Hedgehog spin-vortex crystal stabilized in a hole-doped iron-based superconductor

Magnetism is widely considered to be a key ingredient of unconventional superconductivity. In contrast to cuprate high-temperature superconductors, antiferromagnetism in Fe-based superconductors (FeSCs) is characterized by a pair of magnetic propagation vectors. Consequently, three different types of magnetic order are possible. Of theses, only stripe-type spin-density wave (SSDW) and spin-charge-density wave (SCDW) orders have been observed. A realization of the proposed spin-vortex crystal (SVC) order is noticeably absent. We report a magnetic phase consistent with the hedgehog variation of SVC order in Ni- and Co-doped CaKFe4As4 based on thermodynamic, transport, structural and local magnetic probes combined with symmetry analysis. The exotic SVC phase is stabilized by the reduced symmetry of the CaKFe4As4 structure. Our results suggest that the possible magnetic ground states in FeSCs have very similar energies, providing an enlarged configuration space for magnetic fluctuations to promote high-temperature superconductivity.

patterns on an Fe-As layer in the CaKFe 4 As 4 structure associated with Q 1 = (π, 0), Q 2 = (0, π) magnetic propagation vectors. The upper yellow square in a and e represents the projection of the CaKFe 4 As 4 unit cell. The magnetic unit cells are represented by the central yellow squares in a-d. The brown arrows represent the magnetic moments at the Fe sites and the blue and green arrows the hyperfine field (H hf ) at the inequivalent As1 and As2 sites. In the orthorhombic stripe spin-density wave (SSDW) in a all arsenic atoms experience H hf normal to the plane. In the spincharge-density wave (SCDW), "C 4 phase 4 ", in b every other Fe site supports a moment normal to the plane and the As 2 sites experience an in-plane H hf . In the hedgehog spin-vortex crystal (SVC) phase in c Fe moments display an "all in" or "all out" arrangement centered around the As1 sites generating an out-of-plane H hf = 0 at these sites and H hf = 0 at the As2 sites. In the loops SVC in d the magnetic moments form loops around As1 and neither As site experiences a H hf . e, The chemical structure of CaKFe 4 As 4 . Note the inequivalent As1 and As2 sites adjacent to K and Ca planes, respectively. f, Section of the Fe-As plane with a hedgehog SVC moment arrangement. Spin up currents between the iron atoms, J s (yellow arrows), generate an electric field, E (red arrows), which couples to asymmetric shifts of the two arsenic sites. Unlike in the CaFe 2 As 2 structure, an asymmetric arrangement of arsenic atoms is imposed by the crystallographic symmetry in CaKFe 4 As 4 providing a symmetry-breaking field that favors the SVC-type phases.
An experimental realization of SVC order would demonstrate the diversity of magnetic groundstates in FeSCs, but has not been reported to date.
Application of a suitable symmetry-breaking field may coax the system to condense into one of the three types of magnetic orders as they break distinct symmetries 14 . For example, inplane uniaxial strain favors SSDW as it breaks tetragonal symmetry in the same way and has been widely employed to study this order 19,20 . It is difficult to imagine appropriate, externally applicable symmetry-breaking fields for SCDW or SVC orders. A SVC phase breaks the glide symmetry across the Fe-As planes 14 , which is present in most FeSCs. Consequently, breaking this glide symmetry could favor SVC order 15 . In the recently discovered stoichiometric AeAFe 4 As 4 (Ae = Ca, Sr; A = K, Rb, Cs) superconductors 21 , the alternating Ae and A atom planes inherently break this glide symmetry resulting in two inequivalent As sites (Fig. 1e). Consequently, these compounds could provide a unique opportunity to realize and study a SVC phase. Unfortunately, the parent AeAFe 4 As 4 compounds do not order magnetically 21,22 . Electron count and experimental properties suggests that CaKFe 4 As 4 can be considered analogous to (Ba 0.5 K 0.5 )Fe 2 As 2 (Ref. 22).
In the latter compound, the SSDW is suppressed by hole doping 1 (substituting K into BaFe 2 As 2 ).
