|  Spintronic devices and applications using noncollinear anti-chiral
antiferromagnets
---|---
|  Ankit Shukla,a Siyuan Qian,b and Shaloo Rakheja∗
|  Antiferromagnetic materials have several unique properties, such as a
vanishingly small net magnetization, which generates weak dipolar fields and
makes them robust against perturbation from external magnetic fields and rapid
magnetization dynamics, as dictated by the geometric mean of their exchange
and anisotropy energies. However, experimental and theoretical techniques to
detect and manipulate the antiferromagnetic order in a fully electrical manner
must be developed to enable advanced spintronic devices with antiferromagnets
as their active spin-dependent elements. Among the various antiferromagnetic
materials, conducting antiferromagnets offer high electrical and thermal
conductivities and strong electron-spin-phonon interactions. Noncollinear
metallic antiferromagnets with negative chirality, including Mn3Sn, Mn3Ge, and
Mn3GaN, offer rich physics of spin momentum locking, topologically protected
surface states, large spin Hall conductivity, and magnetic spin-Hall effect
that arises from their topology. In this review article, we introduce the
crystal structure and the physical phenomena, including anomalous Hall and
Nernst effects, spin Hall effect, and magneto-optic Kerr effect, observed in
negative chirality antiferromagnets. Experimental advances related to spin
orbit torque-induced dynamics and the impact of the torque on the microscopic
spin structure of Mn3Sn are also discussed. We then present a potential
antiferromagnetic spintronic device that can serve as a non-volatile memory,
high-frequency signal generator/detector, neuron emulator, and even a
probabilistic bit, depending on the design parameters and the input stimuli,
_i.e._ , amplitude and pulse width of the injected spin current and the
externally applied magnetic field. In this device, spin-orbit torques can be
used to manipulate the order parameter, while the device state can be read via
tunneling magnetoresistance which has been recently experimentally
demonstrated in all-Mn3Sn tunnel junctions at room temperature. We also
present analytic models that relate the performance characteristics of the
device with its design parameters, thus enabling a rapid technology-device
assessment. Effects of Joule heating and thermal noise on the device
characteristics are briefly discussed. We close the paper by summarizing the
status of research and present our outlook in this rapidly evolving research
field.

##

††footnotetext: ∗Electrical and Computer Engineering, University of Illinois
at Urbana-Champaign, Urbana, USA. Tel: 217-244-3616;
<EMAIL_ADDRESS>
##  1 Introduction

The field of spintronics utilizes information stored in the magnetization
vector of magnetically ordered medium. Typically effects, such as spin-
transfer torque (STT) and spin-orbit torque (SOT) in conjunction with a
magnetic field, are used to switch the magnetic state. 1, 2, 3, 4, 5, 6, 7, 8,
9, 10, 11 The non-volatility and scalability of spintronics devices can allow
us to implement information processing devices with reduced resources (_i.e._
, energy, area, time). Spintronics devices can complement the functionality of
silicon CMOS circuits for beyond von Neumann computing applications including
probabilistic and stochastic computing, bio-inspired computing, and Joule
heating free transfer of information. 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26

Typically, the active element in spintronics devices is a ferromagnet (FM) in
which spins on the neighboring ions are aligned in a parallel manner,
resulting in a net macroscopic moment. A two-terminal (2T) device composed of
thin FM layers separated by a non-magnetic insulating spacer, referred to as
the magnetic tunnel junction (MTJ), is considered as the building block in
spintronics computing. The net resistance of the MTJ stack depends on the
relative magnetic orientation of the two FM leads: a low resistance state is
obtained for parallel orientation of the FMs, while a high resistance state is
obtained for anti-parallel orientation of the FMs. In a 2T MTJ, STT is used to
change the orientation of one of the FMs, referred to as the _free_ layer,
while the other FM has a pinned orientation, whose magnetization is
unperturbed in the presence of STT. A three-terminal (3T) MTJ in which the SOT
is used to control the orientation of the free FM layer offers a higher degree
of flexibility and energy efficiency for spintronics computing applications.
The on-off ratio, _i.e._ , the difference between the high and low resistance
states, of the MTJ is read out via the tunneling magnetoresistance (TMR),
which can reach values as high as 631% at room temperature in epitaxial
devices employing MgO as the tunneling barrier 27.

More recently, antiferromagnetic-based spintronics devices have gained
significant attention from the research community. Unlike FMs,
antiferromagnets (AFMs) do not have a macroscopic magnetic moment which
creates both challenges and opportunities for their future computing
paradigms. 28, 29, 30, 31, 32, 33, 34 For example, the lack of dipolar fields
in AFMs can enable their ultra-dense integration in circuits without crosstalk
issues and related security vulnerabilities. However, as AFMs lack a
macroscopic moment, it is difficult to read their magnetic state in device
geometries. In this regard, octupole moment in chirally ordered noncollinear
AFMs (_e.g._ , Mn3Sn, Mn3Ge, Mn3GaN, _etc._) can be used as an order parameter
in an all-AFM MTJs that exhibit finite TMR and have negligible stray fields
and the potential for high-speed operation. The octupole moment is a rank-3
multipole moment of Mn atoms averaged over the entire magnetic unit cell. 35
The octupole moment has the same irreducible representation as the dipole
moment in FMs. Thus, chiral noncollinear AFMs combine the benefits of FMs and
other collinear AFMs, setting the stage for advanced information processing
devices using the octupole moment as the computational state variable. 36, 37

In this paper, we review recent experimental and theoretical progress in
noncollinear AFMs with negative chirality, in which the octupole moment gives
rise to non-degenerate spin-polarized bands at the Fermi level and a finite
TMR at room temperature. The octupole moment in these AFMs can be manipulated
via SOT from a proximal non-magnetic material such as a heavy metal (_e.g._ ,
W, Pt, _etc._), and high-speed order parameter dynamics including chiral
oscillations and switching can be initiated depending on the energy landscape
of the AFM and the input SOT and magnetic field.

We present a discussion of the crystal structure and the observed and
predicted physical phenomena in chiral AFMs in Sec. 2. Details regarding order
parameter dynamics in six-fold and two-fold symmetric anti-chiral AFMs,
particularly in Mn3Sn, are discussed in Sec. 3. These dynamics along with
fully electrical readout of the order parameter, via the tunneling
magnetoresistance (TMR), form the basis of realizing new types of spintronic
devices with chiral AFM serving as the active element. Such devices include
high-frequency signal generators and detectors, bio-mimetic neurons, and non-
volatile memory. The role of Joule heating and thermal noise in influencing
the SOT-driven response of the system is also briefly presented in Sec. 3. As
discussed in Sec. 4, chiral AFM-based spintronic devices can be used to
implement non-von Neumann computing platforms. The paper concludes with a
summary of key findings and outlook in the field of condensed matter physics
and future octupole-based spintronic devices.

##  2 Noncollinear Chiral Antiferromagnets

###  2.1 Crystal Structure

Chiral AFMs are materials with lattice spins oriented in a noncollinear and
chiral configuration. In AFMs like Mn3Sn, Mn3Ga and Mn3Ge, the atomic moments
are arranged in the basal plane of the hexagonal crystal, while in AFMs like
Mn3Ir and Mn3Pt, the atomic spins are in the (111) plane of the face-centered
cubic crystal. Figure 1 shows the crystal and the triangular spin structures
of D019-Mn3Sn and L12-ordered Mn3Ir AFMs. The Mn3Sn (Mn3Ir) AFM has negative
(positive) chirality: hopping between Mn atoms in an anticlockwise (clockwise)
sense, the atomic moment rotates by 120∘ in a clockwise (clockwise) sense.

Fig. 1: Crystal and spin structure in (a) D019-Mn3Sn and (b) L12-ordered
Mn3Ir, respectively.

The noncollinear moments introduce Dzyaloshinskii-Moriya interaction (DMI),
which stabilizes the triangular spin structure with a given chirality. As DMI
is a super-exchange interaction, it is present in crystal structures with
broken inversion symmetry. DMI introduces a term given as
$-\bf{D}_{12}\cdot(\bf{S}_{1}\times\bf{S}_{2})$ ($\bf{D}_{12}$ is the DMI
coefficient and $\bf{S_{1}}$ and $\bf{S_{2}}$ are the atomic spins) in the
Hamiltonian and thus favors a certain orientation of $\bf{S}_{1}$ with respect
to $\bf{S_{2}}$. This means that DMI introduces handedness or chirality to the
spin structure of the material. Although Mn3Sn and Mn3Ir are globally
centrosymmetric crystals, the noncollinearly arranged moments on Mn atoms
break inversion symmetry locally, which introduces DMI in the crystal. Indeed,
neutron diffraction measurements carried out on single-crystal Mn3Sn in 1982
revealed weak ferromagnetism 38 which was later explained in terms of the DMI
mechanism, which leads to canting of moments in the material 39. The chirality
of the spin structure breaks the mirror and time-reversal symmetries and lifts
degeneracies in the materials’ energy-momentum dispersion, which creates band
crossings.

The triangular spin structure in chiral AFMs possess momentum-space Berry
curvature concentrated around hot spots in the reciprocal space, while the
integral of the Berry curvature over the Brillouin zone modifies the magneto-
transport properties of the AFMs. 40 The sign of the Berry curvature flips
with the chirality of the noncollinear AFM. Thus, the sign of magneto-
transport signals that are odd with respect to the Berry curvature also flips
in chiral AFMs. Theoretical calculations have shown multiple Weyl points of
type II in the bulk bandstructure and Fermi arcs on the surface connecting
Weyl points of opposite chirality in Mn3Sn and Mn3Ge. 41, 42 The bandstructure
features in chiral AFMs are described as topological in nature and have
launched the research field of topological spintronics. Table 1 lists the
material properties of various chiral AFMs.

Table 1: List of chiral antiferromagnetic materials $\mathrm{Mn_{3}X}$ and their associated parameters. Here $T_{N}$ is the Nèel temperature, $M_{s}$ is the saturation magnetization, $K_{u}$ is the single-ion uniaxial anisotropy constant, $J_{E}$ is the exchange constant, , is in $\mathrm{MJ/m^{3}}$. Sign of DMI constant, $D_{M}$, which decides the chirality is also mentioned. X | $T_{N}\leavevmode\nobreak\ (\mathrm{K})$ | $M_{s}\leavevmode\nobreak\ (\mathrm{T})$ | $K_{u}\leavevmode\nobreak\ (\mathrm{kJ/m^{3}})$ | $J_{E}\leavevmode\nobreak\ (\mathrm{MJ/m^{3}})$ | $D_{M}$
---|---|---|---|---|---
$\mathrm{Ir}$ 43, 44 | $960\pm 10$ | $1.63$ | $3000$ | $240$ | –
$\mathrm{Pt}$ 45, 43 | $473\pm 10$ | $1.37$ | $10$ | $280$ | –
$\mathrm{Rh}$ 45, 46, 43 | $853\pm 10$ | $2.00$ | $10$ | $230$ | –
$\mathrm{Ga}$ 47, 43, 48, 49 | $470$ | $0.54$ | $100$ | $110$ | +
$\mathrm{Sn}$ 50, 51, 43, 48 | $420-430$ | $0.50$ | $110$ | $59$ | +
$\mathrm{Ge}$ 52, 43, 48, 49 | $365$ | $0.28$ | $1320$ | $77$ | +
$\mathrm{GaN}$ 53, 54 | $345$ | $0.69$ | $10$ | $280$ | –
$\mathrm{NiN}$ 53, 54 | $262$ | $1.54$ | $10$ | $177$ | –

###  2.2 Physical Phenomena

Recent magneto-transport measurements have shown finite anomalous Hall effect
(AHE), 40, 55, 56 the spin Hall effect (SHE), 57 the anomalous Nernst effect
(ANE), 58 and the topological Hall effect (THE) in chiral AFMs. Despite its
vanishingly small net magnetic moment $0.002\leavevmode\nobreak\ \mu_{B}$
($\mu_{B}$ is the Bohr magneton) per Mn atom, a large anomalous Hall
conductivity (AHC) of $20\leavevmode\nobreak\ \mathrm{(\Omega.cm)^{-1}}$ at
room temperature and $100\leavevmode\nobreak\ \mathrm{(\Omega.cm)^{-1}}$ at
low temperatures has been experimentally measured in Mn3Sn 56. Although it is
more challenging to observe AHE in cubic chiral AFMs due to weaker spontaneous
ferromagnetism and stronger magnetic anisotropy, uniaxial stress and magnetic
field were used in Ref. 59 to align the domains in Mn3Pt, leading to a
substantial AHC ($\sim 20-30\leavevmode\nobreak\ \mathrm{(\Omega.cm)^{-1}}$)
in the bulk sample. Similarly, in Ref. 60, strain tuning was used to achieve
AHC exceeding $100\leavevmode\nobreak\ \mathrm{(\Omega.cm)^{-1}}$, while in
Ref. 61, epitaxial strain was shown to alter the AHC in Mn3Pt films by more
than an order of magnitude.

