1. Vortex dynamics in an equilateral triangular arrangement of three magnetic disks
X. M. Cheng,1,2 D. J. Keavney2, D. J. Clarke3, 4, O. Tchernyshyov3, M. Mahoney1, and A. Melikyan5
1 Department of Physics, Bryn Mawr College
2 Advanced Photon Source, Argonne National Laboratory
3 Department of Physics and Astronomy, Johns Hopkins University
4Department of Physics and Astronomy, University of California
5Materials Science Division, Argonne National Laboratory
PEEM image
Motivation
Time-resolved X-ray Photoemission Electron Microscopy (TR-PEEM)
Experimental results of vortex gyration in the tri-disk system
Theoretical calculation of frequency shift due to dipolar interaction
Summary
It is my pleasure to present our studies on vortex dynamics in magnetic tri-disk systems. This work is a collaborative effort of people from Bryn Mawr College, ANL, and JHU.
Here is the outline of my talk. After a brief introduction to the motivation of this work and the imaging method--TRPEEM, I will present both experimental imaging and theoretical calculation of the vortex gyration in the tri-disk system. Finally I will summarize.
Magnetic vortices in micron-sized ferromagnetic disks have been of great interest because of their potential applications in data storage. While the motion of a vortex in a single isolated magnetic disk has been studied extensively, vortex dynamics in multiple-disk planar geometries remains to be fully understood. We report direct time-resolved imaging and theoretical calculations of the vortex states in an equilateral triangular arrangement of three magnetic disks with varied center-to-center spacings. The free-motion trajectories of the vortex cores in the triangular arrangement of three permalloy disks of 2 micron radius were traced using time-resolved x-ray photoemission electron microscopy at beamline 4-ID-C of the Advanced Photon Source. The temporal resolution is 90 ps.
The oscillation amplitude in the tri-disks with 4.5 micron center spacing was smaller than that with 5 micron center spacing.
No significant frequency shift was observed.
Theoretical calculation showed both frequency shift and trajectory change due to dipolar interaction of the disks at varied spacings.

2. Magnetic Vortex Dynamics
Vortex State:
Magnetic vortices have been of great interest
Rich physics
Applications in memory devices
Study on interaction of adjacent vortices lacking
D=4.5µm
D=5µm
A magnetic vortex is a flux-closure state existing in micron- and submicron-sized magnetic disks, which consists of a circulating in-plane magnetization and an out-of-plane vortex core.
Depending on the chirality of in-plane M, a vortex can have ccw and cw two states. Dpending on polarity of the vortex core, a vortex can have up and down two states. Therefore, magnetic vortices have been of great interest because of not only the rich physics in this system, but also their potential applications in memory devices.
So far the motion of a vortex in a single magnetic disk has been studied extensively. However, few studies have been reported on interaction of adjacent vortices, which is very important for data storage application.
Here I will present our research on vortex dynamics in three disks in an equilateral triangular arrangement. Each individual disk has a radius of 2 um, and spacing between the disks are 4.5 um and 5 um
Chirality: CW CCW CW CCW
Polarity: up(+1) up(+1) down (-1) down (-1)
R=2 µm
S.B.Choe et al., Science 304, 420 (2004). J. Raabe et al., PRL 94, (2005).
X. M. Cheng et al, PRB, 79, (2009) K. Guslienko et al., JAP (2002), PRL 96, (2006).

3. X-ray Photoemission Electron Microscopy (X-PEEM)
X-ray Magnetic Circular Dichroism (XMCD) Fe
PEEM
FeNi disk
First, let me describe the imaging method-x-ray PEEM. x-ray comes in, ejecting electrons from the sample surface, these electrons are collected by the electron microscope to form an image.
Magnetic atoms have different absorption spectrum for left and right circularly polarized lights.
Therefore, we can take one image using left polarized light, and a 2nd image using right polarized light. By taking the difference of these two images, topographical and chemical contrast can be eliminated to yield only the magnetic contrast.
The intensity in the difference images is proportional to the dot product of the magnetization vector M , k. Therefore is just the projection of M to the x-ray porpagation direction
Therefore, in the white region, M is paprallel to the X-ray direction, and in the black region, antiparallel, the gray area, it is perpendicular.
This is exactly what a vortex should look like in a PEEM image,
X-rays are incident on the samples at 25° with respect to the surface.
Therefore, in this geometry, PEEM is sensitive only to the magnetization component along the x direction.
The spatial resolution we can achieve is about 100nm
25º y x
Brightness ~ M · k ~ Mx

