Co- and contra-directional vertical coupling between ferromagnetic layers with grating for short-wavelength spin wave generation

The phenomenon of coupling and energy exchange between two identical uniform ferromagnetic films has already been discussed [28, 29] and it was shown with Brillouin light spectroscopy (BLS) for side coupling between two waveguides two years ago [22]. Recently, application of that effect, widely used in photonics from many years [30, 31], has been proposed for signal processing with two coupled magnonic waveguides [32]. In this case, the resonant interaction between SWs propagating in the identical waveguides exists in relatively wide range of frequencies if the distance between them is properly tuned. If the layers are different, the SWs are decoupled and each wave becomes confined to a single layer. However, the existence of the other layer still affects the dispersion of the SWs in a range of frequencies. We can couple them again by introducing the periodicity of the structure. Folding of the bands to the Brillouin zone will result in multiple crossings of the bands that originate from different layers. Then, by tuning the distance between layers, to ensure that crossings lie in the frequency region of strong interaction between layers, we can achieve strong hybridization between the SW bands. To sum up, the criteria of efficient power exchange between different ferromagnetic media (waveguides or layers) are the following: (1) the resonance, i.e. the equality of the frequencies, (2) phase matching—equality of the wavevectors, and (3) strong interaction between layers.

For the efficient energy exchange between SWs in the coupling-in-space mechanism, it is crucial to choose materials with low damping. The best candidates are yttrium iron garnet (YIG) and CoFeB. However, due to the extremely high magnetic contrast between that materials, it is difficult to match all three conditions: the wavelength of the SW in YIG is much shorter than in CoFeB at the same frequency, and it is the wave mainly driven by the exchange interaction. As a result, the mode in YIG creates very weak stray magnetic field around and the layers must be in an unfeasibly close distance to couple. Due to this, we have chosen Py and CoFeB as materials of significant contrast, but small enough to ensure high stray fields for both SW modes.

Considered bilayer system (figure 1) consists of CoFeB 30 nm thick homogeneous film and 30 nm Py film. Py is patterned periodically with grooves of 300 nm width and the depth of either 5 nm (Py5) or 15 nm (Py15). The grating has the lattice constant 600 nm. Such dimensions make the structure feasible for fabrication and SW detection achievable by conventional techniques, such as BLS or SW spectroscopy. We assume, that the layers are infinite along the x₂-coordinate. The separation between layers has been optimized for the sufficient strong interaction at the points of lowest band crossings (discussed latter) and its value is fixed to 200 nm.

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To study coupling in the bilayered structure the Landau–Lifshitz–Gilbert equation has been solved either in frequency-domain (for calculation of the dispersion relations) or in the time-dependent computations, both with the use of COMSOL Multiphysics. The details of calculations are described in [33]. We assumed for the CoFeB (Py) layer the values of the exchange constant of 15 pJ m^–1 (13 pJ m^–1), of the saturation magnetization M_s = 1250 kA m^–1 (860 kA m^–1) and the damping α = 0.004 (0.01). The layers are assumed to be saturated in the x₂ direction by the external magnetic field H₀ = 40 kA m^–1. The results of finite element method have been verified by the micromagnetic simulations performed with MuMax3 [34], which was also used to estimate effectiveness of the SW transfer between the layers and to test filtering properties.

Before presenting the numerical results, we formulate an approach for calculation the coupling between SWs propagating in two different ferromagnetic layers, a and b, which is based on the coupled mode theory widely used in electromagnetism [35, 36]. A dynamic magnetization ${\vec{m}}_{a}$ of the SW mode in the layer a is under influence of the magnetic stray field ${\vec{h}}_{b}$ generated by the SW ${\vec{m}}_{b}$ in the layer b. The energy of this interaction is

$$\begin{eqnarray}&&{E}_{{ba}}=\displaystyle \frac{1}{S}\int {\vec{m}}_{a}^{* }\cdot {\vec{h}}_{b}{\rm{d}}{S},\end{eqnarray} \tag{ 1 }$$

