Magnetic Frustration Driven by Itinerancy in Spinel CoV2O4

TL;DR

This paper investigates how itinerant electrons influence magnetic frustration in the spinel CoV2O4, revealing a complex interplay that leads to non-collinear spin states and spin-glass behavior.

Contribution

It uncovers the role of itinerant electrons in inducing magnetic frustration and complex spin states in CoV2O4, a crossover material between insulating and itinerant regimes.

Findings

Itinerant electrons induce magnetic frustration in CoV2O4.

Non-collinear spin states and spin-glass behavior are observed.

External perturbations can modify the magnetic phases.

Abstract

Localized spins and itinerant electrons rarely coexist in geometrically-frustrated spinel lattices. We show that the spinel CoV2O4 stands at the crossover from insulating to itinerant behavior and exhibits a complex interplay between localized spins and itinerant electrons. In contrast to the expected paramagnetism, localized spins supported by enhanced exchange couplings are frustrated by the effects of delocalized electrons. This frustration produces a non-collinear spin state and may be responsible for macroscopic spin-glass behavior. Competing phases can be uncovered by external perturbations such as pressure or magnetic field, which enhance the frustration.

Equations

$$H=-\frac{1}{2}\sum _{i,j}{J}_{ij}{\mathbf{S}}_i\cdot {\mathbf{S}}_j+\sum _i{D}_i{\left(\widehat{u_i}\cdot {\mathbf{S}}_i\right)}^2$$

$$E\left(\alpha \right)=-12J_{\mathrm{Co}-\mathrm{V}}{S}_{\mathrm{Co}}{S}_{\mathrm{V}}\alpha -4J_{\mathrm{V}-\mathrm{V}}{S}_{\mathrm{V}}^2{\alpha }^2+\text{constant}$$

$$\alpha =-\frac{3J_{\mathrm{Co}-\mathrm{V}}{S}_{\mathrm{Co}}}{2J_{\mathrm{V}-\mathrm{V}}{S}_{\mathrm{V}}}$$

Full text

The interplay between localized-spin and itinerant-electron behavior in geometrically-frustrated systems frust is responsible for many intriguing phenomena such as a spin-liquid state Nakatsuji06 , heavy-fermion behavior Lee13 , spin-ice conduction Udagawa12 , and exotic phases Hanasaki07 ; Iguchi09 ; Kumar10 . While the interplay between the localized spins and itinerant electrons has been investigated intensively on pyrochlores $A_{2}B_{2}$O7 where both $A$ and $B$ sublattices are frustrated Nakatsuji06 ; Lee13 ; Udagawa12 ; Hanasaki07 ; Iguchi09 , it has rarely been explored on spinel systems $AB_{2}$O4, where only the sublattice $B$ is frustrated. In the spinels, the electronic itinerancy on sublattice $B$ can be controlled by chemical pressure on sublattice $A$, which enhances magnetic frustration. On the other hand, magnetic $A$ site ions can suppress the frustration through their magnetic interactions with the localized spins on sublattice $B$. Thus, one can anticipate a rich interplay between localized spins and itinerant electrons.

In the spinel vanadates $A$V2O4, the chemical pressure exerted by a small $A$-site cation can introduce itinerancy. Since it has the smallest magnetic $A$-site cation of any known spinel vanadate, CoV2O4 would be the ideal system to study the interplay between itinerancy and localized spins. In most spinels vanadates ($A$V2O4, $A$=Mn, Fe, Cd, Zn, Mg), non-collinear (NC) spin states are produced by the orbital ordering (OO) of partially-filled $d$-electrons on the V site, which relieves geometric frustration through a tetragonal distortion Kismarahardja1 ; Nishiguchi ; Lee ; Wheeler ; Garlea ; Katsufuji ; MacDougall1 . However, even in a cubic phase without OO, CoV2O4 macroscopically exhibits NC and glassy spin states Kiswandhi ; Huang as well as other magnetic anomalies Kismarahardja . Therefore, the NC spin states in CoV2O4 demand a detailed study.

