# Dynamics and reversible control of the vortex Bloch-point vortex domain wall
in short cylindrical magnetic nanowires

Diego Caso  Pablo Tuero  Javier García  Departamento Física de la Materia
Condensada C03, Universidad Autónoma de Madrid, Madrid 28049, Spain
Konstantin Y. Guslienko  Departamento Polímeros y Materiales Avanzados:
Física, Química y Tecnología, Universidad del País Vasco, UPV/EHU, 20018 San
Sebastián, Spain  EHU Quantum Center, University of the Basque Country,
UPV/EHU, 48940 Leioa, Spain  IKERBASQUE, the Basque Foundation for Science,
48009 Bilbao, Spain  Farkhad G. Aliev<EMAIL_ADDRESS>Departamento
Física de la Materia Condensada C03, Universidad Autónoma de Madrid, Madrid
28049, Spain  IFIMAC and INC, Universidad Autónoma de Madrid, Madrid 28049,
Spain

###### Abstract

Fast and efficient switching of nanomagnets is one of the main challenges in
the development of future magnetic memories. We numerically investigate the
evolution of the static and dynamic spin wave (SW) magnetization in short
(50-400 nm length and 120 nm diameter) cylindrical ferromagnetic nanowires,
where competing single vortex (SV) and vortex domain wall with a Bloch point
(BP-DW) magnetization configurations could be formed. For a limited nanowire
length range (between 150 and 300 nm) we demonstrate a reversible microwave
field induced (forward) and opposite spin currents (backwards) transitions
between the topologically different SV and BP-DW states. By tuning the
nanowire length, excitation frequency, the microwave pulse duration and the
spin current values we show that the optimum (low power) manipulation of the
BP-DW could be reached by a microwave excitation tuned to the main SW mode and
for nanowire lengths around 230-250 nm, where single vortex domain wall
magnetization reversal via nucleation and propagation of SV-DW takes place. An
analytical model for dynamics of the Bloch point provides an estimation of the
gyrotropic mode frequency close the one obtained via micromagnetic
simulations. A practical implementation of the method on a device has been
proposed involving microwave excitation and the generation of the opposite
spin currents via spin orbit torque. Our findings open a new pathway for the
creation of unforeseen topological magnetic memories.

##  I Introduction

In the last decades, numerous advances have been made in the development of
spin based data storage and information processing devices. Low dimensional
structures with a confined magnetic vortex have been in focus as the promising
candidate in the creation of novel types of quantum magnetic memories. Their
functionality as magnetic memories is based on the control of the change in
their topology, mainly between two distinct magnetization configurations. For
example, the control of the polarity of magnetic vortices created in
nanometric disks (i.e. with functionality primarily restricted to the material
plane Kammerer  _et al._ (2011)) through the excitation of spin waves (SW)
Aliev  _et al._ (2009); Awad  _et al._ (2010); Aliev  _et al._ (2011) has been
demonstrated. A great advantage of using the excitation of SWs compared with
spin transfer torque (STT) Chureemart  _et al._ (2011), spin orbit torque
(SOT) Kohno  _et al._ (2020) or magnetic field Sun  _et al._ (2022) for the
topology control is the possibility of fine control of the topology of each
individual element by tuning the excitation frequency. So far, the most widely
used method to drive locally the DWs has been by injection of the spin
currents Wong  _et al._ (2016); Wieser  _et al._ (2010).

Emergent three-dimensional (3D) magnetic structures Raftrey  _et al._ (2022)
allow a greater variety of textures and magnetic topologies to be achieved,
leading to new effects Fernández-Pacheco  _et al._ (2017); Sobucki  _et al._
(2022). An example of this is the creation of tunnels of chiral ”skyrmions”
and ”bobbers”, more complex 3D magnetic textures in nanoribbons Zheng  _et
al._ (2018). Topological singularities such as Bloch points (BP) Feldtkeller
(1965) have been recently found to play a critical role in 3D magnetic
textures hosting skyrmions Birch  _et al._ (2020) and hopfions Raftrey and
Fischer (2021); Tejo  _et al._ (2021). 3D magnetic textures are of special
interest in magnetic media because employment of 3D magnetic topological
solitons like hopfions, domain walls (DW) and Bloch points (BP) can open a a
route to creating new kinds of magnetic information storage and spintronic
devices based on 3D architectures.

Ferromagnetic (FM) nanowires (NWs) are a simple system to confine and explore
3D magnetism that has attracted considerable attention upon the last decades.
Cylindrical nanowires (having typical radii of 30-60 nm) made of ferromagnetic
materials have been proposed as an alternative for the development of numerous
electronic and spintronic devices. Nanowires have a robust microwave (MW)
absorption Dmytriiev  _et al._ (2013), whose frequencies can be adjusted
within a wide range depending on the choice of FM material, which is ideal for
the production of MW devices

One of the most studied  potential applications of ferromagnetic NWs is the
elaboration of magnetic memories. The possibility of arranging them in
hexagonal arrays of high density Ivanov  _et al._ (2016) makes them excellent
candidates for this type of devices, since they allow reaching high
information storage densities Also, their shape allows the creation of vortex
type flux-closure magnetic textures Ivanov  _et al._ (2013); Andersen  _et
al._ (2020), which reduce the magnetostatic interaction between the NWs.
Specifically, the possibility of creating magnetic memories based on DWs has
been studied Wong  _et al._ (2016); Wieser  _et al._ (2010), being inspired by
the racetrack memory proposal Parkin  _et al._ (2008) based on the storage of
information in the form of DWs displaced by pulsed electrical currents.

Although the static properties of long cylindrical nanowires have been well
studied in the recent years Ivanov  _et al._ (2013); Ruiz-Gómez  _et al._
(2018) with some indications on the possibility of magnetization reversal via
BP singularity inside the vortex DW Fernández-Roldán  _et al._ (2018),
investigation of the statics and specially magnetization dynamics in the short
NWs, where the DW and BP emerge, are absent.

