Orbital Transitions and Frustrated Magnetism in the Kagome-Type

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Orbital Transitions and Frustrated Magnetism in the Kagome-Type Copper Mineral Volborthite Zenji Hiroi,*,† Hajime Ishikawa,† Hiroyuki Yoshida,‡ Jun-ichi Yamaura,§ and Yoshihiko Okamoto⊥ †

Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan § Materials Research Center for Element Strategy, Tokyo Institute of Technology, Yokohama, Kanagawa 226-8503, Japan ⊥ Department of Applied Physics, Nagoya University, Nagoya 464-8603, Japan Downloaded via UNIV AUTONOMA DE COAHUILA on July 24, 2019 at 11:36:05 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

ABSTRACT: Volborthite Cu3V2O7(OH)2·2H2O is a copper mineral that materializes a two-dimensional quantum magnet comprising a kagome net of spin-1/2 Cu2+ ions. We prepared single crystals of volborthite using hydrothermal conditions and investigated their crystal structures and magnetic properties. Unusual orbital “switching” and “flipping” transitions were observed: in the former type of transition (switching), the Cu 3d orbital occupied by an unpaired electron changes between the d(3z2−r2) and d(x2−y2) types, and in the latter type of transition (flipping), the d(x2−y2)-type orbitals change their directions. Their origin is ascribed to variations in the orientation of water molecules in the gap between the kagome layers and the accompanying changes of hydrogen bonding. These orbital transitions dramatically modify the magnetic interactions between Cu2+ spins, from the anisotropic kagome type to the formation of spin trimers over the kagome net. The effective spin 1/2 generated on the trimers exhibits a frustrated magnetism, resulting in a rich phase diagram in the magnetic fields. Volborthite is a unique compound showing an exceptional interplay between the orbital and spin degrees of freedom.

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INTRODUCTION

have the same energy. In extended lattices made of such triangles connected via their vertices or edges, such as twodimensional triangular and kagome lattices or a threedimensional pyrochlore lattice, a macroscopic number of states should have the same energy; that is, there is a macroscopic degeneracy. This is called geometrical frustration. On the other hand, even in nongeometrically frustrated lattices, another type of frustration can occur when two or more kinds of magnetic interactions that cannot be satisfied simultaneously compete with each other. For example, in a spin chain with ferromagnetic first-neighbor and antiferromagnetic second-neighbor interactions, the former favors and the latter disfavors a ferromagnetic state. In either case, one specific spin arrangement such as a Néel state with spins pointing up and down periodically is not achieved as a ground state. Instead, exotic ground states can appear: quantum spin liquid states are composed of quantum-mechanically entangled singlet states covering all of the lattice points, as depicted in Figure 1b, and in multipolar phases, higher-order multipole moments rather than conventional magnetic-dipole moments establish a long-range order (LRO). Moreover, unusual

Kagome Antiferromagnet (KAFM). Frustration is one of the most interesting topics in the field of magnetism. When magnetic ions are placed on the lattice points of a trianglebased network, the antiferromagnetic interactions between neighboring spins cannot all be satisfied simultaneously (Figure 1a). Among the possible 23 = 8 states for three spins on one triangle, six states, except two ferromagnetic states,

Figure 1. (a) Kagome lattice with spins on the triangle that couple with each other by the nearest-neighbor antiferromagnetic interaction J pointing up or down (arrows). (b) Example of the spin liquids theoretically predicted to be realized in the spin-1/2 KAFM, in which all spins are paired in a resonating-valence bond state.3 © XXXX American Chemical Society

Special Issue: Paradigm Shifts in Magnetism: From Molecules to Materials Received: April 21, 2019

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DOI: 10.1021/acs.inorgchem.9b01165 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry topological excitations and vortices are expected in frustrated magnets. Therefore, unprecedented physics can emerge from frustrated spin systems.1,2 We have been exploring new compounds that can materialize the KAFM to study the physics of frustration. The kagome lattice was first focused on by Syôzi and Husimi in their statistical−mechanical theory in 1951,4,5 which is obtained by deleting one-fourth of the lattice points periodically from the two-dimensional triangular lattice and composed of triangular and hexagonal motifs (Figure 1a); the original Japanese word kagome means a bamboo-weaving pattern to make a hard basket with a minimal number of bamboo strips. Especially, the spin-1/2 KAFM has attracted much attention because the large quantum fluctuation for the minimum quantum number of the spin in addition to the geometrical frustration can lead to exotic ground states. In spite of the long history of research, a clear picture for the ground state has not yet been established:6 the ground state of the spin-1/2 KAFM may be either a gapless spin liquid, such as that depicted in Figure 1b, or a certain gapped order. Moreover, it has been predicted that, under a magnetic field, four magnetization plateausflat regions in a magnetization versus magnetic field curveappear at 1/9, 1/3, 5/9, and 7/9 of the saturation magnetization.7 These intriguing predictions still remain undiscovered in experiments because of the lack of ideal compounds for the KAFM. The final goal for we “kagomese” who love the KAFM is to realize the model in actual compounds and to uncover the magnetic properties of the KAFM. Orbital Selection and Arrangements in Copper Compounds. A typical magnetic ion with a spin of 1/2 is the divalent copper ion Cu2+ with the 3d9 electron configuration. It is known that many natural minerals containing Cu2+ ions crystallize in layered structures, with the Cu2+ ions forming a kagome net in each layer.6 Chemists prepared such minerals (they are called synthetic minerals, as geologists call only natural ones as such) and, in collaboration with physicists, studied them as candidates for the KAFM: herbertsmithite, ZnCu3(OH)6Cl2 (Figure 2a);8 kapellasite, ZnCu3(OH)6Cl2;9 cadmium kapellasite, CdCu3(OH)6(NO3)2· H2O;10,11 calcium kapellasite, CaCu3(OH)6Cl2·0.6H2O;12 vesignieite, BaCu3V2O8(OH)2 (Figure 2b).13,14 The kagome layers in these synthetic minerals consist of copper−ligand (Cu−X) octahedra connected by their edges, as shown in Figure 2 for herbertsmithite and vesignieite. The divalent copper ion placed in the octahedral crystal field exhibits a strong Jahn−Teller (JT) effect.15 As depicted in Figure 3, the doubly degenerate eg orbitals split into d(x2−y2) and d(3z2−r2) through deformation of the octahedron; one unpaired and two paired electrons occupy the upper and lower levels, respectively, the former of which becomes the spincarrying orbital. This deformation causes a net energy gain for the electrons but a loss of lattice energy. When the former energy exceeds the latter one, this JT deformation should occur. Since the electronic energy gain for Cu2+ is large, typically 12000 K, a JT distortion always takes place and is quite stable.15 Among the two eg orbitals, d(x2−y2) or d(3z2−r2) is selected when the octahedron is elongated or compressed along the z axis, respectively (Figure 3). These deformations are referred to as the (4 + 2) and (2 + 4) coordinations, respectively. Which deformation is selected is not decided by a single octahedron but depends on the three-dimensional network of

Figure 2. Orbital arrangements in the kagome layers of (a) herbertsmithite with the d(x2−y2)-type orbitals and (b) vesignieite with the d(3z2−r2)-type orbitals.

