SCIENCE ADVANCES | RESEARCH ARTICLE CONDENSED MATTER PHYSICS Copyright © 2020 The Authors, some Proof of the elusive high-temperature incommensurate rights reserved; exclusive licensee phase in CuO by spherical neutron polarimetry American Association for the Advancement of Science. No claim to Navid Qureshi1*, Eric Ressouche2, Alexander Mukhin3, Marin Gospodinov4, Vassil Skumryev5,6 original U.S. Government Works. Distributed CuO is the only known binary multiferroic compound, and due to its high transition temperature into the multiferroic under a Creative state, it has been extensively studied. In comparison to other prototype multiferroics, the nature and even the Commons Attribution existence of the high-temperature incommensurate paraelectric phase (AF3) were strongly debated—both ex- NonCommercial perimentally and theoretically—since it is stable for only a few tenths of a kelvin just below the Néel temperature. License 4.0 (CC BY-NC). Until now, there is no proof by neutron diffraction techniques owing to its very small ordered Cu magnetic moment. Here, we demonstrate the potential of spherical neutron polarimetry, first, in detecting magnetic structure changes, which are not or weakly manifest in the peak intensity and, second, in deducing the spin arrangement of the so far hypothetic AF3 phase. Our findings suggest two coexisting spin density waves emerging from an accidental degeneracy of the respective states implying a delicate energy balance in the spin Hamiltonian. INTRODUCTION vector q. This spin rotation plane or spin envelope is defined by two Multiferroic materials combine at least two of the ferroic orders main axes b and , where the latter lies within the a-c plane, at 28° [(anti)ferromagnetism, ferroelectricity, ferroelasticity, and ferroto- from the positive c toward the positive a axis (11). The electric po- roidicity] and have been extensively studied in the last two decades larization along the monoclinic axis b (space group C2/c) may result [see (1, 2) and references therein]. Of particular interest are magneto- from the cycloidal component via the inverse Dzyaloshinskii-Moriya electric multiferroics that have (anti)ferromagnetic and ferroelectric effect (11–14), although the actual mechanism remains under debate order since such a system reveals new opportunities of storing data (15–19) and several other alternative mechanisms, e.g., involving in future memory and multifunctional devices. In so-called type II magnetic frustration and magnetostriction, have been proposed. or improper multiferroics, the onset of a complex magnetic order The direct relation between the chiral spin helix and the electric goes along with breaking of inversion symmetry, and the resulting polarization was demonstrated in an SNP experiment by switching ferroelectric polarization due to atomic shifts is induced by the the chiral domains with an applied electric field (20). Below 213 K magnetic order. In such a case, the electric properties could be ma- down to low temperatures, CuO reveals a commensurate collinear nipulated via a magnetic field and vice versa. In a type I or proper antiferromagnetic structure (AF1) that conserves inversion symmetry multiferroic material, the two orders have different origins, take and is therefore paraelectric. The high transition temperatures place at different temperatures, and are therefore very weakly or not reflect the strong Cu-Cu exchange, which was investigated by both coupled. Complex inversion-symmetry breaking magnetic struc- ab initio methods and inelastic neutron scattering [see (21) and refer- tures in improper multiferroics are often a result of geometrical ences therein]. frustration, and therefore many materials revealing these properties The usual magnetic phase transition sequence of type II multi- have very low transition temperatures into the multiferroic state. ferroics (Fig. 1), e.g., Ni3V2O8 (22, 23) and MnWO4 (24, 25), upon Understanding the properties of high-temperature multiferroics— cooling, is first a high-temperature incommensurate phase (AF3) with for the search or tailoring of new materials pushing the limits even longitudinal amplitude modulation followed by a low-temperature further—has been at the center of recent multidisciplinary research. incommensurate phase (AF2) of complex noncentrosymmetric cy- The appearance of a macroscopic electric polarization induced cloidal nature inducing a ferroelectric polarization. The ground state by an inversion-symmetry breaking magnetic structure at 230 K (AF1) is a commensurate antiferromagnet with the same moment makes CuO (tenorite) the bulk material with the second-highest direction as in the AF3 phase. The variety of magnetic phases is a transition into the multiferroic state (3) after YBaCuFeO5 (4–6). consequence of the temperature-dependent competition between The spin arrangement (7–10) in the so-called AF2 phase was already single-ion anisotropy and exchange interactions (26). investigated 30 years ago, and it was notably the spherical neutron The existence of the AF3 phase in CuO was first promoted in (27) polarimetry (SNP) work by Brown et al. (9) that allowed an unam- based on the praphase concept. Within this approach, it was shown biguous determination of the associated magnetic structure: an in- that the AF2 phase is characterized by two one-dimensional irre- commensurate oblique helix with spins rotating in a plane with slight ducible representations, while both AF1 and AF3 are characterized inclination with regard to the plane perpendicular to the propagation by only one. Furthermore, it was argued that the Cu spins point along the b axis in the AF3 phase. The necessity for an incommen- 1 surate collinear AF3 phase was also justified by the analysis of the Institut Laue-Langevin, 71 Avenue des Martyrs, 38042 Grenoble Cedex 9, France. 2 Université Grenoble Alpes, CEA, IRIG, MEM, MDN, F-38000 Grenoble, France. nonlocal Landau-type free energy based on a mean-field treatment 3 Prokhorov General Physics Institute, Russian Academy of Sciences, ul. Vavilova 38, of a Heisenberg-type Hamiltonian (15). The same moment direction Moscow 119 991, Russia. 4Institute of Solid State Physics, Bulgarian Academy of was predicted, but only half of the Cu spins are believed to order. Sciences, 1784 Sofia, Bulgaria. 