# Covalency, correlations, and inter-layer interactions governing the magnetic
and electronic structure of Mn3Si2Te6

Chiara Bigi  SUPA, School of Physics and Astronomy, University of St Andrews,
St Andrews KY16 9SS, UK  Lei Qiao  Physics Department, International Center of
Quantum and Molecular Structures, Materials Genome Institute, State Key
Laboratory of Advanced Special Steel, Shanghai Key Laboratory of High
Temperature Superconductors, Shanghai University, Shanghai 200444, China
Consiglio Nazionale delle Ricerche (CNR-SPIN), Unità di Ricerca presso Terzi
c/o Università “G. D’Annunzio”, 66100 Chieti, Italy  Chao Liu  Physics
Department, International Center of Quantum and Molecular Structures,
Materials Genome Institute, State Key Laboratory of Advanced Special Steel,
Shanghai Key Laboratory of High Temperature Superconductors, Shanghai
University, Shanghai 200444, China  Consiglio Nazionale delle Ricerche (CNR-
SPIN), Unità di Ricerca presso Terzi c/o Università “G. D’Annunzio”, 66100
Chieti, Italy  Paolo Barone  Consiglio Nazionale delle Ricerche CNR-SPIN, Area
della Ricerca di Tor Vergata, Via del Fosso del Cavaliere 100, I-00133 Rome,
Italy  Monica Ciomaga Hatnean  Current address: Paul Scherrer Institut,
Forschungsstrasse 111, 5232 Villigen PSI, Switzerland  Department of Physics,
University of Warwick, Coventry, CV4 7AL, United Kingdom  Gesa-R. Siemann
SUPA, School of Physics and Astronomy, University of St Andrews, St Andrews
KY16 9SS, UK  Barat Achinuq  Department of Physics, Clarendon Laboratory,
University of Oxford, Oxford, OX1 3PU, United Kingdom  Daniel Alexander Mayoh
Department of Physics, University of Warwick, Coventry, CV4 7AL, United
Kingdom  Giovanni Vinai  Istituto Officina dei Materiali (IOM)-CNR,
Laboratorio TASC, Area Science Park, S.S.14, Km 163.5, 34149 Trieste, Italy
Vincent Polewczyk  Istituto Officina dei Materiali (IOM)-CNR, Laboratorio
TASC, Area Science Park, S.S.14, Km 163.5, 34149 Trieste, Italy  Deepak Dagur
Istituto Officina dei Materiali (IOM)-CNR, Laboratorio TASC, Area Science
Park, S.S.14, Km 163.5, 34149 Trieste, Italy  Federico Mazzola  Istituto
Officina dei Materiali (IOM)-CNR, Laboratorio TASC, Area Science Park, S.S.14,
Km 163.5, 34149 Trieste, Italy  Peter Bencok  Diamond Light Source, Harwell
Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom  Thorsten
Hesjedal  Department of Physics, Clarendon Laboratory, University of Oxford,
Oxford, OX1 3PU, United Kingdom  Gerrit van der Laan  Diamond Light Source,
Harwell Science and Innovation Campus, Didcot, OX11 0DE, United Kingdom  Wei
Ren  Physics Department, International Center of Quantum and Molecular
Structures, Materials Genome Institute, State Key Laboratory of Advanced
Special Steel, Shanghai Key Laboratory of High Temperature Superconductors,
Shanghai University, Shanghai 200444, China  Geetha Balakrishnan  Department
of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom  Silvia
Picozzi<EMAIL_ADDRESS>Consiglio Nazionale delle Ricerche (CNR-
SPIN), Unità di Ricerca presso Terzi c/o Università “G. D’Annunzio”, 66100
Chieti, Italy  Phil D. C. King<EMAIL_ADDRESS>SUPA, School of Physics
and Astronomy, University of St Andrews, St Andrews KY16 9SS, UK

