Confirmation of complex non-collinear Fe(II) spin ordering in the iron oxychalcogenide La$_2$O$_2$Fe$_2$OSe$_2$ by broad range $^{139}$La-NQR

Introduction

The physics of altermagnets forms one of the most vibrant areas in solid state magnetism research in recent years [1, 2, 3, 4, 5]. Altermagnets offer a new perspective for Terahertz spintronics applications of compensated antiferromagnets, since they exhibit electronic band structures with large spin-splitting in the eV range in a zero net magnetization system. The spin-split character of the electron bands relies on a non-relativistic effect, namely an anisotropic electron hopping within the two spin sublattices [6], which is caused by the combination of time-reversal-symmetry breaking and spatial rotation between the spin sublattices. Tetragonal systems with the in-plane Lieb-lattice structural motive like KV$_2$Se$_2$O are inherently proposed candidates for altermagnetic properties with d-wave symmetry [7]. D-wave altermagnets are particularly interesting for applications due to their large 90° anisotropy of the spin-split electronic band structure.

The transition metal oxychalcogenides La$_2$O$_2$Fe$_{2-x}$Mn$_x$OSe$_2$ have $I4/mmm$ symmetry and exhibit the Lieb-lattice structural motive. La$_2$O$_2$Mn$_2$OSe$_2$, showing collinear antiferromagnetic order with Mn$^{2+}$ spins aligned along the c-axis, is proposed as a correlated insulating d-wave altermagnet [8].

The iron system La$_2$O$_2$Fe$_2$OSe$_2$ is an insulator [9]. La$_2$O$_2^{2+}$ layers separate the Fe$_2$OSe$_2^{2-}$ layers stacked along the c-axis, as shown in Figure 1. Below the Néel temperature $T_{\mathrm{N}} = 90$ K the system develops antiferromagnetic order [11, 12, 13]. Fe$^{2+}$ with an atomic $3d^6$ $S = 2$, $L = 2$ high spin ground state and considerable spin-orbit coupling exhibits a strong axial magnetocrystalline anisotropy in the antiferromagnetically ordered state. In contrast to earlier neutron diffraction studies [12, 14], from which a collinear AFM order below 90 K was concluded, a combined local probe study by $^{57}$Fe Mössbauer spectroscopy, $^{139}$La-NMR and $\mu$SR in 2014 proved that the Fe spins align along the O - Fe - O bonds with 90 de-

gree in-plane non-collinear magnetic order on the Lieb lattice [9]. In that study an in-plane Fe order symmetry similar to Sr$_2$F$_2$Fe$_2$OS$_2$ [15] was deduced from the appearance of two different internal magnetic field strength and orientation in the ordered state. A later neutron diffraction study by McCabe et al. [16] supports the non-collinear magnetic order. Figure 2 displays the non-collinear order as deduced by Günther et al. [9] and McCabe et al. [16]. Recently, La$_2$O$_2$Fe$_2$OSe$_2$ attracted further interest due to the report of short-range orthorhombicity both in the paramagnetic phase and in the magnetic phase based on the analysis of the neutron pair distribution function [18].

In 2021 Hussain et al. [17] reported a zero external magnetic field ${}^{139}$La-NQR study on La$_2$O$_2$Fe$_2$OSe$_2$ [17]. The ${}^{139}$La NMR spectra for polycrystalline La$_2$O$_2$Fe$_2$OSe$_2$ show rather broad resonance lines due to the continuous distribution of relative angles formed by the external magnetic field and the principle axes of the local electric field gradient [9]. Those broad lines collapse to discrete lines in the ${}^{139}$La NQR spectra. Below $T_{\mathrm{N}}$ the discrete NQR resonances are expected to shift due to the internal transferred magnetic hyperfine field. In contrast to this expectation, Hussain et al. reported a discontinuous splitting of the resonance signals at the $2\nu_{\mathrm{Q}}$ and $3\nu_{\mathrm{Q}}$ transitions into two subspectra below $T_{\mathrm{N}}$. For one, referred to La 1, the frequency is constant from above $T_{\mathrm{N}}$ down to lowest temperature. The frequency of the second subspectrum, assigned as La 2, rises abruptly at $T_{\mathrm{N}}$ and shows a strong temperature dependence. At 60 K, the La 2 signal crosses the resonance line of La 1 that is fixed in frequency. Below 30 K, also the La 2 spectrum saturates and becomes temperature independent. The authors interpreted these findings as a magnetically induced local lattice anomaly leading to two La-sites with different electric field gradients below $T_{\mathrm{N}}$. In 2023, Yang et al. reported a hydrostatic pressure induced transition from a Mott insulator to a ferromagnetic Weyl metal in La$_2$O$_2$Fe$_2$OSe$_2$ at $\approx35$ GPa [19].

