Revealing complex magnetic interactions in Fe₂P-based compounds: a study using Mössbauer spectroscopy and neutron diffraction

Abstract

The magnetic properties of Fe_2−2xMn_2xP_1−xSi_x (0 ≤ x ≤ 0.5) compounds are studied by neutron diffraction, Mössbauer spectroscopy, and magnetometry. DC magnetization measurements indicate that compounds with 0.2 ≤ x ≤ 0.5 undergo a paramagnetic to ferromagnetic transition, with the Curie temperature increasing as x increases. In contrast, compounds with 0 < x ≤ 0.15 show unclear magnetic ordering in DC magnetization measurements, while AC magnetization measurements display frequency-dependent peaks, indicating glassy spin dynamics. For the x = 0.125 sample, AC magnetization measurements under applied DC fields suggest that the transition at 150 K corresponds to a complex antiferromagnetic (AFM) structure. Mössbauer spectroscopy reveals four distinct regions of hyperfine interactions for different x values, suggesting extreme sensitivity in the magnetic behaviour with Mn and Si substitutions. For 0 < x < 0.15, a drop in the magnetic hyperfine field supports the existence of a complex AFM structure. Neutron diffraction on the x = 0.1 sample confirms an incommensurate AFM structure with a propagation vector q_x = 0.2204(4), consistent with the Mössbauer and magnetization results.

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Johan Cedervall

Johan Cedervall (PhD 2017) is a researcher in inorganic chemistry at Uppsala University, Sweden where he joined in 2023 after a post-doc at ISIS Neutron and Muon Source, UK. His main research interests are within the fields of sustainable magnetic materials, including permanent magnets and magnetic refrigeration. He has expertise within high temperature synthesis, crystallography (including magnetic structure solution) and neutron scattering, which is used to synthesise novel magnetic materials and evaluate their structural and intrinsic magnetic properties. Cedervall has published ∼40 scientific papers and have received research grants from the Swedish Research Council and ÅForsk.

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1 Introduction

Magnetocaloric materials have attracted significant interest as a promising alternative to traditional gas compression methods for magnetic refrigeration.^1–4 Fe₂P-based compounds are among the most promising candidates for magnetic refrigeration due to their large magnetocaloric effect (MCE) resulting from the first-order magnetic transition (FOMT).^5–9 Fe₂P and its derivatives when substituted with Mn and Si exhibit an intriguing phase diagram, Fig. 1. Fe₂P and Mn₂P adopt a hexagonal P6̄2m structure, while the intermediate compound FeMnP crystallizes in an orthorhombic Pnma structure (Co₂P-type). Substituting P with Si in Fe₂P gives rise to another orthorhombic structure. However, single-phase compounds corresponding to Fe₂Si and Mn₂Si cannot be synthesized, as they are in a multiphase region. The center of this phase diagram is occupied by FeMnP_0.5Si_0.5, which retains the Fe₂P structure.

Fig. 1 Quaternary phase diagram of the Fe–Mn–P–Si system. The samples studied in this paper lie on the dotted line.

Fe₂P-based compounds are particularly intriguing due to their tunable transition temperature, non-toxic and non-critical raw materials, and high isothermal magnetic entropy change.¹⁰ These compounds crystallize in a hexagonal P6̄2m space group with two independent metal sites; tetrahedral 3f and pyramidal 3g sites (Fig. 2). Several studies have shown that Mn preferentially occupies the larger pyramidal 3g site in Mn-substituted compounds of Fe₂P.^11–14 The occupation of the non-metal atoms P and Si in the 1c and 2b sites shows no preference in almost all cases. However, XRD refinement and energy calculations of the compound Fe_0.6Mn_1.4P_0.6Si_0.4 using the Korringa–Kohn–Rostoker Coherent Potential Approximation (KKR-CPA)¹⁵ and the neutron diffraction study of Fe_0.70Mn_1.25P_1−xSi_x (x = 0.45, 0.50, 0.55) by Miao et al.¹⁶ suggest that Si atoms preferentially occupy the 2c site. A relation between the Si content and the magnetic hyperfine field at the Fe sites in Fe₂P_1−xSi_x compounds has been established using Mössbauer spectroscopy.¹⁷ It was observed that the magnetic hyperfine field at the tetrahedral and pyramidal Fe sites increases with increasing Si content. Another significant result from Mössbauer studies^12,18 is that the magnetic Fe hyperfine field at the tetrahedral 3f site increases considerably from 11.4 T in Fe₂P to 22.8 T in FeMnP_0.5Si_0.5. The magnetic hyperfine field at the 3g pyramidal site cannot be measured in FeMnP_0.5Si_0.5 due to the absence of Fe atoms at this site. However, Fruchart et al.¹⁵ measured the hyperfine field at the 3g site for the compound Fe_1.4Mn_0.6P_0.6Si_0.4 at 84 K as 19.5 T, which can be compared with 17.2 T in Fe₂P.¹⁸

Fig. 2 Coordination polyhedra of the metal atoms: tetrahedral 3f site (represented as blue polyhedron) and pyramidal 3g site (represented as pink polyhedron).

