## Introduction

With the discovery of superconductivity (SC) in the iron-based compound LaFeAsO1-xFx in 20081, a fascinating new class of materials joined the cuprates as a pristine frontier to further explore unconventional superconductivity and help elucidate the origin of high-temperature superconductors. This was the first family of iron-based superconductors, now referred to as 1111 compounds, but they were followed quickly by the discovery of SC in the related hole-doped Ba1−xKxFe2As22 and electron-doped Ba(Fe1-xCox)2As23,4 (122 family). Both families exhibit in-plane, two-fold rotationally symmetric stripe antiferromagnetic order ($${C}_{2M}^{a}$$) in their respective unsubstituted parent compounds5. Subsequently, superconductivity was discovered in the NaFeAs6 (111) system, as well as the CaAFe4As4 (A = K, Rb, Cs) and SrAFe4As4 (A = Rb, Cs)7 (1144) compounds. These systems all either exhibit antiferromagnetic ordering in their parent compound or become magnetically ordered upon small doping concentrations with superconductivity arising at the phase boundaries as this magnetic order is suppressed by chemical substitution or pressure8. This proximity of SC to magnetic order, analogous to the cuprates and heavy fermions, suggests unconventional superconducting pairing which was indeed demonstrated theoretically9,10,11,12 and experimentally13. Extensive research further led to the discovery of additional exotic magnetic states such as the re-entrant tetragonal $${C}_{4M}^{c}$$ out-of-plane collinear double-Q magnetic phase universally observed in the hole-doped 122s14,15,16,17,18 and the in-plane non-collinear double-Q spin vortex crystal “hedgehog” ($${C}_{4M}^{{ab}}$$) ordering identified in CaKFe4As419,20 as well as suggested for Ba1-xNaxFe2As221. The magnetic phases are shown in Fig. 1b: the notation $${C}_{2M}^{a}$$, $${C}_{4M}^{{ab}}$$, and $${C}_{4M}^{c}$$ refers to the distinguishing attributes of rotational symmetry (with the subscript denoting whether the magnetic ordered state has two-fold or four-fold symmetry) and in- or out-of-plane spins (with the superscript denoting the magnetization direction). The possibility of the iron-pnictides realizing three distinct magnetic phases was proposed by Lorenzana et al.22, and their microscopic origin has been attributed to a range of effects, such as poorer nesting conditions23,24, spin-orbit coupling25,26, disorder27, and quantum fluctuations28. Whether these seemingly unrelated magnetic orders are idiosyncrasies of their respective systems or an indicator of underlying general behavior remains an open question necessitating the exploration of additional frameworks such as the under-investigated 1111 system.

The RFePnO family, where R is a rare earth and Pn is a pnictogen (typically arsenic or phosphorus), exhibits a rich phase diagram. Non-structural probes (e.g., NMR, resistivity, and magnetic susceptibility) have suggested the possible existence of unknown magnetic states in LaFeAs1-xPxO29,30,31,32,33, but determination of magnetic order on the iron sublattice has only been performed in the parent LaFeAsO, where $${C}_{2M}^{a}$$ order was observed34,35,36. The full LaFeAs1-xPxO phase diagram is complex with multiple magnetic states and two disconnected superconducting domes forming a unique pattern over at least four distinct regions37. Such separated islands of superconductivity have not been observed in any other pnictide series, thus making LaFeAs1-xPxO a prime ground for in-depth exploration of these competing states.

At first glance, the unique superconducting features of the LaFeAs1-xPxO system seem a significant departure from the related 122 and 1144 families. However, substitution of fluorine31,33,38 for oxygen was shown to completely suppress30 all magnetic orders and to extend superconductivity over the entire phosphorus-substituted phase diagram. The merging of the two SC domes demonstrates that they are remnants of a single state that had been suppressed by competing magnetic orders. Partial depression of Tc was also seen in the hole-doped 122s in conjunction with the emergence of the $${C}_{4M}^{c}$$ phase39. These observations hint at an underlying universal behavior common to the 1111 and 122 systems, but conclusive comparisons require solving the magnetic structures in the 1111s.

