# Stable and manipulable Bloch point

Marijan Beg<EMAIL_ADDRESS>European XFEL GmbH, Holzkoppel 4, 22869
Schenefeld, Germany  Faculty of Engineering and Physical Sciences, University
of Southampton, Southampton SO17 1BJ, United Kingdom  Ryan A. Pepper  David
Cortés-Ortuño  Bilal Atie  Marc-Antonio Bisotti  Gary Downing  Faculty of
Engineering and Physical Sciences, University of Southampton, Southampton SO17
1BJ, United Kingdom  Thomas Kluyver  European XFEL GmbH, Holzkoppel 4, 22869
Schenefeld, Germany  Ondrej Hovorka  Faculty of Engineering and Physical
Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom
Hans Fangohr<EMAIL_ADDRESS>European XFEL GmbH, Holzkoppel 4, 22869
Schenefeld, Germany  Faculty of Engineering and Physical Sciences, University
of Southampton, Southampton SO17 1BJ, United Kingdom

###### Abstract

The prediction of magnetic skyrmions being used to change the way we store and
process data has led to materials with Dzyaloshinskii-Moriya interaction
coming into the focus of intensive research. So far, studies have looked
mostly at magnetic systems composed of materials with single chirality. In a
search for potential future spintronic devices, combination of materials with
different chirality into a single system may represent an important new avenue
for research. Using finite element micromagnetic simulations, we study an FeGe
disk with two layers of different chirality. We show that for particular
thicknesses of layers, a stable Bloch point emerges at the interface between
two layers. In addition, we demonstrate that the system undergoes hysteretic
behaviour and that two different types of Bloch point exist. These ‘head-to-
head’ and ‘tail-to-tail’ Bloch point configurations can, with the application
of an external magnetic field, be switched between. Finally, by investigating
the time evolution of the magnetisation field, we reveal the creation
mechanism of the Bloch point. Our results introduce a stable and manipulable
Bloch point to the collection of particle-like state candidates for the
development of future spintronic devices.

In recent years, materials with the Dzyaloshinskii-Moriya interaction (DMI)
Dzyaloshinsky1958 ; Moriya1960  came back into the focus of intensive
research, mostly due to the discovery of magnetic skyrmions Rossler2006 ;
Muhlbauer2009  that promise to revolutionise the way we store and process data
Fert2013 . Contrary to the ferromagnetic exchange which tends to align all
spins parallel to each other, the DMI energy is minimised when neighbouring
spins in the crystal lattice are perpendicular. Their mutual competition
results in a twist between neighbouring spins which eventually leads to the
helical magnetic order Bak1980  over the entire crystal lattice. Although the
angle between neighbouring spins, and consequently the helical period, depend
on the relative strength of DM compared to the ferromagnetic exchange energy,
the handedness of the emerged helical order in a helimagnetic material depends
on the material’s chirality - sign of the DM energy constant. So far, studies
have looked mostly at magnetic systems composed of materials with single
chirality. In a search for potential future spintronic devices and new
particle-like magnetisation states, combination of materials with different
chirality may represent an important new avenue for research.

To guide our investigation we start with a thought experiment. In the absence
of chiral interactions, in thin film disks, a vortex configuration can emerge
due to the competition between ferromagnetic exchange and demagnetisation
energies. A vortex can have two different kinds of handedness as well as two
different core polarisations. This results in four possible states of a
magnetic vortex in a disk as shown in Fig. 1 (a). In order to avoid possible
cacophony with the terminology established in previous studies, throughout
this work, we refer to the $+z$ or $-z$ core orientation of the vortex state
as its polarisation, and to the clockwise or counterclockwise
chirality/winding as its handedness. Let us now imagine we have two disks
where vortices with the same handedness were formed, but with different
polarisation. We stack these on top of each other as though they were parts of
a much thicker disk as shown in Fig. 1 (b). This would result in a
magnetisation configuration that is continuous at all points of the sample,
except at the centre, where two vortex cores point in the opposite direction
Miltat2011 . This discontinuity in the magnetisation field is called a Bloch
point Feldtkeller1965 ; Doring1968 . There are two possible discontinuous
polarisations of the cores, and consequently two different types of a Bloch
point, as we show in Fig. 1 (b). A Bloch point has a high magnetic energy,
especially because of the high ferromagnetic exchange energy at the point
where two spins point in the opposite direction. Consequently, the entire
system tends to reduce its energy by expelling the high energy Bloch point out
of the sample. In our system, this would occur by changing the polarisation of
either of the two vortices so that both vortex cores have the same
polarisation but keep the same handedness.

