# Doping-control of excitons and magnetism in few-layer CrSBr

Farsane Tabataba-Vakili  Fakultät für Physik, Munich Quantum Center, and
Center for NanoScience (CeNS), Ludwig-Maximilians-Universität München,
Geschwister-Scholl-Platz 1, 80539 München, Germany  Munich Center for Quantum
Science and Technology (MCQST), Schellingtraße 4, 80799 München, Germany  Huy
P. G. Nguyen  Fakultät für Physik, Munich Quantum Center, and Center for
NanoScience (CeNS), Ludwig-Maximilians-Universität München, Geschwister-
Scholl-Platz 1, 80539 München, Germany  Anna Rupp  Fakultät für Physik, Munich
Quantum Center, and Center for NanoScience (CeNS), Ludwig-Maximilians-
Universität München, Geschwister-Scholl-Platz 1, 80539 München, Germany
Kseniia Mosina  Department of Inorganic Chemistry, University of Chemistry and
Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic  Anastasios
Papavasileiou  Department of Inorganic Chemistry, University of Chemistry and
Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic  Kenji
Watanabe  Research Center for Functional Materials, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Takashi Taniguchi
International Center for Materials Nanoarchitectonics, National Institute for
Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan  Patrick Maletinsky
Department of Physics, University of Basel, Basel, Switzerland  Mikhail M.
Glazov  Ioffe Institute, 194021 Saint Petersburg, Russian Federation  Zdenek
Sofer  Department of Inorganic Chemistry, University of Chemistry and
Technology Prague, Technická 5, 166 28 Prague 6, Czech Republic  Anvar S.
Baimuratov  Fakultät für Physik, Munich Quantum Center, and Center for
NanoScience (CeNS), Ludwig-Maximilians-Universität München, Geschwister-
Scholl-Platz 1, 80539 München, Germany  Alexander Högele  Fakultät für Physik,
Munich Quantum Center, and Center for NanoScience (CeNS), Ludwig-Maximilians-
Universität München, Geschwister-Scholl-Platz 1, 80539 München, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingtraße 4,
80799 München, Germany

###### Abstract

Magnetism in two-dimensional materials reveals phenomena distinct from bulk
magnetic crystals, with sensitivity to charge doping and electric fields in
monolayer and bilayer van der Waals magnet CrI3 [1, 2, 3]. Within the class of
layered magnets, semiconducting CrSBr stands out by featuring stability under
ambient conditions, correlating excitons with magnetic order [4] and thus
providing strong magnon-exciton coupling [5, 6], and exhibiting peculiar
magneto-optics of exciton-polaritons [7, 8, 9]. Here, we demonstrate that both
exciton and magnetic transitions in bilayer and trilayer CrSBr are sensitive
to voltage-controlled field-effect charging, exhibiting bound exciton-charge
complexes and doping-induced metamagnetic transitions. Moreover, we
demonstrate how these unique properties enable optical probes of local
magnetic order, visualizing magnetic domains of competing phases across
metamagnetic transitions induced by magnetic field or electrostatic doping.
Our work identifies few-layer CrSBr as a rich platform for exploring
collaborative effects of charge, optical excitations, and magnetism.

Recent experimental realization of two-dimensional (2D) magnets with
ferromagnetic (FM) order down to the monolayer limit [10, 11] has initiated
extensive research on van der Waals magnets, with observation of magnons [12,
13], magnetic proximity coupling [14, 15], and giant magnetoresistance [16,
17]. With additional electrostatic control of magnetism as in CrI3 [1, 2, 3],
2D magnets promise novel applications in spintronics or magnetic memories,
including high-speed magnetic switching. More recently, antiferromagnetic (AF)
semiconductor CrSBr with a bandgap of $1.5$ eV [18, 19], intralayer FM order
and AF interlayer coupling [20], and high Néel temperatures of $132$ and $140$
K for bulk [21] and bilayer [22] crystals has received particular attention
due to its intriguing magnetic [23, 20] and magneto-optical properties [4,
19], with demonstrations of strongly linearly polarized excitons sensitive to
the magnetic order [4], magnon-exciton coupling [5, 6], exciton-polaritons [7,
8, 9], and large negative magnetoresistance [18, 24, 25]. To date, however,
electric control of magneto-optical phenomena in CrSBr has remained elusive.

In this work, we present an elaborate study of electrostatic control of the
coupled excitonic and magnetic properties of few-layer CrSBr. To this end, we
embed monocrystalline few-layer CrSBr in a field-effect device and perform
cryogenic magneto-optical studies as a function of voltage-induced doping and
in the presence of magnetic fields along the magnetic hard (crystallographic
$c$), easy ($b$), and intermediate ($a$) axes. Upon electron doping, we
observe the emergence of charged exciton complexes in bi- and trilayer
crystals, and investigate their origins both experimentally and theoretically.
The parabolic dispersions in magnetic fields along the $c$ and $a$ axes [4]
allow us to establish a self-consistent description of the neutral and charged
exciton complexes in the presence of coupling between intralayer and
interlayer excitons mediated by hole interlayer tunneling. We utilize this
understanding to demonstrate electric control of the metamagnetic transitions
induced by magnetic field along the $b$-axis, with pronounced exciton energy
jumps correlated with the magnetic order in bi- and trilayer [4]. Finally, we
demonstrate how the coupling between excitons and magnetism can be utilized
for local sensing of magnetic phases which depend on both magnetic field and
charge doping, and extend the technique to optical raster-imaging of magnetic
domains.

Figure 1: CrSBr field-effect device and spectral signatures of doping. a,
Optical micrograph of the few-layer CrSBr crystal, with crystal axes and layer
numbers indicated (ML: monolayer, BL: bilayer, and TL: trilayer; dashed lines
mark the crystal boundaries, the rectangle corresponds to the region in c). b,
Field-effect device layout with few-layer graphene top gate and contact (dark
grey). The CrSBr layers (blue and red indicating spin polarization in
antiferromagnetic order) are encapsulated by two hBN flakes (light grey). c,
DR map of the region marked in a in the energy range of $1.3-1.4$ eV (same
color bar as f, h), with blue areas corresponding to strong exciton resonances
(dotted lines mark the few-layer graphene contact, and black diamonds indicate
the positions on mono-, bi- and trilayer where all data were acquired; the two
rectangles indicate the regions studied in Fig. 4). d, and e, Monolayer DR and
PL spectra at $0$ and $30$ V (top panels) and the corresponding voltage sweeps
(bottom panels), respectively. f \- i, Same as d, e, but for bilayer (f, g)
and trilayer (h, i). X${}_{\text{A}}$, X${}_{\text{B}}$, and
X${}_{\text{B}}^{\prime}$ label neutral and X${}_{\text{B}}^{-}$ charged
exciton transitions.

