Extreme anti-reflection enhanced magneto-optic Kerr effect microscopy

Magnetic and spintronic media have offered fundamental scientific subjects and technological applications. Magneto-optic Kerr effect (MOKE) microscopy provides the most accessible platform to study the dynamics of spins, magnetic quasi-particles, and domain walls. However, in the research of nanoscale spin textures and state-of-the-art spintronic devices, optical techniques are generally restricted by the extremely weak magneto-optical activity and diffraction limit. Highly sophisticated, expensive electron microscopy and scanning probe methods thus have come to the forefront. Here, we show that extreme anti-reflection (EAR) dramatically improves the performance and functionality of MOKE microscopy. For 1-nm-thin Co film, we demonstrate a Kerr amplitude as large as 20° and magnetic domain imaging visibility of 0.47. Especially, EAR-enhanced MOKE microscopy enables real-time detection and statistical analysis of sub-wavelength magnetic domain reversals. Furthermore, we exploit enhanced magneto-optic birefringence and demonstrate analyser-free MOKE microscopy. The EAR technique is promising for optical investigations and applications of nanomagnetic systems.

Second, we present that the EAR technique can work universally on a variety of magnetic media. As shown in Supplementary Figure 2, we were able to demonstrate EAR and Kerr amplitude enhancement for 10nm-thick Ni, Co, and Fe films. Here, the target wavelength is 660 nm. The spectral behaviours of EAR and Kerr amplitude are almost the same regardless of the type of magnetic medium. Detailed information on the thicknesses of the phase-matching and phase-compensation layers for the examined magnetic media is shown in Supplementary Table 2 Table 2 Thickness information of EAR Co/Pt for different target materials of Ni, Co, Fe.
Third, we examined the utilization of the extreme anti-reflection (EAR) platform for the following cases: the atomically thin Co film with a thickness of 0.3 nm and the bulk Co substrate (see Supplementary Figure 3 below). We employed a SiO2/Pt/SiO2 multilayer to support EAR for both cases, as shown in Supplementary Figs. 3(a) and 3(e). First, the atomically thin Co monolayer film placed on top of the EAR multilayer exhibits an extremely enhanced Kerr amplitude up to ~80 degrees at the target wavelength (660 nm), as shown in Supplementary Fig. 3(d). We note that such a configuration, in which the atomically thin magnetic layer is placed on the top of the EAR multilayer, is highly promising for utilizing the compelling magnetic two-dimensional (2D) materials, including CrI3, CrBr3, and FePS3 [1][2][3]. Second, the bulk Co substrate beneath the EAR multilayer can also have an enhanced Kerr amplitude up to ~62 degrees at the target wavelength ( Supplementary Fig. 3(h)). The SiO2/Pt/SiO2 multilayer provides not only the 180-degree out-of-phase but also the identical amplitude required for realising EAR.
Fourth, we examined the applications of the extreme anti-reflection (EAR) platform to the magnetic film with a small Voigt constant (see Supplementary Fig. 3(i)-3(k)). First, we suppose a 10-nm-thick Co layer with a smaller Voigt constant with a factor of ξ, which varies from 10 0 to 10 -2 , than the natural Co layer, keeping its non-magneto-optic permittivity. The weakly-magnetised Co film is coupled with two thin SiO2 spacer layers and a bottom Al mirror, as similar to the cases of the Co/Pt and Pt/Co/Pt/Ta media in the main manuscript. At the target wavelength (660 nm), the magnetic film with a Voigt constant smaller than 10 times (100 times) than the natural Co layer exhibits a Kerr amplitude enhanced up to as large as 79. 1 (27.4) degrees, as shown in Supplementary Fig. 3(k). On the other hand, as shown in Fig. S3(j), the bare magnetic layer, of which the Voigt constant is 10 times (100 times) smaller than the natural Co layer, gives the Kerr amplitude of only about 4.54×10 -2 degrees (4.55×10 -3 degrees), which the MOKE microscope with a common configuration hardly resolves.  [4], this result demonstrates that our technique can be applied in various research areas with less restriction. We clarify that the main target of our study is to present an advanced optical platform for investigating heterostructure films of magnetic transition metals (e.g., Co, Fe, and Ni) and paramagnetic metals (e.g., Pt, Pd, Ir, and Ta) deposited by sputtering process. This is because those heterostructures are the core magnetic systems not only in academic researches like spin orbit torque studies [5] but also in industrial applications such as magnetic random-access memory (MRAM) [6]. It is known that the sputtered magnetic films are rather compatible with arbitrary substrates. For epitaxial magnetic films, layer transfer and wafer bonding techniques can bring a chance to employ extreme anti-reflection. For example, the transfer of an epitaxial yttrium iron garnet (YIG) layer without notable quality degradation from the gadolinium gallium garnet (GGG) substrate to the Si substrate has been demonstrated [7,8]. If a specific bottom substrate for epitaxial growth can never be removed, a similar approach to the simulations in Supplementary Figs. Finally, we considered to apply the EAR platform to the media with in-plane magnetisation in terms of the longitudinal-MOKE (l-MOKE) and transverse-MOKE (t-MOKE) measurements (Supplementary Figure 4).
Here, we examined a 10-nm-thick magnetic layer of which the optical permittivity is identical to the natural Co layer. l-MOKE and t-MOKE, in general, have amplitudes one order smaller than p-MOKE and require oblique incidence of light. We optimised the EAR multilayers for the incidence with an angle of 60 degrees.

