Nematic Fluctuations in Iron-Oxychalcogenide Mott Insu- lators

Nematic fluctuations occur in a wide range of physical systems from liquid crystals to biological molecules to solids such as exotic magnets, cuprates and iron-based high-$T_c$ superconductors. Nematic fluctuations are thought to be closely linked to the formation of Cooper-pairs in iron-based superconductors. It is unclear whether the anisotropy inherent in this nematicity arises from electronic spin or orbital degrees of freedom. We have studied the iron-based Mott insulators La$_{2}$O$_{2}$Fe$_{2}$O$M$$_{2}$ $M$ = (S, Se) which are structurally similar to the iron pnictide superconductors. They are also in close electronic phase diagram proximity to the iron pnictides. Nuclear magnetic resonance (NMR) revealed a critical slowing down of nematic fluctuations as observed by the spin-lattice relaxation rate ($1/T_1$). This is complemented by the observation of a change of electrical field gradient over a similar temperature range using M\"ossbauer spectroscopy. The neutron pair distribution function technique applied to the nuclear structure reveals the presence of local nematic $C_2$ fluctuations over a wide temperature range while neutron diffraction indicates that global $C_{4}$ symmetry is preserved. Theoretical modeling of a geometrically frustrated spin-$1$ Heisenberg model with biquadratic and single-ion anisotropic terms provides the interpretation of magnetic fluctuations in terms of hidden quadrupolar spin fluctuations. Nematicity is closely linked to geometrically frustrated magnetism, which emerges from orbital selectivity. The results highlight orbital order and spin fluctuations in the emergence of nematicity in Fe-based oxychalcogenides. The detection of nematic fluctuation within these Mott insulator expands the group of iron-based materials that show short-range symmetry-breaking.


Introduction
Nematic phases occur in a variety of very different physical systems throughout nature. A nematic phase is formed when a discrete rotational symmetry is broken or reduced, while the translational symmetry is preserved. Nematicity has been studied mostly in the structural organization of soft matter such as liquid crystals. [1][2][3] In these materials it is easy to visualize the nematic state's defining characteristic: the presence of a preferred axis along which liquid crystal molecules are, on average, aligned. This spatial anisotropy is generally described by C 2 rotational symmetry. In the context of cuprate high-temperature superconductivity (HTSC) bond-nematicity was introduced as a competing order with HTSC itself. In recent years, nematicity has become the central focus of efforts to illuminate mechanism(s) of HTSC in iron-based superconductors (FeBS). By now, a variety of iron pnictides have been shown to exhibit electronic nematic states 4,5 and in numerous theoretical pictures, the nematic phase is thought to be intimately related to the superconducting mechanism in iron-based materials. 6 The origin of the nematic phase in iron superconductors is heavily debated. Two scenarios 3 are proposed: 5 One proposal suggests that anisotropic spin fluctuations are necessary precursors of the nematic ordering that has been experimentally observed; 7 the other claims that ferro-orbital ordering involving d xz,yz orbitals is responsible for nematicity. Complicating this debate is the fact that various Fe-based materials exhibit different magnetic order. Nematicity has been mainly studied in the 122 iron pnictides such as BaFe 2 As 2 , but recently nematic behavior was reported in the 1111 material LaAsFeO 1−x F x . 8 An important question emerges. Is nematicity a general phase behavior that exists in other Fe-based materials across the electronic phase diagram? For example, do Fe-based parent materials that are Mott insulators also exhibit nematicity? What role does electron correlation in multi-orbital systems play in the formation of the interlinked nematic and spin density wave (SDW) phase and the competing superconducting phase. 9 Increasing evidence points to an electronic mechanism of nematicity which would place the nematic order in the class of correlation-driven electronic instabilities, like superconductivity and density-wave transitions.
In this work, we report a combined experimental and theoretical investigation of magnetism intrinsic to the correlation-induced 10 Mott insulators La 2 O 2 Fe 2 O(S, Se) 2 (see Figure 1). Neutron powder diffraction was employed to make a magneto-structural comparison of the materials. A short range nematic fluctuating behavior was observed. A critical slowing down of nematic fluctuations observed in the nuclear magnetic resonance (NMR) spin relaxation rate; this result was complemented by Mössbauer spectroscopy data. In contrast to Fe-based superconductors, orbital nematicity does not play a role in the iron oxychalcogenides, since a d xz,yz orbital degeneracy is not present due to an alternating orientation of the FeM 4 O 2 octahedra. Therefore, a different mechanism must be responsible for the nematicity we observe in La 2 O 2 Fe 2 O(S, Se) 2 . Since these 4 are Mott insulators, a strong-coupling view should be relevant. Our magnetic neutron diffraction data reveal the establishment of antiferromagnetic (AFM) ordering and the similarity of magnetic behavior in these materials. Our theoretical modeling explains the nematic fluctuations as being related to quadrupolar fluctuations in a strongly frustrated spin-1 magnet. This agrees well with NMR 1/T 1 data. The quadrupolar fluctuations are closely related to geometric frustration in the non-collinear spin structure, which itself arises as a consequence of strong orbital selectivity in the compounds. Thus, orbital-selective Mott correlations are important for understanding magnetism in the Mott-insulating states of the oxychalcogenides. Such selective-Mott correlations have recently been the focus of attention in the iron-selenides 11-14 and the present study provides a new instance for the relevance of orbital-selective Mott physics in Fe-based materials.