Our field-effect device incorporates a CrSBr flake with mono-, bi-, and
trilayer regions, exfoliated from a bulk crystal grown by chemical vapor
transport (details in Methods). The CrSBr flake shown in the optical
micrograph of Fig. 1a has a characteristic shape, extended along the
crystallographic $a$-axis [4, 26, 27]. Using the conventional dry-transfer
method [28], we fabricated a single-gated device with hBN as dielectric and
few-layer graphene as top gate and contact layer (see schematic in Fig. 1b,
and Methods for fabrication details). To study the sample by low-temperature
differential reflectance (DR) and photoluminescence (PL) spectroscopy at $3.2$
K as a function of electrostatic doping, we identify mono-, bi- and trilayer
regions with strong exciton resonances, marked by diamonds in the DR map of
Fig. 1c.

The spectrum of the neutral monolayer at $0$ V in the top panel of Fig. 1d
shows a feature in DR at $1.342$ eV, which corresponds to a broad absorption
peak (see Extended Data Fig. 1a for absorption spectra determined by Kramers-
Kronig relation) and low-intensity PL (top panel of Fig. 1e). The negligible
energy shift between DR, absorption and PL indicates a momentum-direct exciton
transition [4] labeled as X${}_{\text{A}}$. Our theoretical analysis assigns
the X${}_{\text{A}}$ exciton to the transition between the topmost valence
band ($v$) and the lowest conduction band ($c_{1}$) at the $\Gamma$ point of
the first Brillouin zone, without clear consensus on oscillator strength [4,
19, 4, 29, 23, 19, 30, 20]. We adopt the notion of a nominally dipole-
forbidden transition $v\leftrightarrow c_{1}$ [19] (see Methods for details),
brightened by the asymmetry of our structure and high surface-to-volume ratio
of the monolayer and persisting in the DR spectra of the neutral bi- and
trilayer in the top panels of Figs. 1f and h, respectively (see Extended Data
Fig. 2 for exciton layer assignment).

Figure 2: Magneto-optical spectroscopy of neutral and charged bi- and trilayer
in magnetic field along the $c$-axis. a, b, Evolution of bilayer DR with
magnetic field along the $c$-axis ($B_{c}$) in the neutral and negatively
charged regime at $0$ and $30$ V, respectively. The top panels illustrate the
spin orientation. c, d, Same as a, b, but for the trilayer. e \- h, Calculated
dispersions of the corresponding neutral and charged excitons in magnetic
field along the hard axis.

The assignment of the transition to the top valence and bottom conduction band
is substantiated by the absence of striking signatures upon electrostatic
doping in the FM monolayer, with data in Figs. 1d and e. Due to complete spin-
polarization of the bands, a bound exciton-charge state can not form with
spin-aligned electrons, and we only observe a gradual red-shift of the
X${}_{\text{A}}$ resonance with the gate voltage in Figs. 1d and e, with a
maximum shift of $7$ meV at $30$ V. Although coupling of electrostatic field
and doping effects can not be ruled out in our single-gated device and could
account in part for the shift via the Stark effect, we observe qualitative
signatures of doping in bi- and trilayer for the same voltage range both in DR
(Figs. 1f and h) and PL (Figs. 1g and i).

The most striking spectral features of layer charging in Figs. 1f – i are the
pronounced energy shifts at elevated positive gate voltages around $15$ V,
converting the spectrally narrow peaks X${}_{\text{B}}$ with energy near
$1.370$ eV into X${}_{\text{B}}^{-}$ peaks just below $1.350$ eV. The energy
difference of $19$ meV, as well as the blue-shift of the X${}_{\text{B}}$
resonance upon doping, are reminiscent of negatively charged trions [31, 32]
or Fermi-polarons [33, 34, 35, 36, 37] in monolayer transition-metal
dichalcogenides, indicating enhanced Coulomb interactions due to reduced
dielectric screening in the 2D semiconducting magnet. We assign the neutral
exciton X${}_{\text{B}}$ in Figs. 1f and h, with narrow absorption of $\sim 3$
meV full-width at half-maximum linewidth and negligible Stokes shift, to the
transition between the top valence band and the higher-energy conduction
subband $c_{2}$, which is dipole-allowed for linear polarization along the
$b$-axis [19] (Extended Data Fig. 3 shows linearly polarized out-of-plane
emission). The corresponding PL spectra in Figs. 1g and i stem from hot
excitons, with incomplete relaxation of electron constituents from subband
$c_{2}$ to $c_{1}$.

In the trilayer, the pronounced spectral feature just below X${}_{\text{B}}$,
labelled as X${}_{\text{B}}^{\prime}$ in Figs. 1h and i, deserves a separate
discussion. The transition, red-shifted by $5.4$ meV from X${}_{\text{B}}$, is
strong in DR and clearly present in PL, yet without signatures of doping. Our
theoretical analysis (see Methods) identifies the corresponding exciton as
being localized exclusively in the middle layer, where enhanced screening
changes the energy of X${}_{\text{B}}^{\prime}$ excitons. Remarkably, the
exciton shows no signature of charge-bound states (note the prevalence of its
spectral features at voltages above the X${}_{\text{B}}$ to
X${}_{\text{B}}^{-}$ cross-over in Figs. 1h and i). This observation indicates
that excitons in the middle layer are not subjected to intralayer charge. The
top and bottom layers of the trilayer host field-induced electrons in their
lowest-energy $c_{1}$ states, downshifted by the magnetic layer interactions
with respect to the $c_{1}$ band of the middle layer. We note that a second
sample reproduces all discussed neutral and charged exciton transitions in
mono-, bi-, and trilayers (see Extended Data Fig. 4).

