Frustration wave order in iron(II) oxide spinels

Frustrated magnetic materials can show unconventional correlations such as quantum spin liquid states and monopole excitations in spin ices. These phenomena are observed on uniformly frustrated lattices such as triangular, kagome or pyrochlore types, where all nearest neighbour interactions are equivalent. Here we report incommensurate long-range spin amplitude waves in the spinels Fe2GeO4 and γ-Fe2SiO4 at low temperatures, which indicate that the degree of frustration may itself be a fluctuating quantity that can spontaneously order without a lattice distortion as a ‘frustration wave’. Fe2GeO4 with propagation vector (2/3 + δ 2/3 + δ 0) has ordered Fe2+ moments that vary between fully saturated 4 μB and 0 values, consistent with a frustration wave order. γ-Fe2SiO4 has a more complex (¾ + δ ¾ + δ 0) order that coexists with an ordered spin ice phase. Dynamic orbital fluctuations are proposed to give rise to locally correlated patterns of ferromagnetic and antiferromagnetic interactions consistent with the observed orders. Frustrated magnetic materials provide a great laboratory to study the interplay between classical order and quantum fluctuations. The authors study the frustrated magnetic ground states of two Fe spinel oxides showing that the frustration is a fluctuating characteristic that manifests itself as a “frustration wave”

L ong-range spin order is sometimes avoided in frustrated magnetic materials, leading to unconventional correlations such as quantum spin liquids and ices [1][2][3][4] . Frustrated longrange spin orders are also observed and the degree of frustration for an individual spin or a larger grouping within the ordered lattice may be quantified by the function: 5 where the exchange Hamiltonian for interacting spins S i and S j is -J ij S i .S j . F varies between 0 for unfrustrated spins and 1 for complete frustration. For a collinear spin order on a simple lattice in which exchange couplings are equivalent and are either fully frustrated or unfrustrated, the degree of frustration simplifies to F = N f /N, where N f is the number of frustrated interactions and N is the total number of interactions around each spin. Conventional frustrated systems have constant F at all spins, for example, the canonical pyrochlore-type lattice of corner-sharing tetrahedra of antiferromagnetically interacting moments has F = 1/3 at all sites in the ordered ground states shown in Fig. 1(a, b). The related '2-in 2-out' spin ice order shown in Fig. 1(c) is also uniformly frustrated. The physics of many investigated pyrochlores is thus predicated on the uniformity of the degree of frustration (F) throughout the lattice. Fe 2 GeO 4 and the high-pressure γ-form of Fe 2 SiO 4 are cubic B 2 AO 4 spinels, where orbitally degenerate 3d 6 Fe 2+ cations with S = 2 spins form a pyrochlore-type B-site lattice 6,7 . γ-Fe 2 SiO 4 is also of geophysical interest as one of the main constituents of the Earth's mantle 8,9 . Previous studies have established that both materials have magnetic transitions near 10 K 10-14 , but the lowtemperature spin orders are not reported and preliminary abstracts have differing results 15,16 . Our investigation of their magnetic structures has led to the discovery of frustration wave order as a class of ground states, where spin-spin interactions become spatially non-uniform within a structurally uniform lattice.

Results
Spin order in Fe 2 GeO 4 . Synthesis of the polycrystalline Fe 2 GeO 4 sample and characterisation measurements are described in methods with further details in Supplementary Figs. 1, 5 and 6, Supplementary Tables 1, 3 and 4 and Supplementary Notes 1 and 2. Magnetic susceptibility measurements (Fig. 2a) for Fe 2 GeO 4 reveal two magnetic transitions with a susceptibility maximum at T m1 ≈ 9 K and divergence of field and zero-field cooled susceptibilities at T m2 ≈ 7 K, consistent with a previous report 12 . Alternating Current (AC) measurements show no frequency dependence in the low-temperature features, indicating an absence of the spin-glass behaviour (Fig. 2b). A broad magnetic contribution to the low-temperature heat capacity appears to extend up to around 50 K (Fig. 2c), but the integrated entropy over the two transitions of 5.77 J mol −1 K −1 per Fe 2+ is only 43% of the theoretical value of Rln5 for long-range order of S = 2 spins. Fits to synchrotron powder X-ray diffraction data at 5 K, as well as the neutron data below, show that the crystal structure remains cubic Fd 3 m at low temperatures with no distortion observed (Fig. 2d). This is unusual as spin orders in oxide spinels usually lead to lattice distortions, e.g. ZnV 2 O 4 11,20 all distort from cubic to tetragonal symmetry at orbital or antiferromagnetic ordering transitions. Hence the measurements indicate that the orbital states and a large fraction of the Fe 2+ spins remain dynamic below the two magnetic transitions.