###### Abstract

Mn3Si2Te6 is a rare example of a layered ferrimagnet. It has recently been
shown to host a colossal angular magnetoresistance as the spin orientation is
rotated from the in- to out-of-plane direction, proposed to be underpinned by
a topological nodal-line degeneracy in its electronic structure. Nonetheless,
the origins of its ferrimagnetic structure remain controversial, while its
experimental electronic structure, and the role of correlations in shaping
this, are little explored to date. Here, we combine x-ray and photoemission-
based spectroscopies with first-principles calculations, to probe the
elemental-selective electronic structure and magnetic order in Mn3Si2Te6.
Through these, we identify a marked Mn-Te hybridisation, which weakens the
electronic correlations and enhances the magnetic anisotropy. We demonstrate
how this strengthens the magnetic frustration in Mn3Si2Te6, which is key to
stabilising its ferrimagnetic order, and find a crucial role of both exchange
interactions extending beyond nearest-neighbours and anti-symmetric exchange
in dictating its ordering temperature. Together, our results demonstrate a
powerful methodology of using experimental electronic structure probes to
constrain the parameter space for first-principles calculations of magnetic
materials, and through this approach, reveal a pivotal role played by
covalency in stabilising the ferrimagnetic order in Mn3Si2Te6.

##  I Introduction

While three-dimensional magnets are common-place, long range order is strictly
forbidden to occur in one-dimensional systems [1]. Layered magnetic materials
present a novel environment in which to study the critical dimensionality
between these two extremes, with finite inter-plane coupling expected to have
a strong influence on the development of long-range order, magnetic
anisotropy, and the role of fluctuations [2]. In this respect, Mn3Si2Te6 (MST)
is a particularly intriguing compound. It forms in the $P\overline{3}1c$ space
group (No. 163, Fig. 1(a,b)) [3, 4], containing two inequivalent Mn sites. The
Mn(1) atoms sit at the centre of edge-sharing MnTe6 octahedra. The resulting
Mn atoms would form a triangular lattice. However, $1/3$ of the sites are
occupied by a Si-Si dimer, leaving the Mn(1) atoms in a honeycomb
configuration (Fig. 1(b)), akin to the Cr sites of the layered van der Waals
magnet Cr2Ge2Te6 (CGT) [5, 6, 7]. Unlike CGT, however, extra magnetic ions
(Mn(2)) are situated between the Mn(1) layers, filling the van der Waals gap.
These self-intercalated Mn(2) sites form a triangular lattice, with one Mn(2)
atop every second Mn(1) site, providing a bridging link between the Mn(1)
layers and establishing a more three-dimensional structure. Nevertheless, the
samples cleave easily with a standard top-post method, exposing a flat and
uniform surface.

The Mn ions have been reported to develop a long-range magnetic order below
$\sim 75$ K [8], with the Mn moments aligned ferromagnetically within each Mn
layer, but with antiferromagnetic coupling between neighbouring layers (Fig.
1(a)) [9]. This leads to an overall ferrimagnetic structure [8, 10]. This
speaks to a critical role of the bridging Mn(2) sites, leading to markedly
different interlayer interaction as compared to ferromagnetic CGT [9, 11].
Indeed, the exchange interactions for the first three nearest-neighbours,
$J_{1}$, $J_{2}$ and $J_{3}$ (Fig. 1(b)) are known to be antiferromagnetic
[9]. This creates competing interactions which, combined with the geometrical
arrangement of the Mn-atoms in the MST lattice (side-view of Fig. 1(b)), has
been proposed to result in a high degree of magnetic frustration in this
system.

It remains poorly understood, however, how the overall ferrimagnetic ground
state of MST is stabilised. Moreover, while the Mn ions would nominally be
expected in a ${2+}$ valence state with a quenched orbital moment, this has
recently been questioned, as a huge magnetic anisotropy (up to $13$ T) has
been reported [12, 13, 14]. The intra- and inter-layer magnetic couplings in
MST thus require further exploration, while the underlying electronic
structure from which the magnetic order emerges, and the degree to which
ligand hybridisation vs. electronic correlation effects dominate the magnetic
coupling, remain almost completely unexplored to date.

To address this, here we present a combined theoretical and experimental study
into the electronic structure and magnetic interactions of MST. Our results
point to a significant Mn-Te orbital hybridisation. We show how this promotes
additional exchange coupling terms, inclusion of which is essential to obtain
an accurate description of the magnetic anisotropies and ordering temperature
in MST. Together, our results thus provide key insights on the interplay and
influence of dimensionality, anisotropy, covalency, and correlations on the
magnetic fluctuations of quasi-layered magnetic materials.

