Introduction

The recent discovery of long-range ferromagnetic ordering in bilayer Cr2Ge2Te6 (CGT) and monolayer CrI3 has opened the fascinating new area of two-dimensional (2D) magnetic materials1,2,3. These materials often have different magnetic properties from those of their bulk counterparts and these properties can be strongly dependent on the number of layers. For instance, although bulk CrI3 is ferromagnetic, it is antiferromagnetic in the few-layer limit4,5,6. Furthermore, the inter-layer exchange interaction can lead to antiferromagnetic or ferromagnetic behavior depending on whether the layers are stacked in a monoclinic or rhombohedral manner7,8,9. Indeed, the possibility of engineering van der Waals (vdW) heterostructures from these systems—atomic layer by atomic layer—affords unprecedented control over their magnetism. Moreover, unlike bulk magnets, 2D magnets are highly susceptible to external stimuli, including electric fields, magnetic fields, and strain10. For all of these reasons, 2D magnets have potential applications ranging from high-density data storage and efficient information processing to sensing.

Among these materials, the ferromagnetic semiconductor CGT is especially promising for spintronic and memory storage applications. Both current11 and laser12 induced magnetization switching has been demonstrated in this material. It exhibits spin-orbit torque switching under very low current densities, making it promising for low-power memory devices13. In addition, CGT is a candidate for pressure-sensitive spintronic applications, because of its strong anisotropic spin-lattice coupling14. Although CGT has a low Curie temperature Tc = 68 K in the bulk, which drops to 30 K in the bilayer limit1, several methods have succeeded at increasing Tc, including via doping15,16 and strain17,18. In particular, the application of 2.3% strain has been shown to raise Tc up to room temperature.

Because of these promising properties, its magnetic behavior and—specifically—the dependence of this behavior on the number of layers needs to be understood. Since the first magnetic imaging of CGT, studies have been carried out on samples of different thicknesses19,20,21,22,23. Few-layer CGT with thicknesses between 2 and 5 nm shows soft ferromagnetic behavior with an out-of-plane magnetic easy-axis and no magnetic domains1. In contrast, hysteretic behavior with multi-domain magnetic textures is observed in CGT flakes with thicknesses in the tens of nanometers19,20,21. For samples thicker than about 100 nm, the presence of Bloch-type magnetic stripe domains and skyrmionic bubbles has been reported23. Despite these observations, the evolution of magnetic order with thickness, i.e., the transition from single-domain soft ferromagnetic behavior in few-layers to multi-domain structure in the thicker limit has not been investigated. Also, the type of magnetization textures forming the domains observed in the intermediate thickness range remains unclear.

In this work, we investigate the dependence of CGT’s magnetic behavior on thickness down to the few-layer limit. We image the material’s stray magnetic field, determine the corresponding magnetization configuration, and map its evolution as a function of applied magnetic field. Nanometer-scale magnetic imaging is carried out via a superconducting quantum interference device (SQUID) integrated on a cantilever scanning probe at 4.2 K24. The SQUID-on-lever’s (SOL) ability to simultaneously image both the sample’s topography and stray field allows us to correlate material thickness with magnetic configuration. Furthermore, a comparison of the measured stray-field maps with micromagnetic simulations sheds light on the form of the underlying magnetization configurations and the magnetic interactions which produce them.

Results and discussion

Magnetization per layer

CGT flakes are mechanically exfoliated on a Si/SiO2 substrate. Because CGT is prone to degrade in ambient conditions1, flakes are covered with 10-nm-thick hexagonal boron nitride (hBN) immediately after exfoliation. Figure 1a shows an optical image of a flake with regions of various thicknesses. These regions can be distinguished by their different optical contrast. Their thickness is measured via atomic force microscopy (AFM) (see Supplementary Note 1 and Supplementary Fig. 1), acquired by the SOL scanning probe in non-contact mode. In Fig. 1b, AFM cross-sections taken across the boundaries between regions of different thicknesses show steps in integer multiples of 0.7 nm, which is the thickness of a single atomic layer. The number of layers in each region is then determined using this thickness per layer and a thickness of 1.1 nm for the first layer1. The estimated thickness of the flake ranges from 2 to 16 layers with integer layer steps between adjacent areas. The availability of different thicknesses on the same flake allows us to study the evolution of its magnetic behavior as a function of the number of layers.

