Anisotropy-induced spin disorder in intergrown, ferrimagnetic Fe₇S₈ polytypes

The interplay between structure and magnetic properties in solids is of special importance for the identification, comprehension, and development of possible functionality of materials for scientific and technological applications. In this context, effects emerging due to vacancies that cause changes in crystallographic symmetry and/or magnetic order are of fundamental interest.^1–3 Earth-abundant iron monosulfides of the pyrrhotite group (⁠$Fe1−xS$⁠, $0<x≤0.125$⁠) are an ideal crystallographic testing ground to explore vacancy-induced magnetism, because vacancies can be arranged in different configurations that generate commensurate and incommensurate phases with variable physical and chemical properties.⁴ The iron-deficient Fe₇S₈ endmember occurs in two ferrimagnetic polytypes with threefold and fourfold stacking modulation of full and vacancy layers that stem from the distribution of vacant octahedral sites in a hexagonally closed packing of sulfur and are denoted 3C and 4C pyrrhotite, respectively.^5,6 The mineralogical and physical properties of the latter have been extensively studied because it is a key magnetic remanence carrier in the Earth's crust and extraterrestrial materials for paleomagnetic reconstructions.^7–10 In contrast, 3C pyrrhotite has rarely been reported and it is mainly known from recent gas-hydrate bearing sedimentary deposits.¹¹ Both polytypes have also been synthesized.^6,12–14 In recent years, Fe₇S₈ syntheses for technical applications gained increasing interest and it has been shown that the magnetic properties are sensitive to crystallite size and vacancy order.^12,15–18 Furthermore, superstructure changes within the pyrrhotite group are controlled by vacancy diffusion.^12,19,20 Both polytypes transform into a paramagnetic 1C phase with randomly distributed vacancies at a temperature of approximately 600 K.^4,13 The 1C to 4C transition is reversible, but the relaxation kinetics to reach perfect vacancy ordering occurs over geological time frames.²¹ Moreover, the absence of the natural 3C pyrrhotite in the pre-Holocene (>11 700 years) is probably due to its metastability with respect to 4C pyrrhotite.

The two Fe₇S₈ polytypes derive from a hexagonal NiAs-type substructure and their vacancy order generates two different superstructures (Fig. 1). In the higher-ordered 4C polytype with monoclinic unit cell (space group $C2/c$⁠), the 32 cations occupy the two sites Fe1 and Fe3 in the full layers and Fe2 and Fe4 in the vacancy layers, where only half of the Fe4 sites is filled.^5,22–24 Doubling the unit cell along the c-axis generates a centrosymmetric superstructure with a fourfold modulated stacking sequence. In contrast, the distorted trigonal 3C polytype (space group $P3121$⁠) is non-centrosymmetric with the three sites Fe1, Fe2, and Fe3 in the full layers and Fe4, Fe5, and Fe6 in the vacancy layers [Fig. 1(b)]. The latter two sites are vacant with a statistical probability inferred at about 15% for Fe5 and 85% for Fe6,¹⁴ i.e., two vacancy sites can be next neighbors with a probability of about 12% [Fig. 1(b)].

In the 3C and 4C polytypes, the ferrimagnetism stems from uncompensated sublattices within the stacking sequence, where intra-layer spins are ferromagnetically (FM) and adjacent inter-layer spins are antiferromagnetically (AFM) coupled.^4,25,26 For the 4C pyrrhotite, a theoretical value for the saturation magnetization of $Ms≈21 A m2 kg−1$ was predicted.²⁵ Moreover, this polytype has a characteristic anisotropy with an easy c-plane and a hard c-axis.²⁷ Based on theoretical considerations of the NiAs-type structure by Goodenough,²⁸ the spin–orbit coupling (SOC) due to covalency of Fe–Fe bonds of less than 3 Å can explain the peculiar anisotropy traits. All interlayer Fe–Fe bonds are below this critical value and a magnetic anisotropy energy of $EMA=5.49 meV$ per unit cell has been reported.^22,29 This results in a spin rotation out of the easy plane and in an intrinsic magnetic anomaly at about 32 K, known as Besnus transition.^23,30–32 The spin rotation is incoherent and it can be linked to vacancy order, which generates two anisotropy systems offset by 30°.^33–35 Koulialias et al.³⁴ demonstrated that the coupling between the two systems marks the Besnus transition. Moreover, Haines et al.²⁶ predicted a spin-flop transition for the lesser-ordered 3C polymorph.