Figure 3: Effect of doping on excitons and metamagnetic transitions in
magnetic field along the $b$-axis. a, b, DR as a function of increasing
magnetic field along the $b$-axis ($B_{b}$) at $0$ and $30$ V, respectively.
The top panels indicate the bilayer spin orientation. c, Energy of
X${}_{\text{B}}$ (brown/orange) and X${}_{\text{B}}^{-}$ (purple) at different
gate voltages for upward (downward) magnetic field sweep indicated in dark
(light) colors. The hysteresis loops at each voltage are offset by $50$ meV
for clarity; note that both X${}_{\text{B}}$ and X${}_{\text{B}}^{-}$ coexist
at $15$ and $20$ V. d, Calculated bilayer spin lattice energy as a function of
$B_{b}$ for the AF (black line) and FM (blue and red lines) phases indicated
by the arrows. The brown line shows the upward sweep with jumps corresponding
to the experimental data in a. e, Evolution of the critical magnetic field of
the AF-FM metamagnetic transition with gate voltage (with up- and downward
sweeps indicated by the arrows; the last three data points are for
X${}_{\text{B}}^{-}$). f, Doping-induced metamagnetic switching from AF to FM
at a magnetic field of $0.165$ T, with arrows indicating spin orientations. g
\- l, Same as a \- f, but for the trilayer. i, Hysteresis loops of
X${}_{\text{B}}^{\prime}$. j, Spin lattice energy for trilayer AF (solid
black), FM (solid blue/red), and mixed (dashed blue/red) phases. k, Critical
magnetic fields of the metamagnetic transitions observed in the trilayer. l,
Doping-induced switching in the trilayer from AF to mixed state at a magnetic
field of $0.160$ T.

To elucidate the rich exciton multiplicity in bi- and trilayer crystals, we
performed magneto-optical studies in magnetic fields along the $c$-axis
(perpendicular to the layers), and analyzed the respective exciton dispersions
in the framework of the states introduced above. The data in Figs. 2a and b
show the evolution of DR with magnetic field for the neutral and charged
bilayer at $0$ and $30$ V, respectively, with the corresponding data for the
trilayer shown in Figs. 2c and d. We first note that all states involved
exhibit symmetric energy red-shifts from $0$ T towards increasing absolute
values of the magnetic field, before they level off beyond a saturation field
of $\sim 2$ T [4, 19, 38]. This behavior can be understood by considering spin
canting from the AF state at $0$ T to the FM state at sufficiently high
magnetic fields [4, 38], as indicated by the arrows in the top panels of Figs.
2a – d.

In the neutral bilayer with data in Fig. 2a, the energies of X${}_{\text{A}}$
and X${}_{\text{B}}$ reduce parabolically with the same dispersion to settle
in the saturated FM regime with a broader linewidth and nearly preserved
contrast in DR. The same magneto-dispersion of both excitons is consistent
with different conduction bands and a shared valence band. In the electron-
doped limit at $30$ V of Fig. 2b, X${}_{\text{B}}^{-}$ exhibits a parabolic
energy red-shift similar to X${}_{\text{B}}$ with the same saturation field,
but the dispersion is flatter in the vicinity of $0$ T. The magneto-
dispersions of X${}_{\text{A}}$, X${}_{\text{B}}$, and X${}_{\text{B}}^{-}$ in
the neutral and charged trilayer (with data in Figs. 2c and d) are similar to
the bilayer, whereas the curvature of the dispersion of
X${}_{\text{B}}^{\prime}$ is twice as large and unaffected by doping. The
magneto-dispersions along the $a$-axis reproduce these observations with a
lower saturation field of $1$ T (see Extended Data Figs. 5 and 6).

Figure 4: Optical imaging of magnetism in bi- and trilayer CrSBr. a, Raster-
scan maps of a bilayer region (left rectangle in Fig. 1c; the white areas
delimit the crystal top edge) with bias at $0$ V for five consecutive magnetic
fields along the $b$-axis, with false-color contrast given by the energy of
the X${}_{\text{B}}$ transition in DR with corresponding AF (purple) and FM
(orange) phases indicated on the right of the color bar. b, Same as a, but for
a constant field of $168$ mT and five different gate voltages. c, and d,
Equivalent maps of the trilayer region (right rectangle in Fig. 1c, with the
dashed and dotted lines indicating the bi- to trilayer boundary and the edge
of the graphene contact, respectively; grey pixels correspond to vanishing
exciton DR due to local disorder) for magnetic fields across the metamagnetic
one-flip transition (c) and voltage-controlled magnetism at a constant field
of $156$ mT (d). The pixel color derives from the energies of X${}_{\text{B}}$
in the bilayer and X${}_{\text{B}}^{\prime}$ in the trilayer, respectively,
and the corresponding magnetic states are indicated on the right of the color
bar.

Such negative diamagnetic shifts of neutral and charged exitons are highly
unusual in conventional semiconductors [39]. They can be understood in the
framework of our analysis, invoking interlayer excitons with smaller binding
energy and, consequently, higher absolute energy [40], in addition to
intralayer excitons and their charged counterparts introduced above. With
magnetic field along the $c$-axis causing gradual spin canting from $b$-axis
AF to $c$-axis FM order, spin-conserving hole tunneling sets in to mix intra-
and interlayer exciton states. Our analysis (see Methods) shows in Fig. 2e
that the magneto-dispersions of X${}_{\text{A,B}}$ excitons can be represented
for magnetic fields $B<B_{\text{sat}}$ (with the saturation field
$B_{\text{sat}}$ corresponding to the FM order) as $\mathcal{E}_{\rm
A,B}(B)=\mathcal{E}_{\rm A,B}(0)-tB^{2}$, with the fitting parameter $t\approx
2\>\text{meV/T}^{2}$ related to the interlayer hole tunneling and the energy
splitting between the intralayer and interlayer excitons.

Within the minimal model of Fermi-polarons (see Methods for details), where we
account for the Fermi-sea mediated coupling of intralayer excitons and trions
[36, 37], the magneto-dispersion of X${}_{\text{B}}^{-}$ in Fig. 2f is weaker:
it contains an additional factor of $E_{\text{F}}/E_{\text{tr}}$, given by the
ratio of the Fermi energy $E_{\text{F}}$ to the trion binding energy
$E_{\text{tr}}$. With $E_{\text{F}}=10$ meV and $E_{\text{tr}}=19$ meV
extracted from the data in Fig. 1, the model predicts a slightly flatter trion
dispersion than observed in the experiment of Fig. 2b, possibly because of
disregarded intra-interlayer trion coupling. The dispersions in the neutral
and charged trilayer can be understood along the same lines, as shown in Figs.
2g and h for the same set of fit parameters, by taking into account that
X${}_{\text{B}}$ stems from the outer layers and X${}_{\text{B}}^{\prime}$
exclusively resides in the middle layer. The main difference is that hole
tunneling to both the top and bottom layers is not suppressed, which results
in $\mathcal{E}_{\rm B^{\prime}}(B)\approx\mathcal{E}_{\rm
B^{\prime}}(0)-2tB^{2}$ with twice as large curvature in the dispersion of
X${}_{\text{B}}^{\prime}$ as compared to X${}_{\text{A,B}}$, in full agreement
with experimental data.