Figure 1:  A thought experiment of stacking magnetic vortices to obtain a
Bloch point. (a) Magnetic vortex can have two different chiralities and two
different polarisations, which results in four possible states for a magnetic
vortex. (b) By stacking two disks hosting magnetic vortices with the same
handedness and different polarisation results in a magnetisation
configurations containing a discontinuity, also known as a Bloch point. (c)
Geometry of the system we are going to study in this work.

In the system of two stacked disks we analysed, the reorientation (change of
polarisation) of one of the vortices to expel the Bloch point is possible
because in a magnetic vortex both polarisations are allowed for a particular
handedness. However, if the Dzyaloshinskii-Moriya energy is present in the
system, the polarisation and handedness of a vortex (more precisely, vortex-
like state) are always related to each other and only two possible states of a
magnetic vortex are allowed Beg2015 . In our thought experiment, this would
imply that changing the vortex core polarisation while keeping the handedness
fixed in order to expel the Bloch point from the system is not possible.
Therefore, we can assume that a Bloch point will remain present in the sample.
In this work, we explore this idea.

Using finite element micromagnetic simulations, we demonstrate the existence
and stability of a Bloch point at the interface of two materials with
different chirality. In addition, by exploring the hysteretic behaviour, we
find two different Bloch point configurations and show how by using an
external magnetic field, one can switch between them. Finally, by analysing
the time evolution of a uniformly magnetised system, we reveal the creation
mechanism of a Bloch point. We believe that the existence and stability of a
Bloch point in the studied system, as well as the possibility of its
manipulation, introduces a new particle-like state to the collection of
candidates for the development of future spintronic devices.

##  I Results

###  I.1 Stability

We begin our study by exploring whether a Bloch point emerges in the system
shown in Fig. 1 (c), obtained as a result of a thought experiment we conducted
in the introduction. In order to base our study on realistic material
parameters and therefore encourage the experimental verification of our
predictions, we use cubic B20 FeGe material parameters (see Methods section).
We set the DMI constant in the bottom layer to be negative ($D=-|D|$) and fix
its thickness $h_{\text{bottom}}$ to be $20\,\text{nm}$. In the top layer we
impose a positive DMI constant ($D=|D|$) and vary its thickness
$h_{\text{top}}$ between $2\,\text{nm}$ and $18\,\text{nm}$ in steps of
$1\,\text{nm}$. The diameter of the disk sample is $150\,\text{nm}$ and no
external magnetic field is applied. For each thickness of the top layer, we
begin by initialising the system with a uniform magnetisation configuration in
the positive out-of-plane ($+z$) direction. This initial configuration
corresponds to the fully saturated magnetisation configuration feasible in an
experimental setup. After that, we relax the system by integrating a set of
dissipative time-dependent equations until the condition of vanishing torque
($\mathbf{m}\times\mathbf{H}_{\text{eff}}$) is satisfied. After the relaxed
(equilibrium) state is obtained, we compute a three-dimensional topological
charge in the top and the bottom layer individually using

$Q=\frac{1}{4\pi}\int\mathbf{m}\cdot\left(\frac{\partial\mathbf{m}}{\partial
x}\times\frac{\partial\mathbf{m}}{\partial y}\right)\,\text{d}^{3}r,$ (1)

as suggested by Lee et al. Lee2009 . Because $Q$ is proportional to the volume
(and accordingly the thickness) of individual layers, we normalise it by the
thickness $\tilde{Q}=Q/h$, in order to make the values mutually comparable
between different layers. The only reason we use normalised topological charge
is to track changes in the magnetisation of individual layers in the system.
We show in Fig. 2 (a) how normalised topological charges $\tilde{Q}$ of the
equilibrium state in individual layers depend on the thickness of the top
layer $h_{\text{top}}$. Although the topological charge of the bottom layer
$\tilde{Q}_{\text{bottom}}$ remains relatively constant, a sharp decrease of
$\tilde{Q}_{\text{top}}$ between $8\,\text{nm}$ and $9\,\text{nm}$ thicknesses
of the top layer is evident. Consequently, we assume that depending on the
thickness of the top layer, two different configurations can emerge as
equilibrium states in the studied system. Subsequently, we explore these two
magnetisation configurations.

