Hard magnetic properties in nanoflake van der Waals Fe3GeTe2

Two-dimensional van der Waals materials have demonstrated fascinating optical and electrical characteristics. However, reports on magnetic properties and spintronic applications of van der Waals materials are scarce by comparison. Here, we report anomalous Hall effect measurements on single crystalline metallic Fe3GeTe2 nanoflakes with different thicknesses. These nanoflakes exhibit a single hard magnetic phase with a near square-shaped magnetic loop, large coercivity (up to 550 mT at 2 K), a Curie temperature near 200 K and strong perpendicular magnetic anisotropy. Using criticality analysis, the coupling length between van der Waals atomic layers in Fe3GeTe2 is estimated to be ~5 van der Waals layers. Furthermore, the hard magnetic behaviour of Fe3GeTe2 can be well described by a proposed model. The magnetic properties of Fe3GeTe2 highlight its potential for integration into van der Waals magnetic heterostructures, paving the way for spintronic research and applications based on these devices.


Supplementary Note 1. The material characterization of Fe 3 GeTe 2 crystal Chemical composition
To analyze the chemical composition of synthesized Fe 3 GeTe 2 (FGT) flakes, energydispersive X-ray spectroscopy (EDS) was carried out. As shown in Supplementary Figure 1a     The sudden decrease of remanence indicates that the thermo agitation energy is higher than the perpendicular anisotropic energy, which can also be clearly seen from the evolution of the R xy (B) loop from 160 K to 210 K. (f) R xy (B) curve at 120 K and 160 K. Compared with the 120 K curve, the 160 K curve is indicative of two magnetic phases with increasing temperature.    Magnetic Field (mT)

Supplementary Note 3. Definition of the Curie temperature
The definition of Curie temperature is a vital part of magnetic material measurements.
Generally, based on Curie-Weiss law, we can determine the Curie temperature of a sample through a linear fit to the temperature dependence of inverse magnetization above T C.
However, this method is only accurate when the critical exponent is 1 for the temperaturedependent susceptibility, which is unknown yet for FGT nanoflakes. Here we define the temperature at which the remanence R xy goes to zero as T C [the way similar to 'Nature 546, 265-269 (2017)'], which can give us more accurate T C values.
In our experiments, the sample was firstly cooled down to 2 K under a magnetic field of 1 T (-1T). Then the magnetic field was slowly (5 Oe/s) decreased to 0 Oe at 2 K. Finally we scan temperature from 2 K to 300 K at 3 K/min and get the R xy vs T curve with 1 T (-1 T) remanence. By telling the junction of remanence vs T curve of 1 T and -1 T (Supplementary Because of the non-symmetry in our nanoflake devices, the measured Hall resistance was mixed with the longitudinal magnetoresistance. We processed the data by using (R xyA (+B) -R xyB (-B)) / 2 to eliminate the contribution from the longitudinal magnetoresistance, where R xyA is the half loop sweeping from the positive field to the negative field, R xyB is the half loop sweeping from the negative field to the positive field, and B is the applied magnetic field.
We also measured the R xy (T) at the remanence point for all the samples. To measure the R xy (T) at the remanence point, the magnetic moment of samples was first saturated by a 1 T magnetic field and then the magnetic field was decreased to zero (the remanence point).
Finally, the temperature dependence of the R xy at remanence was measured when the temperature was increased from 2 K to 300 K. In order to eliminate the non-symmetry effect of the device, we measured R xy (remanence) with both 1 T and -1 T saturation. The real R xy (T) at remanence without R xx mixing was calculated using (R xyA (T) -R xyB (T)) / 2. Here R xyA and R xyB are the remenance with 1 T and -1 T saturation, respectively.

Demagnetization effect
The demagnetization effect is significant for FGT nanoflakes with perpendicular magnetic anisotropy. The following sentence is from 'Phys. Rev. B 58, 3223 (1998)', which shows the demagnetization factors for thin film with perpendicular anisotropy. "We approximate the thin film by a homogeneously magnetized ellipsoid of revolution of volume V whose radius R x = R y = R is much larger than the 'film thickness' 2R z . The magnetostatic self-interaction energy is then given by Dµ 0 M 2 V/2, where D ≈ 1 and D ≈ 0 are the demagnetizing factors for in-plane and perpendicular magnetization orientations, respectively." Based on the results above, we can obtain the effective field B eff (magnitude and angle) from the applied magnetic field.
The schematic diagram of the angular relationship of the applied field, magnetization, and perpendicular anisotropy is shown Supplementary Figure 9a.

