Introduction

Compared to the densely allocated frequency bands in current communication systems operating below 6 GHz, next-generation frequency allocations cover a wide frequency range from 7 GHz to 20 GHz1,2. To support these high-frequency bands, bandpass filters must be developed to selectively align to a particular band of interest with minimal insertion loss while strongly rejecting neighboring bands and out-of-band interference. Filter technologies with adjustable center frequencies are an attractive approach to rapidly tune over the entire frequency range using a single filter as opposed to a switched filter-bank of fixed-frequency filters with higher loss and system cost. Furthermore, the filters need to be compact and mass producible at wafer scale to meet the demands of handheld cellular devices.

Coupled micro-electromechanical resonators have been an attractive technology for integration in wireless communication systems as narrowband channel-select filters due to their high-quality factors (Q-factor) and miniaturized footprints. Wang et al.3 has demonstrated high-order micromechanical bandpass filters using one-dimensional (1D) arrays of mechanically-coupled resonators. However, for higher-order filters, long mechanical coupling beams become impractical due to size constraints4, while sensitivity due to fabrication variation causes increased insertion loss and passband distortion5. The high sensitivity of weak electrostatic or mechanical edge-coupled resonators due to structural asymmetries has been leveraged for sensitive parametric mass sensing applications in ref. 4, but prohibits their use in filters. 2D microresonator arrays have shown some success by utilizing weak coupling in one dimension to achieve passband shape and strong coupling in the other dimension to reduce effects of fabrication variation, but suffer from high insertion loss5. Coupled resonator arrays utilizing magnetostatic waves (MSW) have the potential to overcome the weak coupling and extremely narrow bandwidths achieved by electrostatically coupled micromechanical resonators6,7 while introducing a degree of tunability.

The magnetostatic wave resonance can be tuned more than an octave in frequency using a static magnetic field, ensuring that the filter size does not scale to sub-micrometer dimensions at high frequencies. Yttrium iron garnet (YIG) is the most widely used material for MSW devices due to its low Gilbert damping (α = 2.8 × 10−4 for a 100 nm film8) and experimentally reported Q-factors exceeding 30009,10. In state-of-the-art YIG sphere filters11,12,13, polished YIG resonators are attached to a thermally conductive rod and manually aligned to non-planar inductive loops acting as transducers. The assembled sphere and loop structures are then coupled through transmission lines similar to the coupling beams in ref. 3 to synthesize a filter.

In this paper, we show that planar YIG resonators can magnetically couple if they are fabricated in close proximity (Fig. 1), analogous to the electrostatically coupled mechanical resonators in refs. 4,14 or coupled electromagnetic resonators15,16. We demonstrate wafer-scale fabrication of magnetically coupled YIG filters in a highly compact footprint enabled by a surface micromachining process17,18 detailed in Supplementary Information 1. Using multiple coupled resonators introduces additional design flexibility to optimize the unit cell resonator for high Q-factor, coupling, and good spurious suppression with independent control over the passband shape through the resonator coupling factors. A new class of compact coupled-resonator bandpass filters using magnetostatic waves in thin-film YIG is demonstrated.

Fig. 1: Tunable edge-coupled MSW bandpass filters.
figure 1

a Chip microphotograph of multiple MSW bandpass filters and 1-port resonators fabricated on a YIG on GGG chip using YIG micromachining technology17,18. A 4-pole filter (on the left) and a 2-pole filter (on the right) are highlighted in red. A 1-port resonator is highlighted in orange. b Rendering of a 4-pole bandpass filter featuring four YIG resonators with gold electrodes conformally deposited over the etched YIG. Magnetic bias is oriented out-of-plane along the z-axis.

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Results

Bandpass filter design

The bandpass filter shown in Fig. 1 consists of a number of closely-spaced rectangular YIG magnetostatic forward volume wave resonators (MSFVW) with shorted 300 nm thick gold inductive transducers conformally deposited over the outermost resonators. With an out-of-plane DC magnetic bias, the RF magnetic field from the transducers excites MSFVW modes in the YIG mesa. Forward volume waves are a family of highly dispersive modes in a thin films whose lowest order mode is described by the dispersion relation19:

$${\omega }^{2}={\omega }_{0}\left[{\omega }_{0}+{\omega }_{m}\left(1-\frac{1-{e}^{-{k}_{mn}t}}{{k}_{mn}t}\right)\right],$$
(1)

