Design rules for low-insertion-loss magnonic transducers

As shown in the previous chapter, the model is applicable to various types of antennas (e.g., microstrip, U-shape, CPW) regardless of the number of conductors. Additionally, the induced voltage can be efficiently calculated from the magnetization dynamics of the spin-wave medium, independently of the excited spin-wave modes and the direction of the external bias field. The following chapter presents some use cases showing the capabilities of the circuit model. To illustrate the model’s flexibility concerning antenna geometry and spin-wave modes, these use cases utilize diverse samples, highlighting that the applicability of the model remains unaffected by these variations.

Experimental verification

In order to validate the lumped-element circuit model with experimental data, we fabricated a pair of antennas on an 800 nm YIG film on GGG substrate. We extracted the saturation magnetization (\(M_{s}\)) and the damping (\(\alpha\)) parameters by performing ferromagnetic resonance (FMR) measurements on the sample before depositing the antennas. We obtained \(M_{s}=\) 159 kA/m and \(\alpha = 6.9 \times 10^{-4}\). The antennas consist of a double layer made out of 150 nm Au on 10 nm Ti, where the titanium serves as an adhesion layer. CPW dimensions are indicated in Fig. 4b. To excite and detect spin waves, a vector network analyzer was connected via G–S–G Picoprobes to the two antennas, and the scattering matrix S was determined in the 0–20 GHz range, as illustrated in Fig. 4a. An external magnetic field \(H_{ext}\) was applied via an electromagnet in Damon-Eshbach geometry.

To record a reference spectrum without spin-wave contribution (\(S_{ij}^\text {ref}\)) the bias field was shifted far above the expected spin-wave resonance. Subsequently, the field was shifted down to resonance, and the spectrum containing spin waves (\({S}_{ij}\)) was acquired. In the absence of magnetic excitation—i.e. when the spin-wave spectrum is shifted far above the frequency range investigated—\(S_{ij}^\text {ref}\) shows an RLC resonance-like response that is in line with the size of the CPW lines (including the contact pads and the tapering). When the applied field is set around resonance, \(S_{ij}^\text {ref}\) gets modified due to coupling with the spin-wave modes of YIG, resulting in \({S}_{ij}\). To extract the pure spin-wave contribution, we performed de-embedding on the S-parameters. This refers to the process of isolating the intrinsic characteristics of the magnon-antenna interaction by mathematically eliminating the contributions of the surrounding network, allowing for accurate characterization of its true performance. General theory for characterization of the true performance of the device under test (DUT) in RF technology is detailed in¹¹. For the de-embedding we used analytical CPW models of the contact pads and tapering, with fitted parameters on the reference measurement. To compare with the lumped-element circuit model, we converted the de-embedded S-matrix into a Z matrix. Similar approach was applied for the de-embedding in¹⁶.

The device structure was modeled as detailed in section “Model description”. The 800 nm thick YIG film was simulated as a single layer with 200 nm cellsize in the propagation direction, and 500 nm in the direction perpendicular to the propagation. Material parameters for YIG were considered to be: exchange stiffness \(A_{ex} = {3.65}\,{\hbox {pJ}}/{\hbox {m}}\), for saturation magnetization and damping the same values as those obtained from the corresponding ferromagnetic resonance measurements above. In the EM solver an electric current amplitude of 1 mA was applied for the excitation with the antenna. Figure 4c-f show the comparison between de-embedded experimental data and the model for the real and imaginary part of \(Z_{11}\) and \(Z_{21}\) at \(\mu _0 H_{ext} =\) 383 mT. There is a good agreement between the prediction of the model and the experimental data. The modes observable below the main resonance peak can be attributed to the long-wavelength excitation of the contact pads and tapering of the waveguides, as these also lay on top of YIG, and thus generate spin waves with lower wave vectors (these were not included in the simulations). These peaks are barely visible in the transmission data, which is expected considering the curved geometries of the input and output waveguides don’t align well.

Design example for optimized insertion loss

The model can be used to predict the insertion loss (IL) in various device geometries. The insertion loss is the transmission coefficient (\(S_{21}\)), typically expressed in dB. Mathematically, the IL can be expressed by the components of the impedance matrix Z. Intuitively, to achieve high transmission, one has to have small reflections at the input and output ports, and strong coupling between the two. Physically, this coupling is realized via spin waves, and losses in the spin-wave channel can reduce the IL (due to damping and diffraction/scattering).

