1 INTRODUCTION

The propagation of magnetostatic waves (MSW) in magnonic crystals, i.e., MSW waveguides with periodically changed properties is one of extensively studied directions of spin-wave electronics over last decade [1, 2]. A specific feature of propagating MSW in magnonic crystals is the formation in their spectrum of forbidden bands at frequencies fB of the Bragg resonances corresponding to condition

$${{f}_{{\text{B}}}} = f({{q}_{{\text{B}}}} = \pi n{\text{/}}\Lambda ),$$
(1)

where qB is the wave number of MSW at frequency fB, Λ is the period of varying the parameter, and n = 1, 2, …. When fabricating magnonic crystals from yttrium–iron garnet (YIG) films, the formation of the periodic structure can occur due to a change in the geometric parameters, namely, the film thickness [3, 4] or the width of a waveguide [5], the magnetic parameters [6], the placement of the film in a periodic alternating magnetic field [7], or the deposition of a periodic lattice of metallic [8–11] or semiconductor [12] elements on the film surface.

Note that, in the case of formation of lattices of conducting elements on the YIG-film surface, their influence on the SMSW propagation can be manifested either due to the modulation of the wave dispersion under the metal [3–10], or due to the modulation of the SMSW damping coefficient depending on electrical conductivity σ and thickness t of the conductor. For example, in the case of a periodic lattice of metallic strips with their thickness t > ls, where ls is the skin-layer thickness, the magnonic crystals forms due to changing the MSW velocity on the “metalized” film portions. At the same time, at t \( \ll \)ls, the metal strips contribute additional loss on the MSW propagation and can make the observation of the Bragg resonances impossible [13].

It should be noted that the surface metallic lattices are formed using, as a rule, nonmagnetic metals. At the same time, the use of magnetic materials for fabrication of such lattices will allow one to influence the distribution of the internal magnetic field in a YIG film due to the demagnetization effect at the boundaries of the lattice elements. This fact can lead to the appearance of forbidden bands in the SMSW spectrum at frequencies of the Bragg reflections by analogy with [14]. In this case, it should be expected that a change in the direction of the bias field with respect to axes of 1D and 2D metastructures will be accompanied by the appearance of forbidden bands in the transmission spectrum of the metastructure similar to the cases of the noncollinear Bragg diffraction of SMSW in magnonic crystals considered in [15, 16]. In addition, the existence of a magnetic material layer on a YIG film surface can be observed as the features of the SMSW propagation considered for the two-layer magnetic films [17]. The dispersion of the “internal” wave propagating along the interface of magnetic media differs from the case of a free film and can change the dispersion character to the inverse one. This effect should be taken into account when choosing parameters of the lattice material.

The aim of this work is to study the influence of surface periodic metastructures of thin permalloy film at the YIG film surface on the propagation of surface magnetostatic waves (SMSW) in it.

2 EXPERIMENTAL

The magnonic crystals studied in this work were prepared on the basis of the YIG films with the saturation magnetization 4πM ≈ 1750 G with thickness d ≈ 11 μm and the ferromagnetic resonance line width ΔH ≈ 0.5 Oe. The YIG film surface was covered by the permalloy film (Py) with thickness t ≈ 40–90 nm, magnetization 4πM ≈ 8000 G, and resistivity ρ ≈ 40 × 10–8 Ω m using the magnetron sputtering method. Then, the one-dimensional (1D) and two-dimensional (2D) periodic structures were formed from the Py film using photolithography and ion etching. The 1D structures had the form of a lattice of microstrips with width w ~ 5 μm and period Λ ~ 10 μm. The 2D structures have the form of lattices of rectangles with width w ≈ 5 μm, length a ~ 20 μm and periods along axes Λx ≈ 25 μm, Λy ≈ Λ or of squares ≈5 × 5 μm with periods Λx, y ≈ Λ (Insets 1, 2, 3 in Fig. 1, respectively).

