Strong cooperative coupling of pressure-induced magnetic order and nematicity in FeSe

A hallmark of the iron-based superconductors is the strong coupling between magnetic, structural and electronic degrees of freedom. However, a universal picture of the normal state properties of these compounds has been confounded by recent investigations of FeSe where the nematic (structural) and magnetic transitions appear to be decoupled. Here, using synchrotron-based high-energy x-ray diffraction and time-domain Mössbauer spectroscopy, we show that nematicity and magnetism in FeSe under applied pressure are indeed strongly coupled. Distinct structural and magnetic transitions are observed for pressures between 1.0 and 1.7 GPa and merge into a single first-order transition for pressures ≳1.7 GPa, reminiscent of what has been found for the evolution of these transitions in the prototypical system Ba(Fe1−xCox)2As2. Our results are consistent with a spin-driven mechanism for nematic order in FeSe and provide an important step towards a universal description of the normal state properties of the iron-based superconductors.

U nconventional superconductivity is generally believed to be mediated by fluctuations associated with ordering (magnetic, orbital or charge) found in the normal state of the material [1][2][3] . For the high-temperature iron-based superconductors, a leading candidate is the ubiquitous stripe-like antiferromagnetic (AFM) fluctuations associated with the magnetic ordering in the normal state. In some scenarios, these same spin fluctuations also lead to nematic electronic order, identified by a tetragonal-to-orthorhombic (T-OR) structural transition 4,5 . In this context, nematicity corresponds to orientational order between two AFM sublattices that are only weakly coupled because of frustration that arises from large next-nearest-neighbour magnetic interactions [6][7][8] . However, the magnetic origin of nematicity has been recently challenged by the unusual properties of FeSe [9][10][11][12][13][14][15] .
For most iron-based superconductors, including AFe 2 As 2 (A ¼ Ca, Sr, Ba), RFeAsO (R ¼ rare earth) and NaFeAs, stripelike AFM order and nematic order, in the form of a T-OR transition, occur either simultaneously or in relatively close proximity in temperature, and both can be tuned by doping or applied pressure. At ambient pressure, FeSe undergoes a T-OR transition at a temperature of T s ¼ 90 K, but no ordered AFM state has been observed 16 . However, muon spin-resonance (mSR) measurements on polycrystalline samples have demonstrated the emergence of magnetic order in FeSe under applied pressures greater than B0. 8 GPa (refs 17,18), and the magnetic transition temperature, T m , increases with pressure whereas the structural transition is suppressed [19][20][21][22] . Furthermore, superconductivity in FeSe shows a four-fold increase to a transition temperature of T c ¼ 37 K under pressure 23 , accompanying the increase of T m 19,20,22 . Taken together, the absence of magnetic order in the presence of nematic order at ambient pressure and the opposing evolutions of magnetic and nematic order with pressure, call into question the notion of a cooperative relation between magnetism and structure as the origin of the nematic order in FeSe [9][10][11][12][13][14][15] .
Electrical resistivity measurements under applied pressure provide valuable information regarding the existence of phase transitions in the pressure-temperature phase space [20][21][22] , but microscopic measurements under pressure are required to fully elucidate the underlying nature of the transitions, the associated order parameters and the relationship between the magnetism and structure of FeSe. Although direct microscopic information concerning the magnetic structure is best obtained using neutron diffraction, several previous measurements have been unsuccessful in detecting the expected small ordered magnetic moment 18,22 . Therefore, we have used both high-energy x-ray diffraction (HE-XRD) and time-domain Mössbauer spectroscopy through nuclear forward scattering (NFS) to study the structural and magnetic transitions in single-crystalline FeSe under pressure. Distinct structural and magnetic transitions, and a strong cooperative coupling of the respective order parameters are observed for pressures between 1.0 and 1.7 GPa. The transitions merge into a single first-order phase line for pressures 41.7 GPa. The evolution of the phase lines and the order parameters is reminiscent of the prototypical doped system Ba(Fe 1 À x Co x ) 2 As 2 and is consistent with a spin-driven mechanism for nematic order also in FeSe. Our results broaden the base for a universal description of the magnetic, structural and electronic properties of the iron-based superconductors.