where ωm = μ0γmMs, \({\omega }_{0}={\mu }_{0}{\gamma }_{m}{H}_{DC}^{eff}\), t is the film thickness, γm is the gyromagnetic ratio, μ0 is the permeability of free space, kmn is the wave vector, Ms is the saturation magnetization, and \({H}_{DC}^{eff}\) is the effective DC magnetic field. Considering the limits as kmn → 0 and kmn → ∞ in the dispersion relation, ω is restricted within the range19:

$${\omega }_{0}\le \, \omega \, \le \sqrt{{\omega }_{0}\left({\omega }_{0}+{\omega }_{m}\right)},$$
(2)

denoted as the spin wave manifold. When the planar dimensions of the thin YIG film are bounded, the magnetostatic waves reflect off the faces forming a standing waves with wave vectors approximately given by20,21,22,23,24:

$${k}_{mn}=\sqrt{{\left(\frac{\pi m}{l}\right)}^{2}+{\left(\frac{\pi n}{w}\right)}^{2}},\, m,\, n=1,\, 2,\, 3\ldots$$
(3)

where l and w are the length and width of the YIG cavity, respectively. These MSFVW resonances can be further understood through an analogy to Lamb waves in a piezoelectric plate25,26,27,28. An oscillating electric field perturbs the polarization of the piezoelectric film generating a stress field and exciting an acoustic wave. In the ferrimagnetic film, an oscillating magnetic field, conversely, perturbs the static magnetization leading to a precession of spins. Both the piezoelectric and magnetostatic cavities support a discrete number of modes whose wave vectors depend on the cavity dimensions. Nonlinear dispersion relates the wave vectors to the resonant frequencies for both MSFVW and Lamb waves. For MSFVW, this leads to irregularly spaced modes, which all reside in the spin wave manifold.

Unique to MSW, the applied out-of-plane magnetic field shifts the MSFVW dispersion relation in frequency where the tuning rate for the fundamental mode is given by

$$\frac{\partial \omega }{\partial {H}_{DC}^{eff}}={\mu }_{0}{\gamma }_{m}\frac{2{\omega }_{0}+{\omega }_{m}\left(1-\frac{1-{e}^{-{k}_{mn}t}}{{k}_{mn}t}\right)}{2\sqrt{{\omega }_{0}\left[{\omega }_{0}+{\omega }_{m}\left(1-\frac{1-{e}^{-{k}_{mn}t}}{{k}_{mn}t}\right)\right]}},$$
(4)

which simplifies to μ0γm = 2.8 MHz/Oe (for YIG) when kmnt ≪ 1. The magnetostatic scalar potential, ψ, decays exponentially outside the YIG mesa19,29 so the MSFVW resonance in one YIG mesa may couple to adjacent mesas if there is sufficient overlap in their scalar potentials30. Consequently, the spacing of adjacent resonators and their vertical sidewall profile are critical to controlling the inter-resonator coupling strength. For the 4-pole filter in Fig. 1, the spacings between adjacent resonators are 10 μm, 15 μm, and 10 μm and each YIG mesa has length l = 500 μm. The outermost resonators have a width w1,4 = 70 μm while the inner resonators are slightly narrower at w2,3 = 67 μm which was found to marginally improve insertion loss and higher order width mode suppression based on finite element simulation using Ansys HFSS. The 300 nm gold transducers are 10 μm wide over the 3 μm thick YIG mesas. At the probe launches, the gold signal pad is 94 μm wide with a 15 μm gap between the signal and ground pads.

Since each resonator length is much shorter than the electromagnetic wavelength over the tuning range, the 4-pole filter can be modeled with the lumped element circuit shown in Fig. 2. A modified Butterworth-Van Dyke circuit31,32 is typically used to model an acoustic resonance where the mechanical mode is described by a series R-L-C tank circuit and the transducers introduce a shunt plate capacitance. In the case of magnetic resonators, the transducer introduces a parasitic series inductance and the MSFVW is modeled using a parallel R-L-C tank circuit instead. L0 and R0 represent the parasitic inductance and resistance of the gold electrodes. Rm, Cm, and Lm represent the MSFVW resonance of each YIG mesa. Mnm is the inter-resonator coupling between adjacent YIG mesas while MIO represents input/output inductive coupling of the gold electrodes setting the out-of-band rejection level. Similar to a mechanical coupling coefficient, an effective coupling from the electrical to magnetostatic domain can be defined by (5), where fp and fs represent the magnetic resonance and anti-resonance frequencies respectively.

$${k}_{eff}^{2}=\frac{\pi }{2}\left(\frac{{f}_{p}}{{f}_{s}}\right)\cot \left(\frac{\pi }{2}\frac{{f}_{p}}{{f}_{s}}\right)$$
(5)

