Intrinsic 2D-XY ferromagnetism in a van der Waals monolayer
Amilcar Bedoya-Pinto, Jing-Rong Ji, Avanindra Pandeya, Pierluigi Gargiani, Manuel Valvidares, Paolo Sessi, Florin Radu, Kai Chang, Stuart Parkin
IIntrinsic 2D-XY ferromagnetism in a van der Waals monolayer
Amilcar Bedoya-Pinto* , Jing-Rong Ji , Avanindra Pandeya , Pierluigi Gargiani , Manuel Valvidares , Paolo Sessi , Florin Radu , Kai Chang* and Stuart Parkin* NISE Department, Max Planck Institute of Microstructure Physics, Halle, Germany ALBA Synchrotron Light Source, Barcelona, Spain Helmholtz-Zentrum für Materialien und Energie, Berlin, Germany *[email protected], [email protected], [email protected]
Abstract
Long before the recent fascination with two-dimensional materials, the critical behaviour and universality scaling of phase transitions in low-dimensional systems was a topic of great interest. Particularly intriguing is the case of long-range magnetic order in two dimensions, once considered to be excluded in systems with continuous symmetry by the Hohenberg-Mermin-Wagner theorem , but later revisited from the viewpoints of topological vortex order , symmetry breaking fields and finite size effects . Recent experiments on layered magnetic systems show that a sizable out-of-plane magnetic anisotropy is able to stabilize 2D long-range ferromagnetic order, as demonstrated in CrI , CrBr , Fe GeTe and Cr Ge Te , while a spontaneous magnetic ordering has remained elusive for an in-plane 2D magnetic system in the monolayer limit. Here, we construct a nearly ideal easy-plane system, a CrCl monolayer grown on Graphene/6H-SiC (0001), which exhibits ferromagnetic ordering as unambiguously determined by element-specific X-ray magnetic dichroism. Hysteretic behaviour of the field-dependent magnetization is sustained up to a temperature of 10 K, and angular dependent measurements evidence a clear in-plane easy axis, unlike all other van der Waals monolayer magnets reported to date. Moreover, the analysis of the critical exponents of the temperature-dependent magnetization show a scaling behaviour that is characteristic of a 2D-XY system. These observations suggest the first realization of a finite-size Berezinskii-Kosterlitz-Thouless (BKT) phase transition in a quasi-freestanding monolayer magnet with a XY universality class; accessible through the bottom-up rowth of a van der Waals layer with an in-plane hexagonal crystal symmetry and negligible substrate interaction. Main Text
How physical properties change in systems with reduced dimensionalities is a scientific question that has fascinated researchers for decades. Taking advantage of recent technological advances, the possibility to design on-demand atomic-sized entities have led to a plethora of studies of truly 1 and 2- dimensional systems. In this regard, theoretical models of low-dimensional magnetism have been thoroughly revisited hand-in-hand with new experimental findings. It was early noted that the absence of long-range magnetic order in a low-dimensional system (d<3) with continuous symmetry, postulated by Mermin and Wagner , could be lifted by the existence of a sizable magnetic anisotropy. A special case, thereof, are 2D systems with easy-plane anisotropy, where a special long-range order is expected to emerge due to the formation of magnetic vortex-antivortex or chiraly opposing domain-wall bound pairs, theoretically described by the Berezinskii-Kosterlitz-Thouless (BKT) formalism . This implies the occurrence of a phase transition even in a 2D magnetic system with continuous rotational symmetry O(2) . This theory has been later adapted to account for spin-wave interactions with magnetic vortices , symmetry breaking fields and finite size effects , bringing theory much closer to an experimentally realistic scenario. The seminal works of Villain , Jose , Szeto and Bramwell that followed the initial BKT theory allowed the quantification of the order parameters and critical scaling near the phase transition for each scenario. Bramwell and Holdsworth showed by a renormalization group approach that the magnetization of finite-sized 2D easy-plane systems – also called 2DXY- scale with a critical exponent of β= 3π /128 ≈ 0.231, defining a fingerprint of the XY universality class. Perturbations to the rotational invariance of the spins in the plane -such as the symmetry and strength of magnetocrystalline fields- have been shown to strongly influence the critical exponents and the universality class. While a uniaxial in-plane anisotropy (XYh ) drives the system into an Ising-type behaviour (β= 0.125), the effect of a four-fold symmetry (XYh ) is highly dependent on the crystal field strength, with critical exponents ranging between Ising and XY universality classes . In contrast, the hexagonal symmetry (XYh ) barely affects the magnetic behaviour even in the limit of strong crystal fields , such that the system remains in the XY class, enabling the observation of a BKT-type phase ransition. A large number of experiments on quasi 2D-magnetic systems, such as monolayers on surfaces or bulk layered magnets with small interplanar exchange interactions, have contributed to the assessment of the aforementioned theories (comprehensive reviews can be found in Refs. 17,18). However, in those cases, the substrate interaction through bonding and hybridization (monolayers on surfaces) and the (even small) amount of interplanar exchange (layered magnets) were conditions that precluded the realization of an ideal 2D system. Recent advances in the isolation and preparation of crystalline monolayers from innately layered magnets, via exfoliation