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Shoki Nezu, [Thomas Scheike](https://orcid.org/0000-0002-9163-5524), [Hiroaki Sukegawa](https://orcid.org/0000-0002-4034-7848), Koji Sekiguchi

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[Unconventional parametric spin-wave pumping in single-crystal iron films](https://mdr.nims.go.jp/datasets/e0df1466-62c3-4124-84f6-c8af86ca37c3)

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Microsoft Word - Manuscript3.docx1  Unconventional parametric spin-wave pumping in single-crystal iron films 1 Shoki Nezu1, Thomas Scheike2, Hiroaki Sukegawa2 and Koji Sekiguchi3, 4* 2  3 1Graduate School of Engineering Science, Yokohama National University, Tokiwadai 79-5, Yokohama 4 240-8501, Japan 5 2National Institute for Materials Science, Sengen 1-2-1, Tsukuba, Ibaraki 305-0047, Japan 6 3Institute of Advanced Science, Yokohama National University, Tokiwadai 79-5, Yokohama 240-8501, 7 Japan 8 4Faculty of Engineering, Yokohama National University, Tokiwadai 79-5, Yokohama 240-8501, Japan 9  10  11 * Correspondence: Koji Sekiguchi, Institute of Advanced Science, Yokohama National University, 12 Tokiwadai 79-5, Yokohama 240-8501, Japan 13 Telephone: +81-45-339-4147, Fax: +81-45-339-4147 14 E-mail: sekiguchi-koji-gb@ynu.ac.jp   15 2  ABSTRACT 16 Spin waves hold promise for expanding the magnonics research field to include quantum information 17 processing and classical information devices. Parametric pumping is considered a key technique to 18 achieve this important advancement. Recently, a single-crystal iron has shown potential as a spin-19 wave excitation medium; however, parametric pumping in single-crystal iron films has not been 20 investigated. In this study, we explored computationally and experimentally the characteristics of 21 parametrically pumped spin waves in single-crystal iron thin films by respectively using large-scale 22 micromagnetic simulations and a high-precision spectrum analyzer. The results demonstrate 23 unconventional parametric pumping attributed to the competition between the anisotropic and 24 excitation fields, emerging at external magnetic field and low power levels that would be insufficient 25 to induce parametric pumping in isotropic materials. Systematic research on parametric pumping in 26 iron could pave the way for low-energy spin-wave devices, enhancing quantum information device 27 technology. 28 PACS Indexing Codes   29 3  I. INTRODUCTION 30 Intensive research on spintronics revealed the profound importance of spin waves in the 31 development of next-generation information processing. Spin waves can serve as noncharged 32 information carriers and building blocks of broadband (GHz-THz) nanoscale processors.1–3 As 33 building blocks of broadband processors, propagating spin waves have been used to implement 34 successfully various radiofrequency (RF) functionalities, such as transistors,4 logic circuits,5–8 35 multiplexers,9 directional couplers,10 and switches.11,12 In addition to these classical RF functionalities, 36 spin waves can be applied to quantum information processing13,14 as magnons, the quanta of spin 37 waves, exhibit Bose-Einstein condensation (BEC),15,16 and magnon quantum condensates can provide 38 some functionality of quantum bits (qubits)17 even at room temperature. The field of quantum 39 magnonics has received considerable attention recently, prompting vigorous research efforts. These 40 endeavors, including low-temperature quantum18,19 and hybrid magnonics20-22, have paved the way for 41 the exploration of magnons and yttrium iron garnet (YIG) films composed of a well-understood 42 material with small magnetic damping (equally long magnon lifetime). 23–25 However, the research 43 findings on magnonic qubits have not yet been firmly established. The challenges associated with 44 magnonic qubit research arise from the competition between the efficiency of the magnon generation 45 method and the thermal agitation of the environment. To generate a magnon quantum condensate, a 46 nonlinear, high-power RF excitation of spin waves, known as parametric pumping, is essential.26 47 However, parametric pumping unavoidably induces a thermal energy flow in the BEC medium, thus 48 leading to thermal agitation enhancements. It is noteworthy that YIG has a modest saturation 49 magnetization (Ms = 140 kA/m27–29), and the thermal effects on Ms become significant. The spin-wave 50 resonance (the condition for magnon