In this study, we generate a structure of intergrown 4C and 3C polytypes by thermal treatment of a natural pyrrhotite and the link between structural and magnetic traits is used to provide insights into the vacancy ordering of a defect-supported magnetic material.

The initial material is a well-described natural pyrrhotite crystal from Auerbach (Germany) that consists of 4C and 5C^* pyrrhotite phases with a chemical bulk composition of Fe_6.6S₈.^36,37 The thermal conversion was conducted in a Mettler Toledo DSC 1 by heating crystal flakes of $17–19 mg$ to 650 K at 10 K/min followed by cooling to room temperature at the same rate. Powder x-ray diffraction (XRD) with CuK_α radiation was performed at room temperature to identify the phases in the sample before and after the thermal treatment. The TOPAS software was used for the Rietveld refinement of the XRD data.^38,39 Detailed analysis and visualization of the structural data were made with the Crystal Maker software. The behavior of the magnetization in a 3 T field (⁠$M3T$⁠) of both samples between 300 and 5 K was investigated with a Quantum Design Magnetic Property Measurement System (MPMS3). The samples were oriented along the easy c-plane of the 4C pyrrhotite. The complex susceptibility $χ*ω=χ′ω−iχ″ω$ was measured in AC fields with 800 A/m superimposed by a 0.1 T DC bias field at frequencies $f$ between 5 and 200 Hz.

Atomic-scale characterization of the starting, intergrown 4C and 5C^* superstructures,^36,37 revealed an incommensurate 4.91C pyrrhotite, which cannot be chemically differentiated from the intergrown 4C pyrrhotite.³⁷ Rietveld refinement of the 4C phase using the structural model of Powell et al.¹³ yields for the untreated sample the unit-cell parameters $a=11.9134±0.0003 Å$⁠, $b=6.8701±0.0002 Å$⁠, $c=12.9074±0.0004 Å$⁠, and $β=118.047±0.001°$ with a Bragg factor of $RBragg=5.5$⁠. Using the model of Elliot,⁴⁰ the refined monoclinic 5C^* phase (P21/c) has a unit cell with $a=6.8910±0.0004 Å$⁠, $b=28.649±0.0009 Å$⁠, $c=6.8852±0.0003 Å$⁠, and $β=119.9309°±0.004°$ with $RBragg=5.6$⁠. The Rietveld refinement of the diffractogram has a goodness of fit (GOF) of 3.41.

After heating the sample to 650 K, the XRD pattern at room temperature exhibits no 5C^* peaks (Fig. 2). This accords with thermomagnetic data by Koulialias⁴¹ that reveal at $T≈495 K$ an irreversible 5C^* phase transition into a ferrimagnetic phase. In the thermally treated sample, the Bragg peaks and the relatively pronounced background at $2θ<60°$ can be attributed to 4C and newly formed 3C pyrrhotite (Fig. 2). The 4C unit-cell parameters using the same structural model as above are $a=11.9118±0.0004 Å$⁠, $b=6.8723±0.0002 Å$⁠, $c=12.9080±0.0003 Å$⁠, and $β=117.983±0.002°$ with $RBragg=2.2$⁠. These parameters exhibit no significant changes compared to the initial sample. The newly formed 3C pyrrhotite was refined with the model by Keller-Besrest et al.¹⁴ in the space group $P3121$ and yielded unit-cell parameters of $a=6.8848±0.0003 Å$ and $c=17.1472±0.0011 Å$ with $RBragg=2.1$⁠. The background modulations with broad peaks generally stem from poorly ordered crystallites that are small in size and/or reveal a lattice distortion. Taking the structural parameters for 3C pyrrhotite into account when performing the Rietveld refinement, the modulation can be well described by the Keller-Besrest model.¹⁴ However, a better agreement between the measured and refined background, as indicated by the decrease in the GOF from 6.2 to 2.3, is obtained by the model of Nakano et al.,⁶ which rests upon the same space group but describes a slightly different vacancy distribution in the 3C pyrrhotite (Fig. 2). The refinement reveals a 3C polytype with a coherence length of a few nanometers, i.e., of one to three unit cells. To distinguish this perturbed, small-sized phase from the long-range ordered 3C pyrrhotite, it is denoted 3C^* in analogy to the identification of the incommensurate 5C^* pyrrhotite.^36,37 The GOF of 2.3 obtained from the refinement of the XRD pattern shows that the thermally treated sample is well described as an intergrowth of 4C, 3C, and 3C^* polytypes. The concomitant occurrence of the three polytypes indicates the formation of cation vacancy configurations that locally minimize the free energy in the quenched material. In such a process, areas with minimized free energy are separated by boundaries with higher free energy and the small-sized, disordered 3C^* constituent appears to be a likely candidate to form this kind of boundary.

