Studies of magnetic properties of EuSnP single crystals

Abstract

Europium-based compounds exhibit a wide range of intriguing properties due to the element’s ability to exist in two valence states: Eu$^{2+}$, which carries a strong magnetic moment, and non-magnetic Eu$^{3+}$, as well as due to interactions between localized f-electrons and conduction electrons. In this work, we present a comprehensive study of EuSnP single crystals using X-ray diffraction, heat capacity, dc and ac magnetic susceptibility, magnetization, and Mössbauer spectroscopy measurements. EuSnP undergoes an antiferromagnetic transition at $T_\text {N}=21$ K. However, our results indicate that magnetic correlations emerge well above the transition temperature. Mössbauer spectroscopy revealed that Eu valence in EuSnP is 2+ contradicting earlier suggestions of its mixed-valent state. It also shows exceptionally high values of the effective magnetic field (39 T) and large electric field gradient at Eu nuclei likely due to the short distances between Eu atoms and their nearest neighbors. Based on our studies, we propose two possible magnetic ordering schemes of Eu magnetic moments. In the first one, moments are aligned ferromagnetically within each Eu-P plane and there is an antiferromagnetic coupling between Eu-P planes forming a bilayer. In the second one, there is a ferromagnetic order within the entire bilayer and neighboring bilayers are coupled antiferromagnetically. In both scenarios magnetic moments are oriented along or close to the crystallographic c-axis.

Introduction

Compounds containing rare earth elements show a wide range of intriguing properties, which are the result of the interaction of localized f electrons and delocalized conduction electrons. Among rare earths, the europium is particularly interesting because it can have two energetically close, valence states, i.e., Eu$^{2+}$ and Eu$^{3+}$, which results in very different properties of materials. Eu$^{2+}$ with 4$f^7$ electronic configuration (L = 0, S = J = 7/2) has a large magnetic moment while Eu$^{3+}$ with 4$f^6$ configuration (with L = S = 3, J = 0) is not magnetic. In some compounds a change in Eu valence state can occur as a result of a change in temperature¹ or the application of external pressure². There are also materials, e.g. EuNi$_2$P$_2$ or EuZnSb$_2$ that show an intermediate valence state of Eu^3,4. However, one has to be careful with such an interpretation, since Eu$^{2+}$ can oxidize relatively easily creating, e.g., oxide of Eu$^{3+}$ on the surface of the studied material, i.e. a possible presence of Eu$^{3+}$ may not be due to a mixed valence but due to impurities. The Mössbauer spectroscopy, which we used in this study, is one of the very few methods that can quantitatively distinguish between these two valence states of europium.

EuSnP compound has not been studied extensively and there are only a few reports^5,6,7,8,9 of its properties. It has a rather unique crystal structure, P4/nmm space group and NbCrN structure type. To our knowledge, it is the only Eu ternary compound of this structure. It can be viewed as a bilayers formed by Eu-P planes separated by a corrugated Sn layer with very short bonds formed between Eu-P-Sn along the crystal c-axis (i.e. perpendicular to Eu-P planes)⁵. The electrical resistivity of EuSnP has been reported to show a metallic-like temperature dependence and the material undergoes an antiferromagnetic (AF) transition at $T_\text {N}$=21 K⁵. The field dependence of magnetization showed a complicated behavior, including a spin-flop meta-magnetic transition suggesting rather complex magnetic interactions and magnetic structure⁶, which has not been resolved. It has also been suggested that EuSnP may be a mixed valent compound based on the small value of the magnetic entropy obtained from the heat capacity measurements⁶. The presence of Eu-P bilayers has been suggested to be responsible for extremely high increase of $T_\text {N}$ with pressure, which increased to 100 K upon applying 8 GPa pressure^7,8. This rate of $T_\text {N}$ increase with pressure is record high for Eu compounds⁸. No change of Eu valence occurred upon applying this pressure. It is also worth mentioning that a non-magnetic analogue compound SrSnP, which has the same structure, is a superconductor with a critical temperature of 2.3 K¹⁰.

To better understand the properties of EuSnP, we conducted a comprehensive study that included temperature and magnetic field-dependent X-ray diffraction, heat capacity, dc and ac magnetic susceptibility, magnetization, and Mössbauer spectroscopy measurements.

Experimental

Single crystals of EuSnP were grown using the flux method, which we recently thoroughly described¹¹. Starting materials: europium (Onyxmet, dendritic, 99.99 %, in mineral oil), tin (purity 99.999 %), phosphorous (Sigma Aldrich, 99.99 %, red phosphorus) pieces were weighed in the atomic ratio Eu:Sn:P = 1:30:1. The weighed elements were then placed into an alumina crucible, which was subsequently sealed under vacuum within a quartz ampule. The ampule was heated in chamber furnace to $1050^{\circ } C$ within 15 h, and kept at this temperature for 10 h. Then, cooling process was started, and was carried out for 170 h to a temperature equal to ${700^{\circ }}\hbox {C}$. The obtained black-plate single crystals were separated from the flux using a centrifuge. In order to remove any remaining Sn flux from the surface of the obtained single crystals we used amalgam method. Samples quality was checked by scanning electron microscopy (SEM) using a FEI Nova Nano 230 scanning electron microscope with EDX option.

X-ray powder diffraction (XRD) measurements were conducted using a Panalytical Empyrean diffractometer. For the low-temperature XRD studies an Oxford Instruments PheniX closed-cycle helium refrigerator was employed, covering the temperature range from 14 to 300 K. During the measurements the sample position was adjustment against thermal displacement of the sample stage. The temperature was maintained with stability better than 0.1 K. The resulting XRD patterns were refined using the Rietveld-type package FullProf ¹².

The heat capacity measurements were performed using the two-$\tau$ relaxation method with the heat capacity options of a Quantum Design Physical Property Measurement System (PPMS-9) in the temperature range 1.85-296 K. The sample for heat capacity measurements was affixed and thermally linked to the addenda using Apiezon N grease. A background signal from both the addenda and the grease (versus T) were recorded, as well.

Magnetic measurements: ac magnetic susceptibility and dc magnetization were performed using PPMS-9 equipped with the ACMS option. The investigated temperature range was 2-300 K and the externally applied fields were up to $\mu _0H= {9} \hbox {T}$. Due to the small sizes and brittle nature of the obtained samples, the measurements could not be done on one sample, and several crystals were measured for various purposes.