Inspired by this analogy 23 , magnetic order could be induced by electron doping CaKFe 4 As 4 . Here, we report that adding electrons via substitution of Co or Ni for Fe in CaKFe 4 As 4 does induce antiferromagnetism consistent with the hedgehog variation of SVC. We stabilize this phase using the chemical structure as a symmetry-breaking field illustrating the multiple competing, nearlydegenerate magnetic phases in FeSCs. This competition may be an important contributor to hightemperature superconductivity as enhanced magnetic fluctuations can boost pairing.        structures are all modifications of SVC order. There are two motif variations 15 , "hedgehog" with M i Q i (Fig. 1c) and "loops" with M i ⊥ Q i (Fig. 1d). Each variation can be centered on either As1 or As2 and can have four different stacking patterns along the crystallographic c-axis 27 . Only the hedgehog SVC, with the centering depicted in Fig. 1c, is consistent with the H hf at As1 observed by NMR.
We argue that this peculiar, non-collinear magnetic phase is stabilized by the specific crystal structure of CaKFe 4 As 4 . The inequivalence of the staggered As sites (Fig. 1e) generates a symmetry-breaking field that couples to the spin vorticity, M 1 × M 2 , characteristic of SVC order.
One possible mechanism is the electrostatic coupling of the As atoms to spin current loops associated with the magnetic moment motif 14,27 (Fig. 1f). These spin currents generate an electric field analogous to the magnetic field produced by an electric current 29 . The resulting electrostatic force couples the asymmetric shifts of the As atoms to the spin currents. The preference for the hedgehog SVC variation over loops variation is likely a consequence of spin-orbit coupling, which generally favors M i Q i , observed experimentally in the case of the SSDW order of the parent AeFe 2 As 2 compounds 2 , and theoretically in Ref. 11 In conclusion, we confidently identified a hedgehog spin-vortex crystal in Co-and Ni-doped paper disk with Apiezon N grease. An effort was made to keep gaps between crystals to a minimum and the part of the disk not covered by crystals was coated with tungsten powder (Alfa Aesar 99.9% metals basis). This mosaic was positioned so that the gamma ray beam was parallel to the crystallographic c-axes. The sample temperature was maintained using a Janis SHI-850-5 closed cycle refrigerator with vibration damping. The driver velocity was calibrated using an α-Fe foil.
The Mössbauer spectra were fitted using the commercial software package MossWinn 4.0Pre 36 .
Below T N , the Mössbauer spectra can be modeled with a single value of hyperfine field (reported in Fig. 4f) and an increased linewidth with respect to the paramagnetic state. Alternatively, they may be described by a temperature independent linewidth and a distribution of hyperfine fields.
Both approaches suggest that there is a distribution of H hf at the Fe sites in the magnetic phase.
Nuclear magnetic resonance (NMR) measurements of CaK(Fe 0.951 Ni 0.049 ) 4 As 4 (total mass ∼5 mg) were carried out on 75 As (I = 3 2 , γ/2π = 7.2919 MHz/T, Q = 0.29 barns) by using a labbuilt, phase-coherent, spin-echo pulse spectrometer. The spin echo was observed with a sequence of π 2 pulse (2.2 µs) -30 µsπ pulse (4.4 µs). The 75 As-NMR spectra were obtained at a fixed frequency, f = 43.2 MHz, with sweeping the magnetic field up to 8.25 T. The magnetic field was applied parallel or perpendicular to the crystallographic c-axis. 75 As NMR spectra was simulated with two (for H c above T N and H ⊥ c) or three (H c below T N ) sets of peaks with distinct values of H hf and quadrapole interaction, ν Q . The ratio of the integrated area of As1 to As2 is 1:1 in all the spectra. The ratio of As1 with −H hf to +H hf is also 1:1 in the spectra for H c below T N .
The symmetry analysis was performed used the ISOTROPY software suite 37 to systematically analyze the possible magnetic orders of CaKFe 4 As 4 with magnetic space group P4/mmm1'.
A detailed description of the analysis is presented in supplemental materials 27 .