Although the macroscopic spin-glass behavior of Mn1-xCoxV2O4 is enhanced by Co-doping Kiswandhi , CoV2O4 has higher magnetic ordering temperatures than compounds Mn1-xCoxV2O4 with $x<1$. Indeed, CoV2O4 has higher collinear ($T_{\rm CL}$) and NC ($T_{\rm NC}$) spin transition temperatures than any other spinel. This stands in marked contrast to the pyrochlores Hanasaki07 ; Iguchi09 , where spin-glass phases have lower ordering temperatures.

While its earlier characterization was hampered by the difficulty of fabricating single crystals, recent experiments on single crystals of CoV2O4 have reported anomalous physical and magnetic properties Kiswandhi ; Kismarahardja ; Kismarahardja1 ; Huang . This paper clarifies the origin of the NC spin states of CoV2O4 by using density functional theory (DFT) and spin models to interpret neutron-scattering measurements on CoV2O4 single crystals.

In contrast to previous macroscopic measurements Huang ; Kiswandhi , our neutron scattering measurements show CoV2O4 to be an ordered magnet rather than a spin glass. Nevertheless, some latent factors of CoV2O4 can enhance frustration and drive it into a spin glass with the help of external perturbations. Chemically-driven pressure by Co increases itinerancy in CoV2O4. This itinerancy weakens the OO and thus enhances magnetic and structural isotropies. The frustration fostered by that isotropy Ising may be responsible for macroscopic glassy behavior in a magnetic field Huang ; Kiswandhi . Due to the enhanced frustration, external perturbation such as pressure or magnetic field could uncover novel magnetic phases in cubic CoV2O4.

Results

**NC spin states in cubic phase CoV2O4 **

Figure 1 compares the temperature dependence of the (002), (220), and (111) Bragg peaks in cubic CoV2O4 and tetragonal MnV2O4. The temperature dependence of the two end-compounds was recently compared to that of the intermediates compounds Mn1-xCoxV2O4 (x = 0.2, 0.4, 0.6, 0.8) Jie . At the symmetry-allowed Bragg positions of (220) and (111), a ferrimagnetic (FIM) signals occur below $T_{CL}$. While the (002) peak is forbidden by structural symmetry, the observed scattering below $T_{NC}$ indicates the formation of an antiferromagnetic (AFM) component in the $ab$-plane.

The (002) magnetic reflection indicates that the collinear (CL) to NC magnetic transition at $T_{NC}$ coincides with the structural transition at $T_{S}$ in MnV2O4 Garlea . The (220) and (111) Bragg peaks show the FIM CL spin transition (T_{CL}$$\sim150 K) and confirm recent X-ray diffraction and heat capacity measurements Kiswandhi that indicate the disappearance of the structural transition in CoV2O4. However, the additional magnetic transition in CoV2O4 indicated by the (002) peak has an enhanced transition temperature T_{NC}$$\sim75 K compared to T_{NC}$$\sim57 K in MnV2O4. Despite the induced itinerancy, this (002) peak indicates the formation of the two-in/two-out (TI/TO) spin configuration on the V-sublattice. Based on the diffraction data, this spin configuration is similar to the spin configurations in MnV2O4 Magee and FeV2O4 MacDougall . As shown in Supplement, the full widths at half maximum of the Bragg peaks (111) and (220) are not broadened below $T_{NC}$. Consequently, CoV2O4 is an ordered magnet and not yet a spin glass.

Although CoV2O4 is not a spin glass, magnetic frustration is still induced by electronic itinerancy. The ordered magnetic moment refined from the peaks in Fig. 1 is 0.47(3)$\mu_{B}$/V in CoV2O4 which is significantly reduced from 0.95(4)$\mu_{B}$/V in MnV2O4 due to the increased itinerancy. While the moment in CoV2O4 is small, the paramagnetic metallic state was expected down to zero temperature when the inter-vanadium distance $R_{\rm V-V}$=2.97 Å lies below the critical value (2.98 Å) Canosa . Since the TI/TO state originates from OO in tetragonal compounds Magee ; MacDougall , the isosymmetric TI/TO state in cubic CoV2O4 without any OO must have a different origin associated with its itinerancy and frustration.

