Local cation order and ferrimagnetism in compositionally complex spinel ferrites

High entropy oxides (HEOs) are being increasingly reported in various oxide crystal structures, including in the perovskite, fluorite, spinel, and pyrochlore structure motifs.^1–18 HEOs are generally described as single phase solid solution systems exhibiting five or more homogeneously distributed cations in equimolar or near-equimolar ratios on a single crystallographic site. When a positive entropy of formation (ΔS_f) overcomes a positive enthalpy of formation (ΔH_f), the Gibbs free energy of formation (ΔG_f = ΔH_f − TΔS_f) will be negative, making the system an entropy stabilized oxide (ESO). The ESO, (Mg_0.2Ni_0.2Co_0.2Cu_0.2Zn_0.2)O, a five component oxide exhibiting a single rock salt phase, was first introduced by Rost et al. in 2015.¹⁹ HEOs/ESOs draw interest due to inherent opportunities to tailor and combine materials properties. The extent to which their enhanced compositional and configurational complexity impart additional stability, synergistic effects, and emergent/tunable properties to HEO archetypes is an active area of exploration. The magnetic properties of HEOs represent one of the fastest growing research directions in this scientific community. From the perspective of magnetic behavior, the complexity of the local and long-range nuclear and spin structures found in HEOs gives rise to an unusually large number of metal–oxygen–metal interactions.²⁰ The inherent variation in coordination geometry, valence state, spin state, number of cations, and metal cation type each HEO lattice can accommodate presents a rich pallet for exploring the tunability of the structure–property effects in the class. However, relatively few compositions have been studied in detail,²⁰ no doubt due in part to the complex cation and spin order likely to be present.

Spinels crystallizing in cubic space group $Fd3̄m$⁠, with nominal formula AB₂O₄, feature a close-packed oxygen substructure with two thirds of cations occupying octahedral interstices and one third occupying tetrahedral ones. The spinel compounds in this work involve A and B cations in the cation charge state combination of +2 and +3 (charge-balanced spinel oxide families with +4 and +2, and +6 and +1 cation combinations are also known to exist). Spinels are typically classified according to which cations occupy the respective coordination environments, with a degree of inversion, γ, denoting the fraction of A ions occupying the octahedral sites, [A_1−γB_γ]_tet[A_γB_2−γ]_octO₄. In the “normal” spinel structure (γ = 0, [A]_tet[B₂]_octO₄), B³⁺ fully occupies the spinel octahedral site, while A²⁺ fully occupies the tetrahedral site. In the “inverse” spinel structure (γ = 1, [B]_tet[AB]_octO₄), half of the B³⁺ cations fully occupy the tetrahedral site, leaving an equal amount of A²⁺ and B³⁺ atoms occupying the octahedral site. In a “random” spinel structure (γ = 2/3, [A_1/3B_2/3]_tet[A_2/3B_4/3]_octO₄), A²⁺ and B³⁺ atoms take up residence on both sites according to their overall molar ratio, with 1/3 and 2/3 distribution, respectively. Arbitrary intermediate states between the normal and inverse spinels are possible (γ value between 0 and 1). Figure 1 provides a depiction of high entropy spinel AB₂O₄ (with A = 5 equimolar cations and B = a single metal cation), shown for the case of the inverse spinel ferrite structure.

Specific (not random) cation site preferences of the crystal chemistry in spinel oxides are well documented in the literature,^21,22 going back to early reports^23,24 linking cation site distribution and magnetic properties in spinel ferrites. There has been ample research into the magnetic properties of spinel AFe₂O₄ since. AFe₂O₄ (A = Fe²⁺,^25,26 Ni²⁺,²⁷ Cu²⁺,²⁸ Mg²⁺,²⁹ or Co^2+26,30,31) spinels have a variety of important applications in magnetic technologies.^24,32,33 They all have an inverse (either partially or completely) spinel structure. Previous works have shown that, in spinel ferrites, Zn²⁺ prefers to occupy the tetrahedral site while Mg²⁺, Mn²⁺, Mn³⁺, Fe²⁺, Fe³⁺, Ni²⁺, Cu²⁺, and Co²⁺ tend to occupy the octahedral site in spinel ferrites.²⁴ It remains to be seen whether high entropy spinels will follow the trends in site preference observed among the ternary spinel ferrite systems or if the potential for increased configurational entropy will lead to more random cation distributions.