At 300 K, the initial sample has a magnetization in a 3 T field of $M3T=16.8 A m2 kg−1$ [Fig. 3(a)]. Neglecting the magnetic contribution of the antiferromagnetic 5C^* phase and considering the theoretical value of $Ms≈21 A m2 kg−1$ for 4C pyrrhotite,²⁵ the latter is the dominant phase with about 75%. After thermal treatment, the new composition with 4C, 3C, and 3C^* pyrrhotite increased $M3T$ to $18.4 A m2 kg−1$⁠. The 4C and 3C pyrrhotite are both ferrimagnetic, whereas in the 3C^* constituent with an average coherence length of a few nanometers, the ferrimagnetic order is perturbed due to structural disorder. Given this, $M3T$ stems mainly from 4C and 3C pyrrhotite.

For the initial sample, the temperature dependence of $M3T$ between 300 and 50 K can be described by a second-order polynomial with a maximum at about 165 K [Fig. 3(a)]. At lower temperature, the local maximum at $T≈25 K$ is attributed to the Besnus transition.³⁶ This intrinsic low-temperature behavior was explained by the interplay of the Zeeman energy $EZ$ and the magnetocrystalline anisotropy energy $EMA.$³⁴ The maximum in $M3T$ marks the beginning of a pronounced out-of-plane spin rotation where the change in the anisotropy energy $ΔEMA$ exceeds $EZ$⁠. This coincides with the shortening of intra-layer Fe3–Fe3 bonds below the critical value of 3 Å and a change in sign of the anisotropy constant K₃.^13,30,33,34 The $M3T$ curve for the thermally treated sample can also be described by a second-order polynomial between 300 and 50 K. The quadratic terms of both curves are identical. Thus, a vertical and a horizontal shift by 16.5 K to higher temperature and lowering $M3T$ by $1.98 A m2 kg−1$ makes the polynomial curves congruent. The nearly linear difference of the measured curves in this temperature range suggests no spin‐flop for the 3C phase in a field of 3 T.

As mentioned above, the out-of-plane rotation in 4C pyrrhotite results from the interplay of the covalent Fe–Fe bonds and the ordered vacancy arrangements. The monoclinic superstructure consists of chain elements parallel to the c-axis with seven cations that are separated by a vacancy.^23,24 All the Fe–Fe bonds in these chains are covalent and contribute to the SOC. In 3C pyrrhotite, the interlayer Fe–Fe bonds are also covalent but the stochastic occupation of the Fe5 and Fe6 sites fragments the chain segments and generates distortions (Fig. 4). Both effects can influence the Fe–Fe bond lengths. This configuration of the covalent Fe–Fe bonds causes local SOC variations in the lattice, which in turn may explain the absence of long-range, collective out-of-plane spin rotations in 3C pyrrhotite as is characteristic of the 4C polytype.

At $T<50 K$⁠, the peak in $M3T$ indicates the Besnus transition in 4C pyrrhotite, where the two spin rotation systems start to couple.^34,35 Compared to the initial sample, the thermally treated one reveals a broadening of this peak [Fig. 3(a)]. Susceptibility measurements better resolve this effect [Fig. 3(b)]. In both samples, the peak can be fitted with a Lorentzian between 19 and 28 K. The peak broadening is associated with a slight shift from 24 to 23 K. Considering that such a peak occurs for pure 4C pyrrhotite at $T≈32 K,$^23,42 the shift to lower temperature can be explained by a local perturbation in the superstructure due to cation disorder on an atomic level.³⁷ With this in mind, the nearly identical peak positions of the initial and thermally treated samples indicate that the quenching has no critical effect on the vacancy order of the 4C superstructure.

The magnetic susceptibility of the thermally treated sample is nearly double that of the initial one [Fig. 3(b)]. Because Koulialias⁴¹ showed that the heat treatment only marginally enhances the magnetization in a field of 0.1 T, such a drastic increase cannot be attributed solely to the ferrimagnetic 3C or 4C pyrrhotite in the sample. Therefore, the susceptibility increase mainly stems from 3C^* pyrrhotite. In this constituent, the stochastic vacancy distribution and the perturbed stacking modulation make the FM and AFM interactions of the sublattices less constrained with respect to the magnetic orientation of their neighbors and the external field, and this in turn can explain the magnetic relaxation detected in the $χ*T,ω$ measurements. The larger values of $χ′$ indicate an increased spin mobility.

