# Computing and Memory Technologies based on Magnetic Skyrmions

Hamed Vakili  Department of Physics, University of Virginia, Charlottesville
VA 22904 USA  Md Golam Morshed  Department of Electrical and Computer
Engineering, University of Virginia, Charlottesville VA 22904 USA  Timothy Q.
Hartnett  Department of Materials Science and Engineering, University of
Virginia, Charlottesville VA 22904 USA  Wei Zhou  Department of Physics,
University of Virginia, Charlottesville VA 22904 USA  Mohammad Nazmus Sakib
Department of Electrical and Computer Engineering, University of Virginia,
Charlottesville VA 22904 USA  Samiran Ganguly  Department of Electrical and
Computer Engineering, University of Virginia, Charlottesville VA 22904 USA
Chung T. Ma  Department of Physics, University of Virginia, Charlottesville VA
22904 USA  Jun-Wen Xu  Center for Quantum Phenomena, Department of Physics,
New York University, New York, NY 10003 USA  Yassine Quessab  Center for
Quantum Phenomena, Department of Physics, New York University, New York, NY
10003 USA  Kai Litzius  Center for Materials Science and Engineering,
Massachusetts Institute of Technology, Cambridge MA 02139 USA  S. J. Poon
Department of Physics, University of Virginia, Charlottesville VA 22904 USA
Mircea Stan  Department of Electrical and Computer Engineering, University of
Virginia, Charlottesville VA 22904 USA  Prasanna Balachandran  Department of
Materials Science and Engineering, University of Virginia, Charlottesville VA
22904 USA  Andrew D. Kent  Center for Quantum Phenomena, Department of
Physics, New York University, New York, NY 10003 USA  Geoffrey Beach  Center
for Materials Science and Engineering, Massachusetts Institute of Technology,
Cambridge MA 02139 USA  Avik W. Ghosh  Department of Electrical and Computer
Engineering, University of Virginia, Charlottesville VA 22904 USA  Department
of Physics, University of Virginia, Charlottesville VA 22904 USA

###### Abstract

Solitonic magnetic excitations such as domain walls and, specifically,
skyrmionics enable the possibility of compact, high density, ultrafast, all-
electronic, low-energy devices, which is the basis for the emerging area of
skyrmionics. The topological winding of skyrmion spins affects their overall
lifetime, energetics, and dynamical behavior. In this review, we discuss
skyrmionics in the context of the present-day solid-state memory landscape and
show how their size, stability, and mobility can be controlled by material
engineering, as well as how they can be nucleated and detected. Ferrimagnets
near their compensation points are promising candidates for this application,
leading to a detailed exploration of amorphous CoGd as well as the study of
emergent materials such as Mn4N and Inverse Heusler alloys. Along with
material properties, geometrical parameters such as film thickness, defect
density, and notches can be used to tune skyrmion properties, such as their
size and stability. Topology, however, can be a double-edged sword, especially
for isolated metastable skyrmions, as it brings stability at the cost of
additional damping and deflective Magnus forces compared to domain walls.
Skyrmion deformation in response to forces also makes them intrinsically
slower than domain walls. We explore potential analog applications of
skyrmions, including temporal memory at low density - one skyrmion per
racetrack - that capitalizes on their near ballistic current-velocity relation
to map temporal data to spatial data and decorrelators for stochastic
computing at a higher density that capitalizes on their interactions. We
summarize the main challenges of achieving a skyrmionics technology, including
maintaining positional stability with very high accuracy and electrical
readout, especially for small ferrimagnetic skyrmions, deterministic
nucleation, and annihilation and overall integration with digital circuits
with the associated circuit overhead.

###### Contents

  1. I The Memory Landscape
    1. I.1 Introduction: The Role of Nonvolatile Magnetic Memory
    2. I.2 Energy-Delay-Error Trade-off: Skyrmions as small stable mobile magnets
    3. I.3 Road to a Skyrmionic Device
    4. I.4 Skyrmions and Topological Stability
  2. II Phase-Space for static skyrmions
    1. II.1 What determines Skyrmion Size?
    2. II.2 Ferrimagnets for Small, Fast Skyrmions
    3. II.3 Parameter Engineering of Skyrmion Size and Stability
    4. II.4 Nucleating isolated metastable skyrmions
    5. II.5 Skyrmion Stability and Lifetime
  3. III Skyrmion Dynamics
    1. III.1 What determines skyrmion speed?
    2. III.2 Skyrmion Hall Effect and Gateability
    3. III.3 The role of Defects and thermal Diffusion
  4. IV Skyrmionic Device Applications
    1. IV.1 Magnetic Memory for Temporal Computing
    2. IV.2 Reconfigurable Skyrmionic Logic
    3. IV.3 Skyrmionic Decorrelators for Stochastic Computing
    4. IV.4 Artificial Skyrmionic Synapses and Neurons
    5. IV.5 Reservoir Computing using Skyrmions
    6. IV.6 Skyrmion based Spintronic Oscillators
  5. V Technology Challenges
  6. VI Summary/Outlook
  7. VII Acknowledgments
  8. VIII Data Availability
  9. IX Appendix
  10. A Skyrmion topology and Poincaré map classification
  11. B Skyrmion Energy terms
  12. C Dynamics of Skyrmion and DWs

##  I The Memory Landscape

###  I.1 Introduction: The Role of Nonvolatile Magnetic Memory

Digital electronics has been driven for several decades by sustained hardware
scaling and Moore’s law. With the recent slowdown in advances in Complementary
Metal Oxide Semiconductor (CMOS) hardware and rapid growth of software, along
with the migration from cloud computing towards edge devices, there is a
strong impetus to re-examine the limits of computing. In conventional Von
Neumann computer architecture, the processor needs to access data stored in
separate memory banks in order to perform logic operations. Unfortunately, the
speed of memory scaling (1.1$\times$ in every two years) has not kept up with
the drastically increased speed of processor scaling (2$\times$ in every two
years). This increasing gap between memory and processor performance, the so-
called memory wall problem, is considered one of the main bottlenecks to
increasing computer performance.

To mitigate the memory wall problem, a multi-level memory hierarchy is used,
with the most frequently invoked instructions and data sets stored in a high-
speed on-chip cache so that they can be accessed and executed efficiently by
the processor. On-chip cache memory is typically built out of static random
access memory (SRAM). However, SRAM is expensive as it requires at least six
transistors per bit (two sets of CMOS inverter pairs to form a memory latch
plus two access transistors) in order to store binary data with acceptable
reliability. The need for so many transistors per cell also means that the
capacity of the on-chip cache cannot be too large, requiring a large degree of
off-chip access to dynamic random access memory (DRAM). Each DRAM cell
consists of single capacitor-transistor pairs and is significantly smaller
than an SRAM cell, but it is slow and also needs a regular refresh to
replenish data that leaks from the capacitor. Consequently, the number of
clock cycles to access data from the off-chip DRAM also increases, resulting
in overall energy inefficiency. A lot of active research is thus focused on
developing a memory technology that can match the speed performance of SRAM
and the high density of DRAM. Magnet-based nonvolatile memory is an emerging
candidate addressing that technology bottleneck.

Magnet-based non-volatile memory has a rich history, with field-switched
magnetic RAMs (MRAMs) and spin-transfer torque based RAMs (STT-RAMs) now
commercialized Kent and Worledge (2015); Thomas  _et al._ (2017); Slaughter
_et al._ (2017). In magnetic memory, the different directions of magnetization
can be used to store data as digital bits (e.g., up equals one, down is zero).
Magnetic states do not leak away like charges (i.e., magnetic states can be
non-volatile). The energy required to switch a magnet is fairly low, mainly
because internal exchange forces make the magnet act as a single giant spin.
However, energy is dissipated in the overhead/control circuitry due to the
need for currents generated to create the switching fields. Field switched
MRAMs are hard to scale because the scaling of the drive circuit requires
increasing current densities with increasing energy dissipation. STT-RAMs have
better-scaling properties, as the switching is due to spin-polarized currents
directly injected into a magnet across the oxide of a magnetic tunnel junction
(MTJ).

STT-RAM has garnered a lot of attention both from academia and industry due to
its compatibility with CMOS processes and voltages, zero standby leakage (non-
volatility), scalability, high endurance and retention time, and overall
reliability. In fact, STT-RAM has evolved from the discovery of giant
magnetoresistance in the late 1980s Baibich  _et al._ (1988); Binasch  _et
al._ (1989) and theoretical ideas soon after Slonczewski (1989, 1996); Berger
(1996), to a dark horse in the 2000s Kiselev  _et al._ (2003); Gilbert (2004)
to becoming now a commercially viable candidate for on-chip cache memory Kim
_et al._ (2011), with both the integration density of DRAM and comparable
performance to SRAMShao  _et al._ (2021). In particular, its density advantage
compared to volatile transistor-based SRAM can improve the on-chip cache
capacity significantly. Even if just the last level cache capacity in the
multi-level memory hierarchy increases, this reduces the number of off-chip
DRAM accesses. In brief, there is significant motivation to develop a high-
capacity non-volatile on-chip last-level cache.

One of the significant challenges with the widespread adoption of STT-RAMs as
a universal memory technology is the high write energy and low read speed of
STT-RAM cache memory relative to CMOS on-chip cache. It is worth mentioning
that the write latency and energy of STT-RAM can be decreased by trading
against the retention time, the energy barrier between binary states. Longer
access times, along with the high error rates, pose other challenges in using
STT-RAM for on-chip cache (in short, there is an overall energy-delay-error
trade-off) Xie  _et al._ (2017). The error rate of STT-RAM can be mitigated by
adopting intelligent array-level design techniques; however, they do not
maximize STT-RAM benefits. The error rate problem gets exacerbated, and the
robustness of STT-RAM suffers while it is scaled down, requiring stronger
error correction schemes and a compromised energy efficiency benefit.
Requirements like bidirectional write currents, asymmetry in the critical
write currents, single-ended sensing operation, and shared current paths for
read and write impose further challenges in establishing STT-RAMs as a
replacement of last level cache.

Part of the energy hungriness of STT-RAMs arises from the writing of
information, which requires passing current through a high resistance tunnel
barrier. The same current path is used to read, but much lower currents are
used. Structures with orthogonal read-write paths provide a possible way out
of this impasse—a metallic path being used to write information separate from
the read path (which can again be via a magnetic tunnel junction). More
details on the target specifications of SOT and STT RAMs is discussed in Shao
_et al._ (2021)

This brings us to the evaluation of bits encoded by topological (solitonic)
excitations in thin magnetic films, such as line-like domain walls and
circular skyrmions. Such excitations can encode information in ultrasmall
volumes below the thermal superparamagnetic limit that constrains regular
magnets. The solitons can be driven at high speeds along with magnetic
racetracks by modest currents and energy dissipation in heavy-metal
underlayers, generating entirely unique device applications. The high density
of ultrasmall skyrmions stabilized by their topological barriers, as well as
their quasi-ballistic, tunable, and linear dynamics, are particular attributes
that make skyrmionic devices potentially useful in a variety of applications.

The purpose of this review is to go over the physics of isolated skyrmions—the
factors controlling their size, dynamics, lifetime, and switching, the
material classes that support them, and how they may potentially be utilized
for conventional or unconventional computing applications.

The structure of this paper is as follows: In Section I, we go over the memory
landscape; in Sections II and III, we describe the fundamental physics based
and material based limitations and parameter dependences of skyrmions and
domain walls, along with experimental characterization. In Section IV, we
present skyrmion device applications. Lastly, in Section V, we discuss the
challenges and opportunities with select skyrmion devices.

Table 1: Comparison of proposed domain wall-based racetrack memory (RTM) Parkin  _et al._ (2008) with other memory technologies, including spin-transfer torque-based random access memory (STT-RAM), resistive random access memory (RRAM), phase-change memory (PCM), and magnetic random access memory (MRAM). DW-RTM has lower predicted energy consumption, leakage power, and size, but DWs are typically limited by pinning at the racetrack edges, where skyrmions are purported to have a distinct advantage Bläsing  _et al._ (2020). R. Bläsing et al., Proceedings of the IEEE, vol. 108, no. 8, pp. 1303-1321, Aug. 2020; licensed under a Creative Commons Attribution (CC BY) license.  Technology |  Write/Erase Times (ns) |  Write Energy |  Read
Times (ns) |  Read Energy |  Leakage Power |  Endurance |  Non-volatility |  Cell
Size ($F^{2}$)
---|---|---|---|---|---|---|---|---
RTM |  3-250∗ |  Low |  3-250∗ |  Low |  Low |  $\geq$ 1016 |  Yes |  $\leq$ 2
RRAM |  20 |  High |  10-20 |  Low |  Low |  1011 |  Yes |  4-10
PCM |  >30 |  High |  5-20 |  Medium |  Low |  109 |  Yes |  4-12
MRAM |  10-20 |  High |  3-20 |  Low |  Low |  1012 |  Yes |  10-60
STT-RAM |  3-15 |  High |  3-15 |  Low |  Low |  4$\times 10^{12}$ |  Yes |  6-50

###  I.2 Energy-Delay-Error Trade-off: Skyrmions as small stable mobile
magnets

One of the performance metrics of any binary switch is its energy-delay
product, arguing for schemes that lower both their power dissipation and
overall latency. The energy-delay product can be written as
$I^{2}R_{\mathrm{on}}t^{2}=Q^{2}R_{\mathrm{on}}$ with $t=Q/I$, where $Q$ is
the amount of charge needed to switch from off to on and $R_{\mathrm{on}}$ is
the on-state resistance. The charge $Q$ needed to switch a small magnet is set
by (a) the minimum critical charge $Q_{c}$ needed to switch magnetization,
mandated by angular momentum conservation, and (b) an added overdrive
$Q/Q_{c}\gg 1$ required to reduce the dynamic write error rate (WER) or
switching time to application-specific targets. This charge gets smaller for
small volume magnetic excitations, but so does the energy barrier that sets
the thermal stability. Technologies such as Heat Assisted Magnetic Recording
(HAMR) and Bit Patterned Recording attempt to address this specific challenge.
HAMR reduces the anisotropy barrier quickly through spot heating over a
localized size and then allowing it to restore through fast cooling. Bit
Patterned Recording breaks the magnet into tinier lithographically patterned
ordered grains that can individually switch quickly but gain volume stability
through strong inter-grain exchange interactions. A third way to get small
stable magnetic excitations is by nucleating metastable skyrmions, essentially
tiny mobile magnets, held together against thermal fluctuations by their
topological properties.

Let us start with conventional spin-transfer torque (STT), where a current
injected across the oxide of a magnetic tunnel junction (MTJ) gets polarized
by a fixed magnet, and then the spins enter a free magnet and start precessing
incoherently around its magnetization until their transverse component gets
fully absorbed. The transferred angular momentum generates an anti-damping
torque that rotates the free magnetization by acting directly against damping.
We can work out the dynamics of these spins using the Landau-Lifschitz-
Gilbert-Slonczewski (LLGS) Slonczewski (1996); Gilbert (2004) equation with a
stochastic thermal torque, or equivalently a Fokker-Planck equation for the
probability distribution function of the magnetization Xie  _et al._ (2017,
2017). For a small magnet with perpendicular magnetic anisotropy in an MTJ, we
can get the time-dependent WER from an analytical solution to the 1-D Fokker
Planck equation. For large overdrive current $i=I/I_{c}\gg 1$, this works out
to be Munira  _et al._ (2012)

$\mathrm{WER}_{\mathrm{STT}}(t)\approx\frac{\pi^{2}\Delta_{\mathrm{th}}}{4}e^{\displaystyle-2Q/Q_{c}},~{}~{}~{}Q=It,$
(1)

where the critical current $I_{c}$ is set in turn by a critical charge that
satisfies angular momentum conservation between electron and flipped
magnetization

$\displaystyle\underbrace{\Biggl{(}\displaystyle\frac{Q_{c}}{q}\Biggr{)}}_{\displaystyle
N=n\Omega}\frac{\hbar}{2}P\gamma=M_{s}\Omega(1+\alpha^{2})$

where $P$ is the spin polarization of the current, $n$ is the electron
density, $q$ is the charge of an electron, $\gamma=g\mu_{B}/\hbar$ is the
electron gyromagnetic ratio, $\hbar$ is the reduced Planck’s constant, $g$ is
the electron dimensionless magnetic moment or g-factor, $\mu_{B}$ is the Bohr
magneton, $\mu_{0}$ is the permeability of free space, $M_{s}$ is the
saturation magnetization density, $\alpha$ is Gilbert damping, $\Omega$ is the
volume of the flipped magnetic layer and $\Delta_{\mathrm{th}}$ is the thermal
stability factor, $\Delta_{\mathrm{th}}=E_{B}/(k_{B}T)$, the energy barrier
$E_{B}$ for magnetization reversal divided by the thermal energy, Boltzmann’s
constant $k_{B}$ times the temperature $T$. The critical current density is
given by Sun (1999, 2000):

$j^{STT}_{c}=\displaystyle\frac{2q}{\hbar}\frac{\alpha}{P}\mu_{0}M_{s}H_{k}t_{\mathrm{F}},$
(2)

where $t_{\mathrm{F}}$ is the magnetic layer thickness, and $H_{k}$ is the
anisotropy field. The interesting observation is that the critical switching
time, set by the ferromagnetic resonance frequency and the damping
$\tau_{D}=(1+\alpha^{2})/(\alpha\gamma\mu_{0}H_{k})$, increases as we reduce
the damping. However, the critical current $I_{c}=Q_{c}/\tau_{D}=2q\alpha
E_{B}/(\hbar P)$ decreases, such that $Q_{c}$ is independent of anisotropy and
weakly dependent on damping, and is set by angular momentum conservation
alone. Note also that the critical spin current (angular momentum per second),
can be written simply as $I_{s}=(I_{c}/q)\times P\hbar/2=\alpha E_{B}$.

