# Quantum Anomalous Hall and Spin Hall Effects in Magnetic Graphene

Talieh S. Ghiasi<EMAIL_ADDRESS>Kavli Institute of Nanoscience, Delft
University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands  Davit
Petrosyan  Kavli Institute of Nanoscience, Delft University of Technology,
Lorentzweg 1, 2628 CJ Delft, The Netherlands  Josep Ingla-Aynés  Kavli
Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628
CJ Delft, The Netherlands  Tristan Bras  Kavli Institute of Nanoscience, Delft
University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands  Samuel
Mañas-Valero  Kavli Institute of Nanoscience, Delft University of Technology,
Lorentzweg 1, 2628 CJ Delft, The Netherlands  Institute of Molecular Science,
University of Valencia, Catedrático José Beltrán 2, Paterna 46980, Spain
Kenji Watanabe  Research Center for Electronic and Optical Materials, National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Takashi
Taniguchi  Research Center for Materials Nanoarchitectonics, National
Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Eugenio
Coronado  Institute of Molecular Science, University of Valencia, Catedrático
José Beltrán 2, Paterna 46980, Spain  Herre S. J. van der Zant  Kavli
Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628
CJ Delft, The Netherlands

###### Abstract

A promising approach to attain long-distance coherent spin propagation is
accessing quantum Hall topological spin-polarized edge states in graphene.
Achieving this without large external magnetic fields necessitates engineering
graphene band structure, obtainable through proximity to 2D magnetic
materials. In this work, we detect spin-polarized helical edge transport in
graphene at zero external magnetic field, allowed by the out-of-plane magnetic
proximity of CrPS4 that spin-splits the zeroth Landau level. This zero-field
detection of the quantum anomalous spin Hall state is enabled by large induced
spin-orbit and exchange couplings in the graphene that also lead to the
detection of an enhanced Berry curvature, shifting the Landau levels, and
result in an unconventional sequence of quantum Hall plateaus. Remarkably, we
observe that the quantum anomalous Hall transport in the magnetized graphene
persists up to room temperature. The detection of spin-polarized helical edge
states at zero magnetic field and the robustness of the quantum anomalous Hall
transport up to room temperature open the route for practical applications of
magnetic graphene in quantum information processing and spintronic
circuitries.

Decades of research in graphene have unveiled its remarkable charge and spin
transport properties, such as distinctive quantum Hall (QH) charge transport
and long-distance spin propagation neto2009electronic ; han2014graphene . Yet,
harnessing these properties for the development of practical quantum
spintronic devices necessitates targeted modifications to the graphene band
structure. This band structure engineering has been recently achieved non-
invasively by bringing graphene in the proximity of other two-dimensional (2D)
materials sierra2021van , leading to spin Hall wei2016strong ; safeer2019room
, Rashba-Edelstein mendes2015spin ; ghiasi2019charge ; benitez2020tunable ,
anomalous Hall wang2015proximity , and spin-dependent Seebeck
ghiasi2021electrical  effects in graphene. The realization of these phenomena
in proximitized graphene is due to induced spin-orbit and exchange
interactions that cause spin-splitting of the graphene band structure
geim2013van ; yang2013proximity ; dyrdal2017anomalous . These spin-related
effects, however, have been experimentally addressed in the proximitized
graphene majorly by diffusive charge/spin transport, where the spin relaxation
length is limited by spin and momentum scattering mechanisms. On the contrary,
if the transport occurs in the QH regime, one could possibly access spin-
polarized edge states that can allow for long-distance quantum coherent spin
propagation due to their topological protection.

Figure 1: CrPS4 (CPS) magnetic ordering and induced magnetism in graphene. (a)
Top-view ($a-b$ plane) CPS crystal structure schematics showing monoclinic
symmetry with space group C2 and lattice constants $a$ =
$10.871\text{\,}\r{A}$, $b$ = $7.254\text{\,}\r{A}$, and $c$ =
$6.140\text{\,}\r{A}$ diehl1977crystal . The dashed lines indicate the
direction over which the crystal preferentially cleaves, providing flakes with
67.5∘ corner angles, as shown in the optical micrograph of a
$38\text{\,}\mathrm{n}\mathrm{m}$ thick CPS flake (the scale bar is
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$). The CPS is covered with
an hBN/graphene Hall-bar, as described in the supplementary information (SI),
Figure S1. (b,c) SQUID magnetometry of a CPS bulk crystal at various
temperatures, respectively shown in panels b and c for the small and large
range of magnetic field $B_{\mathrm{z}}$ applied perpendicular to the ($a-b$)
plane. The insets are side-view schematics of a bilayer of CPS showing the
layers’ magnetic ordering for each indicated range of $B_{\mathrm{z}}$. (d)
Optical micrograph of device A fabricated with an hBN/graphene/CPS
heterostructure on a SiO2/Si substrate with Ti-Pd electrodes. The scale bar is
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. The CPS crystal used for
the device fabrication is from the same source that is explored by the SQUID
magnetometry (of panels b and c). The thickness of the hBN and SiO2, used as
top-gate and back-gate dielectrics, is 73 and
$285\text{\,}\mathrm{n}\mathrm{m}$, respectively. (e) Transverse resistance
$R_{\mathrm{xy}}=V_{\mathrm{xy}}/I$ measured vs. $B_{\mathrm{z}}$ in the
graphene Hall-bar at $T=$1.8\text{\,}\mathrm{K}$$ with
$I=$0.5\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{A}$$. The light blue curve
is measured from 1.5 to $-1.5\text{\,}\mathrm{T}$ (called trace) and the dark
blue curve (retrace) is measured from $-1.5$ to $1.5\text{\,}\mathrm{T}$. This
measurement is performed at back-gate and top-gate voltages of
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$ and
$V_{\mathrm{tg}}=$0\text{\,}\mathrm{V}$$. The inset is the subtraction of the
trace and retrace measurements ($\Delta R_{\mathrm{xy}}$) showing the
modulation of the $R_{\mathrm{xy}}$ due to the spin-flop transition. (f) Non-
linear component of $R_{\mathrm{xy}}$ associated with the anomalous Hall
effect ($R_{\mathrm{AHE}}$), measured at $T=$300\text{\,}\mathrm{K}$$,
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$ and
$V_{\mathrm{tg}}=$0\text{\,}\mathrm{V}$$. $R_{\mathrm{AHE}}$ is acquired by
subtraction of a linear background (related to the ordinary Hall effect) from
$R_{\mathrm{xy}}$ anti-symmetrized vs. $B_{\mathrm{z}}$ (see also SI Figure
S3). For the linear fit, the $R_{\mathrm{xy}}$ data in the range of
$$8\text{\,}\mathrm{T}$<B_{\mathrm{z}}\leq$9\text{\,}\mathrm{T}$$ is
considered, assuming saturation of the magnetization in that range.

Spin-polarized QH edge states have been detected in graphene through the
Zeeman splitting of the bands, induced by applying a large out-of-plane
magnetic field young2014tunable ; veyrat2020helical . However, in a
graphene/2D magnet heterostructure, the out-of-plane magnetic proximity by
itself should be able to readily bring the graphene transport to the QH
regime, where the induced exchange interaction leads to the spin-splitting of
the QH states, even in the absence of an external magnetic field. This
modified QH transport in graphene due to the co-presence of the induced
magnetism and spin-orbit coupling is conceived as the quantum anomalous Hall
(QAH) effect qiao2010quantum ; tse2011quantum ; qiao2012microscopic ;
qiao2014quantum ; zhang2015quantum ; zhang2015robust ; zanolli2018hybrid ;
hogl2020quantum . Upon broken time-reversal symmetry by the induced magnetism,
there will be a topological protection of the edge states that, based on their
spin polarity, would counter-propagate close to zero energy (so-called helical
states), giving rise to quantum anomalous spin Hall (QASH) effect
wang2017dirac .

Despite several recent experimental efforts in addressing the QH regime in
various hybrids of graphene and magnetic materials song2018electrical ;
song2018asymmetric ; wu2020large ; wu2021magnetic ; chau2022two , direct
experimental evidence for the detection of the QAH and QASH effects in
magnetized graphene is still missing. In fact, recent reports on the QH
transport in some of these graphene-based magnetic heterostructures
wang2022quantum ; tseng2022gate  have shown that the presence of interfacial
charge transfer hinders the exploration of the magnetic proximity effect.
However, here we observe clear evidence for QAH transport in graphene-CrPS4
heterostructures and demonstrate that transport close to zero energy in
magnetic graphene is dominated by the QASH effect, realized by the detection
of the QH helical edge states at zero external magnetic field. In addition, we
measure a distinctive Landau fan diagram with enhanced Landau gaps and non-
trivial sequences of the QH plateaus, providing evidence for a large induced
Berry curvature and breaking of the spin/valley degeneracy in the magnetized
graphene. We further observe that the first QH plateaus stay robust up to room
temperature which is the most sought-after and theoretically predicted
signature for the QAH effect qiao2010quantum ; zhang2015robust ;
hogl2020quantum .

We magnetize graphene by the van der Waals proximity of the interlayer
antiferromagnet, CrPS4 (CPS), which is an air-stable semiconductor with a
bandgap of $\sim$ 1.3 eV lee2017structural  and a Néel temperature of
$T_{\mathrm{N}}\approx$ 38 K peng2020magnetic . Figure 1a (top panel)
illustrates a top view of the CPS crystal structure with the dashed lines
indicating the crystallographic directions along which the crystals
preferentially cleave, as shown in the optical micrograph of an exfoliated
flake. As a result, the CPS flakes acquire a corner angle of
$67.5\text{\,}\mathrm{\SIUnitSymbolDegree}$, the bisector of which is along
the magnetic $a$-axis lee2017structural  (see also supplementary information
(SI), Figure S2). The presence of this characteristic angle allows for the
identification of the CPS crystallographic orientation and guides the
alignment of the graphene Hall bar with one of the CPS magnetic axes (see
Figure 1a, lower panel). In accordance with previous reports peng2020magnetic
, CPS has an anisotropic magnetic behavior which is evaluated using a
superconducting quantum interference device (SQUID) at various temperatures,
shown in Figure 1b and c. The magnetic ordering of the layers in a multilayer
CPS crystal at $B_{\mathrm{z}}=$ $0\text{\,}\mathrm{T}$ is known to be mainly
out-of-plane (with spins oriented along the $c$-axis) angled
$20\text{\,}\mathrm{\SIUnitSymbolDegree}$ towards the $a$-axis (as shown in
the upper inset of Figure 1b). The SQUID measurements at a small
$B_{\mathrm{z}}$ (panel b) for $T<$40\text{\,}\mathrm{K}$$, show a spin-flop
transition when the magnetization of the layers rotate towards the $b$-axis,
whilst holding their respective anti-parallel alignment (lower inset).
Increasing $B_{\mathrm{z}}$ above the spin-flop transition field
($B_{\mathrm{sf}}\sim$0.8\text{\,}\mathrm{T}$$ at $2\text{\,}\mathrm{K}$)
results in canting of the magnetic moments towards the $c$-axis with full
saturation at $B_{\mathrm{z}}=$8\text{\,}\mathrm{T}$$, as shown in Figure 1c.

