Study of the Antiferromagnetic State Nematics in EuFe₂As₂ by Using Spin-Resonance and Magnetic Measurements

Using electron spin resonance spectroscopy and SQUID-magnetometry we obtained direct evidence of the occurrence of magnetic domains in the antiferromagnetically ordered state of a ${\text{EuF}}{{{\text{e}}}_{{\text{2}}}}{\text{A}}{{{\text{s}}}_{{\text{2}}}}$ single crystal. The resonance spectra of europium ions were measured in the temperature range from 4 to 200 K. Using an equation for the resonance field in an antiferromagnet that takes into account the exchange and anisotropy fields, we have performed an analysis of the angular dependence of the spectrum at a temperature of 4.8 K, measured upon the crystal rotation around the c axis. Data analysis showed that ${\text{EuF}}{{{\text{e}}}_{{\text{2}}}}{\text{A}}{{{\text{s}}}_{{\text{2}}}}$ is the antiferromagnet with easy anisotropy plane. Besides, we found in the $ab$-plane the second order axes of easy magnetization for each of the two types of magnetic domains, related to the structural transition and the formation of twins. Magnetic anisotropy caused by the exchange interaction of europium ions with iron ions indicates the occurrence of nematic magnetic ordering in the basal $ab$ plane. An estimate of the magnitude of the exchange field and the anisotropy field is obtained from the angular dependence of the resonance fields.

1 INTRODUCTION

Superconducting iron pnictides (iron based superconductors, IBS) with rare earth ions, such as, for example, EuRbFe₄As₄, attract increased attention due to the interplay of magnetic ordering of the Eu and Fe spin moments with superconducting pairing of delocalized electrons of Fe and As [1–4]. As a result of the mutual influence of various mechanisms, these compounds exhibit a non-trivial superconducting state, as well as magnetic ordering with unusual anisotropic properties. The parent compound EuFe₂As₂ is non-superconducting and stoichiometric, which greatly simplifies the problem and enables studying in detail only its magnetic side. Indeed, according to the results of neutron scattering as well as magnetic resonant X‑ray scattering measurements [5, 6], in EuFe₂As₂ at temperatures below 190 K an antiferromagnetic (AFM) ordering of the spins of Fe 3d electrons occurs in the $ab$ planes (as a commensurate spin density wave, SDW). According to the same data, the layers of Eu atoms with ferromagnetic ordering of spins inside the layer in the $ab$ plane, at ${{T}_{N}} = 19$ K form a layered antiferromagnetic (AFM) three-dimensional type A structure (see below for more detail). A possible magnetic ordering of the Eu spins in the temperature range 19–190 K remains a debatable issue for the time being.

In addition to the mentioned “strong” effects (AFM, SDW) and strong anisotropy of the magnetic ordering of an “easy plane” type, there is a more subtle effect of magnetic twinning in iron pnictides, which leads to nematic superconductivity in superconducting IBS and to anisotropy of magnetic properties in the seemingly isotropic $ab$ plane in EuFe₂As₂. The crystal lattice twinning and the occurrence of magnetic domains at low temperatures in EuFe₂As₂ were detected by neutron scattering [5].

In this paper, using spin resonance measurements, we observed for the first time the nematic structure of the AFM state at temperatures $T < 19$ K and obtained its quantitative characteristics. Due to a combination of magnetic resonance and magnetic measurements in the EuFe₂As₂ crystal, we identified the position and direction of the easy magnetization axes and obtained an estimate of the exchange field ${{H}_{E}}$, as well as the anisotropy field ${{H}_{A}}$ in the temperature range $T <$ 19 K.

2 CRYSTAL STRUCTURE AND MAGNETIC STRUCTURE OF EuFe₂As₂

The crystal structure of EuFe₂As₂ below the temperature ${{T}_{m}} = 190$ K undergoes a phase transition from tetragonal to orthorhombic, (that is the rotational symmetry ${{C}_{4}}$ changes to ${{C}_{2}}$). Due to the very small resulting difference (~0.6%) in the lengths of the elementary vectors a and b, structural twins are formed in which the axes a are rotated by 90° relative to the corresponding axes of the other type of twins. Below ${{T}_{m}}$ the direction of magnetization is set along a axes in each of the structural domains, thereby creating magnetic domains [5]. Due to the large magnetic moment of europium ions and strong spin-lattice coupling, the magnetic field has a significant effect on these domains. Even a weak magnetic field ($H < 1$ T) leads to their redistribution [7–10].