We optimized the geometry for low insertion losses. The schematic and geometrical details are depicted in Fig. 5a and c, respectively. This device is composed of a YIG conduit and two U-shaped antennas. Material parameters for YIG were considered to be: \(M_{sat} = {153.09}\,{\hbox {kA}}/{\hbox {m}}\), \(A_{ex} = {3.65}\,{\hbox {pJ}}/{\hbox {m}}\), and damping coefficient \(\alpha = 3.86\times 10^{-4}\). The 1 \(\upmu {\hbox {m}}\) thick YIG film was simulated as a single layer with 200 nm lateral cellsize in the propagation direction, and 1 \(\upmu {\hbox {m}}\) in the direction perpendicular to the propagation. We used the Damon-Eshbach geometry under an applied external bias field of 440 mT. In the EM solver an electric current amplitude of 1 mA was applied for the excitation with the antenna. From the FEMM simulations, the self impedance of one antenna (\(R_{\Omega }+i\omega L_{0}\)) was obtained to be \(4.95 + 9.56i\,\Omega\), and showed only minor frequency dependency in the observed frequency range. Thus, self impedance was considered to be constant in the further processing. Figure 5d–g depict the Z-matrix as a function of frequency around the resonance frequency, as determined by the model. Notably, the real parts of \(Z_{11}\) and \(Z_{22}\), representing the radiation resistance, closely approach 50 \(\Omega\). The lower peak at the right side of the resonance corresponds to the second strongest excitation mode of the antenna’s excitation spectrum. Consistent with the non-reciprocal characteristics inherent in spin waves within the Damon-Eshbach geometry, \(Z_{12}\) and \(Z_{21}\) exhibit an asymmetric nature, indicating the preferred propagation direction of the spin waves. As per the fundamental theory of two-port networks in RF systems, the insertion loss can be derived from the \(S_{21}\) scattering parameter in dB as follows:

where \(S_{21}\) can be expressed by the Z-parameters, the self impedance of the antennas, and the characteristic impedance of the ports (\(Z_{0}\)):

where \(Z_e = R_{\Omega }+i\omega L_{0}\) is the electrical impedance of the antennas. The resultant insertion loss as a function of frequency is illustrated in Fig. 5b. It is evident that an insertion loss of approximately 5 dB was attained within a bandwidth of approximately 100 MHz. The 5 dB loss is remarkably low and can be attributed to two factors. Firstly, damping can dissipate a significant amount of energy. To understand this effect, the insertion loss at resonance was observed with the damping set to zero, resulting in an insertion loss of 1 dB. Consequently, 4 dB of the total loss can be attributed to damping. The remaining 1 dB is mainly due to propagation away from both antennas, specifically in the non-favorable direction of the Damon-Eshbach mode.

Scaling rules for radiation resistance and insertion loss

One of the most important parameters that determines the efficiency of the spin-wave transducer is the radiation resistance, i.e. the real part of the input impedance of the transducer: \(R_{m} = Re\{Z_{11}\}\). It characterizes the strength of interaction between the antenna and the magnetic film^8,9, representing the real electric power that is transduced from electric to magnonic domain. Ideally, the goal is to get \(R_{m}\) close to 50 \(\Omega\) (the characteristic impedance of the feed waveguide), while minimizing ohmic resistance. This means that most power from the transducer is transferred into spin waves.

The \(R_{m}\) radiation resistance may also be determined from the micromagnetic simulations by energy considerations alone. Since the power delivered to the magnetic film at the input side is

knowing the power flow into spin waves (and the I current through the antenna that generated the spin waves), the value of \(R_{m}\) can be obtained. In a micromagnetic simulation, the magnetic energy density is straightforwardly calculated. If the temporal evolution of the total magnetic energy (\(E_{total}\)) in the spin-wave medium during spin-wave generation is simulated, the slope of the \(E_{total}\) versus time curve quantifies the magnetic power injected into the magnetic film in the form of spin waves. Using the so obtained power flow the value of \(R_m\) easily results from Eq. (12).