5Departament de Física, Universitat Autònoma On the other hand, Monte Carlo studies based on first-principles de Barcelona, 08193 Bellaterra, Barcelona, Spain. 6Institució Catalana de Recerca i Estudis Avançats, 08010 Barcelona, Spain. calculations failed to reproduce its existence (16, 17). A phenome- *Corresponding author. Email: qureshi@ill.fr nological theory of phase transitions in CuO was suggested in (18), Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 1 of 8 SCIENCE ADVANCES | RESEARCH ARTICLE netic moments in the investigated sample and is decisive for this study (see Materials and Methods for more information). RESULTS SNP experiments were carried out using the same sample as in our previous work (11). To access magnetic reflections modulated with the incommensurate propagation vector q = (qh 0 ql), the sample Fig. 1. Usual magnetic phase transition sequence in type II multiferroics. In the was mounted with its b axis vertical (with a precision of approxi- paramagnetic state (P), the magnetic moments are fluctuating (represented by blur) mately 2°) in a two-axis setup. For purely magnetic Bragg reflec- ⎜ ⎟ and condense into the high-temperature incommensurate phase AF3 at TN, which consists of an amplitude modulation with spins pointing along the magnetic easy tions as they are observed in CuO, the polarization matrix P with axis. In the low-temperature incommensurate phase AF2, the spins adopt a cycloidal matrix elements Pf,i takes the following form modulation before they align in a commensurate antiferromagnetic structure (AF1) at low temperature. ⎛ − ​p​ 0​​ ​M​​  2​ − ​Jy​  z​​ − ​Jy​  z​​ − ​Jy​  z​​ ⎞ ​  ─  ​ ─ ​   ​ ​   ​ ─ ​Ix​  ​​ ​Iy​  ​​ ​Iz​  ​​ ​p​ 0​​​(​​​M2​⊥ ​ − ​ ​  ​  ) ​  in which a direct first-order phase transition PM→AF2 was described. y​  M⊥z ​2 ​​​  ​​ ​p​ 0​​ ​Ry​  z​​ ​​​P​  f,i​​  = ​ ​​​     0    ​  ​  ​  ─     ​ ​   ​ ─ ​​ ​​ ​​​​ (1) CuO was argued to exhibit an inverted sequence of symmetry-­ ​Iy​  ​​ ​Iz​  ​​ breaking mechanisms with respect to other type II multiferroics. p​ ​ 0​​​(​​  − ​M​2⊥ ​ + ​ ​  ​​​ )​​ 2 ​p​ 0​​ ​Ry​  z​​ y​  M⊥z Ultrasound velocity (11, 15) and thermal expansion (28) measure- ⎝ ⎠ 0 ​   ​ ─    ​ ─  ​ ments indeed suggest a third magnetically ordered phase just below ​Iy​  ​​ ​Iz​  ​​ the Néel temperature with a temperature stability range of less than 0.5 K. Being close to the elevated Néel temperature, the ordered where ​​M​​  2​ = ​M​  ⊥​​ ​M*⊥​ ​​,​  ​​M​2⊥y​ ​ = ​M⊥​  y​​ ​M*⊥​ y​ ​​, and ​​M2⊥z​ ​ ​ = ​M⊥​  z​​ ​M*⊥z ​ ​ ​​. The off-­ * magnetic moment  is very small, and therefore no neutron diffrac- diagonal terms are R​  ​​ yz​​ = 2Re(​M⊥​  y​​ ​M⊥z ​ ​ ​)​ and ​​Jy​  z​​ = 2Im(​M⊥​  y​​ ​M⊥z​* ​ ​)​, tion study was able to clarify the existence and the nature of this where the latter results from chiral scattering. The respective matrix phase. However, to understand the complete picture of CuO and— elements are normalized to the scattered intensity depending on the especially by comparing it to related compounds—type II multiferroics initial neutron polarization (Ix = M2 + p0Jyz, Iy = Iz = M2); p0 is the in general, the microscopic structure of this AF3 phase is of major polarization of the incoming beam. The reference frame usually used importance since the sequence of magnetic phase transitions and in polarized neutron scattering defines the x direction to be parallel their associated magnetic structures are a result of the competition to Q, z to be vertical, and y to complete a right-handed coordinate between exchange interactions and single-ion anisotropy that govern system. A more detailed description of the SNP technique can be the physics of the system. This provides information to potentially found in Materials and Methods and in (29). develop a Hamiltonian able to better explain the mechanism of For (h0l) ± q reflections in the aforementioned sample geometry multiferroicity in CuO. the z axis is parallel to b and the y axis is in the a-c plane. From Eq. 1, The study presented here uses the SNP technique yielding two it is easy to deduce that the polarization matrix of the (0.5 0 –0.5) key advantages, which presumably makes it the unique microscopic reflection within the collinear AF1 phase (||b) has zero off-diagonal probe to prove and study the AF3 phase: (i) It is extremely sensitive elements and Pxx = Pyy = −Pzz = −1, if one assumes perfect neutron to the direction of the ordered magnetic moments, and (ii) conclu- polarization. Figure 2 shows the temperature dependence of the rocking sive results can be obtained on just a few magnetic Bragg peaks. The curves (Fig. 2A), integrated intensities (Fig. 2B), and the polarization observed intensity in conventional unpolarized neutron diffraction matrix elements Pyy and Pxy (Fig. 2C). It can be seen that for the AF1 is proportional to the square of the magnetic interaction vector M⊥, phase, below 213 K, the Pyy term proves to be in accordance with the and very subtle changes in the spin alignment, especially if they collinear antiferromagnetic structure with ||b. At 213 K, a first-order happen continuously over a very narrow temperature range, can phase transition takes place into the AF2 phase evidenced by the abrupt easily be missed. Furthermore, a unique magnetic structure model change of Pyy to a value of ~−0.1 and also from the jump in the inte- can only be proposed by refining it to tens or hundreds of reflec- grated intensity. At approximately 227 K, the Pyy term starts to rise tions. In the case of the AF3 phase in CuO, it is impossible to mea- and saturates at a plateau of approximately Pyy = 0.2 at 228.8 K before sure such a large number of magnetic reflections with reasonable it drops to zero at 229.3 K. The increase in Pyy with increasing tempera- counting statistics owing to the very small ordered Cu moment. In ture is a clear and unambiguous indication for a larger  component, polarized neutron scattering, the spin of the incident neutron is ro- whereas due to the weak scattered intensity, it is not possible to observe tated by 180° around M⊥ for purely magnetic scattering [and for the a change in the integrated intensities. The temperature stability range case that the real and imaginary parts of M⊥ are parallel, for perpen- of this modified magnetic structure can be estimated