Spin-Wave Channeling in Magnetization-Graded Nanostrips

Magnetization-graded ferromagnetic nanostrips are proposed as potential prospects to channel spin waves. Here, a controlled reduction of the saturation magnetization enables the localization of the propagating magnetic excitations in the same way that light is controlled in an optical fiber with a varying refraction index. The theoretical approach is based on the dynamic matrix method, where the magnetic nanostrip is divided into small sub-strips. The dipolar and exchange interactions between sub-strips have been considered to reproduce the spin-wave dynamics of the magnonic fiber. The transition from one strip to an infinite thin film is presented for the Damon-Eshbach geometry, where the nature of the spin-wave modes is discussed. An in-depth analysis of the spin-wave transport as a function of the saturation magnetization profile is provided. It is predicted that it is feasible to induce a remarkable channeling of the spin waves along the zones with a reduced saturation magnetization, even when such a reduction is tiny. The results are compared with micromagnetic simulations, where a good agreement is observed between both methods. The findings have relevance for envisioned future spin-wave-based magnonic devices operating at the nanometer scale.

Because spin waves are sensitive to the spin texture and the internal field landscape, they are prone to propagate in a conducted way. If the magnetic material presents a texture like a domain wall, Winter modes [11] are excited and channeled along its center since the wall acts as a local potential well for the spin waves [48][49][50][51][52]. Nonetheless, the channeling

Results and Discussion
The system under consideration is shown in Figure 1, where a thin magnetic nanostrip of width w and thickness d is illustrated. The magnetization M lies in the xz-plane and makes an angle ϕ measured from the z-axis. This section presents results for homogeneous ferromagnetic strips (with a constant saturation magnetization) and magnetization-graded strips. The detailed theoretical model for the stripe with nanoscale graduation of the magnetization is presented in the Appendices, where the dynamic matrix method is outlined in Appendix A together with the calculation of the matrix elements associated with the dipolar (Appendix B) and exchange (Appendix C) couplings. Standard values for Permalloy (Py: Ni 80 Fe 20 ) will be used in the calculations. Namely, the saturation magnetization is M s = 800 kA/m, while the exchange constant is A ex = 9.9 pJ/m. Also, the gyromagnetic ratio is γ = 185.66 GHz/T, and the strip thickness is d = 1 nm. It is worth mentioning that such a thickness is not a restriction in this work; similar results are obtained for d < 10 nm. Regarding the discretization of the ferromagnetic strip, such a planar magnonic fiber is divided into small sub-strips, where b = 4 nm is its width (see inset in Figure 1b). This size is less than the exchange length ( ex = 4.96 nm) so that if b < ex , the SW spectra do not noticeably change. (a) Illustration of the coordinate system and the main geometrical parameters of a magnetic strip. In (b), a schematic representation of the dynamic matrix approach is shown, where the system is divided into many sub-strips of width b, which allows including magnetic graduation along the width. The color graduation represents the variation of the saturation magnetization along the width of the system. The transition from a nanostrip to an extended thin film is first studied to identify the nature of the calculated spin-wave modes characterized by the wave vector k. For this purpose, a bias field of µ 0 H = 300 mT is applied along the x axis, while the spin waves propagate along z. The idea here is to study the Damon-Eshbach configuration (M ⊥ k), for which the modes will be notably influenced by the magnetic graduation, as shown below. Otherwise, in backward-volume configuration (M k), the modes are not significantly altered under a variation of the saturation magnetization (not shown). Therefore, in what follows, the equilibrium magnetization always points along x (ϕ = π/2), while the SW propagation is along the long axis z. Open circles in Figure 2a-d show the SW dispersion of an infinite thin film, while lines correspond to the modes of the FM nanostrip evaluated at different widths. By analyzing the spin-wave profiles across the width (upper 3D illustrations in Figure 2), it is observed that the spin excitations depicted in Figure 2a, for w = 80 nm, correspond to edge modes since they are excited with high SW amplitudes at the strip borders. These modes are associated with a reduction of the internal field at the edges of the ribbon, wherein the surface magnetic charges generate a robust demagnetizing field at the edges. To establish a clear picture of the calculated spin-wave modes, they are labeled as SM . As w increases, however, the edge modes become degenerate since the left and right borders are too far away from each other (see Figure 2c,d) so that the dynamic exchange energy is the same for both modes.
Higher-order modes are also observed at low frequencies, as depicted in Figure 2b-d.
