# Magnetism, symmetry and spin transport in van der Waals layered systems

Hidekazu Kurebayashi  London Centre for Nanotechnology, UCL, 17-19 Gordon
Street, London, United Kingdom  e-mail<EMAIL_ADDRESS>Jose H.
Garcia  Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and
BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain
<EMAIL_ADDRESS>Safe Khan  London Centre for Nanotechnology,
UCL, 17-19 Gordon Street, London, United Kingdom  Jairo Sinova  Institut für
Physik, Johannes Gutenberg Universität Mainz, D-55099 Mainz, Germany
Institute of Physics Academy of Sciences of the Czech Republic, Cukrovarnickaa
10, 16200 Praha 6, Czech Republic  Stephan Roche  Catalan Institute of
Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra,
08193 Barcelona, Spain  ICREA–Institució Catalana de Recerca i Estudis
Avançats, 08010 Barcelona, Spain

###### Abstract

The discovery of an ever increasing family of atomic layered magnetic
materials, together with the already established vast catalogue of strong
spin-orbit coupling (SOC) and topological systems, calls for some guiding
principles to tailor and optimize novel spin transport and optical properties
at their interfaces. Here we focus on the latest developments in both fields
that have brought them closer together and make them ripe for future fruitful
synergy. After outlining fundamentals on van der Waals (vdW) magnetism and SOC
effects, we discuss how their coexistence, manipulation and competition could
ultimately establish new ways to engineer robust spin textures and drive the
generation and dynamics of spin current and magnetization switching in 2D
materials-based vdW heterostructures. Grounding our analysis on existing
experimental results and theoretical considerations, we draw a prospective
analysis about how intertwined magnetism and spin-orbit torque (SOT) phenomena
combine at interfaces with well-defined symmetries, and how this dictates the
nature and figures-of-merit of SOT and angular momentum transfer. This will
serve as a guiding role in designing future non-volatile memory devices that
utilize the unique properties of 2D materials with the spin degree of freedom.

Key points:

  * •

Fabrication of 2D vdW magnetic systems offers unprecedented opportunities for
controlling magnetism and spin transport phenomena down to the monolayer
limit.

  * •

Many vdW magnetic systems possess a low-symmetry crystalline structure,
providing an array of exotic spin-orbit Hamiltonians, together with added
richness arising from interface phenomena driven by layer-to-layer registry.

  * •

Understanding the intertwined contribution of spin-spin interaction and
interfacial symmetries is crucial for reaching the upper limit of SOT.

  * •

This exciting research field of spin transport at the frontier of layered
spin-orbit coupling and magnetism will lead to discoveries of new materials,
novel transport effects, topological phenomena and unconventional electron
correlation physics.

Website summary: We present a foresight on how the spintronic properties of
the emerging class of layered materials combining magnetism and strong spin-
orbit coupling can be tailored by proper optimization of chemical interactions
and structural material symmetries. This perspective draws a route to achieve
best performing material design for reaching the upper limit of spin-orbit
torque efficiency in switching magnetization.

## Overall context of magnetism and spin-orbit coupling materials

Figure 1: a Microscope image of an exfoliated CrI3 flake. The numbers in the
image represent expected monolayer numbers calculated by the image
contrast[1]. b Magneto-optical Kerr rotation angle measurements applied to the
different monolayer number regions while sweeping magnetic field. The
monolayer and trilayer regions exhibit a ferromagnetic response, whereas the
bilayer region shows an antiferromagnetic behaviour. c Rashba spin texture in
2D momentum (reciprocal) space kx and ky. For the sake of simplicity, we only
show one spin-subband with electrons at each Fermi surface (Left panel). When
we apply an electric current along the $x$ direction, electrons are
redistributed and as a result, they become spin-polarized due to the spin
texture. This is a source of SOT from the Edelstein effect. d Schematic of a
single-cell of an SOT memory device with vdW magnetic materials. An
information bit is stored by the relative orientation of two vdW magnets,
which can be read by e.g. magnetoresistance through a vertical current through
this stack. An external current is sourced by the write line/path electrode
which is a low-resistive material such as copper. This current causes
magnetization switching due to SOTs in the bottom vdW magnetic layer as
highlighted in the zoom-in figure. The current-supplying bottom electrode
could be another vdW material that acts as an additional spin source for
boosting the switching efficiency. Figure 1a-b are adapted and modified from
Ref. [1].

The realm of two-dimensional (2D) materials has considerably extended recently
with the demonstration of magnetic orders in vdW material systems [2, 1, 3, 4,
5, 6]. Weak mechanical bonding of vdW gaps in these materials offers
unprecedented control of individual layer separations/fabrications by
exfoliation, from which we are able to create truly monolayer magnets[2, 1]
(See Fig. 1a&b as an example) and their heterostructures. This capability in
sample fabrication is not readily within reach by other established thin-film
growth techniques such as molecular beam epitaxy and sputtering, where the
formation of grown layers significantly depends on surface energetics and
kinetics together with lattice-matching between crystal structures of seed and
grown layers. In contrast, exfoliation of vdW materials allows to stack
different vdW monolayers without any thermodynamic constraints, enabling to
access an unlimited number of stack combinations to explore fascinating, novel
electronic structures[7, 8, 9]. This will provide tremendously exciting
opportunities to generate new magnetism, spin-transport and electron-
correlation physics by using 2D building blocks and currently intensive and
extensive research activities on vdW magnetic systems are taking place
worldwide. This class of new layered vdW materials is particularly vast, and
includes transition metal dichalcogenides [10, 11] (TMDs), such as CrSe2,
VSe2, CrTe2, transition metal trihalides [12, 13, 14] (CrI3, CrCl3) and
transition metal phosphorous trichalcogenides [15] (NiPS3, MnPS3, etc.). 2D
magnetism has been measured at low temperatures in Cr2Si2Te6 [16], CrI3 [1]
and Cr2Ge2Te6 [2] mono- and bilayers, or in VSe2 [17], Fe3GeTe2 [4, 18] and up
to room temperature (RT) in single layer MnSe2[19, 20]. In all those
materials, the existence of stable magnetic properties for few layered systems
is believed to be strongly related to their own magnetic anisotropies as well
as the interaction with the substrate that prevents charge density wave
instabilities [21, 22, 23, 24, 25].

In spintronic applications, magnetic moments are informational bits such as
those in hard-disc drives and we process the bits by reversing the moment’s
direction from one to the other between their equilibrium states[26, 27, 28].
Since electronic devices operate either with current- or voltage-regulated
inputs, efficient control of magnetic moments by an electric means is
therefore the primary interest for developing future commercially-viable
functional spintronic devices such as spin random access memory[29, 30]. The
spin-orbit interaction fits to this perfectly because it microscopically
couples the electron motion (i.e. current) and the spin degree of freedom,
hence making it possible to manipulate spins by electric currents (Fig. 1c).
The electrically-generated spin polarization can exert spin torques on local
magnetic moments via the exchange interaction. This phenomenon in solid states
is termed as spin-orbit torque (SOT) which has been discovered and studied
mainly in magnetic multilayers with 3D crystals [31, 32, 33] and new SOT
memory architectures[34] have been proposed and examined in the past several
years. Microscopically, SOT originates from two leading effects, the spin-Hall
effect [35] and Edelstein effect[36]. The efficiency and mechanisms of SOT are
determined by the electronic states of materials as well as interfaces, in
particular inversion symmetry breaking in combination with other crystalline
symmetries[37]. In this regard, 2D vdW materials and heterostructures hold
great potential to produce unique SOT phenomena from their exotic electronic
structures which can be in principle modified by interface engineering.
Furthermore, the efficiency of magnetization switching by SOT is proportional
to magnetic volume[37] and so the smaller the better, ideal to the 2D limit
since the volume is minimised along one dimension. In this extreme monolayer
regime, it is important to stabilize magnetic orders against thermal
fluctuation. This calls for detailed understanding of magnetism at that limit
in terms of their driving mechanisms and the strength of them, which can be
designed and controlled by material choice and external stimuli. When we have
full understanding of both magnetism and SOTs in novel vdW heterostructures,
we can design highly-efficient spintronic memory cells based on vdW materials
such as one shown in Fig. 1d. In this cell, an information bit is stored by
the relative orientation of two vdW magnetic layers separated by another vdW
non-magnetic system in the middle. Tunnelling magneto-resistance effect would
be best in terms of outputting a large voltage difference as a read-out.
Writing actions for this cell is carried out by a horizontal current which
will produce SOT effects to exert magnetic switching of the bottom layer. The
efficiency of switching, in terms of electric power, is a key figure of merit
since the size of a transistor to drive a current for each cell is determined
by this efficiency (a large current implies bigger transistor size) and the
cell density is currently limited by the transistor size. This directly means
that maximising the SOT efficiency leads not only to low-power consumption of
memory cells but also to the high density of data processing. In this sense,
detailed understanding of physical origins and mechanisms of SOTs in any novel
materials is of direct relevance of future ultracompact and low-power spin
memory devices. It is important to note that non-magnetic high-spin-orbit vdW
materials, such as WTe2, could be inserted as an additional spin source for
SOT-driven magnetization switching. In this regard, a whole variety of vdW
heterostructures should be extensively studied for the development of novel
vdW spin memory devices.

This Review particularly focuses on two fundamental aspects of 2D vdW
properties for future spintronic applications. The first is on the exchange
interaction as a source of magnetic order by which 2D vdW magnetic systems
display different exchange interaction mechanisms depending on their
crystalline structures and transport properties from insulators to metals.
Understanding of these underlying mechanisms is useful to visualize different
magnetic orders at play in a variety of materials and provides a good guide
for designing monolayer magnets that are magnetically stable above RT. After
this, we also review recent development on electric and other control of
equilibrium magnetic states, which is the current state-of-the-art in the
relevant field. We then focus on the potential and opportunities of SOC
effects arising within 2D vdW magnetic materials as well as in cases when they
are interfaced to an additional material such as substrates. Their low-
symmetry point groups exhibit a colorful picture of SOT effects and by
applying high-throughput point group analysis, we generate and summarize the
expected symmetry of SOTs in all the class currently available in the
literature. This will aid future SOT experiments using listed materials as
well as those newly discovered but possessing the same crystalline symmetry as
listed. We further catergorize field-like and damping-like torques existing in
these materials as well as more exotic anisotropy-like torques arising from
their low-symmetry properties. These two fundamental aspects have not been the
focus of prior review articles on 2D vdW magnetic systems [38, 39, 40, 41, 42,
43, 44], which extensively cover the recent development of 2D vdW magnetic
systems as well as spintronics with non-magnetic vdW materials[45, 46, 47,
48].

## Different exchange interactions to stabilize magnetic orders in van der
Waals systems

We here briefly summarize that the magnetic order driven by the exchange
interaction of electron spins has different mechanisms, depending on the
conductivity regime of materials. This is very relevant to vdW magnetic
systems since their conductivity spans across a wide range, from insulating to
metallic regimes, possessing a very colorful nature of magnetism due to their
conductivity as shown in Fig.2. This is one of the unique potentials of vdW
magnetic systems as research topics for current/future materials science for
nano-magnetism and spintronic applications. The origin of long-range magnetic
orders in condensed matter arises from the microscopic exchange interaction
between individual spins. The exchange interaction between an array of $n$
spins is usually captured by
$H_{\text{ex}}=-\sum_{i,j}^{n}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$ where
the dot product of $\mathbf{S}_{i}$ and $\mathbf{S}_{j}$ represents the
relative orientation between the two spins. This Hamiltonian describes how
much the spin system gains/loses their total energy by ordering their spin
orientations as a whole.

### The super-exchange coupling in insulators

The exchange interaction model is better suited to describe spins that are
well-isolated, such as in the case of magnetic insulators and low-conducting
semiconductors, where isolated spins interact with each other via non-magnetic
ions (ligands) such as oxygen in their crystal lattices. Such indirect
exchange coupling is often called the super-exchange mechanism [49] and a
schematic to highlight this is shown in Fig.2. This can be explained by the
virtual excitation of electrons in a magnetic cation and subsequent
transmission to the neighbouring cation through the non-magnetic ligand within
a timescale of Heisenberg uncertainty. This virtual process effectively lowers
the total energy of the system when a long-range magnetic order is present.
Magnetic ions for the super-exchange interaction possess the same ionic charge
states.

Many of vdW materials exhibit semiconductor/insulator properties and this
extends to those possessing long-range magnetic ordering. CrX3 (X= I, Br and
Cl) exhibits good insulating properties with a typical bandgap energy of
1.2-3.1 eV [14, 13, 1, 50]. In a single layer of CrX3, each Cr atom is
surrounded by X atoms in an octahedral configuration and the Cr atoms form a
honeycomb lattice. This crystal field splits the Cr $d$ states into $e_{g}$
and $t_{2g}$ manifolds where three electrons in their Cr3+ state occupy the
fully-polarized $t_{2g}$ states with $S$ = 3/2 in the valence bands, following
the first Hund’s rule. These Cr3+ moments are coupled through the super-
exchange mechanism via an X atom with an approximately 90 degree bonding angle
[51, 52]. It is noted that due to the presence of finite antiferromagnetic
coupling between Cr-Cr direct exchange mechanisms in these materials, the
total exchange coupling strength (hence the size of $T_{\rm C}$) is relatively
weak[53]. A comprehensive study for $T_{\rm C}$ of few layers of CrX3 has been
carried out by Kim et al.[54]. Cr2X2Te6 (X=Ge,Si) exhibits
semiconductor/insulator properties due to their electronic bands with about a
0.4 eV gap[2, 16]. Very much like the case of CrX3, a honeycomb Cr3+ network
stabilizes their ferromagnetic order owing to the super-exchange interaction
via Te atoms. Another key group of magnetic vdW insulators is those of MXPS3
(M = V, Mn, Fe, Co, Ni, or Zn)[15], with a bandgap of 1.3 - 3.5 eV[55, 56,
57]. Transition metals X host magnetic moments within the honeycomb lattice,
showing either antiferromagnetic or ferromagnetic order.