We evaluate the possibility of inducing the CPS magnetic properties in
graphene by measuring the charge transport in graphene. We report on two
devices (A and B), each fabricated with a monolayer graphene Hall-bar on a
multilayer CPS flake, fully covered with hBN that is used as a dielectric for
the top-gating (Figure 1d). We apply current along the graphene channel (with
$I+$ and $I-$ electrodes shown in panel d) while measuring the longitudinal
and transverse voltages ($V_{\mathrm{xx}}$ and $V_{\mathrm{xy}}$). In Figure
1e, we show the modulation of the transverse resistance
$R_{\mathrm{xy}}\,=\,V_{\mathrm{xy}}/I$ as a function of $B_{\mathrm{z}}$, at
$V_{\mathrm{tg}}=$0\text{\,}\mathrm{V}$$. Remarkably, $R_{\mathrm{xy}}$
switches at $\sim\pm$0.8\text{\,}\mathrm{T}$$, which is the magnetic field at
which the spin-flop transition occurs in CPS. This is a direct indication that
the magnetic behavior of the CPS is influencing the charge transport in
graphene.

The trace and retrace measurements of $R_{\mathrm{xy}}$ show no hysteresis vs.
$B_{\mathrm{z}}$, except for $B_{\mathrm{z}}$ close to $B_{\mathrm{sf}}$. That
is understood by considering the behavior of the magnetization of CPS
outermost layer ($\vec{M}_{\mathrm{CPS}}$), to which the $R_{\mathrm{xy}}$ is
the most sensitive. The behavior of $\vec{M}_{\mathrm{CPS}}$ is ruled by its
anti-ferromagnetic coupling with respect to the other CPS layers, and yet, the
lack of a neighboring CPS layer on top can cause the $\vec{M}_{\mathrm{CPS}}$
to lag behind the magnetization dynamics of the bulk of CPS, leading to the
observed hysteresis. Thus, by decreasing the $B_{\mathrm{z}}$, the
$\vec{M}_{\mathrm{CPS}}$ switches back to the antiferromagnetic alignment with
respect to the other CPS layers at lower fields of $\sim\,$0.5 T, by which
$R_{\mathrm{xy}}$ in the retrace retrieves its initial value as in the trace
measurement. From the difference of the $R_{\mathrm{xy}}$ in the trace and
retrace measurements, which is about 80$\,\Omega$ (shown in the inset of
Figure 1e), we extract a change of $0.27\text{\,}\mathrm{T}$ in the
$z$-component of the magnetic field sensed by the electrons in graphene, due
to the spin-flop transition of the $\vec{M}_{\mathrm{CPS}}$.

Another indication of the induced magnetism in graphene is the detection of
the anomalous Hall effect which gives rise to a non-linearity of the Hall
voltage. The anomalous Hall component follows the projection of the
magnetization along the $z$-axis. Thus, considering the magnetization
saturation field in CPS above $8\text{\,}\mathrm{T}$, we subtract the linear
component related to the ordinary Hall effect from the measured
$R_{\mathrm{xy}}$. This procedure yields the anomalous Hall signal
($R_{\mathrm{AHE}}$), shown in Figure 1f, measured at
$T=$300\text{\,}\mathrm{K}$$ and $V_{\mathrm{tg}}=$0\text{\,}\mathrm{V}$$. The
observation of such a sizable signal of $R_{\mathrm{AHE}}$ $\sim\,$200
$\Omega$ is a signature of the co-presence of large induced spin-orbit and
exchange interactions nagaosa2010anomalous  in the proximitized graphene up to
room temperature.

Figure 2: Quantum Hall (QH) transport in magnetized graphene. (a) Shubnikov–de
Haas oscillations (SdHOs) of graphene longitudinal resistivity,
$\rho_{\mathrm{xx}}$ measured vs. applied top-gate voltage and magnetic field.
The red dashed line is a linear fit to the maxima of $\rho_{\mathrm{xx}}$
above $8\text{\,}\mathrm{T}$, used as a guide to the eye for the realization
of the non-linear behaviour of the Landau fan diagram close to zero energy.
(b) Landau fan diagram: the red dots are the maxima of the
$\rho_{\mathrm{xx}}$ vs. carrier density ($n$) measured at
$$8\text{\,}\mathrm{T}$<B_{\mathrm{z}}\leq$9\text{\,}\mathrm{T}$$ with the red
dashed lines as linear fits. The black solid lines are the theoretically
expected Landau fan diagram for pristine graphene under an applied magnetic
field, shown for comparison. (c) Total magnetic field ($B_{\mathrm{T}}$)
extracted for the N${}^{\textrm{th}}$ Landau level (LL) derived from the
Landau fan diagram in panel b. Inset: Fan diagram for the magnetized graphene
vs. $B_{\mathrm{T}}$ and $n$, shown for positive magnetic field. (d)
Modulation of $\rho_{\mathrm{xx}}$ and transverse resistance
($R_{\mathrm{xy}}$) vs. applied top-gate voltage and magnetic field, measured
with another set of longitudinal voltage probe that has a transverse pair (see
SI Figure S6). (e) SdHO of the longitudinal conductivity
$\sigma_{\mathrm{xx}}=\rho_{\mathrm{xx}}/(\rho_{\mathrm{xx}}^{2}+R_{\mathrm{xy}}^{2})$
measured at $B_{\mathrm{z}}=$-9\text{\,}\mathrm{T}$$ for $T\,=\,4$ to
$80\text{\,}\mathrm{K}$. The $\Delta\sigma_{\mathrm{xx}}$ plots are obtained
by subtracting a (nearly) linear increase in $\sigma_{\mathrm{xx}}$ vs.
$V_{\mathrm{tg}}$. (f) SdHO amplitude ($\Delta\sigma_{\mathrm{xx}}$) for LL
with $N_{\mathrm{LL}}=-1$ to $-4$ as a function of $T$, normalized to their
maxima. Inset: cyclotron mass, $m_{\mathrm{c}}$, extracted from all the LLs in
panel e, plotted as the ratio with respect to the electron rest mass ($m_{0}$)
vs. $n$. The red curve is a fit to the extracted
$m_{\mathrm{c}}/m_{\mathrm{0}}$ vs. $n$, considering a linear dispersion
relation, resulting in $v_{\mathrm{F}}\approx 0.902\pm 0.005\times
10^{6}\leavevmode\nobreak\ \mathrm{m/s}$ . The black curve is the
theoretically expected behavior for pristine graphene with the Fermi velocity
$v_{\mathrm{F}}=1.1\times 10^{6}\leavevmode\nobreak\ \mathrm{m/s}$. Figure 3:
Quantum spin Hall states in magnetized graphene near zero energy. (a)
Schematics of spin-split Landau levels in magnetized graphene close to the
charge neutrality point, shown for the case of having spin-splitting
($\Delta_{\mathrm{s}}$) much larger than the Coulomb gap ($E_{\mathrm{c}}$). A
schematic of the device and a four-terminal measurement geometry are shown
with the propagation direction of the helical states at the edge. (b)
$V_{\mathrm{tg}}$-dependence of the longitudinal resistivity
($\rho_{\mathrm{xx}}$) at $B_{\mathrm{z}}=0$ to $0.5\text{\,}\mathrm{T}$.
These measurements are performed using the first voltage pair (as in Figure
2a). The plots are offset (except for $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$)
for a clearer demonstration of the SdHOs. The black arrows indicate the
shoulders associated with the SdHO. Inset: Conductance
($G_{\mathrm{xx}}=I/V_{\mathrm{xx}}$) vs $V_{\mathrm{tg}}$ close to the charge
neutrality point (CNP) at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$. (c)
Conductance measured by the second voltage pair (as used in Figure 2d)
considering conductivity matrix
$G_{\mathrm{xx}}^{*}=W\rho_{\mathrm{xx}}/L(\rho_{\mathrm{xx}}^{2}+R_{\mathrm{xy}}^{2})$
vs. $V_{\mathrm{tg}}$, shown for $|B_{\mathrm{z}}|=0$ to
$2\text{\,}\mathrm{T}$. Note that $G_{\mathrm{xx}}^{*}\approx G_{\mathrm{xx}}$
at $B_{\mathrm{z}}=0\,$T. (d) Temperature dependence of transverse resistance
$R_{\mathrm{xy}}$ vs. $V_{\mathrm{tg}}$ measured at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$. The plots (except for the one at
$4\text{\,}\mathrm{K}$) are shown with an offset for clarity. Inset:
temperature-dependence of $G_{\mathrm{xx}}^{*}\approx G_{\mathrm{xx}}$ at zero
energy, measured at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$. (e) Two- and
four-terminal conductance $G_{\mathrm{2T}}=I/V_{\mathrm{2T}}$ and
$G_{\mathrm{4T}}=I/V_{\mathrm{4T}}$ measured at various geometries at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ at $T$ = 1.8 K. The dashed lines
indicate the theoretically expected values for the $G_{\mathrm{2T}}$ and
$G_{\mathrm{4T}}$ considering helical states, color-coded with respect to the
measurement geometry indicated in the legend. The connection numbers can be
found in the device map provided in the schematics.

By cooling down the device, signatures of QH transport become more evident, as
shown in Figure 2. Transport in the proximity-magnetized graphene is measured
at $1.8\text{\,}\mathrm{K}$ for $B_{\mathrm{z}}=-9$ to $9\text{\,}\mathrm{T}$.
The modulation of the resistivity ($\rho_{\mathrm{xx}}$) vs. $B_{\mathrm{z}}$,
i.e., the Shubnikov de-Haas oscillations (SdHOs), is shown in Figure 2a. The
back-gate voltage ($V_{\mathrm{bg}}$) is set to $50\text{\,}\mathrm{V}$ to
shift the Dirac point such that the electron-like Landau levels (LLs) are
reachable by the top-gating. We evaluate the back-gate vs. top-gate dependence
of the LLs as shown in the SI Figure S4, indicating only a shift of the LLs by
electrostatic doping without any significant change in the Landau fan diagram
with $V_{\mathrm{bg}}$ (also see SI Figure S5). This observation reassures the
magnetic proximity effect which is in contrast to recent reports in graphene
in the proximity of CrX3 tseng2022gate  or CrOCl wang2022quantum , where gate-
dependent interfacial charge transfer influences the transport in graphene.

The Landau fan diagram in the magnetized graphene (Figure 2a) appears to be
distinctive from that of pristine graphene, showing a large separation of the
LLs with non-linearity of the first LLs close to zero energy. As indicated by
the divergence from the red dashed line, the SdHO peaks deviate from the
linear behavior for $B_{\mathrm{z}}<3\,$T. This deviation from the
conventional fan diagram could be due to a non-linearity of the graphene
dispersion relation close to the charge neutrality point and/or due to the
non-linear behavior of $\vec{M}_{\mathrm{CPS}}$ vs. $B_{\mathrm{z}}$ close to
the spin-flop transition. Considering a linear dispersion relation for
graphene for $V_{\mathrm{tg}}$ away from the charge neutrality point (as
predicted by DFT calculations for such magnetic heterostructures
zollner2022proximity ), the energy of the N${}^{\textrm{th}}$ LL is
$E_{\mathrm{N}}=\pm\sqrt{2\mathrm{e}\hbar
v_{\mathrm{F}}^{2}|N_{\mathrm{LL}}|B_{\mathrm{T}}}$ with $N_{\mathrm{LL}}$
being the LL number, $e$ the electron charge, $\hbar$ the reduced Planck’s
constant and $v_{\mathrm{F}}$ the Fermi velocity novoselov2005two ;
zhang2005experimental . $B_{\mathrm{T}}$ is the total effective magnetic field
sensed by the electrons which is defined as
$B_{\mathrm{T}}=B_{\mathrm{z}}+B_{\mathrm{M}}$, with $B_{\mathrm{M}}$
originating from the projection of the $M_{\mathrm{CPS}}$ along the $z-$axis.
This relation implies a linear dependency of carrier density $n$ vs.
$B_{\mathrm{z}}$ for each LL when $B_{\mathrm{M}}$ is constant. For pristine
graphene ($B_{\mathrm{M}}=0$), the relation results in the Landau fan shown
with the gray lines in Figure 2b. Using the same equation for the magnetized
graphene, we linearly fit the $\rho_{\mathrm{xx}}$ maxima vs. $n$ for each LL,
for $B_{\mathrm{z}}>\,8\,$T when the CPS magnetization is fully out-of-plane
(shown by the red dashed lines in panel b). Note, that since the graphene is
encapsulated with different materials with various dielectric constants, we do
not rely on the geometrical capacitance for the estimation of $n$. Instead, we
extract $n$ from the dependence of $B_{\mathrm{z}}$ vs. $V_{\mathrm{tg}}$ for
each LL which is comparable with $n$ extracted from Hall measurements that is
a more reliable approach (see SI, section 7 and 8).

