Thickness dependence of the interfacial Dzyaloshinskii–Moriya interaction in inversion symmetry broken systems

In magnetic multilayer systems, a large spin-orbit coupling at the interface between heavy metals and ferromagnets can lead to intriguing phenomena such as the perpendicular magnetic anisotropy, the spin Hall effect, the Rashba effect, and especially the interfacial Dzyaloshinskii–Moriya (IDM) interaction. This interfacial nature of the IDM interaction has been recently revisited because of its scientific and technological potential. Here we demonstrate an experimental technique to straightforwardly observe the IDM interaction, namely Brillouin light scattering. The non-reciprocal spin wave dispersions, systematically measured by Brillouin light scattering, allow not only the determination of the IDM energy densities beyond the regime of perpendicular magnetization but also the revelation of the inverse proportionality with the thickness of the magnetic layer, which is a clear signature of the interfacial nature. Altogether, our experimental and theoretical approaches involving double time Green's function methods open up possibilities for exploring magnetic hybrid structures for engineering the IDM interaction.

(0.14 ~ 0.83 T). When the in-plane magnetic field is larger than 0.5 T, the magnetization is out-of-plane, while it is smaller than 0.5 T, the magnetization is tilted. The schematic magnetization directions are shown as arrows in the left side of each spectrum. The dashed lines indicate Rayleigh scattering came from the interferometer shutter. When t CoFeB < 1.6 nm, the K eff × t CoFeB starts to deviate from the linear behaviour, and DM interaction shows the same behaviour.

Brillouin light scattering (BLS) technique and interfacial Dzyaloshinskii-Moriya (IDM) interaction
In order to determine the interfacial Dzyaloshinskii-Moriya (IDM) interaction, we measure the frequency difference (f) as a function of magnetic field or wavevector of the propagating spin wave (SW) by performing Brillouin light scattering (BLS).
All BLS data are governed by the so-called Damon-Eshbach (DE) mode including the contribution of the IDM interaction 1 : where, 0 f is the SW frequency without the IDM contribution, H ext , K u , A ex , p and 4 sin During the measurements where we vary the k-vector, we fix the applied magnetic field at 0.915 T. As shown in Fig. 3b in the main text and as expected from Supplementary Eq. (1), f varies linearly with the k-vector as described by: where  and D are the gyromagnetic ratio and the IDM energy density, respectively. By linear fitting, we directly extract the IDM energy. We conclude that both magnetic field and kvector dependent measurements are well described by Supplementary Eq. (1) and (2).

Advantages of BLS to determine the IDM interaction energy density
Many different techniques are currently employed to study the IDM interaction, such as ferromagnetic resonance (FMR) 2 , domain wall motion 3,4 , and spin-polarized electron energy loss spectroscopy (SPEELS) 5

Asymmetric SW Dispersion for Out-Of-Plane Magnetization Geometry
Asymmetric SW dispersion is a finger print of the IDM interaction when the magnetization is in-plane configuration. Our experimental conditions are satisfied this conditions. However, Cortés-Ortuño et al. 11 pointed out the asymmetry vanishes when the magnetization is out-ofplane. Therefore, it must be examined in our experiments by reducing the in-plane applied field. BLS SW spectra with various in-plane magnetic fields of the 1.2-nm-thick Co sample are shown in Supplementary Fig. 1. In this figure, the largest peak which occurs around 0 GHz is due to elastically scattered light, so-called "Rayleigh scattering", which is not related with magnetic signals. The peaks around 15 ~ 20 GHz with 0.83 T are typical BLS signals from the Pt/Co/AlO x sample. The closed circles are measured spectra and the open circles are mirror spectra in order to show clearly the frequency differences. We only show rather large fields (> 0.5 T) spectra in the manuscript due to the measurement limitation of our BLS system. The vertical dashed lines which indicate near Rayleigh scattering came from the interferometer shutter, are unavoidable. For the case of H ext < 0.33 T in our data, the spin orientations are changed from in-plane to out-of-plane (the blue arrows schematically indicates the magnetization directions). When the applied magnetic fields are less than 0.33 T, the peak position cannot be determined correctly, because of the shutter. Moreover, when the applied magnetic field is 0.14 T, the SW intensity is too small to confirm the correct peak positions. Therefore, unfortunately, we are not able to obtain meaningful spectra for fields smaller than 0.5 T, and this is the reason why we only show spectra for rather large fields where the magnetization direction is in-plane. Because of the limitation of our measurement system, we cannot determine whether the asymmetric dispersion is vanished for the out-ofplane magnetization or not.