Fig. 1: Magnetization per layer.
figure 1

a Optical image of the CGT flake. Regions of different thickness can be identified from the optical contrast and are indicated with black, yellow, red, orange, aquamarine, blue, green, and purple circles for thicknesses of 2, 3, 4, 6, 8, 10, 14, and 16 layers, respectively. b AFM scans between regions of uniform thickness showing steps of integer layer thickness along the correspondingly colored dashed lines in (a). c Out-of-plane component of the stray magnetic field Bz(x, y) measured above the sample in an applied out-of-plane field, μ0Hz = −133 mT. d Numerical derivative along the x-axis, \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{dx}}}}}}}}}\) of the region indicated by the black dotted square in (c). e Out-of-plane magnetization Mz(x, y) calculated from (c). f \({M}_{z}^{{{{{{{{\rm{avg}}}}}}}}}\) plotted as a function of the number of layers and its corresponding linear fit shown with the red line.

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The magnetic properties of a CGT flake are investigated by imaging the out-of-plane component of the sample’s magnetic stray field, Bz(x, y), in a plane above the sample. Initially, we apply an out-of-plane magnetic field μ0Hz = − 133 mT in order to saturate the magnetization of the flake. The resulting Bz(x, y), shown in Fig. 1c, is peaked at the edges of the flake or at boundaries between regions of different thicknesses, as expected for a sample uniformly magnetized in the out-of-plane direction. The thinnest regions on the bottom-left corner of the image (indicated with black and yellow dots) are 2 and 3 layers thick and show very weak magnetic signatures, visible in \(\frac{d{B}_{z}(x,y)}{dx}\), shown in Fig. 1d.

From Bz(x, y), shown in Fig. 1c, we then determine the sample’s out-of-plane magnetization Mz(x, y), shown in Fig. 1e. By assuming that the magnetization is fully saturated along the z-direction and confined to a 2D plane (the sample is much thinner than the probe-sample distance), we use a reverse propagation method to solve for \({{{{{{{\bf{M}}}}}}}}={{\mathbb{A}}}^{-1}{{{{{{{\bf{B}}}}}}}}\) where \({\mathbb{A}}\) is the transfer matrix25. Figure 1f shows the reconstructed magnetization averaged over areas of uniform thickness, \({M}_{z}^{{{{{{{{\rm{avg}}}}}}}}}\), as a function of the number of layers. In particular, the magnetization of each area increases linearly with the number of layers. From the slope of the linear fit, we obtain a magnetization per layer \({M}_{z}^{{{{{{{{\rm{layer}}}}}}}}}=(10.9\pm 0.8)\,{\mu }_{{{{{{{{\rm{B}}}}}}}}}\,{{{{{{{{\rm{nm}}}}}}}}}^{-2}\), equivalent to a saturation magnetization Msat = (2.2 ± 0.2) μB/Cr atom, which is consistent with previous reports26,27,28. From the non-zero horizontal intercept of the fit, we find that the flake contains n0 = 1.2 ± 0.6 magnetically inactive layers. This effect is likely due to the degradation of the outer surfaces of the flake during the short exposure to air before encapsulation. Hereafter, the number of layers of CGT refers to the number of magnetically active layers, which are roughly one less than the number of physical layers. By considering the geometry of the sample and the number of magnetically active layers in each region, we calculate Bz(x, y) for a saturated flake with Msat = 2.2 μB/Cr. As discussed in Supplementary Note 4 and shown in Supplementary Fig. 6, this calculation matches the measured Bz(x, y) of Fig. 1c.

Thickness-dependent magnetic hysteresis

To investigate the process of magnetic reversal and its dependence on thickness, we map Bz(x, y) as a function of applied magnetic field. Figure 2a–i shows the evolution of the flake’s magnetization reversal with respect to out-of-plane applied field Hz. Starting in the saturated state, initialized at μ0Hz = −133 mT, the applied field is gradually stepped toward zero, into reverse field, and past saturation to 140 mT. Until zero applied field, the magnetization in the thicker regions (10–15 layers) remains almost unchanged, whereas, in the thinner parts of the flake (5–7 layers), domains start to form around μ0Hz = −10 mT (more apparent in Supplementary Fig. 4, discussed in Supplementary Note 3). Upon reversing the direction of Hz, magnetic domains also nucleate in the thicker regions of the flake. With increasing reverse field Hz, these domains spread over the whole flake and result in a complete reversal of the magnetization by μ0Hz ~ 40 mT. Inverting this procedure results in a symmetric reversal process (see Supplementary Figs. 3 and 4).