In Figure 2b, it is demonstrated that the mode SM is excited with high SW amplitude around the strip center, with three nodes located about 1/6, 1/2 and 5/6 of the width. The other higher-order modes follow a typical distribution of a confined system with similar properties. The first five modes are displayed in Figure 2d for w = 1000 nm, where one can observe that the bulk modes (localized around the nanostrip center) match with the one of an extended FM film. The degeneration of the modes shown in Figure 2d is anticipated since the mode quantization due to the geometrical confinement becomes irrelevant for large widths. Indeed, in a standard physical picture, the bulk SW modes in a ferromagnetic nanostrip exhibit different frequencies due to a term of the type k ⊥ = νπ/w eff [87]. Here, k ⊥ is the quantized wave vector along the width, and w eff is an effective strip's width (proportional to w) that considers the dipolar boundary conditions at the strip's edges. Thus, upon increasing w, the wave vector k ⊥ = νπ/w eff tends to zero, and all standing waves are excited close to the ν = 0 mode, which corresponds to the mode of an infinite film. Once the nature of the spin waves in a ferromagnetic nanostrip is understood, the following step is to alter the magnetic properties along the strip's width to induce spin-wave channeling. Note that the edge modes already provide a scenario for localized propagation due to a locally reduced internal field. Nonetheless, creating a sample with perfect edges is not a simple task. Such boundaries typically present an unavoidable roughness or defects that can prevent the natural behavior of the spin waves traveling in such zones. Another option is using domain walls that can channel the spin waves [12,13]. However, domain walls strongly depend on the material features and may be unstable against external magnetic fields. Besides, due to the nature of the formation of the walls, it is not easy to modify its shape, for instance, to bend the wall in arbitrary directions. Therefore, alternative ways to conduct the SWs are of high interest. In this context, magnetic graduation is proposed as a potential alternative for inducing a channelized propagation of spin waves at the nanoscale. Because the spin-wave dynamics strongly depend on the saturation magnetization, a local variation of M s along the nanostrip width will be considered. Of course, spin-wave steering is expected along the zone with reduced saturation magnetization since the magnetic moments experience a local low frequency [88], while the rest of the ferromagnetic strip is excited at a higher frequency. Nevertheless, a quantitative analysis is required because it is necessary to understand how large the magnetic graduation has to be chosen to create a notable SW channeling in the planar magnonic fiber shown in Figure 1b. Besides, it is essential to predict if a channelized mode can be the one with the lowest frequency so that only such mode is excited in a given range of frequencies. If the channelized mode is not the low-frequency one, it will be inevitable to excite it together with other modes, which could be undesirable for magnonic applications. Figure 3 shows the case where the saturation magnetization has been varied along the strip's width (for w = 200 nm), with an M s reduction in the central region (see Figure 3b). Such a decrease in saturation magnetization is quantified by utilizing the parameters ∆M s /M s and ξ. Here, ∆M s corresponds to the difference between its maximum and minimum values, and ξ measures the extension of the graduated zone. In the example shown in Figure 3, the fractional reduction is ∆M s /M s = 0.5 (50%) and ξ = 100 nm. Overall, the SW band structure is notably modified, as shown in Figure 3a. An important aspect is that now the coherent mode, SM (grad) 0 (where "grad" denotes the magnetizationgraded stripe), is localized mainly in the center of the nanoribbon, where M s is reduced (see Figure 3c). The dynamic behavior of the low-frequency mode can be understood from the dependence of the SW frequency on saturation magnetization. In a typical Damon-Eshbach configuration, the frequency of the spin waves is reduced as M s decreases. Therefore, it is expected that in a magnetization-graded nanostructure, the lowest-frequency branch is excited in the zones where the saturation magnetization is reduced since such a zone is energetically compatible with low-frequency magnetic excitations. Of course, spin waves localized in zones with high M s are also feasible, but they are excited at high frequencies, as becomes evident by Figure 3. Besides, modes SM can be excited. At more significant frequencies, it will be unavoidable to excite the channelized mode together with the higher-order ones. Note that width of transition from M s = 800 kA/m to 400 kA/m is about 25 nm. In the case of a more abrupt change of the saturation magnetization, even for a step-like change, the SW dispersion