# Dynamics of bistable Néel domain walls under spin-orbit torque

Eloi Haltz  School of Physics and Astronomy, University of Leeds, Leeds LS2
9JT, United Kingdom  LSPM-CNRS, UPR3407, Sorbonne Paris Nord University,
Villetaneuse, France  Kévin J. A. Franke  School of Physics and Astronomy,
University of Leeds, Leeds LS2 9JT, United Kingdom  Christopher H. Marrows
<EMAIL_ADDRESS>School of Physics and Astronomy, University of Leeds,
Leeds LS2 9JT, United Kingdom

(August 28, 2024)

###### Abstract

Néel magnetic domain walls that are stabilized by achiral energy terms instead
of the usual Dzyaloshinskii-Moriya interaction will be bistable, with the two
possible chiral forms being degenerate. Here we focus on the theoretical study
of the spin-orbit torque driven dynamics of such bistable Néel domain walls.
We find that, for a given domain wall, two propagation directions along a
nanowire are possible, depending on its initial state. These dynamics also
exhibit complex dependence on the spin-orbit torque magnitude, leading to
important transient regimes. Finally, a few ways are proposed for controlled
or random reversal of the domain wall propagation direction. A robust
analytical model which handles all the observed behaviors of such domain walls
is developed and validated by comparing with numerical simulations. The
obtained new dynamics open the way for new uses of domain walls in information
storage and processing devices.

Suggested keywords

## Introduction

The electrical manipulation of magnetic domain walls (DWs) and their potential
technological applications have motivated intense research in magnetism and
spintronics in the last few decades [1]. The possibility of encoding digital
information in the magnetization and manipulating it by shifting the DWs along
a nanowire opens new ways to construct data storage [1, 2], logic devices [3,
4] and, more recently, disruptive processing operations [5, 6, 7, 8, 9, 10].
However, these new uses of DWs require more and more complex dynamics. So far,
different mechanism have been proposed for driving the DW motion in out-of-
plane magnetized wires [11]: applied magnetic fields [12] or electrical
currents via spin-transfer-torques (STTs) [13] or, more efficiently, spin-
orbit-torques (SOTs) for Néel DWs [14].

Driving the motion of a Néel DW under SOT in a ferromagnetic (FM)/heavy metal
(HM) bilayer relies on the combination of two phenomena. First, the spin Hall
effect (SHE) converts the electrical current going through the HM layer into a
transverse pure-spin-current injected in the FM layer. The resulting anti-
damping-like SOT drives Néel DWs in a direction which depends on their
chirality [15, 16, 17]. The second is an effect that stabilizes these DWs in
the Néel structure [18, 19, 20]. Usually, that is achieved by the DMI emerging
from the asymmetry between the interfaces of the FM layer which promotes the
Néel type of the DWs with a given chirality, fixed by the FM/HM stack [17, 21,
16]. In that classic scenario, the sign of the DMI sets the chirality of the
Néel DW and thus fixes the propagation direction for a given SOT drive.

Other phenomena can also be used to stabilize Néel DWs, such as: an additional
in-plane anisotropy [22] or depth gradients of thickness [23], strains [24,
25, 26] or magnetization. Unlike the DMI, these effects stabilize Néel DWs but
without any preferential chirality. One can then speak of bistable Néel DWs
where the chiralities are energetically degenerate. As before, the DW dynamics
depends on the internal structure of the DW so in this scenario, it can be
reversed without changing the driving force if the chirality can be switched.

Here, the shape anisotropy resulting from the demagnetization effect in narrow
magnetic wires is considered, allowing bistable Néel DWs [27, 28]. That effect
is suitable for a multitude of well-known magnetic systems in thin films just
by patterning structures, making it very convenient and versatile. To
manipulate such a bistable Néel DW with SOT, the FM wire can be sandwiched
between two HM layers but of different thicknesses:
HM${}_{\textrm{thick}}$/FM/HM${}_{\textrm{thin}}$, as shown Fig. 1(a). This
cancels out any net DMI because of the symmetric FM interfaces and provides a
non-zero net SHE as the electrical current density in each HM layers are
different but with the same charge/spin conversion. Bistable Néel DWs have
even already been imaged in such stack [28]. These required properties can
also be obtained by introducing a thin metallic spacer layer between the HM
and the FM (such as Cu: HM/Cu${}_{\textrm{thin}}$/FM) which prevents HM/FM
direct interfaces but allows for spin injection though the thin metallic layer
[29, 30]. More generally, the use of an HM with a significant SHE but without
DMI is possible (such as Au, beta-Ta or beta-W).

In the following, the dynamics of these bistable Néel DWs under SOT is
explored. A collective coordinate model is developed to describe the DW
behaviors with two variables, as shown Fig. 1(a): $q$ \- the DW position along
the wire (and so its velocity $v=\dot{q}$) and $\varphi$ \- the angle between
the magnetization at the center of the DW and the current direction (with
$\varphi=0$ or $\pi$ refering to a Néel DW and $\varphi=\pi/2$ or $-\pi/2$
corresponding to a Bloch DW). For every following situation, the initial state
of the DW is defined with two parameters: $Q=\pm 1$ which corresponds to the
orientation of the surrounding magnetic domains ($Q=1$ for an up/down DW and
$Q=-1$ for a down/up DW), and $\varphi^{0}$ the initial orientation of the
magnetization at the center of the DW. The obtained results are in very good
agreement with numerical micromagnetic simulations that we have performed,
validating some assumptions of the analytical model such as the DW shape or
the demagnetization effects [31, 32, 27, 33].

## Results

### DW dynamics under SOT.

First, the DW is set in a certain initial state ($Q$,$\varphi^{0}$) and a
small spin current of $J=3$ GA/m2 is injected. The diagrams in Fig. 1(b) show
in which direction the DW moves and how $\varphi$ evolves under SOT depending
on that ($Q$,$\varphi^{0}$) configuration. For a given $Q$ (rows of the table
in Fig. 1(b)), the DW moves in opposite directions depending on $\varphi_{0}$.
Thus, just by reversing the initial orientation of the DW $\varphi^{0}$, its
motion is reversed. Also, for a given chirality (diagonals of the table Fig.
1(b)) the up/down and down/up DWs move in the same direction. That is the so-
called coherent shift of DWs and it has previously been ensured by the DMI.
Here, since the chirality is not fixed, the up/down and down/up DW with the
same $\varphi^{0}$ move in opposite directions (columns of the table Fig.
1(b)). This leads to the possible expansion or collapse of magnetic domains
under SOT. Usually, that is not possible with current drive in the usual
cases: under SOT with DMI, the DW motion only depends on the chirality; whilst
for the STT, its direction of motion only depends on $Q$ [11]. The DW motion
also induces a tilt of its internal magnetization $\varphi$ which deviates
from a pure Néel state (as shown Fig. 1(b)). That tilting increases with the
SOT magnitude, leading to larger effects at higher current densities.

