Canted standing spin-wave modes of permalloy thin films observed by ferromagnetic resonance

Non-collinear spin structures in materials that combine perpendicular and in-plane magnetic anisotropies are of great technological interest for microwave and spin wave-assisted magnetization switching. [Co/Pt] multilayers are well-known perpendicular anisotropy materials that have the potential to pin the magnetization of a soft magnetic layer, such as permalloy (Py), that has in-plane anisotropy, thereby forming a magnetic exchange spring. Here we report on multilayered [Co/Pt]/Pt/Py films, where an additional ultrathin Pt spacer has been included to control the coupling between the sub-units with in-plane and perpendicular magnetic anisotropy. Vector network analyser (VNA)-ferromagnetic resonance (FMR) measurements were made to obtain a complete picture of the resonant conditions, while the dynamical response of the sub-units was probed by synchrotron-based element- and phase selective x-ray detected FMR (XFMR). For all samples, only slight pinning of the dynamic magnetization of the Py by the [Co/Pt] was noted, and the FMR results were dominated by the 50 nm thick Py layer. Out-of-plane VNA-FMR maps reveal the presence of additional modes, e.g. a perpendicular standing spin-wave (PSSW) state. However, as the magnetic field is reduced below the saturation field, the PSSW state morphs continuously through a series of canted standing spin-wave (CSSW) states into a horizontal standing spin-wave (HSSW) state. The PSSW, CSSW and HSSW states are well described using a multilayer model of the Py film. The observation of CSSW modes is of particular relevance to microwave assisted magnetic recording, where microwave excitation stimulates precession of a soft layer canted out of plane by a pulsed magnetic field.

. In-plane VNA-FMR |S 12 | results for (a) Py (50), and (b) [Co/Pt] 10  as a line running parallel to that of the uniform k = 0 FMR mode. However, in the low field region μ o M B app 0, the PSSW mode morphs into an HSSW mode, rising up in frequency to ∼9. 45 GHz at B app = 0. It is shown that this behaviour can be understood in terms of a multilayer Py model, based on the linearization method used by Nörtemann et al [37], but modified to include the layer-by-layer local dipolar field approach of Bowden et al [42]. The results are in good agreement with the excited states observed in both 50 nm Py and 100 nm Py films. In addition, the ellipticity of FMR precession is highlighted and discussed.
Finally, in section 8, we summarize our results and give suggestions for further exploration.

Experimental results
The samples were deposited on Al 2 O 3 substrates at room temperature using dc magnetron sputtering. The films were grown on a Pt(30) seed layer and capped with a Pt(3) protective layer. The VNA-FMR measurements were performed on a standard CPW with 50 Ω impedance and with a signal line of 500 μm width. An electromagnet mounted on a rotating stage supplied a bias magnetic field either in the sample plane or normal to the sample plane. The measurements were performed by stepping the magnetic field value and sweeping the microwave frequency in the range 0-20 GHz. The samples were placed face down, with the magnetic layers in close proximity to the CPW. For each applied magnetic field value, a reference measurement was made at high field and subtracted in order to remove the background signal. The in-plane VNA-FMR frequency vs applied field map for the 50 nm thick Py reference sample and [Co/Pt] 10 /Pt(1.5)/Py (50) film can be seen in figure 1, where the colour scale represents |S 12 | the magnitude of the measured scattering matrix parameter. In general, the FMR lines for all investigated samples look very similar and follow the expected behaviour for a simple Py film. For directly coupled samples (without a Pt spacer layer) and with thinner Py thicknesses (<50 nm), the results of VNA-FMR measurements change substantially, due to the formation of an imprinted domain and exchange spring structure [43]. Note that the FMR of [Co/Pt] is generally out of range for our setup in terms of accessible frequency/field values [34]. The data shown in figure 1 can be fitted with the usual Kittel formula: Here, (i) γ = 29.4 GHz T −1 , (ii) μ o M = 1.0 T for Py, and (iii) B K is a small in-plane anisotropy field, which gives rise to the small gap v ∼ 0.5 GHz in zero applied field. From the latter we find B K ∼ 0.0003 T (favouring an in-plane axis).
The out of plane FMR results, for the four samples in question, can be seen in figure 2. Note the ten-fold increase in the magnetic field on going from the in-plane to out-of-plane FMR results.