The ANE in which a temperature gradient results in a transverse voltage drop
perpendicular to the heat flow and the magnetization is feasible in chiral
AFMs. A Seebeck coefficient 62 of $0.35\leavevmode\nobreak\ \mathrm{\mu
V.K^{-1}}$ at room temperature and $0.6\leavevmode\nobreak\ \mathrm{\mu
V.K^{-1}}$ at $200\leavevmode\nobreak\ \mathrm{K}$ has been measured in bulk
single-crystal Mn3Sn grown via the Bridgman method. Using _ab initio_
calculations, it was shown that the spin Nernst conductivity (SNC) in Mn3Sn
and Mn3Ga is about five times larger than that of platinum 63. Moreover, the
anomalous Nernst conductivity (ANC) of Mn3Ga at $300\leavevmode\nobreak\
\mathrm{K}$ was found to be $50\times$ that of bcc Fe. The large value of ANC
and SNC in noncollinear AFMs is advantageous for implementing thermoelectric
and spin caloritronic devices. 58

The momentum-dependent spin splitting in Mn3Sn also leads to magnetic SHE 64
which has an anomalous sign change with the octupole polarity. Novel spin
currents in chiral AFMs have also been studied via _ab initio_ methods. In
particular, it was shown that spin currents in noncollinear AFMs have both
longitudinal and transverse spin polarization components. The longitudinal
spin current is similar to the spin-polarized current in FMs. However, the
transverse component is odd under time reversal and is thus distinct from the
conventional SHE, which is even under time reversal. These novel spin currents
65 can help interpret SOT measurements in noncollinear AFMs, while also being
exploited for new spintronics applications.

The first reports of TMR in an all-AFM MTJ appeared in 2023 that showed a TMR
ratio of $2\%$ at $300\leavevmode\nobreak\ \mathrm{K}$ in Mn3Sn/MgO/Mn3Sn 66,
while a TMR of 100% was reported in Mn3Pt/MgO/Mn3Pt 67 junctions. The
electronic bands in Mn3Sn show momentum-dependent spin splitting of the Fermi
surface due to the broken time reversal symmetry. 68, 66 Therefore, in
Mn3Sn/MgO/Mn3Sn junctions, it was shown that the resulting TMR is larger than
that expected from the spin polarization and originates due to the presence of
magnetic octupoles. Theoretical calculations have revealed that for vacuum as
a tunnel barrier and current perpendicular to the Kagome plane, RA product
increased from approximately $0.05\leavevmode\nobreak\
\mathrm{\Omega\dotproduct\mu m^{2}}$ in the parallel configuration to
$0.2\leavevmode\nobreak\ \mathrm{\Omega\dotproduct\mu m^{2}}$ in the
antiparallel configuration. 68 These results are significant as a finite TMR
in AFM-based tunnel junctions can enable fully electrical readout and thus
drive the development of AFM-based spintronics devices.

Imaging the magnetic domains can provide crucial information regarding the
reversal process of the chiral spin structure in a spatially resolved manner.
Using magneto-optic Kerr effect (MOKE) method, Higo _et al._ measured a large
zero-field Kerr rotation angle of $20\leavevmode\nobreak\ \mathrm{mdeg}$ at
$300\leavevmode\nobreak\ \mathrm{K}$ in Mn3Sn. 69 This Kerr angle value in
Mn3Sn is comparable to that found in FMs and is a useful probe to study the
dynamics of magnetic domains in chiral AFMs. In this regard, Uchimura _et al._
measured Kerr rotation angle in thin films of Mn3+xSn1-x ($-0.11\leq x\leq
0.14$) and identified that the symmetry requirements of MOKE are the same as
those of AHE 70. Moreover, they showed that the domain reversal process begins
by nucleation of reversed domains in the media, followed by domain expansion
and preferential propagation along the [$11\bar{2}0$] direction. Like Mn3Sn,
Mn3Ge is also an anti-chiral AFM. However, unlike Mn3Sn, Mn3Ge does not have a
helical phase transition, enabling unique studies of noncollinear magnetic
textures at low temperatures. Additionally, the AHC reported in Mn3Ge can be
three times as high as that in Mn3Sn. Recently, a large polar MOKE signal,
around $8.2\leavevmode\nobreak\ \mathrm{mdeg}$, and a large longitudinal MOKE
signal, around $5.6\leavevmode\nobreak\ \mathrm{mdeg}$, have been measured in
Mn3Ge single crystals 71.

Scanning thermal gradient microscopy (STGM) has also been used to image the
local magnetic structure of Mn3Sn films. STGM technique in the presence of
magnetic fields can form the basis of writing domain patterns into Mn3Sn as
discussed in Ref. 72. This work specifically measures the ANE voltage,
$V_{\mathrm{ANE}}$, for a temperature gradient along the c-axis, which is
established with the help of laser illumination. $V_{\mathrm{ANE}}$ is
governed by the average octupole moment of the domains under the illuminated
laser spot.

##  3 Spintronic Devices Based on Chiral AFMs

To realize an energy-efficient spintronic device based on anti-chiral AFMs, it
is important to investigate methods that can control the octupole moment in a
fully electrical manner. On the readout front, although AHE and anisotropic
magnetoresistance can be utilized, the detected signals are generally weak and
require amplification adding to the energy consumption and area of the
spintronic device. Thus, electrical readout via the TMR is preferred as it
provides a more robust signal and is typically compatible with circuit
implementations utilizing silicon transistors as the peripherals.

A conceptual spintronic device using an anti-chiral AFM is illustrated in Fig.
2. Here, information can be written into the AFM domain via the SOT effect
generated in the heavy metal (HM), while TMR can be utilized to transduce the
magnetic state into a resistance value. In principle, this device can be used
as a high-frequency signal generator/detector, emulator of a biological
neuron, as well as a non-volatile memory element. The specific functionality
depends on the device design, including material growth parameters, presence
of strain, and the characteristics of the input stimuli (_i.e._ , pulse width
and amplitude of the SOT and the magnetic field). To explore the functionality
of the spintronic device, it is important to consider both the order-parameter
writing and reading processes and accordingly quantify the limits and
challenges of the device for future computing applications.

Fig. 2: A conceptual three-terminal spintronic device using a negative
chirality AFM (_e.g._ , Mn3Sn) as the active element. The AFM serves as both
the ‘Free Layer’ and the ‘Pinned Layer’. SOT generated in the heavy metal is
used to manipulate the octupole moment of the free layer. $J_{c}$ and $J_{s}$
are the input charge current density and the spin current density,
respectively. $J_{s}=\theta_{\mathrm{SH}}J_{c}$, where $\theta_{\mathrm{SH}}$
is the spin Hall angle of the heavy metal. If needed, $\mathbf{H}_{a}$ is the
external magnetic field, applied to aid the deterministic switching of the
magnetic octupole, $\mathbf{m}_{\mathrm{oct}}$–this is useful when the AFM
layer is strained resulting in perpendicular magnetic anisotropy. The readout
mechanism is via the TMR such that the parallel orientation of the pinned
layer ($\mathbf{m}_{\mathrm{oct}}$) and the free layer
($\mathbf{m}_{\mathrm{oct}}$) results in a lower resistance state, while the
anti-parallel configuration of the layers results in a higher resistance
state. For a fixed read voltage, $V_{R}$, and load resistance, $R_{L}$, the
load voltage, $V_{L}$, has two different values capturing the low and high
resistance states of the junction. Previous works have utilized Si/SiO2 50,
73, 74, 75, sapphire, 73, 76, 77 and MgO 78, 79, 80 substrates.

In this section, we discuss the control of octupole moment in negative
chirality AFMs via the SOT effect. Progress in SOT manipulation of bulk AFMs
with six-fold symmetric energy landscape and thin-film strained AFMs with two-
fold symmetric energy landscape is presented. In the case of Mn3Sn, a
prototypical negative chirality AFM, the atomic spins are slightly canted
toward the in-plane easy axes, resulting in a small net magnetization, which
is six-fold degenerate in the Kagome plane. 81 In the presence of an in-plane
uniaxial strain, the two-fold degeneracy dominates, while the net
magnetization, compared to the bulk system, is modified. 82

###  3.1 Current-induced dynamics in six-fold symmetric Mn3Sn

Recent experimental works have extensively investigated the SOT-driven
dynamics in polycrystalline films of Mn3Sn, with thicknesses ranging from
$8.3\leavevmode\nobreak\ \mathrm{nm}$ to $100\leavevmode\nobreak\
\mathrm{nm}$, both with and without external magnetic fields. 50, 83, 84, 78,
73, 76, 74, 77 These films, owing to their polycrystalline nature, are
composed of grains with different orientations of the Kagome plane, and
therefore, exhibit different dynamics under the effect of SOT with a fixed
spin polarization. In such polycrystalline films, each grain exhibits a six-
fold symmetric energy landscape, similar to the case of bulk Mn3Sn. The AHE
signal measured in the Hall bar setup is, therefore, an average effect of the
final states of all the different grains.

To explain the observed Hall measurements, one or all of the three different
device configurations shown in Fig. 3 have been explored. In configuration I,
the Kagome lattice is perpendicular to the electric current but parallel to
spin polarization direction, $\mathbf{p}$. That is, $x\parallel[0001]$
($\mathbf{p}\parallel[01\bar{1}0]$). In configuration II, the Kagome lattice
is parallel to the electric current but perpendicular to $\mathbf{p}$, which
corresponds to $x\parallel[01\bar{1}0]$ ($\mathbf{p}\parallel[0001]$).
Finally, in configuration III, both the electric current and $\mathbf{p}$ are
parallel to the Kagome lattice but perpendicular to each other, or
$x\parallel[2\bar{1}\bar{1}0]$ ($\mathbf{p}\parallel[01\bar{1}0]$). In all the
three cases, an in-plane symmetry-breaking magnetic field, $H_{x}$, is applied
parallel to the current. Among the three configurations, II maximizes the
initial SOT efficiency, resulting in relatively low switching power
consumption. 78, 79

Fig. 3: Three different configuration of the Kagome lattice with respect to
the input charge current, $I_{\mathrm{write}}$, and spin polarization,
$\mathbf{p}$. Magnetic field, $H_{x}$, is applied in all the three
configurations. Reproduced with permission from Ref. 50. Fig. 4: (a) Optical
micrograph of the fabricated Mn3Sn/HM bilayer Hall bar devices. Write and read
currents are applied along the x direction under magnetic field $H_{x}$,
$H_{y}$ or $H_{z}$ along the x, y or z direction, respectively. (b) Hall
voltage, $V_{H}$, versus write current, $I_{\mathrm{write}}$, for the Mn3Sn/Pt
($7.2\leavevmode\nobreak\ \mathrm{nm}$), Mn3Sn/Cu ($5\leavevmode\nobreak\
\mathrm{nm}$) and Mn3Sn/W ($7.2\leavevmode\nobreak\ \mathrm{nm}$) devices at
room temperature. The Hall voltage is normalized by the zero-field Hall
voltage, $\Delta V^{\mathrm{field}}_{H}$, obtained from the magnetic field
dependence of each sample. (c) Dependence of $V_{H}$ on $I_{\mathrm{write}}$
($V_{H}$–$I_{\mathrm{write}}$ loops) for the Mn3Sn/Pt
($7.2\leavevmode\nobreak\ \mathrm{nm}$) device under a magnetic field of
$H_{x}=+0.1\leavevmode\nobreak\ \mathrm{T}$ (red) and
$-0.1\leavevmode\nobreak\ \mathrm{T}$ (blue) along the x direction. Reproduced
with permission from Ref. 50.

