Efficient electromagnetic transducers for spin-wave devices

This paper presents a system-level efficiency analysis, a rapid design methodology, and a numerical demonstration of efficient sub-micron, spin-wave transducers in a microwave system. Applications such as Boolean spintronics, analog spin-wave-computing, and magnetic microwave circuits are expected to benefit from this analysis and design approach. These applications have the potential to provide a low-power, magnetic paradigm alternative to modern electronic systems, but they have been stymied by a limited understanding of the microwave, system-level design for spin-wave circuits. This paper proposes an end-to-end microwave/spin-wave system model that permits the use of classical microwave network analysis and matching theory towards analyzing and designing efficient transduction systems. This paper further compares magnetostatic-wave transducer theory to electromagnetic simulations and finds close agreement, indicating that the theory, despite simplifying assumptions, is useful for rapid yet accurate transducer design. It further suggests that the theory, when modified to include the exchange interaction, will also be useful to rapidly and accurately design transducers launching magnons at exchange wavelengths. Comparisons are made between microstrip and co-planar waveguide lines, which are expedient, narrowband, and low-efficiency transducers, and grating and meander lines that are capable of high-efficiency and wideband performance. The paper concludes that efficient microwave-to-spin-wave transducers are possible and presents a meander transducer design on YIG capable of launching \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\lambda = 500}\,$$\end{document}λ=500nm spin waves with an efficiency of − 4.45 dB and a 3 dB-bandwidth of 134 MHz.

For the grating and meander, w = 1 µm is the conductor width. For a meander, p = 2 µm is the distance from center-to-center of each conductor. To ensure the same fundamental frequency as the meander, the grating conductors are separated by 2p.

S1 Radiation Resistance Factors
Figure S1(c) and (d) illustrate the difference between element (r e f ) and array (r a f ) factors for three parallel ( Fig. S1(a)) and anti-parallel ( Fig. S1(b)) currents. For simplicity, a uniform current distribution is assumed for each element, although in reality the current is not uniform in a conductor due to skin and proximity effects. The J sFT of a uniform current distribution (rectangle function) of a single conductor with width w is the sinc function; therefore, the element factor is: where w = 1 µm is the width of each conductor in this example. Taking the first conductor to be centered at the origin, the surface current factor for the parallel (grating) and equal-amplitude currents shown in Fig. S1(a) is where the separation between conductors from center-to-center is 2p = 4 µm. Therefore, the array factor is Taking the first conductor to be centered at the origin, the surface current factor for the anti-parallel (meander) currents shown in Fig. S1(b) is Therefore, the array factor for three anti-parallel and equal elements is In this example, the array factor for a meander is 3 2 , or more generally, N 2 for a grating with N fingers because a meander guides the same current through each finger which results in an array factor that exceeds the element factor. A grating splits the incoming current into various fingers, so while it is capable of frequency selection like a meander, it does not result in the array factor exceeding the element factor. In both cases, the element factor acts as a low-pass filter. Transducers with anti-parallel currents can be designed to exceed the intrinsic and element factor response, and in turn, increase the efficiency at nanoscale wavelengths (section 4.3).