Fig. 1(c) shows the simulated DW dynamics at larger current density $J$ for a
configuration $Q=1$ and $\varphi^{0}=0$. For each value of $J$, the DW
velocity $v$ and angle $\varphi$ (black dots of the top and bottom panels) are
extracted from the micromagnetic simulations after 20 ns of current injection.
For low current, $v$ increases linearly with $J$ and $\varphi$ slightly
deviates from its pure Néel state. When $\varphi$ reach the value of
$\frac{\pi}{4}$ (blue dashed line), the DW velocity exhibits a maximum. For
larger currents, the DW is pushed into its Bloch configuration. Since the SOT
just acts on Néel DWs, the DW slows down. Above a critical current
$J_{\textrm{c}}$ (red dashed line Fig. 1(c)) the DW is locked into its pure
Bloch configuration with $\varphi=\pm\frac{\pi}{2}$ and then $v=0$.

Figure 1: Néel wall motion under SOT. (a) Sketch of a bistable Néel DW in an
asymmetric HM/FM/HM trilayer. (b) Sketch of the propagation direction and
internal dynamics of a bistable Néel DW depending on the orientation of the
surrounding domains (written as $Q=1$ for an up/down DW and $Q=-1$ for a
down/up DW) and on the initial state of the DW ($\varphi^{0}=0$ or $\pi$,
shown as a gray arrow). The tilting of the internal magnetization of the DW is
shown as a black curved arrow. These four cases summarize the results obtained
with micromagnetic simulations. (c) Evolution of the DW velocity $v$ (top
panel) and associated internal angle $\varphi$ (bottom panel) with the current
density $J$ for a DW with $Q=1$, $\varphi^{0}=0$. Black points correspond to
the values extracted from the numerical simulations after 20 ns of current
injection. The yellow full lines show the theoretical evolution of these
quantities in the steady state regime $v^{\infty}$ and $\varphi^{\infty}$
given by Eq. (3) and (4), respectively. The red and the blue vertical dashed
lines correspond to the critical currents $J_{\textrm{c}}=6$ GA/m2 and
$\frac{J_{\textrm{c}}}{\sqrt{2}}=4.25$ GA/m2.

### Collective coordinate model.

To understand the complex dynamics of such DW, it is convenient to use an
analytical collective coordinate model that describes the DW dynamics with
only a few parameters : $q$ and $\varphi$, the central position and the
internal angle of the DW, as defined beforehand and shown Fig. 1(a). The
analytical integration of the LLG equation considering the rigid Bloch DW
ansatz [34] leads to coupled equations of motion [32, 11]:

$\displaystyle\dot{q}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{\alpha\tau_{0}}\left(\alpha^{2}Qj\cos\varphi+\frac{\sin
2\varphi}{2}\right),$ (1) $\displaystyle\dot{\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{0}}\left(Qj\cos\varphi-\frac{\sin
2\varphi}{2}\right),$ (2)

where
$\Delta=\sqrt{\nicefrac{{A}}{{\left(K_{0}-\frac{\mu_{0}}{2}M_{\textrm{s}}^{2}\right)}}}$
is the DW width (defined as positive).
$\tau_{0}=\frac{1+\alpha^{2}}{\alpha\gamma_{0}H_{\textrm{KIP}}}$ is a
timescale associated with the DW dynamics, mainly depending on
$H_{\textrm{KIP}}=\frac{2K_{\textrm{IP}}}{\mu_{0}M_{\textrm{s}}}$, the field
associated to the shape anisotropy strength $K_{\textrm{IP}}$. The
dimensionless current ratio
$j=\frac{J}{J_{\textrm{c}}}=\frac{\frac{\pi}{2}H_{\textrm{SOT}}}{\alpha
H_{\textrm{KIP}}}$ results from the competition between the shape anisotropy,
which tends to stabilize the Néel state of the DW and the SOT which tends to
tilt the DW internal magnetization, accounted as $H_{\textrm{SOT}}$, the
effective field associated to the resulting SOT on the DW. Eq. (1) and (2)
clearly show that $\varphi$ governs the DW dynamics and that $v$ depends on
$\varphi$ roughly as $v\sim\sin 2\varphi$.

For asymptotic steady state motion, the DW velocity and the internal angles
are constant (i.e. $\dot{q}$ and $\varphi$ take constant values that we denote
$v^{\infty}$ and $\varphi^{\infty}$). Two steady state dynamical regimes are
obtained in this case, separated by the critical current density
$J_{\textrm{c}}$:

$\displaystyle v^{\infty}$
$\displaystyle=\begin{cases}\left(S_{1}\right)\;\frac{\Delta}{\tau_{0}}\frac{1+\alpha^{2}}{\alpha}j\sqrt{1-j^{2}}\;&\textrm{if}\;J\leqslant
J_{\textrm{c}}\;(j\leqslant 1),\\\ 0\;&\textrm{if}\;J\geqslant
J_{\textrm{c}}\;(j\geqslant 1),\end{cases}$ (3)
$\displaystyle\varphi^{\infty}$
$\displaystyle=\begin{cases}(\widetilde{\varphi^{0}})+\left(S_{1}\right)\arcsin
j\;&\textrm{if}\;J\leqslant J_{\textrm{c}},\\\
\left(S_{2}\right)\frac{\pi}{2}\;&\textrm{if}\;J\geqslant
J_{\textrm{c}}.\end{cases}$ (4)

$\left(S_{1}\right)$ and $\left(S_{2}\right)$ are equal to $\pm 1$ and give
the DW propagation and tilt angle directions depending on the initial state of
the DW and on the current direction:
$\left(S_{1}\right)=Q\operatorname{sgn}\left(\cos\varphi^{0}\right)$ and
$\left(S_{2}\right)=Q\operatorname{sgn}\left(J\right)$.
$(\widetilde{\varphi^{0}})$ determines whether the DW is mainly left- or
right-handed: $(\widetilde{\varphi^{0}})=0$ if
$\varphi^{0}\in]-\frac{\pi}{2},\frac{\pi}{2}[$ and $=\pi$ if
$\varphi^{0}\in]\frac{\pi}{2},\frac{3\pi}{2}[$. These expressions of
$\left(S_{1}\right)$, $\left(S_{2}\right)$ and $(\widetilde{\varphi^{0}})$
reproduce well the results sketched in the table shown in Fig. 1(b). These
considerations also show that for $J<J_{\textrm{c}}$, $v^{\infty}$ and
$\varphi^{\infty}$ depend of the initial condition $\varphi_{0}$ but not for
$J\geqslant J_{c}$.

The variation of $v^{\infty}$ and $\varphi^{\infty}$ with $J$, plotted as
yellow lines in Fig. 1(c), also reproduce the simulated results with a very
good quantitative agreement in the main part of the current range, far from
$J_{\textrm{c}}$. Eq. (3) and (4) also shows that a maximum of velocity
$\frac{\Delta}{2\tau_{0}}\frac{1+\alpha^{2}}{\alpha}$ is reached for
$J=\frac{J_{\textrm{c}}}{\sqrt{2}}$ for which
$\varphi^{\infty}=\pm\frac{\pi}{4}[\pi]$. Interestingly, that maximum only
depends on the in-plane effective anisotropy but not directly on the SOT.

