A Tunable Magneto-Acoustic Oscillator with Low Phase Noise

A tunable magneto-acoustic oscillator with low phase noise

A. Litvinenko,1, ∗ R. Khymyn,2 V. Tyberkevych,3 V. Tikhonov,1 A. Slavin,3 and S. Nikitov1, 4, 5 1Laboratory of metamaterials, Saratov State University, 410012, Saratov, Russia. 2Department of Physics, University of Gothenburg, 412 96, Gothenburg, Sweden. 3Department of Physics, Oakland University, 48309, Rochester, Michigan, USA. 4Kotelnikov Institute of Radioengineering and Electronics of Russian Academy of Sciences, 125009, Moscow, Russia. 5Moscow Institute of Physics and Technology (National Research University), 141700, Dolgoprudny, Moscow Region, Russia. (Dated: November 17, 2020) A frequency-tunable low phase noise magneto-acoustic resonator is developed on the base of a parallel-plate straight-edge bilayer consisting of a yttrium-iron garnet (YIG) layer grown on a substrate of a gallium-gadolinium garnet(GGG). When a YIG/GGG sample forms an ideal parallel plate, it supports a series of high-quality-factor acoustic modes standing along the plate thickness. Due to the magnetostriction of the YIG layer the ferromagnetic resonance (FMR) mode of the YIG layer can strongly interact with the acoustic thickness modes of the YIG/GGG structure, when the modes’ frequencies match. A particular acoustic thickness mode used for the resonance excitations of the hybrid magneto-acoustic oscillations in a YIG/GGG bilayer is chosen by the YIG layer FMR frequency, which can be tuned by the variation of the external bias magnetic ﬁeld. A composite magneto-acoustic oscillator, which includes an FMR-based resonance pre-selector, is developed to guarantee satisfaction of the Barkhausen criteria for a single-acoustic-mode oscillation regime. The developed low phase noise composite magneto-acoustic oscillator can be tuned from 0.84 GHz to 1 GHz with an increment of about 4.8 MHz (frequency distance between the adjacent acoustic thickness modes in a YIG/GGG parallel plate), and demonstrates the phase noise of -116 dBc/Hz at the oﬀset frequency of 10 KHz. PACS numbers: 85.75.-d, 05.45.Xt, 75.40.Gb, 75.47.-m, 84.30.Qi:

I. INTRODUCTION signiﬁcantly reduce the close-in phase noise, the far-out phase noise still remains determined by the intrinsic pa- One of the most important tasks in the modern com- rameters of the used VCO. munication and radar technology is the development of The phase noise of an oscillator can be estimated using reference oscillators with low phase noise, as the low level an empirical Leeson’s equation[4]: of phase noise translates into a high level of frequency 2 F kT ω0 ωc stability necessary for the improved device performance. L(∆ω) = 10log 1+ 1+ (1) 2P 2Q∆ω ∆ω Also, in digital communication systems phase noise af- s

fects the system bit-error rate, and, therefore, the speed where ω0 is the oscillator central (or ”carrier”) frequency, of data processing. In radar applications, lowering the ∆ω — is the ”offset” frequency, Ps – is the signal power, phase noise leads to the increase of a radar range and F – is the noise factor of the oscillator active element, sensitivity, as it allows to detect a signal reflected from k – is the Boltzmann constant, T – is the ambient abso- the target with a lower power level. lute temperature, Q – is the unloaded resonator quality In many common applications, reference or local tun- factor, and ωc – is the flicker corner frequency [4]. As it able oscillators are based on the yttrium-iron garnet follows from the Leeson’s equation (1), both the ”close- (YIG) resonators, because the frequency of a ferromag- in” and the ”far-out” levels of the phase noise of an os- netic resonance (FMR) in YIG can be easily tuned cillator are, mainly, determined by the quality factor of over a decade by applied bias magnetic field. Also an resonator used in the