$^{151}$Eu Mössbauer spectra were collected at 300 K, 200 K, 100 K, 26 K, 16 K, and 4.2 K using a conventional constant acceleration spectrometer with the ${}^{151}\text{ Sm(SmF3) }$ source. The 21.5 keV $\gamma$-rays were detected with a NaI(Tl) scintillation detector, with an absorber surface density of approximately ${35} \hbox { mg/cm}^{2}$. The spectra were analyzed by means of a least squares fitting procedure. To determine absorption line positions and relative intensities, we used numerical diagonalization of the full hyperfine interactions Hamiltonian. The isomer shift $\delta$ is given relative to the $^{151}\text{ Sm(SmF3) }$ source at room temperature.

Results and discussion

Single crystal characterization

Example single crystals after surface cleaning are shown in Fig. 1a. They are of a black plate-like shape with surface dimensions of up to several millimeters and thickness of about 0.1 millimeter. A SEM image of selected crystal is shown in Fig. 1b where we can see its rectangular shape reflecting features of its crystal structure, which is described below. We also carried out an EDX analysis, which is presented in Fig. 1d and we obtained the following elemental (atomic %) composition: Eu 33.61, Sn 33.69, P 32.70, which is in very good agreement with the expected stoichiometry. Panels c), e) and f) of Fig. 1 present elemental maps and one can see a uniform distribution of Eu, Sn and P. The area marked with a dashed line shows traces of remaining Sn flux on the crystal surface, which corresponds to brighter area in Sn map and darker in Eu and P, which is due to limited penetration depth of the electron beam. Some traces of Sn can remain on the surface despite attempts to remove it, however for our measurements we tried to select crystals as clean as possible.

Fig. 1

a) Photo of example EuSnP crystals, b) SEM image, d) EDX spectra, c), e) and f) SEM images of EuSnP single crystal with individual elemental maps provided by x-ray microanalysis. Reproduced from K. Podgórska et al., Synthesis of europium-based crystals containing As or P by a flux method: Attempts to grow EuAgP single crystals, Solid State Sciences 158 (2024) 107736, Copyright $\copyright$ 2024 Elsevier Masson SAS. All rights reserved.

Fig. 2

Results of X-ray diffraction studies in the temperature range 15-300 K. a) Rietveld refinement (blue line) of diffraction pattern collected at 300 K with Bragg positions marked by bars, reflection at 32$^{\circ }$ of 2$\theta$ originates from remains of Sn flux. b) temperature dependence of a and c lattice parameters, c) temperature dependence of the unit cell volume V with fitted Debye curve 1, d) behavior of a and c parameters in the vicinity of the magnetic ordering temperature with colored areas indicating the uncertainties.

X-ray diffraction (XRD) patterns of EuSnP collected in the temperature 15-300 K range can be fully indexed on the basis of the tetragonal P4/nmm space group (No. 129). The refined XRD pattern at $T = {300} \hbox { K}$ is shown in Fig. 2a, with corresponding lattice parameters a = 4.2979(1) Å and c = 8.7848(2) Å. Small not indexed reflection at 32$^{\circ }$ of 2$\theta$ originates from remains of Sn flux. These values are in good agreement with a previous report⁵. The temperature dependence of lattice constants is shown in Fig. 2b and we observe, as expected, a lattice contraction upon cooling with no deviations from a monotonic behavior down to about 24 K where a parameter first shows a stronger decrease followed by an upturn starting at about 21 K, on the other hand the c parameter starts increasing from about 22 K to 18 K where it becomes nearly constant (Fig. 2d). While these deviations are not large (of about 0.01% of the value at 25 K), they are above the error margin, and they indicate an interplay of magnetic and structural degrees of freedom. Interestingly, deviations from monotonic behavior start at temperatures few degrees higher than magnetic ordering temperature measured by magnetic susceptibility or heat capacity measurements discussed below, which is similar to e.g. trigonal antiferromagnetic EuZn$_2$P$_2$ studied by us recently¹³. Fig. 2c presents the temperature dependence of the calculated unit cell volume, V with a fit in the paramagnetic state approximated by the Debye formula¹⁴:

$$\begin{aligned} V = V_0 + I_C \frac{T^4}{\theta ^3 _D} \int \limits _{0}^{\frac{\theta _\text {D}}{T}} \dfrac{x^3}{e^x - 1} \,dx, \end{aligned}$$

(1)

where $V_0$ is the unit cell volume extrapolated to 0 K, $I_C$ is a slope of the linear part of the V(T) dependent on Grüneisen and compressibility parameters, while $\theta _\text {D}$ is the Debye temperature. From the fit we obtained: $V_0$=169.69 $\pm 0.01$ Å, $I_C = 0.020 \pm 0.001$ Å$^3$/K and $\theta _\text {D}$=193±8 K.

The crystal structure has been described in more detail previously⁵ and here we mainly focus on the local surrounding of Eu atoms as it is important for analysis of Mössbauer spectroscopy results shown below. In the structure, there are three layers repeating along the c-axis. The first one consists of corrugated Sn sheets and between these layers there are two planes consisting of Eu and P atoms. Eu atoms in the planes are surrounded by a square plane of P atoms, which at room temperature are at distances of 3.039 Å and Eu atoms at distances of 4.298 Å. Directly above and below Eu there are P and Sn atoms at distances of 2.964 Å and 3.361 Å, respectively (Fig. 3a).

Fig. 3

a) Local coordination of the Eu atom showing that the c-axis is a fourfold rotation axis. b) Representation of the EuSnP as consisting of Sn-P-Eu quasi molecules.

From the metallic-like dependence of the electrical resistivity⁵ the studied compound can be classified as an intermetallic. Europium in EuSnP has the electronic state of the 4f shell corresponding to that of $\text{ Eu}^{2+}$, which is consistent with the Mössbauer spectroscopy results and magnetic properties measurements presented in this paper. This corresponds to a filling of the 4f shell (from the initial $4f^6$ free atom configuration) by a valence electron (5d). The electronegativity of europium in the Pauling scale has a value between 1.17 (electronegativity for Sm) and 1.20 (electronegativity for Gd), while tin and phosphorus have an electronegativity of 1.96 and 2.19, respectively¹⁵. On the basis of that, we estimate that the Eu–P bond has a polarized covalent character, and the Sn–P bond can be regarded as a covalent one. Therefore, we propose that we deal with a compound with a presence of covalent character bonds, and as a result we should call it mixed europium tin phosphide. Moreover, due to the sizes of the atomic radii of each element in EuSnP (and the possible molecular orbitals overlapping), the maximal bond lengths should not exceed 3 Å. Therefore, there are only two bonds meeting this criterion, i.e., Eu–P and Sn–P (see Fig. 3b). Thus, EuSnP can be considered as consisting of quasi molecules of Sn-P-Eu organized into a crystal lattice.

