) (cid:88) q xx (cid:39) − q yy in a layer [21]Y - (cid:88) * - - - - B (cid:107) [100] M ◦ - * - - ◦ ◦ T ◦ * * ◦ ◦ ◦ ◦ | q xy | (cid:28) | a | , | t z | , | q xx | , and | q yy | S (cid:88) * * (cid:88) (cid:88) (cid:88) (cid:88) | q xy | (cid:28) | a | , | t z | , | q xx | , and | q yy | in a layerZ’ - * * (cid:88) - (cid:88) (cid:88) q xx = − q yy B (cid:107) [110] Y’ - (cid:88) (cid:88) - - - - t x = − t y ( t (cid:107) [1¯10])M’ ◦ - * - ◦ ◦ ◦ q xx = q yy B = 0 ( θ > θ c ) - - - - - (cid:88) (cid:88) q xx = − q yy in a layer B = 0 ( θ < θ c ) ◦ - - - - ◦ ◦ q xx = q yy TABLE II. Cluster multipole decomposition of the spin configurations in each square cupola into the monopole a , toroidalmoment t , and quadrupole tensor q µν . See Eqs. (9)-(11). The symbol ◦ means that the net value for the four cupolas in theunit cell is nonzero in the CMF solutions; the symbol (cid:88) means that the value for each layer is nonzero, but the net valuevanishes because of the cancellation between the layers; the symbol * means that the value for each square cupola is nonzero,but that of each layer vanishes because of the cancellation; the symbol “-” means that the value for each square cupola is zero. Sr Ba g µ B B Sr Ba
Sr Ba g µ B B Sr Ba
Sr Ba g µ B B Sr Ba (c)
Z’Y’ B k [110]
7. The mag-netic field direction is (a) B (cid:107) [001], (b) B (cid:107) [100], and (c) B (cid:107) [110]. q x − y quadrupole type for θ > θ c [Fig. 10(b)] [13, 14](see also Appendix A). This transition occurs mainlybecause of the energy competition between the DM in-teraction and the J exchange interaction, as shown inFig. 10(c); the former energy increases while the latterdecreases for θ > θ c . We note that the J exchangeinteraction also contributes to the stabilization of themonopole-type spin configuration. The competition isalso understood from the spin configurations shown inFigs. 10(a) and 10(b). For θ < θ c , (cid:104) S i (cid:105) × (cid:104) S j (cid:105) is almost parallel to D ij , which is preferable for the DM energy,while the neighboring spin pairs are almost perpendicu-lar to each other, which is unfavorable for the J energy.They are vice versa for θ > θ c .Figure 9(a) shows the phase diagram for B (cid:107) [001].When turning on the magnetic field, the monopole( q x − y quadrupole) type spin configuration continuouslydevelops into that of the phase IV (I) for θ < θ c ( θ > θ c ).While increasing B , the phase IV is extended to the larger θ region, and instead the phase I is narrowed. With a fur-ther increase of B , the phase IV turns into the phase V,while the phase I turns into the phase II before saturationin the region of θ (cid:46) ◦ . The typical spin configurationsare shown in Appendix A and Supplemental Material forRef. [14].Figures 9(b) and 9(c) show the phase diagrams for B (cid:107) [100] and B (cid:107) [110], respectively. Similar to the caseof B (cid:107) [001], by introducing the magnetic field B (cid:107) [100]and B (cid:107) [110], the monopole ( q x − y quadrupole) typespin configuration appears in the phase M (Z) and phaseM’ (Z’) for θ < θ c ( θ > θ c ), respectively. However, thephases M and M’ shrink as B increases, in contrast tothe case of B (cid:107) [001]. For B (cid:107) [110], the phase M’ di-rectly turns into the phase Z’, whereas for B (cid:107) [100],intermediate phases S and T are found before enteringto the phase Z. The typical spin configurations for theseadditional phases are shown in Appendix A. In the inter-mediate θ region, the phase Z (Z’) turns into the phaseY (Y’) before saturation.We summarize in Table I the antiferromagnetic or-der parameter m AF and the interlayer-staggered compo- B // [ ] θ ( o ) 0 1 2 3 0 60 120 180 B // [ ] θ ( o ) 0 1 2 3 0 60 120 180 B // [ ] θ ( o ) 0 1 2 3 0 60 120 180 (a) III I+IIIVV g µ B B
