[1]Vladislav Kataev

[1]Leibniz Institute for Solid State and Materials Research Dresden,
Helmholtzstr. 20, Dresden, D-01069, Germany

2]Institute for Solid State and Materials Physics and Würzburg-Dresden Cluster
of Excellence ct.qmat, TU Dresden, Dresden, D-01062, Germany

# Electron Spin Resonance Spectroscopy on Magnetic Van der Waals Compounds

<EMAIL_ADDRESS>Bernd Büchner<EMAIL_ADDRESS>Alexey
Alfonsov<EMAIL_ADDRESS>*  [

###### Abstract

The field of research on magnetic van der Waals compounds – a special subclass
of quasi-two-dimensional materials – is currently rapidly expanding due to the
relevance of these compounds to fundamental research where they serve as a
playground for the investigation of different models of quantum magnetism and
also in view of their unique magneto-electronic and magneto-optical properties
pertinent to novel technological applications. The Electron Spin Resonance
(ESR) spectroscopy plays an important role in the exploration of the rich
magnetic behavior of van der Waals compounds due to its high sensitivity to
magnetic anisotropies and unprecedentedly high energy resolution that
altogether enable one to obtain thorough insights into the details of the spin
structure in the magnetically ordered state and the low-energy spin dynamics
in the ordered and paramagnetic phases. This article provides an overview of
the recent achievements in this field made by the ESR spectroscopic techniques
encompassing representatives of antiferro- and ferromagnetic van der Waals
compounds of different crystal structures and chemical composition as well as
of a special category of these materials termed magnetic topological
insulators.

###### keywords:

magnetism, ESR, FMR, AFMR, van der Waals compounds

##  1 Introduction

The successful isolation of graphene in 2004 [1] – a monolayer of carbon
exfoliated from bulk graphite – has opened a vast new field of research on
two-dimensional (2D) materials which, besides graphene and its analogs,
include by now III-VI semiconductors, e.g., GaSe, InSe, In2Se3, transition-
metal dichalcogenides, e.g., MoS2, WS2, TiS2 RbSe2, MoTe2, and oxides, e.g.,
TiO2, MoO3, WO3 [2, 3]. Most of these compounds are in the bulk form layered
van der Waals crystals which by virtue of the weak interlayer van der Waals
forces can be relatively easy thinned by exfoliation. In the 2D limit they
feature outstanding physico-chemical, electronic and optical properties often
distinct from the bulk that make these materials extremely interesting for
novel technological applications. More recently the focus of the 2D research
was extended to magnetic van der Waals compounds promising new magneto-
electronic and magneto-optical functionalities (for recent reviews see, e.g.,
Refs. [4, 5, 6, 7]).

Van der Waals magnets appear to be attractive both as model systems for
investigations of fundamental aspects of magnetism in reduced spatial
dimensions and also regarding the practical use in the next-generation
spintronic devices. Though many of them were known already since long time and
their elementary magnetic properties were reported in the literature, magnetic
studies of these compounds experience currently a renaissance due to
improvements in the material synthesis and the progress of the measurement
techniques that make possible to address new scientific questions arising in
the context of the 2D physics.

Among the employed experimental methods the Electron Spin Resonance (ESR)
spectroscopy and related ferromagnetic (FM) and antiferromagnetic (AFM)
resonance techniques, FMR and AFMR, play an important role in obtaining
valuable insights into the static and dynamic magnetic properties of van der
Waals compounds. This became possible due to enormous instrumental development
since the discovery of ESR 80 years ago in terms of the sensitivity, expansion
of the frequency range towards far-infrared region and the use of very strong
magnetic fields. Still studies of magnetic resonance on the monolayers remain
very challenging.

Fortunately, in the absence of the strong covalent bonds between the layers,
which could provide the pathway for the conventional superexchange interaction
[8], the interlayer magnetic coupling in the van der Waals magnets is very
weak. Instead theories suggest that the interlayer coupling originates from
the weak overlap of the ligands’ $p$-electron densities between the layers
giving rise to a kind of super-superexchange between the magnetic ions in the
nighboring planes that is by orders of magnitude smaller that the intraplane
exchange [9, 10]. This implies that important clues about the magnetism of the
monolayers can be obtained from the measurements on bulk crystals which due to
such very weak interlayer coupling are naturally quasi-2D magnets and as such
may demonstrate inherent 2D magnetic phenomena.

This review focuses on the results of magnetic resonance investigations mainly
on the bulk antiferromagnetic van der Waals compounds in the magnetically
ordered and paramagnetic states and to a less extent on their ferromagnetic
counterparts, the FMR studies of which were summarized in a recent review in
Ref. [11].

##  2 Antiferromagnetic van der Waals compounds

AFM van der Waals compounds are of a significant interest both for fundamental
and applied research due to their layered structure with relatively weak
intraplane and even much weaker interplane magnetic interactions as compared
to, e.g., magnetic oxides. This significantly reduces the Néel ordering
temperature $T_{\rm N}$ and makes it possible to influence their magnetic
state by applying moderate magnetic fields and probe magnetic excitations by
GHz and sub-THz resonance techniques. From the fundamental perspective, in van
der Waals antiferromagnets different spin models can be realized based on the
AFM anisotropic Heisenberg, Ising-type, or planar $XY$ Hamiltonians, and their
properties can be studied and compared with theoretical predictions. Regarding
applications, these materials are interesting in view of the antiferromagnetic
spintronic devices featuring ultrafast spin dynamics and being insensitive to
disturbing magnetic fields [12, 13]. For these reasons a significant number of
AFM van der Waals compounds were invstigated with ESR and AFMR spectroscopies.
An overview of the results of these studies will be presented in the following
sections.

###  2.1 CrCl3

Figure 1: Crystal structure of the chromium-based trihalides viewed along the
$ab$-plane (left) and along the $c$-axis (right). Cr$X_{6}$ ($X$ = Cl, Br, I)
octahedra are arranged on a honeycomb plane, the planes are stacked along the
$c$-axis.

In CrCl3, the Cr3+ ($3d^{3},\,S=3/2,\,L=0)$ ion is covalently bonded to six
Cl- ligands forming a regular CrCl6 octahedron. These octahedra share edges
and form a 2D-honeycomb lattice in the $ab$-crystal plane (Fig. 1). The Cr
spins in the $ab$-plane are coupled ferromagnetically with the nearest-
neighbor exchange constant $J^{\rm ab}=-5.25$ K (-0.45 meV) [14] and undergo a
2D-FM transition at $T_{\rm c}^{\rm 2D}\approx 17$ K with the spins lying in
the plane [15, 16]. A large spatial separation between the honeycomb planes
along the $c$-axis amounts to about 5.81 Å and results in a weak
antiferromagnetic inter-layer coupling $J^{\rm c}=18$ mK (1.55 $\mu$V) [14]
that yields a 3D-AFM order at $T_{\rm N}\approx 15$ K [15] (14 K in Ref.
[16]).

ESR studies of CrCl3 have a long history tracing back to the early days of
ESR. It was one of the first compounds whose resonance signal was reported by
Evgenii Zavoisky soon after his discovery of electron paramagnetic resonance
[17, 18]. The interest in this material was much later renewed in the context
of physics of low-dimensional spin systems and functional magnetic properties
of van der Waals magnets. Already in 1991 Chehab et al. [19] pointed out the
prominence of the specific 2D spin dynamics far above the magnetic ordering
temperature of CrCl3. It is characterized by the dominance of the long-wave
modes (wave vector $q\rightarrow 0$) of spin fluctuations manifesting in the
$(3\cos^{2}\theta-1)^{2}$ type of the angular dependence of the ESR linewidth
[20]. It is distinct from the case of 3D spin systems where fluctuations with
all wave vectors are present giving rise to the $(\cos^{2}\theta+1)$ type of
the angular behavior [20].

Magnetic excitations in the 3D-AFM ordered state of CrCl3 were studied by
broad-band AFMR spectroscopy in Ref. [21] for the external magnetic field
applied in the $ab$-crystal plane. Since CrCl3 in the ground state can be
viewed as a two-sublattice collinear antiferromagnet, two types of AFMR modes
can be expected. The first one, the so-called acoustic mode, corresponds to
the in-phase precession of the magnetizations of the AFM-coupled neighboring
FM-ordered Cr planes (Fig. 1) [21]:

$\omega_{-}=\mu_{0}\gamma\sqrt{2H_{\rm e}(2H_{\rm e}+M_{\rm
s})}\frac{H}{2H_{\rm e}}\ .$ (1)

The second, optical mode, is the anti-phase precession of the respective
magnetizations [21]:

$\omega_{+}=\mu_{0}\gamma\sqrt{2H_{\rm e}M_{\rm s}(1-\frac{H^{2}}{4H_{\rm
e}^{2}})}\ .$ (2)

Here, $\mu_{0}$ and $\gamma$ are the vacuum permeability and the gyromagnetic
ratio, and $H_{\rm e}$ and $M_{\rm s}$ are the exchange field and the
saturation magnetization, respectively.

Figure 2: Frequency vs. magnetic field diagram of the AFMR modes in CrCl3 for
the external field $\mathbf{H}$ applied in the $ab$ crystal plane parallel to
the magnetic component of the microwave field
$\mathbf{H}\parallel\mathbf{H_{mw}}$ ($H_{\parallel}$) (a), and perpendicular
to it $\mathbf{H}\perp\mathbf{H_{mw}}$ ($H_{\perp}$) (b). For the
$H_{\parallel}$ geometry both optical and acoustic AFMR modes were observed,
whereas for $H_{\perp}$ only acoustic mode is visible. Red and blue dashed
lines in both panels are fits of the acoustic and optical mode frequencies to
Eqs. (1) and (2), respectively. Precession of the sublattice magnetizations in
the optical and acoustic mode are schematically illustrated in panels (c) and
(d), respectively. (Reprinted with permission from D. MacNeill et al., Phys.
Rev. Lett. 123 (4), 047204 (2019) [21]. Copyright (2019) by the American
Physical Society.)

Owing to the expected smallness of the interlayer magnetic exchange of the Cr
$ab$-planes [14] both modes could be detected at relatively low frequencies
$f<12$ GHz (Fig. 2.) Indeed, fitting of the obtained $f(H)$ frequency
dependences to Eqs. (1) and (2) [Figs. 2(a) and (b)] yielded a small value of
the exchange field $\mu_{0}H_{\rm e}=105$ mT. Consequently, for applied fields
$H>2H_{\rm e}$ the Cr spins become fully polarized along the field direction
and the magnetic system of CrCl3 enters the forced ferromagnetic state.
Similar observation of the optical AFM mode in a CrCl3 crystal coupled to the
superconducting cavity was recently reported in Ref. [22].

