Revealing intrinsic domains and fluctuations of moiré magnetism by a wide-field quantum microscope

Moiré magnetism featured by stacking engineered atomic registry and lattice interactions has recently emerged as an appealing quantum state of matter at the forefront of condensed matter physics research. Nanoscale imaging of moiré magnets is highly desirable and serves as a prerequisite to investigate a broad range of intriguing physics underlying the interplay between topology, electronic correlations, and unconventional nanomagnetism. Here we report spin defect-based wide-field imaging of magnetic domains and spin fluctuations in twisted double trilayer (tDT) chromium triiodide CrI3. We explicitly show that intrinsic moiré domains of opposite magnetizations appear over arrays of moiré supercells in low-twist-angle tDT CrI3. In contrast, spin fluctuations measured in tDT CrI3 manifest little spatial variations on the same mesoscopic length scale due to the dominant driving force of intralayer exchange interaction. Our results enrich the current understanding of exotic magnetic phases sustained by moiré magnetism and highlight the opportunities provided by quantum spin sensors in probing microscopic spin related phenomena on two-dimensional flatland.


Supplementary Note 1. Transmission electron microscopy measurements of a low-twistangle tDT CrI 3 device
We utilize diffraction-based transmission electron microscopy (TEM) measurements to characterize the microscopic lattice structure of low-twist-angle twisted-double-trilayer (tDT) CrI3 encapsulated within hexagonal boron nitride nanoflakes.Thermo Fisher Talos, operated at 200kV and equipped with Gatan OneView camera, was employed for selected area electron diffraction (SAED) and dark-field (DF) TEM imaging.Supplementary Figure 1a shows third-order Bragg peaks ({33 00}) from SAED pattern for a 0.3° tDT CrI3 device.The local mean twist angle over a 860 nm × 860 nm sample area was measured to be 0.4° ± 0.1° by fitting 2D gaussians to the Bragg peaks, giving a moiré period of ~100 nm.Note that the experimentally measured twist angle shows a small deviation (~0.1°) from the targeted value, within the accuracy of the stacking processes.The composite DF-TEM real space image from an ordered region of the specimen (Supplementary Fig. 1b)-obtained by averaging DF-TEM images from three third-order Bragg peaks 120° apart-features the superlattice structures with a periodicity commensurate with the moiré wavelength 1 .The observed distortion from the expected hexagonal lattice pattern is common for low-twist-angle tDT CrI3 samples, which could be induced by lattice strain, relaxation, and local structural inhomogeneity 2 .

Supplementary Note 2. Nitrogen-vacancy measurements of magnetic stray fields
A nitrogen-vacancy (NV) center is a point defect consisting of a substitutional nitrogen atom next to a carbon atom vacancy in a diamond crystal lattice 3,4 .The negatively charged NV center hosts an S = 1 electron spin that serves as a three-level quantum system 3,4 .Supplementary Fig. 2a shows the energy level diagrams for a single NV spin as a function of the magnetic field BNV applied along the NV-axis.At BNV = 0, the ms = ±1 spin states are degenerate at an electron spin resonance (ESR) frequency of 2.87 GHz.As BNV increases, the Zeeman coupling introduces an energy gap 2 BNV between the ms = −1 and ms = +1 NV spin states, which can be optically detected by measuring the spin-dependent NV photoluminescence (PL).shown above denotes the gyromagnetic ratio of an NV spin.
The top panel of Supplementary Fig. 2b shows the optical and microwave sequence for pulsed ESR measurements.NV centers are first initialized to the ms = 0 spin state by a microsecond-long green laser pulse followed by the application of a nanosecond-long microwave π-pulse to induce NV spin transitions.We sweep the frequency f of the microwave pulses and measure the NV fluorescence by a charge-coupled device (CCD) camera during the application of the green laser readout pulse.When f matches the NV ESR frequencies, NV spins will be excited to the ms = ±1 states, which are more likely to relax through a non-radiative pathway back to the ms = 0 ground state and emit reduced PL 4 .The bottom panel of Supplementary Fig. 2b shows a representative NV ESR spectrum recorded at a single camera pixel over the CrI3 sample at 10 K.In the current study, NV ensembles in the diamond substrates show four possible spin orientations with a mirror symmetry axis along the out-of-plane direction (z-axis) 5 .The angle between individual NV orientations and the z-axis is 54° (126°), and the applied external magnetic field Bext is perpendicular to the sample plane.Considering the perpendicular anisotropy of CrI3, the measured NV ESR spectrum shows four-fold degeneracy with only one pair of split NV spin energy levels, from which the out-of-plane magnetic field Bz can be quantitatively obtained as follows 6 : where f± denote the NV ESR frequencies corresponding to the ms = 0 ↔ ±1 spin transitions.After subtracting the contribution of the external magnetic field Bext, the out-of-plane component of stay field Bs emanating from CrI3 samples can be obtained.By measuring the field-induced Zeeman splitting at every pixel of the captured image, we can have the Bs map over the entire field of view as shown in Figs.2d-2g in the main text.In this work, we varied the measurement temperature in a broad range to reveal the stray field Bs response as shown in Supplementary Figs.2c-2f.