Fig. 2: Visualizing the out-of-plane magnetic hysteresis.
figure 2

a–i Bz(x, y) measured above the sample at different μ0Hz as indicated by the value shown in each image. μ0Hz is slowly stepped from −133 mT to 140 mT. Blue and red dashed lines highlight the 7 and 15-layer-thick regions of the flake. Images for more values of μ0Hz can be found in Supplementary Fig. 3, which is discussed in Supplementary Note 3. j \({B}_{z}^{{{{{{{{\rm{avg}}}}}}}}}\) for areas of constant thickness highlighted in (a) plotted against μ0Hz.

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Figure 2j shows Bz(x, y) averaged over two different areas of constant thickness, \({B}_{z}^{{{{{{{{\rm{avg}}}}}}}}}\), as a function of Hz. These local hysteresis loops are shown for the 7 and 15-layer-thick regions indicated in Fig. 2a, though a full set can be found in Supplementary Fig. 5, which is discussed in Supplementary Note 3. Thinner regions of the sample with 7 layers or less, do not show measurable magnetic remanence within the detection limit and have magnetization curves characteristic of a soft ferromagnet. This behavior is consistent with previous observations in 6-layer flakes (2% magnetic remanence), although the measured saturation field of ~40 mT is significantly smaller than the previously reported value of ~0.6 T1. In contrast, regions of the sample thicker than about 9 layers show finite coercivity with a magnetic remanence of 70–80% of the saturation magnetization Msat. This behavior transforms for increasing thickness into bow-tie-shaped hysteresis for 13 layers or more as exemplified by the data shown for the 15-layer-thick region. Noah et al.19 also reported an open hysteresis loop with 100% magnetic remanence for thicknesses less than 10 nm (14 layers) and a bow-tie hysteresis with no magnetic remanence for thicknesses more than 15 nm (21 layers). Bow-tie-shaped hysteresis is a sign of percolating magnetic domains, magnetic vortices, or skyrmion formation during magnetic reversal29,30,31,32. Our measurements give a complete picture of the gradual transition of the magnetic hysteresis loop from soft ferromagnetic to bow-tie shape with increasing thickness, pointing to an evolution of the corresponding magnetic textures with thickness (see Supplementary Fig. 5, discussed in Supplementary Note 3).

Layer-dependent magnetization texture

Our local magnetic measurements confirm that magnetic reversal in few-layer CGT depends on thickness. In order to shed light on the magnetization configurations corresponding to measured stray field patterns, we turn to micromagnetic simulations. We use a model based on the Landau-Liftshitz-Gilbert formalism, which is based on the geometry of the flake and known material parameters (see “Methods” for details). We focus on three areas of the flake representing the behavior observed in the thick (13–15 layers), intermediate (7–9 layers), and thin regions (3–5 layers) of the sample. In our model, we use one less layer compared to the actual number of physical layers in the flake, based on our measurement of approximately one magnetically inactive layer. In Fig. 3a–i, we show \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\) measured above the sample with applied out-of-plane field μ0Hz = 7.5 mT along with corresponding simulations of \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\) and Mz(x, y). \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\propto \frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\) and is measured by oscillating the cantilever on resonance and demodulating the SQUID-on-lever response at this frequency (see “Methods”). It provides a more sensitive measure of small spatial features than Bz(x, y).

Fig. 3: Evolution of magnetic domains with thickness.
figure 3

a \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\) measured at μ0Hz = 7.5 mT over the 13 to 15-layer-thick region of the sample indicated by green and purple dots in Fig. 1a, together with the corresponding simulation of (b) \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\) and (c) Mz(x, y). d–f A similar measurement at the same Hz over the 7 to 9-layer-thick region of the sample, indicated by the aquamarine dot in Fig. 1a, together with corresponding simulations. g–i A final measurement at the same Hz over the 3 to 5-layer-thick region indicated by the red and orange dots in Fig. 1a, together with corresponding simulations. j Simulated local hysteresis for 7 and 15 layers.

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Measured \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\) maps of all three regions agree well with the simulated \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\), which reproduce the shapes, relative amplitudes, and characteristic lengths of the observed patterns. Furthermore, simulated magnetic hysteresis curves, shown in Fig. 3j, for the 7 and 15-layer-thick regions reproduce local hysteresis measurements shown in Fig. 2j. Given this agreement, we can turn to the corresponding simulated Mz(x, y) configurations as the likely magnetization textures present in those regions of the sample.

