Abstract
In magnetic multilayer systems, a large spinorbit 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 nonreciprocal 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.
Introduction
In the presence of spinorbit coupling at interfaces on lowdimensional magnetic heterojunction structures, the effect of structural inversion asymmetry leads to an additional anisotropic exchange term, namely the interfacial Dzyaloshinskii–Moriya (IDM) interaction^{1,2,3,4} (already predicted by Fert in 1980), which is a branch of the Dzyaloshinskii–Moriya (DM) interaction^{5,6}. This interfacial phenomenon has been recently reilluminated and experimentally demonstrated because of its massive potentials to explore new magnetic behaviours such as chiral domain wall (DW) dynamics^{7,8,9,10,11,12,13} and skyrmions^{14,15,16}. To develop this field of DW devices and skyrmionics (the latter with great promises for superior nanoelectronics devices), experimental tools to extract the magnitude and sign of IDM interaction are urgently required. However, contrary to bulktype DM interaction measurements^{17}, recent extensive experiments clearly observed the existence of the IDM interaction, but magnetic field and electric current driven DW dynamics measurements were definitely linked to the perpendicular magnetic anisotropy (PMA)^{8,9,10,11,12,13}. At present, to further explore independent by the underlying physics of the IDM interaction without any other linked material parameters, a radically different experimental approach is strongly required.
In this article we measure the ferromagnetic layer thickness dependent of IDM interaction quantitatively and qualitatively. Inelastic light scattering, socalled Brillouin light scattering (BLS), is performed to observe nonreciprocal spin wave (SW) dispersion relations affected by the IDM interaction^{18}. The detailed explanation about the BLS is shown in Supplementary Note 1. The advantages of BLS to determine the IDM energy density is described in Supplementary Note 2. Our main findings are twofold: first, the inverse proportionality of the IDM energy densities to the ferromagnetic layer thickness shows that the IDM interaction is purely originated from the interfaces, and second, we present a stateoftheart quantummechanical approach to confirm the asymmetric dispersion relations and the inverse proportionality of the IDM interaction. As representative heterostructures, Pt/Co/AlO_{x} and Pt/CoFeB/AlO_{x} are chosen because these multilayer structures are already predicted to have a large IDM interaction^{12}.
Results
SW Frequency differences due to the IDM interaction
Propagating SWs on a magnetic thin film can be localized at the top and bottom surfaces of the ferromagnetic layer when the wavevector k of the SW is perpendicular to the magnetization of the system. This SW mode is namely Damon–Eshbach (DE) mode (often called surface mode)^{19} and it is indeed one of the appropriate physical quantities to investigate the interface effect, especially affected by the IDM interaction. To realize the DE geometry, we first apply an external magnetic field along the inplane as depicted in Fig. 1a. Simply, BLS measures the scattered light from two interfaces, which contains photons at frequencies shifted by the frequencies of excited SWs. In this inelastic process, the photon loses its kinetic energy (Stokes process) to create one of quasiparticles (SW in our study) or gains energy (antiStokes process) by absorbing one. Consequently, these spectral components can determine the frequencies and intensities of SWs existing at the point in the sample where the incident light is focused (see Methods).
Usually, the SW frequencies of Stokes and antiStokes peaks should be at the same position or slightly different due to the PMA energy difference between top and bottom interfaces of the ferromagnet^{20}. However, recent theoretical and numerical calculation proposes a prominent clue that the frequencies and the attenuation lengths of the travelling SWs with opposite wavevectors (±k) are significantly different due to the IDM interaction, and then these characteristics of the SWs are measurable^{21}. For BLS, the frequency difference (Δf) indicates the mismatch between the frequencies of Stokes and antiStokes peaks. We report that a large frequency difference (Δf=1.99 GHz) for Pt/Co(1.2 nm)/AlO_{x} is obviously observed as shown in Fig. 1b. Now, one crucial fact can be emphasized that the GHz range of the Δf is a clear signature of the IDM interaction. The details will be further discussed later.
