RF frequency, 𝝎𝒑= 4 GHz; Bias field, 𝑯𝒅𝒄= 100 Oe

a) Developed an accurate, efficient and fast computational tool that can capture essential physics of parallelRF-excitation of ferromagnets. This is useful in the design of near-future RF devices, and futuristicmagnonic/spintronics devices.

b) Demonstrated computationally the abstract intrinsic processes of three - magnon scattering.

c) Demonstrated in-situ control of the non-linear nature of the material using an additional secondary signalof MHz frequency.

d) Demonstrated exceptional agreement with experimental data with regard to RF threshold fields.

e) Investigated interesting features of magnon dynamics close to RF threshold field. Four-magnon interactionsignored in device level physics are found to have increased role in determining device-behavior.

Excitation of ferromagnets by a microwave signal can lead to interesting non-linear effects. Such phenomenahave garnered considerable research interest recently in both the physics as well as the engineeringcommunities.

Proper understanding of such processes is needed before magnetic-materials could be used for applicationssuch as microwave-signal-processing and novel computation paradigms like magnonics and spintronics.Applications ranging from mobile phones and state-of-the-art satellite systems to critical defense-relatedsystems like RADARs would benefit immensely from the non-linear magnetics based technology.

The advantages of magnetics-based nonlinear devices include: lower energy consumption, better accuracy,and simpler implementation when compared with all-electrical implementations currently in use. However,not sufficient physical understanding exists as of now for the development of design rules that are necessaryfor practical devices.

The main goals of this study are: to develop powerful computational tools and techniques for studyingmagnetic non-linear processes, and use these tools to provide essential predictive capability towards thedesign of magnetic-material-based devices.

As the microwave (or radio-frequency, RF) signal strength increases more magnons are created, resulting innon-linearity for RF powers exceeding the corresponding threshold power.

Magnetic sample with field configurations

Ԧ𝑦

5.1 𝜇𝑚

Ԧ𝑧

Ԧ𝑥

𝒉𝒓.𝒇 𝑯𝒅.𝒄

𝒉𝒓.𝒇 = 𝒎𝒊𝒄𝒓𝒐𝒘𝒂𝒗𝒆 𝒇𝒊𝒆𝒍𝒅

𝑯𝒅.𝒄 = 𝑩𝒊𝒂𝒔 𝒇𝒊𝒆𝒍𝒅

Magnetic-sample and -fields

Three magnon scattering is an energy (𝜔) and momentum (𝑘) conserving nonlinear process intrinsic to the magnetic material.

This work was supported by the U.S. Defense Advanced Research Projects Agency (DARPA) under GrantW911NF-17-1-0100 and by the Center for Micromagnetics and Information Technologies.

The authors acknowledge the Minnesota Supercomputing Institute (MSI) at the University of Minnesotafor providing resources that contributed to the research results reported within this work. This work also usedthe XSEDE P100 and K80 GPU nodes at XStream, Comet and Bridges through allocation TG-ENG180002.

Physical Principles and Methodology

Micromagnetic Solver

Conclusions

Acknowledgements

Motivation

Nonlinearity control using secondary signal

ℎ~ℎ𝑡ℎ

- Agreement with experiment to within 2 dBm(agreements within 5 dBm are considered reasonable).

- Points to the suitability of Micromagnetics for suchnonlinear studies.

- Justifies the newly developed reduced 2D-simulationparadigm as well as magnon-based approach to analysis.

Experimental Conformity

Parameters

Saturation Magnetization : 145𝑒𝑚𝑢

𝑐𝑚3

Intrinsic damping : 3 × 10−5

YIG film thickness : 5.1 𝜇𝑚

Future Worka) Demonstration and study of non-linear damping effects e.g. negative damping.

b) Studies involving two GHz radio-frequencies to look at absorption and intermodulation products.

A. Venugopal, T. Qu, R. Victora, “Non-linear parallel pumped FMR: 3 and 4-magnon processes”, IEEE Microwave Theory and Techniques (MTT), vol.68,Issue 2, Feb. 2020.