Figure 3. Energy diagram of the Cu2+ ion in the 3d9 electron configuration. When an octahedron made of a central Cu2+ (red sphere) and six ligands (sky blue sphere) is compressed or elongated, the high-energy 3d orbital dH is selected to be of the d(3z2−r2) or d(x2−y2) type, respectively.

octahedra. For example, in copper oxide superconductors, all of the octahedra are elongated in the (4 + 2) coordination along the common z axis and connected in the xy plane to form a two-dimensional square lattice made of Cu2+ ions; the B


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Inorganic Chemistry Table 1. Structural Parameters of Four Polymorphs of Volborthite As Determined by XRD Experiments space group T (K) a (Å) b (Å) c (Å) β (deg) V (Å3) Z Cu 3d orbital reference

C2/m RTa 10.606(4)b 5.874(1) 7.213(3) 94.90(3) 447.5(18) 2 d(x2−y2)/d(3z2−r2) 21

C2/m 323 10.657(3) 5.887(1) 7.228(2) 95.035(8) 451.82(6) 2 d(x2−y2)/d(3z2−r2) 22

C2/c 293 10.6118(4) 5.8708(2) 14.4181(6) 95.029(1) 894.79(6) 4 d(x2−y2) 23

I2/a 200 10.6237(3) 5.8468(1) 14.3892(7) 95.3569(1) 889.88(6) 4 d(x2−y2) 35

P21/a 50 10.6489(1) 5.8415(1) 14.4100(1) 95.586(1) 892.13(6) 4 d(x2−y2) 35

a

Room temperature. bLattice constants determined by the powder XRD experiments.

d(x2−y2) orbitals hybridize strongly with the 2p orbitals of the oxide ions in the CuO2 plane to allow a two-dimensional covalent sheet for the high-temperature superconductivity when doped with hole carriers. On the other hand, in KCuF3, the d(x2−z2) and d(y2−z2) orbitals are alternatingly arranged in the xy plane of the three-dimensional perovskite structure. In contrast, the choice of the (2 + 4) coordination seems rare, and thus it is believed that the (2 + 4) coordination may not be intrinsic but just due to a dynamical JT effect between two orientations of the (4 + 2) coordination:16 the former is a structural average of the latter. Nevertheless, we think that the (2 + 4) coordination is possible in some kagome minerals, as exemplified in vesignieite and volborthite. Another typical JT ion is Mn3+ with the 3d4 (t2g3eg1) electron configuration. For Mn3+, the electronic energy gain is relatively small, so a JT distortion may be less stable than that for Cu2+. Thus, an orbital-disordered state with degenerate eg states can appear at elevated temperatures in manganese perovskites.17 Then, an order−disorder transition associated with the orbital degree of freedom may be observed as a function of the temperature or pressure. In contrast, the general belief thus far has been that the large JT stabilization of Cu2+ seems to not allow such an orbital transition. We note that the orbital state of the Cu2+ ion is exceptionally flexible in kagome minerals. To keep the symmetry of the kagome lattice, either the d(x2−y2)- or d(3z2−r2)-type orbital is selected and systematically arranged, such that there is a 3fold axis at the center of each triangle, as shown in Figure 2; a mixture of the two would break this symmetry. In fact, kagome lattices made of only d(x2−y2)-type orbitals are found in herbertsmithite- and kapellasite-related compounds, and those made of d(3z2−r2)-type orbitals are found in vesignieite, although there is a claim that vesignieite comprises not d(3z2− r2) but d(x2−y2).18 It is noted that the 3-fold axis is inevitably lost by a JT distortion when Cu2+ ions are placed in the triangular lattice made of edge-sharing octahedra without depletion. The Cu2+ ion is happily compatible with the kagome lattice. The two kinds of kagome lattices made of only d(x2−y2)- or d(3z2−r2)-type orbitals can appear depending on the size of the hexagonal hole made of edge-sharing octahedra in the kagome layer. Note that the two pictures in Figure 2 are drawn on the same scale: apparently herbertsmithite has larger holes than vesignieite. This difference comes from the size of the interlayer units: large ZnO6 octahedra and small VO4 tetrahedra exist above and/or below each hexagonal hole, respectively. The larger hexagonal hole means that the surrounding Cu−X octahedra are tilted more from normal to the kagome plane. Thus, elongated octahedra with the (4 + 2)

coordination are chosen in herbertsmithite, while compressed ones with the (2 + 4) coordination are chosen for vesignieite with its small hexagonal holes. Volborthite. Volborthite Cu3V2O7(OH)2·2H2O is a copper mineral found in Ural almost 200 years ago and was named after the Russian paleontologist Alexander von Volborth. The physical, chemical, and optical properties were studied by Guillemin.19 The crystal structure of a natural crystal was reported to be monoclinic in 1974 by Leonardsen and Petersen.20 Later in 1990, Lafontaine and co-workers determined the crystal structure in the space group C2/m using both X-ray diffraction (XRD) and neutron diffraction experiments on synthetic powder samples, as listed in Table 1; atomic positions including those of the hydrogen atoms were given.21 This C2/m structure was also observed in a small synthetic crystal of volborthite prepared in 2012.22 The basic crystal structure of volborthite, as shown in Figure 4a, consists of kagome layers made of edge-sharing

Figure 4. (a) Perspective view of the layered crystal structure of volborthite. (b) Plane view of one kagome layer comprising a kagome lattice made of Cu2+ ions carrying a spin of 1/2. (c) Local arrangement of 3d orbitals in the C2/m structure, in which Cu1 has a d(3z2−r2)type orbital and Cu2 has a d(x2−y2)-type orbital. (d) Local arrangement of 3d orbitals in the C2/c structure, in which all three Cu sites have d(x2−y2)-type orbitals.