### The double-exchange coupling in semiconductors

When magnetic ions in a lattice have two types of ionic states, e.g. Mn3+ and
Mn4+, an exchange of the two ionic states depends on the spin state of two
magnetic ions. This exchange is generated by one mobile carrier (either
electron or hole) hopping between the two ions. This hopping is possible when
spin states of the two magnetic ions are parallel, leading to delocalisation
of the mobile carriers and reduction of their total kinetic energy. This is
the double-exchange mechanism [58] that stabilises a long-range magnetic
order. A prototypical example of this is
(La1-xCax)(Mn${}^{3+}_{1-x}$Mn${}^{4+}_{x}$)O3[59]. In this material series,
only one type of Mn ionic states exists for both ends of their compositions
($x=0,1$), which are good insulators, and the super-exchange interaction
stabilises antiferromagnetism in these limits. In their intermediate states,
Mn3+ and Mn4+ coexist (effectively Mn4+ being considered as Mn3+ and one hole)
and a ferromagnetic order outperforms the previously favored antiferromagnetic
one via the double-exchange mechanism. In this regime, the presence of mobile
carriers is confirmed by the conductivity which becomes several orders greater
than those of the insulating compositions.

The double-exchange mechanism has not been discussed extensively in 2D vdW
magnetic systems so far, although this mechanism should be commonly present
for magnets with moderate electric conduction. As discussed later, Cr2Ge2Te6
with chemical and electric doping[60, 61] is one example where the exchange
coupling strength is modified by the double-exchange mechanism, exhibiting
large enhancements of the Curie temperature ($T_{\rm C}$) compared to undoped
Cr2Ge2Te6.

### The band magnetism in metals

When conduction electrons are fully delocalized and well hybridized with
magnetic ions, we enter the itinerant regime where band magnetism is used to
describe the magnetic order. In this mechanism, the total energy of the whole
system with respect to magnetic order is discussed by the energy gain/loss of
electrons close to the Fermi energy with the Pauli paramagnetism picture [62].
Based on this, together with the mean-field model [63], the Stoner criterion
$D(E_{F})U\geq 1$ (here $D(E_{F})$ and $U$ are the density of states (DOS) at
the Fermi level in a non-spin-polarized state and the electron correlation
energy) can be applied[64, 65]. This essentially means that when the Coulomb
energy associated with the electron exchange is strong enough, spontaneous
exchange-energy splitting takes place to drive ferromagnetism. This model is
often employed to discuss the emergence of a magnetic order in metallic
ferromagnets[62].

Magnetic vdW materials with electron conductivity are also available, such as
VX2 (X=Se,Te)[17], MnSe2[19], V5Se8[66], Fe-Ge-Te compounds[4, 18, 67, 68],
Fe0.25TaS2[69] and Cr-X (X=Te,Se) compounds[70, 71, 72]. As shown in Fig. 2,
these metallic 2D vdW materials tend to display higher $T_{\rm C}$ due to the
direct, stronger exchange interaction between individual moments. Some of them
already possess an above-RT $T_{\rm C}$ which is relevant to spintronic device
applications, although a much higher $T_{\rm C}$ than RT is welcomed for
stable device operation in any of these applications. The Stoner criterion is
indeed discussed for some of these materials, for example Fe3GeTe2[73] was
calculated to satisfy the inequality by $D(E_{F})U=1.56\times 0.71\geq 1$.

Figure 2: Experimentally validated critical magnetic transition temperature
fingerprints for various 2D vdWs magnetic systems. a Three different exchange
coupling mechanisms of two individual moments as an underlying origin of the
long-range magnetic ordering for material systems in the insulating and
metallic phases going from left to right. b Two-dimensional vdW magnetic
materials library showing experimentally determined magnetic transition
temperature for materials in insulating and metallic phases. In the insulating
phase, number in brackets corresponds to the bandgap value. The dataset is
characterised in two-ways using the colours and symbols. Colours: Red
(ferromagnetic ordering) and Blue (anti-ferromagnetic ordering). Symbols: Bar
(top and bottom points in the bar highlight the transition temperature for
bulk and mono/few-layers, respectively), $\bigtriangleup$ ($T_{\rm C}$ is
reported for the bulk case), $\bigcirc$ ($T_{\rm C}$ is identified for
mono/few-layers), $\Diamond$ ($T_{\rm C}$ is independent of material
thickness), and $\bigstar$ ($T_{\rm C}$ is enhanced by the external
techniques). The data points in the plot are taken from Refs. [14, 74, 75, 56,
76, 55, 77, 78, 79, 57, 80, 81, 82, 1, 83, 84, 85, 2, 86, 87, 70, 88, 89, 19,
90, 91, 92, 93, 94, 95, 66, 69, 96, 17, 97, 98, 99, 100, 68, 73, 18, 61]

A final remark of this section is on the Mermin-Wagner theorem [101] in which
an absence of long range magnetic order in spin-rotational invariant systems
is introduced. This could be statistically accurate but in reality it is
difficult to achieve this since as soon as we make a real magnet, it
automatically possesses any sort of magnetic anisotropies, be it from the
spin-orbit interaction, anisotropic exchange interaction or magnetic dipole
interaction, all of which opens up a gap for magnon excitations that suppress
thermal fluctuations.

## Control of magnetism and magnetic states by external stimuli

Figure 3: Experimental observation of electric field control of magnetism in
different vdW magnets. a A schematic side view of a field-effect device with a
bilayer CrI3 encapsulated in few-layer graphene and hBN. The left panel is an
optical microscope image for their device with a scale bar of 20 $\mu$m. b
Electric field control of magnetism in measured by ac susceptibility with
different gate voltages as shown. c Coercive force (magenta), saturation
magnetization (purple) (both at 4 K) and Curie temperature (orange) normalized
by their values at zero gate voltage as a function of gate voltage and induced
doping density. d Schematics of a field-effect-transistor device with
Cr2Ge2Te6 and side view of charge accumulation by electric field. e
Experimentally-deduced out-of-plane anisotropy field in Cr2Ge2Te6 as a
function of temperature for different gate voltages, with $T_{\rm C}$ with
gate voltages (inset). f $T_{\rm C}$ of Fe3GeTe2 with different layer
thicknesses measured by different analysis methods shown. g Gate voltage
dependence of $T_{\rm C}$ in a trilayer Fe3GeTe2 and h their hysteresis loops
for different temperatures. i the extracted critical field for spin-flip
transition (black circles) and critical temperature (blue squares) as a
function of pressure in a trilayer CrI3. Figures are adapted from Ref. [102]
(Fig. 3a-c), Ref. [61] (Fig. 3d-e), Ref. [18] (Fig. 3f-h) and Ref. [103] (Fig.
3i) respectively.

As already summarized in the aforementioned section, the transport nature in
magnets is strongly associated with the type of leading exchange interaction
terms and their strength. The exchange interaction can be modified by the
carrier concentration as well as the distance between magnetic ions. Here we
summarize two external control of magnetism and magnetic states, electric
field and strain, applied to 2D vdW magnetic systems.

### Electric field

By fabricating a field-effect-transistor (FET)-type device with a thin-film
magnet, an external control of magnetism by the charge accumulation at the
surface is possible, and leads to a modulation of the Fermi level as well as
surface carrier density. Therefore, this approach is a promising way of
controlling the magnetic properties $in$-$situ$, which has clear relevance to
not only achieving high-$T_{\rm C}$ materials but also gate-tunable spintronic
devices. Electric field control of magnetism and magnetic states has been of
interest within the field of spintronics where atomically-thin magnetic
metals, dilute magnetic semiconductors and magnetic oxides are fundamental and
instrumental materials [104]. 2D vdW magnetic systems can offer added values
to this already-developed research domain as summarised below.

A dramatic change of magnetism by electric field, probed by modification of
$T_{\rm C}$, has been observed in several 2D vdW magnets. Jiang et al.[102]
fabricated parallel-plate capacitor devices with monolayer or bilayer CrI3
together with a ring-shaped local ac-excitation metal patterns (schematic and
microscopy images in Fig. 3a). Magnetic responses from the CrI3 were measured
by magnetic circular dichroism as ac susceptibility (Fig. 3b). A clear change
of $T_{\rm C}$ by applying electrostatic voltages was observed, together with
other sizable changes of the saturation magnetization ($M_{\rm S}$) and
coercive field ($H_{\rm C}$), as summarized in Fig. 3c. Bi-layer CrI3 also
exhibits a linear magneto-electric (ME) effect due to their magnetic point
group with broken spatial-inversion and time-reversal symmetries[105, 106,
107, 82] . Together with a relatively weak spin-flip transition in CrI3, the
linear ME effect can shift the transition field and demonstrate pure electric
switch between ferromagnetic and anti-ferromagnetic states[107, 82] when the
applied magnetic field is set at the point of transition while the gate
voltage is swept. Song et al.[108] demonstrated that tunneling
magnetoresistance (TMR) can be electrically modified by up to one order of
magnitude in four-layer CrI3 by using bistable magnetic states with the same
net magnetization, yet showing different TMR values.

An electrostatic modulation of magnetism has been also demonstrated in
Cr2Ge2Te6 by Wang et al.[109] and Verzhbitskiy et al. [61] By applying the
gate voltage to a 3.5 nm Cr2Ge2Te6 film via a SiO2 solid-state dielectric,
Wang et al. achieved a carrier density of order of 1012 cm-2 for both electron
and hole doing regimes, and observed a moderate change of $M_{\rm S}$ without
any significant change of $T_{\rm C}$. Verzhbitskiy et al.[61] attempted a
similar FET-type device with 20 nm thick Cr2Ge2Te6 with an ionic gel as a
dielectric as shown in Fig. 3d. With this scheme, a much stronger doping
action has been observed, at the level of 1014 cm-2 and $T_{\rm C}$ of
Cr2Ge2Te6 is enhanced from 64 K (bulk)[2] to 200 K. This enhancement is
explained by the activation of the double-exchange interaction mechanism by
introducing a significant number of electrons. In this regime, about one
quarter of Cr3+ magnetic ions become Cr2+, where hoping of electrons via the
Te atoms contributes to the additional exchange mechanism, enhancing the
exchange coefficient as well as $T_{\rm C}$. With this charge accumulation at
the interface, the width of the region where electrons are doped
electrostatically becomes comparable to the vdW gap ($\sim$ 1 nm). Therefore,
we can create an artificial two-dimensional electron gas together with
magnetism within the 20 nm thick Cr2Ge2Te6. Furthermore, the magnetic easy
axis of Cr2Ge2Te6 is switched from out-of-plane to in-plane by this carrier
doping[61].

While CrI3 and Cr2Ge2Te6 are magnetic insulators in their pristine forms, the
electric field modulation of magnetism has been also explored in metallic
itinerant vdW magnets such as Fe3GeTe2 [18]. Deng et al. performed a detailed
study of Fe3GeTe2’s $T_{\rm C}$ with different layer numbers (Fig.3f) and
applied electric fields onto a trilayer Fe3GeTe2 film using solid electrolyte
(LiClO4 dissolved in polyethylene oxide matrix). $T_{\rm C}$ is greatly
enhanced from 100 K to above 300 K by application of around 2 V, as shown in
Fig. 3g&h. This gate-tunability of $T_{\rm C}$ can be understood by the Stoner
model[64, 65] since the energy gain by forming a ferromagnetic phase depends
on the size of DOS at the Fermi level. Theoretical calculations suggest that
by applying the electric field, the Fermi level can be lifted up to a large
DOS peak area in Fe3GeTe2, mainly consisting of Fe $d_{z^{2}}$, dxz and dyz
orbitals[18]. The non-monotonic behaviour of $T_{\rm C}$ as a function of gate
voltage may imply this Fermi level passing by the peak points.

### Strain

Strain is also a good external knob to modify the magnetic ground state of 2D
vdW magnets. In some of these magnetic materials, the interlayer exchange
coupling governs the ground-state spin orientation, leading for instance to
antiferromagnetism in CrI3. Since the exchange coupling is very sensitive to
the distance between two spins (in this case, the vdW gap distance), any
external perturbation modifying this length enables the control of magnetism,
as shown with hydrostatic pressure of few GPa which can induce change of
$T_{\rm C}$ of bulk vdW crystals Cr2Ge2Te6[110], CrI3[111] and VI3[85] as well
as the sign change of the two-fold magnetic anisotropy in Cr2Ge2Te6[112]. In
the thin-film regime, Song et al.[103] and Li et al.[113] both independently
applied hydrostatic pressure of a few GPa on a few monolayer of CrI3 and
observed a monoclinic-to-rhombohedral structural transition, driven by
interlayer antiferromagnetic or ferromagnetic couplings [114, 115, 116, 117].
Similarly, a ferromagnetic-antiferromagnetic transition under applied
pressure, as well as pressure-controlled magnetic state switching between
antiferromagnetism to ferromagnetism have been also demonstrated [103]. Such
monoclinic-to-rhombohedral structural transition can be also observed in
exfoliated CrCl3 crystals with a variation of the interlayer exchange coupling
by up to one order of magnitude, when compared to bulk samples[118]. This
result highlights that sample fabrication processes unintentionally produce
strain that greatly perturbs the magnetic properties of exfoliated magnets.
Whether this monoclinic phase is thermodynamic ground states or meta-stable
states induced by their exfoliation processes is yet to be revealed [118].
Furthermore, $T_{\rm C}$ can be modified by application of strain e.g. to a
trilayer of CrI3 as shown in Fig. 3i.

## Spin-orbit torques in two-dimensional materials

While the magnetic properties have been successfully controlled in a certain
number of 2D vdW magnets, the access to out-of-equilibrium regimes in such
materials by using electric currents and other means presents formidable
challenges, but also holds paramount potential for future spintronic memory
and logic applications[119, 120, 30, 121]. This area of research is growing
fast but to date only a very limited number of reports are available in the
literature [122, 123, 124, 125, 126, 127]. Using electric fields, induced non-
equilibrium spin polarization can result in substantial angular momentum
transfer and magnetization control to a certain degree. This has been achieved
using non-collinearity between two magnets[128] in magnetic tunnel junctions,
or via the intrinsic SOC of materials [37]. In this section, we start with an
overview of the latest experimental demonstrations of magnetization control by
electric currents with 2D vdW materials, and then present a theoretical
analysis of spin torque physics.