or molecular-beam epitaxy , allow for more detailed investigations of low-dimensional critical phenomena. So far, ferromagnetic ordering has been demonstrated in CrI , CrBr and Fe GeTe monolayers , all of them stabilized by a substantial out-of-plane uniaxial anisotropy that places them in the Ising universality class. CrCl , on the other hand, is an in-plane antiferromagnet in the bulk form , resulting from alternating magnetic moments in each individual CrCl layer aligned in the plane. Thus, arises the interesting question, whether a single-layer CrCl would order ferromagnetically –or order magnetically at all-, considering the weak anisotropy and the lifting of interplanar exchange when reaching the monolayer limit. Recent reports on magnetic properties of CrCl exfoliated flakes reach down to the bilayer regime , where the antiferromagnetic interlayer exchange is still preserved, whereas the determination of the magnetic properties of a single monolayer has remained elusive. One reason are the indirect methods involved in the magnetic characterization of the monolayer flakes, which require device fabrication for tunnelling magnetoresistance or Hall-micro magnetometry experiments . Other direct methods such as magneto-optical Kerr-effect, suffer from small signals due to sample dimensions and unfavourable magneto-optical coefficients in CrCl . In this work, we circumvent these difficulties by preparing a homogeneous CrCl monolayer on Graphene/6H-SiC(0001) by molecular-beam-epitaxy (MBE), and by measuring their magnetic properties in-situ via element-specific X-ray magnetic circular dichroism (XMCD). We demonstrate ferromagnetic ordering of the CrCl monolayer with an estimated Curie-Temperature of 10.7 K, and quantify the local moments both on the Cr and Cl atoms. An in-plane easy axis is clearly observed, supported by an unexpectedly large anisotropy energy; and the scaling analysis near the phase transition is consistent with a 2DXY spin system. The key factors for the ealization of a nearly-ideal 2DXY magnetic system rely on a reduced substrate interaction and a weak hexagonal in-plane crystal field present in the van der Waals monolayer. The structure of the CrCl monolayer grown on graphitized 6H-SiC(0001) by MBE features a van der Waals gap to the substrate and the formation of crystalline layer with an in-plane hexagonal lattice, as shown in Figure 1a. The Cr atom is coordinated in an octahedral configuration to the neighbouring Cl atoms, i.e. Cr-Cl bonds are off-plane, and the Cr atoms form a honeycomb lattice. In-situ reflection high-energy electron diffraction (RHEED) patterns show that the CrCl films have a surface perpendicular to the c-axis, and that the film microstructure presents in-plane twisted domains, as seen in the multiple diffraction streaks corresponding to a and √3 a CrCl lattice periodicities along a high-symmetry direction (e.g. Γ-M of Graphene, Figure 1b). This means that a strict 6-fold symmetry is no longer guaranteed on long-range length scales. Figure 1c shows a 3 µm sized scanning tunnelling microscopy (STM) topographic image of a CrCl monolayer, indicating a homogeneous coverage of the graphene substrate. Inside a CrCl grain, the atom resolved STM image (Fig. 1d) displays the CrCl hexagonal lattice superimposed on a clear Moiré pattern, which corresponds to a twist angle of 23.8° (see Extended Data Figure 1) between the CrCl monolayer and the graphene substrate. Various Moiré patterns were observed in different grains, which further supports random in-plane grain orientations (Extended Data Fig.2) consistent with a weak interaction with the graphene substrate. The local electronic properties of the CrCl monolayer, mapped by scanning tunnelling spectroscopy (Figure 1d), reveal a bandgap of ~1.6 eV, which is close to what is predicted from ab-initio calculations , with the Fermi-energy lying in the middle of the gap. This indicates that the electronic properties of the MBE-grown monolayer are intrinsic, with a low concentration of defects/dopants and a negligible charge transfer effect from the substrate. Prior to the in-situ investigation of the magnetic properties of the samples, we performed X-ray absorption spectroscopy measurements to assess the chemical integrity of the CrCl surface after transfer via an ultra-high-vacuum taxi chamber to the synchrotron beamline (see Methods for details). As can be seen in Figure 1e, we did not observe any trace of oxygen contamination but rather a well-defined Cr X-ray absorption line, with sharp multiplet peaks characteristic of a Cr ion in an octahedral configuration. Figure 1.
Structural and electronic properties of a CrCl monolayer. (a) Schematic crystal structure of CrCl /Graphene/6H-SiC layers in top view and cross-section configurations. (b) In-situ RHEED pattern of the substrate and monolayer CrCl grown by MBE, along Γ-M of Graphene (Γ-K of SiC) . Streaks from different high-symmetry directions of CrCl are observed, implying a twisted in-plane orientation of the grains. (c) STM topography of a monolayer CrCl grown on Graphene/6H-SiC(0001), indicating a homogeneous coverage on long length scales. Setpoints: sample bias voltage V = +1.2 V, tunnelling current I = 5 pA. Inset, a magnified topography image, which reveals the grain boundaries. (d) Atom resolved image of the CrCl lattice featuring a moiré pattern (upper panel), and its Fourier transformed image (lower panel). Setpoint, V = +0.1 V, I = 100 pA. The Moiré pattern corresponds to a 23.8° rotation between the hexagonal unit cell of CrCl and graphene. (c) and (d) were acquired at room temperature. (e) dI/dV