CuO4(OH)2 octahedra, which stack such that pairs of VO4 tetrahedra make V2O7 pillars; the V5+ ions there are nonmagnetic. In addition, water molecules occupy the voids among the pillars. There are two Cu sites coordinated octahedrally: Cu1 is located between chains made of Cu2 along the b axis in the kagome net, which is slightly elongated along the a axis (Figure 4b). The local structure is depicted in C


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Inorganic Chemistry Figure 4c:21 Cu1 has a (2 + 4) coordination (Cu1−O2 = 2 × 1.91 Å; Cu1−O3 = 4 × 2.16 Å) and takes the d(3z2−r2)-type orbital, while Cu2 has a (4 + 2) coordination (Cu2−O2 = 2 × 1.90 Å; Cu2−O4 = 2 × 2.05 Å; Cu2−O3 = 2 × 2.38 Å) and takes the d(x2−y2)-type orbital. Both JT deformations are significantly large, and there is no doubt on the selection of these orbitals. In contrast to most kagome minerals, the two kinds of orbitals coexist, probably because they are energetically even and also because the specific three-dimensional structure allows a mixture of the different JT deformations. That said, another room-temperature structure with the space group C2/c was found in a large synthetic crystal of volborthite (Figure 4d),23 which is described later. Volborthite was introduced to the field of magnetism in 2001.24 In the kagome layer, there are strong superexchange interactions between nearby Cu2+ spins via oxide and hydroxide ions (Figure 4c), which, on average, are antiferromagnetic and have a magnitude of ∼100 K estimated from the Curie−Weiss (CW) temperature in the magnetic susceptibility. In contrast, interlayer magnetic couplings must be negligible, as expected from the large distance between spins. Thus, volborthite really looks like a “kagome”. The understanding of magnetic properties of volborthite has advanced with improvement of the sample quality.22−38 The initially studied, low-quality powder samples prepared by a precipitation method showed a broad magnetic transition below 2 K.26 The crystalline quality was improved by hydrothermal annealing, which revealed a sharp magnetic transition at 1.0 K instead of the broad one.27 Moreover, successive transitions were observed as a function of the magnetic field, giving a rich phase diagram.28 Finally, in 2015, large (1−2 mm in size) crystals were grown hydrothermally, and a 1/3 magnetization plateau was observed above 27−28 T.35 Through this stream of research, a large sample dependence of the magnetic properties has been recognized, which caused obstacles in experiments and interpretation and sometimes led to controversy. The purpose of this manuscript is to marshal all of the data about the relationship between the crystal structures and magnetic properties, which reveals an intriguing interplay between orbitals and magnetic interactions in this unique copper mineral, volborthite.

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Figure 5. (a) Preparation of single crystals of volborthite under hydrothermal conditions at 443 K for 5−30 days. (b) Photograph of thick and thin crystals (A and B) obtained after 30 and 15 days, respectively. and the atomic emission from each element was recorded. Thus, the obtained molar ratio of copper and vanadium is 3:1.99, which is reasonably close to the ideal stoichiometry. Structural Analysis. Single-crystal X-ray diffraction (XRD) experiments at 43−340 K were performed using a CCD area detector (Bruker) at the wavelength of the Mo Kα radiation or an imagingplate XRD system (Rapid, Rigaku) at λ = 0.68888 Å in the synchrotron radiation beamline BL8A at the Photon Factory, KEK. A crushed thin crystal of 0.1 × 0.05 × 0.02 mm3 size was examined. The crystal structure was determined by a charge-flipping method using the SUPERFLIP program39 and refined by a least-squares method using SHELX97.40 Magnetic Measurements. Magnetic susceptibility measurements were performed at 1.8−350 K and 0−7 T in a commercial SQUID magnetometer (MPMS3, Quantum Design). One arrowhead crystal (crystal A) was broken into two triangular pieces at the twin plane. One of the thus-obtained monodomain crystals was attached to a sample holder made of quartz with a small amount of varnish. Magnetic susceptibilities at magnetic fields along the principal crystallographic axes were measured. High-field magnetization was measured by the induction method using a pick-up coil in pulsed magnetic fields of up to 75 T, with a duration time of 4 μs, at 1.4 K. The measurements were carried out on a pile of crystals (crystal A) at magnetic fields perpendicular and parallel to the kagome plane. Other measurements were carried out on randomly oriented crystal B and on a polycrystalline sample.

EXPERIMENTAL SECTION

Synthesis. Single crystals of volborthite were prepared by the hydrothermal technique.23,35 CuO (4N), V2O5 (4N), HNO3(aq) (60 wt %, Wako), and 15−400 mL of pure H2O were put into a Teflon beaker of 45−790 mL volume; 1.3 mL of HNO3(aq) was added per 100 mL of H2O. The beaker was sealed in a stainless-steel autoclave vessel (OM Labotech), heated at 443 K for 5−30 days, and then furnace-cooled to room temperature. Transparent yellow-green crystals of volborthite were grown on the solid product composed of polycrystalline volborthite and unreacted starting materials, as depicted in Figure 5a. The arrowhead-shaped crystal is composed of two triangular domains with the surface parallel to the ab plane, i.e., the kagome plane, which is twinned at the (110) interface at the center of the arrowhead. The crystals become thicker with time duration: typical thin and thick crystals grown in 15 and 30 days are shown in Figure 5b, which are called crystals B and A, respectively. It is noted that the crystal size scales the volume of the reaction beaker; the larger the beaker, the larger the crystal. This is probably because the conditions suitable for crystal growth can last for a longer time with larger amounts of starting materials put into a larger beaker. Elemental analysis was performed by the inductively coupled plasma atomic emission spectroscopy method (Horiba JY138 KH ULTRACE). A bunch of thin crystals were dissolved in HNO3(aq),

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RESULTS AND DISCUSSION Orbital Transitions in Volborthite. The crystals synthesized in the present study adopt a monoclinic structure with the space group C2/c,23 different from the previously reported C2/m structure. In these crystals, we observed a large change in the diffraction patterns at around room temperature. Figure 6a shows the temperature evolution of the intensities of selected diffraction peaks measured upon heating after cooling to 250 K, in which drastic changes are observed at 300 K. D


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Figure 6. Temperature dependences of the intensities of selected XRD peaks, labeled with their indices in the monoclinic unit cells, at (a) the orbital flipping transition at Ts1 = 290−300 K from the hightemperature C2/c structure to the low-temperature I2/a structure and (b) the transition to the P21/a structure below Ts2 = 155 K. The solid line for each data set is a guide to the eye.

