FGT films were epitaxially grown on Al2O3 (0001) substrates using a state-of-the-art VEECO MBE system with carefully optimized growth parameters (Methods). During the growing process, the surface crystalline quality was continuously monitored in real time using in situ reflection high-energy electron diffraction along specific crystallographic directions (Supplementary Fig. 1a), which revealed a clear epitaxial relationship between the FGT thin film and the substrate.
Bulk FGT is composed of QL units that exhibit high crystallinity in a trigonal structure with P3m1 symmetry15. Each QL comprises a trilayer heterometallic slab of Fe (labelled Fe I and Fe II) and Ge atoms, sandwiched between two Te atom layers. These Te layers bond adjacent FGT layers through vdW interactions, as illustrated by the structural model overlaid on the cross-sectional TEM image in Fig. 1a. The TEM image directly visualizes the atomic stacking of a bulk-like film where individual layers are clearly resolved and labelled sequentially from the interface. The observed structure shows excellent consistency with the proposed FGT model, except for the first QL adjacent to the Al2O3 substrate, for which a distinct structure as compared with subsequent QLs is observed. Specifically, as displayed in Fig. 1b, the Te–Te spacing within the vdW layer (\({l}_{{\rm{Te}}}\)) is expanded from 0.51 nm in the first QL to 0.56 nm in the second QL and remains nearly unchanged (~0.56 nm) thereafter. By contrast, the vdW gap spacing (\({l}_{{\rm{gap}}}\)) between adjacent QLs follows an opposite trend. \({l}_{{\rm{gap}}}\) between the first QL and second QL is largest (~0.30 nm), while it gradually decreases to ~0.26 nm for the subsequent layers. Notably, in contrast to those in the bulk case, Te and Fe atoms within the first QL are not coplanar along the c-axis. This structural distortion of the first QL is attributed to strain originating from the lattice mismatch between the FGT layer and the underlying substrate Al2O3, a phenomenon commonly observed in thin film growth20,21,22.
Fig. 1: Structural and magnetic characterization of FGT films by TEM, XRD and RMCD.
a, Cross-sectional HAADF-STEM image of a 10-QL FGT/Al2O3 film viewed along the [\(10\bar{1}0\)] direction. The corresponding atomic model is superimposed. The QLs are labelled sequentially beginning with the FGT/Al2O3 interface (first, second, third, fourth and fifth QL). The first QL layer exhibits a distinct structure configuration as compared with the subsequent QLs. The Te–Fe–Te bonds are tilted and non-coplanar along the c-axis. b, Layer-resolved evolution of Te–Te layer spacings within the vdW layers (\({l}_{{\rm{Te}}}\)) and the interlayer vdW gap spacings (\({l}_{{\rm{gap}}}\)) as a function of the QL index. c, Specular 2θ–θ XRD pattern of a 10-QL-thick FGT/Al2O3 film, showing (0 0 L) Bragg reflections as labelled. The intensity is given in counts per second (CPS). d, Thickness-dependent RMCD hysteresis loops (green, 10-QL; blue, 5-QL; red, 2-QL; black, 1-QL), recorded under perpendicular magnetic field geometry (\({H}_{\rm{{z}}}\)) sweeping between −20 kOe and 20 kOe at 2 K. e, Remanence-to-saturation ratio versus film thickness. The 1-QL FGT film exhibits a ratio of 0.65, whereas multilayer FGT thin films (2-QL, 5-QL and 10-QL) approach unity, indicating stronger PMA in thicker samples.
The long-range structural ordering of the 10-QL FGT film was also studied by XRD (Fig. 1c). The (0001)-oriented FGT film is phase-pure, as confirmed by the (0 0 L) family of diffraction peaks and the absence of any additional reflections. We find an out-of-plane lattice constant c = 16.4 Å, in excellent agreement with that of bulk FGT crystals23,24. The composition and thickness of representative FGT films were characterized by Rutherford backscattering spectrometry and X-ray reflection (Supplementary Fig. 1b,c), confirming the composition and high quality of our MBE-grown FGT films.
