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# Magnetic domain walls : Types, processes and applications

G. Venkat  Department of Materials Science and Engineering, University of
Sheffield, S1 3JD, UK  D. A. Allwood  Department of Materials Science and
Engineering, University of Sheffield, S1 3JD, UK  T. J. Hayward  Department of
Materials Science and Engineering, University of Sheffield, S1 3JD, UK

###### Abstract

Domain walls (DWs) in magnetic nanowires are promising candidates for a
variety of applications including Boolean/unconventional logic, memories, in-
memory computing as well as magnetic sensors and biomagnetic implementations.
They show rich physical behaviour and are controllable using a number of
methods including magnetic fields, charge and spin currents and spin-orbit
torques. In this review, we detail types of domain walls in ferromagnetic
nanowires and describe processes of manipulating their state. We look at the
state of the art of DW applications and give our take on the their current
status, technological feasibility and challenges.

##  I Introduction

In recent years, there has been an explosive growth in research into magnetic
nanostructures for memories, sensors and computation. Advantages of magnetic
systems include their inherent non-volatility, fast speed of operation, low
power consumption, nonlinear responses, negligible current leakage, and
compatibility with CMOS fabrication techniques [1]. This has led to magnetic
random access memory (MRAM) products being available in the market as well as
multiple demonstrations and proposals for Boolean and unconventional computing
applications.

In this context, magnetic domain walls (DWs) in patterned magnetic nanowires
have been proposed for many applications, which are introduced below. DWs are
regions in a magnetic material between uniformly-magnetised domains through
which magnetisation changes direction. DWs are typically small (with widths
from a few nanometres to hundreds of nanometres), move relatively quickly
(with reports of speeds of $18\,\rm{km/s}$) [2] and can be manipulated using a
variety of stimuli, as we discuss later.

Magnetic DWs in nanowires have been developed as the information carriers for
memory devices such as the switchable components of MRAM cells [3] and shift
registers, which include the racetrack memory architecture [4] among others
[5, 6]. These memories offer high storage densities with nanowire feature
sizes scaled down to $20\,\rm{nm}$ [7], low read/write energies, low leakage
current and years of data retention [8] in devices without movable parts. The
intense research activities over years has now led to the prospect of
commercialisation by exploiting CMOS fabrication and device/circuit level
integration [9, 10].

DWs in nanowires have also been proposed for logic applications [11, 12].
Control of the motion and interaction of DWs at nanowire junctions has allowed
logic gates [8], adders [13, 14] and shift registers [6] to be demonstrated.
DWs also show complex behaviours when encountering energy barriers, such as at
nanowire junctions. These behaviours are often stochastic in nature and highly
sensitive to thermal perturbations, [15, 16] which was long regarded as a
severe impediment to performing information processing tasks with magnetic
DWs. However, there have been examples of non-Boolean logic devices that make
use of DW stochasticity to realise, for example, neuromorphic computing [17]
and stochastic computing [18] along with other devices such as a random number
generator using DW-based Galton board [19]. The stochastic nature of DW
depinning from nanowire junctions has also been shown to lead to reliable and
complex many-body ensemble behaviour in large arrays of overlapping nanorings
and proposed as the basis for reservoir computing [20].

Beyond information technology applications, DWs in magnetic nanowires have
been used to realise optical based [21] and multiturn [22] magnetic field
sensors. The stray magnetic fields from magnetic DWs in nanowires can also be
used to interact with [23] and position [24] ultracold atoms as well as
control magnetic nanoparticles for biological sensing and cell positioning
applications [25]. There have been proposals [26] and demonstrations [27, 28]
of trapping superparamagnetic beads using DW motion which can potentially be
useful for manipulating biological cells and DNA.

The movement of DWs in magnetic nanowires has been utilised for the majority
of proposals of DW related technological applications and the last two decades
have seen a surge in the study and manipulation of DWs using a variety of
stimuli such as magnetic fields, charge and spin-polarised currents. Novel
spin-related phenomena such as spin-orbit and spin-transfer torques have been
utilised for controlling DW positions and velocities in nanowires in devices
that move the technology from lab demonstrations to being closer to commercial
realisation.

This review discusses the various approaches of creating functionality from
dynamics of DWs in both in-plane and out-of-plane magnetised nanowires, with a
particular focus on the novel computing proposals that have emerged recently.
We will also present fundamental aspects of the nature and control of DWs in
nanowires to assist newcomers to the subject.

The outline of the review is:

  * •

Section 2 details types of magnetic DWs patterned from thin films. This
section also discusses available stimuli to manipulate and control magnetic
DWs in patterned nanowires, including magnetic field, strain, voltage, and
spin-polarised currents. We also discuss how various nanowire-based
structures, including straight nanowires, nanorings, geometric defects, and
nanowire junctions can be used to position DWs and tune their properties.

  * •

Section 3 details DW processes in nanowires with particular attention to the
stochastic features of these behaviours and the associated effects of thermal
perturbations.

  * •

Section 4 concentrates on applications proposed using DWs in magnetic
nanowires, including : memories, Boolean, neuromorphic and reservoir
computing; and non-computing applications, including field sensors, biological
applications, and transporting ultra-cold atoms.

  * •

Section 5 concludes with a summary and future outlook.

##  II Magnetic domain walls

###  II.1 Domain wall structure

Rectangular-cross-section magnetic nanowires typically have thickness from a
few atoms to tens of nanometres, widths from 50 - 500 nm, and can extend in
length to several tens or hundreds of micrometres. They are usually fabricated
using standard lithography techniques such as electron beam lithography [29]
or focussed ion beam milling [30] to pattern thin films into user-defined
designs. We will be considering these rather than nanowires fabricated using
electrodeposition techniques [31, 32] as more control on nanowire features is
possible with lithography and the vast majority of applications proposed using
DWs have considered lithographically patterned magnetic nanowires.

Patterned nanowires made of soft magnetic materials such as
$\rm{Ni}_{81}\rm{Fe}_{19}$ (Permalloy) have a magnetic structure dominated by
shape anisotropy. The extended geometry of such a nanowire means magnetic
domains are usually oriented in plane, along the wire length. This results in
DWs that lie directly across the width of a nanowire to separate oppositely-
oriented magnetic domains. At their simplest, the possible configurations are
referred to as ‘head-to-head’ (H2H) when the DW separates domains with
magnetisation oriented towards the wall and ‘tail-to-tail’ (T2T) when the
domains are oriented away from the wall (Figure 1 (a) and (b)). These DWs most
often have in-plane magnetised (IM) configurations, due to the wires’ width
usually being greater than their thickness. The simplest H2H (and T2T) DW
structure is the ‘transverse’ configuration [33], where moments in the centre
of the wall are perpendicular to the nanowire edge (Figure 1 (i) and (ii)).
This occurs in thinner, narrower nanowires and results in decreased exchange
energy and a high magnetostatic energy. In thicker, wider soft ferromagnetic
nanowires, DWs form ‘vortex’ configurations where the moments rotate
$360^{\circ}$ around a vortex core (Figure 1 (iii) and (iv)). This structure
has a low magnetostatic energy, at the expense of an increased exchange
energy. DW structures with opposite chirality, i.e. the winding direction of
magnetisation through the DW, are degenerate in a perfectly straight, defect-
free nanowire, but lead to different behaviours when encountering asymmetric
wire features or transverse magnetic fields [34]. Asymmetric transverse DWs in
soft ferromagnetic rectangular nanowires have also been predicted [35] and
then imaged using Fresnel-mode electron beam imaging [36].

Figure 1: (a) and (b): DW configurations in the top view of soft ferromagnetic
patterned nanowires, obtained from micromagnetic simulations performed using
Mumax3 [37]. Colour maps the local magnetisation direction and the inset to
each figure part shows a simple schematic of the DW configuration. (c) and (d)
shows schematics of Néel and Bloch type DWs in out-of-plane magnetised
nanowires

The in-plane DW textures shown in Figure 1 can also be described using a
system of magnetic topological defects [38]. For example, transverse DWs can
be described in terms of magnetic edge ‘defect’ states with fractional winding
numbers while vortex DWs can be described in terms of vortex ’defect’ states
with integer winding numbers. This approach provides a simple nomenclature for
capturing DW structure and chirality and can be useful to categorise the
behaviour of DWs and understand their interactions both with each other and
with geometric features in nanowires. DW chirality has also been used
functionally to manipulate DW trajectory [39], measure DW “fidelity” (a
minimum length scale over which structural changes of the DW occur) [40], and
to form the basis of applications such as chirality-based DW memory cells [41]
and logic gates [42].

There has also been much interest in nanowires with perpendicular magnetic
anisotropy (PMA) which are out-of-plane magnetised (OOPM). DWs in OOPM
nanowires are usually much reduced in width of a few nanometres compared to
in-plane-magnetised nanowires [43], which offers improved current-induced
motion efficiency and higher data density [44]. There are usually two types of
DWs in such structures which are the Néel and Bloch types. These DWs usually
have a Bloch structure in wider nanowires and transition to Néel DWs in
narrower structures [45] (shown in Figure 1 (c) and (d)).

###  II.2 Domain wall nucleation

Figure 2: Some of the different nucleation pads used for injecting DWs into
nanowires are (a) rectangular [46], (b) circular [47], (c) attached [48] and
(d) elliptic [49] in shape.

Controlling the nucleation of DWs in nanowires is important for many
experiments and applications in order to test the behaviour of magnetic
features or to represent data.

The simplest approach involves fabricating a large nucleation pad, or
‘injection’ pad, at the end of a wire [50]. Injection pads are typically made
with lateral dimensions from single to tens of micrometres, and with a variety
of shapes, e.g. rectangular, circular, attached or elliptical (shown in Figure
2). The weaker shape anisotropy of the pad compared with the nanowire means
that an externally-applied global magnetic field causes a DW to nucleate
within the pad first, and then be injected into the nanowire at magnetic
fields far lower than the nucleation field of a simple wire end. Injection
pads often lead to a range of DW structures being introduced to an adjoining
IM nanowire as a variety of domain configurations may exist in the larger pad
[51, 52], although DW chirality can be controlled by careful design of an
injection pad and placement of the attached nanowire [53].

DWs in IM nanowires can also be nucleated by using the Oersted field
associated with currents driven through patterned current-carrying wires that
cross over the magnetic nanowire [54, 55]. This approach produces consistent
DW structures using local excitations without the need for a global magnetic
field, while the pulsed nature of the current allows more precise timing of
nucleation events to synchronise with a wider experiment [56].

Stein et al. [57] have shown that combining current pulses of different
polarities and external magnetic fields lead to reproducible creation and
annihilation of DWs in an IM nanowire (shown in Figure 3 (a)). Figure 3 (b)
shows changes in measured resistance after applying multiple current pulses
and the different resistance levels corresponds to various types of DWs being
formed in the nanowire. Specific choices of the polarity and duration of of
the current pulses and polarity of magnetic fields led to stochastic pinning
of DWs in the nanowire region bounded by the current lines as well as the
injection of vortex and transverse walls into the nanowire. In the same year,
Hayashi et al. [58] used short lived current pulses (with $\sim\,\rm{ns}$
lifetimes) to decrease the magnetic fields required to nucleate DWs in IM
nanowires. They found that the nucleation field reduced by around half and the
distribution of the switching fields decreased in width as well and the
authors attributed this to using localised fields generated by the current
lines.

Figure 3: A scanning electron microscopy image of the magnetic nanowire with a
pair of vertically crossing current lines. (b) The variation of the nanowire
resistance after one pulse (green) and after two pulses (red). Taken from
[57].

Injections pads [59] and the Oersted fields from current carrying lines [60]
have also been used to inject DWs in OOPM nanowires. Zhang et al. [61] used
$\Pi$ shaped current lines for increased efficiency of DW injection in Co/Ni
multilayer nanowires by modifying the shape of the current and localising the
Oersted field to inject DWs. Other methods to inject DWs in OOPM nanowires
include controlling the PMA by ion-irradiation [62] and ion milling [63] which
change the physical structure and using Joule heating in particular regions of
the nanowire via current injection [64].

###  II.3 Domain wall motion

There are several approaches to achieve controlled DW motion in patterned
magnetic nanowires. It is well known that, in general, the application of a
magnetic field causes ferromagnetic domains parallel to the field to grow and
oppositely-oriented domains to shrink, with the changes in domain size
mediated by DW motion. This can be understood at a more microscopic level by
examining the Landau-Lifshitz-Gilbert (LLG) equation of motion [65]

$\frac{\rm{d}\mathbf{M}}{\rm{d}t}=\mu_{0}\gamma\left(\mathbf{M}\times\mathbf{H}_{\rm{eff}}\right)+\frac{\alpha}{|\mathbf{M}|}\left(\mathbf{M}\times\frac{\rm{d}\mathbf{M}}{\rm{d}t}\right)$
(1)

Here $\mathbf{H}_{\rm{eff}}$ is the effective local magnetic field that acts
on the magnetisation, $\mathbf{M}$ in the time interval $\rm{d}t$. $\gamma$ is
the electron gyromagnetic ratio, $\mu_{0}$ is the permeability of free space
and $\alpha$ is the Gilbert damping parameter. $\mathbf{H}_{\rm{eff}}$ is
calculated from the derivative of the system energy with changes in
$\mathbf{M}$ [66], which includes any applied external field,
$\mathbf{H}_{\rm{app}}$, as well as other energy contributions, e.g.
magnetostatic, magnetocrystalline, and exchange energy terms. Equation (1) is
most accurate when solved at a scale up to the exchange length of a material,
i.e. the length over which magnetisation remains approximately uniform.

The first term of the left-hand side of Equation (1) shows that the initial
response of $\mathbf{M}$ is to precess around $\mathbf{H}_{\rm{eff}}$. In IM
nanowires and in a 1-D approximation, this means that a magnetic field applied
parallel to the long axis of the nanowire (the longitudinal field component)
will cause DW magnetisation, which is orthogonal to the field, to start to
precess out of plane. This, in turn, creates an out-of-plane demagnetisation
field that acts upon the DW’s remaining in-plane component of magnetisation
and causes it to precess towards alignment with the applied field direction,
expanding the magnetic domain in this process [67]. The second term describes
the damping of the precessional motion, which occurs in all ferromagnetic
materials and, ultimately, causes local magnetisation to align to the local
$\mathbf{H}_{\rm{eff}}$. Similar dynamics occur in OOPM materials with a
magnetic field applied out of plane where the PMA plays a significant role.

The fields required to propagate DWs are usually low ($1-20\,\rm{mT}$) both
for soft magnetic materials, such as Permalloy [68] and for out-of-plane
manipulation in some OOPM materials like Co/Pt [69] although OOPM systems such
as FePt/Pt require considerably higher fields [70]. The magnetic fields are
usually generated by an external electromagnet, and the resulting high power
consumption usually limits the use of field-driven DW motion to laboratory
experiments. There has, therefore, been intense interest in developing
alternative, low-power methods of controlling DW motion and position.