Figure 2:  Normalised topological charge $\tilde{Q}$ dependence in individual
layers, identified equilibrium states, and a Bloch point structure. (a)
Dependence of the normalised topological charge $\tilde{Q}$ in individual
layers on $h_{\text{top}}$. Two identified magnetisation configurations: (b)
for $h_{\text{top}}\leq 8\,\text{nm}$ and (c) for $h_{\text{top}}\geq
9\,\text{nm}$. (d) Bloch point structure and demonstration that the
magnetisation direction covers a sufficiently small closed surface surrounding
the Bloch point exactly once.

To explore the state that emerges for $h_{\text{top}}\leq 8\,\text{nm}$, we
visualise the magnetisation obtained for $h_{\text{top}}=5\,\text{nm}$ in Fig.
2 (b) and for $h_{\text{top}}\geq 9\,\text{nm}$ magnetisation configurations,
we show in Fig. 2 (c) the state obtained with $t_{\text{top}}=15\,\text{nm}$.
We show the magnetisation vector field together with the $m_{z}$ scalar field
after we cut out a slice from the sample in order to be able to look inside
and inspect its centre. We observe that for both states, vortex-like
configurations with clear chiralities and polarisations emerge in individual
layers. In the literature, the vortex-like states formed as a consequence of
DM energy are also called quasi-ferromagnetic Rohart2013 , edged vortex Du2013
, or incomplete skyrmion Beg2015 ; Carey2016 ; Pepper2018  states. In both
states we identified, the polarisation and the handedness of the vortex-like
states in bottom layers are the same and related by the right hand rule -
characteristic for the negative DM energy constant. On the other hand, the
vortex-like states in the top layer differ for two equilibrium states we
obtained.

The vortex-like state in the top layer with $5\,\text{nm}$ thickness shown in
Fig. 2 (b), has the same handedness and polarisation as the thicker bottom
layer. However, this is against the left hand rule which establishes the
relationship between the handedness and polarisation for a positive DMI
constant. Therefore, the DM energy density in the top layer is significantly
higher than in the bottom layer and one can ask why the polarisation of the
vortex-like state in the top layer is not reversed in order to minimise its
energy. Because the energy contribution of the top layer to the total energy
of the system is relatively small due to its thickness, it is energetically
cheaper for the top layer to follow the magnetisation of the bottom layer
(violate the left hand rule, and eventually have the higher energy density),
than to reorient itself, which would result in an energetically more expensive
Bloch point at the interface between two layers.

Now, we look at magnetisation configuration in the system with
$h_{\text{top}}=15\,\text{nm}$, shown in Fig. 2 (c). The chiralities of
vortex-like states in both layers are the same, but their polarisations are
opposite and a Bloch point emerges between them. For $h_{\text{top}}\geq
9\,\text{nm}$ and above, the energy contribution of the top layer to the total
energy of the system becomes significant. Therefore, the top layer now does
not follow the polarisation of the bottom layer, but reverses its polarisation
to follow the left hand rule. It is now energetically cheaper for the system
to host a Bloch point than to tolerate the top layer following the
polarisation of the bottom layer and having high energy density.

The location marked by a circle in the middle of the sample in Fig. 2 (c)
identifies a Bloch point: a noncontinuous singularity in the magnetisation
vector field where the magnetisation magnitude vanishes to zero
Feldtkeller1965 ; Doring1968 . Because micromagnetic models assume constant
magnetisation magnitude, the precise magnetisation configuration at the Bloch
point cannot be obtained using micromagnetic simulations Andreas2014 .
However, it is known how to identify the signature of a Bloch point in such
situations: the magnetisation direction covers any sufficiently small closed
surface surrounding the Bloch point exactly once Slonczewski1975 ;
Thiaville2003 . We illustrate this property in Fig. 2 (d).