TMDs with high spin-orbit coupling (SOC), such as WTe2 and NbSe2, have already
been employed to generate spin torques in ferromagnetic metals[122, 123, 124].
In particular, MacNeill and coworkers observed that the torque symmetry
arising in NiFe/WTe2 bi-layers results from the symmetry lowering at the
interface[122], exhibiting the damping-like symmetry with the spin-
polarization along the out-of-plane direction which cannot be generated by the
spin-Hall effect[37]. Magnetic moments in vdW ferromagnets have been also
manipulated by current-induced spin torques[125, 126, 127]. Interestingly, the
spin-Hall effect in Pt was used to generate spin currents into $10\sim 20$-nm-
thick exfoliated Fe3GeTe2 thin-films, and induce magnetization switching for
low current density of 2.5 x 1011 A/m2[125, 126]. This current density is
actually comparable to the one needed to switch standard 3d transition metal
alloys[37]. A magnetization switching by the spin-Hall effect has been also
discussed in the insulator-Cr2Ge2Te6/Ta system, requiring current density of
$\sim$ 5 x 1011 A/m2, comparable to the one necessary for Tm3Fe5O12[129].
Finally very recently, using current density of about 3.9 x 1012 A/m2, a spin-
orbit torque induced magnetization switching in an all vdW heterostructure was
also reported in Fe3GeTe2/WTe2 stacks [130].

So far, the majority of studies on current-induced spin-torque manipulations
in 2D vdW magnets exploit the properties of conventional heavy metals to
generate the spin source. Due to the intrinsic nature of vdW materials, some
2D vdW magnets lack an inversion center in their crystalline structures, hence
allowing for spin-orbit Hamiltonian with linear and/or cubic in momentum. This
fundamentally means that these materials possess spin textures in momentum
space which can be tailored by electric currents for magnetisation control[36,
37]. While spin-orbit torque arising from spin-textures in momentum space has
been demonstrated and studied in great detail in 3D crystals [31, 131, 132,
133], including the demonstration of damping-like torques due to the Berry-
curvature [134], to date no such result have been reported in 2D vdW magnets.
Since electronic states in 2D vdW materials are inherently very sensitive to
any perturbation caused at interfaces, the 2D vdW material family offers a
fascinating and promising playground to explore novel and efficient current-
induced spin torques with a large number of possible heterostructure
combinations. To guide this research development, we here present a summary of
possible spin-orbit torque symmetries for different point group crystals,
including the contributions emerging at interfaces. We remind the reader that
following Neuman’s principle, it is the point group which governs the allowed
types of linear response tensors, being themselves translation invariant, and
that it is the space group of the material which conditions the band
structure, which can have an effect on the magnitude of the response [135]. We
begin with basic descriptions of the underlying spin physics and then
elaborate a broader picture of torque categories, followed by a discussion
about the theoretical methodologies needed to explore the nature and strength
of the torque phenomena.

### Basics of magnetic torques and spin-orbit interaction

Figure 4: Illustration of spin-orbit torque mechanisms for different spin-
textures and their effects on the magnetic energy profile. a Magnetization
dynamics due to a typical effective SOT field for systems with multiple
orthogonal mirror planes. b Magnetization dynamics for effective SOC fields
with a component parallel to the anisotropy axis. c Magnetic energy density
for a system with perpendicular magnetic anisotropy and a weak in-plane
effective SOC field produced an in-plane spin texture. d Magnetic energy
density for a system with perpendicular magnetic anisotropy and an effective
field with a component parallel to the anisotropy axis, and the spin-textures
that enable these kind of fields.

The magnetization in ferromagnetic materials ${\bf M}$ originates from the
spin angular momentum $\bm{s}$ of lattice atoms. The exchange interaction
enforces the magnetic order but since it is generally isotropic, a source of
magnetic anisotropy is required to dictate the direction of the magnetization
at equilibrium, and to stabilize the magnetic phase. Typically, this
interaction defines two equilibrium positions along the positive and negative
directions of the anisotropy axis, which is characterized by two symmetric
minima in the system’s energy. An external magnetic field with a component
parallel to such an axis will then favor a particular minimum and translates
into a torque ${\bf T}$ on the magnetization, which will align it along the
field direction. Still, large fields are required to induce stable and
reversible switching, a condition which remains a significant technological
challenge. An alternative way is provided by manipulating the SOC to produce
electrically driven torques on the magnetization, known as SOT [136].

SOC is the consequence of the periodic electrical potential due to the nuclei
within the crystal, which is felt by the electrons as an effective magnetic
field ${\bf{B}}_{\rm soc}$ proportional to its angular momentum L[137]. The
characteristics of the coupling between the spin and this effective field
dictate the resulting strength and symmetry properties of the dominant SOC
terms. Additionally, the spin of the conduction electrons propagating through
a magnetic material will couple to its total magnetization through a Zeeman-
like interaction $-(2\Delta_{\rm ex}/\hbar)\bm{s}\cdot{\bf m}$, where $\hbar$
the reduced Planck’s constant[138], ${\bf m}\equiv{\bf M}/|{\bf M}|$ the
normalized magnetization, and $\Delta_{\rm ex}$ the effective exchange
interaction between the localized and itinerant spins. Therefore, in magnetic
systems with SOC, the spin of conduction electrons will be driven by the
torque produced by a combination of the exchange field $\bm{B}_{\rm
ex}\equiv-(2\Delta_{\rm ex}/\hbar){\bf m}$[138, 139] and the effective
momentum-dependent spin-orbit field $\bm{B}_{\rm soc}(\bm{p})$ [137].
Conversely, the action-reaction law states that the magnetization will feel an
opposite momentum-dependent torque due to the electronic spin ${\bf T}_{\rm
soc}(\bm{p})$ and will be then indirectly coupled to spins of the itinerant
electrons which are also affected by the SOC[140]. We should remember however
that since ${\bf{B}}_{\rm soc}$ is a relativistic effect, it does not exists
in the laboratory frame. As a result, any non-zero torque induced by this
field will cancel in equilibrium owing to time-reversal symmetry invariance.
Nevertheless, under the action of an external electrical bias $\Delta V$,
electrons around the Fermi level $E_{F}$ are bestowed with extra energy that
promotes a reorganization of charge favoring electrons with momentum parallel
to the electric field. This non-equilibrium state opens the possibility of a
finite non-equilibirum spin density, an effect first discovered by Edelstein
[36] and referred as inverse spin galvanic [37]. From a different point of
view, such a non-equilibrium spin density can be understood as stemming from
an effective macroscopic SOC field ${\bf{H}}_{\rm soc}(\Delta V)$, that is a
function of the bias and promotes a macroscopic torque ${\bf T}_{\rm soc}={\bf
M}\times{\bf H}_{\rm soc}$. In this sense, the bias provides the energy to
maintain spins alignment with ${\bf H}_{\rm soc}(\Delta V)$, and it will also
affect the magnetization direction through the exchange interaction. The
direction and strength of this field will actually depend on the average of
the momentum-dependent spin polarization, which is usually termed as spin
texture and is computed as the expectation value of the spin operators in
Bloch’s states. In Fig.4c, a sketch of this phenomenon is shown, where the
surface represents the allowed energies ($\,E(\bm{p},\bm{s})\,$) and the
arrows indicate the spin texture pattern. It is clear from the figure that in
the chosen situation, the spin texture compensate after summation over all
allowed momentum. Alternatively, we could also consider the field ${\bf
H}_{\rm soc}$ as an electrically induced non-equilibrium magnetic anisotropy
that promotes the alignment along a different orientation [141].

In systems with an inversion center, the Bloch’s states are doubly degenerated
for any given momentum, leading to two superimposed and opposite spin textures
that will suppress any possible spin torque[142]. This result implies that the
SOTs requires a broken inversion symmetry, as typically achieved through the
creation of an interface. In layered materials though, the inversion symmetry
can be lifted by multilayer stacking [143, 144], strain [123], or even the
interaction with the substrate [125]. The second class of highly symmetric 2D
systems are those possessing a 4-fold rotation axis or higher. At the $\Gamma$
point of those systems, only in-plane fields perpendicular to the layer and
the current such as the one illustrated in Fig.4a are allowed by the in-plane
spin textures (Fig.4c) enforced by the symmetries [145]. In systems with
perpendicular magnetic anisotropy, such a field will gradually reduce the
barrier in Fig.4c without favouring any particular minima, and therefore,
cannot produce deterministic switching. However spin-pseudospin coupling [146]
and other internal degrees of freedom could lead to nontrinvial out-of-plane
textures[147, 148], a possibility that remains largely unexplored in the past.
In fact, currently the role of symmetries is under extensive experimental
study due to its potential for technological applications [149, 122].
Exploiting this out-of-plane spin texture (as depicted in Fig. 4d) is
particularly rewarding since it breaks the energy degeneracy of the moment
pointing $\pm$ $z$ directions by an electric current as shown in Fig. 4b&d.
This leads to deterministic control of the magnetization switching by an
electric current without any needs of external magnetic fields in such
spintronic switching devices.

### Symmetries of the spin-orbit torques in layered materials

The quantitative determination of spin-textures in momentum space requires
knowledge of the microscopic interactions at play, which are complicated to
measure. Still, they lead to the SOT which is a macroscopic observable, or
equivalently, an effective magnetic field ${\bf H}_{\rm soc}$. The underlying
lattice, equilibrium magnetic symmetries and atomic structure fully define the
SOC and, as consequence, will also determine the allowed SOT symmetries.
Moreover, crystal symmetries will also manifests in the torque’s dependence on
the enforced current ${\bf J}$ and the magnetization direction ${\bf m}$
[139]. In general, ${\bf H}_{\rm soc}$ can be written in terms of the current
and magnetization [150, 151, 152, 139].

To illustrate the method, we will limit ourselves only to terms linear in the
current and the magnetization. We already determined that a non-equilibrium
state is mandatory to produce the spin-orbit fields forcing it to be
proportional to current density at lowest order. Its dependence against the
magnetization direction is more complicated since there are two possible
contributions,

${\bf H}^{e}\equiv\Upsilon{\bf J},$ (1)

independent of magnetization and linked to the current density through the
$3\times 3$ matrix $\Upsilon$, and

${\bf H}^{o}\equiv{\bf m}\Lambda{\bf J},$ (2)

which is proportional to the magnetization and is mixed linearly with the
current density through the rank-3 tensor $\Lambda$. The superscript $e/o$ are
included to highlight the even and odd dependence of these fields against
magnetization reversal. These two objects comprises over 36 terms that need to
be determined to fully describe the torque’s symmetries. Through symmetry
analysis, these elements can be further reduced to up to two parameters.

To outline how this symmetry analysis work, we will first start by defining a
point symmetry $\mathcal{R}$, which is a geometric operation that leaves the
positions of the system invariant with respect to a given point, and are
representable as $3\times 3$ unitary matrices if we represent positions as
three-dimensional vectors. The main assumptions of symmetry analysis is that a
representation of these symmetries will also act on all crystal’s physical
properties, which is known as Neumann’s principle[142, 139]. In this sense,
both the current density and magnetization must remain invariant against the
action of all system’s symmetries
$\\{\mathcal{R}_{1},\mathcal{R}_{2},\dots,\mathcal{R}_{N}\\}$, where $N$ the
total number of symmetry operations. This will lead to a set of coupled
equations that will restrict the number of allowed components. In general, one
must evaluate the $N$ symmetries present in the material, but by using group
theory it is possible to reduce it to a much smaller minimal set of symmetries
known as the generating set, that can be combined to span the whole
group[142], and it is sufficient to impose invariance only in that set.

To wrap up and settle the concepts presented here, we provide an example in
Textbox, which illustrates how to use the symmetry analysis to determine the
allowed torques of the $C_{3v}$ point group. This group possesses six symmetry
elements but a generating set of only two, which we choose to be a rotation of
$120^{\circ}$ and an reflection on the $zy$ plane, which are generic for 2D
honeycomb structures and serve as a starting point to describe many different
materials such as: graphene heterostructures [153], hexagonal transition metal
dichalcogenides, Fe3GeTe2[154], CrI3. Materials belonging to higher symmetry
groups such as $D_{6h}$ or $C_{6v}$ will possess fewer allowed torques.

Textbox

The $C_{3v}$ point group of symmetries possess a generating set given by a
$120^{\circ}$ rotation around the $z$-axis and a reflection on the $yz$ plane.
For any possible symmetries $\mathcal{R}$, the matrices $\Upsilon$ and
$\Lambda$ must satisfy the following set of equations [155, 139]
$\Upsilon={\rm det}(\mathcal{R})\,\mathcal{R}\Upsilon\mathcal{R}^{T}$ and
$\Lambda^{i}=\sum_{i^{\prime}=x,y,z}\mathcal{R}_{i,i^{\prime}}\mathcal{R}\Lambda^{i^{\prime}}\mathcal{R}^{T}$,
where $i,j,k\in(x,y,z)$ defining the direction of the axes of a chosen
Cartesian system. Below we show how these tensors change after successive
application of the two symmetries.  The matrix $\Upsilon$ is antisymmetric and
translate into a cross product between the current and the rotation axis ${\bf
H}^{e}=\Upsilon_{yx}{\bf J}\times\hat{\bm{z}}$. The second term is less
straightforward but it is, still, fully defined by just five parameters, that
reduces to three for 2D materials. This is a substantial reduction in the
number of degree of freedom that one would, e.g., use in a phenomenological
fit of the experiment. Also, note that a microscopic calculation would
automatically give such reduction, which can be used as a check, or simply one
can use the symmetry to reduced calculation cost.

A key feature of 2D materials and their combination is the possible
variability of symmetry situations, including interface-induced symmetry
breaking [153, 156, 157, 158, 125]. For instance, stacking different 2D
materials lift the inversion symmetry and can artificially enhance the SOC and
exchange interaction [159, 160, 152, 161, 162, 143]. Accordingly, it is
possible to design and tailor the SOT symmetry by the choice of materials and
the engineering of 2D material stacking. There are several studies that show
how symmetries affect the magnetization dynamics through a 2D layer inserted
at an interface [122, 123, 163, 164, 165, 162, 127, 166]. However, 2D magnets
can also display an internally generated SOT due to their combined SOC, broken
inversion symmetry, and exchange interactions, i.e. not needing an interface
or spin-injection from an adjacent layer, opening the door for _all-in-one_
SOT-memories. To date, the number of synthesized 2D magnetic materials is
still very limited, but novel materials and combination emerge regularly. Most
of the presently known systems belong to the same spatial group and hence will
display the same kind of torques.