DW motion in magnetic nanowires can also be induced by passing electric
current directly through the wires. This approach is popular due to it
offering local addressability, power efficiency, and fast DW motion, and has
resulted in a large number of proposals for DW-based MRAMs and a commercial
memory device [71]. Current control of magnetisation in thin films has been
reviewed extensively from theoretical [72] and experimental [73, 74]
viewpoints. Briefly, the spin angular momentum carried by a spin-polarised
charge current causes an adiabatic ‘spin transfer’ torque (STT) [75, 76, 77]
to be applied to the DW, which causes DW displacement. STT can also have a
non-adiabatic contribution with an unclear origin with contenders being the
force on the conduction electron due to gradient in the _s-d_ exchange field,
linear momentum transfer and spin-flip scattering [74]. STT is very relevant
technologically as it can eliminate the need for magnetic fields and,
therefore, for inefficient solenoids or electromagnets. STT has been proposed
(in simulations) to nucleate DWs at specific locations in nanowires [78] and
Phung et al. [79] showed that they could inject DWs in nanowires with both IM
and OOPM regions (obtained using ion implantation) using STT although the
injection was stochastic in nature which might not be suitable for Boolean
applications. One challenge with SST driven DW motion is that the position of
the electrical contacts necessary for current-induced motion, of course,
create natural limits to the extent of DW motion in nanowires [80].
Furthermore, the high current density for DW motion
($10^{10}-10^{12}\,\rm{A/m^{2}}$) remains a technological challenge, both in
terms of power efficiency and the resultant heating, which might damage the
structure or change its behaviour. This is partly due to current-driven DW
motion having an intrinsic pinning, even for an ideal wire, which leads to a
threshold current both for IM [74] and OOPM systems [54]. The result is that
sustained STT driven DW motion usually requires an external magnetic field
[81, 82, 83, 84]. Various proposals have been made for decreasing the current
densities required for efficient control of DW motion, including using
changing aspect ratios of nanowires [85], different geometries of the current
injection [86], and OOPM materials [87, 88, 89]. Tailoring the PMA in OOPM
systems have been predicted lead to increased efficiency of SST driven DW
motion [90]. There have also been interesting reports of lower switching
current densities in systems involving synthetic ferrimagnets [91] and
antiferromagnets [92], due to the in-plane magnetic interactions in them
leading to narrow DWs.

Figure 4: Types (a) $x$ and (b) $y$ configurations for the SOT due where the
injected current (black dashed arrow) and magnetisation (purple arrow) lie
along and perpendicular to each other respectively. The splitting of electrons
of different spin are shown as well. The ’type-x’ configuration required an
out-of-plane for breaking the symmetry and switching the magnetisaton. Taken
from [93].

Recently, non-magnetic (NM)/ferromagnetic (FM) bi-layers have opened up
multiple schemes of controlling the magnetisation in the magnetic layer. The
spin Hall effect (SHE) [94] describes the generation of a transverse spin
current at the FM/NM interface due to spin-orbit coupling in the NM. The SHE
can significantly affect the DW dynamics in nanowires [95] and can even reduce
the current densities required for DW switching [96, 97]. The torque exerted
by this transverse spin current on the magnetisation in the FM is known as
spin-orbit torque (SOT) and this has been studied intensely in the last few
years [98, 93]. Depending on the relative orientation of the injected current
and the magnetisation in the FM, there can be two configurations of SOT for
switching as shown in Figure 4. The "type-x" configuration require a magnetic
field out-of-plane of the nanowire for breaking the symmetry of the system for
deterministic switching which can be inconvenient for technological
applications. SOT-induced schemes overcome limitations of STT with the added
advantages of reduced power consumption and after device operation. SOT-
induced DW depinning has been predicted [99] and demonstrated [100] in
Ta/CoFeB thin films nanowires, making available an added method of controlling
DWs [101]. While most such studies have been with OOPM systems, there have
also been analytic predictions of SOT driven DW motion in IM systems such as
Pt/CoFeB/MgO [102]. Further novel phenomena such as the Rashba effect and the
Dzyaloshinskii-Moriya interaction (DMI) can also be used to tune DW velocity
[103, 104, 105] by controlling the DW energy using these effects. For
instance, Dao et al. [106] have shown that DWs can be controllably pushed into
nanowires using the chiral interactions induced by DMI between in-plane and
out-of-plane magnetised trilayers of $\rm{Pt}/\rm{Co}/\rm{AlO}_{x}$ in
combination with SOTs induced using currents. The capability of SOT to
nucleate DWs are found to strongly be chirality dependent and can required
current densities less than traditional current induced DW injection methods.
SOT offers a novel and useful control of DWs using electric currents for
device applications, although some challenges such as efficient magnetisation
switching without out-of-plane fields have to be overcome.

DWs in nanowires can also be controlled using applied strain in
magnetostrictive materials, in which strain mediates magnetisation changes via
inverse magnetostriction, also called the ‘Villari effect’ [107]. Metallic
magnetostrictive nanostructures are often coupled to underlying piezoelectric
materials to allow strain in the piezoelectric to be created via electric
potentials applied to patterned contacts before being transmitted to the
magnetic element (Figure 5 (a)). Such systems are termed ‘artificial
multiferroics’, due to their heterogeneous structure and show a variety of
magnetic and DW driven phenomena [108]. Local variations in static strain
across the length of a magnetostrictive nanowire create potential barriers or
wells that can be used to addressably pin or even move DWs along the wire
length [109, 110]. Lei et al. [111] have shown that by applying a static
strain in piezoelectric/multilayer magnetic structures, the magnetic response
can be reproducibly tuned (Figure 5 (b)). Other proposals for manipulating DWs
using strain include applying dynamic strain profiles and Dean et al. [112]
proposed using standing surface acoustic waves (SAWs) along the length of an
IM magnetic nanowire to create an array of DW pinning sites. Subsequently,
Adhikari et al. [113] found that exciting DWs in an array of OOPM Co/Pt
multilayer nanowires using SAWs (created using integrated digital transducers
and shown in Figure 6 (a)) resulted in an increase in the DW depinning
probabilities by approximately a factor of 10 (Figure 6 (b)). Technological
proposals based upon strain-induced control of DWs include memories [114] and
manipulation of magnetic beads in a fluid [115].

Optical pump-probe techniques have also been used to move DWs in IM Co/Ni
films [116] and open up possibilities of ultrafast control and writing of
magnetisation. We therefore see that many varied and technologically relevant
schemes are available for initiating and controlling DW states in both IM and
OOPM nanowires.

Figure 5: A scanning electron microscopy image of the piezoelectric/multilayer
device used for showing effects of strain on the magnetic response. (b) The
variation of the giant magnetoresistance of the device for different voltages
applied to the piezoelectric which leads to a modification of the strain on
the nanowire. Taken from [111]. Figure 6: A schematic of the setup used to
modify DW pinning/depinning using SAWs showing the magnetic stripes and
integrated digital transducers (IDTs). (b) The depinning probability as a
function of increasing SAW voltage for three representative pinning sites.
Taken from [113].

###  II.4 Positioning domain walls

Geometric modifications have been also used to create sites for
pinning/depinning of DWs in nanowires by altering the magnetic energy
landscape. In magnetic nanowires, local geometric modifications invariably
cause change to the magnetostatic energy of a DW, with subsequent
rearrangements in the magnetic configuration of a DW at the modified side
leading to changes in exchange energy [117]. These changes depend upon the
structure and chirality of the DW, and the precise geometry of the modified
region, which has given a lot of scope for investigation.

Figure 7: A schematic of a trilayer nanowire (in black) and DWs in them were
manipulated using Oersted fields created by passing currents through Cu wires
shown at the ends of the nanowire. (b) The variation of the depinning field of
a DW from the notch with the neck width of the notch. A higher depinning field
is observed for a notch with lower neck width. Taken from [118].

The nucleation and pinning of DWs in nanowires are dependent on the defects
introduced during fabrication of the wire. Dutta et al. [119] found that edge
roughness of the wire can trap DWs at length scales less than the resolution
of the fabrication process both in IMA and PMA nanowires. They studied a
system of nanowires in the regime where the DW width was smaller than the
correlation length of the wire roughness (i.e. the length scale defining the
transition between smooth and rough length scales of the nanowire) and found a
discrete distribution of DW traps in the nanowires which limits the precision
of the placement of a DW in the nanowire.

Himeno et al. [118] used artificial necks of varying width in IM NiFe/Cu/NiFe
trilayer submicrometer wires to control the depinning fields of the DWs
(Figure 7). Kläui et al. [120] also used notches in an IM curved ring nanowire
and by injecting currents between various contacts positioned between the
notches, they found multiple transitions in the anisotropic magnetoresistance
(AMR) response from the ring which corresponded to DW pinning at the notches.
Such studies led to further investigations of notches of different shapes [34,
117], modifying the position of notches in the nanowire [121], and proposals
of stepped nanowires for memory applications [122]. These demonstrated in
various ways how the potential landscape and depinning fields of DWs could be
modified.

Asymmetric geometric defects can be used to create additional levels of
control in IM nanowires. DWs of opposite chirality generally interact
differently with defects that are present on one side of a nanowire only,
regardless of whether they are notches [123, 124], protuberances [125], or
adjoining wires at $90^{\circ}$ to the original wire [40]. This has been used
as a means of selecting one chirality over another [126] and the basis of a
logic system using vortex DWs [42, 127]. In all cases, if the geometric
modification is repeated on the other side of the wire, the chirality-
filtering is removed and DWs of the same structural type pass through with
identical depinning fields [56]. Asymmetry along the wire direction of a notch
or protuberance shape creates a different energy pathway for DWs travelling
through the wire in opposite directions. This leads to depinning fields that
depend on the direction a DW is travelling, in effect creating a ‘DW diode’
using both in-plane [128, 129] and OOPM materials [130].

External stimuli can be used to control DW pinning at geometric features. For
example, a DW in an IM nanowire can be pinned through magnetostatic
interactions with an adjacent and close-lying magnetic nanowire [131, 132,
133, 134, 135]. Also, an externally-applied transverse magnetic field can be
used to tune the DW depinning probability function at notches [42], which
gives a greater degree of control. Kim et al. [136] studied the depinning of
DWs at notches of different gap sizes in an OOPM nanowire and demonstrated
that the depinning fields and times could be controlled by changing the gap
sizes. The Oersted field from current-carrying lines patterned across magnetic
nanowires can also be used to pin DWs in an addressable manner [137, 138, 60],
although the Oersted fields created using these lines are usually relatively
low.

Precise understanding of energy pathways allows prediction of DWs experiencing
potential wells or potential barriers. This can be tested experimentally by
pinning a DW and then reversing the applied magnetic field. A potential
barrier will lead to ‘pinned’ DWs moving back away from the defect site under
low fields but a pinned DW will require a similarly-large field to move from
the site in either direction [131]. However, DW depinning from geometrical
modifications is complicated by thermally-activated stochastic processes [16],
which we will explore more later.

DWs have also been positioned by modifying the global wire geometry. Magnetic
fields can cause DW motion in nanowires whenever there is a field component
parallel to the wire direction. Curved nanowires, therefore, provide a natural
limit to the extent of DW motion and they have been used to assist in
exploring the nature of various DW phenomena in nanowires using magnetic
fields and currents in both IP and OOPM systems[139, 140, 141]. An in-plane
rotating magnetic field causes DWs to propagate around curved wire regions in
the directions of same handedness as the field rotation, which has been
exploited for clocking in logic [11] and sensor [22, 142] systems. The
depinning fields in curved nanowires is sensitive to the degree of curvature
of the nanowire and Glathe et al. [143] suggested that DW pinning in a curved
IM nanowire can be better controlled by designing corners as a polygonal line.
Modifying wire geometry can also be used to control DW chirality, with vortex
DWs in curved IM nanowires subject to an externally-applied magnetic field
observed to have chirality that is highly dependent upon the wire width and
magnetic field direction [144]. On the introduction of width gradients in half
rings to form structures having either point of axis symmetry, the authors
could generate vortex DWs with either same or opposite chirality via different
stochastic switching pathways. The stochasticity in these pathways could be
controlled by modifying the width gradient and temperature and the authors
indicated that this could be useful for applications.

Figure 8: (a) ‘Onion’ and (b) ‘vortex’ magnetic states in a soft ferromagnetic
nanowire ring. The DWs in the ‘onion’ state are highlighted by the circles and
the colour wheel maps the colours to magnetisation orientation.

IM ferromagnetic rings offer an interesting combination of having magnetic
properties similar to a straight nanowire locally but with a wider curved
geometry that creates well-defined magnetic states and makes positioning DW
straightforward. The lowest energy magnetic configuration of a soft
ferromagnetic ring is the ‘vortex’ state, in which magnetisation is oriented
circumferentially, i.e. following the local wire direction around the ring,
and no DWs are present (Figure 8 (a)). DWs can be introduced in pairs, with
the so-called ‘onion’ state referring to a configuration with two DWs, one H2H
and one T2T, on opposite sides of the ring (Figure 8 (b)). Other states are
possible [145] but the vortex and onion configurations form the bulk of
interest [146, 147]. Application of an in-plane magnetic field causes DWs to
align on opposite sides of a ring in a radial direction parallel to the field
direction [146]. This means in-plane rotating magnetic fields of sufficient
strength can be used to cause the DW pairs of an onion state to propagate
around the ring in the same direction as the field rotation [148]. Bisig et
al. [149] reported that rotating fields can also be used to set the chirality
of DWs with high fidelity in rings of varying diameters and width with thermal
activation playing a role in the switching process.

The magnetic behaviour of ring arrays can be quite different from that of
individual rings. For instance, the magnetostatic interactions within an array
of close-lying rings can cause a significant difference in the switching
fields of arrays with rings of varying widths and spacings [150]. In fact, the
switching of Cobalt rings arranged in a chain transverse to the switching
field always occurs in pairs [151]. These additional complexities in the
behaviour of multiple elements can be important for using such systems for
non-Boolean computing architectures, as shall be described later on.