oscillator. YIG resonators biased by powerful permanent magnets Thus, the enhancement of the resonator Q-factor is could have rather high resonance frequencies lying in the a key element in the development of new reference GHz frequency range, and demonstrate a relatively low oscillators for information and signal processing[5–8]. arXiv:2011.07648v1 [physics.app-ph] 15 Nov 2020 linewidth, and, therefore, a relatively low level of the This goal, in principle, can be achieved by using res- phase noise, especially at the reasonably large offset fre- onators with low energy losses, such as dielectric[9], quencies from the carrier. Another common method to optoelectronic[10], acoustic[11], and magnetic oscillators reduce the oscillator phase noise is to use voltage con- [12] or the combinations of these oscillator types[13]. trolled oscillators (VCO) stabilized with a phase locked The highest Q-factor, so far, is found in optoelectronic loop (PLL) [1–3], but, although this technique allows to and dielectric resonators, but, unfortunately, these resonator types are, usually, rather bulky and have insuffi- cient thermal stability of their resonance frequency. An alternative is to use the solid-state acoustic resonators ∗ Currently with Spintec, France. Correspondence to: Litvi- that can demonstrate Q-factors that are much higher [email protected] than in magnetic YIG magnetic resonators, while having 2

FIG. 1. a) Scheme of the a simple one-port reflection-based MAR , which was experimentally characterized using a vector network analyzer (VNA); b) Thickness distributions of the magnetic FMR mode and standing acoustic modes in the YIG/GGG bilayer sample; c) S11-parameters of the one-port MAR at different values of the perpendicular-to-plane magnetic bias field. sizes that are much smaller than the sizes of dielectric uration presented in Fig. 1 the whole YIG/GGG struc- and optoelectronic resonators. Unfortunately, the purely ture acts as an effective high-overtone bulk acoustic res- acoustic resonators are not tunable. onator (HBAR)[11, 36, 37]. Note, that HBARs among all A compromise solution would be to use hybrid the known acoustic resonators demonstrate the highest 14 magneto-acoustic resonators (MAR) that can support Q-factor (up to 10 ), making the proposed YIG/GGG hybrid magneto-elastic oscillation modes that combine a MAR design well-suited for the realization of low phase high quality factor of the purely acoustic modes with the noise local oscillators. In this work, using the results excellent tunability of the magnetic modes. It was shown of theoretical analysis of the magneto-acoustic interac- in 1950s-60s that YIG has a considerable magnetostriction and experimental parameters of the YIG/GGG epi- tion constant[14], and that magneto-elastic waves of the taxial parallel-plate structures, we design a novel tun- GHz frequency range can be efficiently excited in mag- able magneto-acoustic oscillator that has a level of phase netic layered films and hetero-structures [15–22]. In the noise, that is much lower than in conventional magnetic 1980s the technological progress in the liquid-phase epi- oscillators based only on the FMR mode of a YIG film. taxy resulted in the development of high-quality (FMR linewidth below 0.5 Oe) YIG films grown on the (almost lattice-matched) mono-crystalline gadolinium gal- II. MAGNETO-ACOUSTIC RESONATOR WITH lium garnet (GGG) substrates. It was also demonstrated A HIGH Q-FACTOR that magnetic oscillations excited in YIG through magnetostriction can effectively excite standing acoustic thick- The scheme of the YIG/GGG MAR is shown in the ness modes in the whole YIG-GGG garnet structure, be- Fig. 1(a). It consists of a parallel-plate straight-edge rect- cause the sound velocities in YIG and GGG are almost angular resonator cut from a monocrystalline epitaxial equal [23–26]. The