A direct comparison of the Landau fan diagram in pristine vs. magnetized
graphene shown in Figure 2b clarifies that the carrier density at each LL is
larger in the magnetized channel, and a linear extrapolation of $n$ to
$B_{\mathrm{z}}=0$ yields to $n\neq 0$ (also see SI, section 8). This implies
a larger energy separation between the LLs due to a higher effective total
magnetic field sensed by the electrons, implying that $B_{\mathrm{M}}\neq 0$.
Following the $E_{\mathrm{N}}$ vs. $B_{\mathrm{T}}$ relation, we extract
$B_{\mathrm{T}}$ for each LL at $B_{\mathrm{z}}=\pm$9\text{\,}\mathrm{T}$$.
The result is shown in Figure 2c, with $B_{\mathrm{T}}$ reaching $\sim\,$20 to
$25\text{\,}\mathrm{T}$ for the first LLs ($N_{\mathrm{LL}}=\pm 1$). This
large difference between $B_{\mathrm{T}}$ and the applied
$B_{\mathrm{z}}=$9\text{\,}\mathrm{T}$$ can be realized considering the
induced Berry curvature in graphene xiao2010berry ; sundaram1999wave , as the
dipolar magnetic field generated by the CPS is expected to be rather small.
The induced Berry curvature acts as an effective magnetic field in momentum
space that deforms the electron motion and modifies the LLs sundaram1999wave .
The energy dependence of the extracted $B_{\mathrm{M}}$ is also supported by
the expected energy dependence of the Berry curvature induced in graphene
zollner2022engineering ; bora2022magnetic .

The behavior of the transverse voltage is also addressed in a simultaneous
measurement of $\rho_{\mathrm{xx}}$ and $R_{\mathrm{xy}}$, shown in Figure 2d,
for another pair of voltage probes. Even though additional features appear in
the Landau fan (absent in panel a, measured with the first voltage pair), the
position of the LLs vs. $V_{\mathrm{tg}}$ is the same (see SI, Figure S6). The
expression
$\sigma_{\mathrm{xx}}=\rho_{\mathrm{xx}}/(\rho_{\mathrm{xx}}^{2}+R_{\mathrm{xy}}^{2})$
is used to consider both longitudinal and transverse voltages in a
conductivity tensor qhe_klitzing , measured at various temperatures up to
$80\text{\,}\mathrm{K}$ (shown in Figure 2e as $\Delta\sigma_{\mathrm{xx}}$,
obtained after standard background subtraction novoselov2005two ). From the
temperature-dependence of $\Delta\sigma_{\mathrm{xx}}$, shown for the first
few hole-like LLs in Figure 2f, we extract the effective mass of the electrons
in the magnetized graphene by fitting $\Delta\sigma_{\mathrm{xx}}$ to the
standard expression $T/\sinh(2\pi^{2}k_{\mathrm{B}}Tm_{\mathrm{c}}/\hbar
eB_{\mathrm{T}})$, with $k_{\mathrm{B}}$ the Boltzmann constant and
$m_{\mathrm{c}}$ the cyclotron mass novoselov2005two ; zhang2005experimental .
The ratio of the cyclotron mass vs. the rest mass of the electrons
($m_{\mathrm{c}}$/$m_{\mathrm{0}}$) extracted from the SdHO at
$$8\text{\,}\mathrm{T}$<|B_{\mathrm{z}}|\leq$9\text{\,}\mathrm{T}$$ for
various $n$ is shown in the inset of Figure 2f. The extracted
$m_{\mathrm{c}}$/$m_{\mathrm{0}}$ ratio shows a deviation from that expected
for the case of pristine graphene with $v_{\mathrm{F}}=1.1\times
10^{6}\,\mathrm{m/s}$ (indicated by the black solid line in the inset)
novoselov2005two ; zhang2005experimental . From the fit to
$m_{\mathrm{c}}/m_{\mathrm{0}}$ vs. $n$, considering graphene’s linear
dispersion relation, we find $v_{\mathrm{F}}\approx 0.902\pm 0.005\times
10^{6}\leavevmode\nobreak\ \mathrm{m/s}$. The estimated $m_{\mathrm{c}}$ and
$v_{\mathrm{F}}$ in the magnetized graphene also hint towards the modulation
of the graphene band structure by the proximity of the CPS.

As the gate-dependence of $B_{\mathrm{M}}$ indicates, the induced magnetism
can have a stronger impact on the LLs closer to the charge neutrality point,
where the spin-splitting is expected to be the largest zollner2022engineering
. The zeroth LL (zLL) in graphene is formed by the wavefunctions from each
valley, occupied by either electrons or holes at zero energy, leading to two
spin-degenerate electron- and hole-like zLLs. By applying a perpendicular
magnetic field, an energy gap ($E_{\mathrm{c}}$) is formed between the
electron- and hole-like bands, originating from Coulomb interactions due to
the confinement of charges in the zLLs. Upon increasing the Zeeman spin-
splitting energy ($\Delta_{\mathrm{s}}$), when
$\Delta_{\mathrm{s}}>2E_{\mathrm{c}}$, a band diagram similar to that shown in
Figure 3a can occur which hosts counter-propagating spin-polarized electron-
and hole-like bands, so-called helical states kane2005quantum ; abanin2006spin
; kharitonov2012edge . These states are different from the chiral ones,
addressed in Figure 2 at $N_{\mathrm{LL}}\,\neq\,0$ which are either electron-
or hole-like and thus propagate in the same direction.

Focusing on the transport close to zero energy, shown in Figure 3b for small
applied magnetic fields ($B_{\mathrm{z}}\leq$0.5\text{\,}\mathrm{T}$$),
remarkably, we observe the emergence of the SdHO-associated shoulders
(indicated by the arrows). This is a signature for the onset of the QAH
transport in the magnetic graphene already at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$. Observing the SdHO implies that the
condition for the Landau quantization, $\mu B_{\mathrm{z}}\gg 1$, with $\mu$
as the charge carrier mobility, is fulfilled qhe_klitzing . Since the carrier
mobility in these devices is rather low
($\mu\sim$2500\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$$,
see SI Figure S9), there must be an additional effective magnetic field of
$B_{\mathrm{M}}\gg$4\text{\,}\mathrm{T}$$ sensed by the charge carriers in the
magnetic graphene to allow for the formation of the LLs. We attribute this to
the finite out-of-plane component of the magnetization of the CPS at
$0\text{\,}\mathrm{T}$, induced in graphene which affects the orbital motion
of the graphene charge carriers.

We further notice that the Dirac curve at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ (in Figure 3b) has a rather wide peak.
When the four-terminal longitudinal measurements are evaluated in terms of
conductance ($G_{\mathrm{xx}}=I/V_{\mathrm{xx}}$), as shown in the inset of
panel b and in Figure 3c (reproduced by different voltage probe pairs), it
becomes evident that the broadened Dirac peak at $0\text{\,}\mathrm{T}$ is a
plateau of conductance at $2\,e^{2}/h$. This quantized conductance at zero
energy is evidence for the presence of counter-propagating spin-polarized
helical states, as depicted in Figure 3a, as a result of the spin-splitting
induced in graphene by the magnetic exchange interaction. In this case, the
channel conductance $G_{\mathrm{xx}}^{*}$ that considers the conductivity
tensor (defined as
$W\rho_{\mathrm{xx}}/L(\rho_{\mathrm{xx}}^{2}+R_{\mathrm{xy}}^{2})$) should go
to zero as soon as transport is dominated by the chiral edge states, since
they propagate only in one direction qhe_klitzing . Thus, Figure 3c shows that
by increasing $B_{\mathrm{z}}$, $G_{\mathrm{xx}}^{*}$ goes towards zero above
and below the zLL, yet it remains close to $2\,e^{2}/h$ at zero energy
implying that $\Delta_{\mathrm{s}}$ is larger than the Coulomb gap also at
$B_{\mathrm{z}}\neq 0$ (see SI section 9 for device B). Note that
$G_{\mathrm{xx}}^{*}\approx G_{\mathrm{xx}}$ at $B_{\mathrm{z}}=0$.

Simultaneous measurement of $R_{\mathrm{xy}}$ in Figure 3d shows that there is
a finite transverse voltage at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$, which
contains the contribution from the anomalous Hall effect as a result of the
finite $z$-component of the $M_{\mathrm{CPS}}$. Remarkably, $R_{\mathrm{xy}}$
shows a double-peak feature at zero energy that merges into a single peak at
higher temperatures, due to thermal broadening. The separation of the two
peaks in the $R_{\mathrm{xy}}$ in terms of energy is similar in magnitude to
the width of the plateau in $G_{\mathrm{xx}}$ at zLL ($\Delta
V_{\mathrm{tg}}\sim$ $2\text{\,}\mathrm{V}$), indicating a correlation between
the longitudinal and transverse voltages, both consistent with the spin-split
picture of the band diagram in Figure 3a. The temperature-dependence of
$G_{\mathrm{xx}}$ at $B_{\mathrm{z}}=0\,$T (inset of Figure 3d) shows that the
zLL conductance plateau deviates from $2\,e^{2}/h$ as we raise the
temperature, flattening out at $2.2\,e^{2}/h$ around $T_{\mathrm{N}}$,
indicating a correlation with the CPS magnetization vs. $T$ (see SI Figure S17
for the full $T$-dependence).

Figure 4: QH plateaus in magnetized graphene. (a) Magnetic field dependence of
transverse conductivity ($\sigma_{\mathrm{xy}}$) and symmetric component of
resistivity vs. $B_{\mathrm{z}}$ ($\rho_{\mathrm{xx}}^{sym}$), shown vs.
$B_{\mathrm{z}}$ at $V_{\mathrm{tg}}=$15\text{\,}\mathrm{V}$$ and
$T=$1.8\text{\,}\mathrm{K}$$. (b) Magnetic field dependence of
$\sigma_{\mathrm{xy}}$, measured $-20<V_{\mathrm{tg}}<15\,$V. The data points
are shown with transparency to visualize the data accumulation happening at
the QH plateaus. The insets show the same data in an enlarged scale for the LL
with smaller filling factors. The red arrows point to the formed QH plateaus
at large magnetic fields. (c,d) Gate-dependence of $\sigma_{\mathrm{xy}}$,
measured at $B_{\mathrm{z}}=\pm 1\,$T (panel c) and at
$B_{\mathrm{z}}=\pm$9\text{\,}\mathrm{T}$$ shown for trace and retrace
measurements (panel d). (e) Temperature-dependence of the longitudinal
conductance ($G_{\mathrm{xx}}^{*}$) and $\sigma_{\mathrm{xy}}$ vs.
$V_{\mathrm{tg}}$ close to charge neutrality point, measured in device A. (f)
$T$-dependence of the $\sigma_{\mathrm{xy}}$ vs. $V_{\mathrm{tg}}$ in device
B. The thickness of top-gate hBN in device A and B is 73 and
$18\text{\,}\mathrm{n}\mathrm{m}$, respectively.

