Abstract
Twodimensional magnetic systems with continuous spin degrees of freedom exhibit a rich spectrum of thermal behaviour due to the strong competition between fluctuations and correlations. When such systems incorporate coupling via the anisotropic dipolar interaction, a discrete symmetry emerges, which can be spontaneously broken leading to a lowtemperature ordered phase. However, the experimental realisation of such twodimensional spin systems in crystalline materials is difficult since the dipolar coupling is usually much weaker than the exchange interaction. Here we realise twodimensional magnetostatically coupled XY spin systems with nanoscale thermally active magnetic discs placed on square lattices. Using lowenergy muonspin relaxation and soft Xray scattering, we observe correlated dynamics at the critical temperature and the emergence of static longrange order at low temperatures, which is compatible with theoretical predictions for dipolarcoupled XY spin systems. Furthermore, by modifying the sample design, we demonstrate the possibility to tune the collective magnetic behaviour in thermally active artificial spin systems with continuous degrees of freedom.
Introduction
According to the Mermin–Wagner theorem, twodimensional magnetic systems with continuous global symmetry are not expected to exhibit longrange order at finite temperatures^{1}. A wellknown example is the XYrotor model, where magnetic moments interact via an isotropic Heisenberg interaction, and it is only quasilongrange order that emerges at a Berenzinskii–Kosterlitz–Thouless transition^{2,3}. Introduction of anisotropic dipolar interactions, however, breaks the continuous symmetry of the system and the Mermin–Wagner theorem no longer applies. However, regardless of the discrete symmetry of the Hamiltonian, twodimensional lattices of dipolarcoupled XY moments (referred to as dXY systems) still feature a continuous groundstate degeneracy, which reflects the frustration due to competing ferromagnetic and antiferromagnetic contributions of the dipolar interaction^{4,5}. Here, it is the thermal fluctuations that select a limited number of spin states from the continuous manifold and lead to a longrange ordered phase at finite temperatures^{5,6,7}. Such an orderbydisorder transition^{8,9} is a rare example in which the magnetic phases in purely classical spin systems are selected by entropic rather than energetic contributions. For interacting dXY moments on a square lattice, a continuous transition to antiferromagnetic stripe order is predicted^{5,6,7,10,11}. The universality class of this transition has not yet been fully resolved, although both Ising^{5,12} and XYh4^{13,14} universality are under discussion. Therefore, magnetostatically coupled spin systems (including dipolar and weaker highermultipole interactions) with continuous degrees of freedom are predicted to feature intriguing physics.
Experimentally, twodimensional dXY systems have remained largely unexplored since, in microscopic systems, dipolar coupling is usually significantly weaker than shortrange exchange interactions. The dipolar interaction naturally dominates, however, in arrays of mesoscopic magnets created with lithographic methods, where the macrospin anisotropy and lattices can be tailored. Such artificial spin systems display a wide variety of static moment configurations, show intriguing fielddriven and thermal magnetic properties, and have been used to explore emergent spin correlations in two dimensions, with a major focus on analogues of the pyrochlore spin ices^{15,16,17,18,19}. Thermally fluctuating systems can be created by incorporating superparamagnetic nanomagnets^{20,21,22} and the equilibrium behaviour in the thermodynamic limit can be measured^{18,19}.
In this work, we experimentally realise squarelattice arrays of interacting nanodiscs, acting as good approximations of dXY moments with continuous spin degrees of freedom, and measure their thermal behaviour with lowenergy muonspin relaxation and soft Xray scattering. Careful comparison of strongly and noninteracting samples provides a means to distinguish the contributions from collective and singleparticle behaviour to the muonspin relaxation signal. For the strongly interacting samples, we observe a phase transition and the development of static magnetism at a temperature where the moments in isolated nanomagnets are still fluctuating. We further demonstrate the emergence of spatial correlations, which is compatible with the longrange order theoretically predicted for dXY spin systems, with the observation of magnetic superstructure peaks by resonant elastic Xray scattering. This realisation of strongly coupled thermally active systems with continuous spin degrees of freedom opens the way for future investigations of dXY spin systems with a variety of arrangements that are predicted to have rich phase diagrams.