In their seminal experimental work, Tsai _et al._ demonstrated magnetic field-
assisted SOT-driven 60∘ deterministic switching between two adjacent stable
states in the Kagome plane of Mn3Sn. 50 To manipulate the order dynamics in
Mn3Sn., charge current pulses of duration $100\leavevmode\nobreak\
\mathrm{ms}$ and amplitude $(10^{6}-10^{7})\leavevmode\nobreak\
\mathrm{A/cm^{2}}$ were applied to a bilayer, which comprises a HM (Pt or W)
layer and polycrystalline Mn3Sn film of thickness $40\leavevmode\nobreak\
\mathrm{nm}$. An optical micrograph of the bilayer device setup to manipulate
and detect the magnetic order in Mn3Sn is shown in Fig. 4(a). In this setup,
an in-plane external magnetic field, $H_{x}$, and a charge current,
$I_{\mathrm{write}}$, are both applied along the x-direction while the spin
polarization, $\mathbf{p}$, corresponding to the spin current generated in the
HM, is along the y-direction. Experiments revealed that both the magnitude and
the switching direction of the Hall voltage, $V_{H}$, at fixed $H_{x}$, were
dependent on the HM layer. This is due to unique positive and negative spin-
Hall angles in Pt and W, respectively, leading to opposite switching
directions in the two cases, as shown in Fig. 4(b). Consequently, the final
steady-state of the octupole moment in Mn3Sn is different for Pt and W. The
maximum change in $V_{H}$ was measured to be about 30% (25%) of $\Delta
V_{H}^{\mathrm{field}}$ ($\Delta V_{H}^{\mathrm{field}}=V_{H}(+H_{z}\to
0)-V_{H}(-H_{z}\to 0)$) for Pt (W). This was attributed to the polycrystalline
nature of the film and the SOT-driven partial 60∘ switching. For a given HM,
the polarity of $V_{H}$-$I_{\mathrm{write}}$ was found to reverse when $H_{x}$
was reversed, as shown in Fig. 4(c). Further, the change in Hall voltage for
the different current directions, _i.e._ , $\Delta
V_{H}^{\mathrm{current}}=V_{H}(+I_{\mathrm{write}}\to
0)-V_{H}(-I_{\mathrm{write}}\to 0)$ was found to increase with $|H_{x}|$.
Finally, no change in $V_{H}$ was observed when the magnetic field was applied
along y- or z-direction.

To investigate the observed Hall voltage in the bilayer device setup, Tsai _et
al._ numerically explored the SOT-driven dynamics in Mn3Sn with each of the
three grain configurations shown in Fig. 3. The numerical simulations revealed
that deterministic switching by 60∘ was possible only in configuration I when
a symmetry-breaking magnetic field was applied perpendicular to the Kagome
plane. $H_{x}$ leads to a small out-of-(Kagome)-plane component of the
octupole moment of Mn3Sn (_i.e._ , $m_{\perp}$ along the x-direction). The
interplay of $m_{\perp}$ along with the appropriate magnitude and direction of
the injected current leads to deterministic switching. Here, the direction of
switching is determined by the vector products
($m_{\perp}\crossproduct\mathbf{p}$), or ($H\crossproduct\mathbf{p}$), which
determine the direction of torques. On the other hand, in configuration II,
numerical simulations revealed the possibility to generate up to THz
oscillations of the order parameter 85, 86. Subsequently, probabilistic
switching to one of the six-stable states was observed when the current pulse
was turned off 85. Configuration III was not studied in detail since it would
not contribute to the AHE in the experiments. This is because the octupole
moment does not have any out-of-Kagome-plane component for the in-plane
magnetic field.

In another work, Takeuchi _et al._ 78 fabricated different epitaxial
$(1\bar{1}00)[0001]$ Mn3Sn films, with thicknesses ranging from
$8.3\leavevmode\nobreak\ \mathrm{nm}$ to $30\leavevmode\nobreak\ \mathrm{nm}$,
on an $(110)[001]$ MgO substrate, and measured the change in the Hall
resistance due to both external magnetic fields as well as charge currents
within a Hall bar device structure. To investigate charge current-driven SOT
dynamics, they considered a device setup, where $\mathbf{p}\parallel[0001]$,
similar to configuration II. Using a large magnetic field of
$|H|=1\leavevmode\nobreak\ \mathrm{T}$, the Kagome lattice was initialized in
either of the two uniform states, where the uncompensated magnetic moments in
all magnetic domains were along the $[1\bar{1}00]$ direction. When injected
with current pulses of duration $100\leavevmode\nobreak\ \mathrm{ms}$ and
amplitude $I$, in the absence of an external field, the Hall resistance,
$R_{H}$, showed fluctuations for $|I|$ above a certain threshold current,
irrespective of the initial state or the direction of current (Fig. 5(a)). The
threshold current in this setup was found to be about an order lower than that
required in the device setup similar to configuration I, where
$\mathbf{p}\parallel[11\bar{2}0]$ (Fig. 5(b)). In the presence of an in-plane
external magnetic field, the latter setup demonstrated the expected
deterministic switching 50 shown in the inset of Fig. 5(b).

Fig. 5: $R_{H}$–$I$ curves of a fabricated Hall device with
$t_{\mathrm{Mn}_{3}\mathrm{Sn}}=8.3\leavevmode\nobreak\ \mathrm{nm}$ for the
configurations of (a) $I\parallel[11\bar{2}0]$ ($\mathbf{p}\parallel[0001]$)
and (b) $I\parallel[0001]$ ($\mathbf{p}\parallel[11\bar{2}0]$), respectively.
The top axis is the current density $J_{\mathrm{HM}}$ corresponding to the
electric current $I$ in the bottom axis. The colours represent the
initializing magnetic fields (the right panel) and the starting points of the
measurements, with the latter indicated by arrows. The inset in (b) shows the
bipolar switching under in-plane magnetic field $H$. Both the electric current
$I$ and magnetic field $H$ are applied along the [0001] direction of Mn3Sn.
Reproduced with permission from Ref. 78.

To study the microscopic origin of the observed fluctuations, Takeuchi _et
al._ numerically explored the magnetic field-free SOT-driven dynamics in
single-domain Mn3Sn, with six minimum energy states, in both configurations I
and II. Firstly, they found that the minimum current required to induce
dynamics in configuration II was indeed smaller than that required for
configuration I. This is because the antidamping-like (ADL) SOT in
configuration II has to overcome the weaker effective in-plane anisotropy. On
the other hand, in configuration I, the ADL SOT acts against the stronger
exchange interaction. Secondly, above the threshold current, the magnetic
octupole demonstrated oscillatory behavior with a non-zero $m_{[0001]}$, which
is the out-of-(Kagome)-plane component $m_{\perp}$. The frequency of these
oscillation was shown to be current-tunable in the 100s of MHz to GHz range.
85, 86 When the current pulse was turned off, $m_{[0001]}$ decreased to zero
and the magnetic octupole settled to the nearest minimum energy state. This
represents a probabilistic, not deterministic, switching behavior since the
particular minimum energy state depends on the state of the oscillating
magnetization, when the current pulse was turned off. For a fixed pulse
duration, therefore, the final state would depend on the pulse magnitude, $I$.
It is likely that the spin state of the single crystal Mn3Sn fabricated by
Takeuchi _et al._ had small non-uniformities between the magnetic domains,
which persisted or changed when injected with current above the threshold.
Consequently, different domains randomly settled into one of the six
degenerate states, leading to the fluctuations in the Hall resistance as the
observed ensemble average effect. This theory was supported by the observation
of reduced fluctuations in the Hall resistance under the application of an
out-of-plane magnetic field. The applied field lifted the degeneracy of the
system making one of the states more favorable to the SOT-driven switching.
The fluctuations in the Hall resistance were also reduced in a wider device
that accommodated more magnetic domains as the non-uniformities were averaged
out.

Although the work by Takeuchi _et al._ was significant in understanding
oscillatory and switching dynamics in $(1\bar{1}00)[0001]$ Mn3Sn thin films,
the oscillating signal or its frequency could not be detected experimentally.
To address these challenges, Yan _et al._ 73 used a diamond microchip with
nitrogen vacancy (NV) centers implanted on the chip’s bottom surface. The NV
centers are optically active defects in diamond and act as spin sensors, and
therefore, enable imaging the effect of charge current on the local spin
structure at microscopic length scales. The diamond chip was placed atop a
bilayer Hall device comprising HM (Pt or W) and $50\leavevmode\nobreak\
\mathrm{nm}$ thick polycrystalline Mn3Sn. When millisecond charge current
pulses of different amplitudes were applied to this device in the absence of
an external magnetic field, the Hall resistance demonstrated characteristic
fluctuations, similar to those reported by Takeuchi _et al._ , if
$I_{\mathrm{write}}$ exceeded a certain threshold current. This can be
observed from Fig. 6(a, b), where the anomalous Hall resistance normalized to
the zero-field Hall resistance, $\frac{(R_{H}-\langle
R_{H}\rangle)}{R_{\mathrm{AHE}}}$, is presented as a function of the write
current pulse number N. Enhanced fluctuations can be observed for
$44\leavevmode\nobreak\ \mathrm{mA}$ while no such fluctuations are noticed in
the case of $2\leavevmode\nobreak\ \mathrm{mA}$, indicating the existence of a
threshold current, which was found to be around $36\leavevmode\nobreak\
\mathrm{MA/cm^{2}}$. While the polycrystalline Mn3Sn could consist of grains
with different configurations, the fluctuations in Fig. 6(a, b) are expected
to be contributed by grains whose Kagome plane is perpendicular to
$\mathbf{p}$ (configuration II). This is because the threshold current to
induce such fluctuations in grains with configuration I is about an order of
magnitude higher 78 and no AHE would be contributed by grains with
configuration III. Thus, the fluctuations reported in Fig. 6(b) are expected
due to the SOT-driven chiral oscillations of the magnetic octupole moment. 78.

Fig. 6: (a, b) Measured normalized Hall resistance, $(R_{H}-\langle
R_{H}\rangle)/R_{\mathrm{AHE}}$, as a function of pulse number (N) for write
current amplitudes of (c) $2\leavevmode\nobreak\ \mathrm{mA}$ and (d)
$44\leavevmode\nobreak\ \mathrm{mA}$. (c, f) 2D images of $B_{z}$ measured
after applying the first current pulse (N = 1) with amplitudes of (e)
$2\leavevmode\nobreak\ \mathrm{mA}$ and (h) $44\leavevmode\nobreak\
\mathrm{mA}$, respectively. (d, g) SOT-induced variation of the stray field,
$\Delta B_{z}$, recorded after applying a $2\leavevmode\nobreak\ \mathrm{mA}$
current pulse 10 (f) and 20 (g) times. (e, h) SOT-induced variation of $\Delta
B_{z}$ recorded after applying a $44\leavevmode\nobreak\ \mathrm{mA}$ current
pulse 10 (i) and 20 (j) times. For (c–h) the black dashed lines mark the
boundary of the patterned Mn3Sn/Pt Hall cross, and the scale bar is
$2.5\leavevmode\nobreak\ \mathrm{\mu m}$. (i) Theoretically calculated spin
rotation frequency $f$ of Mn3Sn as a function of the electric current density,
$J$, flowing in the Pt layer. The experimentally measured $f$ for
$J=42\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$ (the red marker),
$44\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$ (the green marker), and
$48\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$ (the gray marker) agree with the
theoretical calculations. Reproduced with permission from Ref. 73.