To further explore the helical nature of the edge transport at zLL at
$B_{\mathrm{z}}\,=\,0\,$T, we investigate the effect of floating voltage
probes on the transport properties of the devices. Helical charge transport
occurs on both edges of the graphene channel with each edge hosting one spin-
polarized state buttiker2009edge . In this case, the voltage probes present
along the edges alter the conductance between the source and the drain, since
the charges do not maintain their spin when entering a probe. Thus, according
to the Landauer-Buttiker formalism buttiker1986  (see SI, section 12), the
two-terminal conductance $G_{\mathrm{2T}}$ in the presence of two counter-
propagating (helical) edge states is given by the relation:

$G_{\mathrm{2T}}=\frac{e^{2}}{h}\left(\frac{1}{N_{\mathrm{L}}+1}+\frac{1}{N_{\mathrm{R}}+1}\right),$
(1)

where $N_{\mathrm{L}}$ and $N_{\mathrm{R}}$ are the number of floating probes
between the source and drain, along the left and right edges of the conductor.
Similarly, the four-terminal conductance $G_{\mathrm{4T}}$ is given by

$G_{\mathrm{4T}}=G_{\mathrm{2T}}\frac{N_{\mathrm{I}}+1}{N_{\mathrm{V}}+1},$
(2)

where $N_{\mathrm{I}}$ is the number of floating electrodes in between the
source and drain along the edge which hosts the voltage probes, and
$N_{\mathrm{V}}$ is the (minimal) number of floating electrodes in between the
voltage probes roth2009nonlocal . For the measurement of Figure 3c,
$N_{\mathrm{L}}=N_{\mathrm{R}}=N_{\mathrm{I}}=6$, and $N_{\mathrm{V}}=0$,
thus, the $G_{\mathrm{4T}}=G_{\mathrm{xx}}=2\,e^{2}/h$, which is exactly what
is measured at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ at the zLL conductance
plateau. Considering the equations above, we evaluate the transport in various
two- and four-terminal measurement geometries, shown in Figure 3e. The
expected $G_{\mathrm{2T}}$ and $G_{\mathrm{4T}}$ considering the helical edge
states, in the absence of 2D bulk transport, are shown by the dashed lines
that are color-coded with respect to each measurement geometry. The measured
conductance in each geometry approaches the theoretical values at the Dirac
point, where the bulk transport is minimized (and not fully eliminated),
further supporting the contribution of helical states in the magnetized
graphene at 0 T.

We also investigate the QAH transport ($N_{\mathrm{LL}}\neq 0$) by evaluating
the transverse voltage, which is a direct measure of the number of chiral edge
states in the channel in the QH regime. Figure 4a shows the magnetic field
dependence of the transverse conductivity ($\sigma_{\mathrm{xy}}$) close to
zero energy ($V_{\mathrm{tg}}=$15\text{\,}\mathrm{V}$$). Near the SdHOs of
$\rho_{\mathrm{xx}}$, we observe plateau-like features in
$\sigma_{\mathrm{xy}}$ at about $-1$ and $3\,e^{2}/h$ at lower
$B_{\mathrm{z}}$ that reach $-4$ and $5\,e^{2}/h$ for
$|B_{\mathrm{z}}|>$1.5\text{\,}\mathrm{T}$$. This unconventional leveling of
the plateaus, with the smaller filling factors appearing at the smaller range
of $B_{\mathrm{z}}$, can be related to the non-linear and non-monotonic
behavior of CPS magnetic transition, rotating from out-of-plane to in-plane
direction, at this range of the $B_{\mathrm{z}}$ (see also Figure S15). Figure
4b further shows the dependence of $\sigma_{\mathrm{xy}}$ on a larger range of
$B_{\mathrm{z}}$ for $-20<V_{\mathrm{tg}}<$15\text{\,}\mathrm{V}$$, where the
data is shown with transparency of the markers to visualize the accumulation
of data points (resulting in the darker shades) around specific
$\sigma_{\mathrm{xy}}$ values at the QH plateaus (indicated by the red
arrows).

The QH plateaus appear to be asymmetric vs. $B_{\mathrm{z}}$ and have
unconventional sequences, distinct from that expected in pristine graphene
($(4\,e^{2}/h)(N_{\mathrm{LL}}+1/2)$ novoselov2005two ; zhang2005experimental
). This asymmetry that is present up to high filling factors and high
$B_{\mathrm{z}}$ shows different arrangements of the LLs for $\pm
B_{\mathrm{z}}$, depending on the direction of the CPS magnetization. This
remarkable anomaly can be explained by the $B_{\mathrm{z}}$-dependent
interplay between spin-orbit and exchange interactions induced in graphene
zollner2020scattering . In contrast to spin-orbit coupling, exchange
interaction changes sign with $B_{\mathrm{z}}$, thus their overall
contribution breaks time-reversal symmetry and alters the formation of the
zLLs from the K and K’ valleys for positive vs. negative $B_{\mathrm{z}}$.
Moreover, we observe that for the LLs with smaller filling factors, increasing
$B_{\mathrm{z}}$ creates further splitting of the levels by $e^{2}/h$ (shown
more clearly in the insets of Figure 4b). This is an indication of spin and
valley splitting of the LLs when $\vec{M}_{\mathrm{CPS}}$ is pulled towards
the out-of-plane direction by the applied $B_{\mathrm{z}}$. The unconventional
sequence of QH plateaus appears both at small and large magnetic fields which
is presented vs. $V_{\mathrm{tg}}$ in Figure 4c and d, and is also reproduced
in device B (see SI, Figure S16).

We also evaluate the QH transport up to room temperature, with a focus on the
SdHO and QH plateaus close to zero energy shown in Figure 4e. In this range of
$V_{\mathrm{tg}}$, the $G_{\mathrm{xx}}^{*}$ has a maximum at $2\,e^{2}/h$
right at the gate voltage at which $\sigma_{\mathrm{xy}}$ goes to zero,
pointing at the presence of the helical states up to $-9\text{\,}\mathrm{T}$.
By increasing $T$ we observe that the conductance close to zLL decreases (see
also Figure S19 and S20 for the $T$-dependence of resistivity at
$\pm$9\text{\,}\mathrm{T}$$ in both devices). This metallic $T$-dependence
which opposes that of pristine graphene, indicates the presence of a QH
metallic state at zero energy which reassures gapless LLs, consistent with the
presence of the counter-propagating helical states up to large magnetic fields
abanin2007dissipative . We also monitor $\sigma_{\mathrm{xy}}$ vs. temperature
which shows that the plateau-like features close to zero energy persist up to
room temperature. This may be due to the enhanced Curie temperature at the
graphene-CPS interface, as recently reported zhu2023interface . The
$T$-dependence of the QH transport in device B (shown in Figure 4f) also
indicates that the first QH plateau at $\sigma_{\mathrm{xy}}=2\,e^{2}/h$ stays
robust up to 300 K. The Landau plateau at $2\,e^{2}/h$, in particular, is
theoretically predicted to emerge in graphene by the QAH effect
qiao2010quantum ; zhang2015robust  which is expected to be robust in
temperature hogl2020quantum . Note that the charge carrier mobility in these
devices is low, thus the very observation of QH plateaus at room temperature
is due to the robustness of the QAH in graphene, enabled by the magnetic
proximity of the CPS at elevated temperatures.

In conclusion, in graphene-CPS heterostructures, the induced magnetism and
spin-obit coupling and their interplay allow for the emergence of quantum
anomalous Hall and spin Hall effects. The out-of-plane component of the CPS
magnetism induced in graphene enables the onset of the QH transport at zero
external magnetic field. The induced exchange interaction causes the spin-
splitting of zLLs which leads to the emergence of spin-polarized helical
states in the magnetic graphene without the application of an external
magnetic field. Furthermore, QAH transport results in a distinctive Landau fan
diagram with large separations of the LLs, due to the presence of additional
effective magnetic fields as large as 15 T which is a direct consequence of
the enhanced Berry curvature in the magnetized graphene. The broken time-
reversal symmetry in this system lifts spin and/or valley degeneracies and
gives rise to the detection of unconventional QH plateaus. Moreover, in
accordance with theoretical expectations for the QAH effect, the first quantum
Hall plateaus persist up to $300\text{\,}\mathrm{K}$. These observations, in
particular the detection of the helical states at zero external magnetic
field, in addition to the robustness of the QAH effect up to room temperature
open the route for the direct applications of the magnetic graphene in 2D
topological quantum spintronic devices.

Methods

CrPS4 Synthesis. CrPS4 crystals are grown following a solid-state reaction.
Stoichiometric amounts of Cr (99.99 $\%$, Alfa-Aesar), P ($>99.99\%$, Sigma-
Aldrich) and S ($99.99\%$, Sigma-Aldrich) are sealed inside an evacuated
quartz tube (pressure $5\times 10^{-5}$ mbar, length:
$50\text{\,}\mathrm{c}\mathrm{m}$, internal diameter:
$14\text{\,}\mathrm{m}\mathrm{m}$) and placed in a three-zone furnace. A
temperature gradient of $750/650/700^{\circ}$C is kept for 21 days followed by
a quench into water. The obtained crystals are analyzed by powder X-ray
diffraction and energy-dispersive X-ray spectroscopy. The refinement of the
X-ray pattern (ICSD 25059) reveals a monoclinic C face center crystal system
with C121 space group and a unit cell determined by $\alpha$ = $\gamma$ = 90∘
and $\beta$ = 91.99(1)∘ and a = 10.841(9)∘A, b = 7.247(6)∘A and c =
6.100(5)∘A. The amount of elements is Cr:$23.3\pm 0.5\%$, P:$14.8\pm 0.4\%$
and S:$61.8\pm 1.5\%$, in good agreement with the expected ones (Cr:24.6$\%$,
P: $14.7\%$ and S: $60.7\%$). The obtained results are in accordance with the
ones previously reported in the literature peng2020magnetic .

Electrical characterization. The electrical characterization is performed
using DC current source and voltages are measured using Keithleys. DC voltage
source is used for the gates. The measurements are performed using a variable
temperature insert at He atmosphere (in attoDry2100 system).

Acknowledgements

We would like to acknowledge Yaroslav Blanter and Anton Akhmerov for
insightful discussions. This project has received funding from the European
Union Horizon 2020 research and innovation program under grant agreement, No.
863098 (SPRING). J.I.A. acknowledges support from the European Union’s Horizon
2020 research and innovation programme for a Marie Sklodowska–Curie individual
fellowship, No. 101027187-PCSV. E.C and S.M.-V. acknowledge support from the
European Union (ERC AdG Mol-2D 788222, FET OPEN SINFONIA 964396), and the
Spanish MCIN (2D-HETEROS PID2020-117152RB-100 and Excellence Unit “María de
Maeztu” CEX2019-000919-M). K.W. and T.T. acknowledge support from the JSPS
KAKENHI (Grant Numbers 21H05233 and 23H02052) and World Premier International
Research Center Initiative (WPI), MEXT, Japan.