Figure 5: Distributions of magnetic materials predicted using first principle
calculations as a function of their Symmetry. The materials family, which
requires a substrate-induced inversion symmetry breaking to generate torque
components, is identified by the filled blue histograms. Non-centrosymmetric
materials which may manifest SOT are outlined by filled red histograms.

### Material screening and grouping

We here perform a brute-force screening of all possible 2D magnetic materials
in the C2DB database [167] and group them according to their symmetry. To do
this, we selected only those that were predicted to be stable in their
freestanding form according to the database’s stability criteria [167]. We
also discarded all materials without a point group symmetry since we will
restrict the discussion to those materials where symmetry analysis is able to
provide physical insights of the torques based on its point group. The final
result is presented in Fig. 5, where it is clear that hexagonal and trigonal
systems are ubiquitous, in agreement with the trends observed in experimental
synthesis. Table I provides allowed torques for the most common space groups
and gives examples of representative materials that could display them. The
space group is shown in the international notation, and we set the $z$-axis as
the high-symmetry rotational axis and aligned the $x$-axis to one of the
lattice vectors. The table is ordered based on the number of symmetry
operators in each group, or equivalently, from the highest-to-lowest symmetric
groups. In this sense, it is clear that the lowest symmetry group C2/m
requires, in general, 12 parameters to define all possible torques, whenever
the highest symmetry group is limited to three parameters. The table also
reinforces the result presented in Textbox 1 regarding the stringent role of
symmetries concerning the allowed torque components. For example, when there
is a two-fold rotation, it is clear there will be different torques pinned to
the two perpendicular directions that we choose to be the $x$ ad $y$ axes, and
in general, one would expect all torques to be defined by a $2\times 2$
matrix. The presence of vertical mirror planes or in-plane rotational axes
further halves the torque components quantity. Systems with three-fold
rotational symmetry are the best examples to understand the emergence of novel
torques upon symmetry breaking. To better appreciate this mechanism, let us
focus on the torques odd against magnetization reversals of materials with the
spacegroup P$\bar{3}$, which are the simplest to analyse although the result
extends to the those with even dependence. These materials are centrosymmetric
but lose this property and belong to the reduced P3 group when deposited onto
a substrate. The remaining three-fold rotational symmetry then permits only
two kinds of odd torques, one perpendicular and another parallel to the
current. The trigonal structure which only maintains a single vertical mirror
plane will then suppress the torque component parallel to the current. Similar
ideas apply to the odd components of the torques. Therefore, the main message
here is that within a given class of crystals, new torques will appear when
the interaction with the environment starts to disrupt internal crystalline
symmetries. These new torque components will be superimposed to the intrinsic
ones robust to symmetry breaking effects providing a way to optimize the
characteristics of the torque by symmetry engineering of materials. Another
interesting result is that following this approach, we recover most of the
synthetized 2D magnets such as CrI3, CrGeTe3 and Fe3GeTe2, but also some
unexplored families such as the trihalides and the cubic magnets such as FeTe.
In the same table we show the SOT directions allowed by the symmetries for all
materials in the freestanding form (blue color) and under a substrate (red
color). Most of the system present inversion symmetry and as a result, do not
show any torque in their freestanding form and require the presence of a
substrate (modelled here by either removing the mirror plane parallel to the
layer or the inversion operator out of the generating set). Interestingly, we
also identify a set of materials with intrinsic crystal asymmetry leading to
an _intrinsic_ torque in Fe3GeTe2 (P$\bar{6}$m2) and certain magnetic
trihalides like CrBr3 (P$\bar{4}$m2)111 Display intrinsic anisotropy-like SOT,
and magnetic Janus structures such as MnSTe (P$\bar{3}$m1)222Display intrinsic
field-like and damping-like SOT. Additionally, we confirm that the angular
dependence of this internally generated torque changes once the substrate is
incorporated and breaks additional symmetries, with the only exception of the
magnetic Janus TMD [168, 169], for which the angular dependence remains the
same.

### Isotropic and anisotropic torques

The Eqs.(1,2) allow us to consider the SOT as a phenomenon produced by a
current-induced effective magnetic field prescribed by the symmetries and the
SOC. However, despite the apparent simplicity of these expressions, the fields
are function of both the magnetization direction and the applied current, and
can take very complex functional forms as it is evident from the less
symmetric materials reported in Table 1. Nevertheless, given that the
functional form of the allowed torques in isotropic systems (i.e, two-
dimensional electron gas) resembles the one produced by the external effective
magnetic given by ${\textbf{H}}_{\rm eff}\propto{\bf J}\times\hat{z}$, early
studies classified them in two components, a field-like torque that produces
magnetization precession and is characterized by the parameter $\tau_{\rm
FL}$, and a damping-like torque that aligns the magnetization along the field
direction and is proportional to $\tau_{\rm DL}$[170]. These two torques
defines what we term as _isotropic_ torques

${{\bf T}}_{\rm iso}={\bf m}\times\left[\,\tau_{\rm FL}{\bf J}\times\hat{{\bf
z}}+\tau_{\rm DL}{\bf m}\times({\bf J}\times\hat{{\bf z}})+\tau_{\rm z}{\rm
m}_{z}{\bf J}\,\right],$ (3)

given that the strength depends solely on the current magnitude $J$, allowing
the definition of even and odd torques using a single parameter for each.
Here, we use three parameters $\tau_{\rm FL}$, $\tau_{\rm DL}$ and $\tau_{\rm
z}$ to represent the field strength for each symmetry. In fact, by following
the procedure outlined in the Textbox 1, it is straightforward to show that
such a simple description, where the torque is fully characterized by a
_single_ effective field, becomes characteristic of systems with two
orthogonal vertical mirror planes, including totally isotropic systems. In the
less symmetrical 2D systems without these two mirror planes, there will be a
structural anisotropy between two orthogonal directions which would translate
into an anisotropic electrical response. As a consequence, the current
response will be characterized by the two diagonal elements of the
conductivity tensor (i.e $\sigma_{xx}$ and $\sigma_{yy}$), while the torque
response will be generally described by two or more effective magnetic fields.
This means that the strength of the total torque will depend both on the
current orientation and magnitude.

To better grasp these concepts, let us consider first the case of system
possessing a three-fold rotational symmetry and at least a mirror plane
(trigonal). In Texbox 1 we demonstrated how to compute the torque matrices for
the $C_{3v}$ point group symmetry and in Table 1 we combined it with Eq.(1)
and Eq.(2) to express it in vectorial form, leading to Eq.(3). The first two
torques are torques produced by the current-induced spin-polarization along
${\bf J}\times\hat{{\bf z}}$ with field-like and damping-like mechanisms
respectively. Both are described by a single parameter indicating that the
strength of the effective field does not depend on the current orientation.
The last torque term is parallel to the current, becoming non-negligible only
when there is an out-of-plane component of the magnetization ${\rm m}_{z}$.
When the mirror plane is removed, a novel field-like torque emerges that is
perpendicular to both the current and the conventional field-like torque

${{\bf T}}_{\rm FL}^{\rm ani}={\bf m}\times\left[\,\tau_{\rm FL}^{I}{\bf
J}\times\hat{{\bf z}}+\tau_{\rm FL}^{II}{\bf J}\right].$ (4)

As a result, this acts as a superposition of two perpendicular in-plane
effective magnetic fields and thus leads to an anisotropic response as a
function of the current direction.

Lower symmetrical systems, such as those with two-fold rotational symmetry,
possess a structural anisotropy between two orthogonal crystalline directions.
Therefore, the most general relation between the current and the torques
should appear in the form of a $2\times 2$ matrix, which is indeed the case
reported in Table 1,

${{\bf T}}^{\rm ani}_{\rm FL}={\bf
m}\times\left(\begin{array}[]{cc}0&\tau_{\rm FL}^{yx}\\\ \tau_{\rm
FL}^{xy}&0\\\ \end{array}\right){\textbf{J}}$ (5)

where $\tau_{\rm FL}^{yx}$ and $\tau_{\rm FL}^{xy}$ two independent
parameters. Since this field is invariant against two-fold rotation, it is
fundamentally different to that in Eq. (4). For example, for $\tau_{\rm
FL}^{xy}=0$, the field will vanish if the current is applied in the $y$
direction, which will not happen for systems with three-fold symmetry.
Therefore, it is evident here that the existence of two-parameters will change
the effective field strength with the current direction enabling a general
case of anisotropic torques.

### Anisotropy-like spin-orbit torques and low symmetry cases

Other systems such as Fe3GeTe2 do not have inversion symmetry due to the lack
of two orthogonal mirror planes. In these systems, the spin-orbit interaction
allows for a SOT that behaves similarly to the one produced by a magnetic
anisotropy, which is the reason we call it an anisotropy-like SOT. The axis of
the anisotropy is defined by the remaining mirror axis and the direction of
the current. It is similar to a magnetic anisotropy in the sense that one can
associate the following energy density [154]

$E_{\rm AL}=J_{x}m_{x}m_{y}-\frac{1}{2}J_{y}(m_{x}^{2}-m_{y}^{2}),$ (6)

which is quadratic in ${\bf m}$, and it produces the effective torque through
its gradient with respect to the magnetization ${{\bf T}}_{\rm AL}=-\tau_{\rm
AL}\bm{\nabla}_{{\bf m}}E_{\rm AL}$.

Certain systems such as monoclinic crystals with just one mirror axis and a
two-fold rotation, facilitate the formation of torques possessing components
from effective magnetic field proportionals to ${\bf J}\times\hat{{\bf y}}$
and ${\bf J}\times\hat{{\bf x}}$, which are favorable conditions for switching
magnets with an out-of-plane perpendicular magnetic anisotropy.

Table 1

### Measurement symmetry with the predicted ${\bf H}_{\rm SOC}$

Experimentally, we are able to detect these predicted torques via low-
frequency second-harmonic measurements[171] and current-induced FMR
experiments[172, 131]. In both detection schemes, we can consider the current-
induced effective fields that tilt the static magnetization direction by a
free energy change of ${\bf M}\cdot{\bf H}_{\rm SOC}$, or that drive magnetic
precession with a magnetic torque of ${\bf M}\times{\bf H}_{\rm SOC}$. We can
substitute the vector expressions of ${\bf H}_{\rm SOC}$ fields in Table 1 to
predict the torque symmetry for future magnetization-control experiments. For
example, by using ${\bf J}\times\hat{{\bf z}}$ as ${\bf H}_{\rm SOC}$, we have
the torque expression of ${\bf M}\times({\bf J}\times\hat{{\bf z}})$ for spin
dynamics excitation. Assuming that a driving current is flowing along the $x$
direction, i.e. ${\bf J}\parallel\hat{{\bf x}}$, the torque expression reduces
into ${\bf M}\times\hat{{\bf y}}$. This suggests that FMR amplitudes excited
by this ${\bf H}_{\rm SOC}$ display $\cos{\phi}$ dependence when the
magnetization is rotated in-plane with $\phi$ defined by the angle between
${\bf J}$ and ${\bf M}$. In general, measured FMR voltages also have
$\sin{2\phi}$ dependence due to a time-dependent change of resistance arising
from anisotropic magnetoresistance and magnetic precession[173]. As a result,
we expect ${\bf H}_{\rm SOC}(\propto{\bf J}\times\hat{{\bf z}})$ produces
$\sin{2\phi}\cos{\phi}$ dependence when we perform FMR voltage measurements
for in-plane angle rotation of ${\bf M}$. We can apply the same vector
analysis for other effective fields for FMR experiments as well as the low-
frequency second-harmonic experiments.

### Strength of the torques

An important point for comparison with experimental data is how to obtain
theoretically the strength of the SOT parameters. In 2D materials, the SOT
originates solely from the current-induced spin density in the material.
Therefore, in the limit of small currents, the best approach to compute these
parameters is via the Kubo formula within the linear-response approximation
[174]. This approach requires knowledge of the magnetization-dependent
Hamiltonian $H({\bf m})$ and an effective way to compute its associated
retarded Green’s functions $G^{+}(H,E)$. For non-interacting electrons, the
spin-orbit torque is determined by the following expression

${\bf T}^{\alpha}(\mu,T)=-\frac{2\pi\Delta_{\rm ex}{\rm
J}_{\alpha}}{\sigma_{\alpha\alpha}}\int dE\,f(E-\mu,T)\,{\rm ImTr}\left[\,{\rm
Im}G^{+}\,\hat{\rm J}_{\alpha}\,\frac{dG^{+}}{dE}(\hat{\bm{\sigma}}\times{\bf
m})\right],$ (7)

where Im Tr represents the imaginary part of the trace of the operators in the
brackets, $f(E-\mu,T)$ the Fermi-Dirac distribution for a given chemical
potential $\mu$ and temperature $T$, $\hat{{\bf J}}$ the current-density
operator and ${\bf J}$ its macroscopic average, $\sigma_{\alpha\alpha}$ the
conductivity, and $\hat{\bm{\sigma}}$ the Pauli matrices operator. This
approach gives access to the SOTs by making no assumptions on their dependence
against magnetization. Furthermore, Eq. (7) can be further sliced into two
contributions, those resulting from from scattering mechanisms of electrons at
the Fermi level (proportional to the conductivity), and those arising from a
Fermi-sea contribution, which in principle can provide both dissipative and
non-dissipative SOTs, and are primarily scattering independent contributions
[35]. The existence of a SOT proportional to scattering times also evidences
that disorder plays a major role to determine the system’s behavior. In simple
systems, the Kubo formula can be evaluated following a semi-analytic solutions
based on perturbation theory [175, 176], where the disorder is usually
described by point impurities. For low concentration of impurities and when
the single scattering approximation is valid, the Kubo formula reduces to the
Boltzmann equation, which offers a simple way to estimate the behavior of the
torque parameters . However, a quantitative determination for a generic
disorder or a structurally complex system is only attainable through linear
scaling quantum transport methodologies [174], that enable reaching the
required size scales to study the torques in the diffusive and localized
regimes. For ferromagnetic materials, the magnetization can be instead
replaced by the exchange-correlation field, facilitating the use of the Kubo
formula to determine the torques from first principles simulations [177, 178].

