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Micromagnetic simulations using MuMax Magnetic Vortices in Ferromagnetic Nano-cylinders
José Maria Freire Soares e Tavares Pinheiro - up202004194 - M:EF - University of Porto - Department of
Physics and Astronomy
Abstract—MuMax was used to simulate a ferromagnetic nanocylinder. The parameters used in the simulation were A = 13E-12
(J/m) and Ms = 810E3 (A/m), which gave an exchange length of
approximately 15 nm. The values of R and L of the nano-cylinder
were varied from 50 to 220 nm and 10 to 90 nm, respectively, in
10 nm increments. By studying the data calculated by MuMax,
we were able to plot a phase diagram and compare it to those
presented in the referenced literature. Finally, a simple analysis
of the energy functional terms was performed to support the
classification of each geometry state into one of the following
three categories: vortex state, in-plane magnetization, and outplane magnetization.
Index Terms—MuMax, Simulation, Ferromagnetic nanomaterials, Vortex state
I. I NTRODUCTION
M
UMAX provides a robust tool for conducting micromagnetic simulations, enabling the study of the
magnetic behavior of micro- and nanostructures. The present
study aims to introduce Mumax for studying the magnetic
behavior of ferromagnetic nano-cylinders. Through simulating
and visualizing the material response, we aim to map a phase
diagram of a magnetic nano-cylinder and compare it to an
already developed theoretical model. [1] [2]
This software solves the time- and space-dependent magnetization evolution in nano- to micro-scale magnets using
a finite-difference discretization and the Landau-Lifshitz micromagnetic formalism. Mumax is an open-source, GPUaccelerated micromagnetic simulation program developed and
maintained at the DyNaMat group at Ghent University. It includes an easy-to-use browser-based interface, which provides
simple and on-the-fly computations through input scripting.
The Landau-Lifshitz formalism is a rather complex subject.
For our purposes, it’s enough to understand that it is used
to describe the magnetization field motion within a solid
ferromagnetic material when subjected to an external magnetic
field. To conduct the simulations, MuMax numerically solves
the following set of equations: [3]
Z
µ0
M · M · Hdemag
2
V
(1)
∂m
γ =
m × Hef f + αm × (m × Hef f )
(2)
∂t
1 + α2
1 ∂E
Hef f = −
(3)
µ0 Ms ∂m
By doing so, it can calculate the magnetization vector field
at 3-dimensions, as well as the magnetic energy stored in the
E[m] =
d3 rA(∇m)2 − µ0 M · Hext −
nanostructure and its components, including exchange energy,
demagnetization energy, and Zeeman energy. Additionally, it
can assemble visual plots to help the user understand the
formation and evolution of magnetic domains.
In micromagnetic studies, researchers typically focus on
two types of physics: dynamic and static. Dynamic physics
involves time integration and is used to describe phenomena
such as spin waves or domain wall motions. On the other
hand, static physics examines the static magnetic states and is
concerned with minimizing the system’s energy. This approach
provides valuable insights into stable magnetic states, domain
wall profiles, phase diagrams, and hysteresis curves. MuMax
can deal with the two challenges.
To accomplish our goals, we used static simulations. To do
these, MuMax needs to minimize the energy functional, or to
solve Eq.2 without the parcel m×Hef f . This parcel describes
the precession of the magnetization around Hef f . By setting
it to approximately zero, we are essentially stating that the
damping term, m × (m × Hef f ) is much stronger, effectively
canceling out the dynamic cyclotron movement.
As previously stated, we used MuMax to study nanocylinder ferromagnets. This choice wasn’t random, since these
structures have been extensively studied due to their possible
MRAM1 applications. [4] In recent years, there have been
significant advancements in fabrication and observation tools.
Namely, it became possible to observe and excite vortex
states. While current practices in product development focus
on suppressing vortices due to their negative impact on the
net magnetization, there is potential for utilizing vortices in
MRAM applications. [5]
With this in mind, the objective of mapping a phase diagram
should allow us to identify the geometries that support a vortex
state and those that do not. In order to properly understand the
phase transitions, it is required to address the energy of the
micromagnetic system.
The two most important terms in the energy functional
(Eq. 1) are the exchange energy term and the demagnetization
energy term, first and third parcels, respectively.
Demagnetization energy refers to the energy stored in the
magnetic field generated inside the ferromagnet due to its
magnetization - induced by an externally applied field. This
energy is higher when the magnetic moments inside the
material are all aligned, resulting in a stronger demagnetization
field. It also has its upper limits, as it starts to decrease if the
cylinder’s radius becomes too large.
