# Spin-glass states generated in van der Waals magnet Cr2Ge2Te6 by alkali-ion
intercalation

S. Khan  London Centre for Nanotechnology, University College London, London,
WC1H 0AH, United Kingdom<EMAIL_ADDRESS>E. S. Y. Aw  Department of
Physics $\&$ Astronomy, University College London, London, WC1H 0AH, United
Kingdom  L. A. V. Nagle-Cocco  Cavendish Laboratory, University of Cambridge,
Cambridge, CB3 0HE, United Kingdom  A. Sud  WPI-AIMR, Tohoku University,
2-1-1, Katahira, Sendai 980-8577, Japan  S. Ghosh  Institute for Condensed
Matter Physics and Complex Systems, School of Physics and Astronomy,
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  M. K. B. Subhan
Department of Physics $\&$ Astronomy, University College London, London, WC1H
0AH, United Kingdom  Z. Xue  London Centre for Nanotechnology, University
College London, London, WC1H 0AH, United Kingdom  C. Freeman  London Centre
for Nanotechnology, University College London, London, WC1H 0AH, United
Kingdom  D. Sagkovits  London Centre for Nanotechnology, University College
London, London, WC1H 0AH, United Kingdom  A. Gutiérrez-Llorente  Escuela
Superior de Ciencias Experimentales y Tecnología, Universidad Rey Juan Carlos,
Madrid 28933, Spain  I. Verzhbitskiy  Physics Department, National University
of Singapore, Singapore, Singapore.  D. M. Arroo  Department of Materials,
Imperial College London, Exhibition Road, London, SW7 2AZ, United Kingdom  C.
W. Zollitsch  London Centre for Nanotechnology, University College London,
London, WC1H 0AH, United Kingdom  G. Eda  Physics Department, National
University of Singapore, Singapore, Singapore.  E. J. G. Santos  Institute for
Condensed Matter Physics and Complex Systems, School of Physics and Astronomy,
University of Edinburgh, Edinburgh EH9 3FD, United Kingdom  S. E. Dutton
Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, United
Kingdom  S. T. Bramwell  London Centre for Nanotechnology, University College
London, London, WC1H 0AH, United Kingdom  C. A. Howard  Department of Physics
$\&$ Astronomy, University College London, London, WC1H 0AH, United Kingdom
H. Kurebayashi  London Centre for Nanotechnology, University College London,
London, WC1H 0AH, United Kingdom  WPI-AIMR, Tohoku University, 2-1-1,
Katahira, Sendai 980-8577, Japan  Department of Electronic and Electrical
Engineering, University College London, London, WC1E 7JE, United Kingdom
<EMAIL_ADDRESS>
###### Abstract

Tuning magnetic properties in layered van der Waals (vdW) materials has
captured a significant attention due to the efficient control of ground-states
by heterostructuring and external stimuli. Electron doping by electrostatic
gating, interfacial charge transfer and intercalation is particularly
effective in manipulating the exchange and spin-orbit properties, resulting in
a control of Curie temperature ($T_{\text{C}}$) and magnetic anisotropy. Here,
we discover an uncharted role of intercalation to generate magnetic
frustration. As a model study, we intercalate Na atoms into the vdW gaps of
pristine Cr2Ge2Te6 (CGT) where generated magnetic frustration leads to
emerging spin-glass states coexisting with a ferromagnetic order. A series of
dynamic magnetic susceptibility measurements/analysis confirms the formation
of magnetic clusters representing slow dynamics with a distribution of
relaxation times. The intercalation also modifies other macroscopic physical
parameters including the significant enhancement of $T_{\text{C}}$ from 66K to
240 K and the switching of magnetic easy-hard axis direction. Our study
identifies intercalation as a unique route to generate emerging frustrated
spin states in simple vdW crystals.

##  1 Introduction

The discovery of long-range magnetic order in layered van der Waals (vdW)
materials down to individual monolayers [1, 2] has opened up new avenues for
improving the understanding of fundamental concepts in condensed matter
physics [3, 4, 5, 6, 7, 8, 9, 10, 11, 12], together with potential for future
applications [13, 14, 15, 16, 17]. For instance, a clean interface can be
achieved in heterostructures made out of vdW materials and these devices may
then be deployed for efficient switching of magnetisation state by spin-orbit
torques [13, 14], as well as for magneto-optical applications [15, 16, 17].

Among the vdW magnetic materials, the families of chromium halides (i.e. CrI3,
CrBr3 and CrCl3), lamellar ternary chalcogenides (i.e. Cr2Ge2Te6 and
Cr2Si2Te6), metal thiophosphates (MPS3, M = Fe, Mn and Ni) and iron-based
tellurides (i.e. Fe3GeTe2 and Fe5GeTe2), have been intensively and extensively
studied [18, 19, 20, 21, 22]. One common aspect shared by most of these
materials is the magnetic ordering transition temperature, Curie temperature
($T_{\text{C}}$) for ferromagnets, lying below room temperature [14], with a
few exceptions including Fe5GeTe2 [14] and Fe3GaTe2 [23]. In order to fully
realise the potential of 2D magnetic materials for practical applications, a
consequential drive is taking place in either discovering new materials with
above-room-temperature $T_{\text{C}}$, or to elevate $T_{\text{C}}$ of
existing systems by external means. For the latter, taking Cr2Ge2Te6 (CGT) as
an example, its $T_{\text{C}}$ of 66 K [24, 25] is reported to be
significantly enhanced by electrostatic gating [26, 27], strain [28, 29] and
intercalation [7].

The modification of magnetic properties in vdW systems has been primarily
focused on parameters such as $T_{\text{C}}$ and magnetic anisotropy constants
that arise from the exchange and spin-orbit interaction respectively. Electron
doping alters the type and strength of the exchange interaction [31, 32], e.g.
via the change of the Fermi level position for itinerant ferromagnets [33],
and the number of carriers that play a leading role in carrier-mediated
indirect Ruderman–Kittel–Kasuya–Yosida (RKKY) and double-exchange mechanisms
for less-conductive magnets [26]. Doped electrons populate in electronic bands
with a specific orbital character, modifying the spin-orbit property impacting
on magnetic anisotropies [34]. These material properties represent the
homogeneous nature of systems. By contrast, some important yet unanswered
questions in researching vdW magnetic systems are: what arise when a vdW
magnetic system is introduced with inhomogeneous carrier-doping, and what are
resultant parameter changes that we can experimentally access to? To address
this with a model case, we employ Na-ion intercalation into the vdW gaps of
pristine Cr2Ge2Te6 where Na ions are inhomogeneously intercalated within the
gaps. We actively use this nature to intentionally create disorder and study
its consequence to magnetism.