Supplementary Note 2. Optimization of the phase-matching and phase-compensation layers
We employed the anisotropic transfer matrix method based on the Stokes vector and the Mueller matrix [9] to calculate the birefringent magneto-optic reflection of the EAR multilayer. The 4×4 transmission matrix (Mmatrix) contains all possible light reflection and transmission routes inside the multilayer structure. By decomposing the M-matrix into four 2×2 submatrices (G, H, I, and J), we can calculate the Fresnel reflection coefficients (rpp, rps, rsp, and rss) as follows: The M-matrix is given by the serial multiplication of the boundary matrix (Aj) and the propagation matrix (Dj) of the jth layer. Employing Aj and Dj of the six optical layers, (1) Air superstrate, (2) SiO2, (3) AlOx/Co/Pt or AlOx/Pt/Co/Pt/Ta, (4) SiO2, and (5) Al, we can represent the M-matrix as When the yz-plane is the incident plane, the boundary matrix Aj is given as Here, and are the Voight parameter and the refractive index of the jth layer, respectively, and ⃗⃗ = Measured spectra of the non-MO reflection amplitude |rxx|, MO reflection amplitude |rxy|, and Kerr amplitude from the bare and EAR Co/Pt layers.

Supplementary Note 3. Experimental tolerance of the EAR platform to the uncertainties of layer thickness
We experimentally verified the tolerance of our EAR platform to the uncertainties of layer thickness.
Employing spectroscopic ellipsometry, we measured the thicknesses of the top and bottom SiO2 layers and the AlOx/Co/Pt film at nine different positions on the 10 mm × 10 mm sample ( Supplementary Fig. 7(a) and 7(b)).
Here, we considered the AlOx/Co/Pt film as a homogeneous medium with a complex refractive index of 2.25+3.61i and evaluated the effective thickness of the magnetic layer. Due to the non-uniformity of fabrication processes, the thicknesses of the layers change depending on the position, as shown in Supplementary Fig. 7(c).

Supplementary Note 4. Quadratic relation between the MOKE intensity and magnetisation
We consider a vertically magnetised medium of which the magnetisation is saturated to the +z direction.
Under the x-polarised incidence with an electric amplitude E0, the reflected electric field is given as Supplementary Figure 8 shows the calculated MOKE intensity as a function of the magnetisation for the bare Co/Pt and EAR Co/Pt films. The MOKE intensity from the EAR Co/Pt film changes much larger and more sensitively than that from the bare Co/Pt film. Also, as the analyser angle increases, the relation of the MOKE intensity and magnetisation becomes more linear. In the EAR MOKE imaging (Fig. 3 in the main text), we employed an analyser angle of 10 for a high visibility.