generation) in YIG is particularly susceptible to thermal 51 agitations.30–32 52 Recently, an alternative approach to the magnonic qubit has been suggested that utilizes 53 single-crystal iron (Fe) thin films prepared by epitaxial growth on single-crystal substrates as a BEC 54 medium.33,34 The typical saturation magnetization of a single-crystal Fe(001) thin film is Ms = 1.6 55 MA/m, approximately 11 times larger than that of YIG.35,36 While the magnon lifetime in single-56 crystal Fe is shorter compared with that in YIG, the in-plane magnetic anisotropy due to its cubic 57 magnetocrystalline anisotropy of Fe compensates for the group velocity, decay length, and spin-wave 58 amplitude.36,37 The possibility of magnon condensation in single-crystal Fe has been reported under 59 specific conditions; however, the relationship between the efficiency of parametric pumping and the 60 cubic anisotropy axis, and the detailed dynamics of parametrically excited spin waves in single-crystal 61 Fe thin films, remain to be elucidated. An investigation into magnon generation through parametric 62 pumping is crucial for the development of magnonic qubits using single-crystal Fe thin films. 63 In this study, the parametric pumping process of spin waves in single-crystal Fe thin films 64 was investigated in detail using a spectrum analyzer. All-electric detection using a spectrum analyzer 65 is more sensitive and provides better frequency-domain resolution compared with the direct current 66 (DC) spin-Hall effect,38–40 as there is no spin-current conversion. The number of magnons generated 67 by parametric pumping was directly evaluated based on the spectral peak of the spin-wave amplitude 68 as a function of the bias magnetic field and pumping power. Systematic experiments revealed the 69 existence of three different bias fields, i.e., triggering points, for a given pumping frequency. At these 70 triggering points, a larger efficiency (10×) of parametric pumping was achieved. In combination with 71 micromagnetic simulations, the enhancement of pumping efficiency was found to originate from the 72 4  cubic anisotropy of Fe. These results shed light on a promising path for additional magnonic qubit 73 research. 74  75 II. RESULTS AND DISCUSSION 76 Figure 1(a) shows a schematic of the experimental setup. The spin-wave medium was 77 deposited on a MgO(001) single-crystal substrate. A highly (001)-oriented single-crystal Fe film was 78 deposited with a Cr(001) buffer and a MgAl2O4(001) cap. The whole stack structure is Cr (40)/Fe 79 (25)/MgAl2O4 (~2) (thickness in nanometers).35 The deposition was conducted at room temperature 80 using a DC/RF ultra-high vacuum magnetron sputtering system with a base pressure of less than 7 × 81 107 Pa. The Cr buffer and Fe layer were in-situ post-annealed at 700 °C and 300 °C, respectively, to 82 ensure an atomically flat surface and a highly (001)-orientation. The MgAl2O4 cap was prepared by 83 natural oxidation of a Mg (0.45 nm)/Mg19Al81 (1.2 nm) bilayer. The magnetic properties of the single-84 crystal Fe(001) film was characterized using a vibrating sample magnetometer, revealing the 85 saturation magnetization Ms = (1.6 ± 0.1) × 106 A/m and the in-plane magnetic anisotropy field µ0Hani 86 = (66 ± 2) mT.36 The Fe(001) film was patterned into a rectangle using Ar ion milling technique. The 87 dimensions of the rectangle were 180 µm in length and 110 µm in width. The longitudinal axis (y-axis) 88 of the sample was aligned parallel to the hard axis, i.e., Fe[110].  89 A pair of asymmetric coplanar transmission lines was created using a 5 nm Ti adhesive layer 90 and a 200 nm Au layer using vacuum deposition and lift-off processes. The transmission lines were 91 designed with signal width of 1.0 µm and gap width of 0.9 µm, resulting in a characteristic impedance 92 of 50 Ω. The separation distance between the transmission lines was designed to be 5 µm and 93 corresponded to the propagation length of spin waves. Spin waves were parametrically excited by an 94 excitation transmission line launching a microwave field hp at frequency fp using a signal generator 95 (Agilent Technologies 83732B). An external magnetic field Hext was applied parallel to the hard axis, 96 and a signal of pumped spin waves was detected by a detection transmission line connected to a 97 precision spectrum analyzer (Agilent Technologies E4440A). The opposite ends of both the excitation 98 and detection transmission lines were connected to a low-noise ground plane (GND). The measured 99 signal V(f) was composed of the pumped spin-wave amplitude Vsw and background noise Vn, and can 100 be represented as V(f) = Vsw + Vn. The center and span frequencies of the spectrum analyzer were set 101 at fp/2 (half the pumped frequency) and 5 kHz, respectively. Note that the unfavorable electromagnetic 102 interference induced voltage by the excitation antenna at fp was eliminated with this condition as the 103 parametrically excited spin-wave signals possess the frequency of fp/2. 