Here, we present a detailed investigation of structural and magnetic order in 1111 compounds utilizing synchrotron x-ray and neutron diffraction and muon spin relaxation (µSR). We find that unlike any other family of compounds, the LaFeAs1-xPxO series exhibits direct paramagnetic-to-magnetic transitions to all three types of magnetic orders, orthorhombic single-Q $${C}_{2M}^{a},$$ tetragonal in-plane double-Q $${C}_{4M}^{{ab}}$$, (as observed in 1144s), and tetragonal out-of-plane double-Q $${C}_{4M}^{c}$$ (as observed in hole-doped 122s) within the same phase diagram, Fig. 1a. Supplementary Figure 1 displays high-resolution synchrotron x-ray data demonstrating the tetragonal character within experimental resolution of the double-Q magnetic phases (x > 0.35). A key difference to emphasize is that in 122s, the $${C}_{4M}^{c}$$ exists within the overlapping region between the $${C}_{2M}^{a}$$ and SC domes, whereas in LaFeAs1-xPxO the tetragonal magnetic order is observed outside of the $${C}_{2M}^{a}$$. Using a microscopic model in which the isovalent chemical substitution of phosphorus for arsenic is described by an increase in the size of the Fermi surface, see Christensen et al.25, we reproduce the hierarchy of magnetic phases, with $${C}_{2M}^{a}$$ order being favored by small Fermi surfaces, while successively larger Fermi surfaces yield first a $${C}_{4M}^{{ab}}$$ phase followed by a $${C}_{4M}^{c}$$ phase. Here, spin-orbit coupling is a necessary ingredient to solve a degeneracy between the tetragonal magnetic phases, reproducing the observed consecutive reorientations of the magnetic moments.

## Results and discussion

We have conducted a detailed survey of the phase diagram of LaFeAs1-xPxO using high-resolution synchrotron x-ray powder diffraction, neutron diffraction, and µSR. Details of the synthesis and characterization of the polycrystalline samples are presented in the methods section.

We observe a continuous sequence of the three distinct magnetic phases shown in Fig. 1b, which previously have not been observed to occur in a single-phase diagram as a function of substitution in any other pnictide system. The evolution of the magnetic order with increasing phosphorus substitution begins as orthorhombic single-Q $${C}_{2M}^{a}$$ order, followed by two back-to-back tetragonal double-Q magnetic phases: the in-plane $${C}_{4M}^{{ab}}$$ “hedgehog” phase existing over a narrow range around $$x=0.45$$, and the out-of-plane $${C}_{4M}^{c}$$ order covering a wide range of phosphorus content from $$0.5\le x\le 0.8$$. Interestingly, the $${C}_{4M}^{{ab}}$$ phase seems to play the role of an intermediate phase while switching from the in-plane $${C}_{2M}^{a}$$ to the out-of-plane $${C}_{4M}^{c}$$ magnetic states.

### Experimental data and analysis

Our magnetic susceptibility measurements show superconducting transitions in agreement with the literature30,40. Prior to this work, 31P-NMR (phosphorus nuclear magnetic resonance) studies29,30,32 had reported the Néel temperatures (TN) for the $${C}_{2M}^{a}$$ phase for $$x\le 0.2$$ and provided evidence for unknown magnetic order in the substitution range of $$0.4\le x\le 0.7$$. Our µSR measurements show evidence of bulk magnetism for every sample with $$x\le 0.8$$, Fig. 2. Neutron diffraction was used to determine the structural details of the magnetic orders over the entire phase diagram. Supplementary Figure 2 displays refinements of the magnetic structure using various models for three representative samples (x = 0.23, 0.45 and 0.8). Agreement between measured magnetic transition temperatures from neutron diffraction and µSR results was confirmed by performing temperature dependent scans for several of the same samples across both instruments. While our muon spin rotation experiment was crucial in determining the bulk magnetic properties of our samples, an in-depth analysis of the data from which the muon stopping sites could be identified requires short time scales that are inaccessible at the EMU beamline. We hope that this work motivates future muon spin rotation and Mössbauer spectroscopy experiments to determine the bimodal local field distribution for the tetragonal magnetic phases.