Results
Artificial dipolarcoupled XY spin systems
Most previously considered artificial spin systems consist of arrays of elongated magnetic islands with Isinglike degrees of freedom. Here we fabricate arrays of thin permalloy (Ni_{80}Fe_{20}) discs with electronbeam lithography (see Methods) with the aim to mimic the continuous spin degrees of freedom in magnetostatically coupled XY spin systems. Owing to their magnetic shape anisotropy with a large diametertothickness ratio, the magnetisation in the thin permalloy discs will be confined within the plane. If the disc diameter is below ≈150 nm, the disc falls into a singledomain state^{23,24,25} and, to first order, the magnetic far field is that of a pointdipole moment^{26}. Corrections to this approximation are caused, e.g., by quadrupolar or higherorder moments originating from nonuniform magnetisation within the discs^{27}. The effect of these higherorder magnetostatic multipole contributions can be approximated by a modified interaction energy^{26}.
Using these XY macrospins as basic building blocks, one can explore the influence of magnetostatic interactions on the fielddriven behaviour^{28,29,30}, superparamagnetic fluctuations^{22}, and, as shown in this work, on the ordering at thermal equilibrium associated with continuous spin degrees of freedom.
Assuming no other anisotropies are present, the moment is free to rotate within the plane of the thinfilm discs^{22,23,31,32}. In this case, one would expect angular spin fluctuations down to low temperatures. However, in thinfilm nanomagnets, disorder is present due to variations in magnetocrystalline anisotropy and shape, and the inplane moment fluctuations become increasingly constrained as the temperature reduces. This results in a temperature below which the fluctuations of the individual macrospins are slower than the timescale of the experiment, which is referred to as the singleparticle blocking temperature T_{B}^{33,34,35}. Nevertheless, for small imperfections and temperatures larger than T_{B}, the magnetic moments can still explore the full inplane angular range, so that the ensemble is a close approximation of the equilibrium state of a dXY system.
For strong magnetostatic coupling, the mutual interaction between the nanomagnets can lead to the spontaneous transition to longrange ordered equilibrium configurations, which are expected to emerge below a critical temperature T_{C}. To first approximation, this critical temperature is proportional to the dipolar coupling energy determined by the magnetic moment m of the single nanomagnet as well as the lattice parameter a as follows:
For the squarelattice dXY model, a secondorder phase transition from a superparamagnetic to a collinear stripeordered state with antiferromagnetically coupled ferromagnetic spin chains along the x or y lattice directions is predicted^{5,6,7,10,29}. Higherorder multipole moments, leading to additional terms for the interparticle magnetostatic coupling, neither affect the symmetrybreaking mechanism nor the expected longrange order at zero field^{11,29,30}, but can lead to a quantitative change of the transition temperature.
Both temperature scales, T_{B} and T_{C}, will influence the magnetic fluctuations in extended lattices of dXY moments. To separate the contributions of collective and singleparticle behaviour to the temperaturedependent magnetic behaviour, three sample sets are used for comparative measurements (Table 1). Each set consists of a strongly and a noninteracting sample. For both samples in a given set, the nanomagnets have the same disc diameter d and thickness h, and therefore the singleparticle blocking temperatures T_{B} are the same, leaving the lattice periodicity a as the only difference. The nanomagnets in the strongly interacting samples have edgetoedge distances of 20–30 nm and the large magnetostatic interactions will result in a high value of the critical temperature T_{C}. In contrast, the noninteracting samples have twice the lattice periodicity of the strongly interacting samples, effectively reducing the dipolar coupling strength in Eq. (1) by a factor of eight so that T_{B} > T_{C}. Therefore, the noninteracting samples are expected to only give insight into the singleparticle blocking dynamics.
Scanning electron microscopy images of artificial dXY spin systems fabricated by electronbeam lithography (see Methods) are shown in Fig. 1. The samples contain (7–90) × 10^{9} nanomagnets, and thus can be assumed to be in the thermodynamic limit. The singleparticle blocking temperatures determined by μSR experiments lie in the range from 50 to 70 K, below which superparamagnetic fluctuations are slower than the measurement timescale.
Lowenergy μSR experiments
In this work, we measure signatures of emergent longrange order in the artificial dXY spin systems using lowenergy muonspin relaxation (LEμSR). μSR is ideally suited to measure the temperature dependence of magnetic fluctuations in artificial spin systems due to the high sensitivity to the small magnetic fields emanating from the sample as well as to moment fluctuations with frequencies in the range from kHz to GHz, and the possibility to measure in zero magnetic field^{18}. Here, lowenergy spinpolarised muons are implanted close to the permalloy discs in a gold stopping layer and sample the stray fields generated by the nanomagnets. Inhomogeneities in the field distribution and magnetic fluctuations lead to a loss of the polarisation P(t) of the muon ensemble^{36}, which gives information on the phase transitions and magnetic order present in the sample.