While inhomogeneity might be considered negative since it breaks the
periodicity/coherency of crystallinity in real space, it can however generate
new physics that cannot be achieved otherwise. In magnetism, spin-glass states
[35] emerge because of the presence of multiple local energy minima that the
system freezes into, generated by inhomogeneity and frustration/competition
between the ferromagnetic and anti-ferromagnetic exchange interactions.
Hopping between the energy minima originates experimentally measurable
magnetic responses that are time-, temperature- and magnetic-field-dependent
[10]. Characteristic of the phase transition, the spin-glass has an order
parameter defined by Edwards and Anderson [37], together with the critical
temperature of its freezing, $T_{\text{g}}$. The research field of spin-
glasses is vast and diverse, ranging from canonical spin-glass states often
observed in non-magnetic materials with randomly distributed magnetic
impurities [38] to more complex spin-glassy behaviours in frustrated
ferromagnets, called reentrant spin-glasses [39]. Spin glass formation in
highly correlated systems is also known to herald yet more exotic behaviour
such as high-temperature superconductivity in the cuprates [40]. The
realisation of glassy magnetic states in vdW materials therefore suggests
their significant potential for supporting new magnetic and electronic states,
formed by tuning material properties via substitution, intercalation and
exfoliation or heterostructuring.

In this study, we intercalated pristine CGT single crystals with alkali
(sodium, Na)-ions to create magnetically-disordered Na-CGT. As a result, we
discovered a frustrated magnetic state in the vicinity of 35 K by dynamic
magnetic susceptibility measurements suggesting the nature of this transition
to be emergence of spin-glassiness behaviour. Further quantitative analysis
points towards Na-CGT hosting a cluster spin-glass state at low temperatures,
a feature not present in pristine CGT. This emergent frustrated state is
accompanied by a clear increase of electric conductivity, enhanced Tc to $\sim
240$ K and the sign reversal of two-fold magnetic anisotropy term that removes
its out-of-plane magnetic easy axis and creates the magnetic easy plane in Na-
CGT. Our study illustrates that vdW magnetic systems are highly tunable, not
only for the exchange and spin-orbit coupling properties, but also now for
magnetic frustration by intercalation techniques.

##  2 Results $\&$ Discussion

Figure 1: (a) Schematic of the pristine CGT with the uniaxial magnetic
anisotropy along the out of plane orientation. The red arrows represent
magnetic moments along the easy axis. (b) Schematic of the Na-intercalated CGT
with the magnetic easy plane. (c) XRD of pristine CGT and Na-CGT showing a
series of 00l diffraction peaks. The peak around 3 Å-1 in Na-CGT is due to the
beryllium sample holder (see Supplementary Information). (d) The magnified
section for 006 diffraction peak for comparison between the two materials. (e)
Raman spectra for pristine (blue) and Na-intercalated (orange) CGT, showing
the major phonon modes for CGT based on assignments from Ref. 7. (f)
Temperature dependence of resistance in CGT and Na-CGT.

###  2.1 Characterisation of Na-Cr2Ge2Te6

Figure 1 (a $\&$ b) depicts the CGT and Na-CGT systems, respectively, and
illustrates the presence of sodium-ions in the vdW gaps. Single CGT crystals
were intercalated with sodium ions via the ammonia-based method [41] (details
in Methods section). Structural changes in the intercalated CGT are
characterised using X-ray diffraction (XRD) and Raman spectroscopy. XRD
results in Fig. 1 (c $\&$ d) show that the intercalated sample preserves the
original crystalline structure of CGT. The c-axis parameter of pristine CGT is
6.8 Å, which corresponds to the distance between the centre of adjacent
layers. A downshift is observed for all 00l-peaks after intercalation,
indicating an increase in the interlayer spacing by 0.04 Å. This small
increase in interlayer spacing upon intercalation is also observed in other
intercalated compounds [42]. Raman spectroscopy is also performed for a large
number of spatial positions on the sample surface, and the peak structure is
maintained, consistent with the indication from XRD. However, quantitative
analysis of the three most significant Raman modes ($A^{1}_{g}$, $E^{2}_{g}$,
and $E^{5}_{g}$) shows that there are statistically significant changes to
phonon energy and lifetime with intercalation. There are no major changes to
the three main Raman modes that arise from intralayer vibrations of CGT, as
expected, as intercalation should not change the bonding within the layers
themselves. The small changes in peak energy and width (which is determined by
lifetime) are consistent with carrier doping [9, 8] (see Supplementary
Information Fig.S2-S3).

Electrical transport measurements were performed in order to see the effect of
carrier doping by the alkali atoms to the system. Figure 1(f) shows the
temperature dependence of resistance for both CGT and Na-CGT. Unlike the
standard semiconducting behaviour of temperature dependence in pristine CGT
[25, 45], Na-CGT displays a semiconductor-like response down to 80 K, before
showing a metallic-like resistance behaviour in the low temperature regime.
While complex, this clear change in temperature dependence of resistance
further emphasises that electron doping by the intercalation process was
successful.

Figure 2: (a) Temperature dependence of magnetisation $M$ in CGT and Na-CGT
measured during cooling in the absence of a magnetic field. $M$ is normalised
at the lowest temperature data point for each scan. (b)&(c) $M$-$H$ loops of
Na-CGT for ab-plane and c-axis directions measured at 10 K and 200 K,
respectively. (d) Fixed-angle magnetoresistance (MR) results of Na-CGT for
different angle $\theta$ defined from the c-axis direction, for 10 K. (e)
Saturation fields ($\mu_{0}H_{\text{sat}}$) extracted for each MR scan for
different $\theta$ and temperature. (f) A plot of the percentage change of MR
as a function of temperature in Na-CGT for both ab-plane and c-axis.