The gradual spin canting transition along the $c$-axis is contrasted by an
abrupt metamagnetic spin-flip transition [41, 4] along the $b$-axis parallel
to the magnetic moment. At a critical magnetic field near $0.17$ T, the spins
flip from AF to FM order in both the neutral and charged bilayer, when swept
upward from negative to positive values as in Figs. 3a and b. Unlike in
magneto-dispersions along the hard $c$-axis, the critical field along the easy
axis depends hysteretically on the sweep direction (Fig. 3c), which is
indicative of a first order magnetic transition well known from optical
spectroscopy [4] and magneto-transport studies [18, 24, 25].

The critical field is determined by the spin lattice energy, which in turn is
governed by the interplay between the interlayer exchange energy $J$ and
magnetic field interaction [24]. From the data, we estimate the spin lattice
energy for AF and FM states as a function of magnetic field, as shown in Fig.
3d, assuming the crossing points between FM and AF states at the center of the
hysteresis (see Methods for details). In upward direction and with
initialization in the FM state at a negative magnetic field, the spins flip
from FM to AF order and back to FM, as illustrated by the brown trace in Fig.
3d extracted from the data in Fig. 3a. The hysteretic behavior near AF-FM
degeneracy is responsible for abrupt jumps in the brown trace of Fig. 3d.
Crucially, our device allows to tune the critical field of the spin-flip
transition with gate voltage, as evident from the data in Figs. 3c and e. This
effect, also observed in bilayer CrI3 [2, 1], is consistent with an inverse
scaling of the interlayer exchange energy with on-site Coulomb interaction
[42], which reduces the critical field upon negative doping. This implies that
the metamagnetic AF-FM transition near the critical field can be equivalently
induced by doping, as demonstrated in Fig. 3f.

The neutral and charged trilayer exhibit a similarly abrupt AF-FM transition
as the bilayer, however, at larger critical fields near $0.32$ T (Figs. 3g and
h). Surprisingly, we also observe an additional transition into an
intermediate or mixed state near $0.16$ T, which is stable for $30$ mT before
switching back to AF order. In this regime, the X${}_{\text{B}}^{\prime}$
resonance first jumps to lower energy (still above its energy in the FM state)
and then back, while the energy of X${}_{\text{B}}$ remains unaffected. This
intricate signature corresponds to mixed order in the trilayer, resulting in
two adjacent layers with FM and AF order illustrated in the top panel of Fig.
3g.

The corresponding spin lattice energies for AF, FM, and mixed states shown in
Fig. 3j provide some intuition for the spin-flip transitions as a function of
magnetic field. In upward direction, with initialization in the FM state at
negative magnetic field, the trilayer is in an energetically favorable FM
state (solid blue line). With increasing magnetic field, the spins of the
middle layer flip to AF order, which is preserved beyond $0$ T as a metastable
state, where its time-reversal counterpart would be energetically more
favorable. An energy-reducing transition, however, would require the
simultaneous spin-flip in all three layers. With increasing field, the spin
lattice energy of the AF state crosses the FM and the mixed states (solid and
dashed red line, respectively), where the trilayer transitions into a
metastable mixed state. Eventually, as this state becomes less favorable with
increasing magnetic field, a double spin-flip occurs into the energetically
favorable AF order (solid black line) and finally into the FM state (solid red
line). Overall, this intricate sequence of spin-flips in the trilayer with
magnetic field is driven by the minimization of the energy costs for spin-
flips in magnetically ordered layers. As in the bilayer, the critical fields
of the respective transitions in the trilayer depend on the charge doping, as
evident from Figs. 3i, k, l for the AF-FM, FM-AF, and AF-mixed transitions,
shifting towards lower values with increasing doping. Analogously to the
bilayer, the metamagnetic transitions can also be induced by doping, as shown
explicitly for the AF-mixed state transition in Fig. 3l.

With the observation that the exciton energy is correlated with the underlying
magnetic order as in Figs. 2 and 3, the exciton transitions can be employed
for all-optical detection of local magnetic order. In the following, we
demonstrate this feature by mapping out domains of magnetic phases, and their
sensitivity to doping, near metamagnetic transitions. We start out with the
bilayer, initialized in its AF ground state at zero magnetic field and zero
doping (leftmost panel of Fig. 4a), imaged by hyperspectral raster-scan DR
mapping with purple and orange colors representing AF and FM domains,
respectively. With increasing field, the AF domain shrinks and the FM domain
emerges near the critical field at the top edge of the crystal (maps of Fig.
4a at consecutive fields of $164$, $168$ and $172$ mT), until full coverage is
reached at $176$ mT (rightmost panel of Fig. 4a), with the bottom left corner
next to the monolayer flipping last. Remarkably, the characteristic local
nucleation and propagation of FM domains can be induced by doping, as
highlighted for five increasing gate voltages in Fig. 4b at a constant field
of $168$ mT.

We use the technique to visualize the magnetic transition from the AF to the
mixed state in Fig. 4c (see also Extended Data Fig. 7 for large-area mapping
of all trilayer transitions). With increasing magnetic field, the mixed state
nucleates at the edge of the few-layer graphene contact and spreads along the
$a$-axis ($x$-direction). We note that the AF-mixed transition in the trilayer
and the AF-FM transition in the adjacent bilayer happen at the same field
within the resolution limit, which indicates that the flip in the trilayer is
induced by intralayer exchange coupling to the proximal bilayer. Remarkably,
the trilayer mixed state is clearly discerned from the AF state despite
nominally identical magnetization, which renders their discrimination
difficult with conventional methods [18, 4, 22]. As in the bilayer, related
phenomena of phase transitions can be induced by the doping and detected all-
optically, as demonstrated in Fig. 4d.