The equations above set the charge $Q$ required for writing information at a
given speed and error rate. The demand for a low error rate, $\sim 10^{-9}$
for memory and $\sim 10^{-15}$ for logic, will require a significant overdrive
$Q\gg Q_{c}$. In a $100\times 20\times 1$ nm3 Fe magnet of saturation
magnetization $\mu_{0}M_{s}\approx 1.9$ T with about $10^{4}$ spins, assuming
a Gilbert damping of $\alpha=0.01$ and polarization $P=0.7$, we get $Q_{c}/q$
about $5\times 10^{5}$ electrons for destabilization. Assuming an anisotropy
energy $\sim 55$ kJ/m3, we get an energy barrier $E_{b}\approx 30k_{B}T$,
which implies an overdrive with $Q/q\approx 6.5\times 10^{6}$ electrons at an
error rate of $10^{-9}$. At $1$ ns switching time, the current density
$\approx 50$ MA/cm2.

While the critical current is set by the magnet’s energy barrier $E_{B}$ and
the charge by angular momentum conservation, the dissipated energy
$E_{\mathrm{diss}}=I^{2}R_{\mathrm{on}}t$ is several orders of magnitude
higher. This is because the voltage needed to produce the necessary current is
set by the resistance of the oxide tunnel barrier $R_{\mathrm{on}}$, whose
energy barrier is much larger than that of the switching magnet.

The energy dissipated can be reduced in all metallic devices such as proposed
race track memories (Table I), where a spin current is injected into a
metallic magnet from a heavy metal underlayer and the magnetization state read
with a vertical current (e.g., a current passing through a tunnel barrier
between the magnet and a fixed magnet with much higher anisotropy barrier),
thus separating the read and write paths. Spin-orbit interactions in the heavy
metal inject a spin perpendicular to the metal-magnet interface. The
corresponding spin-orbit torque (SOT) can be used to flip a magnet. The
current density can once again be obtained from energy considerations, while
the write error rate obtained using a 2-D Fokker Planck equation with in and
out-of-plane fields. Assuming small in-plane field $H_{x}$ compared to the
anisotropy field $H_{k}$, i.e., $h_{x}=H_{x}/H_{k}\ll 1$ and a magnetization
perpendicular to the heavy metal-magnet interface, the equations once again
simplify Lee  _et al._ (2013, 2014); Xie  _et al._ (2017)

$\displaystyle\mathrm{WER}_{\mathrm{SOT}}(t)$ $\displaystyle\approx$
$\displaystyle e^{\displaystyle-f_{0}t\exp{(-E_{B}/k_{B}T)}},$ $\displaystyle
E_{B}$ $\displaystyle=$ $\displaystyle
U_{0}\left[(1-h_{x})^{2}-2h_{s}(\pi/2-h_{x}-h_{s})\right],$ $\displaystyle
h_{s}$ $\displaystyle=$ $\displaystyle
c_{J}/H_{K},~{}~{}~{}c_{J}=(\hbar/2q)(J\theta_{SH}/M_{s}t_{F}),$
$\displaystyle j^{SOT}_{c}$ $\displaystyle\approx$
$\displaystyle\displaystyle\frac{2q}{\hbar}\frac{\mu_{0}M_{s}t_{F}}{\theta_{SH}}\Biggl{(}\frac{H_{k}}{2}-\frac{H_{x}}{\sqrt{2}}\Biggr{)},$
(3)

where $c_{J}$ is the coefficient of the spin-orbit torque term.

Once again, the current burden is higher for smaller error rates that need
higher anisotropy barriers and for larger flipped magnetization volumes set by
the film thickness. In fact, $j_{c}^{SOT}$ has a similar structure as
$j^{STT}_{c}$ except we replace polarization $P$ with spin Hall angle
$\theta_{SH}$, and remove the damping term $\alpha$, as the torque involves
precessional rather than anti-damping switching (for perpendicularly
magnetized films). $f_{0}$ is the attempt frequency relating to the overall
energy landscape through an entropy term. Plugging typical numbers ($P\approx
0.7,\theta_{SH}\approx 0.2,\alpha\approx 0.3$, same film thickness), we find
$j_{C}^{SOT}/j_{C}^{STT}\approx P/\theta_{SH}\alpha\approx 10$, larger
primarily because the STT anti-damping current is proportional to Gilbert
damping $\alpha$ ($\alpha\ll 1)$, while the SOT in this example acts
perpendicular to the damping torque, like a field-induced torque. However, the
switching time is correspondingly reduced, and once again, the energy-delay
product depends primarily on the switching charge $Q$, set in essence by
angular momentum conservation and acceptable error rates. A further energy
reduction happens as current flows through the lower resistance
$R_{\mathrm{on}}$ of the metal underlayer, which can be more than an order of
magnitude lower than that of the MTJ’s tunnel barrier. Note that the critical
charge $Q_{c}$ depends on the volume of the switching free layer magnet
$\Omega$. It is, therefore, useful to scale the magnet down to reduce the
energy-delay product. However, the energy barrier $E_{B}$ and the thermal
stability factor $\Delta_{\mathrm{th}}$ also decrease, as a result, making the
magnet thermally unstable (this is called the superparamagnetic limit), a
challenge which the aforementioned HAMR and bit patterned recordings
approaches address head-on. Topological excitations such as skyrmions have
added protection coming from their spin texture, which adds an extra barrier
to their erasure, in other words, to continuously deform them to alternate
textures with different topological numbers—for instance—uniformly oriented
spins in their ferromagnetic background.

It is, therefore, intriguing to consider information encoded in these
topological excitations, how we can move them around with an in-plane current,
and read information encoded in them with a vertical read operation. In this
review, we will discuss the physics of isolated skyrmions, for instance, how
to shrink them to a high density while maintaining adequate stability, and the
tradeoffs involved in driving them deterministically and ballistically. We
will argue, for example, that ferrimagnets can be ideal for generating small
Néel skyrmions. We will also show that adjusted for inertial effects when a
current is switched off, the effective speed of skyrmions is less than that of
domain walls for the same set of magnetic parameters. This is because
skyrmions tend to distort during flow, making them ultimately move more like
domain walls. However, domain walls have more opportunity for pinning—in
particular at edges, for moderate speeds, while at higher speeds, the angular
momentum tends to transfer to azimuthal precession of its individual spins. We
discuss potential uses for small, fast skyrmions - isolated skyrmions as
temporal memory bits for unary wavefront-based computing, high density
cascaded skyrmions for reconfigurable logic, skyrmion gases as decorrelators
in stochastic computing, as well as neural and reservoir computing.

###  I.3 Road to a Skyrmionic Device

For a skyrmionic device to be competitive with the existing and emerging
technologies, the size of the skyrmion needs to be small. The small size of
the skyrmion will enable dense memory and logic circuit design. Furthermore,
the device should have properties like low energy cost, high frequency, and
overall material stability with temperature and external magnetic fields. To
elaborate on the various needs:
Size: In a magnetic material, the competition between the different energy
terms (Eq. 7) can stabilize an isolated skyrmion. The phase space of the
material parameters can be calculated numerically and analytically (Section
II)). Depending on the specific memory technology, sub-50 nm skyrmions need to
be stable enough over seconds (e.g., cache memory) to years (e.g., hard drive)
at room temperature (Section II E). Larger skyrmions tend to diffuse more
aggressively, while smaller skyrmions tend to last shorter, pin easier, and
are harder to read electrically. We will show that the saturation
magnetization $M_{s}$ can set the skyrmion size, while the film thickness can
be used to control their lifetime.

Figure 1: 3-D spherical arrangement of spins that generate 2-D skyrmions
through stereographic projection. The north pole becomes the skyrmion core
while the south pole maps to the background spin texture. The figure shows
skyrmions of various winding numbers $N_{sk}$ with same polarization $p=1$
(Eq. 4) and domain angles/helicities $\psi$, along with typical materials
supporting them. (a) Neel skyrmions (e.g. CoGd on Pt), $\psi=0,N_{sk}=1$ (b)
Bloch skyrmions (B20 solids, e.g. FeGe), $\psi=\pi/2,N_{sk}=1$ (c) Hybrid
skyrmions (e.g. B20 on Pt), $\psi=\pi/4,N_{sk}=1$ (d) AntiSkyrmions (D2d
tetragonal inverse Heuslers e.g. MnPtSn), $\psi=\pi/2,N_{sk}=-1$ (e) Higher
winding number skyrmions (Frustrated FM, merging of two skyrmions with
opposite vorticities), $\psi=0,N_{sk}=2$. These excitations map onto the usual
classification of 2-D linear excitations– stars for Néel skyrmions, cycles for
Bloch skyrmions, spirals for hybrids and saddle points for antiskyrmions. In
this picture, skyrmion spins wind anticlockwise as we move anticlockwise
around a circle, while antiskyrmion spins wind in the opposite way. For higher
$m$ values skyrmions create cycloids along the circle while antiskyrmions
trace out astroids. In general, a skyrmion of winding number $N_{sk}$ has
$N_{sk}-1$ kinks (lines of 1800 spin reversal) while an antiskyrmion of
winding number $-N_{sk}$ has $|N_{sk}|+1$ lines of singularity.

Speed and operation time: For a skyrmionic racetrack device, the rate of data
processing is set by the speed at which skyrmions can travel along the
racetrack. The high-frequency operation of a skyrmionic device enables low
energy logic and memory operations. Thus, the speed of the skyrmion is a key
factor for skyrmionic devices to become a competitive technology. In order to
achieve a high speed for skyrmions, small $M_{s}$, low Gilbert damping and
topological damping are required (Section III, Eq. 14). Small topological
damping and size make nearly compensated ferrimagnets promising candidates for
skyrmions. When a very large current is applied to skyrmions, it has been
reported Litzius  _et al._ (2020a) that skyrmions can be distorted or even get
annihilated, which puts an upper limit on skyrmion operating current. A speed
between 100-700 m/s is probably a suitable target for the nearly linear,
quasi-ballistic operation of skyrmion devices.

Material Stability: Due to the thermal dissipation from the applied electric
current when a hypothetical skyrmionic device is operating, the temperature of
the magnetic layer can increase well above the room temperature. In order to
have a reliable operation, the magnetic layer needs to have a sufficiently
high Curie temperature so that the changes in temperature do not affect
skyrmions too much. Additionally, if the device is expected to work under
real-life conditions, it has to be resistant with respect to external magnetic
fields. In the case of ferri- or antiferromagnets, the contribution of
external magnetic fields is negligible. All of these factors will make nearly
compensated ferrimagnets and antiferromagnets particularly important to
skyrmionics.

###  I.4 Skyrmions and Topological Stability

Skyrmions are topological excitations in thin magnetic films and can be viewed
as circular domains with an inverted core relative to the surroundings (Fig.
1)Kang  _et al._ (2016); Finocchio  _et al._ (2016); Everschor-Sitte  _et al._
(2018); Li  _et al._ (2021). Unlike small magnets that tend to get their spins
randomized by thermal fluctuations, a skyrmion is stabilized by spatial
inversion symmetry breaking, generating the Dzyaloshinskii-Moriya interaction
(DMI) of the form $E_{DMI}=\sum_{ij}D_{ij}\cdot(\vec{S}_{i}\times\vec{S}_{j})$
between lattice sites labeled by $(i,j)$. The symmetry breaking (Fig. 2) can
come from bulk inversion asymmetry, such as in B20 solids like MnSe and FeGe,
giving rise to Bloch skyrmions whose spins progressively flip spatially from
up outside to down inside the skyrmion moving in a plane perpendicular to the
radial axis. An alternate way to break the inversion asymmetry is with a heavy
metal underlayer, where the interfacial spin-orbit coupling creates the DMI
resulting in Néel skyrmions, whose spins flip in the plane containing the
radial axis. We can also get antiskyrmions in solids with $D_{2d}$ symmetry,
such as tetragonal Heuslers.

Figure 2: Structures with broken spatial inversion symmetry: (a) An amorphous
ferrimagnets (CoFeGd) on top of a heavy metal (Pt) with interfacial DMI
supporting Néel skyrmions; (b) $D_{2d}$ tetragonal unit cells like Mn2PtSn
supporting antiskyrmions; (c) B20 FeGe supporting Bloch skyrmions.