Supplementary Note 3. Reconstruction of CrI 3 magnetization patterns
In this section, we present the detailed method to reconstruct magnetization patterns of CrI3 samples from the measured stray field maps.Generally, the stray field distribution Bs(R) of a magnetic system is determined by its static magnetization distribution M(R') as follows 7 : where is the magnetostatic Green's function tensor between coordinates R = (x, y, z) and R' = (x', y', z'), and Ω represents the CrI3 sample space.Considering the atomically thin CrI3 flakes used in the current study, we assume a uniform magnetization distribution along the thickness direction of the sample: M(R') = M(x', y').Also, we introduce the , where t M is the thickness of individual atomic layers of CrI3: where . Note that we have neglected the variation of Green's function tensor along the thickness direction.Focusing on the magnetic field at the position of the NV centers (z = d), we take the Fourier transform in the x and y directions, where translational symmetries are present with r = (x, y) and k = (kx, ky): Similarly, we can obtain the Green's function tensor in the Fourier space for , , , Due to the perpendicular anisotropy, CrI3 magnetization is spontaneously aligned along the z-axis Here, we focus on the out-of-plane component of the stray field generated by the magnetic sample: Lastly, we introduce an inverse Fourier transform on Eq. ( 7) to reconstruct the CrI3 magnetization patterns in the real space: Using the method presented above, we successfully reconstructed the magnetization patterns of tDT CrI3 device A at different measurement temperatures as shown in Supplementary Figs.3a-3h.

Supplementary Note 4. Theoretical simulations of stray fields generated by magnetic domains in tDT CrI 3
In this section, we qualitatively simulate the stray field patterns arising from the stacking induced (ferro)magnetic domains formed over arrays of moiré supercells in the ground state of low-twist-angle tDT CrI3.Using the theoretical model proposed in Ref. 8, Supplementary Fig. 4a shows the characteristic hexagonal magnetic lattice structure of two moiré unit cells of tDT CrI3 with opposite ferromagnetic order.The lattice constant of a moiré unit cell is estimated to be ~100 nm based on the targeted small twist angle of 0.3°.The white triangular region denotes the monoclinic (M) interlayer stacking geometry with antiferromagnetic (AFM) order while the remaining areas correspond to the rhombohedral (R) stacking geometry characterized by the ferromagnetic (FM) state 8 .Due to the prominent spatial magnetic "inhomogeneity" and suppressed inter-domain correlations, degeneracy of the ferromagnetic ground states could be lifted by external stimuli such as magnetic field, thermal cycles, and local defects, resulting in stackinginduced (ferro)magnetic domains formed over multiple moiré wavelength at a mesoscopic length scale.In the current simulations, we propose that individual moiré supercells with positive (red) and negative (blue) ferromagnetic order are randomly distributed over a lateral sample area of ~2 μm × 2 μm, as illustrated in Supplementary Fig. 4b.Using Eqs. ( 2)- (7), we are able to simulate the out-of-plane stray field Bs map on a plane with vertical distance d below the proposed magnetic pattern.Considering a diffraction limit of ~0.5 μm for wide-field measurements, we have applied a gaussian filter function to the simulation results to match our measurement resolution as presented in Supplementary Fig. 4c.One can see that the simulated stray field Bs map shows a spatially dependent distribution consisting of individual domains with opposite polarity on a micrometer length scale, in qualitative agreement with our experimental results.
Here, we would like to highlight that the "broad" field of view (~20 μm × 20 μm) enabled by our NV wide field techniques provides an excellent opportunity for investigating the extended moiré domain phases across multiple moiré periods in tDT CrI3, which has not been revealed in the previous study 8 .It is also instructive to note that our measurements were mostly performed under a small external magnetic field (tens of Gausses), which is favorable to the formation of spatially varying domain states in tDT CrI3 especially in the magnetic phase transition regime.