Figure 2 (a) shows temperature-dependent magnetisation along the ab-plane
orientation in both CGT and Na-CGT, and demonstrates the impact of Na
intercalation on magnetisation in the static regime. In contrast to
$T_{\text{C}}$ of CGT $\sim$ 66 K [24], Na-CGT shows its magnetisation values
maintained to a higher temperature up to $\sim$ $240\,K$, implying this as its
$T_{\text{C}}$. Similar $T_{\text{C}}$ enhancement in CGT has been reported by
organic molecule intercalation (to $\sim$ 208 K) [7] and by pressure (to
$\sim$ 250 K) [28] where in all cases metallic transport accompanies the
$T_{\text{C}}$ enhancement. Since $T_{\text{C}}$ scales with the exchange
coupling strength, it is possible to interpret in our sample that doped
electrons enhance the exchange coupling strength and at the same time these
less-localised electrons can contribute to transport properties. In pristine
CGT, electrons are fully localised and the super-exchange mechanism that
arises from virtual electron excitations between two neighbouring Cr ions via
a Te ion with the 90∘ Kanamori-Goodenough mechanism [46]. Local Hund’s rules
within both Te and Cr ions stabilise a ferromagnetic order. Through this
process, electrons have to be virtually excited into the conduction band via
the bandgap, the exchange energy gain by this process is marginal, hence low
$T_{\text{C}}$ in CGT. Doping real electrons to Cr ions changes their ionic
states from Cr3+ into Cr2+, activating the double-exchange mechanism [47] that
bolsters the exchange interaction. It may also be relevant that since Cr2+ is
a Jahn-Teller ion, its substitution for Cr3+ will lead to local distortions
and increased local anisotropy that potentially enhances any spin glass
behaviour that we discuss below.

CGT is known to have uniaxial magnetic anisotropy with the out-of-plane easy
axis [24, 25]. Magnetisation measurements as a function of magnetic field
($M$-$H$ curves) are performed for Na-CGT and the results at 10 K for the ab-
plane and c-axis orientations are shown in Fig. 2 (b). It is evident that
after intercalation, spins prefer to align within the ab-plane compared to the
c-axis in CGT, implying the sign change of the uniaxial anisotropy term upon
intercalation. Figure 2 (c) shows the $M$-$H$ curves of Na-CGT at 200 K,
showing a clear magnetic response for both orientations with the sustained
uniaxial anisotropy. As a proof of the long-range magnetic order, such a
response disappears at 295 K where only a linear $M$-$H$ trend is observed and
shown in Fig.S5(d) Supplementary Information.

Electrically conducting behaviour in Na-CGT allows us to probe the magnetic
response via magnetoresistance (MR) which was impossible in pristine CGT. We
performed angular dependent MR measurements along the sample orientation of
in-plane (ab-plane) to the out-of-plane (c-axis) direction, as shown in Fig.
2(d), showing the clear magnetic anisotropy easy axis (plane) from the c-axis
to easy-plane in Na-CGT since the saturation field becomes larger for more
out-of-plane measurements. Figure 2(e) captures this change in magnetic
anisotropy for different temperatures (calculation of the saturation field
$H_{\text{sat}}$ is highlighted in Fig.S11(b)) of Supplementary Information
where $H_{\text{sat}}$ for c-axis ($0\degree$) increases with decreasing
temperature, and so as the difference in $H_{\text{sat}}$ to the ab-plane
($90\degree$). A sizable change in MR for Na-CGT across a wide temperature
range is shown in Fig. 2(f), which disappears at the temperature where the
long range order of $M$ also vanishes in Fig. 2(a). This measurement reassures
consistency in the new intercalated Na-CGT samples and reproducibility of
experimental data by means of separate probing techniques.

###  2.2 Emergent spin-glass nature

Figure 3: (a) $M$ vs $T$ curves in Na-CGT showing a bifurcation feature with
different field cooling protocols. (b) Temperature dependence of real part of
ac magnetic susceptibility. We define the peak temperature as
$T_{\text{g}}(f)$ that is a function of frequency as shown in the inset. (c)
$\&$ (d) Plots of fitting results using $T_{\text{g}}(f)$ and the function of
the scaling power and VF law, respectively. See more details in the main text.

Another striking property which emerges via the intercalation process is spin-
glass like behaviour in Na-CGT, a feature absent in pristine CGT. It is
important to note that spin-glass states have not been experimentally reported
in other heavily studied 2D magnetic systems [19, 20], such as CrI3, CrBr3,
Fe5GeTe2, as well as CGT with carrier doping by organic molecules [7] and
electrostatic gating [26, 27], and pressure application [28, 29].

Spin-glass behaviour in Na-CGT is identified as the divergence of
magnetisation in temperature between the zero-field cooled (ZFC) and field-
cooled (FC) protocols, at small magnetic fields. As shown in Fig. 3 (a), a
bifurcation between the two cases is observed below a temperature at which a
steep increase in magnetisation takes places. The magnetisation of the ZFC
curve shows a defined broad peak, whereas in the FC protocol, the
magnetisation keeps on increasing as the sample is further cooled. The local
maximum is observed and we define this as $T_{\text{g}}$ [48, 49]. This
bifurcation in the magnetisation is minimised as the size of magnetic field in
increased (see Supplementary Information). Such splitting and irreversibility
in the magnetic memory of the system are recognised as characteristic features
which can arise from various phenomena, including spin-liquid, super-
paramagnetic, disordered anti-ferromagnetic and spin or cluster glass states
[10, 38].

In order to gain further insight in this feature potentially representing a
spin-glass like state in Na-CGT, ac magnetic susceptibility measurements were
performed, where $\chi_{\text{a.c.}}=\chi^{{}^{\prime}}+i\chi^{"}$ [12].
Figure 3 (b) shows the real (in-phase), $\chi^{{}^{\prime}}(T)$, component of
the susceptibility, as a function of temperature at different frequencies
ranging from 50 Hz to 5 kHz. A distinctive cusp-like anomaly is quite evident
in $\chi^{{}^{\prime}}(T)$ at low temperatures, which matches to the same
temperature region of branching-off observed in Fig. 3 (a). In the inset of
Fig. 3 (b), the main feature of $\chi^{{}^{\prime}}(T)$ is a small frequency
shift in the peak-maximum towards higher temperatures as the frequency is
increased. The shift in peak position for a frequency window of Hz-kHz
indicates slow dynamics in the system, and this is contrary to ferromagnetic
and conventional systems where a similar frequency dependence of
$\chi^{{}^{\prime}}(T)$ is found only in the MHz regime [10]. Figure 3 (b)
also highlights that $\chi^{{}^{\prime}}(T)$ for a small magnetic field
disappears at high temperatures. The imaginary part (out of phase),
$\chi^{"}(T)$, of the ac susceptibility shown in Supplementary Information
(Fig.S8), is almost 30 times smaller at the peak maximum than
$\chi^{{}^{\prime}}$. $\chi^{"}(T)$ increases from zero to a finite value near
the expected spin-glass like transition and also displays a clear anomaly with
an inflection point, which is related to the dissipative nature of the system
due to slow magnetic relaxation [11, 38].