4. Pump-probe time-resolved X-ray Photoemission Electron Microscopy
I0(t)
90 ps
100 nm Spatial Resolution
90 ps Temporal Resolution
153 ns
24-bunch mode: Dt
I0(t)
PEEM Optics
B(t)
Pulse
Generator
50Ω
Advanced Photon Source
Argonne National Lab B
FeNi disks
Au Waveguide
10μm
The time resolution is obtained using a pump-probe arrangement.
We patterned the py disk on top of a gold waveguide.
Current pulses were launched down the waveguide using a pulse generator synchronized with the x-ray pulses, providing a time-varying field at the permalloy disks. The x-ray pulses (of the width 90ps and ns periodicity) with an adjustable delay Δt and same freq. was used for imaging.
If the magnetic process is exactly repeatable, we will get a still image. This is like we use a strobe light to look at a rotating object….
In our case, each image was collected for 1-5 min., corresponding to an average over ~108 pulses.
By adjusting the time delay, we can images at different time after the removal of the field . We put these imaged together to make a movie about the super fast vortex dynamics. Here is an example, where the vortex core gyrates CCW.
J(t)
Radius=2 µm

5. Vortex core motion in an isolated disk at 2mT excitation
Radius=2µm B x y k
Brightness ~ Mx
B (mT) 2
Now let’s have a look at our experimental imaging results.
Here is a movie showing the free oscillation of the vortex core in an isolated disk after the 2mT excitation field is removed at t=0. The trajectory is elliptical.
Quantitatively we extracted the core positions and here in this figure I show the y component of the core displacement as a function of time, which can be fit by a damped sinusoidal waveform shown as the red lines.
the freq. of 71 MHz, in agreement with the prediction by LLG equation,
(The oscillation frequency in both directions is 42.2 MHz)
(in which the brightness is proportional to projection of magnetization onto the x-ray light vector direction,)
f=71 MHz, In agreement with theoretical prediction
K. Guslienko et al., JAP (2002), PRL 96, (2006)

6. Vortex core motion in tri-disk system with D=4.5 µm
Disk Spacing D=4.5 µm
Radius=2µm
Here is a movie showing the core motion in a 4.5um-spaced tri-disk system, the extracted core position shifts in y-direction are plotted here, for comparision, we also plot the isolated disk curve in blue. the frequency of y motion in each individual disk is slightly different from that in insolated disk denoted by w0. The ratios are 0.96, 0.89, and 0.92 for the top, bottom left and bottom right respectively.

7. Vortex core motion in tri-disk system with D=5.0 µm
Disk Spacing D=5.0 µm
Radius=2µm
Here is a movie showing the core motion in a 5um-spaced tri-disk system, the extracted core position shifts in y-direction are plotted here, for comparision, we also plot the isolated disk curve in blue. There are also small freq. shifts. The ratios are 1.05, 0.96, and 0.97 for the top, bottom left and bottom right respectively.

8. Calculated frequency shift due to the dipolar interactionx y z
ω/ω0
ω/ω0
k2/k
k2/k
My theoriest colleagues, david and oleg from Hopkins calculated this problem analytically. This is the L they used, in addition to the terms existing in single disk system, they consider dipolar interaction among the three disks characterized by this parameter k2,
For the two possible polarity configurations shown here, they calculated the frequency shift as a function of the strength of the dipolar interaction
w/w2=0.06, D=4um
w/w2=0.03, D=5um
0.702
0.512
1/r^3

9. Comparison between tri-disk system with different spacing
We fitted the t-dependent y-position curves using damped sin waveforms and compare the results from two different spaced tri-disk systems here. The fittings for the 5um spacing system are very good. This actually means the disks act like an isolated disks, showing little interaction.
The fittings for 4.5 um system are not very good. This is a sign of interaction among the 3 disk as what we expected.