As mentioned above, Eu²⁺ ions having high magnetic moments ${{\mu }_{{{\text{Eu}}}}} \approx 7{{\mu }_{{\text{B}}}}$ [11] and coupled by exchange interaction ($J \approx 0.1$ meV [7]) form ferromagnetically ordered layers. The coupling between europium layers is provided by an indirect exchange through the iron ions Fe²⁺, with which europium ions are also connected by a strong exchange [7]. As a result, at temperatures below ${{T}_{N}} = 19$ K, an A-type antiferromagnetic ordering occurs between the europium layers, in which the direction of the layers magnetization rotates by an angle π from layer to layer, remaining within the $ab$ plane. In addition, the spin moments of europium ions are strongly coupled with the crystal lattice, which leads to anisotropy of the magnetic parameters of EuFe₂As₂, although the direct spin-lattice coupling of ${\text{E}}{{{\text{u}}}^{{{\text{2 + }}}}}$ ions is absent due to the absence of an orbital moment in europium ions. In [7, 8], a theoretical description of this anisotropy was proposed through a biquadratic interaction, outside the Heisenberg model, between the Eu localized electrons and electrons introduced into the band from Fe 3d orbitals [6, 7].

To study the magnetism of EuFe₂As₂, various methods were used: measurements of magnetization and magnetoresistance [10, 11]; neutron scattering [5, 12]; magnetostriction and magneto-optics [9]; X‑ray magnetic spectroscopy of circular dichroism [8, 12] and others. Very valuable information about magnetic interactions in this material is provided by the electron spin resonance (ESR) technique [13]. Its advantage is that it allows one to get information about local interactions. In this technique, Eu²⁺ ions serve both as the objects of research and as local probes that “measure” the values of internal fields: magnetic induction B, exchange (molecular) field ${{H}_{E}}$ and anisotropy field ${{H}_{A}}$. To verify the reliability, the values of internal fields obtained from the analysis of ESR data must be compared with independent estimates of the magnetic parameters, such as the exchange integral J and the anisotropy constant K. These estimates are made on the basis of magnetic measurements, which give the volume-averaged values of the magnetic characteristics. In accordance with these considerations, a combination of magnetic resonance and magnetic measurements was used in this work to study the structure of the nematic magnetic ordering.

3 SAMPLES AND EXPERIMENTAL TECHNIQUES

The studied EuFe₂As₂ crystals were grown from a solution in a melt by the method described in detail in [14]. The purity of the initial elements was as follows: Eu 99.95, Fe 99.98, and As 99.9999%. For ESR studies, a ~2.5 × 1.6 × 0.1-mm thin slab was split off from the grown crystal.

The studies were carried out with two X-band ESR spectrometers. One of them, Bruker ER-418s, has a rectangular hollow cavity with a basic ${\text{T}}{{{\text{E}}}_{{102}}}$ mode and an fundamental frequency of about 9.4 GHz. The second spectrometer Bruker Elexsys E580 with cylindrical dielectric resonator ER4118MD5 (TE₀₁₁ mode, fundamental frequency ~9.7 GHz). The cavity is placed in the Oxford CF935 cryostat, which allows one to vary the temperature from 2 to 300 K. This device is equipped with a programmable uniaxial goniometer Bruker ER218PG1 (angle range 0°—360°, setting accuracy 0.5°), which is attached to the cavity. Magnetic field was varied in the range from 0 to 1.7 T.

4 RESULTS AND DISCUSSION

The paramagnetic Fe²⁺ ions present in the EuFe₂As₂ crystal should give signals in the ESR spectrum. However, strong antiferromagnetic exchange between iron ions (${{J}_{{{\text{Fe}} - {\text{Fe}}}}} \approx 30{-} 40$ meV [15]) and their spin relaxation via conduction electrons broaden the resonance signal so much that its detection becomes impossible.

