Polar nematic state in an iron-based superconductor LaFeAsO1-xHx

Sachiko Maki Tokyo Institute of Technology Jun-ichi Yamaura (  jyamaura@lucid.msl.titech.ac.jp ) Tokyo Institute of Technology https://orcid.org/0000-0002-8992-9099 Soshi Iimura Tokyo Institute of Technology Hitoshi Abe High Energy Accelerator Research Organization Hajime Sagayama High Energy Accelerator Research Organization Reiji Kumai High Energy Accelerator Research Organization https://orcid.org/0000-0002-5320-0028 Youichi Murakami KEK Satoru Matsuishi Materials Research Center for Element Strategy Hideo Hosono Tokyo Institute of Technology https://orcid.org/0000-0001-9260-6728


Introduction
Spatial inversion symmetry is intimately involved in the physics of matter, thereby its breaking can trigger exotic quantum phenomena. For example, metallic materials with polar crystallographic structure-polar metals-offer the opportunity to explore interesting physics such as unconventional superconductivity, nonreciprocal transport, and inverse Faraday effect [1][2][3][4] . While over 60 polar metals have been reported to date 5 , a temperature-induced polar metal state as suggested by Anderson and Blount in 1965 is rare because screenings of conduction electrons hamper the formation of a macroscopic electrostatic eld 6 .
Shi et al. reported an example of a ferroelectric-like transition while maintaining the metallic behaviour in LiOsO 3 7 . On rising temperature or doping carrier, such a polar metal will recover its centrosymmetric structure; thus, one would expect a uctuating state, a quantum critical point, and a nematic state adjacent to the polar metal phase [8][9][10] .
Studies of high-T c iron-based superconductors have experienced spectacular growth in condensed matter physics since their discovery [11][12][13] . We believe that the most promising material for studying polar metals is an iron oxypnictide LaFeAsO 1 − x H x , which is a hydrogen-substituted version of the prototype iron-based superconductor LaFeAsO 1-x F x 14 . The LaFeAsO 1 − x H x exhibits a characteristic phase diagram via electron doping, as illustrated in Fig. 1. The diagram has two magnetic parent phases at x ~ 0 (PP1) and x ~ 0.5 (PP2), and two superconducting domes with T c,max = 26 K at x ~ 0.08 (SC1) and T c,max = 36 K at x ~ 0.35 (SC2) [15][16][17][18] . Tetragonal-orthorhombic structural transitions at T s1 (x ~ 0) and T s2 (x ~ 0.5) precede magnetic transitions; the respective orthorhombic structures are centrosymmetric and noncentrosymmetric [18][19][20] . The PP2 structure with the polar point group mm2 serves as a unique parent phase in high-T c materials as the spatial inversion symmetry is broken by temperature 18 . Throughout this paper, we use the term "parent phase" to refer to the magnetic ordered state that breaks lattice C 4 symmetry. The source of SC2 has been discussed in terms of spin-uctuation, orbital-uctuation, and orbital-selective Mott state models 16,21−24 . However, the nematicity or uctuating state in highly electron doped LaFeAsO 1-x H x remains unexplored and warrants investigation.
In this paper, we analyse the average and local structures of LaFeAsO 1 − x H x (0.35 ≤ x ≤ 0.51) by measuring synchrotron X-ray diffraction (XRD) and extended X-ray absorption ne-structure (EXAFS). Our XRD and EXAFS results led us to identify a polar nematic state: a broken lattice C 4 symmetry and a polar uctuated structure that emerge at temperatures far above the parent phase. These phenomena arise from the dynamical and short-range uctuation of the parent phase structure.

Results And Discussion
We rst focus on the change of crystal lattice symmetry in XRD measurements. Figure 2a shows the Xray pro les for 220 T re ection at 300, 120, and 32 K for x = 0.51, where the su x "T" signi es the indexing in the tetragonal system. Though the 220 T pro le is supposed to split into two peaks below the tetragonal-orthorhombic structural transition at T s2 ~ 95 K, the pro le was already broadened at 120 K.