The enhanced susceptibility is linked to relaxation, i.e., to a spin–glass transition, at about 10 K [Figs. 5(a) and 5(b)]. The temperature dependence of the effective relaxation time is related to the peak's maximum frequency $ωmax$ by $τT=1/ωmax$⁠. The relaxation is caused by cooperative spin fluctuations that freeze when the relaxation time exceeds the experimental measurement time at low temperatures. The inflection point of $χ′$ shifts to higher temperatures as the excitation frequency is enhanced and this is well resolved in the $χ″$ peak. From the measured frequency dependence of the relaxation, an apparent activation energy of $Ea=23.24±1.27 meV$ (=2.24 kJ/mol) can be determined using the Arrhenius–Néel equation $τT=τ0expEa/kBT$⁠, where $τ$ is the relaxation time, $τ0$ is a pre-exponential factor, $Ea$ is the apparent activation energy, and $kB$ is the Boltzmann constant [Fig. 5(c)].

An average size of regions with cooperative spin fluctuations is inferred from $Ea$⁠. Such regions are critically determined by the configuration of covalent Fe–Fe bonds and vacancies that induce localized magnetic anisotropy properties. In 4C and 3C pyrrhotite, these bonds are similar (Fig. 1), and therefore, it is assumed that the SOC and magnetocrystalline anisotropy energy in the two polytypes are of the same magnitude. For 4C pyrrhotite, a magnetocrystalline anisotropy energy of $EMA=0.2 meV/Fe$ atom can be inferred.²⁹ Given this, $Ea≈23 meV$ corresponds to cooperative spin fluctuations of about 115 Fe atoms on average, which comprise about three unit cells of the 3C^* pyrrhotite (Fig. 4).

From $χ″$ and the Arrhenius–Néel equation, the relaxation spectrum is determined [Fig. 5(d)], which can be described by the Havriliak–Negami equation,^43,44 $χ*ω=χ∞+Δχ1+i ωτHNmn$⁠, where $χ∞$ is the limiting susceptibility, $Δχ$ is the intensity of the relaxation transition, $τHN$ is the Havriliak–Negami relaxation time, and $n$ and $m$ are shape parameters, which correspond to the slopes in Fig. 5(d). The low-frequency slope is $m$ and the high-frequency slope is $−m×n.$⁴⁵ The obtained slopes $m=0.05$ and $m×n=0.12$ are small compared to those of structural glasses,⁴⁶ which is indicative of a broad relaxation time distribution. Moreover, cooperative spin rearrangements during the glass transition are characterized by the dynamic fragility⁴⁷ $md=1Tg dlogτd T−1Tg=1ln(10) TgEakB$⁠. Considering the slope of the activation diagram at the glass transition temperature $Tg=7.1 K$⁠, which is defined as $τTg=100 s,$^48,49 the dynamic fragility is $md=17$⁠. This suggests a strong glass-forming system that can be well described by an Arrhenius–Néel-like activation behavior.⁴⁷ The broad shape of the relaxation spectrum and the low dynamic fragility indicate a low cooperativity of the relaxations in the glass-transition region, which is most likely caused by a spatial separation of spin fluctuations. Such regions of separated spin fluctuations can be attributed to the disordered 3 C^* constituent, randomly distributed in the pyrrhotite crystal.

In summary, thermally induced diffusion-controlled vacancy ordering in natural ferrimagnetic Fe₇S₈ generates 4C and 3C polytypes that are different in the stacking modulation of the cations in the full and vacancy layers. Highly ordered vacancies yield well-defined anisotropy systems in 4C and to a lesser degree in 3C pyrrhotite. The stochastic vacancy order in the 3C polytype can additionally generate small-sized structures, denoted 3C^*, which result in disordered, magnetic regions with spin‐glass behavior at about 10 K. The 4C, 3C, and 3C^* polytypes can be distinguished by dynamic magnetization measurements in concert with structural analysis, which opens a new avenue to characterize vacancy order in $Fe7S8$⁠. This is essential for their use in paleomagnetic reconstructions of Earth and extraterrestrial systems and can provide further insight into vacancy ordering in defect-supported magnetic materials.

This work was supported by an ETH Research Grant (No. ETH-47 17-1).