Temperature evolution of the high-harmonic magnetic modulations in DyFeO₃

This article is part of a collection of articles related to the International Conference on Neutron Scattering, ICNS2025.

We report on the temperature evolution of magnetic satellites associated with the incommensurate magnetic ordering of Dy³⁺ ions in DyFeO₃ below 4.1 K. Using high-resolution neutron diffraction, incommensurate magnetic reflections, satellites of the 001 reflection with odd harmonics up to the seventh order, were observed and their temperature dependence quantified. Key parameters, such as the modulation length, FWHM and integrated intensities, were analysed across the measured temperature range. The results were compared with a previous study of the soliton lattice in TbFeO₃ which appears under an applied magnetic field. Possible microscopic models underlying the magnetic order of Dy³⁺ are discussed in this context.

1. Introduction

DyFeO₃ is the only known rare earth orthoferrite with an incommensurate ground state (Mareschal & Sivardière, 1969; Ritter et al., 2022; Biswas et al., 2022). Below the Dy Néel temperature ≃ 4 K, it exhibits multiferroic properties such as the giant magnetoelectric effect (Tokunaga et al., 2008; Wang et al., 2016) and the piezomagnetoelectric effect (Nakajima et al., 2015). It was also shown that spin dynamics in DyFeO₃ can be controlled via the inverse Faraday effect (Kimel et al., 2005). Understanding the connection between these properties and the incommensurate ground state requires a thorough characterization of the magnetic order of DyFeO₃ at low temperatures.

DyFeO₃ crystallizes in an orthorhombically distorted perovskite structure, which is described by the Pbnm space group (non-standard setting of Pnma, No. 62) (Eibschütz, 1965; Marezio et al., 1970). Below the Fe³⁺ Néel temperature of ≃ 645 K the magnetic moments of the Fe³⁺ ions establish a structure [first and second irreducible representation (irrep) labels are according to Bertaut (1963) and Cracknell, Davies, Miller & Love (CDML) (Miller & Love, 1967; Cracknell et al., 1979) notations, respectively]. This has magnetic space group (MSG) Pb′n′m, i.e. a nonstandard setting of MSG Pn′ma′, BNS No. 62.448. Below the spin reorientation temperature [30–80 K, exact values vary depending on the study; Ritter et al. (2022) and citations within], the Fe³⁺ ions establish a magnetic order (MSG Pbnm.1, i.e. a nonstandard setting of MSG Pnma.1, BNS No. 62.441). The reorientation of the G component and the disappearance of the F component have both been confirmed by neutron diffraction (Mareschal & Sivardière, 1969; Wang et al., 2016; Ritter et al., 2022) and magnetization studies (Bozorth et al., 1958; Zhao et al., 2014; Wang et al., 2016; Biswas et al., 2022).