where integration is conducted through the area S of the unit cell of the structure (see, figure 1). The fields are normalized according to

$$\begin{eqnarray}&&\displaystyle \frac{1}{S}\int {\vec{m}}_{i}^{* }\cdot {\vec{h}}_{i}{\rm{d}}{S}=1,\end{eqnarray} \tag{ 2 }$$

where i = a, b. The power transferred from the layer b to the layer a at the angular frequency ω is given by

$$\begin{eqnarray}&&{\kappa }_{{ba}}^{(\omega )}=\displaystyle \frac{1}{S}\int \displaystyle \frac{\partial {\vec{m}}_{a}^{* }}{\partial t}\cdot {\vec{h}}_{b}{\rm{d}}{S}={\rm{i}}\omega \displaystyle \frac{1}{S}\int {\vec{m}}_{a}^{* }\cdot {\vec{h}}_{b}{\rm{d}}{S}.\end{eqnarray} \tag{ 3 }$$

The coupling coefficient κ_ab^(ω) which describes energy flow from medium a to b is defined similarly. This formula allows to estimate coupling strength between two waveguides knowing the distributions of stray fields ${\vec{h}}_{i}$ and related SW amplitudes ${\vec{m}}_{i}$ of the separated layers. As ${\kappa }^{(\omega )}\equiv | {\kappa }_{{ab}}^{(\omega )}| =| {\kappa }_{{ba}}^{(\omega )}|$ is a periodic function of the coordinate in grating-assisted coupler [37], the formula (3) describes an average value of the coupling coefficient in the unit cell. The value of the coupling coefficient κ^(ω) defines also the width of the band splitting in a dispersion relation in the frequency scale (figure 2). We can express coupling coefficient also in the wavevector scale by

$$\begin{eqnarray}&&\kappa \equiv {\kappa }^{(k)}={\kappa }^{(\omega )}/v,\end{eqnarray} \tag{ 4 }$$

where $v=({v}_{a}+{v}_{b})/2$, and v_a is the group velocity of SW in the layer a and v_b in the layer b. It is assumed that group velocities are constant close to the modes crossing (in the absence of coupling). By this reformulation of the coupling coefficient from the frequency to the wavevector scale, it is possible to describe the coupled modes at a resonant frequency f₀ as a sum of two modes which wavenumbers differ by Δk = 2κ and one of them is symmetric while the other is antisymmetric. Due to the periodicity in the structure and the folding-back effect the two family of bands (these related to SWs in a and b layers) cross each other. The crossing bands can have the same (figure 2(a)) or opposite (figure 2(b)) sign of the group velocity. In the first case we call them co-directional and in the second contra-directional couplings.

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Considering co-directional coupling (figure 2(a)) at a resonance frequency f₀ = ω₀/2π, the sum of these two waves in the medium a is

$$\begin{eqnarray}&&{m}_{a}(x,t)\propto {{\rm{e}}}^{{\rm{i}}\omega t}[{{\rm{e}}}^{{{\rm{i}}{k}}_{a}x}+{{\rm{e}}}^{{{\rm{i}}{k}}_{b}x}]=2{{\rm{e}}}^{{\rm{i}}({k}_{0}x-\omega t)}\cos (\kappa x),\end{eqnarray} \tag{ 5 }$$

with k_a and k_b being wavevectors in sublayer a and b respectively, while in medium b it is

$$\begin{eqnarray}&&{m}_{b}(x,t)\propto {{\rm{e}}}^{{\rm{i}}\omega t}[{{\rm{e}}}^{{{\rm{i}}{k}}_{a}x}-{{\rm{e}}}^{{{\rm{i}}{k}}_{b}x}]=2{{\rm{e}}}^{{\rm{i}}({k}_{0}x-\omega t)}\sin (\kappa x),\end{eqnarray} \tag{ 6 }$$

where the relative amplitudes were omitted for simplicity. If κ ≪ k₀, i.e. in the weak coupling regime, the interference of that two modes results in the effect of beating with the spatial frequency defined by κ^(k) (figure 2(c)).

