The cross-integration of spin-wave and superconducting technologies is a promising method for creating novel hybrid devices for future information processing technologies to store, manipulate, or convert data in both classical and quantum regimes. Hybrid magnon–polariton systems have been widely studied using bulk Yttrium Iron Garnet (Y3Fe5O12, YIG) and three-dimensional microwave photon cavities. However, limitations in YIG growth have, thus far, prevented its incorporation into CMOS compatible technologies, such as high-quality factor superconducting quantum technology. To overcome this impediment, we have used Plasma Focused Ion Beam (PFIB) technology—taking advantage of precision placement down to the micrometer scale—to integrate YIG with superconducting microwave devices. Ferromagnetic resonance has been measured at milliKelvin temperatures on PFIB-processed YIG samples using planar microwave circuits. Furthermore, we demonstrate strong coupling between superconducting resonators and YIG ferromagnetic resonance modes by maintaining reasonably low loss while reducing the system down to the micrometer scale. This achievement of strong coupling on-chip is a crucial step toward fabrication of functional hybrid quantum devices from spin-wave and superconducting components.

In recent years, there has been large investment into research on cavity magnon–polariton systems with strongly coupled spin-wave and photonic components.1–8 This research is driven in part by the promise of creating new hybrid devices using spin-wave and other technologies, such as superconducting quantum technology.9 Indeed, hybrid ferromagnetic–superconducting systems have been realized using both a 3D electromagnetic cavity9 and planarized 2D resonators.10 This latter work took a large step toward full integration of spin-wave and superconducting technologies; however, the large magnon line-widths of ferromagnetic metals, such as Permalloy (Ni80Fe20), pose deleterious constraints on quantum coherent applications. The ferrimagnetic insulator yttrium iron garnet (Y3Fe5O12, YIG), on the other hand, has a spin-wave damping which is orders of magnitude lower that metallic ferromagnets as well as much longer magnon lifetimes.11–13 Therefore, this material has been the forefront candidate for several applications in the field of spin-wave technology (also known as magnonics14–19). However, despite its exceptional spin-wave qualities, YIG poses issues of its own, specifically its difficult growth and incompatibility standard processing techniques. The growth of high-quality YIG typically requires specialized substrates, such as gadolinium gallium garnet (Gd3Ga5O12, GGG), making it incompatible with CMOS processing techniques.16 Here, we present a method for integrating high-quality YIG with superconducting quantum technologies using plasma focused ion beam (PFIB) technologies. Our primary result is the achievement of strong coupling on chip devices comprising a superconducting resonator and a YIG sample at sub-Kelvin temperatures.

The hybrid superconducting and ferromagnetic devices in this study were fabricated in two stages: the fabrication of planar superconducting devices and the integration of PFIB-processed YIG onto those devices. Superconducting planar devices were fabricated from 100 nm thick niobium nitrate (NbN) films. The films were deposited onto intrinsic silicon substrates at room temperature using reactive sputtering of Nb in a mixture of N2 and Ar gas flow in a similar fashion to other studies.21 NbN material was chosen for its high critical field values, which ensures that the devices remain superconducting and operable under applied field. The superconducting transition temperature was found to be 11 K using four-probe resistivity measurements. Planar devices were fabricated from the NbN films using photo- and electron-beam lithographies and reactive ion etching methods.

YIG samples were processed from a 100 μm thick film, grown on GGG, using a Xe plasma focused ion beam. Processing steps for the YIG samples are shown in Figs. 1(a)–1(d). First, a large section of the film is milled out using a wide beam column. The sample is then tilted by 52° relative to the Xe beam, and a narrower column beam is used to mill beneath the desired section of the film, creating an inverted pyramid shape of YIG [Fig. 1(a)]. Before the sample is fully removed from the surface of the film, the finger of a nanomanipulator approaches the material's edge, where Pt is deposited, connecting the sample to the finger. The pyramid-shaped YIG can then be extracted on the nanomanipulator, as shown in Fig. 1(b). After the sample is fixed to the nanomanipulator, it can be fully removed from the surface and moved to a stage, where finer milling can be performed to achieve the desired shape [Fig. 1(c)]. Afterwards, the sample is removed again by the nanomanipulator and placed on top of prefabricated superconducting microwave devices [Fig. 1(d)]. Currents for Xe milling ranged from 1.8 to 180 nA. For the three samples (nos. 1–3) reported in this work, pieces of YIG were shaped into rough rectangular cuboid shapes with dimensions ranging from 5 to 50 μm. The measured dimensions for each sample are listed in Table I. These pieces were placed, with micrometer precision, on top of NbN microwave devices for spectral measurement. The YIG samples were fixed to these devices using precise Pt deposition. When necessary, ion milling was used to remove excess Pt that could short the microwave devices [an example of this can be seen in Fig. 1(e) as dark lines in the gaps between the central inductive lines and interdigitated capacitors of the superconducting resonator devices].

