Fe3GeTe2 Magnets Research

We investigate the critical fluctuations in Fe3GeTe2 (hereafter abbreviated as FGT), a van der Waals magnet exhibiting a paramagnetic-ferromagnetic phase transition. FGT is an exfoliable magnet that retains robust ferromagnetism with strong out-of-plane anisotropy even at the monolayer limit9,21,22. For bulk FGT, the Tc is approximately 210 K8,23. When thinned down to a monolayer, Tc is significantly suppressed due to enhanced fluctuations24, accompanied by a transition in magnetic properties from 3D to 2D Ising ferromagnetism22,23,25,26.

The investigation of critical fluctuations near the critical point constitutes the primary focus of this work. To achieve this, we developed a cryogenic wide-field microscopy system utilizing ensemble NV centers27, with the experimental configuration depicted in Fig. 1a. The system employs a linearly polarized 532 nm laser to initialize and read out the state of NV centers, while microwave radiation delivered through an antenna enables quantum state manipulation. An in-plane magnetic field was applied to lift the degeneracy of the \(\left\vert \pm 1\right\rangle\) of NV centers and to avoid perturbing the intrinsic anisotropy of the FGT sample. Hexagonal boron nitride (hBN) substrates were transferred above and below the FGT flakes to prevent oxidation and to adjust the distance d between the FGT and NV centers, respectively. More details about the setup are presented in Supplementary Section 1.

NV centers, serving as quantum sensors, have become indispensable tools in condensed matter physics, materials science, and quantum sensing28,29,30,31. The optically detected magnetic resonance technique enables static magnetic field sensing through continuous wave (CW) spectroscopy32, and dynamic decoupling protocols33,34,35,36 establish frequency domain detection channels, particularly effective for resolving varying magnetic signals through spectral engineering37,38. Notably, these methodologies have unveiled critical fluctuations in van der Waals magnets, including Fe3GeTe239, \({{{{\rm{MnBi}}}}}_{2}{{{{\rm{Te}}}}}_{4}{({{{{\rm{Bi}}}}}_{2}{{{{\rm{Te}}}}}_{3})}_{{{{\rm{n}}}}}\)40, CrPS441, CrSBr42, and twisted double trilayer CrI343, primarily through relaxometry near phase transitions. The relaxation time of NV centers is predominantly governed by noise components at frequencies resonant with their energy levels, typically centered around 2.87 GHz. The observed increase in relaxation rate reflects enhanced spin fluctuations near the Tc. Meanwhile, subHertz magnetic domains’ reverse has been observed in our previous work near the Tc44. These results demonstrate NV centers’ capability to detect spin fluctuations across different time scales. Notably, the aforementioned works and ref. 45 have merely reported the existence of critical fluctuations, without exploring their underlying mechanisms, particularly failing to establish connections with scaling theory.

In contrast to previous experiments, our work leverages the quantum coherence of NV centers to probe critical fluctuations. The details of the NV centers and FGT samples are presented in Supplementary Sections 2 and 3. The Hahn-echo pulse sequence (Fig. 1b) exhibits a narrow-band spectral weight function, selectively coupling to noise components whose frequencies match the pulse sequence periodicity46. This frequency-selective coupling modulates the coherence time of NV centers, enabling spectral mapping of magnetic fluctuations.

Multi-mode Imaging for static and fluctuating magnetic fields

Firstly, we perform wide-field imaging measurements to characterize the coherence time of NV centers adjacent to the FGT sample. When T > Tc, although the FGT is paramagnetic and lacks long-range magnetic order, local magnetic fluctuations still suppress the coherence of NV centers. The results are presented in Fig. 2a. The decoherence of NV centers was measured through Hahn-echo pulse sequences, as shown in Fig. 1b. The contrast is defined as the normalized difference between the two frames: C = (Isig − Iref)/Iref, where Isig(Iref) is the gray value of the signal (reference) frame. The decoherence curves are depicted in Fig. 1a, which are obtained by recording contrast as a function of the pulse interval τ. By fitting the decoherence profile using

$$C(t)={C}_{0}\exp (-\chi (t))={C}_{0}\exp (-{(t/{T}_{2})}^{\alpha }),$$

where C0 is a fitting parameter related to contrast; T2 is the coherence time, determined by χ(t) = 1; and α is the stretch exponent related to the properties of noise. The spatial variation of T2 is presented in the inset. The sample comprises three regions. Region ‘a’: without FGT flakes and exhibiting the longest T2; Region ‘b’: with a 70 nm-thick FGT flake and exhibiting intermediate T2; Region ‘c’: with a 90 nm-thick FGT sample and exhibiting the shortest T2. The three curves in Fig. 2a demonstrate that magnetic noise originating from the FGT affects both the T2 and the stretch exponent α. In region ‘a’, we observe α ≈ 1.5, consistent with previous reports in ref. 47. This value decreases significantly in FGT-covered regions ‘b’ and ‘c’. The stretch exponent α is a rather subtle issue, which is affected by the inhomogeneity of the sample47,48,49.

In Fig. 2b, with T < Tc, spontaneous magnetization emerges in the FGT sample. We employ CW spectroscopy (see Supplementary Section 4) to map the static magnetic field distribution. A clear spectral dip at about 2.79 GHz is observed in region ‘a’, while in regions ‘b’ and ‘c’ the stray magnetic field from the FGT broadens the spectral lines and reduces the contrast. In Supplementary Section 5, static magnetic field imaging of Sample #9 was performed at a temperature far below Tc, revealing distinct CW spectral characteristics between domain walls and interiors. Near domain walls, spectral lines are significantly broadened with nearly undetectable optical contrast, while in domain interiors, spectral lines maintain their intrinsic width. As shown in the inset of Fig. 2b, the spectral broadening observed across region ‘b’ originates from reduced magnetic domain sizes near Tc. Quasi-static noise arising from domain reversion may also contribute to this contrast reduction.