A linear fit of the c-axis parameter refined using neutron and x-ray diffraction data is in agreement with Vegard’s law41, thereby confirming the successful systematic substitution of phosphorus for arsenic throughout our solid solution series, Fig. 3a. The unsubstituted parent end-members (LaFeAsO and LaFePO) display structural properties and lattice parameters in agreement with those reported in references42 and43. Both x-ray and neutron diffraction confirmed the tetragonal to orthorhombic structural transition, up to $$x=0.35$$, occurring at a slightly higher temperature than the $${C}_{2M}^{a}$$ magnetic transition, in agreement with previous studies on the 1111 systems42,43,44,45,46,47,48,49. Figure 3a illustrates the continuous suppression with increasing x of the orthorhombic distortion at T = 10 K. The system is fully tetragonal for $$x\,\gtrsim\, 0.35$$.

Identification of the $${C}_{2M}^{a}$$, $${C}_{4M}^{{ab}}$$ and $${C}_{4M}^{c}$$ magnetic orders is easily and unambiguously made by carefully examining the relative intensities and location in reciprocal space of their diffracted magnetic peaks. Very different diffraction angles and relative intensities are expected for the diverse magnetic states. As shown in Fig. 4, for example, the two most intense magnetic peaks for the $${C}_{2M}^{a}$$ phase are observed around 4.05 and 5.4 Å while those of the $${C}_{4M}^{{ab}}$$ phase are at 3.43 and 4.73 Å. Those of the $${C}_{4M}^{c}$$ are at 4.71 and 5.63 Å. Confirmation of the magnetic structures is further revealed from detailed Rietveld refinements using the appropriate models (please also see the Supplementary Figure 2). Examples of neutron diffraction data from each of the three magnetic states are presented in Fig. 4. Data from above (red lines) and below (blue lines) TN are plotted together with their difference (purple lines). The magnetic and structural refinements were performed on the full low-temperature diffraction pattern while magnetic refinements were performed concurrently on the difference pattern. The tested magnetic models indexed the correct peaks and, upon refinement, match intensities and peak profiles appropriately with no extra intensities. The difference plot allows for an easier qualitative visual analysis of the very small magnetic peaks. The refined magnetic moments from neutron diffraction are shown in Fig. 3b. Phase fractions were identified from the nuclear refinement, and the magnetic phase fraction was matched to the main nuclear phase. The refined magnetic moment for $$x=0$$ begins at ~0.6 µB in the $${C}_{2M}^{a}$$ regime, similar to that observed in the 122s, drops continuously to ~0.2 µB and remains roughly constant for compositions with $$x\ge 0.3$$. The refined magnetic moments were congruent with the observed magnetic Bragg reflection intensities, with Bragg peak magnitudes approximately scaling with the square of the moment, further indicating the positive identification of the magnetic phases.

### Theoretical model

To understand the evolution of magnetism in this series, we note that all magnetic states observed here can be described in terms of two order parameters M1 and M2 that are related to the local magnetization m(R) according to:

$$m\left({{{{{\boldsymbol{R}}}}}}\right)={{{{{{\boldsymbol{M}}}}}}}_{{{{{{\bf{1}}}}}}}{\cos }\left({{{{{{\boldsymbol{Q}}}}}}}_{{{{{{\bf{1}}}}}}}\cdot {{{{{\boldsymbol{R}}}}}}\right)+{{{{{{\boldsymbol{M}}}}}}}_{{{{{{\bf{2}}}}}}}{\cos }\left({{{{{{\boldsymbol{Q}}}}}}}_{{{{{{\bf{2}}}}}}}\cdot {{{{{\boldsymbol{R}}}}}}\right)$$
(1)

Here, Q1 = (π,0) and Q2 = (0,π) are the antiferromagnetic wave-vectors in the 1Fe/unit cell. The nature of the magnetic instability is determined by the coefficients of the Landau free-energy expansion $$F\left(M\right)={F}^{(2)}+{F}^{(4)}$$:

$${F}^{\left(2\right)}= \; a\left({M}_{1}^{2}+{M}_{2}^{2}\right)+{\alpha }_{1}\left({M}_{1,x}^{2}+{M}_{2,y}^{2}\right)+{\alpha }_{2}\left({M}_{2,x}^{2}+{M}_{1,y}^{2}\right)\\ +{\alpha }_{3}\left({M}_{1,z}^{2}+{M}_{2,z}^{2}\right)$$
(2)
$${F}^{\left(4\right)}=\frac{u}{2}{\left({M}_{1}^{2}+{M}_{2}^{2}\right)}^{2}-\frac{g}{2}{\left({M}_{1}^{2}-{M}_{2}^{2}\right)}^{2}+2w{\left({M}_{1}\cdot {M}_{2}\right)}^{2}$$
(3)