We then employed RMCD to investigate the magnetic order of these FGT thin films. The RMCD signal (\({I}_{{\rm{RMCD}}}\)), which is the reflected light intensity difference between right- and left-circularly polarized incident light, is proportional to the sample magnetization projected along the out-of-plane direction (\({M}_{{\rm{z}}}\)), that is \({I}_{{\rm{RMCD}}}\propto {M}_{{\rm{z}}}\) (the detailed derivation is outlined in Supplementary Note 1). A schematic of our RMCD set-up is shown in Supplementary Fig. 2. The magnetic field is applied perpendicular to the sample plane. A He–Ne laser (λ = 633 nm) with a power of 50 μW and a beam diameter of approximately 2 mm was used for RMCD measurements. Notably, we did not employ a microscope objective to focus the beam as our films are homogeneously uniform with a size of 10 mm × 10 mm. This configuration allows RMCD measurements to maintain an excellent signal-to-noise ratio even at high magnetic fields (up to 50 kOe) and for monolayer films.
The thickness-dependent RMCD signal (\({I}_{{\rm{RMCD}}}\)) hysteresis loops, measured at 2 K versus the out-of-plane magnetic field \({H}_{\rm{{z}}}\) in the range between −20 kOe and +20 kOe are summarized in Fig. 1d. For the 10-QL FGT thin film, \({I}_{{\rm{RMCD}}}\) exhibits a nearly square-shaped hysteresis loop with a coercive field (HC) of ~5 kOe and a magnetic remanence-to-saturation ratio (\(R={I}_{{\rm{RMCD}}}^{{\rm{r}}}/{I}_{{\rm{RMCD}}}^{{\rm{s}}}\)) of ~0.99, indicating a strong PMA. Figure 1e shows the thickness-dependent variation of R. While for the 5-QL film, R is still ~0.99, it drops to ~0.95 as the film thickness is decreased to 2-QL and further drops to 0.65 for the 1-QL film. In addition, in the 1-QL film, the RMCD signal already starts to increase (decrease) before the applied magnetic field changes sign from negative (positive) to positive (negative), suggesting a non-negligible in-plane magnetization component25,26. These results clearly demonstrate that the MBE-grown 1-QL FGT film exhibits relatively weak PMA, with a finite in-plane component. This is in contrast to exfoliated monolayers of FGT, whose magnetic structure is characterized by the 2D Ising model with strong PMA and R close to 1 (refs. 4,27).
To elucidate the origin of the distinct PMA behaviours in FGT films of different thicknesses, we further study their critical magnetic behaviour using temperature-dependent RMCD measurements. Representative RMCD hysteresis loops recorded by sweeping the magnetic field between −4 kOe and +4 kOe for 1-QL and 2-QL films are shown in Fig. 2a,b, respectively. Data for 5-QL and 10-QL films are shown in Supplementary Fig. 3a,b. As the temperature increases, the \({I}_{{\rm{RMCD}}}^{{\rm{s}}}\) and coercive field (HC) of the hysteresis loops both shrink, and the latter vanishes at about 114 K for the 1-QL sample and at 150 K for the 2-QL sample, respectively, indicating their Curie temperatures in these ranges. Curie temperature (\({T}_{{\rm{C}}}\)) can be extracted more precisely from a power-law fitting of the remnant RMCD signal using \({I}_{{\rm{RMCD}}}^{{\rm{r}}}\propto {\left(1-T/{T}_{{\rm{C}}}\right)}^{\beta }\) with T and β being the ambient temperature and the critical exponent, respectively. As shown in the inset of Fig. 2c and Supplementary Fig. 3c, \({T}_{{\rm{C}}}\) decreases monotonically with decreasing film thickness, in agreement with previous studies4,27. The 1-QL FGT film exhibits \({T}_{{\rm{C}}}^{1\mathrm{QL}}\approx 114.2\,{\rm{K}}\), while thicker samples show progressively higher values up to \({T}_{{\rm{C}}}\approx 215\,{\rm{K}}\).
Fig. 2: Critical behaviour of FGT thin films.
a,b, Representative RMCD hysteresis loops of 1-QL (a) and 2-QL (b) FGT films, recorded at various temperatures with the magnetic field swept between −4 kOe and +4 kOe. c, Temperature dependence of the remnant RMCD signal for a 1-QL (black) and 2-QL (blue) film. Solid red lines represent fits to the power-law using β and \({T}_{{\rm{C}}}\) (Curie temperature) as fitting parameters. Inset: derived \({T}_{{\rm{C}}}\) plotted versus thickness, showing an obvious decrease with decreasing layer number. The uncertainties of the remanent RMCD were estimated from the standard deviation of the zero-field RMCD values obtained by bootstrap resampling of the two hysteresis branches. d, Extracted critical exponent β versus film thickness, indicating a crossover from a 3D Heisenberg to 2D Ising behaviour when the film thickness increases from 1-QL to 2-QL. For thicker samples, the extracted β values continue to increase, approaching the 3D Ising-like regime. Horizontal dashed orange lines represent the theoretical β values for the mean field (\(\beta =0.5\)), 3D Heisenberg (\(\beta =0.365\)), 3D Ising (\(\beta =0.326\)) and 2D Ising model (\(\beta =0.125\)). The uncertainties of β represent the standard errors obtained from nonlinear least-squares fitting.
