Theory

If a magnetic thin film with magnetization M⃗ is placed in an effective external magnetic field He, the magnetization tends to align with the external magnetic field so that magneto-static energy is minimized. If the magnetic moment is perturbed from its equilibrium position, it experiences a restoring torque from the external magnetic field:

where M⃗ is the local magnetic moment, μ0 is the vacuum permeability, and γ is the gyromagnetic ratio. This torque causes the magnetic moment to precess around the external magnetic field as shown in Fig. 1A.

This phenomenon is known as Larmor Precession. In practice, this precession is damped (Fig. 1B), and will relax if there is no other external force. Phenomenologically, this damped precession can be described by the Landau Lifshitz Gilbert (LLG) equation,

where α is the dimensionless Gilbert damping constant, and Ms is the saturation magnetization of the magnetic film. When the magnetic film is exposed to incident microwaves, some are absorbed, exerting an external force on the precessing magnetic moments which drives the oscillation (Fig. 1C). When these incident microwaves are at the same frequency as the natural frequency of the Larmor Precession, a resonance phenomenon occurs and results in a maximum in the absorption of the microwaves. This maximum in absorption can be observed by using a diode as a microwave probe to rectify the reflected microwave signal, providing a theoretical basis for FMR spectroscopy.

where Heff is the effecitve applied magnetic field, d is the thickness and Ms is the saturation magnetization. Conversely, the study of nonlinear effects in Eq. 2 provides a powerful probe into the underlying damping mechanism in FMR. Although the Gilbert Damping term in Eq. 2 accurately describes the damping behavior of many crystals, including YIG, it is phenomenological and provides no insight into the actual damping mechanism. One known damping mechanism that is ubiquitous in FMR is energy loss through the generation of spin waves. Spin waves are the propagation of a relative phase of the local magnetization through a material as detailed in Fig 2.

It is possible for energy from the Larmor Precesssion to be converted into spin-waves in processes known as two-magnon and three-magnon scattering [1], explained in Fig. 3. Two-magnon scattering is caused by impurities in the crystal, and operates with loss of momentum conservation. As such it is considered to be a form of extrinsic damping. [2]. However, three-magnon scattering preserves both energy and momentum and so the damping is intrinsic.

At high power, there is a coupling term between the Larmor Precession and spin wave modes that becomes important and breaks the linearization of Eq. 2 [6]. Therefore, as power is increased, two- and three-magnon damping becomes more important, resulting in a notable broadening of the FMR resonance at higher frequencies and powers. By observing this broadening and the extent to which it deviates from the linearization, we hope to better understand damping mechanisms.

YIG, whose crystalline structure is shown in Fig. 4, is of particular interest here as a material to conduct this study on due to its status as a ferromagnetic insulator. Non-insulating ferromagnetic materials often exhibit high amounts of noise in ferromagnetic resonance due to Eddy Currents within the magnet, which complicates analysis [6]. Since it is an insulator, YIG doesn’t have this issue. Furthermore, of the ferromagnetic insulators, it is known to have a particularly low Gilbert Damping Constant and narrow resonant linewidth, so using YIG will make it easier to observe nonlinear phenomenon and damping effects.