1 Magnetoresistive random-access memory
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Fig. 1. Schematic of a vortex state in a cylindrical ferromagnet [6]
On the other hand, the exchange energy arises from the
exchange interaction, a quantum interaction without a classical
equivalent. It is related to the interaction of electron spins, and,
nowadays, it’s considered as the origin of ferromagnetism. The
exchange energy is at its minimum when the spins are well
aligned and increases as they deviate from alignment. This is
almost the opposite behavior of the demagnetization energy. In
a real micromagnetic system, the exchange energy can never
be zero due to border effects at the nanostructure perimeter.
Lastly, some remarks about the vortex state (see Fig. 1): for
particular ranges of dimensions of cylindrical nanostructures
a curling in-plane spin configuration - vortex - is energetically
favored, with a small spot of the out-of-plane magnetization
appearing at the core of the vortex. Such a system is described by two binary properties: chirality (counter-clockwise
or clockwise direction of the in-plane rotating magnetization)
and polarity (the up or down direction of the vortex core’s
magnetization). Each of these properties represents an independent bit of information in potential high-density nonvolatile
recording media. Moreover, vortex structures provide promising opportunities for memory models as they offer two degrees
of freedom, in contrast to the conventional single degree for
a uniform state.
II. E XPERIMENTS
In order to conduct our simulation, we began by defining
the necessary variables, as well as the geometry and size of
the mesh.
The simulation mesh defines the size of the box around
your magnet. We opted for a box with 64 × 64 × 8 cells. A
cylindrical geometry for the ferromagnet was chosen, and we
wrote a cycle that would run the simulations and change the
volume automatically (Fig.3). The thickness of the cylinder
(in the z direction) ranged from 10 nm to 90 nm, with steps
of 10 nm. The diameter range was from 50 nm to 200 nm,
in increments of 10 nm as well. Additionally, we defined the
exchange constant - A = 13·10−12 (J/m) - and the saturation
magnetization - Ms = 810 · 103 (A/m).
We ensured that the calculated energies and a snapshot of
the computed magnetization field were being saved.
The simulation was conducted in a stationary state, i.e, a
constant external magnetic field was applied and the response
of the cell registered. The initial magnetization of the material
was random.
III. R ESULTS
To process all the data Python and Jupyter Notebook were
used. With the constants previously defined the exchange
Fig. 2. Defined parameters
Fig. 3. For cycle for the simulation
length was calculated to be:
s
Le =
A
10−7 · Ms2
≈ 15 nm
In order to plot the phase diagram, we normalized the
thickness and radius using Le . We expected the values to be
higher (closer to 20 nm), so the range of normalized data fell
a bit short of what would be desirable. By analyzing both
the snapshots and the values for energies and magnetization,
we were able to categorize each geometry into one of three
classes: vortex state, in-plane magnetization, and out-plane
magnetization. The vortex state doesn’t require any further
explanation. In-plane magnetization refers to states where the
majority of magnetization is in the x and y directions of space,
but doesn’t form a chirality pattern. Out-plane magnetization
refers to states where the magnetization is more pronounced
in the z direction, perpendicular to the cylinder basis.
The plot with the numerically calculated data is presented
in Fig. 4. A plot presenting data from [] is displayed in Fig.
5 for comparison purposes.
The classification of magnetic states turned out to be similar
to the one predicted in the literature. Region I includes geometries that can form a stable vortex state. Region II represents
geometries with in-plane magnetization states, and Region III
consists of geometries that favor out-plane magnetization. It is
noticeable that some points are located over the line, or even
in a region not corresponding to their classification. However,
this is not problematic for two reasons. First, the lines indicate
geometries where the magnetic energy is the same in both
states. Line (a) marks the border between in-plane and outplane states that have the same energy. Line (b) is determined
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Fig. 4. Phase diagram with data simulated by MuMax. The solid lines
represent equal equilibrium energies between states. The R and L are
normalized by Le . Regions are identified and explained below.
Fig. 6. Snapshots made by MuMax to observe and study the magnetization
field. This serve as an example, since each geometry had its own snapshot
and could be used for a similar analysis.
Fig. 5. Phase diagram of nano-cylinders. Solid lines represent the equality
of the equilibrium energies. In the shades regions more than one state can be
found. [2]
by equating the energy of the vortex state to the energies of
the other two possible states. Secondly, when comparing the
experimental plot to the literature, we observe that these points
fall into regions where more than one state can be encountered.
To further explain the data and the classification process,
four snapshots are displayed in Figure 6. The first column
presents an in-plane and a vortex state for a thickness of
20 nm and radii of 30 nm and 55 nm, respectively. The
second column shows an out-of-plane and a vortex state
for the indicated dimensions. MuMax also provides energy
values, which are useful for classifying the states (vortex states
have higher exchange energy, while the others have higher
demagnetization energy). Additionally, we can monitor the
magnetization values in each direction.