High entropy spinels have been synthesized by solid state,^34–49 co-precipitation,^50,51 solution combustion synthesis (SCS),^52–59 sol–gel method,^60,61 solvothermal synthesis,⁶² reverse co-precipitation,⁶³ flame spray pyrolysis,⁶⁴ hydrothermal method,⁶⁵ polymerized complex method,⁶⁶ and glycine-combustion method⁶⁷ among other techniques. Examples of reported high-entropy ferrite powders are summarized in Table I.

Research into the magnetic properties and cation site ordering among high entropy spinels is still in its infancy. There is one recent report⁶⁸ demonstrating that the site ordering (Co_0.6Fe_0.4)(Cr_0.3Fe_0.1Mn_0.3Ni_0.3)₂O₄ in A₃O₄ spinel is governed by maximization of the crystal field stabilization energy (CFSE) of the constituent ions through site selectivity rather than by maximization of the configurational entropy (S_config) through random and homogeneous cation disorder.⁶⁸ It is not known whether this generalizes to other high entropy spinel compositions, including compositional variations of the technologically important spinel ferrite (AFe₂O₄). The magnetic properties of spinel ferrites are highly sensitive to the distribution of cations (the degree of inversion, γ) across the tetrahedral and octahedral sublattices.^24,69 For example, this phenomena has been well explored by mixing of Zn-ferrite (a normal spinel) with Co-ferrite (an inverse spinel), resulting in non-magnetic Zn²⁺ swapping for magnetic Fe³⁺ on octahedral sites and increasing the overall magnetization.^70,71 Similar effects may arise in compositionally complex spinel samples, potentially with greater sublattice disorder and/or expanded substitution limits for specific cations.

In 2019, Musicó et al.³⁵ reported on the magnetic properties and trends among a series of high entropy spinel ferrites (compositions AFe₂O₄) and chromates (compositions ACr₂O₄). All reported high entropy spinel ferrites were shown to be room temperature ferrimagnets and, like several other HEO magnetic phases, were reported to generally follow trends found in the majority of the simpler ternary systems involving groups of the constituent cations. For example, the high entropy system, including Zn and Mg (the two non-magnetic cations), had the largest magnetic moment and lowest Néel temperature (T_N) relative to other high entropy spinels. On the other hand, several other HEO ferrites [(Mg_0.2Fe_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄ and (Mg_0.2Mn_0.2Co_0.2Ni_0.2Cu_0.2)Fe₂O₄] were noted to show peculiar magnetic behavior. For example, the low temperature field warming magnetization dips below the field cooling magnetization and for intermediate temperatures indicates an increase in magnetization as temperature is raised. A possible explanation for this peculiar thermal hysteresis is the antiferromagnetic coupling between neighboring elemental components in the cation sublattice. This leads to a key question of whether the reported high entropy spinels contain cation short-range order and small locally ordered magnetic clusters or chemically homogeneous distributions with more dilute clusters, and whether the substructures can be tuned.

The extent to which high entropy spinels will exhibit cation distributions different from bulk ternary spinels that influence their properties remains to be determined. In this work, we explore the influence of local cation ordering upon magnetic properties in a family of AFe₂O₄ high entropy spinels using magnetometry, neutron powder diffraction (NPD), neutron atomic pair distribution function (PDF) studies, and reverse Monte Carlo (RMC) modeling of spin correlations. We draw several connections between observed magnetic properties, the average and local atomic and magnetic structures, and the influence of material processing history in the series.

In this work, the four phase-pure spinel ferrites originally reported by Musicó et al.³⁵ were synthesized by solid state reaction using identical starting materials and processing conditions. We have elected to use the same naming convention as in the original work, with samples labeled F1, F3, F4, and F5, as noted in Table II. Briefly, all the samples were prepared in 5-g batches using mixtures of stoichiometric proportions of dried precursor oxide powders (MgO, NiO, CoO, CuO, ZnO, MnO₂, and Fe₂O₃). The mixtures were ball milled, pressed into pellets, and heat treated according to the details in Table II. Mixtures for samples F1 and F3 were initially heated at 1250 °C for 10 h. Samples F4 and F5 (both containing Mn) were obtained by first heating the mixed powder at 950 °C in order to convert the MnO₂ precursor to Mn₃O₄. After the first heating step, all samples were ground, repressed, and sintered a second time at 1250 °C for 10 h. The heat treatment protocols are the same used for the compositions reported in Musicó et al.,³⁵ though the heat treatment temperatures and times were not included in the original work.