The topological index for a skyrmion is given by its winding number set by the
orientation of the magnetization unit vector $\mathbf{m}$ in the 2-D $(x,y)$
plane

$N_{sk}=\frac{1}{4\pi}\int
dxdy~{}\mathbf{m}\cdot\Bigl{(}\partial\mathbf{m}/\partial
x\times\partial\mathbf{m}/\partial y\Bigr{)}$ (4)

A detailed discussion on the physical origin can be found in the Appendix. The
magnetization vector $\hat{m}$ is characterized by 3-D direction cosines
$(\theta,\Psi)$ while their spatial locations are described in 2-D polar
coordinates $(r,\phi)$. If we assume there is a vortex-like motion, then the
azimuthal angle $\theta$ must not vary around the circumference for a skyrmion
and depend on radial coordinate alone, $\theta=\theta(r)$, while the in-plane
tilt angle $\Psi$ should increase linearly around the circle until it covers
an integer multiple of $2\pi$, so that $\Psi=N_{sk}\phi+\psi$, with vorticity
$N_{sk}$ an integer and $\psi$ the domain angle or helicity. The integral
$N_{sk}$ (Eq. 4) then simplifies to
$[m_{z}(r=\infty)-m_{z}(r=0)]\times[\Psi(\phi=2\pi)-\Psi(\phi=0)]=pN_{sk}$,
where $m_{z}(r)=\cos{\theta(r)}$ and $[m_{z}(r=\infty)-m_{z}(r=0)]$ is the
polarity $p=\pm 1$. The $m_{z}$ difference relates to the radial evolution of
magnetization tilt from core to surrounding (e.g. skyrmions vs merons), while
the $\psi$ difference is the overall winding. For simplicity, in this paper we
take the polarization $p=1$.
It would be useful to see how $\psi$ and $N_{sk}$ can be determined from the
energetics. The DMI energy density can be written in the continuous limit as:

$\displaystyle\displaystyle\frac{\epsilon_{\mathrm{DMI}}}{2\pi
t_{F}}=\begin{cases}D\bm{m\cdot[(z\times\nabla)\times m]},~{}\text{
Interfacial DMI}\\\ D\bm{m\cdot(\nabla\times m)},~{}\text{B20 DMI}\\\
D\Biggl{[}\displaystyle\frac{\partial{m}_{z}}{\partial
y}m_{x}+\frac{\partial{m}_{z}}{\partial x}m_{y}\Biggr{.}\\\
~{}~{}~{}~{}\Biggl{.}-m_{z}\left(\displaystyle\frac{\partial{m}_{y}}{\partial
x}+\frac{\partial{m}_{x}}{\partial y}\right)\Biggr{]},~{}\text{$D_{2d}$
DMI}\\\ \end{cases}$ (5)

Using
$\mathbf{m}=\left(\sin\theta\cos\psi,\sin\theta\sin\psi,\cos\theta\right)$,
the DMI energy terms can be simplified to:

$\displaystyle\epsilon_{\mathrm{DMI}}=\begin{cases}D_{0}\cos\Bigl{[}(N_{sk}-1)\phi+\psi\Bigr{]}\Bigl{(}\partial_{r}\theta+\displaystyle\frac{N_{sk}}{r}\sin{\theta}\cos{\theta}\Bigr{)}\\\
~{}~{}~{}~{}~{}~{}\text{(Interfacial~{}DMI,~{}N\'{e}el~{}skyrmion)}\\\
D_{0}\sin\Bigl{[}(N_{sk}-1)\phi+\psi\Bigr{]}\Bigl{(}\partial_{r}\theta+\displaystyle\frac{N_{sk}}{r}\sin{\theta}\cos{\theta}\Bigr{)}\\\
~{}~{}~{}~{}~{}~{}{\text{(B20~{}bulk~{}DMI,~{}Bloch~{}skyrmion)}}\\\
D_{0}\sin\Bigl{[}(N_{sk}+1)\phi+\psi\Bigr{]}\Bigl{(}-\partial_{r}\theta\cos{2\theta}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\displaystyle\frac{N_{sk}}{r}\sin{\theta}\cos{\theta}\Bigr{)}\\\
~{}~{}~{}~{}~{}~{}{\text{($D_{2d}$~{}DMI,~{}antiskyrmion)}}\end{cases}$ (6)

with $D_{0}=2\pi t_{F}D$ and $t_{F}$ the film thickness. The
${N_{sk}}\sin{\theta}\cos{\theta}/r$ term, as we will see in the next section,
has a much smaller contribution to the energy than the other terms. This would
make $N_{sk}>1$ unfavorable, as the term in exchange energy related to the
winding number changes as $N_{sk}^{2}$. We can then see from the integrals in
the Appendix that for interfacial and B20 DMI $N_{sk}=1$ gives the minimum
energy for $\psi=0(\pi),\pi/2(3\pi/2)$ respectively (depending on DMI sign).
For the antiskyrmion, $N_{sk}=-1$ and $\psi=\pi/2(3\pi/2)$.

Table 2: Selection of skyrmionic materials in bulk, thin films, and
multilayers. Taken with permission from Phd thesis of K. Litzius, Johannes
Gutenberg-Universität Mainz (2018) Litzius (2018)

Material Sample Conduction Tsky/K $\lambda\mathrm{H}$/nm $d\mathrm{SiSky}$/nm
$|D|$ /$\frac{\mathrm{mJ}}{\mathrm{m}^{2}}$ Type Refs. MnSi bulk metal 28 -
29.5 18 — N.A. Bloch Mühlbauer  _et al._ (2009); Neubauer  _et al._ (2009);
Adams  _et al._ (2011); Mühlbauer  _et al._ (2016) MnSi (press.) bulk metal 5
- 29 18 — N.A. Bloch Lee  _et al._ (2009); Ritz  _et al._ (2013); Chacon  _et
al._ (2015) MnSi film ($\sim$ 50 nm) metal <5 - 23 18 — N.A. Bloch Tonomura
_et al._ (2012); Yu  _et al._ (2015) Fe1-xCoxSi bulk semi-metal 25 - 30 37 —
N.A. Bloch Adams  _et al._ (2010); Münzer  _et al._ (2010)
Fe${}_{\mathrm{0\text{.}5}}$Co${}_{\mathrm{0\text{.}5}}$Si film ($\sim$ 20 nm)
semi-metal 5 - 40 90 $\sim$ 75 N.A. Bloch Yu  _et al._ (2010) FeGe bulk metal
273 - 278 70 N.A. N.A. Bloch Huang and Chien (2012) FeGe film ($\sim$ 75 nm)
metal 250 - 270 70 N.A. N.A. Bloch Yu  _et al._ (2011); Huang and Chien
(2012); McGrouther  _et al._ (2016) FeGe film ($\sim$ 15 nm) metal 60 - 280 70
— N.A. Bloch Yu  _et al._ (2011) Cu2OSeO3 bulk insulator 56 - 58 60 — N.A.
Bloch Adams  _et al._ (2012); Seki  _et al._ (2012); Okamura  _et al._ (2016);
Zhang  _et al._ (2016a, b, c) Cu2OSeO3 film ($\sim$ 100 nm) insulator <5 - 57
50 — N.A. Bloch Seki  _et al._ (2012); Rajeswari  _et al._ (2015) Co8Zn8Mn4
bulk metal 284 - 300 125 — N.A. Bloch Karube  _et al._ (2016) Co8Zn9Mn3 bulk
metal 311 - 320 >125 — N.A. Bloch Tokunaga  _et al._ (2015) Co8Zn9Mn3 film
($\sim$ 150 nm) metal 300 - 320 >125 — N.A. Bloch Tokunaga  _et al._ (2015)
GaV4S8 bulk semi-metal 9 - 13 17.7 — N.A. Néel Kézsmárki  _et al._ (2015)
Fe/Ir(111) monolayer N.A. $\sim$ 11 $\sim$ 1 - 20 <5 N.A. Néel Heinze  _et
al._ (2011); Bergmann  _et al._ (2014); Wiesendanger (2016) PdFe/Ir(111)
bilayer N.A. $\sim$ 4.2 $\sim$ 1 - 20 <5 N.A. Néel Bergmann  _et al._ (2014);
Wiesendanger (2016); Romming  _et al._ (2013, 2015); Schmidt  _et al._ (2016);
Hanneken  _et al._ (2016) (Ir/Co/Pt)10 multilayer metal $\lessgtr$300 30 - 90
100 2.0 Néel Moreau-Luchaire  _et al._ (2016) (Pt/Co/MgO) single layer metal
$\lessgtr$300 $\sim$ 500 70 - 130 2.0 Néel Boulle  _et al._ (2016) Pt/Co/Ta
multilayer metal $\lessgtr$300 480 75 - 200 1.3 Néel Woo  _et al._ (2016)
Pt/Co60Fe20B20/MgO multilayer metal $\lessgtr$300 344 <250 1.35 Néel Woo  _et
al._ (2016); Litzius  _et al._ (2017); Lemesh  _et al._ (2018) Ir/Co/Pt
multilayer metal $\lessgtr$300 — 25 - 100 N.A. Néel Litzius (2018)
W/Co20Fe60B20/MgO multilayer metal $\lessgtr$300 460 $\sim$ 250 0.3 - 0.7 Néel
Jaiswal  _et al._ (2017) Pd/Co60Fe20B20/MgO multilayer metal $\lessgtr$300 300
$\lesssim$ 200 0.78 Néel Litzius (2018) Ta/Co20Fe60B20/(Ta)/MgO single layer
metal >300 4200 1000 - 2000 0.33 Néel Zázvorka  _et al._ (2019)
Ta/Co20Fe60B20/MgO multilayer metal $\lessgtr$300 <900 $\sim$ 300 0.33 Néel
Litzius (2018)

The topological property of a skyrmion is characterized by its winding number
$N_{sk}$ (Eq. 4). The 2-D chiral skyrmions can be generated by stereographic
projection of continuous arrows spanning every point of a sphere onto the 2-D
plane (Fig. 1), except a singularity at one of the poles, which is the so-
called Bloch point. As long as the continuous approximation to the magnetic
texture holds, the skyrmions are topologically protected because the winding
number acts as an invariant. Continuous deformation of a skyrmion preserves
its winding number and creates a barrier to phases with alternate winding
numbers (e.g., a homogeneous ferromagnetic background with winding number
zero), as such a deformation amounts simply to rotations on the surface of the
projecting sphere and preserves the winding number. This would allow the
skyrmions to deform or shrink to unpin from the defects at small currents
without undergoing annihilation. Recently, the topological stability of
skyrmion has been directly demonstrated by Je et al. Je  _et al._ , with
longer lifetimes observed for magnetic skyrmions than trivial bubble
structures.

Topological protection has two limitations—one mechanism (intrinsic
annihilation) is due to thermal effects, which can randomly flip spins,
shrinking the skyrmion until it reaches a Bloch point, at which point the
atomic structure of magnets starts to matter, and the skyrmion winding number
can change to zero. The other limit (edge annihilation) is from finite-size
effects when the skyrmion gets too close to the edges of the magnets or to
non-magnetic defects, whereupon the texture of the confined/defect-adjacent
skyrmion no longer maps onto the whole sphere, leading to a similar
annihilation. The difference between edge and intrinsic annihilation is that
for edge annihilation, the texture of the magnet does not pass through the
Bloch point and the change in winding number from 1 to 0 is gradual compared
to intrinsic annihilation. This suggests that as a skyrmion gets larger, the
barrier to intrinsic annihilation becomes larger but not that to edge
annihilation.

Historically, skyrmions were experimentally observed in the form of a periodic
lattice in B20 solids like MnSi. These skyrmion lattices are ground state
configurations that have a strong degree of topological protection
characterized by a fixed winding number in k-space. Due to their topological
nature, skyrmions can shrink to line singularities but are typically unable to
unwind into their background continuum. As we will now see, skyrmions exist as
a result of a competition between exchange anisotropy, demagnetization, and
chiral DMI. Only the DMI part of the energy, specifically its sign, is
responsible for the chirality of skyrmions. At low temperature in an infinite
continuum solid, this imposes a very high barrier that keeps this skyrmion
lattice topologically stable, even while its critical current density
${j_{c}}$ for unpinning and initiating motion is quite small, orders of
magnitude less than a domain wall that tends to snag onto edges.
Unfortunately, it is not obvious how to encode information into a periodic
lattice. The alternate is to work with individual skyrmions nucleated around
defects. However, the thermal stability barrier for these metastable non-
periodic skyrmions is much smaller, allowing them to naturally ‘melt’ into the
surrounding magnetic background unless they are properly stabilized.

Table II shows a list of materials where skyrmions have been seen so far. Let
us now discuss the phase space attributes in these materials that are needed
to stabilize skyrmions.

##  II Phase-Space for static skyrmions

Several parameters set the skyrmion static and dynamic properties -
specifically saturation magnetization $M_{s}$, anisotropy field $H_{k}$,
damping $\alpha$, external magnetic field $H_{\mathrm{ext}}$, DMI $D$ and
exchange stiffness $A_{\mathrm{ex}}$. The phase space can guide us in choosing
materials that can host small skyrmions with long enough lifetimes.
Additionally, a high Curie temperature, set primarily by exchange, is required
to have the required material stability. Figure 3 shows the skyrmion phase
space sandwiched between the background ferro/ferrimagnetic ground state and
the stripe phase.

Figure 3: A sequence of zero-temperature phase space diagrams for various
exchange stiffnesses with interfacial DMI, $D=1$ mJ/m2 and thickness
$t_{FM}=3~{}nm$. As stiffness increases, the favorable region that can host
isolated metastable $\sim 10$ nm skyrmions shrinks, and the borders separating
them from competing phases (ferro/ferrimagnetic background, vs. stripe phase)
move towards lower anisotropy. This is expected, as a smaller anisotropy
compensates for larger exchange. The pictures below the phase diagram show the
evolution of a skyrmion stabilized at zero temperature with parameters listed
above, with subsequently increasing the DMI. Going from the ferromagnetic
phase (left) to the stripe phase (right).

###  II.1 What determines Skyrmion Size?

A skyrmion has a circular core and a transition region (domain wall) to the
background spin texture on the outside. By minimizing energy terms with
respect to skyrmion radius and domain wall transition width, we get an
equation for skyrmion size. The skyrmion energy barrier then can be calculated
from the skyrmion radius and width, as shown in Fig. 4. For a constant
interfacial DMI, there is an optimized thickness that gives the maximum energy
barrier. The points to note are that the energy barrier denoting skyrmion
stability/lifetime is maximized for specific skyrmion radius and film
thickness and that this barrier maximizing (and local energy minimizing)
radius increases with DMI until a destabilization point where there is a phase
transition to a skyrmion lattice. Lowering DMI for an isolated skyrmion lowers
its size but also reduces thermal stability (Fig. 4) by making the metastable
wells shallower and allowing the skyrmion to melt into the homogeneous
magnetic background.

The energy landscape of a ferromagnet is described by the Heisenberg energy,
written in the continuous form as:

$\displaystyle E$
$\displaystyle=t_{F}\int\big{[}A_{\mathrm{ex}}\left(\bm{\nabla}\mathbf{m}\right)^{2}-K_{u}m_{z}^{2}-\mu_{0}M_{s}\mathbf{m}\cdot\mathbf{H}_{\mathrm{ext}}$
(7)
$\displaystyle-\frac{1}{2}\mu_{0}M_{s}\mathbf{m}\cdot\mathbf{H}_{\mathrm{d}}+\epsilon_{\mathrm{DMI}}\big{]}d^{2}\mathbf{r},$

where $A_{\mathrm{ex}}$ is the exchange stiffness related to the exchange
energy by a length unit (e.g. for a simple cubic unit cell of length $a$,
$A_{\mathrm{ex}}=J/a$), $K_{u}$ is the uniaxial anisotropy,
$\mathbf{H}_{\mathrm{ext}}$ is the external magnetic field, $\mathbf{H}_{d}$
is the demagnetization field, $t_{F}$ is the thickness and
$\epsilon_{\mathrm{DMI}}$ is the DMI energy. Minimizing the energy for an
isolated skyrmion, we get the skyrmion size. The result for large skyrmions at
zero magnetic field is Rohart and Thiaville (2013); Bogdanov and Rößler
(2001); Büttner  _et al._ (2018); Wang  _et al._

$R_{sk}=\displaystyle\Delta\sqrt{\frac{A_{\mathrm{ex}}}{A_{\mathrm{ex}}-\Delta^{2}K}}=\frac{\Delta}{\sqrt{1-(D/D_{c})^{2}}},~{}~{}~{}~{}\Delta=\frac{\pi
D}{4K},$ (8)

where $D$ is the DMI. The critical DMI is
$D_{c}=4\sqrt{A_{\mathrm{ex}}K}/\pi$. Here the effective anisotropy includes
contributions from the demagnetization field $K\approx
K_{u}-\mu_{0}M_{s}^{2}/2$. Equation8 shows that a perpendicular magnetic
anisotropy is required to stabalize skyrmion in zero magnetic field.

Note that these equations change a bit with size. For ultra small skyrmions
($R_{sk}\lessapprox$ 10 nm, $\rho=R_{sk}/\Delta~{}\lessapprox$ 2), additional
form factors $f(\rho)$ (Eq. 32) must be taken into consideration, at the cost
of more numerical complexity. In general, the radius takes the form

$R_{sk}=\displaystyle\Biggl{(}\frac{D}{D_{c}}\Biggr{)}^{a}\frac{C_{1}\Delta}{\sqrt{1-C_{2}(D/D_{c})^{b}}}$
(9)

Figure 4: (Left) Schematic of energy terms for a skyrmion. The 2-D topological
contribution to exchange $\propto N_{sk}^{2}\Delta/R$ and the negative DMI
energy $\propto-N_{sk}\pi D$ together produce a local energy minimum for an
isolated metastable skyrmion. (Right) Reducing DMI shifts the minimum to a
lower radius, creating smaller isolated skyrmions while at the same reducing
the energy well and thus their lifetime to dissolve into the background
continuum. Increasing DMI increases skyrmion size, until at $D=D_{c}$ the
skyrmion destabilizes into a stripe phase, indicated in yellow.

It is worth emphasizing that the two terms essential for stabilizing the
skyrmion, both involving $N_{sk}$, are the $\sim-t_{F}DN_{sk}$ Dzyaloshinskii-
Moriya term that gives a negative slope for the energy vs. radius curve, and
the $\sim N_{sk}^{2}\Delta/R_{sk}$ term in the exchange energy that provides a
curvature to its energy curve. Together these two terms allow the creation of
a metastable minimum (Fig. 4). The last term appears solely due to the
circular nature of the skyrmion, i.e., added terms in the cylindrical gradient
arising from variations in cylindrical radial and angular unit vectors from
point to point along the azimuthal direction.