For the $f\ne {f}_{0}$, i.e., for ${k}_{a},{k}_{b}\ne {k}_{0}$, the value of κ may be read from dispersion relations (figure 2) as $\kappa ={k}_{0a}-{k}_{a}={k}_{b}-{k}_{0b}$. The exact solutions for the boundary conditions $| {m}_{a}(0,t)| ={M}_{0}$ (M₀ being maximal SW amplitude), $| {m}_{b}(0,t)| =0$ are [36]

$$\begin{eqnarray}{m}_{a}(x,t) & = & {M}_{0}\left(\cos ({Bx})-{\rm{i}}\displaystyle \frac{{\rm{\Delta }}{k}_{0}}{B}\sin ({Bx})\right){{\rm{e}}}^{-{\rm{i}}{\overline{k}}_{0}x}{{\rm{e}}}^{{\rm{i}}\omega t},\\ {m}_{b}(x,t) & = & {M}_{0}\displaystyle \frac{\kappa }{B}\sin ({Bx}){{\rm{e}}}^{-{\rm{i}}{\overline{k}}_{0}x}{{\rm{e}}}^{{\rm{i}}\omega t},\end{eqnarray} \tag{ 7 }$$

where ${\rm{\Delta }}{k}_{0}=({k}_{0a}-{k}_{0b})/2$, ${\overline{k}}_{0}=({k}_{0a}+{k}_{0b})/2$, and $B=\sqrt{{\rm{\Delta }}{k}_{0}^{2}+{\kappa }^{2}}$.

In case of contra-directional coupling (figure 2(b)) at a resonance frequency f₀, the value of κ becomes imaginary at the vicinity of crossing. The wave is a sum of two modes k₁ = k₀ + iκ and k₂ = −k₀ + iκ which propagate at opposite directions. If we suppose that initially the input wave exists only at the medium a, i.e., $a(x=0)\ne 0$ and $b(x\to \infty )=0$ then we can write expressions for a and b in the simplified form as [36]

$$\begin{eqnarray}{m}_{a}(x,t) & \propto & {{\rm{e}}}^{-\kappa x}\cos ({k}_{0}x-\omega t),\\ {m}_{b}(x,t) & \propto & {{\rm{e}}}^{-\kappa x}\sin ({k}_{0}x-\omega t).\end{eqnarray} \tag{ 8 }$$

The resulting effect is a Bragg reflection: the amplitude of wave a is a decaying function of x (figure 2(d)) and the outgoing wave b propagates in the opposite direction.

The exact solution for $f\ne {f}_{0}$ with the substitution T = iB and the boundary conditions $| {m}_{a}(0,t)| ={M}_{0}$ and $| {m}_{b}(0,L)| =0$, i.e. when the SW mode in the layer b is supposed to have zero amplitude at the distance L is

$$\begin{eqnarray}{m}_{a}(x,t) & = & {M}_{0}\displaystyle \frac{\cosh {Tx}-\tfrac{{\kappa }^{2}}{{T}^{2}}\sinh {TL}\sinh T(x-L)-{\rm{i}}\tfrac{{\rm{\Delta }}{k}_{0}}{T}\sinh {Tx}}{1+\tfrac{{\kappa }^{2}}{{T}^{2}}\sinh {TL}}{{\rm{e}}}^{-{\rm{i}}{\overline{k}}_{0}x}{{\rm{e}}}^{{\rm{i}}\omega t},\\ {m}_{b}(x,t) & = & {M}_{0}\displaystyle \frac{\kappa }{T}\displaystyle \frac{\cosh {TL}\sinh T(x-L)-{\rm{i}}\tfrac{{\rm{\Delta }}{k}_{0}}{T}\sinh {TL}\sinh T(x-L)}{1+\tfrac{{\kappa }^{2}}{{T}^{2}}\sinh {TL}}{{\rm{e}}}^{-{\rm{i}}{\overline{k}}_{0}x}{{\rm{e}}}^{{\rm{i}}\omega t}.\end{eqnarray} \tag{ 9 }$$