### Transient regime.

The non-uniformity of the DW velocity with $\varphi$ has a strong effects in
the transient regime. These are most easily visible around the critical
current, where the simulated DW velocity does not seem to vanish immediately
at $J\geq J_{\textrm{c}}$ as it is expected in the steady state motion. Fig.
2(a) shows the time evolution of the DW velocity during a 50 ns current pulse
of different magnitudes for an initial DW configuration $Q=1$ and
$\varphi^{0}=0$. First, the loading time, just after the current onset is
discussed. The relaxation dynamics is described in the following.

For $J<\frac{J_{\textrm{c}}}{\sqrt{2}}$ (case 1), the velocity, as well the
internal angle (not shown here) tend continuously to the steady state values,
as expected for classical DW motion. For $J>\frac{J_{\textrm{c}}}{\sqrt{2}}$
(case 2), the transient regime is divided into two parts: first the DW
accelerates up to a maximum of velocity with a characteristic time of a few ns
and then slows down, tending towards the steady state values with a much
longer characteristic time. These two parts are due to the continuous
evolution of $\varphi$ from $\varphi^{0}=0$ to the steady state value
$\varphi^{\infty}$ which implies to cross $\pm\frac{\pi}{4}[\pi]$ where the
velocity is at a maximum. The characteristic time of that velocity decay is
much larger, up to tens of ns. For $J>J_{\textrm{c}}$, these two parts of the
transient regime still exist. Even if the asymptotic velocity is zero, the DW
moves for a significant amount of time before being locked into its Bloch
configuration. That explains why the velocities extracted from the simulations
(after 20 ns of current injection; in bold Fig. 2(a)) shown Fig. 1(c) are
slightly above zero at and above $J_{\textrm{c}}$.

Figure 2: Transient regimes of the DW dynamics. (a) The different plots show
the time evolution of the DW velocity during 50 ns current pulses of different
magnitudes: case 1, $J=3$ GA/m${}^{2}\leavevmode\nobreak\
<\frac{J_{\textrm{c}}}{\sqrt{2}}$; case 2, $J=5.5$
GA/m${}^{2}\leavevmode\nobreak\
\in[\frac{J_{\textrm{c}}}{\sqrt{2}},J_{\textrm{c}}]$; case 3, $J=7$
GA/m${}^{2}\leavevmode\nobreak\ >J_{\textrm{c}}$; and case 4, $J=10$
GA/m${}^{2}\leavevmode\nobreak\ \gg J_{\textrm{c}}$. For each plot, black dots
show the values from the numerical simulations. During the current injection
(for $t<50$ ns), the colored lines are the exponential evolution with
characteristic times $\tau$. After the current injection (for $t>50$ ns), the
purple lines show the analytical model considering Eq. (1) and (6). (b)
Evolution of the characteristic time $\tau$ of the transient regime with $J$.
Black dots correspond to the values of $\tau$ obtained with the exponential
fits of the data in (a) and the colored lines show the analytical expression
of $\tau$ for $t\approx 0$ in yellow, and $t\rightarrow\infty$ in blue for
$J\in[\frac{J_{\textrm{c}}}{\sqrt{2}},J_{\textrm{c}}]$ and in red for
$J>J_{\textrm{c}}$. Blue and red dashed vertical lines correspond to
$\frac{J_{\textrm{c}}}{\sqrt{2}}$ and $J_{\textrm{c}}$. (c) shows the DW
motion delay depending on the DW angle when the current is turned off
$\varphi^{0}$. The yellow horizontal line shows the characteristic time
$\tau_{0}$ and the purple curve is the expected variation of the motion delay
with $\varphi^{0}$.

The variation of this characteristic time can be obtained by linearising Eq.
(2) for $t\approx 0$ around $\varphi^{0}$ or $t\rightarrow\infty$ around
$\varphi^{\infty}$. We can obtain that
$\dot{\varphi}\approx\dot{\varphi}\left(\varphi^{0(\infty)}\right)-\frac{\varphi-\varphi^{0(\infty)}}{\tau}$
with $\dot{\varphi}\left(\varphi^{\infty}\right)\rightarrow 0$ and so

$\tau=\frac{\tau_{0}}{Qj\sin\varphi^{0(\infty)}+\cos 2\varphi^{0(\infty)}},$
(5)

the characteristic time of the $\varphi$ variations at the beginning or at the
end of the transient regime (i.e. for $\varphi\approx\varphi^{0}$ or
$\varphi^{\infty}$). Thus, Eq. (5) captures the two different parts of the
transient regime. For the first one, $\varphi^{0}=0$ or $\pi$ gives
$\tau=\tau_{0}$ for any currents and $v_{(t\approx
0)}\propto\left(\varphi_{(t\approx
0)}-\varphi^{0}\right)\propto\left(1-e^{-t/\tau_{0}}\right)$. For the second
part, when it exists (i.e. for $J>\frac{J_{\textrm{c}}}{\sqrt{2}}$), the
characteristic time depends on $J$ explicitly and implicitly through
$\varphi^{\infty}_{(J)}$ in Eq. (5). For
$J\in]\frac{J_{\textrm{c}}}{\sqrt{2}},J_{\textrm{c}}[$,
$\tau_{(J)}=\frac{\tau_{0}}{1-j^{2}}$ and for $J>Jc$,
$\tau_{(J)}=\frac{\tau_{0}}{j-1}$. For each case,
$\left(\varphi_{(t\approx\infty)}-\varphi^{\infty}\right)\propto\left(1-e^{-t/\tau}\right)$
and $\left(v_{(t\approx\infty)}-v^{\infty}\right)\propto e^{-t/\tau}$. The
colored lines in the Fig. 2(a) show the exponential decays of $v$, with the
extracted characteristic times shown as black dots in Fig. 2(b). There are
reproduced well by the analytical $\tau$ versus $J$ curves (yellow :
$J<\frac{J_{\textrm{c}}}{\sqrt{2}}$, blue :
$J\in]\frac{J_{\textrm{c}}}{\sqrt{2}},J_{\textrm{c}}[$ and red :
$J>J_{\textrm{c}}$). This dependence of $\tau$ on the strength of the drive
$J$ leads to non-trivial non-Newtonian dynamics. $\tau$ even diverges at
$J_{\textrm{c}}$, which means that the internal angle tends to
$\pm\frac{\pi}{2}$ but never reaches this value at this critical current.
Conversely, for $J>J_{\textrm{c}}$, $\tau$ is finite, meaning that the DW
reaches its pure Bloch configuration ($\varphi=\pm\frac{\pi}{2}$).