Our work identifies few-layer CrSBr as an intriguing platform to study and
control excitons and magnetism with electrostatic doping. The rich phenomena
observed are intertwined, providing a route to novel devices involving not
only coupled electric and magnetic phenomena, but also adding optical means to
control and exploit magnetism via neutral and charged excitons. The aspects
highlighted in our work indicate the existence of rich magnetic phases in
trilayer crystals beyond just AF or FM order, which can potentially be
utilized in novel devices for spintronics and magnetic memories featuring
layer-selective initialization, control, and read-out by combined
electrostatic and optical means. Finally, the features cooperatively manifest
optical imaging of magnetic order in CrSBr as an efficient and sensitive local
probe of magnetic domains, providing insight complementary to other techniques
[43, 44, 45].

## Methods

CrSBr synthesis:
CrSBr bulk single crystals were synthesized by chemical vapor transport method
using chromium (99.99%, -60 mesh, Chemsavers), sulfur (99.9999%, $1-6$ mm
Wuhan Xinrong New Material Co. Ltd), and bromine (99.9999%, Sigma Aldrich),
combined with a stoichiometry of 1:1:1, sealed in a quartz tube under high
vacuum, and then placed into a two-zone tube furnace. The material was pre-
reacted in an ampoule at 700°C for 25 h until most of the bromine was reacted.
During this procedure one part of the ampoule was kept under 200°C to avoid
pressure disruption of the ampoule. The crystal growth by the vapor transport
method was performed in a two-zone horizontal furnace. First, the source and
growth ends were kept at 800 and 900°C, respectively. After 25 h, the
temperature gradient was reversed, and the temperature at the hot end was
gradually increased from 880 to 930°C for an 8 day period while the growth
zone was keep at 800°C. The high-quality CrSBr single crystals were removed
from the ampoule in an Ar glovebox.

Field-effect devices:
CrSBr, few-layer graphene, and hBN (NIMS) flakes were exfoliated from bulk
single crystals at ambient conditions onto SiO2/Si substrates. Suitable CrSBr
flakes were identified by optical contrast and atomic force microscopy. A
PDMS/PC stamp was used to sequentially pick up the hBN, few-layer graphene,
and CrSBr layers employing the dry-transfer method [28]. Poly-(Bisphenol
A-carbonate) pellets (Sigma Aldrich) were dissolved in chloroform with a
weight ratio of 6. The mixture was stirred overnight at room temperature at
500 rpm using a magneton bar. The well-dissolved PC film was mounted on a PDMS
dome on a glass slide. First, the top hBN layer with a thickness of 64 nm was
picked up with the stamp, followed by the CrSBr flake, few-layer graphene
contact layer, and 84 nm bottom hBN. The stack was released at a temperature
of 190°C onto a pre-patterned SiO2/Si target substrate with Ti/Au metal pads,
then soaked in chloroform solution for 2 min to remove PC residues and cleaned
by acetone and isopropanol. Next, the top gate few-layer graphene flake was
picked up and placed on top of the heterostack, followed by another cleaning
step. The pick-up temperature for CrSBr was around 110°C, for the other flakes
around 100°C. The sample was annealed at 200°C under ultrahigh vacuum for 15
h. Then, Ti/Au contact stripes were fabricated to connect the few-layer
graphene gates to the contact pads using laser lithography and electron-beam
evaporation. The second sample was fabricated in the same way but using hBN
flakes with 28.5 and 26 nm for the top and bottom encapsulating layers,
respectively. Electrostatic doping was controlled by applying a gate voltage
with a programmable DC-source (Yokogawa7651) between the gate and the grounded
contact layer.

Magneto-optical spectroscopy:
Cryogenic PL and DR spectroscopy were performed in back-scattering geometry
with a lab-built confocal microscope in a close-cycle cryostat (attocube
systems, attoDRY1000) with a base temperature of $3.2$ K and a solenoid with
magnetic fields of up to $\pm 9$ T. Magnetic field sweeps along the $b$-axis
in upward (downward) direction were performed by initializing the magnet at
$-500$ mT (500 mT) and then ramping to the target field. We estimate the
magnetic field inaccuracy to 2 mT for sweeps in steps of 2 mT, as deduced from
repeated measurements under nominally identical conditions. Measurements with
magnetic field along the $c$-axis were conducted by positioning the sample
with respect to a low-temperature apochromatic objective (attocube systems,
LT-APO/NIR/0.81) with piezo-units (attocube systems, ANPx101, ANPz101, and
ANSxy100). For measurements along the $b$ and $a$ axes, a custom-built Voigt
adapter was used, consisting of a mirror mounted at 45° and an aspheric lens
(Geltech 350330) glued onto a Ti part. The sample holder was mounted on an
L-shaped adapter with the crystallographic $b$ or $a$ axis of the sample
aligned with the axis of the solenoid. The L-shaped adapter was mounted on
piezo-units (attocube systems, ANPx101, ANPz101, and ANSxyz100) for
nanopositioning and scanning. Momentum-space imaging in 4f and telescope
configuration employed four achromatic doublet lenses (Edmund Optics, VIS-NIR)
and was performed in an attoDRY800 close-cycle cryostat with 4 K base
temperature.

In experiments on sample 1, PL was excited at 870 nm and $100$ $\mu$W with a
Ti:sapphire laser (Coherent, Mira) in continuous-wave mode and Semrock tunable
short-pass and long-pass filters were used (887 nm VersaChrome Edge) in
excitation and detection. DR, defined as $DR=(R-R_{0})/R_{0}$, where $R$ was
the reflectance from the sample and $R_{0}$ was the reference reflectance on
the nearby substrate with hBN, was recorded with a Tungsten-Halogen lamp
(Thorlabs, SLS201L or Ocean Insight, HL-2000-HP). The signal was dispersed by
a monochromator (Roper Scientific, Acton SpectraPro 300i or Acton SP250 or
Teledyne Princeton Instruments, IsoPlane SCT320) with a $300$ grooves/mm
grating and detected by a Peltier-cooled (Andor, iDus 416 or Teledyne
Princeton Instruments, PIXIS 1024) or liquid nitrogen-cooled (Spec-10:100BR)
charge-coupled device. A set of linear polarizers (Thorlabs, LPVIS), half- or
quarter-waveplates (B. Halle, $310-1100$ nm achromatic) mounted on piezo-
rotators (attocube systems, ANR240) were used to control the polarization in
excitation and detection. For sample 2, excitation and detection were
circularly polarized, and PL was excited at 800 nm and $100$ $\mu$W with a
Ti:sapphire laser (SolsTiS, M Squared) with 842 nm short-pass (Semrock 842/SP
BrightLine HC Short-pass Filter) and 808 nm long-pass (Semrock LP Edge Basic
Long-pass Filter) filters in excitation and detection.