HE-XRD and NFS measurements.
The essential results of our study are summarized in Fig. 1. Figure 1a shows that FeSe undergoes a continuous T-OR transition at ambient pressure at T s ¼ 90 K. This is observed as a splitting of the (HH0) Bragg peaks related to the in-plane orthorhombic lattice parameters a OR and b OR . At an applied pressure of 1.5 GPa (Fig. 1e), this structural transition is suppressed to T s ¼ 32 K. However, as the temperature is further lowered there is a subsequent abrupt increase of the splitting at 19 K. When the pressure is increased to 1.7 GPa only the discontinuous T-OR transition remains (Fig. 1f), and T s increases to 35 K at 3.1 GPa (Fig. 1g). To correlate the features observed in the HE-XRD measurements with the evolution of the magnetism in FeSe, Fig. 1c,d show the NFS (time-domain Mössbauer spectra) at two different pressures, P ¼ 2.5 and 4.0 GPa. The small hyperfine field associated with magnetic ordering in FeSe is quite challenging for conventional Mössbauer spectroscopy 23 , but is readily observed in the data from our NSF measurements. The distinctive feature in these spectra is the minimum at intermediate delay times, which is determined by a convolution of the magnetic hyperfine field, quadrupolar splitting and effective sample thickness (see the 'Methods' section). Our fits to these spectra find that the shift in the minima at B30 K (P ¼ 2.5 GPa) and 40 K (P ¼ 4 GPa) is attributable to the presence of a hyperfine field at these temperatures and pressures, which confirms the onset of magnetic order in FeSe under pressure.
Coupled first-order structural and magnetic transitions. The evolution of the structural order parameter, determined from fits to the HE-XRD data, and the evolution of the magnetic hyperfine field are shown in Fig. 2 and summarized in the pressure-temperature phase diagram of Fig. 3. These data provide insight into the relationship between structure and magnetism in FeSe under pressure. At ambient pressure, the orthorhombic distortion dðTÞ ¼ ða OR À b OR Þ=ða OR þ b OR Þ displays a smooth temperature dependence characteristic of a second-order phase transition (Fig. 2a). Starting from B1.0 GPa, however, we observe a small discontinuous increase in d at B19 K (see the inset of Fig. 2a), which is very distinct at P ¼ 1.5 GPa and higher applied pressures, as shown in Fig. 2c. Similar measurements of the canonical BaFe 2 As 2 -type iron-based superconductors clearly established that this discontinuous increase in the orthorhombicity was associated with strong magnetoelastic coupling and a transition to stripe-like magnetic order 24,25 . For FeSe, the magnetic ordering temperature obtained in the mSR measurements 17 on polycrystalline samples is also consistent with the first-order structural transition measured by HE-XRD on our single-crystal sample. In Fig. 2b we show the evolution of the hyperfine field, H hf , extracted from fits to our single-crystal NFS data measured at applied pressures of P ¼ 2.5 and 4.0 GPa (Fig. 1c,d). The relatively small saturated values for H hf of B2.5 and 3.0 T, respectively, correspond to a small ordered moment on the order of 0.2 m B /Fe, consistent with the estimates from the previous mSR experiments 18 and explain the difficulty in detecting a magnetic signal in earlier conventional Mössbauer measurements 23 . We also find that T m and the saturation value of H hf increase with increasing pressure, consistent with the previous mSR studies 17,18 . Although the density of data points is rather low, Fig. 2b suggests that the NFS spectra at P ¼ 4.0 GPa is consistent with a strong discontinuous transition, whereas the magnetic transition at 2.5 GPa could be described as weakly first-order.