When the current is turned off, the DW relaxes in its Néel configuration (as
shown Fig. 2(a)) with $\varphi^{\infty}_{(J=0)}=0$ or $\pi$ depending on the
internal DW angle at this time $\varphi^{0}$. For $J=0$, Eq. (2) gives:

$\varphi(t)=(\widetilde{\varphi^{0}})+\textrm{arccot}\left(\textrm{cot}\left(\varphi^{0}\right)e^{\nicefrac{{t}}{{\tau_{0}}}}\right),$
(6)

with $(\widetilde{\varphi^{0}})=0$ or $\pi$ defined as before. The relaxation
occurs with a characteristic time $\tau_{0}$, independent of $\varphi^{0}$. If
the DW crosses $\varphi=\pm\frac{\pi}{4}[\pi]$ during its relaxation, it
exhibits a velocity peak after a delay of
$-\tau_{0}\,\textrm{log}\left|\textrm{cot}\left(\varphi^{0}\right)\right|$ as
visible Fig. 2(a). Fig. 2(c) shows the evolution of the motion delay whenever
it exists versus the DW angle at the end of the pulse ($\varphi^{0}$ in Eq.
(6)).

The velocity overshoot of the DW for $J>\frac{J_{\textrm{c}}}{\sqrt{2}}$ might
be very convenient for getting out eventual pinning centers as it is the case
for other driving forces [35, 36] however, here, this property is intrinsic to
the DW dynamics. Such spiky and delayed DW dynamics also makes echoes
properties explored for brain-inspired computing [37, 38, 39].

### Brownian motion under large current pulses.

It is interesting to note that $\varphi^{\infty}$ implicitly depends on
$\varphi^{0}$ according to Eq. (4). Thus, the angles
$\varphi^{0}=\pm\frac{\pi}{2}$ are bifurcation points where the asymptotic
angles $\varphi^{\infty}$ towards which $\varphi$ will tend are completely
different for $\varphi^{0}=\pm\frac{\pi}{2}\pm 0^{+}$. In particular,
infinitesimal changes lead to opposite propagation directions of the DW.
However, for large current ($J\gg J_{\textrm{c}}$), required to reach the
reversal points $\varphi=\pm\frac{\pi}{2}$, the SOT tends to keep the DW in
that Bloch configuration (according to Eq. (4) and shown Fig. 1). To achieve
the reversal of the DW propagation taking advantage of these bifurcation
points, we propose to use large current pulses, with magnitudes larger than
$J_{c}$. The SOT pulse sets the DW in its Bloch state
($\varphi=\pm\frac{\pi}{2}$) and after the pulse, the DW relaxed into one of
its Néel state $\varphi^{\infty}_{(J=0)}=0$ or $\pi$ randomly, inducing DW
motions. This internal dynamics is shown Fig. 3(a) for a 20 ns pulse of 15
GA/m2 on a DW with an initial state $Q=1$ and $\varphi$=0. The insets and the
colored highlight indicate if $\varphi\in]-\frac{\pi}{2},\frac{\pi}{2}[$ in
green or $\varphi\in]\frac{\pi}{2},\frac{3\pi}{2}[$ in purple. After the
bifurcation point (shown as a red star) the DW relaxes in one of its Néel
state driving the DW in opposite directions.

As the infinitesimal change of $\varphi$ just plays a role at the end of the
pulse, it is possible to mimic the random reversals of the DW propagation
during a large number of pulse by considering a small random deviation of the
DW internal angle at the end of each pulse. In practice, this is naturally
achieved by the thermal fluctuations or the inhomogeneities of the system. The
Fig. 3(b) shows the DW displacements and internal angle under a series of 20
GA/m2 pulses with 20 ns widths and 20 ns delays. At the end of each pulses
$\varphi$ is deviated by a quantity following a normal distribution with a
standard deviation of $0.1^{\textrm{o}}$. The red stars indicate if the DW
reversal occurs.

Figure 3:  Brownian motion of the DW under large current pulses: (a) shows the
reversal process of the DW at the end of a current pulse with a magnitude
larger than $J_{\textrm{c}}$. The evolution of the DW position is calculated
by numerical integration of the Eq. (1) and(2) (black line) for a pulse of 20
ns width and 20 GA/m2 magnitude. The colored highlight indicate if
$\varphi\in]-\frac{\pi}{2},\frac{\pi}{2}[$ in green or
$\varphi\in]\frac{\pi}{2},\frac{3\pi}{2}[$ in purple. The insets sketch the DW
state before and after the end of the pulse corresponding to a bifurcation
point (indicated as a red star). (b) shows the DW under a series of 20 GA/m2
20 ns pulses with 20 ns delays (as indicated in the inset). To capture the
infinitesimal changes of $\varphi$, at the end of each pulse $\varphi$ is
deviated by an infinitesimal quantity obeying a normal distribution with a
standard deviation of 0.1∘. The red stars indicate when the DW reversal
occurs. (c) shows the probability of the DW motion in the positive direction
(black curve) and of the DW reversal (red curve) calculated during 1,000
pulses for different current densities. (d) shows the square root evolution of
the mean squared displacement $\left<q^{2}\right>$ with the number of pulses
averaged over 200 series of 100 pulses. The yellow line is the linear fit
$\left<q^{2}\right>\propto t$.

Fig. 3(c) shows the probability of the DW displacement in the positive
direction (black curve) and the probability of the propagation reversal (red
curve) versus the current density for 20 ns pulses. Above a critical current
larger than $J_{\textrm{c}}$, the DW displacement in one or the other
direction is completely equiprobable even if the reversal probability is
gradually increasing. During this transition, the DW reversal is quite rare
and the DW moves in the same direction for a large number of pulses. After
this transition, the reversal is completely random and the DW motion obeys the
Brownian law. Fig. 3(d) shows the expected square root evolution of the mean
squared DW displacement with the number of pulses [40]
$\left<q^{2}\right>=2Dt$ with a diffusivity $D=1.4\times 10^{-2}$ $\upmu$m2/20
ns pulse. The current pulse magnitude above which the DW displacement is
Brownian decreases with the pulse duration and with the standard deviation of
the $\varphi$ deviation as these two quantities promote the DW reversal. This
Brownian dynamics does not depends on the current sequence and can be also
obtained for a square- or a sine-wave current with large amplitude (not shown
here). In the following, the reversal of a moving DW under continuous currents
are considered.