interest to magneto-acoustic effects YIG/GGG bilayer magnetized to saturation perpendicu- in garnet hetero-structures has been recently revived in a lar to its plane by a bias magnetic field H0, and excited number of papers where YIG-GGG structures were used by a strip-line antenna connected to a vector network an- either in the transmission line configuration [27–29] or alyzer (VNA). The YIG film in the bilayer has the static with ZnO acoustical transducers which were used for a magnetization 4πMs = 1740 Gs , the FMR linewidth broad-band excitation of acoustic modes in these struc- ∆H0 = 0.5 Oe and the thickness of 9.75 µm, and the tures [30–33]. in-plane sizes of 2x2 mm2. The thickness of the GGG Below, we show that a traditional parallel-plate layer is 364 µm. straight-edge YIG/GGG resonator can be successfully A signal of a given frequency f from the strip-line an- used as a tunable high-Q-factor magneto-acoustic res- tenna excites the FMR mode in the YIG layer (uniform onance element of a local oscillator with a low phase along the YIG film thickness) corresponding to a partic- noise. YIG/GGG films were previously used as hy- ular magnitude of the bias perpendicular bias magnetic brid magneto-acoustic resonators (MAR)[34, 35], where field H0. The FMR mode of the YIG resonator is coupled the YIG film served as an effective, narrow-band and through the YIG magnetostriction to the standing thick- frequency-tunable transducer which can selectively excite ness acoustic modes of the YIG/GGG hetero-structure, an acoustic thickness standing mode of the YIG/GGG and by simultaneous variation of the excitation frequency structure, having a desirable frequency. In the config- f and the bias magnetic field H0 it is possible to align any 3 of the discrete thickness acoustic modes of the YIG/GGG the FMR mode in the absence of the magneto-elastic in- hetero-structure with the YIG resonator FMR frequency teraction Hac = 0. Similarly, we represent the dynamic given by the Kittel formula: acoustic displacement, using the known profiles of the acoustic thickness eigenmodes of the YIG-GGG struc- f = γ(H0 − µ0Ms), (2) ture: where γ = 28.3 GHz/T - is a gyromagnetic ratio of YIG. ˜ In Fig. 1(c) S11-parameter of the MAR is shown at X ˜ −(Γ+iω˜λ)t ξ(z) = ξλ(z)bλ(t)e + c.c. (7) different values of bias magnetic field H0. A distinctive λ feature of S11-parameter of the MAR is a dip with narrow inverse acoustic peaks. A broadband dip belongs Taking into account the following orthogonality relations to the FMR mode of the YIG film, while the narrow for magnetic and acoustic modes acoustic peaks which appear at bias field H0 values of Z d/2 2054, 2153 and 2174 Oe correspond to the high overtone Ms ∗ magnetoacoustic resonances in the YIG-GGG structure. m (µ0 × m)dz = −iA (8) γ −d/2 Note, also that there are frequencies and corresponding Z d/2 bias field values at which acoustic peaks do not appear ˜∗ ˜ 2ρωλ ξλξλ0 dz = Qλδλλ0 , (9) within the FMR dip. This indicates the absence of the −L+d/2 magneto-acoustic coupling. To analyze the magneto-acoustic coupling in the pro- one can rewrite Eqs. (4-5) as: posed structure as a function of the frequency and the acoustic mode number we employ the theoretical descrip- A [˙a(t) + iωa(t) + Γ0a(t)] = iκλb(t) h i ˙ ˜ ∗ (10) tion of the magneto-acouistic interaction in the sample Bλ bλ(t) + iω˜λbλ(t) + Γλbλ(t) = iκλa(t), (Fig. 1) developed in [38, 39]. We start with the density of magneto-elastic energy in the form: with the coupling constant defined by the expression:

W = Wmag + Wel + Wmel. (3) 2 b2γ Z ∂ξ˜ 2 ∗ λ Here W is the density of magnetic energy , κλ = µ0m dz , (11) mag 2ωλρMsLd ∂z which includes contributions from the Zeeman, ex- change and magneto-dipolar interactions; W = el where b = b1111 − b1122. If m and ξ describe plane waves h 2 i ρ (dξ/dt) + ciklmuikulm /2 is the elastic energy with (i.e. magnons and phonons), the coefficient κ defines the bandgap at the point of avoided crossing( hybridization) the tensor of elastic constants ciklm and uik = of their spectra. In our case κ2 defines the part of the (dξi/dxk + dξk/dxi) /2, ξ is an acoustic displacement, oscillator ”energy” involved in the magneto-elastic inter- and ρ is the density of the material; Wmel = 2 action. Please, note that the interaction coefficient κ is biklmMiMkulm/Ms is the magneto-elastic interaction expressed in the units of frequency, i.e. it is defined in with the tensor of magnetostriction constants biklm[40]. Using the expression for the energy in Eq. (3), one can relation to the central (carrier) frequency of the MAR. write two coupled equations for the elastic displacement We assume that the thickness profiles of both the FMR ξ and magnetization M as: magnetic mode and the standing acoustic modes satisfy the ”free” (or ”unpinned”) boundary conditions at the 2 ∂ ξm ∂ ∂W biklm ∂Mi ∂uik both parallel-plate surfaces and at the YIG-GGG inter- ρ 2 = = 2 Mk + ciklm (4) ∂t ∂xl ∂ulm Ms ∂xl ∂xl face: dM = −γ M × Hmag + Hmel , (5) m˜ = 1 dt ˜ (12) ξλ(z) = cos [(z − d/2) πλ/L] , mag mel where H = ∂Wmag/∂M and Hi = 2 For the spatially uniform static magnetization in the YIG 2biklmMkulm/Ms . Below we represent the magnetization vector M as a layer with a sufficiently sharp transition at the YIG-GGG sum of its static and precessional (dynamic) parts, and interface one can write µ0(z) = Θ(d/2 + z)Θ(d/2 − z) the latter is expressed as the FMR mode of the thin YIG and m(z) =m ˜ Θ(d/2 + z)Θ(d/2 − z), where Θ(z) means film: Heaviside theta function. Finally, for the coupling coefficient[41] we obtain: h i −(iω0+Γ)t M = Ms µ0 + ma(t)e + c.c. , (6) 2 2 2 γb πλd κλ = 2 √ 1 − cos , (13) where ω0 and Γ are the angular frequency and damping 2π dλMs c44ρ L parameter of the FMR mode. In our case the static magnetization of YIG µ0 is per- where c44 is the elastic modulus of YIG[42]. pendicular to the film plane, while the dynamic magneti- The theoretically calculated coupling coefficient for the zation m ⊥ µ0 describes the spatial (thickness) profile of YIG/GGG sample used in our experiments is shown in 4

FIG. 2. Coupling coefficients in the MAR as functions of the excitation frequency and the excited acoustic mode number λ: (a) - theoretically calculated coupling coefficient between the FMR mode of the YIG resonator and the thickness acoustic modes of the YIG/GGG structure; (b) - experimentally measured coupling coefficients : σYIG – between the strip-line line and the FMR mode of the YIG resonator(blue squares); σ – between the FMR mode of the YIG resonator and the acoustic thickness modes of the YIG/GGG structure (red circles), σMAR – overall coupling between the strip-line and the acoustic thickness modes (green triangles). The region highlighted in yellow indicates the frequency band where the overall coupling coefficient is suitable for the operation of the oscillator scheme. Note, that the FMR frequency of the YIG resonator is adjusted for a particular acoustic mode by changing the bias magnetic field H0. In the frame (c) experimental S11-parameter at the magnetic field H0 = 2153 Oe and in the frequency band 869 ± 15 MHz is plotted on the Smith charts. The main loop with a diameter ∆ corresponds to the FMR mode, while inner loops correspond to acoustic modes. The central inner loop with a diameter δ corresponds to the acoustic mode with which the FMR frequency is aligned. The values of ∆, δ are used to calculate the experimental values of σYIG, σMAR, σ, while the frequencies f0, f1, f2, f11, f22 are used to calculate the Q-factors QYIG,QMAR, as described in the ”Methods”. At the frequencies where the coupling coefficient δ is close to zero, the inner acoustic loops disappear from the MAR S11-parameter diagram.