from the points dicular real and imaginary parts, the incident neutron polarization of steepest increase and decrease of Pyy and results in roughly 0.5 K. is rotated toward the cross product of Re(M⊥) and Im(M⊥), i.e., Both the change in Pyy and the stability range are clear proofs for the toward Q] independently of the size of the magnetic moment. SNP existence of the AF3 phase in CuO. not only offers direct access to the two components of M⊥ but also The correct description of magnetic domains (see sections S1 allows the full three-dimensional reconstruction of the neutron spin and S2) is crucial for the analysis of polarimetry data. In principle, rotation after the interaction with the sample, which is routinely without any external force, the population of the magnetic domains measured and summarized in the so-called polarization matrix. This can be assumed to be close to equal. However, we observe nonzero yields incomparable precision concerning the direction of the mag- chiral terms Pxy and Pxz within the AF2 phase, implying an imbalance Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 2 of 8 SCIENCE ADVANCES | RESEARCH ARTICLE A tion, since Q = q, and by definition of a cycloid, one of the Fourier 6000 components is parallel to Q and does not contribute to magnetic 5000 scattering, which yields Jyz = 0, or (ii) a sinusoidally modulated magnetic structure. 4000 In the first part of the experiment, we have measured full polar- Counts/min 3000 ization matrices in the AF1 and AF2 phases, which yield excellent agreement (2 = 2.8 for AF1 and 2 = 1.6 for AF2) with the known 2000 magnetic structures (see more details in section S3). The observed 1000 and calculated elements of the polarization matrices are shown in Fig. 3 (A and B). 0 It can be easily shown (see section S4) that the observed Pyy and −167 −166 −165 −164 Pzz terms within the AF3 phase cannot be explained by a single irre- ducible representation. Therefore, spin-density wave (SDW) phases B Integrated intensity (arb. units) 3000 with the ordered moment along the b or the  direction can be ex- cluded as well as a nonchiral spin helix as similarly proposed for 2500 MnWO4 (25). With the information deduced from the temperature 2000 dependence of the Pxy term, we can rule out a helical modulation of the AF3 phase, since this would not explain the disappearance of 1500 the chiral term at the AF2→AF3 phase transition. A purely cycloidal 1000 modulation, as mentioned above, would in principle agree with zero chiral terms; however, there is no structure model, which would satisfy 500 all observed polarization matrix elements measured at T = 228.8 K. 0 We therefore concentrated on sinusoidally modulated magnetic struc- 0.2 C tures and found a solution where the magnetic structure is a collinear SDW of the same symmetry as the AF2 phase with the ordered moment 0 direction between the two easy directions of the system, i.e., between −0.2 the b and the  axis (see refinement results in Fig. 3C). The refined Fourier components of the Cu spin in the AF3 phase are SAF3 = [0.21(2) −0.4 0.27 0.30(1)] B, while the population of the two orientational −0.6 domains has been refined to 49(5)% and 51(5)% (2 = 3.6). From the temperature dependence of the integrated intensity and the calculated −0.8 structure factors for the AF1 and AF3 models, we can estimate the −1 ordered magnetic moment amplitude to be approximately 0.06 B. 200 205 210 215 220 225 230 The resulting magnetic structure, in comparison to AF2, is shown in Fig. 4 will be discussed in the following section. Fig. 2. Temperature dependence of the neutron intensity and polarization measured on the magnetic (000) + q peak. (A) The rocking scans over the in- commensurate ( ≈ −166.2°) and commensurate ( ≈ −164.6°) peak positions at DISCUSSION different temperatures (from 199.5 K shown in blue to 228.8 K shown in red with a linear color gradient in between) revealing the change of the propagation In comparison with other type II multiferroics revealing a subse- vector. The corresponding integrated intensities are depicted in (B). In (C), the Pyy quent condensation of order parameters—one applying to the lon- term indicates the first-order transition from the AF1 to the AF2 phase at 213 K, gitudinal and one to the transversal moment component with respect which is also visible as a jump in the integrated intensities. At 228 K, a moderate to q—the case in CuO is a bit different, since the AF2 structure is increase toward a maximum value of 0.22 indicates the transition into the AF3 closer to a helix than to a cycloid. Nevertheless, if one continues in the phase before the polarization drops to 0 above TN (a close-up of this region is analogy, each of the main axes of the spin envelope in the AF2 phase shown in Fig. 5A). Note that the ordered moment is too small to see a change in should be modulated by a different irreducible representation , the intensities. The black dashed lines mark the phase boundaries, and the red dotted while only one representation is necessary to describe the AF1 line denotes the temperature at which the data collection within the AF3 phase phase. This works well for the AF1←→AF2 phase transition, with the was carried out. b component being modulated according to 1 and the  component being modulated according to 2. However, it is not possible to ex- of chiral domain populations. This favorable situation allowed us to plain the observed data in the AF3 phase with a single irreducible follow the chiral term Pxy as a function of temperature. An optical representation requiring the assumption of a mixed representation. second harmonic generation study revealed that the domain distri- CuO is an interesting system in terms of magnetic anisotropy. bution is stable throughout the entire AF2 temperature range (30). Clearly, the easy axis is defined by the collinear AF1 ground state Consequently, the chiral term is directly proportional to the projec- with the moments pointing along the b axis (see Fig. 4D). However, tion of the spin envelope along the scattering vector, allowing us to within the AF2 phase, the system reveals an easy-plane character, probe a rotation or a squeeze of this spin envelope. As it can be seen since the spins rotate within a circular envelope containing the b axis in Fig. 2C, the chiral Pxy