In particular, simulations of the thick region (13–15 layers), which is indicated by the green and purple dots in Fig. 1a, point to the presence of labyrinth domains, as shown in Fig. 3c. These structures are responsible for the bow-tie hysteresis behavior, which we observe in CGT flakes thicker than 13 layers, shown in Fig. 2j. A closer look at the simulations reveals that the labyrinth domains are separated by Néel-type domain walls in both surface layers, which gradually transform toward Bloch-type domain walls in the interior layers (see Supplementary Fig. 10 and Supplementary Note 5). Simulated Mz(x, y) for the intermediate region with 7 to 9 layers, as indicated by the aquamarine dot in Fig. 1a, reveals long stripe domains (Fig. 3f). Simulated Mz(x, y) for the thin region with 3 to 5 layers also shows stripe domains for 5 layers, but no domains for 3 layers. The absence of domains, in combination with soft ferromagnetic behavior in the very thin limit, is consistent with a previous report by Gong et al.1. The measured and simulated evolution of magnetic domains as a function of increasing applied field is compared in Supplementary Figs. 7–9 and discussed in Supplementary Note 5.

In our simulations, the presence of labyrinth domains in thick CGT as well as their transition to stripes, and—eventually—to regions without domains as thickness is reduced is robust to small adjustments of the material parameters and sample geometry. In fact, the strong layer dependence of the magnetization configuration is a direct result of the similar magnitudes of magnetocrystalline and shape anisotropy in the few-layer samples. As the thickness of the sample changes, the balance between magnetocrystalline anisotropy, which favors out-of-plane magnetization alignment, and shape anisotropy, which favors in-plane alignment, is altered. The fact that this model reproduces the measured layer dependence suggests that the same mechanisms are at work in the experiment.

However, in order to optimally match both simulated hysteresis curves and magnetic images with the corresponding measurements, both the magnetocrystalline anisotropy Ku and the in-plane exchange stiffness Aex must be reduced when simulating the thinner parts of the flake compared to the thicker parts (see “Methods”). The need to tune these parameters to achieve the best agreement suggests a dependence of Aex and Ku on thickness. Although the mechanisms for such an effect remain unclear, a dependence of exchange interactions, and consequently Aex, on the layer number has been predicted theoretically33. Another indication for such dependence is provided by the observation of a rapid drop in Tc as a function of decreasing thickness in CGT1.

Skyrmionic magnetization texture

Simulations of the thick part of the sample (12 to 15 layers thick) reveal labyrinth domains with similar magnetization patterns as were previously observed in flakes in the range of 100 nm or thicker23. In those experiments, it was observed that under increasing Hz, the labyrinth domains transform into bubbles that are mostly homochiral. Such bubbles are topologically identical to skyrmions and are characterized by a topological charge of ±1. Topologically non-trivial spin textures like skyrmions are usually stabilized in non-centrosymmetric magnetic systems via a Dzyaloshinskii-Moriya interaction (DMI)34, which is not present in this system. Nevertheless, the competition between uniaxial anisotropy and magnetostatic energy can give rise to a plethora of magnetic patterns including labyrinth domains, stripes, and skyrmionic bubbles35,36.

To determine whether such skyrmionic bubbles are found in few-layer-thick CGT, we field-cool the sample at μ0Hz = 10 mT. Figure 4a shows a measurement of \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\) over the thick part of the field-cooled sample after the applied field was increased to μ0Hz = 17 mT. Stray field patterns characteristic of labyrinth domains are visible over most of the 15-layer-thick region with some percolating features expanding to neighboring regions. As shown in Fig. 4b, after the field is further increased to μ0Hz = 28 mT, the patterns related to labyrinth domains shrink and transform into bubble-like features.

Fig. 4: Skyrmionic spin texture.
figure 4

a, b \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\) maps of the thick part of the sample at μ0Hz = 17 mT and 28 mT, respectively, after field-cooling in 10 mT. c–f Corresponding \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\) and magnetization Mz(x, y) simulated at μ0Hz = 26 mT and 38 mT, respectively, showing stripe domains shrink into bubbles at higher field. g The simulated magnetization configuration of the magnetic bubbles at μ0Hz = 38 mT, showing a Bloch-type skyrmionic texture in the middle layer, which gradually transforms into a Néel-type texture at the surface layers.