Reflections of the h + k = 2n type, which indicate a C-centered lattice, disappear below 300 K (in the case of l = 2m + 1), and those of the h + k + l = 2n type, which means a body-centered lattice, appear instead. In a cooling experiment, on the other hand, these changes occurred at 290 K, indicating a thermal hysteresis. Thus, a first-order structural transition takes place at Ts1 = 290−300 K. A structural refinement at 200 K reveals a monoclinic structure with the space group I2/a. Note that this nonstandard space group is chosen in order to keep similar monoclinic unit cells through the transition for structural comparison. A second transition is observed at Ts2 = 155 K. As shown in Figure 6b, diffraction peaks of the h + k + l = 2n + 1 type appear below Ts2, indicating a breaking of the body-centered lattice. The transition must be of the second order because the intensity grows with T1/2 without a thermal hysteresis. A structural refinement at 50 K finds a monoclinic structure with the space group P21/a. The structural data of these phases are listed in Table 1. Here we describe what happens, substantially at the Ts1 transition, in terms of the orbital arrangement in the kagome layer. There are three Cu sites in the C2/c structure: Cu1, Cu21, and Cu22. As depicted in Figure 4d, each of the three sites takes the (4 + 2) coordination and thus the d(x2−y2)-type orbital; there is therefore a difference in the orbital selection at the Cu1 site in comparison to the C2/m structure. The d(x2− y2)-type orbitals at the Cu21 and Cu22 sites form a chain along the b axis (Figure 7) through two bridging oxide atoms, O2 and O4. The d(x2−y2)-type orbital of Cu1 is also connected to

Figure 7. Schematic representations of the orbital-flipping transition from the C2/c structure and the orbital-switching transition from the C2/m structure, both to the I2/a structure, in volborthite. Respectively, these occur at 290−300 K in a large arrowhead-shaped crystal35 and at 310 K in a small rugby-ball-shaped crystal.22 Below 155 K, the P21/a structure, which has an orbital arrangement identical with that found in the I2/a structure, is adopted by both crystals. The clovers and cigars represent the d(x2−y2)- and d(3z2−r2)-type orbitals projected along the normal to the kagome layer, respectively. The expected magnetic interactions are distinguished by the types of lines connecting the Cu2+ ions: Jp (thick line), Jq (broken line), Jb(1) − Jb(2) (thin line), and J′ (another broken line in C2/m).

those of Cu21 and Cu22 via O2 (Figure 4d). In addition, there is another connection only for Cu1−Cu21 via the lobe of the d(x2−y2)-type orbital pointing to O31. Thus, the two paths are inequivalent. As a result, there are three kinds of copper chains in the kagome layer: Cu1−Cu21, Cu1−Cu22, and Cu21− Cu22. In contrast, in the low-temperature I2/a structure, the d(x2− 2 y )-type orbitals of Cu1 change their directions at every other row along the b axis (magenta lobes in Figure 7). Thus, orbital flipping takes place at half of the Cu1 sites below Ts1.35 As a result, the two inclined Cu1−Cu2 chains become equivalent, with the d(x2−y2)-type orbitals at Cu1 aligned in a staggered way. Note that the Cu21 and Cu22 sites of the C2/c structure E


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Figure 8. Local structure around the Cu1 site for the (a) C2/m (at room temperature),21 (b) C2/c (293 K),23 and (c) I2/a (150 K)35 structures. A (Cu1)O4(OH)2 octahedron in a kagome layer, and nearby water molecules between the layers, are depicted. Bond lengths in Å are labeled. The positions of the protons of the water molecules have been determined by neutron diffraction in (a) (black spheres), and are estimated in (b) and (c) (red spheres).

the C2/m structure, the positions of the hydrogen atoms were determined by neutron diffraction experiments, as shown in Figure 8a.21 The hydrogen H1 connected to O2 at the vertex of the (Cu1)O4(OH)2 octahedron points upward or downward from the kagome layer and couples with an O5 atom of a water molecule by strong hydrogen bonding (O2−O5 = 2.767 Å); this bond length is close to the typical O−H···O distance of 2.81 Å.41 On the other hand, the O3 ligand atoms of the (Cu1)O4(OH)2 octahedron interact weakly with the H2 atoms of the water molecule (O3−O5 = 2.933 Å). Because there is a mirror plane through Cu1 and O5, H2 should attract the two O3 atoms equally. Thus, four Cu1−O3 distances are same (2.159 Å) and much longer than two Cu1−O2 distances (1.906 Å), selecting the d(3z2−r2)-type orbital at Cu1. For the C2/c and I2/a structures, unfortunately, reliable data on the positions of the hydrogen atoms have not been obtained because of the absence of neutron diffraction data. However, one can estimate the contribution of hydrogen bonding from the determined O−O distances. The C2/c structure lacks the mirror plane found in the C2/m structure, and the O3 site separates into the O31 and O32 sites (Figure 8b). The distances to the O5 atom of the water molecule are shorter for O32, suggesting a stronger hydrogen bond (O5− O32 = 2.941 Å; O5−O31 = 3.073 Å). Then, O32 moves away from Cu1, while O31 comes closer to Cu1 (Cu1−O31 = 1.994 Å; Cu1−O32 = 2.349 Å; Cu1−O2 = 1.942 Å). Thus, the d(x2−y2)-type orbital appears to be selected at Cu1. In contrast, in the I2/a structure (Figure 8c), O31 moves away and comes close to O5, while O32 approaches Cu1 (Cu1− O31 = 2.343 Å; Cu1−O32 = 2.010 Å; Cu1−O2 = 1.937 Å), resulting in a change in the direction of the (4 + 2) coordination and a flipping of the d(x2−y2)-type orbital. This change occurs at every other Cu1 site, which corresponds to the I2/a structure below Ts1. It is reasonable to assume that these changes in the local structure of each octahedron originate from different alignments of the water molecules between the kagome layers. In

become a single site, Cu2, although they keep the arrangement of their d(x2−y2)-type orbitals. This orbital arrangement in the I2/a structure is retained in the lowest-temperature P21/a phase below Ts2. Two kinds of kagome layers become distinct in the P21/a structure, but the bond lengths and angles in each are nearly equivalent.35 Therefore, we discuss the lowesttemperature magnetic properties of volborthite on the basis of the orbital arrangement found in the I2/a structure. A different room-temperature structure, with C2/m symmetry, has been observed in a synthetic powder21 and a small single crystal of volborthite.22 This small crystal was grown under hydrothermal conditions by slow cooling from 453 K; the large crystals in the present study were obtained by maintaining 443 K for many days. In the C2/m structure, Cu1 takes the d(3z2−r2)-type orbital, and there is a mirror plane at Cu1 perpendicular to the b axis. Thus, the kagome lattice is composed of two kinds of copper chains, as shown in Figure 7. Interestingly, the small crystal with the C2/m structure exhibits a first-order structural transition similar to that of the I2/a structure at 310 K found in the C2/c crystal. This transition will be called the orbital-switching transition,22 because at the Cu1 site the d(3z2−r2)-type orbital is transformed to the d(x2−y2)-type orbital. The difference between the two roomtemperature structures may be related to an order−disorder transition of the water molecules between the kagome layers, as will be discussed in the next section. It is also possible that, because of subtle differences in the preparation conditions, they have similar free energies, and one appears as a metastable phase. Origin of the Two Orbital Transitions. In minerals, hydroxide ions (OH)− and water molecules (H2O)0 confined in a cage or sandwiched between layers often play a crucial role in the stabilization of a structure.41 It is therefore plausible that in volborthite the two orbital transitions are related to certain changes in the orientation of the water molecules between the kagome layers and the accompanying variations of hydrogen bonding with the ligands of the CuO4(OH)2 octahedra. For F


