Thermodynamic phase transitions in a frustrated magnetic metamaterial

Materials with interacting magnetic degrees of freedom display a rich variety of magnetic behaviour that can lead to novel collective equilibrium and out-of-equilibrium phenomena. In equilibrium, thermodynamic phases appear with the associated phase transitions providing a characteristic signature of the underlying collective behaviour. Here we create a thermally active artificial kagome spin ice that is made up of a large array of dipolar interacting nanomagnets and undergoes phase transitions predicted by microscopic theory. We use low energy muon spectroscopy to probe the dynamic behaviour of the interacting nanomagnets and observe peaks in the muon relaxation rate that can be identified with the critical temperatures of the predicted phase transitions. This provides experimental evidence that a frustrated magnetic metamaterial can be engineered to admit thermodynamic phases.


Supplementary Figure 2: Selected ZF-μSR spectra for the strongly interacting sample.
Panel a, b: ZF-μSR spectra at temperatures close to the peaks of the longitudinal relaxation rate at 35 K and 145 K, respectively. As explained in the main text, the muon spectra are characterized by a fast relaxation (related to the distribution of both static and dynamic fields) followed by a slower relaxation in the tail at long times (related to dynamics only, shaded regions). Away from the peaks (blue and green points), the tail of the spectra is almost constant indicating that the characteristic time scale of the fluctuations of the local magnetic fields does not match the μSR time scale. However, close to the critical temperatures (red points) the long-time spectrum more strongly relaxes. This is an unequivocal signature of dynamic local fields in the time window of μSR. It should be noted that this effect, which is already clearly visible in the ZF-μSR spectra, is then reflected in the peaks of the longitudinal rate λ L at the magnetic phase transitions. In panel c, we show simple exponential functions fitted to the experimental data. The fits fail to reproduce the ZF-μSR spectra both at short times and at long times, further confirming that a two-component relaxation function is required to analyse the data. In all panels, the error bars represent statistical errors determined by the number of total muon decay events observed for each data point. The implanted muon thermalizes at an interstitial lattice site within a few picoseconds.
Here the muon spin interacts with its magnetic environment and the spin polarization of the muon ensemble ( ) relaxes as a function of time. In a radioactive beta decay, the muon decays via the process → + ̅ + . The positron is finally detected by a pair of plastic scintillators located to the left and right side of the sample (see Figure 2a). Due to parity violation in the μ + decay process, the positron is emitted preferentially along the direction of the muon spin at the moment of the decay. By collecting several millions of positron counts in the detector pair, it is therefore possible to reconstruct the time evolution of the muon spin polarization, which can be analysed to obtain information about the static and dynamic magnetic properties of the sample.

Supplementary Note 2. µSR data analysis
For a detailed introduction to the analysis of μSR data, the interested reader is referred to the recent textbook of Yaouanc and Dalmas de Réotier 7 . The analysis of the time-dependent μSR data has been performed using the free software package MUSRFIT 8 . In a magnetic powder or in situations where the internal fields at the muon sites are randomly oriented for the muon ensemble, the muon polarization function ( ) is given by a sum of two contributions. The so-called transverse relaxation function comprises 2/3 of the spectrum originating from local magnetic fields perpendicular to the initial muon spin and the longitudinal fraction accounts for the remaining 1/3 of the spectrum with the internal field parallel to the initial muon spin. We have therefore fitted the following polarization function to the data: where A is the measured decay asymmetry and is the total asymmetry. In addition, a non-relaxing background contribution due to muons stopping in the Ag sample holder has been subtracted from the data. The relaxation rate of the transverse component λ T is governed by the distribution of static fields sensed by the muon ensemble as well as the relaxation due to dynamic fluctuations of the magnetic fields 7 , if present. The longitudinal relaxation rate λ L is solely given by magnetic dynamics in the time window of μSR. λ L is therefore always smaller (or equal) to λ T . In the fast fluctuation regime both rates become equal and are therefore no longer distinguishable. As the distribution of local fields generated by the nanomagnets in the Au capping layer is not trivial, we considered an empirical model based on a stretched exponential function with the exponent β T . Stretched exponentials are commonly used to describe the muon depolarization in the presence of a distribution of relaxation rates. They are also quite flexible functions, which can fit a wide range of line shapes. Firstly the β T parameter has been kept as a free fit parameter and found to be nearly temperature independent for temperatures below 280 K. In the final analysis, β T has been fitted to one common parameter for all datasets below 280 K and found to be β T = 0.83. In the fast fluctuation regime, β T was found to approach a value close to 1.0 as expected. The In order to validate that the model fits the μSR spectra (see Supplementary Equation 1), we used CONTIN 10,11,12 to compute the inverse Laplace transform of the experimental data.
The algorithm decomposes the raw μSR data into a series of exponentials ( ) ⋅ (where t is time and λ an inverse of time and, in our specific case, a relaxation rate) and computes the distribution s(λ) of their amplitudes, which satisfies Supplementary Equation 2: In addition, these measurements also indicate that, even up to highest measured temperatures, the muons do not diffuse through the gold layer. In order to be sure that muon diffusion does not contribute to the high temperature data, dedicated measurements were performed on the gold layer as described in Section 1.4.