spectrum at the surface of a monolayer CrCl , taken at 1.9 K. The estimated bandgap is 1.6 eV obtained by linearly extrapolating the sharp increase in signal at positive and negative energies to intersect the energy axis. (f) X-ray absorption spectroscopy near the O K and Cr L edge region, ruling out the presence of oxygen in the surface and highlighting a sharp Cr absorption white line. Figure 2 summarizes the key magnetic features of the CrCl monolayer as revealed from X-ray magnetic dichroism (XMCD) measurements: i) sizable local magnetic moments present both in the Cr and Cl toms, ii) a field-dependent magnetization with non-zero remanence and coercive fields typical of ferromagnetic ordering, iii) a weak but detectable magnetic anisotropy favouring an in-plane easy axis, iv) a ferro-to paramagnetic transition between 10 and 12.5 K. As shown in Figure 2a, the X-ray absorption spectra at the Cr L edge taken with different photon helicities (right and left handed, C+ and C- for simplicity) shows a huge difference, yielding values of nearly 100% at high fields (8T). This high XMCD contrast was crucial to performing measurements of the Cr L edge at different magnetic fields (hysteresis loops) with a high level of accuracy. Using sum-rule analysis (Extended Data Figure 3) we extract a total magnetic moment of 2.8 µB/Cr, very close to the theoretically expected value of 3 µB for a trivalent Cr valence (3d ). Although we also found a remarkably high XMCD signal (30%) at the chlorine edge (Figure 2b), sum-rule analysis could not be performed due to the small energy separation of L and L lines resulting from the weak spin-orbit coupling of Cl. Assuming a ferromagnetic super-exchange coupling mechanism between Cr spins over Cl ligands in CrCl , it is very likely that the spin moment of Cl aligns antiparallel to that of Cr. In fact, the small reduction of the Cr magnetic moment (2.8 µB instead of 3 µB) might arise from a certain degree of p-d admixture of Cr and Cl orbitals in the super-exchange path. The resulting exchange coupling is small (J ex ~ 0.6 meV), in line with the low ordering temperature observed in the monolayer samples. Figure 2c displays the low field region of the Cr XMCD hysteresis loops along the easy-axis for different temperatures, showing that the squareness of the hysteresis clearly diminishes as the temperature is raised; and at 12.5 K, both remanence and coercive fields vanish. Field-dependent XMCD data at normal and grazing incidence angles (Figure 2d) evidence a clear in-plane magnetic easy axis. The anisotropy fields extracted from the Cr XMCD hysteresis loops, shown in the inset of Figure 2d, are on the order of 0.5 - 0.6 T (corresponding to anisotropy energies of 0.08 - 0.1 meV). These values are much larger than the ones anticipated from first-principle calculations of a CrCl monolayer , which range from 0.031 to 0.055 meV/Cr; and also appear to be slightly larger than the critical field anisotropy in the antiferromagnetic bulk (0.3 T) and bilayer CrCl (0.23 T). A stable ferromagnetic ordering was not expected a priori when reducing the dimensionality of the few-layer AFM system down to the monolayer, as the removal of the interlayer exchange interaction does not guarantee that the intra-layer ferromagnetic behaviour remains intact. In fact, the magnitudes of the calculated magnetic anisotropy energies (MAE) for a ingle CrCl monolayer were so low that some calculations even predict a marginally favorable perpendicular magnetic anisotropy (PMA) instead of the expected in-plane one. Our observations of a in-plane easy axis in monolayer CrCl provide a clear answer to this puzzle and consolidates its classification as a two-dimensional easy-plane spin system. For completeness, we note that our few-layer CrCl samples grown by MBE show similar magnetic behaviour as those reported in exfoliated flakes , exhibiting signatures of spin-flop transitions at low fields and giant interlayer exchange enhancement in the form of large critical fields (Extended Data Figure 4). Figure 2.
Element-specific magnetic properties of a CrCl monolayer measured by X-ray magnetic dichroism. X-ray absorption spectra with different photon helicities and the resulting difference (XMCD), for the Cr L edge (a) and Cl L edge (b). A large dichroic signal can be observed for both elements. (c) XMCD signal at the Cr L edge as a function of in-plane magnetic field, taken at various temperatures. Remanence and coercive fields disappear at 12.5 K, indicative of a magnetic phase transition. (d) XMCD hysteresis loops taken in grazing (in-plane) and normal (out-of-plane) incidence. A snap of the low-field region is displayed as an inset to visualize the anisotropy fields. To understand the origin of the magnetic anisotropy, we analyse the relevant terms that apply to CrCl . Being a magnetic insulator with a local magnetic moment near 3 µB/Cr, the dominant magnetic nteraction is a 90-degree ferromagnetic super-exchange hopping path over the Cl ligands, as expected from the Goodenough-Kanamori rules . Considering this scenario, besides magnetocrystalline and shape anisotropy, a large effect of the ligand p-orbitals has recently been discussed , in which the spin-orbit strength, the degree of covalency and the distortion angle from the octahedral configuration play an important role. The total anisotropy energy will be thus given by ΔE K (total)= ΔE SI + ΔE D + ΔE ME , defining the magnetocrystalline (single-ion), dipolar and the magnetic exchange anisotropy terms, the latter related to ligand contributions in the Cr 3d- Cl p – Cr 3d super-exchange hopping path. We