Temperaturedependent muonspin depolarisation
To determine the equilibrium properties of artificial dXY systems, we measured temperaturedependent zerofield μSR relaxation spectra with the initial muon spin parallel to the inplane x direction of the square lattice (see Methods). The measured muonspin polarisation function P(t) is a smoothly decaying function with time and can be best described by a sum of a constant and two exponentially damped signals,
with relative fractions g_{i} (where i = 0, slow, or fast) and the corresponding depolarisation rates λ_{slow} ≤ λ_{fast}. Details of the fitting procedure are described in the Methods section.
To distinguish the influence of the collective magnetic behaviour of the nanomagnet ensemble and singleparticle fluctuations on the muon polarisation P(t), comparative measurements were performed for samples with strong and negligible coupling, respectively. Representative μSR spectra P(t, T) for the strongly interacting sample with d = 40 nm and a = 70 , nm and for the noninteracting sample with d = 40 nm and a , = 140 nm (Sample Set 2 in Table 1) are shown in Fig. 2.
For the noninteracting sample, there is almost no damping of the muonspin depolarisation function at temperatures above 60 K and only moderate damping at lower temperatures (Fig. 2a). Here, the polarisation P(t) can be described by a singleexponential decay, i.e. considering only two fractions in Eq. (2) (and thus g_{fast} ≡ 0). The extracted values for the temperaturedependent depolarisation rate \(\lambda _{{\mathrm{slow}}}^{{\mathrm{n}}{\mathrm{.i}}{\mathrm{.}}}(T)\), shown in Fig. 3h, exhibit a steplike increase on reducing the temperature, which is associated with the singleparticle freezing of the individual nanomagnets. A sigmoid fit to \(\lambda _{{\mathrm{slow}}}^{{\mathrm{n}}{\mathrm{.i}}{\mathrm{.}}}(T)\) gives a centre of 49.1 K ± 8.2 K corresponding to the mean value of the blocking temperature T_{B} of the magnetic ensemble measured with muons^{35}.
The measurements of P(t) for the strongly interacting sample (Fig. 2b) show sizeable damping emerging at temperatures well above the singleparticle blocking temperature T_{B}. Furthermore, around 100 K, a rapid contribution to the muonspin depolarisation is observed for times below 0.5 μs, leading to a sizeable signal loss. This contribution is described by the term \(g_{{\mathrm{fast}}}e^{  \lambda _{{\mathrm{fast}}}t}\) in P(t) [Eq. (2)]. Interestingly, the depolarisation rate λ_{fast}(T) shown in Fig. 3c (red diamonds) has a sharp peak around 100 K, defining a critical temperature T_{C}. The value of T_{C} is determined by fitting a piecewise linear function λ_{fast}(T)^{−1} ∝ (T − T_{C}) to the diverging signal of λ_{fast} (red solid lines in Fig. 3c). In contrast, λ_{slow}(T) shown in Fig. 3d (blue dots) is low at high temperatures and, coming from high temperatures, gradually increases below 100 K and saturates below 50 K.
The strong decay of the muonspin polarisation observed in the strongly coupled dXY system, which is not seen in the noninteracting sample, implies that mutual interactions can lead to emergent correlated behaviour at temperatures, where the isolated macrospins would be still superparamagnetic.