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Inorganic Chemistry the C2/m structure, the water molecules must be either quickly vibrating or rotating, or randomly oriented, so as to give an average structure with a mirror plane at Cu1. In the C2/c and I2/a structures, one of the two hydrogen atoms of the water molecule must become static to interact with O32 or O31, respectively. Because the O31−O5−O32 bond angle is 64°, which is much smaller than the 104.5° value of the H−O−H angle in a water molecule, the other hydrogen atom should move away. This change in the orientation of the water molecule causes the orbital-flipping transition between the C2/ c and I2/a structures. The reason why the latter becomes stable at low temperatures is not clear but may be related to the global lattice stability; the total strain or electrostatic energy may be lower when the water molecules in one gap align in a staggered way (I2/a) in comparison to them having a uniform alignment (C2/c). Furthermore, the relationship between the two room-temperature structures, C2/m and C2/c, could be considered to be an order−disorder transition associated with the orientation of the water molecules. Therefore, one may say that the transitions in volborthite are structural transitions of “two-dimensional water ice” sandwiched by the kagome layers. The presence of various orbital states in volborthite indicates that the Cu2+ ions have an exceptionally small JT energy, the result of the flexible layered structure, without strong chemical bonds between layers, and the variable hydrogen bonding from the water molecules in the gap. Note that, in the triangular lattice or in any three-dimensional structure, JT stabilization of Cu2+ is too large to allow such orbital transitions. It is also emphasized that, even though the kagome layer is composed of edge-sharing octahedra extending isotropically in two dimensions, a specific distribution of orbitals gives rise to a variety of networks of Cu2+ and thus anisotropic magnetic interactions between them. Symmetry lowering due to the ordering of molecular units between layers seems to be a common feature in kagome minerals. The kagome symmetry is preserved only when the molecular units such as H2O, NO3, AsO4, and VO4 are randomly oriented at high temperatures, or when there are certain crystalline defects, and is lost when the molecular units become ordered at low temperatures or the defects are removed. This means that the regular kagome lattice requires a certain degree of disorder, which may influence the magnetism by locally modulating magnetic couplings. For highly frustrated systems like the KAFM, especially, the true ground state might easily be masked by such a perturbation. This dilemma makes it a challenge for experimentalists to obtain a perfect KAFM in actual kagome minerals. Magnetic Properties of Volborthite. We will now move to the magnetic properties of volborthite. Figure 9 shows the magnetic susceptibilities at 2−320 K, measured in a magnetic field of 5 T, along the three principal axes of a single-domain crystal, which was obtained by breaking one crystal (A) at the twin boundary. Irrespective of the field direction, these show CW behavior at high temperatures, followed by broad peaks at around 20 K that suggest the development of antiferromagnetic short-range order. No anomaly indicative of a LRO is observed above 2 K; it is noted that previous heat capacity and 51 V NMR experiments revealed successive transitions at lower temperatures of 1.2 and 0.8 K.22,26,28 The differences between the magnetic susceptibilities come from the anisotropic Lande g factor: CW fits to the 270−290 K data (I2/a) yield ga = 2.25, gb = 2.11, and gc = 2.24 for H∥a, H∥b, and H⊥ab, which are

Figure 9. Magnetic susceptibilities of one monodomain crystal (crystal A) of volborthite measured in a magnetic field of 5 T upon heating after zero-field cooling (ZFC) and then upon cooling in the field (FC). The field directions are set parallel (H∥a and b) and perpendicular (H⊥ab) to the kagome layer. The anomalies at the orbital-flipping transition at Ts1 = 290−300 K are expanded in the inset.

typical values for the Cu2+ spin.42 Thus, volborthite is a nearly isotropic Heisenberg spin system. The magnetic susceptibilities exhibit clear discontinuities with thermal hysteresis at 290−300 K, which are obviously due to the orbital-flipping transition at Ts1. Note that their magnitudes are reduced upon cooling through the transition, indicating an enhancement in the average antiferromagnetic interaction. In fact, CW fits to the data at 270−290 and 300− 320 K give CW temperatures of −140 and −90 K, respectively; the antiferromagnetic interactions in the C2/c phase are increased by about 60% in the I2/a phase. Thus, the orbitalflipping transition has a crucial effect on the magnetic interactions. In contrast, there is almost no change in the magnetic susceptibilities at Ts2, although there is a sharp, second-order type peak in the heat capacity (not shown). This means that the magnetic interactions change negligibly at Ts2, which may be consistent with the unsubstantial changes to the orbital pattern and the bond lengths and angles between the I2/a and P21/a phases. Intriguing magnetic properties are observed under high magnetic fields.35,38 Figure 10 shows the magnetization processes, measured under pulsed magnetic fields of up to 75 T, of crystals A and B and a powder sample. At first glance, a large sample dependence can be noted. The magnetization M of the powder sample, which we believed to be of a high quality in our previous study, increases in three stepsfirst at 4.5 T (invisible in the figure), second at 25 T, and third at 46 Tand becomes saturated above 60 T.28 The saturation magnetization is considerably larger than M/Ms = 1/3, where Ms is the magnetization of the state fully polarized by the field: Ms = gSμB ∼ 1.1μB.32 In sharp contrast, M of crystal A, which was measured on a stack of several tens of crystals shown in the inset of Figure 10, exhibits a much larger rise at ∼25 T and a huge plateau at 30− 75 T;35 recent Faraday rotation measurements up to 180 T have shown that the plateau may end at 100 T or may still continue to more than 180 T.37 M/Ms at the plateau coincides with 1/3 when one considers a small field-proportional G


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are segmented and become equivalent, as shown in Figure 7. The magnetic interaction in the chain is −Jp−Jp−Jq−Jq−. Janson and co-workers estimated their magnitudes by fitting the temperature dependence of the magnetic susceptibility below Ts1 to their model.43 Surprisingly, Jp is antiferromagnetic and very large, being around 250 K, while Jq is ferromagnetic and relatively small, around −50 K. The fact that |Jp| ≫ |Jq| is ascribed to the additional magnetic path for Jp via a VO4 tetrahedron located above or below the hexagonal hole of the kagome layer; such a long-range Cu−O−V−O−Cu pathway can give an interaction as large as 300 K.46 Moreover, the interactions in the Cu2 chain are estimated to be Jb(1) = −120 K and Jb(2) = 50 K. As illustrated in Figure 11a, to a first approximation one may identify linear trimers of Cu2+ ions in the I2/a phase, within

