U nderstanding continuous phase transitions near critical points is crucial in fundamental physics. In condensed matter physics and materials science, these transitions, often driven by variations in temperature, pressure, and external fields, reveal complex interactions among lattice structures, electronic states, and magnetic moments. Of particular interest are second-order (continuous) phase transitions, where the order parameter changes smoothly and is often associated with spontaneous symmetry breaking. Near a second-order phase transition, systems exhibit critical phenomena, including divergent correlation lengths, fluctuations, and susceptibilities. These critical phenomena fall into distinct universality classes set by, e.g., the spatial dimension and the symmetry character of the order parameter, such that scaling laws near criticality are independent of microscopic details. Understanding these behaviors provides insights into the underlying physics that governs a wide range of systems.

Over the past decade, nitrogen-vacancy (NV) centers in diamond have emerged as powerful nanoscale quantum sensors capable of probing ferromagnetic [1][2][3] and antiferromagnetic 4,5 ordering, and slow telegraph switching of magnetic domain walls has been observed through time-series fluorescence measurements and through variations in the measured optically detected-magnetic-resonance (ODMR) line width. 6 On the other hand, NV T 1 and T 2 measurements have increasingly been used to measure high-frequency fluctuating magnetic fields that are hard to access with conventional nanoscale magnetic probes like magnetic force microscopy. [7][8][9][10][11][12][13][14][15][16][17][18] The noninvasive visualization of magnetic textures and spin dynamics in response to changes in temperature or external fields is essential to characterizing critical phenomena associated with continuous phase transitions in magnetic systems where correlation scales diverge in time and space near the critical point. Recent studies have pointed to the potential for spin-qubit probes of critical dynamics in spin systems using spin coherence measurements, 18 but all-optical T 1 probes of critical dynamics would unlock the potential for probing materials in high-field environments where T 2 measurements are challenging to implement.

In this study, we investigate nanoscale magnetic ordering and spin fluctuations near the ferromagnetic-paramagnetic (FM-PM) phase transition in a thin film of the high-T c double perovskite strontium iron rhenium oxide (Sr 2 FeReO 6 ) using scanning NV magnetometry and relaxometry. Our results reveal the onset of magnetization characterized by surface magnetic textures that exhibit robust scaling behavior indicative of a second-order phase transition. Concurrently, we observe a pronounced increase in spin fluctuations near T c , which are detected through NV T 1 relaxometry and modeled using Landau-Ginzburg theory. The combined results highlight the pivotal role of spin dimensionality in Sr 2 FeReO 6 and more generally illustrate the importance of complementing bulk dc magnetization measurements with nanoscale probes of spin ordering and spin fluctuations, as these nanoscale energetics and dynamics ultimately drive the functionality of emerging dissipationless spin-based information processing platforms.

We focus here on the high-T c double perovskite Sr 2 FeReO 6 , whose crystal structure is illustrated in Figure 1(a). Sr 2 FeReO 6 is known for its robust ferromagnetism persistent up to 400 K, 19,20 and unlike most metallic ferromagnets, the ferromagnetism in Sr 2 FeReO 6 is attributed to a hybridization mechanism. 21 The ferromagnetic order in Sr 2 FeReO 6 is stabilized by significant exchange splitting of Fe 3d orbitals, driven by strong electron-electron correlations, and spindependent hybridization between Fe 3d and Re 5d orbitals. As a result, the electronic band structure of Sr 2 FeReO 6 near the Fermi level is dominated by spin-polarized Re 5d orbitals, as demonstrated in previous DFT calculations. 22 This combination of half-metallic character, strong electron correlations from 3d orbitals, and large spin-orbit coupling in 5d orbitals makes Sr 2 FeReO 6 not only promising for spintronic applications but also a potential platform for a quantum anomalous Hall insulator phase. 21 Here, a high quality epitaxial thin film of Sr 2 FeReO 6 with thickness ∼35 nm was grown with a sintered Sr 2 FeReO 6 target by pulsed laser deposition on a (001) SrTiO 3 substrate as previously described. 19 Figure 1(a) illustrates the scanning NV experiment conducted here using a Qnami ProteusQ microscope, where single-NV spin states are manipulated via optical and radio frequency (RF) pulses. The NV spin is pumped into the |0⟩ state by an off-resonant 520 nm optical excitation, an RF excitation with frequency around 3 GHz is used to manipulate the ground state spin, and the NV spin can be read out by measuring the NV luminescence intensity, which is larger for the |0⟩ state than the Zeeman-split | ± 1⟩ states. 23 The probe used here incorporates a single negatively charged NV center implanted ∼10 nm below the surface of a (100) diamond nanopillar integrated into a tuning-fork AFM probe. The NV center axis is orientated at an angle of about 55°relative to the surface normal and is thus sensitive to changes in both in-plane and out-of-plane fields.