We shall now discuss the behaviour of DWs at junctions of nanowires. Faulkner
et al. [152] considered an IM three terminal nanowire junction of Permalloy
for controlled DW injection (shown in Figure 9 (a)-(c)). They found that the
fields required for switching the magnetisation of the output arm of the
device was significantly lower when DWs were injected from either one or both
input arms compared to when no DW was injected in either input arm (seen from
the switching fields in the hysteresis loops in Figure 9 (d)-(f)). This was a
consequence of a single DW (in the case of DW injection from one of the input
arms) or linked DWs (when DWs are injected from both input arms) expanding in
the output arm from the junction and causing the magnetisaton to switch. Pushp
et al. [39] subsequently considered an IM Y-shaped nanowire junction and could
reliably inject DWs of a given chirality into each arm of the Y-shaped
structure by varying the strength of the injection field. Such IM junction
based nanowire devices were subsequently proposed for DW based Boolean logic
operations and we shall describe these in Section IV.2. DW behaviours at
junctions have also been considered in OOPM systems and Kwon et al. [153]
considered propagation of Néel DWs in a bifurcated nanowire of multilayer
Ni/Co grown on Pt. The DW injection into each of the two arms was at different
applied fields and this was attributed to the DMI preferring one tilt
direction of the DW surface leading to easier injection in one of the arms. We
thus see that the behaviour of DWs at nanowire junctions offer additional
opportunities of manipulating DW states for developing devices.

Figure 9: FIB images of typical three-terminal structures fabricated, with (a)
zero, (b) one, and (c) two DWs being injected into the junction. Directly
below each image is a hysteresis loop of the output arm corresponding to the
type of structure shown showing the difference in fields at which the
magnetisation switches. Taken from [152].

##  III Domain wall processes in nanowires

DWs propagating in straight, patterned nanowires show a large array of dynamic
behaviours over and above simple motion [68, 154, 155]. These behaviours are
beyond stimulus driven deterministic DW evolution that was considered in the
previous section and are heavily affected by thermally driven phenomena in the
nanowires. We introduce these additional behaviours in brief below.

Figure 10: (a) The measurement setup for a detecting the depinning of a DW in
an IM notched nanowire. (b) The relative probability of a DW depinning from
the notch (black), a vortex DW (red) and transverse DW (blue) being pinned at
the notch. Taken from [156].

We refer to the paper by Hayward et al. for most of the discussions in the
next two paragraphs [16]. DW processes are stochastic and experimental
observations can be classified into three categories [16]:

  * •

DW propagation, even in defect free nanowires, is not deterministic even for
magnetic fields greater than what is required for DW motion

  * •

DWs pinned at engineered pinning sites have rich, multimodal depinning
magnetic field distributions. In fact, it is common to observe the DW
depinning probability increase sigmoidally with applied magnetic field. This
was observed, for example, by Pi et al. [156], who studied DW pinning at a
notch in an IM Permalloy nanowire (Figure 10 (a)) by AMR (Figure 10 (b) black
line).

  * •

DWs move non-deterministically via defect sites when magnetic fields that
would not be expected to be able to induce depinning are applied.

Figure 11: (a) (Top) : A diagram showing the geometry of the nanowire
simulated in [16]. The initial vortex DW configuration is shown in higher
detail. The plots below show the DW velocity and normalised magnetisation in
different regimes of DW dynamics. Taken from [16].

The simple picture of DW motion in a nanowire described in Section II.3 holds
above a threshold field strength, due to overcoming imperfections in a wire,
and with constant mobility $\mu_{\rm{DW}}=\frac{v}{H}$, where $v$ is DW
velocity and $H$ the driving magnetic field [157, 158]. This ‘viscous’ regime
remains until the ‘Walker breakdown’ field, the value of which depends upon
nanowire material, geometry, and DW structure. Above this field, DWs first
enter a regime of oscillatory motion and negative differential mobility,
followed by a turbulent regime that sees the DW velocity mobility become
positive once more [16] (refer to Figure 11). Walker breakdown processes see
dramatic changes to the dynamic structure of DWs, accompanied by the near-
periodic stalling of DW motion and transition of DW configuration between
different types and chiralities. In the nanowire configuration considered in
[16], an oscillatory transition between VDW and TDW types of alternating
chiralities occurs. This stochasticity of DW structure has been explained as
thermally-driven fluctuations in magnetic field driven Walker breakdown
pathways [16]. Furthermore, thermal perturbations affect these different
regimes of DW dynamics quite significantly, especially above Walker breakdown,
when the intrinsic instabilities of the DWs are heavily influenced by
relatively small perturbations [16].

Such DW processes are not restricted to magnetic field stimuli. Similar
dynamics to the field driven dynamics described above have been predicted and
observed with current-driven DW motion in patterned nanowires. Duine et al.
[159] used the stochastic version of the LLG equation which has a stochastic
field component to account for temperature driven perturbations. The authors
found that the variation of the DW velocity with applied current was linear
even without any STT at non-zero temperatures due to these perturbations
indicating that they can play an important role. Subsequently, Hayashi et al.
[83] found that current-driven DWs propagating through an IM Permalloy
nanowire and pinned at a notch would have different transverse and vortex
structure of different chiralities, despite identical experimental conditions
and a single structure being used. They also found that each of the four DW
states observed had substantially different depinning fields but similar
depinning current densities and concluded that with current-induced depinning,
all the DW types may be converting to the same DW state. Higher current
densities again result in DWs experiencing Walker breakdown and turbulent
dynamics, although the Walker threshold current density can be found above the
point where the DW velocity has ceased to respond linearly to current [154].
Hybrid approaches to driving DW motion have also been studied and have shown
that application of a magnetic field lowers current driving thresholds [160,
73]. Furthermore, as seen in Section II.4, the patterned geometry of a
magnetic nanowire plays an important role in determining DW dynamics. For
example, modifying the curvature of nanowires has been predicted to modify the
magnetic-field-driven DW velocities, precession, and oscillation frequencies
[161]. The DW velocity also shows a curvature-dependent oscillatory behaviour
in curved nanowires that is distinct from the oscillatory feature above Walker
breakdown seen in straight wires [162, 163].

In applications for which DW stochasticity is detrimental, Walker breakdown
might need to be minimized and various techniques have been suggested for
this. For instance, doping permalloy nanowires with rare-earth elements such
as holmium can increase the Gilbert damping parameter and thereby suppress
Walker breakdown until much higher magnetic fields [164]. Geometric
suppression of Walker breakdown in nanowires has also been achieved using a
series of cross-shaped wire junctions in a comb-like geometry to interrupt and
reset, at regular intervals, the dynamic pathway of DW dynamics that lead to
magnetic oscillations [165].

We have seen in this and previous sections that DWs in nanowires exhibit rich
dynamics which can be controlled via a variety of stimuli including magnetic
fields, charge and spin-polarised currents, spin-orbit torques and mechanical
strain. DW properties such as nucleation, chirality and position can be
controlled by many of these stimuli as well as other factors such defects in
the nanowires and by modifying the wire geometry. Furthermore, nanowires show
complex behaviour such as stochastic switching of domains involving
probabilistic pinning/depinning of DWs. In the next sections, we shall
describe applications proposed using DWs in nanowires and explain how, while
for some applications these complex effects can be detrimental, for other
novel applications, these probabilistic processes can be highly desirable.

##  IV Applications of domain walls in patterned nanowires

Proposed applications of magnetic DWs in patterned nanowires have mostly
focused on information technologies, originally just for digital memories and
computing, but more recent proposals have included physical realisations of
approaches to unconventional computing. Other uses include magnetic field
sensors, various bio-applications using magnetostatic interactions of DWs with
magnetic beads, and even externally-addressed cold-atom optics, again using
magnetostatic interactions. Here, we review research in each of these areas.
Studies which have theoretically or numerically simulated a device will be a
(Type:Sim) descriptor while an experimentally demonstrative study will be
given a (Type:Expt) descriptor.

###  IV.1 DW memories

The first magnetic DW-nanowire memory proposal envisaged positioning of a DW
between two positions defined by changing lateral dimensions of an IM
ferromagnetic wire [166] (Type:Sim). The element memory might then be defined
either as the magnetisation direction of the central region or the position of
the DW, depending on the readout scheme employed. Positional bistability has
also been created using induced stress patterns in magnetostrictive materials
[109], magnetic fields transverse to the nanowire, particularly when combined
with DW chirality [41] (refer to Figure 12), and the position of contact wires
for current-induced DW motion [167], which was the basis of the first
commercial DW-nanowire MRAM [71].

Figure 12: (a) An SEM image of a nanowire with a notch with axial field
hysteresis loops for different transverse fields to assist depinning. (b) A
proposal for a multiple state memory using the direction of magnetisation in
the wire and orientation of the DW. Taken from [41].

Other proposed memories use more continuous propagation of DWs in nanowires.
Parkin et al.[4, 6] (Type:Expt) proposed a shift register technology known as
‘racetrack memory’ (RM) in 2008 in which digital data are encoded as a series
of DWs or oppositely-magnetised domains along an IM nanowire. This approach
relies upon using current-induced DW motion in order to ensure that all DWs
move in the same direction, thus defining the direction of shift register
operation. This approach also offers relatively simple device integration,
since only single elements for data writing and readout need be incorporated
into an extended wire. Parkin et al [168] (Type:Expt) further proposed
different versions of the racetrack with the latest version (v4) making use of
synthetic antiferromagnets on heavy metals and exploiting spin-orbit driven
effects for driving DWs. These memories promise the advantage of high DW
velocity, low power consumption and lesser fringing fields in the device.

Franken et al. [62] (Type:Expt) engineered effective unidirectional field-
driven DW motion in an OOPM nanowire ring by patterning a series of gradients
in the wire edge profile to make diode structures. These elements prevented
back-propagation of DWs, thus allowing a bidirectional out-of-plane magnetic
field to result in unidirectional DW motion. The ring structure meant that DWs
could continue to circulate controllably for an arbitrary time and fulfil the
requirements of a shift register.

Magnetic tunnel junctions (MTJs) [169] comprising ferromagnetic layers with
free and fixed magnetisation separated by an electrically-insulating layer are
also attractive candidates for memory/logic devices. They are particularly
useful for reading out the states of devices using phenomena such as tunneling
magnetoresistance (TMR) [170]. Lou et al. [171] (Type:Expt) have proposed an
MRAM device operating at multiple levels implemented using MTJs having a DW in
the free layer which are formed due to lithographical imperfections.
Subsequently, Raymenants et al. [172] (Type:Expt) proposed an improvement to
this by introducing a hybrid layer (an additional free layer in an MTJ) to
show a three-operation device for DWs that allows addressable writing, DW
transfer, and readout.

In order to increase the storage densities of memories, it is required to
minimize the size of the magnetic bit. However, the energy barrier to be
overcome for the switching of magnetic domains scales with domain size and
thus smaller magnetic bits are more susceptible to thermal effects that can
lead to stochastic depinning, which is an existing hurdle towards both
dependable switching and long-term information retention in such memories.
Another major active research area is the manipulation of DWs for storing
information. Magnetic fields control of magnetic states are not attractive for
devices due to poor control and high power consumption and so current control
of magnetisation is the way forward. However, the current densities required
for STT/SOT driven DW motion in nanowires is still relatively high which and
their application can lead to significant Joule heating of the device [173].
Furthermore, reading and writing of information requires controlled DW
shifting and precise position control (for alignment with read/write heads)
and these stringent requirements need further study. These challenges are
still to be overcome before the commercialisation of DW based long-term memory
storage devices. The interested reader is referred to others reviews [174,
175, 176] for further information about mechanisms, devices, and materials for
DW memories.

###  IV.2 Boolean computing

Magnetic-field-driven Boolean computing using DWs in IM nanowires was first
demonstrated approximately 20 years ago by Allwood et al[11] (Type:Expt). They
used in-plane rotating magnetic fields to drive DWs around 2D nanowire
circuits (refer to Figure 13), with wire corners providing a means to limit
the extent of DW propagation during any particular phase of field rotation
thus allowing the external field to act as a circuit clock. The behaviour of
DWs at wire junctions allowed nanowire elements to be created that performed
operations usually associated with CMOS logic circuit elements (Figure 14)
[11]. These included a Boolean NOT-gate, which performed an inversion
operation, and an AND/OR-gate, the function of which depended on a dc magnetic
field bias direction. They went on to show how these elements could be
integrated to realise more complex logic circuits, including using a chain of
NOT-gates to create a clocked shift register memory (Figure 13). Allwood et
al. [177] (Type:Expt) further proposed a shift register using a three-terminal
’Y-shaped’ IM nanowire structure with different widths of the nanowires in the
structure. While two of the terminals of the device were used for the NOT
operation described in[11], the third terminal supplied an extra, non-inverted
output and by cascading a series of such structures, they could realised a
shift register operation.

Figure 13: Images of (a) a magnetic nanowire circuit showing a combination of
different gate logic element implementations mentioned in Table 14. (b) A
5-bit magnetic shift register implemented using NOT gates and a fan-out
junction. Taken from [11]. Figure 14: Simple logic elements with their CMOS
and magnetic nanowire implementations. Taken from [11].

Zheng et al. [178] (Type:Expt) extended this idea to show that field driven DW
motion in a Y-shaped permalloy nanowire device which has a "gate" terminal,
the magnetisation of which determines the DW propagation dynamics in the
"source" and "drain" arms, can be used to realise a transistor operation. DW
depinning at the junction is controlled by the direction of the magnetisation
in the gate arm (refer to Figure 15). They proposed in-memory computation
using this element where the junction device can be used to store memory
states as well as perform logic operations (such as OR and NAND) acknowledging
the cross-disciplinary challenges of implementing a spintronic arithmetic
logic unit for feasible in-memory processing of data.

Figure 15: The operation of a three-terminal nanowire device with a junction
where the magnetic state in one arm (’gate’) can be used to control the
pinning of a DW at the junction thus control the magnetic states in the other
arms, realising a transistor operation. The ’calculation’ and ’memory’ stage
show modes of operation to realise both data processing storage. Taken from
[178].

The above applications with IM materials used nanowires with dimensions that
would result in transverse DWs of a random chirality. Vandermeulen et al.
[179] (Type:Sim) proposed a simulated device which used the two chiralities of
transverse DWs in an IM nanowire as Boolean logic levels to realise NOT/AND/OR
logic gates. Around the same time, Goolaup et al. [48] (Type:Expt)
demonstrated an IM nanowire device which could be used to invert or rectify
the chirality states of transverse DWs. They used magnetic force microscopy to
show that the chirality of a T2T transverse DW could be rectified from ’down’
to ’up’ while in the inverter case, the chiralities of both T2T and H2H DWs
could be reversed. The chirality inherent in vortex DWs has also been proposed
as the basis of chirality-encoded DW logic [42, 127]. Omari et al. [127]
(Type:Expt) used the chirality reversal of DWs that propagate through notches
in the wire edges to perform inversion operations. They also showed that two-
input wire junctions could be programmed to operate as logic gates
(AND/NAND/OR/NOR) by controlling the chirality of the DW in the output arm
which in turn was determined by which input arm switched its magnetisation
first.