Fig.2 (a) (black open circles). As it can be seen from the factor QYIG ≈ 200 − 400 the frequency bandwidth of figure, the coupling coefficient κ demonstrates an oscillat- the FMR in a laterally constrained YIG resonator is nar- ing behavior: the overlap integral between the thickness rower, than the frequency spacing between the acous- profiles of the FMR mode and the acoustic modes has lo- tic thickness modes of the YIG-GGG structure ∆fa = 5 cal maxima when there are n/2 acoustic wavelength over Va/(2L) = 4.773MHz, where Va = 3.57 × 10 cm/s is the the thickness of the YIG film, and this integral vanishes velocity of transverse acoustic waves in GGG (for com- to zero when there are (n+1)/2 acoustic wavelength over parison, the transverse acoustic wave velocity in YIG is 5 the thickness of the YIG layer. The oscillations in the Va = 3.85 × 10 cm/s). With this the YIG-film resonator magnitude of the coupling coefficient reduce the operat- can selectively excite a single acoustic shear mode of the ing range of frequencies for the MAR. As it follows from YIG-GGG structure without using any narrow-band ex- the theory, in order to increase the period of oscillations ternal filters. A YIG-GGG MAR can be tuned to excite of the coupling coefficient κ and extend the operating fre- effectively a single acoustic resonance mode having num- quency range of the MAR one has to reduce the thickness bers from 173 to 208 in the [840MHz : 1.0GHz] frequency of the YIG film. band with the step ∆fa = 4.8MHz by changing the mag- The magneto-elastic coupling can be described using nitude of the bias magnetic field H0 applied to the YIG a Darko Kajfez’s method[43] modified for hybrid MAR film. having two resonant subsystems (see description in the ”Methods” section). The experimentally measured oscil- Since the YIG film works as a transducer between the lations in the magneto-elastic coupling have the same pe- electric signals in the strip-line antenna and the acoustic riod as the ones calculated theoretically. This agreement thickness modes, the overall coupling coefficient between between the theory and the experiment confirms that in strip-line and acoustic modes has to be taken into account the proposed structure of a MAR the FMR mode, in- for the oscillator design. The overall coupling coefficient deed, excites the acoustic shear modes of the YIG-GGG is shown in the Fig. 2(b). The detailed description on structure. how to obtain experimental coupling coefficients of the Due to the relatively high value of YIG resonator Q- hybrid YIG-GGG resonator is given in the ”Methods”. 5

III. DESIGN OF MAGNETO-ACOUSTIC low phase noise. Therefore, according to this empiri- OSCILLATOR cal condition the oscillator based on the MAR can oper- ate in the range of frequencies between 0.84 and 1GHz An important parameter for the design of an oscilla- (see the region highlighted in yellow in Fig.2). For fur- tor which employs high overtone resonators is the mode ther design steps we chose the resonance acoustic mode selectivity. In order to get stable oscillations without within a yellow region having number 182, and the Q- modulations and random spurs in the phase noise char- factor of Q = 2497. As it was discussed earlier, the reso- acteristic one has to make sure that when a particular nance characteristic of a one-port reflection-based MAR mode is selected to be resonant, the damping of the adja- has an unusual form of a dip, caused by the FMR in cent modes is sufficiently strong. For the above described the YIG layer with an inverted central peak in the mid- MAR, the selectivity depends on the frequency separa- dle attributed to the resonance acoustic mode with the tion between acoustic modes and on the linewidth of the number 182 of the whole YIG-GGG structure. The anal- FMR mode of the YIG layer. ysis, however, shows that if we use a conventional one- Using our experimental data, we found that the cou- port reflection-based oscillator design, the Barkhausen pling coefficient of the MAR with the resonant acous- stability criterion for the auto-oscillations is only satis- tic mode having number 182 is 3.5 times larger than fied at the points A and B (see Fig. 3 which are situated the corresponding coupling coefficients with the adjacent outside of the central peak of the acoustic resonance for