Supplementary Note 4. Longitudinal field μSR measurements
The zero field μSR measurements presented in the main article indicate that there is To further verify that the magnetic fields seen by the muons are dynamic at high temperatures and static at low temperatures, we performed longitudinal field (LF) μSR measurements on the strongly interacting sample, which allows us to discriminate between static and dynamic fields at the muon site 7 . Using the recently installed spin rotator of the LEM beamline 13 , we rotated the polarization of the incoming muons compared to the zero field measurements, orienting it perpendicular (instead of parallel) to the sample surface.
Then we measured the muon spin relaxation in zero field and in a longitudinal field (parallel to the muon spin) of 60 G. The obtained spectra for 5K and 300 K are shown in Supplementary Figure 5. The zero field spectra are consistent with the data presented in the main article (Figure 2b), where the initial muon spin direction was oriented in the film plane.
Specifically, at 5 K (red curve in Supplementary Figure 5a), the polarization exhibits an initially fast relaxation characterized by , which is followed by an essentially constant tail ( ≈ 0). At 300 K (red curve in Supplementary Figure 5b), the ZF spectrum can be described by a single exponential relaxation function as appropriate for the fast fluctuation regime.
On applying a longitudinal field at 5K (blue curve in Supplementary Figure 5a), we find that the initial muon spin polarization is now essentially conserved over time. This behaviour is a result of the fact that the applied field is much larger than the dipolar fields from the nanomagnets and the dipolar fields are static on the muon timescale 7 . In contrast, at 300 K the applied field has very little influence on the time dependence of the muon spin polarization (compare the red curve with the blue curve in Supplementary Figure 5b). In particular, the fact that the spectrum still relaxes provides unequivocal evidence for magnetic fields which are dynamic in the time window of μSR 7 .
To estimate the fluctuation rate of the dipolar fields from the nanomagnets, we fitted the muon spin polarization data with a dynamic Lorentzian Kubo-Toyabe function 14,15 , which accounts for the longitudinal applied field and local field dynamics in the strong collision approximation. The best-fit curves are plotted in Supplementary Figure 5 as continuous lines.
At 300 K, the fluctuation rate is determined to be 2.9 MHz, which is comparable with the μSR timescale and therefore consistent with the onset of the fast fluctuation regime at 280 K.
In contrast, at 5 K the fluctuation rate is found to be zero within the fit error. This confirms the presence of dynamic local fields at high temperatures in the fast fluctuation regime and the presence of static local fields at low temperatures deduced from the ZF measurements presented in the main article.

Supplementary Note 5. Estimation for nanomagnet moment reorientation timescale
In ref. 16, a barrier energy of 0.925 eV and a prefactor of 10 12 sec -1 were found to describe the dynamics of moment reorientation of an individual nanomagnet reasonably well. Such a barrier energy was also comparable to that resulting from the shape anisotropy. In ref. 16 the volume of a nanomagnet was 470×170×3 nm 3 , which is much larger than that used presently (63×26×6 nm 3 ). Assuming the barrier energy is due to shape anisotropy (which is proportional to magnet volume), this gives a barrier energy estimate for our current system of approximately 40 meV. Using the same prefactor of 10 12 sec -1 , and assuming a simple Arrhenius form, at 40 K (the temperature regime of the lowest critical temperature) the timescale between nanomagnet reorientations is of the order of 0.1 nanoseconds, which is well within the timescale of the muon precession and quite compatible with the temperature scale of the assumed blocking temperature of an isolated nanomagnet. It is emphasized that all of these numbers are order of magnitude estimates.