This approach enables nanoscale resolution, achieving detail down to 50 nm, by mapping surface magnetic textures through ODMR measurements taken at each pixel of the image. In full-B mode, the magnetic field is calculated from the frequency separation between the | ± 1⟩ states observed in the ODMR spectrum, as shown in the left of Figure 1

where Δν represents the frequency separation and γ = 28 GHz/T is the electron gyromagnetic ratio. However, acquiring a full ODMR spectrum at each pixel of an image can be time-consuming; the dual-iso-B measurement protocol provides an alternative high-speed measurement modality in which a single ODMR spectrum is acquired in order to define the frequencies f 1 and f 2 illustrated in the right of Figure 1(c) at the full-width half-maximum of a |0⟩ to | ± 1⟩ transition. The luminescence intensity is then measured at each pixel of the image for RF excitation at f 1 and f 2 . For small changes in magnetic field, the differential photoluminescence intensity ΔPL = PL( f 1 ) -PL(f 2 ) can be mapped to changes in the magnetic field semiquantitatively. Figure 1(d) displays the nanoscale magnetic surface texture of a ferromagnetic Sr 2 FeReO 6 thin film measured in full-B mode (left) and dual-iso-B mode (right). The resolved magnetic domains exhibit structure across measured length scales spanning ∼50 nm to 15 μm (with a 2 μm illustrated field of view in Figure 1(d)),and the measured magnetic field varies by 10-15 G across the full-B map. This dynamic range is sufficiently small to allow for accurate high-speed measurements to be acquired in dual-iso-B mode, as is seen in the qualitative one-to-one relationship between the full-B and dualiso-B maps in Figure 1(d).

The nanoscale resolution of magnetic textures under external stimuli, such as magnetic field and temperature,

Letter provides valuable insights into magnetic ordering in correlated materials that are critical to the development of dissipationless electronic and spintronic devices, as bulk magnetization measurements do not provide a complete picture of spin interactions within and between domains. In the absence of a strong magnetic field, the thermally induced FM-PM phase transition exhibits characteristics of a second-order phase transition. [24][25][26] We probed this transition with temperaturedependent scanning NV microscopy, which was performed in dual-iso-B mode to minimize deleterious effects from drift at elevated temperatures. In Figure 2(a), the FM-PM transition is clearly visualized between 397 and 412 K with the disappearance of any measurable long-range magnetic order.

To quantify this behavior, we performed a contrast analysis by comparing the most and least intense areas (top and bottom 10% average, respectively) from the distribution in each falsecolor map, and we extracted the effective magnetic field strength using an inverse Lorentzian transformation linking count differences to field-induced ODMR shifts. As illustrated in Figure 2(c), the stray magnetic field strength, and the associated surface magnetization rapidly drops as the temperature approaches 400 K. We further compare the extracted magnetic field strength with bulk magnetization measurement data acquired with a superconducting quantum interference device 19 in Figure 2(b). The data in Figures 2(b) and 2(c) show excellent agreement in the onset and gradual increase of the magnetization near the phase transition. Further analysis involved binarizing the images and performing autocorrelations to estimate the average spatial length scale of the magnetic texture (see Supporting Information). Interestingly, this analysis reveals that the average texture size remains constant (∼200 nm) over a broad temperature range from 298 to 400 K [see Figure 2(d)]. We note that this magnetic texture evolution contrasts with first-order transitions driven by external fields where domains gradually grow as longrange magnetic order is established with defects and local disorder serving as nucleation sites.