Heat capacity results

Temperature dependence of the specific heat capacity, $C_{p}$, is shown in Fig. 4. At high temperatures experimental $C_{p}$ reaches a value of 74.2 J/(mol$\cdot$K), which is close to the Dulong-Petit high temperature limit of $C_V=3nR$=74.8 J/(mol$\cdot$K) with n=3. At low temperatures, one can notice a sharp peak at about 20.2 K, which is a signature of antiferromagnetic ordering of Eu$^{2+}$ magnetic moments taking place at this temperature. Far enough above this peak, we observe a typical temperature dependence of the specific heat due to phonon contribution, which in the simplest approach is often modeled using only Debye’s theory of heat capacity with one Debye’s temperature. The result of such a fit (in the temperature range starting from 35 K) is shown in Fig.4 and clearly does not describe the experimental data satisfactorily. Therefore, we use a model that also takes into account a contribution arising from optical phonons (Einstein’s model). The physical necessity of using both contributions comes from the three-element type of the compound studied. We modeled the experimental data (at temperatures sufficiently higher than the magnetic peak) using the formula^16,17:

$$\begin{aligned} C_{\text {ph}}=\dfrac{R}{1-\alpha T} \left[ 9 \left( \dfrac{T}{\theta _\text {D}} \right) ^3 \int \limits _{0}^{\frac{\theta _\text {D}}{T}} \dfrac{x^4 e^x}{ \left( e^x - 1 \right) ^2} \,dx + \sum _{i} \frac{m_i \left( \dfrac{\theta _{\text {E}_i}}{T} \right) ^2 e^{\theta _{\text {E}_i}/T} }{\left( e^{\theta _{\text {E}_i}/T}-1 \right) ^2}\right] , \end{aligned}$$

(2)

where $\theta _\text {D}$ is the Debye temperature, $\theta _{\text {E}_i}$ are Einstein temperatures with $m_i$ as corresponding multiplicities for each individual optical branch, $\alpha$ stands for an anharmonic coefficient, and R is the universal gas constant. There can also be an electronic contribution, $C_{el}=\gamma T$ to the heat capacity. However, $\gamma$ value could not be reliably determined from the low-temperature data because of the presence of $C_{mag}$ and, therefore, $\gamma$ was fixed at zero in the final fit.

Fig. 4

Temperature dependence of the specific heat capacity, $C_p$ (solid black points). The solid grey line is a modeled phonon contribution, $C_{ph}$ using only the Debye’s model. The solid blue line is $C_{ph}$ modeled using the Debye’s and Einstein’s model (Eq.2), the solid orange line is the magnetic contribution ($C_{mag}$) to the heat capacity obtained as a difference between $C_p$ and $C_{ph}$. Inset shows temperature dependence of magnetic entropy, $S_{mag}$ calculated from $C_{ph}$ modeled using the Debye’s and Einstein’s contributions.

In order to facilitate the analysis, the summation over 6 independent optical branches was grouped into 2 branches with threefold degeneracy (i.e. $m_i$=3). The fit to the experimental data was done in the temperature range from 35 K to 281 K, i.e., far from the T$_N$, and it is represented by the solid blue curve in Fig. 4. Parameters obtained from the fit are presented in Table.1. In particular, we obtained Debye temperature $\theta _\text {D}=200 \pm 5$ K, which is in very good agreement with the value obtained from our XRD studies discussed above (193±8 K). The lower Einstein temperature, $\theta _E$=70 K likely corresponds to phonons of heavy Eu and the higher one (312 K) to light P atoms.

Table 1 Debye ($\theta _\text {D}$) and Einstein ($\theta _{\text {E}_i}$) temperatures and anharmonic coefficient ($\alpha$) obtained from the fitting of Eq. (2).

Fig. 5

Temperature dependence of the specific heat capacity, $C_p$ (panel a)) and $C_{\text {mag}}$ (panel b)) measured at different magnetic fields applied along the crystal c-axis.

The magnetic contribution to the specific heat calculated as:

$$\begin{aligned} C_{\text {mag}}(T) = C_p(T) - C_{\text {ph}}(T) \end{aligned}$$

(3)

is presented in Fig. 4 as a solid orange line. One can notice that the peak at $T_\text {N}$ is very sharp and its intensity amounts to nearly 60 $\hbox { J mol}^{-1} \hbox { K}^{-1}$ while from the mean field theory for an antiferromagnet a value of 20 $\hbox { J mol}^{-1} \hbox { K}^{-1}$ is expected¹⁸. Moreover, $C_{\text {mag}}$ remains finite also somewhat above $T_\text {N}$ and vanishes only at higher temperatures (at about 35 K), which is a sign of short range magnetic correlations present above $T_\text {N}$^19,20, which are not accounted for in the mean field theory. A high value of $C_{\text {mag}}$ jump at $T_\text {N}$ is sometimes taken as a sign of the first order transition^21,22. To verify if the transition is of the first-order with a “latent heat”, the peak range at $T_\text {N}$ (around 20 K) was also additionally measured using the method of a single slope analysis of relaxation curves also implemented in the Quantum Design PPMS heat capacity option. However, no significant differences, within experimental uncertainties, were observed between the results obtained by the two methods. In addition to the absence of “latent heat”, no thermal hysteresis was observed at the $T_\text {N}$ peak position in measurements with increasing and decreasing temperatures. Thus, we conclude that the transition is not of the first-order type.

The magnetic entropy $S_{\text {mag}}(T)$ was calculated from the $C_{\text {mag}}(T)$ using the following formula:

$$\begin{aligned} S_{\text {mag}}(T)=\int \limits _{0}^{T} \dfrac{C_{\text {mag}}}{T'} \,dT'. \end{aligned}$$

(4)

Since the measurement of heat capacity was carried out only down to 1.85 K, the $C_{\text {mag}}/T$ data were extrapolated from 1.85 K down to 0 K using a power function in the form of $y=ax^b$ (fitted to the low-temperature experimental points). The calculated magnetic entropy between $T = {0} \hbox { K}$ and 37.4 K was determined by numerical integration of the $C_{mag}/T$ curve and is equal to ${11.2} \hbox { J.mol}^{-1} \hbox { K}^{-1}$, see inset of Fig.4. The theoretical value of $S_{\text {mag}}$ of one mole of particles with spin J in a magnetic field is given by $S_{\text {mag}}= R \ln {(2J+1)}$, which for $J=7/2$ (value for Eu$^{2+}$ ions) amounts to 17.28 ${\hbox {J mol}^{-1} \hbox { K}^{-1}}$. Our experimental value is significantly lower than the theoretical one. Although in the model taking into account only the Debye’s contribution the magnetic entropy obtained is higher, 13 ${\hbox {J mol}^{-1} \hbox { K}^{-1}}$ at $T_\text {N}$ and it increases up to 19 ${\hbox {J mol}^{-1} \hbox { K}^{-1}}$ (higher than expected value), it is only due to the fact that this model does not describe the experimental data properly, underestimating the phonon contribution in the low temperature region (see Fig.4). A previous study on EuSnP also reported low value of magnetic entropy of about 13 ${\hbox {J mol}^{-1} \hbox { K}^{-1}}$⁶. While this value is somewhat higher than ours we note that Fujiwara et al. did not subtract the phonon contribution and simply integrated experimentally measured $C_p$, which naturally results in a higher value of magnetic entropy. Therefore, in this compound, magnetic entropy lower than expected appears to be common. Fujiwara et al. suggested that the reason could be due to intermediate valence state of Eu (i.e. between Eu$^{2+}$ and Eu$^{3+}$), however as we discuss below our Mössbauer spectroscopy results indicate that this is not the case. We would like to note that a lower value of magnetic entropy than the theoretical has been observed in numerous compounds with antiferromagnetic order of Eu moments^{13,19,20,23,24,25,26} and it has been attributed to short range magnetic correlations that are present at temperatures higher than $T_\text {N}$.