Figure 2d summarizes the experimentally determined β values for FGT films with different thicknesses. In general, a larger \(\beta\) is indicative of a more isotropic magnetic interaction. For reference, \(\beta\) values of 0.5 (mean field), 0.369 (3D Heisenberg), 0.326 (3D Ising) and 0.125 (2D Ising) are well established for corresponding idealized systems28,29. These values are indicated as dashed lines in Fig. 2d. For multi-QL films, we obtain β = 0.183 for 2-QL, 0.184 for 5-QL and 0.204 for 10-QL samples (a detailed fitting analysis is provided in Supplementary Note 2). These values of β are below those reported for bulk FGT crystals (~0.25)30,31, and are closer to the value of the 2D Ising model, consistent with the presence of strong PMA.
By stark contrast, the 1-QL FGT film exhibits a larger \(\beta\) of 0.38, close to the value expected for the 3D Heisenberg model. This behaviour also differs remarkably from the exfoliated monolayer FGT flakes, which possess identical structures to the bulk crystal and in turn display strong PMA with a critical exponent β = 0.14 ± 0.02 (ref. 4), characteristic of the 2D Ising model. With increasing thickness, β increases and the system tends to evolve back towards the thicker-sample or bulk-like regime, that is, towards more 3D Ising-like behaviour. At the same time, we note that the extracted exponent for the 25-QL sample (β = 0.237) remains below the ideal 3D Ising value. This value is nevertheless consistent with previous reports of β ≈ 0.25–0.27 for about 18-nm-thick FGT samples measured by RMCD4, as well as that of β ≈ 0.25 reported for bulk FGT from magnetic susceptibility measurements32. Such deviations from the ideal 3D Ising limit are not unusual in finite-thickness samples and may arise from finite-size effects, as also discussed in previous studies.4,33,34.
To uncover the origin of this counterintuitive dimensional crossover in MBE-grown FGT thin films—where the 1-QL film corresponds to a 3D Heisenberg model, while thicker films instead fall within the 2D Ising regime—we further examined the structure of FGT films with nominal thicknesses of 1, 2, 4 and 33 QL using SXRD analysis. As a representative example, the observed (symbols) structure factor magnitudes (\(|{F}_{{\rm{obs}}}\left({\rm{HKL}}\right)|\)) of the reflections along the (1 0 L) direction in reciprocal space of 1-QL, 2-QL and 4-QL FGT films are shown in Fig. 3a–c. Owing to the missing periodicity along the c-axis, the intensity is continuously distributed along the normal component of the momentum transfer, \({q}_{\rm{z}}=L\times {c}^{* }\), with c being the lattice parameter (‘rods’), that is, the index L becomes a real number rather than being limited to an integer. Each dataset typically consists of five rods containing a total of 130–170 reflections. The calculated structure factor magnitudes [|Fcalc(HKL)|] for the models are superimposed by the solid lines. The least-squares refinement was carried out using the Program ‘Prometheus’35. More detailed structural analysis is shown in Supplementary Note 3. The structural models obtained for the 1-QL, 2-QL and 4-QL films from the fitting and viewed along the b-axis are displayed in Fig. 3d–f, respectively.
Fig. 3: SXRD-derived structural models of FGT 1-QL, 2-QL and 4-QL.
a–c, Experimental (symbols) and calculated (lines) structure factor amplitudes shown on a log scale along the (1 0 L) direction in reciprocal lattice units (rec. latt. units) for the 1-QL (a), 2-QL (b) and 4-QL (c) FGT films. In a, the dashed red line represents the calculation for a bulk-like unrelaxed FGT sheet for comparison. The error bars in the data represent the 1σ standard deviations of the experimental structure factor magnitudes. d–f, Models of the atomic structure of the FGT films with thicknesses of 1 QL (d), 2 QL (e) and 4 QL (f). The FGT sheets in the 1-QL and 2-QL films are characterized by densely packed flat atomic sheets (highlighted by grey shading), which in the 4-QL film are present only at the top. vdW-site intercalated Fe atoms are represented by small red spheres and are present only in films with thicknesses of 2 QL or greater.