###  II.2 Ferrimagnets for Small, Fast Skyrmions

A ferrimagnet is a magnetic material with two sublattices of oppositely
oriented but unequal magnetic moments. Calculations have shown that several
ferrimagnets are promising materials for skyrmion device applications,
primarily because we can reduce their saturation magnetization $M_{S}$ near
compensation, which is predicted to reduce the stray field and thus the
skyrmion size while pushing up their speed. These ferrimagnets include
amorphous rare-earth-transition-metal (RE-TM), Mn-based inverse Heuslers
(IHs), and Mn4N. Their small $M_{s}$ and low $K_{u}$, in the order of $10^{4}$
J/m3, fall within the parameter space of hosting ultrasmall ($\sim$10 nm) and
fast skyrmions ($>$100 m/s) at room temperature. Among the parameters
determining skyrmion radius, the exchange constant is usually constrained by
the need for an adequate ordering temperature, an ordered state well above
room temperature.

One of the well-studied classes of ferrimagnets hosting skyrmions is amorphous
rare earth-transition metal (RE-TM) ferrimagnets like CoGd. The structure of
an amorphous RE-TM alloy is shown in Fig. 5. We would normally expect an
amorphous alloy to be isotropic in the out-of-plane and in-plane directions,
resulting in zero perpendicular magnetic anisotropy (PMA). However, amorphous
RE-TM thin films are found to have structural anisotropy, leading to PMA
Harris  _et al._ (1992). In RE-TM, magnetic moments in the RE and TM species
arrange to form two antiferromagnetically-coupled sublattices. The
magnetization of RE-TM ferrimagnets varies with composition and temperature.
At the compensation temperature, moments in the RE and TM sublattices cancel
each other, leading to a vanishing magnetization. Simulations Ma  _et al._
(2019) and experiments Caretta  _et al._ (2018); Quessab  _et al._ (2020) have
reported skyrmions as small as 10 nm in RE-TM with heavy metal interfaces at
room temperature. In atomistic simulations with the Landau-Lifshitz-Gilbert
(LLG) equation, ultrasmall skyrmions are found to be stable at room
temperature in 5 to 15 nm thick CoGd with a compensation temperature near 250
K. For 10 nm thick CoGd, a sub 10 nm skyrmion is predicted for an interfacial
DMI of 0.8 to 1.0 mJ/m2, which is within the range of measured DMIs in
Pt/CoGd/Pt1-xW Quessab  _et al._ (2020). Furthermore, a columnar growth of
skyrmions across the film thickness is seen to arise (Fig. 5) through
computational tomography Ma  _et al._ (2019). Such robustness of structure is
critical to the stability of skyrmionic information-carrying bits.

Figure 5: (Top) Amorphous RE-TM structure from a molecular dynamics
simulation. The calculated RE-TM amorphous structure is a 1.6 nm cube. Red
spheres represent RE atoms, and blue spheres represent TM atoms. The
composition of this structure is RE25TM75 Ma  _et al._ (2019). Out-of-plane
M-H loop of 10 nm CoGd at 300 K. The out-of-loop has a squareness of 1,
demonstrating robust PMA. $M_{s}$ is $\sim$100 kA/m. (Bottom) A simulated
cross-sectional tomograph of an ultrasmall room temperature skyrmion shows a
columnar distribution throughout a 10 nm CoGd sample. Co-sublattice spins are
in the top box, Gd-sublattice in the bottom box, in-plane cross-sections near
the top and bottom interface (left), and out-of-plane cross-sections (right).
C.T. MA et al. Sci Rep 9, 9964 (2019); licensed under a Creative Commons
Attribution (CC BY) license. Ma  _et al._ (2019).

It should be noted that in experimental studies, the ultra-small skyrmions are
much more difficult to image than larger, bubble-like skyrmions. Only a few
techniques are capable of imaging in the nanometer regime and simultaneously
also recording time-resolved image sequences and applying stabilizing fields.
The most prominent techniques currently are X-ray holography, scanning
transmission X-ray microscopy (STXM), full-field transmission soft X-ray
microscopy and X-ray magnetic circular dichroism (XMCD-PEEM) Schütz  _et al._
(2011); Wang  _et al._ (2020). All these methods utilize the X-ray magnetic
circular dichroism (XMCD) effect that leads to a magnetization-dependent
absorption of photons in the material. An additional advantage of this
technique is the element-specific resolution of recorded data, i.e., the
absorption of only one atomic species can be selected to image sublattices and
otherwise compensated materials such as ferrimagnets. While X-ray imaging can
also investigate antiferromagnets directly, this feature also allows us to
obtain data on synthetic antiferromagnets. A holography image showing our
reported 10 nm skyrmions in ferrimagnetic CoGd is presented in Fig. 6 Litzius
_et al._ (2020a).

In experiments, an M-H loop of 10 nm CoGd in Fig. 5 shows a robust PMA, with
an $M_{s}$ close to 100 kA/m. $K_{u}$ is measured to be about 30 kJ/m3. As
discussed earlier, these parameters are very promising for ultrasmall
skyrmions at room temperature. Indeed, $\sim$10 to $\sim$ 100 nm skyrmions
have been reported in RE-TM. $\sim$150 nm skyrmions are reported in
Pt/GdCoFe/MgO through scanning transmission X-ray microscopy (STXM) Woo  _et
al._ (2018). Magnetic force microscopy (MFM) finds sub $\sim$100 nm skyrmion
in Pt/CoGd/W Quessab  _et al._ (2020). The smallest skyrmions in RE-TM are
imaged by X-ray holography, see Fig. 6. Ultrasmall skyrmions of close to 10 nm
are observed in Pt/CoGd/TaOx at room temperature Caretta  _et al._ (2018).
Through current injection, a distribution of skyrmion sizes, from 10 to 50 nm,
can be nucleated in CoGd, as shown in Fig. 6. The distribution in sizes arises
from local variations in magnetic properties. Despite these variations, the
skyrmions are seen to be stable in zero applied field—a critical attribute for
commercial applications. High-speed skyrmions of several hundred m/s are also
reported in RE-TM, which will be discussed in detail in Section III. The
observed small and fast skyrmions make CoGd one of the most promising
materials in realizing skyrmion-based devices.

Figure 6: Imaging of ultra-small skyrmions in ferrimagnets. (a–c) X-ray
holography images of Pt/Co44Gd56/TaOx immediately after injecting a nucleation
current pulse (a), after 20 min (b), and after 40 min (c). (d–e) Image of
skyrmions at different positions obtained after saturating the film and
injecting a nucleation current pulse. The colored squares indicate the absence
of skyrmions in locations present in a and b, respectively. (f) Skyrmions
under zero out-of-plane field $\mu_{0}H_{z}$ (nucleated in a bias field which
was subsequently reduced to zero). In all the images, the light (dark)
contrast indicates magnetization out of (into) the plane. The ring-like
features in some of the images are not a magnetic feature but simply an
artifact from the circular field of view. (g) The histogram shows the
distribution of skyrmion diameters. (h) Magnetic contrast line scans of the
numbered skyrmions in (c) and (e) (grey-filled circles) and fits (red lines).
Reproduced with permission from L. Caretta et al., Nature Nanotech 13,
1154–1160 (2018); Copyright 2018, Nature Publishing Group. Caretta  _et al._
(2018).

Alternatively, epitaxial Mn4N is a RE-free ferrimagnet that seems to be a good
candidate material for skyrmion-based devices. Unlike amorphous RE-TM, Mn4N is
an anti-perovskite with intrinsic perpendicular magnetic anisotropy. With a
deposition temperature greater than 400∘C, Mn4N is thermally stable and will
remain robust during device processing. In comparison, the thermal stability
of amorphous RE-TM remains a question. It has been shown that in GdCo, PMA is
lost abruptly after annealing above 300∘C temperatureUeda  _et al._ (2018). As
shown in Fig. 7(a), the corner Mn atoms (Mn I) and face-centered Mn atoms (Mn
II) couple antiferromagnetically to form the two magnetic sublattices. Since
the two sublattices have a similar but unequal moment, Mn4N has a small
$M_{s}$ of less than 100 kA/m. The intrinsic perpendicular magnetic anisotropy
of Mn4N originates from unequal in-plane (a) and out-of-plane (c) lattice
constants. The experimental $c/a$ ratios of Mn4N thin films are close to 0.99,
and $K_{u}$ less than 100 kJ/m3 Isogami  _et al._ (2020). The measured M-H of
15 nm film in Fig. 7(b) shows robust PMA, $M_{s}$ of 40 kA/m, and an estimated
$K_{u}$ of 80 kJ/m3. While $K_{u}$ of Mn4N is about three times larger than
$K_{u}$ of CoGd, it still falls within the parameter space for room
temperature skyrmions. Although no experiments so far, to our knowledge, have
reported skyrmions in Mn4N, simulation of Mn4N in Fig. 8 (atomistic for $<50$
nm skyrmions, micromagnetic for $>50$ nm), based on these experimental values
of $M_{s}$ and $K_{u}$, find stable room temperature skyrmions. With the
experimental value of anisotropy and DMI ranges from 0 to 7 mJ/m2, the
predicted skyrmion size varies from 10 to 500 nm. Our MFM measurements Fig. 9
show $\sim$ 300 nm skyrmions in 15 nm Mn4N with Pt cap layer. DMI engineering,
discussed in the next section (Sec. II.3), will further allow the tuning of
skyrmion sizes in Mn4N.

Figure 7: (a) Anti-perovskite Mn4N. Atoms at the corner are Mn-I, and atoms at
the face center are Mn-II. Mn-I and Mn-II atoms have nonequivalent magnetic
moments and form the two ferrimagnetic sublattices in Mn4N. (b) M-H loop of 15
nm Mn4N at 300 K. The loops show robust PMA in Mn4N. $M_{s}$ is $\sim$ 40 kA/m
and in-plane anisotropy field is larger than 1.5 T. Reproduced from W. Zhou,
et al., AIP Advances 11, 015334 (2021), with the permission of AIP Publishing.
Zhou  _et al._ (2021) Figure 8: Simulated skyrmion interfacial DMI-$K_{u}$
phase diagram of Mn4N at 300 K. With $K_{u}$ from 0.75 to 1 kJ/m3, and
interfacial DMI from 0 to 7 mJ/m2, various sizes of skyrmion are stabilized as
indicated by the isolines for skyrmion radius in the phase diagram. Figure 9:
Magnetic force microscopy (MFM) image reveals $\sim$ 300 nm skyrmion in Mn4N
(15 nm)/Pt on MgO substrate. The red region corresponds to down magnetic
moments; the dark blue region corresponds to up magnetic moments, which make
up the core of the skyrmion.

Besides Mn4N, Mn-based Inverse Heuslers (IH) are also RE-free ferrimagnets,
and are predicted to host ultrasmall skyrmions at room temperature Xie  _et
al._ (2020). While no experiments have investigated skyrmions in IH to our
knowledge, recent computational works have shown IH to be candidate materials
for skyrmion applications. In addition, the Heusler family of solids are good
candidates for Slater-Pauling half-metals, which suppress interband spin-flip
scattering and support ultra-low Gilbert damping Galanakis  _et al._ (2006). A
schematic diagram of cubic and tetragonal IH is shown in Fig. 10. It is worth
noting that two X atoms are sitting at non-equivalent sites in IH. In Mn-based
IH, Mn is element X. From calculations, several Mn-based IHs are predicted to
host ultrasmall skyrmions at room temperature.

Figure 10: Schematic representation of (a) cubic inverse-Heusler $\mathrm{XA}$
structure and (b) tetragonal inverse-Heusler structure. In the inverse-Heusler
structure, $\mathrm{X_{1}}$ and $\mathrm{X_{2}}$ are the same transition metal
element but they have different environments and magnetic moments.

Figure 11 illustrates the smallest stable skyrmions predicted in 5 nm thick
Mn-based IHs using LLG, assuming varying uniaxial anisotropies (scaled by the
corresponding saturation magnetization, $Q=2K_{u}/\mu_{0}M_{s}^{2}$) for a
given energy barrier $E_{b}$. Out of these IHs, Mn2CuAl, and Mn2CuGa are
predicted to have $M_{s}$ less than 100 kA/m (Appendix A). In zero applied
field, the smallest stable skyrmions with a 50 kBT energy barrier are
predicted to be 6.3 nm in Mn2CuAl and 6.1 nm in Mn2CuGa. In addition to small
$M_{s}$, the Gilbert damping coefficient of Mn-based IH is predicted to be in
the order of $10^{-4}$ to $10^{-3}$ Ma  _et al._ (2017). This is significant
because in addition to small $M_{s}$, low damping $\alpha$ is also important
in obtaining high skyrmion speed, as per Eq. 14. The small $M_{s}$ and small
damping together make Mn-based IHs promising candidates to achieve stable and
fast skyrmions at room temperature. Mn2CuAl , Mn2CuGa, Mn2CoAl and Mn2CoGa
have very large Curie temperature (above 700 K) which would is critical for
reliable room temperature skyrmionics.

Figure 11: Smallest stable skyrmion boundary for inverse Heuslers (zero-
temperature LLG simulations of skyrmion sizes in a confined $100\times 100$ nm
magnetic layer shown here alongside dots on the lines). The lines indicate the
smallest skyrmions with an energy barrier $E_{b}=50k_{B}T$ or $E_{b}=40k_{B}T$
between the ferromagnetic state and skyrmion state. The scatter plot samples
the skyrmion radius in nm along the boundary. The film thickness is assumed
$5\,\mathrm{nm}$ in all calculations. $Q=2K/\mu_{0}M_{s}^{2}$ defines the
effective anisotropy compared to the demagnetization, and $Q>1$ is needed for
a perpendicular system. To show the effect of anisotropy on skyrmion size and
stability, different $Q$ values are chosen for different materials with
different saturation magnetization. Reproduced with permission from IEEE
Transactions on Magnetics, vol. 56, no. 7, pp. 1-8, 2020 Xie  _et al._ (2020).
Copyright 2018 IEEE. The magnetic field $B_{z}$ is parallel to the background,
so the skyrmion shrinks with increasing $B$ field.

In this section, we have discussed several promising ferrimagnets to achieve
small and stable skyrmions at room temperature. Small $M_{s}$, $\lessapprox
100$ kA/m and small to moderate, $K_{u}$ $\lessapprox 300$ kJ/m3 in these
ferrimagnets are keys to stabilizing ultrasmall and fast skyrmions at room
temperature (Table III). Computational models predict ultrasmall skyrmions in
RE-TM, Mn-based Inverse Heusler, and Mn4N. Experiments have measured close to
10 nm skyrmions in CoGd. For Néel skyrmions, while intrinsic properties of the
magnetic layer are important for skyrmion size and stability, the DMI from the
adjacent metal layers is equally crucial. Tuning of DMI will, in fact, have a
strong influence on the properties of skyrmions Soumyanarayanan  _et al._
(2017), and will be discussed in the next section.

Table 3: Materials for hosting skyrmions. CoGd and Mn4N are promising
materials due to their low $M_{s}$. Mn4N has a very large Curie temperature
which would make it stable at room temperature.
Material |  $M_{s}$(kA/m) |  $K_{u}$ (kJ/m3) | Size (nm) |  $T_{C}$ (K)
---|---|---|---|---
FeGe | 300-330 | 14 | Lattice | 280 Xu  _et al._ (2017); Yu  _et al._ (2011)
CoGd | 100 | 50 | 10-150 | 450 Caretta  _et al._ (2018); Quessab  _et al._ (2020); Woo  _et al._ (2018)
Mn2.5Ga | 250 | 1239 | N.A. | 600 Jamer  _et al._ (2014)
MnGa | 247.3 | 1000 | N.A. | 600 Lu  _et al._ (2015)
Mn4N | 87-100 | 88-100 | 100-300 | 710 Şaşioglu  _et al._ (2005)

###  II.3 Parameter Engineering of Skyrmion Size and Stability

To achieve ultra-high density data storage and memory devices, it is desirable
to scale down the size of skyrmions. In addition, the stability and lifetime
of the skyrmion are other important aspects to consider for technological
applications (e.g., volatile or non-volatile memories).