acoustic modes having numbers 181/183. The caulking the mode 182. Therefore, the use of such a design for a advantage for the resonant acoustic mode can be fur- magneto-acoustical oscillator (MAO) will not substan- ther increased if an additional YIG-preselector is used. tially decrease the MAO phase noise figure, since the Moreover, with the increase of the central frequency of phase noise will, mostly, be determined by a relatively the MAR the loaded Q-factor of the magnetic (YIG) low Q-factor of the FMR mode. Moreover, in the sys- resonator grows linearly and, therefore, the effective tems where several competing resonance modes (corre- linewidth of the YIG FMR mode decreases, thus increas- sponding to points A and B on the black curve) can be ing selectivity of the MAR. excited simultaneously it is possible to have mode bistability and chaotic dynamics[44]. In order to take the full advantage of the high Q-factor of a single acoustic resonance mode, a special scheme based on the one-port MAR is designed to satisfy with the Barkhausen stability criterion near the frequency of the acoustic resonance mode. It is done in two steps. First, a circulator is added serially to the one-port MAR forming a two-port circuit which has an S21-parameter absolutely identical to a S11-parameter of the one-port MAR. For a ring oscillator scheme based on such a two- port circuit (see a part of the scheme between point 1 and 3) given adjusted open loop phase and amplification the Barkhausen stability criterion would be satisfied at the same time at the narrow peak corresponding to the frequency of the acoustic resonance and at frequencies FIG. 3. S-parameters of the one-port (see Fig.1) and two- far from the FMR dip. Therefore, in a second step we port composite (Fig.4) MARs: black line - S11-parameter of introduce an additional purely magnetic two-port YIG- the one-port MAR and S21-parameter of a composite two- port MAR; blue line - S -parameter of the FMR-based pre- resonator patterned on the same GGG substrate as a pre- 21 selector-bandpass-filter (having a scratched bottom GGG selector; red line - S21-parameter of the composite two-port MAR. The mode numbers of a corresponding acoustic thick- surface to prevent the formation of the standing acoustic ness modes in the YIG/GGG parallel plate are given in brack- thickness modes) which suppresses the signal at frequen- ets. A, B and C are the points where the Barkhausen criterion cies outside of the FMR resonance. As a result of such a for stable auto-oscillations is satisfied design, the transmission (S21) characteristic of the composite two-port MAR (red line in Fig.3) is approximately The basic principles of the design of an efficient a product of the transmission characteristic of the two- magneto-acoustic oscillator can be understood from the port circuit based on the MAR and a circulator (black analysis of the experimental S-parameter data presented curve in Fig.3), and the transmission characteristic of in Fig. 3. First of all, let us look at the experimentally the FMR-based pre-selector (blue curve in Fig.3). As measured overall coupling coefficient of the one-port sim- a result the transmission characteristic of the composite ple MAR presented in Fig.1 (green triangles in Fig. 2). two-port MAR (Fig.4) has a usual form of a resonance In the experience of practical oscillator design, the min- characteristic with a central maximum. We note, that imal coupling coefficient between the oscillator core and a MAR transmission characteristic with a central maxi- the resonator should be above 0.1 for high stability and mum, similar to the one shown by the red curve in Fig.3, 6 can be obtained by simpler means in a three-layer YIG- thickness mode in a composite two-port MAR is realised. GGG-YIG structure which was used in [35], without use of an additional pre-selector. However, in that case the thickness of the GGG layer can not be adjusted by polish- IV. RESULTS ON LOW PHASE NOISE ing, and the frequency spacing of the acoustic thickness modes mode can not be adjusted after the growth of the YIG layers by liquid epitaxy. Another limitation is that the liquid epitaxy process requires the GGG substrate thickness to be at least 300 µm, while with polishing of one sided YIG-GGG structure this thickness can be re- duced down to 50-100 µm.