The ME behaviors in different phases found in the pre-vious sections can be understood in terms of multipoles.In the present system, the multipoles are defined in acluster form for a square cupola. For the cluster multi-pole description, we define a 3 × M ij ≡ (cid:88) (cid:96) ˜ r i(cid:96) S j(cid:96) , (8)where i, j takes x , y , or z ; ˜ r (cid:96) is the relative coordinate ofsite (cid:96) from the center of the square cupola, and the sumis taken for the four sites in the square cupola. Then,the tensor M ij can be decomposed into the cluster mul-tipoles, i.e., the pseudoscalar monopole a , the toroidalmoment vector t = ( t x , t y , t z ), and the quadrupole ten-sor q ij , which are defined as a = 13 (cid:88) i M ii , (9) t k = 12 (cid:88) i,j ε ijk M ij , (10) q ij = 12 (cid:32) M ij + M ji − δ ij (cid:88) k M kk (cid:33) , (11)0respectively [6], where δ ij and ε ijk represent the Kro-necker delta and the three-dimensional Levi-Civita sym-bol, respectively.We summarize the results of the cluster multipole de-composition in Table II. Here q zz , q yz , and q zx are omit-ted because ˜ r z(cid:96) = 0 for all (cid:96) leads the three relations, a = − q zz , t x = q yz , and t y = − q zx . The nonzerocomponents of the cluster multipoles explain the ME be-haviors in each phase. For example, in the phases I, Z,and Z’, the nonzero P AF in B (cid:107) [100] and [110] is nat-urally expected from the quadrupole of x − y type, q x − y = q xx − q yy . The quadrupole also explains thedivergent behavior of the dielectric anomaly in ε [100] ( T )( ε [1¯10] ( T )) for B (cid:107) [100] ( B (cid:107) [110]) at the N´eel tem-perature, as commonly observed in BaTCPO [13, 14][Figs. 4(a) and 4(e)]. On the other hand, in the phase Y’,the toroidal moment t (cid:107) [1¯10] becomes nonzero in eachlayer, which indicates the free energy has a coupling termbetween E [001] and B [110] . This explains P AF (cid:107) [001] in-duced by B (cid:107) [110]. Similarly, in the phase II, t (cid:107) [110]or t (cid:107) [1¯10] becomes nonzero in each layer, which ex-plains P AF (cid:107) [1¯10] or [110] induced by B (cid:107) [001]. In thephase III, t x or t y becomes nonzero in each layer, whichexplains P AF (cid:107) [010] or [100] induced by B (cid:107) [001].Meanwhile, in the newly-found phases in the small θ region, the net monopole a is activated, together with thequadrupole tensor q xx = q yy (cid:54) = 0. This indicates that thefree energy has a coupling term of E µ B µ with a uniaxialanisotropy, i.e., the coefficient of E z B z is different fromthat of E x B x and E y B y . This explains P (cid:107) B in thephases IV, M, and M’. In the phase T, the net toroidalmoment t (cid:107) [001] is activated, which explains a nonzerocomponent of P perpendicular to both B and t , in addi-tion to a component parallel to B . In the phase V where P = P AF = 0, the nonzero t (cid:107) [001] indicates the freeenergy term of E x B y − E y B x . This means that P or P AF , which is perpendicular to B and [001], is activatedby tilting the magnetic field from B (cid:107) [001] depending onthe way of layer stacking. IV. SUMMARY AND CONCLUDINGREMARKS