With the damping neglected, the efficiency of the energy transfer from the layer a (CoFeB film) to the layer b (Py film) and vice-versa is described in the coupled mode formalism by energy transfer efficiency coefficient:

$$\begin{eqnarray}&&{A}^{2}=\displaystyle \frac{{\kappa }^{2}}{{B}^{2}},\end{eqnarray} \tag{ 10 }$$

which is equal to the ratio of the maximum square amplitudes of the waves in the layers. Energy transfer efficiency coefficient decreases as the value of κ decreases (i.e., out of the resonance) and it also decreases with the increase of phase mismatch Δk₀.

The energy transfer depends also on damping. To estimate its influence, we introduce a damping to our coupled mode calculations by introducing complex wavevectors ${k}_{0a,0b}={k}_{0a,0b}^{r}+{{\rm{i}}{k}}_{0a,0b}^{i}$ to the exact solutions of the coupled modes, equations (7) and (9). The values of kⁱ are calculated from the formulas for surface SW lifetime τ and group velocity v [38]:

$$\begin{eqnarray}&&{k}^{i}=\displaystyle \frac{1}{v\tau }=\displaystyle \frac{4\omega \alpha }{{\omega }_{M}^{2}d}({\omega }_{0}+0.5{\omega }_{M}){{\rm{e}}}^{2{k}^{r}d},\end{eqnarray} \tag{ 11 }$$

where ω₀ = γμ₀H₀, ω_M = γμ₀M_s and d is layer thickness. The complex values of k_0a and k_0b in the co-directional coupling lead to: (1) decrease of the maximum amplitude of SWs and (2) the shift of the distance at which the wave transferred from layer a to the layer b reaches its maximum amplitude. Then, we define the energy transfer efficiency coefficient from the layer a to the layer b for co-directional coupling as [36]

$$\begin{eqnarray}&&{A}^{2}=\displaystyle \frac{{| {m}_{b}({x}_{\max })| }^{2}}{{| {m}_{a}(0)| }^{2}},\end{eqnarray} \tag{ 12 }$$

where x_max is the first intensity maximum in the layer b. For contra-directional coupling, we compare the square amplitude at the output of the layer b to the square amplitude at the input of the layer a:

$$\begin{eqnarray}&&{A}^{2}=\displaystyle \frac{{| {m}_{b}(0)| }^{2}}{{| {m}_{a}(0)| }^{2}}.\end{eqnarray} \tag{ 13 }$$

The line source of the wave in time domain simulations is placed at x₁ = 28 μm in the CoFeB film. CoFeB film exists along the whole x₁-range, while Py spreads out from x₁ = 0 to x₁ = 25 μm. The corrugation of Py film (stripe hole depth) is here c = 5 nm (Py5). Strong damping close to the both ends of CoFeB and left end of Py is introduced to prevent reflections. Figure 4(a) shows magnetization component m₁ after 15 ns of excitation. The amplitude of the SW which goes to the left is decreasing from the maximum value at x₁ > 25 μm to almost zero at x₁ = 14.5 μm. Its wavelength is 1.51 μm. At the same distance, the SW in Py appears to reach maximum amplitude at x₁ = 14.5 μm. Therefore, the energy of the wave in CoFeB has been vertically transformed into the energy of the wave in Py5 completely at a distance of 10.4 μm. Wave induced in Py has the wavelength of 417 nm, that is almost 4 times shorter than those which excites it.