###  I.2 Hysteretic behaviour

In the previous section we demonstrated the existence and stability of a Bloch
point in the proposed system at zero external magnetic field. In all
relaxation simulations so far, we started the relaxation from the uniform
magnetisation configuration in the positive out-of-plane ($+z$) direction
which resulted in the vortex-like state in the bottom layer with the $+z$
polarisation. Furthermore, for $h_{\text{top}}\geq 9\,\text{nm}$, top layer
relaxed in the vortex-like configuration with negative $-z$ polarisation. This
resulted in a head-to-head Bloch point as we showed in Fig. 2 (d). At this
point, we can assume that if we started our relaxations from the uniform
configuration in the $-z$ direction, the polarisations of the bottom and the
top layer would be in $-z$ and $+z$ directions, respectively. Consequently, a
tail-to-tail Bloch point would emerge. Motivated by this assumption and
knowing that uniform magnetisation configurations can be obtained by applying
a sufficiently strong external magnetic field, we now explore the hysteretic
behaviour of the proposed system in order to verify our assumption.

We fix the thickness of the bottom layer with $D<0$ to be $20\,\text{nm}$ as
before and set the thickness of the top layer with $D>0$ to be $10\,\text{nm}$
so that we can expect a Bloch point to emerge. The diameter of the disk is
$150\,\text{nm}$. We apply an external magnetic field in the out-of-plane $+z$
direction and vary it between $1\,\text{T}$ and $-1\,\text{T}$ in steps of
$0.1\,\text{T}$. We simulate the hysteretic behaviour in the standard way by
relaxing the system to an equilibrium state after changing the external
magnetic field, and then using the resulting state as the starting point for a
new energy minimisation. We show the average out-of-plane magnetisation
component $\langle m_{z}\rangle$ as a function of an external magnetic field
$\mu_{0}H$ in Fig. 3 (a). From this plot, we see that hysteretic behaviour of
the simulated system is evident and now we investigate what are the
equilibrium magnetisation states at zero external magnetic field.

Figure 3:  Hysteretic behaviour and two different Bloch point configurations.
(a) Average out-of-plane magnetisation $\langle m_{z}\rangle$ as a function of
external magnetic field $\mu_{0}H$. Two Bloch point configurations at zero
external magnetic field: (b) head-to-head and (c) tail-to-tail.

We start our hysteresis simulations with $1\,\text{T}$ external magnetic
field, which saturates the sample magnetisation in the $+z$ direction. By
reducing the external magnetic field in steps of $0.1\,\text{T}$, we reach
$\mu_{0}H=0$ and plot the equilibrium magnetisation vector field together with
$m_{z}$ scalar field in the $xz$ cross section containing the disk sample
centre in Fig. 3 (b). We see that bottom and top layers relax in vortex-like
magnetisation configurations with the same handedness, but with $+z$ and $-z$
polarisations, respectively. This results in a head-to-head Bloch point as we
observed in the previous section. Now, we keep reducing the external magnetic
field and when we reach $-1\,\text{T}$, the sample is again fully saturated,
but this time in the $-z$ direction. Starting from this magnetisation state,
we increase the field in same steps until we reach $\mu_{0}H=0$, and plot the
$xz$ cross section of the equilibrium state in Fig. 3 (c). Bottom and top
layers are now hosting vortex-like states with the same handedness, but now
with $-z$ and $+z$ polarisations, respectively. Therefore, a tail-to-tail
Bloch point configuration is formed. Finally, we keep increasing an external
magnetic field until we reach the $1\,\text{T}$ value from which we started
the hysteresis loop simulation. We show the evolution of the magnetisation
field in the $xz$ cross section through the entire hysteresis loop in
Supplementary Video 1. We demonstrated that the studied system undergoes
hysteretic behaviour and that two different types of Bloch point
configurations can emerge. In addition, by applying an external magnetic
field, one can manipulate the Bloch point and switch between head-to-head and
tail-to-tail configurations.

###  I.3 Creation mechanism

Now, we explore how the Bloch point is created in the sample. We simulate the
same sample we used in hysteresis simulations, but this time we do not apply
an external magnetic field. We start by initialising the system in the
positive out-of-plane $+z$ direction. After that, we let the magnetisation
evolve towards its equilibrium state and record the magnetisation
configuration every $1\,\text{ps}$. We set the Gilbert damping value to the
experimentally measured value Beg2017  $\alpha=0.28$. In Fig. 4 (a) we show
time evolutions of the normalised topological charge $\tilde{Q}$ as well as
the average out-of-plane magnetisation component $\langle m_{z}\rangle$ for
both bottom and top layers individually.