Supplementary Note 4 The non-linear behaviour of frequency difference of Pt/CoFeB/AlO x
In Fig. 2b in the main text, the frequency difference of Pt/CoFeB/AlO x shows a maximum value at 1.6 nm, while Pt/Co/AlO x shows clear linear behaviour. Physical reason of such nonlinear behavior of Pt/CoFeB/AlO x must be addressed. In order to resolve the un-expected behavior of Pt/CoFeB/AlO x , we plot together K eff × t CoFeB and DM energy density via t CoFeB (thickness of CoFeB) in Supplementary Fig. 2. It is clear that the linear behaviour is broken in the K eff × t CoFeB vs. t CoFeB plot, when t CoFeB < 1.6 nm. Based on our observation, we speculate the interface quality is changed due to the too thin ferromagnetic layer. Such deviation is usually observed in K eff × t vs. t plots for PMA materials (see Supplementary Note 6). The onset of the non-linear behaviour in the frequency difference or DM energy density is exactly the same thickness. Therefore, it implies that the non-linear behaviour in Fig. 2b in the main text is due to the degradation of the interface quality.

Supplementary Note 5 The SW k-vector dependent BLS measurements for a symmetric sample.
In this section, we discuss the BLS measurements for nominal symmetric-interface samples such as Pt (4 nm)/Co (0.6 nm)/Pt (4 nm). As described in the main text, from this nominally symmetric structure we expect negligible or zero IDM interaction. SW k-vector dependent measurements are performed similarly as used for Fig. 4 of the main text and are shown in Supplementary Fig. 3. Open navy circles in Supplementary Fig. 3a indicate the SW dispersion relation. Due to the limited k-vector range, we only observe the symmetric dispersion, which implies a small IDM interaction. Supplementary Fig. 3b shows the correlation ( ) between the frequency differences and SW k-vector. No significant IDM interaction is observed by using BLS. To elucidate, two reasons are suggested; first, our examined system is more symmetric compared to the other reports (Refs. 8 and 13 in main text), and second, the IDM interaction might be small and cannot be detected by BLS as a small f falls within the detection limit. Therefore, a small frequency, which indicates a small or negligible IDM energy density cannot be identified by BLS.
For the BLS measurements, a tandem interferometer with a free spectral range (FSR) of 75 GHz and a 2 8 multichannel analyser is used. The frequency resolution in the measured Stokes and anti-Stokes peaks in the BLS spectra can be determined by using FSR/2 8 GHz. Therefore, the frequency resolution of the BLS setup is approximately 0.29 GHz. Since the correlation between the frequency difference and the IDM energy density is given by 2

Determination of the saturation magnetization and anisotropy energies
In this section, we demonstrate the SW dispersion relation without the IDM interaction. First, in order to define the SW frequency without the IDM interaction, the median value of the Stokes and anti-Stokes peak are taken to determine the perpendicular magnetic anisotropy energy and the saturation magnetization. As illustrated in Supplementary Fig. 4a, the applied magnetic field dependence of SW are measured by BLS for various Co thicknesses (t Co = 1.0, 1.2, 1.4, and 1.6 nm). Since the applied magnetic field is perpendicular to the magnetization creating the surface SW mode, the SW excitation frequencies are given by 12 : where,  is the gyromagnetic ratio (= 2.37×10 11 T -1 s -1 ),  is the angle between the magnetization and the sample plane, K u is the perpendicular uniaxial anisotropy constant, H ext is the external magnetic field, M s is the saturation magnetization, respectively. In this equation, the contributions of dipolar field and exchange energy have been neglected as is justified in the ultrathin limit. Consequently, the measured SW frequencies and the fitted curves show a good correspondence as shown in Supplementary Fig. 4a. For the case of t Co > 1.4 nm, the frequencies of the propagating SWs differ from the thinner thicknesses, which means that the effective uniaxial anisotropy (K eff = 2K s /t-1/2μ 0 M s 2 ) is changed from positive (out-of-plane) to negative (in-plane) values. To elaborate, we plot the anisotropy energy density (K eff ×t Co ) as a function of t Co in Supplementary Fig. 4b. From this plot, we determine the slope and y-crossing, corresponding to the volume anisotropy (-1/2μ 0 M s 2 ) and the surface anisotropy (K s ) 13 , respectively. This allows us to extract M s directly from the BLS measurements. M s is the only necessary physical quantity to convert the measured f to the IDM energy density. The obtained K s is 0.54 mJm -2 and M s is 1100 kAm -1 , which is about 78.5% of the bulk Co value. and the higher order Green's functions are decoupled by the random phase approximations, the set of differential equations for N-atomic ferromagnetic layers can be obtained 22 .
We define the normalized energy quantities as 0 DJ   , uu k K J  , 11 ss k K J  , and sN N k K J  . k x , k y , and a are the x and y component of the SW vector and the lattice constant, respectively. In these calculations, we assume a simple cubic lattice structure, but this model can be extended for bcc and fcc structures 22 . The DM interaction contribution is developed with the number operator, ˆi