Ferromagnetic inter-layer coupling in FeSe_1-xS_x superconductors revealed by inelastic neutron scattering

The FeSe_1-xS_x (x = 0, 0.07 and 0.11) single crystals have been grown by chemical vapor transport technique Li et al. (2020) and co-aligned with a mosaic within 5^∘ for all the sample arrays as shown in Fig. S1 [See Supplementary Materials]. As displayed in Fig. 1(b) S doping causes a slight increase of $T_{\mathrm{c}}$ and a substantial decrease of $T_{\mathrm{s}}$ measured by resistivity and (2, 2, 0) Bragg scattering due to neutron extinction effect Hamilton (1957), and a full suppression is achieved for x $\geq$ 0.18 Coldea (2021). Here and throughout this article, the wave vector $\mathbf{Q}$ is expressed in reciprocal lattice units (r. l. u.) as ($H$, $K$, $L$). Using the orthorhombic notation of the 4-Fe Brillouin zone, the wave vector, expressed in inverse Angstrom, is $\mathbf{Q}$ = [$(2\pi/a)H$, $(2\pi/a)K$, $(2\pi/c)L$] with lattice parameters $a$ $\approx$ $b$ $\approx$ 5.32 $\mathrm{\AA}$ and $c$ $\approx$ 5.52 $\mathrm{\AA}$.

The triple-axis INS experiments with $k$_f = 2.662 $\mathrm{\AA}$^-1 using a focusing pyrolytic graphite (PG) monochromator and analyzer were carried out on the PUMA spectrometer at the Heinz Maier-Leibnitz Zentrum (MLZ), Technische Universität München, Garching, Germany, the 2T spectrometer at the Laboratoire Léon Brillouin, Saclay, France and the IN22 spectrometer using the CryoPAD system at the Institut Laue-Langevin, Grenoble, France. Additional PG filters were placed between the sample and the analyzer to eliminate higher-order contaminations. FeSe_1-xS_x (x = 0, 0.07 and 0.11) single crystals were all mounted in the ($H$, 0, $L$) scattering plane.

The time of flight experiments of FeSe_0.93S_0.07 were performed on the 4SEASONS spectrometer Kajimoto et al. (2011) at the Material and Life Science Experimental Facility (MLF), Japan Proton Accelerator Research Complex (J-PARC), Japan. The 4SEASONS spectrometer has a multiple-$E_{i}$ capability Nakamura et al. (2009); Inamura et al. (2013) with $E_{i}$ = 10, 13.6, 19.4, 30.1, 52.1 and 116 meV, such that neutron scattering events with a series of different incident energies are recorded simultaneously. During the measurements at $T$ = 5 K, 15 K and 90 K, the sample was rotated about the vertical direction over a range $\pm$55^∘ and a step 0.5^∘ steps. Data accumulated at different angles were combined into a four-dimensional dataset, in which we used the Horace software packages for reduction and analysis. After a careful alignment of the measured dataset with the crystallographic coordinate system using all available nuclear Bragg reflections, the entire dataset was downfolded into a minimal, physically independent sector of the three-dimensional momentum space using the point-group symmetry of the system.

Figure 2(a-b) shows the constant-energy maps of spin excitations for FeSe_0.93S_0.07 in the superconducting state ($T$ = 5 K) and nematic state ($T$ = 15 K) at $E$ = 4 meV in the ($H$, $K$) plane where the spin excitations are symmetrically located at $\mathbf{Q}$ = ($\pm$1, 0) and (0, $\pm$1) with an elliptic distribution elongated in the transverse direction. The anisotropic distribution can also be displayed by transverse scan ($K$ scan) and longitudinal scan ($H$ scan) around $\mathbf{Q}$ = (1, 0) in Fig. 2(c-d) as indicated by the arrow in Fig. 2(a-b). Compared with the nematic state ($T$ = 15 K), in superconducting state an enhancement of the spin excitations around $E$ $\approx$ 4 meV is accompanied by a suppression of magnetic response below $E$ $\approx$ 2.5 meV as shown in Fig. 2(e-f), which is a hallmark of the spin resonance mode in agreement with previous INS studies on undoped FeSe Gu et al. (2022); Wang et al. (2016a, b); Ma et al. (2017b); Chen et al. (2019). The elongated distribution along transverse direction of spin resonance in ($H$, $K$) plane is consistent with the previous work on Ba(Fe_0.963Ni_0.037)₂As₂ Kim et al. (2013), CaFe_0.88Co_0.12AsF Ma et al. (2023), ${\mathrm{BaFe}}_{2}{({\mathrm{As}}_{0.7}{\mathrm{P}}_{0.3})}_{2}$ Hu et al. (2016) where the resonance is found to peak sharply at $\mathbf{Q}_{\mathrm{AF}}$ along the longitudinal direction, but broadens along the transverse direction. Other similar constant-energy maps at $E$ = 2$-$6 meV and energy dependence measurements in $E-K$ space at $T$ = 5 K, 15 K and 90 K are displayed in Fig. S2 [See Supplementary Materials]. When entering the tetragonal state ($T$ = 90 K), the magnetic signals at $\mathbf{Q}$ = ($\pm$1, 0) and (0, $\pm$1) become weak and diffusive on a high background (BG), whereas the scattering signals at $\mathbf{Q}$ = ($\pm$1, $\pm$1) are heavily contaminated by (1, 1, 0) phonon as displayed in Fig. S2(m-r).