Magnetometry measurements were performed using a vibrating sample magnetometer (VSM) as part of a Quantum Design PPMS DynaCool system. Temperature-dependent magnetometry (M vs T) was performed using field cooled (FC) and zero field cooling (ZFC) methods. For each sample, a ZFC measurement was first performed by cooling the powder sample to 2 K in the absence of a magnetic field, then applying a 10 mT field and measuring the magnetic moment as the temperature was increased from 2 to 400 K. After the ZFC measurement, the applied field was kept at 10 mT while measuring the magnetization as the temperature was reduced from 400 to 2 K for the FC curve. Magnetic hysteresis (M_H) loops between ±2 T were also measured at temperatures ranging from 2 K and in 25 K steps up to 400 K. All measurements have been normalized to the sample mass.

First order reversal curve (FORC) measurements were performed to evaluate the distribution of magnetic phases and interactions within the samples.^72–74 Following previously published procedures, FORC measurements were performed by first positively saturating the sample, then reducing the magnetic field to a reversal field (H_R). Next, the magnetization is measured as the applied field (H) is increased from H_R back to positive saturation. This sequence was repeated at H_R between the positive and negative saturation. The FORC distribution (ρ) is calculated by applying a mixed second order derivative to the dataset,

where M_S is the saturation magnetization of the sample. FORC measurements were completed at 2 and 300 K for each sample. While H and H_R separately probe up and down-switching events, respectively, FORC distributions are commonly presented with a change of coordinates from (H, H_R) to

where H_C and H_B represent the local coercive and bias (interaction) fields, respectively. As all samples show features prominent along axes in both coordinate systems, FORC distributions are presented in (H, H_R) while also indicating the (H_C, H_B) axes. The FORC distributions also include significant reversible components, which are captured using a constant extension mechanism.⁷⁵ 

Time of flight (TOF) neutron powder diffraction (NPD) data were collected at 300 K on the Nanoscale Ordered Materials Diffractometer (NOMAD) at the Spallation Neutron Source (SNS) at Oak Ridge National Laboratory (ORNL).⁷⁶ For each sample, ∼0.3 g of powder was loaded into a 3 mm diameter quartz capillary, placed in a temperature controlled Ar cryostream, and data were collected for 1 h. Data reduction was completed with the Advanced Diffraction Environment (ADDIE) suite.⁷⁷ The instrument background and sample container contributions were subtracted, data were corrected for detector deadtime, absorption, and multiple scattering effects, and were normalized by the incident flux and total sample scattering cross section. The maximum value of Q vector used for generation of the PDF was 40 Å⁻¹. The coherent neutron scattering lengths vary considerably among the cations in the sample series, with (in order from smallest to largest) b_Mn = −3.73 fm, b_Co = 2.49 fm, b_Cr = 3.635 fm, b_Mg = 5.375 fm, b_Zn = 5.68 fm, b_Cu = 7.718 fm, b_Fe = 9.45 fm, and b_Ni = 10.3 fm. The neutron scattering length of oxygen is b_O = 5.803 fm.

The experimental diffraction data (31° 2θ detector bank) were analyzed with the Rietveld method as embodied in the TOPAS-V7⁷⁸ program. Analysis was completed for each dataset fixing the cation site occupancy according to a normal ([A]_tet[Fe₂]_octO₄), an inverse ([Fe]_tet[A,Fe]_octO₄) [shown in Fig. 1(a)], and a random ([A_1/3Fe_2/3]_tet[A_2/3Fe_4/3]_octO₄) spinel structure model (F$d3̄m$⁠). In all fits, the magnetic scattering was modelled using a magnetic structure model with an I4₁/am′d′ structure featuring one constrained magnetic moment for all tetrahedral sites and one constrained magnetic moment oriented in opposite direction for all octahedral sites [shown in Fig. 1(c)]. Background parameters, scale factors, cubic lattice parameters a_nuc, isotropic atomic displacement parameters (ADPs) (one for oxygen atoms, one for all cations on the tetrahedral site, and one for all atoms on the octahedral site), instrument profile parameters, and the oxygen atom position (u) were refined for the nuclear structure models. For the refinement of magnetic structure, the isotropic ADPs were constrained to be equivalent to those in the nuclear structure, while the lattice parameters a_mag = b_mag were constrained to equal $2anuc/2$ and the lattice parameter c_mag was constrained to equal a_nuc [according to geometry, Fig. 1(b)]. The scale factor of the magnetic phase was constrained to one-half that of the nuclear phase (since V_mag/V_nuc = 1/2). Magnetic moments were refined along the z direction for the magnetic elements sitting on the respective tetrahedral and octahedral sites (constrained as one unique value per lattice site).