Single-ion anisotropy suppressed by itinerancy

First-principles calculations were used to explore the microscopic origin for the complex NC state in cubic CoV2O4. As shown in Fig. 2(a) and (c), the major magnetic anisotropy appears on the V3+ site with a magnitude two or three orders larger than for the $A$-site (Co/Mn). Although the V3+ ions are surrounded by similar octahedra in both CoV2O4 and MnV2O4 , the local [111] single-ion anisotropy (SIA) of V3+ is significantly reduced in CoV2O4 ($-1.2$ meV) compared to that in MnV2O4 ($-4.8$ meV) due to the melting of OO by the pressure-induced itinerancy in CoV2O4. Calculated by DFT, the SIA on the V sites totally disappears with an external pressure around 10 Gpa in CoV2O4 .

While the AFM V-V interaction in a pyrochlore lattice with local [111] SIA favors the all-in/all-out (AI/AO) spin structure, the disappearance of SIA fosters strong magnetic frustration Ising . In MnV2O4 , that frustration was suppressed by OO. However, the frustration reappears in CoV2O4 due to the melting of the OO and the suppression of the easy-axis anisotropy by itinerancy, as shown in Fig. 2(a), (b). The recovered frustration may be responsible for the macroscopic spin-glass behavior Kiswandhi below $T_{NC}$.

The SIA of $A$-site (Fig. 2c), ($A$=Co, Mn) is quite negligible compared to the SIA of V3+. While Mn2+ has a weak easy-plane axis because of the compressed tetragonal structure (c/a$<$1), Co does not exhibit anisotropy because of the isotropic cubic structure. The SIA of Co2+ is much less dependent on pressure than that of V3+ since Co2+ electronic states lie significantly below the Fermi energy ($\epsilon_{F}$) and are thereby electronically encapsulated, as shown in Fig. 3(a). Only V3+ states cross $\epsilon_{F}$. Therefore the pressure-induced itinerancy will only affect the spins on V3+ sites.

Enhanced exchange couplings

As shown in Supplement, the Bragg peaks do not split or broaden with decreasing temperature below 100 K, indicating the absence of a strutural transition. In agreement with this measurement, DFT calculations confirm the structural isotropy ($c/a=1$) of CoV2O4 . As shown in Fig. 3, the $t_{2g}$ ($d_{xy}$=$d_{yz}$=$d_{xz}$) and $e_{g}$ ($d_{z^{2}}$=$d_{x^{2}-y^{2}}$) electronic levels become equally occupied and degenerate in cubic CoV2O4 . The structural and electronic isotropies also produce the same exchange interactions $J^{\rm in}_{\rm V-V}$=$J^{\rm out}_{\rm V-V}$=$-12$ meV between all spins on the tetrahedron as calculated from first principles, Fig. 4. These coupled structural, electronic, and magnetic isotropies foster frustration and the observed NC phase in Fig. 1.

Comparing the densities-of-states of CoV2O4 and MnV2O4 reveals the origin of the enhanced magnetic ordering temperatures in CoV2O4 . The large energy difference ($\sim$ 5 eV) between the occupied V and Mn $d$ states weakens the exchange between Mn and V. By filling the $e_{g}$ minority spin levels as indicated in Fig. 2(d) and 3(a), Co significantly lowers the $t_{2g}$ unoccupied energy level and enhances the exchange interaction between Co and V. DFT calculations reveal that the magnitude of the AFM $J_{AB}$ is twice as large in CoV2O4 (-2.5 meV) as in MnV2O4 (-1.2 meV). As reflected by the neutron-scattering measurements in Fig. 1, the enhanced $J_{A-{\rm V}}$ causes $T_{CL}$ to more than double in CoV2O4 (150 K) compared to MnV2O4 (53K).