Figure 4:  Time evolution and the Bloch point creation mechanism. (a) Time
evolution of the normalised topological charge $\tilde{Q}$ and the average
out-of-plane magnetisation component $\langle m_{z}\rangle$ for the bottom and
top layers individually. (b) Snapshots of the magnetisation configuration in
the $xz$ cross section containing the sample centre at different points in
time.

Starting from $\tilde{Q}_{\text{bottom}}=\tilde{Q}_{\text{top}}=0$ and
$\langle m_{z}\rangle_{\text{bottom}}=\langle m_{z}\rangle_{\text{top}}=1$ at
$t=0\,\text{ps}$, which corresponds to the initial uniform state, all
quantities evolve. In order to explore how the Bloch point is created in the
sample, we show snapshots at different points in time in Fig. 4 (b). We show
the magnetisation vector field together with the $m_{z}$ scalar field in the
$xz$ cross section containing the centre of the sample.

By locating a Bloch point in the magnetisation fields (denoted in
magnetisation field plots) in Fig. 4 (b), we explore its creation mechanism.
Starting from the uniform configuration at $t=0\,\text{ps}$, until
approximately $t=55\,\text{ps}$ the magnetisations in both layers evolve so
that vortex-like states with the same handedness and polarisation are formed.
At $t=58\,\text{ps}$ we identify that the polarisation of the top layer begins
to reverse and a Bloch point is formed at the top boundary of the sample.
Furthermore, Bloch point propagates downwards - towards the interface between
layers, until it eventually reaches its final position at $t=80\,\text{ps}$.
We notice that the Bloch point creation mechanism is based on the polarisation
(core orientation) reversal of the top layer which is the same as the reversal
mechanism for skyrmionic states in confined helimagnetic nanostructures
Beg2015 . More precisely, in the process of reversal of the top layer
polarisation, the Bloch point is created in the same way - at the boundary,
propagating through the thickness of the sample and sitting at the interface
between layers rather than being expelled out of the sample. We show the
entire creation mechanism in the $xz$ cross section of the sample in
Supplementary Video 2.

##  II Discussion

Using finite element micromagnetic simulations, we studied nanostructures
composed of two layers with different handedness. We found that for certain
thicknesses of individual layers, a stable Bloch point can emerge at the
interface between the two layers. By applying an external magnetic field we
found that the system undergoes hysteretic behaviour and two different types
of Bloch point configurations exist. In the first, the magnetisation at the
centre of each layer points towards the interface - the head-to-head
configuration. In the second, the magnetisation points away from the interface
towards the outer surface of the disk - the tail-to-tail configuration. We
demonstrated the switch between these two different types of Bloch point using
an external magnetic field. Finally, we simulated the time evolution in order
to determine the mechanism by which the Bloch point is created, showing that
the Bloch point is created at the boundary of the thinner layer and then
propagates through the thickness of the sample until it reaches the boundary
between layers where it sits. Now, we discuss our findings focusing on four
main topics: (i) the possibility of fabrication, (ii) the suitability of
micromagnetic simulations for the study of Bloch points, (iii) similarities
and differences of our studied system with previous works, and (iv)
implications of our results.

We based our simulations on realistic FeGe material parameters Beg2015  for a
realistically sized disk in order to encourage the experimental verification
of our predictions. In addition, in all of our simulations we used a uniform
magnetisation configuration as an initial magnetisation state which is
feasible to obtain in an experimental setup by applying a strong field. More
precisely, applying an external magnetic field of approximately $1\,\text{T}$,
and then removing it, should allow the system to relax into the magnetisation
configuration containing a Bloch point. However, before our predictions can be
validated, one should be able to grow a helimagnetic thin film which consists
of two layers with different chirality. More precisely, in the growth of a
thin film, it is necessary to be able to switch the Dzyaloshinskii-Moriya
energy constant $D$ once a layer of sufficient thickness has been grown.
Recent research by Spencer at al. Spencer2018  demonstrated that by changing
the Co content $y$ in Fe1-yCoyGe compound grown on Si(111), it is possible not
only to modify the value of the DMI constant, but also to change its sign.
This finding may contribute to the growth of a thin film system, composed of
two layers with different chirality in order to fabricate our proposed system
hosting a stable and manipulable Bloch point.