Figure 10. Magnetizations M measured in pulsed magnetic fields up to 75 T at 1.4 K.35 The samples examined are a polycrystalline sample (powder), a stack of crystals A (shown in the inset), with the magnetic fields both parallel and perpendicular to the ab (kagome) plane, and randomly oriented crystals B.

contribution. On the other hand, crystal B shows intermediate behavior that is between those of the powder and crystal A. The source of this sample dependence will be discussed later. As mentioned in the Introduction, the simple spin-1/2 KAFM has a 1/3 magnetization plateau. However, the expected field range corresponds to half of J, which is 35 T for the J = 100 K value of volborthite.7 Thus, the observed plateau is too large. This fact implies that an appropriate magnetic model for volborthite deviates substantially from the KAFM. As such, the research on volborthite has evolved from the simple kagome model to another one. Beyond the KAFM. A hint for understanding the magnetism of volborthite was obtained using the first-principle calculation based on the density functional theory (DFT) by Janson et al.43,44 With reference to their results, we first consider the magnetic interactions in the C2/c structure. For the orbital pattern shown in Figure 7, there are three kinds of copper chains, each with a different magnetic coupling: the vertical Cu21−Cu22 chain and the inclined Cu1−Cu21 and Cu1−Cu22 chains. According to the DFT calculations, the Cu21−Cu22 chain has the ferromagnetic first-neighbor interaction Jb(1) and the antiferromagnetic second-neighbor one Jb(2), which is characteristic of the edge-sharing CuO2 chain with the d(x2−y2)-type orbitals lying in the plane;44,45 the latter comes from the supersuperexchange pathway via two oxide ions. On the other hand, the Cu1−Cu22 and Cu1−Cu21 chains have only nearest-neighbor interactions, Jp and Jq, respectively. Jp is mediated through one lobe of the Cu1 d(x2− y2) orbital that points to O2, while Jq is mediated through the two lobes that point to O2 and O31 (Figure 4d). Thus, there are three magnetic chainsmade of Jb(1)−Jb(2), Jp, and Jq in the C2/c phase (Figure 7). It should be noted that, in the C2/m phase, Jp = Jq = J′. In the I2/a phase, on the other hand, as a result of the orbital flipping at every other Cu1 site, the two inclined chains

Figure 11. Spin models for the 1/3 plateau state for the I2/a (P21/a) phase of volborthite. (a) Linear spin trimers, within which the spins are coupled by the strongest antiferromagnetic interaction Jp, are generated over the kagome net. They weakly interact with each other by Jq, Jb(1), and Jb(2). (b) Spin model in which each trimer has an effective spin 1/2 (red arrows) and interacts with its neighbors by ferromagnetic first-neighbor J1 and antiferromagnetic second-neighbor J2 and J2′. This model is identical with the spin-1/2 J1−J2 square-lattice model with J2 anisotropy.43

which the three spins are strongly antiferromagnetically coupled by Jp and which interact weakly with each other through other J paths. Replacing each spin trimer by an effective spin 1/2 translates the model to a rhombohedral lattice in which the effective spins are coupled by the ferromagnetic Jb(1), as shown in Figure 11b. In addition, both an antiferromagnetic coupling by Jb(2) along the b axis and another weak coupling along the a axis appear; as a result, the effective spin model of volborthite turns out to be an anisotropic square-lattice model with a ferromagnetic firstneighbor coupling J1 and antiferromagnetic second-neighbor couplings J2 and J2′. According to the DFT calculations, the H


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Inorganic Chemistry magnitudes of these J values are J1 = −34.9 K, J2 = 36.5 K, and J2′ = 6.8 K.43 The huge 1/3 magnetization plateau in volborthite can now be reasonably interpreted as either a classical ferrimagnetic state with up−down−up spins on each trimer (Figure 11a) or a quantum state in which the effective spin 1/2 of every trimer is fully polarized by the magnetic field (Figure 11b). Because Jp = 250 K is relatively large, this polarized state happens to be quite stable and can exist over a wide range of fields, as predicted by calculations to be up to 225 T.43 In summary, the orbital-flipping transition at Ts1 dramatically modifies the magnetic interactions in volborthite from the anisotropic kagome type in the C2/c phase to the J1−J2(J2′) square lattice in the low-temperature phases. Understanding the Sample Dependence. In general, in highly frustrated spin systems such as volborthite in which magnetic LRO is suppressed to very low temperatures, the magnetic correlation length becomes infinitely long upon cooling as the LRO is approached near the quantum critical regime. Thus, one defect in a crystal will seriously affect the surrounding spins over a long distance, and even a small concentration of defects can destroy the true ground state.47 This situation gives rise to interesting phenomena in frustration physics, while also being an obstacle for experimentarists (although theorists can imagine an ideal playground without such a disturbance). In herbertsmithite, which has been assumed to be the most ideal KAFM, the atomic site disorder caused by a mixture of copper and zinc atoms reaches several percentages;48 there is a claim that the spin liquid-like state observed in this system may be the result not of frustration effects but of random coupling caused by the crystalline disorder.49 Therefore, we have to be careful in understanding the experimental results from highly frustrated magnets. Here we consider the enormous sample dependence of the magnetic properties of volborthite. In the temperature evolution of the magnetic susceptibility, shown in Figure 12, the thick crystal A shows a well-defined jump with thermal hysteresis at Ts1, while in the thin crystal B, the transition seems broadened. Note that the susceptibility of crystal B shows a jump upon heating, whereas upon cooling, it gradually decreases and does not match the heating curve even at 200 K, indicating an incomplete transition. Moreover, an anomaly is not observed for the powder sample, probably because it is too broad to be distinguished. In structural refinements, we observed a trace of defects associated with the position of the VO4 unit in crystal B that was absent in crystal A. Therefore, the orbital flipping must be suppressed by crystalline defects, although the orbital pattern of the C2/c structure may be retained locally around defects in the lowtemperature phases. The local failure of orbital flipping should seriously modulate the magnetic interactions. As illustrated in the inset of Figure 12, one misstep in orbital flipping influences nine nearby spins and generates a short chain of seven spins plus two remnant spins instead of three trimers. Because of large magnetic couplings (Jp′) in the short chain, these seven spins do not easily align along the magnetic field; the maximum contribution could be (1/7)μB at H < Jp′. If the two remnant spins are fully polarized, 2μB can be added to give an average of (1/3)μB. However, because these two spins are likely to interact with their surroundings through considerable antiferromagnetic couplings, the total magnetization should be