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In corresponding micromagnetic simulations, we mimic the field-cooling procedure by starting with an arbitrary labyrinth domain, applying μ0Hz = 10 mT, and then letting the system relax to find the minimum energy state. The resulting simulated \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\) and Mz(x, y) in Fig. 4c, e, simulated at μ0Hz = 26 mT, present features similar to those measured in the field-cooled \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\). The corresponding Mz(x, y) shows that stripe domains start from the edge of the flake, form multiple branches, and expand to the thinner regions. These domains are very similar to the domains shown in Fig. 3a–c. As the field is increased in the simulations to μ0Hz = 38 mT, the corresponding \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\) map in Fig. 4d qualitatively matches the bubble-like features in the measured \({B}_{z}^{{{{{{{{\rm{lever}}}}}}}}}(x,y)\). Mz(x, y) maps reveal that, upon the increase of Hz, the underlying domains shrink and transform into bubbles, most of which have skyrmionic magnetization texture, as shown in Fig. 4f. Based on the excellent agreement between experiment and simulation, we can conclude the most probable spin texture behind the bubble-like features shown in Fig. 4b to be skyrmionic bubbles. As was observed in our magnetization simulations of the labyrinth domains, whose helicity transforms through the thickness of the flake, the skyrmionic bubbles also show a modulating helicity from the top to the bottom layer, as schematically shown in Fig. 4g. This behavior has been previously proposed in systems with skyrmions37 and has also been observed in skyrmionic bubbles38. Such bubbles can also be obtained by zero-field-cooling this sample (see Supplementary Fig. 4).

Conclusion

We have investigated the thickness dependence of magnetic ordering in CGT down to the few-layer limit. The measured magnetic hysteresis for different numbers of layers reveals a transition from a bow-tie hysteretic behavior to a soft behavior without remanence for less than 8 layers. Comparison with micromagnetic simulations indicates that complex stray field patterns observed for CGT thicker than 5 layers result from labyrinth and stripe-like magnetization configurations and—under some conditions—from skyrmionic bubbles. These complex magnetic textures emerge from the competition between magnetocrystalline anisotropy and magnetostatic interactions. Our experimental results are reproduced by micromagnetic simulations under the assumption that the magnitude of these two energies is similar. The agreement is further optimized in the thinnest regions of the sample assuming a decreasing magnetocrystalline anisotropy and exchange stiffness with the decreasing number of layers. Although the mechanism for this dependence of the material’s magnetic properties is unclear, the decrease in exchange stiffness with decreasing number of layers has been suggested theoretically33. Nevertheless, we cannot exclude that external factors, which could also vary as a function of sample thickness, including level of oxidation or defect density, could be responsible.

Methods

Cr2Ge2Te6/hBN heterostructure fabrication

The heterostructure is fabricated in ambient conditions using a dry viscoelastic stamping method39. CGT flakes with different thicknesses are first mechanically exfoliated from a single crystal (HQ graphene) onto a SiO2/Si substrate with pre-patterned number markers. A 10-nm-thick hBN flake is exfoliated from the bulk form on the Polydimethylsiloxane (PDMS) thin film and then stacked on top of the CGT flake for passivation. To minimize degradation of the CGT surface, the stacking process is carried out below 120 °C in all steps and handled within less than 10 min20.

SQUID-on-lever

We pattern a nanometer-scale SQUID via focused-ion-beam (FIB) milling at the apex of a cantilever coated with Nb24. The top side of the cantilever is deposited with 50 nm of Nb and 20 nm of Au to create a superconducting film with enhanced resistance against ion implantation effects of subsequent Ga+-FIB milling steps. Before film deposition, FIB milling is used to create a 650-nm-wide plateau on the cantilever’s tip. After film deposition, FIB milling is used to separate two superconducting leads across the cantilever and to create a superconducting loop with a diameter of 80 nm with two constriction-type Josephson junctions on the plateau. The SQUID is characterized and operated at 4.2 K in a semi-voltage biased circuit and its current ISQ is measured by a series SQUID array amplifier (Magnicon). The effective diameter of the SQUID extracted from the interference pattern (see Supplementary Fig. 2 and Supplementary Note 2) is 270 nm. The SQUID attains a DC magnetic flux sensitivity of Sϕ = 1 μΦ0 Hz−1/2 and an AC flux sensitivity at 10 kHz of Sϕ = 0.2 μΦ0 Hz−1/2 and remains sensitive up to μ0Hz > 250 mT (see Supplementary Fig. 2).