After illustrating the anisotropic distribution of spin resonance in ($H$, $K$) plane, we now turn to the $L$-modulation of spin resonance. Figure 3(a-b) shows the temperature difference of spin excitations of undoped FeSe and FeSe_0.93S_0.07 between superconducting state ($T$ = 2 K) and nematic state ($T$ = 15 K) at $\mathbf{Q}$ = (1, 0, $L$) ($L$ = 0, 0.5, 1, 1.5). The spin resonance mode appears at $E_{r}$ $\approx$ 3.5 meV in FeSe, slightly enhanced at $E_{r}$ $\approx$ 4 meV in FeSe_0.93S_0.07. Notably, the spin resonance mode presents a stronger intensity at integer $L$ = 0 and 1 than that at half-integer $L$. This $L$-modulation of spin resonance can be further elucidated by the momentum scans along $\mathbf{Q}$ = (1, 0, $L$) at $E$ = 3.5 meV for undoped FeSe [ Fig. 3(c)] and $E$ = 4 meV for FeSe_0.93S_0.07 [ Fig. 3(d)]. In addition, for a higher S doped sample FeSe_0.89S_0.11, the spin excitations at $E$ = 4 meV in superconducting state are shown in Fig. S5(e) [See Supplementary Materials].

This strongly $L$-modulated spin resonance mode in superconducting state is closely related to the anisotropic distribution of low-energy spin excitations in momentum space in the nematic state ($T$ = 15 K). Except for an intensity enhancement due to spin resonance, the anisotropic distribution of spin excitations in momentum space in the nematic state mimics that in superconducting state as displayed in the constant energy maps in ($H$, $K$) plane [Fig. 2(a-b)] and ($H$, $L$) plane [Fig. 4(a-b)] as well as the $L$ scans in Fig. 4(c). The spin excitations reveal elliptic distribution elongated in the transverse direction and a more elongated ellipse in the $L$ direction both in superconducting and nematic state. In order to quantatively clarify the elliptic distribution of spin excitations in three-dimensional momentum space, momentum scans along $H$, $K$ and $L$ directions are presented in Fig. 2(c-d), Fig. 4(c) and Fig. S3 [See Supplementary Materials] respectively. The intensity of spin excitations in the nematic state ($T$ = 15 K) along the longitudinal ($H$), transverse ($K$) and $L$ directions can be well fitted by Gaussian funtions, revealing the full widths at half maximum (FWHM) value $\kappa_{H}$, $\kappa_{K}$ and $\kappa_{L}$ as shown in Fig. 4(d). Comparing the FWHM along $H$ and $K$ directions, $\kappa_{H}$ and $\kappa_{K}$, respectively, one observes that the latter is slightly larger than the former. With a 6 times larger value $\kappa_{L}$ than $\kappa_{K}$, the $L$ dependence of intensity suggests a much more broadened distribution of spin excitations along $L$ direction, weakened with increasing energy and persisting up to $E$ $\approx$ 15 meV as shown in Fig. S3-S4 [See Supplementary Materials]. According to the FWHM we draw the schematic diagram of three dimensional distribution of spin excitations in momentum space at $T$ = 15 K and $E$ = 4 meV as displayed in Fig. 4(e). The ellipsoids are periodically located at $\mathbf{Q}$ = ($\pm$1, 0, $L$) or (0, $\pm$1, $L$) in a twinned sample where $L$ is integer.

We now discuss the implications of our observed INS results. The feature of integer-$L$ intensity enhancement of spin excitations in FeSe_1-xS_x still survives in the nematic state persisting up to $E$ $\approx$ 15 meV and tells us that the spins in nearest-neighbor FeSe₄ layers are parallel and fluctuate in phase, which is opposite to the antiferromagnetic order in the parent compound of Fe-pnictides with antiparallel spins of the nearest-neighbor layers along $c$-axis. A ferromagnetic-like inter-layer coupling $\mathbf{J}_{\mathbf{\bot}}$ has to be responsible for the integer-$L$ intensity modulation in FeSe_1-xS_x.

The microscopic origin of the absence of the long-range stripe magnetic order in FeSe remains elusive. One scenario is that the stripe magnetic order in FeSe is absent due to the development of other competing instabilities, considering mainly on the in-plane frustrated magnetic interactions Yu and Si (2015); Wang et al. (2015); Glasbrenner et al. (2015); Chubukov et al. (2015); Yamakawa et al. (2016). From another point of view, a realistic three-dimensional antiferromagnetic order requires both the spin correlations in and out of plane. The FWHM of spin excitations obtained by Gaussian fitting along $H$, $K$ and $L$ directions is associated with the spin-spin correlation length $\xi_{a}$, $\xi_{b}$, $\xi_{c}$. The FWHM along $L$ scans is 6 times larger than along the transverse scans as the smaller correlation length $\xi_{c}$ is able to capture the reciprocal space broadening along the long axis $L$. As it is well established that intercalating between the Fe layers can strengthen superconductivity and reduce inter-layer coupling Qiu et al. (2009); Friemel et al. (2012); Pan et al. (2017); Ma et al. (2017a); Ramazanoglu et al. (2013); Qureshi et al. (2012), it seems that superconductivity is a more robust phenomenon in the two-dimensional limit compared to antiferromagnetic order.

In conclusion, the anisotropic momentum distribution of spin resonance in FeSe_1-xS_x mimics the low energy spin excitations in its nematic phase where superconductivity arises, which implies the close relationship between superconductivity and magnetism. The strongly $L$-modulated spin resonance mode inherits the low-energy magnetic scattering in nematic state with maximum intensity at integer $L$. FeSe does not order antiferromagnetically, which is unusual among Fe-based materials. The ferromagnetic nature of the inter-layer coupling might shed light on the absence of magnetic order in FeSe.

The work was supported by the National Key Research and Development of China (Grant No. 2018YFA0704200, 2022YFA1602800), the National Natural Science Foundation of China (Grant No. 12004418) and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB25000000). One of the neutron scattering experiments was performed at the MLF, J-PARC, Japan, under a user program (No. 2018A0019).