BPs were first studied and named by Feldtkeller Feldtkeller (1965), consisting
in exchange-dominated 3D magnetic textures that hide topological singularities
with the defining property that around them every magnetization orientation is
present only one time. BP-DWs, therefore, are complex 3D textures, composed by
two facing magnetic vortices with head-to-head or tail-to-tail vortex cores
that could normally appear in thick NWs, where the DW magnetization is allowed
to curl around the NW axis, minimizing the magnetostatic energy Mermin (1979).
To the best our knowledge, the BP-DW dynamics have been investigated only in
relatively long (the wire length about 10 times of the diameter) NWs
stabilizing the BP via bamboo-like shapes Berganza  _et al._ (2016); Sáez  _et
al._ (2022).

Figure 1:  Sketch of the cylindrical nanowire and its orientation with respect
to the cartesian coordinate axes (a), showing the direction of the applied
microwave magnetic field sinc pulse to excite the spin eigenmodes of the
system. Vertical cut (xz-plane) of the 250 nm long nanowire showing a SV (b)
and BP-DW magnetization configuration (c) with a vortex-like BP-DW in the
middle. Arrows illustrate the in-plane magnetization and the colors are an
indication of the out-of-plane magnetization. Transverse cut (xy-plane) of the
nanowire in the BP-DW state are shown for the planes $z$ = 3LNW/4 (d) and $z$
= LNW/4, (e) (see dashed lines in (c)). Both vortex domains display opposite
vortex polarities. (f) Three dimensional magnetization configuration
surrounding the BP-DW centered in the middle a 250 nm long nanowire. Colors
represent the magnitude of the axial-aligned magnetization.

In this paper, through micromagnetic simulations and analytic theory, we
carried out a detailed investigation of the BP-DW dynamics in relatively short
(length up to three times of the diameter) cylindrical NWs, where the
transition between the SV and vortex DW states confining the BP takes place.
We also investigate the influence of the pinning on the stability of the BP.
Finally, we suggest a device which uses MWs and SOT to control of the
transition between two different magnetic configurations, aiming at the
creation of a topological magnetic memory based on short cylindrical NWs with
fine frequency tuning and a latency time below the ns, i.e similar to one
observed with combined impact of SOT, STT and the voltage control of the
magnetic anisotropy Grimaldi  _et al._ (2020).

##  II Methods

### Simulation details

Micromagnetic simulations were carried out in the open source software MuMax3
Vansteenkiste  _et al._ (2014). The discretization cell size was set at 1.5 nm
$\times$ 1.5 nm $\times$ 1.25 nm, much below the exchange length, which is
approximately 3 nm for FeCoCu Berganza  _et al._ (2016), the material selected
for our NWs. The cell size is chosen based on the spacial scale of the
magnetic topological textures which can appear in the system. We estimate a DW
and vortex core thickness of about 10 nm. The NW diameter was set at 120 nm
and its length is varied between 50 nm and 800 nm (statics) and 50 nm to 400
nm (dynamics). Previously studied NWs mainly were either much longer (more
than $1\,\mu m$ in length) or were almost disk-shaped (circular dots with the
thickness much less than 100 nm). We used typical magnetic parameters of
ferromagnetic FeCoCu NWs in our simulations Fernández-Roldán  _et al._ (2018):

$M_{s}=2\,T;\quad A_{ex}=25\times 10^{-12}\,J/m;\quad\alpha=0.01\quad,$

where Ms is the saturation magnetization of the NW. and Aex and $\alpha$
describe the exchange stiffness constant and the damping of the NW
respectively. $\alpha$ is reduced to 10-4 in the dynamic simulations to better
resolve the modes of the system. For the purpose of simplicity and future
reference in this study, we take the NWs axis parallel to the $z$ axis in the
cartesian coordinate system (see Fig. 1a), i.e., the bases of the NW are
placed parallel to the $xy$ plane. Due to their cylindrical geometry, the NWs
display a considerable shape anisotropy of magnetostatic origin, which favours
the magnetization to be aligned with the NWs axis direction. No additional
magnetocrystalline anisotropies were introduced in our simulations.

In order to uncover the main SW modes, the averaged in-plane magnetization of
the system has been analyzed by means of a Fourier Transform Park  _et al._
(2003) to frequency domain after the application of a homogeneous magnetic
field pulse in the following form:

$\vec{H}_{\text{pulse}}=H_{\text{pulse}}sinc\left(\frac{2\pi
t}{t_{0}}\right)\hat{u}_{y};$ (1)

where H${}_{\text{pulse}}$ is the amplitude of the applied filed pulse, 2 mT,
and t0 is the full width at half maximum (FWHM) of the pulse, 1$\times$10-12
s. The linearity of the excited SW modes has been checked up to the field
magnitude of 10 mT, well above the used field amplitude. The function used for
the pulse is sinc(x) = sin(x)/x, which has given better results than the
Gaussian function, obtaining narrower and better defined peaks in the spin
wave excitation spectrum given by the Fourier Transform.

Figure 2: (a) Total energy density of both studied magnetic configurations (SV
and BP-DW) for NWs length comprehended between 70 and 800 nm and the gap
($\Delta$E = E(BP-DW) - E(SV)) between them in blue. (b) Energy density of the
SV and BP-DW states for NWs length comprehended between 70 and 800 nm
normalized by the parallel saturated state. (c) Sketch of the two level (BP-DW
and SV) system and transitions that exist in the NWs of the studied length
range (100-400 nm).

##  III Numerical results and discussion

We analyze the simulation results in this section. First, the static magnetic
configurations and related energies of NWs without induced pinning are
investigated as a function of the NW’s length. Then, we investigate the main
spin wave modes (in both SV and BP-DW states) and their evolution with the
length of the NW. Finally, the ways of performing transitions between
configurations with different topologies (i.e., SV vs. BP-DW) are presented.

###  III.1 Static magnetization

Depending on the NW’s length and on the initial static magnetization
configuration (ordered or disordered), the NW relaxation could potentially
result in different stable states. If the final states correspond to a local
minima of the total magnetic energy, it means that a relatively small
excitation could make the system evolve into another neighbouring ground
state. In other words, when a system has topologically different but closely
situated in energy metastable states, a relatively small excitation of the
spin wave eigenmodes (involving small microwave power) may trigger the
transition between those two states. We therefore introduce an analysis of the
possible static magnetization configurations with its related energies.