Fig. 1.
figure 1

Experimental dispersion dependences of SMSW: (1) in free YIG film, (2) in YIG film with continuous Py film, (6) in the structure with permalloy strips at θ = 0; calculated dispersion dependences of SMSW for: (3) free YIG film, (4) for the YIG–ideal metal, and (5) YIG–permalloy, respectively. H = 828 Oe.

Full size image

The obtained structures were placed on the input and output microstrip antennas with delay line width s ≈ 50 μm for SMSW. Distance L between the antennas was L ≈ 0.4 cm. We measured the frequency dependences of the modulus of S21( f ) and phase ϕ( f ) of the SMSW transmission coefficient for various angles θ between the direction of external bias field H ≈ 828 Oe and axis x (the inset in Fig. 2). The results of the measurement of the phase incursion were used to calculate the SMSW wave number q using the known [18] relationship q( f ) = –ϕ( f )/L and to built dispersion dependence q = q( f ). It turned out that dependences S21( f ) and ϕ( f ) for the used permalloy films were not qualitatively changed with t. Thus, we will consider the structures based on Py films with thickness t ≈ 40 nm.

Fig. 2.
figure 2

Dependences S21( f ) for (a) free YIG film, (curves s1–s5) structures with the permalloy strips and (curves r1–r5) rectangles, (b) (curves d1–d3)) with squares, (curves 1–2) YIG films with the etched subwavelength surface structure. The numbers near the curves denote angle θ. H = 828 Oe.

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3 RESULTS AND DISCUSSION

The results of the measurement of the SMSW dispersion in a free YIG film and in a YIG film coated by a continuous permalloy film at H = 828 Oe are shown by curves 1 and 2 in Fig. 1. Figure 1 shows also the SMSW dispersion dependences in the free YIG film (dashed line 3) [19]

$${{f}^{2}} - f_{0}^{2} - f_{m}^{2}(1 - \exp ( - 2qd)){\text{/}}4 = 0$$

and in the YIG–ideal metal structure (curve 4) [20]

$${{f}^{2}} - f_{0}^{2} - {{f}_{m}}(f + {{f}_{H}} + {{f}_{m}})(1 - \exp ( - 2qd)){\text{/}}2 = 0,$$

calculated for the parameters of the experiment.

Here, f0 = \(\sqrt {{{f}_{H}}{{f}_{B}}} \), fH = γH, fB = fH + fm, fm = γ4πM, and γ ≈ 2.8 MHz/Oe is the giromagnetic ratio for YIG. Curve 5 in Fig. 1 shows the result of calculating the SMSW dispersion in the YIG–conducting permalloy film structure performed by analogy with [21, 22]. It is seen that these experimental results agree well with the calculated results; in this case, the SMSW dispersion with wave numbers q ≤ 250 cm–1 in the YIG film coated by the continuous permalloy film is close to the case of the metallization. We failed to measure SMSW dispersion for wave numbers q > 250 cm–1 due to significant loss in the structure, similar to the case observed before for metalized YIG films [21] for the spin–electron coupling parameter G < 3. In addition, we note that the absence, at the chosen magnetic field, of features of the dispersion characteristic of “internal waves” in the structures under study is related to a substantial difference in the frequencies of the SMSW spectra in YIG and Py films because of significant difference of the saturation magnetizations [17].

The results of the measurement of the SMSW in YIG films decorated by subwave structures of permalloy strips magnetized along the axis (θ ≈ 0) are shown in Fig. 1 by curve 6. As it is seen, the character of the SMSW dispersion in the subwave structures differs cardinally from that in the case of the structures with continuous Py films and corresponds to the case of free YIG films. Such a behavior of the dispersion can be related to a change in the conditions of screening microwave fields of SMSW in going from the continuous permalloy film to the subwavelength periodic structure. Actually, a decrease in the metallization area and also a change in the magnetic permittivity tensor components of the subwavelength structure lead to an increase in the skin-layer depth and a change in the electron coupling parameter to values G < 1/3.