PDF refinements and model simulations were completed using normal, spinel inverse spinel and the random solution models in the PDFgui⁷⁹ program, and included a scale factor, lattice parameters, quadratic peak sharpening parameters for correlated atomic motion, and isotropic ADPs. Instrument parameters for real-space dampening and broadening were fixed to values determined through refinement of a silicon crystalline standard reference material.

RMC modeling involving both total scattering and Bragg data was conducted using the RMCProfile package.⁸⁰ Fits for both neutron F(Q) and neutron Bragg data were completed. F(Q) is defined by

where Q = k − k′ is the scattering vector of length 4π sin θ/λ for a neutron of wavelength λ scattered at an angle 2θ, and k and k′ are the initial and final wavevectors of the scattered neutron, respectively. S(Q) is the normalized total-scattering structure factor. $b̄i$ is the coherent bound neutron scattering length of species i, averaged over the different isotopes and nuclear spin states of i. c_i is the portion of species i in the material.⁸¹ A 10 × 10 × 10 supercell was first built, based on the unit cell model obtained via Rietveld refinement for each sample. The supercell structural configuration was then optimized in a data-driven manner following the Metropolis approach. Two sets of RMC modeling was completed for each sample, assuming either inverse or random site occupancy (the normal model was excluded because of the poor agreement demonstrated by Rietveld refinements). No atomic position swapping was allowed between tetrahedral and octahedral sites. In the original report by Musicó et al., multiple valence states for Fe and Mn were detected among high entropy spinels via x-ray absorption spectroscopy. Thus, RMC trials were performed for several samples assuming various valence states for Fe and Mn atoms (since valence determines the magnetic form factors assigned). Here, it should be pointed out that the magnetic moment vectors in RMC configurations are unit vectors with magnetic moment magnitude specified (fixed) according to each cation involved. The bond valence sum (BVS) constraint in RMC fitting is implemented in such a way that it effectively constrains the net valence of the whole system to be 0. The findings discussed in this report were found to be independent of the valence settings in RMCProfile. The final configurations were initiated assuming a +2 valence state for all A cations in the AFe₂O₄ compositional series and a +3 valence state for the remaining two Fe cations.

Neutron total scattering structure factor data, F(Q), and Bragg data were used for RMC modeling, since the calculation for magnetic diffuse scattering is performed only in Q-space in RMCProfile. For the inclusion of Bragg data in RMC modeling, peak profiles and background were extracted from Topas refinement and tabulated for RMCProfile to read in—hence the RMC engine only calculates peak intensities. In addition to the experimental data, a minimum distance for each single pair of atoms and an overall bond valence sum were added as model constraints. Given multiple constraints exerted upon the structural model, an automatic weight assigning scheme⁸² was used.

A kernel density estimation (KDE) and the Henze–Zirkler (HZ) statistic were utilized to interpret variation in the distribution of atom sites and magnetic spin correlations resulting from RMC model fits to neutron total scattering data. KDE is a commonly used method for estimating the underlying probability distribution function in a collection of data points with a non-parameterized approach. Here, the standard normal function is used as the kernel and, in practice, that means one would have a standard normal distribution function with the center at each of the data points. The overall distribution function is then composed of all those kernels, while being normalized to give the overall distribution probability of 1. The HZ parameter^83–86 is a parameter characterizing the normality of a distribution and the closer its value gets to 1, the closer that underlying distribution is to a Gaussian distribution. The following equation defines the HZ parameter:

where D_i is the squared Mahalanobis distance of the ith observation to the centroid and D_ij gives the squared Mahalanobis distance between ith and jth observations

β is given by

which includes the particle sample size n_p and sample dimensions d.

The cation distribution in high entropy spinels will be impacted by a number of thermodynamic and crystal-chemical factors. The CFSE of the oxygen ligand field and the S_config resulting from the number of participating cations are two important factors. The electron configuration energy difference between ligand field and isotropic field is defined as CFSE,

The CFSE can be straightforwardly calculated for each participating cation assuming a specific charge state and either an octahedral or tetrahedral geometry, and the net CFSE can be compared for specific distributions of cations to evaluate potential site preferences.

The CFSE for the four compositions in the AFe₂O₄ series are shown in Fig. 2(a), assuming a +2 valence state for all five cations on the A site, and a +3 valence state for the remaining two Fe cations. Regardless of the oxidation state used in calculations, the CFSE of the inverse structure model is the lowest among the model spinel structures while the CFSE of the normal structure model is the highest.