Huge critical magnetic fluctuations near the T c

Next, we investigate the coherence dynamics of NV centers near the critical regime of FGT. As shown in Fig. 3a, the T2 distribution exhibits two dominant peaks: a shorter value of approximately 3.2 μs in regions covered by FGT films and a longer value of 4.7 μs in FGT-free regions. We define the distribution peak position as the characteristic T2 with the FGT sample. Further analysis in Fig. 3b represents the characteristic T2 as a function of temperature for various samples with thicknesses ranging from 10 to 90 nm. All measured T2 decreases monotonically as temperature approaches Tc from above. Although Tc varies with thickness (see Supplementary Table S1) of various FGT sample, the decrease of T2 is consistent. The intrinsic coherence time of the NV centers (bare diamond) is approximately 4.5 μs in our diamond chips47. Thus, the decrease of T2 originates from the magnetic noise generated by critical fluctuations. Further details are provided in Supplementary Section 2.

Spin fluctuation behavior throughout the complete phase transition

In the preceding experiments, the distance d from the FGT layer to the ensemble NV centers layer is about 60 nm50 without inserting hBN flakes. As T → Tc, critical fluctuations intensify dramatically, leading to complete suppression of coherence. The statistical method illustrated in Fig. 3a fails to yield reliable results when T2 is extremely short, so the critical fluctuations at phase transitions can not be detected. We can mitigate this difficulty by inserting the hBN flakes with finite thickness between the diamond chip and the FGT sample. This configuration increases the distance d, thereby reducing the magnetic noise coupling to the NV centers. This protocol enables us to investigate thickness-dependent decoherence rates of NV centers with an individual FGT sample (see Supplementary Section 7). Preliminary estimation indicates magnetic fluctuations decay scaling as \(\langle \delta {B}^{2}\rangle \sim {d}^{-2.5}\).

Therefore, by increasing the distance d, the measured spin noise decreases, enabling the measurement of the T2 of NV centers across Tc. Figure 4a shows three decoherence curves at 190 K, 180 K, and 162 K, corresponding to temperatures above, near, and below Tc. Obviously, the shortest T2 occurs at 180 K, closest to Tc. In the ferromagnetic state (T < Tc), significant magnetic noise variations emerge within the FGT sample, as shown in the inset of Fig. 4a. The data processing strategy illustrated in Fig. 3a becomes inadequate. Instead, we integrate data from all pixels within the region of interest to extract the characteristic coherence time. For T > Tc, both data processing methods yield consistent results, as demonstrated in Supplementary Section 8. This consistency confirms the reliability of our experimental data and reflects global properties rather than localized features.

Analogous to the relaxation rate39,40,41, the decoherence rate is defined as Γ = 1/T2 − 1/T2,0, where T2,0 denotes the intrinsic coherence time of NV centers in the absence of FGT sample. The decoherence rates of NV centers measured with FGT samples #11 and #12 are presented in Fig. 4b with corresponding T2 maps provided in Supplementary Figs. S8 and S9. Despite differences in sample thickness and NV-FGT separation distances, the experimental results for both samples exhibit consistent behavior. The decoherence rate is observed to peak at the critical point, coinciding with the strongest fluctuations. This finding aligns with theoretical predictions from ref. 15. Remarkably, even at NV-FGT separations of hundreds of nanometers, magnetic fluctuations in the MHz frequency range exhibit intensities of hundreds of kHz-far exceeding those reported in prior relaxation-based measurements. This contrast in noise intensity between MHz and GHz frequencies directly reflects the spectral structure of critical fluctuations.

Theoretical model based on scaling theory

Building on these observations, we can now elucidate the mechanisms underlying experimental results. The two questions posed in the Introduction can be resolved. The effective Hamiltonian of the NV centers should be written as18

$$H={\Delta }_{0}\left({\hat{S}}_{{{{\rm{z}}}}}^{2}-\frac{1}{3}\right)+\gamma \left({B}_{0}+\delta B\right){\hat{S}}_{{{{\rm{z}}}}},$$

where Δ0 = 2.87 GHz is the zero-field splitting level, \({\hat{S}}_{{{{\rm{z}}}}}\) is the Pauli operator, B0 is the bias field, and δB is the fluctuation field generated by FGT flakes. The coefficient γ = 28 MHz mT−1 is the electron gyromagnetic ratio. This noise δB induces a decoherence process determined by \(\langle \exp (i\phi (t))\rangle=\exp (-\langle \phi {(t)}^{2}\rangle /2)\), where \(\phi (t)=\gamma \int_{0}^{t}\delta B({t}^{{\prime} })d{t}^{{\prime} }\) is the accumulated phase. According to the fluctuation-dissipation theorem51,52, \(\langle \delta B({t}_{1})\delta B({t}_{2})\rangle=\frac{1}{2\pi }\int\,d\omega \exp (-i\omega ({t}_{1}-{t}_{2}))S(\omega )\), where S(ω) is the power spectral density. Then we have (see Eq. (1))

$$\chi (t) \sim \frac{1}{2}\langle {\phi }^{2}(t)\rangle=\frac{1}{4\pi }\int\,d\omega S(\omega ){W}_{t}(\omega ),$$