The observed spin-glass property has formed within the ferromagnetically
ordered matrix background. This particular type of spin freezing below a
ferromagnetic transition is referred to as the reentrant spin glass state
[39]. Here, the fluctuation of local exchange interaction is sufficient enough
to produce the local energy minima that stabilise frozen spin-glass states
against the ferromagnetic order [52, 53]. Phenomenologically, this condition
is met when the averaged exchange coupling strength for the $i$th and $j$th
site spins <$J_{\text{ij}}^{2}$> and its local fluctuation average
<$J_{\text{ij}}$>2 are comparable to each other [39], where < · · · > denotes
an average over the entire spin system. In Na-CGT, this situation is likely
established due to inhomogeneous Na intercalation that induces an electron-
doping distribution. Since the exchange coupling strength is a function of
carrier density between magnetic ions, such doping inhomogeneity brings about
the fluctuation and hence local competition of the exchange interaction, a key
ingredient for the spin-glass states. While such atomic scale fluctuation is
not directly measured in our study, the clear spin-glass signature observed in
our study strongly indicates its presence.

For more quantitative analysis, a series of analytical procedures is carried
out with phenomenological scaling and power-law models [10, 54] to further
inquest and interpret the data with $T_{\text{g}}$ in $\chi^{{}^{\prime}}(T)$
as a function of frequency. The Mydosh parameter [10], $K$, is measured as the
respective variation of $\chi^{{}^{\prime}}(T)$ peak-maximum position per
frequency, $(\omega=2\pi f)$, defined as, $K=\Delta
T_{\text{g}}/T_{\text{g}}\Delta\log_{10}(f)$. Here, $\Delta T_{\text{g}}$
corresponds to the difference in experimental temperature values of peak-
maximum at high and low frequencies; $\Delta\log_{10}(f)$ is the difference in
excitation frequencies for the given $T_{\text{g}}$ value. K sets a criterion
for differentiating between superparamagetic transition ($K\geq 0.1$) and
spin-glass behaviour ($K\leq 0.1$) [10, 55, 56, 57], and a similar frequency-
dependent shift in $T_{\text{g}}$ has not been reported for anti-ferromagnetic
or ferromagnetic phases at low $\omega$-values [10]. For Na-CGT, $K$ is
experimentally determined to be 0.01 which clearly suggests its spin-glass
nature, sitting between the boundary of canonical and cluster spin glasses[58,
59].

In the next step, the standard theory of dynamic scaling near the phase
transition temperature $T_{\text{g}}(f)$ for given excitation frequencies is
employed [10, 54]. This theory relies on a power law dependence of relaxation
time (experimental attempt time), $\tau=1/2\pi f$, as
$\tau=\tau_{0}[T_{\text{g}}(f)/T_{\text{g}}(0)-1]^{zv}$, where $\tau_{0}$ is
the characteristic relaxation time related to the microscopic flipping time
for dynamical fluctuating entities, such as a single moment or a cluster.
$T_{\text{g}}(0)$ corresponds to the freezing temperature in the limit of
$f\rightarrow 0$, and $zv$ is the critical exponent. The power law equation is
simplified into a linear form using a logarithmic base, and $T_{\text{g}}(0)$
is adjusted in order to get the best linearity [54], where the goodness of fit
is determined from the highest $R^{2}$-coefficient value. Figure 3 (c) shows
the result of this fitting process of the power law. $T_{\text{g}}(0)$ is
found to be 33.7 K, and then using this value, a linear least-squares fitting
method is utilised to extrapolate $\tau_{0}=8.6\times 10^{-10}$ s and
$zv=5.8$. The obtained value of $\tau_{0}$ for Na-CGT lies in the range
($10^{-7}$ to $10^{-11}$ s) typical for cluster spin-glass systems [59, 60]
compared to canonical spin glasses [61, 62] ($10^{-12}$ to $10^{-14}$ s) where
the fluctuation of individuals moments happens at a faster rate. This
indicates slower spin dynamics in Na-CGT at the transition temperature and it
is most likely due to the presence of interacting spin-clusters in the system
instead of individual spins. The value of zv for typical systems with spin-
glass behaviour is reported between 4 and 12 [59, 62], and indeed this is also
the case for Na-CGT, providing another experimental evidence for the cluster
spin-glass nature in Na-CGT.

Figure 4: (a) $\&$ (b) Plots of both the real and imaginary parts of ac
susceptibility ($\chi^{\prime}$ and $\chi"$) as a function of frequency for 35
K and 37 K, respectively. Curves using the best fit parameters calculated by
Eqs. S3 and S4 are shown. (c) $\chi"$-$\chi^{\prime}$ plots for different
temperatures together with the fitted curves generated by the Cole-Cole
equation (Eq.(1)).

For magnetically interacting clusters, a modified form of the Arrhenius law,
known as the Vogel-Fulcher(VF) law [11, 54] is used instead, to analyse ac
susceptibility results. The Arrhenius law describes the relaxation of non-
interacting spins [10, 11], and it is highlighted in Supplementary Information
(Fig.S9) that even though the data seem to fit an Arrhenius law, the fit would
imply unphysical values for the relaxation time and activation energy,
suggesting that a description in terms of non-interacting spins is inadequate.
The phenomenological VF law for the excitation frequency, $f$, takes the
following form, $f=f_{0}\hskip
5.69046pt\textrm{exp}[-E_{\text{a}}/k_{\text{B}}(T_{\text{g}}(f)-T_{0})]$,
where the characteristic frequency, $f_{0}=1/2\pi\tau_{0}$, is obtained using
the characteristic relaxation time from the scaling law, with $E_{\text{a}}$
and $k_{\text{B}}$ being the activation energy and the Boltzmann constant
respectively. The constant, $T_{0}$, is known as the VF temperature and
attributed to inter-cluster interaction strength, and it is constrained having
a value between 0 K to $T_{\text{g}}$ [63]. Figure 3 (d) shows a linear fit to
the experimental data, where the parameters obtained from the least-squares
fitting from the slope and intercept yield $E_{\text{a}}/k_{\text{B}}=97.4$ K
and $T_{0}=29.8$ K, respectively. The agreement of the VF law to the
experimental data and a finite value of $T_{0}$ insinuates interaction among
the clusters, as the ratio between $T_{0}$ and $E_{\text{a}}/k_{\text{B}}$
provides information about the interacting strength between the clusters [11,
54]. In the case of Na-CGT, $T_{0}\sim 0.3\hskip
5.69046ptE_{\text{a}}/k_{\text{B}}$, which lies in the intermediate regime
therefore highlighting a finite interaction between the dynamic entities.

