# Magneto-Optical Tools to Study Effects in Dirac and Weyl Semimetals

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Kerr Rotation as a Function of the Electric and Magnetic Fields

_{3}As

_{2}single crystals are shown in Figure 1b. No Kerr signals from the crystal surface were detected for a weak magnetic field (up to 2000 Gauss), as predicted for a system with both inversion and time-reversal symmetries intact. However, when a direct-current electric bias was applied across the sample, we observed impressively large Kerr rotation signals with increasing current density (from 0 to 144 mA·mm

^{−2}), as seen in Figure 1b. The ROT–MOKE curves are proportional to the cosine of the angle between the E and B fields: its absolute value reaches a maximum when they are parallel (0°) and fell to zero when they were perpendicular (90°). This behavior confirmed our expectation that the complex Hall conductivity of a gyrotropic tensor and the breaking of time-reversal symmetry cause the Kerr effect [48]. However, it is not the only explanation: alternatively, the Kerr effect could be caused by spin polarization provoked by the E and B fields. Indeed, the chiral anomaly causes the Weyl nodes to polarize. Charge transfer between a pair of Weyl nodes takes place across the surface of Fermi arcs, which are known to have a high degree of spin polarization as verified by recent photoemission studies [49,50].

#### 3.2. Magneto-Optical Kerr Signal Combined with Anomalous Hall Effect Measurements

_{2}Te

_{3}cooled to T = 2.9 K. Figure 2A demonstrates the development of the polar Kerr angle, θ

_{K}, and the anomalous Hall resistance R

_{xy}. These data were taken from a typical five-layer region of the Cr–(Bi,Sb)

_{2}Te

_{3}film. The two series are shown as functions of the magnetic field strength (increasing or decreasing) [35]. A roughly square hysteresis was clearly visible in the MOKE and AHE curves, representing ferromagnetic ordering with an easy axis perpendicular to the sample plane. It was considered that the film had a Curie temperature. Figure 2B demonstrates the measurement geometry at T

_{c}≈ 18 K, while Figure 2C shows the longitudinal resistance R

_{xx}as a function of the applied field at 2.9 K. Peaks at ±H

_{c}have been observed in comparable materials and are recognized to increase scattering due to magnetic disorder during coercion [28]. Although other films with the same composition, grown in the same molecular-beam epitaxy chamber, have previously demonstrated quantized conductance at T = 1 K, the absolute resistances measured here show that at T > 2.9 K, the conductivity is dominated by surface and bulk channels rather than quantized edge states. During the MOKE experiments, optical illumination can also affect the magnetization of the film. Figure 2D shows the coercive field H

_{c}as a function of the laser power (spot size ~1 µm, T = 3.2 K). These values are derived from the MOKE hysteresis measurements. We can see that the coercivity decreases with illumination between 20 µW and 2 mW, where local heating is likely to dominate. The trend is well fit by a power law, but plateaus at low powers to a value that agrees with the coercivity extracted from the AHE hysteresis without illumination (red arrow). This suggests that during MOKE measurements, a level of optical illumination below 2 µW has little effect on the magnetic properties of the film.

#### 3.3. Magneto-Optical Kerr Signal as a Function of Pump Intensity

_{s}obtained from a large set of hysteresis curves, measured as a function of laser energy density and pulse time delay. All sample films were grown on glass substrates [46]. The data for various time delays (100 fs, 150 fs, 300 fs, 1 ps, and 3 ps) were all normalized to M

_{0}, which is the value of M obtained when the pump beam is blocked. These were plotted against the energy density of the beam. A typical error bar is displayed beside the data rather than drawing similar bars on all points. At first glance, the data appear noisy, but close inspection discloses some consistent trends. In particular, there was a clear difference between thetemporal behaviors seen at low beam energy densities (<20 mJ/cm

^{2}) and high pump energy densities (>35 mJ/cm

^{2}): at low energy densities, the MOKE response reached a maximum around 300 fs just after the excitation ended, and slowly decayed for longer time delays (1–3 ps). As an alternative, at high energy densities, the MOKE response continued to rise long after the excitation was over. Note that at early times, the values of Ms were equal, and the change in behavior happened at the same excitation density. It is important to note that these observations are in excellent agreement with the behavior of the differential reflection with increasing excitation density. As presented in Figure 3b, the rising differential reflection signal continues toward longer time delays as the pump energy density rises. This correlation between the magnetic MOKE signal and nonmagnetic differential reflection signal suggests that the measured dynamics of the former are not entirely magnetic in origin. Moreover, this observation permits the lines to be drawn that assist as directors in Figure 3a and demonstrate the general trends outlined above.