Strikingly, the induced itinerancy also increases the NC ordering temperature even without OO in CoV2O4. As shown in Fig. 1, $T_{NC}$ significantly increases in CoV2O4 (75K) compared to MnV2O4 (57K). Although it exhibits the higher NC ordering temperature, CoV2O4 also exhibits glassy behavior Kiswandhi ; Huang . While the reduced SIA and induced isotropies foster frustration Ising , the enhanced exchange interaction relieves the frustration and enhances the ordering temperatures. In the series Mn1-xCoxV2O4, the spin-wave gap ($\sim$2meV) remains relatively unchanged with Co-doping ($x$) Jie despite the enhanced magnetic ordering temperatures proportional to $J_{\rm A-V}$. Since the spin-wave gap is proportional to $\sqrt{D_{\rm V}\!\times\!J_{\rm A-V}}$, the increase in $|J_{\rm A-V}|$ is compensated by the reduction in the anisotropy $D_{\rm V}$ in CoV2O4 . By enhancing both competing effects (itinerancy-driven isotropies with reduced SIA and strengthened exchange), Co doping can foster various novel states in CoV2O4.

Novel phases induced by frustration

The comparison between CoV2O4 and MnV2O4 in Fig. 4 reveals the origin of the NC states in CoV2O4 . The key handle to tune the magnetic couplings is the distance between the V atoms ($R_{\rm V-V}$ along the $x$-axis) controlled by chemical doping and external pressure. In MnV2O4 , the OO of the V ions relieves the magnetic frustration of the pyrochlore lattice and stabilizes the TI/TO NC spin state. The AFM Mn-V interactions increase the canting angle while maintaining this TI/TO state (region b). By introducing itinerancy, Co doping promotes isotropic V-V interactions and favors the AI/AO spin state. Within the tetrahedron network, the AI/AO state has two distinct canting angles $\theta$ and $\pi$-$\theta$ compared to the one canting angle $\theta$ of the TI/TO state. Guided by the DFT parameters for CoV2O4, our model calculation indicates that the new two-angle state based on the AI/AO state lies within 0.1 meV/unit-cell of the TI/TO ground state. The isotropic exchange ($J_{\rm V-V}^{\rm in}=J_{\rm V-V}^{\rm out}$) fosters a new two-angle AI/AO structure that can be stabilized by a magnetic field.

External pressure may also increase the degree of frustration. For high external pressure Kismarahardja ; Kiswandhi $\sim$10 GPa, the enhanced itinerancy fully suppresses the local SIA ($D_{\rm V}\sim$0) of V as in Fig. 4(a) and revives the magnetic frustration of the pyrochlore lattice. Although AFM exchange between the Co and V sites then induces the observed isosymmetric TI/TO spin structure, the frustration fostered by itinerancy and the alternative states that compete with the TI/TO ground state are responsible for the measured magnetic anomalies Kismarahardja and spin-glass behavior Kiswandhi .

As shown in Figs. 2(a) and 4(a), high pressure may stabilize a continuum of degenerate states where the two angles ($\theta_{1}$, $\theta_{2}$) rotate without energy cost due to the absence of SIA, as obtained in the Appendix and shown in the energy landscape of Fig. 4(d). This degeneracy can induce spin-glass or spin-liquid-like behavior. Since neutron scattering is limited to relatively low pressures, these novel states should be studied with synchrotron magnetic X-ray scattering.

Magnetic field measurement

Fig. 4(a) provides a guide to uncover the novel states produced by the regenerated frustration. Although all other magnetic couplings (isotropic $J_{\rm V-V}$, reduced $D_{V}$) foster frustration (bold red line), the remnant AFM interaction $J_{\rm Co-V}$ (dotted blue line in Fig 4(a)) still relieves frustration. An external magnetic field ($\vec{H}=H\hat{z}$) can help restore frustration by weakening $J_{\rm Co-V}$, thereby inducing the novel two-angle state of CoV2O4 , as shown in Fig. 5.