The gradual onset of magnetization, coupled with the observed texture evolution, is characteristic of a second-order phase transition, where power law scaling behavior is anticipated. To evaluate this scaling in the measured magnetic domain structure, we extract the radius of gyration (R g ), a parameter that represents the average distance between two points within a given magnetic cluster. Near the phase transition, NV magnetometry scans of the ordered phase (T < T c ) were performed with both large (15 μm) and small (2 μm) fields of view. The bilogarithmic plot of R g against cluster area (A) shown in Figure 3 reveals a power law scaling behavior spanning two and half decades, with the exponent d v = 1.73 ± 0.04. Similarly, a robust power-law scaling is observed in the cluster perimeter (P) as a function of R g , with the exponent d h = 1.34 ± 0.04. Critical phenomena are governed by the concept of universality, where critical exponents depend

Letter on dimensionality and symmetry rather than material-specific details. In our result where SFRO is probed near the critical point (T c ), these critical exponents suggest a three-dimensional universality class for the spin system (see Supporting Information). Characterizing critical exponents is crucial and our results highlight the power of NV magnetometry in understanding critical phenomena in second-order magnetic phase transitions.

On the other hand, magnetic fluctuations, which reflect dynamical deviations from spin equilibrium, can provide a complementary understanding of critical phenomena near continuous phase transitions. Measurements of the NV spin relaxation lifetime T 1 and coherence time T 2 have increasingly emerged as sensitive quantum probes of magnetic noise in an extensive frequency range. 13,[27][28][29] In particular, the change in T 1 relative to the intrinsic T 1 for an NV center interacting with a fluctuating spin system via dipole-dipole interactions is proportional to the dynamical spin structure factor at the intrinsic NV frequency corresponding to the level splitting.

In the all-optical T 1 relaxometry sensing protocol used here, the NV spin state is initially polarized to m s = 0 by an optical pulse. The relaxation process is then monitored by a second optical pulse reading out the luminescence intensity as a function of delay time, as depicted in Figure 4(a). The initialization and read-out laser pulses each have a duration of 5 μs, while the counting windows for both the reference and signal are 300 ns. A similar NV T 1 relaxometry technique has recently been demonstrated to capture the increased magnetic noise near a magnetic phase transition in 2D α-Fe 2 O 3 . 30 In that context, the increase in magnetic fluctuations inferred from the T 1 relaxation is relevant to the in-plane/out-of-plane switching of antiferromagnetic ordering. To study magnetic ordering and the FM-PM continuous phase transition, we first compare T 1 relaxation times for the NV tip brought into proximity with Sr 2 FeReO 6 and mica (respectively) for two reasons. First, the spin relaxation time of NV centers can vary under different tip conditions, as NV centers are known to be sensitive to surface condition and nearby impurities. 27,31 Second, in diamagnetic systems like mica, the magnetic dipole-dipole interaction is absent due to the diminished net magnetic moment of paired electrons. Therefore, the NV T 1 on mica serves as a reference for the intrinsic temperature-dependent NV spin dynamics. Representative T 1 measurements acquired at 300 K on mica and Sr 2 FeReO 6 are shown in Figure 4(b), and the temperaturedependent spin lifetime is shown in Figure 4(c).

Below T c , the NV center T 1 is suppressed for ferromagnetic Sr 2 FeReO 6 compared to the baseline temperature-dependent NV T 1 acquired on mica as a result of magnetic noise generated by the ferromagnetic spin bath. According to the fluctuation-dissipation theorem, the magnetic noise of a system is related to the imaginary part of the dynamical magnetic susceptibility. 23,30,32 At temperatures approaching T c = 405 K, Figure 4(c) highlights the additional increase in the NV center spin relaxation rate compared with the intrinsic spin dynamics measured on the mica substrate, which can be understood as a direct result of the magnetic susceptibility diverging near a second order phase transition. 24,25,33 Near T c , the system is in a state of critical instability where the spins become increasingly correlated over large distances. 26,34 This increased spin-spin correlation leads to enhanced fluctuations as the spins spontaneously align with perturbing fields. NV relaxometry, given its susceptibility to magnetic fluctuations, is therefore a powerful tool for the characterization of continuous magnetic phase transitions.

