## Introduction

Competing electronic phases underlie a number of unusual physical phenomena in condensed matter1,2,3. From superconductivity up to ferromagnetism, when the competition is sizeable a common outcome is phase separation. Compounds that have shown such behaviour are mostly of complex magnetic structures including cuprates1, iron-based superconductors3, ruthenates2, topological kagome magnets4, and manganites5,6. A contrasting case is found in the layered transition metals7,8,9,10,11 where the presence of heavy halide atoms, like in CrI3, stabilises pronounced anisotropy constants resulting in what appears a homogeneous ferromagnetic phase without any separation12,13. Nevertheless, recent experiments14,15,16,17 have unveiled the presence of many subtleties in the magnetism of this compound which a single magnetic transition and a structural phase fail to capture.

Firstly, the magnetic properties of CrI3 depend sensitively on the system structure. It is now understood that whereas bulk CrI3 is ferromagnetic (FM) at 61 K18 with presumably a rhombohedral stacking14,15,19, thin layers can exhibit antiferromagnetic (AFM) coupling at 45 K in monoclinic14,15. However, the monoclinic phase is only observed in bulk at high temperatures. The origin of this puzzling behaviour has not been reconciled since the birth of the field of 2D vdW magnets20,21. Secondly, multiple anomalies can be observed in the temperature dependence of the magnetic susceptibility of bulk CrI3 below 61 K16,17,19. Such anomalies imply that a more complex magnetic ordering involving spins not directly aligned with the easy-axis is likely emerging. Moreover, recent Raman measurements22 appear contradictory with the appearance of a rhombohedral phase for thin layers in both FM and AFM ordering. This is followed by an anomalous phonon mechanism with the deviation of the linewidths as the temperature decreases22. Whether different magnetic phases may exist or competition occurs between structural phases is largely unknown. Here we systematically study the evolution of the magnetic and crystal structures of CrI3 under different temperatures through a synergy of compelling techniques. Such approach has resulted being instrumental to identify, characterize and understand the distinct macroscopic ground states observed in this vdW material with competing magnetic and structural phases.

## Results

### Microscopic details of different magnetic phases: μSR experiments

In a μSR experiment, positive muons implanted into a sample serve as extremely sensitive local microscopic probes to detect small internal magnetic fields and ordered magnetic volume fractions in the bulk of magnetic systems. See details on Supplementary Notes 1-2. Zero-field μSR time-spectra are recorded in a powder sample of CrI3 below (5 K, 30 K, 54 K, and 60 K) and above (65 K and 80 K) the magnetic ordering temperature (Fig. 1a, b). A paramagnetic state is generally characterised by a small Gaussian Kubo-Toyabe depolarization of the muon spin originating from the interaction with randomly oriented nuclear magnetic moments. Conversely, the spectra from 150 K down to 62 K, exhibit a relatively high transverse depolarization rate λT 4.9(2) μs−1. This reflects the occurrence of dense electronic Cr moments and indicates strong interactions between them. In this scenario a novel correlated paramagnetic state may be present in the system at temperatures above the actual Curie temperature.

As the crystal is cooled down, in addition to the paramagnetic signal, an oscillating component with a single well-defined frequency is observed at T 61 K (Fig. 1a, b). Below 50 K, a spontaneous muon spin precession with two well-separated distinct precession frequencies is observed in the μSR spectra and persists down to 5 K. The temperature dependences of the internal fields (μ0Hμ = ω/$${\gamma }_{\mu }^{-1}$$) for the two components are shown in Fig. 2a. The low frequency component shows a monotonic decrease and disappears at TC2 = 50 K. The high frequency component decreases down to 50 K, above which it keeps a constant value within a few Kelvin’s range and then decreases again to disappear at TC1 = 61 K. Thus, the two oscillatory components have clearly different transition temperatures. This implies the presence of two distinct magnetic transitions in CrI3. We also notice that an upturn on both μ0Hμ,1 and μ0Hμ,2 is seen below TC3 = 25 K. Moreover, a strongly damped component appears below TC3 which is seen as some lost of initial asymmetry of the zero field μSR signal. This suggests the presence of another magnetic transition at this temperature. The temperature dependences of the relative weights of the individual components in the total μSR signal are shown in Fig. 2b. The weight of the high frequency component (component I) ω1 gradually increases below TC1 and reaches maximum at TC2, below which the second frequency appears. The third component raises below TC3 = 25 K. The components I and II share the weight of 30 − 70% in the temperature range between 30 K and 50 K. These results portray a clear coexistence of magnetically ordered phases in the temperature domain.