To establish a real-space correlation between the SOT and the chiral
oscillations, Yan _et al._ performed NV wide-field magnetometry after every
write current pulse. Two-dimensional (2D) images of the spatially varying
stray field, generated due to the net magnetization exhibited by Mn3Sn, are
presented in Fig. 6(c-h). The stray field measured after the first write
current pulse (N = 1) in the case of both $2\leavevmode\nobreak\ \mathrm{mA}$
and $44\leavevmode\nobreak\ \mathrm{mA}$ demonstrate multi-domain
characteristic of the polycrystalline Mn3Sn film, as shown in Fig. 6(c, f).
Figure 6(d, e) show that the spatial distribution of the stray field after 10
and 20 current pulses remains largely unchanged for current pulses with
amplitude $2\leavevmode\nobreak\ \mathrm{mA}$. On the other hand, for current
pulses with amplitude $44\leavevmode\nobreak\ \mathrm{mA}$, the stray field
distribution show significant change after 10 and 20 current pulses, as shown
in Figure 6(g, h). In this case the stray field averaged spatially over the
entire Hall cross-region remains constant during the entire measurement
sequence. These results corroborates the role of SOT-driven chiral
oscillations, followed by probabilistic switching, in multi-domain Mn3Sn used
in configuration II. Special care was taken to remove the effects of Joule
heating from these results. Further, utilizing NV relaxometry measurements,
frequency of the current-driven chiral oscillations for write current pulses
above the threshold were also obtained, as shown by symbols in Fig. 6(i).
These extracted data point agree well with the numerical calculations based on
single-domain coupled Landau-Lifshitz Gilbert (LLG) equations, especially for
higher currents. Finally, NV wide-field magnetometry performed in the presence
of an in-plane external magnetic field along the direction of current yielded
stray field, which averaged spatially over the entire Hall cross demonstrated
periodic variation signifying deterministic switching, previously discussed in
Ref. 50.

To get further insight into the dependence of the field-free SOT-driven
dynamics pertinent to configuration II on the intrinsic energy scale of Mn3Sn
and its material parameters, our group has theoretically investigated single-
domain Mn3Sn with six stable states. 87 On the numerical front, we explored
chiral oscillation and probabilistic switching dynamics under the effect of
d.c. and pulsed currents, respectively, and found the results to be similar to
those presented in Ref. 78. Then using a minimal energy landscape model, we
obtained an analytic model of the threshold spin current to excite the
dynamics as
$J_{s}^{\mathrm{th}}=d_{a}\frac{2e}{\hslash}\frac{\quantity(3J_{E}+7\sqrt{3}D_{M})K_{U}^{3}}{9\quantity(J_{E}+\sqrt{3}D_{M})^{3}}$,
where $e$ and $\hslash$ are the charge of an electron and reduced Plank
constant while $d_{a}$, $J_{E}$, $D_{M}$, and $K_{U}$ are the thickness,
exchange energy constant, DMI constant, and uniaxial energy constant of Mn3Sn,
respectively. For d.c. input spin current density $J_{s}\geq
J_{s}^{\mathrm{th}}$, the frequency of oscillation showed a non-linear
(linear) behavior with current for small (large) $J_{s}$. This behavior was
accurately captured by the equation

$\displaystyle\begin{split}f=\frac{\hslash}{2e}\frac{\gamma
J_{s}}{2\pi\alpha_{G}M_{s}d_{a}}\sqrt{1-\quantity(\frac{J_{s}^{\mathrm{th}}}{J_{s}})^{2}},\end{split}$
(1)

where $\gamma$, $\alpha_{G}$, and $M_{s}$ are the gyromagnetic ratio, Gilbert
damping constant, and saturation magnetization, respectively. On the other
hand, the final steady-state, owing to the probabilistic switching behavior,
for current pulses of duration $t_{\mathrm{pw}}$ and amplitude $J_{s}$ was
shown to be dependent on the net input spin charge density,
$J_{s}t_{\mathrm{pw}}$, as

$\displaystyle\begin{split}J_{s}t_{\mathrm{pw}}&=\alpha_{G}\frac{2e}{\hslash}\frac{M_{s}d_{a}}{\gamma}\frac{\frac{(2n-1)\pi}{2}+\tan^{-1}{\quantity(\frac{J_{s}^{\mathrm{th}}/J_{s}}{\sqrt{1-\quantity(J_{s}^{\mathrm{th}}/J_{s})^{2}}})}}{3\sqrt{1-\quantity(J_{s}^{\mathrm{th}}/J_{s})^{2}}}.\end{split}$
(2)

For $J_{s}\gg J_{s}^{\mathrm{th}}$ and small pulse duration, the final state
depends strongly on $J_{s}$ whereas for $J_{s}\gtrsim J_{s}^{\mathrm{th}}$ and
large pulse duration, the final state is controlled by $t_{\mathrm{pw}}$.
Figure 7 shows an excellent agreement between the analytic models of Eqs. 1
and 2 and the numerical data obtained from the coupled solution of LLG
equation. The effects of Joule heating were not considered in this modeling
approach.

Fig. 7: (a) Oscillation frequency as a function of the applied spin current,
$J_{s}$, for different thickness of the Mn3Sn layer. Numerical result from the
solution of coupled LLG equations (symbols) agree very well with the
analytical results obtained from Eq. (1) (lines). (b) The final steady-state
of the magnetic octupole as a function of the pulse duration,
$t_{\mathrm{pw}}$, and the total injected spin charge density,
$J_{s}t_{\mathrm{pw}}$. The overlaid dashed white lines represent the analytic
bounds of Eq. (2). Here, the thickness of the AFM film is
$d_{a}=4\leavevmode\nobreak\ \mathrm{nm}$. Other parameters are
$J_{E}=2.4\times 10^{8}\leavevmode\nobreak\ \mathrm{J/m^{3}}$, $D_{M}=2\times
10^{7}\leavevmode\nobreak\ \mathrm{J/m^{3}}$, $K_{e}=3\times
10^{6}\leavevmode\nobreak\ \mathrm{J/m^{3}}$, and $M_{s}=1.3\times
10^{6}\leavevmode\nobreak\ \mathrm{A/m}$. Adapted from Ref. 87.

Although the aforementioned investigations attributed the current-induced
changes in single crystal as well as polycrystalline Mn3Sn to ADL SOT while
neglecting any significant role of Joule heating, Takeuchi _et al._ evaluated
the temperature rise in the case of devices with configuration I to be quite
large. In addition, their numerical simulations suggested the threshold
current in the case of configuration I to be at least one order of magnitude
higher than that observed experimentally. Furthermore, the spin diffusion
length in Mn3Sn is only about $1\leavevmode\nobreak\ \mathrm{nm}$ 88 but Mn3Sn
films used in the experiments were thicker than $8\leavevmode\nobreak\
\mathrm{nm}$. These peculiarities suggested the contribution of a different
switching mechanism, in addition to SOT, in these experiments. To this end,
Pal _et al._ 76 and Krishnaswamy _et al._ 74 independently investigated the
current-driven dynamics in bilayer of HM and polycrystalline Mn3Sn films of
thicknesses $30\leavevmode\nobreak\ \mathrm{nm}$ to $100\leavevmode\nobreak\
\mathrm{nm}$, in the presence of an in-plane external magnetic field collinear
with current pulse. Both these works showed the demagnetization of Mn3Sn film,
owing to its temperature rise above the Néel temperature, as an important step
in the switching process.

In particular, Krishnaswamy _et al._ showed that the current-induced switching
was a function of the pulse fall time, $\tau$. This can be observed from Fig.
8(a, b), where bistable switching, typical of configuration I, is possible
only for longer fall times, such as $\tau=420\leavevmode\nobreak\
\mathrm{ns}$. On the other hand, shorter fall times
($\tau<400\leavevmode\nobreak\ \mathrm{ns}$) lead to tristable switching,
_i.e._ , a third state with small or zero AHE signal is detected in the steady
state above the critical voltage, whereas below the critical voltage, two
stable states, typical of bistable switching, are obtained. When a current
pulse of duration $10\leavevmode\nobreak\ \mathrm{\mu s}$ is applied, the
temperature rises above the Néel temperature, which was found to be
$390\leavevmode\nobreak\ \mathrm{K}$ in $50\leavevmode\nobreak\ \mathrm{nm}$
thick polycrystalline film, in 10s of $\mathrm{ns}$. When the current is
reduced, the Mn3Sn film cools down below its Néel temperature due to the
diffusion of heat into the substrate. As depicted in Fig. 8(c), for longer
fall times, the current is above a critical value, $j_{c}$, when Mn3Sn film
cools down below the Néel temperature, resulting in bistable deterministic
switching On the other hand, for $\tau=35\leavevmode\nobreak\ \mathrm{ns}$,
the current decreases to values below $j_{c}$ while the temperature of Mn3Sn
film is still above the the Néel temperature, leading to random freeze-out of
Mn3Sn grains, resulting in zero AHE signal. The magnitude and direction of the
detected AHE signal was shown to be dependent on the external magnetic field
as well as the HM, similar to Fig. 4(b, c), thereby making the switching
process a function of heat and SOT. The polycrystalline films of Mn3Sn would
likely consist of grains representing configuration II; however, no coherent
chiral oscillations were observed in experiments, perhaps due to strong
damping or heat-induced demagnetization.

Fig. 8:  (a) Switching loops of Mn3Sn/Pt as a function of pulse voltage with
rise and fall time, $\tau$, increasing from left to right. Short $\tau$ such
as $35\leavevmode\nobreak\ \mathrm{ns}$ leads to tristable switching whereas
long $\tau$ of $420\leavevmode\nobreak\ \mathrm{ns}$ leads to bistable
switching behavior. (b) Corresponding pulse shape. (c) Schematic voltage pulse
with temperature profile indicated by the red shading. Reproduced with
permission from Ref. 74.

On the other hand, Pal _et al._ proposed seeded SOT to explain the observed
field-assisted bistable switching in their $30\leavevmode\nobreak\
\mathrm{nm}$ to $100\leavevmode\nobreak\ \mathrm{nm}$ thick Mn3Sn films. In
this mechanism, the current-induced Joule heating randomizes the magnetic
domains in Mn3Sn as its temperature rises above the Néel temperature, in the
first step. Then the SOT deterministically switches a thin region of Mn3Sn (of
the order of the spin diffusion length) near the HM/AFM interface, if the
current is above a critical value while the thin region cools down. The
interface region then acts as the seed for the deterministic switching of the
rest of the AFM as the film cools down. Switching efficiency showed strong
dependence on the fall time—longer fall times facilitated deterministic
switching—similar to that described in Ref. 74. At a fixed pulse duration, the
critical current density, $J_{c}$, was found to be almost independent of the
in-plane magnetic field, $H_{x}$, as shown in Fig. 9(a), presenting a strong
evidence towards the role of heating Mn3Sn above its Néel temperature for
deterministic switching. On the contrary, the Hall resistance,
$R^{{}^{\prime}}_{xy}$, exhibited a nonmonotonic dependence on $H_{x}$, as
shown in Fig. 9(b). Numerical simulations revealed that the out-
of-(Kagome)-plane component of the net magnetization, $m_{\perp}$, increases
with $H_{x}$. Pal _et al._ attributed the initial increase in
$R^{{}^{\prime}}_{xy}$, for $H_{x}\leq 0.1\leavevmode\nobreak\ \mathrm{T}$, to
the increase in $m_{\perp}$, whereas they explained the subsequent decrease in
$R^{{}^{\prime}}_{xy}$ for higher $H_{x}$ as a consequence of decreasing
energy barrier between the six different minimum energy states. Other
evidences for the role of Joule heating in the observed deterministic bistable
switching included a decrease in $J_{c}$ with an increase in pulse duration
(device temperature) at fixed device temperature (pulse duration). Finally,
unlike pure SOT, where $J_{c}$ would increase with thickness of the magnetic
layer, here $J_{c}$ was found to decrease with the thickness of Mn3Sn layer
while the power density stayed constant.

Fig. 9: (a) Dependence of $J_{c}$ on $H_{x}$ for a pulse duration of
$100\leavevmode\nobreak\ \mathrm{ms}$. Black ($+H_{x}$, $-J_{x}$), red
($+H_{x}$, $+J_{x}$), purple ($-H_{x}$, $-J_{x}$), and blue ($-H_{x}$,
$+J_{x}$) show negative and positive current sweep for positive and negative
$H_{x}$, respectively. (b) Variation of $R^{{}^{\prime}}_{xy}$ on $H_{x}$
during current-induced switching. Reproduced with permission from Ref. 76.

Very recently, Xu _et. al_ 77 utilized current pulses of duration
$2\leavevmode\nobreak\ \mathrm{ms}$ but different amplitudes and investigated
the dynamics in single crystals of Mn3Sn, in the presence of an in-plane
collinear magnetic field. For Mn3Sn in configuration I, they demonstrated a
weak dependence of the critical current on $H_{x}$ while it was found to
decrease with an increase in the film thickness, similar to 76. Thus alluding
to the role of Joule heating in the bistable deterministic switching dynamics
in their devices. They also showed the nonmonotonic behavior of the Hall
resistance with $H_{x}$, suggesting an increase in $m_{\perp}$ with $H_{x}$.
Other expected outcomes such as the dependence of the Hall resistance on the
magnitude and direction of $H_{x}$ were also presented. Unexpectedly, their
Mn3Sn device in configuration II showcased deterministic switching, considered
to be forbidden by previous works. 50, 78, 73, 74 Although they attribute this
behavior to imperfect growth of their Mn3Sn crystals, further experimental and
numerical investigations are required to evaluate the possibility of chiral
oscillation and deterministic switching in configuration II.