104 To understand the parametric pumping process in single-crystal Fe films, the dispersion of a 105 single-crystal Fe film was calculated [Fig. 1(b)] using the parameters Ms = 1.6 MA/m, µ0Hani = 66 mT, 106 thickness d = 25 nm, and µ0Hext = 80 mT. In this saturation condition (Hext > Hani), the magnetization 107 M aligns with the direction of the external magnetic field, and the dispersion simply represents the 108 relationship of the spin-wave resonance frequency f versus the wavevectors ky and kx, which are y- and 109 x-axis components of the wavevector k, respectively. In linear spin-wave excitations, a microwave 110 field hp must be matched with the spin-wave resonance frequency at a given external magnetic field. 111 However, when the amplitude of the microwave field is large enough to overcome spin-wave 112 relaxation, spin-wave excitation becomes possible at microwave fields at frequencies other than the 113 resonant frequency. In this high-power (nonlinear) excitation of the spin waves, the magnetic system 114 allows parametric pumping. There are two typical mechanisms for the parametric excitation of spin 115 5  waves: parallel and perpendicular pumping.41,42 In the case of parallel pumping (M || hp), spin waves 116 at the frequency fp/2 are directly excited by the parallel component of microwave fields at the 117 frequency fp. In the case of perpendicular pumping (M ⊥ hp), spin waves at the frequency fp/2 are 118 excited by the nonresonant magnetization precession, which is induced by the perpendicular 119 component of the microwave field. In both cases, a pair of spin waves with opposite wavevectors (k 120 and k) were generated at the frequency fp/2, obeying the momentum and energy conservation law. 121 Figure 1(c) exemplifies the parametric pumping at µ0Hext = 80 mT. The excitation frequency and the 122 center frequency of the spectrum analyzer were set to fp = 8.90 GHz and fc = fp/2 = 4.45 GHz, 123 respectively. The signals V(f) represent the parametrically pumped spin-wave amplitudes and their 124 dependence on pumping power. At a low input power (Pin = 0.26 mW), the signal only consists of a 125 background noise Vn ~ 10 nV. As the input power increases to Pin = 0.30 mW, a distinct peak with a 126 broad line width emerges at the center frequency fp/2. At higher power levels (Pin = 0.35 mW), the 127 peak becomes sharp, thus indicating the presence of parametric pumping. 128 In a generalized case at a specific Hext, the spin-wave resonance frequency f for the cubic 129 anisotropic Fe film is described by the following equation:36,37,43 130  131 𝑓𝛾 𝜇2𝜋𝐻 𝐻 , 1  134 where the gyromagnetic ratio is γg = 1.76 × 106 T1s1. The in-plane effective magnetic field H1 and 132 normal effective magnetic field H2 can be expressed as 133  135 𝐻 𝐻 cos 𝜙 𝐻 cos 4𝜙2𝐴𝑀𝑘 𝑀 𝑃 sin 𝜙 2  136 𝐻 𝐻 cos 𝜙 𝐻2𝐴𝑀𝑘 𝑀 1 𝑃 3  137 where the saturation magnetization Ms = 1.6 MA/m, the cubic anisotropy field µ0Hani = 66 mT, the 138 exchange constant Aexch = 13 pJ/m, Pk = 1  (1  e|k|d)/|k|d, ϕk represents the angle between the 139 magnetization M and the wavevector k, and ϕeq denotes the angle between the magnetization M and y-140 axis [inset in Fig. 1(a)]. The angle ϕeq = 0 for h ≥ 1, h = Hext/Hani, and 141  142 𝜙𝜋2sin6 / 9ℎ √81ℎ 6/6 / 9ℎ √81ℎ 6/ 4  143 for 0 ≤ h < 1. In this experiment, an external magnetic field was applied parallel to the hard 144 magnetization axis. Therefore, in Fe films with Hext < Hani, the parametric pumping becomes 145 intermediate between parallel and perpendicular pumping, and is known as oblique pumping. At Hext > 146 Hani , the magnetization aligns parallel to the external field. However, since a coplanar transmission 147 line is used for parametric pumping in this study, spin waves are not excited only by pure parallel 148 pumping. Notably, an increase in the external magnetic field induces a transition from parallel to 149 perpendicular pumping44. This