Figure 12. Comparison of the magnetic susceptibilities of crystals A and B at H∥ab and the powder sample around Ts1. The inset illustrates a possible effect that one defect may have on the spin arrangement in the 1/3 plateau state at the orbital-flipping transition: a misstep in orbital flipping at the center site generates a short spin chain instead of three trimers, which causes a reduction or an enhancement of the total magnetization when H is respectively smaller or larger than Hp.

reduced from (1/3)μB. Therefore, a defect can reduce the magnetization before the magnetic field becomes large enough to achieve (1/3)μB. On the other hand, when the magnetic field is further increased, the magnetization cannot be exactly flat at (1/3)μB because the plateau state will be locally broken around the defects. As such, it may exceed (1/3)μB when H becomes large enough to polarize the spins around the defect. Therefore, the magnetization curve of the powder sample is qualitatively explained by this scenario of introduced defects. The amount of defects may not be large in volborthite, compared with other kagome minerals including the percent order of defects. The plateau magnetization of the powder sample in Figure 10 is approximately 5% larger than the powder average of Ms/3. A simple assumption that all nine spins around one defect are forced to align with the magnetic field corresponds to a failure in orbital flipping of about 0.3%. On the other hand, the amount of free spins associated with certain defects, as estimated from the low-temperature Curie tail in the magnetic susceptibility of the powder sample, is only 0.07%.28 Crystal B may have fewer defects and shows the intermediate magnetization curve in Figure 10; crystal A must have even fewer defects because it shows a clean 1/3 plateau state. However, NMR experiments by Yoshida et al., which probe the local structure and are very sensitive to defects, found a certain inhomogeneity, even in crystal A.36 Nevertheless, volborthite is an exceptional system among kagome minerals because an unusually clean crystal has been prepared and thus provides us with a fascinating playground to study frustration physics. Possible Spin-Nematic (SN) Order in Volborthite. An up-to-date magnetic phase diagram for volborthite is shown in Figure 13.38 At zero field, there are two successive transitions at 1.2 and 0.8 K. The corresponding phases I′ and I are replaced by phase II above 4.5 T. Phase II exists in a dome below μ0Hs1 = 22.5 T and T = 2.1 K. The 51V NMR I


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lattice. When the magnetic field, in this case pointing upward, is slightly decreased from Hs, one spin flips downward; this ΔS = 1 excitation is called a “magnon”. With further decreases of the field, many magnons are generated, and they tend to form pairs on nearby sites because the ferromagnetic J1 favors such a pairing. The bound magnon pairs thus produced can migrate in the sea of up spins and eventually undergo Bose−Einstein condensation (BEC) at low temperatures, much like the Cooper pairs of electrons in a superconductor. Because each bound pair carries a quadrupole moment, the resulting BEC is a SN, characterized by an ordered arrangement of quadrupole moments on first-neighbor bonds that mimics the rod- or disklike molecules in a nematic liquid crystal.50 Experiments to search for the exotic SN have been carried out on J1−J2 chain compounds such as LiCuVO451 and NaCuMoO4(OH),52 but no clear evidence has been obtained. Because the magnetic-dipole moments themselves are not ordered in a SN, it is difficult to attain experimental evidence: quadrupole moments are not seen directly by conventional experiments such as neutron scattering and NMR experiments. Hence, SN is actually an elusive “hidden” order. As mentioned above, the effective spin model for volborthite is an anisotropic J1−J2 square-lattice model. Therefore, one may expect a SN just below the saturation state that corresponds to the 1/3 plateau state; in this case, Hs = HP.43 In this context, the N2 phase may be a candidate SN, while the N1 phase, which looks quite similar to the N2 phase in heatcapacity measurements, is more mysterious.38 At present, thermodynamic measurements have revealed the presence of these phases below HP without evidence for the ordering of quadrupole moments. Future experimental and theoretical studies would clarify this interesting issue in volborthite.

Figure 13. Magnetic phase diagram of volborthite.38 At least five phases appear below μ0HP = 27.5 T before reaching the 1/3 plateau state P: phases I and I′ at μ0H < 4.5 T, phase II below μ0Hs1 = 22.5 T, and phases N1 and N2 below and above μ0Hs2 = 25.5 T, respectively. The phase boundaries were determined using heat capacity (HC), magnetocaloric effect (MCE), and magnetization (Mag.) measurements, with the magnetic field perpendicular to the kagome plane.

experiments by Yoshida et al. found incommensurate spin orders for phases I and II, including a spin-density-wave state for phase II.27,31,34 Furthermore, recent 51V NMR, magnetic calorimetry, and heat capacity measurements have revealed two magnetic phases, N1 and N2, between phase II and the 1/3 plateau state (P);36,38 the transitions to the N1, N2, and P phases occur at fields μ0Hs1 = 22.5 T, μ0Hs2 = 25.5 T, and μ0Hp = 27.5 T, respectively. These phases N1 and N2 have been intensively focused on as possible new states of matter. In frustrated spin models with a ferromagnetic first-neighbor interaction J1 and an antiferromagnetic second-neighbor interaction J2 on a one-dimensional chain or a two-dimensional square lattice, a novel quantum state called the spin nematic (SN) has been theoretically predicted to form just before the saturation state, in which, above the saturation field Hs, all spins become polarized with the field. Figure 14 illustrates a simplified SN on a J1−J2 square

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CONCLUSIONS We have observed unique orbital transitionsthe orbitalflipping and -switching transitionsat around room temperature in the kagome-type copper mineral volborthite. In the transition, a portion of the Cu 3d orbitals change their orientations, while in the latter, the d(3z2−r2)-type orbital is transformed to the d(x2−y2)-type orbital. These orbital transitions significantly change the magnetic interactions between the Cu2+ spins, from an anistropic kagome lattice at high temperatures (C2/c or C2/m) to an anisotropic J1−J2 square lattice of effective spin-1/2 trimers generated over the kagome net at low temperatures (I2/a or P21/a). An exotic SN state may also appear under high magnetic fields just below the 1 /3 magnetization plateau state. Volborthite is therefore an exceptional frustrated magnet that shows an intriguing interplay between the orbital and spin degrees of freedom.