The MCE is quantified by measuring isothermal entropy and adiabatic temperature changes in materials. Large values of these parameters indicate the strong potential of a material for magnetocaloric applications. Although a high isothermal magnetic entropy change is observed in the (Fe,Mn)₂(P,Si) system, considerable thermal hysteresis (ΔT_hys = 35 K) is associated with FOMT.¹⁰ The isothermal entropy change (ΔS_mag) of FeMnP_0.5Si_0.5 with an applied magnetic field of 2 T is 16.5 J kg⁻¹ K⁻¹, making it very competitive with other materials for magnetic cooling applicaitons.¹⁹ Corresponding values for other magnetocaloric materials are 5.2, 18.4, 21.7 and 15 J kg⁻¹ K⁻¹, for Gd, Gd₅Si₂Ge₂, LaFe_11.6Si_1.4 and Ni_50.2Mn_35.0In_14.8 (Heusler), respectively.²⁰ The search for optimal Fe : Mn and P : Si ratios is key to obtaining reduced thermal hysteresis while maintaining the giant magnetocaloric effect in the Fe–Mn–P–Si system. So far, the studies on (Fe,Mn)₂(P,Si) compounds were mainly focused on keeping equal amounts of Fe and Mn.^{10,12,21–23} These compounds, which crystallize in the hexagonal Fe₂P-type phase, are explored to understand the magnetic interactions behind the physics of magnetic cooling. In Fe₂P, the magnetic moments are ferromagnetically ordered along the hexagonal c-axis below the transition temperature^7,18 T_c while in FeMnP_0.5Si_0.5 they are aligned along the a-axis.²² However, FeMnP_0.75Si_0.25 shows a complex antiferromagnetic (AFM) ordering.²¹ Therefore, it is important to investigate the compositions between Fe₂P and FeMnP_0.5Si_0.5 to understand the stability of the ferromagnetic state (FM) and ultimately to develop improved design guidelines for magnetocaloric materials.

In this work, a series of stoichiometric compounds Fe_2−2xMn_2xP_1−xSi_x (0 ≤ x ≤ 0.5) were synthesized using the drop synthesis method and studied through neutron diffraction, magnetization measurements, and Mössbauer spectroscopy.

2 Methods

2.1 Synthesis

Master alloys of Fe₂P and FeMnP_0.5Si_0.5 were prepared from stoichiometric amounts of iron (Leico Industries, purity 99.99+%. Surface oxides were reduced in H₂-gas), manganese (Institute of Physics, Polish Academy of Sciences, purity 99.999%), phosphorus (Alfa Aesar, purity 99.999+%) and silicon (Highways International, purity 99.999%) using the drop synthesis method.²⁴ Intermediate samples of Fe_2−2xMn_2xP_1−xSi_x were prepared by mixing stoichiometric amounts of the master alloys. All samples were sealed in evacuated silica ampules and annealed at 1370 K for 2 h, followed by heat treatment at 1270 K for 10 days and rapid quenching.

2.2 Diffraction

X-ray diffraction experiments were performed using a Bruker D8 Advance operating with Cu Kα radiation and equipped with a LynxeyeXE position sensitive detector. Neutron powder diffraction (NPD) was employed using the HRPT instrument²⁵ at SINQ (PSI, Villigen Switzerland). A Ge monochromator (reflection 400) was used that gives a wavelength of 2.45 Å. All samples were measured in a temperature range of 10 to 300 K.

The obtained diffraction data was analyzed using the Rietveld method²⁶ implemented in the FullProf software.²⁷ In the refinements, several structure parameters, including atomic positions, occupancies, and displacement, were allowed to vary. For incommensurate magnetic structures, the magnetic propagation vectors, k, were investigated using k-search implemented in the FullProf Suite. The obtained k-vectors were used as input parameter to find irreducible representations (IR) with the SARAh software.²⁸ The obtained IRs were later implemented in full-pattern magnetic structure refinements.

2.3 Magnetometry

A Quantum Design magnetic property measurement system was used to assess the magnetic properties of samples. The temperature dependent magnetization under field-cooled cooling (FC) and field-cooled warming (FCW) conditions were measured using an applied magnetic field of μ₀H = 10 mT. Temperature- and frequency-dependent AC susceptibility measurements were performed to study the low-frequency relaxation behavior of the compounds with 0.05 ≤ x ≤ 0.15 using an AC magnetic field amplitude of μ₀H_AC = 0.4 mT.

2.4 Mössbauer spectroscopy

Mössbauer measurements were carried out on a spectrometer working with a constant acceleration type of vibrator and a ⁵⁷CoRh source. The spectra were recorded at low temperatures in an Oxford He-flow cryostat and at high temperatures in a Ricor furnace, respectively. The Fe_2−2xMn_2xP_1−xSi_x-samples were mixed with inert BN and enclosed in sealed kapton pockets. The absorbers formed thus had a sample concentration of ∼2–20 mg cm⁻². Calibration spectra were recorded at 295 K using natural Fe metal foil as a reference absorber. The spectra were folded and fitted using the least square Mössbauer fitting program Recoil to obtain the values of the center shift (CS) versus natural α-Fe at 295 K, the electric quadrupole splitting (QS) (in the paramagnetic regime), the electric quadrupole shift (ε) and the magnetic hyperfine field (B_hf) (in the magnetic regime), the full-width at half maxima (W) of the Lorentzian absorption lines and the spectral areas (A).

3 Results

3.1 Phase analysis

All samples are of high purity, with only trace amounts of secondary phases, as shown in Fig. 3 for x = 0.1. Structural refinements show that all samples crystallize in the Fe₂P-type structure, space group (SG) P6̄2m. The evolution of the unit cell parameter, studied using XRD, Fig. S1 in SI, gives a clear trend where a (c) increases (decreases) with increasing amount of x. This also affects the c/a-ratio, which decreases with increasing level of substitution. There is a shift in the unit cell parameters between x = 0.25 and 0.375 due to magnetostriction as samples with x > 0.25 have T_C above RT. The unit cell parameters for the master alloys (Fe₂P and FeMnP_0.5Si_0.5) are in excellent agreement with the values previously reported, which in combination with the NPD results indicate that the substitutions have been successful. The structure refinement to the NPD data confirms that Mn preferably occupies the pyramidal 3g position even in low concentrations, shown as an inset in Fig. 3, in agreement with previous findings.^16,21,22,29 Varying the Fe and Mn occupancies were tested in the refinements, but as it did not improve the refined model they were left in their nominal compositions.