Discussion
Most importantly, the HE-XRD data show that the T-OR transition does not vanish above 1.7 GPa where T s , as inferred from resistivity measurements, would extrapolate to zero 13,20 . Rather, the abrupt onset of the orthorhombicity and the significant coexistence range of the tetragonal and orthorhombic phases (inset of Fig. 2c) demonstrate that the structural and magnetic transitions merge to a simultaneous first-order magneto-structural transition as shown in Fig. 3 and also found, for example, in the CaFe 2 As 2 parent compound 26 . Further, our HE-XRD measurements show that the magnetic order in FeSe breaks the tetragonal symmetry of the lattice in the same manner as the ubiquitous stripe-type magnetic order in the other iron-based materials. To our knowledge, no other AFM order observed in iron-based superconductors or magnetic structures proposed for FeSe 27 break the tetragonal symmetry in this specific way.      The HE-XRD and NFS results for FeSe demonstrate that all features concerning the normal state structure and magnetism, and their coupling, are very similar to other iron-based superconductors, but differ in the details of the temperature and pressure dependencies of the transitions. These results are consistent with a spin-driven mechanism for nematic ordering in FeSe. In Fig. 4, we compare the schematic phase diagrams of BaFe 2 As 2 and FeSe with trends derived from recent theoretical studies of spin-driven nematicity 5 . Figure 4a sketches the sequence of structural and magnetic phase transitions for BaFe 2 As 2 as a function of doping. For Co substitution for Fe in BaFe 2 As 2 (electron doping), the magnetic and structural transitions separate in temperature and both T s and T m decrease with increased substitution 24 . On the other hand, for hole doping, the magnetic and structural transitions are concomitant and decrease in temperature with increased doping. The pressure-temperature diagram for FeSe in Fig. 4b also mirrors the compositional phase diagram described by Fernandes et al. 5 . The character of the phase transitions and their evolution with applied pressure, determined from our measurements, are consistent with this diagram, although T s increases as pressure is lowered, in contrast to the predicted decrease. One must, however, consider the relative strengths of the interactions in such models 5 and how these interactions evolve with pressure. In particular, a stiffening of the elastic constants with increasing pressure can result in a decrease in T s relative to its value at ambient pressure. Similarly, an increase in the magnetic interaction, indicated by the increase in the magnetic ordering temperature and ordered magnetic moment with increasing pressure, can push T s,m for the coupled magneto-structural transition to a higher temperature.
The origin of the large difference between T s and T m at low pressures remains an outstanding question, as well as the presence of nematic order in the absence of magnetic order at ambient pressure. Nevertheless, we note that this is not qualitatively different from the Ba(Fe 1 À x Co x ) 2 As 2 system in which there is a compositional range, close to x ¼ 0.06, where the nematic order is established at B30 K in the absence of any magnetic order 4 . Several possible explanations within a spin-driven scenario have been discussed. One possibility is, for example, that the magnetic transition is suppressed relative to the structural one via frustration 27,28 . However, other scenarios regarding the underlying mechanism for nematicity in FeSe have been put forward 15,[29][30][31] and the HE-XRD and NFS results presented here provide an important benchmark for further theoretical work.

Methods
Sample preparation. The FeSe single crystals were prepared using KCl/AlCl 3 chemical-vapour transport 32 and were thoroughly characterized in previous studies 21,33 . Both transition temperatures, T s and T m , were clearly observed by resistivity measurements under pressure 21 , which facilitates a mapping of the phase diagram [20][21][22] and supports the high-quality of our single crystals. The batch used for the HE-XRD experiment was characterized with DC magnetization and electrical resistivity measurements, and yielded sharp superconducting transitions at T c ¼ 8.7-8.8 K at ambient pressure. Samples for the NFS experiment were prepared similarly 34 using 94% isotopically pure 57 Fe, and had T c values between 8.5-8.8 K.
