Magnetic domains and domain wall pinning in atomically thin CrBr3 revealed by nanoscale imaging

The emergence of atomically thin van der Waals magnets provides a new platform for the studies of two-dimensional magnetism and its applications. However, the widely used measurement methods in recent studies cannot provide quantitative information of the magnetization nor achieve nanoscale spatial resolution. These capabilities are essential to explore the rich properties of magnetic domains and spin textures. Here, we employ cryogenic scanning magnetometry using a single-electron spin of a nitrogen-vacancy center in a diamond probe to unambiguously prove the existence of magnetic domains and study their dynamics in atomically thin CrBr3. By controlling the magnetic domain evolution as a function of magnetic field, we find that the pinning effect is a dominant coercivity mechanism and determine the magnetization of a CrBr3 bilayer to be about 26 Bohr magnetons per square nanometer. The high spatial resolution of this technique enables imaging of magnetic domains and allows to locate the sites of defects that pin the domain walls and nucleate the reverse domains. Our work highlights scanning nitrogen-vacancy center magnetometry as a quantitative probe to explore nanoscale features in two-dimensional magnets.

The sample substrate is glued on a Titanium sample holder which directly contacts with a thin copper plate ( Fig.1 (b)). A heater and resistive thermometer is placed in the copper plate. The coplanar waveguides on the sample substrate and the print circuit board (PCB) are bonded with Au wires. The atomic force microscope works in a frequency modulation mode. The tuning fork is excited electrically and the readout signal is first amplified by using a home-made pre-amplifier placed at the microscope head, and then amplified by a voltage amplifier.
ODMR curve and sensitivity The hyperfine interactions between the 14 N nuclear spin (I = 1) of the NV center and the NV electronic spin results in three transitions, each separated by about 2.16 MHz in the |m s = 0 to |m s = ±1 transitions, as shown in Fig.1 (d). If the microwave applied to the NV center can drive the transition with Rabi frequency much larger than 2.16 MHz, the ODMR curve for each electronic spin transition shows a broad resonance and the hyperfine splitting lines can not be resolved. Fig.2  3 from the resonance frequency. In order to achieve the optimal sensitivity, we reduce the microwave power to resolve the hyperfine splitting lines. In addition, we drive the three hyperfine splitting transitions simultaneously. Therefore, the ODMR curve obtained in this way shows five characteristic resonances lines as shown in Fig.2 (b). The   In the practical applications, we also need to consider the magnetic field measurement dynamic range, which should be in the order of 1 mT for few-layer CrBr 3 . Therefore, most of the images are measured by recording the full ODMR spectrum by driving the NV transition with Rabi frequency of about 5 MHz. A few of the images shown in the Supplementary Information are measured by recording the NV fluorescence with the microwave frequency fixed at the steepest slope of the resonance line corresponding to the external field. With this so called iso-field method, one can achieve the optimal sensitivity, but the dynamic range is limited by the resonance linewidth.
Magnetization reconstruction from stray field map Although it is straightforward to calculate the stray magnetic field given the magnetization distribution, the inverse problem remains a chal-lenge in magnetic source imaging due to the lack of a unique solution in most cases 3,4 . Here we discuss model of stray magnetic field generated by a 2D magnetic and show how to solve the inverse problem by introducing reasonable constraints. Fig.3 shows a 2D material in the xy-plane with coordinate z = 0 and the magnetization distribution M(x, y). The stray magnetic field in a parallel plane with a distance h from the the sample is given by where D(x, y, z) is the dipolar tensor. With the convolution theorem, the integration in Eq. 1 can be rewritten as a product of the 2D Fourier transform of the corresponding quantities in the The 2D Fourier transform of the components of the dipolar tensor reads as, if k = k 2 x + k 2 y = 0 and all components equal to zero if k = 0 . The vanishing of non-zero components of matrix at zero momentum indicates that the uniform magnetization in a infinite large area has no contribution to the stray magnetic field.
Note that the three rows in the matrix in Eq. 3 are not independent. On the one hand, this means that in the momentum space, the three components of the vector stray magnetic field are not On the other hand, we cannot obtain a unique solution of magnetization given the stray magnetic fields in Eq.3 because the matrix is non-invertible. To obtain a unique solution, two additional constraints on the magnetization need to be introduced.
Here, we use the assumption of out-of-plane magnetization, that is M(x, y) = [0, 0, m z (x, y)] T .
With Eq.2 and Eq.3, the stray magnetic field component along z-axis is given bỹ The stray magnetic field component along the NV-axis can be related to the z component as B N V (k x , k y ) = cos(φ) sin(θ)B x (k x , k y ) + sin(φ) sin(θ)B y (k x , k y ) + cos(θ)B z (k x , k y ).
The amplitude of magnetizationm z (k x , k y ) can be obtained by substituting Eq.4 and Eq.5 into Eq.6 and converted to the real space map by an inverse 2D Fourier transform. To reduce the reconstruction noise, a Hanning low-pass filter is used 6 . The direction of the NV axis is determined by adjusting the orientation of a 200 mT external magnetic field to maximumize the NV fluorescence 7 .
The distance h can be obtained by fitting the magnetic field near the edges of the sample 8    The scale bar stands for 1µm for all images.

Magnetic domains in 3L&4L CrBr sample
Magnetic domain evolution This CrBr 3 sample has three-layer and four-layer areas, which are labeled in the AFM image (Fig.9). We measure the magnetic domain evolution in an area which has both 3L and 4L CrBr 3 , as marked by the box #1 in Fig.9. The measurement with the thermally demagnetized sample and full magnetized sample are shown in Fig.10. The depinning field of the 3L CrBr 3 is much lower than the 4L CrBr 3 . The domain reversal in 4L CrBr 3 is affected by domain wall pinning. Discussion on laser heating effect. The magnetization at different laser power of a 3L CrBr 3 area is shown in Fig.11. We observe the magnetization starts to decrease when the laser power is higher than ∼30 µW. We also image the domain structure with laser power from 9 to 24 µW. No obvious domain motion has been observed in this measurement as shown in Fig. 12. All the other results   Fig.9. The external magnetic field is 3 mT along NV axis.

Micromagnetic simulation
Simulations were carried out in a finite differences approach via MuMax3 9 . The simulation cells were set to a size of 1 × 1 × 1 nm 3 . The simulated system had an area of 3.0 × 1.5 µm 2 and a thickness of 2 nm, resulting in 2 layers of cubic cells. Periodic boundary conditions were set to one Figure 13: Stray magnetic field of 3L CrBr 3 at temperatures close to the Curie temperature.
The external magnetic field is 3 mT along NV axis.
repletion in the lateral directions ["setPBC(1,1,0)"]. Temperature was neglected, so assumed to be 0 K. The samples shape was put in using the "ImageShape()" command. Material parameters are stated in the main text. The magnetic damping parameter was set to "alpha=1" for faster relaxation.
Initial magnetization was set using "RandomMag()". The minimum energy state was determined using the "relax()" command. Figure 14: Simulation result of a system with a similar shape to the sample shown in Fig.2 of the main text.