Fig. 3 Neutron diffraction (λ = 2.45 Å) pattern for Fe_1.8Mn_0.2P_0.9Si_0.1 at room temperature. The refined structure model is shown as an inset.

3.2 Mössbauer spectroscopy

3.2.1 Paramagnetic regime. In Fig. 4 the Mössbauer spectra in the paramagnetic regime are presented along with the spectral areas for the two crystallographic sites. The area of Fe(2) at pyramidal 3g sites decreases with increasing x and at x = 0.5, (FeMnP_0.5Si_0.5), all contributions to the pattern come from Fe(1) at tetrahedral 3f sites. This confirms the findings from NPD that Mn atoms are solely located at the pyramidal 3g site all through the substitution series. The fitted results using one Mössbauer pattern each for Fe(1) and Fe(2) are presented in Table S1.

Fig. 4 (a) Mössbauer spectra in the paramagnetic regime for different values of x. The red and blue subspectra emanate from Fe(1) at the tetrahedral 3f and Fe(2) at the pyramidal 3g positions, respectively. (b) Normalized sum of spectral intensities of tetrahedral Fe(1) and pyramidal Fe(2) versus x in Fe_2−2xMn_2xP_1−xSi_x.

A detailed study of the Fe(1) subspectrum for x = 0.5 and x = 0.375 shows some asymmetric doublet behavior that is caused by an atomic disorder in the near surroundings of Fe(1) at the tetrahedral 3f site. It is expected since the four nearest neighbour (nn) atoms P or Si can vary. The nn to the 3f site is two P or Si atoms at the 1b site and two P or Si atoms at the 2c site. Each surrounding would give slightly different Mössbauer subspectra, which may explain the asymmetry and broadening in the spectrum as shown in Fig. 4. It has been argued that Si preferentially occupies the 2c-site,¹⁵ however, by analysing the Mössbauer hyperfine spectra of Fe₂P_1−xS_x with 0.10 ≤ x ≤ 0.16 the P/Si substitution is shown to be almost random.¹⁷ However, the resolution in the present spectra in the paramagnetic regime is too low to resolve any preference of the P/Si occupation.

3.2.2 Magnetic regime. The hyperfine parameters are affected differently due to the various amounts of P, Si, Mn and Fe atoms around each Fe metallic site. The spectra have therefore been fitted with 2 to 4 sextet patterns depending on the x-value (Fig. 5). In the fitting, all subspectra representing Fe at the 3f site have the constraint of having the same central shift CS. Similar constraints have also been used for Fe at the 3g site. These constraints were used primarily to minimize the number of fit variables. The variation of the CS values for different surroundings was furthermore found to be small in the spectrum in the paramagnetic regime for Fe both at 3f and 3g, justifying the constraints used. The average hyperfine results from the fitting of spectra at 85 K are presented in Fig. 6 and Table S2.

Fig. 5 Mössbauer spectra of Fe_2−2xMn_2xP_1−xSi_x at 85 K. Blue and red represent Fe at the pyramidal (3g) and tetrahedral (3f) sites, respectively.

Fig. 6 (a) The average magnetic fields at 85 K for Fe at the tetrahedral 3f site and pyramidal 3g. (b) The center shift CS versus natural α-Fe at 295 K for Fe_3f and Fe_3g at 295 K and 85 K. (c) The electric quadrupole splitting QS at 295 K and electric quadrupole shift 2ε at 85 K in mm s⁻¹. (d) The electric quadrupole ratio at 85 K.

In order to find the polar (θ) and azimuthal (ϕ) angles of the magnetic fields versus the principal axes of the electric field gradient (EFG) tensor at 85 K the factor connecting the two electric quadrupole parameters QS and ε can be calculated using the expression:

where η is the EFG asymmetry parameter and the angles θ and ϕ refer to the situation at 85 K. Given the magnetic state of the compounds at 85 K, QS cannot be directly measured. However, using previously reported values for Fe₂P,³⁰ QS(295 K)/QS(85 K) ≈ 1.15 for both Fe sites. Therefore, the values of K in Fig. 6(d) has been calculated as . The principal z axis of the EFG has been found to be in the ab plane and also a negative sign for QS for both Fe sites in Fe₂P.³¹ The calculated values of K show four different regions: (i) K ∼ – 0.5 for x close to 0, (ii) K ∼ 0 for x ∼ 0.1, (iii) K ∼ – 0.9 for x ∼ 0.25 and (iv) K ∼ 0.9 for x ∼ 0.4–0.5 (Fig. 6(d)).

Region (i) represents the ferromagnetic structure with the magnetic moments along the c-axis. The polar angle is 90° with a small asymmetry parameter η, K would theoretically be −0.5 as also found experimentally here. With only a slight amount of Mn and Si substitution, B_hf drops at both crystal sites and the electric quadrupole ratio drops to zero, making the system enter region (ii).

Region (iv) represents the ferromagnetic structure with the magnetic moments along the a-axis. Since the orientation of the z-axis of the electric field gradient tensor is in the ab-plane, the polar angle will be unknown, as will the azimuthal angle for Fe at the tetrahedral 3f site. Experimentally K ∼ 0.9 is observed, which can be interpreted as having the z-axis along the a-axis with small contributions from the asymmetry parameter (Eq. (1)).

Regions (ii) and (iii) are more difficult to interpret from the experimental values of K. The resonance lines are broad due to the different possible nn configurations of the Fe atoms. The K-value has therefore to be interpreted as a value averaged over all possible nn configurations. In region (ii) the average value for K is close to 0 which may be interpreted as a region with varying polar and azimuthal values, a possible antiferromagnetic helical structure for the magnetic moments. The drop in the magnetic hyperfine field in region (ii) is further an indication of a more complex antiferromagnetic structure. Judging from the B_hf results, region (iii) seems to be dominantly ferromagnetic in character.

