The design and operation of SW ladder filters can be understood through an analogy to piezoelectric microelectromechanical systems (MEMS) acoustic filters34 in Supplementary Information Section II. The transduction of SWs relies on a transverse RF magnetic field instead of an electric field in the case of piezoelectric MEMS resonators. Therefore, the electrical response of the SW transducer is modelled using a series R 0 and an inductive transmission line with impedance Z 0 and physical length P 0 . The SW resonance is modelled as a parallel resonator network (R m , C m , L m ) in series with the transducer as shown in Fig. 1b (ref. 22). Far away from resonance, the SW ladder filter behaves as an inductive divider network. The out-of-band rejection is a function of the number of resonators stages and the impedance contrast between the series and shunt transmission lines. This distributed SW resonator model exhibits a high impedance resonance at f p corresponding to the selected SW mode and a low impedance antiresonance at f s . To realize a band-pass filter, the shunt resonator’s f p must be shifted to roughly align with f s of the series resonator. The resonators in this work rely on forward volume SWs using an out-of-plane magnetic bias. The dispersion of these modes is a function of the effective magnetic field inside the film (\({H}_{{\rm{DC}}}^{{\rm{eff}}}\)), the film thickness t YIG and the SW vector k mn , which depends on the film geometry17,18. A marked shift between the series and shunt resonances can be accomplished by biasing each with a distinct external magnetic field (H DC ). However, realizing two strong, uniform magnetic biases over each SW resonator in a small filter area is exceptionally difficult and would increase the packaged filter volume. Furthermore, the physical separation between the series and shunt resonators would need to increase, resulting in higher routing losses and reduced centre-frequency tuning range. Alternatively, engineering a contrast in the SW wavevectors to obtain the frequency shift required for filters with even a moderate bandwidth is equally as challenging owing to the nonlinear SW dispersion (Supplementary Fig. 1). The effective bias is given by \({H}_{{\rm{DC}}}^{{\rm{eff}}}={H}_{{\rm{DC}}}-{N}_{z}{M}_{{\rm{s}}}\), in which N z M s is the film’s demagnetizing field in the out-of-plane direction and M s = 178 mT is the saturation magnetization for YIG. The demagnetization factor N z is geometry-dependent and approximated using ref. 35 for rectangular prisms. By engineering a contrast in the series and shunt resonator dimensions, the frequency shift owing to the combined difference in k mn and N z enables filters of more than 600 MHz in bandwidth biased by a single external magnetic field. Figure 1a illustrates the SW resonance frequency over the resonator width (with a nominal length and thickness of 600 μm and 3 μm, respectively) for a range of external bias fields, H DC . With a shunt resonator width of 12 μm and a series resonator width of 50 μm, the calculated resonance frequency separation is 1.24 GHz owing to the combined contrast in k mn and N z .
Fig. 1: Design of SW ladder filters with a single magnetic bias. a, Dispersion curve for magnetostatic forward volume waves including demagnetization effects over the YIG resonator width for different magnetic bias fields. With high contrast in the YIG dimensions between the series and shunt resonators, a large resonance frequency separation can be realized using a single external magnetic bias. b, 3D schematic of series and shunt SW resonators in a ladder filter configuration. Each resonator in the filter schematic is described by a two-port distributed SW model based on ref. 22. The series resonator consists of a single, wide YIG mesa (1,000 μm × 50 μm × 3 μm) and a high-impedance Au transmission line. The two shunt resonators are constructed from an array of parallel YIG fins (600 μm × 12 μm × 3 μm) with a low input impedance. c, Modelled SW resonator impedance response and resulting third-order filter response of the cascaded series and shunt resonators. The resonator models are fit from finite element simulations (Ansys HFSS) at 792.3 mT bias field. d, Rendering of a third-order SW ladder filter layout consisting of one series YIG resonator and 12 YIG fins collectively acting as two shunt resonators. The filter is situated between two permanent magnets providing the out-of-plane magnetic bias (H DC ). A cross-section of the filter layout highlights the 10-μm thin GGG membrane and Au ground plane beneath each device. e,f, Fabricated third-order (e) and fifth-order (f) SW ladder filters with footprints of 1.03 × 1.52 mm2 and 1.25 × 1.52 mm2, respectively. The dark region surrounding each filter is the sidewall of the etched GGG cavity beneath each device. g, Chip micrograph showing many functional SW ladder filters, resonators and de-embedding structures. Scale bars, 300 μm (e,f); 2,000 μm (g). Full size image