The dynamic properties of the spin-polarized state in CrCl3 were addressed in
Ref. [23] by high-field ESR spectroscopy covering the frequency range up to
350 GHz. In this case the FMR signal of the Cr spins forcefully aligned by the
external field $\mathbf{H}$ appears anisotropic, shifting differently from the
expected position of the high-temperature paramagnetic resonance signal for
$\mathbf{H}$ applied parallel and normal to the crystal $c$-axis. The measured
field dependence of the FMR modes for $\mathbf{H}\parallel\mathbf{c}$ and
$\mathbf{H}\perp\mathbf{c}$ configurations were analyzed in the framework of a
conventional FMR theory [24, 25, 26]. In agreement with previous studies [27,
16, 21, 28] it has been shown that the observed anisotropy is essentially due
to the platelet-like shape of the measured crystal whereas the intrinsic,
magnetocrystalline anisotropy is practically negligible. This enables one to
classify CrCl3 as an example of an almost ideal isotropic Heisenberg magnet.

It should be noted that even far above the ordering temperature $T_{\rm
N}\approx 15$ K the anisotropic temperature-dependent shifts of the ESR signal
measured at high frequencies such as 90 GHz in Ref. [23] and 120 GHz in Ref.
[29] are still present up to $\sim 3T_{\rm N}$ suggesting that the
corresponding strong magnetic field promotes in-plane ferromagnetic
correlations even in the paramagnetic state, which appears to be a salient
feature of a quasi-2D van der Waals magnet.

Interestingly, Ref. [30] reported that exposure of single crystals of CrCl3 to
visible light affects the parameters of the ESR signal of Cr3+
($3d^{3},\,S=3/2$) ions measured at the X-band frequency of 9.45 GHz , their
intensity, $g$-factor, and the linewidth. This sensitivity to light was
ascribed to the photo-induced electron transitions between the valence band
and localized Cr2+ ($3d^{4},\,S=2$) levels. The similar effect was observed by
the same authors for the ferromagnetic CrI3 (see Sect. 3.1) suggesting a
possible light-control functionality of these compounds in the monolayer
limit.

Besides observing the above discussed uniform acoustic and optical resonance
modes Kapoor et al. [31] were able to detect in a 20 $\mu$m thin single
crystal CrCl3 additional non-uniform modes which were ascribed to the standing
spin waves excited across the thickness of a crystal. They manifest themselves
as additional multiple resonances in the vicinity of the uniform modes. The
distance of each spin-wave resonance from the respective uniform mode scales
linearly with its number as predicted by the theory of standing spin waves in
a material with a nonuniform volume magnetization [32]. The authors could
estimate the lifetime of the resonantly excited magnons to be of the order of
a few ns. Since spin waves can carry information without involving the
electron charge they promise to be an efficient means of low-loss data
transfer in spintronic devices where, as the authors believe, a few-layer
CrCl3 can be used as a suitable medium.

###  2.2 The family of $M_{2}$P2S6

Figure 3: (a) Crystal structure of the $M_{2}$P2S6 ($M$ = Mn, Fe, Ni,
Mn0.5Ni0.5, Cu0.5Cr0.5) compounds of the monoclinic symmetry, space group
$C2/m$. The atoms are colored as $M$ = turquoise, P = pink, and S = yellow.
(b) Individual $M_{2}$P2S6 layer. The $M$S6 octahedra form a honeycomb lattice
with the $C_{3}$ symmetry axis of the octahedra normal to the $ab$-plane
($c^{*}$-axis). The P-P dumbbell occupies the void of each honeycomb.

The family of metal tiosulfates $M_{2}$P2S6 ($M$ = Mn, Fe, Ni, Mn0.5Ni0.5,
Cu0.5Cr0.5) represents an interesting subclass of van der Waals magnets where
the transition metal ions with different spin multiplicities and single ion
anisotropies can occupy the $M$-site in the crystal structure (Fig. 3). The
metal ions are 6-fold coordinated by sulfur ligands and are arranged on a
honeycomb lattice [Fig. 3(b)]. The intra-plane exchange interaction between
the $M$ spins is antiferromagnetic and sizable whereas the coupling between
the planes is weak due to a large spatial interplane separation Thus, the
$M_{2}$P2S6 family offers realization of various quasi-2D AFM spin models
ranging from the Ising and Heisenberg to the $XY$ Hamiltonians. In the
following, ESR works on most frequently studied members of this family with
$M$ = Mn, Fe, Ni, Mn0.5Ni0.5, and Cu0.5Cr0.5 will be briefly reviewed.

####  2.2.1 Ni2P2S6

Ni2P2S6 is an antiferromagnet ordering at $T_{\rm N}=155$ K due to residual,
weak inter-plane coupling. It has been classified as an "easy"-plane
antiferromagnet, i.e., the ordered spins in the honeycomb lattice lie in the
plane where a preferred spin orientation along an in-plane "easy"-axis was
identified [33, 34, 35, 36]. The Ni2+ ion in Ni2P2S6 possesses 8 electrons on
the 3$d$ shell and carries a spin $S=1$. In the presence of a uniaxial
distortion of the surrounding ligand electrical field the spin triplet splits
into a singlet $\mid 0\rangle$ and a doublet $\mid\pm 1\rangle$ giving rise to
a single-ion anisotropy (SIA) of the easy-axis or the easy-plane type
depending on whether the spin doublet or the spin singlet have the lowest
energy [37]. Such a distortion along the $C_{3}$ symmetry axis of the NiS6
octahedron is present in Ni2P2S6 [Fig. 3(b)] and the related single ion
anistropy can potentially contribute to the total anisotropy of the AFM-
ordered Ni spin lattice. The strength and the sighn of SIA parameter $D$ can
be deduced from the anisotropy of the $g$-factor. Given that the orbital
moment of the Ni2+ ion is quenched in fist order, the $g$-anisotropy is
expected to be small and can be resolved only by high-field ESR spectroscopy
owing to its excellent spectral resolution.

Figure 4: High-field ESR results on Ni2P2S6. (a) $\nu(H_{\rm res})$ dependence
of the ESR signals at $T=250$ K for ${\bf H}\parallel{\bf c^{\ast}}$ (blue
squares) and ${\bf H}\perp{\bf c^{\ast}}$ (red circles). Solid lines represent
the fit according to the resonance condition $h\nu=g\mu_{\rm B}\mu_{0}H_{\rm
res}$. Insets: HF-ESR signals (black and blue solid lines) for ${\bf
H}\parallel{\bf c^{\ast}}$ and ${\bf H}\perp{\bf c^{\ast}}$ plotted as a
function of the $g$-factor $g=h\nu/\mu_{\rm B}\mu_{0}H$ at $\nu\approx 410$
GHz and $T=300$ K (top left) and $\nu\approx 330$ GHz and $T=250$ K (bottom
right). Orange dash lines are fits to the Lorenzian line profile. The
$g$-values of the resonance peaks are indicated in the respective plots. (b)
Left vertical scale: $\nu(H_{\rm res})$ dependence of the ESR signal at $T=4$
K for ${\bf H}\perp{\bf c^{\ast}}$ (circles). Solid line depicts the fit of
the data to the easy-plane AFM resonance branch fit function
$\nu=h^{-1}[(g_{\perp}\mu_{\rm B}\mu_{0}H_{\rm res})^{2}+\Delta^{2}]^{1/2}$.
Dashed line corresponds to the paramagnetic resonance condition
$\nu=h^{-1}g_{\perp}\mu_{\rm B}\mu_{0}H_{\rm res}$. Right vertical scale: HF-
ESR signals at selected frequencies. (Reprinted with permission from K.
Mehlawat et al., Phys. Rev. B. 105 (21), 214427 (2022) [38]. Copyright (2022)
by the American Physical Society.)

This has been shown in Ref. [38] reporting the results of the ESR measurements
on single crystals of Ni2P2S6 in the broad frequency and magnetic field range
in the paramagnetic and AFM-ordered states. In the paramagnetic regime far
above $T_{\rm N}=155$ K the Ni2+ ESR signal is almost isotropic. If measured
at frequencies $\nu<150$ GHz the resonance field $H_{\rm res}$ appears
apparently independent from whether the external field is applied in the
$ab$-plane or normal to it [Fig. 4(a), main panel]. However, increasing the
frequency up to 410 GHz enabled one to clearly resolve the anisotropy of the
ESR signal [Fig. 4(a), insets] and quantify the $g$-factor tensor elements
$g_{\parallel}=2.149\pm 0.004$ and $g_{\perp}=2.188\pm 0.004$. Based on this
data the SIA parameter $D$ can be straightforwardly estimated as
$D=\lambda(g_{\parallel}-g_{\perp})/2$ [37]. With the known Ni2+ spin-orbit
coupling constant [39] and the above $g$-factors one gets $D\approx 0.7$ meV.
Its positive sign implies the easy-plane anisotropy of the Ni spins and its
magnitude is in a quantitative agreement with the theoretical prediction [40].

In contrast to an almost isotropic and gapless ESR behavior at $T\gg T_{\rm
N}$, the AFM resonance in the ordered state of Ni2P2S6 is strongly anisotopic
and gapped. The low-energy AFM resonance branch was identified in Ref. [38]
[Fig. 4(b)]. It shows a strong up-frequency shift from the paramagnetic
position following the $\nu(H_{\rm res})$ dependence typical for an easy-plane
antiferromagnet and reveals a zero-field excitation gap $\Delta=260$ GHz (1.07
meV). This gapped spin excitation could not be detected by inelastic neutron
scattering (INS) [36] but becomes observable due to an excellent energy
resolution of the multi-frequency high-field ESR that has been appreciated in
the later INS work in Ref. [41].

####  2.2.2 Mn2P2S6

A counterexample of the easy-plane anisotropic Ni2P2S6 is its sister compound
Mn2P2S6 which is described as a Heisenberg antiferromagnet with an easy-axis
anisotropy [33, 43, 44, 45].