Figure 14. Schematic picture of a SN state expected for a squarelattice frustrated ferromagnet with competing ferromagnetic firstneighbor (J1) and antiferromagnetic second-neighbor (J2) interactions.38 When the magnetic field is slightly decreased from the saturation field (HP for volborthite), “flipped spins” (red arrows) are generated among a background of field-polarized spins (blue arrows). Because of the ferromagnetic first-neighbor interaction, two flipped spins tend to form a bound pair through a J1 bond. These bound pairs undergo BEC at low temperatures, much like the bound pairs of electrons in a superconductor. Because each bound pair carries a quadrupole moment, the resulting BEC is a SN, characterized by an ordered arrangement of quadrupole moments on first-neighbor bonds that mimics the rod- or disk-like molecules in a nematic liquid crystal.50

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Zenji Hiroi: 0000-0003-1190-2344 Hajime Ishikawa: 0000-0002-1303-9460 Jun-ichi Yamaura: 0000-0002-8992-9099 Notes

The authors declare no competing financial interest. J


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(20) Leonardsen, E. S.; Petersen, O. V. The Unit Cell of Volborthite. Am. Mineral. 1974, 59, 372−373. (21) Lafontaine, M. A.; Le Bail, A.; Férey, G. Copper-containing minerals - I. Cu3V2O7(OH)2·2H2O: the synthetic homolog of Volborthite; crystal structure determination from x-ray and neutron data; structural correlations. J. Solid State Chem. 1990, 85, 220−227. (22) Yoshida, H.; Yamaura, J.; Isobe, M.; Okamoto, Y.; Nilsen, G. J.; Hiroi, Z. Orbital switching in a frustrated magnet. Nat. Commun. 2012, 3, 860. (23) Ishikawa, H.; Yamaura, J. I.; Okamoto, Y.; Yoshida, H.; Nilsen, G. J.; Hiroi, Z. A novel crystal polymorph of volborthite, Cu3V2O7(OH)2·2H2O. Acta Cryst. C 2012, 68, i41−i44. (24) Hiroi, Z.; Kobayashi, N.; Hanawa, M.; Nohara, M.; Takagi, H.; Kato, Y.; Takigawa, M. Spin-1/2 kagome-like lattice in Volborthite Cu3V2O7(OH)2·2H2O. J. Phys. Soc. Jpn. 2001, 70, 3377−3384. (25) Fukaya, A.; Fudamoto, Y.; Gat, I. M.; Ito, T.; Larkin, M. I.; Savici, A. T.; Uemura, Y. J.; Kyriakou, P. P.; Luke, G. M.; Rovers, M. T.; Kojima, K. M.; Keren, A.; Hanawa, M.; Hiroi, Z. Muon spin relaxation and susceptibility studies of the pure and diluted spin 1/2 Kagome-like lattice system (CuxZn1‑x)3V2O7(OH)2·2H2O. Phys. Rev. Lett. 2003, 91, 207603. (26) Bert, F.; Bono, D.; Mendels, P.; Ladieu, F.; Duc, F.; Trombe, J.C.; Millet, P. Ground State of the Kagomé-Like S = 1/2 Antiferromagnet Volborthite Cu3V2O7(OH)2·2H2O. Phys. Rev. Lett. 2005, 95, 087203. (27) Yoshida, M.; Takigawa, M.; Yoshida, H.; Okamoto, Y.; Hiroi, Z. Phase Diagram and Spin Dynamics in Volborthite with a Distorted Kagome Lattice. Phys. Rev. Lett. 2009, 103, 077207. (28) Yoshida, H.; Okamoto, Y.; Tayama, T.; Sakakibara, T.; Tokunaga, M.; Matsuo, A.; Narumi, Y.; Kindo, K.; Yoshida, M.; Takigawa, M.; Hiroi, Z. Magnetization “steps” in spin liquids on the kagome lattice. J. Phys. Soc. Jpn. 2009, 78, 043704. (29) Yamashita, S.; Moriura, T.; Nakazawa, Y.; Yoshida, H.; Okamoto, Y.; Hiroi, Z. Thermodynamic Properties of the Kagome Lattice in Volborthite. J. Phys. Soc. Jpn. 2010, 79, 083710. (30) Ohta, H.; Zhang, W.-M.; Okubo, S.; Fujisawa, M.; Sakurai, T.; Okamoto, Y.; Yoshida, H.; Hiroi, Z. Spin dynamics of S = 1/2 kagome lattice antiferromagnets observed by high-field ESR. Phys. Status Solidi B 2010, 247, 679−681. (31) Yoshida, M.; Takigawa, M.; Yoshida, H.; Okamoto, Y.; Hiroi, Z. Heterogeneous spin state in the field-induced phase of volborthite as seen via 51V nuclear magnetic resonance. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 020410. (32) Okamoto, Y.; Tokunaga, M.; Yoshida, H.; Matsuo, A.; Kindo, K.; Hiroi, Z. Magnetization plateaus of the spin-1/2 kagome antiferromagnets volborthite and vesignieite. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 83, 180407. (33) Nilsen, G. J.; Coomer, F. C.; de Vries, M. A.; Stewart, J. R.; Deen, P. P.; Harrison, A.; Rønnow, H. M. Pair correlations, shortrange order, and dispersive excitations in the quasi-kagome quantum magnet volborthite. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 172401. (34) Yoshida, M.; Takigawa, M.; Krämer, S.; Mukhopadhyay, S.; Horvatic, M.; Berthier, C.; Yoshida, H.; Okamoto, Y.; Hiroi, Z. HighField Phase Diagram and Spin Structure of Volborthite Cu3V2O7(OH)2·2H2O. J. Phys. Soc. Jpn. 2012, 81, 024703. (35) Ishikawa, H.; Yoshida, M.; Nawa, K.; Jeong, M.; Krämer, S.; Horvatić, M.; Berthier, C.; Takigawa, M.; Akaki, M.; Miyake, A.; Tokunaga, M.; Kindo, K.; Yamaura, J.; Okamoto, Y.; Hiroi, Z. OneThird Magnetization Plateau with a Preceding Novel Phase in Volborthite. Phys. Rev. Lett. 2015, 114, 227202. (36) Yoshida, M.; Nawa, K.; Ishikawa, H.; Takigawa, M.; Jeong, M.; Krämer, S.; Horvatić, M.; Berthier, C.; Matsui, K.; Goto, T.; Kimura, S.; Sasaki, T.; Yamaura, J.; Yoshida, H.; Okamoto, Y.; Hiroi, Z. Spin dynamics in the high-field phases of volborthite. Phys. Rev. B: Condens. Matter Mater. Phys. 2017, 96, No. 180413. (37) Nakamura, D.; Yamashita, T.; Ishikawa, H.; Hiroi, Z.; Takeyama, S. Extremely wide 1/3 magnetic plateau of volborthite

ACKNOWLEDGMENTS We thank Makoto Yoshida, Yoshimitsu Kohama, and Nic Shannon for fruitful discussion. We also thank Alex Browne for his help in improving the manuscript. This work was partly supported by the Core-to-Core Program for Advanced Research Networks given by the Japan Society for the Promotion of Science.

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