Using Kittel's formula for out-of-plane FMR (in the absence of any crystal field interactions) we expect: In general, for fields in excess of the saturation magnetization μ o M = 1.0 T there is a sharp linear rise in the FMR frequency, as expected. However, for fields below 1.0 T, the FMR frequency is certainly non-zero, and is characterized by a sharp cusp at B app ∼ 1.0 T (see in particular figure 2(c)). Note that the strong absorption below B app ∼ |0.2| T is a background signal arising from the CPW and coaxial cables. In the following section, we provide an explanation of many of the features seen in figure 2. However before leaving this experimental section, consider also the results of XFMR experiments [15,16,44,45]. XFMR measurements were carried out on beamline 4.0.2 of the advanced light source by monitoring absorption/transmission as a function of time delay between a synchronized RF magnetic field (pumping the spin precession) and circularly polarized x-ray pulses (probing the oscillatory magnetization component along the x-ray wavevector). The x-rays were incident at a grazing angle of 40 • . The sample was placed face down on a CPW with a countersunk hole of 500 μm diameter allowing the incident x-ray beam to access the surface of the sample, while the transmitted x-rays were converted to optical radiation through x-ray excited optical luminescence in the Al 2 O 3 substrate, with the emitted light detected by a photodiode mounted behind the sample.
By tuning the x-ray energy to the absorption edge of the element of interest, the XFMR signals from the Py (Fe L 3 edge) and [Co/Pt] (Co L 3 edge) were detected separately, allowing direct measurement of the FMR within each layer. Both the amplitude and phase of precession were extracted by fitting a sine wave to the XFMR delay scans. The amplitudes of the Fe and Co moments, normalized to their static XMCD signals are shown for two samples [Co/Pt] 10 /Py(50) and [Co/Pt] 10 /Pt(1.5)/Py(50), in figures 3(a) and (b), respectively. While for directly coupled layers both [Co/Pt] and Py precess with the same amplitude, it is clear that the Co spins in the [Co/Pt] layer and the Fe spins in the Py film are dynamically decoupled when the Pt spacer exceeds 1.5 nm; thereby providing extra justification for our earlier general statement that the FMR results appear to be dominated by the 50 nm Py film alone. The sole exception appears to be that of the directly coupled [Co/Pt]/Py film shown in figure 2(d), where the cusp in the FMR rises to ∼6 GHz. Initially, it was believed that such behaviour could signal the presence of an exchange spring. However, in the sections 3 and 4 below, a totally different interpretation of figure 2(d), is presented and discussed.

Quantum mechanical description of FMR: single slab
In this section, we concentrate primarily on the out-of-plane results presented in figure 2.
The right-handed coordinate system used in this work can be seen in the inset of figure 4. Initially, the magnetization should lie along the in-plane x-axis, due to (i) the strong demagnetization factor associated with the out-of-plane z-axis, and (ii) a small in-plane anisotropy, discussed earlier. However, when the field is applied along the z-axis the magnetization will rotate out-of-plane making an angle θ with respect to the z-axis.
Next, we follow the quantum mechanical approach pioneered by Van Vleck [36]. Here, the magnetization is envisaged as a single spin J, much in the way of Kittel [3]. We adopt this approach, because it paves the way for a multilayer model (section 6), where the spins are coupled through a Heisenberg exchange interaction.
The time dependence of the angular momentum operator J is given by; Here H is the Hamiltonian, and the angular momentum J is assumed dimensionless. Thus, the necessary commutation relationships, used to evaluate equation (3), take the form J x , J y − = iJ z etc.