The above proposals and demonstrations were with magnetic fields being the
agents to manipulate DWs. We now consider electric current driven proposals
for Boolean logic devices. Incorvia et al. [180] (Type:Expt) demonstrated a
three-terminal DW-based MTJ device for various logic operations including
inverter and buffer operations. A memory cell device was modified to be used
for logic purposes with input information encoded into the position of a
single DW (as a logic ’0’ or ’1’ as shown in Figure 16 (a) and (b)) in a
nanowire using STT and the information state was read out using TMR. They
demonstrated an inverter and a buffer operation (refer to Figure 16 (c) and
(d)) using the device and subsequently realised a three inverter network by
arranging three devices in series.

Figure 16: (a) and (b) A DW in a nanowire read out using the resistance state
of an MTJ on top of the nanowire. The position of the DW denotes a logic ’0’
or ’1’. (c) and (d) The device showing an inverter and buffer operation on
applying clock pulses. Taken from [180].

Manfrini et al. [181] (Type:Expt) used a four-pillar device to propagate DWs
over long distances in the common free layer of four MTJs using STT with the
device state read out using TMR of the output pillar. They then used these
device states to simulate the implementation of a spin torque majority gate
which is a candidate for an efficient spin-logic device [182].

While electric current driven control of DWs in nanowires have many advantages
over field driven control (refer to Section II.3), the challenges associated
with electric field control of DWs (such as high current densities required)
has limited device proposals. While current-driven DW-based NOT-gates were
demonstrated in 2008 using Invar elements [183], it was the use of SOT driven
current to control DWs that has led to progressing the state of the art
involving proposals of current driven DW based logic devices. Baek et al.
[184] (Type:Expt) have shown that a combination of electrical-field-controlled
SOT switching and voltage-controlled magnetic anisotropy switching of the
magnetisation in $\rm{Ta/CoFeB/MgO/AlO}_{\rm{x}}$ can be used to realise a
spintronic logic device. They claimed that compared to a CMOS implementation,
a half-adder implemented using their system would be an order of magnitude
smaller, have lower energy consumption, and offer the flexibility of dynamic
reconfigurability. Subsequently, Luo et al. [12] (Type:Expt) were able to
demonstrate all-electrical control of DWs in OOPM nanowires to perform all
necessary operations for a full logic architecture (Figure 17). This included
development of three-input-one-output wire junctions to create majority gates
with a logical AND/OR function governed by the magnetisation direction of a
central control wire, and half-and full-adder circuits made of cascaded
magnetic nanowire NAND-gates (Figure 17 (c)-(d)). This approach offers the
potential of highly efficient and high-speed processing based on DWs in
magnetic nanowires.

Figure 17: (a) A Y junction device which is demonstrated to be a logic
circuit.(b) MOKE images showing operation of the junction device when driven
using current pulses. (c) A XOR gate demonstrated using NAND gates and (d) a
full adder circuit with magnetic force microscopy images showing different
states in the gate operations. Taken from [12].

We now shift our attention to OOPM materials based proposals and
demonstrations. Alamdar et al. [185] (Type:Expt) drove OOPM Ta/CoFeB/MgO DW-
MTJ heterojunction devices using SOT switching to realise two device inverter
circuits by controlling the DW position in the CoFeB layer, which in turn
determined the output resistance state of the MTJ. They optimised the PMA and
lithography process to obtain a high TMR ratio and suitable average
resistance-area product of the device. Lin et al. [186] (Type:Expt) have
proposed a multistate SOT-DW-MTJ device (refer to Figure 18 (a)) for in-memory
computation applications. Apart from the three typical layer stack of an MTJ
(they considered CoFeB/MgO/CoFeB), the device has a W layer on each side of
the stack to control the position of the DW in the free CoFeB layer using SOT
generated by passing current pulses through the W layer. Along with MOKE
imaging of the DW states in similar devices, they simulated the operation of
buffer, inverter and typical logic gates on single and dual inputs to the
device. They also used micromagnetic and circuit level simulations to show the
operation of a full adder using their device and expect the read/write latency
to be as short as $1.25/0.22\,\rm{ns}$, and average writing energy of
$8.4\,\rm{fJ/bit}$ and a power consumption of $26.25\,\mu\rm{W}$ which they
claim is significantly lower than alternative implementations of devices for
similar applications. This is promising as according to previous estimates by
Xiao et al. [187] the power consumption of a 32 bit STT/SOT driven DW-MTJ
adder circuit is comparable to its CMOS counterpart.

Figure 18: (a) The proposed SOT/DW/MTJ device for in-memory computation. (b)
The proposed hybrid device to make use of multiple DW states for implementing
logic operations. Taken from [186].

Finally, in proposals involving different DW control stimuli, Zhang et al.
[188] (Type:Expt) have supplemented the use of current-induced DW motion with
ultrafast laser pulses of controlled helicity in a nanowire device. The
current pulses had to be timed with the application of optical pulses and
allowed a range of logic operations to be demonstrated at wire junctions, as
shown in Figure 19. Due to requiring lower current densities to drive DWs. the
energy required for this implementation is also lower than tradition current
driven DW motion.

Figure 19: (a) Opto-electronically driven DW motion used for realising: (a)
AND, (b) OR, (c) NAND, and (d) NOR logic gates. Inputs $A$ and $B$ correspond
to current and optical pulses used to drive DWs and $R$ denotes the resistance
state of the wire which serves as the logic gate output. Taken from [188].

The SOT driven control offer the possibility of device operation at reasonable
current densities and this creates industrial potential in these devices.
Moreover, as most of the demonstrations in which individual DW based logic
devices are integrated use CMOS compatible interconnects, this should be
amenable for industrial development. There are a significant number of
challenges that need to be overcome before commercialisation. The individual
logic elements (like a Boolean logic gate) need to scaled down in size for
incorporating advanced logic function implementations in DW based Boolean
logic devices. Moreover, more complex logic and networks need to be
demonstrated and studied for more complex computation and these need to be
tested for repeatability and reliability. They need to be engineering for
precise control of DW manipulation at high speeds of operation. The stability
of DWs becomes are a significant issue here as well and in the next section we
describe device proposals which aims to use this instability of DWs for
computing operations.

###  IV.3 Non-Boolean/Neuromorphic/Unconventional computing

Neuromorphic computing (NC) describes various forms of ‘brain-inspired’
computing [189, 190, 191, 192, 193] and is seen as offering exciting routes to
reducing the power consumption involved in implementing artificial
intelligence algorithms currently implemented using CMOS-based von Neumann
architectures [194, 195]. NC involves replacing part or all of a conventional
computer with bespoke elements that enable computation. One common such
element is a neural network, in which a neuron acts as a processing element
that transforms multiple inputs into a processed output. A synapse is a memory
element that has a multi-weighted transmission response and conducts neuron
outputs, with weights generally trained (using a training algorithm) to solve
a particular task. The response of the neuron to a stimulus, called the
‘activation function’, should have a required degree of nonlinearity to
achieve efficient separation of inputs in classification tasks. Neurons may
also have more advanced functions including 1) an ’integrate-and-fire’
response which is outputting an input signal when its integrated value crosses
a threshold, 2) a ’leaking’ property by virtue of which the energy stored in
the neuron is dissipated over time and 3) ’lateral inhibition’, by which a
firing neuron disables other neurons in the same layer from firing. These
properties will be useful for understanding spintronic
proposals/implementations of NC.

Spintronic implementations of NC [196] have been demonstrated in recent years
using spin-torque nano-oscillators [197], superparamagnetic tunnel junctions
[198], spin waves in a ring resonator [199], and artificial spin ices (ASIs)
[200]. They have also been proposed using interactions in spin-Hall nano-
oscillators [201], ensembles of superparamagnetic particles [202], skyrmions
[203, 204, 205], dipolar-coupled nanomagnets [206], magnetic states of ASIs
[207] and spin waves in a thin film [208, 209] and in coupled nanomagnets on a
thin film [210, 211]. Here, we shall focus on approaches that use magnetic DWs
in nanowires to perform NC. Table 1 shows a comparative analysis of all these
approaches with prospective advantages and challenges.

Type of spintronic element |  Computation performed/proposed |  Advantages |  Challenges
---|---|---|---
Spin torque nano-oscillators and superparamagnetic tunnel junctions |  Magnetic synapse [212], Spoken digit recognition and waveform classification [197], vowel recognition [213]; generation of cursive letters [214], stochastic computing [215], random number generation [198] (Type:Expt) |  Small size, low power consumption, CMOS compatibility, downsizing to potentially atomic dimensions, switching at low current densities (due to aid of thermal perturbations) |  Complicated fabrication of STNOs, Low resistance difference between ON and OFF states of MTJ make reading large arrays non-feasible, high phase noise, trade-off between energy cost and correlation of random number generators
Spin waves in films |  RC properties [199] (Type:Expt), time-multiplexed multi-dimensional input for reservoir computing [216] (Type:Expt), signal estimation [208] (Type:Sim), reservoir metrics and nonlinear autoregressive moving average [209] (Type:Sim), frequency detection [211] (Type:Sim) and evaluation of task agnostic RC metrics [210] (Type:Sim) |  Low power, short wavelengths, nonlinear and anisotropic interactions |  High quality growth of films for low damping, may show best performance at low temperatures
Artificial spin ices |  Time series prediction [200] (Type:Expt), multi-level and deep RC networks [217] (Type:Expt) and pattern classification [207] (Type:Sim) |  Large number of strongly interacting elements with minimal need of interconnects, complex and tunable dynamics |  High resolution fabrication required (elements are typically $100-200\,\rm{nm}$), experimental demonstration challenging due to weak signals
Skyrmions |  skyrmion neurons [218, 219] (Type:Sim), skyrmion synapses [220] (Type:Sim), synaptic pattern recognition [204] (Type:Expt) and RC Pattern recognition [205] |  Complex dynamics and strong interactions in close proximity, relatively low current densities for positioning |  Thermal noise might swamp AMR signals from devices, complex fabrication requirements, stability regions are difficult to achieve on a device.
Spin Hall nano-oscillators |  Synchronisation maps useful for neuromorphic computation [201] (Type:Expt), addition of two neuron outputs and multiplication of neuron output by synaptic weights [221] (Type:Sim) |  Easier fabrication than STNOs, mutual synchronisation possible17 |  Non-volatile storage of oscillator state a challenge, nonlinear and complex interactions between oscillators to be better understood
Superparamagnetic particle ensembles |  Spoken digit recognition and time series prediction [202] (Type:Sim) |  Low power consumption, Large range of response timescales tuned by voltage |  Yet to be fabricated, response reproducibility for a device maybe challenging
Dipolar coupled nanomagnets |  Perceptual organisation in computer vision [206] (Type:Expt) and waveform classification |  Small scale elements and replacement with MTJs can give a technological boost |  Readout done only using imaging techniques21 and electrical readout is envisaged
Domain wall based devices |  DW-based neuron [222] (Type:Expt), DW-based synapse [223, 224] (Type:Expt), RC signal reconstruction, spoken digit recognition and reservoir computing metrics (experimental [225] and simulation [226, 20]) |  Relatively simple fabrication, multiple methods of controlling DW states, easy readout using electrical means, complex dynamical phenomena |  Current and SOT control and CMOS compatibility are yet to be investigated for unconventional computing
Table 1: Unconventional computing implementations using magnetic systems with
potential advantages and challenges.

A DW-based spintronic implementation of a neuron and a synapse was initially
proposed in 2016 [227] (Type:Sim) using MTJs (refer to Figure 20 (a)) with
OOPM materials used for the free and fixed layers with the free layer having a
DW. The neuron implementation was suggested to be achieved by moving the
position of the DW in the free layer using SOT generated by current pulses.
Synaptic behaviour was envisaged via a multi-step conductivity response of the
MTJ obtained by moving the DW in the free layer and between artificial pinning
sites. Subsequently, Hassan et al. [228] (Type:Sim) proposed a neuron with
lateral inhibition by extending the DW-based MTJ proposed by Incorvia et
al.[180] to include an underlying hard IM ferromagnetic track. The integrate
(by passing a current through the DW track) and firing occurred when the DW
reached an MTJ at the end of its track causing the MTJ’s resistance state to
change. Leaking action was obtained by the dipolar field of the ferromagnetic
track on the DW in the MTJ free layer, causing the DW to return to its initial
state after the application of the current pulse. They went on to propose the
arrangement of multiple devices in an array and could obtain lateral
inhibition due to the suppression of DW movement in one track by dipolar
coupling with its neighbouring track. Modification of the spacing between
tracks and the current passing through them could control the degree of
lateral inhibition and the array response was also used to categorise more
than 100 handwritten digits with $94\%$ accuracy.

Further recent proposals of NN functions with DW motion include utilising
magnetocrystalline anisotropy gradients [229] and by using width modulated
wires [230], both of which modify the energy landscapes of DWs for
propagation. The leaking action is provided by a resorting force which is
induced by the modified energy landscape on a displaced DW. Brigner et al.
[229] (Type:Sim) suggest that by introducing anisotropy gradients in a
3-terminal MTJ device, DWs move from regions of higher to lower anisotropy
without external stimuli which leads to a form of leaking action. Upon passing
a current through the DW track, the DW movement causes the integration of the
current response. Similar to the above, field and current free DW movement due
to the width gradient in a nanowire [230] (Type:Sim) causes the simultaneous
implementation of the leaky and integrate actions required for a neuron. Cui
et al. [231] (Type:Sim) simulated current and field-driven DW motion in a pair
of adjacent MTJ-DW devices and obtained lateral inhibition by tuning the
magnetostatic interaction between the devices at the Walker breakdown field.
Wang et al. [232] (Type:Sim) have used micromagnetic simulations of DWs in an
OOPM nanowire in which the DW nucleation is stochastic using STT (due to
thermal effects) and DW dynamics are deterministic using SOT. This led to both
stochastic modifications of synaptic weights and a multilevel output synaptic
response. They could simulate machine learning of breast cancer data with an
accuracy of $95.7\%$ accuracy and estimated an energy consumption of less than
$2\,\rm{fJ}$ for device based on their proposal.