Fig. 5 shows the temperature dependence of $C_p$ and $C_{\text {mag}}$ measured at magnetic fields of up to 7 T. Due to the sample shape, field-dependent measurements were only possible for magnetic field oriented along the crystal c-axis. Typically for an antiferromagnet it is observed that as the magnetic field increases the peak in $C_{\text {mag}}$, first moves to lower temperatures and broadens, and at sufficiently high field vanishes. Clearly this is not what we observe for EuSnP. Fig. 5b, which presents temperature and field dependence of $C_{\text {mag}}$ shows that while there is a shift to low temperatures and broadening the peak splits into two, which is rather unusual behavior. The splitting is already visible at 1 T. While the temperature of the most intense feature remains almost unchanged there is a shoulder at about 17.5 K. In fact, the temperature of the most intense feature remains almost unchanged up to field of 3 T and it only moves to lower temperatures (to about 17.5 K) at 5 T. On the other hand the low temperature feature shifts to 13.9 K at 1.5 T and 8.9 K at 2 T and has already disappeared at 3 T. At 7 T the temperature dependence of $C_{\text {mag}}$ becomes featureless. Similar splitting of $C_{\text {mag}}$ peak has been observed for several other compounds containing Eu and was interpreted as a reorientation of Eu moments induced by applied magnetic field^27,28,29. The temperature and field dependence of the positions of $C_{\text {mag}}$ peaks is presented in Fig.14 together with results of magnetic properties measurements discussed below.

Magnetic measurements

Dc magnetization

The temperature dependence of the molar dc susceptibility $\chi _\text {mol}(T)$ was investigated in field cooling (FC) and zero field cooling modes (ZFC), in external magnetic fields $\mu _0H$ up to ${9} \hbox { T}$ applied parallel or perpendicular to the crystallographic $c$ axis. No hysteretic behavior was observed and the ZFC data are summarized in Fig. 6.

Fig. 6

Temperature dependence of magnetic susceptibility $\chi (T)$ investigated in several external magnetic fields applied (a) parallel and (b) perpendicular to the $c$ axis, for a better visibility the subsequent data are offset along Y axis by 1, the $\chi =0$ level for each curve is marked with a solid line with a color corresponding to each field.

Fig. 7

Temperature dependencies of the molar susceptibility $\chi _\text {mol}(T)$ and the inverted susceptibility $1/\chi _\text {mol}(T)$ investigated in the ZFC mode at the external magnetic field of 9 T applied (a) parallel and (b) perpendicular to the $c$ axis; the solid lines are the fit of the Curie-Weiss law Eq. (5) in the temperature range 150-300 K.

Similarly as previously shown in the literature^5,6,8,9, the temperature dependence of molar dc susceptibility $\chi _\text {mol}(T)$ investigated at magnetic fields $\mu _0H\le {1} \hbox { T}$ has a maximum, see Fig. 6. With an increasing magnetic field, the maximum shifts to lower temperatures, indicating an antiferromagnetic transition. Below $T_\text {N}$ the susceptibility decreases with decreasing temperature, however it drops to zero only for fields applied parallel to the $c$ axis, suggesting that the c-axis is the easy magnetization direction.

Moreover, for both directions ($H\parallel c$ and $H\perp c$) with application of an external magnetic field, we observe a second transition to another antiferromagnetic structure. For $H\parallel c$ the first transition manifests itself as a sharp peak at around ${20} \hbox { K}$ at zero field and shifts to ${11} \hbox { K}$ at ${2} \hbox { T}$, and with further increase of the magnetic field it shifts below ${2} \hbox { K}$. The second transition also originates from the ${20} \hbox { K}$ point at zero field, however is only later visible at around ${16} \hbox { K}$ and ${4} \hbox { K}$ at ${5} \hbox { T}$ and ${7} \hbox { T}$, respectively. For $H\perp c$ a similar behavior as for $H\parallel c$ is observed, i.e., the first transition is refected as the “sharp peak” and the second transition can be distinguished as a “hump” at around ${16} \hbox { K}$ and ${13} \hbox { K}$ at ${2.5} \hbox { T}$ and ${3} \hbox { T}$, respectively. These two transitions are also clearly reflected in the magnetization vs. magnetic field measurements.

As mentioned in Section "Single crystal characterization", despite our greatest efforts, the remnant Sn could not be removed from the crystals’ surface, and as a result, was visible in some of the low field ($\mu _0H<{0.05} \hbox { T}$) measurements of susceptibility as a drop of $\chi _\text {mol}$ below ${4} \hbox { K}$.

The temperature dependence of the inverted susceptibility measured at $\mu _0H= {9} \hbox { T}$ is linear at high temperatures, see Fig. 7, and was fitted in the temperature range 150-300 K according to the inverted form of the Curie-Weiss law (Eq. 5).

$$\begin{aligned} \dfrac{1}{\chi }(T) = \dfrac{1}{C} T - \dfrac{\theta _\text {p}}{C}, \end{aligned}$$

(5)

with $\theta _\text {p}$ being the Curie-Weiss temperature and the Curie parameter C equal to:

$$\begin{aligned} C = \dfrac{N_\text {A}}{3k_\text {B}\mu _0}\mu _\text {eff}^2, \end{aligned}$$

(6)

where $N_\text {A}$ is the Avogadro constant, $k_\text {B}$ is the Boltzmann constant, $\mu _0$ is the permeability of vacuum, and $\mu _\text {eff}$ is the effective magnetic moment (in Bohr magnetons ${\upmu _{\textrm{B}}}$). The fitting parameters are summarized in Fig. 7 and in Tab. 2. The calculated effective moments for both directions of the applied magnetic field are close to the theoretical value of a free $\text{ Eu}^{2+}$ ion. The Curie-Weiss temperatures are positive, indicating strong ferromagnetic interactions between the nearest $\text{ Eu}^{2+}$ ions. Similar values were previously reported in the literature^5,6,8,9.