Supplementary Note 5. NV measurements on a large-twist-angle tDT CrI 3 device
In this section, we show NV magnetometry results measured on tDT CrI3 device B with a large twist angle of ~15° and reduced moiré wavelength of ~ 2.6 nm.Supplementary Fig. 5a shows an optical image of the prepared tDT CrI3 device on a diamond substrate.Supplementary Fig. 5b shows the obtained magnetic stray field map measured at a temperature of 5 K with a perpendicular magnetic field Bext of 71 G.In contrast with the multidomain feature observed in the low-twistangle (~0.3°) tDT CrI3 device A, device B shows a largely uniform magnetic domain over the twisted area, suggesting a pure magnetic order in the ground state.The reconstructed magnetization map is presented in Supplementary Fig. 5c.As reported in the previous studies, it is energetically unfavorable for large-twist-angle tDT CrI3 to host coexisting antiferromagnetic and ferromagnetic orders 9 .Thus, interfacial ferromagnetic interaction is established between the two stacking CrI3 trilayers, resulting in two layers of uncompensated magnetization.It is worth noting that the reconstructed magnetization in the twisted area shown in Supplementary Fig. 5c is approximately twice of that measured for pristine CrI3 trilayer, in consistent with the picture discussed above.

Supplementary Note 6. NV relaxometry measurements of tDT CrI 3
In this work, we use NV relaxometry to detect the longitudinal spin noise arising from tDT CrI3 samples and to investigate their spin fluctuations as a function of temperature and frequency (discussed in the following section).The top panel of Supplementary Fig. 6a shows the measurement protocol for NV relaxometry.NV centers are first initialized to the ms = 0 spin state with a green laser pulse.Spin fluctuations in tDT CrI3 will generate fluctuating magnetic fields, which could couple to proximate NV centers via dipole-dipole interactions [10][11][12] .The fluctuating magnetic fields at the NV ESR frequencies will induce ms = 0 ↔ ±1 spin transitions, leading to an enhancement of NV spin relaxation rate Γ at the corresponding NV ESR frequencies.After a delay time t, a microwave π-pulse at the NV ESR frequency is applied and the corresponding NV spin state occupations are readout by a second green laser pulse.NV spin relaxation rate Γ can be quantitatively obtained by fitting the integrated NV PL with an exponential decay function PL(t) = A0 + A , where A0 and A are constants 11 .Note that Γ includes two components Γ and Γ corresponding to the intrinsic and magnetic fluctuation induced NV relaxations, respectively.The intrinsic component Γ can be measured on NV centers in a diamond area not being covered by the CrI3 device as illustrated in Supplementary Fig. 6a.Γ can be obtained by performing NV relaxation measurements on NV centers underneath the CrI3 device (Supplementary Fig. 6b).Supplementary Fig. 6c shows a set of representative NV spin relaxation spectra for the ms = 0 ↔ −1 spin transition measured on a single camera pixel located on the bare diamond substrate.The external magnetic field is 71 G applied along the out-of-plane direction and the NV ESR frequency is 2.78 GHz.The intrinsic NV spin relaxation rate Γ is obtained to be 4.0 kHz, 3.8 kHz, 4.1 kHz, and 4.2 kHz at 36 K, 47 K, 57 K, and 92 K, respectively.Supplementary Fig. 6d shows NV spin relaxation spectra measured on a single camera pixel over the tDT CrI3 sample area at the same ESR frequency and temperature.Subtracting the intrinsic contribution Γ , spin fluctuation induced NV relaxation Γ = Γ − Γ can be quantiatively obtained.By performing wide-field imaging measurements over the entire sample area, Figs.4a − 4g in the manuscript present a series of Γ maps of the tDT CrI3 device A measured at different temperatures.