FIG. 1.

SEM images showing the PFIB processing steps for YIG. (a) Section of a YIG film that has been milled out using a Xe ion beam. An upside down pyramid structure with an approximate 50 × 50 μm2 base is milled from the surface of the YIG film. (b) The pyramid is attached to a nanomanipulator via Pt deposition on the manipulator tip. Afterward, the connecting bridge [seen in (a)] is milled away and the pyramid is extracted from the film surface as shown. (c) The extracted pyramid from (b) is attached to a metal finger (right object) using Pt deposition. From this position, the pyramid is milled flat into a cuboid shape. The cuboid can be cut into smaller sections (e.g., 50×10×5μm3). (d) The samples are then placed onto the NbN microwave devices using the nanomanipulator. The YIG samples are affixed to the device surfaces by precision Pt deposition. The YIG is milled free from the nanomanipulator by milling briefly near the area connecting the nanomanipulator and the YIG sample. (e) An SEM image of NbN lumped element resonators with YIG placed onto the central inductive line where the magnetic antinode is. The YIG samples are oriented with respect to the shown axes. (f) Simulation of the NbN resonator with zero-field frequency ωp0/2π=12.517 GHz. The field profiles were determined for the bare resonator without YIG and simulated using COMSOL Multiphysics® software.

FIG. 1.

SEM images showing the PFIB processing steps for YIG. (a) Section of a YIG film that has been milled out using a Xe ion beam. An upside down pyramid structure with an approximate 50 × 50 μm2 base is milled from the surface of the YIG film. (b) The pyramid is attached to a nanomanipulator via Pt deposition on the manipulator tip. Afterward, the connecting bridge [seen in (a)] is milled away and the pyramid is extracted from the film surface as shown. (c) The extracted pyramid from (b) is attached to a metal finger (right object) using Pt deposition. From this position, the pyramid is milled flat into a cuboid shape. The cuboid can be cut into smaller sections (e.g., 50×10×5μm3). (d) The samples are then placed onto the NbN microwave devices using the nanomanipulator. The YIG samples are affixed to the device surfaces by precision Pt deposition. The YIG is milled free from the nanomanipulator by milling briefly near the area connecting the nanomanipulator and the YIG sample. (e) An SEM image of NbN lumped element resonators with YIG placed onto the central inductive line where the magnetic antinode is. The YIG samples are oriented with respect to the shown axes. (f) Simulation of the NbN resonator with zero-field frequency ωp0/2π=12.517 GHz. The field profiles were determined for the bare resonator without YIG and simulated using COMSOL Multiphysics® software.

Close modal
TABLE I.

Sample dimensions and their calculated demagnetization factors. Errors on sample dimensions correspond to the deviations from the cuboid shape across each sample. The demagnetization factors were calculated from sample dimensions using equations in Ref. 20. Errors on the demagnetization factors reflect the shape errors. For the sample dimensions, the z direction is oriented parallel to the applied field, along the device transmission line length [see insets of Figs. 2(a) and 2(b)]. The x and y directions are oriented to the respective in-plane and out-of-plane perpendiculars to z axis.