3.2.3 Mössbauer temperature study of the antiferromagnetic region (ii). Here we present a temperature study of the sample with x = 0.125 as a representative for the antiferromagnetic region (ii). The Mössbauer spectra above 177 K could be identified as emanating from a paramagnetic phase, while spectra below 152 K were emanating from a phase where magnetic fields were acting on the Fe nucleus (Fig. 7).

Fig. 7 Representative Mössbauer spectra of Fe_2−2xMn_2xP_1−xSi_x with x = 0.125. Blue subpatterns represent Fe at the pyramidal 3g site and red subpatterns represent Fe at the tetrahedral 3f site. A magnetic transition takes place between 152 K and 177 K.

The spectra in the paramagnetic regime were fitted using two doublets. In the magnetically ordered state, the spectra exhibited significant broadening due to variations in the local environment, where different numbers of neighboring Mn and P/Si atoms influence the magnetic field at the Fe nucleus. This broadening was approximated by four sextets, two representing Fe at the 3g site and two at the 3f site. Table S3 presents the fitted results, while Fig. 7 displays representative spectra.

It is evident that no Fe magnetic moment ordering exists above 150 K for the samples within the time scale of the Mössbauer transition ∼ 10⁻⁷ s and longer. However, the AC susceptibility results indicate slow spin dynamics with glassy behavior at ∼200 K and 320 K (cf. Fig. 9 below); the frequency range 1.7–170 Hz corresponds to a time scale range of 1 – 100 ms. The absence of a magnetic hyperfine splitting of the ⁵⁷Fe Mössbauer spectrum at temperatures T > 150 K can have at least two explanations. One explanation could be that the slow spin dynamics detected in the AC susceptibility measurement originate solely from the Mn magnetic moments. Another relating explanation is that the local Fe magnetic moment disappears at temperatures above the antiferromagnetic ordering temperature. The latter explanation connects to the rare case of mixed magnetism, i.e. the coexistence of strong and weak magnetism, discussed for hexagonal FeMn(P,Si).⁵

The AC susceptibility and Mössbauer results are consistent with the onset of magnetic ordering at ∼150 K. The observed values of ε (∼0.00 mm s⁻¹) are significantly lower compared to QS (∼−0.36 mm s⁻¹ and −0.56 mm s⁻¹ for Fe at the 3f and 3g sites, respectively). This suggests that a magnetic structure with moments aligned along the c-axis can be ruled out, as such an arrangement would yield an electric quadrupole ratio of −0.5 (see the section on the magnetic regime above).

3.3 Magnetic characterisation

The temperature-dependent magnetization of the compounds studied is shown in Fig. 8(a and b). Fe₂P (x = 0) shows a paramagnetic (PM) to ferromagnetic (FM) phase transition with a T_C = 216(2) K, similar to the previously reported value.³⁰ A phase transition from PM to FM is also observed in the samples with 0.2 ≤ x ≤ 0.5. For these four compounds, T_C increases with increasing x, although the increase in T_C levels off at higher x values. Magnetic transition temperatures of Fe₂P-type compounds are often related to the atomic distances between the magnetic atoms situated in the 3f and 3g sites. The positions of the 3f and 3g sites and their shortest interatomic distances (d_3f−3g) were extracted from the neutron diffraction refinements. The values of d_3f−3g and T_C are shown in Table 1. It is clear that an increase of d_3f−3g follows with an increase in T_C for the four compounds with 0.2 ≤ x ≤ 0.5. In Fe₂P, both the 3f and 3g sites are occupied by Fe, which is unlikely for the rest of the compounds. Interestingly, these four compounds exhibit thermal hysteresis (ΔT_hys), i.e. a difference in the magnetic phase transition temperature between the FC and FCW magnetization curves. The presence of ΔT_hys is often used as an indication of a FOMT.¹⁹

Fig. 8 Temperature dependent (a) and (b) magnetization measured under FC and FCW conditions. The direction of temperature change is indicated by arrows. To clearly distinguish the temperature dependent magnetization two different scales for the magnetization are used in (a) and (b).

Table 1 Values of T_C, inter-atomic distances d_3f–3g, and temperature hysteresis ΔT_hys of the compounds with 0.2 ≤ x ≤ 0.5. The values of T_C are taken from the FC magnetization versus temperature curves at μ₀H = 0.01 T

x	T^FCC (K)	d3f–3g (Å)	ΔThys (K)
0	216	2.6272(4)	20
0.2	196	2.6225(7)	20
0.25	266	2.6523(5)	18
0.375	362	2.6551(6)	10
0.5	380	2.6678(4)	14

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The compounds with 0 < x ≤ 0.15 exhibit a much lower magnetization with unclear magnetic ordering (cf. Fig. 8(b)). The temperature-dependent magnetization curves resemble those of materials with glassy behavior. To investigate this behavior further, temperature- and frequency-dependent AC magnetic susceptibility measurements have been performed for the x = 0.125 compound; the results are shown in Fig. 9. The real component of the AC-susceptibility (cf. Fig. 9(a)) exhibits a clear frequency dependence at temperatures above 150 K. The imaginary component reveals several loss peaks (Fig. 9(b)). The presence of several frequency-dependent loss peaks is indicative of different contributions to magnetic dissipation in the temperature range of 150 K < T < 400 K. From a comparison of the AC susceptibility and Mössbauer results, it is concluded that the slow spin dynamics in this temperature range is solely due to the Mn magnetic moments. At the same time, the disappearance of dissipation at T < 150 K is in accordance with the order of AFM at low temperature involving both local Fe and Mn moments, which will be discussed below when describing the results of the magnetic structure analysis.