Finally, the Casimir-du Pré equations are utilised [10, 12], in order to
describe the dependence of both real and imaginary parts of the ac
susceptibility on the excitation frequency at a given temperature. Figures 4
(a) and (b) show good agreement between the experimental data for both
real/imaginary parts and the modified equations (Equations S3 $\&$ S4) for
different temperatures. The slow dynamics, arising in the vicinity of 35 K,
are described by the generalised Debye model that takes into account the
entities (i.e. spin-clusters for Na-CGT) having finite interactions leading to
a system having a distribution of relaxation times due to the presence of
cooperative effects [12]. $\chi"$ is plotted against $\chi^{\prime}$ at a
single temperature value and for different frequencies as shown in Fig. 4 (c),
and we fit them by the following Cole-Cole equation:

$\chi^{{}^{\prime\prime}}(\chi^{{}^{\prime}})=-\frac{(\chi_{0}-\chi_{S})}{2\tan[(1-\alpha)(\pi/2)]}+\sqrt{(\chi^{{}^{\prime}}-\chi_{S})(\chi_{0}-\chi^{{}^{\prime}})+\frac{(\chi_{0}-\chi^{{}^{\prime}})^{2}}{4(\tan[(1-\alpha)(\pi/2)])^{2}}},$
(1)

where $\chi_{0}$ represents the susceptibility in the limit of zero frequency,
known as isothermal susceptibility; $\chi_{\text{S}}$ is the adiabatic
susceptibility for the high frequency limit; $\alpha$ is a dimensionless
parameter to represent its relaxation nature with 0 being the case for single
relaxation time and 1 being for a broad distribution of relaxation times. The
fact that the fit curves show an arc centered away from the origin indicates
the presence of a distribution of relaxation times, compared to a semi-circle
appearing for the system with a single relaxation time [12]. The presence of
such distribution can be attributed to magnetic clusters with different sizes
and anisotropies, each experiencing different relaxation times in the whole
system. Further quantitative analysis can be available in Supplementary
Information. This series of analysis confirms that at low temperatures (in the
vicinity of $\sim$ 35 K), intercalation of CGT by Na-atoms leads to the co-
existence of a cluster spin-glass transition with a background ferromagnetic
state. The analysis point towards a progressive freezing of spin clusters and
slower dynamics.

##  3 Conclusion

In summary, we show the unique feature of Na intercalation processes to
significantly modify the transport/magnetic properties of vdW magnetic
material CGT. We discovered that the intercalated Na-CGT hosts emerging
magnetic states, i.e. the spin-glass states that are absent in pristine CGT.
Such a spin-glass phase is a reentrant spin-glass system that coexists with
the ferromagnetically ordered background. Dynamic susceptibility measurements
analysed by multiple models all suggest the nature of cluster spin-glasses
where spins are interacting to each other to behave collectively with the
distribution of relaxation timescales. This study sheds light on the unique
feature of intercalation to generate magnetically frustrated systems by
inhomogeneous carrier doping. This new way of tuning vdW materials might be
effective beyond the field of magnetism, where inhomogeneity will play an
active role for determining material ground states.

##  4 Methods

Direct vapor transport method was used to synthesise the single-crystalline
flakes of Cr2Ge2Te6. Pre-mixing with a molar ratio of $20:27:153$ of high-
purity elemental Cr ($99.9999\%$ in chips), Ge ($99.9999\%$ in crystals), and
Te ($99.9999\%$ in beads) was carried out and then sealed in a quartz ampule
at high vacuum ($\sim 10^{-5}$ Torr). The ampule was loaded in a single-zone
furnace with increasing the temperature at a rate of 2 K/min reaching up to
1273 K. The furnace was set to slow cooling with the rate of 5 K/hour down to
673 K in order to obtain a low defect crystal growth. Then, nature cooling
process was used with switching off the furnace. From the flux buildup, the
single crystal flakes were separated and then stored in an inert atmosphere in
a glovebox.

CGT was de-gassed in vacuum to a pressure less than 10-6 mbar at 100∘C. Na and
CGT were placed in a custom-made air-tight glass tube at $T\sim-50^{\circ}$C
and submerged in liquid ammonia, with a 1:1 molar ratio of Na:Cr2Ge2Te6. The
solution was maintained for a period of approximately 72 hours to allow for
the intercalation to occur.

X-ray diffraction was performed using a Philips X’pert MPD diffractometer with
Cu Kα radiation ($\lambda=1.5406$ Å) in reflection geometry. For the
measurement on air-sensitive Na-CGT, the sample was loaded into a berylium
dome holder with O-ring type seal inside an Ar glovebox for measurement. Raman
spectroscopy was performed using a Renishaw In-Via microscope equipped with a
488 nm laser and a 2400 I/mm grating, with a x20 magnification. Samples were
measured in an air-free glass sample holder, loaded in a glovebox.

A Quantum Design Magnetic Properties Measurement System (MPMS-3) was used for
dc variable-temperature susceptibility and isothermal magnetisation
measurements. Measurements were taken during warming after cooling in zero
field (ZFC-W) and in the measuring field (FC-W), where the measuring field was
200 Oe. Isothermal magnetisation measurements were performed in an external
magnetic field between $\pm$ 70 kOe. A Quantum Design 9 T Physical Properties
Measurement System (PPMS) using the ACMS-II option was used for ac
susceptibility. Measurements were taken in a static base field with an
oscillating ac driving field, where we used a combination of either 10 Oe and
10 Oe, respectively, or 50 Oe and 15 Oe, respectively. ac measurements were
performed with the driving field oscillating at frequencies of 50, 100, 300,
500, 1000, 2500, and 5000 Hz.

For dc susceptibility measurements where external field is applied
perpendicular to the layers of the crystal. The crystal was held tightly in
place between two pieces of cling-film within a plastic cap which was
suspended in the centre of a straw. For the inplane dc susceptibility
measurements, the crystal was attached to the flat longitudinal edge of a
semi-cylindrical quartz rod by Kapton. For ac susceptibility measurements, the
magnetic field was not applied along a particular axis.