FIG. 5. Experimentally measured phase noise figures for auto- oscillators based on a purely magnetic YIG FMR resonator (blue line) and on a composite two-port magneto-acoustic YIG/GGG resonator (red line). Dotted black line shows the estimation of the phase noise obtained from the Leeson’s formula (1) for a FMR-based magnetic pre-selector resonator (Q = 365) , while the dashed black line shows a similar estimation of a phase noise for the two-port MAR (Fig.4) having the Q-factor Q = 2497. FIG. 4. Scheme of a composite two-port MAR consisting of a YIG/GGG MAR (left part of a YIG layer) connected to a YIG FMR-based pre-selector-filter (right part of the YIG layer). The experimentally measured phase noise figure of an The left part forms a two-port MAR , similar to the one shown auto-oscillator based the composite two-port MAR (see in Fig.1, connected to the purely magnetic YIG-FMR-based Fig.4) having (Q = 2497) is presented in Fig. 5 by a red pre-selector-filter formed by the right part of the YIG layer, curve. In the same figure, for comparison, we show by where the acoustic modes are eliminated by scratching the the blue curve the experimentally measured phase noise bottom surface of the GGG substrate. figure for an auto-oscillator based on a purely magnetic YIG FMR pre-selector oscillator (Q = 365) . The the- Finally, to complete the scheme of MAO a low phase oretical estimations of the phase noise figures in these noise amplifier ABA-54563 is added (see the circuit Fig.4) two auto-oscillators obtained from Eq.(1) are shown in to compensate losses in the oscillator feedback loop. A Fig. 5 by the black dashed line and the black dotted line, variable delay line is used to obtain the correct phase respectively. shift in the loop to satisfy the phase condition of the The phase noise of the MAO based on the compos- Barkhausen stability criterion for oscillations. A vari- ite two-port MAR is -87dBc/Hz at the 1kHz offset and able attenuator is introduced to limit the signal power -116dBc/Hz at the 10kHz offset, which is at least 20 dB at the input of the MAR to suppress the nonlineari- better than the phase noise figure of a conventional auto- ties and avoid the increase of the phase noise. Finally, oscillator based on a YIG FMR resonator. Note, at the a coupler is used to extract the output signal. Given same time, that at large off-set frequencies the phase the adjusted amplification, phase shift in the feedback noise of an auto-oscillator based on a composite two-port loop, and the bias magnetic field applied to the YIG film, MAR degrades, and becomes higher than in the case of the auto-oscillation conditions for the composite two-port a conventional YIG-FMR oscillator. This increase in the MAR are satisfied only at the frequency of the acoustic MAR phase noise is caused, mainly, by the presence of a thickness mode having the number 182 and the effective variable attenuator in the MAR scheme (Fig. 4) which is Q-factor of 2497 (see point ”C” in Fig. 3). Moreover, introduced to avoid saturation of the one-port MAR in the adjacent acoustic thickness mode with the number nonlinear mode. Note, also, that the nonlinearity thresh- 181 is suppressed by 13 dB as compared to the reso- old of the MAR is lower, than of the YIG-FMR two-port nant mode with the number 182. Thus, a single-mode resonator. We would like to mention that an amplifier auto-oscillation regime based on a high-Q-factor acoustic with a lower saturation power level can be used instead 7 of the variable attenuator in the MAO design scheme to U.S. National Science Foundation (Grants No. EFMA- reduce the maximum level of power at the MAR input, 1641989), by the U.S. Air Force Office of Scientific Re- and, consequently, the noise factor of the feedback loop. search under the MURI grant No. FA9550-19-1-0307, This will improve the phase noise figure in the whole and by the Oakland University Foundation. The work range of offset frequencies. of SAN was supported by Agreement with Ministry of Science and Higher Education of the Russian Federation #13.1902.21.0010. V. CONCLUSION We would like to thank Dr. Olivier Klein for the help with precise measurement of the YIG and GGG thickness We have shown that the FMR mode excited in a using interferometry technique. YIG layer of a parallel-plate YIG/GGG hetero-structure can be effectively coupled to the high-Q-factor standing thickness acoustic modes of the YIG/GGG bilayer. The Appendix A: Device fabrication frequency dependence of this magneto-elastic coupling was studied both theoretically and experimentally, and The scheme of the one-port reflection-type MAR is the optimum conditions for this coupling corresponding shown in the Fig. 1. The MAR is manufactured us- to the stable single-mode auto-oscillations were found. ing a two layer parallel-plate YIG/GGG structure with A composite two-port MAR based on the YIG/GGG the YIG thickness of d = 9.75µm, GGG thickness of hetero-structure was developed and practically realized, L = 364µm, and YIG saturation magnetization