and the respective SW profiles do not significantly change (not shown). On the other side, the reduction of the saturation magnetization could be accompanied by the variation of other magnetic parameters, such as the gyromagnetic ratio. If the gyromagnetic ratio is reduced (increased) at the graduated zone, the channelized waves will slightly decrease (increase) their frequency. The reduction to 50% of the saturation magnetization (∆M s /M s = 0.5) is a significant change in the magnetic graduation of the system. Therefore, it is interesting to explore where the SW excitations are concentrated as ∆M s /M s is changed. Figure 4a shows the absolute value of the out-of-plane component of the dynamic magnetization for mode SM (grad) 0 as a function of x, the coordinate along the strip width, and evaluated at k z = 0 and ξ = 100 nm. Here, it is possible to see how the SW localization evolves as graduation diminishes. Under a reduction by 30% (∆M s /M s = 0.3), the mode SM (grad) 0 is still mainly localized within the FM strip center. Nevertheless, when the reduction is only 20% (or less), the magnetization excitation is no longer localized in the strip center since the system tends to the homogeneous case, where the low-frequency mode is located at the strip edges. Also, the case ∆M s /M s = 0.2 depicts a more asymmetric profile along the strip's width (see the orange curve in Figure 4a), which is owed to the hybridization between the modes SM (grad) 0 and SM does not change its frequency notoriously because this mode has a node at the strip center and thus is only slightly influenced by the magnetic graduation.   (localized mainly at the strip center) are excited with SW amplitudes spatially far from each other. Therefore, exploring the channelized modes in the case of a wider ferromagnetic strip is also of interest. In Figure 5, a FM stripe with w = 1000 nm is considered. In the case ξ = 100 nm (see Figure 5a,b), it is evidenced that for the case ∆M s /M s < 0.22, the mode SM . A remarkable result is that an evident SW channeling is feasible even for the case ∆M s /M s = 0.05 (see the green curve in Figure 5b), which is not a trivial result since the graduation of 5% in the saturation magne-tization is not a significant change. Besides, analogous to the previous case (w = 200 nm), as ∆M s increases, the channelized mode reduces its dynamical energy, becoming the lowfrequency mode. It implies that it is attainable to excite only the channelized modes, which have a robust SW amplitude at the strip center, as shown in Figure 5b. Because of the strong localization of the low-frequency modes, it can be expected that the magnetization-graded zones can act as magnonic fibers or effective conduits for spin waves, similar to the SW propagation along domain walls [48,52]. Additionally, in the case of multiple channels, it is expected that the SWs can propagate with negligible interferences.
The evolution of the modes as a function of ξ is shown in Figure 5c, where the cases ν = 0, 1, 2, and 3 are analyzed. Overall, one can observe that as ξ decreases, the lowfrequency mode increases its dynamical energy. Such behavior is associated with the exchange energy that becomes high when the magnetic moments are forced to oscillate in a narrow zone. On the other hand, under a slight increase of ξ, the mode SM (grad) 0 further decreases its frequency. Nonetheless, an additional mode with a node in the center of the graduated zone also reduces its frequency and, hence, is channeled (see blue SW profile in Figure 5d). This last effect is expected because as the graduated zone grows, the lateral standing waves excited within such area are energetically favorable at low frequencies.  Finally, the calculated SW properties are compared with micromagnetic simulations based on the GPU-accelerated code MuMax3 [89]. Two different magnetic thin strips were considered with dimensions of 4096 nm ×200(1000) nm × 1 nm discretized into 2 11 × 2 6 (2 9 ) × 1 cells along the (z, x, y) components, respectively, for two different widths; w = 200 nm and w = 1000 nm. Periodic boundary conditions along the z-direction were applied to simulate a long nanostrip. The graduation of M s was implemented by defining regions in the software with independent parameters. The nanostrip started with the magnetization along the x−direction due to an applied magnetic field of 300 mT (identical to the calculated cases) that saturates the sample. The generation of SWs was implemented through an external pulse in the form of h rf = h 0 sin(2π f c t)ŷ, applied at the center of the sample over a width of 8 nm in z. Here, f c is the cut-off frequency, and h 0 was a hundred times smaller than the field used to saturate the sample. The system evolved for 5 ns. Figure 6 shows the calculated SW dispersions for the cases w = 200 nm and w = 1000 nm, where micromagnetic simulations are carried out at f = 17 GHz and 18 GHz. For the smaller width, w = 200 nm, the