Supplementary Note 7. Inferring intrinsic spin diffusion constant and longitudinal magnetic susceptibility of tDT CrI 3
In this section, we provide the details for extracting the intrinsic spin diffusion constant and static longitudinal magnetic susceptibility of tDT CrI3 from NV relaxometry results.Supplementary Fig. 7a shows the optical image of a low-twist-angle (~0.3°) tDT CrI3 device C transferred onto a [111] oriented single crystalline diamond membrane for relevant measurements.NV ensemble is implanted on the surface of the [111] diamond for wide-field magnetometry measurements 13 .Due to the strong perpendicular magnetic anisotropy, the magnon band gap of CrI3 is larger than the NV ESR frequency under the field conditions of our measurements, thus, the measured NV spin relaxation rate M ESR ( ) f  is mainly driven by the longitudinal spin fluctuations in CrI3, which can be described using a two-magnon noise model 10,12,14 : where , k B is the Boltzmann constant, d is vertical distance between NV centers and the twisted CrI3 sample, tM is the thickness of the magnetic device, T is the temperature, and is the imaginary part of the dynamical longitudinal spin susceptibility.The geometric factor ( ) G  can be expressed as: where  and  are the gyromagnetic ratio of NV centers and CrI3, respectively, and θ = 0° is the angle between the NV spin orientation and the magnetic easy axis of CrI3.Note that the effective NV axis is along the out-of-plane direction for [111] oriented diamond, allowing for application of an external magnetic field in a relatively large range.To obtain the expression of dynamical longitudinal spin susceptibility ESR ''( , ) k f  , we describe the magnetic system studied by a diffusion equation.Assuming U(1) symmetry, the diffusion equation for spins oriented along the direction of the magnetic order parameter s z can be written as 12,15 : Here, we have introduced the spin-relaxation time τs and the spin current s      j , where σ is the spin conductivity, is the spin chemical potential, is the static uniform longitudinal spin susceptibility, and H is an external perturbation thermodynamically conjugate to the spin density.By introducing the intrinsic spin diffusion coefficient / D    , we have: By introducing the Fourier transform     , , Thus, the imaginary part of the dynamical longitudinal spin susceptibility can be written as: Combining Eq. ( 9) and Eq. ( 14), and considering the limit of a long relaxation time ( s    ), the spin relaxation rate of an NV center driven by longitudinal spin fluctuations can be expressed as: Note that we have introduced the magnetic longitudinal susceptibility = in order to express the susceptibility in the conventional unit of emu•cm -3 •Oe -1 .Supplementary Fig. 7b shows NV spin relaxation rate Γ measured on the tDT CrI3 device C as a function of the ESR frequency fESR at 40 K. Here, fESR is systematically changed by varying the magnitude of the external perpendicular magnetic field.By fitting the results using Eq. ( 15), the intrinsic spin diffusion constant D and static magnetic longitudinal susceptibility of the tDT CrI3 sample are obtained to be (4.2 ± 0.3) ×10 -5 m 2 /s and (4.0 ± 0.2) ×10 -2 emu•cm -3 •Oe -1 at 40 K, respectively.magnitude of the stray field emanating from the magnetic device is found to decrease as increasing temperature and eventually reaches zero above the Curie temperature.Supplementary Fig. 9f tracks the temperature dependence of the stray field Bs measured at two local points with opposite domain polarity.Below the magnetic transition temperature, we observed a monotonic decrease of the magnitude of the measured stray field due to the reducing CrI3 magnetization.Notably, Bs decays to zero around the Curie temperature, indicating demagnetization of the sample.Supplementary Figs.10a-10e show the temperature dependence of the NV relaxation rate Γ maps measured for the tDT CrI3 sample D from 21 K to 92 K.The external magnetic field of 71 G is applied along the out-of-plane direction of the sample.The measured relaxation rate maps exhibit a high spatial uniformity over the tDT CrI3 area, consistent with the results presented in the main text.The measured NV spin relaxation rate Γ shows a suppressed value in the low temperature regime (Supplementary Fig. 10a).As the temperature increases, Γ reaches a maximum value around the magnetic transition temperature due to an increase of the magnetic susceptibility (Supplementary Fig. 10d).Supplementary Fig. 10f plots the temperature dependence of Γ measured in the tDT CrI3 sample area, showing a distinct peak value of 9.8 kHz around ~52 K.