Figure 5: Frequency versus magnetic field dependence of the AFM resonance
modes in Mn2P2S6 for the $\mathbf{H}\parallel\mathbf{c^{\ast}}$ field geometry
from Ref. [42]. There are two types of spin excitations with different gap
values at zero magnetic field $\Delta_{1}=102$ GHz and $\Delta_{2}=116$ GHz.
Solid curves correspond to the theoretical description of the resonance
branches of a rhombic biaxial two-sublattice antiferromagnet. The two branches
collapse at the spin-flop field $H_{\rm sf}=36.5$ kOe. Above $H_{\rm sf}$ the
observed branches deviate form the theoretical description which has been
ascribed to their interaction resulting in the coupled vibrations [42].
(Reprinted from M. I. Kobets et al., Low Temp. Phys. 35 (12), 930 (2009) [42]
with the permission of AIP Publishing.)

Its AFM ordering temperature amounts to $T_{\rm N}=77$ K which is twice
smaller than that of Ni2P2S6 suggesting a significantly weaker coupling
between the Mn spins in Mn2P2S6 and a smaller magnetocrystalline anisotropy
that stabilizes a 3D order in van der Waals magnets.

First multi-frequency ESR experiments on single crystals of Mn2P2S6 were
reported by Okuda et al. [46]. From measurements of the AFM resonance at
$T=4.2$ K they concluded that Mn2P2S6 is a uniaxial antiferromagnet with the
zero field spin excitation gap $\Delta\approx 106$ GHz. In the paramagnetic
state, in the temperature range 110 K $<T<293$ K the $(3\cos^{2}\theta-1)^{2}$
type of the angular dependence of the ESR linewidth was observed an ascribed
to the 2D-spin diffusion effect [20]. Later, Kobets et al. [42] reported a
more detailed study of the AFM resonance at $T=4.2$ K that revealed two,
ascending and descending AFM modes with different zero-field gaps amounting to
$\Delta_{1}=102$ GHz a and $\Delta_{2}=116$ GHz, respectively. These modes
collapse at the spin-flop transition field $H_{\rm sf}=36.5$ kOe and evolve
above $H_{\rm sf}$ as interacting modes causing coupled vibrations [42] (Fig.
5). Observation of the two gaps evidences that Mn2P2S6 is in fact a biaxial
antiferromagnet and not a uniaxial one as conjectured in Ref. [46].

Figure 6: Main panel: Temperature dependence of the normalized energy gap
$\Delta(T)/\Delta(3\,{\rm K})=[1-(T/T_{\rm N})]^{b}$ for Mn2P2S6 at different
field regimes rectified from the respective dependences of the AFM resonance
modes B3–B5 whose field dependence at $T=3$ K is schematically shown in the
inset. There, the vertical dashed line separates the spin collinear phase at
$H<H_{\rm sf}$ and the spin flop phase at $H>H_{\rm sf}$. (Reprinted with
permission from J. J. Abraham et al., Phys. Rev. B. 107 (16), 165141 (2023)
[47]. Copyright (2023) by the American Physical Society.)

The frequency versus field diagram of the AFM resonance modes was further
refined in Ref. [47] by extending the frequency and temperature range of the
study. Besides confirming the biaxial nature of the AFM spin-lattice in
Mn2P2S6, the temperature evolution of the AFM modes was studied in some
detail. Of a particular interest was the analysis of the temperature behavior
of the spin excitation gap $\Delta(T)$ at $T<T_{\rm N}$ and at different
magnetic fields. It can be well fitted to the power law
$\Delta(T)\propto[1-(T/T_{\rm N})]^{b}$ (Fig. 6). The exponent $b$ changes
drastically from its value 0.55 - 0.6 in the spin collinear phase at $H<H_{\rm
sf}$ to 0.26 - 0.29 in the spin-flop phase above $H_{\rm sf}$. In a quasi-2D
antiferromagnet this excitation gap usually scales with the sublattice
magnetization $M_{\rm sl}$ which is the AFM order parameter. Thus, $b$ can be
considered as a critical exponent $\beta$ of $M_{\rm sl}$. The value of $b$ in
the spin-collinear phase is close to the mean-field exponent $\beta=0.5$,
whereas in the spin-flop phase it becomes close to the critical exponent
$\beta=0.231$ in the 2D $XY$ spin model [48]. Hence, the spin lattice in
Mn2P2S6 demonstrates an interesting field-driven dimensional crossover of the
spin excitations in the AFM-ordered state that boosts the effective 2D $XY$
anisotropy at $H>H_{\rm sf}$.

Signatures of the 2D spin dynamics in Mn2P2S6 above $T_{\rm N}$ in terms of
the specific angular dependence of the linewidth $\Delta
H\propto(3\cos^{2}\theta-1)^{2}$ initially observed in Ref. [46] at high
frequencies were found also in ESR experiments at a "low" X-band frequency in
Refs. [49, 50]. In particular, signatures of three different spin dynamic
regimes in the paramagnetic state with distinct temperature and angular
dependences of the linewidth alternating from room temperature down to $T_{\rm
N}$ were observed in Ref. [50]: prominent spin correlations of the 2D
character at $T>\sim 150$ K, a 2D $\rightarrow$ 3D crossover spin dynamic
regime at $\sim$ 150 K $>T>\sim 100$ K followed by predominantly 3D spin
correlations by approaching the long-range AFM order at $T_{\rm N}=77$ K. It
should be noted that in the early work in Ref. [51] no signatures of the 2D
spin correlations in the angular dependence of $\Delta H$ were found possibly
due to a much broader width of the ESR signal of the studied crystals as
compared to the narrow resonances observed in Ref. [50].

####  2.2.3 MnNiP2S6

Mn2P2S6 and Ni2P2S6 are distinct in their anisotropic magnetic behavior
featuring easy-axis and easy-plane collinear AFM order, respectively.
Evolution of the magnetic properties in the series (Mn1-xNix)2P2S6 was studied
by static magnetometry in Ref. [44]. The minimum ordering temperature of
$T_{\rm N}\sim 57$ K and the seemingly vanishing magnetic anisotropy was found
for the compound MnNiP2S6. ESR spectroscopic insights into the spin
excitations in the magnetically ordered ground state of this compound and
spin-spin correlations at high temperatures were obtained in Refs. [50, 47].
In contrast to Mn2P2S6, the angular dependence of the ESR linewidth showed the
$\cos^{2}\theta+1$ behavior typical for 3D spin systems possibly due to an
enhanced inter-plane coupling promoted by the Ni substitution. The critical
broadening and the shift of the ESR line was observed already starting at high
temperature $T\sim 200$ K $\gg T_{\rm N}$. It was attributed to a random
distribution of Mn2+ and Ni2+ ions at the $M$ site of the crystal structure of
$M_{2}$P2S6 ($M_{2}$ = MnNi) giving rise to a competition between different
types of order with contrasting magnetic anisotropies (easy-axis versus easy-
plane).

In the ordered state at $T\ll T_{\rm N}$ the measured AFM resonance branches
for the external field applied normal and parallel to the $ab$-plane of the
MnNiP2S6 crystal follow the resonance condition for a hard direction of an
antiferromagnet with two different values of the zero-field excitation gap
$\Delta_{1}=115\pm 9$ GHz and $\Delta_{2}=215\pm 1$ GHz, respectively. This
observation suggests that neither of the two field directions are
energetically favorable for the ordered spins implying that the magnetic
structure of MnNiP2S6 is more complex than the collinear two-sublattice AFM
model applicable to the Mn and Ni counterparts. The above gap values are
intermediate between those found for Mn2P2S6 and Ni2P2S6 suggesting that
substitution of Mn by Ni gradually enhances magnetic anisotropy which is
responsible for the opening of a gap in the spin excitation spectrum.

####  2.2.4 Fe2P2S6

Fe2P2S6 is an antiferromagnet ordering at $T_{\rm N}\approx 120$ K in the zig-
zag-type structure with the Fe spins pointing out of the $ab$-crystal plane
[52]. The Fe2+ ion with 6 electrons on the 3$d$ shell occurs in the high-spin
state with $S=2$, which is the largest spin value in the $M_{2}$P2S6 family.
Moreover, Fe2P2S6 features the largest magnetic anisotropy of the easy-axis
type enabling one to classify it as an Ising antiferromagnet.

Figure 7: Top: Far-infrared transmission spectra of Fe2P2S6 for selected
values of the magnetic field applied perpendicular to the layer planes.
$M_{+}$ and $M_{-}$ are magnetic excitations, $P_{i}$ are the phonon modes,
and $I_{i}$ are other absorption resonances. Bottom: Magnetic field dependence
of the energy positions of hybrid magnon-phonon modes. Crosses and circles
correspond to the far-infrared transmission signals and Raman scattering
peaks, respectively. Lines are the results of the modeling. (Reprinted with
permission from D. Vaclavkova et al., Phys. Rev. B. 104 (13), 134437 (2021)
[53]. Copyright (2021) by the American Physical Society.)

The strong magnetic anisotropy of this material implies that observation of
ESR and AFM resonance requires excitation frequencies beyond the limits of
"conventional" high-frequency ESR techniques usually confined to $\nu\lesssim
1$ THz. Indeed, the AFM resonance excitations in Fe2P2S6 were probed with far-
infrared (FIR) spectroscopy. There, the distinction between the phonon modes
active in this spectral region from magnetic modes were achieved by measuring
the infrared spectra in variable magnetic fields, since only the latter type
of excitations can be field sensitive.

The zero-field AFM excitation gap arising due to the Ising-like magnetic
anisotropy is large and amounts according to Ref. [53] to 122 cm-1 (3.66 THz).
The FIR map of the observed modes is shown in Fig. 7. There, the magnetic
excitations $M_{\pm}$ are field dependent whereas the energy of the phonon
modes $P_{1,2,3}$ is constant in the moderate magnetic fields. Interestingly,
at $\mu_{0}H>12$ T the magnetic mode $M_{-}$ demonstrates the so-called
avoided crossing, first with $P_{1}$ and then in series with $P_{2}$ and
$P_{3}$ modes. This effect is a signature of the formation of the hybrid
magnon-phonon modes which suggests pronounced magneto-elastic coupling in
Fe2P2S6. The magnon-phonon hybridization in Fe2P2S6 was further addressed in
more detail in Ref. [54] where the zero-field hybridization gap between the
otherwise degenerated magnon modes was observed and signatures of magnon-
induced chiral phonons were identified. Interestingly, besides the one-magnon
excitations giving rise to the $M_{\pm}$ modes Ref. [55] reports an
observation of an exotic multipolar magnon that corresponds to the full
reversal of the spin $S=2$ of the Fe2+ ion. Its excitation energy in zero
field amounts to 57.5 meV (13.9 THz) and its Zeeman splitting is 4 times
larger than that of the one-magnon $M_{\pm}$ modes.