Figure 3: Evolution of the main SW mode frequencies changing the nanowire
length from 50 nm to 400 nm obtained after relaxing the BP-DW configuration.
The frequency spectra for the NW lengths below 70 nm correspond to the SV
state since the BP-DW state is not stable at such small NW aspect ratios. The
FFT power is normalized at each nanowire length. Snapshots of the eigenmode
spatial distribution are shown for the main azimuthal modes in the SV state
(the NW length is 60 nm) and the main mode at the length 250 nm at the
nanowire top/bottom surface, as well as a longitudinal cross-section of the
mode spatial distribution, focusing on the Bloch point vicinity (area
delimited by the dashed line).

Fig. 1b shows the vortex state formed after relaxing a 250 nm long NW from a
SV configuration, while Fig. 1c exhibits the BP-DW created after relaxing an
initial configuration of two vortices with opposite polarities, one in each
half of the nanowire (the front view of each vortex can be seen in Fig. 1d and
Fig. 1e). Although the resulting BP-DW state is found to be stable (within
reasonable simulation conditions of torque minimization Vansteenkiste  _et
al._ (2014)) it is well known that in practice, metastable states (such as for
example the double vortex state, see Ref. Aliev  _et al._ (2011)) are
stabilised by the unavoidable natural pinning. We analyze the effect of the
induced pinning on the stabilization of the BP-DW state in the Appendix D.

Our paper mainly concentrates on the intermediate length NWs (100-400 nm in
length and with aspect ratio approximately between 1 and 3) whose static and
dynamic response remained uncovered in the literature. Typical cylindrical
nanowires or dots investigated before were either much longer ($\geq$ 1$\mu$m
) and multiple DWs Ivanov  _et al._ (2016); Andersen  _et al._ (2020) were
created along their length when relaxed, or they remained shorter (length
$\simeq$ radius), approaching a disk geometry where only the singular vortex
state can be stabilized Aliev  _et al._ (2009); Awad  _et al._ (2010). Fig. 2a
shows both magnetic configuration energy densities for NW lengths between 70
nm and 800 nm. Below 70 nm, the BP-DW state is not as stable, as it repeatedly
falls into the SV state due to the magnetostatic energy of the NW, getting
closer to resembling more the frame of a disk under those conditions.

Each configuration energy has been found to be much lower in comparison to the
magnetic energies of a saturated nanowire with a saturated magnetization
parallel to the NW’s axis (see Figure 2b, reinforcing the idea of a two
energy-proximate state system which could reveal transitions from one to the
other state with a given excitation.

Figure 2a shows the difference between the energies of one (SV) and another
(BP-DW) inhomogeneous magnetization states of interest ($\Delta$E = E(BP-DW) -
E(SV)). The energy difference between these states is approximately equal to
5$\%$ of the total energy value. Therefore, these NW lengths seem to be
adequate to make transitions between the single vortex and BP-DW states
induced by external stimuli, since both states have similar energies (see
Figure 2c). This difference initially decreases with length, favoring the
formation of DWs for longer NWs, and single vortex states for short ones.
However, it starts increasing for NWs longer than 450-500 nm, revealing a
minima in the range from 250 nm to 450 nm. Therefore, we can assume that in
this length range the most efficient SV-BP transitions may occur. For the
longest NWs, we have observed that the formation of the two BP-DWs states
inside the NW reduces energy with respect to a single BP-DW state, explaining
the origin of the relative increase of the energy of the BP-DW state for the
longest NWs.

###  III.2 Spin wave modes

In this part we analyze numerically the main spin wave modes in SV and BP-DW
states and study how the DW (and related BP) could be displaced under action
of a MW driving field. In this study we take advantage of metastability of the
BP-DW state in short NWs to explore the possibility to eliminate the BP-DW
completely out of the NW via precessional magnetization motion by using a MW
excitation tuned to one of the most dominant spin wave mode (MM) frequencies.

As we shall see below, our method to manipulate the BP using MWs has arguments
or considerations for and against in contrast of the standard way to apply
magnetic field. The main advantage of the method is the possibility to address
the BP destruction individually by tuning the external frequency, which could
be partially compensated by the need to apply opposite spin currents to
restore the BP. Further below we suggest a practical implementation of the
suggested method.

Figure 3 shows the spin wave modes of a cylindrical NW for an initial BP-DW
configuration for the NW lengths ranging from 50 to 400 nm. Even though the
initial configuration is the BP-DW state, if the NW is short enough however,
this will not be a stable configuration and the system will relax into the SV
state due to the magnetostatic energy contribution. This event is a direct
consequence of the energy gap enhancement between the two states at shorter
NWs lengths, which makes the system no longer metastable for lengths below 70
nm.This phenomenon is also translated to the dynamics of the system shown in
Figure 3: the detected modes for the NW length below 70 nm are different in
nature from the ones above 70 nm. For the NWs that relax into a final BP-DW
state, the frequency of the disclosed modes is progressively reduced with
increasing the length of the NW. The SW modes are often being unfolded into
new modes as the nanowire length increases, providing more complicated
dynamics above about 260 nm in contra-position to the simplicity of the
dynamics in the range of the NW length from 100 to 250 nm.

Figure 4: Spin wave mode spectra of a 250 nm length NW in the BP-DW
magnetization configuration originated from each of the four different methods
explained in the text. Inset of (a) shows the normalized amplitude of the 11.9
GHz mode over the length of the NW.

Figure 4 shows the calculated spin wave modes in a BP-DW configuration of a
250 nm long NW after the BP-DW state was reached using different protocols.
What are in our view the two foremost methods (the ideal case and the method
of the spin currents) to achieve transitions between the two investigated
states will be described in detail in Section III.3. For the opposite fields
and non-magnetic pinning methods detailed explanations are in Appendices A and
B and Appendix D respectively. Most of the main SW modes are presented in the
majority of the reconstructed spectra. The mode located at 11.9 GHz stands out
as the most prominent when the BP-DW results from the ideal case. We have
found that the difference between the spectra is fundamentally based on the
position of the DW, which only in the ideal case (created by hand) turns out
to coincide exactly with the center of the nanowire. Furthermore, the clear
fronting and tailing of the modes observed in the spectras presented in Figure
4 (b, d) is a direct consequence of the asymmetric position of the BP along
the NW.

We have found that BP-DW states could be effectively transformed into SV state
by MW excitation tuned to the dominant SW mode (denominated here as BP-SV
transition and analysed in details separately). Reverse (SV-BP) transition
requires however more detailed investigation as energy of BP-DW state exceeds
one of SV configuration.