phenomenon is attributed to the spatial localization of the pump field, 150 specifically the microwave magnetic field components perpendicular to the film plane at the sides of 151 6  the coplanar transmission line.  152 Figure 2 represents the excitation power dependence of the parametrically pumped spin-153 wave amplitude with a frequency of fp/2 at various external magnetic fields. The excitation frequency 154 was fixed to fp = 8.90 GHz. As shown in Fig. 2(a), by increasing the excitation power from 66 µW to 155 32 mW at µ0Hext = 80 mT, the pumped spin-wave amplitude exhibits a threshold, and the spin-wave 156 amplitude changes the order of magnitude. According to the theory of parametric excitation, the 157 pumped spin-wave amplitude Vsw is proportional to the 𝑃 𝑃 , where Pin and Pth are input and 158 threshold powers, respectively.45,46 The experimental result was analyzed using the following equation, 159  160 𝑉 𝑎 𝑃 𝑃 𝑉 , 5  161 where a is a fitting parameter. The fitting deduced the threshold power Pth = 0.29 mW and shows an 162 excellent agreement as indicated by the broken line. By reducing the external magnetic field to µ0Hext 163 = 64 mT, we examined oblique pumping in the single-crystal Fe film. As shown in Fig. 2(b), we 164 observed that the pumped signal exhibited an additional change at Pin = 0.46 mW and had an 165 amplitude of 103 nV. The pumping at Pth = 11.7 mW increased the amplitude to 105 nV; the change can 166 be explained by equation (5). In the case of µ0Hext = 54 mT, additional changes appear when Pin = 167 0.56 mW and Pin = 4.2 mW, and the pumping of 105 nV appears at the higher threshold Pth = 26.1 mW. 168 Remarkably, in a weaker magnetic field µ0Hext = 40 mT, the spin-wave amplitude was enhanced 169 considerably to 66.2 nV at a very low input power (Pin = 66 µW), even though the pumping observed 170 at 105 nV was lost. 171 The parametric excitation of spin waves in the cubic anisotropic Fe films was systematically 172 investigated by constructing and analyzing the color plot of spin-wave amplitudes as functions of 173 input power Pin and external magnetic field Hext. Figure 3 shows the color plot of spin-wave 174 amplitudes with a frequency of fp/2 at various pumping frequencies fp. The striking characteristic of 175 the V-shaped structure shown in each panel is known as an asymmetric butterfly curve. However, the 176 butterfly curve is not the same as previously reported.47,48 In the case of fp = 6.2 GHz as shown in Fig. 177 3(a), the threshold power gradually decreased from 32 mW to 0.19 mW as the magnetic field 178 increased along the butterfly curve. Once the external magnetic field reached a characteristic field of 179 70 mT (labeled Ⅲ, HⅢ), a minimal increase in Hext of only 4 mT triggered a rapid rise in the 180 threshold power back to 32 mW. This occurred because half the pumping frequency fp/2 deviated from 181 the spin-wave dispersion branch, disabling parametric pumping. Note that there is another remarkable 182 field at µ0Hext = 56 mT (labeled Ⅰ, HⅠ) where parametric pumping is achieved at a low input power (Pin 183 = 66 µW), implying a slightly higher pumping efficiency. At other pumping frequencies fp = 7.6 GHz 184 [Fig. 3(b)], 8.9 GHz [Fig. 3(c)], and 10.2 GHz [Fig. 3(d)], the characteristic fields were also observed 185 and labeled as HⅠ, HⅡ, and HⅢ in each panel. The detailed values of characteristic magnetic fields are 186 summarized in Table 1. These fields correspond to the field where additional changes are observed in 187 Fig. 2.  188 189 7   190 Table 1. Experimentally determined and calculated characteristic magnetic fields. 191 fp (GHz) experiment calculation µ0HⅠ (mT) µ0HⅡ (mT) µ0HⅢ (mT) µ0Hc1 (mT) µ0Hc2 (mT) µ0Hc3 (mT)6.20 56 - 70 53.7 63.1 71.7 7.60 50 68 76 46.5 61.6 74.6 8.90 40 64 80 37.3 59.9 77.7 10.4 30 60 84 18.1 57.5 82.0  192 The origin of characteristic fields could be explained by the parametric instability which was 193 revealed by pioneering works. 44,49,50 The parametric instability indicates that the parametric pumping 194 shows a maximum efficiency when half the pumping frequency fp/2 becomes equivalent to the 195 ferromagnetic resonance (FMR) frequency fFMR. Using equations (1)-(3) and setting k = 0, the FMR 196 frequency of the cubic anisotropic Fe film can be expressed as 197  198 𝑓  𝛾 𝜇2𝜋𝐻 cos 𝜙 𝐻 cos 4𝜙 𝐻 cos 𝜙 𝐻 𝑀 . 6  199 Using equation (6), we calculated the characteristic fields for each pumping frequency; the values are 200 summarized in Table 1. Note that equation (6) has two independent solutions (µ0Hc2, µ0Hc3). 