Beyond the Kubo formula, nonadiabatic time-dependent evolution in realistic
systems is also a great challenge, although there are some recent efforts
along this direction [179].

The Kubo formula provides a method to compute the torque parameters in units
of a torque density (eV/m3). However, since these parameters depend directly
on the driven current density, a current-to-torque conversion efficiency is
usually defined. Such an efficiency definition is quite straightforward in
systems where the torque is mostly due to the spin Hall effect, so it can be
connected to the spin Hall angle which is given by the ratio between the spin
and charge conductivities [35]. Following similar considerations, the torque
conductivity $\sigma_{\tau}$ can be defined as the angular momentum absorbed
by the magnet per second per unit interface area per unit applied electric
field[156], from which the torque efficiency $\chi_{\alpha}^{X}$ is derived
as:

$\chi_{\alpha}^{X}\equiv\frac{2e}{\hbar}\frac{M_{s}t}{\gamma}\frac{\tau_{X}}{{\rm
J}_{\alpha}},$ (8)

where $M_{s}$ the saturation magnetization, $X$ the torque label (FL,DL, etc),
$t$ the thickness of the magnet, which is taken as the Van der Waal gap in 2D
materials. The former expression defines an efficiency for different kind of
torques classified according to their symmetries, but there is an alternative
dimensionless expression which describes the efficiency according to the
torque’s directions expressed as[139]

$\chi_{\alpha\beta}\equiv\frac{e\Delta_{\rm ex}t}{\hbar}\frac{{\rm
T}_{\beta}}{{\rm J}_{\alpha}}.$ (9)

Both expressions allow to quantify a torque efficiency due to a source of
spins independently of geometric factors. For a Cobalt/Platinum system
$\chi_{\alpha}\approx 0.0075$ for a $t\approx 0.85$ nm [180], while in
experiments considering TMD/Py heterostructures the efficiency only reaches
values between $10^{-5}-10^{-3}\,\,\,$ [162]. Another important parameter is
given by $\chi_{\alpha}^{X}/\chi_{\alpha}^{\rm FL}=\tau_{X}/\tau_{\rm FL}$
which measures the relative value of all torques against the dissipative
field-like contribution and generally ranges from 1-10. Notice that in 2D
materials M${}_{s}t$ is actually the magnetization per unit area or sheet
magnetization [107]. From all this discussion, it becomes clear that a lot of
work remains to be done to obtain realistic and quantitative information about
spin torque efficiency in vdW heterostructures combining 2D magnets with
strong SOC materials. Beyond well-controlled ab-initio simulations and the
elaboration of properly designed SOC-dependent tight-binding models, the use
of computational methodologies that can numerically solve the Kubo formula Eq.
(7) for large scale models is a grand challenge, particularly when disorder
effects affecting the spin physics demand non-perturbative treatments. Brute-
force simulations using linear scaling methods demand further developments and
implementations, as well as comparison of their predictions with future
experiments.

As a final remark, the spin-Hall effect[35] should exist in a variety of vdW
material systems, both magnetic and non-magnetic, except in the single
monolayer limit. In order to exploit such spin-charge phenomena, vdW
heterostructures are usually fabricated with one magnetic layer in direct
contact with a spin current source, so as to have a close and efficient
magnetization manipulation. The symmetry of spin-Hall effect is well-
established and the spin-current source layer could be also magnetic, in which
case additional magnetization-dependent anomalous components [181, 182, 183]
might appear.

## Future Outlook

The present review has put in perspective the formidable playground and
plethora of possibilities to create new hybrid materials with emerging spin-
dependent interface properties. Among the vast zoo of novel structures, theory
and modelling based on symmetry features of interfaces bring a pathfinder to
find the most suitable conditions for best controlling spin transport through
spin-orbit or exchange coupling fields. One most urgent quest is the search
for the upper limit of the achievable spin-orbit torque efficiency (at RT and
with scalable materials), which could ultimately enable low-energy and fast
switching of the material magnetization in the frame of SOT memory
technologies[120, 29, 184, 34]. Such is a challenging task which will require
proper combination of theoretical and experimental research within a large
community of material grower, advanced characterization and simulation tools,
as well as optimized device fabrication approaches. The search for materials
with high Curie or Néel temperatures is also a key aspect of this research,
and efforts to guide experiments such as the recently studied monolayer Fe3P
with predicted TC of 420 K[185] should be followed. In this regard, the
exchange coupling strength by controlling molecular orbital alignments in 2D
magnetic semiconductors[53] could be an interesting approach where CrWI6 and
CrWGe2Te6 are predicted to display ferromagnetic order above RT via the purely
super-exchange interaction. Furthermore, high-throughput computation screening
using an existing materials database[186] and other material informatics
methods could be an effective approach to guide us towards discoveries of
novel high-$T_{\rm C}$ vdW magnetic materials. Here we have focused our
attention to the more familiar ferromagnetic cases, but this approach is valid
for predicting torques in non-magnetic vdW systems by using the same space
group analysis as well as for novel types of torques in antiferromagnetic
systems, such as the Néel SOT[187] and other antiferromagnetic specific SOC
effects[188] that are ripe for exploration in these exciting 2D magnetic
materials class. Looking further for more exotic 2D vdW materials with order
parameters, multiferroics is predicted in a certain class of 2D materials[189]
such as in a doped CrBr3 monolayer[190], in a bilayer of halogen-intercalated
phosphorene[191] as well as in a monolayer ReWCl6 with magnetic transitions
induced by electric field[192]. Some vdW magnetic materials, such as
MnBi2Te4[193, 194, 195], host topological electronic states and offer unique
topologically-protected transport phenomena[196]. We anticipate that this
research domain will be rapidly expanding to explore the topological transport
as well as its use in spintronic applications[197]. Besides, nontrivial spin
transport physics is also expected to develop at complex interfaces of vdW
heterostructures, including the generation of skyrmions[198, 199, 200], the
excitation and detection of magnons at the 2D limit[201, 202, 203] and their
interaction with other spin-dependent characteristics of interfaced materials,
as well as a wealth of proximity effects which could reveal unprecedented
manifestations of entanglement and many-body physics, as recently discussed in
the context of topological superconductivity or unconventional
ferroelectricity in magic-angle twisted bi/trilayer graphene and related moiré
superlattices [204]. In conclusion, there is a lot of rooms for further
exploration of spin transport at the frontier of layered magnetism and SOC
materials, and likely many exciting discoveries waiting for emerging to the
surface.