In the first-principles calculations, the effect of electron doping on the magnetic properties of CaKFe 4 As 4 was studied using the virtual crystal approximation (VCA) and the all-electron fullpotential linearised augmented-plane wave basis (Elk FP-LAPW code). 38 The structure parameters were fixed at the measured room-temperature values for undoped samples. 39 Acknowledgements The authors would like to acknowledge W. Straszheim for his expertise and assistance with chemical analysis, D. S. Robinson for experimental support of the x-ray scattering study, and T.
Kong for assistance during early stages of sample preparation. We also acknowledge discussions with E. Data Availability The data that support the findings of this study are available from the corresponding author upon request.

Supplemental Materials Supplemental material referenced in the text can be found on-line.
Competing Interests The authors declare that they have no competing financial interests.
Correspondence Correspondence and requests for materials should be addressed to P. C. Canfield (email: canfield@ameslab.gov).

Supplemental materials
Composition analysis. Figure S1 presents the transition-metal ratio determined by wavelengthdispersive spectroscopy (WDS). In both the Co and Ni substitution series, the measured ratios were lower than the nominal ratios batched in the crucibles. Figure S2 presents  Theoretical analysis of NMR data: hyperfine fields for different magnetic orders Here we derive the internal fields experienced by the As atoms in different magnetic configurations. Since the unit cell is doubled in the double-Q magnetic phases, we consider four As atoms per Fe-As layer (see Fig. S3). The theoretically calculated magnetic hyperfine field, H hf , at As-site As j with j = 1, 2, 3, 4 is obtained from Here, A αβ j,i (α, β = a, b, c) is the nuclear-electron coupling tensor and the sum is over the Fe moments m i of the four Fe-sites belonging to the same plaquette centered at the As site As j (see Fig. S3). The matrix elements A j,i are all related by symmetry such that only A α,β 1,1 are independent parameters. Explicitly, denoting corresponds to a mirror reflection with respect to the ac plane, and refers to a π/2 rotation with respect to the c-axis. The matrix elements A j,i at the other As sites follow from symmetry considerations and are depicted in Fig. S3. In fact, due to tetragonal symmetry only three independent parameters exist, e.g., A aa 1,1 , A ac 1,1 , A cc 1,1 .
Straightforward calculations yield the following hyperfine fields for different double-Q magnetic configurations. For the hedgehog SVC (see Fig. 1c), we find: H hf (As 1 ) = −H hf (As 3 ) = (0, 0, 8mA ca ), H hf (As 2 ) = H hf (As 4 ) = (0, 0, 0), where m is the size of the Fe moment in the ab-plane. In contrast, for loops SVC order (Fig. 1d) the hyperfine field vanishes at all As-sites.
Spin-charge-density wave (SCDW) order (Fig. 1b) with moments along c yields H hf (As 1 ) = 4m c A ac (1, 1, 0) and fields that are rotated by − π 2 with respect to each other from a = 1 to a = 4. for (0, π) wave-vector. Figure S3. Definition and labeling of hyperfine tensors. Definition and labeling of hyperfine tensors A j,i within the magnetic unit cell, which contains four As atoms (As1 in blue, As2 in green) labeled by j. Each As interacts with the magnetic moments on its four Fe neighbors (brown arrows), which are labeled by i, via the hyperfine interaction tensor A j,i . The figure shows that Heat capacity measurements of a CaK(Fe 0.951 Ni 0.049 ) 4 As 4 sample (inset in Fig. 2a) reveal a second-order phase transition at T N . The temperature dependence of the hyperfine fields, H hf (T ), at the Fe-and As1-sites (Fig. 4f) also supports this conclusion. At a second-order phase transition, only one order parameter may become critical and only one generator of the symmetry group may be broken. 42 This restriction requires that the new crystallographic phase has a space group that is a maximal subgroup of the parent space group. If we apply this criteria to the 96 irreducible representations we generate the same list of 24 magnetic structures obtained in the previous paragraph.
Obtaining the same list from independent requirements strengthens our confidence in the result. (2) All the moments reverse every layer (antiferromagnetic stacking). (3 and 4) A double stacking of two aligned layers followed by two reversed layers can be realized without affecting the chemical unit-cell. There are two variations of the latter as the moment motif can either be reversed across the Ca plane or the K plane. Neutron diffraction could indicate which stacking is favored.