Author Contributions

T.S.G. and H.S.J.v.d.Z. conceived and designed the experiments. T.S.G.
performed device fabrication and measurements with the help of D.P. T.S.G. and
D.P. analyzed the data with the help of J.I-A.. T.B. and J.I-A. assisted with
measurements. S.M-V. and E.C. synthesized the CPS crystals and performed the
CPS magnetometry measurements. K.W. and T.T. synthesized the hBN crystals.
T.S.G. wrote the manuscript with the help of D.P. and contributions from all
co-authors.

Supplementary Information

1\. Device Fabrication

Figure S1 shows the steps for the fabrication of device A. Details of the
fabrication steps are stated in the caption. Device B is also fabricated using
the same procedure.

Figure S1: Device fabrication procedure. 1. HBN flake exfoliated on PMMA
(thickness of the hBN flake $\approx\,$$32\text{\,}\mathrm{n}\mathrm{m}$,
shown here for device A). The PMMA layer is initially spin-coated on a water-
soluble layer to assist the suspension and transfer of the PMMA in the next
step. 2. HBN/PMMA transferred on graphene which is exfoliated in advance on a
SiO2/Si substrate. The rectangular-like openings in the PMMA are made by
e-beam exposure, followed by cold developing in H2O:IPA (1:3) solution. 3.
HBN/graphene, etched with O2:CHF3 plasma (power = $40\text{\,}\mathrm{W}$,
bias= 100-150 V) for 80-$110\text{\,}\mathrm{s}$. PMMA is then dissolved in
acetone. 4. Pick up of the hBN/graphene Hall bar with a polycarbonate-PDMS
stamp. 5. Exfoliation of CrPS4 (CPS) on a SiO2/Si substrate and transfer of
the hBN/graphene on the CPS flake, using conventional dry transfer technique
zomer2014fast . 6. PMMA-coated heterostructure, exposed by e-beam lithography
of the electrodes, developed in MIBK:IPA (1:3) solution. After development,
the contact regions are exposed to an O2:CHF3 plasma to enhance the
possibility of making edge contact with graphene. 7. Deposition of Ti(5
nm)-Pd(75 nm) electrodes, followed by lift-off in acetone. After annealing the
device in Ar atmosphere for 1 hr at
$400\text{\,}\mathrm{\SIUnitSymbolCelsius}$, another hBN flake is transferred
on top for the top-gate dielectric (for device A the thickness of this hBN
flake is about $41\text{\,}\mathrm{n}\mathrm{m}$. Thus, the total thickness of
hBN layers, including that of step 1, that would be counted for top-gating
dielectric is about $73\text{\,}\mathrm{n}\mathrm{m}$). 8. The device is
covered with PMMA and has the top-gate area exposed and developed. 9.
Deposition of Ti(5 nm)-Au(100 nm) for the top-gate electrode. The panels are
all related to device A.

2\. CPS magnetization anisotropy

Figure S2: (a) Alignment of graphene channel with respect to the
crystallographic orientation of the CPS. The magnetic $a$-axis is along the
bisector of the 67.5∘ corner angle of the flake. Inset: STM image of CPS
adapted from ref. lee2017structural . The white solid lines are along the Cr
atoms. (b) Magnetic molar susceptibility ($\chi$) of CPS, measured parallel
($\|$) and perpendicular ($\perp$) to the $c-$axis (out-of-plane direction) of
a bulk CPS crystal.

3\. $B_{\mathrm{z}}$-dependence of the longitudinal and transverse resistivity
at room temperature

Figure S3: Longitudinal and transverse resistances measured in device A, at
$T=$300\text{\,}\mathrm{K}$$, $V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$ and
$V_{\mathrm{tg}}=$0\text{\,}\mathrm{V}$$. (a) Data as measured for the
longitudinal resistivity ($\rho_{\mathrm{xx}}$). (b) Transverse resistance
($R_{\mathrm{xy}}$), compared with its anti-symmetrized component versus
$B_{\mathrm{z}}$. (c) Anomalous Hall effect component ($R_{\mathrm{AHE}}$),
acquired by subtraction of a linear fit from the raw $R_{\mathrm{xy}}$ data,
and from the anti-symmetrized $R_{\mathrm{xy}}$ component (in panel b), done
separately for comparison. The linear fit is the average of the linear fits to
the $R_{\mathrm{xy}}$ data for $8<B_{\mathrm{z}}<$9\text{\,}\mathrm{T}$$ and
$-9<B_{\mathrm{z}}<$-8\text{\,}\mathrm{T}$$.

4\. Evaluation of interfacial charge transfer by top-gate vs. back-gate sweeps

In previous studies on graphene in the proximity of 2D magnetic materials,
gate-dependent interfacial charge transfer from the 2D magnet to the graphene
has been reported tseng2022gate ; wang2022quantum . That has given rise to the
observation of unconventional Landau fan diagrams that flatten off vs. gate
due to the suppression of the gate action by the charge transfer. To
investigate the possible gate-dependent interfacial charge transfer in our
graphene-CrPS4 heterostructure, we conducted similar measurements where both
top- and back-gates were swept. Figure S4 show the $\rho_{\mathrm{xx}}$ versus
$V_{\mathrm{tg}}$ and $V_{\mathrm{bg}}$ measured at different magnetic fields
in device A. We observe that increasing the $V_{\mathrm{bg}}$ shifts the
position of the charge neutrality point, while the Landau gaps stay constant.
In Figure S5, we further show the Landau fans measured at
$V_{\mathrm{bg}}=\,0$ and $50\text{\,}\mathrm{V}$, indicating no significant
change in the fan diagrams. As the back gate voltage is only shifting the
position of the charge neutrality point, and the Landau gaps remain constant
for the full $V_{\mathrm{bg}}$ and $V_{\mathrm{tg}}$ range, we can conclude
that the charge transfer from the 2D magnet is minimal in our devices and does
not affect the QH transport in the graphene channel.

Moreover, a recent report on transport in CPS using graphite contacts
wu2023gate  shows the onset of current in CPS at gate electric field above
$0.25\text{\,}\mathrm{V}\mathrm{/}\mathrm{n}\mathrm{m}$ at
$2\text{\,}\mathrm{K}$ with the resistivity of the bulk CPS to be at least 3-4
orders of magnitude larger than that of graphene. Additionally, in contrast to
the large temperature-dependent charge transport reported in CPS wu2023gate ,
our QH measurements at various temperatures (shown in Figure 2e of the main
manuscript) do not show any signature correlating with the CPS thermal
activation. Thus, we confirm the suppression of the gate-activated interfacial
charge transfer between graphene and CPS in the heterostructure. That is of
high importance as it assures that the modulation of the LLs is due to the
magnetic proximity effect, rather than to the charge-related effects.
Furthermore, we note that in our devices there have been several contacts that
have been used to only address the transport in the CPS. Our 2T measurements
in CPS up to RT do not show any current flow in the CPS above the noise level
(pA). This confirms that there is no parallel transport in the CPS layer.

Figure S4: Top- and back-gate dependence of the longitudinal resistivity
$\rho_{\mathrm{xx}}$, measured in device A at
$B_{\mathrm{z}}=$9\text{\,}\mathrm{T}$$ (panel a), and at
$B_{\mathrm{z}}=$3\text{\,}\mathrm{T}$$ (panel b). The sweeping of the back-
gate voltage shifts the Dirac peak and so the Landau levels to lower top-gate
voltages until the shift saturates at
$V_{\mathrm{bg}}\geq$40\text{\,}\mathrm{V}$$. Figure S5: Landau fan diagrams
measured in device A at $T=1.8\,$K (a) at $V_{\mathrm{bg}}=50\,$V (b), and at
$V_{\mathrm{bg}}=0\,$V.

5\. SdHOs detected at different regions of device A

Figure S6: Shubnikov-de Haas oscillations (SdHOs) at
$B_{\mathrm{z}}=$9\text{\,}\mathrm{T}$$ and
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$ detected by the different voltage
probes used for the measurements of longitudinal resistivity
$\rho_{\mathrm{xx}}$ in Figure 2 of the main manuscript, with $V_{1}$ and
$V_{2}$ for panel d, and $V_{2}$ and $V_{3}$ for panel a. This comparison
shows that the SdHOs are the same for different pairs, regardless of possible
inhomogeneities in the sample. The inset shows the SdHOs more clearly for the
range of $-30<V_{\mathrm{tg}}<-10\,$V. The used voltage and current probes are
indicated in the optical micrograph of the sample (scale bar $=10\,\mu m$).

6\. Landau fan diagram in device B

Figure S7: Landau fan diagram measured in device B. Modulation of longitudinal
resistivity (a) and transverse resistance (b) vs. magnetic field and top-gate
voltage, measured in 4-terminal geometry at $T=$1.8\text{\,}\mathrm{K}$$ and
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$.

7\. Carrier density and mobility

Encapsulation of the graphene between hBN and the 2D magnet can change the
environmental dielectric constant which results in an effective capacitance
that is different from the geometrical one. Thus, in a more reliable approach,
we estimate the carrier density from the Hall voltage, measured vs.
$B_{\mathrm{z}}$ at various gate voltages and temperatures (in section 8 we
discuss another approach for the extraction of $n$). At each top-gate voltage,
the Hall component is separated from the anomalous Hall component and is
related to the carrier density according to

$R_{\mathrm{HE}}=\frac{B_{\mathrm{z}}}{ne},$ (3)

where $R_{\mathrm{HE}}=R_{\mathrm{xy}}-R_{\mathrm{AHE}}$. We extract $n$ from
the $B_{\mathrm{z}}$-dependence of $R_{\mathrm{xy}}$ at each
$V_{\mathrm{tg}}$, by taking the slope of $R_{\mathrm{xy}}$ vs.
$B_{\mathrm{z}}$ for $B_{\mathrm{z}}\,>\,$$8\text{\,}\mathrm{T}$. In Figure S8
we show the gate-dependence of the extracted $n$ from the Hall effect for
device A. Using the linear fit between $V_{\mathrm{tg}}$ and $n$, we obtain an
effective dielectric constant of around 2.7, which is less than the dielectric
constant of hBN ($\sim 3.7$ laturia2018dielectric ).

Figure S8: Carrier density ($n$) in device A, derived from the Hall
measurements of $R_{\mathrm{xy}}$, anti-symmetrized vs. $B_{\mathrm{z}}$. The
slopes to the anti-symmetrized $R_{\mathrm{xy}}$ are taken for
$B_{\mathrm{z}}$ in the range of 8 to $9\text{\,}\mathrm{T}$. The relation
between $n$ vs. $V_{\mathrm{tg}}$ is considered to be linear, obtained from
the linear fit of $n$ vs. $V_{\mathrm{tg}}$ in the range of -24 to
$0\text{\,}\mathrm{V}$, shown with the blue solid line. The derived relation
for the gate-dependence of the carrier density for these measurements is
$n[\mathrm{m}^{-2}]=1.99\times 10^{15}(V_{\mathrm{tg}}-14)\;$.

