Tuning the density of zero-field skyrmions and imaging the spin configuration in a two-dimensional Fe3GeTe2 magnet

With the advent of ferromagnetism, two-dimensional (2D) van der Waals (vdW) magnets have attracted particular attention in exploring topological spin textures, such as skyrmions used for next-generation spintronic devices. The discovery of magnetic skyrmions in Fe3GeTe2 (FGT) has sparked interest in investigating the spin configurations of skyrmions in FGT. Here, we used an in situ Lorentz microscope to directly demonstrate the generation and sustainability of Bloch-type skyrmions in a zero magnetic field over a wide temperature range in 2D vdW FGT. By tuning the value of the external magnetic field, the highest-density hexagonal skyrmion lattice emerges after reducing the magnetic field to zero. Moreover, by tilting the FGT nanosheet, we found that the field-free Bloch-type skyrmions in FGT can also represent an invisible contrast when the tilt angle is zero, but a reversed magnetic contrast emerges at a high tilt angle. On the basis of our experiments, we discuss the possible mechanisms for such variable magnetic contrast. These findings offer valuable insights into the spin configurations of skyrmions in 2D vdW FGT and shed light on the identification of spin configurations via Lorentz microscopy. Two-dimensional (2D) van der Waals (vdW) magnet with the advent of ferromagnetism has stimulated particular attention in exploring topological spin texture especially for skyrmions. We here report the real-space observation of tuning the density of field free skyrmions in Fe3GeTe2(FGT), which can serve as a new application for spintronic devices. By tilting the FGT nanosheet, we found that the field-free Bloch-type skyrmions can also represent an invisible contrast when the tilt angle is zero, but emerge a reversal magnetic contrast at high tilt angle. These findings shed light on the identification of the spin configuration in 2D vdW magnet.


Introduction
Two-dimensional van der Waals (2D vdW) materials with long-range ferromagnetic orders presenting diverse novel phenomena [1][2][3][4][5] show promise for fundamental physics and device applications 6,7 . Among these materials, Fe 3 GeTe 2 (FGT) has drawn particular attention due to its strong perpendicular magnetic anisotropy and long-range ferromagnetic order ranging from bulk crystals down to monolayers [8][9][10][11] . More importantly, the Curie temperature (T C ) of FGT atomic layers is controllable via electrostatic gating or patterned microstructures 12,13 . In conjunction with these novel properties, magnetic domain structures have been recently investigated in 2D vdW FGT. Complex magnetic domain structures, including labyrinthine domain structures 10 , bubble domains 14 , double-walled domains 15 , and spike-like domains 16 , have been observed via magnetic force microscopy. Recent investigations of FGT via Lorentz transmission electron microscopy (LTEM) with the application of a magnetic field have led to the discovery of skyrmions. The magnetic field and temperature play a crucial role in skyrmion generation and stability. Rapid cooling below T C with a proper magnetic field has led to field-free skyrmions being observed not only in non-centrosymmetric magnet FeGe 17 and MnSi 18 but also in centrosymmetric magnet MnNiGa 19,20 . Unfortunately, such thermal manipulations of skyrmions in 2D vdW magnets remain elusive. Meanwhile, these field-free skyrmions provide an easy way to identify the spin configurations of skyrmions regardless of the application of a magnetic field.
Very recently, Néel-type skyrmions supported by interfacial Dzyaloshinskii-Moriya interactions were discovered in FGT heterostructures 21,22 . Simultaneously, Bloch-type skyrmions stabilized by magnetic dipole interactions have also been reported in FGT nanosheets 23 . As theoretically predicted, Bloch-type magnetic twists usually emerge in centrosymmetric magnets with competition between dipole interactions and uniaxial magnetic anisotropy, as reported in Mn-Ni-Ga 24 , Ni 2 MnGa 25 , Fe 3 Sn 2 26 , and 2D vdW Cr 2 Ge 2 Te 6 27 . In centrosymmetric FGT with a space group of P63/mmc, Néel-type skyrmions in previous reports seem to be a subject of controversy. Thus, direct identification of the spin configurations of skyrmions in 2D vdW FGT deserves further investigation.
In this work, the generation and sustainability of Blochtype skyrmions in a zero magnetic field over a wide temperature range in 2D vdW FGT were directly demonstrated via LTEM. By tuning the value of the external magnetic field, the highest-density hexagonal skyrmion lattice emerges after reducing the magnetic field to zero. Micromagnetic simulation reveals that compared with stripe domains, the highest-density skyrmion lattice has stronger total energy, indicating that it is in a metastable state. In addition, we found that the field-free Bloch-type skyrmions in FGT can also exhibit no contrast when the tilt angle is zero but present a reversed magnetic contrast at a high tilt angle. On the basis of our experiments, we discuss the possible mechanisms for this disappearing magnetic contrast.