## References

  * [1] Huang, B. _et al._ Layer-dependent ferromagnetism in a van der Waals crystal down to the monolayer limit.  _Nature_ 546, 270–273, DOI: 10.1038/nature22391 (2017).
  * [2] Gong, C. _et al._ Discovery of intrinsic ferromagnetism in two-dimensional van der Waals crystals.  _Nature_ 546, 265–269, DOI: 10.1038/nature22060 (2017).
  * [3] Burch, K. S., Mandrus, D. & Park, J. G.  Magnetism in two-dimensional van der Waals materials.  _Nature_ 563, 47–52, DOI: 10.1038/s41586-018-0631-z (2018).
  * [4] Fei, Z. _et al._ Two-dimensional itinerant ferromagnetism in atomically thin Fe3GeTe2.  _Nature Materials_ 17, 778–782, DOI: 10.1038/s41563-018-0149-7 (2018).
  * [5] Li, H., Ruan, S. & Zeng, Y. J.  Intrinsic Van Der Waals Magnetic Materials from Bulk to the 2D Limit: New Frontiers of Spintronics.  _Advanced Materials_ 31, 1–34, DOI: 10.1002/adma.201900065 (2019).
  * [6] Gong, Y. _et al._ Experimental Realization of an Intrinsic Magnetic Topological Insulator *.  _Chinese Physics Letters_ 36, 076801, DOI: 10.1088/0256-307X/36/7/076801 (2019).
  * [7] Geim, A. K. & Grigorieva, I. V.  Van der waals heterostructures.  _Nature_ 499, 419–425, DOI: 10.1038/nature12385 (2013).
  * [8] Liu, Y. _et al._ Van der waals heterostructures and devices.  _Nature Reviews Materials_ 1, 16042, DOI: 10.1038/natrevmats.2016.42 (2016).
  * [9] Andrei, E. Y. _et al._ The marvels of moiré materials.  _Nature Reviews Materials_ 6, 201–206, DOI: 10.1038/s41578-021-00284-1 (2021).
  * [10] Freitas, D. C. _et al._ Ferromagnetism in layered metastable 1T-CrTe2.  _Journal of Physics Condensed Matter_ 27, 176002, DOI: 10.1088/0953-8984/27/17/176002 (2015).
  * [11] Freitas, D. C. _et al._ Antiferromagnetism and ferromagnetism in layered 1T-CrSe2 with v and Ti replacements.  _Physical Review B - Condensed Matter and Materials Physics_ 87, 014420, DOI: 10.1103/PhysRevB.87.014420 (2013).
  * [12] Zhang, W.-B., Qu, Q., Zhu, P. & Lam, C.-H.  Robust intrinsic ferromagnetism and half semiconductivity in stable two-dimensional single-layer chromium trihalides.  _Journal of Materials Chemistry C_ 3, 12457–12468, DOI: 10.1039/C5TC02840J (2015).  1507.07275.
  * [13] McGuire, M. A., Dixit, H., Cooper, V. R. & Sales, B. C.  Coupling of Crystal Structure and Magnetism in the Layered, Ferromagnetic Insulator CrI3.  _Chemistry of Materials_ 27, 612–620, DOI: 10.1021/cm504242t (2015).
  * [14] McGuire, M. A. _et al._ Magnetic behavior and spin-lattice coupling in cleavable van der Waals layered CrCl3 crystals.  _Physical Review Materials_ 1, 14001, DOI: 10.1103/PhysRevMaterials.1.014001 (2017).
  * [15] Joy, P. A. & Vasudevan, S.  Magnetism in the layered transition-metal thiophosphates MPS3 (M=Mn, Fe, and Ni).  _Physical Review B_ 46, 5425–5433, DOI: 10.1103/PhysRevB.46.5425 (1992).
  * [16] Lin, W., Chen, K., Zhang, S. & Chien, C. L.  Enhancement of Thermally Injected Spin Current through an Antiferromagnetic Insulator.  _Physical Review Letters_ 116, 186601, DOI: 10.1103/PhysRevLett.116.186601 (2016).  1603.00931.
  * [17] Bonilla, M. _et al._ Strong room-temperature ferromagnetism in VSe2 monolayers on van der Waals substrates.  _Nature Nanotechnology_ 13, 289–293, DOI: 10.1038/s41565-018-0063-9 (2018).
  * [18] Deng, Y. _et al._ Gate-tunable room-temperature ferromagnetism in two-dimensional Fe3GeTe2.  _Nature_ 563, 94–99, DOI: 10.1038/s41586-018-0626-9 (2018).
  * [19] O’Hara, D. J. _et al._ Room Temperature Intrinsic Ferromagnetism in Epitaxial Manganese Selenide Films in the Monolayer Limit.  _Nano Letters_ 18, 3125–3131, DOI: 10.1021/acs.nanolett.8b00683 (2018).  1802.08152.
  * [20] O’Hara, D. J., Zhu, T. & Kawakami, R. K.  Importance of Paramagnetic Background Subtraction for Determining the Magnetic Moment in Epitaxially Grown Ultrathin van der Waals Magnets.  _IEEE Magnetics Letters_ 9, 1–5, DOI: 10.1109/LMAG.2018.2867339 (2018).
  * [21] Walker, M. B. & Withers, R. L.  Stacking of charge-density waves in 1T transition-metal dichalcogenides.  _Physical Review B_ 28, 2766–2774, DOI: 10.1103/PhysRevB.28.2766 (1983).
  * [22] Eaglesham, D. J., Withers, R. L. & Bird, D. M.  Charge-density-wave transitions in 1T-VSe2.  _Journal of Physics C: Solid State Physics_ 19, 359–367, DOI: 10.1088/0022-3719/19/3/006 (1986).
  * [23] Feng, J. _et al._ Electronic Structure and Enhanced Charge-Density Wave Order of Monolayer VSe2.  _Nano Letters_ 18, 4493–4499, DOI: 10.1021/acs.nanolett.8b01649 (2018).
  * [24] Yang, J. _et al._ Thickness dependence of the charge-density-wave transition temperature in VSe2.  _Applied Physics Letters_ 105, 063109, DOI: 10.1063/1.4893027 (2014).
  * [25] Coelho, P. M. _et al._ Charge Density Wave State Suppresses Ferromagnetic Ordering in VSe2 Monolayers.  _The Journal of Physical Chemistry C_ 123, 14089–14096, DOI: 10.1021/acs.jpcc.9b04281 (2019).
  * [26] Hirohata, A. _et al._ Review on spintronics: Principles and device applications.  _Journal of Magnetism and Magnetic Materials_ 509, 166711, DOI: https://doi.org/10.1016/j.jmmm.2020.166711 (2020).
  * [27] Grollier, J. _et al._ Neuromorphic spintronics.  _Nature Electronics_ 3, 360–370, DOI: 10.1038/s41928-019-0360-9 (2020).
  * [28] Lin, X., Yang, W., Wang, K. L. & Zhao, W.  Two-dimensional spintronics for low-power electronics.  _Nature Electronics_ 2, 274–283, DOI: 10.1038/s41928-019-0273-7 (2019).
  * [29] Kawahara, T., Ito, K., Takemura, R. & Ohno, H.  Spin-transfer torque ram technology: Review and prospect.  _Microelectronics Reliability_ 52, 613–627, DOI: https://doi.org/10.1016/j.microrel.2011.09.028 (2012).  Advances in non-volatile memory technology.
  * [30] Bhatti, S. _et al._ Spintronics based random access memory: a review.  _Materials Today_ 20, 530–548, DOI: https://doi.org/10.1016/j.mattod.2017.07.007 (2017).
  * [31] Chernyshov, A. _et al._ Evidence for reversible control of magnetization in a ferromagnetic material by means of spin–orbit magnetic field.  _Nature Physics_ 5, 656–659, DOI: 10.1038/nphys1362 (2009).  0812.3160.
  * [32] Miron, I. M. _et al._ Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection.  _Nature_ 476, 189–193, DOI: 10.1038/nature10309 (2011).
  * [33] Liu, L. _et al._ Spin-Torque Switching with the Giant Spin Hall Effect of Tantalum.  _Science_ 336, 555–558, DOI: 10.1126/science.1218197 (2012).  1203.2875.
  * [34] Cubukcu, M. _et al._ Ultra-Fast Perpendicular Spin-Orbit Torque MRAM.  _IEEE Transactions on Magnetics_ 54, 9300204, DOI: 10.1109/TMAG.2017.2772185 (2018).  1509.02375.
  * [35] Sinova, J., Valenzuela, S. O., Wunderlich, J., Back, C. H. & Jungwirth, T.  Spin Hall effects.  _Reviews of Modern Physics_ 87, 1213–1260, DOI: 10.1103/RevModPhys.87.1213 (2015).
  * [36] Edelstein, V. M.  Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems.  _Solid State Communications_ 73, 233–235, DOI: 10.1016/0038-1098(90)90963-C (1990).
  * [37] Manchon, A. _et al._ Current-induced spin-orbit torques in ferromagnetic and antiferromagnetic systems.  _Reviews of Modern Physics_ 91, 035004, DOI: 10.1103/RevModPhys.91.035004 (2019).  1801.09636.
  * [38] Burch, K. S., Mandrus, D. & Park, J.-G.  Magnetism in two-dimensional van der waals materials.  _Nature_ 563, 47–52, DOI: 10.1038/s41586-018-0631-z (2018).
  * [39] Gibertini, M., Koperski, M., Morpurgo, A. F. & Novoselov, K. S.  Magnetic 2d materials and heterostructures.  _Nature Nanotechnology_ 14, 408–419, DOI: 10.1038/s41565-019-0438-6 (2019).
  * [40] Gong, C. & Zhang, X.  Two-dimensional magnetic crystals and emergent heterostructure devices.  _Science_ 363, DOI: 10.1126/science.aav4450 (2019).  https://science.sciencemag.org/content/363/6428/eaav4450.full.pdf.
  * [41] Mak, K. F., Shan, J. & Ralph, D. C.  Probing and controlling magnetic states in 2d layered magnetic materials.  _Nature Reviews Physics_ 1, 646–661, DOI: 10.1038/s42254-019-0110-y (2019).
  * [42] Huang, B. _et al._ Emergent phenomena and proximity effects in two-dimensional magnets and heterostructures.  _Nature Materials_ 19, 1276–1289, DOI: 10.1038/s41563-020-0791-8 (2020).
  * [43] Huang, P. _et al._ Recent advances in two-dimensional ferromagnetism: materials synthesis, physical properties and device applications.  _Nanoscale_ 12, 2309–2327, DOI: 10.1039/C9NR08890C (2020).
  * [44] Wei, S. _et al._ Emerging intrinsic magnetism in two-dimensional materials: theory and applications.  _2D Materials_ 8, 012005, DOI: 10.1088/2053-1583/abc8cb (2020).
  * [45] Han, W.  Perspectives for spintronics in 2d materials.  _APL Materials_ 4, 032401, DOI: 10.1063/1.4941712 (2016).  https://doi.org/10.1063/1.4941712.
  * [46] Avsar, A. _et al._ Colloquium: Spintronics in graphene and other two-dimensional materials.  _Rev. Mod. Phys._ 92, 021003, DOI: 10.1103/RevModPhys.92.021003 (2020).
  * [47] Husain, S. _et al._ Emergence of spinâ€“orbit torques in 2d transition metal dichalcogenides: A status update.  _Applied Physics Reviews_ 7, 041312, DOI: 10.1063/5.0025318 (2020).  https://doi.org/10.1063/5.0025318.
  * [48] Ahn, E. C.  2d materials for spintronic devices.  _npj 2D Materials and Applications_ 4, 17, DOI: 10.1038/s41699-020-0152-0 (2020).
  * [49] Anderson, P. W.  Antiferromagnetism. Theory of superexchange interaction.  _Physical Review_ 79, 350–356, DOI: 10.1103/PhysRev.79.350 (1950).
  * [50] Lado, J. L. & Fernández-Rossier, J.  On the origin of magnetic anisotropy in two dimensional CrI3.  _2D Materials_ 4, 35002, DOI: 10.1088/2053-1583/aa75ed (2017).  1704.03849.
  * [51] Kanamori, J.  Superexchange interaction and symmetry properties of electron orbitals.  _Journal of Physics and Chemistry of Solids_ 10, 87–98, DOI: 10.1016/0022-3697(59)90061-7 (1959).
  * [52] Goodenough, J. B.  Theory of the role of covalence in the perovskite-type manganites [La,M(II)]MnO3.  _Physical Review_ 100, 564–573, DOI: 10.1103/PhysRev.100.564 (1955).
  * [53] Huang, C. _et al._ Toward Intrinsic Room-Temperature Ferromagnetism in Two-Dimensional Semiconductors.  _Journal of the American Chemical Society_ 140, 11519–11525, DOI: 10.1021/jacs.8b07879 (2018).
  * [54] Kim, H. H. _et al._ Evolution of interlayer and intralayer magnetism in three atomically thin chromium trihalides.  _Proceedings of the National Academy of Sciences of the United States of America_ 166, 11131–11136, DOI: 10.1073/pnas.1902100116 (2019).  1903.01409.
  * [55] Wang, F. _et al._ New Frontiers on van der Waals Layered Metal Phosphorous Trichalcogenides.  _Advanced Functional Materials_ 28, 1802151, DOI: 10.1002/adfm.201802151 (2018).
  * [56] Ur Rehman, Z. _et al._ Magnetic isotropy/anisotropy in layered metal phosphorous Trichalcogenide MPS3 (M = Mn, Fe)single crystals.  _Micromachines_ 9, DOI: 10.3390/mi9060292 (2018).
  * [57] Kim, K. _et al._ Suppression of magnetic ordering in XXZ-type antiferromagnetic monolayer NiPS3.  _Nature Communications_ 10, 345, DOI: 10.1038/s41467-018-08284-6 (2019).
  * [58] Zener, C.  Interaction between the d-shells in the transition metals. II. Ferromagnetic compounds of manganese with Perovskite structure.  _Physical Review_ 82, 403–405, DOI: 10.1103/PhysRev.82.403 (1951).
  * [59] Jonker, G. & Van Santen, J.  Ferromagnetic compounds of manganese with perovskite structure.  _Physica_ 16, 337–349, DOI: 10.1016/0031-8914(50)90033-4 (1950).
  * [60] Wang, N. _et al._ Transition from ferromagnetic semiconductor to ferromagnetic metal with enhanced curie temperature in cr2ge2te6 via organic ion intercalation.  _Journal of the American Chemical Society_ 141, 17166–17173, DOI: 10.1021/jacs.9b06929 (2019).  PMID: 31599579, https://doi.org/10.1021/jacs.9b06929.
  * [61] Verzhbitskiy, I. A. _et al._ Controlling the magnetic anisotropy in Cr2Ge2Te6 by electrostatic gating.  _Nature Electronics_ 3, 460–465, DOI: 10.1038/s41928-020-0427-7 (2020).
  * [62] Blundell, S. & Thouless, D.  _Magnetism in Condensed Matter_ , vol. 71 of _Oxford master series in condensed matter physics_ (Oxford University Press, Oxford, 2003).
  * [63] Weiss, P.  L’hypothèse du champ moléculaire et la propriété ferromagnétique.  _Journal de Physique Théorique et Appliquée_ 6, 661–690, DOI: 10.1051/jphystap:019070060066100 (1907).
  * [64] Watanabe, H.  Collective Electron Ferromagnetism, II.  _Journal of the Physical Society of Japan_ 3, 317–322, DOI: 10.1143/JPSJ.3.317 (1948).
  * [65] Stoner, E. C.  Collective electron ferromagnetism II. Energy and specific heat.  _Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences_ 169, 339–371, DOI: 10.1098/rspa.1939.0003 (1939).
  * [66] Nakano, M. _et al._ Intrinsic 2D Ferromagnetism in V5Se8 Epitaxial Thin Films.  _Nano Letters_ 19, 8806–8810, DOI: 10.1021/acs.nanolett.9b03614 (2019).
  * [67] May, A. F., Calder, S., Cantoni, C., Cao, H. & McGuire, M. A.  Magnetic structure and phase stability of the van der Waals bonded ferromagnet Fe3-xGeTe2.  _Physical Review B_ 93, 14411, DOI: 10.1103/PhysRevB.93.014411 (2016).  1508.06959.
  * [68] May, A. F. _et al._ Ferromagnetism Near Room Temperature in the Cleavable van der Waals Crystal Fe5GeTe2.  _ACS Nano_ 13, 4436–4442, DOI: 10.1021/acsnano.8b09660 (2019).