The quadratic part of the expansion, $${F}^{\left(2\right)}$$, in Eq. (2) has two parts: a spin-isotropic term, whose coefficient a determines the magnetic transition temperature, and a spin-anisotropic part, whose coefficients $${\alpha }_{i}$$ select the direction of the magnetization. The quartic coefficients, Eq. (3), determine which magnetic order is favored22,50,51. In particular, the $${C}_{2M}^{a}$$ phase, which corresponds to only one of the order parameters being non-zero, is favored by $$g \, > \, 0$$. On the other hand, $$g \, < \, 0$$ and $$w \, > \, 0$$ favor the $${C}_{4M}^{{ab}}$$ phase, corresponding to |M1| = |M2| and M1 M2, where $$g \, < \, 0$$ and $$w \, < \, 0$$ favor the $${C}_{4M}^{c}$$ phase, parametrized by |M1|=|M2|and M1 M2.

Using a microscopic $$k\cdot p$$ model, a semi-empirical perturbation method for approximating the band-structure first derived by Cvetkovic et al.52, the coefficients of the Landau free energy were derived by Christensen et al.25, providing a direct link between changes in microscopic parameters to changes in the magnetic ground state. Because in this approximation $$w=0$$, the type of tetragonal magnetic phase selected for $$g < 0$$ is determined by the direction of the magnetization, which in turn is determined by the smallest $${\alpha }_{i}$$ coefficient. For in-plane moments, which is the case when $${\alpha }_{1}$$ or $${\alpha }_{2}$$ is the smallest spin-anisotropic coefficient, the $${C}_{4M}^{{ab}}$$ phase is favored, while for out-of-plane moments, which is the case when $${\alpha }_{3}$$ is the smallest coefficient, the $${C}_{4M}^{c}$$ phase is the ground state.

To make contact with our experimental results, we need to model the impact of the isovalent chemical substitution on the band structure. To this end, we note that Dynamical Mean-Field Theory (DMFT) and Density Functional Theory (DFT) calculations indicate that LaFeAsO is more strongly correlated than LaFePO53,54. One known effect of the correlations is to shrink the size of the Fermi pockets. This seems to be the case, for instance, in the related isovalently-substituted compounds Ba(Fe1−xRux)2As2 and Fe(Se1-xSx)55,56. Thus, to proceed, we assume that the effect of phosphorus substitution is to increase the size of both electron and hole Fermi surfaces in our model (while keeping the total electronic occupation constant) and compute the Landau coefficients as a function of the change in the Fermi momentum $$\frac{\triangle {k}_{F}}{{k}_{F}}$$. While other effects may also contribute to the elucidation of our experimental data, we are confident that our simplified model provides useful insight to understand them. This is the main goal of the comparison between the theoretical and experimental results, and not to rule out other possible scenarios. Full details of our theoretical model are presented in the Supplementary Note 1 and Supplementary Figure 3.

Our results for g and $${\alpha }_{i}$$ are shown in Fig. 5, using the general expressions derived by Christensen et al.25. For relatively small Fermi surfaces, corresponding to the small x region of the phase diagram in our experiment, the magnetic ground state is the $${C}_{2M}^{a}$$. As the Fermi surface area increases, corresponding to increasing phosphorus in our experiment, g becomes negative and the $${C}_{4M}^{{ab}}$$ phase is preferred, since $${\alpha }_{1}$$ is the smallest coefficient. Upon further increasing $${k}_{F}$$, $${\alpha }_{3}$$ becomes the smallest coefficient, and the $${C}_{4M}^{c}$$ phase becomes the leading magnetic instability. We verified that this evolution of the coefficients is generic for a wide range of temperatures.

