Electromagnetic Approach to Cavity Spintronics

Rair Macêdo, Rory C. Holland, Paul G. Baity, Luke J. McLellan, Karen L. Livesey, Robert L. Stamps, Martin P. Weides, and Dmytro A. Bozhko

Phys. Rev. Applied 15, 024065 – Published 25 February 2021

Abstract

The fields of cavity quantum electrodynamics and magnetism have recently merged into cavity spintronics, investigating a quasiparticle that emerges from the strong coupling between standing electromagnetic waves confined in a microwave cavity resonator and the quanta of spin waves, magnons. This phenomenon is now expected to be employed in a variety of devices for applications ranging from quantum communication to dark matter detection. To be successful, most of these applications require a vast control of the coupling strength, resulting in intensive efforts to understanding coupling by a variety of different approaches. Here, the electromagnetic properties of both resonator and magnetic samples are investigated to provide a comprehensive understanding of the coupling between these two systems. Because the coupling is a consequence of the excitation vector fields, which directly interact with magnetization dynamics, a highly accurate electromagnetic perturbation theory is employed that predicts the resonant hybrid mode frequencies for any field configuration within the cavity resonator. The coupling is shown to be strongly dependent not only on the excitation vector fields and sample’s magnetic properties but also on the sample’s shape. These findings are illustrated by applying the theoretical framework to two distinct experiments: a magnetic sphere placed in a three-dimensional resonator and a rectangular, magnetic prism placed in a two-dimensional resonator. The theory provides comprehensive understanding of the overall behavior of strongly coupled systems and it can be easily modified for a variety of other systems.

Received 25 October 2020
Revised 8 January 2021
Accepted 25 January 2021

DOI:https://doi.org/10.1103/PhysRevApplied.15.024065

Physics Subject Headings (PhySH)

Light-matter interaction Magnons Optical & microwave phenomena Photonics Polaritons Spintronics

Optical microcavities

Ferromagnetic resonance

Atomic, Molecular & OpticalQuantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Rair Macêdo ^1,*, Rory C. Holland¹, Paul G. Baity¹, Luke J. McLellan ¹, Karen L. Livesey^2,3, Robert L. Stamps⁴, Martin P. Weides ¹, and Dmytro A. Bozhko ^1,3

¹James Watt School of Engineering, Electronics & Nanoscale Engineering Division, University of Glasgow, Glasgow G12 8QQ, United Kingdom
²School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
³Center for Magnetism and Magnetic Materials, Department of Physics and Energy Science, University of Colorado Colorado Springs, Colorado Springs, Colorado 80918, USA
⁴Department of Physics and Astronomy, University of Manitoba, Winnipeg, Manitoba MB R3T 2N2, Canada

^*Rair.Macedo@glasgow.ac.uk

Article Text (Subscription Required)

Click to Expand

Supplemental Material (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 15, Iss. 2 — February 2021

Subject Areas

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Driving field $h$ exciting magnetization $m$ confined to (a) a ferromagnetic sphere and (b) a ferromagnetic rectangular prism. (c) Response of $m$ represented through the susceptibility tensor component $χ_{x x} (ω)$ . The solid lines are for the sphere and the dashed lines are for the rectangular prism. The lines for $χ_{x x} (ω)$ are calculated using the magnetic parameters for YIG: $μ_{0} M_{s} = 0.1758$ T, $γ / 2 π = 28$ GHz/T, and $μ_{0} H_{0} = 0.178 T$ . Note that the resonance $ω_{0}$ is not the same for the two systems due to different contributions of the demagnetizing field. For a sphere, $ω_{0} / 2 π = 4.58$ GHz as all demagnetizing factors are equal to $\frac{1}{3}$ due to symmetry [4]. For a rectangular prism, $ω_{0} / 2 π = 6.95$ GHz and the demagnetizing fields are calculated [32] using the dimensions $10 \times 5 \times 50 μ m^{3}$ , which yield $D_{x} = 0.3266$ , $D_{y} = 0.6115$ , and $D_{z} = 0.0618$ .
Reuse & Permissions
Figure 2
(a) Behavior of magnetization excited by a linearly polarized excitation such as shown in (b) when the sample is placed inside of a rectangular microwave resonator. We show a cross-sectional field configuration at $z = 3.75$ mm generated by capacitive coupling (simulated with comsol 5.5). Experimental $| S_{11} |$ spectra (color maps) and perturbation theory (dashed lines) for the hybridized cavity magnon-polariton modes, close to $ω_{c} = ω_{0}$ , for a YIG sphere (0.5 mm diameter) placed at positions (c) $A$ ( $y = 2.5$ ), (d) $B$ ( $y = 10$ ), and (e) $C$ ( $y = 15$ mm). In panel (f) we give a full map of $ω_{gap}$ for any given $x - y$ position. (g) Experimental points and theoretical lines of $ω_{gap}$ as the sample is moved within the microwave cavity (along $y$ and at $x = 27$ mm) through positions $A$ , $B$ , and $C$ [see panel (b)].
Reuse & Permissions
Figure 3
(a) Diagram of the setup used here where a two-dimensional coplanar waveguide resonator generates an oscillating magnetic field that couples to oscillating magnetization in a magnetic thin-film stripe. A static field $H_{0}$ is applied along the sample’s long axis (along $z$ ) and the oscillating magnetic field $h_{c}$ at the sample position has only a component along the $x$ direction—denoted as $h_{c x}$ . A full schematic of the magnetic sample and oscillating magnetization with respect to the oscillating and static fields is given in (b). (c) Anticrossing for the hybrid magnon-resonator modes calculated close to $ω_{c} = ω_{0}$ . Here, we considered the magnetic sample to be a Py ( ${Ni}_{80} {Fe}_{20}$ ) rectangular prism ( $14 \times 0.03 \times 900 μ m^{3}$ ) and the resonance frequency of the resonator is $ω_{c} / 2 π = 5.0$ GHz. The solid lines are for no damping [using Eq. (14)] and the dashed lines take damping into account.
Reuse & Permissions
Figure 4
(a) Scattering parameter $| S_{11} |$ calculated from the quality factor $Q_{p}$ from perturbation theory as a function of both the input frequency $ω$ and externally applied magnetic field $H_{0}$ . (b) Comparison between experimental and theoretical $| S_{11} |$ spectra at $ω_{c} = ω_{0}$ . The solid line is for the theory [vertical cut in (a)] and the dashed line is for experimental data [a vertical cut in Fig. 2]. Here we consider the system to be slightly overcoupled with $β = 1.05$ ; the dissipations for the two systems are taken to be $ω_{c}^{″} = 1.3 \times 10^{- 3}$ and $ω_{0}^{″} = 10^{- 4}$ . Scattering parameters for a two-dimensional microwave resonator coupled to a YIG thin-film stripe are also given using perturbation theory (c) compared to experimental data (d).
Reuse & Permissions
Figure 5
Anticrossing for the hybridization between a YIG sphere and the dark mode of a reentrant resonator. The solid lines are the eigenfrequencies for a sample placed at the center, between the posts. The dashed lines are the eigenfrequencies obtained using the strength of the field nearer the posts. The diagram above shows the resonator geometry and the field profile at $ω_{c} = 20.8$ GHz calculated using comsol 5.5.
Reuse & Permissions

Physical Review Applied