Magnetic field dependence
To precisely and systematically investigate this interface effect by means of BLS, two different methods (magnetic field dependence and incident angle dependence) as a function of the thickness of the ferromagnet (t_{FM}) are proposed in this work. We now discuss the details of two different methods successively. The DE SW frequencies (dispersions) including the IDM interaction are given as^{21}:
where f_{0} is the SW frequency without the IDM contribution, H_{ext}, K_{u}, A_{ex}, γ, p and are the external magnetic field, the magnetic anisotropy, the exchange stiffness, the gyromagnetic ratio, the polarity of the magnetic field (p=±1) and the wavevector of the SW, where θ is the incident angle of the light, respectively. Therefore, the definition of the frequency difference is simply derived as , where M_{s} and D are the saturation magnetization and the IDM energy density, respectively. This SW dispersion apparently shows that the Δf is invariant while the magnetic field increases (or decreases). So, the fielddependent measurement allows us to minimize the uncertainties of the measured Δf. The measured SW frequencies of Stokes and antiStokes peaks as a function of the magnetic field for Pt/Co(1.2 nm)/AlO_{x} are shown in Fig. 2a. Two SW frequencies increase with increasing applied magnetic field, but the Δf (the averaged Δf is 2.18 GHz) is indeed a constant (see the inset in Fig. 2a). From these fielddependent measurements, we can convert to the IDM energy densities because the SW wavevector is fixed at k_{x}=0.0167, nm^{−1} (θ=45°). It must be noted that the minimum applied inplane field is 0.5 T is large enough to pull the magnetization in the plane. Therefore, the observed Δf is only for the inplane magnetization, and we cannot conclude that Δf will vanish or not when the magnetization is out of plane. Due to the limitation of BLS measurement setup, it is hard to determine Δf for the outofplane magnetization (see additional Supplementary Note 3 and Supplementary Fig. 1).
In many magnetic systems, interface effects can be identified by an inverse proportionality to the ferromagnetic layer thickness such PMA^{22}, exchange bias^{23}, switching current density of spin transfer torque^{24}, the effective field of the interlayer exchange coupling^{25} and so on. In this point of view, we systematically measure Δf as a function of the thickness of the ferromagnets (Co and CoFeB) and nine data points with different magnetic fields are averaged for each thickness. As shown in Fig. 2b, Δf approaches to the origin when for both Co and CoFeB samples by which we consequently confirm that the IDM interaction for our asymmetric structures is purely originated from the interface. For the thinner CoFeB cases (t_{CoFeB}<1.6 nm), the frequency differences deviate from the inverse proportionality. It implies that the nonlinear behaviour in Fig. 2b is due to the degradation of the interface quality (see Supplementary Note 4 and Supplementary Fig. 2).
SW propagation direction dependence of Δf
We now demonstrate another proof that the asymmetric frequency differences indeed emerge from the interface. In recent previous theoretical work, CortésOrtuño^{26} claims that the frequency differences Δf by the DM interaction can be expressed as:
where α indicates the angle between k_{} and M, and Δf_{0} is a frequency difference at α=π/2. The physical interpretation of equation (2) is that the frequency differences Δf is created by the energy differences of two propagating SWs for both interfaces. Since the IDM interaction introduces these energy differences, this equation is another clear evidence of the DM interaction, especially for the case of the interface effect. Figure 3 shows the angular dependence of the frequency differences between the angle of SW kvector and the direction of M. Figure 3a indicates the case of α=±π/2 (usual BLS measurement geometry, that is, ) and α=0° (90° rotation from usual BLS measurement geometry, that is, k_{}//H). It is clearly shown that Δf (=+1.71, −1.73 GHz) are finite and opposite sign for α=±π/2, while Δf=0.11 GHz for α=0° is less than BLS limitation (∼0.29 GHz, see the Supplementary Note 5 and Supplementary Fig. 3a,b). The systematic angular dependent measurements are shown in Fig. 3b and we overlap the sinusoidal curve from equation (2). As expected, they are in good agreement with each other.
SW kvector dependence
Furthermore, we measure the dispersion relation of SW (frequency versus wavevector) by varying the incident angle θ of the probing light, which determines the selected SW’s wavevector . We note that the magnetic field and k_{x}dependent measurements span two different branches of equation (1), and those two independent measurements can provide more reliable results in the present study. The dependence of f_{DE} on k_{x} are plotted in Fig. 4a for various Co thicknesses. The solid lines correspond to linear fit to the experimental results. For all Co thicknesses, the f_{DE} linearly decreases with increasing k_{x}. Following the equation (1), f_{0} and Δf varies quadratically and linearly with k_{x}, respectively. However, for the limited range of k_{x} (−0.03 nm^{−1}<k_{x}<+0.03 nm^{−1}), the f_{0} are almost constant, accordingly, such significant variation in f_{DE} results from those in Δf. Therefore, these asymmetric and linear dispersion relations can be regarded as the direct evidence that the Δf in our measurement is a consequence of IDM interaction. Recently, the asymmetric SW dispersion relation has been experimentally observed by using spinpolarized electron energy loss spectroscopy in double layer Fe films^{27}. For comparison, we also examined the dispersion relation for Pt(4 nm)/Co(0.6 nm)/Pt(4 nm) representing a symmetric structure, where the IDM interaction at the bottom and top interfaces of the FM are known to be approximately of the same magnitude but with the opposite sign, thus leading to zero IDM interaction. Interestingly, for the symmetric structure, no significant IDM interaction is observed (see Supplementary Note 5). Fig. 4b shows the Δf versus k_{x} for selected Co thickness, t_{Co}=1.1, 1.2, 1.5 and 1.6 nm. For each film thickness, one obtains a clear linear dependence. From the slopes, we can extract IDM energy density using the relation of equation (1) together with the gyromagnetic ratio γ and the saturation magnetization M_{s} deduced from the further BLS measurements.