Fermi-polaron model:
We develop the Fermi-polaron model for multilayer CrSBr based on the approach
used previously for transition metal dichalcogenide monolayers [35, 36, 37],
taking into account the spin-polarized band structure of CrSBr. We recall that
both monolayer and multilayer CrSBr are described by a centrosymmetric
$D_{2h}$ point symmetry group. According to Ref. [19], in the set of axes with
$z\parallel c$ (normal to the monolayer, magnetic hard axis), $y\parallel b$
(magnetic easy axis), and $x\parallel a$ (magnetic intermediate axis) the
orbital Bloch function of the topmost valence band $v$ (we account for a
single valence band $v$ owing to its significant separation from the lower-
lying bands) in the monolayer is transformed according to the $B_{3g}$ (or
$\Gamma_{4}^{+}$, i.e., as $yz$) irreducible representation, and for two
nearest conduction bands $c_{1}$ and $c_{2}$ according to $A_{g}$
($\Gamma_{1}^{+}$ as $x^{2}+y^{2}+z^{2}$) and $B_{1u}$ ($\Gamma_{3}^{-}$ as
$z$), respectively. Intralayer ferromagnetic spin-spin interactions result in
the complete spin polarization of the Bloch states with the spins aligned
along the $b$ ($y$) axis in the monolayer. In multilayers, spins are aligned
antiferromagnetically along the positive and negative directions of the
$b$-axis in neighboring layers. Optical transition $v\leftrightarrow c_{1}$ is
forbidden in the dipole approximation and the transition $v\leftrightarrow
c_{2}$ is allowed for light polarized along the easy axis $b\parallel y$. Note
that the predicted order of $c_{1}$ and $c_{2}$ bands varies in the
literature, cf. Refs. [4, 19, 46], due to the small $c_{1}-c_{2}$ energy
splitting and, correspondingly, its strong dependence on the ab initio and
model parameters.

The lowest-energy optical transition in the monolayer, labeled as
X${}_{\text{A}}$ is linearly polarized along the $b\parallel y$-axis, broad in
absorption and faint in PL, without a sizable Stokes shift. The latter feature
indicates a momentum-direct exciton transition, which, however, should be
nominally forbidden by dipolar selection rules. We speculate that due to the
asymmetry of the structure (inequivalence of $z\to-z$) enhanced by the large
surface-to-volume ratio in the monolayer, the dipolar selection rules are
compromised and the $v\leftrightarrow c_{1}$ transition becomes weakly
allowed. X${}_{\text{A}}$ also prevails in the spectra of the bi- and
trilayer, where its features in absorption and PL are contrasted by the much
more pronounced and spectrally narrow resonances of X${}_{\text{B}}$, which we
assign to the dipole-allowed $v\leftrightarrow c_{2}$ transition with linear
polarization along the $b\parallel y$-axis.

With this understanding, we consider the manifold of relevant neutral and
negatively charged exciton states as in the Extended Data Fig. 2. We assume
that the difference of the binding energies of the intra- and interlayer
excitons exceeds by far the $c_{1}-c_{2}$ conduction band splitting. We also
neglect bound charge complexes of interlayer excitons (negative interlayer
trions) due to much smaller binding energies [40]. Furthermore, we neglect
interlayer electron and hole tunneling at zero magnetic field where single-
particle tunneling between the same bands is spin-forbidden, while at finite
fields the tunneling $c_{1}\leftrightarrow c_{2}$ is suppressed by the
conduction band splitting. At finite magnetic field along the $c$-axis, the
spins in the neighboring layers are canted, and hole tunneling between the
layers becomes allowed [4].

In this framework, the bilayer system Hamiltonian breaks into two identical
blocks describing electrons either in the top or in the bottom layer:

$H_{\text{BL}}=\begin{pmatrix}H&0\\\ 0&H\end{pmatrix}.$ (1)

Each block accounts for both $vc_{1}$ intra- and interlayer (A) excitons and
$vc_{2}$ intra- and interlayer (B) excitons, as well as their coupling with
the corresponding Fermi-sea in the bottom conduction subband $c_{1}$:

$H=\begin{pmatrix}E_{\text{A}}&sB&0&0&0\\\ sB&E_{\text{IA}}&0&0&0\\\
0&0&E_{\text{B}}&sB&\sqrt{E_{\text{tr}}E_{\text{F}}}\\\
0&0&sB&E_{\text{IB}}&0\\\
0&0&\sqrt{E_{\text{tr}}E_{\text{F}}}&0&E_{\text{B}}-E_{\text{tr}}\end{pmatrix},$
(2)

where $E_{\text{A(B)}}$ are the bare energies of A(B)-excitons (in absence of
tunneling and electron doping); $E_{\text{IA(IB)}}=E_{\text{A(B)}}+\Delta$ are
the energies of the interlayer A(B)-excitons with the difference in the
binding energies $\Delta$. The coefficient $s$ quantifies the magneto-induced
interlayer hole tunneling; $E_{\text{F}}$ is the Fermi level and
$E_{\text{tr}}$ is the trion binding energy. The Hamiltonian in Eq. (2)
corresponds to the non-self-consistent approximation for the exciton-electron
scattering matrix element, and we set the coupling parameter to be
$\sqrt{E_{\text{tr}}E_{\text{F}}}$, omitting a numerical coefficient $\sim 1$
as well as the Fermi-sea effect on the trion binding energy [47].