### Propagation reversal under $H_{z}$.

As for a DW motion driven by continuous current
$\dot{\varphi}_{(\varphi=\pm\frac{\pi}{2})}=0$ (according to Eq. (2)), the
controlled reversal of its propagation requires additional torques acting on
$\varphi$. This could be achieved by using an in-plane field or field-like
torques. However, this will not allow repeatably reversing the propagation
without changing the polarity of these torques. This implies knowing the DW
state at any moment to reverse its propagation. On the other hand, the use of
an out-of-plane field $H_{z}$ induces a torque on the magnetization
independent of $\varphi$ (only depending on $Q$). In a presence of $H_{z}$ Eq.
(1) and (2) become:

$\displaystyle\dot{q}$ $\displaystyle=$
$\displaystyle\frac{\Delta}{\alpha\tau_{0}}\left(Q\alpha^{2}\left[j\cos\varphi+h_{z}\right]+\frac{\sin
2\varphi}{2}\right),$ (7) $\displaystyle\dot{\varphi}$ $\displaystyle=$
$\displaystyle\frac{1}{\tau_{0}}\left(Q\left[j\cos\varphi+h_{z}\right]-\frac{\sin
2\varphi}{2}\right),$ (8)

with the dimensionless out-of-plane field $h_{z}=\frac{H_{z}}{\alpha
H_{\textrm{KIP}}}$ corresponding to the ratio between $H_{z}$ which tends to
rotate $\varphi$ in the $QH_{z}$ direction (whatever the value of $\varphi$)
and $H_{\textrm{KIP}}$ which tends to keep the DW in its Néel state. In
particular, when the field is applied,
$\dot{\varphi}_{(\varphi=\pm\frac{\pi}{2},H_{z}\neq 0)}\neq 0$ which allows
for overcoming the reversal limits $\varphi=\pm\frac{\pi}{2}$. Note that even
if $v$ changes while $H_{z}$ is applied, the proper reversal is governed by
the DW state when the field is turned-off.

Figure 4: Propagation reversal under $H_{z}$: (a) and (b) show the propagation
reversal of a DW driven by continuous SOT (with $J=3$ GA/m2) under out-of-
plane field pulses in the ($q$,$\varphi$) space for $Q=1$ and respectively,
$\varphi^{0}\in]-\frac{\pi}{2},\frac{\pi}{2}[$ (a) and
$\varphi^{0}\in]\frac{\pi}{2},\frac{3\pi}{2}[$ (b). For both (a) and (b),
pulse width of duration $t_{H_{z}}=3$ ns and magnitude $\mu_{0}H_{z}=$2, 4 and
7 mT and are applied for, respectively, the cases 1, 2, and 3. Black dots and
lines correspond to the numerical simulations. The spacing between dots
corresponds to a 1 ns time interval. The colored lines correspond to the
numerical integration of the $q$-$\varphi$ model given by Eq. (7) and (8). The
color shows if $\varphi\in]-\frac{\pi}{2},\frac{\pi}{2}[$ in green, or if
$\varphi\in]\frac{\pi}{2},\frac{3\pi}{2}[$ in purple, as indicated in the
insets. The black arrows indicate the DW propagation directions. (c) and (d)
show the reversal phase diagram in the $(H_{z},t_{H_{z}})$ space for a DW
driven by $J=3$ GA/m2. Colored dots indicate the final state of the DW after
10 ns relaxation following the field pulse obtained by the micromagnetic
simulations: $\varphi\in]-\frac{\pi}{2},\frac{\pi}{2}[$ in green and
$\varphi\in]\frac{\pi}{2},\frac{3\pi}{2}[$ in purple. The black circles
correspond to the configurations shown in (a) and (c). The background color
shows the same results but obtained by the numerical integration of the
$q$-$\varphi$ model (Eq. (7) and (8)). Black lines show the analytical
hyperbolas, which separate the reversal regions obtained with the linearized
model.

Fig. 4(a) and (b) show some examples of propagation reversal driven by field
pulses for pulse width of $t_{H_{z}}=3$ ns and different magnitudes
($H_{z}>0$) plotted in the $(q,\varphi)$ space for $Q=1$. For every
$(H_{z},t_{H_{z}})$ pulse, the DW stabilizes itself in its steady state under
a continuous current of $3$ GA/m2 (during 5 ns), a pulse of field is applied
during $t_{H_{z}}$, and the DW relaxes in its new state during 5 ns. To avoid
the initial transient regimes, we consider $\varphi^{0}$ as
$\varphi^{0}=\varphi^{\infty}\approx 0.2\pi\in]-\frac{\pi}{2},\frac{\pi}{2}[$
or $\varphi^{0}=\varphi^{\infty}\approx
0.8\pi\in]\frac{\pi}{2},\frac{3\pi}{2}[$ for (a) or (b), respectively. The
black arrows indicate the propagation direction of the DW. In both (a) and
(b), the black dots correspond to the evolution of the DW state obtained from
the micromagnetic simulations with the space between them corresponding to 1
ns time intervals. The colored lines correspond to the numerical integration
of the $q$-$\varphi$ model accounting a field pulse (given by Eq. (1) and (2)
in absence of the field and Eq. (7) and (8) during the field pulse). The color
of these lines depends on $\varphi$: green for
$\varphi\in]-\frac{\pi}{2},\frac{\pi}{2}[$ and purple for
$\varphi\in]\frac{\pi}{2},\frac{3\pi}{2}[$. The plots shown in Fig. 4(a) and
(b) clearly demonstrate that the applied field can be used to overcome the
reversal limit which flips the propagation direction of the DW after the field
for, possibly, the two states of the DW.

This field-induced reversal strongly depends on the magnitude and width of the
field pulse. To handle that dependence, the above simulation has been repeated
for different $(H_{z},t_{H_{z}})$. Fig. 4(c) and (d) show the final state of
the DW after the field pulse for each $(H_{z},t_{H_{z}})$ (obtained with the
micromagnetic simulations - dots - and by numerically integrating the
$q$-$\varphi$ model - background color). The configurations detailed in (a)
and (b) are circled in black in (c) and (d). To obtain the reversal
conditions, Eq. (8) can be linearized by considering the effect of the out-of-
plane field on $\varphi$ is much larger than the combined effects of the SOT
and the in-plane anisotropy giving
$\varphi\approx\varphi^{0}+\frac{Qh_{z}}{\tau_{0}}t$. The conditions for a DW
reversal can thus be defined as
$\left(\frac{4n+1}{2}\pi-\varphi^{\infty}\right)<Qh_{z}\frac{t_{H_{z}}}{\tau_{0}}<\left(\frac{4n+3}{2}\pi-\varphi^{\infty}\right)$
for $\varphi^{0}\in]-\frac{\pi}{2},\frac{\pi}{2}[$ (the cases shown in Fig. 4
(a) and (c)), and
$\left(\frac{4n-1}{2}\pi+\varphi^{\infty}\right)<Qh_{z}\frac{t_{H_{z}}}{\tau_{0}}<\left(\frac{4n-3}{2}\pi+\varphi^{\infty}\right)$
for $\varphi^{0}\in]\frac{\pi}{2},\frac{3\pi}{2}[$ (the cases shown in Fig. 4
(b) and (d)). $n$ corresponds to the number of flips in multiple flipping
processes.

The analytic hyperbolic lines delimiting these regions in the
$(H_{z},t_{H_{z}})$ space are plotted in black in Fig. 4(c) and (d). They
match well to the simulated cases except for low fields and low pulse widths
where the $\dot{\varphi}\sim H_{z}$ assumption is not well justified. For
larger $H_{z}$ and $t_{H_{z}}$, the regions where the DW propagation is
reversed are more and more condensed, which is equivalent to a quasirandom
flipping process (similar to the coin reversal in a heads-or-tails process
[41]).