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The coupling coefficient read out from the dispersion relation (figure 3(a)) is κ = 0.131 μm⁻¹ which gives the value of transfer length ${L}_{\mathrm{tr}}=\pi /2\kappa =11.9\,\mu {\rm{m}}$, it is a little longer than obtained from time-dependent simulations. The same coupling coefficient calculated from equation (3) gives the value of 0.145 μm⁻¹ and thus L_tr = 10.9 μm.

When the damping is introduced, the effect of energy exchange between layers is strongly suppressed. As it is shown in figure 4(b), the short wave in Py5 is still excited by the wave in CoFeB, but neither clear maximum amplitude in Py nor corresponding minimum in CoFeB are visible. The damping length for Py film is of about 2–3 μm, thus it is much smaller than the length at which transfer from CoFeB occurs. At a given distance, Py film looses more energy than it gets from CoFeB and full transfer of energy and its accumulation cannot take place.

We can recover accumulation of energy in Py layer with damping by increasing coupling coefficient between layers. For this purpose we increase the corrugation depth of Py to c = 15 nm (Py15). The dispersion relation for Py15/CoFeB is shown in figure 3(b). It is clearly seen, that the separation between branches increased significantly at the poins of crossings, which indicates increase of coupling strength. The calculated value of κ (equations (3) and (4)) is 0.58 μm⁻¹ which gives L_tr = 2.7 μm. Note that the resonant frequency f₀ is 11.9 GHz for the Py15/CoFeB.

The time-dependent simulations are presented in figure 4(c). The wave in CoFeB vanish much faster with the distance, because it transfers the energy to Py in shorter time. The excited wave in Py grows and the amplitude reaches maximum value at x₁ = 23 μm. Therefore, the energy is clearly accumulated in Py despite damping. At further distance it is seen that small amount of energy is transferred back to CoFeB and again back to Py15.

The crossing shown in figure 2(b) from the dispersion relation in figure 3(b) at 10.6 GHz represents contra-directional coupling between SWs in CoFeB and Py of 15 nm deep grooves. The coupling strength estimated from dispersion relation is 0.476 μm⁻¹, while the calculated from the equation (3) based on the stray field amplitude and the magnetization amplitude is 0.435 μm⁻¹, that gives the transfer lengths of 3.34 μm and 3.6 μm, respectively.

We performed time domain simulations with the 10.6 GHz excitation at the source line in the system similar to the discussed above for co-directional coupling. The difference is that we introduced the fragment of the homogeneous Py layer at the x₁ between 20 and 25 μm with the strong damping close to the right edge. This allows us to observe the wave going out of the periodic structure toward the right direction.

Figure 5(a) shows the results of simulations after 5 ns of excitation. Clearly, the wave of the wavelength of about 700 nm is excited in Py15 by the wave of the wavelength 3.85 μm in CoFeB. However, the amplitudes of both waves decay with the distance. The distance at which amplitude decreases by the factor 1/e is about 4 μm. At the output (homogeneous part of Py) we observe the wave which propagates to the right, as shown in figure 5(a). Therefore, the energy of the wave in CoFeB which propagates to the left has been vertically transferred to the energy of the wave in Py, which propagates to the right.

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With the damping introduced (figure 5(b)), the amplitudes of both waves in Py15 and CoFeB decrease. The amplitude of the outgoing wave in Py is also suppressed. However, the transfer of SW from CoFeB to Py film is still visible.

The excitation of the SWs in Py film by a wave propagating in CoFeB is a resonant effect, thus it occurs at maximal efficiency at the particular frequency f₀. The efficiency decreases with moving far away from f₀ and also is suppressed by the damping, always present in the ferromagnetic film. To elucidate influence of those two effects on the transfer efficiency we use the model presented in section 2.2. The efficiency of the energy transfer (A²) from CoFeB to Py film determined from equation (10) for the case without damping, from equation (12) for co-directional crossing with damping, and from equation (13) for contra-directional crossing with damping, are shown in figure 6.