In this study, we report simulations of zero external magnetic field NQR/NMR spectra for all possible transitions of the ${}^{139}$La ($I=7/2$) nucleus further supported by additional NQR measurements in a wide frequency range between 2.8 and 20 MHz in the temperature range 4.2 to 100 K. We show that neither a splitting nor a strong change of the EFG at the La-sites below $T_{\mathrm{N}}$ is needed to explain the experimental observations. Instead, all measured NQR spectra, including those from ref. [17], are well modelled assuming two La-sites with different transferred magnetic hyperfine fields generated by the non-collinear magnetic ordering of the Fe-spins.

Synthesis

Polycrystalline samples of La$_2$O$_2$Fe$_2$OSe$_2$ were prepared by a solid-state reaction using La$_2$O$_3$, Fe, and Se powders, following a similar procedure to that reported by Günther et al. [9]. Initially, the La$_2$O$_3$ powder was dried at 1000 °C for 12 h to remove moisture. All precursors were weighed in the stoichiometric ratio, ground in an agate mortar, and pressed into 5 mm-diameter pellets in an Ar-filled glovebox (MBraun, Germany). The pellets were sealed in a 15 mm-diameter quartz tube under vacuum and heated in a GERO high-temperature furnace with the following sequence: ramp to 600 °C at 2 °C min$^{-1}$ and dwell for 12 h, ramp to 800 °C at 1 °C min$^{-1}$ and dwell for 1 h, ramp to 1100 °C at 2 °C min$^{-1}$ and dwell for 96 h. After sintering, the furnace was cooled to room temperature. To improve homogeneity, the resulting pellets were reground and subjected to a second sintering step at 1100 °C for 72 h. Structure and phase purity were checked by powder X-ray diffraction and scanning electron microscopy along with energy-dispersive X-ray spectroscopy.

Experiment

Pulsed ${}^{139}$La nuclear quadrupole resonance (NQR) measurements have been performed on La$_2$O$_2$Fe$_2$OSe$_2$ using a LapNMR Pulse Generator, an AMT M3205 Pulse Amplifier, and the Tecmag TNMR software. The NQR frequency has been varied from 7.5 MHz to 20 MHz in 0.1 MHz steps for different temperatures from 4.2 K to 100.4 K. Additionally, measurements at frequencies between 2.8 MHz and 3.5 MHz in steps of 0.05 MHz have been performed at 100.4 K. No signal has been observed in that frequency region below $T_{\mathrm{N}}$. Each time spectrum has been multiplied by a gaussian envelope function with a width $\sigma=8\,\mu\mathrm{s}$ centered at the position of the echo (Gaussian apodization) to suppress detected noise appearing over the whole time frame. The time spectra were then Fourier transformed to obtain the frequency dependent intensities. only the frequencies of the used excitation frequency $\pm100$ kHz have been taken into account.

The resulting spectra are shown in Figure 3. The dashed lines correspond to the theoretical predicted resonances (see section IV). Above the ordering temperature of 90 K only one resonance line 9.4 MHz could be identified between 7.5 MHz and 14 MHz. This is the ${}^{139}\mathrm{La}$ $3\nu_\mathrm{Q}$ signal as been identified by Hussain et al. [17]. Below the ordering temperature three additional resonance peaks appear in the investigated frequency regime. These lines are equidistantly separated by $\nu_\mathrm{Q}$ at all measured temperatures but exhibit an order parameter like shift as a function of temperature. In contrast, the frequency of the original peak, which is considerably reduced in intensity, is temperature independent within the frequency resolution.

The separation of the three additional resonance peaks by exactly $\nu_\mathrm{Q}$ and their order parameter like homogeneous shift as a function of temperature clearly confirm the presence of a magnetic field at the La position. Not all seven expected resonance peaks have been observed due to frequency limitations of the LC circuit. The strength of the magnetic saturation field at $T \approx0$ K was determined by the absence of resonance peaks above 16.3 MHz to 1.127(10) T. A second La position, that does not observe any magnetic field of comparable strength, is needed to explain the temperature independent resonance peak. Magnetically induced local lattice anomalies (suggested by Hussain et al. [17]) are not sufficient

to explain the appearance of additional resonance peaks, which are split exactly by $\nu_\mathrm{Q}$.

Simulations

According to Günther et al. [9] two different La sites exist in the magnetically ordered state, one, La 2, observing a larger magnetic field along the $c$ axis and thus parallel to the main component of the EFG $V_{zz}$ and the other, La 1, observing a small magnetic field in the $a-b$ plane (crf. black arrows in Figure 2). To compare the observed spectra with the suggested model, ${}^{139}\mathrm{La}$ NMR spectra have been simulated for both cases $\vec{B} \parallel\vec{V}_{zz}$ and $\vec{B} \perp\vec{V}_{zz}$ for various strengths of magnetic field $B$ assuming $\nu_\mathrm{Q}=3.147$ MHz (see Figure 4). The employed Hamiltonian to calculate energies and eigenstates is

Relative intensities are then calculated by treating the excitation pulse of the NMR experiment as a small perturbation and calculating $\langle f \vert\cos\varphi\sin\vartheta I_x + \sin\varphi\sin\vartheta I_y + \cos\vartheta I_z \vert i\rangle$, where $\lvert i\rangle$ and $\lvert f\rangle$ are eigenstates of $\mathcal{H}$. To simulate powder spectra, intensities have been calculated 2000 times for random directions of the excitation pulse to account for the random orientation of the grains of material. The results are shown in Figure 4.