10. Summary
Elliptical vortex core trajectories in tri-disk systems imaged by TR-PEEM
Shift in frequency of vortex core motion observed and calculated
The observed frequency shift in the 5 µm spaced tri-disk may result from slight difference in disk shape
The observed frequency shift the 4.5 µm spaced tri-disk might be due to the interaction among disks

11. ?

12. Vortex Core Motion at 2mT excitationB
Here is the result for the vortex core motion with 2mT excitation field, which shows a damped elliptical motion.

13. Vortex Core Motion at 4mT excitationB
1 mT
2 mT
4 mT
This slides shows the results for 4mT excitation case. It shows a core trajectory with a even larger ellipicity, which is almost a liner motion
Therefore in our 6um Py,disk, as the excitation field increases, we observed both circular motion similar to Choe et al’s result, and almost linear motion, similar to Raabe et al’s result.
We also note that there is an amplitude limit for the free vortex oscillation ,that is the oscillation amplitude never exceeds about 20% of the radius, regardless of the bias field magntitude.
If you focus on the first several frames of the movie, you might be able to see some delicate structures, which provides clues for the origin of the observed field dependence. I will discuss in more details in the next slide.

14. Images average over 108 pulses
Micromagnetic Simulation: Vortex Core Trajectories
1mT
2mT
4mT
Vortex motion following a single pulse: Circular regardless of field magnitude
Free core circulation limit: ~0.2 R
Initial core displacement<0.2R Circular core trajectory
Initial core displacement>0.2R Transient domain state,
Randomization of core polarity
Cancellation of x component of vortex motion
The micromagnetic simulations also shows that the motion following a single pulse is circular rather than linear, regardless of the magnitude of the field pulse.
And that amplitude of the core circulation does not exceed ~0.15 Ms, corresponding to core displacement of about 0.25 R, in good agreement with the experimentally observed free oscillations amplitude limit, 0.2 R.
When the initial core displacement smaller than 0.2R, the integrity of the vortex state is maintained throughout the measurement, showing a circular core trajectory
When the initial core displacement larger than 0.2R, the vortex core equation of motion is not valid any more, distortions of the core region occur, promoting the transient domain state and subsequent randomization of the core polarity, resulting in cancellation of the X-component of vortex motion, giving the illusion of elliptical or linear motion.
The transition between the linear and non-linear regimes appears to be gradual, given that the trajectory we observe at 2 mT has a slight ellipticity, whereas at 1 mT the core trajectory is circular.
, while the phase of the Y-component of motion is maintained with respect to the field pulse, thereby preserving the oscillations.
Thus the existence of this transient domain state in the initial motion from highly displaced vortices suggests a mechanism to reconcile the observations of linear and circular trajectories in the literature. The interpretation of past experimental results has been based on the assumption that the integrity of the vortex state is maintained throughout the measurement. While this is true for low-amplitude excitations, the assumption breaks down for large-amplitude excitations.
Thus we conclude that the true motion a well-formed magnetic vortex will always correspond to a circular (or elliptical) core trajectory but that a threshold exists, in this case a drop in field of ~2 mT, beyond which significant distortions of the vortex core occur.
Bias of core polarity
Images average over 108 pulses

15. Summary of Vortex Dynamics
Vortex core polarity reversed by non-resonant in-plane magnetic field
A narrow window exists for predictable core reversal above this field regime randomization of core polarity observed
Transient domain state observed in both experiments and simulations
Imaged core trajectory changed from circular to more elliptical as excitation field increases
A critical core displacement exists at ~20% of the disk radius
<0.2R Circular core trajectory
>0.2R Transient domain state,
Randomization of core polarity
Cancellation of x component of core motion
we use time-resolved PEEM and micromagnetic simulation to examine the vortex dynamics in Ni80Fe20 patterned disks.
We demonstrate that as exciation field increase, the imaged core trajectory is changed from circular to more elliptical.
We show that a critical trajectory exists at ~20-25% of the disk radius, beyond which the initial motion is characterized by significant distortion of the vortex, leading to transient domain states and instabilities in the core polarization, which influence the subsequent core trajectories.