To gain further insight into the effect of magnetic fluctuations on the muonspin depolarisation, we performed μSR experiments on the strongly interacting sample using a longitudinal measurement geometry (see Methods). Here we apply a magnetic field that suppresses the influence of the internal static fields on P(t), leaving fluctuations as the only source of muonspin depolarisation^{36}. Comparative measurements for three different temperatures are shown in Fig. 4a–c. The difference between P(t, μ_{0}H = 0 mT) and P(t, μ_{0}H = 6 mT) is associated with the contribution of the static magnetic moment configurations to the muonspin depolarisation. We calculated the phenomenological parameter g_{static}(T) from the integrated area between the curves (i.e. the shaded region in Fig. 4a–c) and, from the temperature dependence of g_{static}(T) (Fig. 4d), it can be seen that static magnetism develops below 100 K. Furthermore, the fluctuation rates f_{fluct}(T) (Fig. 4e) and the width of the internal magnetic field distribution δ(T) (Fig. 4f) can be determined by simultaneous fitting of the data obtained with and without applied magnetic fields with the socalled dynamicexponential Kubo–Toyabe function^{37,38}. The additional measurements shown in Fig. 4e and f, compared to those shown in Fig. 4d, are supplementary measurements obtained at 6 mT. Here, the P(t, T, 6 mT) curves were fitted with the same Kubo–Toyabe function but, because there are no zerofield measurements at the same temperature, the fit function is less constrained, and the errors of the fitting parameters are larger. In summary, at high temperatures we observe high fluctuation rates f_{fluct} (Fig. 4e), that rapidly decrease below 100 K, and vanish below 40 K.
Tailoring of magnetic properties by sample design
To elucidate the role of magnetostatic coupling on the collective magnetic behaviour, we performed equivalent zerofield muonspin depolarisation measurements for the three different sets of strongly and noninteracting samples listed in Table 1. These measurements, along with the fitted depolarisation rates are summarised in Fig. 3. For the strongly interacting samples, there is an apparent divergence of λ_{fast}(T) (red diamonds in Fig. 3a, c, e) at a critical temperature T_{C}, below which the value of λ_{slow}(T) (blue dots in Fig. 3b, d, f) gradually increases. For the noninteracting samples, there is a steplike increase in the depolarisation rate \(\lambda _{{\mathrm{slow}}}^{{\mathrm{n}}{\mathrm{.i}}{\mathrm{.}}}(T)\) (cyan circles in Fig. 3g–i), corresponding to the blocking temperature T_{B}. Below T_{B}, the values of \(\lambda _{{\mathrm{slow}}}^{{\mathrm{n}}{\mathrm{.i}}{\mathrm{.}}}(T)\), λ_{slow}(T) and λ_{fast}(T) level off, indicating that magnetic fluctuations vanish as the moments freeze out at lower temperatures. In all strongly interacting samples, there is a temperature window between T_{B} and T_{C} (shaded area in Fig. 3) where collective magnetic order emerges.
We observe qualitative trends for decreasing nanomagnet diameter d and lattice periodicity a (i.e. going from left to right in Fig. 3): For the noninteracting samples, there is an increase in the singleparticle blocking temperature T_{B} with decreasing dot diameter d (Fig. 3g–i). For the strongly interacting samples, the values for T_{C}, g_{fast} and the experimental maximum values of λ_{slow} and λ_{fast} all increase across the series. Also, going across the series as the lattice periodicity a decreases, we observe an increase in the critical temperature T_{C}, which is in line with Eq. (1).
We can conclude that there are three (two) temperature regimes for the magnetic behaviour of strongly interacting (noninteracting) dXY systems: At high temperatures the magnetic response is governed by fast superparamagnetic fluctuations, whereas at low temperatures we observe the freezing of magnetic configurations due to singleparticle blocking. Between these limiting regimes, mutual interactions can modify magnetic fluctuations and spinequilibrium configurations, leading to correlations and longrange ordered states in the considered artificial dXY spin systems. In the following we will discuss the μSR signatures of each regime, and how they relate to the magnetic behaviour of the sample determined by the energy scales k_{B}T_{C} and k_{B}T_{B}.
Hightemperature superparamagnetic regime
At high temperatures, i.e. T > T_{C} for strongly interacting, and T > T_{B} for noninteracting samples, the superparamagnetic fluctuation rates of the single nanomagnets are much higher than the muon precession frequency, i.e. the Larmor frequency. Consequently, the timeaveraged magnetic field at the muon site is small (socalled motional narrowing in the fastfluctuating regime^{36,38}) leading to the small, almostconstant muonspin depolarisation rates λ_{slow}(T > T_{C}) and \(\lambda _{{\mathrm{slow}}}^{{\mathrm{n}}{\mathrm{.i}}{\mathrm{.}}}\left( {T > T_{\mathrm{B}}} \right)\).