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Figure 6(a) shows A² as the function of frequency for the co-directional coupling. Without damping (black-solid line), the efficiency of the energy transfer is 1 at f₀ = 12.95 GHz. The energy is transferred completely from a wave of wavelength λ = 1510 nm in CoFeB layer to the wave of λ = 417 nm in permalloy (Py5) at a distance of about 11 μm. As the frequency is changed, the efficiency decreases as the consequence of decoupling and phase mismatch (an asynchronous state). With damping taken into account, the efficiency of the process goes down to only 0.04 at resonance (blue-dotted line). The efficiency is increased to 0.3 at the resonance for Py with deeper groves, i.e., in the Py15/CoFeB structure, where the coupling coefficient reaches value of 0.58 μm⁻¹ at f₀ = 11.9 GHz. For the contra-directional crossing with resonance at f₀ = 10.5 GHz (in Py15/CoFeB), the energy transfer efficiency is 1 for the synchronous state without damping at f₀. It decreases when moving the frequency out of the resonance (see, figure 6(b), solid line). The damping decreases maximum energy reached in Py to almost 40 percent of the initial energy in CoFeB at the resonance (green dash–dotted line).

Interestingly, the damping in the structure does not decrease the efficiency only, but also shifts the position of the maximum amplitude in the case of co-directional coupling. This results, that for no damping the maximum amplitude is reached at L_tr ≈ 11 μm, while with the damping it is for x_max = 3.7 μm (both for Py5/CoFeB).

Patterning of the Py film may results on deterioration of its magnetic properties, especially, the process may increase magnetic losses. Therefore, we performed analysis of the transfer efficiency between the layers as a function of Py damping parameter, α_Py, keeping the value of α in CoFeB constant. In figure 7(a) the dependence of the distance x_max at which the wave excited at f₀ in Py by the wave in CoFeB reaches its maximum amplitude is shown. As it is seen, this distance decreases exponentially as the damping in Py increases. The value of x_max indicates at which point it is optimal to cut the periodicity of Py layer to avoid back-coupling. However, the total energy transferred also decreases with damping, what is shown in figure 7(b). An increase of the α from 0.01 to 0.02 decreases A² from 0.4 to 0.2 for contra-directional coupling and this decreases A² from 0.3 to 0.15 for co-directional coupling in Py15/CoFeB bilayer. Additionally, we provide a dependence of A² on the damping parameter in CoFeB for the damping parameter in Py fixed to 0.01. The dependence follows exponential decay of efficiency with the damping and it is qualitatively similar to the dependence for permalloy. Note, that the efficiency A² is better for for contra-directional coupling than for co-directional coupling and for the same value of α in spite of the fact that the coupling coefficient is smaller. This is mainly, because of the lower resonant frequency and wavenumber for contra-directional coupling, which leads to the lower effective damping (equation (11)).

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Next, we have analized how the energy transfer efficiency coefficient depends on the separation between layers and the depth of thickness modulation in permalloy layer. Figure 8(a) shows the values of A² for the value of grooves in Py fixed to c = 15 nm. The value of A² reaches maximum of about 0.3 and 0.4 for the co-directional and contra-directional couplings, respectively, for the distance between the layers of 200 nm. Figure 8(b) shows that the efficiency of energy transfer between layers (fixed distance of 200 nm) increase with the increase of the thickness modulation. That strictly reflects the dependence of coupling coefficient κ on the layers separation and thickness modulation.

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Finally, we analyzed with MuMax3 the transmission of the broadband signal (wave packet) excited in CoFeB to the Py layer. The results are shown in figures 3(c) and (d) by the color map. Very good agreement with the previous computations has been obtained, proving the correctness of the investigations. Moreover, as indicated by the intensity map, the signal transmits between the layers only at the vicinity of the crossings. That suggests, a system may operate not only as a short-wavelength SW generator, but also as a frequency filter.