Early studies (Nowik & Williams, 1966; Gorodetsky et al., 1967; Gorodetsky et al., 1968; Belov et al., 1968; Berton & Sharon, 1968; Schuchert et al., 1968) described the Dy³⁺ magnetic structure in DyFeO₃ below  ≃ 4 K as a com­men­surate ordering. A neutron powder diffraction study by Mareschal & Sivardière (1969) showed that the Dy³⁺ ordering is incommensurate, as evidenced by a slight splitting of the 001 reflection. However, some recent reports still use the commensurate description, but in the form of a representation (Tokunaga et al., 2008; Rajeswaran et al., 2013; Zhao et al., 2014; Nakajima et al., 2015; Wang et al., 2016; Stanislavchuk et al., 2016; Hoogeboom et al., 2021). The Dy³⁺ incommensurate ordering in DyFeO₃ was not discussed for half a century but has been recently addressed by Ritter et al. (2022) and Biswas et al. (2022) with neutron powder diffraction. The authors of those reports proposed three different models of the Dy³⁺ magnetic ordering with modulation vector (0, 0, δ): (i) a transverse spin-density wave [magnetic superspace group (MSSG) Pbnm.1(0, 0, γ)s00, 62.1.9.4.m441.1; Ritter et al., 2022], (ii) an elliptical-based helical ordering [MSSG Pbn2₁.1(0, 0, γ)s0s, 33.1.9.2.m144.1; Ritter et al., 2022] and (iii) a spin-density wave coexisting with commensurate ordering [MSSG Pn′(α0γ)0, a nonstandard setting of MSSG Pb′(α, β, 0)0, 7.1.2.1.m26.1; Biswas et al., 2022]. Model (iii) is highly improbable, as it results in Dy³⁺ magnetic moments ranging from 6.6 to 14.2 µ_B, with the latter value exceeding the expected magnetic moment of the isolated Dy³⁺ ion, i.e. 10 µ_B. In the helical model (ii) (Ritter et al., 2022), half of the Dy magnetic moments rotate clockwise and the other half rotate anticlockwise. This results in a change in the angle between Dy magnetic moments in neighbouring helices from one unit cell to the next. This seems rather unlikely and was not discussed further by Ritter et al. (2022). In fact, it was shown (Fabrykiewicz et al., 2021) that helical ordering with one global chirality is only possible in the Sohncke magnetic groups, i.e. space groups without inversion centres, mirrors or glide planes. This makes the chiral magnetic ordering incompatible with the Pbnm space group of DyFeO₃. In addition, the Dy³⁺ ions show a very strong Ising-like anisotropy (Zvezdin & Matveev, 1979), which supports the validity of the spin-density wave model (i). However, all these models introduce a modulation of the magnetic moment amplitude of Dy. This would be typically caused by the Kondo effect, a screening of the 4f orbital by conduction-band electrons, but DyFeO₃ is insulating and does not allow for such an effect.

Therefore, high-resolution single-crystal neutron diffraction on DyFeO₃, as already mentioned by Mareschal & Sivardière (1969) and recently readdressed by Ritter et al. (2022), is highly needed to characterize the Dy³⁺ incommensurate ordering in detail. A recent study by Nikitin et al. (2025) discusses this issue and, based on the point-charge model of the crystal electric fields, suggests collinear ordering of Dy moments. Further, it presents a numerical model of the domain structure in DyFeO₃ that explains the multiple satellites of the 001 reflection and follows its evolution in the applied field. Our results are in line with those of Nikitin et al. (2025). In this study, we focus on the temperature evolution of the domain distribution based on the characteristics of the satellite reflections, that is, their intensities and widths. We provide analysis of the key parameters as a function of temperature and interpret them in the context of soliton lattice formation.

2. Methods

A single crystal of DyFeO₃ was grown using the crucible-free zone melting method with optical heating in a mirror furnace FZ-4000 (Crystal System Co., Japan). The resulting 6 mm diameter and 17 mm length cylindrical crystal with the b axis aligned with the cylinder axis was cut into two half-cylinders, such that the large flat surface is the (001) crystal face.

Neutron diffraction measurements (Schmalzl et al., 2024) were carried out on the IN12 cold-neutron three-axis spectrometer at the Institut Laue–Langevin (ILL), with a wavelength of 5.463 Å and a velocity selector upstream from the monochromator. A high-resolution diffraction setup was used, with a pyrolytic graphite (PG) (002) monochromator and a PG(002) analyser operating in optimal vertical focusing and no horizontal focusing mode. Horizontal collimation was achieved with 20′ collimators placed before and after the sample. The sample was mounted in an ILL Orange Cryostat with the H0L reflections in the scattering plane. The sample was oriented on the neutron Laue diffractometer OrientExpress at the ILL.

The high resolution provided by cold neutrons was required to separate the satellite reflections. However, the 1/e penetration depth of DyFeO₃ for 5.463 Å neutrons is 0.18 mm, which results in collection of the signal effectively only from the crystal surface. The flat sample surface perpendicular to the c axis allows us to maximize the intensity of the 001 reflection by reducing the shadowing effect.