The stabilization of the $${C}_{4M}^{{ab}}$$ and $${C}_{4M}^{c}$$ phases splits the single superconducting dome, as seen by the recovery of a single SC dome with fluorine doping30, into two disconnected domes. However, the highest temperature for any type of electronic order remains very stable across the phase diagram. Although the $${C}_{4M}^{c}$$ is a key feature occurring in a narrow region within the $${C}_{2M}^{a}$$ dome in the hole-doped 122s, it was not observed outside this dome as is the case in LaFeAs1-xPxO. The existence of a narrow $${C}_{4M}^{{ab}}$$ range, sandwiched between the much wider $${C}_{2M}^{a}$$ and $${C}_{4M}^{c}$$ domes, implies a strong competition between the three states and superconductivity. The $${C}_{4M}^{{ab}}$$ phase has been observed in the 1144 system, which is an example of an A-site-ordered doubling along the c-axis of the 122 structure;57 this establishes the host 122/1144 structure is capable of providing access to the $${C}_{4M}^{{ab}}$$ or the $${C}_{4M}^{c}$$ phase with such tuning. Our results uncover a previously hidden universality in magnetic ordering in close proximity to superconductivity. While it has been well-established that magnetism and superconductivity are intimately associated in the iron-pnictides, we demonstrate here that the $${C}_{4M}^{{ab}}$$ and $${C}_{4M}^{c}$$ are similarly related to SC. This universality is supported by a recent µSR study in which Sheveleva et al. 21 suggested the existence of the $${C}_{4M}^{{ab}}$$ phase in Ba1-xNaxFe2As2.

As to what is responsible for the progression of the diverse magnetic states, we note that the isovalent substitution of phosphorus is expected to modify the Fermi surface area55,56,58. The theoretical model attributes the evolution of the magnetic ground states to the changes in the size of the Fermi surface which, in turn, is assumed to be driven by a suppression of correlations upon increasing the P content. Besides changes in the electronic structure, the emergence of disordered scattering centers caused by variations in ionic radius of the substituents also play a role. The disappearing and reappearing character of the superconducting domes suggests a tuning parameter that, rather than increasing monotonically, first increases and then decreases with $$x$$ across the phase diagram; disorder could adopt this pattern as substitution across the phase diagram mirrors the ratio of whichever is dominant between arsenic and phosphorus around the $$x=0.5$$ concentration.

Charge doping by fluorine or oxygen vacancies in this system has been shown to quickly suppress $${C}_{2M}^{a}$$ with a net positive effect on superconducting $${T}_{c}$$31,59,60$$.$$ Disorder (via isovalent chemical substitution), however, has a dual destructive effect on both $${T}_{c}$$ and $${T}_{N}$$ but suppresses $${C}_{2M}^{a}$$ at a faster rate than SC. This is because magnetism is suppressed61,62 by both inter and intraband scattering while only interband scattering is pair-breaking for the unconventional $${s}^{\pm }$$ superconducting state9,63,64,65. Our results for LaFeAs1-xPxO are in agreement with this trend, since the ground state for $$x=0$$ is $${C}_{2M}^{a}$$. Introducing disorder by increasing $$x$$ gradually suppresses the $${C}_{2M}^{a}$$ phase, allowing a relatively narrow SC dome to emerge between $$x=0.25$$ and $$x=0.45$$. At this crossroads, we observe a suppression of both $${T}_{N}$$ and $${T}_{c}$$. We note similar suppression of $${T}_{N}$$ at the intersection between the $${C}_{4M}^{{ab}}$$ and $${C}_{4M}^{c}$$ at $$x\approx 0.5$$ and between the $${C}_{4M}^{c}$$ and SC at $$0.7 \, < \, x \, < \, 0.8$$. There is a clear symmetry in the phase diagram with the maximum $${T}_{c}$$ of both superconducting domes ($$x\approx 0.27$$ and $$x\approx 0.8$$) being approximately midway between the minimally disordered end members and the maximum disorder at $$x=0.5$$. The $${C}_{4M}^{c}$$ and the $${C}_{4M}^{{ab}}$$ phases appear to be more resilient to disorder than the $${C}_{2M}^{a}$$, in agreement with theoretical predictions27.