Theoretical fitting by modified Stoner-Wohlfarth model
Three different energies, the magnetic anisotropic energy, the Zeeman energy due to the interaction between the applied magnetic field and magnetic moments, and the thermal agitation energy determine the magnetic behavior of the FGT nanoflakes.

Fitting to the R xy hysteresis loops at different temperatures
From the M R /M S ratio (~ 1), we know that all the spins in FGT nanoflakes align perpendicular to the sample surface at the remanence. Except the regime near the coercive field, an FGT nanoflake behaves like a single domain particle. Therefore, we can use the Stoner-Wohlfarth model to describe the magnetic behavior of a FGT nanoflake in the magnetic field regime away from the coercivity. As shown in Supplementary Figure 9a, the angles between the applied magnetic field and the direction of the perpendicular anisotropy and the magnetic moment are θ and φ, respectively. The direction of the magnetic field means the direction of the POSITIVE magnetic field.
Based on the Stoner-Wohlfarth model, the energy of a FGT nanoflake at temperature T can be written as where K A is the magnetic anisotropic energy, V S is the volume of the sample, M S (T) is the magnetic moment of a unit volume FGT at temperature T, and B is the applied magnetic field.
With an applied magnetic field B at known θ, we can easily calculate the φ value using the In anomalous Hall measurements, the R xy is proportional to the value of M Z (the magnetization perpendicular to the sample surface). Therefore and Supplementary Figure 7(d-h). After the K A (T) value is obtained, the angular dependent coercivity can then be fitted based on a modified Stoner-Wohlfarth model at temperature T.

Fitting to the angular dependence of coercivity based on a modified Stoner-Wohlfarth model
When the magnetic field is swept to a negative field, the B field in Supplementary Equation 3 and Supplementary Equation 4 is negative.
To fit the angular dependence of coercivity, we make two assumptions,

Supplementary Equation 4 can have two kinds of solutions, the stable state (low energy)
and the unstable state (high energy state), as shown in Supplementary Figure 9b. With an increasing magnetic field B (more negative B value), the energy difference ∆E between the meta-stable state 1 and the unstable state 2 decreases. At a certain B field, the thermal agitation energy is large enough to overcome the ∆E in a standard experimental time and the magnetic moment will flip to a stable state in the opposite direction. As the FGT nanoflake shows a nearly square-shaped R xy loop (magnetic loop), we can assume that this B field is the coercive field.
2. When the first domain flips to the opposite direction under an applied magnetic field, other un-flipped magnetic moments will generate an effective field on the magnetic moment in the first flipped domain.
An important issue for the assumption 1 is to determine the ratio ∆E/k B T to realize the experimentally observable flipping of domains. Neel and Brown proposed that the relaxation time τ for the system to reach thermodynamic equilibrium from the saturated state can be written as where f 0 is a slowly variable frequency factor of the order 10 -9 sec -1 . Assuming τ = 100 Sec,

Supplementary Note 5. Mean field and spin wave fittings
To fit the temperature dependent remane nce in Fig. 3c, we tried the mean field theory (the Brillouin function) and the spin wave theory.

The Curie-Weiss mean field theory
Due to the interaction between magnetic moments, an internal field can be written as where n w is a parameter describing the strength of the internal field and M sp is the spontaneous magnetization.
Where M 0 is the magnetic moment at zero K, ℑ is Brillouin function, k B is Boltzman constant, N the total number of unit magnetic moments. From the above equations, we easily Now we calculate the value of T C Therefore, we obtain We obtain Combine Supplementary Equations 11, 12 and 16, we can easily calculate the M sp vs (T/T C ) curves for different J values.

Spin wave model
The temperature dependence of magnetic moments of a three dimensional spin wave is

Supplementary Note 6. The effect of the surface amorphous oxide layer
Though we tried to minimize the exposure of samples to ambient conditions, but a significant oxide layer could still form quickly on the top of the nanoflakes, which has been confirmed by cross-sectional electron microscopy images shown in Supplementary Figure 10a.
We Moreover, we can see that FGT is a promising material whose magnetism can survive in ambient environment for a certain time.
From our experiments, we conclude that the effect of oxide layer includes: 1.
The switch of magnetic moment in the square shape loop of FGT with oxide layer is not as sharp as that in ultra clean FGT flakes, which is due to the pinning effect of the oxide layer.

2.
The coercivity of FGT slightly increases after the oxidization, which is also due to the domain wall pinning effect.