Samplex (μm)y (μm)z (μm)DxDyDz
#1 50 ± 6 15 ± 1 45.8 ± 0.3 0.187 ± 0.004 0.61 ± 0.01 0.204 ± 0.002 
#2 10.8 ± 0.5 6.1 ± 0.3 36.3 ± 0.4 0.335 ± 0.004 0.570 ± 0.007 0.0958 ± 0.0003 
#3 12.5 ± 0.5 6.5 ± 0.5 47.5 ± 0.5 0.326 ± 0.005 0.59 ± 0.01 0.0818 ± 0.0003 
Samplex (μm)y (μm)z (μm)DxDyDz
#1 50 ± 6 15 ± 1 45.8 ± 0.3 0.187 ± 0.004 0.61 ± 0.01 0.204 ± 0.002 
#2 10.8 ± 0.5 6.1 ± 0.3 36.3 ± 0.4 0.335 ± 0.004 0.570 ± 0.007 0.0958 ± 0.0003 
#3 12.5 ± 0.5 6.5 ± 0.5 47.5 ± 0.5 0.326 ± 0.005 0.59 ± 0.01 0.0818 ± 0.0003 

Ferromagnetic resonance (FMR) spectra were measured on YIG samples 1 and 2. The samples were placed on top of NbN coplanar waveguide transmission lines with a 10 μm width, 6 μm gap geometry [see insets of Figs. 2(a) and 2(b)]. The devices were then placed inside Cu boxes for microwave measurement and mounted inside a superconducting solenoid within an adiabatic demagnetization refrigerator [see the supplementary material Fig. S1(a) for a diagram of the measurement setup]. The solenoid field was calibrated using an electron spin resonance measurement of a known material22 [see supplementary material Fig. S1(b) and text]. Microwave transmission, S21, was measured by a vector network analyzer (VNA) sending signals to ports on the NbN transmission line. The VNA signals were attenuated by 20 dB at 70, 4, and 0.5 K stages before reaching the transmission line. At the 4 K stage, a high-electron-mobility transistor (HEMT) provided 40 dB amplification to signals after passing through the chip. An isolator placed before the HEMT prevented contamination of transmission signals from spurious reflections back to the transmission line. For some measurements, additional 45 dB amplification was provided by a room temperature amplifier. VNA excitations ranged from −30 to 0 dBm, and no power dependence was observed in this range for any samples. Ferromagnetic resonance (FMR) was then measured at 3.2 K and 80 mK by applying a static magnetic field parallel to the transmission line via the superconducting solenoid. For all measurements presented, the signals are normalized with respect to the microwave background signal.

FIG. 2.

(a) Spectral measurement of sample 1 on top of a 10 μm wide NbN transmission line at 3 K. Inset: SEM image of sample 1. The x and z axes of the sample are labeled. The y axis (unlabelled) is oriented perpendicular to the NbN film surface. (b) Measurement of sample 2 on top of a 10 μm wide NbN transmission line at 2.9 K. Inset: SEM image of the sample 2 with labeled axes. (c) FMR resonances of sample 1 at μ0H=160 mT. The FMR linewidths show no significant change from 2.9 K to 80 mK.

FIG. 2.

(a) Spectral measurement of sample 1 on top of a 10 μm wide NbN transmission line at 3 K. Inset: SEM image of sample 1. The x and z axes of the sample are labeled. The y axis (unlabelled) is oriented perpendicular to the NbN film surface. (b) Measurement of sample 2 on top of a 10 μm wide NbN transmission line at 2.9 K. Inset: SEM image of the sample 2 with labeled axes. (c) FMR resonances of sample 1 at μ0H=160 mT. The FMR linewidths show no significant change from 2.9 K to 80 mK.

Close modal

As shown in Figs. 2(a)–2(c), the FMR lines appear in the spectrum as absorbing resonances whose frequency increases with applied field. The spectra have a strong dependence on the sample shape and size. For example, sample 1, with the largest volume, exhibits several spin-wave modes [Fig. 2(a)]. We associate these additional resonances with a combination of magnetostatic surface spin waves and forward and backward volume magnetostatic spin waves.15 However, identification of the modes is difficult due to the imperfect cuboid sample shape and potential effects on the spin waves from the coplanar waveguide geometry of the superconducting transmission line.23 On the other hand, if the width and thickness (i.e., x and y dimensions) of the YIG are reduced, such as for sample 2, then modes become more isolate, leaving only one dominant mode in the spectrum [Fig. 2(b)]. We associate this resonance in the spectrum of sample 2 with the Kittel mode.24 Our main focus is on the Kittel mode, whose resonance frequency is described according to the Kittel equation24 