The electrical transport and magnetotransport measurements are performed by
the four-point measurement technique [64, 65], as to characterise the
electronic properties of the sample. A Keithley 2450 source meter is used to
apply the current to the outer most connections and contact free resistance is
measured by the two inner electrical connections. A copper sample box is used,
inside which the sample is mounted. The sample stage is then placed on a probe
copper arm inside of a closed cycle helium cryostat (Oxford Instruments), with
a variable temperature range of 5 - 300 K. The probe arm of the cryostat sits
between the poles of an electromagnet, which can produce a homogeneous
magnetic field in the sample region.

## Acknowledgements

S.K. and H.K. acknowledge support from the UK Engineering and Physical
Sciences Research Council (EPSRC) via EP/T006749/1. E.S.Y.A. would like to
thank the EPSRC for studentship funding from the Centre for Doctoral Training
in Advanced Materials Characterisation (EP/S023259/1). L.A.V.N-C acknowledges
a scholarship EP/R513180/1 to pursue doctoral research from EPSRC. A.S. thanks
JSPS Postdoctoral fellowship for research in Japan (P21777). Magnetic
measurements at the University of Cambridge were made on the EPSRC Advanced
Characterization Suite (funded under EP/M0005/24/1). E.J.G.S. acknowledges
computational resources through CIRRUS Tier-2 HPC Service (ec131 Cirrus
Project) at EPCC (http://www.cirrus.ac.uk) funded by the University of
Edinburgh and EPSRC (EP/P020267/1). EJGS acknowledges the Edinburgh-Rice
Strategic Collaboration Awards and the EPSRC Open Fellowship (EP/T021578/1)
for funding support. A.G.L. acknowledges financial support from the Spanish
Ministry of Science, Innovation and Universities under grant No. PRX19/452.

## Author contributions statement

S.K. and H.K. conceived the idea of the project. I.V. and G.E. synthesised and
provided pristine CGT crystals. E.S.Y.A. and C.A.H. intercalated Na-ions in
the CGT crystals. E.S.Y.A. performed Raman and XRD experiments. L.A.V.N-C.
conducted dc-magnetometry and ac susceptibility experiments with supervision
of S.E.D. S.K. performed the electrical transport and magnetotransport
experiments. Data analysis and interpretations were carried out by S.K.,
E.S.Y.A., L.A.V.N-C., H.K. and all other co-authors. S.K., E.S.Y.A.,
L.A.V.N-C., A.S. and H.K. wrote the manuscript with input from the other co-
authors.

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Supplementary Information for “Spin-glass states generated in van der Waals
magnet Cr2Ge2Te6 by alkali-ion intercalation"

##  S1 X-ray Diffraction

In order to distinguish background peaks from CGT 00l diffraction patterns, a
graphite reference was also measured in a beryllium dome. The peaks shown in
Figure S1 (a) corresponds to graphite peaks from literature, with the
exception of the peak at 44∘ which is present in both diffraction patterns.
Hence, the peak at 44∘ can be attributed to the sample environment, as opposed
to Na-CGT.

To rule out presence of some common impurities in pristine and intercalated
CGT, calculated XRD patterns with Cr-Te compounds were compared with the
experimental data. As shown in Figure S1 (b), there is no overlapping of
significant peaks between CGT and Cr-Te compounds.

Figure S1: X-ray diffraction peaks: (a) XRD patterns of Na-CGT (red) and
graphite (black), both performed inside a beryllium dome sample holder. The
graphite reference shows that the peak at 44∘ in both diffraction patterns can
be attributed to the holder S[1, 2].(b) Experimental XRD patterns of CGT, Na-
CGT, compared with calculated patterns for CrTe S[3], CrTe3 S[4]. This
comparison rules out the presence of Cr${}_{\text{x}}$Te${}_{\text{y}}$
compounds in the intercalated CGT.

##  S2 Raman spectroscopy analysis

The analysis of the Raman spectroscopy data was performed using the Python 3
S[5] package SciPy S[6]. Using the `curve_fit` function, a Lorentzian peak-
shape was fitted to each of the modes studied, defined as:

$I(x)=\frac{A}{\pi}\frac{\Gamma/2}{(x-x_{0})^{2}+(\Gamma/2)^{2}}$ (S2)

where $x_{0}$ is the centre of the peak, $\Gamma$ is the full width at half
maximum (FWHM), and $A$ is proportional to peak height.

| Centre (cm-1) | FWHM (cm-1)
---|---|---
Material | $A_{g}^{1}$ | $E_{g}^{2}$ | $E_{g}^{5}$ | $A_{g}^{1}$ | $E_{g}^{2}$ | $E_{g}^{5}$
CGT | 135.17(11) | 109.85(8) | 233.09(18) | 2.6(2) | 3.5(6) | 3.4(7)
Na-CGT | 136.03(16) | 109.8(6) | 234.2(5) | 3.8(4) | 4.4(11) | 7.1(13)
Table S1: The average peak position and full width half maximum (FWHM) for the
$A_{g}^{1}$, $E_{g}^{2}$, and $E_{g}^{5}$ modes for pristine and Na-
intercalated CGT, with standard deviation as error presented in brackets to
the same number of decimal places as the value. 25 and 121 spectra were used
to calculate the values for pristine and Na-intercalated CGT, respectively.

Raman spectra from map scans of the two samples, with spatial resolution 10
$\mu$m, were studied quantitatively in this way. A fit example is shown in
Figure S2. The spectra for Na-intercalated CGT exhibited significant
background throughout the measured range, which was not present in the spectra
for the pristine sample. To account for this, a 12-order polynomial was fit to
the regions of the data where there were no peaks. This background was
subtracted before peak fitting occurred.

Figure S2: Plots showing the fits for CGT and Na-CGT. Figure S3: Histograms
showing the peak position (a,c,e) and peak FWHM (b,d,f) of the $A_{g}^{1}$,
$E_{g}^{2}$, and $E_{g}^{5}$ Raman modes.

For each spectrum in the map scans, the fit was performed to obtain $x_{0}$
and FWHM for each of the $A^{1}_{g}$, $E^{2}_{g}$, and $E^{5}_{g}$ peaks. This
was used to calculate an average and standard deviation, which is tabulated in
Table S1. The full distributions are shown in Figure S3.

