Introduction

Long-range spin order is sometimes avoided in frustrated magnetic materials, leading to unconventional correlations such as quantum spin liquids and ices1,2,3,4. Frustrated long-range 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

$$F = \raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} \left( {1-{\sum} {J_{ij}{\mathbf{S}}_i.} {\mathbf{S}}_j/{\sum} {\left| {J_{ij}} \right|\left| {{\mathbf{S}}_i} \right|\left| {{\mathbf{S}}_j} \right|} } \right)$$

where the exchange Hamiltonian for interacting spins Si and Sj is –JijSi.Sj. 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=Nf/N, where Nf 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.

Fig. 1
figure 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-Fe2SiO4 in Fig. 5e

Fe2GeO4 and the high-pressure γ-form of Fe2SiO4 are cubic B2AO4 spinels, where orbitally degenerate 3d6 Fe2+ cations with S = 2 spins form a pyrochlore-type B-site lattice6,7. γ-Fe2SiO4 is also of geophysical interest as one of the main constituents of the Earth’s mantle8,9. Previous studies have established that both materials have magnetic transitions near 10 K10,11,12,13,14, but the low-temperature spin orders are not reported and preliminary abstracts have differing results15,16. 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 Fe2GeO4

Synthesis of the polycrystalline Fe2GeO4 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 Fe2GeO4 reveal two magnetic transitions with a susceptibility maximum at Tm1 ≈ 9 K and divergence of field and zero-field cooled susceptibilities at Tm2 ≈ 7 K, consistent with a previous report12. 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 Fe2+ 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 \(Fd\ \bar{\it3}\ m\) 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. ZnV2O417, LiMn2O418, MgCr2O419 and Co2GeO411,20 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 Fe2+ spins remain dynamic below the two magnetic transitions.

Fig. 2
figure 2

Magnetic and structural characterisation of Fe2GeO4. a Magnetic susceptibility in an applied field of 0.5 T with inset is showing the low-temperature region where magnetic transitions occur at Tm1 ≈ 9 K and Tm2 ≈ 7 K. The Curie–Weiss fit to points between 150 and 400 K gives an effective paramagnetic moment of 4.25 μB, consistent with high-spin 3d6 Fe2+ 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 Cp = γT + βT3 + δT5. The inset shows the low-temperature region with discontinuities at Tm1 and Tm2 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−1K−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

Sharp magnetic diffraction peaks indicative of long-range spin order appear below the magnetic transition at Tm1 ≈ 9 K with an additional weak peak observed below Tm2 ≈ 7 K, as shown in Fig. 3a. These peaks were indexed by very similar propagation vectors ki = (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi \({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi 0) for peaks appearing below Tmi (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 kij apply to different sites Fej (j = 1 or 2), ki1 = (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\)-δi 0) and ki2 = (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi \({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi 0). A good fit to the peaks that  are observed below Tm1, as shown in Fig. 3b where refined δ1 ≈ –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 Tm2 is fitted by an additional order of small perpendicular moment components that describe a canting of the above magnetic structure. Temperature variations of the moment amplitudes μi and propagation vector shifts δi are shown in Fig. 3d and the maximum amplitudes of the two modulated moment components are μ1 = 3.94(3) and μ2 = 0.92(7) μB at 1.8 K. The maximum resultant amplitude is μ = 4.05(8) μB, in agreement with the ideal value of 4 μB for high-spin Fe2+. Ordered moments are modulated between 0 (fully frustrated) and 4 μB (unfrustrated) values. The average ordered moment magnitude is 64% of the ideal value, so the equivalent of approximately one-third of the spins remain dynamic below the magnetic ordering transitions, qualitatively consistent with the substantial reduction of magnetic entropy.