ωFMR=γμ0[Heff+(DxDz)Ms]·[Heff+(DyDz)Ms].
(1)

Here γ is the gyromagnetic ratio, μ0 is the permeability of free space, Ms is the saturation magnetization, and Dx, Dy, and Dz are demagnetization factors, which account for the effect of the sample shape. Heff=H+Hk is the effective field consisting of the external field H produced by the superconducting solenoid and applied along the z direction and the field Hk associated with the YIG magnetocrystalline anisotropy. The field-dependent resonance frequency for sample 2 can be fit to this equation to determine the saturation magnetization of the PFIB-processed YIG. First, however, the demagnetization factors are calculated for the cuboid shape of the sample by using equations supplied in Ref. 20. Using these factors, which are listed in Table I, the saturation magnetization is determined to be 234 ± 4 mT in reasonable agreement with other measurements on YIG near zero temperature [see supplementary material Fig. S3(b) and Refs. 6 and 13]. The magnetocrystalline anisotropy, which depends on Ms, is determined to be μ0Hk30 mT and will be discussed in more detail below. However, due to the weakness of the FMR signal, a determination of the coupling strength is not possible, and therefore, an extraction of the magnon linewidths from the FMR data is unreliable. Nevertheless, from fits of the resonance signals of samples 1–3, we can determine a linewidth range of 15–40 MHz. Furthermore, as shown in Fig. 2(c), we find that the FMR modes show little to no change between 2.9 K and 80 mK, indicating the saturation of the linewidth in the zero-temperature limit.

FIG. 3.

Spectral measurement of the YIG-resonator device (sample 3) as a function of external applied field H. Several avoided crossings occur as each YIG FMR mode crosses the NbN resonator frequency. The black dashed lines represent fits of the branches of the largest anticrossing based on Eq. (2). The gap between the two branches corresponds to twice the effective coupling. For the largest anticrossing, the effective coupling is estimated to be g/2π=63±5 MHz.

FIG. 3.

Spectral measurement of the YIG-resonator device (sample 3) as a function of external applied field H. Several avoided crossings occur as each YIG FMR mode crosses the NbN resonator frequency. The black dashed lines represent fits of the branches of the largest anticrossing based on Eq. (2). The gap between the two branches corresponds to twice the effective coupling. For the largest anticrossing, the effective coupling is estimated to be g/2π=63±5 MHz.

Close modal

The main result of this work is the achievement of strong coupling between the FMR modes of the PFIB-processed YIG and the photonic mode of a superconducting resonator. A scanning electron microscopy (SEM) image of the lumped-element resonator devices, which consist of LC circuits comprised of interdigitated capacitors connected by an inductive line, is shown in Fig. 1(e). Using the PFIB processing techniques, YIG was placed on the center of the inductive line of the shown resonators. At this position, the field profile derived from microwave simulations, as shown in Fig. 1(f), confirms that the YIG is exposed to a uniform magnetic excitation field parallel with the chip surface and directed across the line. This ensures the strongest coupling of both resonant systems as well as that the FMR condition is met by having the excitation field perpendicular to the externally applied field from the solenoid, which is applied parallel to the inductive line of the resonator. Of the devices shown in Fig. 1(e), we focus on the YIG-resonator device, which exhibited well-defined signatures of strong coupling; the spectra of other select devices on the same chip are shown in Fig. S2 of the supplementary material. The device of interest, which is called sample 3, has a zero-field resonance frequency of ωp0/2π=12.517 GHz with a full-width at half-maximum linewidth of Γ/π=9.2 MHz. At zero field, the loaded quality factor is 1250 after placement of the YIG. The resonance exhibits Fano-like25,26 behavior [see Fig. S3(a) in the supplementary material], indicating the presence of interference. Such interference could be due to imperfections created by the placement of YIG or coupling of the resonator to stray resonances in the background transmission.