term of the (000) + q reflection becomes and the  direction. This second direction is also apparent when a negligible just when entering the AF3 phase, which can be explained large enough magnetic field is applied along the b axis at low tem- by two types of magnetic structures: (i) a purely cycloidal modula- peratures (11), which results in a spin-flop transition aligning the Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 3 of 8 SCIENCE ADVANCES | RESEARCH ARTICLE 1 A B C obs cal 0.5 0 −0.5 −1 Fig. 3. Results of the refinement to the SNP data. (A) The commensurate AF1 phase at 200 K, (B) the helical AF2 phase at 215 K, and (C) the so far uncharacterized AF3 at 228.8 K. Circles represent the observed polarization values, while squares show the refinement for the different magnetic Bragg reflections labeled on the horizontal axis. The color code identifying the different Pf,i elements is given as an inset in (C), e.g., a blue symbol with red edge means Pxz. All observed (obs) and calculated (cal) values are listed in table S1. spins close to  (see Fig. 4C). Now, the results concerning the AF3 Within each of the coexisting SDW phases according to 1 or 2, phase suggest a collinear arrangement in between those two special respectively, the population of the configurational domains does not directions at approximately 51° from the b axis toward the  direction. affect the respective polarization matrices (see section S2). There- A single-phase scenario with mixed representations 1 + 2 can fore, the phase volumes  can be refined by weighting the theoretical arise due to a competition of various contributions to the magnetic polarization values Pf,i and scattered intensities I of the phases de- anisotropy—including terms of eighth order with respect to the order scribed by 1 and 2 for both observed reflections [calculated with parameters along the b and the  directions, which would reveal a Mag2Pol (31) based on the magnetic structure models assuming coincidence of their temperature instability, i.e., an accidental de- equally sized spins] according to generacy leading to a higher-order multicritical point. In the usual interpretation of Landau theory, the combination of two represen- ​​​  ​​  1​​​ ​P​f,i ​ ​  1​  ​​​ ​I​​  ​​  1​​​ + ​​​  ​​  2​​​ ​P​f,i ​ ​  2​  ​​​ ​I​​  ​​  2​​​ ​​P​ f,i​​ = ​─────────────       ​​ (2) tations is only allowed for the successive condensation of two order ​​​  ​​  1​​​ ​I​​  ​​  1​​​ + ​​​  ​​  2​​​ ​I​​  ​​  2​​​ 1 parameters. However, we note that the AF3-AF2 phase transition For our sample at T = 228.8 K, this yields  = 35(1)% and might be accomplished via a change of the phase shift φ between 2 = 65(1)% (2 = 2.8). The calculated [observed] values are Pyy = 0.21 components of the order parameters belonging to 1 and 2, i.e., [0.23(2)] and Pzz = −0.21 [−0.17(2)] for the (000) + q reflection from φ = 0 () in AF3 to φ = /2 (−/2) in AF2, which could and Pyy = −0.27 [−0.24(4)] and Pzz = 0.27 [0.27(5)] for the (200) − occur by means of either a first-order transition or two second-order q reflection. ones through intermediate values, corresponding to low-symmetry A similar phase separation of different irreducible representations magnetic modes (four chiral and rotational domains)—as we will was theoretically predicted in the Er2Ti2O7 pyrochlore compound further test. In the former case, the first-order transition could ex- in which the long-range order—resulting from the order-by-disorder pand to some finite temperature range due to a phase coexistence. phenomenon—is driven from the 2 to the 3 state upon Y doping A second interpretation of our data is possible, which again in- as a consequence of the competition of structural disorder with volves both the b and the  direction to be energetically equivalent, thermal and quantum fluctuations (32, 33). A frozen mosaic of which is motivated by the circular spin-rotation envelope within the 2 and 3 domains was indeed observed experimentally (34). AF2 phase. The AF3 phase can be described as a coexistence of two In the two-phase scenario, the transition into the multiferroic SDW phases, one with ||b (Fig. 4D) and the other with || (Fig. 4C). AF2 phase would presumably take place by a continuous increase of Two SDW phases, corresponding to each of the irreducible repre- the respective perpendicular component within each phase forming sentations, may appear at the same temperature as a result of a first-­ an elliptical spin envelope, which becomes circular at approximately order phase transition between them, which implies their coexistence 227.5 K evidenced by the saturating chiral polarization term and the in a certain temperature range. This phase coexistence takes place corresponding value of Pyy. In the one-phase scenario, the transi- due to a certain combination of parameters of magnetic interactions tion can be interpreted as a simultaneous increase and rotation of and their temperature dependences in CuO, which results in a unique the spin envelope—close to the AF3 phase being defined by a large thermodynamic path along the line of the first-order transition be- axis along the [0.21(2) 0.27 0.30(1)] direction with a small axis per- tween two SDW sinusoidal phases. This scenario is congruent with pendicular to it within the b plane (φ ≈ 0)—toward the main the experiment (see section S5). The phase volume of the first con- anisotropy axes of the helical structure (φ = /2), for which two figuration would result in Pyy = −0.935, whereas the second config- possibilities exist: either the SDW moment direction rotates toward uration yields Pyy = 0.935. However, due to the different magnetic the b axis or toward the  direction. Note that a deviation of the order interaction vectors of the two configurations, the resulting observed parameter from being parallel or perpendicular to the monoclinic axis polarization would not average to 0 for all magnetic Bragg reflec- results in the presence of four magnetic domains during the second-­ tions, even for a homogeneous distribution of the two separated phases. order phase transition (see section S2). The simulation results are Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 4 of 8 SCIENCE ADVANCES | RESEARCH ARTICLE A B C D Fig. 4. Magnetic structure models. (A) Helical magnetic structure in the AF2 phase