Fig. 3
figure 3

Low-temperature neutron diffraction results for Fe2GeO4. 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) + ki for ki = (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi \({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi 0). The five magnetic peaks with black labels that appear below Tm1 = 8.9 K have a propagation vector k1, while the weak 000 peak indicated in pink appears below Tm2 = 6.6 K with vector k2. 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 (k1) and pink (k2), and the structural reflections are in green. c The k1 order below Tm1 with sinusoidal modulation of the Fe1 (blue) and Fe2 (red) moment amplitudes in the [\(1\bar{1}0\)] and [110] directions, respectively. Moments vary between 0 (fully frustrated) and 4 μB (fully ordered) values. The additional k2 order below Tm2 adds a small modulated tilting of the moments shown separately in Supplementary Fig. 6. d Temperature variations of the magnetic moments μ and propagation vector contributions δ for Fe1 and Fe2 with canting in the ab-plane. Fits of the critical law μ = μ0(1 − T/Tm)β to the moment variations are shown, where β1 = 0.35(6) and Tm1 = 8.6(2) K, and β2 = 0.3(1) and Tm2 = 7.2(2) K for the two transitions. Error bars are standard deviations

Frustration wave picture for Fe2GeO4

Amplitude-modulated spin-density wave (SDW) order of local moments is relatively common in metallic magnets, where exchange couplings are coupled to the Fermi surface vectors, as described in RKKY theory. However, amplitude modulation of moments between zero and fully saturated values as observed in Fe2GeO4 is highly unusual in non-metallic materials, as even complex spin textures such as helimagnets, spin vortices or skyrmions have uniform moment amplitudes, while spin directions change. Elliptical spiral structures in frustrated systems can modulate moment amplitudes over part of the available range, e.g. in FeTe2O5Br21, and ‘idle spin’ orders provide a special case where some spins remain disordered due to frustration of their interactions with surrounding uniformly ordered spins, an example is observed below 0.7 K in the pyrochlore Gd2Ti2O722. The only close SDW analogue to Fe2GeO4 we are aware of is the spin order in Ca3Co2O623, where chains of collinear moments are modulated between 0 and 5.0 μB moments for S = 2 Co3+ moments with a sizable orbital contribution.

Strongly frustrated systems based on orbitally non-degenerate ions such as S = 5/2 Fe3+ in FeTe2O5Br and S = 7/2 Gd3+ in Gd2Ti2O7 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 of very rare collinearly ordered components with full amplitude modulation to lowest temperature in Fe2GeO4 and Ca3Co2O6, both of which are based on high-spin 3d6 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 Fe2GeO4 are shown in Fig. 4. High-spin Fe2+ cations have the degenerate t2g4eg2 ground state with one doubly occupied and two half-occupied t2g orbitals, and t2gt2g magnetic exchange interactions occur across the shared edges of FeO6 octahedra, as well as more weakly through the 90° Fe-O-Fe pathway. Only one of the three t2g orbitals on each Fe2+ cation overlaps with another in the 90° Fe-O-Fe pathway, hence three electronic possibilities exist. Direct t2g1t2g1 interactions are antiferromagnetic (JAF), but the t2g2t2g1 interactions are ferromagnetic (JF), in keeping with Goodenough–Kanamori exchange rules24,25 or the Kugel–Khomskii approach26, as shown in Fig. 4a. We assume that coulombically unfavourable t2g2t2g2 configurations are avoided at low temperatures, as observed in the orbitally ordered ground state of magnetite27. This corresponds to a local orbital ordering constraint. Each tetrahedron of four Fe2+ spins thus has two antiferromagnetic t2g1t2g1 and four ferromagnetic t2g2t2g1 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 JAF ≈ –JF is a 3 up/1 down configuration in which two spins are unfrustrated (F= 0), while the other two are partially frustrated (F= 1/3). This demonstrates how large variations in F between neighbouring spins may arise as a result of the local orbital ordering configuration. The alternative arrangement with antiferromagnetic couplings on opposite (O) tetrahedral edges has all spins equally frustrated (F= 1/3). There are 24 equivalent A-type configurations, but only six for O-types, hence (neglecting any long range orbital correlations) 80% of Fe4 tetrahedra are A-type at any instant in the orbitally fluctuating state in the absence of longer-range orbital correlations and 16% of Fe2+ spins are in a locally unfrustrated (F= 0) environment at the apices between two A-type 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.