Spectral measurements of the device were performed using the same methods as used to measure FMR of samples 1 and 2. As shown in Fig. 3, when field is applied to the chip, the YIG FMR frequency increases, crossing the resonator frequency. When the two resonances coincide, coupling lifts the degeneracy of the system, resulting in an avoided crossing of modes in the spectrum. For our device, we observe several avoided crossings corresponding to coupling between the resonator photons and different spin-wave modes. The frequency separation between the branches of the avoided crossings is related to the strength of coupling, and the two branches can be described generally by an equation of the form1,27

ω±=12[ωp+ωFMR±(ωpωFMR)2+4g2].
(2)

Here, ωp is the resonator photon frequency and g is the coupling constant. Figure 3 shows fits of the branches of the largest avoided crossing using Eq. (2). We use a least squares fitting procedure to independently fit the two branches. The saturation magnetization Ms and g are used as fit parameters, while ωp/2π is fixed at 12.494 GHz. Although the resonator frequency has a field dependence due to the proliferation of vortices in the superconductor,10,28 the field range for the avoided crossing is small enough that an adequate fit can be accomplished without including this effect. From fits of the two branches result, we find that μ0Ms=205±5 mT and g/2π=63±5 MHz for the largest avoided crossing, where the errors reflect the disagreement between the two independent fits of each branch.

The coupling constant itself has been quoted as a function of the spin density, ρ, taking the form1 

g=γ2μ0ωpρVm2Vp,
(3)

where Vm is the volume of the magnetic sample and Vp is the resonator's mode volume. Here, we take the spin density to be ρ=2×1022 cm−3 (appropriate for good-quality YIG1) and Vp=1.59×105 cm3 (estimated using the finite-element simulations package COMSOL Multiphysics®). Substituting these values into Eq. (3), we find g=70±3 MHz, which is in reasonable agreement with the fits of the experimental data. The coupling g can also be theoretically derived using the electromagnetic perturbation theory (see the supplementary material of Ref. 27). In this case, g is defined in terms of the magnetization rather than spin density. Using μ0Ms=205 mT, the coupling is predicted to be g=68±3 MHz, in close agreement with the value based on an assumed spin density.

Without the inclusion of magnetocrystalline anisotropy, the saturation magnetization of the PFIB-processed YIG shows consistent values of μ0Ms300 mT for samples 2 and 3. Since the Kittel mode cannot be reliably identified for sample 1, we cannot provide a proper estimate of Ms. Nevertheless, in comparison with other studies on YIG in the sub-Kelvin regime,6,12,29 the extracted values from our samples would initially seem much higher than previously reported. While demagnetization factors are taken into account, the high Ms values imply that other effects, such as magnetocrystalline anisotropy, should be considered. At 4.2 K, the first and secondary cubic magnetocrystalline anisotropy constants30 for YIG are K1=2480 and K2=118 J/m3. Although the samples are cuboid in shape, the internal anisotropy fields can be approximated by a relation for a (111)-oriented 2D film.31,32 While the exact calculation of these fields depends on the in-plane crystallographic orientation, which is lost during the PFIB processing, an order of magnitude approximation is μ0Hk2K1/Ms2K2/Ms. By including this Ms-dependent field term in Eq. (1), we determine μ0Hk30 mT and saturation magnetization values μ0Ms=234±4 mT (sample 2) and 205 ± 5 mT (sample 3). These values of Ms are more in line with the saturation magnetization μ0Ms=240±1 mT found for an unstructured YIG film [see Fig. S3(b) in the supplementary material] and is also in the range of previously reported values for pure YIG at 4 K.13,33 Unfortunately, the PFIB-processed samples' magnetic moments were below the sensitivity of the available SQUID (superconducting quantum interference device) magnetometer. Previous studies indicate that ion implantation can significantly decrease the magnetization of YIG,33 so without other means of investigation (e.g., Brillouin light scattering34) we cannot make a strong argument about presence of these effects in our samples. However, a thorough study of anisotropies and magnetization behaviors after PFIB fabrication is beyond the scope of this article and is a subject for a future work.