201 Comparing the calculated characteristic fields with the experimentally observed field, it is obvious 202 that µ0Hc2 and µ0Hc3 agree with HII and HIII, respectively. The threshold power Pth at HII and HIII 203 increases as pumping frequency increases, exhibiting a parametric instability feature. 204 Furthermore, we focused on the magnetic field labeled HI, where enhanced spin waves were 205 observed at an extremely low input power. For instance, when fp = 8.90 GHz, the spin-wave amplitude 206 was enhanced at Pin = 66 µW. Note that the characteristic field HI decreases as the pumping frequency 207 increases. As the characteristic field appeared at a lower magnetic field, the possibility of three-208 magnon scattering was investigated. Three-magnon scattering refers to a nonlinear process in which 209 three magnons interact with each other, and involves two processes: splitting, where one magnon 210 splits into two, and confluence, where two magnons merge into one. Using equation (6) and the 211 condition fFMR = fp and k ~ 0, the characteristic fields µ0Hc1 were calculated for each excitation 212 frequency. As shown in Table 1, the magnitudes of the characteristic field HI were in good agreement 213 with the calculations, with the only exception being the case of fp = 10.4 GHz. The reason for this 214 mismatch could be that experimentally, the external magnetic field was limited in the range 30 mT ≤ 215 µ0Hext ≤ 96 mT.  Our observations consequently reveal an unconventional parametric pumping, 216 emerging at external magnetic field and power levels that would be insufficient to induce parametric 217 pumping in isotropic materials. 218 To understand the dynamics of these unconventional efficient parametric pumping in the Fe 219 films, we performed micromagnetic simulations by numerically solving the Landau–Lifshitz–Gilbert 220 equation, ∂m/∂t = –γgµ0m × Heff + Gm × ∂m/∂t, using MuMax3.51 In this equation, m represents the 221 unit vector along the magnetization, Heff encompasses the effective magnetic field components, 222 8  including exchange, magnetostatic, and external magnetic fields, and G denotes the Gilbert damping 223 constant. The thin Fe film was simulated with a unit cell 10 nm × 10 nm × 25 nm. The grid size was 224 specified as (Nx, Ny, Nz) = (600, 600, 1) with periodic boundary conditions denoted as (Px, Py, Pz) = 225 (512, 256, 0). The material parameters for the single-crystal Fe film were: cubic anisotropy Kc = 54.8 226 kJ/m3, saturation magnetization Ms = 1.6 MA/m, Gilbert damping constant G = 0.002, and exchange 227 stiffness constant Aexch = 13 pJ/m. Detailed descriptions are available in the Supplementary Materials. 228 The color plots of the parametrically pumped spin-wave intensities with a frequency of fp/2 229 were reproduced in simulations using the pumping frequencies fp = 6.2, 7.6, 8.9, and 10.4 GHz, and 230 the simulated results are shown in Fig. 4. As observed, the butterfly curves (shown in red and green 231 color) are not identical to the experimental results; the butterfly curve values did not decrease to Pth = 232 0.19 mW but to 2.3 mW. Except for this point, the simulation results reproduced the experimentally 233 obtained pumping characteristics. As shown in Figs. 4(a)-(d), HⅠ, HⅡ, and HⅢ are characteristic fields, 234 and the threshold power Pth was reduced. These simulation values of HⅠ, HⅡ, and HⅢ are in agreement 235 with the experimental results and the calculated characteristic fields µ0Hc1, µ0Hc2, and µ0Hc3. Similar 236 to the experimental results, the unconventional characteristic field HI decreases as pumping frequency 237 increases. Notably, in the cases of fp = 8.9 GHz and fp = 10.4 GHz, the simulations predicted the 238 existence of another characteristic field HⅣ. As shown in Figs. 4(c) and 4(d), the values of HⅣ were 239 close to the magnetocrystalline anisotropy field Hani. However, this field was not observed 240 experimentally. This could be the reason that experimentally, in a backward volume configuration, 241 spin waves cannot propagate for 5 μm in single-crystal Fe thin films at Hext = Hani, while the 242 simulation focused the parametric excitation process underneath the excitation transmission line 243 (within 2 μm length).37 244 Figure 5 exemplifies spin-wave generation under parametric pumping in simulations. Each 245 panel is the frequency