Strong coupling of quantized spin waves in ferromagnetic bilayers

Zhizhi Zhang, Huanhuan Yang, Zhenyu Wang, Yunshan Cao, and Peng Yan

Phys. Rev. B 103, 104420 – Published 12 March 2021

Abstract

We formulate a strong-coupling theory for perpendicular standing spin waves (PSSWs) in ferromagnetic bilayers with the interlayer exchange coupling (IEC). Employing the Hoffmann boundary condition and the energy-flow continuity across the interface, we show that the PSSWs are still quantized but with nonintegral quantum numbers, similar to the case with a surface pinning but in sharp contrast to that with a free surface. The magnon-magnon coupling is characterized by the spectrum splitting which is linear with the IEC in the weak-coupling region but getting saturated in the strong-coupling limit. Analytical predictions are verified by full micromagnetic simulations with good agreement.

Received 28 October 2020
Revised 7 February 2021
Accepted 1 March 2021

DOI:https://doi.org/10.1103/PhysRevB.103.104420

Physics Subject Headings (PhySH)

Exchange interaction Ferromagnetism Micromagnetism Spin waves

Bilayer films

Landau-Lifschitz-Gilbert equation Micromagnetic modeling

Condensed Matter, Materials & Applied PhysicsQuantum Information, Science & Technology

Authors & Affiliations

Zhizhi Zhang , Huanhuan Yang, Zhenyu Wang , Yunshan Cao, and Peng Yan ^*

School of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu 610054, China

^*Corresponding author: yan@uestc.edu.cn

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Issue

Vol. 103, Iss. 10 — 1 March 2021

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Images

Figure 1
(a) Schematic of the in-plane magnetized bilayer with thicknesses of $d_{1}$ and $d_{2}$ , respectively. The inset depicts the PSSW profiles across the two films. (b) Frequencies of the lowest four resonant modes of YIG ( $n = 0, 1, 2, 3$ ) and the Py uniform mode ( $n = 0$ ) as a function of $H_{0}$ when $J_{int} = 0$ . Solid curves are calculated based on Eq. (3). Circles correspond to the micromagnetic simulations. Red pentagrams indicate the resonance of two layers.
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Figure 2
(a) The PSSW spectra in the bilayer structure with different $J$ . The square, circle, diamond, and pentagram symbols represent the Py dominant mode and the YIG PSSW modes ( $n = 3$ , 4, and 5), respectively. (b) The PSSW spectra as a function of $J$ . (c) Spatial profiles of PSSW in the bilayer with $J = 0.20 mJ / m^{2}$ [labeled by the dashed green line in (b)]. The dashed white line indicates the interface.
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Figure 3
(a) Spatial distribution of $| m_{z} |$ of the Py dominant mode in the bilayer with the IEC strength $J = - 0.18 mJ / m^{2}$ at 5.5 GHz (upper panel) and $J = 0.20 mJ / m^{2}$ at 8.0 GHz (lower panel). The patched gray region represents the interface. Squares and circles represent the data in Py and YIG layers, respectively, obtained from micromagnetic simulations. Solid blue and green curves represent the fitting results. Since the spin-wave amplitudes in YIG are too small, we magnify them by 6 or 3 times, as labeled in each part of the figure. (b) Wave vectors in YIG (upper panel) and Py (lower panel) layers as a function of the ferromagnetic IEC strength ( $J > 0$ ). Solid curves represent the numerical solutions of Eqs. (3) and (11) by the graphic method (see Appendix pp2). (c) Decay length of the dynamic magnetization in the Py layer as a function of the antiferromagnetic IEC strength ( $J < 0$ ). The green line represents the analytical formula (13). (d) Frequency splitting as a function of $J$ . The patched green (yellow) part represents the linear (saturated) region. The solid green curve in (d) represents the formula (15). The dashed blue curves in (c) and (d) represent the numerical solutions by the graphic method (see Appendix pp2). All symbols represent the results from full micromagnetic simulations.
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Figure 4
(a) Spatial distribution of $| m_{z} |$ for the YIG ( $n = 4$ ) mode at 10.4 GHz (upper panel) and the YIG ( $n = 5$ ) mode at 13.1 GHz (lower panel) in the bilayer with $J = 0.6 mJ / m^{2}$ . Since the spin-wave amplitude in YIG ( $n = 4$ ) mode is too small, we magnify them by three times, as labeled in the figure. (b) Wave vectors in YIG (upper panel) and Py (lower panel) layers. (c) Frequencies of the YIG ( $n = 4$ and 5) modes as a function of $J$ . The magenta and green colors represent the YIG $n = 4$ and 5 modes, respectively. Diamond and pentagram symbols correspond to the micromagnetic simulation results. Solid curves represent the numerical solutions by graphic method (see Appendix pp2).
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Figure 5
Sum of the wave number $(k_{1} d_{1} + k_{2} d_{2}) / π$ in the coupled bilayer as a function of $J$ . Symbols are micromagnetic simulation results. Solid curves represent the data based on numerical solutions by graphic method (see Appendix pp2).
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Figure 6
Dispersion relations (blue curves) [Eq. (3) with $ω_{1} = ω_{2}$ ] and boundary conditions (orange curves) [Eq. (11)] to graphically determine $k_{1}$ and $k_{2}$ : with $J = 0.1 mJ / m^{2}$ (a), $J = 1.0 mJ / m^{2}$ (b), $J = - 0.03 mJ / m^{2}$ (c), and $J = - 0.2 mJ / m^{2}$ (d), where $k_{1}$ is a real number; and with $J = - 0.03 mJ / m^{2}$ (e) and $J = - 0.2 mJ / m^{2}$ (f), where $k_{1}$ is purely imaginary number. The square, triangle, hexagon, circle, rhombus, and pentagram symbols represent solutions of the Py dominant mode and the YIG PSSW modes ( $n = 1$ , 2, 3, 4, and 5), respectively.
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