####  2.2.5 CuCrP2S6

The quartenary compound CuCrP2S6 is structurally distinct from other members
of the $M_{2}$P2S6 family in that nonmagnetic Cu1+ and magnetic Cr3+ ions are
alternatively arranged on the common honeycomb plane and Cu1+ ions are
displaced up and down the plane resulting in the local electric dipole
moments. This material undergoes an antiferroelectric phase transition at 140
K [56, 57] and an AFM phase transition at $T_{\rm N}=30$ K [58, 59]. Thus,
CuCrP2S6 represents an interesting realization of interpenetrating Cu-based
antiferroelectric and Cr-based antiferromagnetic lattices. According to the
static magnetic data CuCrP2S6 is an easy-plane antiferromagnet with an
additional in-plane easy-axis along the $a$-direction [60, 34]. So far, little
is known about ESR and AFM resonance properties of this compound. A room
temperature ESR signal at the X-band frequency was measured in Ref. [58] and
attributed to the Cr3+ ions. Recently, Wang et al. [60] reported mutifrequency
ESR experiments in a range $\nu=3-13$ GHz. At $T>T_{\rm N}$ a single
paramagnetic $\nu(H)$ resonance branch was observed, whereas at $T<T_{\rm N}$
two AFM branches were found for the $\mathbf{H}\parallel\mathbf{a}$ field
geometry demonstrating a typical resonance response of an antiferromagnet with
the magnetic field applied along the easy-axis. The first branch has a zero-
field excitation gap of $\sim 14$ GHz and softens towards the spin-flop field
of $\mu_{0}H_{\rm sp}\sim 0.4$ T. The second branch ascends out of $H_{\rm
sp}$ towards stronger magnetic fields and reaches the maximum frequency of 13
GHz at 0.6 T.

###  2.3 CrSBr

In CrSBr the magnetic Cr3+ ($S=3/2$) ions are coordinated by S and Br ligands
forming the distorted CrS2Br2 octahedra. The interconnected octahedra build up
a 2D spin-3/2 square lattice in the $ab$-crystal plane. The planes are stacked
along the $c$-axis. The intralayer magnetic exchange is ferromagnetic while
the interlayer coupling is antiferromagnetic yielding a long-range AFM order
at $T_{\rm N}=132$ K [61]. The ordered spin lattice exhibits a triaxial
anisotropy with the easy $b$-axis, an intermediate $a$-axis, and the hard
$c$-axis [61].

Figure 8: AFM resonance branches of CrSBr at $T=5$ K $\ll T_{\rm N}$ for the
in-plane orientation of the applied magnetic field normal to the easy $b$-axis
($H_{\perp}$) (a) and parallel to it ($H_{\parallel}$) (b). The modes of the
oscillating sublattice magnetization vectors – optical, acoustic, right-hand
(RH), and left-hand (LH) – are illustrated on the right. (Reprinted with
permission from T. M. J. Cham et al., Nano Lett. 22 (16), 6716 (2022) [62].
Copyright (2022) by the American Chemical Society.)

The specific type of anisotropy in CrSBr was reflected in the frequency versus
magnetic field dependence of the AFM modes investigated in detail in Ref.
[62]. For the field applied in the $ab$-plane along the intermediate $a$-axis,
i.e., normal to the easy $b$-axis ($H_{\perp}$) two gapped modes were observed
[Fig. 8(a)]. One mode is the acoustic mode corresponding to the in-phase
oscillations of the sublattice magnetization vectors $\mathbf{M_{1}}$ and
$\mathbf{M_{2}}$ of the two-sublattice collinear antiferromagnet tilted away
from the easy axis by the applied field. The other mode is the optical mode
corresponding to the out-of-phase oscillations of these vectors (see the
sketches in Fig. 8). Turning the field direction parallel to the easy $a$-axis
($H_{\parallel}$) changes the behavior of the AFM modes [Fig. 8(b)]. At small
fields $\mathbf{M_{1}}$ and $\mathbf{M_{2}}$ are almost aligned along the easy
axis and can oscillate in the right-hand (RH) and left-hand (LH) manner (see
the sketches in Fig. 8). The presence of biaxial anisotropy in the $ab$-plane
causes the avoiding crossing of the modes at zero field where they appear as
symmetrical and antisymmetrical combination of RH and LH oscillations.
Increasing the field strength up to $H_{\parallel}=\pm 0.4$ T causes
discontinuity of the $\nu(H_{\parallel})$ dependence due to the flop of
$\mathbf{M_{1}}$ and $\mathbf{M_{2}}$. Above this field only one mode is
present corresponding to the joint precession of the two vectors to be
eventually aligned along the field direction in the forced ferromagnetic (FM)
state.

The observed AFM resonance branches were successfully modeled with the two
coupled Landau-Lifshitz equations yielding the interlayer exchange field
$\mu_{0}H_{\rm e}=0.395$ T, and the hard and intermediate-axis anisotropy
parameters $\mu_{0}H_{\rm c}=1.3$ T and $\mu_{0}H_{\rm a}=0.383$ T,
respectively. The authors of Ref. [62] notice that $H_{\rm e}$ in CrSBr is
significantly larger than that, e.g., in CrCl3 ($H_{\rm e}=0.1$ T, see Sect.
2.1) which may explain a higher AFM ordering temperature of CrSBr.

The out-of-plane AFM modes in CrSBr for $\mathbf{H}\parallel\mathbf{c}$ were
probed in Ref. [63]. As expected for the hardest anisotropy axis, both
acoustic and optical modes soften towards the saturation field $\mu_{0}H_{\rm
sat}\approx 1.7$ T where the sublattice magnetization vectors $\mathbf{M_{1}}$
and $\mathbf{M_{2}}$ are fully polarized in the field direction in the forced
FM state. Furthermore, since CrSBr is a direct-gap semiconductor [61], it
could be shown in Ref. [63] that the AFM spin waves may coherently modulate
the electronic structure resulting in the magnon-exciton coupling. By means of
the time-resolved pump-probe spectroscopy it was demonstrated that the energy
of the excited electron-hole pair (exciton) is modulated at the frequencies of
the acoustic and optical AFM modes due to a spin-dependent interlayer
electron-exchange interaction.

AFM resonance excitations in CrSBr were further quantified in Ref. [64] where
the field dependence of the two modes was measured in the three principle
crystallographic directions. The authors have developed the microscopic model
based on a general spin Hamiltonian compatible with the orthorhombic symmetry
of CrSBr to describe the experimental data that enabled a precise
determination of the AFM interlayer exchange $J_{\perp}=0.069$ K and the
single-ion anisotropy parameters $D=0.396$ K and $E=0.207$ K.

The ESR response in the paramagnetic state of CrSBr was reported in Ref. [65].
Signatures of the 2D FM spin correlations at elevated temperatures were
observed in the specific angular dependence of the linewidth $\Delta
H\propto(3\cos^{2}\theta-1)^{2}$ [20] where $\theta$ is the angle between the
$c$-axis and the applied magnetic field. From the analysis of the angular and
temperature dependences of the ESR parameters it was concluded that CrSBr can
be described as a 2D magnet with the XY anisotropy, and as such it might
provide a model system for the observation of topological phase transitions
and of the formation of bound vortex and antivortex pairs.

###  2.4 CrPS4

Similar to the case of CrSBr, the Cr3+ ions in the van der Waals compound
CrPS4 form a ferromagnetic 2D square spin lattice in the $ab$-crystal plane
whereas the planes are weakly AFM coupled. However, in contrast to CrSBr, the
Cr spins in CrPS4 order below $T_{\rm N}=34$ K normal and not parallel to the
$ab$-plane [66] so that this material can be classified in first approximation
as an easy-axis antiferromagnet. The weak interlayer coupling enabled Li et
al. [67] to probe AFM excitations in the ordered state of CrPS4 at frequencies
up to 25 GHz were both acoustic (in-phase) and optical (out-of-phase) modes of
precession of the sublattice magnetization vectors $\mathbf{M_{1}}$ and
$\mathbf{M_{2}}$ could be observed. It turns out that the excitation spectra
for the in-plane magnetic field applied along $a$\- and $b$-axes are different
suggesting that the magnetic anisotropy of CrPS4 is not uniaxial but
orthorombic. Interestingly, application of the magnetic field at intermediate
orientations between these two crystal axes gives rise to the hybridization of
the two modes. While for $\mathbf{H}\parallel\mathbf{b}$ these modes are not
entangled and cross each other, at intermediate angles hybridization yields an
anticrossing gap amounting to several GHz. For $\mathbf{H}$ applied normal to
the $ab$-plane another interesting effect, a change of the sense of precession
of $\mathbf{M_{1}}$ and $\mathbf{M_{2}}$ referred to as chirality switching,
was observed at a critical field given by the difference of the magnetic
anisotropy fields in the $a$ and $b$ directions suggesting an intimate
relation of this effect to the orthorombic magnetic anisotropy in CrPS4.

##  3 Ferromagnetic van der Waals compounds

Ferromagnetic semiconducting or metallic van der Waals compounds appear to be
particularly attractive for applications in magneto-electronic devices where
they can be used for the generation of spin polarized currents, as an element
of field-effect transistors, spin filters, etc. An important prerequisite for
that is stabilization of ferromagnetic order in these compounds in the limit
of a few or single layers which can be maintained if a significantly strong
easy-axis magnetic anisotropy is present in these materials. Quantitative
information on magnetic anistropy and magnetization dynamics can be obtained
from measurements of ferromagnetic resonance. Since in this kind of materials
magnetic layers are only weakly coupled, reliable predictions on the magnetic
behavior in the two-dimensional limit can be made from FMR measurements on
bulk crystals, whereas recent progress in the detection techniques makes it
possible to study FMR directly on thin flakes cleaved from a bulk crystal.

An overview of the currently available FMR techniques and their applications
to various ferromagnetic van der Waals compounds has been recently published
in Ref. [11]. Here, along with mentioning selected FMR studies reviewed in
Ref. [11] we will discuss also works which were not in the scope of that
review, in particular those addressing besides FMR also ESR in the
paramagnetic state.