##  II Methods

###  II.1 DFT

We performed density functional theory (DFT) simulations, using the Vienna ab
initio Simulation Package (VASP) [15, 16]. The generalized gradient
approximation (GGA) based on the Perdew-Burke-Ernzerhof (PBE) functional [17]
was employed to treat the exchange-correlation interaction. We considered the
localized 3$d$ electron correlation of Mn atoms by using the GGA+$U$ method
[18, 19], with an effective Hubbard $U$ parameter chosen to be $1$ eV, and
other effective $U$ choices of $0$, $2$, and $3$ eV also tested and compared.
The projector-augmented-wave (PAW) potentials [20, 21] were used to describe
the electron-ion interaction. The energy cutoff was selected to be $550$ eV
for the plane-wave basis set. The Brillouin zones were sampled using a 6 × 6 ×
4 k-grid mesh in the $\Gamma$-centered scheme. The forces convergence
threshold on each atom was chosen as $0.005$ eV/Å and the self-consistent
calculations were stopped when the energy difference was smaller than
$10^{-7}$ eV per atom. We optimized ionic positions with ground state
ferrimagnetic magnetic configuration, and experimental lattice constants were
used and fixed in all calculations.

###  II.2 Monte Carlo Calculations

A standard Metropolis algorithm has been used for Monte Carlo (MC)
simulations, with 105 MC steps for equilibration and 5$\times 10^{5}$ MC steps
for averaging. Starting from the crystallographic unit cell, comprising four
Mn(1) and two Mn(2) sites, we performed calculations on a
16$\times$16$\times$8 supercell with $N_{s}=12288$ spins. The transition
temperature can be estimated from the peak appearing in the temperature
evolution of the specific heat per spin, evaluated as

$\displaystyle C_{v}$ $\displaystyle=$
$\displaystyle\frac{k_{\mathrm{B}}\beta^{2}}{N_{s}}\left[\langle
E^{2}\rangle-\langle E\rangle^{2}\right]$ (1)

where $E$ is the energy calculated using model Eq. (4), $k_{B}$ is the
Boltzmann constant and $\beta=1/k_{\mathrm{B}}T$, while $\langle...\rangle$
indicates statistical averages. We also define the ferromagnetic $\bm{F}$ and
ferrimagnetic $\bm{f}$ order parameters as:

$\displaystyle{\bm{F}}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(\sum_{i=1,4}\bm{S}_{i}+\sum_{i=1,2}\bm{s}_{i}\right)$
$\displaystyle{\bm{f}}$ $\displaystyle=$
$\displaystyle\frac{1}{6}\left(\sum_{i=1,4}\bm{S}_{i}-\sum_{i=1,2}\bm{s}_{i}\right)$
(2)

where $\bm{S}_{i}$ and $\bm{s}_{i}$ label here the spins on Mn(1) and Mn(2)
sites, respectively, within the unit cell, with associated (generalized)
susceptibilities for the magnetic order parameters $OP=F,f$ calculated as:

$\displaystyle\chi^{OP}$ $\displaystyle=$ $\displaystyle\beta
N_{s}\left[\langle OP^{2}\rangle-\langle OP\rangle^{2}\right]$ (3)

Results of Monte Carlo calculations are summarized in Fig. 5. We used four
sets of parameters in our simulations: two sets correspond to the full model
parametrization summarized in Tab. 1 and Tab. 2, while two sets consist in
simplified model with isotropic
$J_{1}^{\mathrm{iso}},(J_{2}^{\mathrm{iso}}+J_{5}^{\mathrm{iso}}),J_{3}^{\mathrm{iso}}$
exchange interactions and single-ion anisotropy terms. In all cases, we find a
transition to an ordered ferrimagnetic phase characterized by ferromagnetic
layers of Mn(1) and Mn(2) spins antiferromagnetically aligned and lying in the
basal plane. The critical temperature is inversely proportional to the value
of U, getting lower as the Hubbard parameter is increased, as expected. On the
other hand, the full model parametrization entails a stronger magnetic
frustration arising from longer-range antiferromagnetic interactions within
the honeycomb layers, that reduces the transition temperature by about 20%
with respect to the simplified model. The range of the estimated critical
temperatures settles between 89 and 118 K for $U$ between 2 and 1,
respectively.

###  II.3 Sample growth and characterisation

Mn3Si2Te6 single-crystals were grown by the chemical vapour transport method
using Iodine as a transport agent, as described in Ref. [4]. X-ray Diffraction
(XRD) reported in Supplementary Fig. S1 show only $(00l)$ Bragg peaks of the
expected MST structure, indicating that the facets of the obtained crystals
(see inset in Fig. S1) are parallel to the $ab$-plane. The cell parameters
extracted from fitting the XRD data are $a=b=7.07700(6)$ Åand $c=14.25081(2)$
Å. A Quantum Design Magnetic Property Measurement System was used to measure
the magnetization between 1.8 and 300 K in applied fields up to 5 T. A Quantum
Design Physical Property Measurement System was used to measure electrical
resistivity between 20 and 300 K.