The structure of the 33-QL sample is essentially bulk-like (Supplementary Note 3) and consistent with previous reports36 in terms of crystal symmetry (P3m1), as well as the comparable concentrations of Fe vacancies in the QL layers and vdW-site Fe atoms (~10%). Reducing the film thickness to the 2D limit induces a number of modifications. Most prominently, there is a general tendency to form flat dense atomic layers accompanied by an overall compression of the FGT sheets. In particular, the 1-QL and 2-QL film structures exhibit substantial lattice modifications from the bulk structure, as demonstrated in Fig. 3d,e. In these ultrathin films, we found densely packed flat layers consisting of three atoms (Fe–Ge–Fe) instead of only two (Fe–Ge) (highlighted by the grey rectangles in Fig. 3d,e). Such structural rearrangements are common in ultrathin films and are generally attributed to the minimization of the kinetic and electrostatic energy by smoothing the surface charge density contour and depolarization, respectively37,38.
The extent of this structural relaxation in the monolayer limit is remarkably pronounced. As shown in Fig. 3a, the structure factor magnitudes calculated along the (1 0 L) rod for a hypothetical bulk-like 1-QL thick FGT sheet (Fig. 3a, dashed red line) deviate strongly from our experimental data, confirming the distinct reconstruction process in the 1-QL film. Such reconstruction gives rise to three inequivalent Fe atomic layers, introducing an additional out-of-plane component to the magnetic exchange interaction and thereby enabling 3D Heisenberg-like magnetic behaviour even in the single-QL limit, analogous to the 3D Heisenberg-like regime observed in ultrathin magnetic films only a few ångströms thick34,39,40,41,42,43.
While no vdW gap exists in the 1-QL film, the signature of Fe intercalation emerges starting from the 2-QL film. Such self-intercalation occurs naturally in vdW materials during their MBE growth, where the inserted atoms act as internal agents that reconstruct in-plane bonding and help stabilize the lattice structure16,44,45. The estimated occupancy of intercalated Fe atoms (about 50%) within the vdW gap in 2-QL films is substantially higher than that in thick films or in bulk crystals, where it typically amounts to 10–15%36. Notably, in thick films or in bulk crystals, the vdW gap is compressed and the Fe atoms within the vdW layer reside near their centre. These structural modifications become rapidly less important with increasing film thickness. As shown in Fig. 3f, the 4-QL film already exhibits an almost bulk-like structure, with the exception of the topmost layer, where substantial vertical atomic shifts are observed involving two (highly disordered) Te sites. The average concentration of intercalated Fe remains relatively high and in the 30–40% regime. These self-intercalated Fe atoms reinforce the easy-axis exchange anisotropy along the out-of-plane direction46, thereby enhancing PMA and driving the system towards the 2D Ising regime. In addition, structural disorder generally increases as the film thickness decreases, as reflected by the mean-squared displacement amplitudes rising from 0.05 in the 33-QL sample to 0.20 Å2 in 1-QL film (see, for example, Supplementary Tables 2 and 3). This disorder probably originates from local atomic displacements at grain boundaries and at the film–substrate interface (Fig. 1a).
Such self-intercalation of Fe atoms also contributes to the magnetic behaviour, most prominently at high magnetic fields. The corresponding RMCD measurements can be found in Fig. 4a,b for 10-QL and 1-QL and Supplementary Fig. 12a,b for 2-QL and 5-QL films. All multilayer FGT thin films (2-QL, 5-QL and 10-QL) exhibit an unsaturated magnetization persisting up to room temperature. On a quantitative basis, \({I}_{\mathrm{RMCD}}\) was linearly fitted versus the magnitude of the magnetic field in the high field regime. The extracted slopes continue to increase towards room temperature for the 2-QL, 5-QL and 10-QL samples, with a similar temperature evolution but different magnitudes (Fig. 4c).
