S18_Inducing Ferromagnetic Resonance d(FMR) in Yttrium Iron Garnet (YIG) to observe Gilbert Damping and Nonlinear Effects

Ferromagnetic Resonance (FMR) in Yttrium Iron garnet (YIG) to observe Gilbert Damping and Nonlinear Effects

Aaron T. Breidenbach and Aditya M. Pidaparti

Abstract

A planar stripline was used to direct microwaves into a thin film crystal of Yttrium Iron Garnet (YIG) in an external magnetic field to observe ferromagnetic resonance (FMR). From the resonant response of the sample, the Gilbert damping coefficient of thin film YIG (∼100nm) was found to be (1.9±0.2)∗10^-3. Additionally, the input microwave power was increased to observe nonlinear FMR phenomena. The presence of two magnon nonlinear broadening at high power was qualitatively confirmed, but the YIG sample was determined to be too thin to observe a three magnon nonlinearity.

Introduction

One current area of significant technological interest is the development of fast Magnetic Random Access Memory (MRAM) for computers. Traditional forms of RAM utilize electric signals to read and write information, requiring constant power to maintain a memory state, making the memory volatile. MRAM is an alternative, but currently requires a strong current to flip bits. Inducing spin-currents, a net flow of electrons that are spin biased, to apply a torque to the unit cell (known as spin-transfer torque, STT) in MRAM reduces the current to flip a bit allowing MRAM to be both more precise and use less power than traditional RAM. In addition, magnets are non-volatile in nature, since the induced magnetization will remain even after power is turned off, which means that MRAM could feasibly even be used as an ultra-high speed replacement for disk drives if produced cheaply enough.

MRAM, however, requires a magnetic material which has a strong enough STT response to flip the magnetization of a bit. Ferromagnetic Resonance (FMR) has long been used as a tool to probe the magnetic properties of materials that are potential candidates for MRAM [4]. It is a spectroscopic technique, which is preformed by injecting microwaves into a magnetic sample to drive a precession of the net magnetic moment about an applied external field.

Theoretically, the Gilbert Damping of a pure ferromagnetic crystal should be very low. However, in practice, impurities in the crystal provide extrinsic mechanisms that can increase the damping of a magnetic sample to be much higher than what it would intrinsically be for a pure crystal. The effects of extrinsic damping become much more prominent at high input microwave power where nonlinear effects also become important in governing the precession of the magnetic moment. As such the goal of this experiment is two- fold: (1) FMR will be used to find the Gilbert Damping Constant of YIG to determine its applicability to MRAM applications, and (2) the input microwave frequency and power will be increased in order to better understand damping effects in the nonlinear FMR regime.

Theory

FMR is a spectroscopic probe of magnetic materials, and is preformed by measuring the response of a sample when a magnetic field H is applied. This applied field causes the magnetization M of the material to precess due to a net magnetic torque. This is shown below in Fig. 1.

Figure 1: (A) Larmor precession of the magnetic moment about the effective magnetic field. (B) Larmor precession with Gilbert damping. (C) Damped Larmor precession which is driven with microwave radiation.

This phenomenon, Larmor precession, is damped (Fig. 1B) and will relax in the absence of external input. In a thin film, the precession is described by the Landau Lifshitz Gilbert (LLG) equation (Eq. 1),

(Equation 1)

where α is the dimensionless Gilbert damping constant, and Ms is the saturation magnetization of the magnetic film. When the film is exposed to incident microwaves, some are absorbed, which exerts a torque on the precessing magnetic moment. This drives oscillation (Fig. 1C), and as the incident waves approach the natural frequency of the system, maximum absorption and resonance occurs. For a thin film with negligible crystalline anisotropy and an in-plane external magnetization (a good approximation of our samples), the resonant frequency fr is described by the Kittel Equation (Eq. 2),

(Equation 2)

where d is a first order correction for a small amount of crystalline anisotropy.

It is worth noting that Eq. 1 is intrinsically nonlinear, but, it can be linearized if the power input is small. This linearization of the differential equation produces a simple relationship between the resonant linewidth, ∆H, and Gilbert Damping Constant [10],

(Equation 3)

where ∆H0 is the inhomogenous broadening of the resonance at zero frequency, fr is the resonant frequency, and γ is the gyromagnetic ratio. Therefore, one can extract the Gilbert Damping from a linear fit to a plot of linewidth as a function of microwave frequency.

While the damping term in Eq. 1 works for many crystals including YIG, it is purely phenomenological. A known damping mechanism in FMR energy loss is via generation of spin waves, or magnons. As shown in Fig. 2, spin waves are the propagation of a relative phase of local magnetization through a material.

f

Figure 2: Propagation of a spin wave through a material, which provides a damping mechanism in FMR. The wave is drawn though the ends of the local magnetization vectors.[2]

These SWMs (known as magnons in their quasiparticle form) have an associated wave vector, k. FMR can be considered as the k=0 magnon, and a qualitative f(k) dispersion relation is shown in Fig. 3.