For the Py film in question, the Hamiltonian is given by: The individual terms are as follows. The Zeeman (ZM) interaction: The crystal field Hamiltonian: (Note that the anisotropy parameter K A must be negative to favour the in-plane x-axis.) The dipolar field contribution arising from all the dipoles in the film takes the form: Here: Note that equation (8) contains both the demagnetizing term and the Lorentz field. Some readers may question this approach, given that the Lorentz field per se cannot contribute to the torque on the magnetization of the slab. However, in sections 5 and 6, we generalize the single slab model discussed here, to a multilayer model. In the latter, the Lorentz field μ 0 M 3 , only holds if the spins of all the layers are collinear (see Bowden et al [42]). Thus, for both k = 0 standing spin wave modes and exchange springs, it is necessary to generalise equation (8). Finally, we note that while the above QM approach holds only for T = 0 K, the results obtained by this method are the same as those given by Kittel. For the Zeeman interaction, anisotropy, and local dipolar field contributions, in question, we find: (Note the presence of the anti-commutators.) (iii): Next, we assume all the moments are in the z-x plane, at an angle θ with respect to the z-axis. Consequently: Thus: Summing up therefore: Note that we have followed the usual convention in defining the anisotropy field: Alternatively, in terms of magnetic moments: Equation (16) is the master equation, referred to earlier. Note that the torque is solely about the y-axis. We are now in a position to determine the equilibrium angle of magnetization θ, and hence the elementary ground state excitations. In equilibrium, the torque must be zero. Thus: Next, we use the master equation (16), now equipped with the equilibrium angle θ, to determine the elementary excitations using the process known as linearization μ x → μ 0 x + dμ x e iωt etc (see Van Vleck [36] and Nörtemann et al [37]). Here, the dμ x etc, are small time-dependent deviations away from the equilibrium values. Thus, on collecting the first order terms (linearization), we find: Consequently, we have a solution for dμ x etc, provided: Some results for the Py(50) film can be seen in figures 4(a) and (b). In general, the excitation spectra in figure 4(b) replicate the general features found in the out-of-plane FMR results shown in figure 2. In particular, the presence of a small anisotropy field B K ensures that the FMR frequency is non-zero for fields B z app < μ 0 M. However, the most striking feature of the FMR calculation of figure 4(b) is that of the sharp downturn in frequency (cusp), to zero, at B z . We believe that this may be an example of the Goldstone theorem: in a system with a continuous broken symmetry, there may exist excitations with arbitrary low energy. At the cusp, the spins approach the angle of 0 • very rapidly. Explicitly: Consequently, as θ → 0, the derivative dθ dB z app → ∞, and concomitantly ω → 0. In physical terms, the spins become very mobile at the cusp. Thus, it is energetically very easy to set-up a uniform k = 0 spin-wave. Alternatively, we can describe the above behaviour as a 'soft-mode'. In this regard, we note that Kuanr et al [46] have reported soft mode behaviour, in nanosized Py strips. However, the latter gave no interpretation of their soft mode, which is not the same as that under discussion here. On course, sharp features in any characteristic curve, invite mode mixing of one kind or another. However, before examining such possibilities (e.g. magnetostatic modes), it is probably wise to examine how the cusp behaves with slight misalignment of the applied field (applied along the normal to the plane). As we shall see, it turns out that alignment of the magnetic field perpendicular to the Py film is critical.

FMR with slight misalignment of out-of-plane applied field
First, we acknowledge that this problem has been addressed, many years ago, by Tannenwald and Seavey [47] (see their figure 2). However, the latter do not include an in-plane anisotropy. So, unlike figure 4(b) above, resonance was not observed by Tannenwald and Seavey [47], until the applied field B z app , normal to the plane, exceeds μ 0 M.
Suppose the applied field is slightly misaligned in the z-x plane. The master equation (16) is transformed to: Thus, we have a new solution for the equilibrium angle, which requires a numerical solution: Note that this solution readily reverts to that of equation (12), when B x app = 0. Turning now to the dynamics we find: Consequently, we have a solution for dμ x etc, provided: The results of this calculation, for small misalignments of the applied field, can be seen in figures 4(c)-(d). In summary therefore, the out-of-plane VNA-FMR k = 0 more results can be well-understood in terms of small misalignments of the applied field, with respect to the film normal. In appendix B more VNA-FMR diagrams can be seen, showing how the cusp changes with slight field adjustments about the normal to the film.
This completes our discussion of the k = 0 uniform mode. However, as noted earlier, the out-of-plane 2D VNA-FMR maps also contain additional information about k = 0 modes. However, before we examine these modes, it is advantageous to detail a multi-layer model of the Py film, which can be used to accurately model the properties of the PSSW, CSSW, and HSSW modes.

Multilayer Py model
Essentially, we follow the multilayer model described by Utsumiya et al [48] (and references therein), used to interpret the exchange spring behaviour in pinned L1 0 -FePt/Py exchange-spring films. However, we differ from former in that local dipolar fields are included, using the layer by layer model of Bowden et al [42]. In addition, we also make use of the 'double differential energy matrix' detailed by Bowden et al [49]. The eigenvalues of this matrix allow a quick identification of whether a particular energy state is stable or otherwise.