From this table it can be seen that there is a statistically significant
increase in peak position for the $A^{1}_{g}$ and $E^{5}_{g}$ modes; within
error, the $E^{2}_{g}$ mode does not change with intercalation. This
consistent, statistically-significant change demonstrates that intercalation
has taken place. However, it is a surprising result. With intercalation of an
organic molecule in CGT, the Raman peaks decreased in energy due to the
addition of electrons to the anti-bonding orbitals which weakened the bonds
S[7]; this is also common to electron-doped layered materials, for instance
gated S[8] or intercalated graphite S[9]. We speculate that this may occur to
increased thermal conductivity of Na-intercalated CGT, leading to better
dissipation of the heat from the laser.

We also observe an increase in peak-width with intercalation. This is similar
to the result for intercalated graphite S[9]. It likely results from the
reduced lifetime of the phonon due to electron-phonon coupling, though an
alternative explanation could be spatial inhomogeneity within the sample. We
note that the degree of the peak broadening depends on the different modes,
implying its complex origins including spatial disorder/defects and electron-
phonon coupling. The intercalation of an organic tetrabutyl ammonium (TBA)
molecule into CGT S[7] led to the emergence of a new phonon mode while no new
mode emerges in our Na intercalated CGT. We attribute this difference that TBA
has a Raman-active mode associated with its intra-molecular vibrations,
whereas Na is a single atom.

##  S3 DC-SQUID Measurements

As highlighted in the main text in Figure 2(a), the $M$-$T$ curve in Na-CGT
shows a different behaviour compared to CGT. In Figure S4, the magnetisation
response as a function of temperature is compared along the two orientations
of out of plane (c-axis) and in-plane (ab-plane) for the Na-CGT sample. It is
evident from this measurement that the magnetic ordering temperature has been
enhanced and the value of $M$ goes to zero in the vicinity of 240 K (inset of
Figure S4).

Figure S4: $M$ vs $T$ curves in Na-CGT for two different orientations measured
during cooling in the absence of a magnetic field.

In Fig. S5, the dependence of magnetisation as a function of external magnetic
field ($M$-$H$ curves) is shown for various different temperatures. Figure S5
(a) shows $M$-$H$ curves for Na-CGT at 10 K for the ab-plane and c-axis
orientations. Along the ab-plane, a saturation magnetic moment of 3.05
$\mu_{\textrm{B}}$/Cr is obtained whereas along the c-axis it is found to be
2.87 $\mu_{\textrm{B}}$/Cr. It is clear from the experimental $M$-$H$ curves
that it requires less external magnetic field for the magnetisation to
saturate along the ab-plane orientation than c-axis. Hence, it is evident from
these measurements that a change in the magnetic anisotropy takes place after
intercalation of Na-ions into CGT, and the spins prefer to align in the in-
plane orientation compared to the out of plane orientation.

Figure S5 (b $\&$ c) show the $M$-$H$ curves of Na-CGT at the elevated
temperature of 200 K where a clear magnetic response is observable, together
with the uniaxial magnetic anisotropy. The magnetic response disappears at 295
K as shown in Fig. S5(d) where a linear trend is observed. This confirms that
the ferromagnetic ordering phase transition to a paramagnetic phase takes
place in Na-CGT.

Figure S5: $M$-$H$ loops in Na-CGT at different temperatures of (a) 10 K, (b)
100 K, 200 K and (d) 295 K. Inset in (a), (b) $\&$ (c) zooms in close to
0-magnetic field position.

Finally, in Figure S6, $M$-$H$ curves for CGT and Na-CGT for two different
orientations are compared at 10 K. It is clearly visible that the magnetic
state of CGT is modified post intercalation and a change in magnetic
anisotropy takes place.

Figure S6: $M$-$H$ loop Comparison between CGT and Na-CGT for two
orientations, (a) c-axis and (b) ab-plane, at 10 K.

##  S4 Magnetic Susceptibility

The temperature dependence of the magnetisation in Na-CGT with small external
magnetic field (20 mT) for the out of plane orientation is shown Figure S7 (a)
and the bifurcation of magnetisation is also seen here with different field
cooling protocols (similar to ab-plane orientation shown in the main text.
Figure S7 (b) $\&$ (c) show $M$-$T$ curves for external field of 100 mT and
this highlights that as the magnetic field is increased from 20 mT to higher
field values, the branching between zero field-cooling (ZFC) and field-cooling
(FC) protocols starts to disappear.

Figure S7: (a) $M$ vs $T$ curves of Na-CGT for different measurement protocols
with external magnetic field along the c-axis orientation, inset shows the
zoomed-in divergence at low temperatures. (b) $\&$ (c) show that the
bifurcation features start to disappear in both different orientations as the
magnetic field is increased.

As mentioned in the main text, the ac magnetic susceptibility is expressed in
complex form as $\chi_{a.c.}=\chi^{\prime}+i\chi"$, where the real part
,$\chi^{\prime}$, is the in-phase component and the imaginary part, $\chi"$,
is the out of phase component S[10]. Figure S8 shows the temperature
dependence of $\chi"$ for different excitation frequencies, and even though
the amplitude is smaller compared to the real part shown in Fig. 3(b) in the
main text, the anomaly feature is clearly present in $\chi"$, supporting the
spin-glass picture. In typical spin-glass systems, relative low amplitude of
$\chi"$ is suggestive of distribution of relaxation mechanism instead of a
single relaxation time S[11].

Figure S8: Temperature dependence of the imaginary part of the ac
susceptibility for different excitation frequencies in Na-CGT. Figure S9:
Arrhenius Law fitting to the peak-maximum dependence on excitation frequency
in the real part of ac susceptibility, using Eq. S3. The best fit parameters
are shown in the figure.

Figure S9 shows the ln$\tau$ vs 1/$T$ plot of extracted from the peak-maximum
$T_{\text{g}}$ in $\chi^{\prime}$ measurements. We used the following
Arrhenius-type equation S[10, 11] to extract the characteristic activation
energy $E_{a}$ as well as timescale $\tau$ given by:

$\tau=\tau_{0}\hskip
5.69046pt\textrm{exp}\left[\frac{E_{a}}{k_{B}[T_{g}(f)]}\right]$ (S3)

where $\tau=1/2\pi f$ and $k_{B}$ is Boltzmann constant. As pointed out in the
main text, even though the experimental data can be fit using Eq.(S3), the
best fit parameters of $E_{\text{a}}/k_{\text{B}}$ = 8035 K and
$\tau_{0}=4\times 10^{-98}$ are both unphysical. The relaxation mechanisms in
Na-CGT can thus not be described in terms of non-interacting spins, suggesting
the interacting nature of Na-CGT’s magnetic clusters S[11].