Additionally, these reversal regions are not exactly the same for
$\varphi^{0}\in]-\frac{\pi}{2},\frac{\pi}{2}[$ and
$\varphi^{0}\in]\frac{\pi}{2},\frac{3\pi}{2}[$. Thus, it is possible to
reverse only the DW appertaining to one of these cases, setting all the DW
with the same $\varphi$, as shown in the second panel in Fig. 4(a) and (b). At
the opposite, a pulse with conditions that correspond to a reversal for the
two cases flips all the DWs, independently of their initial angle, as shown
third panel in Fig. 4(a) and (b). The shift between the reversal regions for
$\varphi^{0}\in]-\frac{\pi}{2},\frac{\pi}{2}[$ and
$\varphi^{0}\in]\frac{\pi}{2},\frac{3\pi}{2}[$ increases with the current
density as $\varphi^{\infty}$ is much closer to the reversal point for one
type of DW than for the other (as visible in the table Fig. 1(b)).

### Step of $K$.

Even if an out-of-plane field allows the propagation reversal, its effects are
strongly dependent on the pulse width and magnitude, and significantly disturb
the DW propagation with multiple reversals. To achieve similar effects in a
much more controlled way, it is possible to consider a spatial variation of
the out-of-plane anisotropy $K_{(x)}$ (with $x$ the position along the wire).
That can be directly associated with an effective local out-of-plane field
$H^{K}_{z(q)}=-\frac{\Delta}{\mu_{0}M_{s}}\partial_{q}\tilde{K}_{(q)}$ where
$\tilde{K}_{(q)}=\frac{\int
K_{(x)}(1-m_{z(x,q)}^{2})dx}{\int(1-m_{z(x,q)}^{2})dx}$ is the effective
variation of the out-of-plane anisotropy with $q$, the position of the DW
along the magnetic wire. Here, $m_{z}$ is the $z$-component of the local unit
magnetization vector $\hat{\bm{m}}$. This field vanishes where the anisotropy
is uniform and it has a maximum where $K$ changes most rapidly with position
$x$.

Fig. 5(a) shows the effect of a step of anisotropy of magnitude $\delta K$ at
the position $x_{\textrm{step}}$ on a moving DW with $Q=1$ and
$\varphi^{0}=\varphi^{\infty}$ for different $J$ and different step
magnitudes. The evolution of the DW dynamics is represented in the
$(q,\varphi)$ space and the arrows indicate the DW propagation directions. The
black dots correspond to the simulated evolution of the DW dynamics with a
time interval between them of 1 ns. The gray curve in the background shows the
anisotropy variation along the wire. As visible in the different cases, the DW
can be either stopped (case 1), transmitted (case 2 or 4), or reflected (case
3), depending on $J$ and $\delta K$.

As for a regular out-of-plane field, $H_{\delta K}$ acts on $\varphi$ as
described by the Eq. (8), but here the strength of the field experienced by
the DW depends on its position along the wire $q$. For high positive $\delta
K$, if $H_{z(q)}^{K}\propto-\delta K$ pushes $\varphi$ down to zero, the DW
propagation is reversed (since $v\sim\sin 2\varphi$) and the field experienced
by the DW decreases. That pushes the DW back to its steady state with
$\varphi^{\infty}>0$, reversing again its propagation direction. The
converging repetition of that process leads to the pinning of the DW before
the anisotropy step as shown in the first panel of Fig. 5(a) (case 1).

If $\varphi$ does not cross zero while the DW cross the region where the
anisotropy changes, the DW propagation is not reversed and the DW is
transmitted as shown in the second panel of Fig. 5(a) (case 2). That occurs
for lower $\delta K$, where $\varphi$ is less affected by the anisotropy step,
and for larger $J$ where $\varphi^{\infty}$ is farther from $0$ and the DW is
faster, experiencing the effective field for a shorter period of time. For
negative $\delta K$, $H_{z}^{K}$ pushes $\varphi$ farther from zero. For low
negative $\delta K$ and low $J$, the DW crosses the anisotropy step without
crossing the reversal limit $\varphi=\frac{\pi}{2}$, and so it is transmitted,
as shown in the third panel (case 2) in Fig. 5(a).

For larger $J$, $\varphi^{\infty}$ is larger and even a small $\delta K$
allows the DW to cross this reversal limit. The DW moves then farther from the
anisotropy step and experiences $H_{z}^{K}$ less and less. The DW is therefore
reflected by the anisotropy step as shown in the fourth panel of Fig. 5(a),
case 3.

For large and negative $\delta K$, the proximity of the anisotropy step can
lead to multiple reversals of $\varphi$. If the current is too small, the DW
is either reflected (for an odd number of reversals) or transmitted (for an
even number of reversals; as shown in the fifth panel of Fig. 5(a), case 4).
For larger current, the DW reverses just one time and the DW is reflected
similarly to case 3. These different effects of the anisotropy step are
summarized in Fig. 4(b). The colored points corresponds to the $(J,\delta K)$
configurations where the DW is either stopped (gray points), transmitted
(green points), or reflected (purple points).

Figure 5: Propagation reversal at an anisotropy step: (a) shows the
propagation reversal of a DW at the anisotropy step at the position
$x_{\textrm{step}}$ in the $(q,\varphi)$ space for
$\varphi^{0}=\varphi^{\infty}\approx 0.2\pi$ and $Q=1$ and for $(J,\delta
K)=(3,0.01)$, $(3,0.005)$, $(3,-0.005)$, $(3,-0.02)$, and $(1.5,-0.025)$ in
units of (GA/m2, MJ/m3). The gray shape in the background (relative to the
right axis) shows the x-variation of the anisotropy. Black dots and lines
correspond to the numerical simulations with a spacing between the dots
corresponding to a 1 ns time interval. The colored lines correspond to the
numerical integration of the $q$-$\varphi$ model considering the anisotropy
step as a local field $H_{z}^{K}(q)$. Its color shows whether
$\varphi\in]-\frac{\pi}{2},\frac{\pi}{2}[$ in green and
$\varphi\in]\frac{\pi}{2},\frac{3\pi}{2}[$ in purple. The black arrows
indicate the DW propagation direction. (b) shows the reversal phase diagram in
the $(J,\delta K)$ space. Colored dots indicate the final state of the DW
after its interaction with the anisotropy step obtained by the numerical
simulations: transmission in green, stoppage in gray, and reflected in purple.
The black circles correspond to the configurations shown in (a). The
background color show the same results but obtained by the numerical
integration of the $q$-$\varphi$ model. Black lines show the frontiers between
the different behaviors considering an approximated square field as discussed
in the main text. (c) shows a sketch of the different approximations of the
effective field $H_{z}^{K}(q)$: the gray shape in the background (relative to
the right axis) shows the variation of the anisotropy, the blue curve
(relative to the left axis) shows the variation of $H_{z}^{K}(q)$ (by assuming
a rigid DW) and the dashed blue curve shows the square approximation of
$H_{z}^{K}(q)$ used for the simplified $q$-$\varphi$ model.