Sample synthesis
High-quality Fe 3 GeTe 2 single crystals were grown using the Te self-flux method from a mixture of pure elements Fe (99.99%), Ge (99.9999%), and Te (99.995%) with a composition of Fe 2 GeTe 4 28 . The mixture was then sealed in an evacuated quartz tube and heated to 1000°C. The melt was held at 1000°C for 3 h and then cooled slowly to 680°C at a rate of 1°C/h, and the excess Te flux was removed by spinning the tube. The typical size of the single crystals was~5 × 5 × 0.2 mm, with a cleavable layer in the ab plane. The composition of the single-crystal FGT was further checked by scanning energy dispersive X-ray spectroscopy, confirming that the atomic ratio was 3:1:2 (Fig. S1).

Lorentz TEM measurement
The nanosheets for Lorentz TEM observation were fabricated from a single crystal by using a focused iron beam. The magnetic domain was directly observed by using a Tecnai F20 in Lorentz TEM mode and a JEOL 2100F Lorentz TEM, both equipped with liquid nitrogen and low-temperature holders (~90 K) to study the temperature dependence of the magnetic textures. The objective lens was turned off when the sample holder was inserted, and the perpendicular magnetic field was induced to the nanosheet by increasing the objective lens in a small increment. For tilting experiments, the crystalline orientation of the nanosheets was first checked by selected-area electron diffraction (SAED) to ensure compliance along the [001] direction. After marking the value of this angle, a series of tilting experiments were performed. In addition, the specific field cooling (FC) manipulation is shown as follows. First, the sample was heated higher than the Curie temperature T C~1 50 K. Second, the perpendicular magnetic field was applied by increasing the objective lens current gradually in a very small increment. Third, the temperature of the sample was cooled gradually from 150 K to 93 K. Finally, at 93 K, the small perpendicular magnetic field was turned off.

Micromagnetic simulations
Micromagnetic simulation was performed via the Object Oriented MicroMagnetic Framework (OOMMF) code based on the LLG function 29 . The slab was 2000 × 2000 × 100 nm with a rectangular mesh of size 5 × 5 × 5 nm. The saturation magnetization was chosen as Ms = 2.23 × 10 5 A/m at 100 K, and the magnetic anisotropy constant Ku was 0.5 × 10 5 J/m 3 . To make analytic progress, we considered an initially random spin state as the paramagnetic state. A field cooling procedure was performed as follows. Initially, the slab was set as a random state, and an external field was applied along the z-axis. After that, the perpendicular field was reduced to zero, and the slab was relaxed to an equilibrium state with each step of 30 ns.

