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Received 28 Nov 2013 | Accepted 24 Jul 2014 | Published 5 Sep 2014 DOI: 10.1038/ncomms5796 Ferroic of magnetic toroidal order

Anne S. Zimmermann1, Dennis Meier2 & Manfred Fiebig2

Electric dipoles and violate spatial inversion symmetry, and magnetic dipoles and break time-inversion symmetry. Breaking both symmetries favours magnetoelectric charge- coupling effects of enormous interest, such as , skyrmions, polar superconductors, topological insulators or dynamic phenomena such as electromagnons. Extending the rationale, a novel type of ferroic order violating space- and time-inversion symmetry with a single order parameter should exist. This existence is fundamental and the inherent magnetoelectric coupling is technologically interesting. A uniform alignment of magnetic vortices, called ferrotoroidicity, was proposed to represent this state. Here we demonstrate that the magnetic vortex pattern identified in LiCoPO4 exhibits the indispensable hallmark of such a ferroic state, namely hysteretic poling of ferrotoroidic domains in the conjugate toroidal field, along with a distinction of toroidal from non-toroidal poling effects. This consolidates ferrotoroidicity as fourth form of ferroic order.

1 HISKP, Universita¨t Bonn, Nussallee 14-16, 53115 Bonn, Germany. 2 Department of Materials, ETH Zu¨rich, Vladimir-Prelog-Weg 4, 8093 Zurich, Switzerland. Correspondence and requests for materials should be addressed to M.F. (email: manfred.fi[email protected]).