High-Energy X-ray Diffraction. HE-XRD measurements were performed on the six-circle diffractometer at station 6-ID-D at the Advanced Photon Source, using 100.3 keV x-rays and a beam size of 100 Â 100 mm 2 . A sample with dimensions 120 Â 120 Â 20 mm 3 was loaded into a membrane-driven copper-beryllium diamond anvil cell (DAC). A tungsten gasket with an initial thickness of 120 mm was pre-indented to a thickness of 70 mm, and a 660 mm hole was laser-drilled to accommodate the sample and pressure calibrants (ruby spheres and silver foil), as shown in Fig. 1b. Helium gas was used as the pressure-transmitting medium and loaded at P ¼ 0.5 GPa. The pressure was initially determined by the fluorescence lines from ruby spheres at ambient temperature, and, during the diffraction measurements, was determined in situ by analyzing selected Bragg peaks from the silver foil. The DAC was mounted on the cold finger of a He closed-cycle refrigerator and temperature-dependent measurements were performed between T ¼ 5 and 300 K for various pressures. The pressure was always changed at temperatures well above 120 K. Extended regions of selected reciprocal lattice planes and the powder diffraction pattern of silver were recorded by a MAR345 image plate system positioned 1.474 m behind the DAC, as the DAC was rocked by up to ±3.2°about two independent axes perpendicular to the incident x-ray beam. High-resolution diffraction patterns of selected Bragg reflections were also recorded by using a Pixirad-1 detector positioned 1.397 m behind the DAC while For a, the control parameter is partial element substitution (doping) for the case of BaFe 2 As 2 . For b, the control parameter is pressure for the case of FeSe. T s , T s*,m and T s,m denote the structural transition temperature, the magnetic ordering temperature associated with the discontinuous change in the orthorhombicity, and the joint structural and magnetic transition temperature, respectively.
rocking around one of the two axes perpendicular to the x-ray beam. The images in Fig. 3 show typical examples of diffraction patterns measured using the Pixirad-1 detector, and demonstrate the excellent mosaic of the single crystal under applied pressure, as evidenced by the well-split pattern of the (HH0) Bragg peaks (in tetragonal notation) due to the orthorhombic distortion 35 . The orthorhombic lattice parameters were determined by fitting the Bragg peak positions after integrating the data over the transverse scattering directions. This procedure was used for both the data recorded by the Pixirad-1 detector and the data recorded by the MAR345 image plate system.
Nuclear forward scattering. In NFS, highly monochromatic synchrotron radiation from an electron bunch excites the 57 Fe nuclei in the sample and the decay curve is measured as a function of time. In the presence of a hyperfine interaction that splits the 57 Fe nuclear energy levels, oscillations in the scattered intensity with time (quantum beats) are observed and can be directly compared with conventional Mössbauer spectroscopy 36 . NFS spectra were collected at beamline 16-ID-D at the Advanced Photon Source with an incident energy monochromated to the 57 Fe nuclear resonance at 14.4125 keV, with a resolution of 2 meV, and a cross-section of 35 Â 50 mm 2 of the beam. A 57 FeSe single crystal of dimensions 50 Â 50 Â 18 mm 3 was loaded into a membrane-driven copperberyllium DAC with 600 mm culet anvils, which allowed us to collect NFS data at P ¼ 2.5 and 4.0 GPa. A non-magnetic Cu-Be gasket was pre-indented to 55 mm thickness, and a hole of diameter 270 mm was laser-drilled to accommodate the sample and pressure calibrant (ruby spheres). Helium was loaded as the pressure-transmitting medium to enable hydrostatic pressure conditions. The DAC was mounted on the cold finger of a helium-flow cryostat, which achieved temperatures down to T ¼ 11 K. The intensity of the resonantly scattered photons in the forward direction, with 153.4 ns separation between the individual bunches, were recorded by an Avalanche Photo Diode detector. The program CONUSS 37 was used to analyze the spectra and to determine the magnitude of the hyperfine field. The NFS spectrum at ambient temperature was well modelled without a magnetic hyperfine field, fitting only the effective sample thickness and quadrupolar splitting, which yielded a value of 0.20(5) mm s À 1 at P ¼ 0.8 GPa consistent with earlier reports 23 . The low-temperature spectra were satisfactorily fit by including a hyperfine magnetic field. The two angles defining the direction of the hyperfine magnetic field with respect to the incident beam were also determined in addition to the (slightly pressure and temperature dependent) effective thickness of the sample. The values for the quadrupolar splitting were fixed to 0.15 and 0.20 mm s À 1 at P ¼ 2.5 and 4.0 GPa, respectively.
Data availability. All relevant data are available from the corresponding authors upon request.