###  III.3 Creation of the BP-DW state (SV-BP transition)

We turn now to investigate in details the SV-BP transition using as an example
the 250 nm long NW, were the difference of the total magnetic energies between
single vortex and DW configurations is close to minimum (see Figure 2a).
Achieving control of the SV-BP transition could the first step in getting a
full control over the 3D magnetization textures and their topology in short
ferromagnetic nanowires.

####  III.3.1 BP-DW created by hand

It was detailed above how a BP-DW could be generated in the middle plane of
the nanowire with magnetization imposed as two perfect vortex states with
opposite polarities followed by a relaxation. This is achievable with the
precise control of the magnetization that the simulation code grants but
challenging to carry out in practice. Due to the high symmetry that this
system presents, with the BP-DW perfectly centered in the middle plane of the
NW, we shall further refer it as the ”ideal” case. Figure 4a shows the
corresponding frequency spectrum of the spin wave modes in the ideal case,
presented with a dominant mode at 11.9 GHz and followed by a 16.3 GHz high-
intensity mode. The 16.3 GHz mode seems to be relevant only in the ideal case
and it has a much smaller amplitude or even does not appear at the rest of the
(less symmetric) cases considered. A detailed analysis of the modes over the
NW length reveals that these modes mainly excite the vortices itself, not the
BP directly, which is what triggers the BP-DW motion (see inset of Figure 4a
showing the analysis of the 11.9 GHz mode). See Ref.Vid  videos 1 and 2
showing the 11.9 and the 16.3 GHz modes excited in the ideal case.

####  III.3.2 BP-DW created by spin current injection

Having verified different ways to create the BP-DW configuration from the
initial SV state, we shall further concentrate on, in our view, the most
practical method which involves driving opposite spin currents into a
ferromagnetic nanowire in order to impose the formation of the BP-DW state. In
this section we show and describe the means to create DWs in 180-300 nm
nanowires using spin currents.

Figure 5: Cross-section sequence of the mz magnetization component of the 250
nm long NW, showing the used method of spin polarized currents to reconstruct
the BP-DW state. In (a) a current of 9 $\times$1012 A/m2 is applied on each
half of the NW, with opposite directions, to the initial SV state
configuration. After 4 ns the steady BP state is reached (b). Switching off
the currents and letting the system relax returns it into the DW state (c).
(d) Average axial-aligned magnetization component evolution in a 250 nm length
NW with the spin polarized current density of 9 $\times$1012 A/m2. The three
dynamic regimes are presented in the transition. Inset images represent a
snapshot of the longitudinal cross-section of the axial $M_{z}$ component in
the NW in each one of the regimes.

The application of spin polarized currents with a polarization degree P = 1
with opposite directions from each of the ends of the nanowire towards its
interior, with an initial magnetic configuration of a single vortex (see
Figure 5a), has been studied in details. The following procedure has been
implemented in order to evaluate the minimum spin current density needed in
order to successfully achieve a SV-BP transition: we leave spin currents
simulations for long enough time (7 ns) to ensure the evolution of the system
if the injected currents are strong enough. Then, the volume averaged mz
component of the magnetization in the NW is evaluated over time, as it is an
indicative of any transitions involving the polarity of the vortex core. In a
typical spin current injection that promotes the system to transform from SV
to BP-DW state (see Figure 5d) there are three clear regimes: first, an
oscillation of the SV magnetization induced by the injection of the spin
currents. When the oscillations become large enough, the SV state is no longer
manifested. Two things can happen once this point is reached: if the currents
are too strong, then the system does not evolve into a steady state but rather
becomes uncontrollable, this is characterized by large and sharp ups and downs
in the average mz magnetization. However, if the spin currents strength is
precisely tuned, then there is an extra step in between that displays a steady
BP-DW state, before turning again into an uncontrollable system (see Figure
5d). This state expresses itself as a durable, almost constant, close to zero
average out-of-plane magnetization. See Ref.Vid  video 3 showing the spin
current injection to a 250 nm NW to see the full procedure.

Figure 6:  (a) Critical spin current density (Jc) needed to create a steady BP
state and time of destruction of the BP-DW with the main mode at 0.1 mT
plotted as a function of the NW length. Inset of panel (a) depicts the
probability to detect a 1 V-DW propagation reversal mode against the NW
length.(b) Time required for the MW excitation to start to displace the DW, in
logarithmic scale, plotted against the excitation field amplitude, for
different lengths of the NW. For the NW length of 250 nm, the main mode (MM)
and gyrotropic mode (GM) have been studied. Inset corresponds to a zoom into
the main mode excitations for the NW length ranging from 150 to 250 nm. The
point at 300 nm in the time of destruction in (a) is a lower end estimation
following its trend on (b).

The protocol shown in Figure 5(a, b, c) for obtaining BP-DW by using spin
currents is as follows: first, we apply opposite spin currents from each NW’s
end towards its interior, then the currents are removed once the steady BP-DW
state is reached (see Figure 5b). By letting the nanowire relax from the
steady BP-DW state, the desired BP-DW state could be obtained. Figure 5b shows
the magnetization of the nanowire after the application of spin currents of 9
$\times$ 1012 A/m2 for 4 ns, from which state the NW is relaxed to a BP-DW
configuration.

Using this protocol and the appropriate current intensities, it is indeed
possible to reconstruct the desired BP-DW state. For all studied NW lengths
the spin currents needed to generate the BP-DW were within the range $9\times
10^{12}$ A/m2 and $14\times 10^{12}~{}\text{A/m}^{2}$ (see Figure 6a).
Interestingly, close to L=250 nm a minimum in the critical spin current
density $J_{c}$ required to obtain BP-DW state emerges.

Our numerical experiments revealed that the efficiency of the spin current
induced reversal SV-BP transition substantially increases in the vicinity of
NW length $L$ = 250 nm (see drop of the minimum spin current needed shown in
Figure 6a). In order to figure out the possible origin on this unexpected
behaviour, we have investigated the variation of the magnetization reversal
modes as a function of the NW length.