In general, the original master equation is now replaced by a matrix of coupled equations, which can be used to determine the equilibrium set of angles {θ i , φ i }. Once the latter are available, the elementary FMR excitations are subsequently determined using the linearization procedure of Nörtemann et al [37]. In addition, we make use of local variables, as opposed to an overall global Cartesian frame of reference. This approach confers two advantages. One, the number of incremental variables is reduced, from three variables ∂μ i x , ∂μ i y , ∂μ i z to just two per spin: {∂θ i , sin θ i ∂φ i }. Two, it is easy to determine the ellipticity of the precession of the spins, about their equilibrium positions.
First, we note that the single-particle terms, namely the Zeeman and the anisotropy interactions, have already been detailed [see equations (9) and (10), respectively]. Thus, we only need to consider terms that involve (i) magnetic exchange, and (ii) dipolar interactions between differing layers.
For thin films, with nearest neighbour interactions, we recast the exchange interaction H Ex into the 1D-planar form: Here, we make the following assumptions: (i) The spins occupy a cubic lattice. (ii) The integer k defines a plane within the thin film (along the z-axis normal to the plane of the film). (iii) All the spins within a plane (k-fixed, ρ = 1, 4) are parallel J k J k+ρ . (iv) Only the nearest-neighbour interactions are included. Note that the inclusion of the factor 1/2 avoids double counting. However, slight modifications are necessary at the top and bottom layers of the film. Finally, if all the spins are parallel, the exchange term reduces to H Ex (k) = −6λ ex J 2 , as expected.
For the magnetic exchange we find: Here, only those terms in the Hamiltonian that contain J k can contribute to the right-hand side of equation (3). On evaluating the necessary commutation relationships, for the in-plane terms we find: Whereas for the out-of-plane terms: In summary therefore: Turning now to the local dipolar fields (Dip) in the Cartesian reference frame: Here, the local dipolar field B Loc is given by: In this equation, the dipolar constants are: These numbers are taken from [42]. Note that if all the spins are parallel: A result, used earlier, in our discussion of a single slab. Finally, we need to convert from the Cartesian frame into local polar coordinates. In place of the Cartesian set of vectors î ,ĵ, k, , we define a set of local unit vectors: Note: (i) εr coincides with the vector r (in our case a given magnetic moment μ), (ii) the unit vector εφ does not involve a change in θ (it is a rotation about the z-axis), and (iii) if we set θ = φ = 0, the set εθ, εφ, εr coincides with the Cartesian set î ,ĵ,k, . Also, it is easy to show that: So εθ, εφ, εr is a right-handed set of vectors. Moreover, we can easily transform between the two different sets with the unitary transformation U. Explicitly: ⎛ ⎜ ⎝ εθ εφ εr All the above is in agreement with Varshalovich et al [50]. The final step involves 'linearization'. Following Van Vleck [36] and Nörtemann et al [37], we set: Here j k is a small, time-dependent variation about the DC component J 0 k . In terms of local coordinates: i.e. small orthogonal rotations of the spin about its equilibrium position. Proceeding in this manner, we find, after some manipulation, that the dynamical equations can be expressed in the matrix form: Here Ψ is a column matrix containing the ∂θ k , sin θ k ∂φ k (k = 1, N) amplitudes, where N is the number of layers, and M is a 2N × 2N matrix. Thus, the resonance frequencies ω are given by the eigenvalues of U (iM) U −1 , where U is a similarity transformation. In passing, we also note the matrix M is gyroscopic, with eigenvalues of the form ± ω k . This has to be the case, given that the solution must hold for spins with a different sign for the gyromagnetic constant γ.

Excited states in Py films
In figure 5 an expanded diagram of the VNA-FMR diagram of figure 2(b) can be seen, with arrows indicating the presence of PSSW, CSSW, and HSSW excited states.
In applied fields B x app > μ 0 M (∼1 T), a faint straight line is evident, running parallel to the principle k = 0 uniform mode, but shifted upwards in frequency by about 2.6 GHz (it is especially evident in figures 2(b) and (c)). This is a characteristic feature of a PSSW state [6]. In fields exceeding 1.02 T, the spins point normal to the film. However, pinning on the top and bottom of the film leads to a half-wavelength excitation: Here, (i) hν 0 is the energy associated with the uniform mode, and (ii) k = π t (λ = 2t), where t is the thickness of the film. Thus, in fields exceeding 1.02 T, the PSSW (M ⊥ ) mode should run parallel to the normal mode, but shifted upwards by Dk 2 . Using Weber's Py results for a 500 nm film, we find D = 6.14 × 10 −7 Hz m 2 and μ 0 M = 1.053 T (see also appendix B of Bowden et al [51]). With these figures, the calculated upwards frequency shift of the PPSW mode is 2.63 GHz for a 50 nm Py film. This estimate agrees well with the data in figure 5.