follow the rationale by Kim et al. [Ref. 33] applied to CrI to quantify the relevant energies. First, the single-ion anisotropy arises from the anisotropy of the orbital moment L, with an energy scale of ΔE SI = - ξ ΔL S/4, where ξ is the spin-orbit coupling strength. This term fully vanishes if the orbital moment is quenched, as expected in ionic-like Cr compounds. However, the Cr 3d spin-orbit (LS) coupling admixes the t and e g orbitals to recover a non-zero value of the orbital moment L. In fact, as extracted from sum-rule analysis (Extended Data Fig. 3), we obtain L=-0.08 µB for normal (out-of-plane) and L=-0.12 µB and grazing incidence (in-plane) XMCD spectra, respectively, yielding a measurable orbital anisotropy ΔL= 0.04. Using ξ (Cr 3d) = 30 meV , we obtain ΔE SI = 0.14 meV in favour of an in-plane easy axis. As for the dipolar term, the in-plane easy axis is also favoured by the shape anisotropy and is of the order of ΔE D =µ m / 4πr = 0.02 meV for Cr spins . The last term (ΔE ME ) derived in [Ref. 33] and highlighted as a key factor to account for the large out-of-plane anisotropy in CrI , is strongly dependent on the ligand spin-orbit coupling strength ξ P , the degree of covalency (Δ) and the angular deviation δθ from the Cr-ligand plane given by the ideal crystal structure (degree of trigonal distortion), and follows a power-law scaling ΔE ME ~ (ξ P ) a (δθ) b /(Δ) c . Compared to the parent compounds CrI and CrBr , it is evident that CrCl attains the lowest magnetic exchange anisotropy due to the low spin-orbit strength of the Cl ligand and smallest degree of covalency (largest bandgap). Considering values of ξ P =70meV, Δ=3.5eV and δθ=2° and using four-site multiplet cluster calculations, the calculated ΔE ME in Ref. 33 resulted in negligible values (1 µeV) for CrCl . In our case, based on the experimental STS data (Fig. 1d) the bandgap of CrCl is estimated to be 1.6 eV, resulting in a factor of 0.45 in the charge transfer energy Δ and a factor of 4 (using the exponent c =1.73) increase of ΔE ME with respect to the calculations, being still of the order of a few µeV. The main message here is that only a substantial rigonal distortion (δθ) will result in a sizable ΔE ME for our MBE-grown CrCl layers. In fact, we observe by angular-dependent X-ray absorption measurements (Extended Data Figure 5) a maximum in the pre-edge intensity of the Cr L edge at θ=25° degrees instead of 35.3° to the normal plane, the latter angle expected from the ideal octahedral structure. This measurement is sensitive to the admixed Cr 3d-orbitals (e.g. d ) within the Cr-Cl bonding plane, and is therefore a good tool to map any angular deviation. We believe that this substantial trigonal distortion (δθ = 10°) could be induced in our samples during the nucleation and growth process, and increase not only ΔE ME by an order of magnitude (~40 µeV) but also impact the anisotropy of the orbital moment (ΔL=0.04) that enters the single-ion anisotropy term (ΔE SI ) and which is the dominant contribution to explain the observed anisotropy fields in the XMCD hysteresis loops. In order to characterize the magnetic behaviour below and above T c and find insights into the nature of the phase transition, field scans of the Cr L edge XMCD were acquired between -8T and 8T and up to temperatures of 35 K, as shown in Figure 3a. Concomitant with the disappearance of the hysteresis at 12.5 K (Figure 2d), the M(H) curves evolve into a softer S-shape above T c and move towards a linear dependence, entering the paramagnetic regime. As for the study of the critical behaviour at the phase transition, the magnetization is a good order parameter and scales as M= M (1-T/T c ) β , β being the critical exponent that determines the universality class. In two-dimensional systems, the isotropic Heisenberg system (n=3) does not have any spontaneous order, such that the Ising (n=1) and XY (n=2) models are the relevant scenarios for CrCl , where n is the dimensionality of the spin degree of freedom. Though often understood in terms of an out-of-plane easy axis, the Ising universality can also be found in in-plane magnetized systems , as long as there is only one preferred (uniaxial) magnetization direction. To perform the analysis of the critical exponent β, we rely on the evolution of the remanent magnetization M r (XMCD signal at zero field) as a function of temperature. Figure 3b shows the fitting of the data yielding values of β= (0.235±0.02), matching well with the expected value of the 2DXY model . It is worth noting that the scaling holds sufficiently away from T c (up to T reduced =1 - T/Tc = 0.65), a wide range observed in a few other metallic mono-and few-layer systems on crystalline substrates as well as in layered magnets . We also fixed the exponent β= 0.125 in the fitting to discard any kind of Ising-type universality (green line in Figure 3b), resulting in a rather poor agreement. The error bars n M r arise from the uncertainty in the determination of the XMCD signals at zero field. An average of the M r values of the trace and retrace scans (Extended Data Figure 6) have been taken to improve the accuracy. The extrapolated T c , considering the ideal 2DXY model, amounts to 10.7 K, which appears reasonable. However, finite-sized effects are known to contribute to a rounding of the M(T) curve , shifting non-zero magnetization values towards higher temperatures, such that the value inferred from the ideal 2DXY fit constitutes a lower bound for T c . To further assess the 2DXY behaviour in our samples, a second analysis procedure, the so-called Arrott-Noakes plots , have been