Experimental realisations of NC based elements using DW-based MTJ devices have
concentrated on synaptic behaviour and Siddique et al. [223] (Type:Expt)
showed both linear and nonlinear activation functions by using patterned MTJ
elements on an underlying DW-carrying $\rm{CoFeB}$ nanowire (which was the MTJ
free layer). DW positions were changed by SOT induced by currents passed
through the free layer and read out via the resistance states of output MTJs.
This approach allows the precise form of activation function to be specified
by the relative size of successive MTJ regions along the DW nanowire conduit,
for example, using successively larger and then smaller MTJs creates a
sigmoidal response (refer to Figure 20 (b)). Furthermore, it is relatively
fast, being operated with $8\,\rm{ns}$ current pulses, and energy efficient,
since each pulse consumed $<16\,\rm{pJ}$. Leonard et al. [224] (Type:Expt)
have demonstrated a DW-based synapse read out using a single MTJ. The device
consisted of an MTJ stack with an OOPM $\rm{CoFeB}$ free layer and an
underlying Ta layer for writing magnetic states using SOT. The $\rm{CoFeB}$
layer had a DW track patterned notches so as to control DW nucleation and
pinning. The MTJ resistance was read out using TMR via electrical contacts and
they considered trapezoidal and rectangular shaped DW tracks for different
functionalities. Since the trapezoidal device had a varying tack width and the
threshold voltage to depin the DW is inversely proportional to the track
width, it showed deterministically switched resistance states. Also, while it
had 9 intermediate notches, the authors were able to obtain only 4 resistance
states from it and they attributed this to inconsistencies in the fabrication
process. The rectangular device, on the other hand, had constant width and
notch configurations and showed probabilistically switched resistance states.
The authors subsequently simulated machine learning of a Fashion-MNIST dataset
(using the trapezoidal geometry) and of an CIFAR-100 image recognition (using
the rectangular geometry). Importantly, the authors also compare the
performance of their DW-based synapse with other proposed synapse
implementations in terms of write energy, update duration and write noise.
Leonard et al. [222] (Type:Expt) have also shown DW-based device with MTJ
readout that used a similar stack configuration and SOT driven stimulus as
used in [224]. The device had a voltage dependent activation function and they
used this response to implement machine learning using a noisy version of the
Fashion-MNIST dataset and claimed that such devices could be used to implement
robust networks for NC. The interested reader can refer to the recent review
by Hu et al. [233] for more details on such devices.

Kumar et al. [234] have proposed a DW based synapse using a meandering OOPM
nanowire configuration. The magnetic states were written using a combination
of an OOP magnetic field and SOT via an underlying Tungsten layer. This layer
was grown using a novel procedure involving sub layers grown at different
pressures and this reduced the roughness of the Tungsten on contact with the
magnetic layer and aided spin current propagation. This led to low SOT current
densities being required $\sim 10^{6}\,\rm{A}/\rm{m}^{2}$ and a low power
consumption of $0.4\,\rm{fJ}$ to move a DW by $\sim 19\,\mu\rm{m}$. The
authors varied the distance between neighbouring segments in the meandering
nanowire to realise synapses of varying degree of stochasticity.

The stochastic dynamics of DWs at notches in IM nanowires used have also been
used to to demonstrate functional binary stochastic synapses by Ellis et al.
[235] (Type:Expt). They used magnetic field driven nucleation and Oersted
fields of current pulses for DW dynamics in an IM permalloy nanowire with an
artificial notch to realise the sigmoid-like passing probability of a DW using
MOKE. They went on to simulate and demonstrate machine learning of handwritten
digit recognition in the device using a gradient learning rule that adjusted
synaptic stochasticity and energy throughput depending on the number of
measurements.

All these explorations indicate that there is promise in realising NC devices
and circuits using DW-based devices. There is now a need to scale the elements
of this implementations to realise more complex circuits and realise further
novel functionalities. The realisation of neurons and the other proposed
foundational NC elements are necessary for this purpose. Furthermore, the
possible fabrication and behavioural inconsistencies possible with DW tracks
and nanostructures need to be resolved for technological implementations.

Figure 20: (a) A combination of a biological synapse and neuron with its
proposed spintronic equivalent. Taken from [227]. (b) Linear and nonlinear
activation functions realised using an MTJ based device. Taken from [223]

.

Apart from proposals for neural network elements, other forms of
unconventional computing have also been explored. One such paradigm is
stochastic computing where complex computations can be realised using streams
of random bits [236]. In this context, the inherent uncertainties of DW
dynamics can also be useful for stochastic computing paradigms [237].
Hernandez et al. [19] (Type:Expt) have used the stochastic pinning and
propagation of DWs at nanowire junctions DWs to realise a Galton board, an
archetypal experiment in statistics. The original Galton board consisted of
balls moving down a slanted surface encountering a grid of obstacles which led
to a Gaussian diffusion random walk of the balls. In [19], a magnetic field
was used drive a single DW in a branched network of IM magnetic nanowires to
recreate a Galton board (Figure 21 (a)). The distribution of DWs across the
eight output wires approximated a normal function (Figure 21 (b)), although
the final position of successive DWs was highly uncorrelated. The authors
passed the binarised output sequence through the NIST Statistical Test Suite
for Random and Pseudorandom Number Generators and it passed all 13 tests
indicating a high degree of uncorrelation. Interestingly, removing the central
wires in a separate structure resulted in a flattened DW distribution across
output wires (Figure 21 (c) and (d)). This highlights how this simple approach
could be adapted to different tasks. One might envisage adaptive, addressable
tuning of DWs at wire junctions offering training functionality in
reconfigurable magnetic nanowire Galton board networks.

Another random number generator was demonstrated by Narasimman et al. [238]
(Type:Expt) who reported a $65\,\rm{nm}$ CMOS readout circuit for an OOPM Hall
cross in which the magnetic texture took random orientations due to the
thermal and demagnetization effects. They changed the magnetic textures using
pulsed currents and detected these textures using the anomalous Hall effect by
a modulation and amplification scheme. Their device consumed a power of
$126\,\mu\rm{W}$ when driven using a $1.2\,\rm{V}$ supply and could function
with voltage drifts of $440\,\mu\rm{V}$.

Figure 21: SEM micrographs of a magnetic nanowire Galton board which are (a)
complete and (c) with the central element missing. The distribution of outputs
of (b) the complete board which approached a binomial distribution and (d) the
board with an element missing. Here the distribution is flattened indicating
tuning of the board response. Taken from [19]

The final unconventional computing paradigm that we shall consider is that of
reservoir computing (RC), which is a bio-inspired computational paradigm like
NC and has emerged as a prime contender for processing time-varying signals
[239, 240]. A reservoir computer typically consists of an input layer, a
reservoir layer and an output layer (refer to Figure 22). A conventional
reservoir consists of a recurrent neural network with fixed internal weights.
The recurrence in the network, the neuron interactions, and the behaviour of
individual neurons themselves combine to provide a time-dependent nonlinear
reservoir response to input, which makes RC well-suited for analysing time-
varying data. However, unlike traditional recurrent neural networks, only the
output layer has to be trained, which makes training very efficient.

Figure 22: (a) A typical implementation of RC, highlighting the layered
architecture: an unchanged reservoir layer with individual neurons has time-
varying signals fed into it via weighted connections. A trainable output layer
then provides a weighted sum of nodal states from the reservoir. Taken from
[226]

.

The fixed internal weights of the reservoir implies that it can be replaced
using any dynamical system and this has led to many _in-materio_ RC schemes,
which involve using a physical system’s complex response to a stimulus as the
reservoir transformation. Ideally, the physical reservoir should offer a means
to represent data, nonlinear sensitivity to input, a memory of previous input
that recedes over time (‘fading memory’), and have readable output. In
practice, it should also be scalable and exhibit reproducible device dynamics.
Magnetic systems are attractive candidates for reservoirs due to their
inherent hysteretic and nonlinear response to stimulus, e.g. magnetic field or
SOT. There are also a host of methods for representing, inputting, and reading
data, as we have described here for DW-based nanowire systems. This promise is
reflected in the diversity of studies that have proposed magnetic devices for
RC, including using skyrmions [205], magnetic states in ASIs [207], and
superparamagnetic particles [202], and demonstrations using STNOs [197] and
spin waves in ASIs [200]. The interested reader can refer to the review by
Allwood et al. [241] for further details about RC using nanomagnetic devices.

We shall focus on proposals and demonstrations of RC which use DW dynamics.
Dawidek et al. [20] (Type:Expt) showed that the stochastic nature of DW
pinning and depinning in arrays of IM permalloy nanorings are suitable for RC.
The arrays were driven by rotating magnetic fields with the DW behaviour in
the array changing according to the field strength. Low-strength magnetic
fields left DWs pinned at ring junctions, while strong fields caused all DWs
to pass through junctions. The stochastic depinning resulting from
intermediate fields gave rise to DW annihilation, from a moving DW meeting a
DW pinned at a junction, which leads to the formation of magnetic vortex
states in some nanorings. However, DWs subsequently moving through a junction
within a vortex-state ring repopulated the nanoring with two DWs. A dynamic
equilibrium between these DW loss and gain phenomena led to the array
magnetisation (via the DW population) becoming a field-dependent emergent
property of the ring arrays, as determined by polarised-neutron magnetometry
(Figure 23 (a)). Imaging array magnetisation using X-ray photoelectron
emission microscopy (X-PEEM) revealed a changing and rich magnetic state
population in these arrays (Figure 23 (b)). The authors also simulated machine
learning of spoken digit recognition with these arrays and obtained $99.4\%$
accuracy for classification accuracy for a single speaker and upto $89\%$
accuracy for eight speakers. Subsequently, Vidamour et al. [226] (Type:Sim)
used a phenomenological model of the ring array system to optimise the
dynamics and performance of these arrays for different classification tasks by
tuning the scaling and input rate of data into the reservoir. They used task
agnostic metrics for quantifying the determine the capabilities of these
arrays for computation and showed the association of these metrics with
performance in different tasks.

Vidamour et al. [225] (Type:Expt) then went on to report the fabrication of
electrical devices using these arrays (Figure 23 (c)) with these arrays and
used their anisotropic magnetoresistance (AMR) response for machine learning.
They now experimentally evaluated the task agnostic kernel and generalisation
rank metrics for assessing the suitability of the array for RC and used these
metrics for identifying the operating parameters of the arrays. They went on
to demonstrate the performance of various tasks of varying computational
requirements of non-linearity and memory (including signal transformation,
spoken digit recognition and nonlinear autoregressive moving average series
prediction) with the array and achieved state of the art performance.

Figure 23: (a) The variation of the magnetisation of the ring array with
rotating magnetic field amplitude showing a strongly nonlinear response. (b)
An X-ray photoemission microscopy image of the ring array showing the
different magnetic states existing at an intermediate rotating field
amplitude. The arrows indicate the direction of the magnetisation sensitivity.
(c) The variation of state populations in the array with number of field
rotations for different rotating field amplitudes. The very different
evolution for different fields is indicative of the ’fading memory’ in the
system. Taken from [20]

Further proposals of RC include that by Ababel et al. [242] (Type:Sim) who
proposed using the oscillations of a single in the potential barrier between
two protrusions in an IM Ni nanowires to perform RC. They encoded the data
input into the amplitude of a driving magnetic field at $500\,\rm{MHz}$ and
used the DW position between two ‘anti-notch’ potential barriers along a
nanowire length to represent output. The complex DW propagation dynamics in
this simple one-DW system proved capable of performing a number of
classification tasks, including sine / square wave differentiation, spoken
digits, and handwritten digits. Subsequently, Hon et al. [243] (Type:Sim)
performed RC using micromagnetic simulations of an array of nanowires with
$\lambda$-shaped junctions (refer to Figure 24). The DW motion, dynamics, and
depinning behaviour at junctions under a clocking magnetic field were used to
simulate short term memory (STM) and parity check (PC) tasks using RC
methodology. The devices would appear to be robust to changes in temperature,
with similar performance estimated at 0 and 300 K.

Figure 24: (a) A schematic of a grid of $\lambda$-type junctions used for
acting as a reservoir. Also shown is the schematic of a single junction with
associated dimensions. The method of inputting data into the grid using a
clock field (denoted by the ellipse). The states for an input of ’0’ and of
’1’ are also shown. (b) The variation of the short term memory (STM) and
parity check (PC) of the system at different temperatures. Taken from [243]

The temporal signal processing utility of RC lends itself to analysing delayed
feedback dynamics and a typical example of this are the Mackey-Glass equations
[244] which are nonlinear, time-delayed differential functions that give rich
dynamics, originally envisaged as simulating the dynamics of regulated
biological systems. Mackey-Glass oscillators obey these equations and allow
oscillatory and chaotic dynamics to be explored in a simple manner. This is
particularly exciting for RC as the nonlinear transformations using ‘edge-of-
chaos’ dynamics should offer rich and tunable performance. Williame et al.
[245] (Type:Sim) demonstrated Mackey-Glass oscillator behaviour in simulations
of STT-driven DW motion in a nanowire with an elliptical protrusion, although
they commented that, in practice, current-driven heating of the device and
fabrication uncertainties might affect device performance.

These proposals and demonstrations show that DW dynamics in nanowires devices
possess the capabilities required for RC in a variety of tasks. Future work in
magnetic implementations of RC will no doubt see a wider exploration of
suitable systems, and control and tuning of these systems as well as further
demonstrations of machine learning in a variety of tasks and environments.

###  IV.4 Other applications

The global market for magnetic field sensing was USD $\$2.21$ billion in 2018
and projected to rise to $\$4.22$ billion by 2026 [246]. Although this is
dominated by Hall sensors, field sensing is a common application of magnetic
materials [247] and magnetoresistive sensors made up almost 15% of the global
market in 2018. Some proposals of magnetic field sensing have involved DW
devices.

Wolfe et al. [21] (Type:Expt) proposed a sensor based on the Faraday effect in
an IM magnetic garnet thin film ridged waveguide. When a single DW, created
using a gradient magnetic field, crosses the path of light incident on the
film, the Faraday rotation changes and thus the field causing the
magnetisation change in the film may be detected. Diegel et al. [22] suggested
a four-bit DW-based multi-turn counter, primarily for automotive and
industrial applications. The sensor layout consists of four rectangular-shaped
spirals, two winding clockwise and two anti-clockwise, of soft ferromagnetic
nanowire multilayers that exhibit GMR. Electrical connections formed a half-
bridge configuration that gave stability against temperature variation. The
spiral centres had DW injection pads that acted as a source or sink of DWs in
the magnetic free layer, depending on the rotation direction of the external
magnetic field. This gave opposite sensitivity in the two spiral pairs to the
sense of field rotation and caused either an increase or decrease in each
spiral’s free-layer DW population. The GMR arrangement meant that these
differences could be measured easily to determine the net whole number of
turns of magnetic field in either rotation direction.

DWs have also been used for temperature sensing. Klingbeil et al. [248]
(Type:Expt) considered meandering DW formation in Bi-substituted rare earth
iron garnet films and measured out-of-plane hysteresis loops using a Faraday
effect based setup. They monitored the change in the overall magneto-optical
response, the DW nucleation fields and the magneto-optical susceptibility (the
derivative of the magneto-optical response with field) as temperature was
varied. Using appropriate calibration, they could monitor changes in
temperature over a $20-140\,^{\circ}\rm{C}$ range with an accuracy of
$0.1\,^{\circ}\rm{C}$ and with a temperature drift of better than
$0.15\,^{\circ}\rm{C}$.