cross-section of spin-wave dispersion, as shown in Fig. 1(b). The pumping 246 frequency and power were fixed at fp = 8.9 GHz and Pin = 0.22 mW, respectively. Figures 5(a) to 5(d) 247 represent spin-wave intensities at the pumping frequency fp = 8.9 GHz, while Figs. 5(e) to 5(h) 248 represent intensities at the pumped frequency fp/2 = 4.45 GHz. The pink lines show the theoretical 249 dispersion relations. At the weak field µ0Hext  36 mT (HⅠ), spin wave generation was dominated by 250 the red region on the dispersion branch at fp = 8.9 GHz and k ~ 0 [Fig. 5(a)]. Although no dispersion 251 branch exists at fp/2 = 4.45 GHz [Fig. 5(e)], the presence of scattered spin waves is undeniably evident, 252 as manifested by the diffuse green stripe region extending along ky. These scattered spin waves likely 253 originate from the pump source, which consists of spin waves with a frequency of fp = 8.9 GHz. As 254 the external magnetic field is increased, a theoretical dispersion emerges at fp/2 = 4.45 GHz and µ0Hext 255 = 60 mT (HⅡ) [Fig. 5(f)]. This signifies the initiation of parametric spin-wave generation. Additionally, 256 weak spin waves with finite wavevectors (k ≠ 0) are generated on the branch [Fig. 5(b)] of the ∞-257 shaped structure at fp = 8.9 GHz through a second-order parametric process. When the external 258 magnetic field exceeds the crystalline magnetic anisotropy field µ0Hext > µ0Hani = 66 mT, spin-wave 259 generation is dominated at fp/2 = 4.45 GHz [Figs. 5(g) and 5(h)] with 2.63- and 2.34-times stronger 260 intensity than at fp = 8.9 GHz [Figs. 5(c) and 5(d)]. 261 The unconventional, efficient spin-wave generation at HⅠ was caused by the competition 262 between the anisotropic and excitation RF fields. The external magnetic field HⅠ (HⅠ < Hani) was not 263 strong enough to pin the direction of magnetization, and the magnetization was tilted by the RF field. 264 The dispersion changed as a function of the excitation field. Figure 6 shows the transition of the 265 9  dispersion relation as a function of the input RF power Pin. The pumping frequency and external 266 magnetic field were fixed at fp = 8.9 GHz and µ0Hext  36 mT (HⅠ), respectively. As shown by Figs. 267 6(a) and 6(e), at the low input power of Pin = 1.68 mW, the spin-waves were generated at fp = 8.9 GHz 268 along the theoretical ∞-shaped dispersion indicated by pink lines, i.e., fixed magnetization angle ϕeq 269 calculation. However, as the input power increases from 2.20 mW [Figs. 6(b) and 6(f)], 12.43 mW 270 [Figs. 6(c) and 6(g)], and 18.22 mW [Figs. 6(d) and 6(h)], the strong generation points shown in red 271 and yellow deviate from the ∞-shaped dispersion. Additionally, as input power increases, the pumped 272 spin waves at fp/2 = 4.45 GHz become dominant even when µ0Hext  36 mT (HⅠ). These simulation 273 results demonstrate that the efficient spin-wave generation at HⅠ is caused by the modulation of the 274 magnetization angle due to the RF field. 275 Finally, we analyzed the modulation of the magnetization angle in the Fe thin film beneath 276 the excitation transmission line in simulations. Figure 7(a) represents a snapshot of the magnetization 277 directions in the 3 μm × 3 μm region, 2.5 ns after the RF excitation. The external magnetic field and 278 input power were µ0Hext  36 mT (HⅠ) and Pin = 18.2 mW, respectively. As observed, the static 279 component of magnetization at 2.5 ns varies across different locations in the Fe thin film, thus 280 forming a multidomain-like structure. By deducing the magnetization angle |ϕ| between the 281 magnetization m and the y-axis of each cell and averaging over the 3 μm × 3 μm region, the ϕavg of 282 magnetization was derived using the equation: ϕavg = Σ|ϕ|/(NxNy). As shown in Fig. 7(b), the ϕavg 283 becomes smaller than the theoretical magnetization angle ϕeq = 26.3°, and the ϕavg changes from 25.8° 284 to 15.1° as the input power increases from 0.32 mW to 31 mW. Note that each ϕavg represents an 285 averaged value of the inhomogeneous distribution of magnetization angle, while the theoretical 286 analysis yields a uniform magnetization angle ϕeq over the region of interest. Considering this RF 287 modulation of ϕavg, the dispersion transition was revisited in Figs. 7(c), 7(d), and 7(e). When the input 288 power was Pin = 0.22 mW [Fig. 7(c)], spin waves were generated at the pumping frequency fp = 8.9 289 GHz. According to the evaluation listed above, the angle was ϕavg = 25.1°, and the