The entropy contribution made by the compositional mixture on the tetrahedral and octahedral lattice sites will differ for normal, inverse, and random spinels. The ideal S_config per cation for HEOs with multiple cations occupying different sublattice sites can be calculated⁶⁸ by

where R is the universal gas constant, a^x is the number of sites on the x sublattice, $fix$ is the fraction of elemental species occupied on the respective sublattice, and N is the number of elements in a given sublattice. For example, for the normal spinel AFe₂O₄, Eq. (9) can be evaluated as

As shown in Fig. 2(b) the random spinel offers the highest entropy contribution, regardless of the number N of cations on the A sublattice site of AFe₂O₄ (from 1 to 8, with Fe not included), followed by the inverse and normal spinel configurations. The S_config calculations for N from 1 to 8 with Fe from the B site included are also completed in Fig. S1 in the supplementary material. Since F3 and F4 do not have Fe on the A site their S_configs are larger than those of F1 and F5 when random and inverse spinel models are considered, respectively. It is worth noting that the S_configs for the compositions in the series are all smaller than 1.5R, meaning the spinels do not meet the classic definition of a high entropy oxide (or even medium entropy oxide). In thermodynamic terms, these compositions are more accurately classified as compositionally complex oxides (CCOs).

To summarize, the CFSE and S_config terms for AFe₂O₄ spinels provide competing drivers for cation site selectivity, favoring inverse vs random cation configurations, respectively. Observed cation distributions could very well be influenced by the sample-specific processing procedures.

Results of temperature- and field-dependent magnetometry for the samples in our study are shown in Figs. 3 and 4(a), respectively. For all of the samples, the black dashed ZFC curve shows a continuous increase with increasing temperature, indicating the enhanced role of thermal fluctuations by reorienting the magnetization with the magnetic field. Also, the ZFC curve is well separated from the FC curve, with no apparent inflection, and the FC curve is smooth and persistently non-zero, indicating the system does not have a blocking temperature and is not superparamagnetic in this temperature range. This is consistent with the previous work by Musicó et al.,³⁵ which identified the Néel temperature (T_N) of >400 K; the Néel temperature of other spinel ferrites includes 858 K for Fe₃O₄ and 725 K for CoFe₂O₄. These conclusions are also consistent with the M_H loops, shown in Fig. 4(a), which show that every sample has a very small, but non-zero, coercivity (≈2–5 mT) at 300 K.

The field-dependent magnetometry measurements of F1, F3, F4, and F5 samples in this study are shown in Fig. 4(a), the measurements completed by Musicó et al.³⁵ are reproduced in Fig. 4(b) for comparison. This comparison shows significant differences in the M_H loops, with the samples in this work having a nearly constant (compositionally independent) magnetic saturation (M_S) and an exceedingly small coercivity in all of the samples. By comparison, the samples from Musicó et al.³⁵ show a variable M_S between the different samples, with F3 being particularly large, while the others are similar to the current work. Also the coercivity in Ref. 35 is generally larger than in the current work, especially for F5. The two batches of samples were synthesized in the same laboratory, by the same researcher, using the same batch size, grinding and sintering procedure, and precursors from the same supplier. The only known change in sample preparation involved a replaced heating element for the furnace used. Additionally, the magnetometry measurements of samples prepared for Musicó et al.³⁵ were completed soon after synthesis, whereas and magnetometry measurements of samples prepared in the present work was completed two years after synthesis. The observed difference in magnetic behavior among samples of the same composition suggests a potential high sensitivity to synthesis and processing conditions, and/or ageing, in high entropy spinels, and deserves further investigation. For consistency, it is important to note that the M_H loops in this work were field cooled in 10 mT, while Ref. 35 uses 100 mT. This difference would manifest if the system has an antiferromagnetic phase coupled to the ferromagnetic phase or high anisotropy phase with a saturation field larger than the 2 T applied here. In these cases, the hysteresis loop would be vertically and horizontally shifted in Fig. 4 due to the exchange bias effect. The absence of these features suggests that the cooling field is not responsible for the observed differences.

The differences in the M_H plots maybe the result of a strong sensitivity to local compositional distributions and microstructure. This sensitivity would manifest in the magnetic reversal behavior of the sample, which can be probed using the FORC technique. The calculated FORC distributions are shown in Fig. 5 at 2 K [(a), (c), (e), and (g)] and 300 K [(b), (d), (f), and (h)] for all samples.