  * [69] Morosan, E. _et al._ Sharp switching of the magnetization in Fe0.25TaS2.  _Physical Review B_ 75, 104401, DOI: 10.1103/PhysRevB.75.104401 (2007).
  * [70] McGuire, M. A. _et al._ Antiferromagnetism in the van der Waals layered spin-lozenge semiconductor CrTe3.  _Physical Review B_ 95, 144421, DOI: 10.1103/PhysRevB.95.144421 (2017).  1701.08621.
  * [71] Wang, Y. _et al._ Magnetic anisotropy and topological Hall effect in the trigonal chromium tellurides Cr5Te8.  _Physical Review B_ 100, 24434, DOI: 10.1103/PhysRevB.100.024434 (2019).
  * [72] Yan, J. _et al._ Anomalous hall effect of the quasi-two-dimensional weak itinerant ferromagnet cr 4.14 te 8.  _EPL (Europhysics Letters)_ 124, 67005, DOI: 10.1209/0295-5075/124/67005 (2019).
  * [73] Zhuang, H. L., Kent, P. R. & Hennig, R. G.  Strong anisotropy and magnetostriction in the two-dimensional Stoner ferromagnet Fe3GeTe2.  _Physical Review B_ 93, 134407, DOI: 10.1103/PhysRevB.93.134407 (2016).
  * [74] Cai, X. _et al._ Atomically thin CrCl3: an in-plane layered antiferromagnetic insulator.  _Nano letters_ 19, 3993–3998 (2019).
  * [75] Wiedenmann, A., Rossat-Mignod, J., Louisy, A., Brec, R. & Rouxel, J.  Neutron diffraction study of the layered compounds MnPSe3 and FePSe3.  _Solid State Communications_ 40, 1067–1072 (1981).
  * [76] Long, G. _et al._ Isolation and characterization of few-layer manganese thiophosphite.  _ACS nano_ 11, 11330–11336 (2017).
  * [77] Le Flem, G., Brec, R., Ouvard, G., Louisy, A. & Segransan, P.  Magnetic interactions in the layer compounds MPX3 (M= Mn, Fe, Ni; X= S, Se).  _Journal of Physics and Chemistry of Solids_ 43, 455–461 (1982).
  * [78] Ghazaryan, D. _et al._ Magnon-assisted tunnelling in van der waals heterostructures based on CrBr3.  _Nature Electronics_ 1, 344–349 (2018).
  * [79] Samuelsen, E., Silberglitt, R., Shirane, G. & Remeika, J.  Spin waves in ferromagnetic CrBr3 studied by inelastic neutron scattering.  _Physical Review B_ 3, 157 (1971).
  * [80] Lee, J.-U. _et al._ Ising-type magnetic ordering in atomically thin FePS3.  _Nano letters_ 16, 7433–7438 (2016).
  * [81] Liu, P. _et al._ Exploring the magnetic ordering in atomically thin antiferromagnetic MnPSe3 by raman spectroscopy.  _Journal of Alloys and Compounds_ 828, 154432 (2020).
  * [82] Huang, B. _et al._ Electrical control of 2D magnetism in bilayer CrI3.  _Nature Nanotechnology_ 13, 544–548, DOI: 10.1038/s41565-018-0121-3 (2018).  1802.06979.
  * [83] Zhong, D. _et al._ Van der waals engineering of ferromagnetic semiconductor heterostructures for spin and valleytronics.  _Science advances_ 3, e1603113 (2017).
  * [84] Chattopadhyay, T., Brückel, T. & Burlet, P.  Spin correlation in the frustrated antiferromagnet MnS2 above the néel temperature.  _Physical Review B_ 44, 7394 (1991).
  * [85] Son, S. _et al._ Bulk properties of the van der Waals hard ferromagnet VI3.  _Physical Review B_ 99, 41402, DOI: 10.1103/PhysRevB.99.041402 (2019).  1812.05284.
  * [86] Khan, S. _et al._ Spin dynamics study in layered van der waals single-crystal Cr2Ge2Te6.  _Physical Review B_ 100, 134437 (2019).
  * [87] Lin, M.-W. _et al._ Ultrathin nanosheets of CrSiTe3 : a semiconducting two-dimensional ferromagnetic material.  _Journal of Materials Chemistry C_ 4, 315–322, DOI: 10.1039/C5TC03463A (2016).
  * [88] Lv, H., Lu, W., Shao, D., Liu, Y. & Sun, Y.  Strain-controlled switch between ferromagnetism and antiferromagnetism in 1T-CrX2 (X= Se, Te) monolayers.  _Physical Review B_ 92, 214419 (2015).
  * [89] Sun, X. _et al._ Room temperature ferromagnetism in ultra-thin van der waals crystals of 1T-CrTe2.  _Nano Research_ 13, 3358–3363 (2020).
  * [90] Chu, J. _et al._ Sub-millimeter-scale growth of one-unit-cell-thick ferrimagnetic Cr2S3 nanosheets.  _Nano letters_ 19, 2154–2161 (2019).
  * [91] Xie, L. _et al._ An atomically thin air-stable narrow-gap semiconductor Cr2S3 for broadband photodetection with high responsivity.  _Advanced Electronic Materials_ 2000962 (2020).
  * [92] Zeugner, A. _et al._ Chemical aspects of the candidate antiferromagnetic topological insulator MnBi2Te4.  _Chemistry of Materials_ 31, 2795–2806 (2019).
  * [93] Kan, M., Adhikari, S. & Sun, Q.  Ferromagnetism in MnX2 (X= S, Se) monolayers.  _Physical Chemistry Chemical Physics_ 16, 4990–4994 (2014).
  * [94] Itoh, H. & Miyahara, S.  Magnetic susceptibility and thermal expansion of MnSe2 with pyrite structure.  _Journal of the Physical Society of Japan_ 42, 470–472 (1977).
  * [95] Lei, S. _et al._ High mobility in a van der waals layered antiferromagnetic metal.  _Science advances_ 6, eaay6407 (2020).
  * [96] Zhang, Y. _et al._ Ultrathin magnetic 2d single-crystal CrSe.  _Advanced Materials_ 31, 1900056 (2019).
  * [97] Zhang, H., Liu, L.-M. & Lau, W.-M.  Dimension-dependent phase transition and magnetic properties of VS2.  _Journal of Materials Chemistry A_ 1, 10821–10828 (2013).
  * [98] Zhang, L.-Z. _et al._ Critical behavior and magnetocaloric effect of the quasi-two-dimensional room-temperature ferromagnet Cr4Te5.  _Physical Review B_ 101, 214413 (2020).
  * [99] Liu, Y. _et al._ Magnetic anisotropy and entropy change in trigonal Cr5Te8.  _Physical Review B_ 100, 245114 (2019).
  * [100] Seo, J. _et al._ Nearly room temperature ferromagnetism in a magnetic metal-rich van der waals metal.  _Science advances_ 6, eaay8912 (2020).
  * [101] Mermin, N. D. & Wagner, H.  Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models.  _Physical Review Letters_ 17, 1133–1136, DOI: 10.1103/PhysRevLett.17.1133 (1966).
  * [102] Jiang, S., Li, L., Wang, Z., Mak, K. F. & Shan, J.  Controlling magnetism in 2d cri3 by electrostatic doping.  _Nature Nanotechnology_ 13, 549–553, DOI: 10.1038/s41565-018-0135-x (2018).
  * [103] Song, T. _et al._ Switching 2D magnetic states via pressure tuning of layer stacking.  _Nature Materials_ 18, 1298–1302, DOI: 10.1038/s41563-019-0505-2 (2019).  1905.10860.
  * [104] Matsukura, F., Tokura, Y. & Ohno, H.  Control of magnetism by electric fields.  _Nature Nanotechnology_ 10, 209–220, DOI: 10.1038/nnano.2015.22 (2015).
  * [105] McGuire, M. A.  Crystal and magnetic structures in layered, transition metal dihalides and trihalides.  _Crystals_ 7, DOI: 10.3390/cryst7050121 (2017).  1704.08225.
  * [106] Sivadas, N., Okamoto, S. & Xiao, D.  Gate-Controllable Magneto-optic Kerr Effect in Layered Collinear Antiferromagnets.  _Physical Review Letters_ 117, 267203, DOI: 10.1103/PhysRevLett.117.267203 (2016).  1607.02156.
  * [107] Jiang, S., Shan, J. & Mak, K. F.  Electric-field switching of two-dimensional van der Waals magnets.  _Nature Materials_ 17, 406–410, DOI: 10.1038/s41563-018-0040-6 (2018).
  * [108] Song, T. _et al._ Voltage Control of a van der Waals Spin-Filter Magnetic Tunnel Junction.  _Nano Letters_ 19, 915–920, DOI: 10.1021/acs.nanolett.8b04160 (2019).
  * [109] Wang, Z. _et al._ Electric-field control of magnetism in a few-layered van der Waals ferromagnetic semiconductor.  _Nature Nanotechnology_ 13, 554–559, DOI: 10.1038/s41565-018-0186-z (2018).
  * [110] Sun, Y. _et al._ Effects of hydrostatic pressure on spin-lattice coupling in two-dimensional ferromagnetic Cr2Ge2Te6.  _Applied Physics Letters_ 112, 72409, DOI: 10.1063/1.5016568 (2018).
  * [111] Mondal, S. _et al._ Effect of hydrostatic pressure on ferromagnetism in two-dimensional CrI3.  _Physical Review B_ 99, 180407, DOI: 10.1103/PhysRevB.99.180407 (2019).  1901.00706.
  * [112] Lin, Z. _et al._ Pressure-induced spin reorientation transition in layered ferromagnetic insulator Cr2Ge2Te6.  _Physical Review Materials_ 2, 51004, DOI: 10.1103/PhysRevMaterials.2.051004 (2018).
  * [113] Li, T. _et al._ Pressure-controlled interlayer magnetism in atomically thin CrI3.  _Nature Materials_ 18, 1303–1308, DOI: 10.1038/s41563-019-0506-1 (2019).  1905.10905.
  * [114] Jiang, P. _et al._ Stacking tunable interlayer magnetism in bilayer CrI3.  _Physical Review B_ 99, 144401, DOI: 10.1103/PhysRevB.99.144401 (2019).  1806.09274.
  * [115] Jang, S. W., Jeong, M. Y., Yoon, H., Ryee, S. & Han, M. J.  Microscopic understanding of magnetic interactions in bilayer CrI3.  _Physical Review Materials_ 3, 31001, DOI: 10.1103/PhysRevMaterials.3.031001 (2019).  1809.01388.
  * [116] Sivadas, N., Okamoto, S., Xu, X., Fennie, C. J. & Xiao, D.  Stacking-Dependent Magnetism in Bilayer CrI3.  _Nano Letters_ 18, 7658–7664, DOI: 10.1021/acs.nanolett.8b03321 (2018).  1808.06559.
  * [117] Soriano, D., Cardoso, C. & Fernández-Rossier, J.  Interplay between interlayer exchange and stacking in CrI3 bilayers.  _Solid State Communications_ 299, 113662, DOI: 10.1016/j.ssc.2019.113662 (2019).  1807.00357.
  * [118] Klein, D. R. _et al._ Enhancement of interlayer exchange in an ultrathin two-dimensional magnet.  _Nature Physics_ 15, 1255–1260, DOI: 10.1038/s41567-019-0651-0 (2019).
  * [119] Kent, A. D. & Worledge, D. C.  A new spin on magnetic memories.  _Nature Nanotechnology_ 10, 187–191, DOI: 10.1038/nnano.2015.24 (2015).
  * [120] Dieny, B. _et al._ Opportunities and challenges for spintronics in the microelectronics industry.  _Nature Electronics_ 3, 446–459, DOI: 10.1038/s41928-020-0461-5 (2020).  1908.10584.
  * [121] Grollier, J. _et al._ Neuromorphic spintronics.  _Nature Electronics_ 3, 360–370, DOI: 10.1038/s41928-019-0360-9 (2020).
  * [122] MacNeill, D. _et al._ Control of spin-orbit torques through crystal symmetry in WTe2 /ferromagnet bilayers.  _Nature Physics_ 13, 300–305, DOI: 10.1038/nphys3933 (2017).  1605.02712.
  * [123] Guimarães, M. H., Stiehl, G. M., MacNeill, D., Reynolds, N. D. & Ralph, D. C.  Spin-Orbit Torques in NbSe2/Permalloy Bilayers.  _Nano Letters_ 18, 1311–1316, DOI: 10.1021/acs.nanolett.7b04993 (2018).  1801.07281.
  * [124] Shi, S. _et al._ All-electric magnetization switching and Dzyaloshinskii–Moriya interaction in WTe2/ferromagnet heterostructures.  _Nature Nanotechnology_ 14, 945–949, DOI: 10.1038/s41565-019-0525-8 (2019).
  * [125] Alghamdi, M. _et al._ Highly Efficient Spin-Orbit Torque and Switching of Layered Ferromagnet Fe3GeTe2.  _Nano Letters_ 19, 4400–4405, DOI: 10.1021/acs.nanolett.9b01043 (2019).  1903.00571.
  * [126] Wang, X. _et al._ Current-driven magnetization switching in a van der Waals ferromagnet Fe3GeTe2.  _Science Advances_ 5, DOI: 10.1126/sciadv.aaw8904 (2019).
  * [127] Ostwal, V., Shen, T. & Appenzeller, J.  Efficient Spin-Orbit Torque Switching of the Semiconducting Van Der Waals Ferromagnet Cr2Ge2Te6.  _Advanced Materials_ 32, 1–7, DOI: 10.1002/adma.201906021 (2020).
  * [128] Ralph, D. & Stiles, M.  Spin transfer torques.  _Journal of Magnetism and Magnetic Materials_ 320, 1190–1216, DOI: 10.1016/j.jmmm.2007.12.019 (2008).
  * [129] Avci, C. O. _et al._ Current-induced switching in a magnetic insulator.  _Nature Materials_ 16, 309–314, DOI: 10.1038/nmat4812 (2017).
  * [130] Shin, I. _et al._ Spin-orbit Torque Switching in an All-Van der Waals Heterostructure (2021).  2102.09300.
  * [131] Fang, D. _et al._ Spin-orbit-driven ferromagnetic resonance.  _Nature Nanotechnology_ 6, 413–417, DOI: 10.1038/nnano.2011.68 (2011).
  * [132] Ciccarelli, C. _et al._ Room-temperature spin-orbit torque in NiMnSb.  _Nature Physics_ 12, 855–860, DOI: 10.1038/nphys3772 (2016).  1510.03356.
  * [133] Yoshimi, R. _et al._ Current-driven magnetization switching in ferromagnetic bulk Rashba semiconductor (Ge,Mn)Te.  _Science Advances_ 4, DOI: 10.1126/sciadv.aat9989 (2018).
  * [134] Kurebayashi, H. _et al._ An antidamping spin-orbit torque originating from the Berry curvature.  _Nature Nanotechnology_ 9, 211–217, DOI: 10.1038/nnano.2014.15 (2014).
  * [135] Bradley, C. J. & Cracknell, A. P.  _The Mathematical Theory of Symmetry in Solids: Representation theory for point groups and space groups_ (Oxford University Press, 1972).
  * [136] Manipatruni, S. _et al._ Scalable energy-efficient magnetoelectric spin–orbit logic.  _Nature_ 565, 35–42, DOI: 10.1038/s41586-018-0770-2 (2019).
  * [137] Bandyopadhyay, S. & Cahay, M.  _Introduction to Spintronics_ (CRC Press, 2008), first edn.
  * [138] Kurz, P., Förster, F., Nordström, L., Bihlmayer, G. & Blügel, S.  Ab initio treatment of noncollinear magnets with the full-potential linearized augmented plane wave method.  _Physical Review B_ 69, 024415, DOI: 10.1103/PhysRevB.69.024415 (2004).
  * [139] Freimuth, F., Blügel, S. & Mokrousov, Y.  Spin-orbit torques in Co/Pt(111) and Mn/W(001) magnetic bilayers from first principles.  _Physical Review B_ 90, 174423, DOI: 10.1103/PhysRevB.90.174423 (2014).  1305.4873.
  * [140] Haney, P. M. _et al._ Current-induced order parameter dynamics: Microscopic theory applied to Co/Cu/Co.  _Physical Review B_ 76, 024404, DOI: 10.1103/PhysRevB.76.024404 (2007).
  * [141] Garate, I. & MacDonald, A. H.  Influence of a transport current on magnetic anisotropy in gyrotropic ferromagnets.  _Physical Review B_ 80, 134403, DOI: 10.1103/PhysRevB.80.134403 (2009).  0905.3856.
  * [142] Dresselhaus, M. S., Dresselhaus, G. & Jorio, A.  _Group theory_ , vol. 53 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2008).