###  II.4 X-ray and UV spectroscopies

For our spectroscopic measurements, samples were cleaved in ultra-high-vacuum
(UHV) using a top-post method. X-ray absorption spectroscopy (XAS) and x-ray
magnetic circular dichroism (XMCD) measurements were performed using the
electromagnet end station of the I10 beamline at Diamond Light Source, UK. The
pressure was better than $10^{-9}$ mbar, and measurements were performed
across the Mn $L_{2,3}$ absorption edge in total electron yield (TEY)
detection, thus probing $\sim\\!6$ nm depth from the sample surface. Spectra
were measured at $10$ K with left- (CL) and right-circularly (CR) polarized
x-rays in both normal- and grazing-incidence (i.e., at an angle of
$20^{\circ}$ from the $ab$-plane) geometries (see Supplementary Fig. S2). An
applied field of $1.4$ T collinear with the beam axis was applied to magnetize
the sample. The main error source on the XAS/XMCD quantitative data analysis
arises from the step-edge background subtraction. To estimate this, each
spectrum was analysed using different step-edge background choices (mainly
varying the energy range of interest as this was found to affect the
background shape the most) and we extracted the standard error deviation from
the set of obtained values. The $m_{s}$ sum rules have been corrected for Mn
$jj$-coupling [22] and the temperature dependent XMCD asymmetry has been
calculated from the dichroic signal at the $L_{3}$ absorption peak as

$A=\frac{I_{\mathrm{CR},L_{3}}-I_{\mathrm{CL},L_{3}}}{I_{\mathrm{CR},L_{3}}+I_{\mathrm{CL},L_{3}}}.$

The error on the asymmetry has been estimated from the XMCD residual value
obtained for several measurements performed well above the transition
temperature.

Resonant photoemission spectroscopy (resPES) measurements were performed at
the APE-HE beamline (Elettra synchrotron, Italy) at a base pressure lower than
$10^{-10}$ mbar, with the sample temperature kept at $\sim 107$ K. XAS across
the Mn $L_{2,3}$ edge was performed in linear horizontal polarization and in
TEY detection to determine the relevant energies for resPES. Measurements were
acquired by a Scienta Omicron R3000 hemispherical electron energy analyser.
The binding and photon energies were calibrated with the Fermi edge of a gold
reference sample, assuming a work function of $4.4$ eV. For better comparison
with the experiment, our P-DOS DFT calculations were convolved with a Voigt
function to emulate lifetime broadening of $0.01$ eV and an experimental
resolution $0.3$ eV.

The ARPES data were acquired at the APE-LE beamline (Elettra synchrotron) with
a Scienta DA30 hemispherical electron energy and momentum analyzer, with the
samples held at a temperature of $77$ K. The base pressure in the chamber was
better than $10^{-10}$ mbar. Energy and angular resolution were better than
$40$ meV and $20^{\circ}$, respectively.

##  III Results and Discussion

###  III.1 Magnetic order

Consistent with prior studies [9, 8, 23], our single-crystal samples (see
Supplementary Fig. S1) exhibit finite net magnetization onsetting below a
transition temperature ($T_{\mathrm{C}}$) of $\sim 79$ K (Fig. 1(d)). $M$ vs.
$H$ magnetization curves performed at $T=2$ K (Fig. 1(c)) indicate a large
magnetic anisotropy with a magnetic hard axis along the $c$ axis: the
magnetization fully saturates to $1.42\pm 0.02$ $\mu_{\mathrm{B}}$/Mn in the
$ab$-plane ($H\perp{}c$) for less than $2$ T magnetic field, while $5$ T is
insufficient to saturate the moment in the $H~{}\parallel~{}c$ configuration.
Consistent with this large anisotropy, Fig. 1(d) shows that the magnetic molar
susceptibility is more than a factor of two larger for the magnetic field
aligned perpendicular vs. parallel to the $c$-axis.