Supplementary Note 6. Absence of muon diffusion in the evaporated Au overlayer
In μSR, a finite longitudinal relaxation rate λ L is due to the temporal fluctuation of the magnetic field experienced by the individual implanted muons. In principle, this can have two different origins. Either the local magnetic field at the muon changes its direction as a function of time (dynamic magnetism) or the muon moves due to thermally activated diffusion between interstitial lattice sites in the background of an otherwise static magnetic environment (muon diffusion). These two effects are difficult to disentangle without a priori knowledge of the material under investigation.
In the present study, we investigate the magnetic properties of the artificial kagome spin ice system by implanting the muons in an Au capping layer and measuring static and/or dynamic magnetic fields generated by the stray fields of the nanomagnets. For interpreting the high temperature data correctly, it is important to check whether or not the muon diffuses through the gold layer in the investigated temperature range.
Previous measurements on a high quality bulk Au crystal revealed long range (nm) thermally activated muon diffusion above approximately 100 K 17 . Performing a systematic study on thin metallic films 3 with the same low energy muon apparatus at the Paul Scherrer Institute used in the present work, it was also found that in the best quality epitaxial Au films muon diffusion sets in at approximately 100 K. Nevertheless, it was demonstrated that sputtered films showed no muon diffusion up to 300 K, indicating that the diffusion appears to only occur in high quality crystalline materials.
In the present study, the Au capping layers were prepared by thermal evaporation, which results in polycrystalline films. Therefore we can expect that muon diffusion is also with a being the distance between the maximum of the muon stopping profile and the Cr layer, d = 0.288 nm the jump distance between two octahedral interstitial lattice sites in Au where the muon is hopping, the attempt frequency, and the activation energy.
Supplementary Equation 3 has been derived for Cr layers on both sides of the Au film. For our geometry with a single Cr layer, we can expect the relaxation rate to be approximately a factor of two smaller.
We measured the relaxation rate by μSR in a weak transverse field of 100 G, which was applied perpendicular to the film and the initial muon spin direction. A representative μSR spectrum is shown in Supplementary Figure 6b. The oscillatory component is due to the precession of the muon spin around the applied field and the muon spin polarization function ( ) can be fitted with the following function: where is the maximally observable asymmetry of the muon decay, is the Larmor Clearly, the simulated curves for muon diffusion are far from the measured data.
In conclusion, we have confirmed that there is no long range (nm) muon diffusion in the Au capping layer of our samples in the measured temperature range between 5 K and 300 K. This absence of diffusion is also in accordance with the observation of a Gaussian-like nuclear relaxation at high temperatures in the weakly interacting sample ( Supplementary   Figure 4b), which at the same time also excludes other types of diffusion, such as grain boundary diffusion at high temperatures.

Supplementary Note 7. Monte Carlo Simulations
We used Monte Carlo simulations to compute the equilibrium configurations of the nanomagnet moments. The conventional way to model the individual nanomagnets in artificial spin ice is to approximate them as single Ising macrospins arranged on a finite kagome lattice 16,19 . The approximation is justified because the nanomagnets are small enough to be single-domain and the thickness of few nm's ensures that the exchange interactions between the microscopic spins dominate, keeping them aligned parallel to each other.
Nevertheless, it has recently been shown using micromagnetic simulations 20 that, when the nanomagnets are brought close together, the magnetization at the tips of the nanomagnets slightly curls to minimize the dipolar interactions. As a consequence, and also due to the spatial extent of the nanomagnets, the short range interactions between nanomagnets are enhanced. In our simulations, we have effectively accounted for these effects by following an approach similar to Budrikis and coworkers 19 . We employed the macrospin model on a kagome lattice and considered modified interactions, J n , between n-th nearest-neighbours <i,j> n . The Hamiltonian is the following: where m i is the macrospin unit vector for the magnet i , r ij is the unit vector connecting the magnets i and j, and r n is the distance between the n-th nearest-neighbours. Interactions up to fifth nearest-neighbours were required to obtain the low temperature transition to the LRO