Sharp magnetic diffraction peaks indicative of long-range spin order appear below the magnetic transition at T m1 ≈ 9 K with an additional weak peak observed below T m2 ≈ 7 K, as shown in Fig. 3a. These peaks were indexed by very similar propagation vectors k i = ( 2 = 3 + δ i 2 = 3 + δ i 0) for peaks appearing below T mi (i = 1 or 2). Representation analysis shows that the single Fe B lattice position is split into magnetically distinct Fe1 and Fe2 sites. The magnetic intensities from each transition are fitted by a double-k model, in which different propagation vectors k ij apply to different sites Fej (j = 1 or 2), k i1 = ( 2 = 3 + δ i -2 = 3 -δ i 0) and k i2 = ( 2 = 3 + δ i 2 = 3 + δ i 0). A good fit to the peaks that are observed below T m1 , as shown in Fig. 3b where refined δ 1 ≈ -0.025(1), can only be obtained using a model in which ordered moment amplitudes are modulated, as displayed in Fig. 3c. The sublattices of Fe1 and Fe2 spins are mutually perpendicular and each has collinear antiferromagnetic chains of spins pointing parallel to their propagation direction. The additional magnetic peak observed below T m2 is fitted by an additional order of small perpendicular moment components that describe a canting of the above a b c Fig. 1 Frustrated magnetic order on a uniform pyrochlore lattice. Spins lie at the apices of a lattice of corner-sharing tetrahedra, and all nearest-neighbour couplings are of equal magnitude, as represented by the blue lines. a A ground-state configuration for collinear spin order when the nearest-neighbour couplings are antiferromagnetic. This ground state is highly degenerate, but lattice distortion may stabilise one configuration. Four antiferromagnetic couplings are satisfied in each tetrahedron and two are frustrated as indicated by 'sad face' symbols; this construct is useful for the more complex frustrated configurations shown in Fig. 4. b Non-collinear spin order for situation (a) can give rise to the 'all-in all-out' configuration, where all magnetic moments point towards or away from the centre of each B-site tetrahedron, preserving cubic symmetry and with each antiferromagnetic coupling partly frustrated. c Strong dipolar interactions can lead to a related '2-in 2-out' spin ice order, with two spins pointing in and two pointing out of each tetrahedron when spin-spin couplings are weakly ferromagnetic. This is also shown as one of the magnetic phases for γ-Fe 2 SiO 4 in Fig. 5e magnetic structure. Temperature variations of the moment amplitudes μ i and propagation vector shifts δ i are shown in Fig. 3d 23 , where chains of collinear moments are modulated between 0 and 5.0 μ B moments for S = 2 Co 3+ moments with a sizable orbital contribution. Strongly frustrated systems based on orbitally non-degenerate ions such as S = 5/2 Fe 3+ in FeTe 2 O 5 Br and S = 7/2 Gd 3+ in Gd 2 Ti 2 O 7 can stabilise spin arrangements of varying amplitude to minimise exchange energy between unfavourably oriented moments and gain entropy from the thermally fluctuating components at non-zero temperatures. However, the observation a Magnetic susceptibility in an applied field of 0.5 T with inset is showing the low-temperature region where magnetic transitions occur at T m1 ≈ 9 K and T m2 ≈ 7 K. The Curie-Weiss fit to points between 150 and 400 K gives an effective paramagnetic moment of 4.25 μ B , consistent with high-spin 3d 6 Fe 2+ spins and a Weiss temperature of θ = -19.6 K b Real part of the AC susceptibility in an oscillating magnetic field with amplitude 9 Oe and frequencies as shown. No frequency dependence of the features that would evidence spin-glass behaviour is observed. c Heat capacity variation with the lattice contribution fitted by the polynomial C p = γT + βT 3 + δT 5 . The inset shows the low-temperature region with discontinuities at T m1 and T m2 marked. The magnetic contribution is evident up to 55 K, but the integrated entropy in the 2-55 K range of 5.77 J mol −1 K −1 is only 43% of the theoretical Rln(2S + 1) = 13.38 J mol −1 for S = 2. Error bars are standard deviations. d Fit of the cubic spinel model to the synchrotron X-ray diffraction profile (λ = 0.1917 Å) at 5 K with the region containing (400) and (440) reflections that are sensitive to a tetragonal lattice distortion shown in the inset. No peak splittings or broadenings that would evidence a lattice distortion are observed of very rare collinearly ordered components with full amplitude modulation to lowest temperature in Fe 2 GeO 4 and Ca 3 Co 2 O 6 , both of which are based on high-spin 3d 6 ions with unquenched orbital contributions, suggests that an additional factor operates in these materials. We propose that dynamic correlations of the orbital and spin states in these materials give rise to modulations of F that match the periodicity of the SDW, hence a 'frustration wave'.