Table 2 Curie-Weiss temperatures $\theta _\text {p}$ and effective magnetic moments $\mu _\text {eff}$ determined from fitting of Eq. 5 to the experimental data and the theoretical value of $\mu _\text {eff}$ for a free $\text{ Eu}^{2+}$ ion.

Fig. 8

Field dependent magnetization $\mu _\text {at.}(\mu _0H)$ investigated at $T = {2} \hbox { K}$ for fields applied parallel and perpendicular to the $c$ axis; the dashed line marks the theoretical value of magnetization for a free $\text{ Eu}^{2+}$ ion $\mu _\text {sat}^\text {th} = {7}\,{\upmu _{\textrm{B}}}$.

Comparing the field dependence of magnetization per one $\text{ Eu}^{2+}$ ion $\mu _\text {at.}(\mu _0H)$ at $T = {2}\hbox { K}$ investigated with external magnetic fields applied parallel and perpendicular to the $c$ axis one can easily see the anisotropic behavior, cf. Fig. 8. For both directions, magnetization is zero at $\mu _0H= 0$ and at first increases linearly with increasing magnetic field. However, the slope is very different for the two directions, i.e., for $H \perp c$ the increase is approximately ${1}\,{\upmu _{\textrm{B}}}$ per ${1} \hbox { T}$, while for $H \parallel c$ the increase is two orders of magnitude smaller. No hysteretic behavior and the initial linear increase are expected for an antiferromagnetic system.

The $\mu _\text {at.}(\mu _0H)$ for $H\parallel c$ is characterized by two step-like metamagnetic transitions, the first one at around 2 T indicates a spin-flop transition. Above 7 T, where the second transition occurs, $\mu _\text {at.}$ values saturate to ${7}\, {\upmu _{\textrm{B}}}$, which is the theoretically expected value of saturation magnetization for a free $\text{ Eu}^{2+}$ ion. Therefore, this transition is a crossover to a field-induced ferromagnetic (FIF) state, where the magnetic moments of the $\text{ Eu}^{2+}$ ions are ordering ferromagnetically along to the external magnetic field.

For the $H \perp c$ direction two metamagnetic transitions are also visible, however they are not as pronounced as for the $H\parallel c$. The second transition at around 6 T is also a crossover to the FIF state. Although, not all of the magnetic moments are exactly parallel to the external magnetic field, since the $\mu _\text {at.}$ is not saturated and increases monotonously, reaching only ${6.85}\, {\upmu _{\textrm{B}}}$ at ${9} \hbox { T}$, suggesting that full saturation could possibly be reached in higher fields.

Fig. 9

Field dependencies of magnetization measured in external magnetic fields applied (a) parallel and (b) perpendicular to the $c$ axis investigated in several temperatures.

Similar features are observed for $\mu _\text {at.}(\mu _0H)$ investigated at other temperatures, see Fig. 9. The metamagnetic transitions are less pronounced and shift to lower fields with increasing temperatures. While at 300 K, the $\mu _\text {at.}(\mu _0H)$ is typical for a paramagnetic state (shows a small linear increase with the application of an external magnetic field). The field-dependent magnetization at ${30} \hbox { K}$ increases nonlinearly reaching quite high values at ${9} \hbox { T}$, more than ${4}{\upmu _{\textrm{B}}}$, despite being above the antiferromagnetic order temperature ($T_\text {N}\approx {21}\hbox { K}$). This large ferromagnetic component along the direction of the applied magnetic field is more than a half of the magnetic moment of $\text{ Eu}^{2+}$, which is not expected for a paramagnetic state at the temperature significantly higher than the magnetic ordering temperature and indicates the occurrence of the FIF state. This is consistent with the heat capacity and ac susceptibility data described in Sections "Heat capacity results" and "Ac susceptibility", respectively.

Spin-flop transition occurs in other Eu compounds, but they typically have a lower symmetry compared to that of EuSnP. A classic example is one-dimensional EuSb$_2$ or compounds with uniaxial anisotropy such as EuGa$_2$As$_2$ or Eu$_3$Ge$_2$As$_4$, which all have a monoclinic crystal structure^30,31,32 or orthorhombic Eu$_2$InSnP$_3$³³. However, two-step-like metamagnetic transitions in M(H) dependence are less common. They have been observed in monoclinic EuAs$_3$³⁴, orthorhombic EuPdSb³⁵, Eu$_5$In$_2$As$_6$³⁶, EuPtIn³⁷ or trigonal EuAg$_4$As$_2$³⁸. All of these compounds, not only have lower symmetry, and in some cases with nonequivalent Eu lattice sites, but also show two Neél temperatures. Having this in mind, EuSnP with its relatively simple crystal structure appears to be quite unique.

Ac susceptibility

Fig. 10

Temperature dependencies of (a, b) real $\chi '(T)$ and (c, d) imaginary $\chi ''(T)$ parts of the ac susceptibility investigated in ZFC mode, in external magnetic fields applied (a, c) parallel and (b, d) perpendicular to the $c$ axis; note that for panel (d) the Y axis has a different scale than panels (a-c); for a better visibility the consecutive curves are offset (a-c) by 5 and (d) by 0.5 along the Y axis, the $\chi '=0$ and $\chi ''=0$ levels are marked for each curve with a solid line with a color corresponding to each field. The gray arrows indicate subsequent transitions in panel (a) the two antiferromagnetic (1) and (2) at 2.5 T and the ferromagnetic (3) at 5 T and for panel (b) the two antiferromagnetic (1) and (2) at 3 T and the ferromagnetic (3) at 5 T.

The temperature dependence of ac susceptibility was investigated in the FC and ZFC modes using the ac driving field $H_\text {ac}= {10}\hbox { Oe}$ ($H_\text {ac}\approx {795,8} \hbox { A/m}$) with the frequency $f = {997} \hbox { Hz}$ applied parallel or perpendicular to the crystallographic $c$ axis. The real and imaginary parts of the ZFC data are presented in Fig. 10. Additionally, in Fig.11 we show a comparison between $\chi '(T)$ and the magnetic component of the heat capacity $C_\text {mag}(T)$ without the application of an external field. We observe a very good agreement of the antiferromagnetic transition temperatures (the same peak positions) in both types of measurements. Similarly as for the dc susceptibility data, the results of ac susceptibility confirm an anisotropic magnetic character of the EuSnP compound. A diamagnetic signal detected for low fields ($\mu _0H< {0.05}\hbox { T}$) measurements below ${4}\hbox { K}$ corresponds to the remnant Sn from the flux.