####  3.1.1 SOT-induced dynamics in two-fold symmetric Mn3Sn

Recently, Higo _et. al_ demonstrated that a $30\leavevmode\nobreak\
\mathrm{nm}$ thick film of Mn3Sn, used in a bilayer heterostructure comprising
HM and Mn3Sn, grown epitaxially on $(110)[001]$ MgO substrate, exhibited
perpendicular magnetic anisotropy (PMA). 79 As shown in Fig. 10(a), the Mn3Sn
layer was found to be oriented as MgO $(110)[001]\parallel$ Mn3Sn
$(01\bar{1}0)[0001]$ with perpendicular magnetic octupole moment along the
$[01\bar{1}0]$ direction. The lattice mismatch between the Mn3Sn layer and the
MgO substrate, for a thin HM layer, led to an in-plane tensile strain of about
$0.2\%$ in the Kagome plane along the $[2\bar{1}\bar{1}0]$ direction.
Consequently, the six degenerate minimum energy states, with the octupole
pointing along the six-equivalent $\langle 2\bar{1}\bar{1}0\rangle$ direction,
as shown in Fig. 10(b), reduced to two degenerate minimum energy states shown
in Fig. 10(c).

When write current pulses of duration $100\leavevmode\nobreak\ \mathrm{ms}$
and amplitude $I_{\mathrm{write}}$ were applied to this bilayer structure, in
the presence of a symmetry-breaking magnetic field, $H_{x}$, parallel to the
current, the Hall voltage, $V_{H}$, displayed bidirectional deterministic
switching behavior above a critical current, as shown in Fig. 10(d) (left
axis). The corresponding critical current density was found to be
$J_{c}^{w}=14\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$. Strikingly, the entire
volume of the Mn3Sn film was found to switch due to the applied current pulse,
as shown by the ratio on the right axis of Fig. 10(d), where $\Delta
V_{H}^{\mathrm{field}}$ is the Hall voltage change due to a magnetic field
applied in the out-of-plane direction, $H_{z}$. This ratio was limited to only
$(25-30)\%$ in all previous works. 50, 78, 73 No switching was observed for
magnetic fields perpendicular to the Kagome plane, $H_{y}$, as it did not
break the two-fold symmetry. This, however, changed for devices, where the
Kagome plane was rotated from the current direction ($x$) by an angle. This is
because $H_{y}$ was not perpendicular to the rotated Kagome plane anymore,
instead it broke the symmetry, and enabled deterministic switching. Finally,
in the absence of any external field, $V_{H}$ showed fluctuations, similar to
those described in Ref. 78, likely representing probabilistic switching
dynamics.

Fig. 10: (a) Schematic illustrating the bidirectional switching of the
perpendicular magnetic octupole (orange arrows) of Mn3Sn by the SOT. The three
Mn moments on the kagome easy plane are shown by arrows with different
colours. Red and yellow spheres represent Mn atoms at z = 0 and 1/2,
respectively. An electrical current $I$ (green arrow) flowing in the W layer
generates a spin current (green spheres) whose polarization vector (red arrow)
is perpendicular to the kagome plane and induces the SOT on the Mn3Sn layer.
$\varepsilon$ (pink arrows) and H (magenta arrows) schematically show the
epitaxial, uniaxial tensile strain and the magnetic field, respectively. (b)
Anti-chiral spin structure on the kagome bilayers with no strain
($\varepsilon=0$), which has six degenerate states with the octupole pointing
along the six-equivalent $\langle 2\bar{1}\bar{1}0\rangle$ direction. (c)
Anti-chiral PMA binary states on the kagome bilayers stabilized by the
application of the uniaxial tensile strain ($\varepsilon>0$). (d) The Hall
voltage, $V_{H}$ (left) and volume fraction of the current controllable
$V_{H}/|\Delta V_{H}^{\mathrm{field}}|$ (right) versus write current
$I_{\mathrm{write}}$ of the W/Mn3Sn heterostructure at room temperature. The
bias magnetic field of $0.1\leavevmode\nobreak\ \mathrm{T}$ is applied along
the x direction. Reproduced with permission from Ref. 79.

It is well known that in the case of FMs and collinear AFMs the experimentally
measurable magnetic order, _i.e._ , magnetization vector and Néel vector,
respectively, rotate in the same direction as the constituting atomic spins.
89, 90 Although Higo _et. al_ demonstrated a field-assisted SOT-induced
deterministic switching dynamics in PMA Mn3Sn, 79 similar to that in FMs,
there was no discussion regarding the switching direction of the
experimentally measured magnetic octupole. To address this important issue,
Yoon and Zhang _et. al_ considered a heterostructure, comprising HM layer (Pt)
and $8.8\leavevmode\nobreak\ \mathrm{nm}$ thick PMA Mn3Sn film, and applied a
symmetry-breaking in-plane magnetic field along a current pulse of duration
$100\leavevmode\nobreak\ \mathrm{ms}$ and amplitude $I$. 80 The measured Hall
resistance, $R_{H}$, in configuration II (Fig. 11(a)), showed deterministic
switching (Fig. 11(b)), for currents above a certain threshold current
($J_{\mathrm{co}}\approx 15\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$), similar
to Ref. 79. However, only $30\%$ of the magnetic domains were found to switch
for the SOT-driven dynamics, which is different from the $100\%$ switching
shown on the right axis of Fig. 10(d). They attribute the low ratio of
switched magnetic domains to possible non-uniform current distribution in the
Hall cross device or tilting of the octupole moment from the perpendicular
orientation. They observed almost no change in $R_{H}$ for configuration I,
even for currents twice as large as that used in configuration II. This
observation is similar to that reported in Refs. 78, 73, where the threshold
currents for devices in configuration I were found to be about an order higher
than those for configuration II (see also Sec. 3.1). Finally, the temperature
rise for $15\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$ was found to be only
about $4\leavevmode\nobreak\ \mathrm{K}$, thus eliminating any contribution of
Joule heating from the switching dynamics, unlike Refs. 76, 74, where the
temperature increased by $140\leavevmode\nobreak\ \mathrm{K}$ for a
$1\leavevmode\nobreak\ \mathrm{ms}$ long current pulse of magnitude
$17\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$, resulting in deterministic
switching (Fig. 9(b)). The small temperature rise could be due to a thinner
Mn3Sn layer ($8.8\leavevmode\nobreak\ \mathrm{nm}$) as compared to the
$30\leavevmode\nobreak\ \mathrm{nm}$ thick Mn3Sn layer used by Pal _et. al_ ,
where the critical current was found to decrease with the thickness. Another
reason for the wide variation in the temperature rise could be the different
substrates in the two device setups—sapphire in Ref. 76 and (110)MgO in Refs.
79, 80.

To compare the aforementioned switching dynamics in configuration II to that
in FMs, Yoon and Zhang _et. al_ measured the field-assisted SOT-driven changes
in $R_{H}$ of a W/CoFeB bilayer. As shown in Fig. 11(c), the polarity of the
switched magnetization in the CoFeB FM was found to be opposite to that of the
octupole moment in Mn3Sn (Fig. 11(b)), for the same polarization of the
injected spin current. This surprising observation, pertinent to the opposite
rotation of the octupole moment as compared to the direction of rotation of a
ferromagnetic moment, was termed as the “handedness anomaly”. Through careful
examination of the measurements, they concluded that the effect of the field-
like torque on the SOT-induced dynamics in the Mn3Sn film was negligible.
Also, owing to the epitaxial nature of the Mn3Sn film, the likelihood of self-
generated spin-polarized current 91 in Mn3Sn was negligible. Consequently, the
observed SOT-driven switching in PMA Mn3Sn could be completely attributed to
the ADL torque. This indicates that the direction of the ADL SOT on the
magnetic octupole is opposite to its direction on the magnetization of a FM,
and therefore, opposite to the direction of the ADL SOT on the constituting
sublattice vectors of Mn3Sn.

Fig. 11: (a) Schematic of the transport measurement reported in Ref. 80. The
in-plane spins (yellow spheres and green arrows) are injected from Pt to
Mn3Sn. The sublattice moments and the octupole moment are defined as
$\mathbf{m}_{(A,B,C)}$ and $\mathbf{m}_{\mathrm{oct}}$ with angles of
$\varphi_{(A,B,C)}$ and $\varphi_{\mathrm{oct}}$, respectively. An optical
micrograph of the fabricated Hall bar with a channel width Wch and a length
Lch of $15-30\leavevmode\nobreak\ \mathrm{\mu m}$ and $90\leavevmode\nobreak\
\mathrm{\mu m}$, respectively, is shown on the top of the stacks with the
definition of the current and voltage directions. (b) $R_{H}-I$ curves for
different magnetic field, applied parallel to the injected current, $I$, in
the x-direction. (c) $R_{H}-I$ curve measured in W($5\leavevmode\nobreak\
\mathrm{nm}$)/CoFeB($1.45\leavevmode\nobreak\ \mathrm{nm}$)/MgO stacks
deposited on a Si substrate, where the bottom W layer injects spins with the
same polarization onto the CoFeB layer as those from the top Pt layer onto the
Mn3Sn layer. The schematics in (b) and (c) illustrate the switched magnetic
states, represented by the octupole moment and the magnetization vector,
respectively, in Mn3Sn and CoFeB, respectively. Reproduced with permission
from Ref. 80.

To get a deeper insight into the ADL SOT-induced dynamics in PMA Mn3Sn in
configuration II, Higo _et. al_ numerically investigated solutions of single-
domain coupled LLG equations for a range of input currents and magnetic
fields. They showed that, in the steady state, the octupole moment either
stayed near the initial minimum energy state or ground state, switched to the
other ground state, or exhibited chiral oscillations. The final steady-states
were found to be a function of the initial ground state, the direction and
magnitude of the input stimuli. Yoon and Zhang _et. al_ , on the other hand,
developed an effective energy landscape model of the magnetic octupole moment,
wherein the in-plane epitaxial tensile strain manifested as perturbations in
exchange and uniaxial anisotropy energy interactions. This results in clear
two-fold and four-fold effective anisotropies, in addition to the intrinsic
six-fold anisotropy. They utilized this model to accurately fit and describe
the quasi-static second-harmonic Hall measurements of PMA Mn3Sn, used in
configuration II, as a function of the direction of the symmetry-breaking
magnetic field and the input current, keeping their respective magnitudes
fixed. The dependence of the different steady-states on the intrinsic energy
landscape and the material parameters of PMA Mn3Sn, however, was not discussed
in either of these works. The time scales associated with switching and
oscillation dynamics, and their relation to the material properties as well as
the external stimuli, were also not addressed.

