Theory of Electrically Tunable Thin-Film Electromagnetic Crystals Based on a Coplanar Waveguide

Artificial magnetic materials with a spatial periodic modulation of their physical properties or geometry, known as magnonic crystals, are promising for new microwave devices such as phase shifters [1, 2], various spin-wave logic elements [3, 4], high-sensitivity magnetic sensors [5, 6], and others [7, 8]. Recent advances related to the thin-film technology have resulted in fabrication of the multilayered multiferroic structures that combine advantages of the ferrites and ferroelectrics. Owning to the dual tunability of wave spectra by both electric and magnetic fields, such structures were widely used in microwave devices [9]. Different kinds of ferrite-ferroelectric all-thin-film structures were suggested based on coplanar or slot transmission lines [10–12]. In these structures, spin-electromagnetic waves (SEW) are originated from the electrodynamic coupling of the electromagnetic wave propagating in the transmission line with the spin waves (SW) propagating in the ferrite film. However, up to now the periodic multiferroic structures based on coplanar waveguides (CPW) were investigated only experimentally [10]. The purpose of this work is to develop a theory of hybridized spin-electromagnetic waves in the thin-film regular and periodic multiferroic waveguiding structures based on CPW. In order to distinguish the periodical multiferroic waveguides from known ferrite magnonic crystals, as well as from photonic crystals, below we name them as electromagnetic crystals (EMC). A studied EMC structure is shown in Fig. 1. It is composed of several layers enumerated with index j, namely, a sapphire substrate (j = 1), a barium strontium titanate (BST) ferroelectric film (j = 2), an epitaxial yttrium iron garnet (YIG) film (j = 3), and a gadolinium gallium garnet substrate (j = 4). The periodic segments of the CPW form the thin-film EMC. Here the central metal strip of width h and two conducting ground planes are positioned in the plane z between ferroelectric and ferrite layers. The segments of narrow $w_{1}$ and wide $w_{2}$ slots has the period λ. Application of control voltage U to the periodic CPW electrodes provides a reduction of the ferroelectric film permittivity $\varepsilon _{2}$ and so maintains an electric tunability. The hybridized waves are considered to propagate along x-axis, i.e. along CPW, which is magnetized to saturation by a uniform magnetic field H along z-axis. Due to the symmetry of the fundamental CPW mode its dispersion relation can be found through analytical solution of the full set of Maxwell’s equations utilizing the method of approximate boundary conditions described in details in Ref. [12]. The obtained dispersion relation for a regular CPW was used for a numerical calculation of the transmission characteristics of the EMC using the transfer-matrix method [13]. Note that this method takes into account the insertion losses and is suitable for the finite-length periodic structures. Following the outlined theory, Fig. 2(a) illustrates the effect of hybridization of two principal electromagnetic modes for the regular CPW and regular slot-line structure. One can clearly see that the area of the maximum hybridization of SEWs in the CPW shifts to the higher wavenumbers in comparison to the slot-line structure. This behavior is determined by a sufficient reduction of the phase velocity of the microwave electromagnetic waves due to additional “magnetic wall” boundary condition applied at the central metal strip. Note that the calculations were carried out for the typical parameters of YIG and BST films commonly used in microwave devices (see, e.g., Ref. [9]). A reduction of the CPW slot width w shifts the SEW dispersion characteristic to the higher wave numbers. Consequently, the SEW formed in the EMC accumulate the different phase shifts in different segments of the periodic structure at a fixed frequency (see Fig. 1). The band-gaps appear at the frequencies where this phase shift is a multiple of π. As a result, a major part of the spin-electromagnetic wave power will be reflected from the electrodes of EMC at the frequencies that satisfy to the Bragg condition. This effect is visible on the transmission characteristic shown in Fig. 2(b). The characteristic was calculated for the EMC with number of periods $N \quad =10$ and $\Lambda \quad =1$ mm with the use of the transfer matrix method. In this case, the width of the first band gap (denoted by I in Fig. 2(b)) is 24.6 MHz at a level of 3 dB from the maximum loss at 33 dB. In addition, this figure illustrates electric tuning of the EMC band gap positions for a reduction of the ferroelectric film permittivity $\varepsilon _{3}$ from 1500 to 750. In summary, a novel EMC based on CPW were studied. In particular, it was found that a high microwave signal rejection of more than 30 dB appears for the periodic structures. Furthermore, the electric tuning for the first band-gap reaches values of 10.45 MHz by the dielectric permittivity reducing of the ferroelectric film by half. All these advantages make this kind of EMC perspective for development of new microwave devices. The work in SPbETU was supported in part by the Russian Science Foundation, Grant 14-12-01296P. The work in LUT was supported by the Academy of Finland.