^{2}for any time interval. For all measurements in the picosecond and sub-ps ranges, a residual magneto-optical difference was shown at the maximum possible pump energy density. The saturation of the MOKE response at high pump energy densities is shown in more detail in Figure 3.

#### 3.4. Magneto-Optical Kerr and Faraday Imaging

^{2}. Above this fluence, however, we observed purely optical, helicity-dependent reversal. If F was even a little higher than this threshold value, then magnetization reversal was detected following laser pulses of both helicities, and even a linearly polarized pulse can trigger the reversal. Nevertheless, the size of the reversal area appears to be slightly larger if the helicity of the laser pulse favors the helicity-dependent reversal at a lower energy density. This is shown in the last row in Figure 5b. At laser energy densities greater than ~4.7 mJ/cm

^{2}, a multi-domain state forms. As demonstrated in Figure 5b, the Gaussian spatial profile of high-energy density laser pulses (F > 4.5 mJ/cm

^{2}) can be separated into four regions, corresponding to the four types of effect that the pulses can have on the magnetic medium: no reversal; helicity-dependent, purely optical reversal; helicity-independent reversal; and the formation of a multi-domain state. Note that the area containing helicity-dependent magnetization reversal is much smaller than the laser spot.

## 4. Discussion

_{2}laser making excitation near the Weyl node energies [54]. Recent experiments have shown that it is possible to produce a femtosecond pulse < 1 ps until a wavelength of 15 µm [55].

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The experimental MOKE setup and (

**b**) the rotational MOKE data of Cd

_{3}As

_{2}bulk crystals. Magnetic field rotation in the y–z plane is parallel to the sample surface. The input laser is p-polarized, with a wavelength of 670 nm. The rotational MOKE signals of the Cd

_{3}As

_{2}crystals under different current densities and a constant magnetic field accepted a cosine-function dependence on y. Here, y is defined as the angle between E and B. The magnetic field was fixed at 2000 Oe (from [34]).

**Figure 2.**Synchronized MOKE and AHE measurements in a film of Cr–(Bi,Sb)

_{2}Te

_{3}at T = 2.9 K (from [35]). (

**A**) Magnetic hysteresis loops presenting an anomalous Hall resistance R

_{xy}(red squares) and a Kerr angle θ

_{K}(blue triangles). They depend on the magnetic field strength. MOKE was done with a spot size of ≈1 µm and a laser power of ≈1 µW. (

**B**) Diagram of the experimental setup. MOKE measurements were made on the unused leg of the Hall bar to avoid optical effects on transport. (

**C**) Magnetic hysteresis with a longitudinal resistance R

_{xx}. (

**D**) Coercive field H

_{C}of the film from MOKE hysteresis measurements as a function of the MOKE laser power. The red arrow specifies the H

_{C}value obtained from AHE hysteresis without illumination (from [35]).

**Figure 3.**(

**a**) The normalized spontaneous magnetization in response to the laser energy density for few time delays in the picosecond and subpicosecond range. (

**b**) The differential reflection signal as function of the time delay for two pump energy densities: 13.3 mJ/cm

^{2}and 35 mJ/cm

^{2}(from [46]).

**Figure 4.**(

**a**) Schematic of the time-resolved MOKE microscope. The numbers “1” and “2” represent the alternative shutter positions for a fast pump–probe. (

**b**) The magnetic reversal in a permalloy sample (from [47]).

**Figure 5.**(

**a**) Plan of the single-shot time-resolved Faraday magneto-optical microscopy experiment and (

**b**) steady-state magnetooptical images of the Gd

_{26}FeCo sample at room temperature. The images were taken after a single 100-fs σ

^{+}- or σ

^{−}-polarized laser pulse with energy density F. Before each pulse, the sample was returned to a single-domain state by applying a uniform magnetic field pulse (from [51]).

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Cheskis, D.
Magneto-Optical Tools to Study Effects in Dirac and Weyl Semimetals. *Symmetry* **2020**, *12*, 1412.
https://doi.org/10.3390/sym12091412

**AMA Style**

Cheskis D.
Magneto-Optical Tools to Study Effects in Dirac and Weyl Semimetals. *Symmetry*. 2020; 12(9):1412.
https://doi.org/10.3390/sym12091412

**Chicago/Turabian Style**

Cheskis, Dima.
2020. "Magneto-Optical Tools to Study Effects in Dirac and Weyl Semimetals" *Symmetry* 12, no. 9: 1412.
https://doi.org/10.3390/sym12091412