Results
The 2D vdW FGT is a ferromagnet with a strong uniaxial magnetic anisotropy and an easy axis along the caxis, while it exhibits absolute isotropic characteristics in the ab plane 30 . Owing to competition between the uniaxial magnetic anisotropy and magnetic dipole interaction, stripe domains separated by Bloch walls spontaneously stabilized without an external magnetic field, as schematically shown in Fig. 1a. In Fig. 1b, c, we show the magnetic domain at 93 K along two orthogonal imaging directions. The corresponding crystalline orientation and crystal structure were examined by using SAED and scanning transmission electron microscopy with high-angle annular dark-field atomic resolution, which coincides with previous reports 23, 31 . In the case of the bc plane, the magnetic contrast induced by the spins inside the domain formed alternating dark or bright stripes, which were further identified by the line profile of magnetic contrast. For magnetic domains observed in the ab plane, the flake was tilted to reduce the diffraction contrast and induce magnetic contrast, which is similar to an approach reported in the previous work 23 . Clearly, the LTEM contrast is only generated by the domain wall, resulting in bright-dark or dark-bright stripes, as verified by the line profile shown in Fig. 1c. The different magnetic domains in the two orthogonal imaging planes demonstrate that the strong magnetocrystalline anisotropy of FGT persists despite additional shape anisotropy terms arising in the cross-section TEM sample. It is noteworthy that with decreasing width of the domain wall, the magnetic contrast in the ab plane gradually vanished, which will be discussed later.
The evolution of the magnetic stripe as a function of temperature and magnetic field in the bc plane was studied in the Supporting Information (Fig. S2); here, we focus on the magnetic domain in the ab plane. The flake along the [001] direction was tilted to reduce the diffraction contrast and induce magnetic contrast. Figure 2a-c shows the magnetic domain as a function of temperature observed via LTEM under a zero field. As the temperature decreases to 147 K, slightly below T c~1 50 K, magnetic stripes emerge. Upon further cooling, the stripes become wider and more distinctive (Fig. S3). The period of the domain as a function of temperature is shown in Fig. 2h. With the application of an external magnetic field, the stripes gradually transformed into bubbles (details seen in Supporting Information, Fig. S4). In Fig. 2d-f, we show the temperature-dependent LTEM images of bubbles at temperatures ranging from 147 K to 93 K under their corresponding critical magnetic field. Clearly, the critical magnetic field increases with decreasing temperature (Fig. S5). Simultaneously, the size of bubbles represents a similar variation, as shown in Fig. 2i. The magnetic domain texture is further analyzed with magnetic properties in FGT, as shown in Fig. 2g. The a.c. susceptibility measurements in the H//c and H//ab directions were performed to identify the magnetic transition. For the 150 K transition, the peak in each direction appears, which is related to the Curie temperature. In contrast, at the 118 K transition, the peak only appears in the black curve (in the ab plane) but remains almost absent in the H//c direction.
Interestingly, the period of the magnetic stripe exactly represents a significant change above 118 K and gradually remains constant at lower temperatures. The varied periodicity may result from the pinned magnetic domain wall 32 or a spin-flop transition 14 , which needs further investigation.
To obtain a field-free skyrmion lattice with the highest density, we tuned the magnitude of the cooling field (H CF ), as shown in Fig. 3a-c. The detailed cooling procedure and specific LTEM images are shown in the Methods and Fig. S6. When the magnetic field is lower than 100 Oe or higher than 700 Oe, a stripe emerges. With the magnetic field increasing from 100 Oe to 700 Oe, the size of skyrmions first decreases and then increases (Fig. 3g). Meanwhile, the density of skyrmions represents an inverse variation reaching a maximum value at H CF~4 00 Oe (Fig. 3b). A stronger magnetic field will force nucleation sites to agglomerate, thereby reducing skyrmion density and enlarging skyrmion size. Clearly, the skyrmion density is apparently enhanced via FC manipulation compared with the random distribution induced solely by the magnetic field. In addition, the spin configuration of field-free skyrmions is further analyzed, which represents dark to bright or bright to dark rings in the underfocused images identified with Bloch-type domain walls separating the spin-up and spin-down domains 23 .