The orbital interactions and resulting magnetic exchange interactions that can give rise to frustration wave order in Fe 2 GeO 4 are shown in Fig. 4. High-spin Fe 2+ cations have the degenerate t 2g 4 e g 2 ground state with one doubly occupied and two half-occupied t 2g orbitals, and t 2g -t 2g magnetic exchange interactions occur across the shared edges of FeO 6 octahedra, as well as more weakly through the 90°Fe-O-Fe pathway. Only one of the three t 2g orbitals on each Fe 2+ cation overlaps with another in the 90°Fe-O-Fe pathway, hence three electronic possibilities exist. Direct t 2g 1 -t 2g 1 interactions are antiferromagnetic (J AF ), but the t 2g 2 -t 2g 1 interactions are ferromagnetic (J F ), in keeping with Goodenough-Kanamori exchange rules 24,25 or the Kugel-Khomskii approach 26 , as shown in Fig. 4a. We assume that coulombically unfavourable t 2g 2 -t 2g 2 configurations are avoided at low temperatures, as observed in the orbitally ordered ground state of magnetite 27 . This corresponds to a local orbital ordering constraint. Each tetrahedron of four Fe 2+ spins thus has two antiferromagnetic t 2g 1 -t 2g 1 and four ferromagnetic t 2g 2 -t 2g 1 interactions along its edges, and these have two distinct arrangements as shown in Fig. 4b. The configuration where antiferromagnetic couplings are on adjacent (A) edges of the tetrahedron is notable, as the simplest collinear ground state for comparable interaction strengths J AF ≈ -J F is a 3 up/1 down configuration in which two spins are unfrustrated (F = 0), while a Magnetic scattering profiles obtained by subtracting the 25 K D20 data from profiles between 2.5 and 9.5 K, recorded in~0.3 K steps and at 12 and 15 K. h k l labels correspond to magnetic satellite reflections at (hkl) + k i for k i = ( 2 =3 + δ i 2 =3 + δ i 0). The five magnetic peaks with black labels that appear below T m1 = 8.9 K have a propagation vector k 1 , while the weak 000 peak indicated in pink appears below T m2 = 6.6 K with vector k 2 . b Fit of the crystal and magnetic structures at 2 K to high-resolution D2B data at 2 K (λ = 1.59 Å). The inset shows the fit in the low-angle region of D20 data (λ = 2.41 Å). Two weak impurity peaks are labelled with asterisks. Magnetic reflection markers are in violet (k 1 ) and pink (k 2 ), and the structural reflections are in green. c The k 1 order below T m1 with sinusoidal modulation of the Fe1 (blue) and There are 24 equivalent A-type configurations, but only six for O-types, hence (neglecting any long range orbital correlations) 80% of Fe 4 tetrahedra are A-type at any instant in the orbitally fluctuating state in the absence of longer-range orbital correlations and 16% of Fe 2+ spins are in a locally unfrustrated (F = 0) environment at the apices between two Atype tetrahedra. This shows that dynamically correlated orbital fluctuations (dynamic orbital order) can lead to large local fluctuations in F, and long-range magnetic and frustration wave order at low temperatures. We propose that frustration wave order in Fe 2 GeO 4 arises from exchange interactions between ordered spin components in one sublattice via the dynamic components of their neighbours in the other. This is illustrated using the ordered Fe1 spins, which are represented by a commensurate approximant ( 2 = 3 -2 = 3 0) model for simplicity in Fig. 4c (the small incommensurability most likely results from next nearest-neighbour Fe-O…O-Fe magnetic couplings that are neglected here). Ferromagnetic chains of fully ordered Fe1 up spins (+S) and partially fluctuating Fe1 down spins with ordered components of -S/2 are linked by Fe2 spins that are ordered in a perpendicular direction, so no Heisenberg exchange occurs. However, fluctuating components of Fe2 spins can couple to Fe1 spins as shown in Fig. 4d. The observed order is consistent with A-type tetrahedra leading to unfrustrated interactions around the Fe1 (+S) chain, while other tetrahedra fluctuate between A and O configurations leading to some frustrated interactions at the other Fe1 (-S/2) chains. The Fe2 spin components fluctuate between up and down states in the various configurations and have no static order parallel to Fe1 spins. The orbital states fluctuate in a highly correlated manner but without leading to localisation and orbital order. Similar orbital pictures can be drawn for any sampled region in the full incommensurate structure (Fig. 3c). The modulation of the F consistent with the