\({k}_{eff}^{2}\) is a function of the resonator design determined by the ratio of Lm to L0. Similar to an acoustic filter, \({k}_{eff}^{2}\) sets a bound on the maximum achievable filter bandwidth and impacts the passband ripple. The representative 1-port resonator (highlighted in Fig. 1) exhibits a measured effective coupling and quality-factor of \({k}_{eff}^{2}=1.53\%\) and Q = 2206 at 3962 Oe with frequency response shown in Fig. 3. From the measured resonator impedance response, R0, L0, Rm, Cm, and Lm can all be extracted through separate fittings near the magnetostatic resonance and outside the spin wave manifold where the resonator behaves as an inductor. Using the same resonator parameters, the measured filter response excluding spurious modes can be fit to the lumped model in Fig. 2 by tuning MIO, M12, and M23 under the assumption that each resonator has the same resonant frequency and the filter is symmetric. Figure 5b shows the frequency response of the lumped model fitted to the measured S21 for a 4-pole filter along with the extracted model parameters.

Fig. 2
figure 2

Lumped circuit model of an edge-coupled 4-pole MSW bandpass filter with electrically short transducers.

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Fig. 3: Measured resonator impedance.
figure 3

Frequency response of the 1-port resonator highlighted in Fig. 1a at 3962 Oe showing a Q = 2206 and \({k}_{eff}^{2}=1.53\%\).

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Experimental results

Filter s-parameters are measured using an Agilent PNA-L N5230A network analyzer with a pair of ground-signal (GS) probes from 3205 Oe to 5303 Oe corresponding to a center frequency tuning over 5.6 GHz as shown in Fig. 4a. A single-pole electromagnet powered by a constant current source provides the required out-of-plane magnetic bias (pictured in Fig. 4b) and a single-axis Gauss meter is used to map the source current to the applied field. Prior to each measurement, the device under test is aligned to the center of the electromagnet’s pole to ensure field uniformity. Figure 5a,b show the frequency response near the passband for a 2-pole and 4-pole filter at 3864 Oe. Outside of the spin wave manifold, no magnetostatic waves may propagate, so the filter behaves as two coupled inductors. Consequently, the out-of-band rejection is governed by the inductive coupling strength between the input and output electrodes. For the 2-pole filter with an electrode spacing of 67 μm, the rejection is  − 25 dB while for the 4-pole filter with a spacing of 229 μm, the rejection is  − 35 dB. At 3864 Oe, the 2-pole filter shows an insertion loss (IL) of  − 3.55 dB and a 3 dB bandwidth of 57.0 MHz while the 4-pole filter has an IL of 6.94 dB and a 3 dB bandwidth of 17.0 MHz. From Fig. 5a and 5b, higher-order magnetostatic spurious modes are visible to the right of the passband with a frequency separation of 29–40 MHz. As described in ref. 20, the current distribution along the transducer length can excite either even or odd ordered modes. Based on the resonator dimensions and electrically short transducer, the frequency spacing of these spurs agree well with odd ordered length modes. Figure 6 shows the linear center frequency tuning at a rate of 2.7 MHz/Oe and the 3 dB bandwidth over the applied bias for the 4-pole filter. With a well-calibrated bias field, the extrapolated center frequency tuning line should intersect—ωm at 0 Oe. However, the Gauss meter used for the field calibration is thicker than the fabricated chip, so the reported bias is underestimated by approximately 219 Oe. The filter’s tuning was also measured using a neodymium permanent magnet mounted on a 3-axis stage to precisely calibrate the field accounting for any thickness difference between the chip and sensor. In this setup, the extrapolated 0 Oe intersection is at ωm = μ0γm ⋅ 1751 Oe, which agrees well with the saturation magnetization of liquid phase epitaxy (LPE) YIG.

Fig. 4: Filter tuning response and experimental setup.
figure 4

a Measured 4-pole MSW bandpass filter frequency response at different out-of-plane magnetic biases from 3205 Oe to 5303 Oe. b Experimental setup showing the fabricated filter chip resting on the pole of an electromagnet, two GS probes connected to one device under test, and an optical microscope used for probe landing and device alignment.

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Fig. 5: Measured filter passbands.
figure 5

Frequency responses for the a 2-pole and b 4-pole filters highlighted in Fig. 1 near the passband at 3864 Oe. b also shows a comparison of the fitted lumped element model in Fig. 2 with measured S21. The fitted circuit assumes all four resonators are identical and excludes the spurious passbands caused by higher order MSFVW modes.

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Fig. 6: Measured 3 dB bandwidth and center frequency of the 4-pole MSFVW filter.
figure 6

The center frequency tunes rate of 2.7 MHz/Oe.