focusing on a part of a Cu-zigzag chain and showing a right-handed rotation (emphasized by the spiral) of the magnetic moments along the propagation vector. The circular envelope is depicted around the spins, where the b direction is represented in yellow and the  direction is in red. (B to D) Collinear sinusoidally modulated magnetic structures with magnetic moments along a direction in between b and  modulated by 1 + 2 (B), along the  direction modulated by 2 (C), and along the b axis modulated by 1 (D). The amplitude modulation is emphasized by sine waves. shown in Fig. 5A, and the different scenarios are shown schematically A 0.4 in Fig. 5B. The transition of the spins was assumed to be linear in 0.3 temperature, and at each temperature, the expected spin configuration was used together with the previously derived domain populations 0.2 (one-phase scenario) or phase volumes (two-phase scenario) to cal- 0.1 culate the polarization matrix and extract the Pyy and Pxy elements 0 (note that a nonlinear temperature variation of the spins would in −0.1 fact only shift and/or compress/expand the calculated curve on the −0.2 temperature axis). The resulting values are shown together with the same data points as in Fig. 2B focusing on the AF2→AF3 transition. −0.3 It can be seen that the two-phase scenario captures the temperature −0.4 226.5 227 227.5 228 228.5 229 229.5 dependence of the polarization values very well, while the one-phase scenarios reveal a substantially different tendency for the Pyy terms. Note that a superposition of the two one-­phase scenario curves would B coincide with the two-phase scenario curve; however, this would imply the necessity of including an additional magnetic domain for which there is no justification from symmetry considerations: (i) Those two domains cannot be related by any symmetry operator due to the special direction of the spin component in the AF3 phase, and (ii) the max- imum number of possible domains is four due to the group-subgroup relation of the nuclear and magnetic symmetry. Last, we will discuss the possibility of a temperature-dependent population of chiral domains in the transition from the AF2 to the AF3 phase. In that case, the thermodynamic imbalance of chiral T = 227.8 K T = 228.0 K T = 228.2 K T = 228.4 K T = 228.6 K T = 228.8 K domains in the AF2 phase would gradually be leveled due to thermal Fig. 5. Simulated phase transition between the AF2 and AF3 phases. (A) The data points are the same as in Fig. 2B. Dashed and dotted lines represent the one- fluctuations when approaching AF3. The polarization data in the phase scenario for the two different possibilities: The SDW moment direction moves AF3 phase at 228.8 K can then be equally well explained with an elliptic toward the b axis, or the SDW moment direction moves toward the  direction, helix, SAF3 = [i*0.15 0.195(4) i*0.22], yielding a 28% compression of respectively, with a simultaneous increase of the perpendicular component. Solid the spin envelope along b. Alternatively, a coexistence of the AF2 lines show the expected polarization values for a coexistence of two SDW structures circular helix [53(3)%] with a || SDW is equivalent. These models in the AF3 phase with a gradual increase of the respective perpendicular compo- suggest a transition from a circular helix (1 + 2) to a phase de- nent for decreasing temperature. (B) Perspective view on the spin envelopes at six scribed by 2, which, however, cannot be reconciled with the tem- different temperatures (shown in red and yellow for components along  and b, perature dependence of the Pyy term. On approaching a pure 2 state, respectively) in comparison to the circular envelope deep within the AF2 phase one would expect a steadily increasing value in the temperature range (shown in gray). Rows 1 and 2 depict the simultaneous rotation and expansion (for 228.8 K < T < TN, but it can be seen in Fig. 5A that Pyy reaches a the two cases mentioned above) of this envelope when going from the AF3 to the AF2 phase in the one-phase scenario. In the last row, the two-phase scenario is plateau of approximately 0.2, which is defined by the two data points sketched, showing two envelopes at each temperature representing the two coex- just below 229 K revealing similar error bars at the same counting isting SDW phases. In each of those phases, the component perpendicular to the time, before it drops to 0. The existence of a pure 2 state would SDW moment direction increases upon cooling until the circular spin envelope is imply a significantly larger value of Pyy at T = 228.87 K, which is still reached in both phases at T = 227.8 K. inside the AF3 phase as compared to the data point at 229.1 K with a considerably larger error bar due to the vanishing ordered mag- netic moment. using neutron scattering methods. Our data suggest that the associ- ated magnetic structure is incompatible with an SDW phase corre- sponding to a single-order parameter being modulated by a single CONCLUSIONS irreducible representation. Instead, we have revealed that the AF3 Using SNP, we have derived unambiguous proof for the existence of phase has two spin components, which lie along the  and b direc- the elusive AF3 phase in CuO, which, to date, could not be observed tions belonging to two different irreducible representations. The Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 5 of 8 SCIENCE ADVANCES | RESEARCH ARTICLE magnetic structure is best described as a coexistence of two SDW phases with the ordered moment direction along either of the main axes of the AF2 phase modulated by one irreducible representation. Our results reveal unique microscopic information concerning the AF3 phase and the magnetic anisotropy in CuO with the  di- rection appearing energetically slightly more favorable just below the Néel temperature. The latter could be considered as a bicritical or tetracritical point of the corresponding phase diagrams in the parameter space of critical thermodynamic coefficients. This adds an extremely important missing piece to the puzzle of the complex behavior of magnetic phases in this extensively studied multiferroic compound. The full description of the magnetic symmetry and the accidental degeneracy of order parameters supply valuable ingredients for further theoretical studies, which