commensurate approximant ( 2 = 3 -2 = 3 0) model for Fe 2 GeO 4 is shown in Fig. 4c. The Fe1 (+S) spins are always unfrustrated while those at Fe2 sites are highly frustrated with pyrochlore-like F = 1/3 values, and the partly frustrated Spin Order in γ-Fe 2 SiO 4 . The high-pressure spinel γ-Fe 2 SiO 4 was also studied to investigate the chemical pressure effects of replacing Ge in Fe 2 GeO 4 by smaller Si. High-pressure synthesis of the polycrystalline γ-Fe 2 SiO 4 sample and characterisation measurements are described in methods with further details in Supplementary Figs. 2, 3, 4 and 7, Supplementary Tables 2, 5, 6 and 7 and Supplementary Notes 1 and 3. Two magnetic ordering transitions are observed in low-temperature neutron diffraction profiles of γ-Fe 2 SiO 4 , as shown in Fig. 5a and b. Magnetic diffraction peaks appearing below T m1 = 12 K are indexed by propagation vector k 1 = ( 3 = 4 + δ 1 3 = 4 + δ 1 0), where δ 1 ≈ 0.030 (1), and are fitted by double-k magnetic structures, similar to that of k 1 -Fe 2 GeO 4 . Two modulated spin components are present and their combinations can describe a canted arrangement (Fig. 5c) or an elliptical helical order (Fig. 5d). These fit the magnetic intensities equally well, and the ordered moment amplitudes are modulated in both cases, so it is not clear whether this is a frustration wave or a more conventional elliptical spin order. Further magnetic peaks that emerge below T m2 = 8 K for γ-Fe 2 SiO 4 are indexed on a commensurate k 2 = (1 0 0) vector and are fitted by an ordered spin ice model (Fig. 1c, 5e), in which all moment amplitudes are equal. Spin ice ordering is very rare in transition metal oxide spinels, but is reported in the V 3+ sublattice of FeV 2 O 4 , although this phase is tetragonally distorted with both Fe 2+ -V 3+ and V 3+ -V 3+ magnetic interactions operating 28 . The observation of a spin ice phase competing with the modulated wave state reveals a fine energy balance between these two classes of ground state in γ-Fe 2 SiO 4 . Long-range spin ice orders in pyrochlore oxides such as Sm 2 Mo 2 O 7 and Nd 2 Mo 2 O 7 result from weak exchange coupling and large dipolar interactions coupled with local anisotropy 29 . Local variations of ferromagnetic and antiferromagnetic couplings driven by the correlated orbital fluctuations may also help to stabilise the spin ice phase of γ-Fe 2 SiO 4 .

Discussion
In conclusion, the unusual magnetic structure of Fe 2 GeO 4 evidences a previously unrecognised class of ground states for orbitally degenerate spins on frustrated lattices, in which the degree of frustration orders spatially across structurally equivalent sites, resulting in large amplitude modulations of the moment in the magnetically ordered phases. This arises because the exchange interactions depend on the d-orbital occupancy, so that a coupling of spins and orbitals can give rise to a long-range modulation of the exchange interactions and hence the frustration function. Weak coupling of Fe 2+ orbital states to the lattice appears to be important for avoiding structural distortions that probably destabilise frustration wave orders in other orbitally degenerate materials. The γ-Fe 2 SiO 4 analogue has a more complex modulated order that may be frustration wave driven, competing with a spin ice phase. Frustration waves lead to spatial organisation of statically ordered and highly correlated, but dynamic orbital and spin components that may give rise to novel excitations and quantum phenomena in these and other materials. Further exploration of the complex spin orders in Fe 2 GeO 4 and γ-Fe 2 SiO 4 using single crystals and of their excitations by inelastic neutron scattering and other spectroscopies will thus be worthwhile. a Magnetic scattering profiles obtained by subtracting 25 K D20 data from profiles between 2.5 and 14 K, increasing in~0.6 K steps. Blue and green arrows, respectively, show changes in diffraction intensity at the T m1 = 13 K and T m2 = 8 K transitions. b Fit of the crystal and magnetic structures at 2 K to D20 data at 2 K with 90°takeoff angle. The inset shows the fit to prominent magnetic peaks in the low-angle region for data with 42°takeoff angle to give high resolution. Magnetic reflection markers are in violet (k 1 ) and pink (k 2 ), and structural reflections are in green. c Canted model for the k 1 = ( 3 =4 + δ 3 =4 + δ 0) order is observed below T m1 , showing Fe1 (blue) and Fe2 (red) moments.