Fig. 4
figure 4

Frustration wave model for orbital and spin correlations in Fe2GeO4. a Nearest-neighbour Fe2+–Fe2+ t2g orbital and magnetic interactions. The t2g1t2g1 interaction (left) leads to antiferromagnetic coupling JAF between cation spins shown in the lower corners and is represented by the light blue bar below and in (b) and (d). The t2g2t2g1 interaction (right) shows t2g2 orbital order and associated ferromagnetic coupling JF, respectively, represented by red and green bars below and in (b) and (d). b The two configurations for orbital order and magnetic interactions within the tetrahedra of four Fe2+ cations. Four edges have orbital order and ferromagnetic couplings and two are antiferromagnetic. Tetrahedra with antiferromagnetic couplings on adjacent (A, left hand image) edges have a 3 up/1 down spin ground state for JAF ≈ -JF, where the two spins adjacent to the frustrated interaction are partially frustrated, but the others are unfrustrated. The alternative configuration has antiferromagnetic couplings on opposite (O, right hand image) edges and is uniformly frustrated with ferromagnetic and antiferromagnetic ground states of comparable energy for JAF ≈ -JF, the latter is shown here. c (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\)\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) 0) approximant model for the Fe1 spin order in Fe2GeO4 shown in Fig. 3c, projected on the [110] plane of the cubic spinel structure. Fe1 moments form ferromagnetic chains perpendicular to the image plane; one-third have fully ordered up spins (Fe1: + S) and two-thirds are partially fluctuating down spins (Fe1:-S/2). Perpendicular ordered spins at Fe2 sites (red) are not shown. Values of the frustration index F in the different spin layers are shown. d Two fluctuating configurations for orbital and spin orders within the representative unit circled in (c). Static Fe1 spin components (blue) are coupled through dynamic components of the Fe2 spins (red). The central tetrahedron is always A-type and the two blue (Fe1: + S) spins have no frustrated interactions to nearest-neighbours and are fully ordered. The top and bottom tetrahedra fluctuate between A and O configurations and have (Fe1:−S/2) spins with some frustrated interactions. Fe2 spin components fluctuate between up and down in these and other configurations and have no static ordered component parallel to Fe1 spins

We propose that frustration wave order in Fe2GeO4 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 (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\)\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) 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 (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\)\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) 0) model for Fe2GeO4 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 Fe1 (–S/2) spins have intermediate values, shown as F= 1/6 on the assumption of equal fluctuations between A and O configurations of their local spin tetrahedra.

Spin Order in γ-Fe2SiO4

The high-pressure spinel γ-Fe2SiO4 was also studied to investigate the chemical pressure effects of replacing Ge in Fe2GeO4 by smaller Si. High-pressure synthesis of the polycrystalline γ-Fe2SiO4 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-Fe2SiO4, as shown in Fig. 5a and b. Magnetic diffraction peaks appearing below Tm1 = 12 K are indexed by propagation vector k1 = (\({\raise.5ex\hbox{$\scriptstyle 3$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$}}\) + δ1 \({\raise.5ex\hbox{$\scriptstyle 3$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$}}\) + δ1 0), where δ1 ≈ 0.030(1), and are fitted by double-k magnetic structures, similar to that of k1-Fe2GeO4. 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.