constructed from the temperature dependent data. In this representation, the magnetization and susceptibility power-law scaling are visualized together across the phase transition, and a consistent set of the critical exponents β and γ, corresponding to the magnetization and susceptibility, can be deduced. This approach is widely used to distinguish between the various magnetic interaction models (e.g. Heisenberg, Ising, XY, Potts) in 3- and 2-dimensions, each of which have a specific set of critical exponents. One complication, specifically occurring for the 2DXY model, is that the susceptibility χ(T) does not follow a power-law but rather an exponential scaling as T c is approached from the high-temperature side (leading to divergence). In real samples, however, finite-size effects and existing in-plane anisotropy fields preclude the divergence of χ(T) and a peak close to T c is observed instead . With this modified χ(T) behaviour, some attempts to fit data of easy-plane systems –such as in monolayer Fe/W(001) - to a power-law dependence, resulted in abnormally high critical exponents γ=[3.6 – 5] . On the other hand, Rogiers et.al. worked out the high-temperature series expansions of the spin ½ 2DXY model, finding a finite-temperature phase transition (consistent with the BKT conjecture) but with a conventional power-law critical behaviour. A numerical estimate for the critical exponent γ=2.4 ± 0.3 was given. On this basis, we pursue the determination of the critical exponent γ by an Arrott-plot analysis of the temperature dependent XMCD data, as shown in Figure 3c. A proper determination of the critical exponents can be recognized by the following features of the M^(1/β) vs (B/M)^γ plot: i) A linear behaviour at high fields for all temperatures near the phase transition, ii) a positive y-intercept for T>T c and a negative one for T Scaling behaviour and critical exponents. (a) Field-dependent Cr L XMCD signals for a wide range of temperature across the phase transition, including the paramagnetic regime. (b) XMCD values at zero magnetic field (remanence) as a function of temperature. The scaling fits with 2D Ising (green) and 2DXY models (red) are shown. (c) Modified Arrott-Plots for the temperature-and field dependent XMCD data. A consistent set of critical exponents is inferred (β=0.235, γ= 2.2), matching with the 2DXY model predictions. Extrapolation of the linear fits at high-fields is drawn as guide to the eye to visualize the y-intercept evolution across the phase transition. Taking into account the analysis presented above, we have demonstrated that a CrCl monolayer, featuring van der Waals coupling to the graphene substrate and hexagonal symmetry in the plane but without strict single crystalline long-range order, constitutes a nearly ideal realization of a 2DXY magnetic system. The van der Waals nature at the CrCl /substrate interface minimizes effects such as hybridization, bonding, and substrate-driven crystalline anisotropy fields, factors that can drive the system to an Ising or anisotropic Heisenberg system; or even cause a departure from a 2-dimensional behaviour by strong substrate hybridization. The in-plane hexagonal crystal symmetry (XYh6) of CrCl , on the other hand, is the least perturbative for the XY model and is additionally diminished in our xperimental system due to the in-plane twisting of crystalline domains. As compared with bulk layered magnets, our monolayer system has no interplanar exchange coupling J ex,inter which is also a perturbation of ideal 2DXY behaviour and dimensionality. Hence, our quasi-freestanding, atomically-thin CrCl layer grown by MBE, constitutes an appealing model system that can, in turn, be treated accurately by theoretical calculations. Interesting follow-up studies would be to gently turn on in-plane anisotropy fields by choosing different rigid substrates with strong crystalline fields or a given step terrace morphology, and see how this affects the 2DXY behaviour of the monolayer. In that way, recipes to attain a crossover to Ising-type (uniaxial anisotropy) behaviour may be developed, a useful pathway to design on-demand magnetic anisotropies for functional spintronic devices based on 2D materials. On a more fundamental side, the finite-size effects of the BKT phase transition can be studied by modifying the grain size and percolation behaviour of the MBE-grown monolayers. These tuning knobs are a powerful tool to reach beyond the state-of-the-art and current understanding of exfoliated van der Waals magnets. Our demonstration of a CrCl ferromagnetic monolayer grown by MBE is thus a critical step to achieving exquisite control of the magnetic properties of innate 2D systems by thin film growth engineering. References Hohenberg, P.C. Existence of Long-Range Order in One and Two Dimensions. Phys. Rev. , 383–6 (1967). 2. Mermin, N. D. & Wagner, H. Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett. , 1133–1136 (1966). 3. Berezinskiǐ, V. Destruction of Long-range Order in One-dimensional and Two-dimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems. Sov. J. Exp. Theor. Phys. , 610 (1972). 4. Kosterlitz, J. M. & Thouless, D. J. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys. C Solid State Phys. , 1181–1203 (1973). 5. Kosterlitz, J. M. The critical properties of the two-dimensional xy model. J. Phys. C Solid State Phys. , 1046–1060 (1974). 6. José, J. V., Kadanoff, L. P., Kirkpatrick, S. & Nelson, D. R. Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model. Phys. Rev. B , 1217–1241 (1977). . Szeto, K. Y. & Dresselhaus, G. Two-dimensional XY model with multiple symmetry-breaking fields. , 3186–3193 (1985). 