The experimental and theoretical values of g for sample 3 show relatively good agreement with only 10% difference. Since shape variation is already accounted for in the uncertainty on the theoretical value, it cannot completely account for the difference. Instead, the 10% difference indicates that either Vp is slightly underestimated in the simulation or the effective moment density ρ is reduced due to damage from the PFIB processing. Although neither Vp nor ρ can be measured directly, there is some hint when comparing the spin-density and electromagnetic perturbation theory based predictions of g. Since the electromagnetic perturbation theory uses Ms, which is determined from fitting, the general agreement between spin-density and electromagnetic perturbation predictions of g indicates that ρ is close to the expected value for ideal YIG, even though the determined values of Ms are different for samples 2 and 3. Therefore, it seems probable that surface damage from PFIB processing does not strongly affect the effective spin density and that a misestimation of Vp is responsible for the disagreement between experimental and theoretical determinations of g. Furthermore, this also implies that the difference between Ms for samples 2 and 3 may depend on other factors, such as Hk and the crystalline orientation, which is unaccounted for.

Based on the experimental determination of g and assumed spin-density for the first avoided crossing, the single-spin coupling constant can be determined as g0/2π=g/2πN=7.2 Hz, where N=ρVm. In addition, sample 3 exhibited at least two more crossings at higher fields, indicating the presence of strong magnon–photon coupling to higher order spin-wave modes. Indeed, the cooperativity6C=g2/Γκ can be calculated using the coupling g and resonator and magnon linewidths (Γ and κ, respectively) to confirm the strong coupling regime for at least two of the avoided crossings. While the respective linewidths can be difficult to extract precisely from the data, they are estimated as κ=40±5 MHz and Γ=6.4±0.8 MHz from fits of S21 at μ0H=337.5 mT (see the supplementary material). Therefore, the cooperativity is approximately C15 for the first avoided crossing, confirming the sample is firmly in the strong coupling regime where C1. Furthermore, using the resonator linewidth information and following the method described in the supplement, the average number of magnons nm and photons np in the system can be estimated as np=nm=(6±2)×103.

While the first avoided crossing has a coupling g/2π=63±5 MHz, each subsequent avoided crossing shrinks in gap size and coupling with g/2π=18±2 MHz for the second crossing at μ0H=362 mT and g/2π=10±10 MHz for the third at μ0H=380 mT. Though the exact nomenclature of these higher-order width or thickness spin-wave modes is undetermined, in the context of Eq. (3), the smaller coupling rates imply that the effective ρ is heavily reduced for these modes (i.e., fewer spins participate in coupling) in comparison with the Kittel mode, in which the spins precess together as a singular macrospin. The reduced coupling rates also imply a steadily reducing cooperativity, and therefore, it can be concluded that future applications relying on higher order spin-wave modes will need to achieve even larger coupling rates to remain in the strong coupling regime. Nevertheless, the study here shows that such coupling rates may be possible using PFIB-processed YIG.

Therefore, we have established a general method for creating on-chip hybrid devices with high-quality spin-wave and superconducting components using PFIB technology. Using these methods, we fabricated several devices and demonstrated strong coupling between YIG FMR modes and a superconducting resonator mode. This serves as a crucial step toward creating hybrid quantum devices with spin-wave and superconducting components. The techniques used here can be used to create any number of photon–magnon devices and circuits, such as hybrid logical gates,35 low-temperature multiplexers,36 or spin-based quantum memory.

See the supplementary material for further experimental, device characterization, and analysis details, along with corresponding figures.

Authors would like to thank I. I. Syvorotka for the fruitful discussions. This work was supported by the European Research Council (ERC) under the Grant Agreement No. 648011. D. A. Bozhko acknowledges support from UCCS Committee on Research and Creative Works. R. Macêdo acknowledges support from the Leverhulme Trust, and the University of Glasgow through LKAS funds. R. C. Holland was supported by the Engineering and Physical Sciences Research Council (EPSRC) through the Vacation Internships Scheme. S. McVitie and W. Smith acknowledge support from EPSRC Grant No. EP/M024423/1. R. H. Hadfield acknowledges support through ERC Grant No. 648604 and a Royal Society Leverhulme Trust Senior Research Fellowship.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Supplementary Material