With the aid of the Py multilayer model of section 5, it is possible to calculate, not only the normal k = 0 mode, but also all the excited states. In figure 5, we show the calculated normal mode and the first excited state, in a Py(50) film, with (i) the field applied 1.2 • off the film normal, and (ii) no artificial surface pinning, other than the fact that the local dipolar field for the top and bottom layers differs from those in the bulk by ∼5%. The strength of the Py exchange interaction λ ex was adjusted to give a splitting of 2.63 GHz between the normal mode (k = 0) and the PSSW, above 1.02 T, in the out-of-plane results (see the two straight lines in figure 1). The number of layers used for the 50 nm-thick Py film was set at N = 188 and λ ex = 214 K. For clarity, the calculated data are superimposed on the experimental data only for positive magnetic field values. The comparison shows that there is very reasonable agreement between the experiment and theory. Clearly, the PSSW morphs into an HSSW of some sort, cutting the frequency axis at ∼9. 3 GHz.
The multilayer model not only gives the frequencies of all the excited states but also their eigenvectors. A schematic picture of the in-plane HSSW and out-of-plane PSSW excited states, and the transition between the two, can be seen in figure 6.
Note that the exited states are characterised by (i) a node in the middle of the film, and (ii) maximum displacements at the top and bottom of the film, but phase shifted by 180 • with respect to each other (asymmetric). Moreover, for the higher energy modes, the multi-layer model predicts two stationary nodes, three nodes etc, all inside the film. All of the states are characterized by λ 1 = 2t, λ 2 = t, λ 3 = 2t 3 etc. In particular, none of the calculated modes shows pinning at the surfaces of the film. The same is also true for the calculated PSSW modes above 1.0 T. The results of the multilayer model therefore suggest that the rf excitation field must be non-uniform across the magnetic film thickness [23][24][25][26][27][28]. This situation is examined in more detail in the following section. However, in passing we note two things. Firstly, the HSSW state shown in figure 5 (9.3 GHz in zero applied field) is the same as that obtained when the field is applied in-plane, and reduced to zero. Secondly, the frequency of the HSSW state in zero field (9.3 GHz) is much greater than that of the PSSW (2.63 GHz) despite both states being characterized by a wavelength λ = 2t. This difference in frequency is due entirely to the action of the demagnetization field, normal to the plane of the film, which forces the precession of the spins into an elliptical orbit. Indeed, an explicit equation for the ellipticity of precession is readily obtained, using Kittel's equation for a single slab with the applied field in-plane: Note that ε → 1 when μ o M → 0. This is equivalent to switching-off the demagnetization field. The ellipticity is also unity for out-of-plane fields, when B z app > μ o M + B K . Finally, we note that the above results represent the first observation of the transition between PSSW and HSSW modes in extended thin films by FMR. Similar dispersions of SSW states have been observed in laterally structured ferromagnets such as Py wires [52][53][54]. However, due to the confined geometry of the nanostructures, the PSSWs are often obscured by the presence of additional spin waves modes such as quantized magnetostatic forward volume modes and dipole-exchange modes [54].

Excitation of asymmetric SSW states
The excitation of antisymmetric spin-waves states has attracted much attention [25][26][27][28]. In particular, calculations by Kostylev [26] show that the in-plane dynamic rf-field at a given thickness is the film is generated, primarily, by two components: (i) the primary rf-exciting field, from the CPW, and (ii) a secondary rf-field, generated by the eddy currents in the film. These two contributions give rise to an overall asymmetric rf-field, with (i) a much reduced skin depth, than simple calculations would suggest, and (ii) the potential to excite asymmetric SSW modes (see for example figure 2(c) in reference [26], and subsequent discussion). In addition, pinning at one of the surfaces can also induce asymmetry. Clearly, given that the Py layers sit atop a [Co/Pt] layer, characterized by perpendicular anisotropy, such a situation exists in this work. However, here, we must urge caution. Firstly, if the Pt spacer exceeds 1.5 nm, the Py film is effectively magnetically decoupled from the [Co/Pt] layer (see figure 3(b)). Secondly, even when the Pt spacer is removed, both the in-plane and out-of-plane Py/FMR results, basically follow the Kittel-like behaviour. Nonetheless, asymmetry in the samples is present, together with asymmetry in the effective rf-field. Thus, asymmetric SSW-states can be excited, and witnessed [25].