Previous studies of the phase diagram of the magnetization reversal modes in
cylindrical NWs showed that the reversal modes are indeed strongly dependent
on the NW size Proenca  _et al._ (2021). Their aspect ratio determines the
type of DW formed (transverse, T-DWs, or vortex DW, V-DWs) and the number of
nucleated DWs Proenca  _et al._ (2021). Although the saturation magnetization
used in our work is higher than one used in the work by Proenca et al. Proenca
_et al._ (2021), the expected limits of the transverse to vortex DWs mediated
reversal modes transition should, in principle, appear for lower nanowire
diameters than ours. Indeed, for NWs with a diameter of 120 nm investigated in
this work, our simulations confirm that all magnetization reversal modes are
V-DWs. However, the reversal modes are quite sensitive to the length of the
NWs, providing variations on the presence of the different observed reversal
modes even for the limited range of the studied NW lengths (see inset to
Figure 6a).

In particular, for the length range between 220 nm and 280 nm, we observed two
possible reversal modes in our simulations, related to a spiral rotation of
two V-DWs (2 V-DW) or, in other cases, the propagation of one V-DW (1 V-DW).
Interestingly, the maximum in the probability of the 1 V-DW propagation
magnetization reversal coincides with the maximum efficiency in the spin
current injection (located at a nanowire length 250 nm, see Fig. 6a). We
attribute this effect to a generally higher vortex symmetry in this NW length
range which matches to the highly symmetric spin current injection implemented
to accomplish the backward SV-BP transition. Details of the estimation of the
probability of the V-DW mediated magnetization reversal (depicted in the inset
of Figure 6a) are provided in Appendix C.

###  III.4 Optimization of the control over magnetization topology

In the previous sections SW modes in nanowires for the BP-DW configurations
have been disclosed (Figures 3, 4). One could expect that these configurations
under a MW excitation at a frequency close to one of the main eigenfrequencies
of the nanowire would ultimately result in a transition between the BP-DW (a
metastable state) and the SV state (a stable state). To verify this
hypothesis, a sinusoidal excitation magnetic field has been applied to each
magnetization configuration as follows:
$\vec{H}_{\text{excitation}}=H_{\text{MW}}\sin(2\pi ft)\hat{u}_{y}$. Where
H${}_{\text{MW}}$ is the excitation amplitude and $f$ is the frequency of the
SW mode used to excite the NW. The MW field applied throughout the whole NW
has been directed perpendicularly to the NW axis, similarly to the sinc pulse
used before to disclose the SW eigenmodes of the system (see Figure 1a).

The result of applying the MW field is as follows: after a certain time (which
depended on the intensity of the excitation and the used SW mode frequency),
the BP-DW moves towards one of the two NW’s ends and the SV state is created
in the NW (see Ref.Vid  video 4 showing the BP-DW state destruction with the
application of a MW excitation field with the frequency of the main SW mode).
Figure 6b shows dependence of the time of application of the MW microwave
excitation required to remove BP-DW as a function of the excitation amplitude
for different NW lengths. The 11.9 GHz mode has been chosen for this study as
the main SW mode (MM), since it is notably present in all SW excitation
spectra shown in Figure 4. The time of destruction becomes effectively larger
if the excited SW mode is not the main in amplitude. For example, if the 250
nm BP-DW nanowire is excited with a frequency close to its gyrotropic mode
frequency (GM, located at 0.69 GHz for 250 nm NW length), which is directly
related to the BP position oscillations and should induce a more localized
excitation of this magnetization texture, then the time of BP-DW destruction
becomes much larger (see Figure 6b), even though, over time, the transition is
accomplished. Latency times below the ns are accomplished in our system by
exciting the most predominant SW mode (11.9 GHz) with MW fields exceeding 5 mT
(see Appendix E).

Figure 6b shows that the BP-DW to SV state transition takes longer to occur as
the amplitude of excitation decreases. For MW excitation amplitudes smaller
than those shown, the displacement of the DW was not observed for times below
10 ns. It can be seen that the transition occurs for relatively short times
even with small MW field amplitudes ($\approx\ $ 0.2 mT ). Figure 6b also
shows a big leap in the efficiency of the BP-DW destruction changing the NW
length from 250 nm to 300 nm. This effect may be attributed to a transition
from a regime in which several SW modes of similar amplitude are involved in
the dynamics of the system, 300 nm long NWs, as opposed to a regime in which
the dynamics relay on one dominant main mode, 250 nm and shorter NWs (see
Figure 3). This recipe, therefore, presents a way of moving the domain walls
in a nanowire with relatively small energy cost. The BP velocities in the
destruction process reach $\approx$ 400 m/s, demonstrating such high
velocities even with a small applied field.

Figure 6a compares the time required to remove the BP-DW using a MW excitation
of 0.1 mT of the main SW mode and the minimum spin current required to restore
BP-DW in the NW length range, where such reversible transitions were found to
be possible to perform. The energy required for the BP-DW destruction-and-
restoration cycle can be obtained from these two graphs. Simulations suggest
that the needed time for the BP-DW destruction notably increases in the NW
length range 250-300 nm due to topological differences arising from the fact
that a single BP-DW is more stable at lengths in this mentioned range (see
Figure 6b). However, for the NW lengths from 150 to 250 nm the time of
destruction is almost maintained. The reconstruction with the spin currents is
most effective at the NW length of 250 nm, for which there is a minimum of the
spin current density $J_{c}$ necessary to reconstruct the BP-DW. We conclude
that the optimal length for the SV to BP-DW cycle efficiency is close to 250
nm.

Figure 7: Time of destruction of the BP-DW state with an MW excitation field
with the amplitude of 1 mT and the frequency of GM and corresponding amplitude
of the GM for different stopping times after the application of a 9 $\times$
1012 A/m2 spin current density in a 250 nm nanowire plotted against the
average out-of-plane magnetization component. The point at almost 40 ns is a
lower estimation of the destruction time.

###  III.5 Optimization of the control over the magnetization topology using
the gyrotropic mode

Up until this point, the destruction of the BP-DW has been performed
administrating a small MW excitation with the frequency of the detected main
SW mode of 11.9 GHz. The magnitude of this mode in comparison with the others
allows the destruction to be efficient both in time and in required amplitude
of the applied MW field (see Figure 6b). However, the mode spectra generated
after the reconstruction of the BP-DW via spin currents displays a
redistribution of the modes amplitudes. Once the BP-DW state is created by
injection of opposite spin currents, the gyrotropic mode becomes also the main
eigenmode of the system (see Figure 4c), meaning that it might be now more
efficient to destroy the new BP-DW state by using the gyrotropic mode
frequency as the MW excitation frequency.