Figure 2 c, d shows the temperature dependences of the transverse λT and the longitudinal λL depolarisation rates, respectively, of components I and II. The λT is a measure of the width of the static magnetic field distribution at the muon site, and also reflects dynamical effects (spin fluctuations). The λL is determined by dynamic magnetic fluctuations only. For both components, λT is higher than λL in the whole temperature range, indicating that magnetism is mostly static in origin. However, λL1 has a higher overall value than λL2, implying that the magnetic order with TC1 = 61 K contains more dynamics. The presence of three transitions are clearly substantiated by the anomalies, seen in λT and λL (Fig. 2c, d). Namely, the λT1 starts to increase below TC1 and peaks at TC2, then decreases and tends to saturate. Nevertheless, it increments again below TC3. λT2 also exhibits an increase below TC3. Similarly, λL1 goes to high values for T < TC1, saturates at T < TC2 and then enlarges again for T < TC3, followed by a peak at lower temperature.

We note that it is not possible to discriminate in the analysis the contribution of strongly damped components and a high frequency component into λL1 for T < 30 K and thus its peak at low temperatures could be due to the contribution from feature III. The increase of the dynamic longitudinal muon spin depolarization rate for T < 30 K, accompanied by a peak at lower temperatures, is a signature of a slowing down of magnetic fluctuations. These results imply that magnetic transitions at TC1, TC2 and TC3 are influencing each other and they are strongly coupled, in spite of the fact that they are phase separated. Even though temperature is the main driving force to the appearance of these three phase transitions, their origin as well as their influence on the underlying magnetic properties of CrI3 are still open questions to be further investigated. These findings point to a unconventional thermal evolution of the magnetic states in a 2D vdW magnet.

### Macroscopic magnetic properties: SQUID magnetometry

To support the picture of multiple magnetic phases in CrI3, we carried out SQUID magnetometry measurements on polycrystalline and single crystal samples (Fig. 3). See Supplementary Note 3 for details. The magnetisation was measured at zero-field- (ZFC) and field-cooled (FC) conditions where the sample was cooled down to the base temperature in a weak external field and magnetization recorded upon warming. The most prominent anomaly in the thermal variation of the magnetic susceptibility onsets at 61 K as shown by DC measurements reflected in Fig. 3a, b. However, we also find signatures of additional magnetic transitions with distinct characteristics under different orientations of the magnetic field. Remarkably, there is a shoulder in both the FC and ZFC traces at around 50 K, which shows up very prominently in the in-plane magnetisation of the crystal (Fig. 3b) and much more subtly in the out-of-plane orientation (Fig. 3a).

The real part of the AC measurements (Fig. 3c, d) shows that the 61 K transition is independent of the AC drive frequency. Interestingly, there is no peak in the imaginary component of the AC magnetization in both orientations. This transition has so far been treated as a ferromagnetic long-range order phase transition19. However, the insensitivity of the imaginary component questions the type of magnetic order on the system. The second feature at 50 K is particularly visible in the thermal dependence of the in-plane AC magnetic moment of the crystal (Fig. 3d). This can be attributed to the second magnetic phase transition at TC2. The relative height of this feature with respect to the main 61 K transition (Fig. 3d) is maximum at lowest fields. As with the main anomaly, this peak also exhibits no frequency dependence, which indicates that this phase transition is of long-range order nature. The DC magnetisation for the in-plane orientation at low temperature is non-zero, implying that some component of the magnetization exists in the crystallographic ab-plane. For both field orientations, the hysteresis is nearly zero from 61 K down to 50 K, whereas it suddenly increases below 50 K. This observation remarks the notion of a strong magnetic anisotropy along the c-axis for CrI3, and reveals the presence of some in-plane magnetic moment.

Moreover, the imaginary component of the AC magnetisation for the in-plane orientation in Fig. 3d exhibits a slight increment below ~ 30 K with a reduction of the real component. The fact that the most significant effect across ~ 30 K was seen in the imaginary part of the AC susceptibility indicates that the transition at this temperature is related to the slow in-plane magnetic fluctuations. These results lay the foundation of three different temperature phase domains, which are consistent with the μSR results and can be considered as an independent piece of evidence for the presence of multi magnetic phases in CrI3.