theoretical 290 dispersion branch, shown by the white broken line, was located at 8.20 GHz and did not reach the 291 parametric pumped frequency fp/2 = 4.45 GHz. When the power increased to Pin = 2.20 mW [Fig. 292 7(d)], the angle was modulated to the value of ϕavg = 21.4°, and the theoretical dispersion branch was 293 reduced to 6.66 GHz. When the power was Pin = 18.2 mW [Fig. 7(e)], the angle decreased further to 294 ϕavg = 17.4°, and the branch frequency became fp/2 = 4.45 GHz. As shown in Fig. 7(e), the strong 295 generation points (shown in red color) are switched to approximately fp/2 = 4.45 GHz, thus 296 demonstrating parametric pumping. The inhomogeneous distribution of magnetization under the 297 radiation of the RF field induced the unconventional parametric pumping process. Under the 298 condition that the propagating direction of the spin waves aligns with the easy axis, we will lose the 299 competition between the anisotropic and excitation radiofrequency fields, disabling the observation of 300 unconventional parametric pumping. The anisotropic field, aligned in the same direction as the 301 external magnetic field, simply strengthens the effective field, resulting in conventional parametric 302 pumping observed in isotropic materials. 303  304  305 III. CONCLUSION 306 The parametric pumping process in cubic magnetic anisotropic Fe films was experimentally 307 10  investigated using a high-precision spectrum analyzer. By detecting spin-wave intensities as a 308 function of both the external magnetic field and excitation power, three distinct characteristic fields 309 (HⅠ, HⅡ, and HⅢ) were identified at which parametric pumping occurred at low power levels. The 310 characteristic field HⅠ induces unconventional pumping at fFMR = fp using the remarkably low input 311 power of 66 µW. The characteristic fields HⅡ and HⅢ originated from the conventional parametric 312 pumping at fFMR = fp/2. Large-scale micromagnetic simulations reproduced the experimental results 313 and revealed that the competition between the anisotropic and excitation fields modulated the 314 magnetization angle. The efficient unconventional pumping originated from the modulation of the 315 magnetization angle. The advantage of single-crystal iron films lies in their noteworthy saturation 316 magnetization, which is one order of magnitude higher than that of YIG films. This substantial 317 saturation magnetization enables GHz/THz operation and enhances the temperature stability of 318 magnonic devices. As demonstrated in this study, the in-plane magnetic anisotropy of single-crystal 319 iron films facilitates unconventional parametric pumping at remarkably low power levels, unlocking 320 the potential of these films as spin-wave media and transforming the landscape of magnonics. The 321 details of parametric pumping characteristics presented herein will contribute to future research efforts 322 on magnonic qubits.  323  324 IV. ACKNOWLEDGEMENTS 325 This work was supported by Grants-in-Aid for Scientific Research (19H00861, 18H05346, and 326 22K18321) from the Japanese Society for the Promotion of Science (JSPS). S. N. acknowledges the 327 support of a Grant-in-Aid for JSPS Fellows (23KJ0989). K. S. acknowledges the support of Grants-in-328 Aid for Scientific Research (20H05652). 329  330 V. DATA AVAILABILITY 331 The data supporting the findings of this study are available from the corresponding author upon 332 reasonable request. 333  334 VI. AUTHOR CONTRIBUTIONS 335 S. N. and K. S. planned the experiments. S. N., T. S., H. S., and K. S. designed and prepared the 336 samples. S. N. performed parametrically excited spin-wave measurements and the micromagnetic 337 simulations. S. N. and K. S. wrote the manuscript. All authors discussed the results. 338  339 VII. COMPETING INTERESTS 340 The authors declare no competing financial interests. 341  342 REFERENCES 343 11  1. A.Hirohata, K. Yamada, Y. Nakatani, I. L. Prejbeanu, B. Diény, P. Pirro, and B. Hillebrands, 344 Review on spintronics: Principles and device applications, J. Magn. Magn. Mater. 509, 166711 345 (2020). 10.1016/j.jmmm.2020.166711. 346 2. S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. Molnar, M. L. Roukes, A. Y. 347 Chtchelkanova, and D. M. Treger, Spintronics: A spin-based electronics vision for the future, 348 Science 294, 1488–1495 (2001). 10.1126/science.1065389, Pubmed:11711666. 349 3. K. Sekiguchi, The Basis of Magnon Transistors, AAPPS Bull. 28, 2 (2018). 