The Casimir-du Pré equations analysis method is utilised as mentioned in the
main text S[10, 12]. These equations describe the dependence of both real and
imaginary parts of the ac susceptibility on the excitation frequency at a
given temperature, and are used in the description of slow magnetic relaxation
that takes the form of a Debye model. In the Debye model, a system relaxes for
an entity (polarisation or magnetisation) and assumes a non-interacting nature
between the components of that entity. However, as shown above for Na-CGT,
this entity belongs to clusters with finite interaction strength which makes
it difficult to describe the experimental data by a Debye model and leads to
the system having a distribution of relaxation times due to the presence of
cooperative effects. A modification of the Casimir-du Pré equations is carried
out to take into account the broad distribution of relaxation times, referred
to as the generalised Debye model S[12]. A detailed description of this model
can be found elsewhere S[12]. This introduces a parameter, $0<\alpha<1$, which
determines the spread of relaxation times, with two extremes of zero being the
case for single relaxation time and the unity being for a broad distribution
of relaxation times. The frequency dependence of ac susceptibility,
($\omega=2\pi f$), in the generalised Debye model takes the following form
S[12]:

$\chi(\omega)=\chi_{S}+\frac{(\chi_{0}-\chi_{S})}{1+(i\omega\tau)^{1-\alpha}}$
(S4)

where $\chi_{0}$ represents the isothermal susceptibility and $\chi_{S}$ is
the adiabatic susceptibility S[12]. The Casimir-du Pré equations are modified
by considering the following distribution function of $\tau$:

$g(\tau)=\frac{1}{2\pi}\frac{\sin(\alpha\pi)}{\cosh[(1-\alpha)\ln(\tau/\tau_{c})]-\cos(\alpha\pi)}.$
(S5)

Figure S10: (a) Temperature dependence of extracted median relation time
$tau_{c}$ obtained from fit procedure carried out in Fig. 4 (a) $\&$ (b) in
the main text. (b) $\alpha$ extracted by fitting experimental data by three
different approaches, i.e. individual fits to each susceptibility component
and Cole-Cole plot fitting. (c) Distribution function of $\tau$ calculated by
S5 with $\alpha$ for each temperature.

As such, the real and imaginary parts of the complex ac susceptibility become:

$\chi^{\prime}(\omega)=\chi_{S}+\frac{(\chi_{0}-\chi_{S})}{2}\left[1-\frac{\sinh[(1-\alpha)\ln(\omega\tau_{c})]}{\cosh[(1-\alpha)\ln(\omega\tau_{c})]+\cos[(1/2)(1-\alpha)\pi]}\right]$
(S6)
$\chi^{\prime\prime}(\omega)=\frac{(\chi_{0}-\chi_{S})}{2}\left[\frac{\sin[(1/2)(1-\alpha)\ln(\omega\tau_{c})]}{\cosh[(1-\alpha)\ln(\omega\tau_{c})]+\cos[(1/2)(1-\alpha)\pi]}\right],$
(S7)

where $\tau_{c}$ is the average relaxation time. We use these equations to fit
the experimental data of the real and imaginary parts of ac susceptibility
(e.g. those shown in Fig. 4). The temperature dependence of $\tau_{c}$ and
$\alpha$ is extracted by the fitting and shown in Figs. S10 (a)-(b)
respectively. There is a clear increase in the average relaxation time in the
vicinity of 35 K. This coincides with the earlier findings of bifurcation of
magnetisation in different field-cooling protocols and the shift of peak-
maximum in ac susceptibility measurements and it signifies slower dynamics as
the freezing of the clusters takes place. The Cole-Cole analysis can also
enables to determine $\alpha$ by the equation:

$\chi^{{}^{\prime\prime}}(\chi^{{}^{\prime}})=-\frac{(\chi_{0}-\chi_{S})}{2\tan[(1-\alpha)(\pi/2)]}+\sqrt{(\chi^{{}^{\prime}}-\chi_{S})(\chi_{0}-\chi^{{}^{\prime}})+\frac{(\chi_{0}-\chi^{{}^{\prime}})^{2}}{4(\tan[(1-\alpha)(\pi/2)])^{2}}}.$
(S8)

$\alpha$ extracted by the three different approaches, i.e. $\chi^{\prime}$,
$\chi"$ and Cole-Cole analysis, all show qualitatively similar temperature-
dependent trends with comparable values, as shown in Fig. S10(b). The $\alpha$
values between 0.2 to 0.8 further support the picture of multiple channels of
the spin-glass relaxation presented in the main text. Finally, we show the
calculated $g(\tau)$ by extracted $\alpha$ from the Cole-Cole analysis in Fig.
S10(c). This visualises the statistical distribution of relaxation times for
magnetic clusters in Na-CGT.

##  S5 Magneto-transport

Here we show technical details and further analysis of magnetoresistance(MR)
measurements measured in our Na-CGT. Figure S11(a) shows the MR measurements
for Na-CGT at 10 K while the magnetic field is swept from positive to negative
and then negative to positive directions. A peak-shaped resistance change is
observed upon sweeping the external magnetic field around zero. The MR value
plotted in Fig. 2(f) is deduced by the relative resistance difference between
the high field and zero field values. We believe that this is anisotropic
magnetoresistance where the resistance change depends on the angle between the
current direction and magnetic moment orientation. On this assumption, the
resistance represents the moment orientation and can be used for analysis of
magnetic switching behaviour. In order to more reliably identify the
completion of magnetisation switching, we use the derivative form of MR as
shown in Fig. S11(b). This approach is also effective to remove any
contribution from linear MR components.

The magnitude of the uniaxial magnetic anisotropy can be estimated by the
difference in saturation fields $H_{\text{sat}}$ for the out of plane (OOP)
and in-plane (IP) orientations, as shown in Figure S11 (c). The finite
$H_{\text{sat}}$ difference representing the presence of magnetic anisotropies
is another proof of the presence of long-range magnetic order. The temperature
dependence is good agreement with the $T_{\text{C}}$ estimated from different
measurement techniques such as magnetometry.

Figure S11: (a) Typical MR measurements of Na-CGT measured during the magnetic
field sweep for 10 K. Resistance is normalised at the high field limit for
each scan. The inset highlights the MR behavior close to zero-field, showing a
small hysteresis. (b) Plot of derivative values of MR for determining the
saturation field $H_{\text{sat}}$ from each scan. (c) Temperature dependence
of the difference in $H_{\text{sat}}$ for two orientations in Na-CGT to
represent the temperature dependence of the uniaxial magnetic anisotropy.

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