8. Bramwell, S. T., Holdsworth, P. C. W. & Rothman, J. Magnetization in ultrathin films: Critical exponent β for the 2D XY model with 4-fold crystal fields. Mod. Phys. Lett. B , 139–148 (1997). 9. Bramwell, S. T. & Holdsworth, P. C. W. Magnetization and universal sub-critical behaviour in two-dimensional XY magnets. J. Phys. Condens. Matter , (1993). 10. Szeto, K. Y., Chen, S. T. & Dresselhaus, G. Temperature dependence of the magnetic susceptibility of CoCl2-graphite intercalation compounds. Phys. Rev. B , 4628–4638 (1985). 11. Huang, B. et al. Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit. Nature , 270–273 (2017). 12. Chen, W. et al. Direct observation of van der Waals stacking-dependent interlayer magnetism. Science , 983–987 (2019). 13. Fei, Z. et al. Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2. Nat. Mater. , 778–782 (2018). 14. Gong, C. et al. Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals. Nature , 265–269 (2017). 15. Villain, J. Theory of one- and two-dimensional magnets with an easy magnetization plane. II. The planar, classical, two-dimensional magnet. J. Phys. , 581–590 (1975). 16. Bramwell, S. T. & Holdsworth, P. C. W. Magnetization: A characteristic of the Kosterlitz-Thouless-Berezinskii transition. Phys. Rev. B , 8811–8814 (1994). 17. Taroni, A., Bramwell, S. T. & Holdsworth, P. C. W. Universal window for two-dimensional critical exponents. J. Phys. Condens. Matter , 275233 (2008). 18. Vaz, C. A. F., Bland, J. A. C. & Lauhoff, G. Magnetism in ultrathin film structures. Reports Prog. Phys. , 056501 (2008). 19. Kuhlow, B. Magnetic Ordering in CrCl3 at the Phase Transition. Phys. Status Solidi , 161–168 (1982). 20. McGuire, M. A. et al. Magnetic behavior and spin-lattice coupling in cleavable van der Waals layered CrCl3 crystals. Phys. Rev. Mater. , 1–10 (2017). 21. Cai, X. et al. Atomically Thin CrCl3: An In-Plane Layered Antiferromagnetic Insulator. Nano Lett. , 3993–3998 (2019). 22. Klein, D. R. et al. Enhancement of interlayer exchange in an ultrathin two-dimensional magnet. Nat. Phys. , 1255–1260 (2019). 23. Kim, H. H. et al. Evolution of interlayer and intralayer magnetism in three atomically thin chromium trihalides. Proc. Natl. Acad. Sci. U. S. A. , 11131–11136 (2019). 4. Wang, Z. et al. Determining the phase diagram of atomically thin layered antiferromagnet CrCl3. Nat. Nanotechnol. , 1116–1122 (2019). 25. Kim, M. et al. Micromagnetometry of two-dimensional ferromagnets. Nat. Electron. , 457–463 (2019). 26. A. Molina-Sánchez, G. Catarina, D. Sangalli and J. Fernandez-Rossier, Magneto-optical response of chromium trihalidemonolayers: chemical trends. J. Mater. Chem. C (2020) DOI:10.1039/D0TC01322F 27. Webster, L. & Yan, J. A. Strain-tunable magnetic anisotropy in monolayer CrCl3, CrBr3, and CrI3. Phys. Rev. B , 144411 (2018). 28. Zhang, W. B., Qu, Q., Zhu, P. & Lam, C. H. Robust intrinsic ferromagnetism and half semiconductivity in stable two-dimensional single-layer chromium trihalides. J. Mater. Chem. C , 12457–12468 (2015). 29. Yang, B., Zhang, X., Yang, H., Han, X. & Yan, Y. Nonmetallic Atoms Induced Magnetic Anisotropy in Monolayer Chromium Trihalides. J. Phys. Chem. C , 691–697 (2019). 30. Goodenough, J. B. An interpretation of the magnetic properties of the perovskite-type mixed crystals La1-xSrxCoO3-λ. J. Phys. Chem. Solids , 287–297 (1958). 31. Kanamori, J. Superexchange interaction and symmetry properties of electron orbitals. J. Phys. Chem. Solids , 87–98 (1959). 32. Lado, J. L. & Fernández-Rossier, J. On the origin of magnetic anisotropy in two dimensional CrI3. 2D Mater. , 1–8 (2017). 33. Kim, D. H. et al. Giant Magnetic Anisotropy Induced by Ligand LS Coupling in Layered Cr Compounds. Phys. Rev. Lett. , 207201 (2019). 34. Back, C. H. et al. Experimental confirmation of universality for a phase transition in two dimensions. Nature , 597–600 (1995). 35. Qiu, Z. Q., Pearson, J. & Bader, S. D. Magnetic phase transition of ultrathin Fe films on Ag(111). Phys. Rev. Lett. , 1646–1649 (1991). 36. Huang, F., Kief, M. T., Mankey, G. J. & Willis, R. F. Magnetism in the few-monolayers limit: A surface magneto-optic Kerr-effect study of the magnetic behavior of ultrathin films of Co, Ni, and Co-Ni alloys on Cu(100) and Cu(111). Phys. Rev. B , 3962–3971 (1994). 37. Rau, C. Ferromagnetic order and critical behavior at surfaces of ultrathin epitaxial films. Appl. Phys. A Solids Surfaces , 579–587 (1989). 38. Als-Nielsen, J., Bramwell, S. T., Hutchings, M. T., McIntyre, G. J. & Visser, D. Neutron scattering investigation of the static critical properties of Rb2CrCl4. J. Phys. Condens. Matter , 7871–7892 (1993). 39. Arrott, A. & Noakes, J. E. Approximate equation of state for nickel near its critical temperature. Phys. Rev. Lett. , 786–789 (1967). 0. Elmers, H. J., Hauschild, J., Liu, G. H. & Gradmann, U. Critical phenomena in the two-dimensional XY magnet Fe(100) on W(100). J. Appl. Phys. , 4984 (1996). 41. Atchison, J., Bhullar, A., Norman, B. & Venus, D. Finite-size Kosterlitz-Thouless transition in 2DXY Fe/W(001) ultrathin films. Phys. Rev. B , 125425 (2019). 42. Rogiers, J., Grundke, E. W. & Betts, D. D. Spin one-half XY model - Analysis of high temperarture series expansions of some thermodynamic quantities in two dimensions. Can. J. Phys. , 1719–1730 (1979). 43. Elmers, H. J. et al. Submonolayer magnetism of Fe(110) on W(110): Finite width scaling of stripes and percolation between islands. Phys. Rev. Lett. , 898–901 (1994). 44. Farle, M., Baberschke, K., Stetter, U., Aspelmeier, A. & Gerhardter, F. Thickness-dependent Curie temperature of Gd(0001)/W(110) and its dependence on the growth conditions. Phys. Rev. B , 11571–11574 (1993). 