Figure 3: The dispersion relation for SWMs of a ferromagnetic material at different sample thicknesses. It is important to note that there is always an anomalous region in this dispersion where df/dk is negative. This always leads to a dip in this dispersion relation that allows a degenerate k ̸= 0 mode with the same frequency (and energy) as the k=0 FMR mode.

There are several possible processes for energy from the Larmor Precession to be converted into SWMs, but the two and three magnon dispersion mechanisms were our focus. Two magnon dispersion is a process in which the k=0 FMR mode converts to a wavevector that is degenerate in frequency (and therefore energy). Since there is always some dip in the f(k) dispersion relation in FMR (see Fig. 3), this degenerate magnon always exists. Since two magnon scattering involves the generation of a non-zero wavevector mode with finite momentum from the zero wavevector mode, momentum is not conserved and hence it requires the presence of crystal imperfections to absorb momentum. Since the two magnon mode requires imperfections in the crystal to be excited, it is considered an extrinsic damping mechanism.

Qualitatively, the signature of two magnon scattering is an enhancement in the resonant linewidth at some finite input microwave frequency which is much larger than the linear broadening contribution of Eq. 4. Mathematical treatment of this non-linearity is considerably more complex, and can be described as follows

(Equation 4)

where H' is the field derivative of the Kittel equation, Λ_k represents defect-mediated coupling between the FMR and k^th magnon, δ_⍺ is a Lorentzian function which can be reduced to the Dirac delta function in the low damping limit, and f_k is the dispersion relation. Note that the correlation length, ε, is representative of the average distance between defects in the crystal and could be compared to average defect lengths found with microscopy were a fitting to Eq. 4 to be done to extract this material parameter.

On the other hand, three magnon dispersion is intrinsic and therefore conserves both energy and momentum. The k=0 FMR magnon is converted to two magnons, each at a frequency f_FMR/2. However, the size of the dispersion relation dip is a function of sample thickness. Therefore, below a critical width it is impossible to excite the three-magnon dispersion process. At a low enough input power, only a small amount of energy will be transferred to the three magnon mode that is consistent with the amount of dissipation the three magnon mode can provide. However, at some critical power, the energy of the uniform FMR mode will be large enough to overcome the three magnon dissipation and the energy lost to spin waves will grow exponentially [8]. Therefore, the characteristic signature of three magnon scattering is a sharp decrease in the amplitude of the FMR resonance above some critical power.

YIG is of particular interest here as a material to conduct this study on due to its status as a ferromagnetic insulator. Metallic ferromagnetic materials often exhibit high amounts of noise in ferromagnetic resonance due to eddy currents within the magnet, which complicates analysis [8]. 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 therefore narrow resonant linewidth, so using YIG will make it easier to observe nonlinear linewidth broadening.

Setup

Figure 4: Experimental setup schematic.

Above schematically depicts the experimental setup. A YIG device is placed on a coplanar stripline which directs microwave radiation (1-10GHz) into it. The YIG and stripline are between electromagnets which apply a steady magnetic field. Between these electromagnets are smaller modulating coils, which are attached to a function generator and produce a time variance in the magnetic field. Next to the YIG setup between the coils is a Hall probe to measure the field. Additionally, a microwave diode sits next to the sample as a probe to the absorption of microwaves by the YIG sample. The output voltage from the diode is sent through a lock-in amplifier, which uses the modulation from the waveform generator as a reference to filter out the unrelated (not from magnetic interactions) microwave background detected by the diode. Finally, all data and measurement control devices are connected to a computer so the experiment can be controlled by LabView software.

Data Collection and Analysis

Data is collected by setting the incident microwaves at a fixed power and frequency and sweeping the applied field around the resonant field (per Eq. 2) to measure the resonant response. Multiple of these field sweeps can be done at different microwave input frequencies to extract data necessary to fit Eq. 2 and Eq. 3. Additionally the input microwave power for a given field sweep can be increased when looking for nonlinearities.

Even though the introduction of the time varying component of the magnetic field causes the field derivative of the absorption spectrum to be measured instead of the direct microwave absorption spectrum, it is still possible to derive the Gilbert damping coefficient, FMR resonance field, and FMR resonance linewidth directly. Since the microwave absorption spectrum is known to be Lorentzian [4] one can extract a resonance linewidth by taking the distance between the two symmetric peaks that occur in the measured field derivative of the absorption spectrum, which correspond to the inflection points of the absorption spectrum. Furthermore, the zero derivative point between the two peaks corresponds to the resonant magnetic field in the original absorption spectrum. Fig. 5 shows the raw field derivative of the absorption spectrum for several different excitation microwave frequencies, with call-outs to clearly label how these material parameters are extracted. The process of finding these magnetic parameters is repeated at several different injected microwave frequencies.