Intermediate correlated regime
The temperature window T_{C} ≥ T ≥ T_{B} (shaded areas in Figs. 3, 4), which we can define for the strongly interacting samples only, is characterised by the emergence and growth of static magnetic correlations. Experimental signatures related to this regime are the gradual increase of the muonspin depolarisation rate λ_{slow}(T_{C} ≥ T ≥ T_{B}) (Fig. 3b, d, f), the emergence of a large additional contribution to the muonspin depolarisation λ_{fast}(T) that peaks at T_{C} (Fig. 3a, c, e), the rapid reduction of the fluctuation rate f_{fluct}(T) (Fig. 4e, i.e. the system is slowing down), and the increase of the parameter g_{static}(T) (Fig. 4d). As the fast muonspin depolarisation described by λ_{fast}(T) is not observed for the noninteracting samples, it cannot be caused by the presence of local disorder or singleparticle behaviour. Therefore, our μSR results demonstrate that mutual interactions in strongly interacting artificial dXY spin systems lead to quasistatic correlations at temperatures where the isolated nanomagnets would still be in the superparamagnetic fluctuating regime.
We now focus on the temperature dependence of the muonspin depolarisation rates λ_{slow}(T) and λ_{fast}(T) in more detail and relate this to the expected behaviour of a secondorder phase transition from the paramagnetic into a stripeordered phase predicted for the squarelattice dXY spin system^{5,30}. As the muonspin depolarisation is both influenced by the width of the field distribution experienced by the muon ensemble as well as magnetic fluctuations within the sample, both (quasi)static and dynamic properties of the emergent correlations can contribute to the observed μSR spectra.
To relate the static magnetic order to the measured μSR spectra, we calculated the timedependent muonspin depolarisation for emerging magnetic order using a meanfield description of the dXY order parameter^{6,12} and randomised muonspin stopping positions in accordance with the implantation profile in the 80 nmthick gold layer (see Methods and Supplementary Note 1). The depolarisation function P(t) obtained from these meanfield simulations yields two depolarisation rates that vanish above T_{C} and display a squarerootlike temperature dependence below T_{C}. Although based on a greatly simplified model, the meanfield simulations give reasonable results for the depolarisation rates: As shown in Fig. 3b, d, f, the magnitude of the slower of the two depolarisation rates obtained by our meanfield description (solid blue line) agrees well with the experimentally determined values of λ_{slow}(T) (blue dots) for the smaller nanomagnetic discs (Fig. 3d, f), and the theoretical values only deviate by a factor of two for the largest considered discs with diameter of d = 70 nm (Fig. 3b).
The qualitative agreement between the experimentally observed values of λ_{slow}(T) and the calculated muonspin depolarisation indicates that static antiferromagnetic order is the main contribution to λ_{slow}(T). The temperature dependence of λ_{fast}(T), which peaks at T_{C}, however, is not reproduced by our simulations since it probably originates from critical correlations emerging at the phase transition, which are not captured by a meanfield description.
Furthermore, it is interesting to note that the measured value of λ_{fast}(T_{C}), seen in Fig. 3, is astonishingly large (up to 15 μs^{−1}), being more than an order of magnitude larger than λ_{slow}(T) (maximum values around 0.7 μs^{−1}). This is unusual for a μSR signal, as dynamic depolarisation rates due to a critical slowing down at a phase transition are usually smaller than the depolarisation caused by quasistatic fields^{36,38}. Also, the rapid depolarisation described by λ_{fast}(T) contributes to the net muonspin depolarisation with a small fraction g_{fast} only (Table 1), indicating the emergence of a new type of magnetic environment at T_{C} in a small fraction of the sample. Here, we suggest that the rapid muonspin depolarisation originates from the critical slowing down in the vicinity of the phase transition, where longlived correlations can originate from magnetic clusters forming around T_{C}. The relaxation time scales of these correlated regions diverge close to T_{C}, and thus can feature broad, random magnetic field distributions that are static in the μSR time window, and thus lead to a rapid depolarisation of the muon spin. With the increase of static magnetic order upon further cooling, λ_{slow}(T) increases (Fig. 3b, d, f), and the effect of thermal disorder on the muonspin relaxation, and thus λ_{fast}(T) (Fig. 3a, c, e), becomes less relevant.
Lowtemperature blocked regime
Finally, at low temperatures T < T_{B} the system is dominated by singleparticle blocking, where fluctuations of the individual nanomagnets become much slower than the timescale of the measurement. This behaviour is corroborated by the vanishing fluctuation rate f_{fluct} in dynamic μSR measurements (Fig. 4e), the saturation of the static magnetic fraction of the muon signal g_{static} (Fig. 4d), and the constant lowtemperature value of the slow depolarisation rates λ_{slow}(T) and \(\lambda _{{\mathrm{slow}}}^{{\mathrm{n}}{\mathrm{.i}}{\mathrm{.}}}(T)_{}^{}\) for strongly and noninteracting samples, respectively (Fig. 3).