In the weak doping regime ($E_{\text{F}}\ll E_{\text{tr}}$) and moderate
magnetic fields ($|sB|\ll\Delta$), we obtain the energies of the A-excitons
(decoupled from the Fermi-sea) and Fermi-polarons formed by B-excitons in the
form:

$\mathcal{E}_{\text{A}}(B)\approx E_{\text{A}}-tB^{2},$ (3a) $\mathcal{E}^{\rm
RP}_{\text{B}}(B,E_{\text{F}})\approx E_{\text{B}}-tB^{2}+E_{F},$ (3b)
$\mathcal{E}^{\rm AP}_{\text{B}}(B,E_{\text{F}})\approx
E_{\text{B}}-E_{\text{tr}}-E_{\text{F}}-\frac{E_{\text{F}}\Delta}{E_{\text{tr}}(\Delta+E_{\text{tr}})}tB^{2},$
(3c)

with $t=s^{2}/\Delta$. In this regime, the A-exciton and B-exciton (repulsive
polaron, RP) have the same magneto-dispersion, while the attractive Fermi-
polaron (AP) has a smaller dispersion in magnetic field due to the additional
term $E_{\text{F}}/E_{\text{tr}}\ll 1$. In the main text we use the notation
$\mathcal{E}_{\text{B}}(B)=\mathcal{E}^{\rm RP}_{\text{B}}(B,0)$.

Due to a sizable spread in the calculated conduction band splitting
$c_{1}$–$c_{2}$ [4, 19], we take $E_{\text{A,B}}$ as two fitting parameters.
By fitting the experimental data to the eigenvalues of the Hamiltonian in Eq.
(2), we obtain $E_{\text{A}}=1344$ meV, $E_{\text{B}}=1370$ meV, $t=2.2$
meV/T2, $B_{\text{sat}}=1.9$ T, $E_{\text{tr}}=19$ meV, and $E_{\text{F}}=10$
meV. In Figs. 2e, f, we show the eigenenergies of X${}_{\text{A}}$,
X${}_{\text{B}}$, and X${}_{\text{B}}^{-}$ in the neutral and negatively
charged regime, respectively.

In the trilayer, we also assume suppressed interlayer tunneling of electrons
as well as the exciton as a whole quasiparticle. The relevant states are shown
in Extended Data Fig. 2c, and the Hamiltonian is written as:

$H_{\text{TL}}=\begin{pmatrix}H&0&0\\\ 0&H^{\prime}&0\\\ 0&0&H\end{pmatrix},$
(4)

where the blocks of the states related to the top pair of layers and the
bottom pair of layers are identical to the bilayer Hamiltonian in Eq. (2),
while the Hamiltonian $H^{\prime}$ is different, describing electrons in the
middle layer.

We assume that the states in the middle layer have a different energy due to a
different dielectric environment as compared to the outer layers, and label
such excitons with a prime (see Extended Data Fig. 2c). Based on experimental
observations (Figs. 1h and 2c) that (i) X${}_{\text{B}}^{\prime}$ has no
charge-bound state upon doping, and (ii) no clear signature of
X${}_{\text{A}}^{\prime}$ is observed, we disregard these states in our model
(Extended Data Fig. 2c). Consequently, the middle block $H^{\prime}$ in the
Hamiltonian of Eq. (4) reduces to:

$H^{\prime}=\begin{pmatrix}E_{\text{B}}^{\prime}&sB&sB\\\
sB&E_{\text{IB}}^{\prime}&0\\\ sB&0&E_{\text{IB}}^{\prime}\end{pmatrix},$ (5)

where $E_{\text{IB}}^{\prime}=E_{\text{B}}^{\prime}+\Delta$ is the energy of
the interlayer B-excitons in the middle layer (note the multiplicity of two
for such states).

Thus, the spectrum of the trilayer is a superposition of the bilayer spectrum
(Eqs. (3)) and the spectrum of the middle layer that for $|sB|\ll\Delta$
approximates to:

$\mathcal{E}_{\text{B}}^{\prime}(B)\approx E_{\text{B}}^{\prime}-2tB^{2}.$ (6)

We note that the curvature of its dispersion in magnetic field is twice that
of the bilayer excitons described by Eq. (3b), as the two interlayer excitons
couple with the intralayer exciton. By extracting the splitting at $0$ T (AF
state) in the neutral regime (as in Figs. 1h and 2c) we determine
$E_{\text{B}}^{\prime}=1365$ meV. The eigenenergies of X${}_{\text{A}}$,
X${}_{\text{B}}$, X${}_{\text{B}}^{\prime}$, and X${}_{\text{B}}^{-}$ in the
neutral and negatively charged regimes are shown in Figs. 2g,h, respectively.

Estimation of the spin lattice energy:
To understand the spin-flip transitions in magnetic field along the $b$-axis
as in Figs. 3a, g, we estimate the spin lattice energy by the following
Hamiltonians for bi- and trilayers [42]:

$\displaystyle H^{(s)}_{\text{BL}}$ $\displaystyle=Js_{1}s_{2}+\mu
B_{b}(s_{1}+s_{2}),$ (7) $\displaystyle H^{(s)}_{\text{TL}}$
$\displaystyle=J(s_{1}s_{2}+s_{2}s_{3})+\mu B_{b}(s_{1}+s_{2}+s_{3}),$ (8)

where $B_{b}$ is the magnetic field along the $b$-axis, $s_{1,2,3}=\pm 1$ are
the spin orientations along the $b$-axis for the respective layers, $J$ is the
positive interlayer exchange between neighboring layers, and $\mu$ is the
magnetic dipole moment of one layer. We assume that each layer exhibits
intralayer FM order and neglect the intralayer exchange energies in the
Hamiltonians of Eq. (7) and (8). The calculations are shown in Figs. 3d and j
for bi- and trilayer, respectively.