###  3.1 Cr$X_{3}$ ($X$ = Cl, Br, I)

In chromium-based trihalides Cr$X_{3}$ ($X$ = Cl, Br, I) the six-fold
octahedrally coordinated Cr3+ ions with $S=3/2$ and $L=0$ form the edge-
sharing network in the $ab$ crystal plane resulting in a 2D magnetic honeycomb
lattice. The planes are stacked along the $c$-axis and are held together via
weak van der Waals forces (Fig. 1). The Cr spins are ferromagnetically coupled
in the plane. The coupling between the planes is ferromagnetic for $X$ = Br
and I and is antiferromagnetic for $X$ = Cl. A 2D in-plane FM order in CrCl3
takes place at $T_{\rm c}\approx 17$ K followed by the a inter-plane AFM order
at $T_{\rm N}\approx 15$ K [15]. In contrast, both CrBr3 and CrI3 undergo a
transition to a 3D FM order at $T_{\rm c}=37$ K [68] and 68 K [69],
respectively. ESR spectroscopy of the antiferromagnet CrCl3 was overviewed in
Sect. 2.1. In the following a summary of the magnetic resonance studies of
ferromagnetic members of the Cr$X_{3}$ family will be presented.

Probably the first FMR experiment on single crystals of CrBr3 was reported in
1962 by Dillon [70]. From the temperature dependence of the resonance field
measured at a frequency of about 20 GHz the temperature dependence of the
easy-axis uniaxial magnetocrystalline anisotropy was determined amounting to
$K=9.4\cdot 10^{5}$ erg/cm3 at the minimum temperature $T=1.5$ K. Recent
broad-band FMR study in Ref. [71] covering the frequency range $1-40$ GHz
arrived at the similar result. There, the dispersion in small magnetic fields
of the in-plane FMR mode was successfully fitted in the frame of the multi-
domain FMR theory.

Critical spin dynamics by approaching the ordering temperature $T_{\rm c}$
from above was analyzed in Refs. [72, 73, 74] from the temperature dependent
critical broadening of the ESR linewidth $\Delta H_{\rm crit}\propto[(T-T_{\rm
c})/T_{\rm c}]^{-p}$ at 0.36 GHz [72], 9.4 GHz [74] and 240 GHz [73].
Interestingly, despite a large difference in the excitation frequencies and
corresponding external magnetic fields the critical exponent $p$ converges to
the range $p\sim 0.5-0.9$ in these works. In purely 2D magnets the critical
exponent $p$ is expected to be mostly larger than 1 and to sensitively depend
on the type of the exchange interaction, FM or AFM, and should be distinct for
Heisenberg, Ising or $XY$ magnets [20]. However, in real systems the finite
interlayer interaction hampers the 2D critical behavior by this reducing $p$
in most experimental cases (FM or AFM) to $p<1$ [20], as it apparently the
case for CrBr3, too.

In fact, the two-dimensionality of the spin correlations in CrBr3 manifests in
the $(3\cos^{2}\theta-1)^{2}$ type of the angular dependence of the ESR
linewidth as observed in Refs. [73, 74]. This kind of dependence is due to the
dominating secular part of the dipole-dipole interaction which for the case of
2D FM correlations has the angular form $(3\cos^{2}\theta-1)$ [20]. In 3D
magnets the angular dependence of the linewidth is typically of the form
$\Delta H\propto(\cos^{2}\theta+1)$ [20].

In CrI3, FMR was first probed by Dillon and Olson [69] in 1965 at three
excitation frequencies of 86, 91 and 99 GHz and at a temperature of 1.5 K. The
analysis of the data yielded the easy-axis uniaxial magnetocrystalline
anisotropy constant $K=3.1\cdot 10^{6}$ erg/cm3. Its value appears three times
larger as the one for the sister compound, CrBr3. Since magnetic anisotropy is
an important contributor to the stabilization of magnetic order in quasi-2D
systems, this result is consistent with a much higher FM ordering temperature
$T_{\rm c}$ in CrI3 as compared to CrBr3. These early experiments were later
significantly extended regarding the frequency and temperature range where the
magnetic resonance excitations were probed [71, 75], providing more accurate
estimates of the relevant magnetic parameters. While the work in Ref. [71]
focuses on the low-frequency domain 1–40 GHz, Ref. [75] reports FMR results in
a frequency range 30–330 GHz in magnetic fields up to 15 T. Remarkably, in the
latter work measurements at $T>T_{\rm c}$ revealed the persistence of a zero
field excitation gap and quasi-static anisotropic internal fields up to
$T\approx 1.3\,T_{\rm c}$ thereby suggesting that a significant easy-axis
anisotropy maintains 2D short-range magnetic order in CrI3 above the bulk FM
ordering temperature.

Detailed insights into the structure of the magnetic anisotropy of CrI3 were
obtained in Ref. [76] by studying the angular dependence of the FMR signals at
$\nu=120$ and 240 GHz and at $T=5$ K. The symmetry-based theoretical analysis
of the data revealed a very strong intra-plane Ising-like Kitaev interaction
[77] of $K\sim 60$ K (-5.2 meV), 25 times stronger than the isotropic
Heisenberg exchange $J\sim-2.3$ K (-0.2 meV). It is argued that this
particularly strong anisotropy explains the robust FM order in a monolayer of
CrI3 at $T_{\rm c}^{\rm ML}=34$ K [78]. Indeed, in Ref. [79] an optical
pump/magneto-optical Kerr probe technique was used to detect sub-THz magnetic
resonance modes in CrI3 bilayers under an in-plane magnetic field. These modes
were successfully modeled with Landau-Lifshitz-Gilbert equations yielding the
parameters, such as the zero-field excitation gap, anisotropy field, and the
saturation field, rather similar to the results of the FMR measurements on
bulk single crystals.

Finally, the photosensitivity of the ESR signal of CrI3 was observed by
Singamaneni et al. [30] manifesting in the change of its intensity,
$g$-factor, and the linewidth under application of visible light. The authors
observed the similar effect also for the antiferromagnetic CrCl3 and ascribe
both observations to the photo-induced electron transitions between the
valence band and localized Cr2+ ($3d^{4},\,S=2$) levels (see Sect. 2.1).

###  3.2 Cr5Te8

Cr5Te8 is a half-metallic van der Waals ferromagnet with a high ordering
temperature $T_{\rm C}$ in the range 245–255 K depending on the degree of the
Cr deficiency [80, 81]. The Cr atoms are octahedrally coordinated by the Te
anions and within the $ab$-layers the CrTe6 octahedra share common edges [80].
At $210\,{\rm K}<T<T_{\rm C}$ the magnetic easy axis lies in the $ab$-plane
and reorients normal to the plane below 210 K [81]. Details of this
reorientation transition were addressed in the FMR study performed in Ref.
[81] at an X-band frequency of 9.45 GHz. Angular dependent measurements of the
FMR signal were carried out in the temperature range 180–260 K and analyzed in
the frame of the phenomenological FMR theory [24, 25, 26]. It was concluded
that the reorientation of the effective magnetic anisotropy axis taking place
in the range from $T_{\rm C}$ to 210 K is due the temperature-driven
competition between the 2-nd order intrinsic magnetocrystalline anisotropy of
Cr5Te8 and the shape anisotropy of the platelet-like sample.

###  3.3 Cr${}_{2}X_{2}$Te6 ($X$ = Si, Ge)

Figure 9: Layered crystal structure of the Cr${}_{2}X_{2}$Te6 ($X$ = Si, Ge)
compounds of the rhombohedral symmetry, space group $\bar{R}3$ (left). In the
individual layers CrTe6 octahedra share edges and form a honeycomb lattice
with the pair of $X$ atoms in the center of each honeycomb (right).

The main structural features of the van der Waals compounds Cr${}_{2}X_{2}$Te6
($X$ = Si, Ge) bear similarities with the $M_{2}$P2S6 family (see Sect. 2.2).
The octahedrally coordinated Cr${}^{3+}\,(S=3/2)$ cations form honeycomb
layers in the $ab$-plane stacked along the $c$-axis (Fig. 9). However, the
magnetic properties are very different. In both Cr2Si2Te6 and Cr2Ge2Te6 the Cr
spins are coupled ferromagnetically in the planes and between the planes and
order FM out-of-plane at $T_{\rm C}=32$ K and 61 K, respectively [82, 83]. In
the ordered state Cr2Si2Te6 demonstrates 2D Ising-like magnetic behavior [82]
while it is of the anisotropic Heisenberg-like type in Cr2Ge2Te6 [83].

The intrinsically 2D character of the spin dynamics in Cr2Si2Te6 was
characterized by a multi-frequency ESR/FMR study in the range 13–20 GHz in
Ref. [84]. The authors focused on the temperature and angular dependence of
the $g$-factor shift of the resonance signal both above and below $T_{\rm c}$.
They found out that the $g$ shift follows the $(3\cos^{2}\theta-1)$ dependence
in a broad temperature range while varying the angle $\theta$ which the
applied field makes with the normal to the $ab$-plane. As has been shown
before by Nagata et al. (see, e.g., Ref. [85]) and further elaborated in Ref.
[84] such kind of dependence arises due to an anisotropic spin-spin
correlations in a 2D magnet.

Much more attention was given to magnetic resonance investigations on
Cr2Ge2Te6. This interest was stirred up by the discovery of intrinsic
ferromagnetism in the few layers thin flakes of Cr2Ge2Te6 that order below
$\sim 30$ K [86]. Since according to the Mermin-Wagner theorem stabilization
of magnetic order in a 2D spin system could only be possible in the presence
of easy-axis anisotropy [87], the aspects of magnetic anisotropy in Cr2Ge2Te6
were extensively studied by ESR and FMR techniques.

First efforts to quantify the magnetic anisotropy energy in bulk crystals of
Cr2Ge2Te6 were undertaken by Zhang et al. [88] with FMR experiments in a
limited frequency range $\leq 20$ GHz. From the analysis of the in-plane FMR
branch $\nu(H_{\rm ab})$ the uniaxial anisotropy constant $K_{\rm u}$ was
estimated to reach $3.65\times 10^{5}$ erg/cm3 at the minimum temperature
$T=5$ K. This work was followed by a comprehensive ESR and FMR investigation
of the magnetic anisotropy carried out in Ref. [89] over a wide frequency and
temperature range. Angular and frequency dependences of the resonance modes at
$T<T_{\rm C}$ were modeled using the phenomenological FMR theory [24, 25, 26].
This enabled an accurate determination of the easy-axis type of $K_{\rm
u}=(4.8\pm 0.2)\times 10^{5}$ erg/cm3. Density functional calculations
performed in the same work could successfully reproduce this result assuming
the presence of a substantial electronic correlation strength $U\sim 2$ eV.
Furthermore, multifrequency ESR measurements in the paramagnetic state of
Cr2Ge2Te6 [89] revealed the onset of spin-spin correlations far above $T_{\rm
C}$ suggesting an intrinsic 2D character of ferromagnetism manifesting even in
the bulk form of the Cr2Ge2Te6 compound. Similar results albeit with somewhat
smaller value of the estimated $K_{\rm u}$ were obtained in the FMR study at
$T<T_{\rm C}$ in Ref. [90]. The observed deviation of the $g$-factor tensor
elements of Cr3+ ions to the values larger than $g=2$ were ascribed in Ref.
[90] to the transferred spin-orbit coupling of the $p$ orbitals of the heavy
Te ligands hybridized with the Cr 3$d$ orbitals. The contribution of orbital
magnetism to the $g$-tensor was also proposed in the FMR work in Ref. [91]
where the enhanced spin-orbit coupling was likewise conjectured to be the
reason for the observed dumping of the FMR signal.