In this work, we used finite elements to discretise the continuous
magnetisation fields. Because of the imposition of a constant magnetisation
saturation in micromagnetics, this method is inaccurate to determine the Bloch
point structure Andreas2014 . This is because at the Bloch point the
magnetisation vanishes to zero Feldtkeller1965 ; Doring1968 . However, it is
well known that micromagnetics can provide us enough information to determine
whether the Bloch point is present in the sample and its position, by
analysing the magnetisation field on a sufficiently small closed surface
surrounding the Bloch point. More precisely, it is necessary for the
magnetisation direction to cover a sufficiently small closed surface
surrounding the Bloch point exactly once Slonczewski1975 ; Thiaville2003 .
Using this condition we were able to determine the existence and the position
of a Bloch point. However, because of the imposed constant magnetisation
saturation, we did not attempt to determine its internal structure, which was
not the topic of this work. As an example of suitability of micromagnetics to
determine the existence of a Bloch point, Rybakov et al. Rybakov2015
predicted, using micromagnetic simulations, the existence of a chiral bobber -
a hybrid particle emerging at the surface of chiral magnets composed of both
smooth magnetisation configuration and a Bloch point. The existence of a Bloch
point as a part of chiral bobber was then experimentally confirmed in FeGe
Ahmed2018 ; Zheng2018 , demonstrating the validity of the micromagnetic
predictions.

To some extent similar concept was introduced by Zhang et al. Zhang2016 ,
where two ferromagnetic layers hosting skyrmions were antiferromagnetically
coupled in order to suppress the skyrmion Hall effect, when skyrmions are
driven using spin-polarised currents. However, in that system, no Bloch point
can emerge between two skyrmions with different polarisations because they are
physically separated by an insulating spacer of non-zero thickness. On the
other hand, we achieve different polarisations (in our case, of vortex-like
states) by stacking helimagnetic layers with different chirality. Because of
that, there is no insulating spacer between the two layers with different
chiralities and a Bloch point can be identified.

Our discovery of a stable and manipulable Bloch point in a planar magnetic
system could add the Bloch point to the collection of already existing and
well studied particle-like magnetic configurations that could possibly change
the way we store and process data. Now, we discuss and speculate about the
possibilities of how the Bloch point we explored in this work can be employed
in spintronic applications. We found two different Bloch point configurations
that can be switched using an external magnetic field, thus one can propose
that a single information bits (0 or 1) can be encoded using the type of a
Bloch point (HHBP or TTBP) – similar to the bit-patterned media. However, from
the hysteresis simulations, we saw that the average out-of-plane component of
magnetisation $\langle m_{z}\rangle$ at zero field is almost zero due to the
symmetry of the magnetisation field. This poses a difficulty for reading such
information bits. A possible solution is to increase the thickness of the
bottom layer, which does not affect the stability of the Bloch point, but
increases $\langle m_{z}\rangle$ at zero field. Furthermore, our initial
studies show that the Bloch point does not emerge only is disk geometries, but
also in a wide variety of different geometries, such as nanostrips. In this
case, the manipulation of a Bloch point is not only limited to its type, but
also its position in the sample. This manipulation can be achieved using an
external excitation, for instance, external magnetic fields or spin-polarised
currents. However, a Bloch point coupling to such external excitations is
beyond the scope of this work and should be a topic of further research.

All results obtained in this work can be reproduced from the repository in
Ref. Repository, , which contains micromagnetic simulation, data analysis, and
plotting scripts.

##  III Methods

In our simulations, magnetisation dynamics is governed by the Landau-Lifshitz-
Gilbert (LLG) equation Landau1935 ; Gilbert2004

$\frac{\partial\mathbf{m}}{\partial
t}=-\gamma_{0}^{*}\mathbf{m}\times\mathbf{H}_{\text{eff}}+\alpha\mathbf{m}\times\frac{\partial\mathbf{m}}{\partial
t},$ (2)

where $\gamma_{0}^{*}=\gamma_{0}(1+\alpha^{2})$, with $\gamma_{0}=2.21\times
10^{5}\,\text{m}\,\text{A}^{-1}\text{s}^{-1}$ and $\alpha\geq 0$ is the
Gilbert damping. We compute the effective magnetic field
$\mathbf{H}_{\text{eff}}$ using