A thorough analysis of the different regimes of operation in strained PMA
Mn3Sn films, and their dependence on the material parameters and external
stimuli, have been presented in our recent work. 92. Our numerical simulations
revealed that in the absence of a magnetic field the magnetic octupole either
stays in the initial well or oscillates around the spin polarization,
depending on the magnitude of applied current. On the other hand, for a non-
zero magnetic field, the magnetic octupole can be deterministically switched
to the other state, if the injected current satisfies
$J_{c}^{\mathrm{th1}}<J_{c}<J_{c}^{\mathrm{th2}}$, where
$J_{c}^{\mathrm{th1}}$ and $J_{c}^{\mathrm{th2}}$ represent the lower and
upper current bounds, respectively. For currents above $J_{c}^{\mathrm{th2}}$,
the octupole exhibits chiral oscillations, whose frequency is current tunable.
No switching was observed for currents below $J_{c}^{\mathrm{th1}}$, instead
stationary states in the initial ground states are obtained. Our results
showed some differences from those reported by Higo _et. al_. To investigate
these differences, we utilized an effective energy landscape model of the
magnetic octupole moment, which clearly depicts the dependence of the energy
barrier separating the two minimum energy states on the in-plane tensile
strain. 51, 93, 80 We found that $J_{c}^{\mathrm{th1}}$ and
$J_{c}^{\mathrm{th2}}$ were the minimum currents required to overcome the
maximum effective in-plane anisotropy in the two energy wells. Since the
external magnetic field breaks the symmetry, $J_{c}^{\mathrm{th1}}$
($J_{c}^{\mathrm{th2}}$) decreases (increases) with the magnitude of the
field. We then derived analytic expressions of the threshold currents, which
were found to be independent of damping constant but linearly proportional to
the magnitude of the field, the final stationary steady states as well as the
switching time as functions of the input stimuli and the material parameters.
Our models show excellent agreement against numerical data, and therefore,
provide important insight into the dynamics of the material, which would be
useful to experimentalists and designers. Finally, we varied the Gilbert
damping constant and found that for damping constants below a certain value
($\alpha\lesssim 2\times 10^{-3}$ for the set of material parameters
considered in our work), the dynamics is similar to that reported by Higo _et.
al_. For these low values of damping constants, analytic expressions deviate
from the numerical results. To resolve these differences, careful analysis
involving the exchange interaction due to the out-of-(Kagome)-plane component
of magnetic octupole is required. 80

###  3.2 Joule heating in the presence of SOT

As discussed in Refs. 76, 74, 77, Joule heating likely plays an important role
in the current-induced switching dynamics in the bilayer devices, comprising
HM and Mn3Sn, especially in configuration I. Temperature rise above the Nèel
temperature could also be the reason behind the failure in detection of chiral
oscillations, in both six-fold and two-fold symmetric Mn3Sn, in configuration
II. 74 In general, Joule heating induced temperature rise as well as the rate
of change of the temperature depend on the properties of the conducting and
substrate layers in addition to the input current pulse. In our recent work,
87 we considered a one-dimensional (1D) heat flow model 94 to investigate the
current-induced temperature change in the device setup shown in Fig. 2.

Although the current pulse is applied to the HM, a part of the current leaks
into the metallic Mn3Sn layer, leading to heat generation in both the layers.
To evaluate the current distribution between the two layers a parallel
resistor model was employed. Per this model, the total generated heat depended
on the individual resistivity and dimension of both layers. Both the HM and
Mn3Sn were assumed to be at the same temperature and any spatial variation in
the temperature was neglected. The generated heat was assumed to flow through
the substrate towards the bottom electrode. Consequently, the temperature at
the junction between the bilayer and the substrate, $T_{j}$, was found to be
strongly dependent on the properties of the substrate such as its thermal
conductivity ($\kappa_{\mathrm{sub}}$), thickness ($d_{\mathrm{sub}}$), mass
density ($\rho_{\mathrm{sub}}$), and specific heat capacity
($C_{\mathrm{sub}}$). Therefore, the rise in the junction temperature, $\Delta
T_{j}(t)$, for current density pulse of magnitude $J_{c}$ and duration
$t_{\mathrm{pw}}$, was calculated as 94

$\displaystyle\begin{split}&\Delta T_{j}(t)=\Delta
T_{M}\left(\quantity(1-\frac{8}{\pi^{2}}\sum_{k=\mathrm{odd}}\frac{\exp(-k^{2}t/\tau_{\mathrm{th}})}{k^{2}})\right.\\\
&\left.-\theta(t-t_{\mathrm{pw}})\quantity(1-\frac{8}{\pi^{2}}\sum_{k=\mathrm{odd}}\frac{\exp(-k^{2}(t-t_{\mathrm{pw}})/\tau_{\mathrm{th}})}{k^{2}})\right),\end{split}$
(3)

where
$\tau_{\mathrm{th}}=\frac{4d_{\mathrm{sub}}^{2}\rho_{\mathrm{sub}}C_{\mathrm{sub}}}{\pi^{2}\kappa_{\mathrm{sub}}}$
is the thermal rate constant, $\theta(\cdot)$ is the Heaviside step function,
and

$\Delta
T_{M}=\frac{d_{\mathrm{sub}}J_{c}^{2}}{\kappa_{\mathrm{sub}}}\frac{\quantity(\rho_{\mathrm{Mn_{3}Sn}}d_{\mathrm{Mn_{3}Sn}}+\quantity(\frac{\rho_{\mathrm{Mn_{3}Sn}}}{\rho_{\mathrm{HM}}}\frac{d_{\mathrm{HM}}}{d_{\mathrm{Mn_{3}Sn}}})^{2}\rho_{\mathrm{HM}}d_{\mathrm{HM}})}{\quantity(1+\frac{\rho_{\mathrm{Mn_{3}Sn}}}{\rho_{\mathrm{HM}}}\frac{d_{\mathrm{HM}}}{d_{\mathrm{Mn_{3}Sn}}})^{2}}$
(4)

is the maximum possible rise in $T_{j}$ due to $J_{c}$. Here,
$\rho_{\mathrm{Mn_{3}Sn}}$ ($\rho_{\mathrm{HM}}$) and $d_{\mathrm{Mn_{3}Sn}}$
($d_{\mathrm{HM}}$) are the electrical resistivity and thickness of Mn3Sn (HM)
layer.

Figure 12 shows $\Delta T_{M}$ as a function of $d_{\mathrm{sub}}$ and
$J_{c}$. Here, the thickness of Mn3Sn (W) film is
$d_{\mathrm{Mn_{3}Sn}}=4\leavevmode\nobreak\ \mathrm{nm}$
($d_{\mathrm{W}}=7\leavevmode\nobreak\ \mathrm{nm}$) 79 and
$\rho_{\mathrm{Mn_{3}Sn}}=330\leavevmode\nobreak\
\mathrm{\mu\Omega\leavevmode\nobreak\ cm}$
($\rho_{\mathrm{W}}=43.8\leavevmode\nobreak\
\mathrm{\mu\Omega\leavevmode\nobreak\ cm}$). 79 The substrate is assumed to be
MgO and its relevant material parameters 95 are
$\kappa_{\mathrm{sub}}=40\leavevmode\nobreak\ \mathrm{W/(m\leavevmode\nobreak\
K)}$, $\rho_{\mathrm{sub}}=3580\leavevmode\nobreak\ \mathrm{kg/m^{3}}$, and
$C_{\mathrm{sub}}=930\leavevmode\nobreak\ \mathrm{J/(kg\leavevmode\nobreak\
K)}$. In Fig. 12, an increase in $J_{c}$, increases the generated heat in the
metallic layers, which in turn, increases $\Delta T_{M}$. On the other hand,
an increase in $d_{\mathrm{sub}}$, increases the thermal resistance in the
direction of heat flow, thereby slowing down the heat removal from the system
and increasing $\Delta T_{M}$. The dashed black curve, corresponding to
$\Delta T_{M}=120\leavevmode\nobreak\ \mathrm{K}$, represents the maximum
temperature rise before Mn3Sn gets demagnetized. Consequently, no chiral
oscillations would be observed in the region above the dashed curve. Here, the
room temperature and the Nèel temperature were assumed to be
$300\leavevmode\nobreak\ \mathrm{K}$ and $420\leavevmode\nobreak\ \mathrm{K}$,
respectively. Although not shown here, for a fixed $J_{c}$ and
$d_{\mathrm{sub}}$, $\Delta T_{M}$ would increase for thicker
$d_{\mathrm{HM}}$ or $d_{\mathrm{Mn_{3}Sn}}$ since the generated heat would
increase with the respective thickness. This explains the observed reduction
in critical current density with an increase in AFM thickness. 76 The
generated heat is expected to be lower if Pt was used as the HM since it has
lower resistivity. 50 Finally, for the same values of $J_{c}$ and
$d_{\mathrm{sub}}$, $\Delta T_{M}$ would be higher in the case of both
sapphire and SiO2 substrates. This is due to their lower
$\kappa_{\mathrm{sub}}$, 96 and therefore, higher thermal resistance ($\propto
1/\kappa_{\mathrm{sub}}$).

Fig. 12: Maximum possible rise in the junction temperature, $\Delta T_{M}$,
for different thickness of substrate MgO, $d_{\mathrm{sub}}$, and input charge
current density, $J_{c}.$ $\Delta T_{\mathrm{M}}$ increases with both
$d_{\mathrm{sub}}$ and $J_{c}$, as expected. The dashed black line corresponds
to $\Delta T_{M}=120\leavevmode\nobreak\ \mathrm{K}$, which is the maximum
temperature rise before the antiferromagnetic order in Mn3Sn changes. Here,
the thickness of the AFM film is $d_{a}=4\leavevmode\nobreak\ \mathrm{nm}$.
Adapted from Ref. 87.

Other factors that affect $\Delta T_{j}(t)$ are $\tau_{\mathrm{th}}$ and
$t_{\mathrm{pw}}$, which decide the rate of change of the temperature and the
total generated heat, respectively. For fixed $J_{c}$ and $d_{\mathrm{sub}}$,
$\Delta T_{j}(t)$ reaches to about 95% of the respective $\Delta T_{M}$ in
$t\approx 3\tau_{\mathrm{th}}$; therefore, applying
$t_{\mathrm{pw}}\ll\tau_{\mathrm{th}}$ could ensure lower $\Delta T_{j}(t)$.
In the case of $d_{\mathrm{sub}}=3\leavevmode\nobreak\ \mathrm{\mu m}$,
$\tau_{\mathrm{th}}\approx 304\leavevmode\nobreak\ \mathrm{ns}$ while
$J_{c}=76.8\leavevmode\nobreak\ \mathrm{MA/cm^{2}}$ leads to $\Delta
T_{M}=120\leavevmode\nobreak\ \mathrm{K}$ for the MgO substrate. As shown in
Fig. 13, $\Delta T_{j}(t)\approx 16\leavevmode\nobreak\ \mathrm{K}$ for
$t_{\mathrm{pw}}=10\leavevmode\nobreak\ \mathrm{ns}$ ($\ll\tau_{\mathrm{th}}$)
but increases to about $70\leavevmode\nobreak\ \mathrm{K}$ for
$t_{\mathrm{pw}}=200\leavevmode\nobreak\ \mathrm{ns}$. Short input pulses
($t_{\mathrm{pw}}\ll\tau_{\mathrm{th}}$) could also enable the application of
$J_{c}$ above the limit imposed by the dashed black curve in Fig. 12, and
hence facilitate faster switching of the magnetic octupole, if applicable.

Fig. 13: Temperature rise of HM-Mn3Sn bilayer as a function of time for the
charge current pulse of amplitude $J_{c}=76.8\leavevmode\nobreak\
\mathrm{MA/cm^{2}}$ but different durations, as indicated in the legend. Here,
$d_{\mathrm{sub}}=3\leavevmode\nobreak\ \mathrm{\mu m}$,
$\tau_{\mathrm{th}}\approx 303\leavevmode\nobreak\ \mathrm{ns}$, and $\Delta
T_{M}=120\leavevmode\nobreak\ \mathrm{K}$ for MgO substrate. $\Delta T_{j}(t)$
is small if $t_{\mathrm{pw}}\ll\tau_{\mathrm{th}}$ as in the cases of
$10\leavevmode\nobreak\ \mathrm{ns}$, and increases with $t_{\mathrm{pw}}$.

The heat generated in the bilayer, due to a fixed $J_{c}$, would reach the
bottom electrode faster for a thinner substrate; therefore,
$\tau_{\mathrm{th}}$ decreases with $d_{\mathrm{sub}}$; consequently, $\Delta
T_{j}(t)$ reaches $\Delta T_{M}$ faster for lower $d_{\mathrm{sub}}$. This is
expected since the maximum possible $\Delta T_{M}$, at a fixed $J_{c}$, is
smaller for lower $d_{\mathrm{sub}}$ (Fig. 12). Finally, for the same value of
$d_{\mathrm{sub}}$, $\tau_{\mathrm{th}}$ would be higher in the case of both
sapphire and Si/SiO2 substrates, due to their lower thermal diffusivity
$\alpha=\frac{\kappa_{\mathrm{sub}}}{\rho_{\mathrm{sub}}C_{\mathrm{sub}}}$.
This slow heat dissipation could be behind the dependence of deterministic
switching on fall time, as reported in Ref. 76, 74, where sapphire and SiO2
were used as substrate.