The frequency and the chosen spin-wave mode are pivotal in determining the transmitted power. To illustrate this relationship, Fig. 6 depicts the radiation resistance of a YIG conduit-microstrip antenna system for the three common spin-wave modes: forward volume (FV), backward volume (BV), and Damon-Eshbach (DE), in case of sinusoidal excitation by the antenna. These investigations were conducted through the model presented above. For the microstrip we assumed a thickness of 500 nm and a width of 5 \(\upmu {\hbox {m}}\), and its length matches the entire width of the YIG stripe. Excitation frequencies ranging from 2 to 30 GHz were employed, with corresponding external magnetic bias fields configured to induce spin-wave propagation at a consistent wavelength of 10 \(\upmu {\hbox {m}}\). The required bias-field values for a constant wavelength were calculated from the analytical dispersion curves and subsequently verified by extracting the wavelengths numerically from the micromagnetic simulations (see Supplementary). The excitation current was set at a magnitude of 1 mA. The YIG stripe featured a width of 10 \(\upmu {\hbox {m}}\) and a thickness of 1 \(\upmu {\hbox {m}}\). Material parameters for YIG in the micromagnetic simulations were considered to be: \(M_{sat} = {140}{\hbox {kA}}/{\hbox {m}}\), \(A_{ex} = {3.65}\,{\hbox {pJ}}/{\hbox {m}}\), and damping coefficient \(\alpha = 5\times 10^{-4}\). The 1 \(\upmu {\hbox {m}}\) thick YIG film was simulated as a single layer with 200 nm lateral cellsize both in the propagation direction, and in the direction perpendicular to the propagation. We observed a strong frequency dependence of the radiation resistance in all three geometries. For the two volume modes (FV and BV) we see approximately an order of magnitude lower radiation resistance compared to the surface mode. Additionally, the radiation resistance seems to scale faster in the DE case. To better illustrate the underlying mechanisms, we also plotted the group velocity for the three geometries, which also significantly differ from each other, especially the DE group velocities are much lower. The group velocity plays an important role in the radiation resistance, as it limits the power flow away from the antenna. The group velocity was determined as the derivative of the dispersion curve. For simple geometries, analytical dispersion relations are typically applicable^17,18, whereas for more complex cases, they can be extracted from micromagnetic simulations using broadband excitation. In the present study, given the simplicity of the structures, we observed no significant differences between the results of the two approaches. Consequently, all plots depicting the group velocity are based on the analytical method. The energy density under the antenna will be inversely proportional to the group velocity, with faster wave propagation removing energy quicker from the vicinity of the antenna. The lower resulting amplitude under the antenna induces a smaller voltage in the exciting antenna, thus, presents a lower radiation impedance. We can remove this effect from the radiation resistance curves in Fig. 6 by multiplying with the group velocity, which reveals linear tendencies for all three cases. This is expected, as the power carried by spin waves scales linearly with the frequency¹⁹.

In general, we expect the radiation resistance to scale linearly with the volume of the excited material. In case of scaling the length of the antenna this is rather trivial, assuming a long, linear wavefront. To see if this holds true in case of the thickness scaling as well, we simulated the radiation resistance for YIG conduits with progressively increasing thickness. To eliminate the influence of antenna geometry, spin waves were excited in such a manner that, instead of a finite antenna, a sinusoidal excitation of 0.5 mT was applied along the short centerline of the conduit. The \(\textbf{B}\) vector of 0.5 mT was rotated in-plane at a fixed frequency of 6 GHz within each mesh cell of the short centerline, with corresponding external magnetic bias fields configured to induce spin wave propagation at a consistent wavelength of 10 \(\upmu {\hbox {m}}\) in the FV mode. The YIG conduit featured a width of 10 \(\upmu {\hbox {m}}\) with a thickness varied between 50 nm and 5 \(\upmu {\hbox {m}}\). Simulations were done with periodic boundary conditions to remove edge effects. Material parameters for YIG in the micromagnetic simulations were considered to be: \(M_{sat} = {140}{\hbox {kA}}/{\hbox {m}}\), \(A_{ex} = {3.65}\,{\hbox {pJ}}/{\hbox {m}}\). The YIG film was simulated as a single layer of varied thickness with 200 nm lateral cellsize both in the propagation direction, and in the direction perpendicular to the propagation. Since no field of an actual antenna was used for the excitation, but rather an “artificially” created field of 0.5 mT, the radiation resistance was estimated from micromagnetic simulations based on the temporal evolution of the total energy in the conduit using Eq. (12) and assuming a current amplitude of 0.5 mA. In order to be able to trace the total energy injected into the film at any moment in time the damping coefficient must have been set to zero. Figure 7 illustrates the radiation resistance obtained as a function of thickness, revealing a significant dependency on the YIG thickness. Since the thickness of the spin-wave medium was varied, the group velocity was not constant and could also contribute to the observed radiation resistance. To further investigate this, the group velocity was calculated and plotted in Fig. 7 for each thickness using the analytical considerations outlined in¹⁸. Additionally, Fig. 7 also depicts the product of the radiation resistance and the group velocity for each thickness. By removing the effect of the group velocity the results again fits well with the expected linear scaling with the thickness.