In conclusion, we have investigated the magnetoelec-tric behavior of SrTCPO composed of antiferromagneticsquare cupolas by the combination of experimental mea-surements and theoretical analyses. In experiments, bythe help of stable single crystal growth, we obtained thefull magnetization curves at low temperature (1.4 K upto 45 T) for three different field directions, B (cid:107) [001], B (cid:107) [100], and B (cid:107) [110], and the dielectric constantas functions of temperature and the magnetic field (upto 18 T) for B (cid:107) [100] and B (cid:107) [110]. The magneti-zation curves show magnetization jumps, whose criticalfields depend on the field direction, similar to those ofisostructurals BaTCPO [9, 14] and PbTCPO [15]. Thedielectric constant shows an anomaly at the critical fields. We found several differences between SrTCPO and pre-viously studied BaTCPO and PbTCPO; in particular,the ratio of the critical field to the saturation field ismuch larger in SrTCPO for B (cid:107) [001]. To understandthe experimental observations, we studied a spin modelby using the the CMF method, following the previousstudies for BaTCPO [14] and PbTCPO [15]. We foundthat the model well explains all the data for SrTCPO, in-cluding the finite- T phase diagrams, by tuning the modelparameters. The agreements strongly support the valid-ity of the simple microscopic model and our analyses forthe isostructural series of A TCPO.We have also investigated further interesting ME be-haviors by extending the model parameter space. Al-though we did not find any additional phases by lin-early interpolating the parameters between the Sr andBa cases, we unveiled a variety of unprecedented phases,including ferroelectric ones, by changing the DM anglewith the parameter set for the Sr case. We investigatedthe ME behaviors in all the phases, and rationalized themby using the cluster multipole decomposition. We foundthat the spin configurations in the additional phases fora small DM angle acquire the cluster form of not onlyquadrupole, which was already identified for the previousstudies, but also monopole and toroidal moments. Thus,our results indicate that the antiferromagnetic squarecupola could host all the multipoles giving rise to thelinear ME effect. A smaller θ is expected to be possiblyrealized by compressing the cupola in the [001] direction,e.g., by an external pressure and chemical substitutions.Our findings would stimulate further material investiga-tion in the family of A TCPO and the materials composedof the Cu-based square cupolas [22–24] for such intrigu-ing ME behaviors.
ACKNOWLEDGMENTS
The authors thank M. Toyoda and K. Yamauchi forfruitful discussions. This work was supported by JSPSGrant Numbers JP17H01143, JP16K05413, JP16K05449and by the MEXT Leading Initiative for Excellent YoungResearchers (LEADER). The ESR work was carried outat the Center for Advanced High Magnetic Field Sci-ence in Osaka University under the Visiting Researcher’sProgram of the Institute for Solid State Physics, theUniversity of Tokyo. Measurements of dielectric prop-erties in a magnetic field were performed at the HighField Laboratory for Superconducting Materials, Insti-tute for Materials Research, Tohoku University (ProjectNo. 18H0014). Numerical calculations were conductedon the supercomputer system in ISSP, The Universityof Tokyo. K.K., M.A., M.H., S.K., T.K., and Y.M. arepartially supported by JSPS Core-to-Core Program, A.Advanced Research Networks.1