To obtain the mobility $\mu$ of charge carriers in the graphene channel, we
refer to the relation $\sigma_{\mathrm{xx}}=ne\mu$. The $\sigma_{\mathrm{xx}}$
is plotted versus $V_{\mathrm{tg}}$, shown in Figure S9a, for device A. By
considering the $n$ versus $V_{\mathrm{tg}}$ relation, described earlier, and
by the linear fits to the $\sigma_{\mathrm{xx}}$ vs. $V_{\mathrm{tg}}$, we
obtain
$\mu_{\mathrm{h}}=$2300\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$$
and
$\mu_{\mathrm{e}}=$1450\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$$
for device A, and
$\mu_{\mathrm{h}}=$4100\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$$
and
$\mu_{\mathrm{e}}=$920\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$$
for device B at $T=$1.8\text{\,}\mathrm{K}$$.

We further extract the mobility at various temperatures, shown in Figure S9b.
In previous reports mobility_Vs_T1 ; mobility_vs_T2 , pristine graphene
encapsulated with hBN shows a slight decrease in mobility upon increasing the
temperature, whereas, in graphene on SiO2, the mobility decreases more rapidly
due to higher electron-phonon interaction. In our samples, the mobility stays
constant up to $T\approx$90\text{\,}\mathrm{K}$$, especially for the
electrons, and then decreases more rapidly at higher temperatures. Apart from
the contribution from electron-phonon scattering, the temperature dependence
in our devices could also be affected by the enhancement of the thermal
fluctuations of the magnetic moments in CPS at temperatures above the Néel
temperature. In device B, we extract the mobility for holes to be
$\mu_{\mathrm{h}}=4100$ and
$835\text{\,}\mathrm{c}\mathrm{m}^{2}\mathrm{/}\mathrm{V}\mathrm{s}$ at 1.8 K
and RT, respectively.

Figure S9: (a) Gate-dependence of the conductivity ($\sigma_{\mathrm{xx}}$) at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$, measured at
$T=\,$$1.8\text{\,}\mathrm{K}$ and $V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$
in device A. We extract the mobility for the electrons and holes separately by
the linear fits to the data above and below the charge neutrality point. (b)
Temperature-dependence of the mobility extracted for each carrier type in
device A.

8\. Estimation of total magnetic field from Landau fan diagrams

The energy of the Landau levels, $E_{N}$, in pristine graphene is given by the
relation:

$E_{N}=\sqrt{\alpha|N_{\mathrm{LL}}|B_{\mathrm{T}}},$

where $\alpha=2e\hbar v_{F}^{2}$ and $N_{\mathrm{LL}}$ is the Landau level
index. Considering the relation above and the dispersion relation in graphene,
we obtain:

$B_{\mathrm{T}}=\frac{\Phi_{0}n}{4|N_{\mathrm{LL}}|},$ (4)

where $\Phi_{0}$ is the flux quantum and $n$ is the carrier density. The
position of the LLs depends on the total magnetic field applied to the
graphene, which in our magnetized system is the sum of the external magnetic
field $B_{\mathrm{z}}$ and the field originating from the CPS magnetization
$B_{\mathrm{M}}$ induced by the CPS, i.e.
$B_{\mathrm{T}}=B_{\mathrm{z}}+B_{\mathrm{M}}$. To extract the
$B_{\mathrm{M}}$ from the Landau fan diagram, we investigate the measurements
at $|B_{\mathrm{z}}|>$8\text{\,}\mathrm{T}$$. In this range of
$B_{\mathrm{z}}$, the CPS magnetization is supposed to be fully saturated out-
of-plane, meaning that the change in the position of the SdHOs is only
determined by the change in $B_{\mathrm{z}}$. With this consideration, the
equation
$B_{\mathrm{T}}=B_{\mathrm{z}}+B_{\mathrm{M}}=\frac{\Phi_{0}n}{4|N_{\mathrm{LL}}|}$
linearly relates $B_{\mathrm{z}}$ and $n$ in the range
$|B_{\mathrm{z}}|>$8\text{\,}\mathrm{T}$$.

As mentioned in section 7, one reliable approach to extract $n$ is by
considering the Hall resistance at $B_{\mathrm{z}}>$8\text{\,}\mathrm{T}$$.
However, we can also extract $n$, from the $B_{\mathrm{z}}$ vs.
$V_{\mathrm{tg}}$ dependence of the SdHO peaks for each LL, considering the
following relation:

$\alpha(V_{\mathrm{tg}}-V_{\mathrm{cnp}})=\frac{2eN}{\hbar\pi}(B_{\mathrm{z}}+B_{\mathrm{M}})$
(5)

assuming a linear relation between $n$ and $V_{\mathrm{tg}}$, we make a linear
fit to the $n$ (extracted for each LL) vs. $V_{\mathrm{tg}}$, shown by the red
line in Figure S10a. The $n$ extracted by this approach is comparable with
that extracted from Hall measurements (shown by the blue line). In Figure S10b
we compare the $B_{\mathrm{M}}$ obtained using the $n$ extracted by both
approaches. For the extraction of the $B_{\mathrm{M}}$ in the main manuscript,
we stick to the second approach described here for obtaining $n$, which gives
a lower bound for the estimation of the $B_{\mathrm{M}}$. Note that if $n$ is
extracted from the geometrical capacitance, it would give rise to much larger
$B_{\mathrm{M}}$ values.

Figure S10: (a) carrier density extracted for each LL from the
$B_{\mathrm{z}}$ vs. $V_{\mathrm{tg}}$ dependence of the SdHO peaks at
$B_{\mathrm{z}}>$8\text{\,}\mathrm{T}$$ (shown by the black circles). The red
line is the linear slope of $n$ vs. $V_{\mathrm{tg}}$, considering the black
circles. The blue line is $n$ vs. $V_{\mathrm{tg}}$ relation extracted from
Hall voltage at various temperatures, as shown in Figure S8. (b) The extracted
magnetic field originating from the CPS magnetization ($B_{\mathrm{M}}$),
estimated considering the $n$ extracted by the two approaches shown in panel
a. This panel is shown only for positive $B_{\mathrm{z}}$. The red dots are
the ones we report in Figure 2c of the main manuscript for
$B_{\mathrm{z}}=$9\text{\,}\mathrm{T}$$.

Figures S11 and S12 show linear fits for $B_{\mathrm{z}}$ vs. $n$ for each LL,
given by the red dashed lines for $B_{\mathrm{z}}>0$ and $B_{\mathrm{z}}<0$,
determined for device A and B. Direct comparison of the black solid with the
red dashed lines clarifies that the $B_{\mathrm{T}}\neq B_{\mathrm{z}}$ in the
magnetized graphene. With $B_{\mathrm{M}}$ being the only unknown in Equation
5, we can obtain the total field $B_{\mathrm{T}}$ for each LL which is shown
in Figure 2c of the main manuscript (for device A) and in Figure S12c (for
device B). Note that all the measurements compared for devices A and B are
performed at the same $V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$.

Figure S11: (a) and (b) show the position of the resistivity peaks in the
SdHOs (red circles) for $B_{\mathrm{z}}>$8\text{\,}\mathrm{T}$$ and
$B_{\mathrm{z}}<$-8\text{\,}\mathrm{T}$$, respectively (obtained from Figure
2a of the main manuscript). Considering the saturation of the CPS
magnetization in this range of $B_{\mathrm{z}}$, the Landau fans evolve
linearly by the applied magnetic field. The red dashed lines are the linear
fits to the SdHO peaks in the magnetized graphene. The solid black lines are
the Landau fan diagram expected in pristine graphene following Equation 5 with
$B_{\mathrm{M}}=0$. The panels are related to device A. Figure S12: (a) and
(b) show the position of the resistivity peaks in the SdHOs (red circles)
obtained from Figure S7 for $B_{\mathrm{z}}>$8\text{\,}\mathrm{T}$$ and
$B_{\mathrm{z}}<$-8\text{\,}\mathrm{T}$$, respectively, measured in device B.
The red dashed lines are the linear fits to the SdHO peaks in the magnetized
graphene. Black solid lines show the fan diagram expected for pristine
graphene. (c) Extracted total effective magnetic fields that give rise to the
position of the LLs.

9\. Helical states in device B

We also evaluate the presence of helical states in device B, by focusing on
the gate dependence of the conductance close to the charge neutrality point.
We start with four-terminal geometry as shown in the schematics of Figure
S13a. The gate-dependence of the four-terminal longitudinal conductance
($G_{\mathrm{xx}}^{*}$) for positive and negative ranges of $B_{\mathrm{z}}$
is shown in panel b and c. We realize that, unlike the observations in device
A, the conductance close to the charge neutrality point, measured in this
geometry at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$, does not reach the
theoretical value of 2e2/h expected for the helical states. This could be due
to a larger contribution or dominance of the bulk transport as compared with
edge transport at $0\text{\,}\mathrm{T}$ in this device, because of a smaller
$B_{\mathrm{M}}$ at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ (see Figure S12).
However, by applying the $B_{\mathrm{z}}$ as small as
$\pm$1\text{\,}\mathrm{T}$$ the channel is brought more in the QH regime where
the transport is dominated by the edge states close to the charge neutrality
point. With that, we observe that the $G_{\mathrm{xx}}^{*}$ reaches 2$e^{2}/h$
at the Dirac point and stays at that value for the whole range of
$|B_{\mathrm{z}}|>$1\text{\,}\mathrm{T}$$. Thus, this observation also
reassures the emergence of the helical states in the graphene/CPS
heterostructure.

Figure S13: (a) Optical micrograph and schematics of device B, and the
measurement geometry. The scale bar is 10 $\mu$m. The circles with ‘x’
represent the electrodes through which there was no electrical connection. (b,
c) Gate-dependence of the channel conductance $G_{\mathrm{xx}}^{*}$, measured
in device B at positive and negative $B_{\mathrm{z}}$, shown per
$0.5\text{\,}\mathrm{T}$. The $G_{\mathrm{xx}}^{*}$ is calculated considering
the conductivity tensor that includes both $V_{\mathrm{xx}}$ and
$V_{\mathrm{xy}}$ measured in a 4-terminal geometry (as also shown for device
A in Figure 3c of the main manuscript).

We further conducted two-terminal measurements in device B, applying
$1\text{\,}\mathrm{m}\mathrm{V}$ AC voltage and measuring current, shown in
the schematics of Figure S14a. The 2T conductance ($G_{\mathrm{2T}}$) measured
at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ for different distances shown in
the device schematics, indicates that the $G_{\mathrm{2T}}$ at the charge
neutrality point gets close to the theoretically expected values for the
helical states (shown by the dashed lines). Yet, the $G_{\mathrm{2T}}$ is not
fully matching with the expected values, as also observed in device A (Figure
3e) which once again indicates the finite contribution of bulk transport at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$. It is worth noting that the distance
between the source and the drain in these configurations is relatively long
and the device has shown to be inhomogeneously doped along the channel. With
that and considering the low mobility of the charge carriers, coherent spin-
polarized edge transport over such long distances is unlikely to occur, which
can explain the discrepancy between the theory and the experiment.

Figure S14: (a) Schematics of device B with different two-terminal measurement
geometries used for evaluation of conductance in panel b. The circles with ‘x’
represent the electrodes through which there was no/poor electrical
connection. (b) Gate-dependence of the two-terminal conductance
($G_{\mathrm{2T}}$) at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$, color-coded
with respect to the geometries shown in panel a. The dashed lines are the
expected values for the $G_{\mathrm{2T}}$, accounting for the presence of
helical states.