The FORC distributions all show their primary feature aligned along the H_B axis with a small feature in the H_C direction that is more prominent at low temperatures. A small bump in the H_R direction is also seen at 2 K but is absent at room temperature. While these features are present for all of the samples, they vary in intensity and prominence, with the attributes being lowest for sample F3.

The FORC distribution encodes a map of the switching events within a particular hysteretic system. The bias–field axis identifies the shift from H = 0 of a particular set of switching events. For a system without exchange bias, this shift can be attributed to interactions within the sample. The FORC distribution shown here has a wide spread in H_B, indicating strong demagnetizing interactions, e.g., interactions that drive the system away from a saturated configuration.^74,87 These interactions can be the result of dipolar coupling or exchange-motivated frustration. The presence of long-range ferrimagnetic magnetic ordering without a blocking temperature, and with only a very small coercivity, suggests the interactions are likely due to frustration. At low temperatures, the protrusion along the H_C axis suggests the intrinsic anisotropy has increased compared to the room temperature measurements. The protrusion along the H/H_R axes indicate some of the sample reverses by a domain nucleation/propagation mechanism.⁸⁸ The appearance of the domain reversal mechanism indicates that large areas of the sample have become magnetically coupled and can support domain growth through their boundaries. Complementary to this, the higher temperature samples may consist of magnetic islands or regions that are coupled by non-magnetic or antiferromagnetic phases.

The neutron diffraction patterns of all four samples appear visually similar [Fig. 6(a)] and can be indexed to a cubic spinel $Fd3̄m$ structure model with corresponding I4₁/am′d′ ferrimagnetic ordering model, with no detected secondary phases. Distinguishing cation order is challenging with Bragg diffraction data. The scattering length of involved elements, the degree of cation inversion, and the magnetic order on the cation sublattice may all contribute to observed variation in peak intensities. As a demonstration, the calculated neutron diffraction patterns for nuclear and magnetic scattering assuming inverse, random, and normal spinel structure models with assumed ferrimagnetic spin order for each sample are shown in Fig. S2 in the supplementary material. It can be seen in Fig. 6(a) that the nuclear Bragg peak intensities are higher for F1 and F3 samples relative to F4 and F5 samples. The intensities of dominant magnetic peaks [for example, the (331) and (111) reflections] appear similar for all compositions.

Separate Rietveld refinements assuming normal, random, and inverse spinel models were completed for each sample (supplementary material, Table S1 for F1, Table S2 for F3, Table S3 for F4, Table S4 for F5), each incorporating the ferrimagnetic structure shown in Fig. 1(c). A summary of Rietveld refinement results is presented in Table III. The Rietveld refinement results reveal distinct preference in cation site ordering. For all samples, the normal spinel structure model resulted in poor agreement and nonsensical magnetic moments. The combination of inverse spinel and ferrimagnetic model fits the data better than those of random and normal spinel models for F1 and F3 samples [the F1 result is shown in Fig. 6(b)], while the combination of random solution spinel and ferrimagnetic model fits the data better than normal and inverse models for F4 and F5 samples [the F4 result is shown in Fig. 6(c)]. The ferrimagnetic scattering contributes 100% to the hkl peak (331), a large portion to the peaks (531), (222), (111), and some percentage to the (422), (400), and (220) peaks. The intensity of peaks with combination contributions from nuclear and magnetic structures are not precisely fit with either random or inverse structure; the average structures may lie somewhere in between. Since the nuclear Bragg peaks are overlapped with the magnetic Bragg peaks and there are a high number of participating cations, Bragg diffraction data alone is not enough to determine the detailed structure and cation site preferences of the high entropy spinels.

A comparison of the neutron PDFs for the high entropy spinels is provided in Fig. 7(a), displayed between 1.7 and 10 Å. All pair correlations have been numbered and identified by category with a color key to aid discussion. First, it is immediately apparent that the local atomic configurations of the high entropy spinels in the present study are similar; in particular, the location and peak shapes of nearest-neighbor and next-nearest neighbor pair correlations are coincident across the four compositions. This immediately rules out significant local distortion of specific A site cations. Next, there are specific sets of peaks in the data that vary systematically from sample to sample at low r: the peaks highlighted by magenta (peak 1, 5, 9, 12, 15, 17), green (peak 2, 6, 10, 13), red (peak 4), and pink (peak 8) dashed lines representing A–O, A_oct–A_oct/O–O, O–O, and A_oct–A_oct/A_tet–A_tet/A–O correlations, respectively, are matched in amplitude; the peaks highlighted by sky blue (peak 3, 11, 16) and orange (peak 7, 14), representing A_oct–A_tet/A_tet–A_tet/A–O and A_oct–A_tet/A_tet–O correlations, respectively, differ in amplitude. Inspection reveals that samples F1 and F3 exhibit similar and higher amplitudes of specific correlations relative to samples F4 and F5. PDF data comparisons for every 10 Å across the whole range (1.7–50 Å) are shown in supplementary material, Fig. S3.