### Coexistence of structural phases: Temperature-dependent synchrotron X-ray diffraction

The behaviour observed on the critical temperatures involves a volume-wise interplay between various magnetic states, providing an important constraint on theoretical models. One possible interpretation of the data is that below TC1 there is an evolution of the magnetic order in specific volumes of the crystal, which coexists with a correlated paramagnetic state. This interpretation is supported by the temperature dependent measurements of the total magnetic fraction Vm (Fig. 4a). The magnetic fraction Vm does not acquire the full volume below TC1 = 61 K. Instead, it gradually increases below TC1 and reaches ~ 80% at TC2 = 50 K. An additional increase of Vm by 10 − 15% takes place below TC3 = 25 K, at which the third strongly damped component appears and reaches nearly 100%. The magnetism below TC3 does not give extra coherent precession but it causes the strong depolarization of the μSR signal, reflected in the lost of the initial asymmetry. This indicates that 10 − 15% volume is characterised by highly disordered magnetic state.

The volume-wise evolution (Fig. 4a) of the magnetic order across TC1, TC2, and TC3 in CrI3 strongly suggests the presence of distinct magnetic states in separate volumes of the system. We quantify this via the volume fraction VP of the sample obtained from Rietveld refinement of the synchrotron X-Ray powder diffraction data (Fig. 4b). See details in Supplementary Note 4. We observe that the material is composed of a mixture of rhombohedral (R) and monoclinic (M) phases19 on a broad thermal range. The quantification of VP confirmed the absence of a single-phase structural scenario below the first-order crystallographic transition temperature happening in our system at around 150 K. The high-temperature monoclinic phase (C2/m) is not entirely substituted by the low-temperature rhombohedral structure (R$$\overline{3}$$). Indeed, the two-phase coexistence region is not restricted to the narrow interval previously proposed19. We found evidence of the persistence of a residual volume of the sample in the monoclinic phase in all measured temperature range, down to our base temperature (10 K) (Fig. 4c-f). Most interestingly, some peak widths and intensities show significant discrepancies with that expected from the original structural dichotomic model19. For instance, the intensity of the monoclinic peak (130)M (Fig. 4d) is barely affected by the temperature over the whole spectrum. Conversely, other peaks such as (-131)M and (002)M (Fig. 4c), and (400)M and (-262)M (Fig. 4e), are gradually suppressed at lower temperatures but do not disappear completely. It is worthwhile highlighting their Lorentzian-shape with anomalously wide half-width indicating that the monoclinic phase persists down to low temperatures in the form of short-ranged domains not much bigger than hundreds of Angstroms in size. We observed that the two coexisting structural phases share essentially the same value for the a-parameter, i.e., a = 6.844 Å at 80 K in both monoclinic and rhombohedral, suggesting intergrowth and a composite-like microstructure. Still, the goodness of fit parameters of the Rietveld refinement at low temperatures (below 150 K) portrays significant discrepancies with the original structural models, precluding the extraction of more quantitative conclusions regarding the high-to-low temperature phase conversion and urging for a more in-depth revision of the crystal structure resolution. The origin of the continuous variation of the fractions of monoclinic and rhombohedral (Fig. 4a) over a broad range of temperature is rather unclear and calls for additional works on synthesis, characterisation and crystal phase modelling.

Interestingly, the most compelling temperature-dependent structural evolution concerns the rhombohedral reflections which describe characteristic temperature-dependent features that are closely correlated with the magnetic critical temperatures observed (Fig. 4g–j). Firstly, the evolution of some structural reflections suggests a sizeable increase of intensity near ~ 50 K, such as in (223)R (Fig. 4h). Secondly, a significant number of rhombohedral peaks (though not all) display a maximum of intensity at ~ 25 K and then an intensity drop is clearly observed below that temperature (Fig. 4g, i, j). These relative intensity changes depict significant structural changes in the crystal, and correlate well with the magnetostructural effects in CrI3 around the magnetic transitions TC1, TC2 and TC3. Additional discussions are included in Supplementary Note 5 on several other reflection peaks at different temperatures which further support the coupling between structural phases and magnetism. It is worth mentioning that we have discarded the presence of concomitant phases by chemical analysis and by the instability of the refinement process in the presence of chromium oxides, hydrates and diiodide phases.