The tetragonal "C 4 " in the hole-doped (Ae 1−x A x )Fe 2 As 2 is reported to be a SCDW with caxis moments like that shown in Fig. 1b. Surprisingly, a SCDW is orthorhombic in CaKFe 4 As 4 . In the AeFe 2 As 2 structure all As sites are symmetry equivalent (green and blue As atoms would be equivalent in Fig. 1 and have the same distance from the iron plane). In AeFe 2 As 2 's paramagnetic were not resolved it is likely that the magnetic order is an incommensurate variety of SVC.
Ginzburg-Landau theory of magnetic orders in the presence of a symmetry-breaking crystalline field Magnetic order with propagation vectors Q 1 = (π, 0) and Q 2 = (0, π) is described by where R denotes an Fe site in the lattice and M i are N = 3 component magnetic order parameters.
The new ingredient in the case of the AeAFe 4 As 4 compounds is the difference in the heights of the two As atoms staggered above and below the Fe plane, which effectively breaks the glide plane symmetry of the Fe-As layers. The symmetry breaking field generated by this crystal structure can be parametrized in terms of a vector η = ηẑ that couples linearly to the vector spin chirality according to: 23 The emergence of this term can be derived from symmetry considerations. As discussed in details in Ref. 14, inside the SVC phase, besides long-range magnetic order, a composite order parameter also condenses: M 1 × M 2 , called spin-vorticity-density wave. This "chiral" order parameter, which lives on the plaquettes of the Fe-As layer, is related to the spin-vortices that appear in the SVC phase; it preserves time-reversal symmetry and has wave-vector Q 1 + Q 2 = (π, π) in the Feonly square lattice. Importantly, as shown in Ref. 14, this order parameter has the same symmetry as a spin-current-density wave with the same propagation vector (also known as a triplet d-density wave), manifested as staggered spin loop currents along the plaquettes (see Fig. 1f of the main text). While charge loop currents generate magnetic fields, spin loop currents generate electric fields 29 . In particular, the staggered configuration of spin loop currents creates staggered electric fields of opposite directions above and below the Fe-plane. Because the As atoms located above and below the Fe-plane are themselves staggered, all As atoms either experience an electric field pointing up or down (see Fig. 1f of the main text). Consequently, the two As atoms, As1 and As1, become inequivalent, and their heights with respect to the Fe plane also become different.
Thus, the symmetry breaking field induced by the crystal, η, triggers immediately a spinvorticity density-wave -similarly to how uniaxial strain triggers nematic order. Although the spin-vorticity density-wave does not trigger magnetic order, the fact that this order parameter is only non-zero inside the SVC phase implies that, for large enough |η| > η c , the magnetic ground state necessarily becomes SVC. Indeed, including F η does not change the size of M i and the energies E α for SSDW and SCDW orders since the spin vorticity vanishes for these two phases.
In contrast, 4u . The critical value η c that enforces an SVC ground state in the case that E SVC (η = 0) > E GS ≡ min{E SSDW , E SCDW } follows from a straightforward calculation: Therefore, the closer the energy of the SVC state is to the energy of the magnetic ground state (in the absence of the symmetry breaking field), the smaller the value of the symmetry breaking field required to make the SVC state the ground state.
ab initio. In the virtual crystal approximation (VCA) method, homogeneous doping is imposed by replacing Fe with an artificial nucleus of a fractional charge between that of Fe and Co. The k-point mesh was set to (5×5×5) dimensions. For each dopant concentration, we study the relativistic total energies of two magnetic orders imposed on the Fe sublattice, namely, the stripe order along the aaxis and the "hedgehog" SVC order with Fe spins lying in-plane and centered on As-sites near the K-plane, as pointed out in the main text. These two configurations were suggested to represent the main competing fluctuations in the undoped compound, based on the NMR measurements and ab initio findings 26 Figure S5. Magnetic structure vs additional electron. Energy difference between the "hedgehog" spin-vortex crystal and stripe-type phases of CaKFe 4 As 4 vs electron doping per transition-