will help to better understand this system with outstanding position in the family of magnetically induced ferroelectrics and multiferroics in general. MATERIALS AND METHODS Spherical neutron polarimetry Fig. 6. Comparison between SNP and conventional unpolarized neutron In addition to the general law in magnetic neutron scattering that scattering. A simple Néel order with q = (0.5 0 0) on a primitive single-domain only the component of the magnetic structure factor perpendicular cubic lattice with one magnetic atom in the unit cell is assumed. The magnetic to the scattering vector Q (generally denoted as the magnetic inter- moment of fixed size (shown as blue arrows projected onto the a-c plane on the action vector M⊥) contributes to magnetic scattering, neutrons with left-hand side sketches) is chosen (A) along the b axis, (B) 20° inclined from b their polarization axis parallel to M⊥ will undergo a non–spin-flip toward c, and (C) 20° inclined from b toward the 〈101〉 direction. The sketches scattering process, while neutrons with polarization perpendicular on the right-hand side show the corresponding local reference frames for a Q = (000) + q reflection together with the magnetic interaction vector M⊥, the incident to M⊥ will be scattered with a spin-flip. Magnetic structures like neutron polarization Pi (Pi||y in this illustration) and the final neutron polarization helices or cycloids reveal chiral scattering, which is polarization Pf. For each magnetic moment configuration, the modulus of the expected M⊥ dependent and will create or annihilate polarization along the scat- (compared to the magnetic structure factor FM) and the two components of Pf tering vector Q and can therefore be revealed by analyzing the final are given. Note that unpolarized neutrons do not reveal any difference between neutron polarization along the x direction (in polarized neutron cases (A) and (B) and that a 3% change of M⊥ in (C) can also be accounted for by a experiments, a local reference frame is defined for every Bragg reflec- 3% smaller ordered moment. SNP clearly differentiates between the three cases— tion, where Q||x, z is vertical and y completes the right-handed set). with a precision of approximately 1 to 2% of the respective components Pf,y and The direct access to the two components of M⊥ is the key advantage Pf,z—independently of the ordered moment size. of SNP compared to conventional unpolarized neutron scattering, which is visualized in Fig. 6. Furthermore, it is possible to fully re- The additional terms in comparison to Eq. 1 are N2 = NN* (N is construct the three-dimensional rotation of the neutron spin after the nuclear structure factor), ​​R​ ny​​  = 2Re(N ​M*⊥​  ​)​ ​  z​​  = 2Re(N ​M*⊥z y​  , ​​Rn ​  ​)​​  ,​​ * * the interaction with the sample including the created or annihilated J​ ny​​  = 2Im(N ​M⊥​  ​) y​  ​, and ​​Jn ​  z​​  = 2Im(N ​M⊥z ​  ​)​  ​, while ​​Ix​  ​​  = ​N​​  2​ + ​M​⊥2 ​ y​  + ​ 2 polarization. The final neutron spin Pf is related to the initial one by M​⊥z ​  p​  ix​​ ​Jy​  z​​​, ​​Iy​  ​​  = ​N​​  2​ + ​M​⊥2 ​ + ​  ​ + ​ y​  M⊥z​2 ​ + ​ ​  p​  iy​​ ​Rn​  y​​​, and ​​Iz​  ​​  = ​N​​  2​ + ​M​⊥2 ​ y​  + ​ M​ 2 ​ + ​ ​  p ​  ​​ ​ R ​  . ​​​ In the absence of nuclear scattering, i.e., for purely ​​P​  f​​ = ​PP​  i​​ + P′​ (3) ⊥z iz nz magnetic peaks, N = Rny = Rnz = Jny = Jnz = 0. By further assuming where ​P​is a rotation matrix acting on the initial neutron spin the same degree of incident neutron polarization (pix = piy = piz = p0) Pi and P′ is the created/annihilated polarization. By aligning the initial and analyzer efficiency for the three directions as well as correcting neutron polarization along the directions x, y, or z and by analyzing the data for the latter (pfx = pfy = pfz = 1), one obtains the polariza- the component of the final neutron spin along these directions, one tion matrix shown in Eq. 1. can summarize nine measurements in the so-called polarization The polarized neutron diffraction experiment was carried out at matrix, which is better defined as a pseudomatrix, since it combines the hot neutron polarized diffractometer D3 (ILL, Grenoble) using the rotation and the created/annihilated polarization. For the general a wavelength of  = 0.85 Å. A standard orange cryostat was placed case, i.e., including nuclear, magnetic, and interference terms as well within the cryogenic polarization device in a two-axis geometry. as different degrees of polarization and analyzer efficiency for the Note that an applied electric field along the b axis would have pro- different directions, the polarization matrix is duced a majority chiral domain. However, since it is not known ⎜ ⎟ how an electric field affects the stability of the AF3 phase, and since it might stabilize the AF2 phase, we have decided to study the sample ⎛ p​ ​ix​​(​N​​ 2​− ​M​​ 2​) −​J​yz​​ − ​p​iy​​J​nz​​− ​J​yz​​ p​ ​iz​​J​ny​​− ​J​yz​​ ⎞ in its thermodynamic equilibrium. The sample was mounted with ​p​fx​​─────────── ​  ​Ix​  ​ ​​ ​p​fx​​─ ​  ​Iy​  ​ ​​ p​ ​fx​​─​  ​I​z ​ ​​ its b axis along the vertical axis of the diffractometer. The orienta- p​ ​ix​​J​nz​​+ ​R​ny​​ p​ ​iy​​(​N​​ ​+ ​M⊥​  ​ y​− ​ ​  ​​) + ​R​ny​​ p​ ​iz​​R​yz​​+ ​R​ny​​ 2 2 2 ​​P​f,i​​ = ​ ​​​           ​p​fy​​​  ​Ix​ ​​ ​ ​p​fy​​   ​  M⊥z ​​ ​​ (4) tion was achieved using neutron Laue diffraction (OrientExpress, ILL) ​ ─ ​​ ​ ─────────────── ​Iy​ ​​ ​ ​ ​ ​ p​ ​fy​​​  ─ ​I​z​​ ​ ​ ​ ​ with a precision of approximately 2°. Guide fields in the order of a few − ​p​ix​​J​ny​​+ ​R​nz​​ p​ ​iy​​R​yz​​+ ​R​nz​​ p​ ​iz​​(​N​​ ​− ​M​⊥ ​ y​+ ​ ​  ​​) + ​R​nz​​ 2 2 2 tens of milligauss were used to maintain the neutron polarization ⎝ ​p​fz​​─ ⎠ M⊥z ​  ​Ix​  ​ ​​ ​p​fz​​─ ​  ​Iy​  ​ ​​ ​p​fz​​──────────────    ​  ​I​z ​ ​​ along the beam path and to rotate it within directions perpendicular Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 6 of 8 SCIENCE ADVANCES | RESEARCH ARTICLE to the direction of flight. The precession coils—decoupled from the to eighth order. As a result, various phase diagrams in the parame- outer guide fields and the zero-field sample chamber by an inner and ter space of temperature-dependent coefficients for quadratic terms outer superconducting Meissner shield—allow the three-dimensional of the order parameters in the potential were