###  3.3 Role of Thermal Fluctuations in Mn3Sn

Recent experimental 50, 78, 73, 76, 74, 79, 77, 80 and theoretical works, 97,
87, 98, 92 as discussed in Secs. 3.1 and 3.1.1, have made significant progress
in elucidating the current-driven dynamics in Mn3Sn with six equivalent
minimum energy states as well as in Mn3Sn with PMA, both in the presence and
absence of a symmetry breaking magnetic field. The promising results make
Mn3Sn a strong candidate for CMOS compatible, compact and low power memory
devices as well as GHz-THz signal sources. A practical realization of such
devices, however, requires an investigation into the effect of thermal
fluctuations, at different temperatures, on the long-term stability of a Mn3Sn
memory element and on the linewidth of a coherent signal source. Therefore, a
detailed investigation of the thermal stability factor, $\Delta$, which
depends on the energy barrier between the stable states, corresponding to thin
Mn3Sn films is needed.

Towards this, Sato _et. al_ , in a very recent work, 99 evaluated the thermal
stability of $20\leavevmode\nobreak\ \mathrm{nm}$ thick epitaxial films of
Mn3Sn of varying diameters $D$, ranging from $175\leavevmode\nobreak\
\mathrm{nm}$ to $1000\leavevmode\nobreak\ \mathrm{nm}$, each of which
exhibited PMA due to tensile strain from the MgO substrate. They applied
magnetic field pulses of duration $1\leavevmode\nobreak\ \mathrm{s}$ but
different amplitudes in-the-(Kagome)-plane perpendicular to the interface, and
explored the switching probability of the bistable magnetic order. Then using
the Nèel-Arrhenius thermal activation model and assuming an attempt frequency
of $1\leavevmode\nobreak\ \mathrm{GHz}$, typically used for FMs, they
evaluated, from the switching probability curves, the thermal stability factor
of the different samples. They found that epitaxial Mn3Sn films, characterized
by dominant two-fold and weak six-fold anisotropies, of diameters below $\sim
300\leavevmode\nobreak\ \mathrm{nm}$ exhibited single-domain behavior, _i.e._
, $\Delta\propto D^{2}$. On the other hand, for $D\gtrsim
300\leavevmode\nobreak\ \mathrm{nm}$, the epitaxial films are expected to be
multi-domain in nature since $\Delta$ showed negligible change with $D$. At
$D=300\leavevmode\nobreak\ \mathrm{nm}$, $\Delta$ is evaluated to be in the
range of about $100-150$. This result, however, is sensitive to the value of
the attempt frequency, and needs further analysis for higher attempt
frequencies.

Kobayashi _et. al_ investigated the dependence of the critical switching
voltage on the duration of the applied pulse, $t_{p}$, for a bilayer
comprising Pt and $40\leavevmode\nobreak\ \mathrm{nm}$ thick Mn3Sn, grown on
Si/SiO2 substrate. 100 A symmetry-breaking magnetic field, $H_{x}$, was also
applied parallel to the current/voltage pulse. The Hall resistance measured as
a function of the applied voltage, for $t_{p}$ in the range of
$10\leavevmode\nobreak\ \mathrm{\mu s}$ to $1\leavevmode\nobreak\ \mathrm{ms}$
and different $H_{x}$, showed tristable switching behavior, similar to Fig.
8(a), indicating clearly that Joule heating played a role in switching
dynamics. 76, 74, 77 The voltage at which the Hall resistance changed to zero,
$V_{\mathrm{critical}}$, was found to decrease monotonically with an increase
in $t_{p}$, as expected. 76 Nevertheless, ignoring the effects of Joule
heating, and using a switching-time model, pertinent to FMs in the thermally-
activated regime, given as 101

$V_{\mathrm{critical}}=V_{\mathrm{c0}}\quantity[1-\quantity(\frac{1}{\Delta}\log{\quantity(\frac{t_{p}}{t_{0}})})^{1/n}],$
(5)

where $V_{\mathrm{c0}}$ is the zero temperature critical voltage, $1/t_{0}$ is
the attempt frequency while $n$ is the exponent of the thermally-assisted
switching dynamics, Kobayashi _et. al_ estimated $\Delta=131.1\pm 0.5$ at
$300\leavevmode\nobreak\ \mathrm{K}$. Here, it was assumed that
$t_{0}=1\leavevmode\nobreak\ \mathrm{ns}$ and $n=1$, which are typically used
for FMs. 101 The resonant frequency in noncollinear chiral Mn3Sn was recently
measured to be approximately $0.9\leavevmode\nobreak\ \mathrm{THz}$ 102 while
a previous work measured the attempt frequency in MnIr, which is a collinear
AFM, in the THz regime. 103 Another recent work theoretically calculated the
attempt frequency in Mn3Ir to be greater than $10\leavevmode\nobreak\
\mathrm{THz}$. 104 The attempt frequency in Mn3Sn, therefore, could also be in
the THz regime. 105 Consequently, assuming $t_{0}=1\leavevmode\nobreak\
\mathrm{ps}$, the estimated value of $\Delta$ increases by about 6.9,
resulting in $\Delta\approx 138$. Including the effects of Joule heating in
the calculation is expected to lead to further changes in $\Delta$.

Previous theoretical works involving STT-driven dynamics in single-domain PMA
FMs 106, 107 showed that a careful Fokker-Plank- or ensemble LLG-based
approach was required to understand the switching behavior in the long pulse
regime. In particular, these works found $n$ to be equal to 2, and discussed
the importance of its accuracy for an accurate estimation of $\Delta$.
Similarly, solving an ensemble of coupled LLG equations, in the presence of
Gaussian random thermal noise, 108 could help decipher $n$, accurately, in the
case of SOT-driven switching dynamics in Mn3Sn nanomagnets with six-fold as
well as two-fold symmetry. This would then enable an accurate estimation of
$\Delta$ as well as the long-term retention capability of single-domain Mn3Sn
nanodots.

###  3.4 Readout of order parameter

As discussed in Sec. 2.2, several methods may be used to detect the state of
the octupole moment in Mn3Sn and related anti-chiral AFMs. Of these methods,
TMR offers the potential of providing a large readout signal that would be
compatible with CMOS circuitry. In anti-chiral AFMs, the ferroic order of the
cluster magnetic octupole breaks the macroscopic time-reversal symmetry
leading to a finite TMR. 68, 66 For the setup shown in Fig. 2, the low and
high resistance states correspond to angles $0^{\circ}$ and $180^{\circ}$
between the magnetic octupoles of the pinned and free Mn3Sn layers,
respectively. 66 The octupole moment of the pinned layer,
$\mathbf{m}^{\mathrm{P}}_{\mathrm{oct}}$, is fixed and not expected to change
for the currents applied to the device. On the other hand, the dynamics of the
order parameter of the free layer, $\mathbf{m}_{\mathrm{oct}}$, depends on the
magnitude of $H_{a}$ and $J_{c}$, as described in detail in Secs. 3.1 and
3.1.1. The different steady-state response of $\mathbf{m}_{\mathrm{oct}}$,
with respect to $\mathbf{m}^{\mathrm{P}}_{\mathrm{oct}}$, is then measured as
$V_{L}$ across the resistive load, $R_{L}$, for a given read voltage, $V_{R}$.
Here, $V_{R}$ is sufficiently small, such that the current due to it does not
disturb the steady states of either $\mathbf{m}^{\mathrm{P}}_{\mathrm{oct}}$
or $\mathbf{m}_{\mathrm{oct}}$.

Chen _et. al_ reported a TMR of 2% at room temperature 66 in a trilayer device
comprising epitaxial Mn3Sn($12\leavevmode\nobreak\
\mathrm{nm}$)/MgO($3.3\leavevmode\nobreak\
\mathrm{nm}$)/MgO($42\leavevmode\nobreak\ \mathrm{nm}$). The resistance area
(RA) product of this device, however, was extremely large on the order of
$10^{8}\leavevmode\nobreak\ \mathrm{\Omega\leavevmode\nobreak\ \mu m^{2}}$
while the corresponding resistance states were also quite large (about
$2\leavevmode\nobreak\ \mathrm{M\Omega}$), making the device CMOS
incompatible. Although lowering the RA product reduced the device resistance,
it also reduced the already small TMR of the trilayer device. Similarly,
reducing the thickness of the tunnel barrier lowers both the resistance and
the TMR. The $c$-axis of the epitaxial Mn3Sn film was at an angle of about
$30^{\circ}$ with the normal direction of the film. Therefore, the current due
to $V_{R}$ passes through the trilayer at an angle of about $60^{\circ}$ with
respect to the Kagome plane. Theoretical calculations revealed that with
vacuum as the tunnel barrier and current perpendicular (parallel) to the
Kagome plane, TMR increased to about $130\%$ ($150\%$) for a barrier thickness
of about $0.45\leavevmode\nobreak\ \mathrm{nm}$ ($0.65\leavevmode\nobreak\
\mathrm{nm}$). In both the cases, the RA product was found to be $\leq
1\leavevmode\nobreak\ \mathrm{\Omega\leavevmode\nobreak\ \mu m^{2}}$ in the
parallel as well as the antiparallel orientations of the two octupole moments.
Calculations suggested that both the TMR and RA product would increase to
higher values with the thickness of the barrier, at least for thicknesses
smaller than $1.1\leavevmode\nobreak\ \mathrm{nm}$.

As shown by Dong _et. al_ in their theoretical work, 68 large TMR of about
$300\%$ at room temperature, for vacuum as a tunnel barrier
($0.6\leavevmode\nobreak\ \mathrm{nm}$ thick) and current perpendicular to the
Kagome plane of the AFM tunnel junction, is possible. In this calculation, the
RA product increased from approximately $0.05\leavevmode\nobreak\
\mathrm{\Omega\dotproduct\mu m^{2}}$ in the parallel configuration to
$0.2\leavevmode\nobreak\ \mathrm{\Omega\dotproduct\mu m^{2}}$ in the
antiparallel configuration. On the other hand, when
$\mathbf{m}_{\mathrm{oct}}$ was at $60^{\circ}$ and $120^{\circ}$ with respect
to $\mathbf{m}^{\mathrm{P}}_{\mathrm{oct}}$, the RA product was found to be
approximately $0.07\leavevmode\nobreak\ \mathrm{\Omega\dotproduct\mu m^{2}}$
and $0.11\leavevmode\nobreak\ \mathrm{\Omega\dotproduct\mu m^{2}}$,
respectively. Assuming same values of RA product for the device setup of Fig.
2, where current is parallel to the Kagome plane, we estimated
$V_{L}=91\leavevmode\nobreak\ \mathrm{mV}$ ($71\leavevmode\nobreak\
\mathrm{mV}$) in the parallel (antiparallel) configuration, for
$V_{R}=100\leavevmode\nobreak\ \mathrm{mV}$, $R_{L}=50\leavevmode\nobreak\
\mathrm{\Omega}$ and the area of the tunnel barrier $\mathcal{A}=100\times
100\leavevmode\nobreak\ \mathrm{nm^{2}}$. Finally, the relative orientations
of $60^{\circ}$ ($120^{\circ}$) leads to intermediate value of $V_{L}$ as
$88\leavevmode\nobreak\ \mathrm{mV}$ ($82\leavevmode\nobreak\ \mathrm{mV}$).
Further improvement in $V_{L}$ could be achieved by increasing $V_{R}$ to a
higher value as long as the current flowing through the tunnel junction does
not disturb the octupole moments. Improvements in the RA product of the tunnel
junction without compromising the TMR could also help enhance the values of
$V_{L}$. The TMR of $300\%$ with vacuum as the tunnel barrier, for a fixed RA
product, likely represents the upper limit. Calculations showed that a more
realistic tunnel junction with monolayer hexagonal hafnia reduced the TMR to
$124\%$. In the case of AHE, the Hall voltage was found to be on the order of
a few $\mathrm{\mu V}$ for 10s of $\mathrm{\mu A}$ of read current, 87 which
is quite small, making TMR as the better choice for readout.