The simplest picture in which to describe this magnetic order is to consider
fully localised Mn spins, with Mn in a 2+ oxidation state, and hence nominal
$3d^{5}$ configuration [24, 23]. Meanwhile, Te2- would be expected to
contribute a filled $p^{6}$ valence band, leading to semiconducting transport
(Fig. 1(e), Refs. [9, 8, 12, 25]). Nonetheless, a pronounced kink in the
measured resistivity at the magnetic ordering temperature points to a non-
negligible coupling between the magnetism and charge carriers in the system,
beyond the simple picture outlined above. To investigate this further, we
measured x-ray spectroscopy at the Mn $L_{2,3}$ edge, as shown in Fig.2. The
x-ray absorption measurements (yellow curve in Fig. 2) shows a spectrum
broadly reminiscent of those observed in other nominally Mn2+ compounds [26,
27]. The branching ratio $L_{3}$/($L_{3}$+$L_{2}$) exceeds the expected $2/3$
statistical value (i.e., $0.768\pm 0.004$), consistent with a nominal Mn high-
spin state and with the observed semiconducting transport behavior [28, 29].
We stress, however, that a picture of fully localised Mn moments is an
oversimplification. Hints of this are already detectable in its XAS features,
where the characteristic multiplet structure is smeared out and less
pronounced than in typical ionic Mn2+ compounds [30]. This provides a first
spectroscopic indication that in fact there is a non-negligible hybridisation
of the Mn with the chalcogen ions, a point we return to below.

Our x-ray absorption data show a clear circular dichroism at low temperature,
confirming that the magnetism of MST primarily proceeds via ordering of the Mn
spins. The x-ray magnetic circular dichroism (XMCD) signal obtained at $10$ K
in a grazing incidence geometry (with a $1.4$ T field applied at $20^{\circ}$
to the $ab$-plane) is approximately twice as large as for a normal incidence
geometry where the field is applied along the $c$-axis, in good agreement with
the magnetic susceptibility measured by SQUID (Fig. 1.(d)). Temperature-
dependent XMCD spectra measured in remanence, after the sample was field-
cooled to $10$ K in $1.4$ T planar field (inset of Fig. 2), yield an XMCD
asymmetry which closely follows the bulk demagnetization (Fig. 1(d),
reproduced in green in the inset of Fig. 2). In good agreement with the bulk
$T_{\mathrm{C}}$, the XMCD asymmetry vanishes at $T\approx 75$ K becoming
comparable to our measurement error (see Fig. S2 in Supplementary Material for
the full temperature-dependent XMCD spectra).

Having established that our spectroscopic measurements probe bulk-like
magnetic properties in MST, we turn to the insights on the magnetic and
electronic structure which they advance. The main spectrum arises from the
multiplet structure of the Mn $d^{5}$ states that are split by the crystal-
field interaction. In addition, we observe a distinct pre-peak feature of
$(-,+)$ shape, marked by a black arrow in the bottom graph of Fig.2. Such a
pre-edge feature in other Mn-based compounds has been ascribed to transitions
from the Mn $2p$ core level to unoccupied $p$-$d$ hybridized valence states
[31], and thus indicates hybridization with the Te 5$p$ valence states here.
Indeed, the shape of our measured XMCD is similar to that found previously for
Ga1-xMnxAs, where the ground state of Mn is found as a hybridised state with
16% $d^{4}$, 58% $d^{5}$, and 26% $d^{6}$ character, yielding a $d$-count of
$\sim\\!5.1$ electrons/Mn atom.[27]. To consolidate these findings, we applied
quantitative sum-rule analysis to our XMCD data-set, to extract the atom-
specific orbital and spin magnetic moments ($m_{l}$ and $m_{s}$, respectively)
[32, 33, 22]. Our results indicate a smaller than expected ordered spin moment
of $m_{s}=0.546\pm 0.004~{}\mu_{\mathrm{B}}/\mathrm{Mn}$ \- likely due to a
finite out of plane component of the applied field for our experimental
geometry, thus not reaching saturation for our available fields. Nonetheless,
constitent with prior calculations [9], we find a small but positive orbital
moment of $m_{l}=0.022\pm 0.004~{}\mu_{\mathrm{B}}/\mathrm{Mn}$ (in agreememt
with the DFT value of 0.027 $\mu_{\mathrm{B}}$ per Mn), with
$m_{l}/m_{s}=0.040\pm 0.005$, further confirming the deviation from a purely
high-spin localised picture of the Mn atoms in this system. The positive sign
of this ratio points directly to a $d$ shell which is more than half filled,
entirely consistent with our above qualitative assessment of the spectral
lineshapes observed. Indeed, an orbital to spin ratio of 0.04 tallies with a
$d$-count of $\sim$5.1 electrons/Mn, with the small excess of electrons
compared to Mn2+ implying the presence of holes in the Te 5$p$ band.