Finally, the dependence of radiation resistance on the spin-wave wavelength has also been investigated (Fig. 8). After removing the effect of the group-velocity change, we observe no significant change in the radiation resistance. The micromagnetic simulations and their analyses were conducted in the same manner as for the thickness dependence. In this case, the thickness was held constant at 1 \(\upmu {\hbox {m}}\), while the wavelength was varied by adjusting the external magnetic field according to analytical considerations. The geometry of the simulated structure, aside from the constant thickness, and all other simulation parameters remained identical to those used in the thickness study.

As seen in section “Design example for optimized insertion loss”, even a few-micron scale YIG device can be designed for low insertion loss, meaning the transducers themselves operate at close to 100% power efficiency. Magnetic losses, while not high, are responsible for 4 dB out of 5 dB insertion losses in the above case study.

Downscaling magnonic devices further will, in general, decrease the radiation resistance of the transducer and typically, at the same time, will increase ohmic losses. This can be circumvented by higher frequency operation, which is usually advantageous, or selecting configurations with lower group velocity, this, however, will increase the delay of the magnonic device. We expect that a similar effect can be achieved with designing structures that retain spin-wave energy under the antenna, akin to a cavity resonator, or a magnonic matching network. These, in general will be limited by the maximum linear amplitude of spin waves, the damping coefficient, and the bandwidth requirements. For sub-micron scale magnonic devices a distributed power-delivery system seems to be more advantageous, i.e. a long antenna that powers many small magnonic inputs. In this case the radiation resistance scales with the number of devices.

Limitations of applying matching networks

Magnonic transducers cannot always be designed to match desired impedance characteristics. To maximize signal power transfer and efficiency, one viable solution is to employ a matching network¹¹, ensuring that the transducer’s impedance matched to the impedance of the connected circuitry. If the transducer is intrinsically designed to match the system impedance, as presented in section “Design example for optimized insertion loss”, the inclusion of matching networks can be omitted. This simplification facilitates the miniaturization of the device and streamlines the design process. However, for specific designs (e.g. for wide-band applications, or very small scale devices), due to the scaling rules presented in section “Scaling rules for radiation resistance and insertion loss”, it is not always possible to achieve the required radiation impedance. In such cases a matching network can eliminate reflection of the signal from the antenna due to mismatch, but there are some caveats.

Matching networks, in general, can be built from lumped circuit elements, or transmission-line segments. The former requires inductors and capacitors, which are both difficult to realize in a small footprint, depending on the required value (design specific). The latter requires transmission-line segments with lengths comparable to the electromagnetic wavelength, which will be much larger than the magnonic device, negating the gain of miniaturization in the magnonic domain. The designs presented in this paper show that an antenna length of less than 100 \(\upmu {\hbox {m}}\) can generate spin waves with wavelengths of a few micron with excellent efficiency. This is a characteristics that is hard to improve with a matching network. If shorter wavelengths are required, wavelength conversion might be applicable^20,21.

Although achieving perfect matching is desirable at the input, it is not the only condition to achieve a low insertion loss. The transduction efficiency is further limited by the ohmic losses in the antenna, to \(R_{m}/(R_{\Omega }+R_{m})\). That is, the \(R_{m}\) radiation resistance must be large compared to the \(R_{\Omega }\), which is uselessly dissipating the signal away as heat. Matching networks will eliminate reflections, but they cannot decrease the ratio of resistive losses to the power delivered to spin waves. Changing the length of the antenna does not affect this ratio, as both \(R_{m}\) and \(R_{\Omega }\) scales linearly with the antenna length, however scaling down the conductor cross section or the magnetic film thickness will both increase the ratio of resistive losses.