45. Resnick, D. J., Garland, J. C., Boyd, J. T., Shoemaker, S. & Newrock, R. S. Kosterlitz-thouless transition in proximity-coupled superconducting arrays. Phys. Rev. Lett. , 1542–1545 (1981). 46. Mondal, M. et al. Role of the vortex-core energy on the Berezinskii-Kosterlitz-Thouless transition in thin films of NbN. Phys. Rev. Lett. , 1–5 (2011). 47. Baek, D. H., Chung, J. W. & Han, W. K. Critical behavior of the p(2×1)-O/W(110) system. Phys. Rev. B , 8461–8464 (1993). 48. Hadzibabic, Z., Krüger, P., Cheneau, M., Battelier, B. & Dalibard, J. Berezinskii-Kosterlitz-Thouless crossover in a trapped atomic gas. Nature , 1118–1121 (2006). 49. Rapini, M., Dias, R. A. & Costa, B. V. Phase transition in ultrathin magnetic films with long-range interactions: Monte Carlo simulation of the anisotropic Heisenberg model. Phys. Rev. B , 1–7 (2007). 50. Kim, S. K. & Chung, S. B. Transport Signature of the Magnetic Berezinskii-Kosterlitz-Thouless Transition. arXiV (2020). https://arxiv.org/abs/2003.08956 Methods Epitaxial growth and in-situ structural characterization. The growth and room temperature STM experiments of monolayer CrCl films were carried out in a home-built multi-chamber ultra-high vacuum system, with multiple MBEs and an Omicron VT-STM-XT attached to a rotational central distribution chamber. The base pressure is 7 × 10 -10 mbar in the MBE chambers and 1 × 10 -10 mbar in the STM chamber. In order to prepare the graphene substrate, an n-doped 6H-SiC(0001) substrate with a Si termination was first degassed at 500°C by passing direct current through it until base pressure was resumed. Then the substrate was annealed at 950°C in a Si flux for 3 min, during which a Si-rich (3 × 3) reconstruction was formed. Finally, the substrate was annealed at 1350°C for 10 min to generate onolayer or bilayer graphene on the surface. CrCl was evaporated from a Knudsen cell containing anhydrous CrCl powders (99.9%) in an h-BN crucible. During the growth, the temperature of the evaporator and the substrate were set at 320°C and 150°C, respectively. The growth is stopped when stripes of the substrate totally vanished in the RHEED pattern. The as-prepared sample was immediately transferred to the STM chamber without breaking the ultra-high vacuum environment. Mechanically sharpened PtIr alloy tips calibrated on Au(111) surfaces were used in the room temperature STM measurements. Low temperature STM spectra were acquired using electrochemically etched tungsten tips whose spectroscopic properties had been calibrated using the Ag(111) surface state. Because of their different intensities in the dI/dV curves, conduction and valence bands have been measured using different set-point conditions (1V, 50pA and -2V, 100pA for conduction and valence band, respectively). The dI/dV signal was acquired using a lock-in technique, with a modulation of 20 meV rms at a frequency of 733 Hz applied to the tip. X-ray absorption and magnetic circular dichroism spectroscopy. The X‐ray absorption spectroscopy measurements were carried out at the BOREAS beamline at the ALBA Synchrotron, Barcelona, and at the VEKMAG beamline at BESSY, Berlin. The surface quality of the CrCl monolayers is fully preserved from our growth chamber to the end station, due to a compatible ultra-high-vacuum mobile transfer system (Ferrovac UHV suitcase, p base < 1x10 -10 mbar). The XAS/XMCD measurements (Main Figure 2 and Figure 3) were performed at the VEKMAG end-station installed at the PM2 beamline of the synchrotron facility BESSY II, HZB. This end station provides a vector magnetic field option with a maximum magnetic field up to 9 Tesla in the beam direction, 2 Tesla in the horizontal plane and 1 Tesla in all directions for a temperature range of 2 K–500 K. The XAS spectra were recorded by means of total electron yield (TEY) for the L and L resonant energies of the Cr chemical element. The TEY is measured by recording the drain current as a function of the x-ray photon energy normalized by a Ta grid x-ray monitor mounted in a magnetically shielded environment as the last optical element before the sample. The TEY is known to be surface sensitive, providing information over the escape length of the electrons which exhibits a mean free path of about 3 nm. As such, the surface magnetic properties are provided in a selective manner by this recording channel. XAS/XMCD experiments at BOREAS beamline (Extended Data Figures 4 and 5) were performed in a high-field vector magnet endstation HECTOR in a base pressure of p<=10 -10 mbar. The x-rays were produced by an APPLE-II type undulator with fully circular or linear polarization. The absorption signal was measured as the total electron yield (TEY) signal determined by the sample-to-ground drain current. The sample absorption signal was normalized by the impinging photon flux determined by the TEY signal on a freshly evaporated gold mesh, placed between the las optical element and the sample. The drain current signals were detected by a Keithley 428 current amplifier. The sample temperature was varied between 2K and 300K as measured on the cold finger of the cryostat. A magnetic field of up to T was applied along the beam direction in order to magnetically polarize the sample in the XMCD experiments, and the XMCD signal was determined by the difference of the XAS signal in the two opposite helicities. 1. Barla, A. et al. Design and performance of BOREAS, the beamline for resonant X-ray absorption and scattering experiments at the ALBA synchrotron light source. J. Synchrotron Radiat. , 1507–1517 (2016). 