Fig. 9 Frequency and temperature dependent real and imaginary components of the AC magnetic susceptibility without applied fields (a) and (b), and with superimposed DC applied fields (c) and (d) respectively, for the x = 0.125 compound.

In order to get more insight into the glassy spin dynamics, Fig. 9(c) and (d) show AC susceptibility versus temperature results for fixed AC frequency (1.7 Hz and 170 Hz) along with superimposed DC magnetic fields (0–5 mT). The real component of the AC-susceptibility (cf. Fig. 9(c)), including its frequency dependence is suppressed by the superimposed DC magnetic field. Even more noticeable is the suppression of the imaginary component of the AC susceptibility at T > 150 K. In the presence of the 1 mT DC field, the loss peak at T ≈ 320 K vanishes, while the loss peak at T ≈ 200 K is much reduced (cf. Fig. 9(b)). Both loss peaks vanish in the presence of 5 mT DC field, while the relaxation behavior at T < 150 is unaffected by the superimposed DC magnetic fields (cf. Fig. 9(b)). From this it can be concluded that the spin dynamics of the antiferromagnetically ordered phase is not affected by the superimposed DC magnetic field, while the time scales of the glassy spin dynamics at T > 150 K are strongly DC magnetic field dependent with a shortening of the time scales with increasing DC magnetic field. A similar frequency dependent behaviour has been observed for the x = 0.05, 0.1, and 0.15 compounds which are discussed in SI.

To estimate the effective paramagnetic moment of the compounds (x = 0, 0.2, and 0.25) which show a clear PM-FM transition below 400 K, the temperature dependent inverse DC-susceptibility has been fitted to the Curie-Weiss (CW) law (Fig. S3(a) in SI).³² The calculated effective moments are 5.96 μ_B, 9.9 μ_B, and 10.4 μ_B per formula unit for the x = 0, 0.2, and 0.25 samples, respectively. The extracted values for the x = 0.2, and 0.25 samples are likely overestimated, since the x = 0.2 sample exhibits a Griffiths phase like behaviour^33,34 and x = 0.25 is not completely paramagnetic even at 400 K (Fig. S3(b)). Moreover, the field dependent magnetization at 400 K indicate that the samples which did not show a clear PM-FM transition (e.g. x = = 0.1 and 0.125) are also not completely paramagnetic at 400 K, as indicated in Fig. S3(b) revealing a non-linear magnetic field dependence at low fields. Therefore, among the studied compounds, in the measured temperature range, the CW law is only valid for the x = 0 sample, yielding a calculated effective moment of 5.96 μ_B/f.u. and a Weiss temperature of 278.3 K.

3.4 Neutron diffraction results

Fe_2−2xMn_2xP_1−xSi_x (x = 0.1 and 0.125) samples were examined using NPD measurements at various temperatures ranging from 300 K to 10 K. The nuclear structure was determined at room temperature (300 K), while the magnetic structure was determined at 10 K. The x = 0.125 sample has a small amount (0.7(2)%) of the secondary phase Fe₃Si. The magnetic propagation vector was determined using the k-search program in the FullProf software. Several possible propagation vectors were obtained from the k-search, which were then tested to find the propagation vector that best describes the structure.

Magnetic structures for the samples x = 0.1 and 0.125 were determined using simulated annealing and representation analysis (SARAh). Magnetic structure refinement was performed in FullProf, utilizing the basis vectors from SARAh. The neutron diffraction patterns acquired below 150 K reveal a prominent magnetic peak, 000 ± k, for both samples, indicating an incommensurate antiferromagnetic (AFM) structure with a non-zero propagation vector. Some weak satellite peaks were also observed in the Q range of 1.7–2.3 Å⁻¹. With the proposed k-vector, three magnetic models—moments aligned along the b-axis, c-axis, and in the ab-plane—were found to best fit the observed data. The primary distinction between these models lies in their fit to the weak satellite peaks, 001 ± k and 101 ± k. Fig. 10(a) presents the observed and calculated NPD patterns for the model with moments aligned along the b-axis at 10 K for x = 0.1. The magnetic contributions of the three models, along with the total calculated intensity (nuclear and magnetic contributions) for the b-axis model, are shown in Fig. 10(b), here shown for x = 0.1. The comparison reveals that the models with moments along ab and b fit the 001 ± k peak more accurately than the c-axis model, while the c-axis model fails to produce any intensity at the 101 ± k peak position. Since the model with moments along the a-axis (not shown here) does not provide a good fit to the observed NPD pattern, it suggests that the magnetic moments are more likely to align along the b-axis rather than within the ab-plane. The x = 0.125 sample shows the same behavior in the NPD patterns.

Fig. 10 (a) Observed and calculated neutron diffraction patterns of Fe_1.8Mn_0.2P_0.9Si_0.1 in the antiferromagnetic state, for the model with moments along b axis. (b) Refinement comparison of magnetic models with moments along the c-axis, b-axis, and within the ab-plane, shown for the enlarged magnetic satellite peaks. Dashed lines represent the magnetic contribution of each model, while the total calculated intensity for the model with moments along the b-axis is shown in as a green line. The hkl indices of the main Bragg reflections and their corresponding magnetic satellites are indicated.