##  4 Computing applications

Fig. 14: The dynamics of octupole’s azimuthal angle versus time (a) applied
d.c. current $J\ =\ 2.7\times 10^{6}\leavevmode\nobreak\ \mathrm{A/cm^{2}}$
where the threshold current density $J_{s}^{\mathrm{th2}}\ =\ 2.6\times
10^{6}\leavevmode\nobreak\ \mathrm{A/cm^{2}}$. (b) Applied a.c. current $J\ =\
0.8J_{s}^{\mathrm{th2}}+\lambda J_{s}^{\mathrm{th2}}\cos(2\pi
f_{\mathrm{ac}}t)$ and $f_{\mathrm{ac}}=1\leavevmode\nobreak\ \mathrm{GHz}$
and (c) $f_{\mathrm{ac}}=5\leavevmode\nobreak\ \mathrm{GHz}$. (d) The
octupole’s thermal fluctuation in a two-fold low-barrier configuration, T =
300 K. (e) Averaged thermal flipping time versus volume at various
temperatures. (f) The octupole’s response to an alternating spin current
signal at 300 K.

Their rich bandstructure physics and the range of dynamical responses in the
presence of SOT and magnetic fields make Mn3Sn and other negative chirality
materials an excellent choice for new spintronic computing applications. Of
particular interest in this regard are biologically inspired computing and
probabilistic computing paradigms. As we discuss below, the SOT-induced
incoherent oscillations of the order parameter in Mn3Sn have striking
resemblance to the time-domain response of spiking biological neurons. On the
other hand, low-energy barrier Mn3Sn thin films or nanodots can exhibit
thermally induced switching which can be used for implementing p-bits, which
are an important building block for stochastic computing hardware and
Boltzmann machines.

###  4.1 Spiking neurons

In pursuit of enhanced energy efficiency compared to artificial neural
networks (ANNs), spiking neural networks (SNNs) 109 more closely mimic the
spiking behaviors of biological brains. The fundamental computing units of
SNNs, known as spiking neurons, accumulate potential in response to input and
fire when this potential surpasses a predefined threshold. This “Integration-
Firing” dynamics is essential to the functionality of a spiking neuron.
Implementing these neurons with beyond-CMOS technology could further improve
the energy efficiency and operational frequency of SNNs 110.

Mn3Sn is a promising material for building spiking neurons, owing to a Dirac-
comb-like nonlinear dynamics for high damping and currents near the threshold
current. 86 As illustrated in Fig. 14(a), for a two-fold anisotropy Mn3Sn,
when the input spin current density slightly exceeds a critical value, denoted
by $J_{s}^{\mathrm{th2}}$, the in-plane component of the magnetic octupole
measured via the TMR, $\propto\cos{\quantity(\varphi_{\mathrm{oct}})}$,
($\varphi_{\mathrm{oct}}$ is the azimuthal angle of the magnetic octupole)
demonstrates spiking behavior in the time domain. Previous works have shown
that the rate of firing of these neurons can be controlled if the d.c. current
is superimposed with a small a.c. current such the net current just exceeds
$J_{s}^{\mathrm{th2}}$. Furthermore, the co-application of a subthreshold
direct current (d.c.) stimulus (e.g., $0.8J_{s}^{\mathrm{th2}}$) with an a.c.
component (e.g., $\lambda J_{s}^{\mathrm{th2}}\cos(2\pi f_{\mathrm{ac}}t)$)
can engender a richer spectrum of spiking behaviors. Both the amplitude and
the frequency of the a.c. signal can control the nature of the spikes, _i.e._
, it might be possible to generate single spikes as well as bursts of spikes
as shown in Fig. 14(b)&(c). Such neuron emulators could then be coupled to
each other to get interesting dynamics, where the rate of firing of the second
neuron could be controlled by that of the first. If the coupling is
bidirectional in nature, unique dynamics, which can then be used for
computation, e.g. collective state computing and oscillatory hopfield network
111 is possible.

In our prior work, we estimated the performance of SNNs comprised of
$\mathrm{Mn_{3}Sn}$-based neurons and ferromagnetic-domain-wall-based synapses
in executing inference tasks. 112 Our benchmarking framework included models
of neurons, synapses, and interconnects, revealing that this network
outperformed CMOS-based networks in both energy efficiency and operational
frequency. For instance, in conducting a complete inference on a small LeNet
dataset, the $\mathrm{Mn_{3}Sn}$-based SNN required only
$1.21\leavevmode\nobreak\ \mathrm{ns}$ and consumed a mere
$4.23\leavevmode\nobreak\ \mathrm{nJ}$, in contrast to the analog-CMOS SNN,
which took about $29\leavevmode\nobreak\ \mathrm{ns}$, while also expending
$29\leavevmode\nobreak\ \mathrm{nJ}$ energy. Such remarkable performance
estimations underscore $\mathrm{Mn_{3}Sn}$’s potential in non-von Neumann and
beyond-CMOS computing paradigms.

###  4.2 Probabilistic computing

While traditional computing is characterized by its pursuit of deterministic
outputs, probabilistic computing emerges as a novel paradigm integrating
randomness and uncertainty into the computing process. Insights from
neuroscience suggest that the human brain optimizes for energy efficiency at
the expense of reliability by incorporating stochasticity in computation. This
approach proves particularly effective in addressing specific problems such as
factorization, satisfiability, and combinatorial optimization. 16 In this
context, the fundamental computing unit, known as the p-bit, functions akin to
a controllable random number generator, with its output being a function of
external input (_i.e._ , SOT).

Low barrier MTJs are proposed as viable candidates for p-bits in diverse
applications, including invertible logic gates and Boltzmann machines, 14 as
well as Gaussian random number generators. 113 Their suitability arises from
the ease of inducing stochastic behaviors with the aid of thermal noise. In
Section 3.3, we have discussed both field- 99 and voltage-driven 100
probabilistic switching, highlighting the potential applications of
$\mathrm{Mn_{3}Sn}$ in p-bit development.

We further simulate the thermal stability and fluctuation dynamics of
$\mathrm{Mn_{3}Sn}$, as depicted in Fig. 14(d-f). Specifically, Fig. 14(d)
illustrates the intrinsic thermal fluctuations of the octupole’s azimuthal
angle within a bi-stable configuration. At 300 K, the octupole oscillates
between two equilibrium states in the absence of external fields and currents,
with an average switching time $\tau=14\leavevmode\nobreak\ \mathrm{ps}$ for a
volume of $160\leavevmode\nobreak\ \mathrm{nm^{3}}$. As shown in Fig. 14(e),
the flipping time exhibits an exponential dependence on size and is estimated
to be around $\mathrm{ps}$ to $\mathrm{ns}$ scale, indicating its potential
for high-frequency p-bit applications. Figure 14(f) illustrates the response
of $\varphi_{\mathrm{oct}}$ to an external alternating current. In this
scenario, $\varphi_{\mathrm{oct}}$ undergoes oscillations between two
equilibrium states, as dictated by the applied current. Notably, these
transitions exhibit a slight delay of approximately $100\leavevmode\nobreak\
\mathrm{ps}$. Further investigations into its thermal dynamics and the
development of effective reading and writing methodologies are anticipated to
advance its application potential.

##  5 Conclusions and outlook

Noncollinear AFMs with Kagome lattices exhibit properties such as spin
momentum locking, topologically protected surface states, large spin Hall
conductivity, and magnetic spin-Hall effect arising from the topology. In
particular, research pertaining to the physics and applications of the
negative chirality noncollinear AFM, Mn3Sn, have intensified in the last few
years. Mn3Sn can only be stabilized in excess of Mn, whereas with a lower Mn
concentration the system gets contaminated with Mn2Sn. Below its Neél
temperature of $420-430\leavevmode\nobreak\ \mathrm{K}$, the negative
chirality 120 degrees spin structure is stabilized due to the combination of
exchange coupling and DM interaction between the Mn moments. With Mn
deficiency, a helix phase emerges at intermediate temperatures where Mn
moments form a spiral spin structure propagating along its c-axis. At much
lower temperatures ($\leq 50\leavevmode\nobreak\ \mathrm{K}$), spin glass like
texture is also feasible. Due to their many intriguing transport properties
including large spin Hall effect, anomalous Hall effect, and FM-like spin
polarized charge current, negative chirality AFM metals can enable the
development of AFM spintronic devices and applications along a similar route
as their FM counterparts.

In this paper, we presented recent developments, both on theoretical and
experimental fronts, related to understanding Mn3Sn’s non-trivial
bandstructure properties, which lead to strong magnetotransport signatures in
the material. Anomalous Hall effect in Mn3Sn, comparable to that in elemental
FMs like Co, Ni, Fe has been experimentally reported. Magnetic switching of
Mn3Sn has been demonstrated in bilayers comprising the AFM and heavy metal and
current densities ($\sim(10^{6}-10^{7}\leavevmode\nobreak\ \mathrm{A/cm^{2}}$)
comparable to that used for switching FMs in FM/heavy metal bilayers have been
obtained. In this structure, readout voltages $>1\leavevmode\nobreak\
\mathrm{mV}$ could be detected by optimizing the annealing conditions and the
seed layer during the fabrication process. Large magneto-optic Kerr signal,
around $20\leavevmode\nobreak\ \mathrm{mdeg}$ at $300\leavevmode\nobreak\
\mathrm{K}$, has been reported in Mn3Sn. Spin-torque ferromagnetic resonance
measurements have been used to quantify the field-like torque due to the
magnetic spin Hall effect and values greater than that of conventional heavy
metals like Pt, Ta, and W have been found in Mn3Sn. NV relaxometry
measurements have been performed to determine the frequency of chiral
oscillations of the order parameter in Mn3Sn as a function of the input
current. More recently, a finite TMR, around $2\%$ at $300\leavevmode\nobreak\
\mathrm{K}$, has been experimentally reported in all-AFM MTJs with Mn3Sn
electrodes and Mgo tunnel barrier. This discovery sets the stage for
developing advanced CMOS-compatible spintronic devices using Mn3Sn as the
active magnetic element. Studies of strained Mn3Sn thin films with two fold
symmetric energy landscape can be used as memory elements or high-frequency
signal generators and detectors depending on the magnitude of input spin
current and the magnetic field.

Theoretical models of critical input current to induce different dynamics with
and without magnetic fields in six-fold and two-fold symmetric Mn3Sn can be
valuable to gain physical insight into the effect of material parameters on
the order parameter dynamics. Likewise, models of oscillation frequency and
switching time versus input current can shed light on the performance metrics
and scaling limits of Mn3Sn-based spintronic devices. The effect of Joule
heating in Mn3Sn cannot be ignored, especially when the input current
significantly exceeds the threshold current for a given dynamical response.
However, by employing pulsed SOT in the short pulse regime, the issue of Joule
heating can be reduced. Understanding the impact of thermal noise on the order
parameter in Mn3Sn is crucial for realizing non-volatile memory as well as
p-bits, which serve as the fundamental unit of probabilistic computing
hardware. Toward this, thermal stability of Mn3Sn thin films with
perpendicular anisotropy has been studied experimentally as well as
theoretically by employing the Néel-Arrhenius model. However, further research
in this area is needed to quantify Mn3Sn’s attempt frequency and the effective
temperature in the presence of SOT.

To move the research field further and to develop practical spintronic
applications using anti-chiral AFMs, experimental techniques to probe time-
domain dynamics without Joule heating are needed. Advances in improving TMR in
all-AFM tunnel junctions are also critical for ensuring robust readout signals
that are compatible with CMOS circuitry. Additional work is needed to
elucidate the role of thermal noise in Mn3Sn thin films of various dimensions.
Theoretical calculations will be indispensable to shed light on the exotic
bandstructure physics of these materials and provide a theoretical rigor to
experimental investigations. Meanwhile, the state of analytical models must be
further advanced to enable technology-device co-design and benchmarking.
Development of prototypes of antiferromagnetic spintronic devices that are
CMOS-compatible can allow this field to rapidly advance and lead to novel
computing paradigms with superior performance compared to their ferromagnetic-
only counterparts.

## Author Contributions

Ankit Shukla: methodology, investigation, formal analysis, investigation, data
curation, and writing – original draft. Siyuan Qian: investigation, data
creation, and writing – review. Shaloo Rakheja: supervision,
conceptualization, methodology, resources, funding acquisition, and writing –
review and editing. All authors discussed the results and the implications of
this manuscript. All authors have given approval to the final version of the
manuscript.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

This research was partially supported by the NSF through the University of
Illinois at Urbana-Champaign Materials Research Science and Engineering Center
DMR-1720633. The authors also acknowledge the support of AFRL/AFOSR, under
AFRL Contract No. FA8750-21-1-0002, NSF Grant No. CCF-2021230 and ZJU-UIUC
Joint Research Center on Dynamic Research Enterprise for Multidisciplinary
Engineering Science (DREMES).

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