The stray fields from DWs have also been used to control secondary systems. In
particular, there have been several proposals and demonstrations of
controlling magnetic nanoparticles (MNPs) for biological applications.
Vavassori et al. [249] (Type:Expt) considered a micron-sized square ring of IP
permalloy with a DW positioned at diagonally opposite corners. When a
superparamagnetic bead was also positioned at one of the corners and the
corresponding DW was displaced using a magnetic field, a dipole moment was
induced in the bead and the stray field due to this caused the field required
to displace the DW to increase. They simulated this shift in field and also
measured a $12\,\rm{Oe}$ shift in the AMR response of the square ring when
beads were dispersed on it compared to when they weren’t. Furthermore,
simulations showed that by reducing the width of the square ring, the shift in
field could be increased. Bryan et al. [26] (Type:Sim) used finite element
micromagnetic simulations to model the magnetostatic interactions between a
superparamagnetic bead and a H2H DW in a nanowire and observed the behaviour
of the DW around the stationery bead. They then used analytical formulations
to model how the hydrodynamic drag on the bead affects the DW movement. They
found that that the DW was pinned around the superparamagnetic bead below a
threshold magnetic field which was proportional to the bead diameter squared.
Even from small beads, this effect is sufficient to reduce the domain wall
mobility by five orders of magnitude.

Figure 25: (a) A zig-zag wire structure which was used to position DWs and
superparamagnetic beads with magnetic fields. (b) Optical images showing the
shifting of protein coated magnetic beads by controlling DW movement in the
nanowires structure. Taken from [250]

.

Shortly afterwards, Donolato et al. [250] (Type:Expt) controlled the movement
of protein carrying beads of different sizes exploiting the positioning of DWs
in zig-zag wires of permalloy using pulsed magnetic fields (refer to Figure
25). The beads tended to follow the DWs with some drift as the speed of DW
motion is considerably faster than the maximum measured bead velocity of
$15\,\mu\rm{m/s}$. The modification of the field pulse sequence was used to
change the direction of bead movement as well as hold the bead at the DW
constriction for analysis (refer to Figure 25). The authors also demonstrated
continuous control of bead movement by considering DWs in a ring structure and
applyign rotating magnetic fields. Small field steps of $1\,^{\circ}$ were
used to obtain fine control of nanoparticle positioning with an accuracy of
$100\,\rm{nm}$. Bryan et al. [27] (Type:Expt) considered different types of
nanowires (single planar, two perpendicular and curved) and simulated traps in
them to identify the trapping stray field produced. They went on to fabricate
these wire and used X-ray imaging to identify that the beads were trapped in
the nanowire corners where the DWs were pinned. Subsequently, Rapoport et al.
[28] considered a closed loop curvilinear permalloy track and simulated the
motion of vortex DWs through curvilinear junctions to find that the DWs
actually split into a H2H and T2T DW (with opposite stray field polarities)
upon exiting the junction. They used these DWs to move a superparamagnetic
bead at a junction and to sort beads of different sizes.

Similar concepts have also been extended to controlling the movement of atomic
size particles. Allwood et al. [24] (Type:Sim) used finite element
micromagnetic simulations to show that above a DW in a nanowire, a magnetic
field trap for a single ${}^{87}\rm{Rb}$ atom could be formed. Furthermore,
they expected that the position of this trap could be modified by applying
moving the DW. Subsequently, West et al. [23] considered an array of
undulating IM permalloy nanowires and nucleated DWs in them at remanance after
saturating the wires orthogonal to the their length. The resulting large array
of tiny magnets worked as a magnetic mirror which could reflect
${}^{87}\rm{Rb}$ atoms via the fringing field of the magnets. They could
switch the magnets on and off reproducibly by switching the magnetisation
direction of the array of nanowires.

These studies motivate the use of DW based devices as novel magnetic field,
temperature and atomic particle sensors. While they might still be far from
commercial application, the versatility of DW dynamics to respond to a variety
of stimuli caused by different agents offers new insight into their potential
uses.

##  V Future outlooks

Magnetic DWs have rich and varied behaviour and a wide variety of applications
have been proposed utilising them. Here we have reviewed the various types of
DWs in magnetic thin films and nanowires with both in-plane and out-of-plane
anisotropy. We have reviewed the methods to create and manipulate DW dynamics
with a variety of methods and highlighted how rich the behavioural and
stimulus space of DWs is in for potential applications. We have also looked at
different applications that have been suggested using them and reviewed the
state of the art in them.

The state of the art with CMOS based devices for Boolean memories and
computation is capable of producing $7\,\rm{nm}$ transistor devices which are
facing issues of repeatable device manufacturing and performance [251]. In
order to be a viable alternative for CMOS devices, DW based devices thus need
to be studied and tailored to operate at similar sizes and operate at high
speeds with low operational power and latency. Over the last couple of
decades, one of the biggest obstacles inhibiting DW proposals for devices from
hitting the market in a large scale has been that current densities required
for applications involving Oersted field or STT are typically too large for
technological feasibility. The current densities required for STT have been
declining and with the advent of SOT, there is encouraging signs that we will
soon reach reasonable current densities for technological realisations. The
material science research required for identifying candidates for high SOT
efficiency is key to making more progress in this area. Furthermore, one of
the challenges with SOT based devices is that it is still quite sensitive to
fabrication conditions and interface qualities. This is a significant step
that has to be overcome for reproducible and reliable devices. Another
significant problem for Boolean memory and computing applications
stochasticity of DW behaviour especially as the magnetic element size is
decreased for higher scalability and memory density. Again, material science
seems to be important here and the key is to identify material systems that
have energy barriers for thermal driven magnetisation changes are high enough
that reduction in storage or processing bit size is possible.

However, the stochasticity of DWs lend themselves naturally to non-Boolean
applications as has been discussed in Section. IV.3. However, before
commercial realisation, some important points must be addressed. In the
approach where neural network components (like neurons and synapses) are being
realised with various DW devices, larger networks and more advanced
functionalities need to be demonstrated if these devices are to compete with
other existing or proposed technological implementations of NC. Furthermore
the issue of reproducability and robustness is something that has not been
actively pursued in the community and this is going to be important for
technological realisation. In the other forms of unconventional computing that
we discussed (like reservoir computing), one of the key pursuits should be to
identify applications and tasks which can’t be easily (or economically)
performed with existing CMOS based systems. This will require demonstrating
advantages of the DW-based implementations in terms of niche applications,
energy throughput or other relevant metric. We are quite excited to see how
this research field progresses and where it take us.

###### Acknowledgements.

GV, DAA and TJH acknowledge funding from the Horizon 2020 FET-Open SpinEngine
(Agreement no 861618), the EPSRC MARCH project EP/V006339/1, the Leverhulme
grant RPG-2019-97 and the EPSRC project EP/S009647/1. We would also like to
acknowledge C. Swindells, I. Vidamour and A. Welbourne for discussions.