###  III.2 Electronic structure

To address the role and extent of this implied covalency, we combine resonant
photoemission spectroscopy (resPES) measurements of the valence band
electronic structure with calculations from density-functional theory. Fig.
3(a) shows the valence band photoemission measured while scanning the photon
energy across the Mn $L_{3}$ absorption edge. The data show a prominent Mn-
derived state located at $\sim 3.6$ eV binding energy, whose photoemitted
intensity closely follows the $L_{3}$ XAS signal. This can thus be attributed
as a nominally localised Mn component, which carries the majority of the spin
moment in this system. Nonetheless, we find that there are increases in
spectral weight through the Mn resonance not just for this peak, but more
broadly for the entire valence band region. This implies that Mn becomes
hybridised throughout the valence bands, as is evident from the Mn partial
density of states (P-DOS) extracted as the difference of the PES measurements
performed on- and off- resonance shown in Fig. 3(b).

This directly points to a strong hybridisation between Mn and Te orbitals in
this system, which in turn can be expected to weaken the electronic
correlations in MST. Such correlations are expected to play a crucial role in
the exchange-coupling among the Mn spins [13, 9, 12, 34], and a quantification
of the extent of electronic correlation vs. covalency in MST is strongly
required. To this end, we compare the experimental Mn and Te P-DOS (the later
extracted from off-resonant measurements) with the results of DFT+$U$
calculations in Fig.3(b). While the Te P-DOS only weakly changes with
increasing Hubbard-$U$ parameter used in the calculation, the intense Mn-
derived peak shifts rapidly away from the Fermi level, while its shape also
becomes modified. The prominent peak in the on-resonance resPES data provides
a robust experimental feedback, allowing best matching the Hubbard-$U$
parameter for MST from comparison with our Mn P-DOS calculations. Consistency
between the experimental measurements and theoretical calculations is found
only for $U$ in the range of $1-2$ eV. This is substantially lower than
typical $U$-values of about $4$ eV found for Mn2+ oxides [35] and even other
chalcogenides such as MnSe2 [36] and MnTe2 [37], and lower than previously
utilised for calculations of MST [9, 25]. A deep physical interpretation of
the $U$ value within the DFT+$U$ approach is not straightforward, due to the
lack of formal correspondence between DFT+$U$ and the full many-body approach
characteristic of the Hubbard model. Nonetheless, the small value of the
Hubbard parameter here points to a significant reduction of correlations in
MST, due to pronounced ligand hybridisation.

In the occupied states, our calculations ($U=1$ eV, Fig. 4(a-d)) indicate a
set of flat bands visible at an energy of $\sim\\!3-4$ eV below the valence
band top. These match well the peak in the Mn P-DOS visible in our res-PES
measurements discussed above, as well as a corresponding non-dispersive
feature visible in our measured dispersions from angle-resolved photoemission
spectroscopy (ARPES, Fig. 4(e),(f)). Our orbitally-projected calculations
indicate that these derive dominantly from the Mn states. However, consistent
with our x-ray spectroscopic measurements, there is a strong hybridisation of
the Mn states throughout the valence region. Interestingly, the two
inequivalent Mn sites hybridise differently with the ligand valence states.
Mn(1) (Fig. 4(a)), which sits within the MnSiTe3 layer (Fig. 1(a)), has
significant weight throughout the entire valence band bandwidth, while Mn(2),
which sits in the interstitial sites between the layers, has a strongly
localised Mn peak, exhibiting less intermixing with the Te $p$-orbitals across
the rest of the valence band.

The Te states, meanwhile, contribute rather dispersive hole-like bands across
a broad bandwidth of $\sim 5$ eV (Fig. 4(c)). Our orbitally-resolved
calculations indicate that the valence band maximum (VBM) is mostly
contributed by in-plane Te $p_{xy}$-states and Mn(1), with wavefunctions thus
mainly localised within the honeycomb layer. On the other hand, Te $p_{z}$
states provide a channel for inter-layer hybridization between the Mn(1) and
Mn(2) ions. While many states are observed in our calculations due to the
large unit cell of MST, the photoemission spectral weight enhances two such
dispersive states (Fig. 4(e-g)), with the hybridized hole-like dispersion
appearing rather broad in the ARPES measurements. This is consistent with them
having a three-dimensional character, while the surface sensitivity intrinsic
to ARPES, combined with the large $c$-axis lattice constant (see supplementary
Fig.S1), leads to substantial $k_{z}$ broadening. This is also reflected in
the surface-projected DFT calculations shown in Fig. 4(h). Aside from the
surface-projected band features, broad background intensity is seen across the
whole valence band energy range (up to $-6$ eV), broadly consistent with our
experimental results.