The IDM energy densities
Next, we convert the measured Δf to the IDM energy densities for our asymmetric structures as shown in Fig. 5. For Co samples, the measured IDM energy densities (D_{H} and D_{k} indicate the IDM energy densities from the field dependence and SW wavevector dependence, respectively) are in excellent agreement each other (see Fig. 5a). Figure 5b shows the measured IDM energy densities for the CoFeB sample. In this case, the effective magnetic anisotropy for all thicknesses is inplane. The maximum IDM energy density is obtained to be about D=1.2 mJ m^{−2} (D=0.7 mJ m^{−2}) for Pt/Co(1 nm)/AlO_{x} (Pt/CoFeB(1.6 nm)/AlO_{x}). Recall that the saturation magnetization (M_{s}) is one unique material parameter to convert the IDM energy density in equation (1). The saturation magnetization M_{s} (equal to 1,100 kA m^{−1} for Co and 948 kA m^{−1} for CoFeB) is determined by BLS measurement as well (see Supplementary Note 6 and Supplementary Fig. 4a,b).
Numerically calculated SW dispersion relation
Two types of BLS measurements (magnetic field dependence and k_{x}vector dependence) are presented so far. From these measurements, we found the inverse proportionality of Δf, which is a typical signature of the interfacial nature and the asymmetric dispersion relation. One of the main goals of the present work is to demonstrate the SW dispersion relation affected by the IDM interaction and the inverse proportionality not only by experiment but also theoretically. In previous work, theoretical evidences based on atomicscale models^{28,29,30} have been reported. Udvardi et al.^{28} predict reciprocal SW dispersion relations for the specific crystallographic orientation in the Fe/W(110) by using firstprinciple calculations, without dipole–dipole interaction and external field, and Costa et al.^{26} provide dynamic susceptibilities (SW frequencies, life times and amplitudes) for ± SW vectors in the one or two monolayer (ML) of Fe on the W(110) based on multiband Hubbard model. CortésOrtuño and Landeros demonstrate reciprocal SW dispersion relations for different crystallographic classes. Here we introduce the numerical calculations for asymmetric SW dispersion relations and inverse proportionality by means of the double time Green’s function technique, it is useful to study the thicknessdependent SW dispersion relations. This technique is well developed in statistical physics^{30} and magnetism^{31,32}. The Hamiltonian with the IDM interaction for the finite thickness ferromagnetic layer in terms of the spin operators is given by^{32,33}:
where g is the Landé gfactor, J_{ij} and D_{ij} are the isotropic interatomic Heisenberg and anisotropic DM exchange energies between the ith and jth spins, and K_{u} and K_{s} are the bulk and surface uniaxial anisotropy energies. In this model, we use different definition of coordinate system, we set the film normal along the zaxis. 〈i,j〉,〈i,j〉’, and 〈i,j〉” denote the summations of the nearest neighbours. Here 〈i,j〉 is summation of all spins, 〈i,j〉’ is for top and bottom interfaces, and 〈i,j〉” is only at the bottom interface where we assumed a heavy metal is placed. Therefore, we assume that the DM interaction exists only at the bottom interface. The SW dispersion relations can be obtained by solving equation (3). The detailed explanations and full derivations are shown in Supplementary Note 7 and Supplementary Fig. 5a,b.
