Thermal instability in a ferrimagnetic resonator strongly coupled to a loop-gap microwave cavity

Cijy Mathai, Oleg Shtempluck, and Eyal Buks

Phys. Rev. B 104, 054428 – Published 20 August 2021

Abstract

We study the nonlinear response of a ferrimagnetic sphere resonator (FSR) strongly coupled to a microwave loop-gap resonator (LGR). The measured response in the regime of weak nonlinearity allows the extraction of the FSR Kerr coefficient and its cubic damping rate. We find that there is a certain range of driving parameters in which the system exhibits instability. In that range, self-sustained modulation of the reflected power off the system is generated. The instability is attributed to absorption-induced heating of the FSR above its Curie temperature.

Received 2 May 2021
Revised 19 July 2021
Accepted 9 August 2021

DOI:https://doi.org/10.1103/PhysRevB.104.054428

Physics Subject Headings (PhySH)

Ferrimagnets

Cavity resonators Microwave techniques

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Cijy Mathai , Oleg Shtempluck, and Eyal Buks

Andrew and Erna Viterbi Department of Electrical Engineering, Technion, Haifa 32000, Israel

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Issue

Vol. 104, Iss. 5 — 1 August 2021

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Images

Figure 1
FSR-LGR coupling: (a) A sketch of the FSR made of YIG having a radius of $R_{s} = 1 mm$ that is integrated inside the aluminum cylindrical LGR having a gap width of $0.3 mm$ . The sphere is held by ceramic ferrules (CFs). A sapphire wafer (labeled as S) is inserted into the gap to increase the capacitance. (b) The numerically calculated magnetic field energy density distribution (normalized with respect to the maximum value) corresponding to driving at the resonance frequency $ω_{e} / (2 π) = 3.3 GHz$ . (c) A VNA reflectivity $| S_{11} |^{2}$ measurement as a function of magnon frequency $ω_{s}$ (proportional to the externally applied magnetic field). The coupling coefficient $g_{eff}$ is extracted from the theoretical fit (white dashed lines) following Eq. (2).
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Figure 2
Reflection coefficient $| S_{11} |^{2}$ in dB units for three values of MW input power $P_{p}$ . (a)–(c) present the experimental data corresponding to MW input powers $P_{p}$ of $- 20, - 5$ , and $+ 10$ dBm, respectively. The second row [(c)–(e)] shows the corresponding theoretical fits that are obtained from Eq. (3). The theoretical fit parameters are $γ_{2 e} = 1.5 MHz, γ_{e} = 4 MHz, γ_{s} = 1 MHz, K_{M} = 6.325 nHz, δ_{e} = 35 MHz$ , and $γ_{3 s} = 0.001 nHz$ . To obtain a proper fit, $N_{s}$ and $g_{eff}$ are taken as variable values varying as a function of $P_{p}$ . For $P_{p} = - 20, - 5$ , and 10 dBm, $N_{s}$ values are taken as $1 \times 10^{19}, 5 \times 10^{19}$ , and $8 \times 10^{19} m^{- 3}$ , and $g_{eff}$ values are taken as 14, 14, and $12 MHz$ , respectively.
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Figure 3
Thermal instability. (a) Spectral density $I_{SA}$ of the signal reflected off the LA, as a function of the detuning frequency $f_{d}$ , for the driving frequency $f_{p} = 3.2224 GHz$ and normalized driving power $P_{p} / P_{c} = 1.288$ specified by the black cross overlaid in (c). (b) Spectral density $I_{SA}$ in dB as a function of the driving frequency $f_{p}$ and detuning frequency $f_{d}$ for $P_{p} / P_{c} = 1.7$ [indicated by the overlaid horizontal dashed line in (c)]. (c) The SB spacing frequency $ω_{SM} / (2 π)$ in $MHz$ as a function of driving frequency $f_{p}$ and normalized driving power $P_{p} / P_{c}$ . The overlaid blue (red) dashed line represents the threshold condition $E_{F} = E_{cF}$ ( $E_{P} = E_{cP}$ ). The following values are assumed for the calculations: $ω_{+ F} / 2 π = 3.317 GHz, ω_{+ P} / 2 π = 3.314 GHz, γ_{+ F} = 1.3 \times γ_{+ P}, σ_{F} / w_{TF} = 2.6 \times σ_{P} / w_{TP}, (K_{+ F} / γ_{+ F}) (w_{TF} / σ_{F}) = 0.5$ , and $K_{+ P} = 0$ .
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Figure 4
Limit cycle. (a) Numerical integration of the equations of motion (4) and (5) is performed with the following parameters: $Im (w_{F} - w_{P}) = - 0.1, Re (w_{F}) = - 1, Re (w_{P}) = - 1.5, σ_{F} = 0.01, σ_{P} = 0.02$ , and $w_{TF} = w_{TP} = 0.01$ . The values of driving detuning frequency $Im (w_{F})$ and driving amplitude $w_{1} = w_{1 F} = w_{1 P}$ are indicated by the black cross in (b). The LC is shown in (a) as a closed curve in the complex $B$ plane, in (c) as a periodic function of $Θ - 1$ vs the normalized time $τ$ , and in (d) as a periodic function ${| B |}^{2}$ vs $τ$ . The plane of driving frequency and driving amplitude is shown in (b). No steady state solution exists in the region between the blue and red curves (labeled as A).
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