Neutron PDF refinements in Figs. 7(b) and 7(c) show the effect of changing the cation ordering model (inverse, normal, random configuration models) for F1 and F4 samples, respectively. Fits for F3 and F5 are shown in the supplementary material, Fig. S4, and provide similar results to F1 and F4, respectively. It is highly likely the unfit residual is in part due to magnetic spin correlations that are unaccounted for in the models.^89,90 At present, there are no PDF data refinement programs that allow for shared site occupancy and magnetic PDF refinement; thus, we have elected to present the nuclear structure fits alone here and address magnetic spin correlations separately. The 15th pair amplitude (an A–O correlation at ∼8.75 Å) is shown to be sensitive to the degree of inversion in spinel ferrite models. Overall, for F1 and F3, the random model and inverse model provide a very similar fit quality and cannot be easily distinguished with PDF refinement. For sample F4 and F5, the fit quality for a random model is superior to that for an inverse model. These results correlate well to conclusions drawn from Rietveld refinement. However, no further conclusion can be made from PDF refinement due to the high numbers of cations present per site and the limitation of existing software for exploring local nuclear and magnetic cation ordering together.

As noted above, quantitative real-space refinement of the series of data is complicated by the presence of magnetic spin correlations and the large number of cations involved on the spinel tetrahedral and octahedral sublattices. Large box modeling of PDF data with nuclear and magnetic spin correlations was pursued to provide additional insight into structure–property characteristics in the series. Combined fits of F(Q) (supplementary material, Figs. S5 and S6) and Bragg (supplementary material, Figs. S7 and S8) intensities were completed for all samples assuming an inverse spinel model and a random solution spinel model (trials with a normal spinel model were not tried as they were ruled out conclusively by Rietveld analysis and small box PDF modeling). The agreement of fits to experimental data is high for both model types for all four samples. Table S5 (for F1 and F3) and Table S6 (for F4 and F5) in the supplementary material list the resulting anisotropic atomic displacement parameters (ADPs) on cation tetrahedral sites, cation octahedral sites, and oxygen sites of each RMC refinement for F1, F3, F4, and F5. For F4 and F5 samples, resulting ADP values for the tetrahedral sites and for oxygen positions in the random solution spinel model fits are found to be nearly half those found for inverse spinel model fits. Such an anomaly, combined with Rietveld refinement results, indicate a strong preference for a random model over an inverse one for F4 and F5 samples. Thus, the following discussion is based on analysis of the model configurations obtained from an inverse spinel model for samples F1 and F3 and a random solution spinel model for samples F4 and F5.

With the magnetic spin configuration obtained via RMC modeling for all samples, all magnetic moment vectors contained in the supercell were mapped into a single unit cell and then projected onto the XY plane. Results are shown visually in the supplementary material Figs. S9 and S10, and the magnetic spin distribution of F1 and F3 samples is more dispersed on both tetrahedral and octahedral sites than F4 and F5. In all samples, the magnetic spin distributions of octahedral sites are more disperse than those found on tetrahedral sites. Overall, less collinear spin order and stronger displacements from atom sites are observed for F1 and F3 samples. Two types of statistical analyses were completed to quantify the findings. First, a KDE analysis was conducted, capturing the proportion of magnetic moments found within a specified radius (with the variation ranging from 0 to 1). In principle, the faster such a proportion approaches 1, the stronger the collinearity in magnetic ordering. The results are presented in Fig. 8(a) for the tetrahedral site magnetic vector distribution and (b) for the octahedral site magnetic vector distribution, indicating a significantly more collinear spin arrangement in F4 and F5 samples than in F1 and F3 samples. This result is consistent with the smaller net magnetic moment in F1 and F3 samples—see Table S7 in the supplementary material. Additionally, in all samples, tetrahedral site spin correlations are found more collinear than octahedral site spin correlations.