  * [143] Zollner, K. _et al._ Scattering-induced and highly tunable by gate damping-like spin-orbit torque in graphene doubly proximitized by two-dimensional magnet Cr2Ge2Te6 and monolayer WS2.  _Physical Review Research_ 2, 1–12, DOI: 10.1103/physrevresearch.2.043057 (2020).  1910.08072.
  * [144] Zihlmann, S. _et al._ Large spin relaxation anisotropy and valley-Zeeman spin-orbit coupling in WSe2/Gr/hBN heterostructures.  _Physical Review B_ 97, 075434, DOI: 10.1103/PhysRevB.97.075434 (2018).
  * [145] Manchon, A., Koo, H. C., Nitta, J., Frolov, S. M. & Duine, R. A.  New perspectives for Rashba spin–orbit coupling.  _Nature Materials_ 14, 871–882, DOI: 10.1038/nmat4360 (2015).  1507.02408.
  * [146] Tuan, D. V., Ortmann, F., Soriano, D., Valenzuela, S. O. & Roche, S.  Pseudospin-driven spin relaxation mechanism in graphene.  _Nature Physics_ 10, 857–863, DOI: 10.1038/nphys3083 (2014).
  * [147] Ok, S. _et al._ Custodial glide symmetry of quantum spin Hall edge modes in monolayer WTe2.  _Physical Review B_ 99, 121105, DOI: 10.1103/PhysRevB.99.121105 (2019).
  * [148] Garcia, J. H. _et al._ Canted Persistent Spin Texture and Quantum Spin Hall Effect in WTe2.  _Physical Review Letters_ 125, 256603, DOI: 10.1103/PhysRevLett.125.256603 (2020).
  * [149] Liu, L. _et al._ Symmetry-dependent field-free switching of perpendicular magnetization.  _Nature Nanotechnology_ 16, 277–282, DOI: 10.1038/s41565-020-00826-8 (2021).
  * [150] Garello, K. _et al._ Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures.  _Nature Nanotechnology_ 8, 587–593, DOI: 10.1038/nnano.2013.145 (2013).
  * [151] Belashchenko, K. D., Kovalev, A. A. & Van Schilfgaarde, M.  First-principles calculation of spin-orbit torque in a Co/Pt bilayer.  _Physical Review Materials_ 3, 11401, DOI: 10.1103/PhysRevMaterials.3.011401 (2019).  1810.11003.
  * [152] Dolui, K. _et al._ Proximity Spin-Orbit Torque on a Two-Dimensional Magnet within van der Waals Heterostructure: Current-Driven Antiferromagnet-to-Ferromagnet Reversible Nonequilibrium Phase Transition in Bilayer CrI3.  _Nano Letters_ 20, 2288–2295, DOI: 10.1021/acs.nanolett.9b04556 (2020).
  * [153] Kochan, D., Irmer, S. & Fabian, J.  Model spin-orbit coupling Hamiltonians for graphene systems.  _Physical Review B_ 95, 165415, DOI: 10.1103/PhysRevB.95.165415 (2017).  1610.08794.
  * [154] Johansen, Ø., Risinggård, V., Sudbø, A., Linder, J. & Brataas, A.  Current control of magnetism in two-dimensional fe3gete2.  _Physical Review Letters_ 122, 217203, DOI: 10.1103/PhysRevLett.122.217203 (2019).  1812.06096.
  * [155] Seemann, M., Ködderitzsch, D., Wimmer, S. & Ebert, H.  Symmetry-imposed shape of linear response tensors.  _Phys. Rev. B_ 92, 155138, DOI: 10.1103/PhysRevB.92.155138 (2015).  1507.04947.
  * [156] MacNeill, D. _et al._ Thickness dependence of spin-orbit torques generated by WTe2.  _Physical Review B_ 96, 054450, DOI: 10.1103/PhysRevB.96.054450 (2017).  1707.03757.
  * [157] Gupta, V. _et al._ Manipulation of the van der Waals Magnet Cr2Ge2Te6 by Spin-Orbit Torques.  _Nano Letters_ 20, 7482–7488, DOI: 10.1021/acs.nanolett.0c02965 (2020).
  * [158] Cheng, C., Sun, J. T., Chen, X. R., Fu, H. X. & Meng, S.  Nonlinear Rashba spin splitting in transition metal dichalcogenide monolayers.  _Nanoscale_ 8, 17854–17860, DOI: 10.1039/c6nr04235j (2016).
  * [159] Shao, Q. _et al._ Strong Rashba-Edelstein Effect-Induced Spin–Orbit Torques in Monolayer Transition Metal Dichalcogenide/Ferromagnet Bilayers.  _Nano Letters_ 16, 7514–7520, DOI: 10.1021/acs.nanolett.6b03300 (2016).
  * [160] Zhang, W. _et al._ Research Update: Spin transfer torques in permalloy on monolayer MoS2.  _APL Materials_ 4, 32302, DOI: 10.1063/1.4943076 (2016).
  * [161] Husain, S. _et al._ Emergence of spin–orbit torques in 2D transition metal dichalcogenides: A status update.  _Applied Physics Reviews_ 7, 041312, DOI: 10.1063/5.0025318 (2020).
  * [162] Hidding, J. & Guimarães, M. H.  Spin-Orbit Torques in Transition Metal Dichalcogenide/Ferromagnet Heterostructures.  _Frontiers in Materials_ 7, DOI: 10.3389/fmats.2020.594771 (2020).
  * [163] Lv, W. _et al._ Electric-Field Control of Spin-Orbit Torques in WS2/Permalloy Bilayers.  _ACS Applied Materials and Interfaces_ 10, 2843–2849, DOI: 10.1021/acsami.7b16919 (2018).
  * [164] Stiehl, G. M. _et al._ Layer-dependent spin-orbit torques generated by the centrosymmetric transition metal dichalcogenide $\beta$-MoTe2.  _Physical Review B_ 100, DOI: 10.1103/PhysRevB.100.184402 (2019).  1906.01068.
  * [165] Xie, Q. _et al._ Giant Enhancements of Perpendicular Magnetic Anisotropy and Spin-Orbit Torque by a MoS2 Layer.  _Advanced Materials_ 31, 1–9, DOI: 10.1002/adma.201900776 (2019).
  * [166] Liang, S. _et al._ Spin-Orbit Torque Magnetization Switching in MoTe2/Permalloy Heterostructures.  _Advanced Materials_ 32, 1–6, DOI: 10.1002/adma.202002799 (2020).
  * [167] Haastrup, S. _et al._ The Computational 2D Materials Database: High-throughput modeling and discovery of atomically thin crystals.  _2D Materials_ 5, 042002, DOI: 10.1088/2053-1583/aacfc1 (2018).  1806.03173.
  * [168] Lu, A. Y. _et al._ Janus monolayers of transition metal dichalcogenides.  _Nature Nanotechnology_ 12, 744–749, DOI: 10.1038/nnano.2017.100 (2017).
  * [169] Zhang, J. _et al._ Janus Monolayer Transition-Metal Dichalcogenides.  _ACS Nano_ 11, 8192–8198, DOI: 10.1021/acsnano.7b03186 (2017).  1704.06389.
  * [170] Gambardella, P. & Miron, I. M.  Current-induced spin–orbit torques.  _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ 369, 3175–3197, DOI: 10.1098/rsta.2010.0336 (2011).
  * [171] Hayashi, M., Kim, J., Yamanouchi, M. & Ohno, H.  Quantitative characterization of the spin-orbit torque using harmonic Hall voltage measurements.  _Physical Review B_ 89, 144425, DOI: 10.1103/PhysRevB.89.144425 (2014).
  * [172] Liu, L., Moriyama, T., Ralph, D. C. & Buhrman, R. A.  Spin-torque ferromagnetic resonance induced by the spin Hall effect.  _Physical Review Letters_ 106, 36601, DOI: 10.1103/PhysRevLett.106.036601 (2011).  1011.2788.
  * [173] Mecking, N., Gui, Y. S. & Hu, C.-M.  Microwave photovoltage and photoresistance effects in ferromagnetic microstrips.  _Physical Review B_ 76, 224430, DOI: 10.1103/PhysRevB.76.224430 (2007).
  * [174] Fan, Z. _et al._ Linear scaling quantum transport methodologies.  _Physics Reports_ 903, 1–69, DOI: 10.1016/j.physrep.2020.12.001 (2021).  1811.07387.
  * [175] Lee, K.-S. _et al._ Angular dependence of spin-orbit spin-transfer torques.  _Physical Review B_ 91, 144401, DOI: 10.1103/PhysRevB.91.144401 (2015).  1409.5600.
  * [176] Sousa, F., Tatara, G. & Ferreira, A.  Skew-scattering-induced giant antidamping spin-orbit torques: Collinear and out-of-plane Edelstein effects at two-dimensional material/ferromagnet interfaces.  _Physical Review Research_ 2, 43401, DOI: 10.1103/physrevresearch.2.043401 (2020).
  * [177] Xue, F., Rohmann, C., Li, J., Amin, V. & Haney, P.  Unconventional spin-orbit torque in transition metal dichalcogenide-ferromagnet bilayers from first-principles calculations.  _Physical Review B_ 102, 14401, DOI: 10.1103/PhysRevB.102.014401 (2020).  2005.01109.
  * [178] Mahfouzi, F., Mishra, R., Chang, P. H., Yang, H. & Kioussis, N.  Microscopic origin of spin-orbit torque in ferromagnetic heterostructures: A first-principles approach.  _Physical Review B_ 101, 60405, DOI: 10.1103/PhysRevB.101.060405 (2020).
  * [179] Nikolić, B. K. _et al._ First-principles quantum transport modeling of spin-transfer and spin-orbit torques in magnetic multilayers.  In _arXiv_ , 1–35, DOI: 10.1007/978-3-319-50257-1_112-1 (Springer International Publishing, Cham, 2018).  1801.05793.
  * [180] Fan, X. _et al._ Quantifying interface and bulk contributions to spin–orbit torque in magnetic bilayers.  _Nature Communications_ 5, 3042, DOI: 10.1038/ncomms4042 (2014).
  * [181] Taniguchi, T., Grollier, J. & Stiles, M. D.  Spin-transfer torques generated by the anomalous hall effect and anisotropic magnetoresistance.  _Phys. Rev. Applied_ 3, 044001, DOI: 10.1103/PhysRevApplied.3.044001 (2015).
  * [182] Baek, S.-h. C. _et al._ Spin currents and spin–orbit torques in ferromagnetic trilayers.  _Nature Materials_ 17, 509–513, DOI: 10.1038/s41563-018-0041-5 (2018).
  * [183] Iihama, S. _et al._ Spin-transfer torque induced by the spin anomalous hall effect.  _Nature Electronics_ 1, 120–123, DOI: 10.1038/s41928-018-0026-z (2018).
  * [184] Bhatti, S. _et al._ Spintronics based random access memory: a review.  _Materials Today_ 20, 530–548, DOI: 10.1016/j.mattod.2017.07.007 (2017).
  * [185] Zheng, S. _et al._ High-Temperature Ferromagnetism in an Fe3P Monolayer with a Large Magnetic Anisotropy.  _Journal of Physical Chemistry Letters_ 10, 2733–2738, DOI: 10.1021/acs.jpclett.9b00970 (2019).
  * [186] Torelli, D., Thygesen, K. S. & Olsen, T.  High throughput computational screening for 2d ferromagnetic materials: the critical role of anisotropy and local correlations.  _2D Materials_ 6, 045018, DOI: 10.1088/2053-1583/ab2c43 (2019).
  * [187] Wadley, P. _et al._ Electrical switching of an antiferromagnet.  _Science_ DOI: 10.1126/science.aab1031 (2016).  https://science.sciencemag.org/content/early/2016/01/13/science.aab1031.full.pdf.
  * [188] Jungwirth, T. _et al._ The multiple directions of antiferromagnetic spintronics.  _Nature Physics_ 14, 200–203, DOI: 10.1038/s41567-018-0063-6 (2018).
  * [189] Seixas, L., Rodin, A. S., Carvalho, A. & Castro Neto, A. H.  Multiferroic Two-Dimensional Materials.  _Physical Review Letters_ 116, 206803, DOI: 10.1103/PhysRevLett.116.206803 (2016).  1601.06438.
  * [190] Huang, C. _et al._ Prediction of Intrinsic Ferromagnetic Ferroelectricity in a Transition-Metal Halide Monolayer.  _Physical Review Letters_ 120, 147601, DOI: 10.1103/PhysRevLett.120.147601 (2018).
  * [191] Yang, Q., Xiong, W., Zhu, L., Gao, G. & Wu, M.  Chemically Functionalized Phosphorene: Two-Dimensional Multiferroics with Vertical Polarization and Mobile Magnetism.  _Journal of the American Chemical Society_ 139, 11506–11512, DOI: 10.1021/jacs.7b04422 (2017).
  * [192] Xu, M. _et al._ Electrical Control of Magnetic Phase Transition in a Type-I Multiferroic Double-Metal Trihalide Monolayer.  _Physical Review Letters_ 124, 67602, DOI: 10.1103/PhysRevLett.124.067602 (2020).
  * [193] Li, J. _et al._ Intrinsic magnetic topological insulators in van der Waals layered MnBi2Te4 -family materials.  _Science Advances_ 5, eaaw5685, DOI: 10.1126/sciadv.aaw5685 (2019).
  * [194] Wu, J. _et al._ Natural van der Waals heterostructural single crystals with both magnetic and topological properties.  _Science Advances_ 5, DOI: 10.1126/sciadv.aax9989 (2019).
  * [195] Hu, C. _et al._ A van der Waals antiferromagnetic topological insulator with weak interlayer magnetic coupling.  _Nature Communications_ 11, 97, DOI: 10.1038/s41467-019-13814-x (2020).
  * [196] Deng, Y. _et al._ Quantum anomalous Hall effect in intrinsic magnetic topological insulator MnBi2Te4.  _Science_ 367, 895–900, DOI: 10.1126/science.aax8156 (2020).  1904.11468.
  * [197] Mellnik, A. R. _et al._ Spin-transfer torque generated by a topological insulator.  _Nature_ 511, 449–451, DOI: 10.1038/nature13534 (2014).
  * [198] Yang, M. _et al._ Creation of skyrmions in van der Waals ferromagnet Fe3GeTe2 on (Co/Pd)n superlattice.  _Science Advances_ 6, DOI: 10.1126/sciadv.abb5157 (2020).
  * [199] Wu, Y. _et al._ Néel-type skyrmion in WTe2/Fe3GeTe2 van der Waals heterostructure.  _Nature Communications_ 11, 3860, DOI: 10.1038/s41467-020-17566-x (2020).  1907.11349.
  * [200] Park, T.-E. _et al._ Néel-type skyrmions and their current-induced motion in van der Waals ferromagnet-based heterostructures.  _Physical Review B_ 103, 104410, DOI: 10.1103/PhysRevB.103.104410 (2021).  1907.01425.
  * [201] Zhang, X.-X. _et al._ Gate-tunable spin waves in antiferromagnetic atomic bilayers.  _Nature Materials_ 19, 838–842, DOI: 10.1038/s41563-020-0713-9 (2020).
  * [202] McCreary, A. _et al._ Distinct magneto-Raman signatures of spin-flip phase transitions in CrI3.  _Nature Communications_ 11, 3879, DOI: 10.1038/s41467-020-17320-3 (2020).  1910.01237.
  * [203] Cenker, J. _et al._ Direct observation of two-dimensional magnons in atomically thin CrI3.  _Nature Physics_ 17, 20–25, DOI: 10.1038/s41567-020-0999-1 (2021).
  * [204] Huang, B. _et al._ Emergent phenomena and proximity effects in two-dimensional magnets and heterostructures.  _Nature Materials_ 19, 1276–1289, DOI: 10.1038/s41563-020-0791-8 (2020).

## Acknowledgements

H. K. and S. K. acknowledge supports from EPSRC via EP/T006749/1 and also help
by Oscar Lee for producing graphical images. S. R. and J. H. G. acknowledge
funding from the European Union Seventh Framework Programme under Grant No.
881603 (Graphene Flagship). ICN2 is funded by the CERCA Programme/Generalitat
de Catalunya and supported by the Severo Ochoa programme (MINECO Grant. No.
SEV-2017-0706).

## Author contributions

The authors contributed to all aspects of the article.

## Competing interests

The authors declare no competing interests.

## Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in
published maps and institutional affiliations.