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Considering the total loss (\(1-{\left\vert {S}_{11}\right\vert }^{2}-{\left\vert {S}_{21}\right\vert }^{2}\)) for the 2-pole filter biased at 3652 Oe and measured far away from all magnetostatic resonances, an average of 43% of the input power is dissipated in the thin 300 nm gold transducers, radiated, or absorbed by the YIG on GGG substrate. Based on finite element simulations, the loss is primarily attributed to the resistance of the gold transducers. A second sample was fabricated with 3 μm electroplated gold to reduce resistive losses. Figure 7 compares the measured insertion loss and total loss for the same 2-pole filter with different gold thicknesses. As expected, the mean out-of-band loss shows significant improvement from 43% to only 12%. The average loss within the 3dB bandwidth exhibits a slight bias dependent improvement up to a 14.5% reduction. The insertion loss improvement reflects the change in mean in-band loss with a maximum improvement for the 2-pole filter from 4.43 dB to 2.92 dB around 3660 Oe. From Fig. 7b, the total loss peaks at the resonance frequency of each magnetostatic mode. Using an HFSS simulation with 3 μm thick gold, each material’s contribution to the total loss at the fundamental coupled modes is calculated. This analysis reveals that the remaining loss is concentrated at the interface between the YIG mesa and the gold transducer. By increasing the resonator length, both the simulated interface loss and insertion loss can be reduced.

Fig. 7: Filter comparison with different metal thicknesses.
figure 7

a Measured S21 and b total loss for 2-pole filters with 300 nm and 3 μm thick gold transducers biased at 3652 Oe and 3660 Oe, respectively. Frequency is plotted relative to the center frequency to account for the slight difference in bias strength.

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The linearity of the MSW bandpass filter is evaluated by measuring the input referred 3rd order intercept point (IIP3) in the passband as well as the stopband both below and above the passband at a bias of 3652 Oe. The nonlinearity measurements are performed using two Keysight E8257D signal generators with a frequency separation of Δf = 15 MHz. The higher of the two-tone frequencies for each region are listed in Table 1. A wideband power divider combines the two tones while an Agilent PXA spectrum analyzer measures the resultant spectrum. A two-stage calibration is performed to remove all cable and system losses at every tone frequency and input power level. Far away from the passband, the filter is expected to be linear and no intermodulation products were observed. Based on the maximum output power of the signal generators and noise floor of the spectrum analyzer, the lower stopband IIP3 is estimated to be greater than 37.95 dBm at 3652 Oe. The passband shows the greatest nonlinearity with an IIP3 of  −4.75 dBm at 3652 Oe, as shown in Fig. 8. The measured IIP3 in each frequency region is summarized in Table 1.

Table 1 Summary of the 4-pole Filter IIP3 measurements
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Fig. 8: Two-tone IIP3 measurement in the passband of a 4-pole filter at 3652 Oe bias.
figure 8

The two input tones are separated by 15 MHz with the high tone at 5.799 GHz.

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Discussion

In this paper, we have demonstrated a novel edge-coupled highly-tunable magnetostatic bandpass filter using state-of-the-art micromachining fabrication techniques. The designed 2-pole and 4-pole filters have been tuned over an octave from 4.5 GHz to 10.1 GHz showing a consistent passband shape, and the filter linearity was characterized in three distinct frequency regions. These initial designs show performance comparable to other state-of-the-art frequency tunable bandpass filters, as summarized in Table 2, but offer a high degree of design flexibility with adjustable bandwidth. Our micromachining process enables precise control over the YIG mesa shape and spacings to synthesize miniaturized MSW channel-select filters analogous to electromagnetic coupled-resonator filter design.

Table 2 Performance comparison with other tunable bandpass filters
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Methods

Filter fabrication

Starting with a 3 μm thick YIG film epitaxially grown on a 500 μm GGG substrate, the YIG mesas are patterned through ion milling with a thick SPR220-7.0 photoresist mask. The gold electrodes are deposited through glancing angle e-beam evaporation and patterned through a bi-layer liftoff process. Additional details of the fabrication process are provided in Supplementary Information 1.

Measurement setup

Small signal s-parameters of the MSFVW filters are measured on chip using a pair of non-magnetic GS probes and an Agilent PNA-L N5230A network analyzer. An out-of-plane magnetic bias is generated by a single-pole electromagnet and calibrated using a single-axis Gauss meter (both provided by DexinMag). Two RF tones for the non-linearity measurements are provided by Keysight E8257D signal generators and combined using a wideband power divider. The output spectrum is measured by an Agilent PXA spectrum analyzer.