As discussed in the introduction, singleparticle blocking in artificial XY spin moments only occurs because the magnetic nanodiscs are not perfect, and the value of T_{B} gives an estimate of the anisotropies present in the system. We find that, across the set of samples considered in this work, the blocking temperature T_{B} increases as both the disc diameter d as well as the lattice periodicity a, decrease (Fig. 3g–i). This can be explained by the fact that the nanofabrication of denser arrays and smaller discs leads to a larger variation in particle size and shape.
For the noninteracting samples, the arrangement of blocked moments is expected to be completely random, governed by the directions of the local anisotropy of the nanoparticles, and the static random local fields lead to a more effective depolarisation of the muon spin than the fluctuating fields at hightemperatures. As moment fluctuations freeze out, the relaxation rate \(\lambda _{{\mathrm{slow}}}^{{\mathrm{n}}{\mathrm{.i}}{\mathrm{.}}}(T)\) gradually increases upon cooling, and saturates for T < T_{B}.
To understand the effect of singleparticle blocking on the muonspin relaxation in the strongly interacting samples, we have to consider that magnetic correlations are already present in the vicinity of T_{B} < T < T_{C}. Here, due to the anisotropy generated from the coupled nanomagnets on the square lattice, and as the moment fluctuations slow down considerably below T_{C}, one can expect that the local moments freeze below T_{B}. However, small anomalies in the relaxation rate λ_{slow}(T) (Fig. 3d, f) indicate that the magnetic order becomes slightly less correlated for T < T_{B}. This partial loss of order can be explained by heterogeneous freezing, i.e. a reorientation of the XY moments from the x and y lattice direction favoured in the longrange ordered phase to locally defined random anisotropy axes.
Emergence of longrange order
Since lowenergy muonspin relaxation measurements can only establish the presence of a transition to a phase characterized by a local static magnetic field, but not its corresponding longrange order, we have performed auxiliary soft Xray resonant magnetic scattering measurements at low and high temperatures, with the Xray energy at the Fe L_{3} absorption edge. Here, squarelattice artificial dXY spin systems with a lattice periodicity of a = 120 nm and disc diameter and disc height of d = 90 nm and h = 5 nm, respectively, were investigated. The magnetic scattering patterns obtained with linearly polarised Xrays (see Methods for experimental details) display a clear temperature dependence: At high temperature featureless weak diffuse magnetic scattering intensity is observed (Fig. 5b), whereas at low temperatures streaklike features emerge (Fig. 5a). These features are indicative of ordering, and their streaklike appearance is due to the reflection scattering geometry. The observation of magnetic scattering intensity centred at halfinteger positions in reciprocal space at low temperatures corresponds to emergent longrange order that results in a doubling of the structural lattice periodicity, which is compatible with the expected stripelike ground state of the squarelattice dXY spin system.
Discussion
Here, we extend the work on artificial spin systems from the study of the thermodynamics of Ising degrees of freedom^{18,19,20,21,22} to thermally active magnetic metamaterials with inplane XY anisotropy.Our results provide robust evidence of a phase transition as well as longrange order compatible with the theoretically predicted behaviour of a dXY model on a square lattice.
To achieve the continuous inplane magnetic degrees of freedom, we choose nanoscale permalloy discs that, due to their shape and size, exhibit a macrospin moment confined to the sample plane. The blocking temperature T_{B} in all investigated samples is at least a factor of two smaller than the transition temperature T_{C}. Therefore, the effect of any imperfections of the discs are small enough, so that at T_{C} the magnetic moments can still explore the full inplane angular range and therefore the ensemble is a close approximation of a lattice of XY moments.
Although the magnetostatic interactions between the discs are unlikely to be of purely dipolar character, higherorder multipole contributions are not expected to alter the qualitative zerofield equilibrium behaviour of the squarelattice dXY spin system^{11,27}. Our experimental μSR results, in combination with the soft Xray scattering measurements, clearly demonstrate that, upon cooling, our strongly coupled artificial XY spin systems develop magnetic correlations and exhibit a phase transition to a longrange ordered state in agreement with predictions for the 2D dXY model.