## References

## References

  * Vedmedenko  _et al._ [2020] E. Y. Vedmedenko _et al._ , “The 2020 magnetism roadmap,” Journal of Physics D: Applied Physics 53, 453001 (2020).
  * Varga, Zhukov  _et al._ [2007] R. Varga, A. Zhukov, _et al._ , “Supersonic domain wall in magnetic microwires,” Physical Review B 76, 132406 (2007).
  * Khvalkovskiy  _et al._ [2018] A. Khvalkovskiy _et al._ , “Magnetic memory with a switchable reference layer,” Journal of Applied Physics 124, 133902 (2018).
  * Parkin  _et al._ [2008] S. S. Parkin _et al._ , “Magnetic domain-wall racetrack memory,” Science 320, 190–194 (2008).
  * Gaidis  _et al._ [2011] M. C. Gaidis _et al._ , “High density planar magnetic domain wall memory apparatus,” (2011), US Patent 8,009,453.
  * Hayashi  _et al._ [2008] M. Hayashi _et al._ , “Current-controlled magnetic domain-wall nanowire shift register,” Science 320, 209–211 (2008).
  * Fukami  _et al._ [2013] S. Fukami _et al._ , “20-nm magnetic domain wall motion memory with ultralow-power operation,” in  _2013 IEEE International Electron Devices Meeting_ (IEEE, 2013) pp. 3–5.
  * Sun  _et al._ [2018] G. Sun _et al._ , “Memory that never forgets: emerging nonvolatile memory and the implication for architecture design,” National Science Review 5, 577–592 (2018).
  * Bläsing  _et al._ [2020] R. Bläsing _et al._ , “Magnetic racetrack memory: From physics to the cusp of applications within a decade,” Proceedings of the IEEE 108, 1303–1321 (2020).
  * Businesswire [2022] Businesswire, “Everspin Announces New STT-MRAM EM128LX xSPI Memory,”  (2022).
  * Allwood  _et al._ [2005] D. A. Allwood _et al._ , “Magnetic domain-wall logic,” Science 309, 1688–1692 (2005).
  * Luo  _et al._ [2020] Z. Luo _et al._ , “Current-driven magnetic domain-wall logic,” Nature 579, 214–218 (2020).
  * Trinh  _et al._ [2012] H.-P. Trinh _et al._ , “Domain wall motion based magnetic adder,” Electronics letters 48, 1049–1051 (2012).
  * Roohi  _et al._ [2016] A. Roohi _et al._ , “A tunable majority gate-based full adder using current-induced domain wall nanomagnets,” IEEE Transactions on Magnetics 52, 1–7 (2016).
  * Metaxas  _et al._ [2007] P. Metaxas _et al._ , “Creep and flow regimes of magnetic domain-wall motion in ultrathin Pt/Co/Pt films with perpendicular anisotropy,” Physical Review Letters 99, 217208 (2007).
  * Hayward [2015] T. Hayward, “Intrinsic nature of stochastic domain wall pinning phenomena in magnetic nanowire devices,” Scientific reports 5, 1–12 (2015).
  * Jin  _et al._ [2019] T. Jin _et al._ , “Synaptic element for neuromorphic computing using a magnetic domain wall device with synthetic pinning sites,” Journal of Physics D: Applied Physics 52, 445001 (2019).
  * Ma  _et al._ [2018] X. Ma _et al._ , “An area and energy efficient design of domain-wall memory-based deep convolutional neural networks using stochastic computing,” in  _2018 19th International Symposium on Quality Electronic Design (ISQED)_ (IEEE, 2018) pp. 314–321.
  * Sanz-Hernández  _et al._ [2021] D. Sanz-Hernández _et al._ , “Tunable stochasticity in an artificial spin network,” Advanced Materials 33, 2008135 (2021).
  * Dawidek  _et al._ [2021] R. W. Dawidek _et al._ , “Dynamically driven emergence in a nanomagnetic system,” Advanced Functional Materials 31, 2008389 (2021).
  * Wolfe and Lieberman [1991] R. Wolfe and R. Lieberman, “Fiber optic magnetic field sensor based on domain wall motion in garnet film waveguides,” Applied Physics Letters 58, 1733–1735 (1991).
  * Diegel  _et al._ [2009] M. Diegel _et al._ , “A new four bit magnetic domain wall based multiturn counter,” IEEE Transactions on Magnetics 45, 3792–3795 (2009).
  * West  _et al._ [2012] A. D. West _et al._ , “Realization of the manipulation of ultracold atoms with a reconfigurable nanomagnetic system of domain walls,” Nano letters 12, 4065–4069 (2012).
  * Allwood  _et al._ [2006a] D. Allwood _et al._ , “Mobile atom traps using magnetic nanowires,” Applied Physics Letters 89, 014102 (2006a).
  * Klein  _et al._ [2013] T. Klein _et al._ , “Interaction of domain walls and magnetic nanoparticles in giant magnetoresistive nanostrips for biological applications,” IEEE Transactions on Magnetics 49, 3414–3417 (2013).
  * Bryan  _et al._ [2010a] M. T. Bryan _et al._ , “The effect of trapping superparamagnetic beads on domain wall motion,” Applied Physics Letters 96, 192503 (2010a).
  * Bryan  _et al._ [2010b] M. T. Bryan _et al._ , “Switchable cell trapping using superparamagnetic beads,” IEEE Magnetics Letters 1, 1500104–1500104 (2010b).
  * Rapoport and Beach [2017] E. Rapoport and G. S. Beach, “Architecture for directed transport of superparamagnetic microbeads in a magnetic domain wall routing network,” Scientific Reports 7, 1–9 (2017).
  * Martín  _et al._ [2002] J. I. Martín _et al._ , “Fabrication and magnetic properties of arrays of amorphous and polycrystalline ferromagnetic nanowires obtained by electron beam lithography,” Journal of Magnetism and Magnetic Materials 249, 156–162 (2002).
  * Basith  _et al._ [2011] M. Basith _et al._ , “Direct comparison of domain wall behavior in Permalloy nanowires patterned by electron beam lithography and focused ion beam milling,” Journal of Applied Physics 110, 083904 (2011).
  * Coey and Hinds [2001] J. Coey and G. Hinds, “Magnetic electrodeposition,” Journal of Alloys and Compounds 326, 238–245 (2001).
  * Schloerb  _et al._ [2010] H. Schloerb _et al._ , “Magnetic nanowires by electrodeposition within templates,” Physica Status Solidi (b) 247, 2364–2379 (2010).
  * McMichael and Donahue [1997] R. D. McMichael and M. J. Donahue, “Head to head domain wall structures in thin magnetic strips,” IEEE Transactions on Magnetics 33, 4167–4169 (1997).
  * Bogart  _et al._ [2008] L. Bogart _et al._ , “The effect of geometrical confinement and chirality on domain wall pinning behavior in planar nanowires,” Journal of Applied Physics 104, 033904 (2008).
  * Nakatani  _et al._ [2005] Y. Nakatani _et al._ , “Head-to-head domain walls in soft nano-strips: a refined phase diagram,” Journal of Magnetism and Magnetic Materials 290, 750–753 (2005).
  * Petit  _et al._ [2010] D. Petit _et al._ , “Magnetic imaging of the pinning mechanism of asymmetric transverse domain walls in ferromagnetic nanowires,” Applied Physics Letters 97, 233102 (2010).
  * Vansteenkiste  _et al._ [2014] A. Vansteenkiste _et al._ , “The design and verification of MuMax3,” AIP advances 4, 107133 (2014).
  * Tchernyshyov and Chern [2005] O. Tchernyshyov and G.-W. Chern, “Fractional vortices and composite domain walls in flat nanomagnets,” Physical Review Letters 95, 197204 (2005).
  * Pushp  _et al._ [2013] A. Pushp _et al._ , “Domain wall trajectory determined by its fractional topological edge defects,” Nature Physics 9, 505–511 (2013).
  * Lewis  _et al._ [2009] E. Lewis _et al._ , “Measuring domain wall fidelity lengths using a chirality filter,” Physical Review Letters 102, 057209 (2009).
  * Atkinson  _et al._ [2008] D. Atkinson _et al._ , “Controlling domain wall pinning in planar nanowires by selecting domain wall type and its application in a memory concept,” Applied Physics Letters 92, 022510 (2008).
  * Omari and Hayward [2014] K. Omari and T. Hayward, “Chirality-based vortex domain-wall logic gates,” Physical Review Applied 2, 044001 (2014).
  * Ravelosona  _et al._ [2006] D. Ravelosona _et al._ , “Domain wall creation in nanostructures driven by a spin-polarized current,” Physical Review Letters 96, 186604 (2006).
  * Tudu and Tiwari [2017] B. Tudu and A. Tiwari, “Recent developments in perpendicular magnetic anisotropy thin films for data storage applications,” Vacuum 146, 329–341 (2017).
  * DeJong and Livesey [2015] M. DeJong and K. Livesey, “Analytic theory for the switch from bloch to néel domain wall in nanowires with perpendicular anisotropy,” Physical Review B 92, 214420 (2015).
  * Bryan [2009] M. T. a. o. Bryan, “Transverse field-induced nucleation pad switching modes during domain wall injection,” IEEE Transactions on Magnetics 46, 963–967 (2009).
  * Cui  _et al._ [2018] B. Cui _et al._ , “Magnetization switching through domain wall motion in Pt/Co/Cr racetracks with the assistance of the accompanying Joule heating effect,” Physical Chemistry Chemical Physics 20, 9904–9909 (2018).
  * Goolaup  _et al._ [2015] S. Goolaup _et al._ , “Transverse domain wall profile for spin logic applications,” Scientific reports 5, 1–8 (2015).
  * Thomas  _et al._ [2005] L. Thomas _et al._ , “Observation of injection and pinning of domain walls in magnetic nanowires using photoemission electron microscopy,” Applied Physics Letters 87, 262501 (2005).
  * Atkinson  _et al._ [2003] D. Atkinson _et al._ , “Magnetic domain-wall dynamics in a submicrometre ferromagnetic structure,” Nature Materials 2, 85–87 (2003).
  * Cowburn  _et al._ [2002] R. Cowburn _et al._ , “Domain wall injection and propagation in planar Permalloy nanowires,” Journal of Applied physics 91, 6949–6951 (2002).
  * Bryan  _et al._ [2006] M. Bryan _et al._ , “Multimode switching induced by a transverse field in planar magnetic nanowires,” Applied Physics Letters 88, 032505 (2006).
  * McGrouther  _et al._ [2007] D. McGrouther _et al._ , “Controlled domain wall injection into ferromagnetic nanowires from an optimized pad geometry,” Applied Physics Letters 91, 022506 (2007).
  * Koyama  _et al._ [2011] T. Koyama _et al._ , “Observation of the intrinsic pinning of a magnetic domain wall in a ferromagnetic nanowire,” Nature Materials 10, 194–197 (2011).
  * Fukami  _et al._ [2011] S. Fukami _et al._ , “Current-induced domain wall motion in perpendicularly magnetized cofeb nanowire,” Applied Physics Letters 98, 082504 (2011).
  * Hayward and Omari [2017] T. Hayward and K. Omari, “Beyond the quasi-particle: stochastic domain wall dynamics in soft ferromagnetic nanowires,” Journal of Physics D: Applied Physics 50, 084006 (2017).
  * Stein  _et al._ [2012] F.-U. Stein _et al._ , “Generation and annihilation of domain walls in nanowires by localized fields,” Applied Physics Letters 100, 192403 (2012).
  * Hayashi  _et al._ [2012] M. Hayashi _et al._ , “Microwave assisted resonant domain wall nucleation in Permalloy nanowires,” Applied Physics Letters 101, 172406 (2012).
  * Cayssol  _et al._ [2004] F. Cayssol _et al._ , “Domain wall creep in magnetic wires,” Physical Review Letters 92, 107202 (2004).
  * Hayashi  _et al._ [2006a] M. Hayashi _et al._ , “Dependence of current and field driven depinning of domain walls on their structure and chirality in Permalloy nanowires,” Physical Review Letters 97, 207205 (2006a).
  * Zhang  _et al._ [2016] S. Zhang _et al._ , “Highly efficient domain walls injection in perpendicular magnetic anisotropy nanowire,” Scientific reports 6, 1–8 (2016).
  * Franken, Swagten, and Koopmans [2012] J. Franken, H. Swagten, and B. Koopmans, “Shift registers based on magnetic domain wall ratchets with perpendicular anisotropy,” Nature Nanotechnology 7, 499–503 (2012).
  * Tanigawa  _et al._ [2008] H. Tanigawa _et al._ , “Current-driven domain wall motion in cocrpt wires with perpendicular magnetic anisotropy,” Applied physics express 1, 011301 (2008).
  * Narayanapillai  _et al._ [2014] K. Narayanapillai _et al._ , “Thermally assisted domain wall nucleation in perpendicular anisotropy trilayer nanowires,” Journal of Physics D: Applied Physics 47, 105005 (2014).
  * Gilbert [2004] T. L. Gilbert, “A phenomenological theory of damping in ferromagnetic materials,” IEEE Transactions on Magnetics 40, 3443–3449 (2004).
  * Hillebrands and Ounadjela [2003] B. Hillebrands and K. Ounadjela,  _Spin dynamics in confined magnetic structures I_ , Vol. 83 (Springer Science & Business Media, 2003).
  * Porter and Donahue [2004] D. G. Porter and M. J. Donahue, “Velocity of transverse domain wall motion along thin, narrow strips,” Journal of Applied Physics 95, 6729–6731 (2004).
  * Beach  _et al._ [2005] G. S. Beach _et al._ , “Dynamics of field-driven domain-wall propagation in ferromagnetic nanowires,” Nature Materials 4, 741–744 (2005).
  * Emori  _et al._ [2012] S. Emori _et al._ , “Time-resolved measurements of field-driven domain wall motion in a submicron strip with perpendicular magnetic anisotropy,” Journal of Applied Physics 111, 07D304 (2012).
  * Attané  _et al._ [2006] J. Attané _et al._ , “Thermally activated depinning of a narrow domain wall from a single defect,” Physical Review Letters 96, 147204 (2006).
  * Fukami  _et al._ [2009] S. Fukami _et al._ , “Low-current perpendicular domain wall motion cell for scalable high-speed MRAM,” in  _2009 Symposium on VLSI Technology_ (IEEE, 2009) pp. 230–231.
  * Tatara  _et al._ [2008] G. Tatara _et al._ , “Microscopic approach to current-driven domain wall dynamics,” Physics Reports 468, 213–301 (2008).
  * Boulle  _et al._ [2011] O. Boulle _et al._ , “Current-induced domain wall motion in nanoscale ferromagnetic elements,” Materials Science and Engineering: R: Reports 72, 159–187 (2011).
  * Beach, Tsoi, and Erskine [2008] G. Beach, M. Tsoi, and J. Erskine, “Current-induced domain wall motion,” Journal of Magnetism and Magnetic Materials 320, 1272–1281 (2008).
  * Berger [1978] L. Berger, “Low-field magnetoresistance and domain drag in ferromagnets,” Journal of Applied Physics 49, 2156–2161 (1978).
  * Zhang and Li [2004] S. Zhang and Z. Li, “Roles of nonequilibrium conduction electrons on the magnetization dynamics of ferromagnets,” Physical Review Letters 93, 127204 (2004).
  * Thiaville  _et al._ [2005] A. Thiaville _et al._ , “Micromagnetic understanding of current-driven domain wall motion in patterned nanowires,” EPL (Europhysics Letters) 69, 990 (2005).
  * Sbiaa and Piramanayagam [2014] R. Sbiaa and S. Piramanayagam, “Multi-level domain wall memory in constricted magnetic nanowires,” Applied Physics A 114, 1347–1351 (2014).
  * Phung  _et al._ [2015] T. Phung _et al._ , “Highly efficient in-line magnetic domain wall injector,” Nano letters 15, 835–841 (2015).
  * Kläui  _et al._ [2005a] M. Kläui _et al._ , “Controlled and reproducible domain wall displacement by current pulses injected into ferromagnetic ring structures,” Physical Review Letters 94, 106601 (2005a).
  * Yang  _et al._ [2007] S. Yang _et al._ , “Spin-transfer-torque-driven domin-wall dynamics in Permalloy nanowires,” Physical Review B 75, 220403 (2007).
  * Beach  _et al._ [2006] G. Beach _et al._ , “Nonlinear domain-wall velocity enhancement by spin-polarized electric current,” Physical Review Letters 97, 057203 (2006).
  * Hayashi  _et al._ [2006b] M. Hayashi _et al._ , “Influence of current on field-driven domain wall motion in Permalloy nanowires from time resolved measurements of anisotropic magnetoresistance,” Physical Review Letters 96, 197207 (2006b).
  * Emori and Beach [2011a] S. Emori and G. S. Beach, “Roles of the magnetic field and electric current in thermally activated domain wall motion in a submicrometer magnetic strip with perpendicular magnetic anisotropy,” Journal of Physics: Condensed Matter 24, 024214 (2011a).
  * Yamaguchi  _et al._ [2006] A. Yamaguchi _et al._ , “Reduction of threshold current density for current-driven domain wall motion using shape control,” Japanese journal of applied physics 45, 3850 (2006).
  * Boone  _et al._ [2010] C. Boone _et al._ , “Rapid domain wall motion in Permalloy nanowires excited by a spin-polarized current applied perpendicular to the nanowire,” Physical Review Letters 104, 097203 (2010).
  * Fukami  _et al._ [2008] S. Fukami _et al._ , “Micromagnetic analysis of current driven domain wall motion in nanostrips with perpendicular magnetic anisotropy,” Journal of Applied Physics 103, 07E718 (2008).
  * Jung  _et al._ [2008] S.-W. Jung _et al._ , “Current-induced domain wall motion in a nanowire with perpendicular magnetic anisotropy,” Applied Physics Letters 92, 202508 (2008).
  * Komine  _et al._ [2011] T. Komine _et al._ , “Reduction of intrinsic critical current density for current-induced domain wall motion by using a ferrimagnetic nanowire with perpendicular magnetic anisotropy,” Journal of Applied Physics 109, 07D503 (2011).
  * Emori and Beach [2011b] S. Emori and G. S. Beach, “Enhanced current-induced domain wall motion by tuning perpendicular magnetic anisotropy,” Applied Physics Letters 98, 132508 (2011b).