To provide a more quantitative understanding, we apply a mean-field Landau-Ginzburg theory, which effectively captures the magnetic fluctuations near T c . This theory incorporates the interaction between a NV center and a spin bath based on different spin models. As described in greater detail in the Supporting Information, we treat Sr 2 FeReO 6 as an infinite slab of thickness D with magnetic fluctuations measured by a NV center at distance d above its surface. For the Ising universality class, the NV relaxation rate takes the general form where τ = T -T c is the relative temperature, ω 0 is the intrinsic frequency of the NV center, γ is a damping coefficient, and α is an unknown Landau parameter. In the XY universality class, the same relaxation rate becomes

T T d d D 1 1 1 ( ) 7 2 sin max( , 2 ) 7 2 cos max( , 0) 1 3 3 2 2 2 2 0 2 2 2 2 2 0 2

where θ is an in-plane rotation angle between the NV direction and the given ferromagnetic domain. (Note that the relaxation rate in eq 2 becomes independent of θ in the paramagnetic phase, τ > 0, as expected.) Unlike in the Ising universality class, the NV relaxation rate in the XY universality class remains finite in the ferromagnetic phase even for large relative temperatures, |τ| ≫ γω 0 /α, which is consistent with the measured temperature-dependent spin dynamics plotted in Figure 4(d). Indeed, while the height-dependent spin relaxation rate plotted in the inset of Figure 4(c) is consistent with both Ising and XY universality classes due to the same

+ scaling, the temperature-dependent behavior in the ferromagnetic phase [see Figure 4(d)] points to the XY universality class.

To further examine the magnetic texture dependent T 1 , we conducted an iso-T 1 scan where the decay in intensity is monitored at a given delay time as a function of NV center position (see Supporting Information). The T 1 time for an NV center interacting with local magnetic textures can be estimated assuming a single exponential decay behavior across the scanning area. One might naively expect T 1 to vary with the spatially varying magnetic texture as a result of reduced crossrelaxation 35,36 as the magnetic field lifts the spin degeneracy. However, no statistically significant correspondence was found between T 1 and the mapped magnetic texture. This discrepancy may arise from the complex magnetic structure of Sr 2 FeReO 6 , which exhibits magnetic anisotropy between inplane and out-of-plane directions, as indicated by bulk magnetization measurements (see Supporting Information). This complexity hinders a straightforward correlation between stray field profiles and magnetic domain morphology, thus complicating the comparison of spin noise contributions from domain walls and domains. Spatially resolved T 1 measurements were also examined at elevated temperatures, including T ∼ T c , where the iso-T 1 maps showed no apparent spatial dependence on weak magnetic textures.

In conclusion, we have studied the continuous phase transition of the high-T c ferromagnetic oxide Sr 2 FeReO 6 using scanning NV magnetometry and relaxometry, which provide quantitative, minimally perturbative probes of spin ordering and spin dynamics. The power-law scaling behavior observed in the magnetic texture is a powerful way of understanding the phase transition and suggests that the critical point is in a 3D universality class. With the magnetic fluctuations complementarily sensed by NV relaxometry across the phase transition, we can then argue that the ferromagnetic Sr 2 FeReO 6 thin film can be modeled as a 3D XY spin system on length scales comparable to the film thickness. We note that a dimensional crossover might happen if the film thickness is further reduced to the length scale of the spin-spin interactions. The nature of the phase transition in this scenario would pose an interesting question as there is no long-range magnetic order for spin systems with continuous symmetries in the 2D limit.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.4c05401.

Image analysis, magnetization analysis, cluster analysis, modeling of magnetic fluctuations, spatially resolved spin-lifetime measurements, and universality class analysis. (PDF)

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AUTHOR INFORMATION Corresponding Author Benjamin J. Lawrie -Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, United States; orcid.org/0000-0003-1431-066X; Email: lawriebj@ornl.gov