Interestingly, the obtained results indicate that the effectiveness of using
gyrotropic mode to realize the BP-DW to SV state transition strongly depends
on the BP-DW symmetry with respect to center of the nanowire. As mentioned
above, to obtain the BP-DW, the opposite spin current injection should be
stopped in the steady BP-DW regime shown in Figure 5d. However, even in this
regime, the BP-DW does not completely resemble the configuration obtained in
the ideal case, but it is rather dependent on the exact moment of relaxation
to result on a more or less symmetric outcome. As Figure 7 shows, the time of
destruction of the spin current reinstated BP-DW state decays as the system
becomes more asymmetric. Simultaneously, the amplitude of the GM excited to
accomplish the BP-DW to SV state transition increases with the asymmetry of
the BP-DW position with respect to the center of the nanowire.

##  IV Analytical model for dynamics of the Bloch point

In order to provide analytical estimation for the lowest (gyrotropic) spin
wave frequency in a nanowire we consider a BP-DW in a cylindrical nanowire
with a length $L$ and radius $R$. We use the cylindrical coordinate system
($\rho$, $\phi$, z) with the O$z$ axis coinciding with the nanowire axis. We
assume that the domain wall is a vortex type and the $\rho$-magnetization
component is equal to zero, $m_{\rho}$ = 0. The $m_{z}$ magnetization
component we choose in the azimuthally symmetric trial form suggested by
Thiaville et al. Thiaville  _et al._ (2003)

$m_{z}(\rho,z)=\frac{z}{\sqrt{\rho^{2}+z^{2}}}\frac{b^{2}}{b^{2}+\rho^{2}}$
(2)

where $b$ is the BP size (order of 10 nm). To calculate the BP excitation
gyrotropic frequency, we use Landau-Lifshitz-Gilbert (LLG) equation of motion,
which describes the evolution of magnetization M(r,t) in a ferromagnet in the
presence of an effective magnetic field Heff and damping. It is written in the
following form:

$\frac{d\textbf{M}}{dt}=-\gamma\textbf{M}\times\textbf{H}_{eff}+\frac{\alpha}{M_{s}}\textbf{M}\times\frac{d\textbf{M}}{dt},$
(3)

where $\gamma$ is the gyromagnetic ratio, $\alpha$ is the dimensionless
damping constant and $M_{s}$ is the saturation magnetization. The effective
field Heff = -$\delta$E/$\delta$M is a combination of the external magnetic
field, exchange field and the demagnetizing field, where E is the total
magnetic energy density. It was shown by Thiele Thiele (1973) that for
description of the dynamics of magnetic domains the LLG equation can be
rewritten in another form that allows simplifying the calculations. The
Thiele’s approach is applicable to describe dynamics of stable magnetization
configurations (magnetic solitons) that can be characterized by a position of
its center X(t) (a collective coordinate) that can vary with time. To consider
BP gyrotropic dynamics we assume that magnetization as a function of the
coordinates r = ($\rho$, z) and time can be written in the form M(r,t) = M(r,
X(t)). The three-dimensional vector X(t) represents oscillations of the Bloch
point center position. Then, neglecting damping, we can rewrite the LLG
equation as:

$G_{\alpha\beta}\frac{dX_{\beta}}{dt}=-\frac{\delta W}{\delta X_{\alpha}},$
(4)

where W is the total magnetic energy, $\alpha$, $\beta$ = $x,y,z$. Introducing
the unit vector of magnetization m=M/Ms, the components of the global (volume
averaged) gyrotensor G, can be defined as follows:

$G_{\alpha\beta}=\frac{M_{s}}{\gamma}\int
d^{3}\textbf{r}\left(\frac{\partial\textbf{m}}{\partial
X_{\alpha}}\times\frac{\partial\textbf{m}}{\partial
X_{\beta}}\right)\textbf{m}$ (5)

The dependence M(r,X(t)) is unknown, and, therefore, we make the simplest
assumption M(r,X(t)) = M(r \- X(t)), that corresponds to the BP rigid motion
around the equilibrium position X = 0. We assume that metastable equilibrium
BP position is located on the nanowire axis $Oz$ at the point $z=0$. The
corresponding vortex type DW containing this BP is located in the $z=0$ plane
separating the nanowire in two symmetrical domains with magnetization directed
upwards ($z>0$) and downwards ($z<0$). The global gyrovector given by Eq. (5)
can be calculated at the BP position X = 0 using the standard approach as

$G_{\alpha\beta}=\frac{M_{s}}{\gamma}\int d^{3}\textbf{r}F_{\alpha\beta}^{e},$
(6)

where F${}_{\alpha\beta}^{e}$ is the emergent electromagnetic field tensor
F${}_{\alpha\beta}^{e}$ = $\left(\frac{\partial\textbf{m}}{\partial
X_{\alpha}}\times\frac{\partial\textbf{m}}{\partial
X_{\beta}}\right)\textbf{m}$ Guslienko (2016).

The tensor Gαβ is antisymmetric and, therefore, it can be represented via a
dual vector G, known as gyrovector. Substituting the magnetization
distribution (2) to Eq. (6) one can calculate the global gyrovector components
in the cylindrical coordinates as G = ($G_{\rho}$, 0, 0). The $\rho$-component
for the case $L>>R$ is

$G_{\rho}=-4\pi b\frac{M_{s}}{\gamma}atan\left(\frac{R}{b}\right).$ (7)

The global gyrovector is finite and is a function of the BP radius $b$. The
equilibrium value of $b$ can be found from minimization of the total magnetic
energy of centered BP consisting of the exchange and magnetostatic energies.
We note that that the nullification of the global gyrovector is a property of
infinite systems with specific asymptotic behavior of the magnetization at the
infinity (m=const). Such magnetization textures are localized 3D solitons,
toroidal hopfions, for instance Tejo  _et al._ (2021). The Bloch point is a
non-localized 3D magnetic topological soliton and may reveal a non-zero global
gyrovector in restricted cylindrical NW geometry given by Eq. (7). The finite
global gyrovector allows to calculate the BP gyrotropic frequency. The non-
zero transverse ($G_{\rho}$) gyrovector component leads to expelling of the BP
to the NW circumference at any value of the BP-DW velocity directed along the
NW length due to the gyroforce. The SV state magnetization does not depend on
the thickness coordinate $z$ except some small area near the nanowire faces.
Therefore, SV state is essentially two-dimensional and described by 2D
magnetic topological charge, which is proportional to the $z$ component of the
vortex gyrovector, $G_{z}=-2\pi L\frac{M_{s}}{\gamma}$. The global gyrovector
of the BP-DW is a result of more complicated 3D magnetization distribution
given by Eq. (2). The ratio of the BP and SV gyrovector absolute values,
$\frac{G_{\rho}}{G_{z}}=\frac{\pi b}{L}$, is small for the nanowire lengths
100 - 300 nm investigated in the present article.