Supplementary Figure 8 Calculated MOKE intensity depending on the magnetisation in the (a) bare and (b) EAR
Co/Pt films. Here, the extinction efficiency of the analyser is 10 3 .

Supplementary Note 5. Visibility of MOKE microscopy
The visibility of MOKE measurement is defined as the ratio between the difference and sum of the MOKE intensities from two fully magnetised media with opposite magnetisations ( = +1 and = −1 ).
Employing the quadratic equation (Eq. S1), the MOKE visibility can be calculated as

Supplementary Note 7. Avoidance of thermal effects on the magnetic domains
The 1-nm-thin Co medium exhibits the ferromagnetic ordering of the vertical magnetisation below the critical temperature ( c ). The power of the incident light should be low enough not to heat the magnetic film and influence the ferromagnetic ordering. We examined the behaviour of magnetic domain reversal depending on the power of the incident light to the EAR Co/Pt film. The incident light with a power below ~200 W does not change the temporal behaviour of magnetic domain reversal and the final MOKE intensity (Supplementary Fig.   10a). On the other hand, the incident light of >200 W shows notable heating effects; the speed of magnetic domain reversal becomes faster as the incident power increases (Supplementary Fig. 10b). In the measurements of Fig. 4 in the main text, we set the incident power to 110 W that is low enough to avoid the thermal effects but high enough to employ almost the full dynamic range of the photoreceiver.

Supplementary Note 8. Detection of a single Barkhausen jump event
In our measurement setup, the output power fluctuation of the employed laser diode dominates in the signal fluctuation. The intrinsic electric noise fluctuation of the photodetector (Newport 2151 fW-detector) is smaller than the resolution of the employed analog-to-digital (AD) converter (National Instruments PCI-6111). In Fig. S11, which shows the detection of a single Barkhausen jump event in the EAR Co/Pt film, we can also identify the standard deviation of the signal fluctuation ( ) and the resolution of the AD converter ( ). The signal fluctuation, which cannot be observed when the laser is turned off, is ~1.3 times larger than the resolution of the AD converter. Here, the signal fluctuation corresponds to the area of ~35.8 2 nm 2 (=1.28×10 -3 m 2 ). In Fig. 4 in the main manuscript, we collected the stepwise signal changes by Barkhausen jumps larger than the signal fluctuation. As an example, the stepwise signal change (I) of the single Barkhausen jump event in Fig. S11 corresponds to the area of ~69.7 2 nm 2 . Meanwhile, the electric signal change corresponding to the optical diffraction limit, (/2) 2 , is about 568 mV, which significantly larger than and .
We clarify that the accuracy of measuring the difference between the averaged values of each step of the stepwise behaviour is given by the resolution of the employed analog-to-digital (AD) converter. As shown in Supplementary Figure 11 Detection of a single Barkhausen jump event in the EAR Co/Pt film and identification of the signal fluctuation (δI fluc ) and the resolution of the AD converter (δI ADC ).

Supplementary Note 9. Electric voltage difference from the detector between up-and down-saturated magnetisation
Supplementary Figure12 shows typical raw data of the measured electric voltage from the detector between up-and down-saturated magnetisation for the bare Co/Pt and EAR Co/Pt films. Here, for a clear comparison, we controlled the electric voltage of up-saturated magnetisation to be the same in both cases, but the visibility does not depend on the signal intensity. The EAR Co/Pt film produces the electric voltage difference from the detector of ~7.93 V, which is ~15.86 times larger than that of the Co/Pt film (only ~0.50 V). In the bare Co/Pt film, the significant non-MO background reflection causes the high voltage signal at the down-saturated magnetisation.