Hybrid imaging

The SOL scanning probe is capable of simultaneously performing AFM and scanning SQUID microscopy (SSM). It operates in a custom-built scanning setup under high vacuum in a 4He cryostat. Non-contact AFM is carried out using a fiber-optic interferometer to measure the cantilever displacement and a piezo-electric actuator driven by a phase-locked loop (PLL) to resonantly excite the cantilever at f0 = 285.28 kHz to an amplitude Δz = 16 nm.

Since the current response of the SQUID-on-lever is proportional to the magnetic flux threading through it, this response provides a measure of the z-component of the local magnetic field integrated over the loop. We measure a calibration curve (ISQ vs. μ0Hz) before and after each scan, ensuring that the SQUID response is linear for a field range larger than that produced by the flake. Because of the asymmetry in the SQUID, the response is non-zero and monotonic even at μ0Hz = 0 mT, although the response is weaker than at other fields and must be calibrated by fitting with a higher order polynomial. By scanning the sample using piezoelectric actuators at a constant tip-sample spacing of 200 nm, we map Bz(x, y). We can also measure Blever ∝ dBz/dz by demodulating the SQUID-on-lever response at the cantilever oscillation frequency. Due to spectral filtering, the resulting signal contains less noise than DC measurements of Bz. The spatial resolution of the SSM is limited by the tip-sample spacing and by the 270-nm SQUID-on-lever effective diameter. Maps of Bz(x, y) and Blever(x, y) are taken using a scanning probe microscopy controller (Specs) at a scan rate of 330 nm s−1, 338 ms per pixel, and typically take several hours.

Micromagnetic simulations

We simulate the flake’s magnetization configuration using the Mumax3 software package40,41. The software utilizes the Landau-Lifshitz-Gilbert micromagnetic formalism with finite-difference discretization. To mimic the layered structure of CGT we make use of the finite difference mesh by setting the thickness of a mesh cell to the thickness of a CGT layer. The geometry, estimated from optical images, and corresponding number of layers, is reproduced for three different parts of the studied CGT flake. The structure is discretized into cells of size 3.5nm × 3.5 nm × 0.7 nm. The saturation magnetization is chosen to be Msat = 2.2 μB/Cr, based on the linear fit of the magnetization presented in Fig. 1f. In order to reproduce the observed presence of domains in the thicker and their absence in the thinner parts of the flake, we assume that the magnetocrystalline anisotropy Ku of the system is similar to its shape anisotropy Ks42,43 which can be approximated from that of an infinitely extended plate: \({K}_{{{{{{{{\rm{s}}}}}}}}}=1/2\,{\mu }_{0}{M}_{{{{{{{{\rm{sat}}}}}}}}}^{2}\approx 13,600\,\,{{\mbox{J}}}\,\,{{{\mbox{m}}}}^{-3}\)35. The intra-layer exchange stiffness Aex is estimated based on the Curie temperature Tc1,44. Both Ku and Aex were further optimized from this starting point to match our both measured hysteresis curves and magnetic images. Two values were used for the Ku = 13,700 J m−3 for the simulation of the thicker part of the sample (11–15 layers) and Ku = 12,800 J m−3 for the thinner parts (3–9 layers). This change was implemented since domains start forming at different Hz for the thinner and thicker parts. The intra-layer exchange stiffness was also modulated for the three different simulated regions with Aex = 2.5 ⋅ 10−13 J m−1, 1.8 ⋅ 10−13 J m−1 and 1.4 ⋅ 10−13 J m−1 for the thicker (12–15 layers), intermediate (7–9 layers) and thinner (3–5 layers) regions, respectively. This was again done to match measured magnetic hysteresis and respective magnetic field maps. The interlayer exchange stiffness is assumed to be 3 % of the intralayer stiffness, estimated based on the ratio of the interlayer and intralayer exchange interactions between Cr atoms45. Small adjustments on the interlayer exchange stiffness do not change the magnetic behavior of the simulated flakes.

In order to generate simulated maps of Bz(x, y) and \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\), we use the magnetization maps generated by Mumax3. We calculate Bz(x, y) at a height of 200 nm above the sample, corresponding to our SQUID-sample distance. \(\frac{{{{{{{{\rm{d}}}}}}}}{B}_{z}(x,y)}{{{{{{{{\rm{d}}}}}}}}z}\) is calculated assuming dz = 16 nm, corresponding to the oscillation amplitude of the cantilever. Finally, we apply a Gaussian blurring of 2σ = 270 nm to approximate the point-spread function of the SQUID sensor.