Fig. 11

Comparison of (a) temperature dependencies of ac magnetic susceptibility $\chi '(T)$ and magnetic component of heat capacity $C_\text {mag}(T)$ without application of an external field, and (b) temperature dependencies of first derivative of product of ac susceptibility and temperature $d(\chi ' T)/dT(T)$; the arrows point to characteristic temperatures: superconducting transition of Sn and the Neél temperature of EuSnP.

Observing closely the evolution of $\chi '(T)$ and $\chi ''(T)$ without and with application of the external dc magnetic field, one can distinguish three types of behavior, which correspond to the three regions in the magnetic phase diagram (see Section "Magnetic phase diagram", Fig. 14). Without application of an external magnetic field, the real part of ac susceptibility $\chi '(T)$, investigated using the $H_\text {ac}$ parallel and perpendicular to the $c$ axis, exhibits a peak at about $T_\text {N}= {21}\hbox { K}$ associated with the antiferromagnetic transition. Similarly as for the dc susceptibility, for $H_\text {ac}$ parallel to the $c$ axis, $\chi '(T)$ decreases nearly to zero, while for ”perpendicular fields”, the decrease is only partial. This indicates that the $c$ axis is the easy magnetization direction, which is consistent with results of our other measurements and with literature⁸.

The single maximum is visible up to (external, $\mu _0H$) fields of $\mu _0H$=0.5 T (20 K) and 1 T (20 K) for $H\parallel c$ and $H\perp c$, respectively. With an increase of the external magnetic field (applied parallel and perpendicular to the $c$ axis) the single maximum splits into three maxima, two of them shift to lower temperatures, and one shifts to higher temperatures. This indicates that the two of them are associated with antiferromagnetic phase transitions. While the first antiferromagnetic transition (1), for the $H\parallel c$ direction, is visible as the peak that shifts at first slowly from 21 K (in zero field) to 19.7 K (3 T), then more rapidly to 17 K (5 T) and finally to 4 K at 7 T; the second (2) antiferromagnetic transition is only clearly distinguished as a pronounced peak at 19 K in 1 T field and shifts to ${13}\hbox { K}$ and 7 K in 1.5 T and 2 T, respectively. Similarly, for the $H\perp c$ direction, the first transition (1) shifts slowly from 21 K (zero field) to 18 K (3 T), while the second (2) transition is visible as a broad maximum at 16 K at 2 T and shifts as far as to 10 K at 3 T.

The third maximum (3), with ferromagnetic-like behavior, is visible as a very broad maximum (instead of a negative, a positive curvature of the $\chi '(T)$ appears above $T_\text {N}$) above 2 T and 3 T for $H\parallel c$ and $H\perp c$, respectively. This observation is a proof that the magnetic moments of the $\text{ Eu}^{2+}$ ions align along the applied magnetic field.

The imaginary part of susceptibility $\chi ''(T)$ does not have a maximum, instead at $T_\text {N}$ a slope change is observed and $\chi ''(T)$ increases more rapidly with decreasing temperature. Furthermore, for magnetic fields applied parallel to the $c$ axis the imaginary part of susceptibility at low temperatures is of the same order of magnitude as $\chi '(T)$, which is very high. Meanwhile, for fields perpendicular to the $c$ axis, $\chi ''(T)$ is typical, i.e., $\chi ''(T)$ is one order of magnitude smaller than $\chi '(T)$. The imaginary part of susceptibility corresponds to the energy dissipation in the system. Therefore, high values of $\chi ''(T)$ indicate high energy loss in the system, which may be a sign of a long relaxation time of the magnetic moments. There can be various reasons for this behavior. It can suggest that magnetic moments are easily disrupted from the equilibrium state by the driving field (even in high external magnetic fields) and they need a long time to fully return to the equilibrium state. Other explanations include the possibility of some magnetic frustration or instability in the system, for example, the magnetic order is only a metastable state. It can also be a sign of weak coupling between EuSnP–EuSnP quasi molecules with simultaneous strong polarization of the magnetic moment of the $\text{ Eu}^{2+}$ ion within the EuSnP molecule. This high dissipation of energy is most likely reflected in $C_p(T)$ as the very high peak at $T_\text {N}$.

The spin flop metamagnetic transition manifests itself also in the ac susceptibility, where for $\mu _0H={2}{\hbox { T}}\parallel c$ the $\chi '(T)$ exhibits a huge maximum and simultaneously the $\chi ''(T)$ increases rapidly and is of the same order of magnitude as $\chi '(T)$. Such a behavior is not observed for $H\perp c$ as could be expected from a lack of spin-flop transition along this direction.

$^{151}$Eu Mössbauer spectroscopy results

Fig. 12

$^{151}$Eu Mössbauer spectra and fits (solid lines) obtained at room temperature on single crystals and powdered sample in panel a) and b), respectively. Panel c) shows geometry of the experiment, where $V_{ii}$ are components of the electric field gradient tensor and B is the effective magnetic field at the Eu site. In the paramagnetic state the angle $\theta$ is between incoming $\gamma$-rays and $V_{zz}$.

In order to obtain more information regarding magnetic properties, particularly regarding orientation of Eu magnetic moments, we conducted $^{151}$Eu Mössbauer spectroscopy measurements in the paramagnetic and antiferromagnetic states. Typically, Mössbauer spectroscopy measurements are performed in transmission geometry on a powder sample mixed with a filler (B4C in our case). A measurement on powdered single crystals is shown in Fig.12b. One can see two signals, a stronger one centered at about -10 mm/s, which is typical for Eu$^{2+}$ ions^13,39,40,41, and a weaker one at about 0 mm/s, which is due to non-magnetic Eu$^{3+}$ ions^39,42. In some compounds, Eu oxidizes relatively easily on the surface of the sample to the extent that for surface-sensitive techniques single crystals have to be cleaved⁴³. Since oxidation might also occur during grinding, we wanted to verify whether the signal from Eu$^{3+}$ originates from EuSnP or is the result of the single-crystal grinding procedure. We also noticed that single crystals are sensitive to air and disintegrate after a couple of months if they are not kept in a protective atmosphere. Therefore, we decided to try to perform measurements on the single crystals as well. Because they are very thin flat flakes, they could be relatively easily fixed to a flat sample holder. The result of the measurement on single crystals is presented in Fig.12a. One can immediately notice that the signal from Eu$^{3+}$ is much weaker compared to a measurement on powdered crystals and its intensity is only a few percent relative to Eu$^{2+}$ and is estimated more precisely from a measurement at 4 K, which is discussed below. We therefore decided to continue temperature-dependent measurements on single crystals.