$\mathbf{H}_{\text{eff}}=-\frac{1}{\mu_{0}M_{\text{s}}}\frac{\delta
E[\mathbf{m}]}{\delta\mathbf{m}},$ (3)

where $E[\mathbf{m}]$ is the total energy functional, and contains several
energy contributions:

$E=\int\left[w_{\text{ex}}+w_{\text{dmi}}+w_{\text{z}}+w_{\text{d}}\right]\,\text{d}^{3}r.$
(4)

The first term $w_{\text{ex}}=A\left[(\nabla m_{x})^{2}+(\nabla
m_{y})^{2}+(\nabla m_{z})^{2}\right]$ with material parameter $A$ in Eq. 4 is
the energy density of the symmetric exchange contribution. The unit vector
field $\mathbf{m}$, with Cartesian components $m_{x}$, $m_{y}$, and $m_{z}$,
represents the normalised magnetisation field
$\mathbf{M}=M_{\text{s}}\mathbf{m}$, where $M_{\text{s}}$ is the saturation
magnetisation. The second term
$w_{\text{dmi}}=D\mathbf{m}\cdot\left(\nabla\times\mathbf{m}\right)$ with
material parameter $D$ is the Dzyaloshinskii-Moriya (DM) energy density. This
form of DM energy density is obtained by including Lifshitz invariants for
materials of crystallographic classes T and O. The coupling of magnetisation
to an external magnetic field $\mathbf{H}$ is defined by the Zeeman energy
density term $w_{\text{z}}=-\mu_{0}M_{\text{s}}\mathbf{H}\cdot\mathbf{m}$,
with $\mu_{0}$ being the magnetic constant. The $w_{\text{d}}$ term is the
demagnetisation (magnetostatic) energy density. Because demagnetisation energy
cannot be neglected in the study of FeGe helimagnetic nanostructures Beg2015 ;
Vousden2016 ; Carey2016 ; Beg2017  we include its contribution in all
simulations. We assume the simulated material is isotropic and neglect the
magnetocrystalline anisotropy energy contribution. The specific boundary
conditions Rohart2013  have been validated by a series of standard problem
simulations proposed by Cortés-Ortuño et al. Cortes2018 . The FeGe material
parameters we use are: Beg2015  saturation magnetisation
$M_{\text{s}}=384\,\text{kA}\,\text{m}^{-1}$, exchange energy constant
$A=8.78\,\text{pJ}\,\text{m}^{-1}$, and Dzyaloshinskii-Moriya energy constant
$|D|=1.58\,\text{mJ}\,\text{m}^{-2}$. The relevant length scales are the
exchange length
$l_{\text{ex}}=\sqrt{2A/\mu_{0}M_{\text{s}}^{2}}=9.73\,\text{nm}$ and helical
length $L_{\text{D}}=4\pi A/D=70\,\text{nm}$. We discretise the finite element
mesh so that the maximum spacing between two neighbouring mesh nodes is below
$l_{\text{max}}=3\,\text{nm}$.

We implemented this micromagnetic model in the finite element method framework
and developed a micromagnetic simulation tool Finmag Finmag . For the low-
level finite element operations, we use the FEniCS project Logg2012  and for
the adaptive step time integration we use Sundials/CVODE solver Hindmarsh2005
; Cohen1996 .

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###### Acknowledgements.

This work was financially supported by the OpenDreamKit – Horizon 2020
European Research Infrastructure project (676541), EPSRC Centre for Doctoral
Training grant EP/L015382/1, and the EPSRC Programme grant on Skyrmionics
(EP/N032128/1).

##  IV Author contributions

M.B. and H.F conceived the study, and M.B. performed finite element
micromagnetic simulations. B.A., R.A.P, and D.C.-O contributed by running
additional micromagnetic simulations using different micromagnetic simulation
tools to confirm and validate main results. M.-A.B., M.B. D.C.-O., R.A.P.,
T.K, G.D., and H.F. developed the micromagnetic finite element based simulator
Finmag. M.B. with the assistance of H.F., R.A.P., and O.H. prepared the
manuscript.

##  V Competing interests

The authors declare no competing financial or non-financial interests.