Experimental and Theoretical Results
Neutron Diffraction Neutron powder diffraction experiments were performed on powder samples of La 2 O 2 Fe 2 O(S, Se) 2 using the C2 high-resolution diffractometer at the Canadian Nuclear Laboratories in Chalk River, Ontario. The C2 diffractometer is equipped with an 800 wire position sensitive detector covering a range of 80 degrees. Data were collected in the 2θ angular range from 5 • to 117 • using a Si (5 3 1) monochromator at a wavelength λ of 1.337Å. The lattice parameters of We verified the consistency of the data with the non-collinear 2-k magnetic structure previously obtained for La 2 O 2 Fe 2 OSe 2 . 15 The high-temperature paramagnetic (PM) phase is compared to the 2-k antiferromagnetic phase in Fig. 7. The Sarah suite of programs 16 was used to analyze the representations and provide the magnetic basis vectors for refinement with Fullprof. 17,18 The magnetic cell is commensurate and is doubled, in both a and c, with respect to the structural cell.
The ordering is associated with k 1 = (1/2, 0, 1/2) and k 2 = (0, 1/2, 1/2), and the single Fe site on (1/2, 0, 0) in the nuclear I4/mmm cell is described by two distinct orbits governing the two (1/2, 0, 0) and (0, 1/2, 0) Fe sites that are independent in the magnetically ordered state. However, we note that for powder samples the diffraction pattern is indistinguishable from the pattern assigned as the collinear AFM3 model in the literature. 15,19 Neutron diffraction shows that the magnetic structures of La 2 O 2 Fe 2 O(S, Se) 2 are similar with a distinction in the AFM onset temperatures T N .
The magnetic peak intensity over a temperature range from 4 to 300 K, shown in Figure 1, is proportional to the square of the magnetic order parameter. The magnetic peak intensity data reveals the onset of AFM ordering at 90.1(9) ± 0.16 K and 107.2(6) ± 0.06 K for La 2 O 2 Fe 2 OSe 2 and La 2 O 2 Fe 2 OS 2 , respectively. The T N value for La 2 O 2 Fe 2 OSe 2 is in excellent agreement with values in the literature. 10,15,19 Nuclear Pair Distribution Function Nuclear pair distribution function (PDF) measurements on is the scattering vector and r is the interatomic distance. PDF is a total scattering method i.e., meaning that both Bragg and diffuse scattering intensity data is simultaneously collected. Therefore, the technique is sensitive to deviations from the average structure. 21 Our PDF data were generated from the total scattering experiments using neutrons of a wavelength band from 0.1 to 2.9Å. The maximum momentum transfer used for the Fourier transform was 31.4Å −1 . The instrument parameters Q damp = 0.017Å −1 and Q broad = 0.019Å −1 were fixed for all of our fits and structural refinements of the data were performed using the PDFGUI 22 program and DIFFPY-CMI 23 suite.
We performed data refinements using an orthorhombic model of La 2 O 2 Fe 2 O(S, Se) 2 with a sliding fit range from (1.5-21.5)Å to (29.5-49.5)Å in 1Å steps to investigate the temperature dependent structure on these various length scales. This resulted in 29 fits for each temperature at which PDF data were collected. We have extracted the orthorhombicity parameter δ = (a − b)/(a + b) for both length scale and temperature series sequential refinements where a, b are lattice parameters. Figure 2 shows the refined orthorhombicity δ extracted from fits of neutron PDF data. Structural models obtained from refining PDF data are strictly valid 24 within the refined length scale; this specificity condition enables the probing of local structure on different length scales by varying the refinement range. By conducting such fittings, we observed higher orthorhombicity for short-range fitting (1.5 -21.5)Å compared to long-range fitting (30.0-50.0) 7 Å . This suggests that by tracking the orthorhombicity parameter we observe local scale distortions from tetragonal to orthorhombic structure. These spatially limited distortions represent fluctuating nematic order 25, 26 over a broad range of temperatures. It should be emphasized that neutron powder diffraction, from the same samples, provides clear evidence that the average, long-range structures of La 2 O 2 Fe 2 O(S, Se) 2 remain tetragonal throughout the high and low temperature regimes. 27 However, the PDF results provide evidence of local symmetry breaking that results in nematic C 2 regions of fluctuating order. 25,26,28,29 Mössbauer spectroscopy In order to investigate the magnitude and the orientation of the ordered static Fe magnetic moment with respect to the electric field gradient (EFG) principal axis within the distorted FeO 2 S 4 octahedra, and to study the modification of the electronic surroundings via the electric quadrupole interaction, 57 Fe Mössbauer spectroscopy was performed on La 2 O 2 Fe 2 OS 2 and is discussed in comparison to similar work on La 2 O 2 Fe 2 OSe 2 . 30 The 57 Fe nucleus probes the onsite magnetism of the Fe ion, therefore any significant changes in orbital and spin degrees of freedom should be reflected in the 57 Fe Mössbauer spectra. Mössbauer spectra were analyzed by using the full static Hamiltonian with nuclear spin operators I z , I + = I x + iI y , and I − = I x − iI y . Here B is the hyperfine field at the 57 Fe site while Q, g I , and µ N indicate the nuclear quadrupole moment, g factor and magneton, respectively. The polar angle θ and the azimuthal angle φ describe the orientation of the Fe hyperfine field B with respect to the EFG z axis. The azimuthal angle φ was set to zero to avoid cross correlation. This is justified, because the asymmetry parameter η is small and the polar angle θ is close to zero for both systems. For both samples, the whole temperature series was fitted simultaneously to determine η, then η was fixed, and then the spectra were fitted individually.