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resently, three forms of long-range order, all with individual and electric fields in field-cooling experiments in MnPS3 and transformation properties under space- and time-inversion, LiFeSi2O6 (refs 21,22). These results, however, are insufficient for Phave been demonstrated to be of primary ferroic nature. verifying a ferroic state since key aspects such as hysteretic sub- We distinguish ferromagnetism ( þÀ), ferroelectricity ( Àþ) Curie poling of domains and the distinction between and ferroelasticity ( þþ), where the first and second sign toroidal and non-toroidal contributions to the magnetoelectric represent conservation ( þ ) or violation ( À ) of space- and time- effect were not shown. In LiCoPO4, a was inversion symmetry, respectively. Existence of a fourth form of reoriented via a nonlinear magnetic-field effect, that is, not by ferroic order violating space- and time-inversion symmetry coupling to the toroidal order parameter23. Reports on ( ÀÀ) is compelling and has been proposed. The concept of ferrotoroidic poling are thus virtually non-existent. ferrotoroidicity as this ‘missing piece’ is the topic of this work and Here we establish the ferroic nature of toroidal order, using the ferroic complement to the space- and time-inversion LiCoPO4 as model case. By laser-optical second-harmonic symmetry-breaking phenomena mentioned in the introductory generation (SHG), we measure the hysteretic response of 1–6 paragraph . P LiCoPO4 to a conjugate toroidal field, including the spatially N In general, a toroidal ( ÀÀ)-like moment T ¼ i¼1 riÂsi is resolved evolution of the domains throughout the poling cycle. formed by NZ2 spins per unit cell. In LiCoPO4, our reference We then verify that this response is solely determined by the compound for the following discussion, we have two pairs of cooperation of the toroidal-field-establishing magnetic and spins yielding oppositely oriented toroidal moments of different electric fields and not by either of these fields alone. Most amplitude so that a net moment T ¼ (0,Ty,Tz) survives. importantly, we separate the toroidal from any non-toroidal A spontaneous uniform alignment of toroidal moments is the contribution to the poling process and find a dominance of the basis of a ferrotoroidic state. In refs 7,8, it was shown that the toroidal field. toroidal moment per unit cell, the ‘toroidization’, is a reasonable choice as order parameter. In particular, it does not diverge or Results depend on the choice of origin for the distance vectors r . In spite i Magneto-crystallographic structure of LiCoPO . LiCoPO is of all this, the concept of ferrotoroidicity as additional form of 4 4 chosen as model compound for our experiments because its ferroic order remains questionable because cyclic hysteretic magnetoelectric properties, its toroidal unit-cell structure and the poling between different single-domain states in a conjugate distribution of its domains have been well characterized7,19,23–25. field has never been shown. This reversibility is crucial as LiCoPO has the olivine crystal structure and possesses mmm10 exemplified by crystals with regions of opposite optical activity9. 4 point symmetry in the paramagnetic state. Below 21.8 K, the Since no conjugate field exists that would allow to exert a direct Co2 þ magnetic moments form a collinear magnetic order. The gyrotropic force promoting the crystal towards single-domain resulting point symmetry is 20 that allows a toroidal moment optical activity, there is no ‘ferrogyrotropy’. T ¼ (0,T ,T ). (Note that one of these vector components was The possibility of toroidal order in crystals has been considered y z originally denoted as ‘antiferromagnetic’19, a misconception that since 1984 (refs 8,10–13). A variety of candidate materials were was corrected later on7,26.) As Fig. 1a shows, the net toroidal identified by their linear magnetoelectric properties: induction of moment in LiCoPO arises as a superposition of two an electric polarization by a magnetic field (P pa H ) and of a 4 i ik k complementary toroidal moments, each generated by N ¼ 2 by an electric field (M pa E )14–17. Minimization k ik i spins. These two contributions to T possess opposite orientation, of the free energy of a system with macroscopic toroidization 7 but a different amplitude, so that they do not compensate and reveals that the toroidal moment is uniquely related to a a reveal a net moment T 0. As detailed in ref. 24, Tz is associated magnetoelectric effect according to aij ¼ eijkTk with ^e as the Levi- Civita . An antisymmetric contribution to the magnetoelectric effect thus indicates the presence of a toroidal moment13 (more on this issue in the Supplementary Discussion). a b d LiCoPO Just like all ferroelectrics are pyroelectric, but not vice versa, such 4 − − + + magnetoelectric response is, however, insufficient for evidencing + ferrotoroidicity. For evidencing the ferroic nature of an ordered 18 ~ + − − − + state, two verifications are obligatory . (i) The existence of r Tz Tz domains as regions with different orientation of the order r c − parameter; and (ii) hysteretic orientation of these domains Tz − − + + between opposite single-domain states in the conjugate external field, as discussed above. In 2007, the first criterion was fulfilled x 19 − + + + + by relating the domain structure in LiCoPO4 to ferrotoroidicity . y Verification of the more fundamental second criterion and, thus, the ultimate step that could establish ferrotoroidicity as fourth Figure 1 | Toroidal moment and ferrotoroidic order. (a) Origin of the form of ferroic order are still due. toroidal moment Tz in LiCoPO4 shown for the magnetic unit cell (rectangle). Using the analogy to the established types of ferroic order one Two pairs of N ¼ 2Co2 þ spins, highlighted in purple and green, can specify the properties of the conjugate field required for respectively, give opposite contributions to Tz. The circumferences ferrotoroidic poling, that is, vector-like nature and the simulta- associated to the respective pairs show that because of ~ror, these two 8 neous violation of space- and time-inversion symmetry . A way contributions do not cancel. This leads to the net toroidal moment Tz to generate such a toroidal field S is to apply magnetic and electric parallel to the crystallographic z axis that breaks the crystallographic mirror fields together and consider SpE Â H. A side effect of this symmetry perpendicular to the z axis. Note that a toroidal moment TyooTz approach is that the magnetic and electric fields not only establish is also present. As discussed in the text, however, it is neglected here and the toroidal field S but in addition may also act on the sample on shown separately as the Supplementary Fig. 1. (b) Simplified three- their own or by non-toroidal contributions to the linear dimensional sketch of the toroidal moment in a.(c) Sketch as in b, but for magnetoelectric effect20. In the maybe closest approach, the opposite orientation of the toroidal moment. (d) Periodic continuation experiments revealed a sensitivity of the polarized neutron of the unit cells sketched in b and c into a ferrotoroidic crystal with diffraction yield to the signs of simultaneously applied magnetic domains in the xy plane.