Figure 5: An example plot of the field derivative of the microwave absorption spectrum as a function of applied field for a range of frequencies, along with guiding lines to illustrate how the linewidth and resonance field are extracted.

Results

The methods established in the previous section were used to find the resonant frequency as a function of field so that a fit to the Kittel Equation (Eq. 2) can be preformed. As can be seen in Fig. 6, this fit is very precise with negligible error in the frequency and resonant field and allows the values of γ/2π, 4πMs, and d to be extracted as 27.5 ± 0.1 GHz/T, 1.76 ± 0.04 kOe, and (3.0 ± 0.1) ∗ 10⁻² respectively. Since these values are in reasonable agreement with other literature on YIG [4], this measurement demonstrates that the setup is working properly and that FMR is being observed.

Figure 6: Resonance frequency as a function of applied field at an input microwave power of -12 dBm so that a fit to the Kittel Equation (Eq. 2) could be performed.The error in frequency and the resonant field is negligible, so error bars are not included.

With the value of γ/2π determined, it is now possible to determine the Gilbert damping constant, α, through a linear fit to linewidth vs resonant frequency through Eq. 3 in a low power (-12 dBm), low frequency limit as is shown in Fig. 7. When this is done, the values of ∆H₀ and α are determined to be (1.9 ± 0.2) ∗ 10⁻³ and 0.5 ± 0.1 Oe respectively. This damping is consistent with values found in other thin film YIG experiments [3].

Figure 7: Linewidth as a function of resonant frequency at an input power of -12dBm so that a fit to Eq. 3 could be performed. The extracted values of the Gilbert Damping Constant ⍺ and ∆H₀ are shown on the above plot.

With success demonstrated in the low-power, low-frequency linear FMR regime, the power and frequency are increased to look for nonlinearities. At an increased power of 0 dBm, the characteristic two magnon broadening of the FMR linewidth at higher frequencies is observed, as is shown in Fig. 8. A fitting to Eq. 4 was then attempted, which could be reduced down to two free parameters, ε and H', through the use of extracted values from the linear regime and by the assumption that Λ_k is constant with respect to wavevector and can be absorbed into H' [9]. A full χ2 minimization could not be completed due to computational difficulties, but a reasonable fit could be done when varying these parameters by eye. This rough fit is shown in Fig. 8 and a defect correlation length of ε ≈ 30 nm was extracted from this.

Figure 8: Linewidth as a function of frequency at 0 dBm. At high frequencies, the characteristic two magnon non-linear broadening is present in the large enhancement in linewidth at finite frequency followed by a relative plateau in the broadening. A preliminary fit to Eq. 4 (done by eye) is shown in the red line, from which the approximate value of ε was derived.

Moving on, the FMR amplitude as a function of power was observed in order to look for the three magnon nonlinearity. In Fig. 9, the normalized FMR amplitude can be seen as almost constant with power with only a slight decline in signal size for powers ∼15-30 dBm. We eliminated three magnon dispersion as a cause qualitatively to small to be the right order of magnitude from what was expected.

Most likely, the YIG sample used was too thin for the three magnon nonlinearity to occur. While the sample used was originally 3 μm and much thicker than the ∼500 nm three magnon thickness threshold found through simulation, the sample was etched in processing, likely past the three magnon threshold thickness

Figure 10: Normalized amplitude of the FMR signal as a function of input power. It's roughly constant as a function of power in the range from -12 to 30 dBm with only a very slight signal decline at the highest powers. This demonstrates that the three magnon non-linearity was not present in this sample, which would manifest itself as an exponential decline in signal at some threshold power.

Conclusion

This project was successful in measuring linear regime FMR and extracting material parameters that are consistent with other FMR literature on YIG. Additionally this project was able to demonstrate qualitatively the presence of a two magnon nonlinear linewidth broadening at higher input microwave powers along with a preliminary quantitative fitting. Finally, the YIG sample studied here was determined to be too thin for the excitation of the three magnon FMR mode to be possible.

Looking forward, further work can be done in this project to develop a better sample processing technique which allows one to etch a YIG sample to a desired aspect ratio while maintaining its thickness, so that three magnon scattering could potentially be observed on a new sample. Additionally, further work can be done in order to optimize the two magnon nonlinear linewidth broadening fit as well as comparing the extracted correlation length ε to defect lengths found with microscopy.

Acknowledgements

We would like to thank Bill Peria, Tao Qu, and Tim Peterson for their mentorship, as well as Paul Crowell for managing the us and the experiment.

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