To check the effect of a magnetic field on CoV2O4, we carried out further elastic neutron-scattering measurements on the Co-rich single-crystal spinel Co0.8Mn0.2V2O4, which preserves the cubic structural and magnetic isotropies as in Fig. 4(a) but exhibits stronger scattering intensity than CoV2O4 due to the larger size of the single crystal. The V3+ AFM components in the $ab$-plane increase with the magnetic field ($H_{z}>$ 3 T), as indicated by the increased intensity of (020) (see Fig. 5(a)). At $H_{z}<$ 3 T, the increased intensity of (220) reflects the reorientation of the magnetic domains; at $H_{z}>$ 3 T the (220) intensity is saturated, indicating that all magnetic domains are fully oriented and that the FIM components are constant. This is consistent with magnetization measurements on a polycrystalline sample, which provide a saturation field of $\sim$2 T Huang . Although our measurements can not disentangle the magnetic components along [001] ($M_{z}^{\rm Co}$ and $M_{z}^{\rm V}$), we do not expect $|M_{z}^{\rm V}|$ to decrease with field above 3 T because that would require $M_{z}^{\rm Co}$ to further increase. So we can safely assume that both $M_{z}^{\rm V}$ and $M_{z}^{\rm Co}$ are constant above 3 T. Since the AFM components of V3+ in the $ab$-plane ($M_{ab}^{\rm V}$) continue to grow above 3 T, the canting angle ($\theta={\rm tan}^{-1}[M_{ab}^{\rm V}/M_{z}^{\rm V}]$) of the V3+ spins must increase with the magnetic field along [001].

Using the spin model (Eq. 1) combined with DFT parameters (Fig 4(a)), we confirm the increase in the canting angle with magnetic field in Fig. 5(c). The two-angle AI/AO state has an energy within 0.1meV/unit-cell of the the one-angle TI/TO ground state in the Co-rich region. We predict that this new state is stabilized by a large magnetic field of about 140 T, as shown in Fig. 5(c),(d). Although only the one-angle TI/TO state was previously reported in vanadate compounds ($A$V2O4, $A$=Zn, Mn, Fe), various competing states appear in CoV2O4 due to frustration. It is likely that those states can be revealed by a magnetic field or pressure.

Of course, the critical magnetic field ($H_{z}$ = 140 T) is too large for neutron scattering measurements. However, the first-order phase transition from the one-angle to the two-angle state may be captured by magnetic susceptibility measurements. Moreover, various methods can be employed to reduce the critical field. Since external pressure suppresses SIA and revives frustration as discussed in the previous section, pressure may also reduce the critical magnetic field. Contrary to the usual expectation, a magnetic field may strengthen frustration and noncollinearity in CoV2O4 by weakening the only exchange coupling ($J_{\rm Co-V}$) that hampers frustration.

Discussion

It is natural to wonder if cations significantly smaller than Co2+ such as Be2+ can be substituted on the $A$-site to induce even more itinerancy and consequent frustration. However, a non-magnetic $A$-site cannot support localized V-spins so the system would become paramagnetic Canosa . Because strong magnetic interactions between the $A$ and $B$ sites is required to maintain the localized V spins, Co is the only candidate $A$-site cation to support localized spins with enhanced $J_{\rm Co-V}$ while inducing itinerancy on the $B$ site.

Compared to other vanadates ($A$V2O4), the frustration in magnetically and structurally isotropic CoV2O4 explains its NC and macroscopic spin-glass properties. Since the AFM interaction between Co and V is the only factor that relieves the magnetic frustration, weakening the AFM interaction by a magnetic field or further reducing the SIA by external pressure can rekindle the frustration and reveal alternative states. Among spinel vanadates, CoV2O4 is uniquely located at the crossover between localized and itinerant behavior. Consequently, many exotic properties and new phases can be produced by restoring the frustration of the pyrochlore lattice.