3. Results

A representative L scan across the 001 reflection at 1.75 K is shown in Fig. 1(a). It reveals a comb of satellite reflections indexed with the modulation vector (0, 0, δ) with δ = 0.0197 (1) reciprocal-lattice units (r.l.u.), revealing the odd harmonics of the underlying modulation. Satellites up to the seventh order are resolved, as well as the zero-order commensurate contribution. Selected scans in the temperature range 1.75–4.86 K are shown in Fig. 1(b). For each temperature, a set of nine Gaussians, representing the individual satellites together with a flat background, were fitted. Eight of these represent magnetic satellite contributions at ±δ, ±3δ, ±5δ and ±7δ with respect to the 001 reflection. The positions of the satellites were fitted with a single global δ parameter, multiplied by the order of the satellites. Left- and right-hand satellites of the same order were restricted to share the same intensities and FWHMs. The final Gaussian, centred at L = 1, represents the commensurate contribution.

Figure 1 Neutron scattering intensities on the (00L) points in reciprocal space of single-crystal DyFeO3. They are shown on a logarithmic scale. (a) Representative scan at 1.75 K. Fits of odd magnetic satellite reflections up to seventh order are shown with coloured Gaussians, and the background is shown with a grey rectangle. (b) Selected scans in the temperature range 1.75–4.86 K.

The temperature dependence of the normalized (to the value of 1 at 1.62 K) integrated and first-, third- and fifth-order satellite reflection intensities is shown in Fig. 2(a). Fig. 2(b) presents the modulation vector length as a function of temperature. Below  = 4.1 K the intensities of all satellites are increasing. The higher the order of the satellite, the weaker the increase in intensity with decreasing temperature, as visual­ized in Fig. 2(a). The saturation of the intensity of the first-order satellite reflections occurs at the characteristic saturation temperature  ≃ 2.8 K. In contrast, the higher-order satellite reflections are not saturated down to a temperature of 1.62 K. Below , the modulation vector length decreases with decreasing temperature, reaching a minimum of 0.0187 (1) r.l.u. at , and then increases with further decreasing temperature as displayed in Fig. 2(b).

Figure 2 Temperature dependence of key parameters describing the DyFeO3 incommensurate magnetic ordering. (a) Normalized integrated intensities of the sum and first- (I1), third- (I3) and fifth-order (I5) satellite reflections. (b) Modulation vector (0, 0, δ) length. The saturation temperature of the first-order satellites, = 2.8 K, corresponds to the minimum of the modulation vector length δ = 0.0187 (1) r.l.u. Note that the intensities of the third- and fifth-order satellite reflections are not saturated down to the lowest temperatures measured.

4. Discussion

Incommensurate magnetic order in orthoferrites is uncommon, as apart from DyFeO₃ an incommensurate ordering with high-order magnetic satellite reflections has only been reported for TbFeO₃ (Artyukhin et al., 2012). While the incommensurate order seems to be the ground state of DyFeO₃, in TbFeO₃ it appears as a pocket phase in the temperature range ∼2.7 < T < ∼3.5 K when a magnetic field stronger than 1 T is applied along the c axis. A comparison of DyFeO₃ and TbFeO₃ scans along the modulation vector directions is provided in Fig. 3(a). The incommensurate order in TbFeO₃ is described as a soliton lattice, rooted in the magnetic ordering of the Tb sublattice. Within one unit cell, Tb ions order in an antiferromagnetic state, with A_x = 6.5 µ_B and G_y = 5 µ_B labelled with the order parameter η. On larger length scales the antiferromagnetic domains arise with a very regular length of ∼360 Å (67 unit-cell lengths) along the b-axis direction. Thus, the order parameter η changes between ±1 in alternating domains, with sharp jumps marking domain walls with a small extent in space. This domain structure results in satellite reflections with high-order harmonics, called the soliton lattice due to its regular size. As the magnetic order of the Dy sublattice in DyFeO₃ is antiferromagnetic with moments lying within the ab plane (Ritter et al., 2022; Biswas et al., 2022; Mareschal & Sivardière, 1969), it is qualitatively similar to the case of Tb in TbFeO₃.