Appendix A: Spin configurations in the small θ region Figure 11 shows typical spin configurations in thephases IV, V, M, T, S, M’, Y’, and Z’ in the small θ re-gion, obtained by the CMF method. Spin configurationsof other phases (I, II, III, Y, and Z) have been reported in Ref. [14]. Appendix B: Phase diagram for BaTCPO withB (cid:107) [110]
Figure 12 shows the phase diagrams computed withthe parameter set for BaTCPO [Eq. (2)] and B (cid:107) [110]by the CMF method for comparison to those for SrTCPOin Figs. 9(c) and 6(c). The phase diagrams for the othertwo field directions were reported in Ref. [14]. [1] I. E. Dzyaloshinskii, Sov. Phys. JETP , 628 (1960).[2] D. N. Astrov, Sov. Phys. JETP , 708 (1960).[3] T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima,and Y. Tokura, Nature , 55 (2003).[4] M. Fiebig, T. Lottermoser, D. Meier, and M. Trassin,Nature Reviews Materials , 16046 (2016).[5] A. A. Gorbatsevich and Y. V. Kopaev, Ferroelectrics , 321 (1994).[6] N. A. Spaldin, M. Fiebig, and M. Mostovoy, J. Phys.Condens. Matter , 434203 (2008).[7] Y. V. Kopaev, Physics-Uspekhi , 1111 (2009).[8] N. A. Spaldin, M. Fechner, E. Bousquet, A. Balatsky,and L. Nordstr¨om, Phys. Rev. B , 094429 (2013).[9] K. Kimura, M. Sera, and T. Kimura, Inorg. Chem. ,1002 (2016).[10] K. Kimura, M. Toyoda, P. Babkevich, K. Yamauchi,M. Sera, V. Nassif, H. M. Rønnow, and T. Kimura,Phys. Rev. B , 134418 (2018).[11] N. W. Johnson, Canad. J. Math , 169 (1966).[12] P. Babkevich, L. Testa, K. Kimura, T. Kimura, G. S.Tucker, B. Roessli, and H. M. Rønnow, Phys. Rev. B , 214436 (2017).[13] K. Kimura, P. Babkevich, M. Sera, M. Toyoda, K. Ya-mauchi, G. S. Tucker, J. Martius, T. Fennell, P. Manuel,D. D. Khalyavin, R. D. Johnson, T. Nakano, Y. Nozue,H. M. Rønnow, and T. Kimura, Nat. Commun. , 13039(2016). [14] Y. Kato, K. Kimura, A. Miyake, M. Tokunaga, A. Mat-suo, K. Kindo, M. Akaki, M. Hagiwara, M. Sera,T. Kimura, and Y. Motome, Phys. Rev. Lett. ,107601 (2017).[15] K. Kimura, Y. Kato, K. Yamauchi, A. Miyake, M. Toku-naga, A. Matsuo, K. Kindo, M. Akaki, M. Hagiwara,S. Kimura, M. Toyoda, Y. Motome, and T. Kimura,“Magnetic structural unit with convex geometry: a build-ing block hosting an exchange-striction-driven magneto-electric coupling,” (2018), arXiv:1807.10457.[16] S. S. Islam, K. M. Ranjith, M. Baenitz, Y. Skourski, A. A.Tsirlin, and R. Nath, Phys. Rev. B , 174432 (2018).[17] T. Moriya, Phys. Rev. , 91 (1960).[18] I. A. Sergienko, C. S¸en, and E. Dagotto, Phys. Rev. Lett. , 227204 (2006).[19] We found a factor 2 missing in the plot of m AF inRef. [14].[20] We found a mistake in the definition of P AF in Ref. [14]: q i should be omitted.[21] Table SI in the Supplemental Material of Ref. [14] showsa remark q xx = − q yy for Z phase. To be precise, thisremark should be q xx (cid:39) − q yy .[22] S.-J. Hwu, M. Ulutagay-Kartin, J. A. Clayhold,R. Mackay, T. A. Wardojo, C. J. O’Connor, andM. Krawiec, J. Am. Chem. Soc. , 12404 (2002).[23] G. Giester, U. Kolitsch, P. Leverett, P. Turner, and P. A.Williams, Eur. J. Mineral. , 75 (2007).[24] E. R. Williams, K. Marshall, and M. T. Weller, Crys-tEngComm , 160 (2015). (a) B = 0 for θ < θ c (b) (c) phase IV(d)(e) phase V(f) (g) phase M(h) B
5, (e,f) B = 1 .
5, (g,h) B = 0 .
3, (i,j) B = 0 .
4, (k,l) B = 0 .
5, (m,n) B = 0 . B = 0 .
8, and (q,r) B = 1 . T g µ B B B // [ ] θ ( o ) 0 1 2 3 0 60 120 180 (a) g µ B B
FIG. 12. (a) Ground-state and (b) finite- T phase diagramsin the magnetic field parallel to [110] ( B (cid:107)(cid:107)