The obtained frequency and field relations have been used to check the validity of the model described in Günther et al. [9]. According to the model one La site is observing a large magnetic field (about 1 T at 50 K) below $T_N$ mainly parallel to the $c$ axis, thus being parallel to the principal axis of the EFG $V_{zz}$. The direction of $V_{zz}$ being parallel to the $c$ axis is a consequence of the crystals symmetry, as the rotational axis parallel to $c$ through the La atom has 4-fold symmetry. This is consistent both with the distance of all three resonance peaks appearing below $T_N$ as well as their frequency. A second La site observes a small magnetic field in the $a-b$ plane according to the model. The expected magnetic field would be smaller by a factor of approximately 30 taking only dipole field calculations into account [9]. The temperature dependent resonance peak positions according to the model of Günther et al. [9] are shown in Figure 3. The temperature dependence of the transferred magnetic hyperfine fields at the La sites has been assumed to be proportional to the magnetic field at the iron position determined by $^{57}$Fe Mössbauer spectroscopy (see the Supporting Information). Assuming the resonance peak at $3\nu_Q = 9.44$ MHz to be temperature independent, only two free parameters are remaining: the low temperature saturation field at the high field La site and $\nu_Q$.

Our simulations show that there is no detectable change of frequency for the $3\nu_Q$ resonance peak expected below 100 mT and even for the $2\nu_Q$ resonance peak any splitting might not be observable for fields below 50 mT. Therefore, from the measurements shown here and the data published in Hussain et al. [17] it can be concluded that the magnetic field is smaller than 50 mT, which is in agreement with the proposed value of about 35 mT for $T \approx0$ K [9]. To investigate the magnetic field strength at that La site, additional measurements at $f \approx\nu_Q$ below $T_N$ would be needed (see lower left part of Figure 4(b)). These require an improved low pass signal filtering.

Summary

In this study, we report simulations of zero external magnetic field NQR/NMR spectra for all possible transitions of the $^{139}$La ($I=7/2$) nucleus and additional NQR measurements in a wide frequency range between 2.8 and 20 MHz in the temperature range 4.2 to 100 K. We show that neither a splitting nor a strong change of the EFG at the La-sites below $T_N$ is needed to consistently explain the experimental observations. Instead, all measured NQR spectra, including those from ref. [17], are well modelled assuming two La-sites with different transferred magnetic hyperfine fields generated by the non-collinear magnetic ordering of the Fe-spins. The results of the model calculations and the measurements are in very good agreement. We conclude that the magnetic structure in $\mathrm{La}_{2}\mathrm{O}_{2}\mathrm{Fe}_{2}\mathrm{OSe}_{2}$ is the one shown in Figure 2 as proposed by Günther et al. [9] and McCabe et al. [16]. Combining the different nuclear probe methods NMR and Mössbauer spectroscopy is a powerful tool to separate collinear single-$\vec{q}$ from non-collinear multi-$\vec{q}$ magnetic structures in polycrystalline specimen of $\mathrm{La}_{2}\mathrm{O}_{2}\mathrm{Fe}_{2}\mathrm{OSe}_{2}$, since for ${}^{139}\mathrm{La}$ and ${}^{57}\mathrm{Fe}$ the electric quadrupole interaction is of similar order of magnitude as the magnetic Zeeman interaction at the nuclear sites.

ACKNOWLEDGMENTS

This research was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through projects B01 and C02 of the SFB 1143 (project-id 247310070), by the Würzburg-Dresden Cluster of Excellence ct.qmat, EXC 2147 (project ID 455319354), and by individual grant WU595/14-1. J.A.G and S.W. are indebtedly thankful to Nico Grässler for his experimental support and very fruitful discussions as well as R. Kluge and C.G.F. Blum for technical support (all IFW Dresden).

SUPPORTING INFORMATION - ${}^{57}\mathrm{Fe}$ MÖSSBAUER

Figure 5 presents the temperature dependent magnetic hyperfine field obtained from ${}^{57}\mathrm{Fe}$ Mössbauer spectroscopy measurements. The data is described utilizing a phenomenological

phenomenological $B_{\mathrm{Hyp}} = B_{0}\left(1-\left(\frac{T}{T_{\mathrm{c}}}\right)^{\alpha}\right)^{\beta}$ model. Values of $\alpha= 1.6(4)$, $\beta= 0.134(11)$, $T_{\mathrm{c}} = 89.99(5)\,\mathrm{K}$ are obtained. The critical exponent $\beta= 0.134$ is close to that of the magnetic square planar Ising model ($\beta= 1/8$). The exponent $\alpha$ describes the behavior for $T \to0$ (Bloch law). The value $\alpha= 1.6(4)$ is close to the expectation $\alpha= 2$ for an antiferromagnet.