For the refinement of the nuclear structure at 300 K, several profile parameters, atom occupancy, and atomic displacement parameters were refined. Fe atoms were found to occupy only the tetrahedral 3f site, while the pyramidal 3g site is occupied by the Fe and Mn atoms. These occupancies were initially refined and subsequently fixed in the final refinement. In the magnetic structure refinement at 10 K, the same parameters as those used in the 300 K refinement were refined, with atomic occupancies fixed. A symmetric Gaussian size broadening was used to fit the magnetic reflections at low temperature. Furthermore, the magnetic moment parameters and the propagation vector were refined. Table 2 shows the results of the NPD refinement at 300 K and 10 K for the x = 0.1 sample. The unit cell parameters at 10 K; a = 5.8996(1) Å and c = 3.4507(1) Å are slightly smaller than lattice parameters at 300 K; a = 5.9156(1) Å and c = 3.4553(1) Å. The parameter a decreases continuously with decreasing temperature, while c initially decreases until 150 K, after which it starts to increase down to 10 K (Fig. S4). The magnetic ordering becomes antiferromagnetic in this case, likely due to the increase in the c/a ratio below 150 K.

Table 2 Atomic coordinates, atomic displacement parameters and occupancy of Fe, Mn, P and Si atoms at 300 K and 10 K (antiferromagnetic state) obtained from refinements of neutron powder diffraction data for the sample x = 0.1. The full coordinates of the atomic sites are 3f: (x 0 0), 3g: , 2c: , 1b: . For 300 K refinement, R_Bragg = 2.57, R_wp = 6.58, R_p = 4.49, χ² = 6.30. For 10 K refinement, Prop. vector q_x = 0.2204(4), R_Bragg = 2.51, R_mag = 3.39, R_wp = 6.71, R_p = 4.33, χ² = 17.2

Atom	Site	300 K			10 K
		x	Biso (Å)	Occ	x	Biso (Å)	Occ	M (μb)
FeI	3f	0.25601(9)	0.43(2)	1	0.2559(1)	0.20(3)	1	0.7(2)
FeII/MnII	3g	0.5935(1)	0.43(2)	0.8/0.2	0.5923(2)	0.20(3)	0.8/0.2	2.3(2)
PI/SiI	2c		0.80(4)	0.9/0.1		0.79(6)	0.9/0.1	—
PII/SiII	1b	0	0.80(4)	0.9/0.1	0	0.79(6)	0.9/0.1	—

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Fig. 12 shows the proposed magnetic structure model for the x = 0.1 sample. The magnetic moments propagate sinusoidally along the a-axis with an incommensurate propagation vector of k_x = 0.2204(4). The moments of atoms at the 3f and 3g sites are aligned along the b-axis. The magnetic moments of Fe(1) and Fe/Mn(2) atoms are 0.7(2) μ_B and 2.3(2) μ_B respectively, at 10 K. From Mössbauer spectroscopy, a sinusoidal model would give broader Mössbauer spectra due to magnetic fields ranging over a wider interval, which is not directly observed here; however the suggested model is the best from the neutron diffraction analysis.

Previous studies have shown that Fe moments at 3f sites typically have lower magnetic moments than Mn at 3g sites.^21,35,36 As the temperature decreases, k_x increases for both samples (x = 0.1 and 0.125), with the x = 0.125 sample showing a larger k_x (0.2254(5)) than the x = 0.1 sample at the same temperature (Fig. S5). The magnetic moments at the 3f and 3g sites also increase upon cooling, with the 3g site showing a slightly larger increase than the 3f site, here shown for x = 0.1 sample. The unusual peak shape observed for the 000 ± k in the NPD pattern could be attributed to a disorder in the magnetic arrangement or multiple propagation vectors within the magnetic structure. However, incorporating additional k-vectors did not result in significant changes to the magnetic structure. The temperature-dependent magnetization curve for the compounds 0 < x ≤ 0.15 from magnetometry suggests that the samples are not in a purely paramagnetic state even at 300 K. The AC susceptibility measurements for the x = 0.125 sample reveal a glassy spin dynamics at T > 150 K and a possible AFM transition below 150 K. This can be correlated to the observation of a hump with slight intensity above 150 K for the 000 ± k peak in the NPD data (Fig. 11) shown here for x = 0.1 sample, indicating short-range or fluctuating magnetic correlations which has been observed before in Fe–Mn–P–Si system.³⁷ With decreasing temperature, this feature evolves into a slightly broadened antiferromagnetic peak at 10 K, indicating that the system is moving from short-range to long-range order with decreasing temperature, which is not unusual.³⁸

Fig. 11 Heat map of the evolution of the diffraction data for x = 0.1 sample in the Q region 0 to 3.3 Å⁻¹ on cooling. The pure magnetic reflection 000 ± k (k = (0.2203 0 0)) has an increase in intensity below ≈150 K and shows broadening compared to the nuclear Bragg peaks in the figure.

Fig. 12 (a) Magnetic structure depicting the propagation of magnetic moments along a-axis. Fe I atoms sitting at the 3f site are shown inside polyhedra (c) Unit cell with magnetic moments pointing along the b-axis.

The broadness of the 000 ± k peak below 150 K in the NPD pattern likely arises from the system not attaining infinite magnetic correlation even at low temperatures. It is interesting to observe that a slight substitution of Mn at one of the Fe sites alters the magnetic structure, leading to moments aligning along the b-axis from the c-axis alignment in Fe₂P. However, more precise information on the nature of the magnetic correlations would require a polarized neutron scattering study.

An incommensurate AFM ordering has previously been reported for the (FeMn)₂(PSi) system with the composition FeMnP_0.75Si_0.25.²¹ For this composition, Fe moments align along the a-axis, while Mn moments align along the b-axis. In Fe₂P-based compounds, the chemical disorder of the 3f and 3g sites and the lattice parameter ratio c/a play a crucial role in determining the ground-state magnetic order.³⁹

4 Conclusions

The magnetic behaviour for substitutions in Fe₂P has been demonstrated with neutron diffraction, Mössbauer spectroscopy and magnetometry. It is shown that Fe₂P is extremely sensitive to distortions of the structure, here exemplified by substitutions.