10\. Magnetic field dependence

Here we show the magnetic field dependence of $\rho_{\mathrm{xx}}$ and
$\sigma_{\mathrm{xy}}$ in device A measured at two other gate-voltages close
to the charge neutrality point, $V_{\mathrm{tg}}=\,$9 and
$10\text{\,}\mathrm{V}$, to be compared with the one shown in Figure 4a of the
main manuscript for $V_{\mathrm{tg}}=\,$$15\text{\,}\mathrm{V}$. We
consistently observe small plateau-like features at small $B_{\mathrm{z}}$
range that can be correlated with the low-field magnetization behavior of CPS
that can affect the filling factor.

Figure S15: (a) Symmetrized component of $\rho_{\mathrm{xx}}$ vs.
$B_{\mathrm{z}}$. (b)
$\sigma_{\mathrm{xy}}=\rho_{\mathrm{xy}}/(\rho_{\mathrm{xy}}^{2}+\rho_{\mathrm{xx}}^{2})$
vs. $B_{\mathrm{z}}$. Measurements are performed at
$T=\,$1.8\text{\,}\mathrm{K}$$ at $V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$
and $V_{\mathrm{tg}}=\,9$ and $10\text{\,}\mathrm{V}$ in device A.

In Figure S16 we show the transverse conductivity in device B, to be compared
with that of device A (shown in Figure 4b of the main manuscript). We observe
the QH plateaus in device B also appear with an unconventional sequence,
similar to that of device A, together with a large asymmetry with respect to
$B_{\mathrm{z}}$.

Figure S16: (a) Dependence of $\sigma_{\mathrm{xy}}$ on $B_{\mathrm{z}}$ shown
for $V_{\mathrm{tg}}=$ -10 to 1 V (hole doping). (b) The same data as in panel
a spanned to highlight the QH plateaus with small filling factors. Note that
the sign of the transverse voltage probes in the measurements in device B is
opposite to that of device A, which causes the sign of $\sigma_{\mathrm{xy}}$
to be opposite to that of device A for the same carrier type.

11\. Temperature dependence

In this section, first, we provide further information on the transport in
devices A and B at $B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ and then we
elaborate on the measurements at $B_{\mathrm{z}}=\pm$9\text{\,}\mathrm{T}$$.
In Figure S17 we show the $T-$dependence of the channel conductance at the
zeroth LL (Dirac point) in device A.

Figure S17: Temperature dependence of the conductance value
$G_{\mathrm{xx}}=G_{\mathrm{xx}}^{*}$ at the charge neutrality point
($V_{\mathrm{tg}}=V_{\mathrm{cnp}}$), measured in device A at
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$ and $B_{\mathrm{z}}=0\,$T up to 300
K. Note that we account for the shift of the $V_{\mathrm{cnp}}$ vs. $T$.

We further investigate the temperature dependence of $\rho_{\mathrm{xx}}$ in
device B, which is shown in Figure S18a measured at $B=$0\text{\,}\mathrm{T}$$
for $T=$1.8 to $300\text{\,}\mathrm{K}$. The gate-dependence of
$R_{\mathrm{xy}}$, shown in Figure S18b also exhibits a double-peak feature at
the Dirac point ($V_{\mathrm{tg}}\approx$0\text{\,}\mathrm{V}$$), similar to
the same measurement conducted in device A (shown in Figure 3d of the main
manuscript). Note that the second bump in $R_{\mathrm{xy}}$ at
$V_{\mathrm{tg}}\approx$2.5\text{\,}\mathrm{V}$$ is associated with the
inhomogeneity in the channel.

Figure S18: (a) Temperature-dependence of the longitudinal resistivity
$\rho_{\mathrm{xx}}$ versus $V_{\mathrm{tg}}$ measured in device B at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ and
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$. (b) $T$-dependence of the Hall
resistance $R_{\mathrm{xy}}=V_{\mathrm{xy}}/I$ versus $V_{\mathrm{tg}}$ at
$B_{\mathrm{z}}=$0\text{\,}\mathrm{T}$$ and
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$. The curves are shown with an offset
along the y-axis for clarity.

Figure S19 and Figure S20 show the temperature dependence of
$\rho_{\mathrm{xx}}$ versus $V_{\mathrm{tg}}$ for device A and B. We observe
that the charge neutrality point shifts to higher gate voltages with
temperature, as the doping of the graphene channel slightly changes. The
Landau levels become less pronounced as the temperature increases due to the
thermal broadening. In both devices, at the Dirac point ($N_{\mathrm{LL}}=0$),
$\rho_{\mathrm{xx}}$ increases by increasing temperature. This metallic
behavior is an indication of the presence of a gap-less zeroth Landau level
which supports the band alignment shown in Figure 3a of the main manuscript,
and so the presence of helical states. Moreover, the oscillation in
$\rho_{\mathrm{xx}}$ related to the first Landau level, in both devices,
persists up to close to room temperature. While the SdHOs merge into the
background resistance (from the bulk of the graphene channel), the one related
to $N_{\mathrm{LL}}=1$ remains as a kink or shoulder near the Dirac point
(e.g. shown in Figure S19f) which indicates the robustness of the first LL
against temperature. This is further evaluated through the $T-$dependence of
$\sigma_{\mathrm{xy}}$.

Figure S19: Temperature dependence of the channel resistivity
$\rho_{\mathrm{xx}}$, measured up to 300 K versus $V_{\mathrm{tg}}$ at
$B_{\mathrm{z}}=-9\,$T (a), and at $B_{\mathrm{z}}=9\,$T (b) in device A with
V1 and V2 voltage probes (see Figure S6). (c) The maximum value of
$\rho_{\mathrm{xx}}$ (at the Dirac point) versus temperature, at
$B_{\mathrm{z}}=\pm 9\,$T. (d) and (e) show the temperature dependence of
$\rho_{\mathrm{xx}}$ in device A with V3 and V4 voltage probes (see Figure
S6), for $B_{\mathrm{z}}=$-9\text{\,}\mathrm{T}$$ and $9\text{\,}\mathrm{T}$,
respectively. (f) $\rho_{\mathrm{xx}}$ measured at
$T=$275\text{\,}\mathrm{K}$$, shown for
$B_{\mathrm{z}}=\pm$9\text{\,}\mathrm{T}$$. The black arrow points at the
shoulder associated with the SdHO related to the first LL, persisting up to
elevated temperatures. All measurements are performed at
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$.  Figure S20: Temperature dependence
of the longitudinal resistivity versus $V_{\mathrm{tg}}$ measured in device B,
(a) at $B_{\mathrm{z}}=$-9\text{\,}\mathrm{T}$$ and (b) at
$B_{\mathrm{z}}=$9\text{\,}\mathrm{T}$$. All measurements are performed at
$V_{\mathrm{bg}}=$50\text{\,}\mathrm{V}$$.

Figure S21 shows the temperature dependence of $\sigma_{\mathrm{xy}}$ versus
$V_{\mathrm{tg}}$ for device A (a, b) and device B (c, d) at
$B_{\mathrm{z}}=\pm$9\text{\,}\mathrm{T}$$. For both devices, the plateaus
corresponding to higher Landau level indexes become less pronounced as the
temperature increases. Both devices show a sequence of $2\,e^{2}/h$ and
$-3\,e^{2}/h$ across the Dirac point (Figure S21a and c). As the temperature
increases, the plateau at $2\,e^{2}/h$ remains for device B up to
$300\text{\,}\mathrm{K}$ (panel c), whereas for device A (inset of panel a),
the QH plateau gradually changes to $e^{2}/h$, yet the plateau at
${-3\,e^{2}/h}$ persists up to high temperatures. Note that the sign
convention of the voltage probes in device A and B has been opposite to each
other during the measurements, thus the results of device A at -9 T should be
compared with that of device B at +9 T.

According to theoretical predictions, if the quantum anomalous Hall effect is
present in graphene, it is expected to be robust against disorders and thus
should persist up to elevated temperatures hogl2020quantum . Thus, the
persistence of the QH-related features in the device up to room temperature in
such devices which have rather poor mobility is only reassuring that the
transport is ruled by the quantum anomalous Hall effect, made possible by the
CPS paramagnetic state at RT. Figure S22 shows the temperature dependence of
$B_{\mathrm{T}}$ extracted at $B_{\mathrm{z}}=\pm$9\text{\,}\mathrm{T}$$ (as
explained in section 8), obtained for device A. The $T-$dependence shows that
the $B_{\mathrm{T}}$ is changing by about or less than 5 percent for the shown
$T$ range. This can be related to the fact that the ferromagnetic ordering in
CPS persists up to elevated temperatures when it is interfaced with graphene
zhu2023interface .

Figure S21: Temperature-dependence of the transverse conductivity versus
$V_{\mathrm{tg}}$ measured in device A (a,b) and device B (c,d), at
$B_{\mathrm{z}}=\pm$9\text{\,}\mathrm{T}$$.  Figure S22: (a) and (b) show the
temperature dependence of $B_{\mathrm{T}}$ obtained according to Equation 5
for each LL, at $B_{\mathrm{z}}=-9$ and $9\text{\,}\mathrm{T}$ respectively.
The panels are all related to device A.

12\. Landauer-Büttiker formalism

To model the role of the edge channels on the measured signals, we have used
the Landauer-Büttiker equation buttiker1986 :

$I_{i}=\frac{\mathrm{e^{2}}}{\mathrm{h}}\sum_{j}\left(T_{ji}V_{i}-T_{ij}V_{j}\right),$
(6)

where $I_{i}$ is the current in terminal $i$, $T_{ij}$ the transmission
probability from electrode $i$ to electrode $j$, and $V_{j}$ the voltage at
electrode $j$. To account for all the electrodes in device A, for instance, we
have modeled for a device geometry with 14 electrodes that are connected in a
closed loop where contact 14 is between 13 and 1 (as shown in the schematics
of Figure 3e of the main manuscript). In this case, the $i=14$ component of
Equation 6 is linearly dependent on the other components ($i=$ 1 to 13). To
determine $V_{1}$ to $V_{14}$, we have written Equation 6 for the terminals 1
to 13 and replaced the $i=14$ component with the condition
$V_{\mathrm{I_{-}}}=0$, where $V_{\mathrm{I_{-}}}$ is the potential at the
grounded contact. The resulting system is shown here:

$\begin{bmatrix}I_{1}\\\ I_{2}\\\ \vdots\\\
I_{14}\end{bmatrix}=\begin{bmatrix}m+n&-n&0&\cdots&0&-m\\\
-m&m+n&-n&\cdots&0&0\\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\\
0&0&0&\cdots&0&0\end{bmatrix}\begin{bmatrix}V_{1}\\\ V_{2}\\\ \vdots\\\
V_{14}\end{bmatrix},$

which we write as $I_{\mathrm{mat}}=MV_{\mathrm{mat}}$, where
$I_{\mathrm{mat}}$ describes the input currents, $V_{\mathrm{mat}}$ the output
voltages, and $M$ is the matrix obtained from the above considerations. $m$
and $n$ are the number of edge modes propagating from electrode $i$ to
electrode $i-1$ and to $i+1$, respectively. The last row of $M_{14,j}=0$
$\forall j\neq 6$, $M_{14,6}=1$ and $I_{i}=0$ $\forall i\neq 1$ or $6$,
$I_{1}=-I_{6}=I_{0}$. Finally, we calculate $V_{\mathrm{mat}}$ using
$M^{-1}I_{\mathrm{mat}}=V_{\mathrm{mat}}$, where $M^{-1}$ is the inverse of
$M$, and obtain the potentials in the 14 terminals as a function of $m$ and
$n$. Using the Landauer-Büttiker formalism, we acquire the equations 1 and 2
in the main text for two- and four-terminal conductance at the Dirac point
where $m=1$ and $n=1$, to consider for a co-presence of an electron-like and
hole-like band.