By assuming $\delta K$ is quite small compared to $K$, the DW width $\Delta$
is considered as constant, independent of the anisotropy step [34]. A sharp
step of anisotropy at a position $x_{\textrm{step}}$ with a magnitude of
$\delta K$, is thus directly associated to an out-of-plane field
$H_{z\left(q-x_{\textrm{step}}\right)}^{K}=-\frac{\delta
K}{2\mu_{0}M_{\textrm{s}}}\cosh^{-2}\left(Q\frac{q-x_{\textrm{step}}}{\Delta}\right)$,
as plotted in Fig. 5(c). By numerically integrating the coupled space and time
evolution of $q$ and $\varphi$ in Eq. (7) and (8), considering the above
$H_{z\left(q-x_{\textrm{step}}\right)}^{K}$, it is possible to reproduce the
simulated results with a very good agreement. The lines in Fig. 5(a) show the
time evolution of $q$ and $\varphi$ by considering this effective local field
with the colors indicating the DW state:
$\varphi\in]-\frac{\pi}{2},\frac{\pi}{2}[$ (in green) and
$\varphi\in]\frac{\pi}{2},\frac{3\pi}{2}[$ (in purple). The background colors
in Fig. 5(b) show the final velocity of the DW obtained with this approach:
stopped in gray, transmitted in green, and reflected in purple.

The DW reversal conditions can even be obtained by considering $H_{\delta K}$
as a uniform field of magnitude $H_{z}^{K}=-\frac{\delta
K}{\mu_{0}M_{\textrm{s}}}$ for
$q\in[x_{\textrm{step}}-\Delta,x_{step}+\Delta]$ and 0 outside this region (as
plotted Fig. 5(c)). Therefore, the reflection, transmission, or pinning of the
DW is given by the DW final state for $q=x_{\textrm{step}}\pm\Delta$ after the
DW interaction with the anisotropy step. Since $H_{\textrm{SOT}}\ll H_{z}^{K}$
and $\alpha H_{\textrm{KIP}}\ll H_{z}^{K}$, Eq. (7) and (8) can be linearized
such as :
$\dot{q}\approx\frac{\Delta}{\alpha\tau_{0}}\left(-Q\alpha\frac{\delta
K}{2K_{IP}}+\frac{\sin 2\varphi}{2}\right)$ and
$\dot{\varphi}\approx-\frac{Q\delta K}{2\alpha K_{IP}\tau_{0}}$. The frontiers
between the different regions of the diagram in Fig. 5(b) can thus be
extracted. For
$\varphi_{\left(q=x_{\textrm{step}}-\Delta\right)}=\varphi^{\infty}\in]-\frac{\pi}{2},\frac{\pi}{2}[$,
the frontier between the regions 1 and 2 in Fig. 5(b) is given by
$\varphi_{\left(q=x_{step}+\Delta\right)}\leq 0$. This gives $\delta
K_{[1/2]}=QK_{\textrm{IP}}\left(\frac{\sin^{2}\varphi^{\infty}}{2+\alpha\varphi^{\infty}}\right)$.
The frontiers between region 2 and regions 3 and 4 is given by
$\varphi_{\left(q=x_{\textrm{step}}+\Delta\right)}\geq\frac{\pi}{2}$ which
leads to $\delta K_{[2/3]}=\delta
K_{[2/4]}=-QK_{\textrm{IP}}\left(\frac{\cos^{2}\varphi^{\infty}}{2-\alpha\left(\frac{\pi}{2}-\varphi^{\infty}\right)}\right)$.
The frontier between regions 3 and 4 is given by two conditions:
$\varphi_{\left(q=x_{\textrm{step}}+\Delta\right)}\geq\frac{\pi}{2}$ as
before, but also $\varphi_{\left(q=x_{\textrm{step}}-\Delta\right)}\leq\pi$
corresponding to the two reversal process case. This gives $\delta
K_{[3/4]}=-QK_{\textrm{IP}}\left(\frac{\sin^{2}\varphi^{\infty}}{\alpha\left(\pi-\varphi^{\infty}\right)}\right)$.
These frontiers are shown by black lines in Fig. 5(b) and match both the
simulated and the numerical results with a good agreement. Since these
calculations are based on the linearization of Eq. (8), they are more accurate
if $\varphi$ only deviates slightly from
$\varphi_{\left(q=x_{\textrm{step}}-\Delta\right)}=\varphi^{\infty}$, such as
the frontier between 1 and 2 for low current and the one between 2 and 3 for
$J$ close to $J_{\textrm{c}}$. This results show that by adapting the
anisotropy step $\delta K$ to the current driving the DW, it is possible to
select one of the processes. Such locally variable anisotropy can be
controlled by using voltages [42, 43], and can lead to a three-state DW
transistor.

Figure 6: DW-based oscillators: (a) shows an example of the DW oscillations
for a DW between two anisotropy steps (similar to the case 3 in Fig. 5(a) with
a continuous current of $J=3$ GA/m2 and $\delta K=\pm$0.02 MJ/m3 and for an
initial state $Q=1$ and $\varphi^{0}=0$. Black dots, black lines, the gray
profile, and colored lines have the same meanings as in Fig. 5(a). (b) shows a
sketch of a DW oscillator based on a finite magnetic wire where the continuous
electrical current is injected in the bottom HM layer. (c) shows the different
behaviors of the DW in such devices: DW stopped, destroyed or reflected by the
track extremities, depending on the current density for respectively:
$J=1.5,\,3,\,4.5,\,7$ GA/m2 in the ($q$,$\varphi$) space. The background
indicates the shape of the finite magnetic track.

In the case of a DW between two anisotropy steps corresponding to the region 3
of the diagram Fig. 5(b), the DW will be reflected at each of the two steps
leading to the oscillations of the DW propagation as shown Fig. 6(a),
analogous to confinement in a Fabry-Perot cavity.

### Finite magnetic track.

A similar effect can be obtained at the extremities of a finite magnetic wire,
as shown Fig. 6(b) and (c). The black points in Fig. 6(c) correspond to the
simulated trajectory of the DW in $(q,\varphi)$ space and the background color
corresponds to the shape of the magnetic element (visible Fig. 6(b)). As
previously, the extremities of the magnetic wire act on the DW internal angle
$\varphi$. For low current, the DW stopped in its Bloch configuration (here
$\varphi=0$) just before the end of the wire, as shown in the first case of
the Fig. 6(c). If the current increases, the DW goes off the end of the track
before being in its Bloch state and the magnetic wire is then fully saturated
as shown in the second case of the Fig. 6(c). For a sufficiently large
current, the end-of-track effect pushes the DW across its reversal points
before it leaves the wire and the DW is then reflected along the track. A
similar process occurs at the other extremity and then the DW oscillates
continuously from one to the other extremity, as in the third case of Fig.
6(c). If $J>J_{\textrm{c}}$ and the DW is in the middle of the wire, far from
the track extremities, the DW is stopped as expected. However, if the DW is
close to the track end, it will be attracted and then stopped in its Bloch
state after small oscillations of $\varphi$ around $\frac{\pi}{2}$ as in the
fourth case of the Fig. 6(c).