Accounting for the azimuthal symmetry of the system the moving BP energy for
small BP position oscillations near the equilibrium in the NW center can be
written as:

$W(\textbf{(}X))=W(0)+\frac{1}{2}\kappa(X^{2}+Y^{2})+\frac{1}{2}\kappa_{z}Z^{2}$
(8)

Solving the Thiele equation of motion (4) for the BP center position X(t) one
can find the gyrotropic oscillation frequency

$\omega_{G}=\frac{\sqrt{\kappa\kappa_{z}}}{G_{\rho}}.$ (9)

The stiffness coefficients $\kappa$, $\kappa_{z}$ were calculated numerically,
see Table I. The equilibrium value of $b$ is 7.2 nm for the given nanowire
parameters. Using the data from Table I and Eq. (9) one can calculate the BP
gyrotropic frequency to be $\omega_{G}$/2$\pi$ = 0.45 GHz without pinning and
$\omega_{G}$/2$\pi$ = 2.20 GHz with pinning.

| Without pinning | With pinning
---|---|---
Transverse disp. | 1.2336$\times$10-22 | 2.67208$\times$10-22
Axial disp. | 8.08524$\times$10-20 | 8.76542$\times$10-19
Table 1: Second derivative of the BP energy (in units of J/nm2) with respect
to the BP’s displacements under applied field in transverse and axial
directions in the nanowire. The simulated nanowire has 120 nm of diameter and
250 nm of length. Used magnetic parameters are: $\mu_{0}M_{s}$ (saturation
magnetization) = 2 T, $A_{ex}$ (exchange stiffness constant) = 25 pJ/m and
$\alpha$ (damping) = 0.01. Pinning is made by increasing two times the
saturation magnetization in the middle of the NW where BP is located.  Figure
8: Schematic of the suggested device to implement the described transition
mechanism. (a) Electric current is simultaneously applied through two Pt
electrodes contacting the opposite NW ends in the SV state, triggering two
opposite spin polarized currents through NW by SOT, effectively switching the
NW to the BP-DW state. (b) MW excitation is applied perpendicularly to the NW
in the BP-DW state to trigger the transition into the SV state.

##  V Device implementation

A suitable device to implement the proposed reversible BP-DW manipulations
could include two heavy metal (Pt) electrodes located on the ends of the
FeCoCu NW. These electrodes would be able to generate the spin polarized
current via SOT once electric current through Pt is applied, effectively
restoring the BP-DW state (see Figure 8a). We note that the SOT is known to
provide much higher energy efficiency and lower latency times than STT Saha
_et al._ (2022). Additionally, the device would avoid possible heating in the
NW since the spin current and charge current are independent separated events,
which would not happen with STT.

For the BP-SV transition (see Figure 8b), external frequency-tuned MW is
needed, one way to achieve this is placing the sample on a coplanar waveguide
(CPW) and apply magnetic pulses. Another way is to take advantage of the two
Pt electrodes and use them as a waveguide. In the latter case the MW
excitation will have a predominant axial direction. This is however is not an
issue since we have checked that the BP-SV transition can indeed be also
achieved using an axial excitation and with similar latency times as in the
case of a perpendicular MW excitation.

##  VI Conclusions

In summary, we have investigated numerically and by analytical theory static
and dynamic (spin wave modes) properties in short cylindrical magnetic
nanowires, where the ground (single vortex) and metastable (vortex domain wall
with a Bloch point) states have similar energies. We have demonstrated
effective reverse control over the 3D magnetization texture topology switched
between these two states by using a low power microwave short microwave
excitation. To destroy the BP-DW state the MW field frequency was chosen to be
close to one of the dominant SW mode frequencies. The opposite spin polarized
currents were used to reestablish the BP-DW magnetization configuration.
Finally, an implementation of the mechanism has been proposed on a practical
device involving the generation of the opposite spin currents via SOT. Our
results path a way towards the creation of a novel type of energy efficient
magnetic memories based on short ferromagnetic cylindrical nanowires.

##  VII acknowledgments

The work in Madrid was supported by Spanish Ministry of Science and Innovation
(RTI2018-095303-B-C55, PID2021-124585NB-C32, TED2021-130196B-C22) and
Consejería de Educación e Investigación de la Comunidad de Madrid
(NANOMAGCOST-CM Ref. P2018/NMT-4321) grants. F.G.A. acknowledges financial
support from the Spanish Ministry of Science and Innovation, through the María
de Maeztu Programme for Units of Excellence in R&D (CEX2018-000805-M) and
”Acción financiada por la Comunidad de Madrid en el marco del convenio
plurianual con la Universidad Autónoma de Madrid en Linea 3: Excelencia para
el Profesorado Universitario”. J.C. thanks Spanish MECD for the fellowship.
K.G. acknowledges support by IKERBASQUE (the Basque Foundation for Science).
K.G. work was supported by the Spanish Ministry of Science and Innovation
under grant PID2019-108075RB-C33/AEI/1013039/501100011033 and by the Norwegian
Financial Mechanism 2014–2021 trough project UMO-2020/37/K/ST3/02450.

## Appendix

##  Appendix A BP-DW created by opposite magnetic fields

The first proposed method that could be experimentally achievable consists of
applying two homogeneous magnetic fields in the direction of the NW axis to
the vortex state, with opposite directions in each of the two halves (further
called opposite magnetic fields method). If the nanowire has its main axis in
the direction of the Z axis and its center is at the origin of coordinates, we
would apply the following magnetic field:

$\vec{H}=\left\\{\begin{matrix}H\hat{u}_{z}\quad\,\,\,\,z<0\\\
-H\hat{u}_{z}\quad z>0\end{matrix}\right.$ (10)

The protocol to follow with this method is as follows: (i) apply the magnetic
field $\vec{H}$ to the nanowire (ii) relax the nanowire with the applied field
(iii) remove the field and relax the nanowire.