The control of the magnitude of the magnetic anisotropy energy of Cr2Ge2Te6 by
hydrostatic pressure and its monitoring by FMR spectroscopy was achieved by
Sakurai et al. [92]. The in- and out-of-plane FMR branches $\nu(H_{\rm ab})$
and $\nu(H_{\rm c})$ were measured at $T=4.2$ K in the frequency range 50–260
GHz and variable applied pressures in the range $P=0-2.39$ GPa [Fig. 10(a-b)].
It has been shown that increasing the pressure reduced the zero field
excitation gap signaling a reduction of the uniaxial magnetic anisotropy
constant $K_{\rm u}$ with respect to its value of $4.8\times 10^{5}$ erg/cm3
at the ambient pressure. Eventually, at the highest $P=2.39$ GPa $K_{\rm u}$
reduced almost twice to $2.6\times 10^{5}$ erg/cm3 [Fig. 10(c)]. At this
pressure the studied sample showed seemingly isotropic FMR behavior with the
gapless and degenerated $\nu(H_{\rm ab})$ and $\nu(H_{\rm c})$ resonance
branches because the reduced $K_{\rm u}$ was fully compensated by the negative
easy-plane shape anisotropy of the platelet-like sample [Fig. 10(c)].

Figure 10: (a-b) FMR out-of-plane and in-plane branches $\nu(H_{\rm res})$ for
Cr2Ge2Te6 at $P=0$ GPa (a) and 2.01 GPa (b). With increasing pressure the two
branches come closer together due to the reduction of the magnetic anisotropy.
(c) Pressure dependence of the uniaxial magnetic anisotropy constant $K_{\rm
u}$ obtained from the fit of the FMR branches (red circles) and from their
simulations based on the FMR theory [24, 25, 26] (blue squares). Horizontal
dashed line denotes the modulus of the negative easy-plane shape anisotropy
constant $K_{\rm shape}$ given by the platelet-like shape of the measured
crystal. By approaching the maximum pressure $P=2.39$ GPa $K_{\rm shape}$
practically compensates $K_{\rm u}$ yielding isotropic FMR response.
(Reprinted with permission from T. Sakurai et al., Phys. Rev. B. 103 (2),
024404 (2021) [92]. Copyright (2021) by the American Physical Society.)

Unlike the above cited works dealing with bulk single crystals of Cr2Ge2Te6,
Zollitsch et al. [93] succeeded to detect and analyze the FMR modes in the
frequency range 12–18 GHz in ultra-thin flakes of this material with the
lateral size in the $\mu$m range and thicknesses varying from 153 nm down to
11 nm (15 monolayers). The key element of their setup was a superconducting
resonator with a high quality factor and strong oscillating microwave field. A
strong coupling regime between the spin wave modes in the sample and microwave
photons in the resonator was achieved resulting in the so-called light-matter
hybrid modes or polaritons. The analysis of these hybrid modes enabled
important conclusions that the magnetic anisotropy constant $K_{\rm u}$
practically did not depend on the thickness of the flake and that the Gilbert
dumping parameter in a 15 monolayers thin flake $\alpha\lesssim 0.021$
appeared to be comparable with conventional ferromagnetic films, such as FeNi
(permalloy), altogether making Cr2Ge2Te6 in the 2D limit a promissing
candidate for spintronic applications.

###  3.4 FexGeTe2 ($x$ = 3, 4, 5)

The family of Fe-rich van der Waals compounds FexGeTe2 ($x$ = 3, 4, 5) feature
metallic conductivity, sizable magnetic anisotropy and high ferromagnetic
ordering temperatures $T_{\rm C}$ approaching and even exceeding room
temperature. Therefore, they are considered for the use in room-temperature
spintronic devices. For this reason complex magneto-transport properties of
FexGeTe2 are currently in the focus of intensive research while there are only
a few reports on FMR studies on these materials.

Ref. [94] reports FMR experiments at the X-band frequency of 9.48 GHz on a
single crystal of Fe3GeTe2 with $T_{\rm C}\approx 204$ K. Broad FMR absorption
line of the order of $\sim 2$ kOe was observed whose angular and temperature
dependence was analyzed with the conventional FMR theory [24, 25, 26]. As a
result the effective dumping parameter was estimated amounting to $\alpha_{\rm
eff}\sim 0.58$. The authors conjecture that its much larger value as, e.g.,
for Cr2Ge2Te6 (see Sect. 3.3) could be due to the itinerant character of
ferromagnetism of Fe3GeTe2, enhanced magnon scattering or spatial
magnetization inhomogeneity across the sample.

Broadband FMR spectroscopy on a single crystal of Fe5GeTe2 with $T_{\rm
C}=332$ K was carried out in the frequency range 5–150 GHz at various
temperatures between 10 and 300 K in Ref. [95]. The data were analyzed using
the standard theoretical FMR approach. The authors find a significant
anisotropy of the $g$-factor suggesting contribution of the orbital magnetism
possibly due to the hybridization of the Fe 3$d$ orbitals with the 5$p$
orbitals of Te. Interestingly, the effective dumping constant was evaluated to
be $\alpha_{\rm eff}=0.035$ at $T=300$ K and $\alpha_{\rm eff}=0.007$ at
$T=10$ K, i.e., drastically smaller as the one reported for Fe3GeTe2 in Ref.
[94] and comparable with $\alpha_{\rm eff}$ for Cr2Ge2Te6 and permalloy (see
Sect. 3.3).

Very recently Pal et al. [96] reported detailed multi-frequency FMR results on
a single crystal of Fe4GeTe2 which orders at $T_{\rm C}\approx 270$ K. This
material attracts significant attention with respect to its transport
properties featuring emergent electronic phases [97] and regarding a peculiar
temperature driven reorientation of the direction of the spins at $T_{\rm
SR}\sim 110$ K from in-plane at $T>T_{\rm SR}$ to out-of-plane at $T<T_{\rm
SR}$ [98, 99]. FMR measurements were carried out in a frequency range 75–350
GHz, at temperatures between 300 and 3 K for the in-plane and out-of-plane
orientation of the applied magnetic field. An analysis of the temperature
evolution of the FMR $\nu(H)$ branches made it possible to disentangle
different contributions to the internal anisotropy field, to single out and
quantify the temperature dependence of the intrinsic magnetocrystalline
anisotropy $K_{\rm u}(T)$ of Fe4GeTe2. It turns out to be always of the easy-
axis type in the entire temperature range but strongly increasing in the
magnitude below $\sim 150$ K and overcoming the (extrinsic) easy-plane shape
anisotropy of the platelet-like crystal at $T_{\rm SR}$ which causes the spins
to turn from the in-plane to the out-of-plane direction. Characteristic
temperatures for the evolution of $K_{\rm u}(T)$ were found to be in a close
correspondence with the emergence of electronic transitions observed in Ref.
[97] suggesting remarkable intertwined magnetic and electronic behaviors in
Fe4GeTe2.

##  4 Magnetic van der Waals topological insulators

Figure 11: Crystal structure of Bi2Te3 in two projections. The Bi atoms are
bonded to the Te atoms in the octahedral fashion and build up the quintuple
layer Te–Bi–Te–Bi–Te. These layers are stacked along the $c$-axis.

Topological insulators (TIs) are a special class of band insulators where by
virtue of the strong spin-orbit coupling the order of the electronic bands in
the bulk of the material is inverted (for the reviews see, e.g., Refs.[100,
101, 102]). Since the spin-orbit coupling is present only in the interior, the
order of the bands should be restored outside the material. Thus the bands
inevitably cross at the Dirac point at the surface yielding robust, so-called
topologically protected conducting electronic surface states. These surface
electrons have unique properties distinct from the "trivial" two-dimensional
metals. They are characterized by the gapless Dirac-cone linear energy
dispersion and the spin-momentum locking which suppresses the scattering of
the Dirac electrons and makes the surface of a TI highly conducting. Due to
these special properties the energy gap at the Dirac point can only be opened
by breaking the time reversal symmetry by a magnetic field [103]. Such a field
can be generated by doping a TI with magnetic atoms or a TI may itself possess
an intrinsic regular magnetic lattice. Opening of the gap would enable new
exotic phenomena such as an anomalous quantum Hall effect or topological
magnetoelectric effect [102, 103] which bring magnetic TIs into the focus of a
very active current research.

The most well-known bulk magnetic TIs are van der Waals compounds such as
magnetically doped Sb2Te3, Bi2Te3 and Bi2Se3, and intrinsically magnetic
(MnBi2Te4)(Bi2Te3)n. Their magnetic properties were addressed in several ESR
studies which will be summarized in the following.

###  4.1 Magnetically doped topological insulators

The van der Waals compounds Bi2Se3 and Bi2Te3 termed tetradymites are
isostructural and crystallize in a lattice of rhombohedral symmetry, space
group $R\bar{3}m$. The building blocks of the structure are quintuple
Te–Bi–Te–Bi–Te layers in the $ab$-crystal plane stacking along the $c$-axis
(Fig. 11). In fact, these materials are not really bulk insulators but rather
doped semiconductors of $n$\- and $s$-type for Bi2Se3 and Bi2Te3, respectively
[104]. Thus, besides featuring the surface conducting states [100] they are
also conducting in the bulk.

The bulk electronic properties of Bi2Se3 were probed in Refs. [105, 106] by
ESR of small amount of Gd3+ ions doped to Bi2Se3. The ESR spectrum (Fig. 12)
features a typical fine structure due to the small splitting of the $S=5/2$
ground state multiplet of the Gd3+ ion by the crystal field [37]. The shape of
the lines is asymmetric of the Dysonian type, typical for metallic samples
with dimensions larger than the penetration depth (skin depth) of the
microwaves [107].