Second, the HZ statistic was calculated to quantify the non-Gaussian behavior of the oxygen coordinate distributions and magnetic moment vectors within the XY plane. Different starting cation distribution configurations are found to influence the HZ statistic values for F1 and F3 samples; however, values determined for F1 and F3 are generally larger than those found for F4 and F5. Overall, in Figs. 9(a) and 9(c), it can be observed that F1 and F3 samples exhibit a more non-Gaussian behavior in oxygen atom distributions and in magnetic moment vector distributions on the octahedral site than F4 and F5 samples. (HZ parameters for F4 and F5 samples are much closer to 1, i.e., no obvious non-Gaussian distortion could be observed.) Meanwhile, if focusing on magnetic moment vector distributions on the tetrahedral site [Fig. 9(b)], all four samples show HZ parameter values close to 1, indicating a nearly Gaussian distribution. These comprehensive results here suggest strong coupling between non-collinear magnetic correlations originating primarily from octahedral site spins and oxygen site displacements, especially in F1 and F3 samples. Variation in the level of spin–lattice coupling among high entropy spinels of similar composition demonstrates a potentially tunable interaction in the more extended family.

We have determined the average ferrimagnetic structure in a family of AFe₂O₄ high entropy spinels with neutron Bragg diffraction analysis and found evidence for a variation in the tetrahedral and octahedral site occupancies of A site cations with local structure analysis of neutron total scattering data. These high entropy spinels are not found to have strictly random cation configuration structures (highest S_config), but rather, F1/F3 samples in this study were found to have configurations closer to an inverse spinel. The distinct populations of Fe and other magnetic cations on tetrahedral and octahedral sites in the samples manifests in variation in the prevalence and propagation of local spin clusters, as revealed by FORC analysis of magnetometry data. RMC modeling reveals spin non-collinearity in all four samples, enhanced in cases presenting a larger degree of cation inversion.

While identifying the cause for the variation in cation disorder among similar compositions falls outside the scope of the present study, we bring attention to a potential competition between crystal field stabilization effects (which promotes inverse spinel ferrite configurations for the compositions studied) and the entropy stabilization present in truly random spinel ferrite configurations. How much the relative effects of these phenomena are influenced by sample synthesis conditions or other factors, potentially creating samples in one of many different metastable states, remains unknown. Overall, this work highlights that it may prove extremely challenging to achieve a true solid solution with ideal/homogeneous cation mixing in compositionally complex ceramics with complex energy landscapes. We suggest great care be taken in this community to precisely describe reaction conditions, etc. when reporting HEO properties and drawing conclusions regarding the effects of configurational entropy on resultant products. Additionally, experimental characterization of the local to long-range crystal–chemical structure in HEOs may be key to determining and understanding their physical properties. The limitations of the characterization methods explored herein indicate future efforts are needed toward combined experimental probes and advanced data modeling. Hard and soft x-ray absorption near edge spectroscopy (XANES), soft x-ray magnetic circular dichroism (XMCD), and ⁵⁷Fe Mössbaruer spectroscopy (with and without external magnetic field) are examples of techniques that may provide additional insights.

See the supplementary material for additional configurational entropy calculations, average structure results from Rietveld refinement, and local structure results from small-box PDF refinement and large box RMC modeling.

K.P. acknowledges financial support from the National Science Foundation (Grant No. DMR-2145174). This research used beamline 1B (NOMAD) of the Spallation Neutron Source, a DOE Office of Science User Facility operated by the Oak Ridge National Laboratory. We acknowledge Dr. Jue Liu for helpful discussion regarding Rietveld refinements and Namila Liyanage, Nan Tang, and Tianyu Li for helpful discussion regarding magnetometry results. D.A.G. and C.K. were supported by the U.S. Department of Energy (Grant No. DE-SC0021344) Partial funding for open access to this research was provided by University of Tennessee′s Open Publishing Support Fund.

The authors have no conflicts to disclose.

Xin Wang: Conceptualization (equal); Formal analysis (equal); Investigation (lead); Methodology (equal); Writing – original draft (lead); Writing – review & editing (equal). Brianna Musicó: Conceptualization (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Corisa Kons: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Peter Metz: Formal analysis (equal); Methodology (equal); Writing – review & editing (equal). Veerle Keppens: Conceptualization (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Dustin A. Gilbert: Conceptualization (equal); Funding acquisition (equal); Methodology (equal); Supervision (equal); Writing – review & editing (equal). Yuanpeng Zhang: Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Katharine Page: Conceptualization (lead); Funding acquisition (equal); Investigation (equal); Methodology (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.