Unlike iron, europium ions remain paramagnetic (disordered) over a wide temperature range, and the ESR signal from these ions is well observed (see, for example, [13, 14, 16, 17]). Its main parameters are as follows: the resonance signal position corresponds to the g-factor 1.97, the width varies from 0.07 to 0.13 T, the resonance line shape is asymmetric (Dyson line shape [18]). The asymmetry is caused by the skin effect and is associated with the admixture of dispersion into the absorption signal. Therefore, to obtain the exact value of the resonance field ${{H}_{R}}$ and the resonance signal width $\delta H$ its modelling is required with a function representing a mixture of absorption and dispersion of the Lorentz shape [18, 19].

In addition, due to the large signal width, comparable to the magnitude of the resonance field, it is necessary to take into account the contribution caused by a circularly polarized microwave field with the direction of rotation opposite to the main microwave field forming the resonance signal. The analytical expressions proposed in [20] to approximate the ESR signal under the described conditions were used in our work to determine the signal parameters.

Figure 1 shows temperature dependence of the resonance field ${{H}_{R}}(T)$, whose value was determined by the method described above. It can be seen that when approaching the magnetic transition temperature ${{T}_{N}}$ from above, the ESR signal shifts towards lower fields. This indicates the influence of a demagnetizing field, the magnitude of which is proportional to the sample magnetization.

Fig. 1.

(Color online) Temperature dependence of the resonance field of Eu²⁺ ions in the EuFe₂As₂ crystal whose ab plane is parallel to the magnetic field. The inset shows an example of the ESR signal observed at a temperature of 92 K.

According to the theoretical calculations of C. Kittel [21], in parallel orientation (when the external field is parallel to the plane of a thin sample), the demagnetizing field is summed with the external one, which leads to a decrease in the magnitude of the external field required to create resonance condition $\omega = \gamma \tilde {H}$ (where $\gamma = g{{\mu }_{{\text{B}}}}{\text{/}}\hbar$ is the gyromagnetic ratio, g is the g factor of the paramagnetic ion, μ_B is the Bohr magneton, $\hbar$ is the reduced Planck constant, and $\tilde {H}$ is the total field). In our case, the parallel orientation corresponds to the condition $H\parallel ab$. This condition is implemented in the experiment, the results of which are shown in Fig. 1.

After the transition to an ordered state with a decrease in temperature below $T < {{T}_{N}}$ the resonance signal is greatly broadened, but remains observable. In addition, it is bifurcated (see Fig. 2). This splitting corresponds to the one observed earlier in [14], in which the resonance frequency was measured as a function of magnetic field at a constant temperature. Both signals shift rapidly towards high fields with lowering temperature.

Fig. 2.

Electron spin resonance spectra of Eu²⁺ ions obtained at a temperature of 4.8 K and for different orientations of the EuFe₂As₂ crystal in the magnetic field. The crystal was rotated around its c axis, which is perpendicular to the magnetic field. The origin of the angle corresponds to the position H || a for one of the domains. The incidental signals in the darkened area are caused by resonator parts and do not depend on the angle.

Figure 1 shows the position of only the low-field resonance signal in the temperature range $T < {{T}_{N}}$. Here, shifting to high fields does not correspond to the action of a demagnetizing field and indicates the emergence of a different mechanism in the ${{H}_{R}}(T)$ dependence. There are reasons to assume, that in this region the position of the signal is determined by the anisotropy field ${{H}_{A}}$, the value of which increases with decreasing temperature [8]. The presence of the two signals indicates that there are two groups of Eu²⁺ ions with different local environments. If the shift of the signals is determined by the anisotropy field, then the value of ${{H}_{A}}$ is different for two different groups of ions, or they have different axes of anisotropy. The latter assumption agrees well with the presence of magnetic domains [10] associated with structural twins whose axes a (and axes of magnetic anisotropy) are rotated by 90° relative to each other, as noted above.