Given the multi-layer Py model, discussed in section 6, it is a relatively easy matter to examine the effects of pinning by the [Co/Pt] layer. Here, we briefly discuss a simple model, where the [Co/Pt] layer is replaced by an extra 5 layers of Py, but with an anisotropy which favours the normal to the film, specifically to B K (pin) = +0.1 T, shown schematically in figure 7(b). Here, the frequency of the uniform mode decreases from ν = 1.5 to 0.991 GHz, while that of the first standing spin wave mode increases to ν = 9.032 GHz. In this example, the amplitude of precession of the uniform mode increases by some 15%, as the pinning layer is approached (not shown). This is understandable, given that the bulk of the Py film wishes to remain in-plane, while pinning layers want to point out-of-plane. A schematic picture of the first excited state can be seen in figure 7(a). Note the definite movement of the node towards the pinning layer (bottom).
Thus, the mode in question is no longer purely antisymmetric. In principle therefore, even a uniform rf-field can excite such a mode. However, as described in appendix A, VNA-FMR results on a 100 nm Py film, deposited directly on top of a MgO substrate, show clear evidence of an excited HSSW state. In this case, it is hard not to accept that such states must be driven by an asymmetric rf-field (see e.g. reference [26]). Note that the in-plane magnetic anisotropy and in-plane magnetic field (as in appendix A), there is no need to differentiate between HSSW, CSSW and PSSW modes as the magnetization remains always in the sample plane, and such states are commonly defined as PSSW modes in the literature.  Note also that for the directly coupled [Co/Pt]/Py structures, no evidence of PSSWs can be observed (see figure 2(d)), even though the pinning at the bottom Py interface is expected to be strongest for this sample. The absence of PSSWs for the [Co/Pt]/Py sample may be due to the formation of a domain structure within the Py layer [43].
Finally, it should be acknowledged that the above theory could be generalized to include Pincus modes (Pincus 1960 [55], see also Kishine et al [56]). Here, soft surface pinning at the top and bottom of a film, magnetized perpendicular to the film, allows excitation of asymmetric PSSW modes in uniform rf-fields. Such Pincus modes require the Larmor frequency of the surface spins to match that of PSSW excited state. Clearly, this is an avenue for further exploration. For example, one can ask the question: can Pincus modes can be extended to include CSSW states? Indeed, this may not be possible, given that both the effective anisotropy, and energy of the CSSW state change, as magnetic moments move progressively into the plane of the film.

Conclusions and discussion
In this paper, both FMR and XFMR results for Py and [Co/Pt]/Py films have been presented and discussed. In general, the FMR results appear to be dominated by the Py film, showing little in the way of polarization by the underlying [Co/Pt] multilayer. In particular, by using XFMR, we have demonstrated that the dynamic coupling between [Co/Pt] and Py layers is completely suppressed when separated by a 1.5 nm thick Pt spacer layer. Most importantly, we have shown that out-of-plane VNA-FMR maps reveal the presence of standing spin waves modes in the whole investigated magnetic field range. For example, a PSSW state, with λ = 2t, above the cusp at B z app μ 0 M, has been clearly identified. Moreover, as the magnetic field is reduced below μ 0 M, this state morphs, continuously, via a series of CSSW states, and, eventually into an in-plane HSSW state, which increases in frequency to ∼9.5 GHz. All of the PSSW, CSSW and HSSW modes are characterized by λ = 2t, but differ significantly in frequency. This is primarily due to the demagnetizat ion factor of −1 associated with a thin film. For fields applied out-of-plane, the magnetic field seen by the Py spins is given by B eff = B app − μ 0 M. However, as the field is reduced, the spins move increasingly into the plane of the film, where the effective field is given by B eff = B app + |B K | B app + |B K | + μ o M . Here, the diminution of the applied field is more than made up by the contribution originating from μ o M. Finally, the observed VNA-FMR results for both k = 0 and k = 0 have been interpreted using a multi-layer Py film model, based on (i) the linearization procedure of Van Vleck [36] and Nörtemann et al [37], and (ii) the local-dipolar field model for thin films (Bowden et al [51]). Finally, our results provide better understanding of CSSW modes and should be of particular interest for MAMR, where CSSW modes may be optimized to reduce the energy required to reorient the magnetization.

Data availability statement
The data that support the findings of this study are available upon reasonable request from the authors. Figure B1. VNA-FMR 2D maps, with the applied field aligned with the surface normal (middle panels), and slightly off the film normal (left and right panels).