obtained. manipulation of the neutron spin. The neutron spin analysis was carried out using polarized 3He spin-filter cells whose time-dependent SUPPLEMENTARY MATERIALS efficiency was monitored with calibration measurements by period- Supplementary material for this article is available at http://advances.sciencemag.org/cgi/ ically monitoring the Pzz term of a nuclear reflection, which, for per- content/full/6/7/eaay7661/DC1 Section S1. Symmetry analysis in the extended little group (paramagnetic symmetry) fect neutron polarization and spin-filter efficiency, should always be 1. Section S2. Order parameters in magnetic domains The actual observed value is a product of the expected value, the Section S3. Confirmation of the AF1 and AF2 phases initial neutron polarization (p0 = 0.935 on the D3 diffractometer), Section S4. Incompatibility of the AF3 phase with a single irreducible representation and the spin-filter efficiency [pHe(t) between 0.85 and 0.95 for this Section S5. Theoretical explanation of the proposed AF3 phase experiment]. The time-dependent decay of the monitored values Fig. S1. Order parameter in magnetic domains. Fig. S2. Temperature dependence of magnetic Bragg intensities. revealed a half-time of approximately 100 hours for the used cells, Fig. S3. Theoretical phase diagrams. which were replaced every 24 hours by a new one revealing a flip- Table S1. Transformation properties of the complex order parameters S1 and S2 in the ping ratio of approximately 16. All data shown have been corrected extended little group (paramagnetic group). for this time-dependent decrease of the spin-filter efficiency, but Table S2. Basis vectors Snx of the irreducible representations n. Table S3. Observed and calculated polarization matrix elements in the AF1, AF2, and AF3 not for the incident polarization, since if chiral scattering is present, phases. not all polarization matrix elements depend on the neutron polar- ization in the same way (see Eq. 1 or 4). The measuring procedure for every single element of the polar- REFERENCES AND NOTES 1. S.-W. Cheong, M. Mostovoy, Multiferroics: A magnetic twist for ferroelectricity. Nat. Mater. ization matrix consists in counting the spin-up (n+) and spin-down 6, 13–20 (2007). neutrons (n−) in the detector by flipping the analyzed polarization 2. N. A. Spaldin, R. Ramesh, Advances in magnetoelectric multiferroics. Nat. Mater. 18, typically every second, which yields Pf,i = (n+− n−)/(n++ n−). The 203–212 (2019). polarization values have been corrected by measuring the background 3. T. Kimura, Y. Sekio, H. Nakamura, T. Siegrist, A. P. Ramirez, Cupric oxide as an induced-multiferroic with high-Tc. Nat. Mater. 7, 291–294 (2008). on both sides of the Bragg peak. 4. B. Kundys, A. Maignan, C. Simon, Multiferroicity with high-TC in ceramics of the The search for a phase with a very narrow temperature stability YBaCuFeO5 ordered perovskite. Appl. Phys. Lett. 94, 072506 (2009). range requires considerable instrumentational effort. Therefore, be- 5. M. Morin, E. Canévet, A. Raynaud, M. Bartkowiak, D. Sheptyakov, V. Ban, M. Kenzelmann, fore starting any measurement, the temperature was stabilized for E. Pomjakushina, K. Konder, M. Medarde, Tuning magnetic spirals beyond room 30 min even for temperature changes of 0.1 K, and we could assure temperature with chemical disorder. Nat. Commun. 7, 13758 (2016). 6. T. Shang, E. Canévet, M. Morin, D. Sheptyakov, M. T. Fernández-Díaz, E. Pomjakushina, a temperature stability throughout the measurements of a few hun- M. Medarde, Design of magnetic spirals in layered perovskites: Extending the stability dredths of a kelvin. range far beyond room temperature. Sci. Adv. 4, eaau6386 (2018). To determine the magnetic structures in the different phases, full 7. J. B. Forsyth, P. J. Brown, M. B. Wanklyn, Magnetism in cupric oxide. J. Phys. C: Solid State Phys. or partial polarization matrices were recorded on different magnetic 21, 2917–2929 (1988). 8. B. X. Yang, T. R. Thurston, J. M. Tranquada, G. Shirane, Magnetic neutron scattering study Bragg reflections. The analysis of the polarimetry data was performed of single-crystal cupric oxide. Phys. Rev. B 39, 4343–4349 (1989). using the software Mag2Pol (31), which is able to refine magnetic 9. P. J. Brown, T. Chattopadhyay, J. B. Forsyth, V. Nunez, Antiferromagnetism in CuO studies structure models with symmetrical constraints and multiple mag- by neutron polarimetry. J. Phys. Condens. Matter 3, 4281–4287 (1991). netic domains. 10. M. Ain, A. Menelle, B. M. Wanklyn, E. F. Bertaut, Magnetic-structure of CuO by neutron-diffraction with polarization analysis. J. Phys. Condens. Matter 4, 5327–5337 (1992). Single-crystal neutron diffraction 11. Z. Wang, N. Qureshi, S. Yasin, A. Mukhin, E. Ressouche, S. Zherlitsyn, Y. Skourski, J. Geshev, The temperature dependence of selected magnetic reflections was V. Ivanov, M. Gospodinov, V. Skumryev, Magnetoelectric effect and phase transitions measured on the single-crystal neutron diffractometers D10 (ILL, in CuO in external magnetic fields. Nat. Commun. 7, 10295 (2016). Grenoble) and D23 (CEA CRG ILL, Grenoble). Both were equipped 12. H. Katsura, N. Nagaosa, A. V. Balatsky, Spin current and magnetoelectric effect in noncollinear magnets. Phys. Rev. Lett. 95, 057205 (2005). with a four-circle cryostat, and a wavelength of 2.36 Å was selected 13. I. A. Sergienko, E. Dagotto, Role of the Dzyaloshinskii-Moriya interaction in multiferroic from a pyrolytic graphite monochromator. perovskites. Phys. Rev. B 73, 094434 (2006). 14. M. Mostovoy, Ferroelectricity in spiral magnets. Phys. Rev. Lett. 96, 067601 (2006). Symmetry analysis 15. R. Villarreal, G. Quirion, M. L. Plumer, M. Poirier, T. Usui, T. Kimura, Magnetic phase The magnetic structure models, in particular the coupling of Fourier diagram of CuO via high-resolution ultrasonic velocity measurements. Phys. Rev. Lett. 109, 167206 (2012). coefficients between magnetic moments in the unit cell, as well as the 16. G. Giovannetti, S. Kumar, A. Stroppa, J. van den Brink, S. Picozzi, J. Lorenzana, High-Tc derivation of the order parameters in the different symmetry-allowed ferroelectricity emerging from magnetic degeneracy in cupric oxide. Phys. Rev. Lett. 106, magnetic domains were carried out using symmetry/representation 026401 (2011). analysis. Illustrative and pedagogic examples for analog considerations 17. G. Jin, K. Cao, G.