The topological phase in which the formation of skyrmions is possible highly
depends on the competition of the different energies in the magnetic system as
discussed in Sec. I.4. Similarly, the radius of the skyrmion, as defined in
Eqs. 8 and 9, is determined by the saturation magnetization, the DMI, the
effective anisotropy including exchange and stray fields and the exchange
interaction. Therefore, the tunability of the skyrmion size and stability is
set by these parameters. A straightforward approach to this problem, as
discussed in Sec. II.2, is to choose materials with a low saturation
magnetization (to minimize the stray fields) plus a sizable DMI, such as in
ferrimagnetic alloy thin films or multilayers Woo  _et al._ (2018); Caretta
_et al._ (2018); Quessab  _et al._ (2020) and synthetic antiferromagnets Dohi
_et al._ (2019); Legrand  _et al._ (2020).

Alternatively, several methods have been developed to engineer the DMI. Before
we jump into the various approaches to engineer DMI, it is worth mentioning
the physics of DMI briefly. In systems where magnetic thin films are grown on
heavy metal layers, the broken spatial inversion symmetry leads to the
antisymmetric interaction DMI as mentioned earlier in Section I.D. It is well
established that the strong spin-orbit coupling (SOC) of HM layers adjacent to
the magnetic layers has a dominant effect on the DMI. For Pt/Co bilayers,
first-principles calculations demonstrate that DMI is almost purely
interfacial, and the source of the DMI is found to be the SOC energy in the
adjacent HM layer Yang  _et al._ (2015). While HM SOC is crucial, it is not
the only ingredient that contributes to the DMI. In Au-based systems, for
instance, SOC alone fails to explain the smaller magnitude of the DMI despite
the large SOC of Au. The other important ingredient that determines the DMI is
the presence of HM metal d states around the Fermi level, which in turn
controls the relative overlap between the d states of HM and magnetic
elements, in other words, the orbital hybridization. Theoretical study based
on a series of 3d transition metals monolayer deposited on several 5d HM metal
substrates shows that the 3d-5d orbital hybridization plays a significant role
in both the sign and magnitude of the DMI Belabbes  _et al._ (2016). In the
case of Au, the 5d states are far below the 3d states leading to a smaller
degree of orbital hybridization that produces a smaller DMI value. Moreover, a
recent study demonstrates that in HM/FM bilayers, the DMI magnitude and sign
depend both on the orbital hybridization and the transitions between HM
orbitals close to the K points in the BZ, namely, $d_{yz}\rightarrow d_{xy}$
and $d_{xz}\rightarrow d_{x^{2}-y^{2}}$ Jadaun  _et al._ (2020). To summarize,
at HM/FM interfaces, the DMI emerges from the complex interplay between (i)
inversion symmetry breaking, (ii) SOC of the HM metal layers, (iii) orbital
hybridization between d states of HM layers and magnetic layers, and (iv)
transitions between the HM d orbitals.

In metallic magnetic systems, interfacial DMI is commonly used to create
skyrmions Boulle  _et al._ (2016); Woo  _et al._ (2016), so that we can tune
the DMI by controlling the interfaces. A wide range of heavy metals with large
spin-orbit coupling in proximity with a magnetic layer can induce DMI, as
reported in first-principles calculations Belabbes  _et al._ (2016). A common
approach to tuning the DMI relies on the overall chirality. Indeed, the bottom
and top interfaces of a magnet both contribute to the net DMI. To maximize the
net DMI, a magnetic layer can be grown on a heavy metal with a strong DMI,
e.g., Pt/Co, and capped by a layer that has a weak DMI, such as oxides (e.g.,
MgO Woo  _et al._ (2016); Boulle  _et al._ (2016) or TaOx Caretta  _et al._
(2018); Arora  _et al._ (2020)), or a strong negative DMI, e.g., Co/Ir. Using
this additive method, an interfacial DMI close to 2 mJ/m2 was reported in
Ir/Co/Pt multilayers, allowing the nucleation of sub-100 nm skyrmion Moreau-
Luchaire  _et al._ (2016); Legrand  _et al._ (2017). Similarly, one of the
highest surface DMI constant of about 2 pJ/m was measured in an ultrathin
Pt/Co(0.5 nm)/MgO stack, which was shown to host 130-nm skyrmions Boulle  _et
al._ (2016).

Another method to control the DMI involves the use of heavy metal alloys,
providing a finely tuned inversion symmetry breaking mechanism at the
interface Quessab  _et al._ (2020); Zhu  _et al._ (2019). In a recent study,
Quessab et al. demonstrated the ability to control the interfacial DMI in a
thin ferrimagnetic alloy by varying the capping layer composition Quessab  _et
al._ (2020). By gradually changing the W composition (x) in Pt/CoGd(5
nm)/Pt1-xWx, the stack symmetry was progressively broken to induce DMI, as
seen in Fig. 12(a). The DMI energy was tuned over a large range, from a weak
DMI to an interfacial DMI of 0.23 mJ/m, which allowed the formation of sub-100
nm skyrmions. Recently, Morshed et al. investigated
Pt/CoGd/$\text{Pt}_{1-x}W_{x}$ theoretically using first-principles
calculations and found that the reduction of SOC at the metal layer near the
capping layer interface and the simultaneous constancy at the bottom interface
as a function of W composition are responsible for the saturating behavior of
the DMI as shown in Fig. 12(a) Morshed  _et al._ (2021). They also predicted
tuning of the DMI by introducing a series of HM in the capping layer of
Pt/CoGd/X where X=Ta, W, Ir, and found that W in the capping layer favors
higher DMI while Ir produces lower DMI Morshed  _et al._ (2021). In another
study, Zhu et al. have also reported the tuning of the interfacial DMI in a
heavy metal/ferromagnet structure Pd1-xPtx/FeCoB/MgO Zhu  _et al._ (2019). The
DMI increased a factor of 5 as $x$ varied from 0 to 1. Control of the spin-
Hall efficiency was achieved as well, and a maximum spin-Hall angle of 0.60
was observed for Pd0.25Pt75, which could be of use for energy-efficient
skyrmion dynamics induced spin-orbit torque.

Figure 12: (a) Evolution of the DMI energy as a function of the W composition
in Pt/CoGd(5 nm)/Pt1-xWx (Y. Quessab et al. Sci Rep 10, 7447 (2020); licensed
under a Creative Commons Attribution (CC BY) license. Quessab  _et al._
(2020)). (b-c) magneto-optical Kerr images of FeCoB/TaOx before and after
applying a negative gate voltage at 0 $\mu$T and 30 $\mu$T, (d) variation of
the DMI energy in FeCoB/TaOx as a function of the gate voltage (Reproduced
with permission from Nano Letters. 2018, 18, 8. Copyright 2018 American
Chemical Society 4871–4877Srivastava  _et al._ (2018)).

Electric field control of the interfacial DMI has been recently demonstrated
Srivastava  _et al._ (2018); Herrera Diez  _et al._ (2019). Srivastava et al.
reported a 130% variation of the DMI energy via voltage gating in
Ta/FeCoB/TaOxSrivastava  _et al._ (2018) (see Fig. 12(d)). By applying a
negative voltage (-20 V), the authors were also able to shrink the size of
stripe domains and skyrmion bubbles, as depicted in Fig. 12(b) and 12(c),
respectively. It was observed that ionic gating modifies not only the DMI but
also the anisotropy and the saturation magnetization Herrera Diez  _et al._
(2019). Interestingly, ion (He+) irradiation was also shown to lead to an
increase of the interfacial DMI and domain wall velocity in Ta/CoFeB/MgO due
to disorder introduced at the interface Diez  _et al._ (2019).

Computational methods, such as density functional theory (DFT) Giannozzi  _et
al._ (2009) and atomistic simulations Evans  _et al._ (2014), can play an
important role in parameter engineering and tuning. These methods can not only
be used to rationalize an observation and uncover insights but also have
predictive capabilities. We demonstrated the potential of DFT to calculate the
saturation magnetization ($M_{S}$), magnetocrystalline anisotropy ($K_{U}$),
and exchange coupling ($J$) for bulk ferrimagnetic Mn4N. The parameters
calculated from DFT then feed the atomistic spin dynamics code, which in turn
predicts the Néel temperature ($T_{N}$) for this system. The saturation
magnetization ($M_{s}$) is the total magnetic moment per unit volume, which
can be calculated $M_{s}$ using the formula, $M_{s}=\mu_{B}/V$, where
$\mu_{B}$ is the atomic magnetic moment (in Bohr magnetons) and $V$ is the
unit cell volume (in cm3). Both $\mu_{B}$ and $V$ can be obtained from spin-
polarized collinear DFT calculations. We can also calculate $K_{u}$ using the
formula,

$K_{u}=E^{[001]}-E^{[100]}$ (10)

where $E^{[001]}$ and $E^{[100]}$ are the total energies using non-collinear
DFT calculations with the inclusion of spin-orbit coupling, where the spins
are constrained in the [001] and [100] directions, respectively. The classical
Heisenberg model suggested by Kübler et al. Kübler  _et al._ (1983) for the
Heusler alloys was used to calculate $J$.

Table 4: The DFT calculated $M_{s}$, $K_{u}$ and $J$ parameters for the bulk
ferrimagnetic Mn4N using the experimental lattice constant value of 3.868 Å.
Our 0 K DFT calculations predict an in-plane [100]-direction as the easy-axis
magnetocrystalline anisotropy.

Compound |  $M_{s}$(kA/m) |  $K_{u}$(kJ/m3) |  $J$(meV)
---|---|---|---
Bulk Mn4N | 121.94 | 1376.11 | 74.2

The calculated values for $M_{s}$, $K_{u}$ and $J$ using the experimental unit
cell parameter value of 3.868 Å is given in Tab. 4. The predicted normalized
magnetization vs Temperature curve is shown in Fig. 13. The DFT parameters
yielded a $T_{N}$ of 740 K from our atomistic simulation, which is in
excellent agreement with the experimental $T_{N}$ of 745 K Takei  _et al._
(1962).

Figure 13:  Prediction of $T_{N}$ for bulk Mn4N from atomistic simulation
using the DFT-calculated atomic magnetic moments, $K_{U}$, and $J$ as input
(see Table 4).

###  II.4 Nucleating isolated metastable skyrmions

The emergence of skyrmions in a material depends on the stability of the
skyrmion phase. In some materials, the DMI stabilizes skyrmionic lattices that
coincide with the ground state and are already present without nucleating them
explicitly. Indeed, in epitaxial B20 FeGe films, it has been shown that
skyrmions can appear at zero field, as evidenced by remanent topological Hall
resistivity over a large range of temperatures Gallagher  _et al._ (2017) or
by electron holography in a FeGe nanodisk at 95 K Zheng  _et al._ (2017).
Another example is in Fe/Ni/Cu/Ni multilayers, for which the interlayer
exchange coupling allows the stabilization of zero-field and room-temperature
skyrmions Chen  _et al._ (2015). However, for technological applications, it
is usually desirable to create, move and annihilate skyrmions as a way of
writing information. Therefore, materials where the skyrmion is a metastable
state are more attractive, and in this case, several methods can be employed
to nucleate multiple skyrmions, with proposed device applications Müller
(2017); Finizio  _et al._ (2019); Je  _et al._ (2021).

Magnetic Fields: A common nucleation method relies on the use of a magnetic
field that can be applied externally or locally (e.g., via an Oersted loop
around a nanostructure Woo  _et al._ (2016)). In ferro- and ferri-magnetic
films and multilayers with large demagnetization field, the ground state is
formed by stripe domains that can be further shrunk into skyrmion bubbles Woo
_et al._ (2016); Lemesh  _et al._ (2018); Woo  _et al._ (2018, ); Juge  _et
al._ (2019). The application of an external magnetic field has also been shown
to lead to the formation of antiferromagnetic skyrmions in synthetic
antiferromagnets Legrand  _et al._ (2020); Dohi  _et al._ (2019).
Nevertheless, external magnetic fields are not compatible with the
implementation of data storage or memory devices.

Spin Currents: On the other hand, several studies have reported the nucleation
of metastable skyrmions by injecting an in-plane current in the presence of an
out-of-plane magnetic field to provide stability Caretta  _et al._ (2018);
Lemesh  _et al._ (2018); Song  _et al._ (2020a); Legrand  _et al._ (2017).
Caretta et al. demonstrated the nucleation of skyrmions in a nearly
compensated ferrimagnetic CoGd alloy with a diameter as small as 10 nm at room
temperature with a train of bipolar nanosecond pulses of a maximum amplitude
of 1.7 $\times$1012 A m-2 Caretta  _et al._ (2018). Creation and annihilation
of a single magnetic skyrmion was achieved in a ferrimagnetic Pt/GdFeCo/MgO
multilayer using 10 ns current pulses with a current density of 2.5
$\times$1010 A m-2 Woo  _et al._ . Furthermore, some studies have reported
that current pulses can be used to nucleate skyrmions without applying an
external magnetic field Lemesh  _et al._ (2018); Woo  _et al._ (2017). Lemesh
et al. found that in a Pt/CoFeB/MgO multilayer at 379 K, zero-field skyrmions
could be nucleated upon injection of bipolar nanosecond current pulses Lemesh
_et al._ (2018). In a similar material, Woo et al. reported the emergence of
zero-field skyrmions at room temperature after sending 20 ns current pulses
for 5 s with a repetition rate of 3.33 MHz and a current density of 1.6
$\times$1011 A m-2 Woo  _et al._ (2017).

Joule Heating: So far, the mechanism for the nucleation of metastable
skyrmions has not yet been fully elucidated. Nevertheless, the use of current
pulses, which are often applied over an extended period and with a large
current density, suggests a nucleation mediated by thermal effects due to the
Joule heating. This was suggested by Legrand et al. Legrand  _et al._ (2017)
and later confirmed by the temperature-dependence study of current-induced
skyrmion nucleation by Lemesh et al. Lemesh  _et al._ (2018). Note that these
studies report the nucleation of skyrmion arrays in the hottest area of the
used wire devices and usually do not reliably nucleate single skyrmions. A
connection to single skyrmion nucleation processes has been provided by
Büttner et al. Büttner  _et al._ (2017) that were able to produce similar
results by application of millions of low current-density pulses that led to
the formation of skyrmions on material inhomogeneities. Wang et al. have shown
thermal generation and manipulations of skyrmions Wang  _et al._ (2020).

The general appearance of a current-induced skyrmion nucleation set-up is
shown in Fig. 14. The current pulses first modulate the alignment and
formation of stripe domains before breaking those into skyrmions at higher
current densities. The involvement of thermal effects leads to a random
contribution to the splitting of domains so that larger numbers of current
pulses at lower current densities may suffice to nucleate a single skyrmion at
a defect while at higher current densities require far fewer pulses to be
applied to the sample, resulting in a lower nucleation latency.

Figure 14: Observation of current-induced stripe and skyrmion nucleation in
X-ray holography. (a) Schematic overhead view of the sample and connection to
the pulse electronics. The sample consists of a magnetic wire that is directly
connected to the pulse set-up. The center of the wire is imaged
holographically (see (b)) with a circular FOV. (b) Imaging set-up. The
magnetic wire was grown on a transparent Si3N4 membrane. The backside of the
membrane is covered in gold, and two holes are cut by focused ion beam
exposure to provide an interference pattern. The structure is exposed to
circularly polarized X-rays. The interference of the transmitted beam of the
two holes forms the hologram, from which an image of the out-of-plane
magnetization in the wire can be reconstructed. (c) Nucleation of skyrmions
and stripes with unipolar pulse trains. The injection of millions of pulses
leads to the formation of stripe domains at low and skyrmions at high current
densities. The black-and-white contrast denotes magnetization pointing in the
+z and –z directions, respectively. Reproduced with permission from F. Büttner
et al. Nature Nanotech 12, 1040–1044 (2017). Copyright 2016, Nature Publishing
Group Büttner  _et al._ (2017).