of M0 = and the phase noise figure in the auto-oscillator based 1740/(4π)Oe. The chemical-mechanical polishing tech- on the developed MAR was substantially improved, in nique was used to create the highly parallel surfaces of comparison with the auto-oscillator based on the conven- YIG/GGG structure with the wedge angle less than 200, tional YIG FMR two-port resonator. We would like to which ensures the formation of high overtone acoustic stress, that the advantage of using shear acoustic modes thickness resonances in the YIG/GGG structure. The in the developed MAR over the longitudinal acoustic fabricated sample had the lateral dimensions of 1x2.5mm. waves in conventional HBARs is caused by the fact that shear acoustic modes are insensitive to the amorphous load. This property provides a significant simplification Appendix B: Device characterization of technological requirements for the manufacturing of the proposed composite MARs. The developed MAR and the YIG pre-selector FMR In summary, we have designed a composite magneto- resonator were characterized using Keysight (Agilent acoustic auto-oscillator with a low phase noise based on a E8361A) vector network analyzer. The phase noise of the parallel-plate YIG/GGG bilayer. The relatively narrow developed composite MAO was measured using a signal FMR linewidth of the YIG layer provides the possibil- source analyzer (Keysight E5052B). ity for selective resonance excitation of a single acoustic The experimental method used for the characterization thickness mode of the YIG/GGG structure and, there- of the magneto-acoustic coupling in the MAR was a mod- fore, makes possible a significant improvement of the Q- ified Darko Kajfez’s method. This method is based on the factor of the resulting MAR. The phase noise figure of the analysis of the resonance loops in an S11-parameter graph anto-oscillator based on the developed composite MAR is plotted on a Smith chart. This method gives dimension- -87dBc/Hz at 1kHz offset and -116dBc/Hz at 10kHz off- less values of the loaded Q-factors, and coupling coef- set which is at least 20dB better than the performance of ficients that define interaction of electrical sub-systems the conventional auto-oscillating scheme based on a con- in the filter theory. To obtain the S11-parameter of the ventinal the YIG FMR two-port resonator. The designed MAR we used an electrical setup shown in Fig.1a. The MAO can be used in the frequency-agile data transmis- strip-line antenna used in our experiments had the width sion where abrupt frequency-hoping is employed. As an of 0.5 mm, which is comparable to the sample size, and outlook, we note that the microwave circulator used in ensures an efficient excitation of the FMR mode. In order our scheme of the composite two-port MAR can be re- to obtain the loaded Q-factors and coupling coefficients placed with a matching circuit or an active component of the composite MAR some additional geometrical con- to make the developed MAR CMOS-compatible. structions were used on the Smith chart. First, to get the parameters of a magnetic resonator, three lines have to be drawn from the node of the big loop, as shown in Fig. ACKNOWLEDGMENTS 1c: one line goes through the center of the loop, while two other lines go at the angle of 45 degrees to each side This work was supported by the Grant of the Gov- of the center line. This allows us to extract from the ernment of the Russian Federation for supporting sci- S-parameter measurement the values of the correspond- entific research projects supervised by leading scientists ing frequencies f1, f2 and f0 as the intersection points at Russian institutions of higher education (Contract between the auxiliary lines and the S-parameter curve. No. 11.G34.31.0030) and supported in part by the The interval f2 − f1 defines the FMR linewidth, and is 8 used to calculate the loaded Q-factor of the YIG mag- where 2R=2 is a radius of the Smith Chart. The overall netic resonator: coupling between the strip-line and the acoustic subsys- tem can be obtained using a similar formula: f0 QYIG = , (B1) f2 − f1 δ Similar geometric constructions were made with the σ = , (B4) MAR 2R − δ largest inner loop, in order to obtain the frequencies f11, f21, which define the width of the resonant acoustic mode of the YIG/GGG structure, and its loaded Q-factor: In order to compare the theoretically calculated κ with the experimental data we introduced a coupling coeffi- f0 QMAR = , (B2) cient σ, which provides a measure of coupling between f21 − f11 the magnetic and acoustic sub-systems: In order to calculate the coupling coefficients between the subsystems in the MAR one should measure the diameter δ of the main resonance loop and the largest inner loop. σ = , (B5) The coupling between the strip-line and the YIG FMR ∆ − δ magnetic resonator is defined by the expression: ∆ Detailed explanation and derivation of the above pre- σ = , (B3) sented expressions can be found in [43]. YIG 2R − ∆

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