Shifting of the two signals upon the rotation of the crystal around the c axis (during rotation, the field constantly remains in the $ab$ plane) occurs independently of each other (see Fig. 2). The shape of the spectrum is periodically repeated. The signals diverge to a maximum distance of 0.6—0.7 T at the angles $\varphi = n\pi {\text{/}}2$ (where n is an integer, $\varphi = 0$ corresponds to $H\parallel a$ for one of the twins).

When $\varphi \approx n{\kern 1pt} \pi {\text{/}}4$ the signals converge and merge into one. If we assume that each of the two types of domains has its own second-order magnetic anisotropy axis directed along the crystallographic a axis, then we can imagine the angular dependence of the resonance fields of the two signals as a superposition of two identical functions with a period of 180° shifted by 90° relative to each other, as shown in Fig. 3. It should be noted that [13] reported the presence of a very weak ($\Delta H \ll 10$ mT) 90° modulation of the resonance field at $T > {{T}_{N}}$. The authors of [13] explained its presence by the influence of crystal fields.

Fig. 3.

Angular dependence of the resonance fields of the ESR signals of Eu²⁺ ions recorded at a temperature of 4.8 K when EuFe₂As₂ crystal is rotated around the c axis. The field direction lies in the $ab$ plane. Half-filled squares are experimental values, solid curves are calculated using expression (2) with parameters ${{H}_{0}} = 0.33$ T and ${{H}_{E}}{{H}_{A}} = 0.34$ T².

The crystal structure of EuFe₂As₂, in which europium ions form layers (see, for example, [5]), admits the presence of easy magnetization planes. The 180‑degree in-plane anisotropy suggests that besides the “easy planes” at $T < {{T}_{N}}$ the axes of easy magnetization are also formed, coinciding with the longer a axis, as follows from the neutron scattering data [5]. The existence of magnetic domains leads to doubling of the number of easy magnetic axes, and, consequently, the repetition period of the spectrum becomes equal to 90° (Fig. 3).

To describe the angular dependence and estimate the magnitude of the anisotropy field, we used Eq. (4.3.11) from the book by A.G. Gurevich [22]. This equation establishes the relationship of the resonance field with ${{H}_{A}}$ and ${{H}_{E}}$ for the case of anisotropy within the easy magnetization plane:

$${{\left( {\frac{\omega }{\gamma }} \right)}^{2}} = H_{R}^{2} - 4{{H}_{E}}{{H}_{A}}\cos (2\varphi ).$$

(1)

In the left part of this equation, ${{H}_{0}} = \omega {\text{/}}\gamma$ is the magnetic field that creates the Zeeman splitting of energy levels equal in magnitude to the electromagnetic field quantum $\hbar \omega$ ($\omega = 2\pi \nu$, ν is the frequency of the spectrometer). In the right part there is a combination of fields involved in creating the resonance condition: external field, in which the ESR signal is observed, ${{H}_{R}}$; the product of the exchange field ${{H}_{E}}$ by the anisotropy field ${{H}_{A}}$ and by the cosine of the doubled angle φ between the directions of the anisotropy axis and the external field. This equation can be rewritten to highlight the observed value ${{H}_{R}}$ explicitly:

$${{H}_{R}} = \sqrt {H_{0}^{2} + 4{{H}_{E}}{{H}_{A}}\cos (2\varphi )} .$$

(2)

Note that Eq. (2) has no solution when $\cos (2\varphi ) < 0$ and $H_{0}^{2} < {\text{|}}4{{H}_{E}}{{H}_{A}}\cos (2\varphi ){\text{|}}$, i.e., in low fields.

Using Eq. (2), we were able to describe the upper part of the angular dependence of the resonance field, where the conditions written above are met (see Fig. 3). By fitting the parameters so that the calculated curve coincides with the experimental dependence, we can find the product of the exchange field ${{H}_{E}}$ and the intraplane anisotropy field ${{H}_{A}}$. At a temperature of 4.8 K, it turned out to be approximately 0.34 T². In order to extract the anisotropy field from this product, it is necessary to use the results of magnetic (non-resonance) measurements, as well as literature data on the magnitude of the exchange interaction.