When plotting the energies for geometries with a thickness
of 20 nm as a function of the radius, we observe in Fig. 7
that the demagnetization energy is dominant for smaller radii.
However, for R > 40 nm, a change occurs, and the exchange
interaction takes over. This suggests that for radii greater than
40 nm, a vortex state will be present rather than an in-plane
magnetized state. This trend holds true for all geometries:
the appearance of vortex states coincides with a decrease in
demagnetization energy.
When analyzing the data for thicknesses of 60 nm (Fig. 8),
we notice a similar pattern, but with an important detail. The
exchange energy is higher for every radius. Initially, it may
Fig. 7. Plot of the energies as a function of the radii for 20 nm thickness.
seem that every state is a vortex state. However, upon analyzing the magnetization values, we observe that for smaller radii,
Mz is predominant (see Fig. 3), reaching a maximum absolute
value of 0.6 A/m. Therefore, we classify this state as an out-ofplane magnetization state. Upon examining the corresponding
snapshot in Fig. 6, we observe a weak chirality pattern in
the magnetization field, which explains the higher exchange
energy. As the radius increases, the Mz values approach 0,
promoting the formation of a vortex state.
The values for the Zeeman energy obtained by MuMax
were all approximatelly zero, so we didn’t consider them for
analyzis purposes.
Regarding Fig. 4, we can use the data from the plot to
estimate the single domain radius (REQ ) and the absolute
single domain radius (RS ). REQ is the radius, for a given
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innovative MRAM systems, sensors, and more. Additionally,
it provided a rough estimate for the single-domain radius.
The analysis of snapshots and energy plots led to an
understanding of the dominant types of energy involved in
the system - exchange energy and demagnetization energy.
The balance between the two energies determines the system’s
minimal energy state, which is characterized by a certain state
depending on its geometry.
It would have been interesting to extend the intervals of
R and L to obtain more data points closer to zero in each
dimension. Approaching R = 0 nm could have allowed the
observation of the system’s behavior as it tended to a nanowire
configuration. On the other hand, if L tended to 0 nm, a thin
film configuration would have been approached. In neither
case, a vortex state would be expected. Instead, more states of
uniform magnetization, either in-plane or out-of-plane, would
be anticipated.
Fig. 8. Plot of the energies as a function of the radii for 60 nm thickness.
V. R EFERENCES
R EFERENCES
[1] Konstantin L. Metlov, Konstantin Yu. Guslienko, “Stability of magnetic
vortex in soft magnetic nano-sized circular cylinder,” Journal of
Magnetism and Magnetic Materials, no. 242–245 (2002) 1015–1017,
2002. [Online]. Available: https://shorturl.at/fpuy1
[2] Konstantin L. Metlov, “Map of metastable states for thin
circular magnetic nano-cylinders,” 2018. [Online]. Available:
https://arxiv.org/pdf/0707.2938.pdf
[3] J. M. Johnathan Leliaert. Mumax3 workshop - session 1. Video format.
[Online]. Available: https://shorturl.at/kBUW4
[4] P. W. Tai Min, Yimin Guo, “Vortex magnetic random access memory,”
U.S. Patent US 7,072.208 B2, Jul. 4, 2006.
[5] Qinglei Meng, “Magnetic vortex,” 2008. [Online]. Available:
https://shorturl.at/ktwI5
[6] Roman ANTOS, YoshiChika OTANI, and Junya SHIBATA, “Magnetic
vortex dynamics,” 2008. [Online]. Available: https://shorturl.at/csvCF
Fig. 9. Plot of the Mz as a function of the radii for 60 nm thickness.
thickness, for which E(vortex) = E(in/out-plane). For the same
thickness, if r < REQ , the energy of that state should be lower
than the energy of a vortex state. RS is a limiting value, i.e.,
for r < RS the vortex state is unstable, hence prohibited.
To make this estimation, we can use the solid lines dividing
the three considered states. However, in comparison with the
expected plot from Fig. 5, it is safe to say that this estimation
provides a rough measurement for both REQ and RS . [1]
IV. C ONCLUSION
MuMax software was used to perform micromagnetic simulations at a basic level, showcasing the power and multiple
capabilities of MuMax.
The simulation involved a ferromagnetic nano-cylinder system. A phase diagram regarding its magnetization states was
created based on the visual and numerical data provided by
MuMax. This helped in determining which geometries supported a vortex state, which is essential for the development of
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