-C. Guo, L. He, Origin of ferroelectricity in high-Tc magnetic ferroelectric CuO. Phys. Rev. Lett. 108, 187205 (2012). on other compounds are presented in (35). 18. P. Tolédano, N. Leo, D. D. Khalyavin, L. C. Chapon, T. Hoffmann, D. Meier, M. Fiebig, Theory of high-temperature multiferroicity in cupric oxide. Phys. Rev. Lett. Theory 106, 257601 (2011). Theoretical analysis of the possible magnetic states compatible with 19. I. N. Apostolova, A. T. Apostolov, S. G. Bahoosh, J. M. Wesselinova, S. Trimper, the propagation vector q = (qx 0 qz) and phase transitions between Multiferroicity in the dielectric function of CuO. Phys. Status Solidi Rapid Res. Lett. 7, 1001–1004 (2013). them was performed on the basis of Landau theory by minimization 20. P. Babkevich, A. Poole, R. D. Johnson, B. Roessli, D. Prabhakaran, A. T. Boothroyd, of the thermodynamic potential built as its invariant (with respect Electric field control of chiral magnetic domains in the high-temperature multiferroic CuO. to the paramagnetic group) expansion for the order parameters up Phys. Rev. B 85, 134428 (2012). Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 7 of 8 SCIENCE ADVANCES | RESEARCH ARTICLE 21. H. Jacobsen, S. M. Gaw, A. J. Princep, E. Hamilton, S. Tóth, R. A. Ewings, M. Enderle, 34. J. Gaudet, A. M. Hallas, D. D. Maharaj, C. R. C. Buhariwalla, E. Kermarrec, N. P. Butch, E. M. H. Wheeler, D. Prabhakaran, A. T. Boothroyd, Spin dynamics and exchange T. J. S. Munsie, H. A. Dabkowska, G. M. Luke, B. D. Gaulin, Magnetic dilution interactions in CuO measured by neutron scattering. Phys. Rev. B 97, 144401 (2018). and domain selection in the XY pyrochlore antiferromagnet Er2Ti2O7. Phys. Rev. B 94, 22. G. Lawes, M. Kenzelmann, N. Rogado, K. H. Kim, G. A. Jorge, R. J. Cava, A. Aharony, 060407(R) (2016). O. Entin-Wohlman, A. B. Harris, T. Yildirim, Q. Z. Huang, S. Park, C. Broholm, 35. L. C. Chapon, An introduction to the use of representation analysis for studying A. P. Ramirez, Competing magnetic phases on kagomé staircase. Phys. Rev. Lett. 93, magnetoelectrics and multiferroics. EPJ Web Conf. 22, 00013 (2012). 247201 (2004). 23. G. Lawes, A. B. Harris, T. Kimura, N. Rogado, R. J. Cava, A. Aharony, O. Entin-Wohlman, Acknowledgments: We acknowledge the ILL beam time on the instruments D3, D10, and D23. T. Yildirim, M. Kenzelmann, C. Broholm, A. P. Ramirez, Magnetically driven ferroelectric Funding: This work was partially supported by the Russian Science Foundation (16-12-10531) order in Ni3V2O8. Phys. Rev. Lett. 95, 087205 (2005). and the Bulgarian National Science Fund BNSF DN-08/9. Financial support from the Spanish 24. O. Heyer, N. Hollmann, I. Klassen, S. Jodlauk, L. Bohatý, P. Becker, J. A. Mydosh, T. Lorenz, Ministry of Economy, Competitiveness and Universities, through the “Severo Ochoa” D. Khomskii, A new multiferroic material: MnWO4. J. Phys.: Condens. Matter 18, L471–L475 Programme for Centres of Excellence in R&D (SEV-2015-0496) and the MAT2017-85232-R (AEI/ (2006). FEDER, EU), and from Generalitat de Catalunya (2017 SGR 1377) is acknowledged. The open 25. I. Urcelay-Olabarria, J. M. Perez-Mato, J. L. Ribeiro, J. L. García-Muñoz, E. Ressouche, access fee was covered by FILL2030, a European Union project within the European V. Skumryev, A. A. Mukhin, Incommensurate magnetic structures of multiferroic MnWO4 Commission’s Horizon 2020 Research and Innovation programme under grant agreement studied within the superspace formalism. Phys. Rev. B 87, 014419 (2013). Nº731096. Author contributions: N.Q. conceived and led the project. N.Q. carried out the SNP 26. M. Kenzelmann, A. B. Harris, A. Aharony, O. Entin-Wohlman, T. Yildirim, Q. Huang, S. Park, experiment with participation of E.R., V.S., and A.M. N.Q. analyzed the polarimetry data. N.Q. G. Lawes, C. Broholm, N. Rogado, R. J. Cava, K. H. Kim, G. Jorge, A. P. Ramirez, Field and E.R. performed the unpolarized single-crystal neutron diffraction experiment and dependence of magnetic ordering in Kagomé-staircase compound Ni3V2O8. analyzed the data. N.Q. wrote the manuscript with assistance from E.R., V.S., and A.M. A.M. Phys. Rev. B 74, 014429 (2006). supplied the theoretical work. M.G. grew the single-crystal sample. V.S. initiated the 27. V. P. Sakhnenko, N. V. Ter-Oganessian, Praphase concept for the phenomenological collaboration on CuO. All authors contributed to the scientific discussion and edited the description of magnetoelectrics. Crystallogr. Rep. 57, 112–117 (2012). manuscript. Competing interests: The authors declare that they have no competing interests. 28. A. Rebello, Z. C. M. Winter, S. Viall, J. J. Neumeier, Multiple phase transitions in CuO Data and materials availability: All data needed to evaluate the conclusions in the paper are observed with thermal expansion. Phys. Rev. B 88, 094420 (2013). present in the paper and/or the Supplementary Materials. Additional data related to this paper 29. P. J. Brown, Spherical neutron polarimetry, in Neutron Scattering from Magnetic Materials, may be requested from the authors. The data will be accessible through their DOI (http://doi.ill. T. Chatterji, Ed. (Elsevier, 2005), chap. 5, pp. 215. fr/10.5291/ILL-DATA.5-54-259 and https://doi.ill.fr/10.5291/ILL-DATA.EASY-418). 30. T. Hoffmann, K. Kimura, T. Kimura, M. Fiebig, Second harmonic generation spectroscopy and domain imaging of the high-temperature multiferroic CuO. J. Physical Soc. Japan 81, Submitted 16 July 2019 124714 (2012). Accepted 2 December 2019 31. N. Qureshi, Mag2Pol: A program for the analysis of spherical neutron polarimetry, flipping Published 14 February 2020 ratio and integrated intensity data. J. Appl. Cryst. 52, 175–185 (2019). 10.1126/sciadv.aay7661 32. V. S. Maryasin, M. E. Zhitomirsky, Order from structural disorder in the XY pyrochlore antiferromagnet Er2Ti2O7. Phys. Rev. B 90, 094412 (2014). Citation: N. Qureshi, E. Ressouche, A. Mukhin, M. Gospodinov, V. Skumryev, Proof of the elusive 33. A. Andreanov, P. A. McClarty, Order induced by dilution in pyrochlore XY antiferromagnets. high-temperature incommensurate phase in CuO by spherical neutron polarimetry. Sci. Adv. 6, Phys. Rev. B 91, 064401 (2015). eaay7661 (2020). Qureshi et al., Sci. Adv. 2020; 6 : eaay7661 14 February 2020 8 of 8