Figure 12: (a) X-band ($\nu=9.5$ GHz) Gd3+ ESR spectrum for
(Bi0.994Gd0.006)2Se3 and (b) its resonance field angle dependence at $T=4\,$K
with the applied magnetic field H parallel to the [001] direction. In panel
(b) the field is rotated towards the $ab$-plane. (c) Q-band ($\nu=34$ GHz)
Gd3+ ESR spectrum. Solid lines are simulations based on the crystal field
Hamiltonian. (Reprinted with permission from J. C. Souza et al., Phys. Rev. B.
106 (23), 235109 (2022) [106]. Copyright (2022) by the American Physical
Society.)

The width of the individual lines of the spectrum $\Delta H(T)$ depended
approximately linear on temperature due to the so-called Korringa relaxation
mechanism of localized spins on conduction electrons with the slope [107]:

$b=d(\Delta H)/dT\propto[D(E_{\rm F})J_{\rm ce}]^{2}\ .$ (3)

Here, $D(E_{\rm F})$ is the density of electronic states at the Fermi energy
and $J_{\rm ce}$ is the constant of exchange interaction between the localized
spin and the spin of a conduction electron. Such a behavior was reasonably
ascribed to the interaction of Gd spins with bulk conduction electrons whereas
a significant increase of this slope below 40 K and of the related shift of
the $g$-factor $\Delta g=D(E_{\rm F})J_{\rm ce}$ [107] observed in Ref. [105]
was speculatively attributed to the contribution of the highly conducting
surface state which considering the small skin depth of the studied sample
might be sizable. This effect was not discussed anymore in the later work in
Ref. [106] where the authors extended the ESR experiments to the Q-band
frequency of 35 GHz. As compared to the X-band, the Q-band Gd3+ ESR spectrum
showed a strong enhancement of the central line with respect to the other
components of the fine structure possibly indicating a collapse of the fine
structure (Fig. 12). It was conjectured, that the conduction electrons
surrounding Gd3+ ions might be stronger localized at a smaller magnetic field
corresponding to the X-band frequency and became more mobile in a higher
magnetic field corresponding to the Q-band frequency and thus screen the
crystal field responsible for the fine structure splitting. This was discussed
as a possible manifestation of the weak antilocalization effect [108]
previously observed in the magneto-transport measurements on thin layers of
Bi2Se3 [109].

With the idea to influence the topological surface states in Bi2Te3 Hor et al.
[110] synthesized a series of Bi2-xMnxTe3 single crystals with the
concentration of Mn2+ dopants in the range $x=0.005-0.09$. Static magnetometry
results evidenced ferromagnetic phase transition for samples with $x\geq 0.04$
at $T_{\rm C}\approx 12$ K and surface sensitive angular resolved
photoemission spectroscopy measured at a slightly higher temperature revealed
modification of the surface states in these samples. The occurrence of FM
order at such moderate concentration of magnetic dopants was rather
unexpected. As a possible reason for it a long-range Ruderman-Kittel-Kasuya-
Yosida (RKKY) interaction [111] between Mn spins mediated by bulk mobile
charge carriers was proposed and first-principle calculations gave theoretical
estimates of its strength [112, 113]. To verify theoretical predictions
detailed X-band ESR measurements on these crystals were carried out in Ref.
[114]. A single exchange-narrowed resonance line ascribed to Mn2+ ions was
observed for all samples and its temperature evolution was studied in the
range from 80 K down to $T_{\rm C}$. Above $\sim 25$ K the linewidth depended
approximately linear on $T$ with the Korringa slope $b$ [Eq. (3)] varying only
slightly with $x$ around $b\sim 1$ mT/K. At lower temperatures the signal
demonstrated strong critical broadening of the 2D character due to the slowing
down of the FM correlations in a quasi-2D magnet [20]. Magneto-transport
measurements perfomed in the same work enabled one to determine the charge-
carrier density and from it to make an estimate of the density of states
$D(E_{\rm F})$ and using the experimental value of $b$ to estimate, according
to Eq. (3), the exchange constant $J_{\rm ce}\approx 0.5-0.7$ eV. Finally, the
RKKI exchange constant $J_{\rm ij}$ between Mn spin sites $i$ and $j$ was
evaluated according to [111]:

$J_{\rm ij}=\frac{3m^{*}V^{2}}{4\pi h^{2}}\cdot J_{\rm
ce}^{2}\cdot\frac{\sin(2k_{\rm F}r)-2k_{\rm F}r\cos(2k_{\rm F}r)}{r^{4}}\ .$
(4)

Here, $m^{*}=0.35m_{\rm e}$ is the effective mass [115], $V$ is the unit cell
volume, $k_{\rm F}$ is the Fermi vector, and $r$ is the distance between the
Mn sites $i$ and $j$ which was estimated from the concentration of Mn in the
samples. $J_{\rm ij}$ appeared to be ferromagnetic with the magnitude $|J_{\rm
ij}|\approx 2-3$ meV (23–35 K). This value agreed well with the theoretical
predictions in Refs. [112, 113] validating the long-range RKKY interaction to
be responsible for the FM order of the Mn-doped Bi2Te3.

The development of the FM correlations by approaching $T_{\rm C}\approx 12$ K
in the Bi1.95Mn0.05Te3 single crystal was further studied by X-Band ESR in
Ref. [116]. Measurements of the angular dependence of the Mn ESR signal
revealed a practically isotropic behavior at $T>30$ K. By lowering the
temperature towards $T_{\rm C}$ the resonance field was getting progressively
anisotropic deviating from the high-temperature paramagnetic position to
smaller values for the external field applied along the $c$-axis and to the
higher values for the field applied in the $ab$-plane. Based on these
observations the authors provided qualitative arguments for the easy-axis type
of FM order in Mn-doped Bi2Te3 in agreement with the ESR results in Ref.
[114]. In addition to the main Mn ESR line the authors reported an additional
weak signal in the spectrum at a low field of $\sim 1000$ Oe for the in-plane
field direction which position does not change with temperature and is not
sensitive to $T_{\rm C}$. This signal was attributed to a small amount of Mn
clusters forming in the host material due to the limited solubility of Mn.

In the subsequent works in Refs. [117, 118, 119] the same group of authors
reported X-band ESR results on single crystals of the 3D topological insulator
Bi1.08Sn0.02Sb0.9Te2S [120] where they observed signals attributed to
intrinsic magnetic centers of different origin in the material formally
composed of nonmagnetic elements. Based on the measurements of the temperature
and angular dependences of the ESR spectra it was concluded that one type of
the centers could be induced by the nonmagnetic structural defects around
which superparamagnetic nano-sized inclusions might arise. The other type of
centers contributed to the ESR spectrum was conjectured to be the bulk charge
carriers organized in small electron and hole droplets.

Besides Mn-doped Bi2Te3, magnetic doping of the sister compound Bi2Se3 was
also considered to be promising for the alteration of the band structure of
the topological surface states. That moderate Mn doping can result in a long-
range FM order in a thin film of Bi2Se3 was shown by Savchenko et al. [121]
who fabricated single-crystalline films of this material with thicknesses in
the range 0.3–0.7 $\mu$m nominally doped with 6–8 % of Mn. These films
demonstrated an FM transition at $T_{\rm C}=5.3$ K and the properties of the
FM ordered state were examined by X-band ($\nu=9.4$ GHz) ESR/FMR spectroscopy.
Strong temperature and angular dependences of the FMR signal were observed and
analyzed with the standard FMR theory. It was found that the films feature an
easy-plane magnetic anisotropy which increases with increasing the Mn
concentration and reaches the value of $K_{\rm u}=-3720$ erg/cm3 at the
highest doping level.

###  4.2 Intrinsically magnetic topological insulators

Figure 13: Crystal structure of the (MnBi2Te4)(Bi2Te3)n family with $n=0$
(left) and $n=1$ (right). MnBi2Te4 on the left is composed of the stacked
septuple Te–Bi–Te-Mn–Te–Bi–Te layers. In MnBi4Te7 on the right the neighboring
septuple layers are separated by the quintuple Te–Bi–Te–Bi–Te layers.

While magnetically doped topological insulators were known for already some
time, intrinsic magnetic TIs, i.e., stoichiometric compounds comprising a
regular long-range ordered magnetic lattice, were discovered only recently in
the family of (MnBi2Te4)(Bi2Te3)n van der Waals compounds [122, 123]. The
crystal structures of the two mostly studied so far materials MnBi2Te4 and
MnBi4Te7 are shown in Fig. 13. They are closely related to the structure of
the nonmagnetic TI Bi2Te3 (Fig. 11). MnBi2Te4 is built up of septuple (ST)
Te–Bi–Te-Mn–Te–Bi–Te layers stacked along the $c$-axis. The ST layer differs
from the quintuple (QT) Te–Bi–Te–Bi–Te layer of Bi2Te3 by the presence of an
additional Mn–Te atomic sublayer. In MnBi4Te7 magnetic ST layers are separated
by nonmagnetic QT layers. MnBi2Te4 and MnBi4Te7 order antiferromagnetically at
$T_{\rm N}=24$ K and 13 K, respectively [122, 124]. The magnetic moments of
the Mn2+ ($S=5/2$) ions in the ST layers order out-of-plane ferromagnetically
while the intelayer coupling between the QT layers is antiferromagnetic. Both
compounds are self-doped magnetic semiconductors and feature topological
surface electronic states [122, 124].

The spin dynamics in the AFM-ordered and paramagnetic states of MnBi2Te4 and
MnBi4Te7 and its possible impact on the surface electronic states was
addressed by the AFMR/ESR spectroscopy in Refs. [122, 124, 125, 126].
Multifrequency ESR measurements in the range 75–330 GHz made it possible to
map the AFMR excitation branches in the ordered state and study their
evolution by raising the temperature above $T_{\rm N}$ [124, 126].

Figure 14: (a) Resonance branches $\nu(H)$ at $T=4$ K $\ll T_{\rm N}$ for
MnBi4Te7 (a) and for MnBi2Te4 (b). Typical absorption signals of the MnBi4Te7
compound at selected frequencies are shown in panel (a). In both panels
symbols are experimental data and solid and dashed lines are results of
theoretical modeling. The green solid line in panel (b) denotes the
paramagnetic resonance branch $\nu=(g\mu_{\rm B}/h)H$. (Reprinted with
permission from A. Alfonsov et al., Phys. Rev. B. 104 (19), 195139 (2021)
[125]. Copyright (2021) by the American Physical Society.)