  * Lepadatu  _et al._ [2017] S. Lepadatu _et al._ , “Synthetic ferrimagnet nanowires with very low critical current density for coupled domain wall motion,” Scientific reports 7, 1–10 (2017).
  * Yang  _et al._ [2015] S.-H. Yang _et al._ , “Domain-wall velocities of up to $750\,\rm{m}\rm{s}^{-1}$ driven by exchange-coupling torque in synthetic antiferromagnets,” Nature Nanotechnology 10, 221–226 (2015).
  * Song  _et al._ [2021] C. Song _et al._ , “Spin-orbit torques: Materials, mechanisms, performances, and potential applications,” Progress in Materials Science 118, 100761 (2021).
  * Dyakonov and Perel [1971] M. I. Dyakonov and V. Perel, “Current-induced spin orientation of electrons in semiconductors,” Physics Letters A 35, 459–460 (1971).
  * Seo  _et al._ [2012] S.-M. Seo _et al._ , “Current-induced motion of a transverse magnetic domain wall in the presence of spin Hall effect,” Applied Physics Letters 101, 022405 (2012).
  * Ryu  _et al._ [2013] J. Ryu _et al._ , “Current-driven domain wall motion with spin Hall effect: Reduction of threshold current density,” Applied Physics Letters 102, 172404 (2013).
  * Haazen  _et al._ [2013] P. Haazen _et al._ , “Domain wall depinning governed by the spin Hall effect,” Nature Materials 12, 299–303 (2013).
  * Ramaswamy  _et al._ [2018] R. Ramaswamy _et al._ , “Recent advances in spin-orbit torques: Moving towards device applications,” Applied Physics Reviews 5, 031107 (2018).
  * Zhang  _et al._ [2017] Y. Zhang, S. Luo, X. Yang, and C. Yang, “Spin-orbit-torque-induced magnetic domain wall motion in Ta/CoFe nanowires with sloped perpendicular magnetic anisotropy,” Scientific reports 7, 1–10 (2017).
  * Lo Conte  _et al._ [2014] R. Lo Conte _et al._ , “Spin-orbit torque-driven magnetization switching and thermal effects studied in Ta$\backslash$CoFeB$\backslash$MgO nanowires,” Applied Physics Letters 105, 122404 (2014).
  * Yoon  _et al._ [2017] J. Yoon _et al._ , “Anomalous spin-orbit torque switching due to field-like torque–assisted domain wall reflection,” Science advances 3, e1603099 (2017).
  * Kohno  _et al._ [2020] R. Kohno _et al._ , “Domain wall propagation by spin-orbit torques in in-plane magnetized systems,” Physical Review B 102, 020410 (2020).
  * Miron  _et al._ [2011] I. M. Miron _et al._ , “Fast current-induced domain-wall motion controlled by the Rashba effect,” Nature Materials 10, 419–423 (2011).
  * Martinez  _et al._ [2013] E. Martinez _et al._ , “Current-driven domain wall motion along high perpendicular anisotropy multilayers: The role of the Rashba field, the spin Hall effect, and the Dzyaloshinskii-Moriya interaction,” Applied Physics Letters 103, 072406 (2013).
  * Ajejas  _et al._ [2017] F. Ajejas _et al._ , “Tuning domain wall velocity with Dzyaloshinskii-Moriya interaction,” Applied Physics Letters 111, 202402 (2017).
  * Dao  _et al._ [2019] T. P. Dao _et al._ , “Chiral domain wall injector driven by spin–orbit torques,” Nano Letters 19, 5930–5937 (2019).
  * Hristoforou and Ktena [2007] E. Hristoforou and A. Ktena, “Magnetostriction and magnetostrictive materials for sensing applications,” Journal of Magnetism and Magnetic Materials 316, 372–378 (2007).
  * Meier [2015] D. Meier, “Functional domain walls in multiferroics,” Journal of Physics: Condensed Matter 27, 463003 (2015).
  * Dean  _et al._ [2011] J. Dean _et al._ , “Stress-based control of magnetic nanowire domain walls in artificial multiferroic systems,” Journal of Applied Physics 109, 023915 (2011).
  * Yu  _et al._ [2022] G. Yu _et al._ , “Strain-driven magnetic domain wall dynamics controlled by voltage in multiferroic heterostructures,” Journal of Magnetism and Magnetic Materials 552, 169229 (2022).
  * Lei  _et al._ [2013] N. Lei _et al._ , “Strain-controlled magnetic domain wall propagation in hybrid piezoelectric/ferromagnetic structures,” Nature communications 4, 1–7 (2013).
  * Dean  _et al._ [2015] J. Dean _et al._ , “A sound idea: Manipulating domain walls in magnetic nanowires using surface acoustic waves,” Applied Physics Letters 107, 142405 (2015).
  * Adhikari  _et al._ [2021] A. Adhikari _et al._ , “Surface acoustic wave assisted depinning of magnetic domain walls,” Journal of Physics: Condensed Matter 33, 31LT01 (2021).
  * Novosad  _et al._ [2000] V. Novosad _et al._ , “Novel magnetostrictive memory device,” Journal of Applied physics 87, 6400–6402 (2000).
  * Sohn  _et al._ [2015] H. Sohn _et al._ , “Electrically driven magnetic domain wall rotation in multiferroic heterostructures to manipulate suspended on-chip magnetic particles,” Acs Nano 9, 4814–4826 (2015).
  * Sandig  _et al._ [2016] O. Sandig _et al._ , “Movement of magnetic domain walls induced by single femtosecond laser pulses,” Physical Review B 94, 054414 (2016).
  * Bogart  _et al._ [2009] L. Bogart _et al._ , “Dependence of domain wall pinning potential landscapes on domain wall chirality and pinning site geometry in planar nanowires,” Physical Review B 79, 054414 (2009).
  * Himeno  _et al._ [2003] A. Himeno _et al._ , “Dynamics of a magnetic domain wall in magnetic wires with an artificial neck,” Journal of Applied Physics 93, 8430–8432 (2003).
  * Dutta  _et al._ [2017] S. Dutta _et al._ , “The spatial resolution limit for an individual domain wall in magnetic nanowires,” Nano letters 17, 5869–5874 (2017).
  * Kläui  _et al._ [2003a] M. Kläui _et al._ , “Domain wall pinning in narrow ferromagnetic ring structures probed by magnetoresistance measurements,” Physical Review Letters 90, 097202 (2003a).
  * Huang and Lai [2009] S.-H. Huang and C.-H. Lai, “Domain-wall depinning by controlling its configuration at notch,” Applied Physics Letters 95, 032505 (2009).
  * Al Bahri  _et al._ [2019] M. Al Bahri _et al._ , “Staggered magnetic nanowire devices for effective domain-wall pinning in racetrack memory,” Physical Review Applied 11, 024023 (2019).
  * Petit  _et al._ [2008] D. Petit _et al._ , “Domain wall pinning and potential landscapes created by constrictions and protrusions in ferromagnetic nanowires,” Journal of Applied Physics 103, 114307 (2008).
  * Eastwood  _et al._ [2011] D. Eastwood _et al._ , “Chirality-dependent domain wall pinning in a multinotched planar nanowire and chirality preservation using transverse magnetic fields,” Journal of Applied Physics 109, 013903 (2011).
  * Bryan  _et al._ [2007] M. Bryan _et al._ , “Symmetric and asymmetric domain wall diodes in magnetic nanowires,” Applied Physics Letters 91, 142502 (2007).
  * Goolaup  _et al._ [2011] S. Goolaup _et al._ , “Dependence of pinning on domain wall spin structure and notch geometry,” in  _Journal of Physics: Conference Series_ , Vol. 266 (IOP Publishing, 2011) p. 012079.
  * Omari  _et al._ [2019] K. A. Omari _et al._ , “Toward chirality-encoded domain wall logic,” Advanced Functional Materials 29, 1807282 (2019).
  * Allwood  _et al._ [2004] D. Allwood _et al._ , “Domain wall diodes in ferromagnetic planar nanowires,” Applied Physics Letters 85, 2848–2850 (2004).
  * Himeno  _et al._ [2005] A. Himeno _et al._ , “Propagation of a magnetic domain wall in magnetic wires with asymmetric notches,” Journal of Applied Physics 97, 066101 (2005).
  * Luo  _et al._ [2021] Z. Luo _et al._ , “Field-and current-driven magnetic domain-wall inverter and diode,” Physical Review Applied 15, 034077 (2021).
  * Hayward  _et al._ [2010a] T. Hayward _et al._ , “Pinning induced by inter-domain wall interactions in planar magnetic nanowires,” Applied Physics Letters 96, 052502 (2010a).
  * Hayward  _et al._ [2010b] T. Hayward _et al._ , “Direct imaging of domain-wall interactions in $\rm{Ni}_{80}\rm{Fe}_{20}$ planar nanowires,” Physical Review B 81, 020410 (2010b).
  * O’Brien  _et al._ [2009] L. O’Brien _et al._ , “Near-field interaction between domain walls in adjacent Permalloy nanowires,” Physical Review Letters 103, 077206 (2009).
  * O’Brien  _et al._ [2011] L. O’Brien _et al._ , “Dynamic propagation and nucleation in domain wall nanowire devices,” Journal of Physics: Condensed Matter 24, 024222 (2011).
  * Purnama  _et al._ [2011] I. Purnama _et al._ , “Current-induced coupled domain wall motions in a two-nanowire system,” Applied Physics Letters 99, 152501 (2011).
  * Kim  _et al._ [2009] K.-J. Kim _et al._ , “Depinning field at notches of ferromagnetic nanowires with perpendicular magnetic anisotropy,” IEEE Transactions on Magnetics 45, 4056–4058 (2009).
  * Jang  _et al._ [2009] Y. Jang _et al._ , “Current-induced domain wall nucleation and its pinning characteristics at a notch in a spin-valve nanowire,” Nanotechnology 20, 125401 (2009).
  * Jang  _et al._ [2012] Y. Jang _et al._ , “Current-driven domain wall motion in heterostructured ferromagnetic nanowires,” Applied Physics Letters 100, 112401 (2012).
  * Ho  _et al._ [2018] L. A. Ho _et al._ , “Field-driven single domain wall motion in ferromagnetic nanowires,” RSC advances 8, 14539–14551 (2018).
  * Vernier  _et al._ [2004] N. Vernier _et al._ , “Domain wall propagation in magnetic nanowires by spin-polarized current injection,” EPL (Europhysics Letters) 65, 526 (2004).
  * Garg  _et al._ [2017] C. Garg _et al._ , “Dramatic influence of curvature of nanowire on chiral domain wall velocity,” Science advances 3, e1602804 (2017).
  * Diegel, Mattheis, and Halder [2004] M. Diegel, R. Mattheis, and E. Halder, “$360^{\circ}$ domain wall investigation for sensor applications,” IEEE Transactions on Magnetics 40, 2655–2657 (2004).
  * Glathe and Mattheis [2012] S. Glathe and R. Mattheis, “Magnetic domain wall pinning by kinks in magnetic nanostripes,” Physical Review B 85, 024405 (2012).
  * Schönke  _et al._ [2020] D. Schönke _et al._ , “Quantification of competing magnetic states and switching pathways in curved nanowires by direct dynamic imaging,” ACS Nano 14, 13324–13332 (2020).
  * Castano  _et al._ [2003] F. Castano _et al._ , “Metastable states in magnetic nanorings,” Physical Review B 67, 184425 (2003).
  * Kläui  _et al._ [2003b] M. Kläui _et al._ , “Vortex formation in narrow ferromagnetic rings,” Journal of Physics: Condensed Matter 15, R985 (2003b).
  * Vaz  _et al._ [2007] C. A. Vaz _et al._ , “Ferromagnetic nanorings,” Journal of Physics: Condensed Matter 19, 255207 (2007).
  * Negoita  _et al._ [2013] M. Negoita _et al._ , “Domain walls in ring-shaped nanowires under rotating applied fields,” Journal of Applied Physics 114, 013904 (2013).
  * Bisig  _et al._ [2015] A. Bisig _et al._ , “Dynamic domain wall chirality rectification by rotating magnetic fields,” Applied Physics Letters 106, 122401 (2015).
  * Kläui  _et al._ [2005b] M. Kläui _et al._ , “Domain wall coupling and collective switching in interacting mesoscopic ring magnet arrays,” Applied Physics Letters 86, 032504 (2005b).
  * Welp  _et al._ [2003] U. Welp, V. Vlasko-Vlasov, G. Crabtree, J. Hiller, N. Zaluzec, V. Metlushko, and B. Ilic, “Magnetization reversal in arrays of individual and coupled Co-rings,” Journal of Applied physics 93, 7056–7058 (2003).
  * Faulkner  _et al._ [2003] C. C. Faulkner _et al._ , “Controlled switching of ferromagnetic wire junctions by domain wall injection,” IEEE Transactions on Magnetics 39, 2860–2862 (2003).
  * Kwon  _et al._ [2017] J. Kwon _et al._ , “Asymmetrical domain wall propagation in bifurcated PMA wire structure due to the Dzyaloshinskii-Moriya interaction,” Applied Physics Letters 110, 232402 (2017).
  * Mougin  _et al._ [2007] A. Mougin _et al._ , “Domain wall mobility, stability and Walker breakdown in magnetic nanowires,” EPL (Europhysics Letters) 78, 57007 (2007).
  * Tretiakov  _et al._ [2008] O. Tretiakov _et al._ , “Dynamics of domain walls in magnetic nanostrips,” Physical Review Letters 100, 127204 (2008).
  * Pi  _et al._ [2011] U.-H. Pi _et al._ , “Static and dynamic depinning processes of a magnetic domain wall from a pinning potential,” Physical Review B 84, 024426 (2011).
  * Schryer and Walker [1974] N. L. Schryer and L. R. Walker, “The motion of $180^{\circ}$ domain walls in uniform dc magnetic fields,” Journal of Applied Physics 45, 5406–5421 (1974).
  * Lee  _et al._ [2007] J.-Y. Lee _et al._ , “Dynamic transformations of the internal structure of a moving domain wall in magnetic nanostripes,” Physical Review B 76, 184408 (2007).
  * Duine, Núñez, and MacDonald [2007] R. Duine, A. Núñez, and A. MacDonald, “Thermally assisted current-driven domain-wall motion,” Physical Review Letters 98, 056605 (2007).
  * Koyama  _et al._ [2012] T. Koyama _et al._ , “Current-induced magnetic domain wall motion below intrinsic threshold triggered by Walker breakdown,” Nature Nanotechnology 7, 635–639 (2012).
  * Cacilhas  _et al._ [2020] R. Cacilhas _et al._ , “Controlling domain wall oscillations in bent cylindrical magnetic wires,” Physical Review B 101, 184418 (2020).
  * Moreno  _et al._ [2017] R. Moreno _et al._ , “Oscillatory behavior of the domain wall dynamics in a curved cylindrical magnetic nanowire,” Physical Review B 96, 184401 (2017).
  * Thomas  _et al._ [2006] L. Thomas _et al._ , “Oscillatory dependence of current-driven magnetic domain wall motion on current pulse length,” Nature 443, 197–200 (2006).
  * Broomhall and Hayward [2017] T. Broomhall and T. Hayward, “Suppression of stochastic domain wall pinning through control of Gilbert damping,” Scientific reports 7, 1–12 (2017).
  * Lewis  _et al._ [2010] E. Lewis _et al._ , “Fast domain wall motion in magnetic comb structures,” Nature Materials 9, 980–983 (2010).
  * McMichael  _et al._ [2000] R. D. McMichael _et al._ , “Domain wall traps for low-field switching of submicron elements,” Journal of Applied Physics 87, 7058–7060 (2000).
  * Numata  _et al._ [2007] H. Numata _et al._ , “Scalable cell technology utilizing domain wall motion for high-speed MRAM,” in  _2007 IEEE Symposium on VLSI Technology_ (IEEE, 2007) pp. 232–233.
  * Parkin and Yang [2015] S. Parkin and S.-H. Yang, “Memory on the racetrack,” Nature Nanotechnology 10, 195–198 (2015).
  * Zhu and Park [2006] J.-G. J. Zhu and C. Park, “Magnetic tunnel junctions,” Materials today 9, 36–45 (2006).
  * Ikeda  _et al._ [2008] S. Ikeda _et al._ , “Tunnel magnetoresistance of 604% at 300 K by suppression of Ta diffusion in CoFeB/MgO/CoFeB pseudo-spin-valves annealed at high temperature,” Applied Physics Letters 93, 082508 (2008).
  * Lou  _et al._ [2008] X. Lou _et al._ , “Demonstration of multilevel cell spin transfer switching in mgo magnetic tunnel junctions,” Applied Physics Letters 93, 242502 (2008).
  * Raymenants  _et al._ [2021] E. Raymenants _et al._ , “Nanoscale domain wall devices with magnetic tunnel junction read and write,” Nature Electronics 4, 392–398 (2021).
  * Yamaguchi  _et al._ [2005] A. Yamaguchi _et al._ , “Effect of Joule heating in current-driven domain wall motion,” Applied Physics Letters 86, 012511 (2005).
  * Kumar  _et al._ [2022] D. Kumar _et al._ , “Domain wall memory: Physics, materials, and devices,” Physics Reports 958, 1–35 (2022).
  * Kang  _et al._ [2019] W. Kang _et al._ , “A comparative cross-layer study on racetrack memories: Domain wall vs skyrmion,” ACM Journal on Emerging Technologies in Computing Systems (JETC) 16, 1–17 (2019).
  * Mittal [2016] S. Mittal, “A survey of techniques for architecting processor components using domain-wall memory,” ACM Journal on Emerging Technologies in Computing Systems (JETC) 13, 1–25 (2016).
  * Allwood  _et al._ [2006b] D. Allwood _et al._ , “Magnetic domain wall serial-in parallel-out shift register,” Applied Physics Letters 89, 102504 (2006b).
  * Zheng  _et al._ [2020] X. Zheng _et al._ , “Paradigm of magnetic domain wall-based in-memory computing,” ACS Applied Electronic Materials 2, 2375–2382 (2020).
  * Vandermeulen  _et al._ [2015] J. Vandermeulen _et al._ , “Logic and memory concepts for all-magnetic computing based on transverse domain walls,” Journal of Physics D: Applied Physics 48, 275003 (2015).
  * Currivan-Incorvia  _et al._ [2016] J. A. Currivan-Incorvia _et al._ , “Logic circuit prototypes for three-terminal magnetic tunnel junctions with mobile domain walls,” Nature communications 7, 1–7 (2016).
  * Manfrini  _et al._ [2018] M. Manfrini _et al._ , “Interconnected magnetic tunnel junctions for spin-logic applications,” AIP Advances 8, 055921 (2018).