To understand the mechanism stabilizing the skyrmion lattice after field cooling, we performed micromagnetic simulation via the OOMMF code. The input magnetic parameters were used as reported in previous work 23 . In this approach, we take a random spin configuration as the paramagnetic state. Figure 3d-f shows the representative remanent state after the field cooling manipulation. The variation in the size and density of the skyrmions is the same as that observed in the LTEM experiments, as shown in Fig. 3h. At H CF = 400 Oe, the skyrmions with a homogeneous circular shape are closely packed, forming hexagonal lattice skyrmions. As the magnetic field varies, individual skyrmions stretch to deform the lattice arrangement. The corresponding formation of these spin configurations in centrosymmetric magnets should originate from the competition of the exchange energy, the magnetic anisotropy energy, and the demagnetizing energy, as shown in Fig. 3i. As the external magnetic field increases, the strength of the dipole interaction exhibits a reversal variation compared with the other two interactions, reaching a minimum value at H CF = 400 Oe. Owing to the high density of skyrmions, an increasing number of domain walls reduces the demagnetizing energy; however, it simultaneously gives rise to the exchange energy. Meanwhile, various magnetic moments deviate from the easy axis, resulting in an increasing uniaxial magnetocrystalline anisotropy. The magnetic field-dependent total energy is obtained; obviously, it represents a maximum value with the highest density of skyrmions, which indicates that, compared with the stripe domain, the thermalequilibrium skyrmion lattice is in a metastable state. Figure 4 represents the externally applied temperature (T)-magnetic field (H) phase diagram with the ZFC process and FC process measured by systematic Lorentz TEM observations at various T and B. The formation and annihilation magnetic field for skyrmions are plotted by white dashed lines. As shown in Fig. 4a, the phase diagram for skyrmions is limited. The magnetic field to generate skyrmions increases with decreasing temperature. After FC manipulation, the high-density yellow-colored area with zero-field skyrmions covers the entire temperature range toward T C, as shown in Fig. 4b.
On the basis of field-free skyrmions, we further investigate the spin configuration observed in the FGT flake at various oblique angles, as shown in Fig. 5. The parameter α is defined as the angle between the electron beam direction and the [001] direction of the flake. Based on the direct LTEM observation, at a zero-degree tilt, no magnetic contrast appeared in the defocused LTEM image. With increasing oblique angles, the magnetic contrast gradually appeared. At α = 7°, the sharp contrast of dark to bright or bright to dark rings in the images identified with Bloch domain walls, which is further demonstrated by the line scan positions for contrast profiles of a single skyrmion with two peaks clearly observed in the bottom of Fig. 5b. At a higher tilting anglẽ ±20°, the skyrmion represents half-black/half-white magnetic contrast and a reversal contrast with reversing tilting direction. This can be easily distinguished from the contrast profile. Clearly, only one peak was observed at a high tilted angle (±20°), and the intensity of the profile Fig. 3 Variation in the magnetic state at room temperature at different magnetic fields after FC manipulation. a-c Representative underfocused LTEM images of skyrmion bubbles taken at 93 K after a field cooling manipulation. Micromagnetic simulations of the domain field cooling process as a function of magnetic fields of 100 Oe (d), 400 Oe (e), and 800 Oe (f). g Magnetic field-dependent mean size and density of skyrmion bubbles obtained from LTEM data. h The calculated mean size and density of skyrmions under different magnetic fields. i Field dependence of the energy terms in the simulation after a field cooling process.
represents mirror symmetry with a reversing tilting angle. The same experiment was performed on the magnetic stripe domain, as shown in Fig. S7.

Discussion
We now discuss the possibility that leads to the varied observations of magnetic contrast in the FGT thin plate. LTEM is sensitive to the in-plane magnetic component, which further induces the phase shift of the electron beam. Note that magnetic contrast was not observed at the zero tilt because of the symmetric deflection of magnetic texture, such as in a Néel type skyrmion 33,34 , which can result from a small magnetization 35 or a narrow domain wall 36 of a Bloch-type twist. Based on the above observation, we exclude the existence of a Néeltype magnetic domain in the FGT flake. Through a The flake was cooled to a specific temperature, and then an external magnetic field was applied. b An external magnetic field was applied and then slowly cooled down. The points represent different magnetic states: hollow triangle -stripe, dot -skyrmion and hollow rectangleferromagnetic state.