Method

Sample preparation

Single crystals of CoV2O4, Co0.8Mn0.2V2O4 and MnV2O4 were grown by the traveling-solvent floating-zone (TSFZ) technique. The feed and seed rods for the crystal growth were prepared by solid state reaction. Appropriate mixtures of MnO, CoCO3, and V2O3 were ground together and pressed into 6-mm-diameter 60-mm rods under 400 atm hydrostatic pressure, and then calcined in Ar at 1050 *∘*C for 15 hours. The crystal growth was carried out in argon in an IR-heated image furnace (NEC) equipped with two halogen lamps and double ellipsoidal mirrors with feed and seed rods rotating in opposite directs at 25 rpm during crystal growth at a rate of 20mm/h.

Neutron-scattering experiments

Single-crystal neutron diffraction was performed to determine the crystal and magnetic structures using the four-circle diffractometer (HB-3A) at the High Flux Isotope Reactor (HFIR) of the Oak Ridge National Laboratory (ORNL). A neutron wavelength of 1.003Å  was used from a bent perfect Si-331 monochromator chak11 . High magnetic field single-crystal neutron diffraction experiments were performed on the cold neutron triple-axis spectrometer (CTAX) at HFIR, ORNL. The incident neutron energy was selected as 5.0 meV by a PG (002) monochromator, and the final neutron energy was also set as 5.0 meV by a PG (002) analyzer. The horizontal collimation was guide-open-80′-open. Contamination from higher-order beams was removed using a cooled Be filter. The scattering plane was set in the (H,K,0) plane and the magnetic field was applied perpendicular to the scattering plane. The nuclear and magnetic structures were refined with the program FULLPROF rod93 . Due to the domain re-orientation effect, intensities of both (220) and (020) diffractions increase sharply in small magnetic fields, but the (220) diffraction is saturated above about 3 T. The intensity of the (020) diffraction, corresponding to the magnetic component of V in the $ab$-plane, inceases linearly with field.

First-principles calculations

First-principles calculations were performed using density-functional theory within the local spin-density approximation with a correction due to on-site Hubbard interaction (LSDA+$U$) as implemented in the Vienna ab initio simulation package (VASP-5.3) Kresse . We used the Liechtenstein Lich implementation with on-site Coulomb interaction $U$ = 6.0 eV and on-site exchange interaction $J_{\rm H}$ = 1.0 eV to treat the localized 3d electron states in Co, Mn, and V; this choice of $U$ is close to that chosen in previous work on CoV2O4 Kaur and MnV2O4 Nanguneri ; Sarkar . The spin-orbit interaction was included. The projector augmented wave (PAW) potentials PAW1 ; PAW2 explicitly include 13 valenced electrons for Mn ($3p^{6}3d^{5}4s^{2}$), 9 for Co ($3d^{8}4s^{1}$), 13 for V ($3s^{2}3p^{6}3d^{4}4s^{1}$), and 6 for oxygen ($2s^{2}2p^{4}$). The wave functions were expanded in a plane-wave basis with an energy cutoff of 500 eV. To evaluate the on-site single-ion anisotropy (SIA) interaction $D$, only one cation of interest was kept while the surrounding magnetic atoms were replaced by neutral and isoelectronic Ca2+ and Al3+ cations for Co2+/Mn2+ and V3+, respectively. This is the same technique that was successfully used for BiFeO3 wein12 and CaMn7O12 zhang13 .

Microscopic spin model

Spin states in spinels can be described by the following model Hamiltonian,

[TABLE]

which contains six inequivalent sublattices. Isotropic exchange constants $J_{\rm Co-V}$ describe nearest-neighbor interactions between the Co and V sites. $J_{\rm Co-Co}$ and $J_{\rm V-V}$ describe nearest-neighbor interactions between Co-sites and V-sites, respectively. The easy-axis anisotropy is assumed to be zero for the Co-sites, while for the B-site spins, the easy-axis anisotropy $D_{\rm V}$ is along the local $<$111$>$ direction. The azimuthal directions of each vanadium spin is constrained, but the canting angle $\theta_{i}$, described in Fig. 4, is allowed to vary between 0 and $2\pi$. Since $\theta_{i}$ may have a unique value in adjacent planes, both the two-in-two-out and all-in-all-out configurations are possible. These angles are equal to the polar angle when $\theta_{i}$ is between 0 and $\pi$, while the polar angles equals $2\pi-\theta_{i}$ and the azimuthal angle changes by $\pi$ when $\theta_{i}$ is greater than $\pi$.

