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 Fe 2 GeO 4 and  - Fe 2 SiO 4 at low temperatures which indicate that that the degree of frustration may itself be a fluctuating quantity that can spontaneously order without a lattice distortion as a ‘frustration wave’. Fe 2 GeO 4 with propagation vector (⅔+  ⅔+  0) has ordered Fe 2+ moments that vary between fully saturated 4  B and zero values, consistent with a frustration wave order.  - Fe 2 SiO 4 has a more complex (¾+  ¾+  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.


Introduction
Long 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 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 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 = ⅓ at all sites in the ordered ground states shown in Figs. 1(a) and (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.
Fe2GeO4 and the high pressure -form of Fe2SiO4 are cubic B2AO4 spinels where orbitallydegenerate 3d 6 Fe 2+ cations with S = 2 spins form a pyrochlore-type B-site lattice 6,7 . -Fe2SiO4 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 10,11,12,13,14 , but the low temperature 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.

Spin order in Fe2GeO4
Synthesis of the polycrystalline Fe2GeO4 sample and characterisation measurements are 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 report. 12 AC (Alternating Current) measurements show no frequency-dependence in the low temperature features indicating an absence of 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 ͞ 3m 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. ZnV2O4 17 , LiMn2O4 18 , MgCr2O4 19 and Co2GeO4 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 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 = (⅔+i ⅔+i 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 = (⅔+i -⅔-i 0) and ki2 = (⅔+i ⅔+i 0). A good fit to the peaks observed below Tm1, as shown in Fig. 3b where refined 1 ≈ ˗0.025(1), can only be obtained using a model in which ordered moment amplitudes are modulated, as displayed in Fig. 3c.

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 FeTe2O5Br, 21 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 Gd2Ti2O7. 22 The only close SDW analogue to Fe2GeO4 we are aware of is the spin order in Ca3Co2O6, 23 where chains of collinear moments are modulated between 0 and 5.0 μB moments for S = 2 Co 3+ moments with a sizeable orbital contribution.
Strongly frustrated systems based on orbitally non-degenerate ions such as S = 5/2 Fe 3+ in FeTe2O5Br and S = 7/2 Gd 3+ 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 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 the degree of frustration 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

Spin order in -Fe2SiO4
The high pressure spinel -Fe2SiO4 was also studied to investigate the chemical pressure effects of replacing Ge in Fe2GeO4 by smaller Si. High-pressure synthesis of the polycrystalline - their combinations can describe a canted arrangement (Fig. 5c) or an elliptical helical order (Fig. 5d).
These fit the magnetic intensities equally well and 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 Tm2 = 8 K for -Fe2SiO4 are indexed on a commensurate k2 = (1 0 0) vector and are fitted by an ordered spin ice model (Figs. 1c and 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 FeV2O4 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 -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 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 -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 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 -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 -Fe2SiO4 using single crystals, and of their excitations by inelastic neutron scattering and other spectroscopies will thus be worthwhile.

Sample synthesis and characterisation
Fe2GeO4 and olivine-type α-Fe2SiO4 were synthesised as polycrystalline powders by grinding

Powder neutron diffraction
Powder neutron diffraction (PND) data were collected at the ILL facility in Grenoble. High resolution profiles for a 3g 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, 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; and 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 of sample. Data acquired with high take-off angle (90°) were used for crystal structure refinement to confirm cubic symmetry, whereas low take-off 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 (¾,0,¼).
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 = (⅔+i ⅔+i 0). The first order is incommensurate with 1 ≈ ˗0.025(1), but assuming 1 = 2 does not fit peak positions for the second phase correctly and refining the propagation vector shift independently gives 2 ≈ 0. Hence this order appears to be commensurate with vector k2 = (⅔ ⅔ 0), as reported elsewhere, 15 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.
Magnetic peaks that emerge below Tm2

Data Availability
Data that support the findings of this study have been deposited at https://datashare.is.ed.ac.uk/handle/10283/838.     profiles obtained by subtracting 25 K D20 data from profiles between 2.5 and 14 K, recorded 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 2K 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. A weak impurity peak is labelled with an asterisk.