Designing ladder filters with appreciable bandwidth and low loss also requires the component resonators to have high quality factors (Q-factors) at both f p and f s and sufficiently high coupling factor, \({k}_{{\rm{eff}}}^{2}\) (defined in Supplementary Information Section III). Widely available single-crystal YIG films grown on lattice-matched gadolinium gallium garnet (GGG) substrates using liquid-phase epitaxy gives high Q-factors at f p owing to the material’s low inhomogeneous broadening and low Gilbert damping factor on the order of 10−4 (refs. 36,37,38). Recently reported experimental Q-factors in this material platform include Q = 1,313 at f p = 11.6 GHz (ref. 20), Q = 2,206 at f p = 6.79 GHz (ref. 39) and Q = 200–350 at f p = 4–11 GHz (ref. 40). The Q-factor at f s is primarily a function of the transmission line design and can be maximized using low-resistivity Au transducers22. Until recently, effective resonator coupling has been limited to \({k}_{{\rm{eff}}}^{2} < 3{\rm{ \% }}\) (refs. 40,41,42) using conventional fabrication techniques that allow only topside electrodes for YIG-on-GGG resonators. The effective coupling for SW resonators strongly depends on the confinement of the transverse RF magnetic field within the YIG film19. To push the experimentally demonstrated \({k}_{{\rm{eff}}}^{2}\) up to 18%, Tiwari et al.19 introduce a backside anisotropic GGG wet-etch process to place a ground plane in close proximity (<20 μm) below the YIG film with through-GGG vias to wrap the transducer around the YIG resonator.
Building off the high-coupling resonators in ref. 19, Fig. 1b shows a rendering of the series and shunt component resonators for the SW ladder filter. The series resonator consists of a 1,000 μm × 50 μm × 3 μm ion-milled YIG mesa with a bottom ground plane 10 μm beneath the YIG film. The transducer uses 3-μm electroplated gold to minimize resistance (R 0 ). On the basis of the study in ref. 22, the transducer width is 60% of the YIG width to strongly suppress spurious SW modes. The shunt resonator consists of a parallel array of six narrow YIG fins each measuring 600 μm × 12 μm × 3 μm. The relatively narrow shunt resonator reduces the demagnetization field and increases k mn , resulting in a higher resonance frequency at the same externally applied bias. Using six YIG fins in parallel provides a balance between a low off-resonance impedance for strong out-of-band rejection and a high shunt resonator Q-factor for low insertion loss. Consistent with the series resonator, the 3-μm-thick Au shunt transducer is 60% of the YIG width and a ground plane is 10 μm away. Notably, the wide transducers and close proximity of the bottom ground plane reduce the impedance of the component resonator. As a result, there is also an engineering design trade-off between the filter impedance matching to 50 Ω, maximum bandwidth and spur suppression. Figure 1c illustrates the SW ladder filter operation by plotting the input impedance of the series and shunt resonators against the modelled filter response for a third-order one-series, two-shunt SW ladder filter. Further details on resonator and filter design, modelling and measured performance are available in Supplementary Information Section III. Figure 1d illustrates a third-order SW ladder filter layout combining the individual series and shunt resonators from Fig. 1b. The filter is situated in a uniform out-of-plane magnetic field. A cross-section of the filter layout highlights the selectively thinned GGG substrate enabling the placement of a close-proximity ground plane beneath the filter (required for high resonator \({k}_{{\rm{eff}}}^{2}\)). To experimentally validate these SW ladder design concepts, they are fabricated using a 15 × 15-mm YIG-on-GGG chip. Figure 1e,f shows micrographs of a third-order and fifth-order fabricated SW filter. Figure 1g highlights a micrograph of the fabricated YIG-on-GGG chip demonstrating 100% yield across all filters and resonators.