The ground state spin configuration was found by minimizing the classical energy for a given set of parameters. To avoid local minima, this was accomplished by calculating the classical energy on a grid with $\theta_{i}=0$ to $2\pi$ and finding the two angles with the lowest energy. This process was repeated for values of the external magnetic field ranging from 0 to 173 T.

The inelastic neutron cross section for undamped spin waves was calculated using the $1/S$ formalism outlined in Ref. [Haraldsen2009, ] and the qppendices of Ref. [Fishman2013, ]. For direct comparison with experimental intensities, the effects of the magnetic form factor and the instrumental resolution were included in the calculation. The coefficients for Co2+, and V3+ are from Ref. [neutrondatabooklet, ]. The resolution function was approximated as a Gaussian in energy with a full width at half-maximum of 1.5 meV. Effects from finite resolution in $\bm{Q}$ were not considered.

While DFT can provide guidance for the values of the isotropic exchange interactions, LSDA+$U$ overestimates the experimental moment (SV=0.23(7)) of CoV2O4 measured by neutron scattering. Our spin model uses the magnetic moment (SV=0.25), which is within the experimental uncertainty. In addition, parameters calculated with DFT were adjusted to reproduce the measured canting angle of CoV2O4 ($\theta$ = 20.8 $\pm$ 1.7∘) in zero field. Care was also taken to avoid a long-range spiral configuration Tomiyasu04 that was not observed in our neutron diffraction measurements. The final set of parameters used for CoV2O4 are $J_{\rm Co-Co}=0.5\textrm{meV}$, $J_{\rm Co-V}=-2.5\textrm{meV}$, $J_{\rm V-V}=-11.0\textrm{meV}$, $D_{\rm V}=-2.7\textrm{meV}$, $S_{\rm Co}=$ 1.50 and $S_{\rm V}=0.25$.

Canting angles of the new phase with ${\textbf{\emph{D}}_{\rm V}}$=0.0

At high pressures, we take $D_{\rm V}$ = 0.0 in Eq.(1). Then, the total energy per unit magnetic unit cell is

[TABLE]

where $\alpha=\cos\theta_{1}+\cos\theta_{2}$. The total energy is at a minimum if

[TABLE]

which limits the allowed combination of $\theta_{1}$ and $\theta_{2}$. When $\theta_{1}=\theta_{2}$ and $D_{\rm Co}$ = $D_{\rm V}$ = 0.0, this condition is equal to the expression for $\theta$ in ref. Nanguneri .

Acknowledgements

The research at HFIR, ORNL, were sponsored by Department of Energy, Office of Sciences, Basic Energy Sciences, Materials Sciences and Engineering Division (J.H.L., S.O., R.F.) and Scientific User Facilities Division (J.M., S.E.H., M.M.). The research at UNIST was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT $\&$ Future Planning (2.150639.01). J.M. thanks the support of the Ministry of Science and Technology of China (2016YFA0300500). S.E.H. acknowledges support by the Laboratory’s Director’s fund, ORNL. H.D.Z thanks the support from NSF with grant NSF-DMR-1350002. The authors acknowledge valuable discussions with G. MacDougall.

Author contributions

J.H.L. conceived the original idea and carried out first-principles calculations. S.E.H. and R.S.F. performed spin-wave simulations. J.M., H.C., T. H, and M.M. measured and analyzed neutron scattering Bragg peaks. H.D.Z. synthesized the samples. J.H.L., S.E.H., J.M., H.B.C., T.H., S.O., M.M, R.S.F. discussed the results. J.H.L. and R.F. wrote the manuscript.

Additional information

Competing finalcial interests: The authors delcare no competing financial interests.