simulated SW propagation evidences the channeling of the waves at f = 17 GHz, which is in concordance with the model's prediction. Here, it is possible to observe that the edge modes are not excited at such a low frequency. Nevertheless, if the frequency of the field h rf increases (18 GHz, for instance), the dynamic state of the system is compatible with the propagation of a channelized mode superimposed with the edge modes. This behavior is also seen for the wider sample (w = 1000 nm), wherein the purely channelized mode is excited at a low frequency (17 GHz), while both the channelized and the edge modes are excited at a high frequency (18 GHz). These results demonstrate the validity of the model because it can predict the necessary conditions for spin waves steering along the nanostrip. Besides, under certain conditions, the profile of magnetic graduation chosen here allows to excite only the channeled modes since a notable reduction of the frequency is also reported. The results presented here apply to ultrathin nanostructures, where the dynamic magnetization does not significantly vary across the nanostrip thickness. Therefore, it is required that d ≤ ex so that at such thickness ranges, the exchange interaction is dominant, and the magnetic moments will oscillate parallel with each other along the strip thickness. Besides, if the graduation profile along the strip width changes, the localization of the magnetic excitations also changes. For instance, if the graduation is active just on one edge of the nanostructure, a robust localization of an edge mode will be induced in such a boundary (not shown). On the other hand, the experimental viability of creating graduated samples is currently feasible. For instance, the realization of vertical graduation along the thickness in thin films is achieved in epitaxial compositionally graded alloy films by co-sputtering, keeping the power of one material fixed while changing that of the other to achieve the intended composition profile [77,84,90]. Combinatorial material deposition can also be employed to fabricate samples with lateral gradients. A review of this method is found in Ref. [91], which typically results, however, in extended thin films. Further, lithographic masks can be used to manipulate the structures locally, allowing to achieve lateral gradients within single mesoscopic or nanoscopic structures. Some of the co-authors of this paper successfully used such a route on FeAl-alloys [92]. A direct way to locally modify magnetic properties employing a focused ion beam has been demonstrated in Refs. [93,94] and could be employed similarly to realize the graded nanostructures suggested in this paper.

Conclusions
The spin-wave spectra of magnetization-graded ferromagnetic nanostrips have been studied. A channelized spin-wave propagation has been predicted by modifying the saturation magnetization along the strip's width. The spin-wave steering as a function of the fractional decrease of the saturation magnetization has been studied. It is demonstrated that spin waves can be conducted in magnetization-graded ferromagnetic strips, even when the saturation magnetization is slightly reduced. Besides, under the increase of the saturation magnetization contrast, the channelized spin-wave mode becomes the lowfrequency one, allowing for exciting only such a mode. Micromagnetic simulations reveal the predicted channeled spin-wave propagation, validating the findings obtained with the theoretical calculations. These results are relevant for future applications associated with magnonic fibers and spin-wave steering at the nanometer scale.

Conflicts of Interest:
The authors declare no competing interests.

Appendix A. Dynamic Matrix Method
According to the system shown in Figure 1, an orthonormal basis e 1 , e 2 , e 3 is defined in relation to the basisx,ŷ,ẑ by a ϕ-rotation around the y-axis, so that e 3 points along the equilibrium configuration (see Figure 1a). Unless otherwise stated, spatial components are assumed to be relative to this basis, i.e., Λ ij = e i · Λ · e j . Of course, it is expected that the magnetic moments at the lateral edges of the strip are canted due to an increase in the demagnetizing field, which is remarkable when the magnetization is along the width of the nanostructure (ϕ = ±π/2). Nonetheless, in the current approach, it will be assumed that the external field H Z is strong enough to make the equilibrium magnetization fully parallel to it. To take into account the modulation of the dynamic magnetization along the strip's width, the dynamic matrix method is used [95][96][97]. This method dictates the subdivision of the strip into N wire-shaped cells with small cross-sections of width b and thickness d, referred to as sub-strips, which are coupled by exchange and dipolar interactions. Then, by increasing the number N of subdivisions, the continuous system is described (see Figure 1b), which is validated through a convergence test.