Finally, as shown by our comparison of three different sample sets, the relevant temperature scales, i.e. the critical temperature T_{C} and the singleparticle blocking temperature T_{B}, can be influenced by careful sample design. Therefore, due to their tunability, where the shape, size, and interdot distances can be precisely tailored, these artificial spin systems enable investigations of ordering phenomena and critical behaviour in the thermodynamic limit of magnetostatically coupled XY systems with controlled amounts of disorder, such as random displacements^{39,40} or the presence of vacancies^{5,41,42}, for which more complex phase diagrams are predicted^{43}.
Methods
Sample fabrication
The samples were manufactured with electronbeam lithography. Here a polymethyl methacrylate polymer layer was spincoated on a silicon substrate. Then the desired pattern was exposed with a Vistec EBPG 500Plus electronbeam writer over areas up to 3 cm^{2}. On the developed resist, a thin permalloy (Ni_{80}Fe_{20}) film was deposited using electronbeam evaporation at base pressure of 3 × 10^{−7} mbar, and capped with a 4 nm layer of gold to prevent oxidation. Following ultrasoundassisted liftoff, the patterned arrays were coated with a continuous 80 nmthick gold layer, acting as a stopping layer for the muons. SQUID magnetometry confirmed that the patterned nanomagnets support thermally activated moment fluctuations above 70 K. The samples for soft Xray resonant magnetic scattering experiments were fabricated in a similar manner, and capped with 3 nm Al to avoid oxidation.
Lowenergy muonspin relaxation
Experiments were performed at the lowenergy muon LEM beamline of the Swiss Muon Source^{44,45}. The lowenergy (15 keV) spinpolarised muons (μ^{+}) are implanted into the gold layer, stopping above the arrays of dXY nanomagnets^{18}. Here, the muons act as local probes, randomly sampling the local stray fields emanating from the nanomagnets. Stray fields that are transverse to the initial muon spin cause them to precess, and magnetic fluctuations and field distributions lead to a loss of the spin polarisation of the ensemble. The asymmetry of the muon decay leads to an asymmetry between forward and backwarddetected decay positrons A(t), which is related to the (normalised) muon polarisation function via P(t) = A(t)/A_{0}, with the initial asymmetry A_{0}. The relaxation of the depolarisation function P(t) is determined by the variance and fluctuations of the randomly sampled field distribution of the artificial spin system^{36}.
To characterise the zerofield thermodynamic behaviour of the artificial dXY system, μSR measurements with the initial muon spin parallel to one of the inplane lattice directions, i.e. x, were performed upon cooling. To decouple the static from the dynamic magnetic response, a longitudinal geometry was used, with the initial spin polarisation parallel to the outofplane direction z and a field of 0mT or 6mT applied along z. At different temperatures between 300 and 5 K, μSR spectra with 5–30 × 10^{6} events per time spectrum were measured.
μSR fitting procedure
The model for the zerofield muonspin polarisation in Eq. (2) considers three contributions to P(t) = A(t)/A_{0}. For the fitting routine, the parameters A_{0}, g_{0}, g_{slow} and g_{fast} are held constant for each sample, and only λ_{slow} and λ_{fast} are fitted to each measured P(t, T). Fitting was performed with musrfit^{46} and Python lmfit^{47}. The value of the initial asymmetry A_{0} is determined at high temperatures T > T_{C}. The fraction g_{0} combines the constant signal due to muons that are not stopped in the sample with the nonrelaxing longitudinal contribution (as only transverse field components cause the muon spin to precess) to P(t). The value of g_{0} is determined in the static magnetic state for \(T \ll T_{\mathrm{B}}\). The slow relaxation fraction g_{slow} is the dominant contribution to the muonspin depolarisation. For noninteracting samples, the fast fraction is not considered, i.e. g_{fast} ≡ 0. In contrast, for strongly interacting samples, the value of g_{fast} is determined for temperatures where the rapid depolarisation is most pronounced, i.e. close to T_{C}.