We regard the T s2 transition as static and long-range order of the orthorhombic distortion, the temperature of which was estimated using the lattice constant 18 and the resistivity 16 anomalies. Moreover, the pro le with a shoulder at 32 K exhibits inequivalent intensities of two re ections split from 220 T , manifesting the presence of polar structure below T s2 ( Supplementary Information, Fig. S1). We proceed to examine the local structure of LaFeAsO 1-x H x from the As K-edge EXAFS measurement for x = 0.51, 0.45, 0.42, and 0.37. Figure 3a, b shows the representative k 2 -weighted EXAFS oscillation k 2 χ(k) and R-space magnitude of the Fourier transformation (FT), respectively, at 9.7 and 250 K for x = 0.51. In the radial direction without phase-correction, the peak amplitudes around R = 2.1 and 2.9 Å correspond to As-Fe and As-La/As-O shells, respectively. We analysed the bond distances from the t to the rst As-Fe shell with an R-range of 1.85-2.30 Å based on the high-temperature P4/mmm structure. Figure 3c plots the temperature dependence of the As-Fe distances, which decrease on cooling within the hightemperature range (> 100 K). However, for x = 0.51, 0.45, and 0.42 the upturns in bond distances on cooling were observed at respective temperatures of 95, 90, and 30 K, which correspond closely to T s2 .
The elongation of bond distance below T s2 arises from the negative thermal expansion of the c-axis, as reported in previous XRD measurement 18 .
Next, we employed a Fourier-ltered back transformation in EXAFS, which enables the detection of unknown tiny distortions within a speci c coordination sphere 26,27 . Figure 4 illustrates the Fourier-ltered EXAFS amplitudes for As-Fe (1.60-2.45 Å) shells of R-spectra (EXAFS oscillations in Supplementary  Information, Fig. S2). This examination found in ection points or "kinks" in each of pro les at the low temperature side. The amplitudes at the lowest temperature and at 250 K are plotted in the gure insets with solid blue and dashed black lines, respectively. The data at 250 K is normalised by the peak position and height of the object pro les used for the baseline. We de ne the difference amplitude between the object pro les and the baseline as the phenomenological formula k 2 χ dif (q) = k 2 χ(q)(T)−s 0 k 2 χ(s 1 q)(250 K), where s 0 and s 1 are the scale factors ( Supplementary Information, Fig. S3). We evaluated the kink positions q beat indicated by arrows as the peaks in the wavenumber derivatives of k 2 χ dif (q). The q beat values were roughly 11.2, 11.0, 11.6, and 11.3 Å -1 respectively for x = 0.51, 0.45, 0.42, and 0.37 at the lowest temperature. The kink is due to the beat produced from the phase difference of EXAFS oscillations with the difference of bond distances ΔR in the shell. Based on the relation ΔR = π/(2q beat ) 26 , we can estimate ΔR of the As-Fe distances to be 0.140, 0.143, 0.135, and 0.139 Å for x = 0.51, 0.45, 0.42, and 0.37, respectively.
The uniform As-Fe distance in the high-temperature phase splits into two long and two short distances in PP2 along with the loss of inversion symmetry 18 , leading to a small ΔR; this was not observed in PP1 19,20 . Note that ΔR at 32 K for x = 0.51 in PP2 was previously determined to be 0.14 Å from XRD study 18 , which agrees with our EXAFS result. We therefore ascribe the kinks in the EXAFS amplitude to the presence of local polar structure, and regard the k 2 χ dif (q) as the fraction of the polar structure. The amplitude is largest at 30 K for x = 0.51 in PP2; its value drops with decreasing x and/or rising temperature, but can still be observed far outside of PP2.
Our results from XRD and EXAFS are summarised in Fig. 1 together with the previously observed phase diagram 15,16,18 . The green area signi es the local distortion with broken inversion symmetry. The phase diagram shows that the lowering of lattice symmetry appears below T * , and dynamical and short-range polar structure emerges over wide ranges in temperature and doping. Taking account of the gradual changes in H and beat amplitude, their transformations may be regarded as cross-over phenomena instead of a phase transition.