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to the loss of the mirror plane mz perpendicular to the z axis. Hysteretic poling. Figure 2a shows the response of LiCoPO4(100) 0 0 À 1 This reduces the point symmetry from mmm1 to mmm . Ty at 17 K to which a fixed electric field Ey ¼ 11 kV cm and a reflects the rotation of the spins by y ¼ 4.6° away from the y axis. tunable magnetic field m0Hx are applied. A strikingly pronounced This further reduces the magnetic point symmetry to 20 with hysteresis is observed in the integrated SHG intensity when 7 respect to the x axis. A calculation reveals Ty/Tz ¼ siny. Thus, cycling the magnetic field in a ±2-T interval. The hysteresis is À 1 Tz44Ty, which suggests a much more effective coupling between centred and reveals a coercive field of 0.9 T or 10 T kV cm . Tz and the toroidal field. In contrast to ref. 19, we therefore focus We note that even in a field-cooling run through the ordering on the Tz-related subset of domain states, effectively omitting Ty temperature, a magnetic field (m0Hx ¼ 7 T) or an electric field À 1 and y, which is like approximating the magnetic point symmetry (Ey ¼ 11 kV cm ) applied alone leaves the sample in a multi- 0 0 2 by the higher symmetry mmm where Ty ¼ y ¼ 0. domain state; Fig. 2b,c. A series of SHG images tracing a hysteresis is shown in Fig. 2d. We see that with the cycling field, the sample alternates between opposite single-domain states of Second-harmonic generation. An established probe for imaging different brightness. In the intermediate multi-domain state domains is SHG, a nonlinear optical process denoting the emis- domains possess a lateral size in the order of 100 mm. Just like in sion of light at frequency 2o from a crystal irradiated with light at other ferroic compounds nucleation and growth of domain walls frequency o. This is expressed by the equation appear to determine the domain reversal. The progressing Pð2oÞ¼e0w^EðoÞEðoÞ, in which E(o) and P(2o) are the electric domain walls, however, do not possess the preferred orientation field components of the incident light and the nonlinear polar- often imposed by field energy minimization in other ferroics. For ization, respectively, with the latter acting as source of the SHG the ferrotoroidic order this may be expected, although, because its 2 wave of intensity ISHGp|P(2o)| (ref. 27). The nonlinear field energy was calculated to be smaller than that associated to susceptibility w^ characterizes the host material. Following the magnetic, electric or elastic order7,8. Neumann principle, symmetry determines the set of tensor In Fig. 3 the relation between the magnetic and the electric field a components wijk 0. Ferrotoroidic order affects this symmetry is quantified. In Fig. 3a,b, we see typical hysteresis loops recorded and leads to polar, time-non-invariant (‘c-type’)28 contributions at fixed magnetic and tunable electric field and vice versa. In / w^ðTÞ which couple, to the first order, linearly to the order separate runs, both the sign and the amplitude of the fixed field parameter T, so that w^ðÀTÞ¼ Àw^ðTÞ. Thus, SHG light from was varied as presented in Fig. 3c. The result is a spectacular opposite ferrotoroidic domain states exhibits a relative 180° confirmation that solely the product of the magnetic and the shift that allows to distinguish these domain states in an electric field determines the shape of the hysteresis loops. In all interference experiment29. In the Methods section, we describe graphs in Fig. 3c, the single-domain state associated to the low the SHG technology in detail and in the Supplementary Methods, SHG yield is encountered once a field of þ 0.25 T kV cm À 1 is we discuss the time non-invariance and the non-zero components exceeded. In turn, decreasing the field below À 0.25 T kV cm À 1 of the SHG tensor w^ (refs 30,31). leads to the single-domain state associated to the high SHG yield.

 a 0H (T) d –2 –1 0 1 2 0 T 0.5 T 0.65 T 11 kV cm –1 1.0

0.7 T 0.8 T 1 T

0.5 SHG intensity

17.0 K 0 T −0.2 T −0.35 T

–20 –10 0 10 20 –1 μ0H⋅E (T⋅kV cm )

bc −0.45 T −0.5 T −1 T H only E only z

0.2 mm y

Figure 2 | Poling and domain structure of a LiCoPO4 (100) sample in magnetic and electric fields. (a) SHG intensity at 17.0 K in dependence of À 1 a static magnetic field m0Hx tuned between þ 2 Tand À 2 T in the presence of a static electric field Ey ¼ 11 kVcm . SHG at 2:o ¼ 2.385 eV is emitted from the sample (Bwzzz) and superimposed with a SHG reference wave from a quartz plate (Bwref) as detailed in the Methods section. Here and in all following figures, error bars indicate the statistical error of the photon count. SHG intensities are arbitrarily scaled with respect to the largest measured value  1. À 1 (b,c) Multi-domain states emerging after cooling the sample in a magnetic or electric field only (b: m0Hx ¼ 7T,Ey ¼ 0; c: Ey ¼ 11 kVcm , m0Hx ¼ 0). Since contributions from wref were not used in these runs, opposite domains are artificially shaded in different colours. (d) Typical progress of the domain structures of a LiCoPO4 (100) sample throughout a magnetic-field-driven hysteresis loop at constant electric field. Images were taken at 17.7 K, 10 kVcm À 1, and 2:o ¼ 2.330 eV. (Note that the coercive field is temperature dependent, varying reproducibly by about an order of magnitude between 17.0 and 17.7 K (not shown)).

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a b As a result, all hysteresis loops lie upon one another. As further 0.5 T 1.2 kV cm –1 confirmation, magnetic and electric fields applied alone do not 1.0 17.7 K 1.0 17.7 K change the multi-domain state (Fig. 2b,c). This excludes an interference from nonlinear magnetic- or electric-field effects23.