Acknowledgements: We thank J. Förste for help with the Python control of the
setup. This research was funded by the European Research Council (ERC) under
the Grant Agreement No. 772195 and the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) within Germany’s Excellence Strategy under grant
No. EXC-2111-390814868. F. T.-V. acknowledges funding from the Munich Center
for Quantum Science and Technology (MCQST) and the European Union’s Framework
Programme for Research and Innovation Horizon Europe under the Marie
Skłodowska-Curie Actions grant agreement No. 101058981. A. R. acknowledges
funding by the Bavarian Hightech Agenda within the Munich Quantum Valley
doctoral fellowship program. A. P. was supported by the Onassis foundation -
Scholarship ID: F ZS 045-1/2022-2023 and Bodossaki foundation Scholarship. K.
W. and T. T. acknowledge support from JSPS KAKENHI (grant No. 19H05790,
20H00354 and 21H05233). P. M. acknowledges financial support by the ERC
consolidator grant project QS2DM and by SNF project No. 188521. Theoretical
analysis by M. M. G. was supported by RSF project 23-12-00142. Z. S. was
supported by ERC-CZ program (project LL2101) from Ministry of Education Youth
and Sports (MEYS). A. S. B. acknowledges funding by the European Union’s
Framework Programme for Research and Innovation Horizon 2020 under the Marie
Skłodowska-Curie grant agreement No. 754388 (LMUResearchFellows) and from
LMUexcellent, funded by the Federal Ministry of Education and Research (BMBF)
and the Free State of Bavaria under the Excellence Strategy of the German
Federal Government and the Länder.
Contributions: K. M., A. P. and Z. S. synthesized CrSBr crystals, K. W. and T.
T. provided hBN crystals. H. P. G. N. fabricated the field-effect devices. F.
T.-V., H. P. G. N. and A. R. performed the experiments. A. S. B. developed the
theoretical model with support from M. M. G. F. T.-V., H. P. G. N., A. R., P.
M., M. M. G., A. S. B. and A. H. analyzed and discussed the data. F. T.-V., M.
M. G., A. S. B and A. H. wrote the manuscript. All authors commented on the
manuscript.
Corresponding authors:
F. T.-V. (f.tabataba@lmu.de), A. S. B. (anvar.baimuratov@lmu.de), and A. H.
(alexander.hoegele@lmu.de).
Data availability: The data that support the findings of this study are
available from the corresponding authors upon reasonable request.
Code availability: The codes that support the findings of this study are
available from the corresponding authors upon reasonable request.
Competing interests: The authors declare no competing interests.

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Extended Data Fig. 1: Differential reflectance, absorption, and
photoluminescence spectra. a, Monolayer DR (top), absorption (middle), and PL
spectra (bottom) at 0 and 30 V. b, and c, Same as a but for the bilayer and
trilayer, respectively. X${}_{\text{A}}$, X${}_{\text{B}}$, and
X${}_{\text{B}}^{\prime}$ label neutral and X${}_{\text{B}}^{-}$ charged
exciton transitions. Extended Data Fig. 2: Neutral and charged excitons
considered in mono-, bi- and trilayer. a – c, Intra- and interlayer excitons
and intralayer, interband charge-bound states in mono- (a), bi- (b), and
trilayer (c). The dashed lines in b and c separate the different blocks in the
interaction Hamiltonians in Eqs. (1), (2), (4), and (5) for electrons in
layers L1, L2, and L3, respectively. Electrons (holes) are shown in blue
(red), $c_{1}$ and $c_{2}$ refer to the electron conduction subbands.
X${}_{\text{A}}$ (X${}_{\text{B}})$ are intralayer excitons with the electron
in the $c_{1}$ ($c_{2}$) subband, and X${}_{\text{IA}}$ and X${}_{\text{IB}}$
are respective interlayer excitons. Prime (′) denotes states of L2 in the
trilayer with a different dielectric environment and without bound states with
electrons in the $c_{1}$ band. Extended Data Fig. 3: Momentum-space and linear
polarization data. a – c, Momentum-space PL dispersions at 0 V of mono- (a),
bi- (b), and trilayer (c). d – f, PL spectra of mono- (d), bi- (e), and
trilayer (f) under linearly polarized excitation along the $b$-axis with co-
and cross-polarized detection. Insets in e and f show the integrated PL
intensity as a function of detection polarization angle together with a
$\text{sin}^{2}$ function. Extended Data Fig. 4: Magneto-dispersions along the
hard $c$-axis and doping-dependence for sample 2. a, Optical micrograph of the
CrSBr flake of sample 2, showing mono-, bi-, and trilayer regions, as well as
the crystallographic axes. b, and c, Energies of X${}_{\text{B}}$,
X${}_{\text{B}}^{\prime}$, and X${}_{\text{B}}^{-}$ as a function of magnetic
field along the $c$-axis ($B_{c}$) extracted from 0 and 16 V differential
reflectance (DR) magneto-dispersions for bi-, and trilayer, respectively. Spin
orientations are indicated in the top panels. d, and e, Monolayer absorption
and photoluminescence (PL) spectra (top) and voltage sweeps (bottom),
respectively. f – i, Same as d, e, but for the bilayer (f, g) and trilayer (h,
i). X${}_{\text{A}}$, X${}_{\text{B}}$, and X${}_{\text{B}}^{\prime}$ label
neutral and X${}_{\text{B}}^{-}$ charged exciton transitions. The absorption
was determined by applying Kramers-Kronig relation to the differential
reflectance. PL and DR for the monolayer were not measured at the same
positions on the sample.  Extended Data Fig. 5: Bilayer magnetic field
dependence along $c$-, $b$-, and $a$-axes. a – f, Differential reflectance (a,
c, e) and photoluminescence (b, d, f) magneto-dispersions with magnetic fields
along the hard $c$ (a, b), easy $b$ (c, d), and intermediate $a$ axes (e, f)
at 0 V. X${}_{\text{A}}$, and X${}_{\text{B}}$ label neutral exciton
transitions. Top panels in a, c, and e indicate the bilayer spin orientation.
Extended Data Fig. 6: Trilayer magnetic field dependence along $c$-, $b$-, and
$a$-axes. a – f, Differential reflectance (a, c, e) and photoluminescence (b,
d, f) magneto-dispersions with magnetic fields along the hard $c$ (a, b), easy
$b$ (c, d), and intermediate $a$ axes (e, f) at 0 V. X${}_{\text{A}}$,
X${}_{\text{B}}$, and X${}_{\text{B}}^{\prime}$ label neutral exciton
transitions. Top panels in a, c, and e indicate the trilayer spin orientation.
Extended Data Fig. 7: Large-area magnetic domain imaging. a – l, Spatial maps
of a large area of the sample showing the energies of X${}_{\text{B}}$ and
X${}_{\text{B}}^{\prime}$ in bi- (BL) and trilayer (TL) regions, respectively,
for different $b$-axis magnetic fields ranging from 0 mT (a) to 350 mT (l),
extracted from hyperspectral differential reflectance maps. The color bar
indicates the energies of the different spin orientations. Outlines of mono-,
bi- and trilayer are marked with black dashed lines and the few-layer graphene
contact is indicated by green dotted lines. White areas indicate regions away
from bi- and trilayer. Grey areas signal the absence of excitonic resonances.