The lack of long-range magnetic order in fully-substituted LaFePO ($$x=1$$) explains the relatively robust superconducting dome on this side of the phase diagram when compared to the As side. The last feature left to account for is the relative suppression of $${T}_{c}$$ for phosphorus content above $$x=0.8$$. A reduction in $${T}_{c}$$ is in line with LaFePO being a better metal with weaker electronic correlations than LaFeAsO66. However, in a system with such fierce phase competition driving a complex phase diagram, other factors may be influencing the superconducting behavior.

## Conclusions

We have presented the results of neutron, x-ray, and µSR on a systematically substituted series of LaFeAs1-xPxO polycrystalline samples along with the most complete phase diagram to date, accounting for the evolution of the magnetic structures and their relative placement with respect to superconductivity in this system. Our results highlight the simultaneous delicate balance and fierce competition between magnetism and superconductivity in a system in which all magnetic states known to the iron-pnictide family exist in the same compositional phase diagram. Our observation of $${C}_{2M}^{a}$$, $${C}_{4M}^{{ab}}$$, and $${C}_{4M}^{c}$$ in the 1111 and previously reported in the 122 and 1144 compounds uncovers a long-sought underlying generality of magnetic ordering in the iron-pnictides and the role it plays in relation to superconductivity. With a better understanding of the nature of the ground states and a theoretical model to explain their evolution, we have developed a framework through which to view a unified understanding of the interplay between superconductivity and magnetism in systems with different crystal structures and distinct types of chemical substitution.

## Methods

### Synthesis

Due to the toxic and carcinogenic properties of arsenic together with the air sensitivity of precursors, all synthesis was conducted inside an Ar-atmosphere glovebox with <5 ppm O2 and <0.1 ppm H2O. Initial synthesis of the powder samples was attempted following the procedures outlined by Lai et al.30. However, due to its limitations in only being able to substitute up to $$x=0.66$$, we developed a comprehensive recipe capable of producing high quality samples covering the full LaFeAs1-xPxO phase diagram as demonstrated by neutron and synchrotron diffraction.

Powder samples were prepared using LaAs, La2O3, LaP, Fe, Fe2O3, FeAs, Fe2As, FeP, and Fe2P precursors. Immediately prior to use, La2O3 was dried by heating overnight to 900 °C and placed inside an Ar-atmosphere glovebox after cooling to ~400 °C. Synthesis was carried out by thoroughly grinding precursors to a fine and well-mixed powder that was loaded into alumina crucibles. The alumina crucibles were flame sealed under vacuum inside quartz tubes. The samples were annealed at 1100 °C for up to 40 h. After each annealing cycle, the samples were checked for homogeneity by laboratory XRD until complete. Initial processes required up to four annealing cycles to achieve target purity, but the final procedure using Fe, Fe2O3, FeAs and FeP achieved homogeneity after only two cycles.

### Experimental details

Quality of the samples was monitored after each synthesis step via laboratory x-ray diffraction on a PANalytical XRD. Magnetic and superconducting properties were measured in 10 Oe magnetic fields using a Quantum Design Magnetic Properties Measurement System (MPMS).

High resolution x-ray diffraction was performed for samples with $$x=0$$ to $$x=0.6$$ on beamline 11-BM-B at the Advanced Photon Source at Argonne National Laboratory. High resolution neutron diffraction was performed for samples ranging from $$x=0.23$$ to $$x=1$$ on POWGEN at the Spallation Neutron Source at Oak Ridge National Laboratory. High flux neutron diffraction was carried out for samples with $$x=0$$ to $$x=0.8$$ on WISH at ISIS at Rutherford Appleton Laboratory. µSR was performed for samples of $$x=0.23$$ to $$x=1$$ on EMU at ISIS at RAL. The asymmetry from our measured temperature-dependent muon decay curves was quantified using Mantidplot67 by a combined fit of a flat background and stretched exponential function with the form $${G}_{z}\left(t\right)={A}_{0}+{A}_{1}{e}^{{\left(-\lambda t\right)}^{{{{{{\rm{\beta }}}}}}}}$$. This asymmetry was plotted as a function of temperature and the magnetic transition temperature TN was extracted from second derivatives. Rietveld refinements were performed using the GSAS EXPGUI software suite68 and GSAS-II69.