Within error bars, both samples La
In the magnetically ordered state a clear magnetic sextet is observed in data, indicating the presence of a long range ordered (LRO) state. The nature of this LRO state is the same for both systems in terms of the magnitude of the ordered moment (magnetic hyperfine field B hyp (T )) and its temperature dependence. The lines in Figure 3b) indicate fits of the sublattice magnetization M (t) by using the equation the magnetic hyperfine field data. The magnetic critical exponents β crit can be estimated as a fit parameter (see Table 1).
The 57 Fe Mössbauer spectroscopy provides clear evidence that in both systems the angle between the strongest EFG component ( local z main axis) and the magnetic hyperfine field B, θ, is equal to 0 in the magnetically ordered regime, i.e., the magnetic hyperfine field B is oriented parallel to the local z axis of the EFG principal axis system. Nuclear Magnetic Resonance (NMR) Figure 4 shows the representative 139 La field sweep NMR spectra on La 2 O 2 Fe 2 OS 2 . Each spectrum consists of three pairs of satellites, and one split central transition. For lower symmetries, such as tetragonal systems, in the presence of finite EFG at the probed nucleus with nuclear spin I = 7/2, one would expect such NMR spectra. Upon lowering T , the spectral shape remains unchanged down to 109 K (the line in the temperature range 109-105 K could not be resolved). This indicates the AFM ordering within the system. In the ordered state the spectral shape changes considerably because of the development of the static internal magnetic field associated with Fe ordered moment. The field swept NMR spectra can be well described by diagonalizing the full nuclear Hamiltonian where B is the external magnetic field, z axis being the principle axis of EFG and γ 2π = 6.0146   For T < 30 K, we find a smooth crossover from gap-like behavior to a gapless regime, where 1/T 1 T 3 . Therefore, two distinct regimes are observed where 1/T 1 T 3 for T << ∆, while

13
The NMR 1/T 1 behavior for 30 K < T follow an unusual T 3 power-law form which suggests the activation of an additional spin fluctuation channel. We now address the origin of this behavior within the framework of an S = 1 Heisenberg model with appreciable geometric frustration, an additional bi-quadratic term, and single-ion anisotropy as is appropriate for Fe-based systems.
Theory Motivated by earlier local density approximation + dynamical mean-field theory (LDA+DMFT) calculations in which we estimated an S = 1 on each Fe site, we begin with the spin S = 1 bilinear-bi-quadratic model of the square lattice. This model has been widely employed in context of magnetism and nematicity in Fe-based SCs. [39][40][41] The Hamiltonian is where S i is a spin-1 operator on site i. J ij = J 1a , J ij = J 1b such that J 1a is ferromagnetic (FM) and along the square-diagonal, and K is the coefficient of the biquadratic interaction for S = 1 along with D being the single-ion anisotropy term that fixes the orientation of the spin. It is known that this model admits various ordered phases, depending upon the relative values of the J ij and K for a given D: 42 a q = (π, 0) collinear antiferromagnetic (CAF) phase, a q = (π/2, π) AFM * phase, a q = (π, π) Néel AFM phase, and a q = (π, 0) anti-ferro-quadrupolar are shown in Figure 6.
Within the temperature range 30 < T < T N (∼ 100 K), the experimental data for NMR 1/T 1 fits AT 2 exp(−∆/T ) to a very good approximation, with ∆ estimated to be approximately 100 K. Moreover, at low temperatures T < 30 K, we also find that the 1/T 1 T 3 , indicating gapless spin excitations in the spin fluctuation spectrum. This is consistent with earlier findings on the Se compound. 30