The shape of the track extremity has only small effects on these results. The
possibility to make DW walls oscillate in such small elements is also a new
way for small magnetic oscillators with a triangular wave time evolution, and
for which the frequency $f=v/2l$ is just governed by the DW velocity $v$ and
the track length $l$.

## Conclusion

The possibility to stabilize Néel DWs with an achiral mechanism allows for the
bistability of these magnetic textures, where both chiralities are
energetically degenerate. This gives rise to new forms of SOT-driven DW
dynamics. Two propagation directions for such DWs are now possible, only
depending on their initial state. By increasing the current density, the DW
exhibits a non-monotonic velocity law with a maximum of velocity and a
critical current above which the DW is locked in its pure Bloch state and does
not move. This non-uniformity of the velocity law also induces important
transient regimes and out-of-equilibrium dynamics. Finally, the possibility of
controlled or random reversal of the DW chirality, and therefore, of its
propagation direction, has been demonstrated under certain conditions (large
current pulses, out-of-plane field, steps of magnetic anisotropy or the
extremities of finite length magnetic tracks). Of particular note are the
ability to reflect or transmit DWs at a step in the anisotropy $K$, with the
potential for control through voltage-controlled magnetic anisotropy, and new
forms of DW oscillator in which a DW is continuously reflected between
anisotropy seps or the ends of a magnetic nanowire. All these completely new
behaviors, simulated numerically, have been perfectly reproduced with a
suitable analytical model. All these new dynamics offer very interesting
possibilities for future DWs-based devices that can possibly mix storage and
processing.

## Methods

Micromagnetic simulations. The micromagnetic simulations have been performed
using the Mumax3 open-source software allowing a finite difference approach to
the magnetization dynamics [44]. The magnetic wire is discretized into 2 nm
$\times$ 2 nm $\times$ 1 nm cells. A saturation magnetization of
$M_{\textrm{s}}=1\times 10^{6}$ A/m, an exchange stiffness of $A=3\times
10^{-11}$ J/m, an out-of-plane uniaxial anisotropy of $K_{0}=1\times 10^{6}$
J/m3, a gyromagnetic ratio of $\gamma_{0}=0.2\times 10^{6}$ Hz/(A/m) and a
Gilbert damping parameter of $\alpha=0.1$ have been considered [22, 45]. The
chosen a 32 nm width and a 1 nm thickness for the wire leads to an effective
uniaxial shape anisotropy along the magnetic wire of magnitude
$K_{\textrm{IP}}=14$ kJ/m3, which stabilizes bistable Néel DWs. We define the
$x$ direction as along the wire, $y$ transverse to it, and $z$ normal to the
wire plane, as shown in Fig. 1(a). The spin current $J_{\textrm{s}}$ is
injected into the magnetic wire with a spin direction
$\bm{\mathbf{s}}=\hat{\bm{\mathbf{y}}}$ for all the results we present. For
simplicity, we assume a charge/spin conversion ratio of unity so that the
charge and spin currents have equal magnitudes.

$q-\varphi$ model. The analytical collective coordinates model describes the
DW dynamics with two parameters : $q$ and $\varphi$ are the central position
and the internal angle of the DW, as shown Fig. 1(a). It is obtained by the
analytical integration of the LLG equation considering the rigid Bloch DW
ansatz [34] :
$\bm{\mathbf{m}}=\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right)$
with $\theta(x)=2\arctan(e^{Q\frac{x-q}{\Delta}})$, $\phi(x)=\varphi$
uniformly along the wire and $\Delta>0$ the DW width. It leads to:

$\displaystyle\alpha\frac{\dot{q}}{\Delta}+\dot{\varphi}$ $\displaystyle=$
$\displaystyle
Q\gamma_{0}\left(\frac{\pi}{2}H_{\textrm{SOT}}\cos(\varphi)-\frac{\partial_{q}\sigma}{2\mu_{0}M_{\textrm{s}}}\right),$
(9) $\displaystyle\frac{\dot{q}}{\Delta}-\alpha\dot{\varphi}$ $\displaystyle=$
$\displaystyle\gamma_{0}\left(\frac{\partial_{\varphi}\sigma}{2\Delta\mu_{0}M_{\textrm{s}}}\right),$
(10)

where
$H_{\textrm{DL}}=\frac{\hbar}{2e}\frac{1}{\mu_{0}M_{\textrm{s}}}\frac{J\theta_{\textrm{SH}}}{t}$
is an effective field associated with the resulting SOT on the DW. $\sigma$ is
the surface energy density integrated along the infinite and homogeneous
magnetic wire:

$\sigma=2\Delta
K_{\textrm{IP}}\sin(2\varphi)-2\mu_{0}M_{\textrm{s}}H_{\textrm{z}}q+\sigma_{0},$
(11)

with $\sigma_{0}$ depending on the exchange stiffness $A$ and the effective
out-of-plane anisotropy $K=K_{0}-\frac{\mu_{0}}{2}M_{\textrm{s}}^{2}$
(independent of $q$ and $\varphi$), $H_{\textrm{z}}$ the eventual applied or
effective out-of-plane field and $K_{\textrm{IP}}$ the easy-axis shape
anisotropy resulting from the demagnetization effects. Thus,
$\frac{\partial_{q}\sigma}{2\mu_{0}M_{\textrm{s}}}=-H_{\textrm{z}}$ and
$\frac{\partial_{\varphi}\sigma}{2\Delta\mu_{0}M_{\textrm{s}}}=H_{\textrm{KIP}}\frac{\sin(2\varphi)}{2}$
with $H_{\textrm{KIP}}=\frac{2K_{\textrm{KIP}}}{\mu_{0}M_{\textrm{s}}}$ the
field associated with the magnitude of the in-plane anisotropy.

For a uniform magnetic wire, in the absence of field, $\partial_{q}\sigma=0$
and then Eq. (2) gives an implicit analytical expression of $\varphi_{(t)}$
variations such as :

$e^{-t/\tau_{0}}=\frac{\sqrt[2\left(Qj-1\right)]{\frac{1-\sin\phi}{1-\sin\phi^{0}}}}{\sqrt[2\left(Qj+1\right)]{\frac{1+\sin\phi}{1+\sin\phi^{0}}}\,\sqrt[\left(j^{2}-1\right)]{\frac{Qj-\sin\phi}{Qj-\sin\phi^{0}}}},$
(12)

depending on $\varphi^{0}$, the internal angle at the considered initial time
$t=0$.

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###### Acknowledgements.

This project has received funding from the EPSRC under grant number
EP/T006803/1 and the European Union’s Horizon 2020 research and innovation
program under the Marie Sklodowska-Curie grant agreement No 750147. K.J.A.F.
acknowledges support from the Jane and Aatos Erkko Foundation. The main part
of these numerical simulations were performed on MAGI, the computing platform
of the University Sorbonne Paris Nord, managed by Nicolas Greneche. We thank
Yves Roussigné (from the LSPM) for fruitful discussions.