## References

  * (1) Neto, A. C., Guinea, F., Peres, N. M., Novoselov, K. S. & Geim, A. K.  The electronic properties of graphene.  _Reviews of Modern Physics_ 81, 109 (2009).
  * (2) Han, W., Kawakami, R. K., Gmitra, M. & Fabian, J.  Graphene spintronics.  _Nature Nanotechnology_ 9, 794–807 (2014).
  * (3) Sierra, J. F., Fabian, J., Kawakami, R. K., Roche, S. & Valenzuela, S. O.  Van der Waals heterostructures for spintronics and opto-spintronics.  _Nature Nanotechnology_ 16, 856–868 (2021).
  * (4) Wei, P. _et al._ Strong interfacial exchange field in the graphene/EuS heterostructure.  _Nature Materials_ 15, 711–716 (2016).
  * (5) Safeer, C. _et al._ Room-temperature spin Hall effect in graphene/MoS2 van der Waals heterostructures.  _Nano Letters_ 19, 1074–1082 (2019).
  * (6) Mendes, J. _et al._ Spin-current to charge-current conversion and magnetoresistance in a hybrid structure of graphene and yttrium iron garnet.  _Physical Review Letters_ 115, 226601 (2015).
  * (7) Ghiasi, T. S., Kaverzin, A. A., Blah, P. J. & Van Wees, B. J.  Charge-to-spin conversion by the rashba-edelstein effect in two-dimensional van der Waals heterostructures up to room temperature.  _Nano Letters_ 19, 5959–5966 (2019).
  * (8) Benítez, L. A. _et al._ Tunable room-temperature spin galvanic and spin Hall effects in van der Waals heterostructures.  _Nature Materials_ 19, 170–175 (2020).
  * (9) Wang, Z., Tang, C., Sachs, R., Barlas, Y. & Shi, J.  Proximity-induced ferromagnetism in graphene revealed by the anomalous Hall effect.  _Physical Review Letters_ 114, 016603 (2015).
  * (10) Ghiasi, T. S. _et al._ Electrical and thermal generation of spin currents by magnetic bilayer graphene.  _Nature Nanotechnology_ 16, 788–794 (2021).
  * (11) Geim, A. K. & Grigorieva, I. V.  Van der Waals heterostructures.  _Nature_ 499, 419–425 (2013).
  * (12) Yang, H.-X. _et al._ Proximity effects induced in graphene by magnetic insulators: first-principles calculations on spin filtering and exchange-splitting gaps.  _Physical Review Letters_ 110, 046603 (2013).
  * (13) Dyrdał, A. & Barnaś, J.  Anomalous, spin, and valley Hall effects in graphene deposited on ferromagnetic substrates.  _2D Materials_ 4, 034003 (2017).
  * (14) Diehl, R. & Carpentier, C.-D.  The crystal structure of chromium thiophosphate, CrPS4.  _Acta Crystallographica Section B: Structural Crystallography and Crystal Chemistry_ 33, 1399–1404 (1977).
  * (15) Young, A. _et al._ Tunable symmetry breaking and helical edge transport in a graphene quantum spin Hall state.  _Nature_ 505, 528–532 (2014).
  * (16) Veyrat, L. _et al._ Helical quantum Hall phase in graphene on SrTiO3.  _Science_ 367, 781–786 (2020).
  * (17) Qiao, Z. _et al._ Quantum anomalous Hall effect in graphene from rashba and exchange effects.  _Physical Review B_ 82, 161414 (2010).
  * (18) Tse, W.-K., Qiao, Z., Yao, Y., MacDonald, A. & Niu, Q.  Quantum anomalous Hall effect in single-layer and bilayer graphene.  _Physical Review B_ 83, 155447 (2011).
  * (19) Qiao, Z., Jiang, H., Li, X., Yao, Y. & Niu, Q.  Microscopic theory of quantum anomalous Hall effect in graphene.  _Physical Review B_ 85, 115439 (2012).
  * (20) Qiao, Z. _et al._ Quantum anomalous Hall effect in graphene proximity coupled to an antiferromagnetic insulator.  _Physical Review Letters_ 112, 116404 (2014).
  * (21) Zhang, J., Zhao, B., Yao, Y. & Yang, Z.  Quantum anomalous Hall effect in graphene-based heterostructure.  _Scientific Reports_ 5, 10629 (2015).
  * (22) Zhang, J., Zhao, B., Yao, Y. & Yang, Z.  Robust quantum anomalous Hall effect in graphene-based van der Waals heterostructures.  _Physical Review B_ 92, 165418 (2015).
  * (23) Zanolli, Z. _et al._ Hybrid quantum anomalous Hall effect at graphene-oxide interfaces.  _Physical Review B_ 98, 155404 (2018).
  * (24) Högl, P. _et al._ Quantum anomalous Hall effects in graphene from proximity-induced uniform and staggered spin-orbit and exchange coupling.  _Physical Review Letters_ 124, 136403 (2020).
  * (25) Wang, X.-L.  Dirac spin-gapless semiconductors: promising platforms for massless and dissipationless spintronics and new (quantum) anomalous spin Hall effects.  _National Science Review_ 4, 252–257 (2017).
  * (26) Song, H.-D. _et al._ Electrical control of magnetic proximity effect in a graphene/multiferroic heterostructure.  _Applied Physics Letters_ 113 (2018).
  * (27) Song, H.-D. _et al._ Asymmetric modulation on exchange field in a graphene/BiFeO3 heterostructure by external magnetic field.  _Nano Letters_ 18, 2435–2441 (2018).
  * (28) Wu, Y. _et al._ Large exchange splitting in monolayer graphene magnetized by an antiferromagnet.  _Nature Electronics_ 3, 604–611 (2020).
  * (29) Wu, Y. _et al._ Magnetic exchange field modulation of quantum Hall ferromagnetism in 2d van der Waals CrCl3/graphene heterostructures.  _ACS Applied Materials and Interfaces_ 13, 10656–10663 (2021).
  * (30) Chau, T. K., Hong, S. J., Kang, H. & Suh, D.  Two-dimensional ferromagnetism detected by proximity-coupled quantum Hall effect of graphene.  _npj Quantum Materials_ 7, 1–7 (2022).
  * (31) Wang, Y. _et al._ Quantum Hall phase in graphene engineered by interfacial charge coupling.  _Nature Nanotechnology_ 1–8 (2022).
  * (32) Tseng, C.-C. _et al._ Gate-tunable proximity effects in graphene on layered magnetic insulators.  _Nano Letters_ (2022).
  * (33) Lee, J. _et al._ Structural and optical properties of single-and few-layer magnetic semiconductor CrPS4.  _ACS Nano_ 11, 10935–10944 (2017).
  * (34) Peng, Y. _et al._ Magnetic structure and metamagnetic transitions in the van der Waals antiferromagnet CrPS4.  _Advanced Materials_ 32, 2001200 (2020).
  * (35) Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P.  Anomalous Hall effect.  _Reviews of Modern Physics_ 82, 1539 (2010).
  * (36) Zollner, K. & Fabian, J.  Proximity effects in graphene on monolayers of transition-metal phosphorus trichalcogenides MPX3 (M: Mn, Fe, Ni, Co, and X: S, Se).  _Physical Review B_ 106, 035137 (2022).
  * (37) Novoselov, K. S. _et al._ Two-dimensional gas of massless Dirac fermions in graphene.  _Nature_ 438, 197–200 (2005).
  * (38) Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P.  Experimental observation of the quantum Hall effect and Berry’s phase in graphene.  _Nature_ 438, 201–204 (2005).
  * (39) Xiao, D., Chang, M.-C. & Niu, Q.  Berry phase effects on electronic properties.  _Reviews of Modern Physics_ 82, 1959 (2010).
  * (40) Sundaram, G. & Niu, Q.  Wave-packet dynamics in slowly perturbed crystals: Gradient corrections and berry-phase effects.  _Physical Review B_ 59, 14915 (1999).
  * (41) Zollner, K. & Fabian, J.  Engineering proximity exchange by twisting: Reversal of ferromagnetic and emergence of antiferromagnetic Dirac bands in graphene/Cr2Ge2Te6.  _Physical Review Letters_ 128, 106401 (2022).
  * (42) Bora, M., Behera, S. K., Samal, P. & Deb, P.  Magnetic proximity induced valley-contrasting quantum anomalous Hall effect in a graphene-CrBr3 van der Waals heterostructure.  _Physical Review B_ 105, 235422 (2022).
  * (43) von Klitzing, K.  The quantized Hall effect.  _Reviews of Modern Physics_ 58, 519–531 (1986).
  * (44) Kane, C. L. & Mele, E. J.  Quantum spin Hall effect in graphene.  _Physical Review Letters_ 95, 226801 (2005).
  * (45) Abanin, D. A., Lee, P. A. & Levitov, L. S.  Spin-filtered edge states and quantum Hall effect in graphene.  _Physical Review Letters_ 96, 176803 (2006).
  * (46) Kharitonov, M.  Edge excitations of the canted antiferromagnetic phase of the $\nu$= 0 quantum Hall state in graphene: A simplified analysis.  _Physical Review B_ 86, 075450 (2012).
  * (47) Büttiker, M.  Edge-state physics without magnetic fields.  _Science_ 325, 278–279 (2009).
  * (48) Büttiker, M.  Four-terminal phase-coherent conductance.  _Physical Review Letters_ 57, 1761 (1986).
  * (49) Roth, A. _et al._ Nonlocal transport in the quantum spin Hall state.  _Science_ 325, 294–297 (2009).
  * (50) Zollner, K. _et al._ Scattering-induced and highly tunable by gate damping-like spin-orbit torque in graphene doubly proximitized by two-dimensional magnet Cr2Ge2Te6 and monolayer WS2.  _Physical Review Research_ 2, 043057 (2020).
  * (51) Abanin, D. A. _et al._ Dissipative quantum Hall effect in graphene near the Dirac point.  _Physical Review Letters_ 98, 196806 (2007).
  * (52) Zhu, W. _et al._ Interface-enhanced room-temperature curie temperature in CrPS4/graphene van der waals heterostructure.  _Physical Review B_ 108, L100406 (2023).
  * (53) Zomer, P., Guimarães, M., Brant, J., Tombros, N. & Van Wees, B.  Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride.  _Applied Physics Letters_ 105, 013101 (2014).
  * (54) Wu, F. _et al._ Gate-controlled magnetotransport and electrostatic modulation of magnetism in 2D magnetic semiconductor CrPS4.  _Advanced Materials_ 2211653 (2023).
  * (55) Laturia, A., Van de Put, M. L. & Vandenberghe, W. G.  Dielectric properties of hexagonal boron nitride and transition metal dichalcogenides: from monolayer to bulk.  _npj 2D Materials and Applications_ 2, 6 (2018).
  * (56) Gannett, W. _et al._ Boron nitride substrates for high mobility chemical vapor deposited graphene.  _Applied Physics Letters_ 98 (2011).  242105\.
  * (57) Zhu, W., Perebeinos, V., Freitag, M. & Avouris, P.  Carrier scattering, mobilities, and electrostatic potential in monolayer, bilayer, and trilayer graphene.  _Physical Review B_ 80, 235402 (2009).