Comparing panels a) and b) of Fig.12 we also notice a different shape of the signal from Eu$^{2+}$. In fact in the paramagnetic state in the presence of the electric quadrupole interaction for $^{151}$Eu there are 8 allowed transitions and corresponding lines in the spectrum. Their position is given by parameters such as: the isomer shift, $\delta$, the largest main component of the electric field gradient tensor (EFG), $V_{zz}$, and the quadrupole coupling, $\epsilon$, which is a result of the interaction of EFG with a nuclear electric quadrupole moment, $Q_g$, and, in magnetically ordered state, the effective magnetic field at the nucleus, B. Since hyperfine interactions (quadrupolar electric and magnetic) are angular dependent, in general one can obtain information on the orientation of the magnetic field, and thus Eu magnetic moment, with respect to direction of $V_{zz}$, see Fig.12c for geometry of the Mössbauer spectroscopy experiment. Another parameter is the angle $\theta$ between the incoming $\gamma$ rays and $V_{zz}$ (in the paramagnetic state) or B (in the magnetically ordered state). In the case of a powder sample with randomly oriented crystallites the spectrum averaged over the whole range of $\theta$ values is obtained^44,45.

From the room temperature spectrum measured on single crystals we obtained $\delta$ = -10.75 mm/s, a typical value for Eu$^{2+}$ ions^13,39,40,41, but one of the highest among ternary 1:1:1 Eu based compounds⁴⁶. When it comes to EFG on Eu$^{2+}$, it is expected to be rather small, which is due to the electronic configuration of $\hbox {Eu}^{2+}$, i.e. half-filled 4$f^7$ with the $^8\hbox {S}_{7/2}$ ground state, so that the 4f electron contribution to the EFG is expected to be zero. Therefore, the main contribution to EFG is expected to come from the neighboring ions, called the “lattice” contribution, which depends on the local symmetry. The local surrounding of the Eu atom is discussed above and shown in Fig.3. It results in axial symmetry, with c-axis becoming a fourfold rotation axis (4mm point symmetry group). This will cause an axially symmetric electric field gradient tensor with the largest main component, $V_{zz}$, along the c-axis⁴⁷. The presence of different atoms above and below Eu will result in a different charge distribution, and due to this, one might expect significant value of the electric field gradient at the Eu nucleus. We obtained a value of $V_{zz}=110\cdot 10^{20}$ V/m$^2$, which is 2-3 times bigger compared to compounds, in which the local environment of Eu has a higher symmetry^13,40,41. $V_{zz}$ increases with decreasing temperature, as expected due to the lattice contraction and the corresponding decrease of interatomic distances ⁴⁸, see inset of Fig.13a. The isomer shift, $\delta$, is rather temperature independent, it changes only by 4% in the entire temperature range.

Fig. 13

$^{151}$Eu Mössbauer spectra and fits (solid lines) measured at 26 K (panel a), 16 K (panel b) and 4 K (panel c). Green solid line represents the signal from the phase containing Eu$^{3+}$ ions. Inset in panel a) shows the temperature dependence of $V_{zz}$.

In the magnetically ordered state the magnetic field present at Eu nucleus causes further splitting of nuclear energy levels, which leads to several groups of peaks observed at 16 K and 4 K, see Fig.13b, c. From the fit at 4 K we estimate that the relative amount of the phase containing Eu$^{3+}$ amounts to 4.4%. Therefore, we conclude that such a small amount of the Eu$^{3+}$ containing phase cannot account for a 30% smaller magnetic entropy, as was found above from the heat capacity results, and as previously suggested⁶. This small amount of Eu$^{3+}$ sometimes occurs in Mössbauer spectra of Eu compounds, which predominantly contain Eu$^{2+}$ ions, and is taken as a result of surface oxidation^39,40,49,50. On the other hand, in mixed-valent compounds, the amount of Eu$^{3+}$ and Eu$^{2+}$ is typically comparable^51,52,53,54 and presence of Eu$^{3+}$ within the compound of interest is observed also by other techniques.

We find that the angle $\beta$ between B and $V_{zz}$ is 0.5$^{\circ } \pm$ 4.5$^{\circ }$ and 7.3$^{\circ } \pm$ 1.5$^{\circ }$ at 16 K and 4 K, respectively. Since $V_{zz}$ is oriented along the c-axis we find that Eu magnetic moments are oriented along or very close to the c-axis.

The effective magnetic field B increases with the temperature decrease, as expected, and is equal to 32.5±0.5 T and 39.0±0.7 T at 16 K and 4 K, respectively. These values are exceptionally high⁴⁶. One of the possible reasons can be the contribution of Eu 5d electrons, which participate in spin coupling in many rare-earth compounds and intermetallics⁵⁵. In the case of some Eu containing families of compounds it has been shown that the effective magnetic field increases with decrease of inter-atomic distances⁴⁹, very high field observed in EuSnP may be related to very small distances between Eu and surrounding atoms, 3.039 Å to P atoms in the plane and only 2.964 Å to P located above Eu (see Fig.3). Unfortunately, there are no other equiatomic 1:1:1 Eu compounds with the same crystal structure and type as EuSnP, and we cannot make a direct comparison to compounds with the same local Eu surrounding. Although there are 1:2:2 Eu compounds with the same space group, they have different structure type, CeBe$_2$Ge$_2$ versus NbCrN of EuSnP resulting in different local surrounding of Eu e.g. lack of other atoms in the Eu plane. There are reports of the effective fields for EuCu$_2$Sb$_2$ (29 T)⁵⁶, EuPd$_2$Sb$_2$ (19.5 T)⁵⁷ and EuZn$_2$Sn$_2$ (29 T)⁵⁸, which all have the shortest distance between Eu and other atoms of about 3.4 Å. Among 1:1:1 phosphides, EuPtP, EuLiP, EuCuP, for which values of B are reported, have hexagonal crystal structures⁴⁶ and have B of 31.9 T, 25.8 T, and 30 T, respectively^46,59,60. For these compounds the shortest distances are in the range 3.10 Å - 3.14 Å. For an orthorhombic arsenide EuLiAs a value of B=22.7 T was reported⁴⁶, the shortest distances here are 3.13 Å. Even if we go beyond the 1:1:1 family, there are only very few compounds with higher values of B, e.g. EuRh$_2$Si$_2$ (43.5 T)⁶¹ or EuNi$_2$Ge$_2$ (39.8 T)⁶². For this family of ThCr$_2$Si$_2$-structure type compounds it has been shown that the effective magnetic field increases with decrease of Eu-transition metal distance⁴⁹.