Laser Pulses: Alternatively, ultrafast nucleation of skyrmions using single
ultrashort laser pulses was theoretically predicted Koshibae and Nagaosa
(2014) and demonstrated in a CoFeB thin film Je  _et al._ (2018) and a
ferrimagnetic TbFeCo alloy Finazzi  _et al._ (2013) as well as TmIG Büttner
_et al._ (2020). Although the relatively thick films in the TmIG study Büttner
_et al._ (2020) show negligible Dzyaloshinskii-Moriya interactions, the
results from Ding  _et al._ (2019) suggest that in ultrathin rare earth iron
garnet films, in which interfacial DMI has recently been found, fast thermal
excitations might be used to controllably nucleate chiral magnetic skyrmions.
Interestingly, Je et al. and Büttner et al. did not find any dependence on the
light polarization; in fact, nucleation occurs even with linearly polarized
light, which does not carry any angular momentum, thus demonstrating that
heating could be a sufficient stimulus to create skyrmions Je  _et al._
(2018). Furthermore, it was demonstrated in a Pt/Co/Ir ultrathin multilayer
that skyrmion bubbles in the micrometer scale could be formed using surface-
acoustic waves (SAWs) in the presence of a small out-of-plane magnetic field
Yokouchi  _et al._ (2020). It was found that the bubble size was independent
of the SAW wavelength. The skyrmion density increased with the RF power and
was maximum for SAWs with a wavelength of 16 $\mu$m, which is roughly twice
the size of the observed bubbles. Remarkably, the authors argued that the SAWs
did not generate a significant temperature increase that could explain the
formation of the skyrmion bubbles Yokouchi  _et al._ (2020), which still keeps
the mechanism for laser-induced skyrmion nucleation an open question.

Figure 15: A schematic illustration of the VCMA gate.

Voltage Pulses: Another method to nucleate or annihilate skyrmions is by using
the Voltage Controlled Anisotropy (VCMA) Nakatani  _et al._ (2016); Schott
_et al._ (2017); Bhattacharya  _et al._ (2020) (Fig. 15). In this method a
skyrmion is nucleated or annihilated by applying electric field pulses. The
applied electric field can change the perpendicular magnetic anisotropy (PMA)
according to the following equation:

$\displaystyle K_{u}(V)=K_{u}(0)-{\xi V}/{t_{ox}t_{F}}$ (11)

where $\xi$ is the VCMA coefficient, $V$ is the applied voltage, $t_{ox}$ is
the oxide thickness and $t_{F}$ the film thickness. By lowering the
anisotropy, the overall barrier for flipping of magnetic moments is also
lowered, which would nucleate the skyrmion. Along a racetrack, the preferred
locations of skyrmion nucleation would be at the defect sites, which already
have lower anisotropy compared to their surrounding. Since the VCMA
coefficient is not very large, the required voltage for skyrmion nucleation
and annihilation is expected to be large ($\sim$7-20 V) Schott  _et al._
(2017); Bhattacharya  _et al._ (2020) which would make its integration with an
electric circuit challenging. This problem may be alleviated by building
memory elements out of reshuttling skyrmions as in wavefront-based temporal
memory, where occasional nucleation events can be used to replenish their
finite lifetime at an energy cost that is readily amortized over multiple
compute cycles. By using ion irradiation Zhang  _et al._ (2016d) it is
possible to engineer locations of the aforementioned defect sites in a
controlled way, which, as we discuss in Section V, is critical for skyrmionic
device reliability. Alternatively, voltage control for magnetic exchange bias
has also been proposed for skyrmion nucleation Li  _et al._ (2017); Yang  _et
al._ (2017). The exchange bias (EB) is a property of a coupled
antiferromagnetic (AFM)–ferro(ferri)magnetic (FM, FiM) system that occurs due
to exchange coupling at the interface Wu  _et al._ (2010, 2013). Electric
field control of EB has been used to achieve a deterministic and reversible
switching of the magnetization in the FM (FiM) layer which can be used for the
precise creation of single skyrmions Guang  _et al._ (2020).

###  II.5 Skyrmion Stability and Lifetime

As reported earlier, a number of different methods Caretta  _et al._ (2018);
Woo  _et al._ (2018); Quessab  _et al._ (2020) including STXM, MFM, and X-ray
holography, have imaged 10 to 150 nm skyrmions in CoGd. The time needed for
these measurements suggests lifetimes of at least several hours. Since decade-
long lifetimes are needed for many applications, computational estimates are
crucial to providing guidance on skyrmion lifetimes.

To calculate skyrmion lifetimes, the transition-state theory (TST) for spin
transition Henkelman  _et al._ (2000) has been employed. To use the TST, the
minimum energy path needs to be calculated. Several studies have used the
geodesic nudged elastic band (GNEB) method Bessarab  _et al._ (2012, 2015) to
compute the minimum energy path for skyrmion annihilation. From the MEP, the
energy barrier and transition rate can be calculated. In the GNEB method, the
two endpoints of the annihilation process along the energy landscape are set
as the uniformly magnetized ferromagnetic and skyrmion states. These two
states are then connected via a path which is divided into replicas (or
‘images’) of the system. Each image is connected to its neighboring images by
a spring force. The total force from the springs as well as the force from the
gradient of the energy landscape in the magnetization space is then calculated
for each image. The GNEB approach matches are a more conventional nudged
elastic band (NEB) approach, except in the former, the spring constants are
also modified so that the highest energy image climbs to the saddle point of
the energy-magnetization landscape where the spring force disappears, and only
the energy minimization force remains. The system is iterated based on the
calculated dynamical equation until the path converges to the minimum energy
path (MEP) with the desired accuracy. From the calculated MEP, the attempt
frequency $f_{0}$ and the energy barrier $E_{b}$ are calculated and thence the
lifetime is extracted, $\tau=f_{0}^{-1}e^{E_{b}/kT}$. In the study by Wild et
al. Wild  _et al._ (2017) a variation of more than 30 orders of magnitude in
the attempt frequency was reported, which could suggest a dominant role for
the entropy effects compared to the energy barrier. Recently, spin-polarized
scanning tunneling microscopy has been used to locally probe skyrmion
annihilation by individual hot electrons Muckel  _et al._ (2021). Such
measurements can be quite useful for a better understanding of skyrmion
lifetime under practical conditions. Recently, room temperature lifetimes of
up to 1 s were predicted for a 4 nm skyrmion in only a 7 layer ferromagnetic
film Hoffmann  _et al._ (2020). This result is very promising for long-
lifetime skyrmions at room temperature. According to the analytical model on
skyrmions Büttner  _et al._ (2018), thicker films and larger skyrmions
increase the energy barrier to annihilate a skyrmion, which leads to longer
lifetimes. This is especially encouraging for ferrimagnets, such as CoGd and
Mn4N. As discussed earlier in Sec. II.2, intrinsic PMA allows the use of
thicker films in these materials, so a lifetime of a decade may have already
been achieved—and is certainly realistic. Our simulations with GNEB suggest
that for 10 nm skyrmions, long lifetimes of up to 10 days are possible with
CoGd thin films (Fig. 16). For larger skyrmions $>20$ nm with thick films
($>15$ nm), years-long lifetimes seem feasible. Ultimately, there is a trade-
off between film thickness and drivability because spin-orbit torque acts near
the magnet-heavy metal interface. This brings us to a critical evaluation of
skyrmion dynamics.

Figure 16: Simulated skyrmion lifetime by GNEB in CoGd films with $M_{s}$ of
50 kA/m. For 10 nm skyrmions, as indicated by the red curve, lifetimes range
from microseconds in 1.5 nm thick CoGd film to near 10 days in 15 nm thick
CoGd film. For larger skyrmions (20 nm), a year long lifetime is found in 15
nm thick CoGd film (red circle).

##  III Skyrmion Dynamics

To have a low latency skyrmion-based device, fast skyrmions at small energy
costs are required. An efficient way to drive a skyrmion or a domain wall is
using spin-orbit torque. As the picture shows (Fig. 17), a flowing charge
current in a heavy metal underlayer creates a vertical separation of spins
along the axis perpendicular to the metal-magnet interface. The injected spins
create an effective magnetic field causing precession of the skyrmion spins
around it, leading to a flow of the skyrmion whirring forward like a buzz saw.
For a Néel skyrmion, there will also be a precessional phase lag in the
transverse direction, leading to a net Magnus force that drives a Néel
skyrmion at an angle with respect to the charge current. The actual
expressions can be readily obtained from the Thiele approximation to the spin
dynamics Thiele (1973); Hrabec  _et al._ (2017); Tomasello  _et al._ (2014).

###  III.1 What determines skyrmion speed?

Figure 17: Current in a heavy metal underlayer separates spins through spin-
orbit torque, and the injected spins lead to precessional torque on individual
spins that drives a domain wall or a skyrmion forward. A finite tilt in domain
angle $\psi$ causes the topological excitation to see a transverse Magnus
force as well depending on whether it is Néel ($\psi=0,\pi$), Bloch
($\psi=\pm\pi/2$) or hybrid.

We start with the LLG micromagnetic equation for the normalized magnetic
moment, in presence of an applied SOT applied to different units $i$
(sublattices or layers) Slonczewski (1996)

$\displaystyle t_{F_{i}}\frac{M_{s_{i}}}{\gamma_{i}}\frac{d\bm{m}}{dt}$
$\displaystyle=-M_{s_{i}}\bm{m}\times\bm{H}_{\mathrm{eff}}+\alpha
t_{F_{i}}\frac{M_{s_{i}}}{\gamma_{i}}\bm{m}\times\frac{d\bm{m}}{dt}$
$\displaystyle-
a_{j}\>\bm{m}\times(\bm{m}\times\bm{P})-\eta\>a_{j}\>(\bm{m}\times\bm{P})$
(12)

where $\bm{P}$ is the polarization of the injected spins from the heavy metal
to the magnet, $a_{j}=(\hbar/2q)\theta_{\mathrm{SH}}j_{\mathrm{HM}}$ is the
adiabatic spin-orbit torque coefficient that arises when conduction electrons
follow the local texture of the spatially varying magnetization (i.e., the
inertial term of a hydrodynamic equation), $\alpha$ is Gilbert damping, $\eta$
is the non-adiabaticity parameter, and $\theta_{\mathrm{SH}}$ is the spin Hall
angle that relates transverse spin current density to longitudinal charge
current density $j_{\mathrm{HM}}$, related through the strength of the spin-
orbit coupling in the heavy metal.

Using the Thiele approximation (see Appendix Subsection D) we get the
following equation for ith layer in a multi sublattice magnet Büttner  _et
al._ (2018)

$\displaystyle\bm{F}_{ex_{i}}+\Sigma_{j}F_{ij}+4\pi
N_{i}\;t_{F_{i}}\;M_{s_{i}}(\hat{z}\times\bm{v}_{sk})/\gamma_{i}$
$\displaystyle-
t_{F_{i}}\;M_{s_{i}}\mathcal{D}.(\alpha\bm{v}_{Sk})/\gamma_{i}-\frac{4\hbar}{2q}\;I\;\theta_{{SH}}R(\psi)\bm{}{j}_{\mathrm{HM}}=0$
(13)

We have taken $\bm{P}=\bm{j}_{\mathrm{HM}}\times\hat{z}$ to generalize to any
$\bm{j}_{hm}$ direction.
$\bm{F}_{ex_{i}}=-\mu_{0}M_{s_{i}}\int\bm{H}_{\mathrm{eff}}.\nabla\bm{m}$
denotes forces from edges, defects, skyrmion-skyrmion repulsion and varying
fields (for example anisotropy gradient), and is zero when
$\bm{H}_{\mathrm{eff}}$ is uniform. $F_{ij}$ on the other hand arises from
interactions between neighboring layers or sublattices. The third term arises
from the inertial term to the right of the LLG equation, with $N_{i}=\pm 1$
the skyrmion winding number. The fourth term arises from the Gilbert damping
term in the LLG equation, and involves $\mathcal{D}$ the dissipation
tensor—assumed isotropic with diagonals ${\cal{D}}_{xx}$ set by the
dissipation tensor, ${\cal{D}}_{ij}=\int
dxdy(\partial_{i}\bm{m}\cdot\partial_{j}\bm{m}$). For a 2$\pi$ model this term
matches the exchange integral to within a constant (Eq. 32),
${\cal{D}}_{xy}={\cal{D}}_{yx}=0$, ${\cal{D}}_{xx}\approx R_{sk}/2\Delta\times
f_{ex}=\rho f_{ex}(\rho)/2$. $\gamma_{i}$ is the gyromagnetic ratio,
$t_{F_{i}}$ is the thickness of sublattice i. $f_{ex}$ is the fitting term
coming from the exchange integral. $M_{s_{i}}$ saturation magnetization of the
ith film. The final term arises from the adiabatic spin orbit torque, where
$I=\frac{1}{4}I_{d}(\rho,\Delta)$ is the integral of DMI energy Eq. 31,
$I_{d}=\int_{0}^{\infty}dr[r\partial_{r}\theta+N_{sk}\sin{\theta}\cos{\theta}]=\pi
R_{sk}f_{\mathrm{DMI}}$. This integral is, in effect, a shape factor of the
skymion. $R(\psi)=\begin{pmatrix}\cos{\psi}&\sin{\psi}\\\
-\sin{\psi}&\cos{\psi}\end{pmatrix}$ is the rotation matrix acting on the 2-D
unit vectors and involves the domain angle $\psi$ ($\psi=0,\pi$ for Néel
skyrmions, $\pm\pi/2$ for Bloch, and in between for hybrid), the current
density $\bm{j}_{\mathrm{HM}}$ in the HM layer assumed to be along the $x$
direction (polarization in $-\hat{y}$ direction). The non-adiabatic SOT term
vanishes upon volume integration.

Solving for $\bm{v}_{sk}$, we get

$\bm{v}_{sk}=\displaystyle\frac{4\pi
B\theta_{SH}R(\psi-\theta_{0})}{\sqrt{\alpha^{2}S(T)^{2}{\cal{D}}_{xx}^{2}+(4\pi
S_{N}(T))^{2}}}\mathbb{j}_{hm},$ (14)

where $B=\pi\hbar I/{2q}$,
$S(T)=\Sigma_{i}{M_{s_{i}}(T)t_{F_{i}}}/{\gamma_{i}}$ is the spin angular
momentum summed over volume,
$S_{N}(T)=\Sigma_{i}{N_{{sk}_{i}}M_{s_{i}}(T)t_{F_{i}}}/{\gamma_{i}}$ is the
topological spin angular momentum and $\tan{\theta_{0}}=4\pi\langle
N_{sk}\rangle/\alpha{\cal{D}}_{xx}$.

Figure 18: Non-linearity of drive and deformation as the origin of skyrmion
speed reduction. (a) Simulated skyrmion and DW speeds as a function of applied
current density. In contrast to Thiele model descriptions considering
skyrmions as rigid particles (which expect skyrmion motion to follow a linear
trend), the speeds of DWs and skyrmions are, in fact, practically identical
and follow the 1-D DW model. (b) DW and skyrmion displacement (black line)
increase even after the current is switched off due to inertial effects caused
by the canting of spins in the wall (blue curve). This effect coupled with
non-rectangular current pulses can prolong the displacement of skyrmions in an
experiment. (c) Experimental observation of skyrmion and DW speeds and 1-D
model taking into account the pulse shape. No significant difference is
visible Litzius  _et al._ (2020a).