Figure 9(a, b, c) shows the evolution of the mz magnetization component in the
nanowire throughout these steps, that ultimately lead to the creation of a DW
state from an initial SV state. This method however has two drawbacks. The
first one is the intensity of the applied magnetic field H0, which must be
sufficient to be able to reverse the magnetization in one of the halves of NW
while maintaining that of the other half. In this case, a limit value for the
field amplitude of 0.6 T has been found, below which the final state remained
the single vortex. The second drawback resides in the difficulty of creating
strong magnetic fields with such a specific spatial asymmetry on a small
scale.

Figure 9:  Cross-section sequence of the mz magnetization component of the 250
nm long NW, showing the used method of opposite magnetic fields to reconstruct
the BP-DW state. (a) Displays the initial SV state before the opposite
magnetic fields applied. After two 0.6 T opposite fields, the mz magnetization
component is almost completely saturated on each half of the NW (b). After
relaxing the NW the BP-DW is reached (c).

##  Appendix B BP-DW created by pinning

In order to reduce the amplitude of the magnetic field necessary to create the
BP-DW in the NW the introduction of BP-DW pinning has been investigated. This
method dwells in the addition of a defect in the system trying to localize the
BP-DW on it. In this case, a layer of non-magnetic material has been
introduced in the middle plane of the nanowire (see modes of the system in
Figure 4d). Here, the same procedure has been followed as in the opposite
magnetic fields method with the exception that a non-magnetic layer with Ms =
0 and thickness of 2.5 nm has been introduced in the center of the 250 nm long
nanowire. We found similar results as in the opposite magnetic fields method
without pinning (see Appendix A), however, the value of the critical magnetic
field needed for the DW to be generated was reduced from 0.6 T to 0.45 T.

Although the pinning of BP-DW we explored approaches us to the appropriate
goal, it still presents drawbacks similar to the opposite magnetic fields
method. One could try to reduce the critical field to generate the BP-DW even
more by adding a thicker pinning layer, but high accuracy in the symmetry of
the applied opposite fields would still be needed. Besides, as we describe in
Appendix D, extra pinning could complicate the goal of achieving low energy
manipulation of the magnetic SV-BP topologies. In the next section we describe
what we believe is a most practical way to realize the SV-BP transition, that
is by injection of opposite spin currents.

Figure 10: Hysteresis loops for a 250 nm long NW in the case of each of the
two presented reversal modes. Insets show the cross-section of the NW for two
different types of detected magnetization reversal modes at H=0: spiral
rotation of two V-DWs (2 V-DW) and propagation of one V-DW (1 V-DW). Figure
11: (a) Energy gap between the BP-DW and the SV state for the different
pinning introduced in the NW. Average spin wave modes spectra of the different
pinned systems are depicted in (b, c, d, e).

##  Appendix C Estimation of the reversal modes and their probability
calculated from the obtained hysteresis loops

To obtain the data presented in the inset of Figure 6a in the main text,
several hysteresis loops were performed saturating up to 1T of axial field
(see Figure 10). At H=0, when the field is being reduced, the static magnetic
configuration reveals the reversal modes of the system. This process was
executed for the NW lengths within the range of 220-280 nm depicted in the
inset of Figure 6a.

However, the type of reversal mode in this range was found to be uncertain,
i.e., the NW hysteresis loop in axial magnetic field is characterized by a
particular reversal mode (either the spiral rotation of two V-DWs or the
propagation of one V-DW) each time a hysteresis loop was performed for a
specific length. Meaning that different reversal modes can be presented for
the same system in separated hysteresis runs. This is due to the fact that the
two observed reversal modes are close in energy for most of the studied NW
lengths. In order to describe the probabilistic character of the reversal mode
topology, repeated simulations of ten hysteresis loops were performed for each
NW length. The inset of Figure 6a represents the probability to obtain the 1
V-DW propagation reversal mode vs the length of the NW.

##  Appendix D Influence of pinning on the control over the magnetization
topology

The investigated BP-DW is formed as a metastable state above SV ground state
(Fig. 2). This facilitates the switching from the BP-DW to SV state under
rather small fine tuned microwave field as well as the backward switching by
using reasonable spin current densities (Fig.6). We have investigated the
efficiency of such switching not only by varying the NW length (see Fig. 6)
but also through the implementation of different strategies for pinning of the
BP-DW for the specific NW length of 250 nm for which switching of the magnetic
texture topology was found to be less energetically costly. Specifically, we
introduced BP-DW pinning in such NWs by using a modulated bamboo-like geometry
Berganza  _et al._ (2016); Sáez  _et al._ (2022) (geometric pinning), enhanced
local saturation magnetization $M_{s}$ (magnetic pinning) or the
implementation of a non-magnetic layer (non-magnetic pinning). We found that
the energy gap between the two metastable states is reduced if the BP-DW
position pinning is introduced (see Fig. 11a), as a consequence of an enhanced
stability of the BP-DW state under the different pinning conditions. Our
simulations reveal that the BP-DW stabilization is better achieved through a
magnetic pinning, rather than geometrically modulating the middle of the NW,
even though the geometrical modulation is introduced in 25 nm NW segment and
the magnetic pinning is introduced only in a 2.5 nm segment.

Pinning of the BP position is an additional obstacle to perform the state
switching via a microwave field. Thus, MW excitation of the main SW modes in
the studied amplitude range (up to 2 mT) could only destroy the BP-DW state in
the case of the bamboo-like pinning, with the times of transition being only
slightly above those of an ideal case 250 nm length NW (see Fig. 6b). For any
of other implemented pinning methods (see Fig. 11a), the transition from BP-DW
to SV state is forbidden.

##  Appendix E Latency times in the high MW excitation limit

Figure 12: Time of destruction of the BP-DW state (latency time) in a 250 nm
NW under a frequency-tuned to the main spin wave mode observed (11.9 GHz) MW
excitation against the amplitude of the driving field.

Latency times below the nanosecond are achieved in the BP-DW to SV state
transition for high MW driving fields (see Figure 12).

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