Magnetic phase diagram

Fig. 14

Magnetic phase diagram of EuSnP constructed based on magnetic and heat capacity measurements. For both directions of applied magnetic field several magnetic phases can be distinguished: paramagnetic (P), antiferromagnetic (AF), antiferromagnetic after a metamagnetic transition (AF1 and AF2) and a field induced ferromagnetic state (FIF). The characteristic points for Neél temperature ($T_\text {N}$), crossover field from AF1 (or AF2) to FIF ($H_\text {cr}$), and crossover temperature from P to FIF ($T_\text {FIF}$) were determined from: the maxima from Fisher’s method applied to ac and dc susceptibility (max. in AC Fisher and max. in DC Fisher, respectively), minimum in the second derivative of dc magnetization (min. in $\textsf{d}^{\textsf{2}}\mathsf {M/dH}^{\textsf{2}}$), maximum in the first derivative of dc magnetization (max. in DC $\mathsf {dM/dH}$), maximum in heat capacity (max. in HC), and maximum in the real part of ac susceptibility (max. in AC $\chi '(\textsf{T})$).

The magnetic phase diagram, summarizing the results of our dc and ac susceptibility, magnetization, and heat capacity measurements, is shown in Fig.14. The determination of individual transition points was performed using the same methods described in our previous paper¹³. The “max. in DC $\mathsf {dM/dH}$” points on the phase diagram were determined by taking the maximum in dM/dH vs. H.

In the constructed phase diagram we can distinguish several magnetic phases. At zero magnetic field, EuSnP changes its state from a paramagnetic (P) to an antiferromagnetic (AF) state (where the magnetic moments of $\text{ Eu}^{2+}$ are along the $c$ axis). When external magnetic field is applied parallel to the $c$ axis, EuSnP undergoes a spin flop transition and the system is in a spin flop antiferromagnetic state (AF1). For $H \perp c$, the system undergoes a metamagnetic transition and a new antiferromagnetic state is formed (AF2). Between the paramagnetic state and the AF1 (or AF2) state we observe a crossover state. This new state is the field induced ferromagnetic state (FIF). The transition between the antiferromagnetic state (AF1 or AF2) and the FIF occurs at characteristic crossover fields $H_\text {cr}$. Although FIF state is similar to the paramagnetic (P) state (parallel alignment is not directly caused by the interactions between magnetic moments), a certain specific temperature $T_\text {FIF}$ (for a given field) can be defined at which the moments attain a forced ferromagnetic alignment.

Summary and conclusions

EuSnP has a relatively simple, but rather uncommon, crystal structure (NbCrN structure type). To our knowledge, it is the only Eu compound with 1:1:1 stoichiometry of this type of structure. It features Eu atoms in Eu-P planes and two of such neighboring planes are separated by a Sn layer. The distances between Eu and its nearest neighbors are quite short, which has consequences in electronic properties as, e.g., evidenced by high values of the effective magnetic field (39 T) and large electric field gradient at Eu nuclei shown by Mössbauer spectroscopy. It has been suggested that Eu in EuSnP might be in a mixed-valence state. However, our studies show that the electronic state of the 4f shell of Eu corresponds to that of Eu$^{2+}$ ion. Although Mössbauer spectroscopy shows the presence of a small amount of Eu$^{3+}$, it is due to impurities, the amount of which was increasing upon powdering of single crystals.

The EuSnP compound is an antiferromagnet with a $T_\text {N}$ of 21 K. However, magnetic correlations start at temperatures significantly higher than the ordering temperature, as indicated by smaller than expected magnetic entropy and also dc and ac susceptibility results, e.g., the field dependence of magnetization at ${30}\hbox { K}$ (above $T_\text {N}$) increases nonlinearly and at ${9}\hbox { T}$ attains quite high values (see Section "Dc magnetization" Fig. 9). Heat capacity measurements also show a very high peak of its magnetic part, $C_{\text {mag}}$, which for the external magnetic field oriented along the c-axis was splitting, suggesting a reorientation of magnetic moments. The presence of metamagnetic transitions, also for the external field perpendicular to the c-axis, was evidenced by measurements of magnetization and ac susceptibility as well. Magnetization versus field curves show unusual two-step like behavior suggesting that magnetic order and its field dependence are complex. For the field applied along the c-axis a spin-flop transition occurs at about 2 T. Based on our measurements, we constructed magnetic field-temperature phase diagrams, which show transitions between different magnetic phases of EuSnP.

Fig. 15

Possible order of Eu magnetic moments at zero magnetic field.

Considering that the sign of the paramagnetic Curie temperature is positive, this indicates ferromagnetic correlations between the nearest neighbors moments. Moreover, from the Mössbauer spectroscopy we show that at zero external magnetic field $\text{ Eu}^{2+}$ magnetic moments are aligned along or close to the c-axis. Taking these results into account and looking at the crystallographic structure, we can find at least two plausible interpretations of the possible magnetic structure.

In the first interpretation, one can see that Eu atoms are arranged in Eu-P bilayers separated by Sn layer (Fig.3). Interestingly, the shortest Eu-Eu distances within the Eu-P plane (4.278 Å) and the distance to the nearest Eu atoms located in the second plane of the bilayer (4.237 Å) are almost the same. The mechanism responsible for the ferromagnetic coupling within a bilayer could be due to the superexchange trough orthogonal Eu-P-Eu bonds formed by Eu-P in the plane to Eu located along the c-axis in the second plane. The Eu-P bond along the c-axis is the shortest one formed by Eu (2.964 Å) while the Eu-P distance within the plane is 3.039 Å. According to Goodenough-Kanamori-Anderson rules^63,64 in such an orthogonal case one can obtain a ferromagnetic coupling. Thus, in this scenario, we propose that at zero magnetic field, magnetic moments are aligned along the c-axis or inclined at a small angle (of 7$^{\circ }$ at 4 K), as indicated by Mössbauer results. Since the four-fold symmetry of the c-axis corresponds to four equivalent Eu sites in the crystallographic structure, the resulting magnetic structure could be a tilted one within the bilayers and adjacent bilayers would couple antiferromagnetically (Fig.15a). We cannot exclude more complicated magnetic structure as e.g. helical or cycloidal ones, but in the case of the axial symmetry the Mössbauer spectroscopy is not sensitive to rotation of magnetic moment about the c-axis. This could explain a complex shape of M(H) curves.

The alternative interpretation, is that the magnetic moment in a single Eu-Sn-P quasi-molecule is first of all determined by the polarization of the electron densities in this molecule (due to the chemical bonds). Therefore, in each Eu-P plane Eu magnetic moments are coupled ferromagnetically with a possible small inclination from the c-axis, as mentioned above, while the coupling to the second Eu-P plane of the bilayer is antiferromagnetic (Fig.15b).

However, any tentative hypothesis would require a proof from other studies, for example: neutron diffraction, synchrotron radiation spectroscopy, muon spin rotation spectroscopy, or NMR spectroscopy studies.