Meanfield calculation of the muonspin depolarisation
We derived the timedependent μSR depolarisation function \(P(t,\mathbf{x},T)\) for a muon implanted at position \(\mathbf{x}\) by calculating its precession in the static magnetic field generated by the meanfield order parameter ϕ(T) of an antiferromagnetic stripeordered phase^{36}:
Here the static field \(\left {\mathbf{B}(\mathbf{x},T)} \right \propto \phi (T) \cdot M_{{\mathrm{dot}}}{\mathrm{/}}\left \mathbf{x} \right^3\) determines the precession dynamics with antiferromagnetic order parameter ϕ(T), disc moment M_{dot}, and distance \(\mathbf{x}\) between muon and XY moment. \({\theta}={\angle}({\mathbf{S}}_{\mu} (0), {\mathbf{B}})\) denotes the angle between field direction \(\mathbf{B}\) and initial muonspin direction \({\mathbf{S}}_{\mu} (0)\), and γ_{μ}/(2π) = 135.54 MHz T^{−1} is the muon gyromagnetic ratio. In the experiment, the muon is implanted at random positions. Therefore P(t, T) was averaged over the lateral spread of the muon beam and the implantation depth of the muons in the stopping layer. The latter is weighted by depth probabilities obtained from TRIM.SP depth profiles of muons implanted in an 80 nm gold layer on a silicon substrate^{48}. P(t, T) can be fitted by a twoexponential decay, yielding two relaxation rates λ_{1}(T) and λ_{2}(T). Both λ_{1}(T) and λ_{2}(T) follow the squareroot temperature dependence of the meanfield order parameter, and the magnitude of the slower relaxation rate compares well with the measured values of λ_{slow}(T) (see solid line Fig. 3b, d, f). Further details on the calculations are documented in the Supplementary Note 1.
Scattering experiments
Soft Xray resonant magnetic scattering measurements were performed using the endstation RESOXS^{49} at the SIM beamline^{50} of the Swiss Light Source, Paul Scherrer Institute. The setup with the smallangle reflection geometry is described elsewhere^{19}. In order to be sensitive to the weak magnetic scattering signal, a mask was used to block out the highintensity structural scattering peaks. To enhance the magnetic signal, the Xray energy was tuned to the iron L_{3} absorption edge (708 eV), and the scattering signals collected in the low and hightemperature phase were measured using a 2D Xray area detector (Princeton Instruments PME chargecoupled device (CCD) camera with pixel size of 20 μm; sampledetector distance was 175 mm). Scattering patterns obtained with the Xray energy tuned to 690 eV (a few electron volts below the Fe L_{3} edge) were used to verify the magnetic origin of the scattered intensity and for normalisation purposes. The magnetic signal shown in Fig. 5 was obtained by subtracting the scattering signals taken with two perpendicular Xray polarisations, i.e. horizontal and vertical linear polarisations.
Data availability
The datasets analysed during the current study are available in the Zenodo repository https://doi.org/10.5281/zenodo.1252365, that also contains a list of the LEM run numbers for which the raw data can be obtained from the SμS data repository (http://musruser.psi.ch).
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Acknowledgements
μSR experiments were performed at the LEM (μE4) beamline, Swiss Muon Source SmuS, and soft Xray scattering experiments were performed at the SIM beamline, Swiss Light Source, both at the Paul Scherrer Institute, Switzerland. This work was funded by the Swiss National Science Foundation (SNSF project grants 200021155917, 200021159736, and 200021172774). D.L. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie Grant Agreement No 701647. J.R.L.M. is grateful to the Swiss National Center of Competence in Research, Molecular Ultrafast Science and Technology (NCCR MUST). We thank Vitaly Guzenko and Dario Marty for help with the sample fabrication, Urs Staub for providing support at the SIM beamline, and Peter Holdsworth and Luca Anghinolfi for helpful discussions.
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The experiment was conceived by S.L.L., L.J.H., O.S. and H.L. N.L. fabricated the samples; O.S. assisted with the sample fabrication; N.L., H.L., S.L.L., S.H. and O.S. performed the μSR experiments; A.S., T.P. and Z.S. provided the μSR beamline support; N.L., S.H. and H.L. performed the analysis; D.S., P.M.D. and N.L. performed the simulations; V.S., D.L. and J.L. assisted with the Xray scattering experiments; N.L., D.S., S.H., P.M.D., H.L., S.L.L. and L.J.H. were involved in the preparation of the manuscript. All authors contributed to the manuscript.
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Leo, N., Holenstein, S., Schildknecht, D. et al. Collective magnetism in an artificial 2D XY spin system. Nat Commun 9, 2850 (2018). https://doi.org/10.1038/s41467018052162
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