0.5 0.5 Toroidal and non-toroidal poling. In the final step, we need to SHG intensity distinguish ferrotoroidic from non-ferrotoroidic contributions to the magnetoelectric poling process. Only the antisymmetric –2 0 2 –0.5 0.0 0.5 product of magnetic and electric fields establishes the toroidal 8 E (kV cm –1)  H (T) field , which may thus be represented by the coefficients 0 tor tor non non aij ¼Àaji . Any symmetric contributions (aij ¼Àaji ) c relate to conventional non-toroidal magnetoelectric poling such 17.7 K as in Cr2O3 (ref. 32) and need to be separated. 1.0 The insets to Fig. 4a,b show two field configurations that allow us to distinguish toroidal and non-toroidal contributions. We 8 apply magnetic and electric fields (ExHy EyHx)/2 so that we 8 tor couple to the components axy ayx. The associations axy ¼ tor ðaxy À ayxÞ=2 and axy ¼ðaxy þ ayxÞ=2 are obvious. Exemplary hysteresis loops obtained on a LiCoPO4 (001) sample are shown +E 1 in Fig. 4. Note that Fig. 4a shows the first example of purely toroidal hysteretic poling of a ferrotoroidic domain structure. +E 2 The non-toroidal poling, too, can drive the sample into a single- SHG intensity 0.5 −E domain state (Fig. 4b), but here the coercive field is a different 3 one. With loops as in Fig. 4, the ratio of the coercive fields +H 1 was determined in 420 runs performed at three different voltages with the result shown in Fig. 4c. We find −H tor non 1 jjEH co:=jjEH co: ¼ 1:21 Æ 0:14, which is a first quantitative indication that the toroidal and the non-toroidal poling fields –1 0 1 indeed couple to different material properties.  H⋅E ⋅ –1 25 0 (T kV cm ) From literature it is known that, almost independent of tor non temperature, ayx ¼ 2axy, which leads to ga :¼ axy =axy ¼ 1=3. If Figure 3 | Prevalence of the product of magnetic and electric fields in the toroidal poling were as effective as the non-toroidal poling in the poling procedure. (a,b) SHG hysteresis loops obtained on LiCoPO4 terms of the field energy F / a^EH that is required to switch the (100) at 17.7 K and 2:o ¼ 2.339 eV under (a) static magnetic field LiCoPO4 sample to a single-domain state, we would expect a ratio Hx ¼þ0.5 T and tunable electric field Ey, and (b) static electric field tor non À 1 jjEH co:=jjEH co: ¼ 3 of the coercive fields (to compensate for Ey ¼þ1.2 kVcm and tunable magnetic field Hx.(c) Various combinations ga ¼ 1/3). The strikingly different experimental value of 1.21 in signs and amplitudes of static and tunable magnetic and electric fields À 1 À 1 shows that the toroidal field is much more effective in promoting ( þ E1 ¼þ1.2 kVcm , þ E2 ¼þ1.8 kVcm , À E3 ¼À1.0 kV/cm, the ferrotoroidic single-domain state. Apparently, the þ H1 ¼þ0.5 T, À H1 ¼À0.5 T). In all these cases, the coercive field and ferrotoroidic and the non-ferrotoroidic contributions to the the domain state are solely determined by the product (Ey Á Hx) with À 1 magnetoelectric field product connect to different types of m0|EH|co. ¼ 0.25 T kVcm as coercive field. lines are guides to the mechanisms pinning the domain walls and thereby determining eye. the width of the hysteresis.

acb 1.0 17.0 K 17.0 K 1.0 2 co. non | EH /|

tor 1 co. tor Non | SHG intensity SHG intensity EH | E E H 0.5 0.5 x y x y H 0 –20 0 20 –20 0 20 2610  ⋅ ⋅ –1  ⋅ ⋅ –1 0H E (T kV cm ) 0H E (T kV cm ) E (kV cm –1)

Figure 4 | Distinction between ferrotoroidic and non-ferrotoroidic contributions to the magnetoelectric poling procedure. (a) Purely toroidal hysteresis À 1 loop. (b) Non-toroidal hysteresis loop. a and b were obtained at 2:o ¼ 2.330 eV and 17.0 K on LiCoPO4 (001) with E ¼ 12 kVcm . The respective experimental field configurations are sketched as insets. (c) Ratio of the coercive fields obtained by toroidal and non-toroidal poling at different electric fields. Error bars result from 420 consecutive runs.

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Acknowledgements Additional information This work was supported by the SFB 608 of the Deutsche Forschungsgemeinschaft and Supplementary Information accompanies this paper at http://www.nature.com/ by a Sinergia grant (No. CRSII2_147606/1) of the Swiss National Science Foundation. naturecommunications Helpful exchange with N.A. Spaldin and C. Ederer is appreciated. We thank H. Schmid and J.-P. Rivera for many clarifying discussions and for providing the samples. Competing financial interests: The authors declare no competing financial interests. Reprints and permission information is available online at http://npg.nature.com/ Author contributions reprintsandpermissions/ A.S.Z. performed the experiments; and M.F. and D.M. supervised the work and wrote the How to cite this article: Zimmermann, A.S. et al. Ferroic nature of magnetic toroidal manuscript. order. Nat. Commun. 5:4796 doi: 10.1038/ncomms5796 (2014).

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