Figure 6a shows numerically calculated SW dispersion relations for a ferromagnetic ML with the IDM interaction term δ_{0}=SD/J_{ex}a, where S, a and J_{ex} are spin number 1/2, the lattice constant and the exchange energy, respectively. A parabolic SW dispersion relation (black line) is obtained when δ_{0}=0, when δ_{0} is nonzero, parabolic SW dispersion relations are shifted as given by equation (1) and shown by the red and blue lines for different strengths of δ_{0} ferromagnetic ML. As mentioned above, the SW vector of our BLS setup is limited from 0.0099 to 0.0205, nm^{−1}; the small range is indicated by the green box in Fig. 6a. That is the reason that we obtained only linear behaviours of f_{DE} in Fig. 4a and one can be pointed out that this numerical result can sufficiently support our experimental data. Finally, the inverse proportionality of the IDM energy density as a function of the thickness of the ferromagnetic layer (t_{FM}) is shown in Fig. 6b and the inset indicates asymmetric SW dispersion relations for δ_{0}=0.1. In this calculation, we consider the thickness of the ferromagnetic layer from 2 ML to 20 ML. These full numerical SW dispersion relations reflect our experimental observations very well. First, the SW frequencies at k_{x}=0 increase with increasing t_{FM}. The experimental results show the same trend in Fig. 4a. Since the SW frequency is related with the interface PMA energy, it must be increased with increasing t_{FM} (see Supplementary Fig. 5a). Second, the parabolic SW dispersion relations have an additional linear k_{x}. Because the coefficient of a linear k_{x} term is proportional to the D_{k}, we can directly extract D_{k} from the SW dispersion relations. Very recently, there is another numerical and theoretical approach about the interface exchange boundary conditions for the classical linear dynamics of magnetization^{34}. This profound and accurate prediction also shows the inverse proportionality of the frequency difference and the results are consistent with our experimental and numerical data.
Discussion
In conclusion, using a versatile lightscattering technique, we have observed the IDM interaction in the inversion symmetry broken systems. The quantitative magnetic layer thicknessdependent measurements and careful analysis show the inverse proportionality of the frequency differences and confirm that the IDM interaction is a pure interfacial effect with maximum energy density of 1.2 mJ m^{−2} for Co with Pt underlayer. Furthermore, two different measurement methods, the magnetic field dependence and SW wavevector dependence, allow us to obtain identical results. These findings take us a step closer to boosting the IDM interaction leading to (meta) stable skyrmion states for future data and memory devices. Finally, our numerical calculations confirm the asymmetric SW dispersion relations due to the IDM interaction and the inverse proportionality.
Methods
Thin film deposition
The sample of Pt(4 nm)/Co(0–2 nm)/AlO_{x}(2 nm) and Pt(4 nm)/Co_{48}Fe_{32}B_{20}(0–2 nm)/AlO_{x}(2 nm) were prepared on Si/SiO_{2} substrates using DC magnetron sputtering with a base pressure of ∼7 × 10^{−8} mbar. To investigate the thickness dependence of IDM interaction, the ferromagnetic layers were grown in a wedge shape over 2 cm wide wafers with the help of an in situ moving shadow mask. AlO_{x} layer was obtained from plasma oxidation of 2nmthick Al layer as deposited on top of the ferromagnetic layers. The plasma oxidation process was carried out for 10 min in an in situ isolated chamber with a 0.1 mbar background pressure of oxygen and a power of 15 W.
Brillouin light scattering
The samples are pasted on an angle controlled sample holder for the BLS measurement. The BLS spectra are measured by using a (3+3) pass tandem Fabry–Perot interferometer and a ppolarized (300 mW power and 532 nm wavelength) single longitudinal mode LASER is used as a light source. The DC external magnetic field is applied parallel to the film surface and perpendicular to the scattering plane. The backscattered light from the sample is focused and collected. The spolarized light is passed through the interferometer and the photomultiplier tubes^{35}. All measurements are performed at room temperature. We use the applied magnetic field (0.01–1.18 T) and incident angle of light (25°–60°) corresponding to k_{x}=0.0099–0.0205, nm^{−1} for magnetic field dependence and dispersion relation measurements, respectively. The accumulation time for each spectrum was about 60 min.
Additional information
How to cite this article: Cho, J. et al. Thickness dependence of the interfacial Dzyaloshinskii–Moriya interaction in inversion symmetry broken systems. Nat. Commun. 6:7635 doi: 10.1038/ncomms8635 (2015).
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Acknowledgements
This work is supported by the research programme of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO), and National Research Foundation of Korea (Grant No. 2013R1A1A2011936) and KIST institutional programme (2E24882).
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C.Y.Y. and J.S.K. conceived the project; sample fabrication was done by J.S.K. and Y.Y.; the measurements were performed by J.C., N.H.K., N.J.J.v.H. and A.S.; data analysis and manuscript preparation were done by J.S.K., J.C., R.L., D.S.H., Y.Y. and C.Y.Y.; numeral calculation was done by C.Y.Y; all authors discussed the results.
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Supplementary Figures 15, Supplementary Notes 17 and Supplementary References. (PDF 1120 kb)
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Cho, J., Kim, N., Lee, S. et al. Thickness dependence of the interfacial Dzyaloshinskii–Moriya interaction in inversion symmetry broken systems. Nat Commun 6, 7635 (2015). https://doi.org/10.1038/ncomms8635
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