3/8 S2 EM simulation setup
Ansys HFSS has a built-in Polder susceptibility tensor for modeling ferrites based on the small signal approximation to the Landau-Lifshitz equation. HFSS requires the input of the saturation magnetization, M s , ferromagnetic resonance linewidth, ∆H, the frequency at which the resonance linewidth is defined, f m , and either a uniform or non-uniform effective bias field, H e . Anisotropy, exchange and non-linearity are not modeled. The ferrite sample is assumed to be in saturation.
For a bias alongẑ, the Polder permeability tensor, [μ], used by HFSS is is the permeability of vacuum, γ is the gyromagnetic ratio, and f is the frequency.
HFSS is not capable of modeling the exchange interaction. However, when designing transducers using closed-form expressions, the Polder susceptibility tensor can be modified easily to include the exchange field by replacing is the phenomenological exchange constant, and A is the exchange stiffness constant. Designing with closed-form expressions is further advantageous because of the reduction in computational resources. For example, when simulating the a = b = 5 µm CPW of section 4.2 using the methodology without a film outlined in section 3, a single frequency required 6 GB of RAM to solve for 104,400 tetrahedral elements in 137 seconds, whereas simulating the CPW with a film, a single frequency required 21.34 GB of RAM to solve for 234,600 tetrahedral elements in 367 seconds. The simulation computer utilizes a Intel(R) Xeon(R) Silver 4214R processor with 2x 12 cores. Simulating the CPW transducer with the film (the numerical approach) more than doubled the simulation time and mesh elements and more than tripled the memory usage because both the metal and the film have to be meshed. While this appears to be a modest saving, consider a more complicated structure, such as the N = 9 meander transducer of section 4.3. When simulating the meander without a film, a single frequency required 24 GB of RAM to solve for 403,800 tetrahedral elements in 613 seconds, whereas simulating the meander with a film, a single frequency required 106.8 GB of RAM to solve for 794,440 tetrahedral elements in 2,465 seconds. Simulating the meander transducer with the film (the numerical approach) more than quadrupled the simulation time and memory usage and nearly doubled the number of mesh elements. The simulation time and memory usage increased by a greater factor than the number of mesh elements increased because modeling a permeability tensor, even if for the same number of mesh elements, requires greater computational resources than modeling a constant permeability. This illustrates that the theoretical approach to designing transducers is particularly compelling when modeling more complex transducers, which are required to obtain efficient and wideband transduction.
Because HFSS does not have a built-in absorbing boundary layer (ABL) for spin waves, an ABL was manually specified on both sides of the transducer by defining a spatially-varying linewidth. Although the software requires a linewidth, the ABL function is presented here in terms of the damping parameter, α, since it is a fundamental material property. Two different functions were utilized: an exponential function and a hyperbolic tangent function 1 . For the CPW transducers in sections 3 and 4.2, an exponential taper was utilized: where α max = 0.0005, and α min = 10 −8 , and ABL = 100 µm is the width of the absorbing layer on either side of the transducer's center. Since α min is so low in the film directly below the transducer, it has no noticeable effect on the impedance of the transducer.
For the meander and grating transducers in section 4.3, a hyperbolic tangent function was utilized for defining a spatially varying linewidth: where α max = 0.001, α min = 10 −7 , ABL = 2.5 µm is the width of the absorbing layer on each side of the transducer, and Y offset depends on the transducer size (number of fingers) and ensures that the ABL begins after, not directly underneath, the transducer. These two profiles are plotted in Fig. S2.
Eq. 8 Eq. 7 2Y offset Figure S2. (a) Profile of damping parameter using the exponential function given in Eq. 7. (b) Profile of damping parameter using the exponential function given in Eq. 8. In this example, Y offset = 2.25 µm for the N = 9 meander.

5/8
ℎ Figure S3. L-section MN corresponding to Tab. S1  Table S1 provides the shunt inductance, L sh and series capacitance, C s , values of the L-section matching network (Fig. S3) used for the transducers of section 4.3 and Fig. 6 in the paper. The inductor and capacitor each have a quality factor of 100.

S4 Dispersion Relations
The dispersion curves for transducers modeled in sections 3, 4.2 and 4.3 of the main text are given in Fig. S4. A YIG film (M s = 140 kA/m) with thickness of 100 nm was utilized for all transducers. The exchange field was not included in these curves since these transducers are validated with HFSS, which is not able to model the exchange interaction. Figure S4(a) plots the first four thickness modes for the CPW transducer designed in section 3 of the main article with a bias of B a = 0.3763 T. Because there is very little coupling to higher-order modes at low wavenumbers, only the fundamental mode (n = 0) is considered in the main text. Figure S4(b) plots the fundamental mode's dispersion relation for the a = b = 1 mum CPW of section 4.2 (red curve), the a = b = 2, 3, 5 µm CPW of section 4.2 (blue curve), and the meander/grating transducers of section 4.3 (green curve). The targeted LTE-U band is highlighted in grey. The following dispersion relation is used for forward volume waves 2 tan where d is the film thickness, k is the magnitude of the in-plane wavenumber, n is the thickness mode number, χ d is the diagonal component of the susceptibility tensor and is a function of angular frequency, ω.