Although MnBi4Te7 orders AFM at $T_{\rm N}=13$ K, the interlayer coupling
between magnetic ST layers separated by nonmagnetic QT layers is such weak
that an application of a small magnetic field of the order of some tens of mT
polarizes the Mn spins and turns the system into the forced FM state. In this
regime MnBi4Te7 demonstrates the resonance response typical for an easy-axis
ferromagnet [Fig. 14(a)] for which analytical solutions do exist [127]:

$\displaystyle{\rm H\parallel c-axis:}$ $\displaystyle
h\nu=\text{g}\mu_{B}\mu_{0}(H+|H_{a}|)$ (5) $\displaystyle{\rm H\parallel ab-
plane:}$ $\displaystyle h\nu=\text{g}\mu_{B}\mu_{0}\sqrt{H(H-|H_{a}|)}\ .$ (6)

Fitting the data to Eqs. (5) and (6) yields the value of the total anisotropy
field $H_{a}$. After a proper subtraction from $H_{a}$ of the demagnetizing
field due to the platelet-like shape of the sample the intrinsic
magnetocrystaline energy of MnBi4Te7 was evaluated to be $E_{\rm a}=-0.18\pm
0.01$ meV per Mn in one magnetic ST layer.

The AFM excitation spectrum of MnBi2Te4 is more complex [Fig. 14(b)]. At
fields smaller than $\sim 5$ T the resonance branches are typical for a
uniaxial easy-axis antiferromagnet. For $\mathbf{H}$ along the easy axis
($c$-axis) there are two branches, one ascending (L2) and one descending (L3),
that collapse at the spin-flop field $\mu_{0}H_{\rm sf}=3.5$ T, above which a
new ascending resonance branch L1 appears. For $\mathbf{H}$ in the hard plane
($ab$-plane) there are two non-linear dispersing modes L4 and L5.
Surprisingly, the L1 and L3 branches cross each other at a threshold field of
$\sim 6$ T above which they follow a linear field dependence. This observation
is at odds with the canonical spin-wave theory according to which the branches
L1 and L4 should asymptotically approach the paramagnetic line $\nu=(g\mu_{\rm
B}/h)H$ but never cross it under the condition that the AFM exchange energy
$E_{\rm exch}$ is much larger than the magnetic anisotropy energy $E_{\rm a}$
[127]. This puzzle was solved by numerical modeling of the branches on the
basis of a linear spin-wave theory with the second quantization formalism
[127, 128]. An excellent agreement with experiment was achieved [solid red and
black lines in Fig. 14(b)] yielding the AFM interplane exchange energy $E_{\rm
exch}=0.65\pm 0.01$ meV per Mn in one ST layer and $E_{\rm a}=-0.21\pm 0.01$
meV per Mn in one ST layer. The unusual crossing of the L1 and L4 branches
appeared to be due to the similar strength of $E_{\rm exch}$ and $E_{\rm a}$
in MnBi2Te4 and the closeness of these energy scales to that of the applied
magnetic field. Thus, such a strong applied magnetic field can completely
decouple the magnetic ST layers and turn the formerly AFMR modes L1 and L4
into the linear dispersing FM excitations according to Eqs. (5) and (6)
[dashed red and black straight lines in Fig. 14(b)]. The similarity of the
anisotropy energies of MnBi2Te4 and MnBi4Te7 suggests that it is an intrinsic
property of the magnetic ST layers little affected by the insertion of
nonmagnetic QT layers.

Temperature dependent measurements of the resonance signals evidence the 2D
character of the FM in-plane correlations in the weakly coupled magnetic ST
layers in MnBi4Te7 persisting far above the 3D ordering temperature $T_{\rm
N}=13$ K. Specifically, the zero field excitation gap of the FM modes does not
close at $T_{\rm N}$ but vanishes only at $\sim 30$ K. The resonance field
deviates from its isotropic paramagnetic value up to the same temperature. In
contrast, the zero field excitation gap for MnBi2Te4 closes at $T_{N}=24$ K
and the resonance field of the signals measured at small applied fields
rapidly approaches the isotropic paramagnetic value above $T_{\rm N}$ as
expected for a 3D magnet with significant inter-layer coupling. However,
measurements at fields stronger than the threshold field of $\sim 6$ T, i.e.,
in the regime where the magnetic ST layers become decoupled, the 2D FM
correlations remain prominent up to temperatures of $\sim 2T_{\rm N}$
manifesting in the anisotropic shifts of the resonance signals.

Figure 15: Temperature dependence of the ESR linewidth at $\nu=9.56$ GHz and
at two orientations of the applied magnetic field for MnBi4Te7 (a) and for
MnBi2Te4 (b). Dashed lines are the fits to the Korringa dependence
$\Delta(H)=\Delta H_{0}+bT$. Vertical dashed lines indicate the magnetic
ordering temperatures $T_{\rm N}$ and the onset of the Korringa relaxation
regime $T^{\ast}$. (Reprinted with permission from A. Alfonsov et al., Phys.
Rev. B. 103 (18), 180403 (2021) [126]. Copyright (2021) by the American
Physical Society.)

The dynamics of the Mn spins in the paramagnetic state of MnBi2Te4 and
MnBi4Te7 and its possible relation to the surface electronic structure was
studied with the X-band ESR spectroscopy in Refs. [122, 126]. Since both
compounds are doped semiconductors demonstrating bulk metallic conductivity
one expected the Korringa mechanism to be the main channel of relaxation of Mn
spins similar to, e.g., the Mn-doped Bi2Te3 (see, Sect. 4.1). Indeed, upon
increasing the temperature above $T_{\rm N}$ the critical broadening of the
ESR line due to the short-range FM correlations ceases and the linear in
temperature Korringa relaxation regime characteristic of an uncorrelated
metallic paramagnet sets in at $T^{\ast}\sim 60$ K (Fig. 15). The data above
$T^{\ast}$ was fitted to the Korringa dependence $\Delta(H)=\Delta H_{0}+bT$,
where $\Delta H_{0}$ is the residual linewidth and $b$ is the Korringa slope
defined in Eq. (3). In most of the metallic compounds the parameter $b$ was
found to be independent of the direction of the applied field. Evidently, that
was also the case for MnBi4Te7 where the Korringa slope was almost the same
for both field orientations with $b_{\rm c}\sim b_{\rm ab}\sim 9$ Oe/K [Fig.
15(a)]. However, the Korringa slope for MnBi2Te4 was found to be surprisingly
anisotropic with $b_{\rm c}=7$ Oe/K$\,\gg b_{\rm ab}=2.4$ Oe/K [Fig. 15(b)].

To understand this striking observation the authors have carried out density-
functional-theory (DFT) calculations of the bulk electronic structure. The
results show that the density of states of the bulk charge carriers in a
certain range of the carrier concentrations depends on the direction of the Mn
magnetic moments. This non-trivial property causes the anisotropy of the
dynamics of the localized Mn moments reflected in the anisotropy of the
Korringa slope $b$ which, due to the specific details of the band structure,
is very strong in MnBi2Te4 and is much less pronounced in MnBi4Te7. This
finding helped to find an explanation of the unexpected temperature evolution
of the band structure of topological surface states observed in MnBi2Te4 by
angle-resolved photoemission spectroscopy (ARPES) [122]. The energy gap opened
at the surface Dirac point at $T<T_{\rm N}$ due to the static AFM order of the
Mn spins did not close completely at $T_{\rm N}$ and persisted up to much
higher temperatures. In view of the ESR results this was ascribed to the
strong anisotropy of the Mn spin fluctuations caused by their anisotropic
relaxation on bulk charge carriers that produces a nonzero instantaneous out-
of-plane magnetic field acting on the surface electronic states on a timescale
much longer than that in the ARPES experiment which then detects a kind of a
dynamically opened gap at the Dirac point.

The sensitivity of the density of states to the particular orientation of the
Mn moments is obviously related, among other factors, to the strong spin-orbit
coupling in particular due to the presence of the heavy element Bi in
MnBi2Te4. Indeed, recent ESR experiments in Ref. [129] on the new candidate
material for ferromagnetic topological insulator MnSb2Te4 with mich lighter Sb
instead of Bi documented a completely isotropic Korringa behavior of the Mn
ESR linewidth and the DFT calculations showed that the band structure is
insensitive to the orientation of the Mn moments.

##  5 Conclusion

In conclusion, this article summarized the achievements of the ESR, AFMR and
FMR studies on quasi-2D van der Waals magnets in the elucidation of important
fine details of the magnetically ordered ground state of these compounds, in
particular a precise determination of the type, the symmetry and the magnitude
of the intrinsic magnetocrystalline anisotropy. Also, these investigations
provided comprehensive insights into the low-energy magnon excitations in the
ordered phase and the details of the correlated spin dynamics in the
paramagnetic state. Owing to the very weak interlayer magnetic coupling many
of these materials demonstrate in the bulk form signatures of intrinsic 2D
behavior both in the static and dynamic sectors. Therefore, the obtained
results on the studied bulk single crystals enable reliable predictions for
the monolayers of these materials in particular in terms of the stability of
the long-range magnetic order and the damping of the spin wave excitations,
which are the decisive criteria for the use of monolayers in the spin-
electronic devices. Furthermore, the ESR spectroscopy contributed to the
significant progress in the understanding of the interplay between magnetism
and electronic structure in semiconducting and metallic van der Waals
compounds which affects the magnetic anisotropy and thus the type of magnetic
order and also gives rise to novel magneto-electronic effects which are
realized, e.g., in magnetic topological insulators.

Acknowledgments

The authors gratefully acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG) through Projects No. 447482487, 499461434,
455319354, and within the Collaborative Research Center SFB 1143 “Correlated
Magnetism - From Frustration to Topology” (project-id 247310070), and the
Dresden-Würzburg Cluster of Excellence (EXC 2147) “ct.qmat - Complexity and
Topology in Quantum Matter” (project-id 390858490).

## Declarations

Funding

Deutsche Forschungsgemeinschaft, projects 447482487, 499461434, 455319354,
247310070, and 390858490.

Conflict of interest

The authors declare no conflict of interest.

Ethics approval

not applicable

Availability of data and materials

not applicable

Authors’ contributions

All authors jointly developed the conception of the review. V.K. wrote the
first draft of the manuscript with active participation of A.A. and B.B. All
authors edited its final version.

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