Fig. 4: Evidence of self-intercalation of Fe atoms in FGT thin films.
a,b, RMCD measurements of 10-QL (a) and 1-QL (b) FGT films at various temperatures with magnetic field swept between −50 kOe and 50 kOe. Dashed red lines indicate linear fits in the high-field region. c, Temperature dependence of the RMCD slopes extracted from the high-field fits. The extracted slopes exhibit a pronounced temperature dependence that increase continuously up to room temperature for the 2-QL (light green), 5-QL (orange) and 10-QL (yellow) FGT thin films, with similar temperature evolution but differing magnitudes. By contrast, 1-QL film shows much smaller slope across the entire temperature region, except for a distinct peak at around 114 K, close to its \({T}_{{\rm{C}}}\) (inset), suggesting a fundamentally different magnetic response from multilayer FGT films. The uncertainties of the data in c represent the standard errors obtained from linear least-squares fitting. d, High-field susceptibility obtained from linear fits to the simulated data based on Weiss molecular field theory. A well-defined peak appears near \({T}_{{\rm{C}}}\), indicating that the exchange interaction becomes comparable to the thermal energy (\({k}_{{\rm{B}}}T\)) in this temperature range, thereby facilitating field-induced spin alignment. FM, ferromagnetic phase; PM, paramagnetic phase.
The observation of unsaturated magnetization at high magnetic field up to room temperature was also confirmed by Hall transport measurements (Supplementary Fig. 13a,b). The temperature dependence of the remanent Hall resistance follows a trend similar to that of the remanent RMCD signal in both samples (Supplementary Fig. 14), further supporting the consistency between the transport and optical probes. It should be noted that the slope of resistivity versus applied magnetic field \(\left({\rho }_{{xy}}={R}_{0}{H}_{\rm{z}}+{R}_{{\rm{s}}}{M}_{\rm{z}}\right)\), where \({R}_{0}\) and \({R}_{{\rm{s}}}\) are the ordinary and anomalous Hall coefficients respectively, is usually used to estimate the carrier density under the assumption that \({M}_{\rm{z}}\) is saturated at high magnetic field47,48,49. However, this approach becomes unreliable, especially for those materials with intrinsic high carrier densities or unsaturated magnetizations. We normalized the fitted slope of both \({\rho }_{{xy}}\) and RMCD signal by the Hall resistivity \({\rho }_{{xy},{H}_{\rm{z}}=0}^{50{\rm{K}}}\) and RMCD signal \({I}_{\mathrm{RMCD},{H}_{\rm{z}}=0}^{50{\rm{K}}}\) at zero field and at 50 K for comparison (Supplementary Fig. 13c). Compared with \({\rho }_{{xy}}\), \({I}_{{\rm{RMCD}}}\propto {M}_{\rm{z}}\) and does not have a linear field dependence associated with \({R}_{0}\). The nearly identical temperature trends of the high-field slopes obtained from \({\rho }_{{xy}}\) and RMCD for both 5-QL and 10-QL FGT films suggest that the anomalous Hall coefficient (\({R}_{{\rm{s}}}\)) dominates over the ordinary Hall coefficient (\({R}_{0}\)) in FGT systems in the high-field region. Thus, the increase in Hall resistivity at high applied magnetic fields originates from unsaturated magnetization, which gives rise to the anomalous Hall effect (\({R}_{{\rm{s}}}{M}_{\rm{z}}(H_{\rm{z}})\)) rather than an ordinary Hall effect (\({R}_{0}{H}_{\rm{z}}\)) (see more details in Supplementary Note 4). By contrast, the slope of 1-QL film remains almost zero across the examined temperature range, except for an observable peak at around 114 K, close to its TC (Fig. 2c, inset). This behaviour reveals a fundamentally different magnetic response from that of the multilayer films, probably arising from the absence of a vdW gap that could accommodate intercalated Fe atoms. To gain further insight into the high-field behaviour for the 1-QL film, we simulated the temperature-dependent magnetization–field (M–H) curves using the Weiss molecular field theory (see more details in Supplementary Note 5). The calculated curves reproduce the experimental trend well, where the high-field susceptibility obtained from linear fits to the simulated data exhibits a well-defined peak near \({T}_{{\rm{C}}}\) (Fig. 4d and Supplementary Fig. 18a). This peak is intrinsic and originates from the comparable magnitudes of exchange interaction and thermal energy (\({k}_{{\rm{B}}}T\)) near \({T}_{{\rm{C}}}\), which facilitates spin alignment under an external field. The same feature was verified in our RMCD measurements of a reference ferromagnetic thin film sample of SrRuO3, confirming its intrinsic origin (Supplementary Fig. 18b,c).