Now we consider the SMSW transmission spectra in the subwavelength structure as the magnetic field direction is changed with respect to the axes of the lattice of permalloy microparticles. Figure 2a shows the results of the measurement of dependences S21( f ) for the magnonic crystal with permalloy strips (curves s1–s4) and rectangular (r1–r4) in the case of YIG film free from metal (YIG curves) for which the band of the measured signal occupies the range from f0 (long-wave boundary of SMSW) to f1 (they are shown by the arrows in Fig. 2a). It is seen that, in the frequency range f0 – f3, the SMSW propagation level in the magnonic crystal with permalloy strips decreases by approximately 20 dB (curve s1), whereas the strip breaks during formation of rectangular elements enable one us to decrease the introduced loss to 5‒10 dB (curve r1). From the comparison of curves s1‒s3 and r1–r3, it is seen that the high-frequency part of dependence S21( f ) has a wide forbidden band, the central frequency of which decreases as angle θ increases. However, in this case the gap depth in the forbidden band decreases and, at θ = 60°, the losses of SMSW in the magnonic crystal with the strips and also with the rectangles almost correspond to the case of the YIG film. Note that the existence of another period Λr = a + w (inset 2 in Fig. 1) in the 2D structure from the rectangles leads to the appearance of the absorption band at θ ~ 60° (curve r4 in Fig. 2b). The increase in angle θ to 90° leads to an increase in the SMSW losses on the structure with the strips in the entire observed frequency band and, with the rectangles, in the short-wave range.

As SMSW is propagated in the structure with metallic squares, the differences from the case of the strips and the rectangles are a lower level of the losses introduced at θ = 0 (curve d1 in Fig. 2) and a smaller width of the forbidden band Δf (curve d2 in Fig. 2b for θ = 30°); in this case, the angular dependence of characteristic S21( f ) corresponds to the 90° symmetry of the structure (at θ = 60°, dependence S21( f ) coincided with curve d2).

It should be noted that, for all the metastructures studied in this work, the measured dispersion dependences for SMSW propagating in them coincided with the case of the free YIG film (for example, curve 6 in Fig. 1 was obtained for the structure with the strips at θ = 0). The absence of the Bragg forbidden bands in the S21( f ) dependences is due to the choice of the surface structure periods. The point is that the microstrip antenna width s determines the minimum wave length λmin of the excited MSW

$${{\lambda }_{{\min }}} > 2s.$$
(2)

It is seen from comparison of Eqs. (1), (2) that, as the lattice period decreases to the value Λ < s, the Bragg resonances are behind the spectrum of the excited MSW, i.e. the structures for which Λ < λmin are subwavelength ones.

The described transformation of dependence S21( f ) as angle θ is changed is similar to the behavior of the absorption band in the band of observing SMSW that propagates in the YIG film with the subwavelength lattice made of 1-μm-deep grooves with period 7 μm etched on the film surface [23]. For comparison, curve 1 in Fig. 2b shows dependence S21( f ) for SMSW propagating in the 7.7-μm-thick YIG film with the etched surface structure in the form of the lattice made of 5-μm-wide grooves arranged with the period 10 μm at θ = 40°. At the same time, we did not observe the resonance features for the considered structures at frequencies near to f0 observed in [23] at θ = 0 and θ = 90° (curves 2 and 3, respectively, in Fig. 2b).

We call attention that the wide transmission bands considered above have a “smeared” low-frequency boundary (for example, curve s3 in Fig. 2a), while the high-frequency boundary has a sharp character: its slope dS12( f )/df is higher than 3 dB/MHz. As a fixed frequency f is chosen near this boundary, the level of the signal transmission signal is changed by the shift of the band of SMSW observation by δf ~ 7 MHz, which is easily achieved using the elastic deformation of the YIG film and can be used to design the modulator of the level of the signal transmission similar to [24].

4 CONCLUSIONS

Thus, it was shown in this work that the deposition of subwave metastructures made of 40-nm-thick permalloy film on the surface of the yttrium–iron garnet film makes it possible to form the forbidden band in the frequency dependence of the transmission coefficient of the surface magnetostatic wave propagating in the film; the forbidden band parameters are dependent on the configuration of the metastructure and the direction of the external magnetic field with respect to the symmetry axes of the structures.