Methods
(d) Alternative elliptical helix model for the k 1 order is showing the planes of rotation for the moments. (e) The additional k 2 = (1 0 0) ordered spin ice phase is observed below T m2 , showing the tetrahedra of 2-in-2-out moments box furnace at 900°C for 60 h and then slow-cooling for 12 h. α-Fe 2 SiO 4 was transformed to spinel-type γ-Fe 2 SiO 4 in a Walker-type multi-anvil press in BN capsule, pressurising at 6 GPa at 900°C for 20 min before quenching. Laboratory X-ray diffraction using a Bruker D2 diffractometer confirmed the formation of cubic Fd 3 m normal spinels with Fe 2+ only at the octahedral Bsites. Physical measurements were performed using a Quantum Design MPMS XL7 SQUID magnetometer for DC susceptibility and a Quantum Design PPMS for AC susceptibility and heat capacity measurements.
Powder neutron diffraction. Powder neutron diffraction (PND) data were collected at the ILL facility in Grenoble. High-resolution profiles for a 3 g sample of Fe 2 GeO 4 were collected on instrument D2B at wavelength λ = 1.59475 Å with 10' collimation at 2 K and at full flux at 2, 6, 10, 50, 100, 200 and 300 K. Refinements were performed on high-resolution integrated data from the central region of the detector. Additional PND data were collected from D20 with λ = 2.41 Å at 1.8, 2.5, 12, 15 and 25 K; ramp collection between 2.5 and 9.5 K in~0.3 K steps was used to follow the evolution of the magnetic structure. PND data for γ-Fe 2 SiO 4 were collected between 2 and 300 K from D20 (λ = 2.41) on 120 mg sample. Data acquired with high takeoff angle (90°) were used for crystal structure refinement to confirm cubic symmetry, whereas low takeoff angle (42°) data were used for magnetic structure determination. The structural and magnetic refinements were performed with the Rietveld refinement routines implemented in FullProf, using the k-search and the BasIreps software for magnetic symmetry determination and analyses 30,31 . Crystal and magnetic structures were visualised with FPStudio in the FullProf suite and with the VESTA software 32 . Representation analysis shows that the single Fe B lattice site is split into magnetically distinct sites, Fe1 at (½,½,½) and Fe2 at ( 3 =4,0,¼). Details on the representation analysis and the basis vectors can be found in Supplementary Notes 2 and 3.
Magnetic diffraction peaks appearing below the two Fe 2 GeO 4 transitions have very similar propagation vectors k i = ( 2 =3 + δ i 2 =3 + δ i 0). The first order is incommensurate with δ 1 ≈ ˗0.025(1), but assuming that δ 1 = δ 2 does not fit the peak positions for the second phase correctly and refining the propagation vector shift independently gives δ 2 ≈ 0. Hence this order appears to be commensurate with vector k 2 = ( 2 =3 2 =3 0) as reported elsewhere 15 , but observation of more peaks will be needed to confirm its nature. The k 2 spin components are perpendicular to the k 1 moments shown in Fig. 3c, but modelling these in the xy-plane or z-direction gave equally good fits.
γ-Fe 2 SiO 4 has two quite different magnetic phases. The k 1 = ( 3 =4 + δ 1 3 =4 + δ 1 0) phase appearing below T m1 = 12 K is similar to that of k 1 -Fe 2 GeO 4 , but has two amplitude-modulated spin components. Magnetic peaks that emerge below T m2 = 8 K are indexed on a commensurate k 2 = (1 0 0) vector and the intensities are fitted by the ordered spin ice model, showing that this is a separate magnetic phase. Absolute values of the moments cannot be determined without knowledge of the magnetic phase proportions. The maximum amplitude in the canted description of the incommensurate phase with vector k 1 of 2.77(9) μ B and the 1.29(4) μ B moment from the spin ice phase with k 2 sum to 4.1(1) μ B at 1.8 K , indicating that the sample comprises 68% of the k 1 and 32% k 2 phase, both with ideal (maximum) moment values of 4.0 μ B . Moment magnitudes vary between 0 and 4.0 μ B in the canted model and between 1.8 and 3.6 μ B for the elliptical helical model for the k 1 phase.

Data availability
Data that support the findings of this study have been deposited at https://doi.org/ 10.7488/ds/2439. Received: 9 January 2018 Accepted: 19 September 2018