Individual sub-strips have different magnetic properties, so the n-th cell possesses a magnetization M n (r, t). In this paper, Zeeman, exchange and dipolar interactions are considered, hence H n = H Z + H n ex + H n dip . The magnetization dynamics of each sub-strip is modeled by the Landau-Lifshitz equationṀ n (r, t) = −γM n (r, t) × H n (r, t), for 1 ≤ n ≤ N, with γ being the absolute value of the gyromagnetic ratio and the dot indicating a time derivative. Both magnetization and effective fields are written in terms of static (zeroth order) and dynamic (first-order) components, omitting higher-order terms, which enable linearization of the equations of motion. The dynamic components are assumed to behave like monochromatic plane waves, with the wave vector k confined to the z-axis. Thus M n (r, t) = M n eq + m n e i(k·r−ωt) and H n (r, t) = H n eq + h n e i(k·r−ωt) , where m n , h n ∈ C 3 are complex vectors perpendicular to the equilibrium configuration (M n eq H n eq H Z ), and M n eq = M n s e 3 , with M n s being the saturation magnetization. Also, the angular frequency is ω = 2π f , where f is the frequency of the spin waves. By keeping terms up to first order in m n and h n , the following system of linear equations is obtained, A convenient linear structure is used for the dynamic effective field, namely, . Concerning the canonical basis of C 2N , the matrix representation of T is given by where 1 ≤ i, j ≤ 2 and 1 ≤ n, p ≤ N are integers uniquely defined by q = (2 − i)N + n and l = (j − 1)N + p, for each 1 ≤ q, l ≤ 2N. Here, δ ij stands for the Kronecker delta symbol. It is worth mentioning that analytical expressions for the tensor T ql are not found in the literature since the dynamic dipole-dipole interaction is not easy to handle in the case of a ferromagnetic nanostrip. For instance, expressions in terms of numerical integrations are given in Ref. [97] for the dynamic dipolar fields. In this manuscript, analytical expressions for the dipolar contribution are derived. Although somewhat extensive, these analytical terms allow obtaining the magnonic dispersion and magnetization profiles in a short time, representing an essential advantage over other numerical procedures or micromagnetic simulations. The detailed derivation of the dipolar and exchange contributions is presented in the following.

Appendix B. Dipolar Interaction
The sub-strip geometry motivates the following definitions, which can be thought of as nothing more than a handy notation. The projection operator over the system's bounded direction is P =x ⊗x +ŷ ⊗ŷ, and the axial radius is ρ = P · r. The operator p integrates a function over the p-th sub-strip cross section (b×d rectangle), and R denotes the integral over the real line along the long z-axis. Thus, one can note that Ω p = p R . Expressions of the form n p η(ρ − ρ ) are considered, where the p-integral is over the primed coordinates, whereas the n-integral runs over the non-primed coordinates. Also, η is a well-behaved function with a singularity at the origin. More explicitly, it can be seen that for n = p, and for n = p. Here, dS = dxdy stands for the differential cross section area, and (δx np , δy np ) represents the relative position of the n-th sub-strip with respect to the p-th sub-strip. At this point, introducing the shorthand notation n p η may seem unnecessary. However, as the manuscript moves forward, its role of simplifying the discussion on otherwise tedious mathematical details will become evident. For example, it is interesting to consider the effect of changing the order of derivation and integration. If n = p, the operators commute. However, if n = p, it follows from the above equation that The magnetic field due to the p-th sub-strip is given by Then, the average magnetic field over the n-th sub-strip cross section due to the Here the tensor field K is defined by with K l denoting the l-th Bessel function of the second kind. Equation (A8) follows by noting that ∇|r − r | = −∇ |r − r |, commuting the operators R ∇ = ∇ R , using the identity ∞ −∞ e iaz dz/ √ z 2 + b 2 = 2K 0 (|ab|) for a, b ∈ R, invoking the product rule ∇e ik·r η(ρ) = e ik·r (ik + ∇)η(ρ) in succession, and canceling out the exponentials. Equation (A9) is obtained by considering the derivatives K 0 = −K 1 and K 0 = (K 0 + K 2 )/2, the identity 2K 1 (x) = x(K 2 − K 0 )(x), and using the commutation property shown in Equation (A5), where the resulting integrals are evaluated by taking K 1 (x) = 1/x + O(x). Note that K is symmetric and traceless, so that the demagnetizing tensor Λ np dip is also symmetric, with trace 1 if n = p and trace 0 otherwise [97,98]. Also, K xz (x, y) = K yz (y, x).
Thus, in solving n p K(ρ − ρ ), only four components need to be considered: K xx , K xy , K xz and K zz . Here, the spatial components are chosen concerning thex,ŷ,ẑ basis.
Finally, if the system is divided into many sub-strips, the continuous approach establishes that J = 2A ex /b, where A ex is the exchange constant of the magnetic material defined in the continuum limit.