350 4. A. V. 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A continuous wave was 485 launched into the excitation antenna, and parametrically excited spin waves were observed by the 486 detection antenna with the spectrum analyzer. The external magnetic fields were applied parallel to 487 the hard axis direction of the Fe films (y-direction). The inset within the top-right depicts the utilized 488 coordinate system. The angles ϕk and ϕeq represent the angle between the magnetization M and the 489 wavevector k, and M and y-axis, respectively. b. Illustration of the dispersion relation of spin waves 490 and the parametric pumping process. Spin waves with a frequency of fp/2 and wavenumber of ±k are 491 generated by a pumping field with a frequency of fp. c. The amplitudes of parametrically pumped 492 spin-waves at different excitation powers Pin = 0.26, 0.30, and 0.35 mW. The pumping frequency and 493 external field were set to fp = 8.90 GHz and µ0Hext  80 mT, respectively. The center and span 494 frequencies were set to fc = fp/2 = 4.45 GHz and f2  f1 = 5 kHz, respectively. 495  496 15   497 FIG. 2 Generation of parametrically excited spin waves measured at fp/2 = 4.45 GHz. Excitation 498 power dependencies of parametrically pumped spin-wave amplitudes at the pumping frequency fp = 499 8.90 GHz and the various external magnetic fields: a. 80 mT, b. 64 mT, c. 54 mT, and d. 40 mT. The 500 broken lines correspond to the theoretical fittings. At each excitation power measurement, five 501 independent measurements were performed. The data points represent the mean values, and the error 502 bars represent the standard deviations of the measurements. 503  504  505  506 FIG. 3 Experimental threshold characteristics of parametric pumping. The amplitudes of 507 parametrically excited spin waves with a frequency of fp/2 were measured at three distinct pumping 508 frequencies: a. 6.2 GHz, b. 7.6 GHz, c. 8.9 GHz, and d. 10.4 GHz. The external magnetic field and 509 excitation power were changed in the range 30 < Hext < 96 mT and 66 µW < Pin < 32 mW.  HⅠ, HⅡ, 510 and HⅢ represent the characteristic fields that exhibit parametric pumping at low input power levels. 511  512 16   513 FIG. 4 Simulated threshold characteristics of parametric pumping. The intensities of 514 parametrically excited spin waves with a frequency of fp/2 were simulated at four distinct pumping 515 frequencies: a. 6.2 GHz, b. 7.6 GHz, c. 8.9 GHz and d. 10.4 GHz. The external magnetic field and 516 excitation power were changed in the range 30 < Hext < 96 mT and 66 µW < Pin < 32 mW. HⅠ, HⅡ, HⅢ, 517 and HⅣ represent the characteristic fields that exhibit parametric pumping at low input power levels. 518  519  520 FIG. 5 Spin-wave generation under parametric pumping conditions. Spin-wave intensities at the 521 pumping frequency fp = 8.90 GHz (a-d) and the pumped frequency fp/2 = 4.45 GHz (e-h) in 522 simulations. Each panel represents the frequency cross-section of spin-wave dispersion. The input 523 power was Pin = 0.22 mW. The external magnetic fields were, a, e. 36 mT, b, f. 60 mT, c, g. 68 mT, 524 and d, h. 76 mT. The pink lines show the theoretical dispersion relations. 525  526 17   527 FIG. 6 Spin-wave generation under unconventional pumping conditions. Spin-wave intensities at 528 the pumping frequency fp = 8.90 GHz (a-d) and the pumped frequency fp/2 = 4.45 GHz (e-h) in 529 simulations. Each panel represents the frequency cross-section of spin-wave dispersion. The external 530 magnetic field was µ0Hext = 36 mT. The excitation powers were, a, e. 1.68 mW, b, f. 2.20 mW, c, g. 531 12.43 mW, and d, h. 18.22 mW. The pink lines delineate the theoretical dispersion relations at µ0Hext 532 = 36 mT. 533  534  535  536 18  FIG. 7 Modulation of magnetization angle under unconventional pumping conditions. a. 537 Simulated snapshot of magnetization directions in the 3 μm × 3 μm region underneath the excitation 538 antenna at 2.5 ns after the radiofrequency (RF) excitation with fp = 8.9 GHz, Pin = 18.2 mW and µ0Hext 539 = 36 mT. b. Averaged magnetization angle ϕavg computed by varying the input power from 32 μW to 540 31 mW. The black dotted line is the theoretically obtained uniform magnetization angle ϕeq = 26.3°. c-541 e. Transition of spin-wave dispersions by RF excitation at Pin = 0.22, 2.20, and 18.2 mW. The white 542 broken lines show the dispersion relations calculated with the averaged magnetization angle ϕavg. 543  544  545  546