The velocity equation has a simple interpretation—we can write it as

$\displaystyle qn_{e}{v}_{sk}=R(\psi-\theta_{0}){j}_{hm},$ (15)

where the electron density $n_{e}$ is again set by angular momentum
conservation of the form:

$\displaystyle n_{e}\displaystyle\frac{\hbar}{2}\theta_{SH}$ $\displaystyle=$
$\displaystyle\frac{M_{s}}{\gamma}{t_{F}}\sqrt{(4\pi
N_{sk})^{2}+\alpha^{2}{\cal{D}}_{xx}^{2}},$ (16) $\displaystyle=$
$\displaystyle\frac{M_{s}t_{F}\alpha{\mathcal{D}}_{xx}}{\gamma}\sec{\theta_{0}}.$

The new rotation matrix $R(\psi-\theta_{0})$ tells us that there is a
deviation $\Theta_{SkH}$ from the current path due to the Magnus force or the
skyrmion Hall effect:

$\frac{v_{sk,y}}{v_{sk,x}}\equiv\tan{\Theta_{SkH}}=\Biggl{(}\frac{4\pi\langle
N_{sk}\rangle\;\cos{\psi}-\alpha\>\mathcal{D}_{xx}\;\sin{\psi}}{4\pi\langle
N_{sk}\rangle\;\sin{\psi}+\alpha\>\mathcal{D}_{xx}\;\cos{\psi}}\Biggr{)},$
(17)

With the average topological index $\langle N_{sk}\rangle=S_{N}/S$. We see
that even in the absence of dissipation, there is a topological damping term
proportional to $N_{sk}$ that limits the skyrmion speed compared to a domain
wall. which depends on the magnitude of the topological term $N_{sk}$ relative
to the effective damping $\alpha{\cal{D}}_{xx}$ (Gilbert damping times shape
factor)—one term tries to move the skyrmion perpendicular to the current,
while the other tries to bring it back to the current. The rotation matrix
depends on the domain wall angle $\psi$, trying to move along the current for
Néel skyrmions ($\psi=0,\pi$) and perpendicular for Bloch skyrmions
($\psi=\pm\pi/2$).

Let us now compare these skyrmion velocities with the simpler 1-D DW equations
within the Thiele approximation:

$v_{\mathrm{DW}}=\frac{\pi}{2}\frac{D_{int}j_{\mathrm{HM}}}{\sqrt{(S_{0}(T)j_{hm})^{2}+(\alpha
S(T)j_{0})^{2}}}.$ (18)

With $S_{0}(T)=|S_{1}(T)-S_{2}(T)|$, $S=|S_{1}(T)+S_{2}(T)|$ and
$j_{0}=2qt_{F}D_{int}/\hbar\theta_{SH}\Delta_{0}$. For speeds close to the
magnon speeds, relativistic effects have to be taken into account. In
Appendix. B, we explain the relativistic dynamics of the skyrmions and DWs and
the relativistic corrections to Eqs. 14,18 in detail.

Note that Eq. 18 suggests that ferromagnetic domain wall speeds saturate to a
constant $\propto D_{int}/M_{\mathrm{eff}}$ for large current densities
$j_{\mathrm{HM}}$ that sit in both the numerator and the denominator of the
velocity term $\bm{v}_{\mathrm{DW}}$, leading to an overall damped motion
(Fig. 18). This may imply that skyrmions could quickly overcome the speeds of
a corresponding straight, 1-dimensional domain wall in the same material.
However, it is important to realize that skyrmion speeds are furthermore
dependent on internal spin dynamics, which start to dominate at high currents
and are not captured by the Thiele equations. A simple example of this is the
change in size and shape of a skyrmion, which occurs under the influence of a
current Litzius  _et al._ (2017, 2020b). Especially at higher current drives,
these effects can dominate the skyrmion dynamics, leading to an elongation of
the skyrmion and the formation of a domain wall-like section within the
otherwise round skyrmion. These, in turn, cause them to be driven by the same
dynamics as domain walls. Together with other effects such as added
topological damping, it becomes apparent that a skyrmion cannot move faster
than a domain wall under identical conditions.

It is also important to point out that by applying a proper 1-D domain wall
model that accounts for the drive’s pulse shape and inertial effects caused by
domain wall spin canting (see Fig. 18), skyrmions near the relativistic speed
limit (Appendix Subsection D) move with the same speeds as domain walls. This
has been observed both in micromagnetic simulations as well as current-driven
experiments on ferrimagnetic CoGd (Fig. 18). The relativistic speed limit
increases strongly for antiferromagnets ($M_{\mathrm{eff}}=0$) with zero
topological damping, while the limitation set by skyrmion distortion will
still remain in place and prevent skyrmions from moving faster than domain
walls.

Even nanoscale skyrmions like those reported by Romming et al. Romming  _et
al._ (2013), with arguably the highest rigidity of all skyrmions (due to small
size) are not impervious to the effect, as Fig. 18(d) shows: The mobility of
the skyrmions (blue, orange and green data) initially follows the linear
mobility curve for undistorted skyrmions that can be derived from the Thiele
equation Büttner  _et al._ (2018). However, after a critical current density
is reached, the skyrmion cannot withstand the enormous torques any longer and
collapses long before reaching the 1-D domain wall limit (purple, red) in the
material. The gray shaded area gives a phenomenologically determined stability
regime for skyrmions, outside of which the skyrmion starts to loose its
rigidity. Note that the slope of the Thiele mobility increases with decreasing
magnetic field up to about 1 T, where it becomes too low to stabilize
skyrmions in the material.

###  III.2 Skyrmion Hall Effect and Gateability

One of the most defining characteristics of skyrmions is their topological
property. As they are topologically non-trivial and can be mapped by a
continuous transformation onto a sphere, these structures experience an
uncommon response to applied currents. In contrast to topologically trivial
domain walls, skyrmions exhibit the so-called skyrmion Hall effect; a force
that drives them at an angle, the skyrmion Hall angle ($\Theta_{SkH}$) with
respect to the applied driving current (Fig. 19).

Figure 19: Sketch of the skyrmion Hall angle. Skyrmions, due to their
topological nature, do not move parallel to a drive but at an angle, called
the skyrmion Hall angle $\Theta_{SkH}$. This angle lets them drift towards the
edge of a device, complicating their application. For compensated ferrimagnets
and antiferromagnets, the two oppositely oriented sub-lattices can negate one
another and reduce the skyrmion motion to one parallel to the driving current.
Reproduced with permission from Phd thesis of K. Litzius, Johannes Gutenberg-
Universität Mainz (2018). Open Access 2018. Litzius (2018).

This curious property of skyrmion dynamics can complicate the applicability of
skyrmions in data storage devices, especially in racetrack devices, as that
relies on reproducibly moving skyrmions back and forth along the track. While
this motion is technically possible and has in fact been reported before
Litzius  _et al._ (2017), in the presence of strong current pulses along a
narrow track, needed for fast storage and high bit density, skyrmions can
crash into the track edges and get annihilated. We can predict the skyrmion
Hall angle for any combination of skyrmion type, chirality, core polarization,
and drive (Eq. 17). Fig. 21 gives a summary of the most common combinations
shown as different quadrants for STT vs. SOT drives, and for Néel vs. Bloch
skyrmions, for different magnetizations, current directions, for different
signs of the DMI $D$.

Figure 20: Three mechanisms that change the skyrmion Hall angle. (a) Pinning
sites in the material randomize the skyrmion trajectories at low current
densities and thus reduce the skyrmion Hall angle. This effect is mostly
present in the creep, and depinning regime. C Reichhardt and C J Olson
Reichhardt 2016 New J. Phys. 18 095005; licensed under a Creative Commons
Attribution (CC BY) license. Reichhardt and Olson Reichhardt (2016). (b) At
high drives, skyrmions can be deformed by the field-like component of the
spin-orbit torque, thus changing their effective size and the skyrmion Hall
effect. Reproduced with permission from K. Litzius et al. Nature Phys 13,
170–175 (2017). Copyright 2017, Nature Publishing Group Litzius  _et al._
(2017). (c) Additionally, bubble skyrmions can experience DW oscillations and
size deformations that can alter the skyrmion Hall angle. Reproduced with
permission from K. Litzius et al. Nat Electron 3, 30–36 (2020). Copyright
2020, Nature Publishing Group Litzius  _et al._ (2020b). Figure 21:
Characteristic motion directions for different skyrmion types and drives. The
arrow originating at the initial skyrmion position indicated by the red dot
gives the trajectory of the skyrmion. $j_{C}$ gives the charge current
direction and the domain wall orientation of the skyrmions is indicated by OW
(outward), IW (inward), CW (clockwise),and CCW (counter-clockwise). K.
Litzius, Ph.D. thesis, Johannes Gutenberg-University Mainz, 2018. Open Access
2018. Honda and Tanaka (2017); Litzius (2018) Figure 22: Temperature and drive
dependence of the skyrmion Hall angle. No significant temperature dependence
of the skyrmion Hall effect was found when plotted against the skyrmion
velocity. Note that the latter can experience temperature dependencies due to
randomization of the torque and conductivity. Two regimes, corresponding to
creep and depinning/flow regime, respectively, are visible. Reproduced with
permission from K. Litzius, et al. Nat Electron 3, 30–36 (2020). Copyright
2020, Nature Publishing Group Litzius  _et al._ (2020b).

From the previous models, one might expect a constant skyrmion Hall angle for
any drive since it appears not to depend on the applied current density.
However, a series of experiments Litzius  _et al._ (2017); Jiang  _et al._
(2017); Juge  _et al._ (2019) have recently shown that this, in fact, is not
the case and that the skyrmion Hall angle in ferromagnets depends strongly on
the applied drive. Several mechanisms have been proposed to explain this
deviation from the Thiele model. Most notably, the theory by Reichhardt &
Reichhardt Reichhardt and Olson Reichhardt (2016) that proposes pinning
effects to randomize the skyrmion trajectories in the creep and depinning
regime (see Fig. 20(a)). Other theories have proposed mechanisms that cause
skyrmion deformations in the flow regime and could also contribute to a change
in the skyrmion Hall angle Litzius  _et al._ (2017, 2020b). These deformations
could either originate in a field-like spin-orbit torque contribution (Fig.
20(b)) Litzius  _et al._ (2017), or in material inhomogeneities that induce
random fluctuations in the skyrmion, which change the dynamics in a non-rigid
way that is not included in the Thiele model (Fig. 20(c)) Litzius  _et al._
(2020b).

Aside from the drive dependence of the skyrmion dynamics, no direct influence
of thermal effects on the skyrmion Hall angle has been found (Fig. 22) over a
wide range of temperatures.

Since the skyrmion Hall angle is typically an undesirable property of
skyrmionics, much work has been put into reducing its influence. The most
promising approach is to employ compensated ferrimagnetic and anti-
ferromagnetic materials, in which the skyrmion Hall angle cancels out due to
the existence of sub-lattices of opposite polarization ($c_{\mathrm{p}}$). In
the former, a reduction of the skyrmion Hall angle has already been observed
experimentally Woo  _et al._ (2018), which makes these material classes
currently the most promising candidates for skyrmion-based data storage
devices. There are other ways to create dynamic compensation points, typically
by using graded material compositions, which can lead to automatic self-
focusing of propagating skyrmions into quiescent lanes where the Magnus force
cancels (Fig. 27).

###  III.3 The role of Defects and thermal Diffusion

Due to thermal effects, skyrmions in a race track can undergo diffusive
displacement. If the racetrack is defect-free with little to no pinning sites,
a skyrmion would be extremely susceptible to such thermal effects. This kind
of random walk can be beneficial in stochastic applications of skyrmions,
discussed later—but potentially destructive for some applications of skyrmions
that encode analog information into their coordinates, as discussed in Sec.
IV. Furthermore, skyrmions show inertia-driven drift shortly after a current
pulse is removed, rather than stopping immediately (Fig. 18(b)). One way to
control such undesirable motion is by engineering confinement barriers such as
point defects with a different anisotropy or notches etched into the racetrack
(missing material). This would increase the periodic pinning of skyrmions
uniformly along the racetrack, instead of a random creep. But it should be
noted that the pinning must be small enough for the skyrmion to be movable by
moderate magnitudes of current.

The mean squared displacement (MSD) of skyrmion due to diffusion can be
described as Troncoso and Álvaro S. Núñez (2014); Zhao  _et al._ (2020)
$\langle(r(t)-r(0))^{2}\rangle=4D_{diff}t$ where $r(t)$ is skyrmion position
after time $t$. $D_{\mathrm{diff}}$ is the diffusion constant:

$D_{\mathrm{diff}}=k_{B}T\displaystyle\frac{\alpha\mathcal{D}_{xx}}{(4\pi
S_{N})^{2}+(\alpha\mathcal{D}_{xx})^{2}}$ (19)

To calculate the localization time with a certain probability, we can use the
Gaussian distribution:

$P_{\mathrm{acc}}=\int_{-L/2}^{L/2}dr\displaystyle\frac{1}{\sqrt{8\pi
tD_{\mathrm{diff}}}}e^{\displaystyle-r^{2}/(8tD_{\mathrm{diff}})}.$ (20)

For a desired accuracy ($P_{\mathrm{acc}}$) and localization time (t) and
length (L), we can get the required $D_{\mathrm{diff}}$. To get a sense of
magnitude, if we require that the center of a skyrmion remains localized to
within $\pm$ 40 nm with probability $1-10^{-8}=0.99999999$ for 1 hr, the
diffusion constant must be less than $10^{-10}$ $\mu\text{m}^{2}/\text{s}$.
The table below (Table 5) shows the relation between diffusion constant and
confinement lifetime.

Table 5: Table of diffusion constants and corresponding $\pm 40$ nm skyrmion
localization times at $1e^{-8}$ inaccuracy.
$D_{diff}~{}(\mu m^{2}/s)$ | Localization time (t)
---|---
$1e^{-2}$Zhao  _et al._ (2020) | Volatile (ms)
$1e^{-8}$(CoGd) Beach  |  $10^{3}$ (s) (Cache)
$1e^{-10}$ | 1 Day
$1e^{-12}$ | 1 Year
42 kBT Barrier | 10 Years (Hard drive)

By placing a notch (Fig. 23), it is possible to constrain a skyrmion’s
position. The skyrmion’s positional lifetime (meaning the time it takes to go
from one side to the other) can be calculated from transition state theory,
$\tau_{p}=f_{0}^{-1}e^{E_{b}/k_{B}T}$, where $f_{0}$ is the attempt frequency.
Approximately an $E_{b}$ of 30 $k_{B}$T will result in positional lifetime of
seconds for an attempt frequency $f_{0}$ of $10^{10}$ Hz Bessarab  _et al._
(2018); Suess  _et al._ (2019). In the presence of a notch, the skyrmion needs
to shrink in size in order to go past it, making it energetically unfavorable.
The geometry and composition of the defect should be optimal so that the
skyrmion can be localized in equilibrium and driven past it with a modest
drive current without being annihilated at the notch or the racetrack edge.

As seen in Fig. 23, the skyrmion positional lifetime can be long enough
(years) for long-term memory applications and at the same have a moderate
unpinning current. In Fig. 23, the material parameters are $M_{s}=100$ kA/m,
$A_{ex}=12$ pJ/m, $K_{u}=50$ kJ/m3, 200 nm racetrack width, 5 nm thickness and
100 nm notch radius. The notch is created by removing relevant parts of the
racetrack in our model. By using a few of these notches consecutively, we find
that it is possible to localize the skyrmion, in effect discretizing the
skyrmion position. The skyrmion would still execute random rattles about its
localized position between two neighboring notches, but it can not go over to
the next block without a current pulse.

As was mentioned in section III.F, to determine $E_{b}$, the minimum energy
path (MEP) between two sides of the notch needs to be calculated. To find the
MEP here, we used the string method E  _et al._ (2002). Similar to the MEP
calculations for the skyrmion lifetime, an initial path needs to be determined
by hand. As discussed earlier, in the skyrmion lifetime calculations the
initial path is between the skyrmion and the ferromagnetic background ground
state. For the positional lifetime, however, the skyrmion path is between two
sides of the notches. The path is divided into multiple images of the system
to define the initial path. Then at each iteration, the path is moved toward
the minimum energy and reparametrized. This is repeated until the path
converges close enough to the MEP.

Figure 23: a) The racetrack geometry with four identical semicircular notches
(radius 100 nm) separated by 200 nm from each other. The space between notches
would be the places where the skyrmions would be positioned. b) Calculated
minimum energy path for skyrmion passing a defect using the parameters
mentioned in the main text. By using the string method Suess  _et al._ (2019),
the MEP from one side of the defect to the other side is calculated. The
minimum unpinning current is calculated to be a 4 ns current pulse of
$1.4\times 10^{11}$ A/m2

.