In the iron-based superconductors BaFe 2 (As 1-x P x ) 2 and Sr 1-x Na x Fe 2 As 2 , nematic states arise at speci c doping levels from parent to superconducting phase, while breaking C 4 magnetic and/or lattice symmetries 28,29 . Hence, we view the lowering of lattice symmetry at temperatures T s2 ≤ T ≤ T * revealed by our XRD measurements as a nematic state. In the same way, we suggest that our nematic state involves an electronic or magnetic nematicity. Moreover, the local polar structure was identi ed over a wider temperature and doping range. Since the polar structure below T s2 entails the breaking of lattice C 4 symmetry, both of these phenomena that occur above T s2 should be intertwined. We thus propose that the T s2 ≤ T ≤ T * region can be called a polar nematic state, although it is di cult to identify the vanishing temperature for beat amplitude. Let us now consider the interplay between the local polar structure and the superconductivity. Lowering of lattice symmetry was unobserved via XRD for x = 0.40 and 0.35 in SC2, whereas the local polar structure was detected via EXAFS for x = 0.37. Since the orthorhombicity in PP2 rapidly reduces with decreasing x from x = 0.51 18 , the presence of a minute lattice distortion may have been experimentally undetectable via XRD. Regardless, the aforementioned electric-eld gradient anisotropy from NMR 30 was evident even in the superconducting phase. Hence, we suggest that the polar nematic state is linked with the superconducting phase 30 . We consider SC2 to be derived from PP2 by the introduction of holes, namely as x decreases from ~ 0.5. Thus, in relation to superconductivity, the polar structure may have an effect on T c or the paring mechanism. As an example, noncentrosymmetric superconductors can give rise to exotic pairing states with spin-singlet and spin-triplet mixtures 1,31 . In contrast, the superconductivity in LaFeAsO 1-x H x emerges after recovering inversion symmetry, where one might expect fluctuation-related phenomena instead. Intriguing theories have been proposed by Anderson and Blount 6 , and Ydlium et al. 32 positing that ferroelectric-like soft phonons enhance T c or drives the superconductivity. Moreover, an oddparity superconductivity derived from parity uctuation has been predicted in the vicinity of inversion symmetry breaking 33,34 . Since polar structure is also observed in SmFeAsO 1-x H x with T c = 55 K 35,36 , further insight awaits from the more detailed investigation of local physical properties in this system.
In conclusion, the average and local structures for highly electron doped LaFeAsO 1-x H x with bipartite parent phases were investigated using XRD and EXAFS measurements. The second parent phase (x 0 .5) entails the time-reversal and the spatial inversion symmetries broken. We have demonstrated that a dynamical state with broken lattice C 4 symmetry and polar structure-a polar nematic state-emerges in a wide temperature/doping range above the parent and the superconducting phases. This observation reported in this study is the intriguing result because the electronic, magnetic, and lattice instabilities should be weak generally in the highly electron-doped region. We conclude that EXAFS serves as a good probe for the detection of local polar structure that could help map further studies of nematicity.

Methods
Powder samples of LaFeAsO 1 − x H x were prepared using a high-pressure solid-state reaction as described in a previous study 15 . Synchrotron XRD and As K-edge transmission EXAFS were carried out over the whole temperature range on beamlines BL-8A/8B/9C at the Photon Factory at the High Energy Accelerator Research Organization (KEK). For XRD measurements, a very ne powder sample was enclosed in a capillary with a diameter of 0.1 mm, which was irradiated with an X-ray beam. The sample was continuously rotated during the exposure. Two-dimensional XRD images were obtained using a diffractometer with a curved imaging plate (R-AXIS, Rigaku Corp.) at wavelength λ = 1.0993 Å. The images were integrated to yield 2θ-intensity data using the DISPLAY software (Rigaku). For EXAFS measurements, all crystalline reagents were fastidiously mixed with BN powder. This process is key to obtaining high-quality data up to the high-k region. EXAFS oscillations and R-space magnitude of the Fourier transformation were extracted from the raw data using the Athena program. The R-space data were tted to the theoretical signals calculated by FEFF8 code using IFEFIT on the Arthemis platform 37