2. Noll, T., Radu, F. The mechanics of the VEKMAG experiment. Proceedings of Mechanical Engineering Design of Synchrotron Radiation Equipment and Instrumentation Conference (MEDSI´16), , 370–373 (2017). Author contributions A.B-P., K. C. and S.S.P.P. conceived the study, and A.B-P. was the lead researcher. A.B-P. carried out the full magnetic characterization by XAS/XMCD, analysed the data and wrote the manuscript. J.J. grew the samples for the beamtimes under the guidance of A.B-P. and K.C, and performed the in-situ RHEED and STM characterization. K.C. initiated and optimized the substrate preparation and epitaxial growth. P.S. performed the low-temperature STM characterization. J.J., A.K.P and F.R. assisted with the XMCD measurements in BESSY, whereas P.G. and M.V. assisted and performed part of the XAS/XMCD experiments in ALBA. All authors discussed the data and commented on the manuscript. S.S.P.P. supervised the entire project. Acknowledgements A.B-P. thanks the HZB and CELLS-ALBA for the allocation of synchrotron radiation beamtime under proposals 192-08773-ST (HZB) and 2019093862 (CELLS-ALBA). F.R and A.B-P. acknowledge the financial support for the VEKMAG project and for the PM2-VEKMAG beamline by the German Federal Ministry for Education and Research (BMBF 05K10PC2, 05K10WR1, 05K10KE1) and by HZB. MV and PG acknowledge additional beamtime through the ALBA IHR program and funding via Mineco grant FIS2016-78591- C3-2-R (AEI/FEDER, UE). We warmly thank Chen Luo, Kai Chen, Sangeeta Thakur and Steffen Rudorff for technical support at BESSY. upplementary Material Extended Data Figures Extended Data Figure 1. The Fourier transform of a simulated moiré pattern between CrCl and graphene. The twist angle is 23.8°. Extended Data Figure 2. Atom resolved STM topography images (upper panels) and the corresponding Fourier transformed images (lower panels) with different twist angles between monolayer CrCl and graphene, acquired on the same sample. Extended Data Figure 3. XMCD sum-rule analysis from spectra taken at 2K and 8T. (a,b) XMCD spectra and integral (area under curve) over Cr L and L edges at normal (NI) and grazing (GI) incidence, determining the p and q values for the sum rule analysis. (C) Isotropic XAS (I + + I - + I ) at normal incidence and its area under curve to obtain the r value. The extracted spin (S) and orbital (L) moments amount to S=2.75 µB (2.82 µB) and L=-0.08µB (-0.12 µB) in NI (GI), respectively. A correction factor of 2.19 has been applied to the spin expectation value to account for the admixture of the Cr L and L edges in the integral calculations (p-value). The calculation of the orbital magnetic moment (integral over both L and L ) is not affected by the mixing of spectral weight. Extended Data Figure 4. XMCD characterization of a few-layer CrCl sample, taken at grazing incidence (B in-plane). (a) XMCD spectra at saturation fields (6T) show a very similar lineshape and XMCD ratio (close to 100%) as in the monolayer case. (b) XMCD hysteresis loop, taken at 3.5K, showing two kinks corresponding to the spin-flop transition (SP-F) and critical field (CF) of the collinear antiferromagnetic state. The high values of the critical fields (~2T) denote an enhancement of interlayer exchange in the antiferromagnetic phase. Extended Data Figure 5 . Angular dependent XAS data of the CrCl monolayer. (a) Cr L edge as a function of polar angle θ. A maximum of the pre-edge intensity, related to admixed Cr 3d-bonding states, is expected when θ passes through the Cr-Cl plane (shaded area in the inset). A close-up of the pre-edge XAS (panel b) highlights the angular intensity evolution, featuring an intensity maximum around 24-25 degrees from the surface normal (panel c). This corresponds to a distortion angle of 10° with respect to the theoretical (relaxed) Cr-Cl plane (θ=35.3°). Extended Data Figure 6. Low-field region of the XMCD hysteresis loops at different temperatures. An average of the remanence values for the trace and retrace curves have been taken for the scaling analysis. Extended Data Figure 7. Modified Arrott-plots with critical exponents characteristic of other two-dimensional magnetic models. (a) 2D Ising model, (b,c) Anisotropic 2D Heisenberg model , with critical exponents γ=2.2 [Ref.3] and γ=2.8 [Ref.4], respectively. Unlike the 2DXY case, the scaling analysis clearly fails to describe our data. The y-intercept yield negative values, the linearity is poor and the change of curvature does not match with the phase transition temperature. -400 -200 0 200 400-4-2024 X M CD d i ff, no r m B (mT) B c = 45 mT a Extended Data Figure 8. Modified Arrott-plots with critical exponents of three-dimensional magnetic models. (a) 3D Heisenberg, (b) 3D Ising, (c) 3D XY, (d) Tricritical mean-field model. As anticipated from our well-characterized, quasi free-standing 2D monolayer, all of the 3D models fail to describe our data. References (Extended Data) 1. Thole, B. T., Carra, P., Sette, F. & Laan, G. Van Der. X-Ray Circular Dichroism as a Probe of Orbital Magnetization. Phys. Rev. Lett. , 1943–1946 (1992). 2. Dreiser, J. et al. X-ray magnetic circular dichroism (XMCD) study of a methoxide-bridged DyIII-CrIII cluster obtained by fluoride abstraction from cis-[CrIIIF2(phen)2]+. J. Phys. Chem. A , 7842–7847 (2012). 3. Binder, K. & Landau, D. P. Critical properties of the two-dimensional anisotropic Heisenberg model. Phys. Rev. B , 1140–1155 (1976). 4. Elmers, H. J., Hauschild, J. & Gradmann, U. Critical behavior of the uniaxial ferromagnetic monolayer Fe(110) on W(110). Phys. Rev. B , 15224–15233 (1996)., 15224–15233 (1996).