##  IV Skyrmionic Device Applications

The original racetrack idea Parkin  _et al._ (2008) relied on a sequence of
domain walls separating regions of opposite polarization, encoding the 0 and 1
bits. These domain walls on racetracks can have very high density, and the
racetracks, in principle, can be stacked in 3-D. However, domain wall motion
is restricted by pinning and creep. For skyrmions, information is contained in
their absence or presence, but it relies once again on the ability of
individual skyrmions to execute the particulate, quasi-ballistic,
deterministic driven motion. While conventional densely sequenced
DW/skyrmions-based memory is a possible application, there are other novel
applications that can leverage the size and mobility advantages of such
topological magnetic textures. We now discuss some of the device concepts
involving skyrmions, including key operations needed such as read, write,
erase and move. These applications could be key to building native memories
and accelerators for applications such as temporal logic that leverages
temporally encoded data for processing, stream-based computing that uses
continuously streaming bits and is particularly oriented towards signal
processing, for building spiking neural networks, ultra-compact reservoir
computing, and nano-oscillators. While all of these can be built using
conventional CMOS technology, skyrmions may provide scalability due to the
compactness of the circuits and low power consumption due to the physics of
the skyrmion motion mapping onto the application model.

###  IV.1 Magnetic Memory for Temporal Computing

Figure 24: Temporal memory circuit illustration and concept: panel (a) shows
the concept of N=2 cells single column temporal memory capable of storing and
playing temporal wavefronts. A magnetic tunneling junction is placed for
skyrmion detection. Panel (b) shows the detailed description of the temporal
memory circuits in each cell. The Control lines consist of source line (SL),
the bit line (BL), write line (WL), recovery write line (RWL), read
enable(RE), recovery line (RD), and erase line (ERASE). H. Vakili et al., IEEE
Journal on Exploratory Solid-State Computational Devices and Circuits, vol. 6,
no. 2, pp. 107-115, 2020; licensed under a Creative Commons Attribution (CC
BY) license. Vakili  _et al._ (2020). Note that only a single skyrmion per
racetrack is used, with multi-bit information stored in the position of the
skyrmion. Having a single skyrmion per racetrack increases its reliability and
control over a packed, interacting array of skyrmions. Figure 25: (a) An
example of two wavefronts with disparate duty cycle fed to the temporal memory
cell. (b-e) Micromagnetic simulation of four operational phases of the
temporal memory in magnetic racetracks. The figures show a background of up-
spin magnetic material (black) with a down-spin skyrmion (white); color
indicates the in-plane spin orientation. The location of the skyrmion at the
beginning of each phase is shown, while the arrows denote the position of the
skyrmion after finishing that phase. The magnetic tunnel junction (MTJ) placed
for skyrmion detection is denoted by the red circle in the first column. The
patterned area in the racetracks represents an energy barrier. Amplified
repulsion from the edges prevents the pinning of skyrmions at the edges and
skyrmion annihilation and helps in maintaining their expected trajectory
towards the read MTJ. H. Vakili et al., IEEE Journal on Exploratory Solid-
State Computational Devices and Circuits, vol. 6, no. 2, pp. 107-115, 2020;
licensed under a Creative Commons Attribution (CC BY) license. Vakili  _et
al._ (2020).

A recently proposed temporal computing scheme called race logic Madhavan  _et
al._ (2014, 2017); Tzimpragos  _et al._ (2019) has shown acceleration of
dynamic programming and machine learning algorithms. Information
representation in race logic is different than the conventional Boolean
approach as it represents multi-bit information at the arrival times of the
digital rising edge of a wavefront in a wire Madhavan  _et al._ (2014). Such a
temporal coding scheme potentially achieves orders of magnitude energy
efficiency Madhavan  _et al._ (2014) compared to traditional data encoding
while accelerating the dynamic programming algorithm. However, in Boolean
computing, the analog temporal data needs to be converted to the digital
domain. This domain translation imposes a significant area and energy penalty
from analog-to-digital converter circuits, limiting the computational
capability. Thus, there is a need for a native analog temporal memory to store
and retrieve the analog temporal data to enable complicated processing with
the temporal coding scheme.

Assuming a quasi-ballistic motion of skyrmion with an applied drive current,
the time duration of a drive current determines the distance traveled by the
skyrmion on the racetrack. Therefore, we can design a scheme of storing the
temporal data in the position of a single skyrmion along the racetrack. This
temporal data can represent timing delays in race logic for pattern matching
applications, as well as thresholds for regular or automata decision trees for
parallel processing of near-sensor serial input data Tzimpragos  _et al._
(2019); Madhavan  _et al._ (2020). However, an adequate lifetime of skyrmions
(Table V) will be needed to save on nucleation energy, as pre-written
skyrmions can be reused rather than nucleating and destroying them repeatedly.
Such a scheme allows for occasional replenishment of skyrmions through added
voltage-gated nucleation events, whose cost is readily amortized over several
(ranging from hundreds to millions) operating cycles.

In skyrmion-based temporal memory Vakili  _et al._ (2020) (Fig. 24), an
arriving wavefront launches a pre-fabricated skyrmion positioned below the MTJ
at the beginning of the racetrack. By turning on transistor $T_{1}$, a drive
current is initiated in the heavy metal underlayer. This drive current is shut
off when the wavefront completes its duty cycle. Thus, the analog arrival time
stamp of the rising edge of the input signal gets encoded onto the eventual
position where the skyrmion comes to rest and is set by the duty cycle
(fraction of the total time when the signal is active) of the input signal.
The write operation is then the launch of the skyrmion and its subsequent
placement by a current whose duration is set by the wavefront arrival time.
Due to the non-volatile nature of the magnetic skyrmion, there is no need to
store the arrival time repeatedly, as it can be replayed back as needed.
During this read operation, the skyrmion is driven to the read MTJ, consuming
time set by the duration of the rising edge. This information, however, is
lost once the skyrmion reaches the read MTJ at the end of the main racetrack.
Thus, to replay the temporal data repeatedly, the proposed design Vakili  _et
al._ (2020) needs a recovery operation to avoid destructive read operation.
The recovery operation needs an extra racetrack called the recovery racetrack
(see Fig. 25) whose job is to store the positional value of the temporal data
lost during the read operation by launching a recovery skyrmion as soon as the
read is executed, much like passing the baton during a relay race. The
proposed circuit wavefront is designed in such a way that the skyrmions in
both the main and recovery racetrack travel the same distance during the read
and recovery operations through synchronization of the drive currents. Due to
the matched speed of the skyrmions in both racetracks, the position of the
skyrmion representing the temporal data is restored and is ready to be
replayed back repeatedly (if the speeds do not match perfectly, this too can
be accounted for in a calibration phase). The recovery racetracks only
participate during the read and recovery operations. A reset operation is
needed to store a new wavefront with a different duty cycle. The read MTJ
stacks placed at the beginning and end of the racetracks provide a feedback
signal to stop skyrmion movement after successful detection.

Figure 26 shows the breakdown of energy dissipated in the circuit, resolved
into individual components such as the 8 transistors $T_{1}\ldots T_{8}$,
reference resistance $R_{0}$ (needed for voltage swings for reading Fig. 24),
buffer used in the read circuitry (the inverter resistance), the main and
recovery track metal lines, and the read MTJ for a 640 nm racetrack at 50$\%$
duty cycle, assuming 45nm (dark bars) and 16nm (light bars) CMOS transistors.
We can see that the read and recovery operations cost the most energy, and the
bulk of this energy comes from the overhead transistor switching costs ($\sim$
1 pJ overall) compared to the loss in the metal lines ($\ll 10$ fJ). Notably,
doing the full operation with CMOS memory instead of its skyrmionic
counterpart would consume well over a hundred transistors to account for
counters, storage latches, and decoders, at considerably larger energy cost
and overall footprint. Furthermore, there is increased efficiency due to our
ability to playback the wavefront duty cycle at low damping (Energy-delay
product another 32X lower than 1 GHz CMOS) for one cycle. If we do 10-100
reads per cycle, the product is almost 2500X lower. Finally, there is energy
saving from repeated read operations due to the non-volatility of the
skyrmions.

Figure 26: Energy breakdown of each circuit elements and total energy for 50 %
duty cycle. A set of 640 nm racetracks used to capture wavefront and provide
the energy breakdown. The high overdrive voltage of transistors consume the
largest energy in all operations. Both the racetracks and the associated
transistors are active during the read and recovery operation, resulting in
consuming most energy. However, joule heating in the main (TrackM) and
recovery (TrackR) racetracks contributes only a small fraction of the overall
energy. Simulation results are shown in darker and brighter color bars for 45
nm and 16 nm CMOS transistors, respectively. H. Vakili et al., IEEE Journal on
Exploratory Solid-State Computational Devices and Circuits, vol. 6, no. 2, pp.
107-115, 2020; licensed under a Creative Commons Attribution (CC BY) license.
Vakili  _et al._ (2020).

###  IV.2 Reconfigurable Skyrmionic Logic

A reconfigurable hardware fabric offers increased performance per Watt at a
lower non-recurring design and fabrication cost than the Application Specific
Integrated Circuits (ASICs). Field-Programmable Gate Arrays (FPGAs) are medium
to large integrated circuits that can be programmed to repeatedly perform any
desired digital logic without any extra engineering cost. A matrix of
programmable logic blocks is interconnected by a routing fabric which is also
programmable Kuon  _et al._ (2008). Static Random Access Memory (SRAM) Cells
store the logic function data configurations in modern FPGAs. Moreover, SRAM
cells also store the select bits from the multiplexers (MUXs). Though FPGAs’
re-programmability and manufacturing simplicity are advantageous, the use of
high cost per bit and volatile SRAM cells leads to large area overhead.

Figure 27: (a-c) Self-converging skyrmions can be used to design
reconfigurable logic design. a) By varying the anisotropy, the skyrmion will
end up in different lanes (b) Changing the anisotropy can be achieved by using
VCMA gates, Which can be used to get a step-like function vs. $K_{u}$. (d)
Skyrmionic programmable logic device using controllable convergence lane. The
programmable feature of the design allows it to be programmed for computing
different logic functions such as AND, NAND, OR, NOR, XOR and XNOR operations.
Level shifter (high to low) converts the applied voltage at the BLB to
necessary step values of voltage needed at the large VCMA gate for lane
selection purposes. The small VCMA gates placed in each lane control the
motion of skyrmion, and the magnetic tunneling junction placed at the end of
each lane detect the presence/absence of skyrmion. Reprinted with permission
from H. Vakili et al., Phys. Rev. B 102, 17442, 2020. Copyright 2020 by the
American Physical Society.Vakili  _et al._ (2020); Sakib  _et al._ .

Nowadays, FPGAs are used in a wide variety of applications such as compute
accelerators, automotive, consumer electronics, etc., as the feature size
reduction of CMOS technology is paving the way towards an increased number of
logic blocks in FPGAs. Moreover, FPGAs are also used/offered by several cloud
providers to accelerate bioinformatics, web search, and machine learning
workloads Caulfield  _et al._ (2016). However, the operating of these devices
is generally carried out under strict performance, power, and reliability
constraints. In this context, reconfigurable logic using skyrmionic racetracks
carries the potential to bring non-volatility to FPGAs, thus obliterating the
security concerns of programming the FPGA only at power ON. Moreover,
reconfigurable logic using skyrmionic racetracks can be a promising
alternative to the high-cost SRAM cells in FPGAs, as they can be much more
densely packed than the SRAM-based designs exhibit reasonable propagation
delay.

Skyrmions and domain walls need spin current for their operations and can be
used for ultra-small, low power, and nonvolatile memory or logic operations.
Both skyrmions and DWs can have reasonably long lifetimes and high speeds.
Skyrmions tend to have lower pinning than DWs, so the energy requirements can
be lower. Nevertheless, one major difference between DW and skyrmion is that
the DW can only move in the direction of the racetrack (1-D movement). In
contrast, the skyrmion can move along both axes of the racetrack (2-D
movement). By exploiting this extra degree of freedom and self converging
nature of skyrmion discussed shortly, a reconfigurable logic device can be
designed with only one nucleation site. In Zhang  _et al._ (2015a) by using
the merging, duplication, and skyrmion’s ability to deform to a double domain
wall and back, a logic device has been proposed, which demonstrates some of
the interesting dynamical behavior of skyrmions.

Skyrmion-based reconfigurable logic has been proposed in Luo  _et al._ (2018)
and in Zhang  _et al._ (2019). However, the device Luo  _et al._ (2018); Zhang
_et al._ (2019) needs extra fine-tuning of the parameters to get the
reconfigurability which is not the case for the device proposed in Sakib  _et
al._ . By leveraging the interaction between the skyrmions combined with the
electric-field-controlled magnetic anisotropy (VCMA) effect, a reconfigurable
logic gate is proposed in Zhang  _et al._ (2019). However, both the
reconfigurable logic gates Luo  _et al._ (2018); Zhang  _et al._ (2019) need
at least two nucleation sites and more than one energy-hungry skyrmion
nucleation to do logic computation.

Figure 27 shows an example of self-converging skyrmions, created using a
compositionally graded structure that generates a compensation point where the
transverse Magnus force cancels, and the skyrmion moves linearly along a 1-D
racetrack. In the example shown, such compensation is achieved by using a
varying DMI (e.g., PtxW1-x heavy metal underlayer) with a B20 magnet so that
an injected skyrmion picks up growing interfacial DMI along with its current-
driven transport and evolves into a Bloch-Neel hybrid that naturally reaches
such a compensation. To make the converged lanes align with a pre-fabricated
racetrack, we can place intermediate repulsive lanes with high anisotropy and
further control electrostatically using a gate that tunes the voltage-
controlled magnetic anisotropy (VCMA) across the interface between the magnet
and a MgO overlayer. The lane selectivity with tuned anisotropy is shown in
the inset at zero temperature (finite temperature gives stochastic wiggles
around the transition points).

The reconfigurable logic circuit works as follows. Based on the desired
operation (AND/NAND, OR/NOR, XOR/XNOR), the corresponding voltage is applied
to the large global VCMA gate. This would put the skyrmion into the
appropriate lane. The small VCMA gates in each lane control the subsequent
movement of a skyrmion. For the case of AND/NAND operation, the lane 1 VCMA
gate is ON only when both inputs are 1, while for OR/NOR, the lane 2 gate is
OFF only when both inputs are 0. For XOR/XNOR, the lane 3 VCMA gate is on only
when exactly one of the inputs is 0, and the other is 1 (i.e., the two inputs
are different). For the logic operation, the skyrmion is moved along the
appropriate lane to the read MTJ at the end. Then for recovery, a current in
the opposite detection is applied for a certain amount of time (depending on
the lane length) to put the skyrmion back to its starting piston in the lane.
The corresponding circuit needed for the operations is shown in Fig 27(d)

Figure 28: Skyrmionic AND-OR logic device (a) Device structure with two input
lanes and two output lanes. The two electrode gates (with voltages
$V_{\mathrm{g1}}$ and $V_{\mathrm{g2}}$) are designed to modulate skyrmion
motion through the VCMA effect. (b) Profile of driving current density
($j_{\mathrm{SOT}}$) and positive voltages applied to achieve device function.
(c-e) Three cases of inputs and their corresponding simulated processes and
outputs. In the snapshots of the micromagnetic simulations, the color red
represents spin up ($m_{z}=+1$), blue represents spin down ($m_{z}=-1$), and
white represents spins that are horizontal in the plane ($m_{z}=0$). Skyrmion
trajectories are indicated in the magnetization snapshots at
$t=$8\text{\,}\mathrm{n}\mathrm{s}$$. Reprinted with permission from H. Zhang
et al., Phys. Rev. Applied 13, 054049, 2020. Copyright 2020 by the American
Physical Society. Zhang  _et al._ (2020a).

A skyrmionic AND-OR gate is proposed by using two joint racetracks with a
notch (Fig. 28(a)) under the assistance of voltage control magnetic anisotropy
(VCMA) Zhang  _et al._ (2020a). By accurately controlling the gate voltage
$V_{\mathrm{g1}}$, $V_{\mathrm{g2}}$ and the current density
$j_{\mathrm{SOT}}$ with time (Fig. 28 (b)), the motion of the skyrmions can be
guided along the racetracks. In Fig. 28(c)-(d), when only a single skyrmion
arrives at the notch, it will go to the bottom track no matter where it
originated. In Fig. 28(e), when both tracks have a skyrmion, they will both