From Mössbauer spectroscopy, a significant drop in the magnetic hyperfine field for 0.05 < x < 0.15 as compared to x = 0 is observed. The drop in hyperfine field comes from the change of the magnetic structure, from ferromagnetic along the c-axis for Fe₂P, to an incommensurate antiferromagnetic structure. The magnetic hyperfine field at the tetrahedral 3f site has been found to increase by a factor of ∼ 2 between x = 0 and x = 0.5. In fact, the Fe magnetic field at the tetrahedral 3f site is, at x = 0.375, larger than the Fe field at the pyramidal 3g site. This is in contrast to the situation in pure Fe₂P. These changes in the hyperfine field have a counterpart in the magnetic moments. Further Mössbauer studies are needed on samples with x closer to 0 to reveal the exact nature of the antiferromagnetic structure adopted in region (ii). From magnetometry, it is observed that the samples with a small percent Mn substitution (0.05 < x < 0.15) exhibit glassy spin dynamics in the temperature range 150 K < T < 400 K, which arises solely from the Mn moments. The AC susceptibility with superimposed DC fields for a sample in this region, x = 0.125, suggests an AFM behavior below 150 K. Samples with a higher substitution for Mn (0.2 < x < 0.5) exhibit a PM-FM transition in DC susceptibility, the same as observed for Fe₂P. The implication of a complex AFM structure from Mössbauer spectroscopy and magnetometry is confirmed by NPD measurements for x = 0.1 and 0.125 samples revealing an incommensurate AFM structure with propagation vectors k_x = 0.2204(4) and 0.2254(5) respectively at 10 K. The AFM 000 ± k peak for both samples exhibits a slight intensity above 150 K, suggesting the presence of short-range magnetic correlations. Furthermore, the magnetic diffraction peaks are broader than the nuclear peaks below 150 K, indicating that the system does not attain infinite antiferromagnetic correlation even at lower temperatures.

To conclude, the effect on simultaneous substitutions of Mn and Si in Fe₂P to Fe_2−2xMn_2xP_1−xSi_x have been investigated with regard to their magnetic properties. The system show mainly ferromagnetic features with magnetic transition temperatures correlated to the interatomic distances between the two iron sites in the structure. However, a region 0.05 ≤ x < 0.2 have been found to have antiferromagnetic behavior, originating from an incommensurate magnetic structure propagating along the crystallographic a-axis with a propagation vector k_x = 0.2204(4).

Author contributions

Karthika K. Thilakan: conceptualization, data curation, formal analysis, investigation, methodology, visualization, writing – original draft. Sagar Ghorai: data curation, formal analysis, investigation, methodology, visualization, writing – original draft. Wei Liu: data curation, formal analysis, investigation, writing – review and editing. Lennart Häggström: conceptualization, data curation, formal analysis, investigation, methodology, visualization, writing – original draft, writing – review and editing. Fredrik Lindgren: data curation, investigation, methodology, writing – review and editing. Vladimir Pomjakushin: data curation, investigation, writing – review and editing. Premysl Beran: data curation, formal analysis, investigation, methodology, writing – review and editing. Oliver Gutfleisch: formal analysis, writing – review and editing. Peter Svedlindh: data curation, investigation, methodology, writing – review and editing. Johan Cedervall: conceptualization, data curation, formal analysis, investigation, methodology, visualization, writing – original draft, writing – review and editing.

Conflicts of interest

There are no conflicts to declare.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request. Researchers interested in accessing these datasets for replication or further studies can contact the corresponding author to request the necessary materials.

The SI contains extensive analysis of the data from Mössbauer spectroscopy and frequency-dependent AC magnetic susceptibility. See DOI: https://doi.org/10.1039/d5ta03047a.

Acknowledgements

Financial support from the ÅForsk Foundation (grant no. 22-378), the Swedish Foundation for Strategic Research (SSF) through SwedNess (grant no. GS1715-0008) and WISE – Wallenberg Initiative Materials Science for Sustainability is gratefully acknowledged. S. G., W. L., and O. G. thankfully acknowledge the financial support of the German Research Foundation (DFG) in the framework of the Collaborative Research Centre Transregio 270 (CRC-TRR 270) and the European Union, through project CoCoMag (grant no. 101099736). This work is based on experiments performed at the Swiss spallation neutron source SINQ, Paul Scherrer Institute, Villigen, Switzerland.