Figure 3 (a) Comparison of the (00L) scan of DyFeO3 at 1.75 K with the (0K1) scan of TbFeO3 at 3.3 K and 2 T magnetic field applied along the c axis from Fig. 2(b) of Artyukhin et al. (2012). (b), (c) and (d) Temperature dependence of intensities Ip, and (e), (f) and (g) temperature dependence of FWHMs Hp, of (b) and (e) third-, (c) and (f) fifth- and (d) and (g) seventh-order (pth) satellite reflections of DyFeO3 incommensurate ordering (circles), normalized to the corresponding values of the first-order satellite reflection. In panels (b)–(g), uncertainties are given as grey ribbons that extend to one standard deviation in either direction. Analogous values extracted from Fig. 2(b) of Artyukhin et al. (2012) are displayed as squares for comparison.

Despite these similarities, there are significant differences between the rare earth magnetic orderings in DyFeO₃ and TbFeO₃. Incommensurate ordering seems to be the ground state for DyFeO₃, but for TbFeO₃ it requires an external magnetic field. The modulation vector of DyFeO₃ is along the c axis, while that of TbFeO₃ is along the b axis. In both DyFeO₃ and TbFeO₃, the rare earth magnetic moments lie in the ab plane, which means that, in DyFeO₃, Dy³⁺ magnetic moments are perpendicular to the modulation vector, and in TbFeO₃, Tb³⁺ magnetic moments have components both perpendicular and parallel to the modulation vector. Across the magnetic phase transition in TbFeO₃, the length of the incommensurate part of the modulation vector length δ increases from zero to δ ≃ 0.015 r.l.u.; in DyFeO₃, however, it only depends slightly on temperature, 0.0187 (1) ≤ δ < ∼0.02 r.l.u. [Fig. 2(b)]. For TbFeO₃ (Artyukhin et al., 2012), the intensities of higher-order satellites decrease very slowly with satellite order, while their FWHMs increase very slowly with satellite order, as shown in Fig. 3. This is different in DyFeO₃, where much larger dependencies of satellite intensities [Figs. 3(b)–3(d)] and widths [Figs. 3(e)–3(g)] on satellite order were observed.

As noted in the Results section, the saturation of the intensity of the first-order satellite reflections in DyFeO₃ occurs at the characteristic temperature ≃ 2.8 K, which corresponds to the temperature of the δ(T) minimum marked with a dashed vertical line in Fig. 2. This satellite intensity measures the volume of the magnetically ordered Dy regions. The ordered regions grow in size from to , and marks the temperature when magnetic Dy order spans the whole crystal. Note that within a Dy magnetically ordered region the soliton lattice exists with its signature regular domain structure. As the modulation length is inverse to the domain size, the domains within the soliton lattice grow, reaching a maximum length of ∼204 Å at the saturation temperature . However, at the Dy magnetic order is not settled. While the intensity of the first-order satellite is saturated between and the base temperature 1.62 K, the intensity of the third- and fifth-order satellites increases with decreasing temperature [Fig. 2(a)], suggesting subtle changes in the soliton lattice. According to the analysis of the intensity ratios of the magnetic satellites, we propose that in this temperature range this concerns the fine structure of the domain walls.

The intensities of the high-order satellites reduce with increasing temperature faster than those of the first satellite. In Figs. 3(b)–3(d) dashed lines mark the values expected for the relation I_p/I₁ = 1/pⁿ for n = 1, 2, 4. The value n = 2 corresponds to the square modulation of the order parameter, while larger values of n may describe a situation where the order parameter changes smoothly across the domain wall, such that the domain walls extend in length. The domain walls form a buffer layer between adjacent antiferromagnetic domains with a reduced magnetization of the Dy sublattice. Figs. 3(b)–3(d) show that the measured satellite intensities fall between the values n = 2 and n = 4 and become close to the square mod­ulation (n = 2) at low temperatures. With decreasing tem­perature, the relation tends towards lower n values, suggesting that the domain walls reduce in size. Further, the satellite intensities for TbFeO₃ fall close to the n = 1 case. An extended analysis (Artyukhin et al., 2012) shows that this calls for an extension of the simplistic model employed here but is qualitatively consistent with the square modulation of the order parameter.