Figure 4 shows the magnetization of the investigated EuFe₂As₂ crystal versus applied field measured at $T = 4.2$ K. This dependence allows us to estimate the saturation magnetization value ${{M}_{{\text{s}}}}$ and obtain the value of the exchange field using the relationship ${{H}_{E}} = \lambda {{M}_{{\text{s}}}}$ $\left( {\lambda = \frac{J}{{\mu _{{\text{B}}}^{2}}}} \right)$ [22]. We used the exchange integral value $J = 0.093$ meV from [7], where it was calculated from magnetic measurements on the EuFe₂As₂ crystal, similar to ours. Assuming the saturation magnetization value ${{M}_{{\text{s}}}} = 6.7{{\mu }_{{\text{B}}}}$ (Fig. 4), we obtained ${{H}_{E}} \simeq 10.5$ T and, respectively, ${{H}_{A}} \simeq 32$ mT. In order to understand how real the value ${{H}_{A}}$ obtained is and to compare it with the literature data, it is necessary to recalculate the anisotropy field into the anisotropy constant K using the relationship ${{H}_{A}} = \frac{K}{{{{M}_{{\text{s}}}}}}$. This gives the value $K \simeq 1.2 \times {{10}^{{ - 5}}}$ eV at $T = 4.8$ K. This result is quite comparable to the anisotropy constant estimated in [7] based on magnetization measurements at temperature 4.2 K: $K = 7.4 \times {{10}^{{ - 6}}}$ eV.

Fig. 4.

Magnetization of the EuFe₂As₂ crystal (in units of Bohr magnetons per formula unit) versus the applied field at a temperature of 4.2 K. The field is parallel to the $ab$ plane. It can be seen that the saturation of magnetization at $H \approx 1$ T is preceded by a prolonged metamagnetic transition (in the range of 0.4–0.7 T) of the “spin canting” type [11].

We note that the small value of the anisotropy field compared to the exchange field (${{H}_{A}}{\text{/}}{{H}_{E}} \simeq 0.003$) agrees well with the studied situation. Namely, the exchange interaction occurs in the layer of europium ions, directly between them. In contrast, the anisotropy field is formed indirectly, with the participation of iron ions located in adjacent layer, via the biquadratic interaction [8]. Therefore, anisotropy can be large in the direction of the c axis and small in the plane.

In conclusion, in this paper we studied magnetic interactions that determine the magnetic structure of the EuFe₂As₂ crystal in a magnetically ordered state at $T < 19$ K. The study was performed using magnetometry and ESR spectroscopy at a frequency of about 10¹⁰ Hz. The anisotropy field arising in this temperature range causes a large shift of the resonance signal to high fields. In addition, the signal is split into two components corresponding to the two groups of magnetic domains rotated 90° relative to each other. In order to disentangle the features of the magnetic structure of EuFe₂As₂, we measured (at $T = 4.8$ K) and analyzed angular dependence of the resonance field when the sample was rotated around the c axis (whereas an applied field maintained $H\parallel ab$).

Based on the performed analysis, the following conclusions are made:

(i) the EuFe₂As₂ compound at temperature $T <$ 19 K is the antiferromagnet with an easy magnetization plane coinciding with the crystallographic $ab$ plane;

(ii) in this easy plane there is also an anisotropy with a period of 180°, pointing at the existence of an easy magnetization axis parallel to the crystallographic a axes;

(iii) the presence of magnetic domains, associated with crystallographic twins, leads to the emergence of two angular dependencies with a period of 180° shifted by 90° relative to each other. The presence of a second-order axes in each domain can be considered as a nematic ordering.

The theoretical approximation of the angular dependence allowed us to estimate the exchange field ${{H}_{E}} \simeq 10.5$ T and anisotropy field ${{H}_{A}} \simeq 32$ mT at $T = 4.8$ K. The latter value agrees well with the estimate of the anisotropy factor made on the basis of the magnetometry data [7].