Notes and references

1. E. Brück, J. Phys. D: Appl. Phys., 2005, 38, R381.
2. V. V. Khovaylo, V. V. Rodionova, S. N. Shevyrtalov and V. Novosad, Phys. Status Solidi B, 2014, 251, 2104–2113.
3. E. L. França, A. O. D. Santos, A. A. Coelho and L. M. D. Silva, J. Magn. Magn. Mater., 2016, 401, 1088–1092.
4. K. A. Gschneidner and V. K. Pecharsky, Int. J. Refrig., 2008, 31, 945–961.
5. N. H. Dung, Z. Q. Ou, L. Caron, L. Zhang, D. T. Thanh, G. A. D. Wijs, R. A. D. Groot, K. H. Buschow and E. Brück, Adv. Energy Mater., 2011, 1, 1215–1219.
6. H. Yibole, F. Guillou, L. Zhang, N. H. V. Dijk and E. Brück, J. Phys. D: Appl. Phys., 2014, 47, 075002.
7. L. Lundgren, G. Tarmohamed, O. Beckman, B. Carlsson and S. Rundqvist, Phys. Scr., 1978, 17, 39.
8. N. T. Trung, Z. Q. Ou, T. J. Gortenmulder, O. Tegus, K. H. Buschow and E. Brück, Appl. Phys. Lett., 2009, 94, 102513.
9. F. Guillou, G. Porcari, H. Yibole, N. V. Dijk and E. Brück, Adv. Mater., 2014, 26, 2671–2675.
10. D. T. C. Thanh, E. Brück, N. T. Trung, J. C. Klaasse, K. H. Buschow, Z. Q. Ou, O. Tegus and L. Caron, J. Appl. Phys., 2008, 103, 07B318.
11. M. Bacmann, J.-L. Soubeyroux, R. Barrett, D. Fruchart, R. Zach, S. Niziol and R. Fruchart, J. Magn. Magn. Mater., 1994, 134, 59–67.
12. M. Hudl, P. Nordblad, T. Björkman, O. Eriksson, L. Häggström, M. Sahlberg, Y. Andersson, E. K. Delczeg-Czirjak and L. Vitos, Phys. Rev. B, 2011, 83, 134420.
13. N. H. Dung, L. Zhang, Z. Q. Ou, L. Zhao, L. V. Eijck, A. M. Mulders, M. Avdeev, E. Suard, N. H. V. Dijk and E. Brück, Phys. Rev. B, 2012, 86, 045134.
14. B. Malaman, G. L. Caër, P. Delcroix, D. Fruchart, M. Bacmann and R. Fruchart, J. Phys.: Condens. Matter, 1996, 8, 8653.
15. D. Fruchart, S. Haj-Khlifa, P. de Rango, M. Balli, R. Zach, W. Chajec, P. Fornal, J. Stanek, S. Kaprzyk and J. Tobola, Crystals, 2019, 9, 37.
16. X. F. Miao, L. Caron, P. Roy, N. H. Dung, L. Zhang, W. A. Kockelmann, R. A. de Groot, N. H. van Dijk and E. Brück, Phys. Rev. B, 2014, 89, 174429.
17. P. Jernberg, A. Yousif, L. Häggström and Y. Andersson, J. Solid State Chem., 1984, 53, 313–322.
18. R. Wäppling, L. Häggström, T. Ericsson, S. Devanarayanan, E. Karlsson, B. Carlsson and S. Rundqvist, J. Solid State Chem., 1975, 13, 258–271.
19. S. Ghorai, J. Cedervall, R. Clulow, S. Huang, T. Ericsson, L. Häggström, V. Shtender, E. K. Delczeg-Czirjak, L. Vitos, O. Eriksson, M. Sahlberg and P. Svedlindh, Phys. Rev. B, 2023, 107, 104409.
20. T. Gottschall, K. P. Skokov, M. Fries, A. Taubel, I. Radulov, F. Scheibel, D. Benke, S. Riegg and O. Gutfleisch, Adv. Energy Mater., 2019, 9, 1901322.
21. V. Höglin, M. Hudl, L. Caron, P. Beran, M. H. Sørby, P. Nordblad, Y. Andersson and M. Sahlberg, J. Solid State Chem., 2015, 221, 240–246.
22. V. Höglin, M. Hudl, M. Sahlberg, P. Nordblad, P. Beran and Y. Andersson, J. Solid State Chem., 2011, 184, 2434–2438.
23. K. Katagiri, K. Nakamura and H. Wada, J. Alloys Compd., 2013, 553, 286–290.
24. B. Carlsson, M. Gölin and S. Rundqvist, J. Solid State Chem., 1973, 8, 57–67.
25. P. Fischer, G. Frey, M. Koch, M. Könnecke, V. Pomjakushin, J. Schefer, R. Thut, N. Schlumpf, R. Bürge, U. Greuter, S. Bondt and E. Berruyer, Phys. B, 2000, 276–278, 146–147.
26. H. M. Rietveld, J. Appl. Crystallogr., 1969, 2, 65–71.
27. J. Rodriguez-Carvajal, Phys. B, 1993, 192, 55–69.
28. A. Wills, Phys. B, 2000, 276–278, 680–681.
29. M. J. Neish, M. P. Oxley, J. Guo, B. C. Sales, L. J. Allen and M. F. Chisholm, Phys. Rev. Lett., 2015, 114, 106101.
30. J. Cedervall, M. S. Andersson, E. K. Delczeg-Czirjak, D. Iuşan, M. Pereiro, P. Roy, T. Ericsson, L. Häggström, W. Lohstroh, H. Mutka, M. Sahlberg, P. Nordblad and P. P. Deen, Phys. Rev. B, 2019, 99, 174437.
31. T. Ericsson, L. Häggström, R. Wäppling and T. Methasiri, Phys. Scr., 1980, 21, 212.
32. S. Mugiraneza and A. M. Hallas, Commun. Phys., 2022, 5, 95.
33. S. Ghorai, V. Shtender, P. Ström, R. Skini and P. Svedlindh, J. Alloys Compd., 2022, 895, 162714.
34. S. Ghorai, S. A. Ivanov, R. Skini and P. Svedlindh, J. Phys.: Condens. Matter, 2021, 33, 145801.
35. Z. Q. Ou, L. Zhang, N. H. Dung, L. V. Eijck, A. M. Mulders, M. Avdeev, N. H. V. Dijk and E. Brück, J. Magn. Magn. Mater., 2013, 340, 80–85.
36. X.-F. Miao, S.-Y. Hu, F. Xu and E. Brück, Rare Met., 2018, 37, 723–733.
37. X. Miao, L. Caron, J. Cedervall, P. Gubbens, P. Dalmas de Réotier, A. Yaouanc, F. Qian, A. Wildes, H. Luetkens and A. Amato, et al., Phys. Rev. B, 2016, 94, 014426.
38. A. Singh, S. Mohapatra, P. Khare, N. Ganguli, A. Wildes, V. Siruguri and S. Kaushik, Mater. Res. Express, 2019, 6, 066107.
39. G. Li, W. Li, S. Schönecker, X. Li, E. K. Delczeg-Czirjak, Y. O. Kvashnin, O. Eriksson, B. Johansson and L. Vitos, Appl. Phys. Lett., 2014, 105, 262405.

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