Magnon Dynamics near ℎ𝑡ℎ

Magnon number plot showing the dominant magnons for 𝒉~𝒉𝒕𝒉

Landau-Lifshitz-Gilbert equation (LLG)

𝑳𝑳𝑮:𝒅𝑴

𝒅𝒕= −

𝛾

1 + 𝛼2𝑴×𝑯𝒆𝒇𝒇 − 𝛼

𝛾

1 + 𝛼2𝑴× (𝑴×𝑯𝒆𝒇𝒇)

(Damping term)(Precession term)

Equation of motion

𝑴 = 𝑴𝒂𝒈𝒏𝒆𝒕𝒊𝒛𝒂𝒕𝒊𝒐𝒏𝑯𝒆𝒇𝒇 = 𝑵𝒆𝒕 𝒎𝒂𝒈𝒏𝒆𝒕𝒊𝒄 𝒇𝒊𝒆𝒍𝒅

Magnetization at each cell evolves under LLG

GPU Nodes (used mainly for simulations) [% Project contribution]

K40 [30%]

V100, V100-4 [10%]

V100-8 [30%]

Regular nodes (used for magnon analysis)[% Project contribution]

small, amdsmall [30%]

Additional analysis tools developed allow the interpretation of real-space magnetization in

terms of magnons. This allows for convenient extraction/formulation of

properties important for device-engineering.

Real space magnetizations

Non-linear phenomena

𝒉𝒕𝒉: Threshold-field

Landau-Lifshitz-Gilbert Micromagnetic solver is based on CUDA (parallel programming language) and relies on modern GPUs for computing magnetization dynamics under RF excitation.

Salient features of the solver:(a) GPU specific optimization for the new mangi V100 GPUs has resulted in an additional 1.5x speed-up. (b) 100% GPU utilization by the code on K40s available at MSI. (c) 100x faster than CPU code for practical designs.(d) Single CPU-GPU code design helps achieve high throughput on MSI.(e) Double precision.

Most important parameters are: effective magnetic-field (𝑯𝒆𝒇𝒇) and magnetization (𝑴).

Effective magnetic field comprises of:

(a) Neighbor independent fields: Applied (DC and RF), thermal fields.

These allow for a high degree of parallelism.

(b) Neighbor dependent fields : Exchange, demagnetization fields.

Magnon-dynamics Studies in Ferromagnets including Effects of Secondary Signal

Aneesh Venugopal, Tao Qu, R. H. Victora

Department of Electrical and Computer Engineering, University of Minnesota Twin-Cities

Demonstration of 3-magnon Scattering Process

Upon excitation with a 3 GHz RF signal, two 1.5 GHz magnons are produced.

A novel method that allows in-situ control of the non-linearity has been developed.

Conventionally, non-linearity control is achieved by changing the thickness of the sample, however, this has the obvious disadvantage of having to physically replace the magnetic sample.

This has limited the use of magnetic-materials for various applications that require control of non-linearity e.g. noise reduction in communication systems.

A method is developed that uses an RF signal (RF2) in addition to the primary RF signal (RF1) to reduce the non-linearity.

The additional RF signal needed is much lower in frequency (~MHz) and amplitude compared to the

primary signal (GHz), both highly advantageous for device-applications.

MSI (~55%)XSEDE* (~35%)

L.W**(~10%)

Resource contribution to the project

≡

Angular precession frequency

𝜔 𝝎𝝎

𝝀Spatial

Wavelength

𝑘 =1

𝜆

Spin-wave

Number of magnons with frequency 𝜔 and wavelength 𝑘

𝑁(𝑘, 𝜔)

Quasi-particle: Magnon

Wave-particle duality

a)

b)

Unit cell: 𝑯𝒆𝒇𝒇, 𝑴

*Extreme Science and Engineering Discovery Environment**Local lab workstations

Acting alone a 6 GHz signal, results in an exponential

increase in the number of magnons (blue). However, with

an additional signal of 10 MHz of intensity 8 Oe, no

increase is seen (brown).

Threshold-field of the primary signal (ℎ𝑅𝐹1) can be

controlled using frequency as well as intensity of the

secondary signal (ℎ𝑅𝐹2).

Dominant magnons frequency and momentum resolved.

Magnons with same energy but opposite wave-vectors have identical numbers.

Nu

mb

er of m

agno

ns (arb

. log)

𝒉~𝒉𝒕𝒉

Growth dynamics of k =0 and k≠0 magnons

Growth dynamics of 3 GHz and rest of the magnons

(a) Demonstrated magnon pair creation and subsequent pair interactions that lead to increased importance of 4-magnon scattering in device physics. (Akin to Cooper-pairs in superconductors.)

(b) Demonstrated direct excitation of 𝒌 ≠ 𝟎 magnons.

RF field >= ℎ𝑡ℎ

RF field < ℎ𝑡ℎ

Magnon scattering

Type equation here.

𝐻𝑑𝑐=

15

0 O

e (1

um

th

ick

sam

ple

)

Three-magnon process

𝜔𝑝 = 3.0 𝐺𝐻𝑧

𝝎𝒌 = 𝟏. 𝟓 𝑮𝑯𝒛

𝜔=𝐸𝑛𝑒𝑟𝑔

𝑦

ℏ(𝐺𝐻𝑧)

Num

ber o

f magnons (a

rb. lo

g)