A recent report on single-crystal neutron diffraction measurements of DyFeO₃ (Nikitin et al., 2025) shows a temperature evolution of the 001 magnetic satellite reflections similar to Fig. 2. The authors use a similar model to that of Artyukhin et al. (2012) and show that magnetic domains exhibit a log-normal distribution of size with a strongly reduced magnetization at the domain walls. According to Nikitin et al. (2025), the widths and intensities of satellite reflections correspond to the variance of the distribution of domain lengths and spatial extent of the domain walls, respectively. While Nikitin et al. (2025) applied their model to follow the evolution of the soliton lattice as a function of the applied field, we can utilize it to follow the evolution as a function of temperature. This shows that the domain length distribution does not change significantly with temperature, as evidenced by the satellite reflection width in Figs. 3(e)–3(g), while the spatial extent of the domain walls reduces strongly with decreasing temperature, as the satellite reflections increase with decreasing temperature [Figs. 3(b)–3(d)]. This is consistent with our findings, which are based solely on an analysis of reflection widths and intensities.

The DyFeO₃ modulation vector length varies between different studies. Reports on single crystals give 0.0149–0.0169 r.l.u. (Nikitin et al., 2025) and 0.0187–0.0205 r.l.u. [Fig. 2(b), this study], while reports on powder samples give 0.0173 (5) r.l.u. at 2 K (Biswas et al., 2022) and 0.0265–0.0290 r.l.u. (Ritter et al., 2022). The ranges in length correspond to the investigated temperature ranges. Note that the values reported by Ritter et al. (2022) are significantly (∼75%) greater than the correponding values in the work of Nikitin et al. (2025). Moreover, the reported is inconsistent; it was reported for the range 3.7–4.7 K [Nikitin et al. (2025), Biswas et al. (2022) and Ritter et al. (2022), and citations therein]. As such, the magnetic properties of DyFeO₃ seem to be sensitive to the sample form, preparation method, presence of impurities or temperature/field treatment protocol.

The Fe³⁺ magnetic order below is still under debate. The same Fe³⁺ magnetic order above and below was reported by Wang et al. (2016) and Ritter et al. (2022), while Biswas et al. (2022) reported a order below . Additionally, on the basis of measurements of the thermal conductivity, magnetization and electric polarization, Zhao et al. (2014) reported the unresolved magnetic structure labeled `Fe_III', stabilized by an applied magnetic field below , and suggested that it could be a ground state of the Fe³⁺ magnetic sublattice below ∼500 mK. Here, in the investigated temperature range of 1.62–4.86 K, the 001 commensurate reflection is present as shown in Fig. 1(b). Its intensity is relatively weak around and above but it was not visible in a test measurement at a different wavelength, proving that it originates from multiple scattering and showing the high crystalline quality of the sample. Below the temperature of saturation of the first-order satellite reflections, = 2.8 K, the commensurate reflection 001 is relatively strong. In the Pbnm space group, the 001 nuclear reflection is extinct. This commensurate peak could originate from the presence of a weak A_x component of Fe³⁺ magnetic ordering, allowed in the phase, or might be caused by the weak commensurate component of the Dy³⁺ ordering.

5. Conclusions

We report on the incommensurate magnetic ordering of DyFeO₃ below 4.1 K using high-resolution neutron diffraction. We have investigated the characteristics of the magnetic satellites of the 001 reflection as a function of temperature. The results were compared with the soliton lattice model in TbFeO₃. We have analysed the formation of the soliton lattice as a function of temperature, showing that first the magnetically ordered Dy regions grow in size to populate the whole volume of the crystal, while later the domain walls reduce in size down to the lowest temperature measured. The magnetic properties of DyFeO₃ seem to be sensitive to the sample form, preparation method or presence of impurities. Understanding the reasons for these differences could lead to the possibility of precisely controlling the DyFeO₃ magnetism.