Fig. 5
figure 5

Low-temperature neutron diffraction results for γ-Fe2SiO4. 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 Tm1 = 13 K and Tm2 = 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 (k1) and pink (k2), and structural reflections are in green. c Canted model for the k1 = (\({\raise.5ex\hbox{$\scriptstyle 3$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$}}\) + δ \({\raise.5ex\hbox{$\scriptstyle 3$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$}}\) + δ 0) order is observed below Tm1, showing Fe1 (blue) and Fe2 (red) moments. (d) Alternative elliptical helix model for the k1 order is showing the planes of rotation for the moments. (e) The additional k2 = (1 0 0) ordered spin ice phase is observed below Tm2, showing the tetrahedra of 2-in-2-out moments

Further magnetic peaks that emerge below Tm2 = 8 K for γ-Fe2SiO4 are indexed on a commensurate k2 = (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 V3+ sublattice of FeV2O4, although this phase is tetragonally distorted with both Fe2+–V3+ and V3+–V3+ magnetic interactions operating28. 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-Fe2SiO4. Long-range spin ice orders in pyrochlore oxides such as Sm2Mo2O7 and Nd2Mo2O7 result from weak exchange coupling and large dipolar interactions coupled with local anisotropy29. Local variations of ferromagnetic and antiferromagnetic couplings driven by the correlated orbital fluctuations may also help to stabilise the spin ice phase of γ-Fe2SiO4.

Discussion

In conclusion, the unusual magnetic structure of Fe2GeO4 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 Fe2+ 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-Fe2SiO4 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 Fe2GeO4 and γ-Fe2SiO4 using single crystals and of their excitations by inelastic neutron scattering and other spectroscopies will thus be worthwhile.

Methods

Sample synthesis and characterisation

Fe2GeO4 and olivine-type α-Fe2SiO4 were synthesised as polycrystalline powders by grinding stoichiometric quantities of Fe (-22 mesh, 99.998%, Alfa Aesar), GeO2 / SiO2 (99.999% Alfa Aesar) and Fe2O3 (99.999% Sigma Aldrich) powders and pressing them into a pellet. The reactions were carried out in evacuated silica tubes, heating in a box furnace at 900 °C for 60 h and then slow-cooling for 12 h. α-Fe2SiO4 was transformed to spinel-type γ-Fe2SiO4 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\({\bar{\it{3}}}\)m normal spinels with Fe2+ only at the octahedral B-sites. 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 Fe2GeO4 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-Fe2SiO4 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 analyses30,31. Crystal and magnetic structures were visualised with FPStudio in the FullProf suite and with the VESTA software32. Representation analysis shows that the single Fe B lattice site is split into magnetically distinct sites, Fe1 at (½,½,½) and Fe2 at (\({\raise.5ex\hbox{$\scriptstyle 3$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$}}\),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 Fe2GeO4 transitions have very similar propagation vectors ki = (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi \({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) + δi 0). The first order is incommensurate with δ1 ≈ ˗0.025(1), but assuming that δ1=δ2 does not fit the peak positions for the second phase correctly and refining the propagation vector shift independently gives δ2 ≈ 0. Hence this order appears to be commensurate with vector k2 = (\({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) \({\raise.5ex\hbox{$\scriptstyle 2$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 3$}}\) 0) as reported elsewhere15, but observation of more peaks will be needed to confirm its nature. The k2 spin components are perpendicular to the k1 moments shown in Fig. 3c, but modelling these in the xy–plane or z-direction gave equally good fits.

γ-Fe2SiO4 has two quite different magnetic phases. The k1 = (\({\raise.5ex\hbox{$\scriptstyle 3$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$}}\) + δ1 \({\raise.5ex\hbox{$\scriptstyle 3$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 4$}}\) + δ1 0) phase appearing below Tm1 = 12 K is similar to that of k1-Fe2GeO4, but has two amplitude-modulated spin components. Magnetic peaks that emerge below Tm2 = 8 K are indexed on a commensurate k2 = (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 k1 of 2.77(9) μB and the 1.29(4) μB moment from the spin ice phase with k2 sum to 4.1(1) μB at 1.8 K, indicating that the sample comprises 68% of the k1 and 32% k2 phase, both with ideal (maximum) moment values of 4.0 μB. Moment magnitudes vary between 0 and 4.0 μB in the canted model and between 1.8 and 3.6 μB for the elliptical helical model for the k1 phase.