## Discussion

We would like to emphasise that while TC1 has been previously observed19, with some unclear evidences for TC217, TC3 has never been noticed with macroscopic probes. As mentioned above, we relate the transition across TC3 to the slowing down of the spin fluctuations. The magnetic moments fluctuate at a rate which is slower than the nearly instantaneous time scale of other techniques, e.g., neutron scattering23. Thus, neutron scattering and specific heat would hardly be sensitive to TC3. μSR as a local probe and sensitive to small ordered fraction and slow fluctuations, combined with AC susceptibility, uncovers the novel TC3 transition. Furthermore, the full width at half maximum (FWHM) of the (1,1,0) peak measured in neutron diffraction on bulk CrI323, shows a smooth increase below 60 K with sudden variations at 50 K and within the range of ~ 32 − 25 K. This is similar to the temperature dependence of the magnetic fraction VM (Fig. 4a) extracted from μSR measurements. As a volume-integrating probe in reciprocal space, neutron scattering techniques are sensitive to both the ordered moment and its volume fraction, but these two contributions cannot be separated from the measured scattered intensity. Hence, some additional transitions bellow TC2 can be missed in the temperature dependence of the neutron scattered intensity. In particular, when two magnetic states are in microscopic proximity with each other (phase separation). In μSR however we can separately measure the internal field and the ordered volume fraction. The local probe features of μSR make this technique an excellent complementary approach to neutron diffraction and magnetization measurements.

The strong interplay involving distinct structural phases and competing magnetic orders found in CrI3 raised several implications on the understanding of past and ongoing investigations on this material. In light of our findings, it is not surprising that thin layer CrI3 assumed a monoclinic structure and consequently an AFM ordering12,14,15 as that is one of the phases stabilised in bulk. Indeed, when bulk CrI3 is exfoliated in a glove-box at room-temperature, monoclinic is the phase present in the structure. This phase does not change to rhombohedral as several groups had observed in different samples, devices, and conditions15,17,20,24,25,26. The coexistence of rhombohedral and monoclinic in bulk CrI3 may suggest that both FM and AFM couplings are present over the entire crystal with no preference whether layers are more exposed to the surface or internal to the system24,27. The mixing of both structural phases is also a strong asset for breaking the inversion symmetry in centrosymmetric materials28 and consequently the appearance of chiral interactions (i.e., Dzyaloshinskii-Moriya)23,29,30. In systems where the competition between single-ion anisotropy and dipolar-filed is not substantial, geometrical faults may contribute to the appearance of topologically non-trivial spin textures. Furthermore, the observation of the coexistence of two structures in bulk CrI3 provides an interesting framework for further theoretical and experimental investigations. Such as in terms of crystal prediction at different temperatures and phase-mixing (e.g., random structure search), stacking order organisation at low energy cost, and spin-lattice mechanisms for unknown magnetic phases.

## Methods

### CrI3 bulk crystal growth

Chromium triiodide crystals were grown using the chemical vapour transport technique. Chromium powder (99.5% Sigma-Aldrich) and anhydrous iodine beads (99.999% Sigma-Aldrich) were mixed in a 1:3 ratio in an argon atmosphere inside a glovebox. 972 mg of the mixture were loaded into a silica ampoule with a length, inner diameter and outer diameter of 500 mm, 15 mm and 16 mm respectively. Additional details in Supplementary Notes 1.

### μSR experiment and analysis

The μSR method is based on the observation of the time evolution of the spin polarization $$\overrightarrow{P}$$(t) of the muon ensemble. In μSR experiments an intense beam (pμ= 29 MeV/c) of 100 % spin-polarized muons is stopped in the sample. Currently available instruments allow essentially a background free μSR measurement at ambient conditions31. Additional details in Supplementary Notes 2.

### SQUID magnetometry

Magnetization curves and zero-field-cooled/field-cooled susceptibility sweeps were carried out in a SQUID magnetomoter (Quantum Design MPMS-XL-7) on single crystals of CrI3 were the relative orientation of the basal plane of the sample with the external magnetic field (both AC and DC) is controlled. Additional details in Supplementary Notes 3.

### Synchrotron X-ray diffraction measurements

Synchrotron X-ray powder diffraction (SXRPD) measurements were performed at the the new Material Science Powder Diffraction beamline at ALBA Synchrotron. Powder Diffr., 28, S360–S370 (2013), proposal no. 2021014858. (Barcelona, Spain) using the multi-crystal analyser MAD detector system. Additional details in Supplementary Notes 4.