Finally, we note that there is a strong demarcation in the contribution of the
Mn(1) and Mn(2) states to the conduction band electronic structure as can be
readily seen in Fig.4(a,b). The lowest energy states are contributed almost
entirely from the Mn(1) site, with some intermixing of the Te, while the Mn(2)
states are located approximately $0.70-0.85$ eV higher in energy. The
different chemical environments of Mn(1) and Mn(2) may again account for their
different contributions to the conduction bands, with the lowest conduction
band minimum found to be located almost entirely in the honeycomb layer.

###  III.3 Exchange interactions

The above comparison between DFT and our spectroscopic results strongly
constrains the relevant $U$ parameter to relatively small values in MST. This,
in turn, has a sizeable influence on the magnetic interactions and ordering
tendencies in this system. To explore this, we have developed a magnetic
model, where the exchange coupling constants have been estimated by mapping
the ab initio total energies onto a classical Heisenberg model, that can be
generally expressed as:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{ij}\bm{S}_{i}\cdot\mathcal{J}_{ij}\cdot\bm{S}_{j}+\sum_{i}\bm{S}_{i}\cdot\mathcal{A}_{i}\cdot\bm{S}_{i}$
(4)

Exchange interactions between classical spins at sites $i,j$ are described
here by the tensor $\mathcal{J}_{ij}$, including anisotropic effects, while
the second term accounts for the magnetic single-ion anisotropy (SIA). The
full exchange tensor can be decomposed into its isotropic part
$J^{\mathrm{iso}}_{ij}=\frac{1}{3}\mathrm{Tr}{\mathcal{J}}_{ij}$, an
antisymmetric term
${\mathcal{J}}^{A}_{ij}=\frac{1}{2}({\mathcal{J}}_{ij}-{\mathcal{J}^{T}}_{ij}$)
and a symmetric term
${\mathcal{J}}^{S}_{ij}=\frac{1}{2}({\mathcal{J}}_{ij}+{\mathcal{J}^{T}}_{ij})-J^{\mathrm{iso}}_{ij}\bm{I}$.

In order to account for the covalency in MST, we have considered up to fifth
nearest-neighbor isotropic exchange interactions in our model. Consistent with
previous studies, we label the first and second Mn(1)-Mn(2) nearest-neighbour
coupling as $J^{\mathrm{iso}}_{1}$ and $J^{\mathrm{iso}}_{3}$, while we
consider two additional in-plane Mn(1)-Mn(1) exchange interactions:
$J^{\mathrm{iso}}_{4},J^{\mathrm{iso}}_{5}$ (see Fig. 1(b)), as well as the
next nearest-neighbour $J^{\mathrm{iso}}_{2}$[9, 13]. The exchange parameters
have been extracted using the four-state energy mapping method[38, 39]: this
is a supercell approach that allows extracting the full exchange tensor
describing the coupling between a selected pair of magnetic sites at a given
distance, while the interaction with all other magnetic sites is canceled out
by a tailored choice of four magnetic configurations (details can be found in
the appendices of Ref. [39]). The four-state method also allows us to extract
the single-ion anisotropy, i.e., a site-dependent local quantity, instead of
simply the total magnetic anisotropy energy. This is crucial for MST,
comprising two inequivalent Mn sites that may in principle display different
SIA. The anisotropic part of the exchange tensor of nearest-neighbor
Mn(1)-Mn(2) ($\mathcal{J}_{1}$) and Mn(1)-Mn(1) ($\mathcal{J}_{2}$) were also
considered: noticeably, a non-negligible Dzyaloshinksii-Moriya interaction is
found between Mn(1)-Mn(2) magnetic moments, with the Dzyaloshinskii vector
parallel to the $c$-axis ($D_{z}$), while the Mn(1)-Mn(1) exchange tensor only
displays a symmetric anisotropic part. We emphasize that the inclusion of i)
the exchange coupling in tensorial form, ii) different single-ion anisotropies
for the two Mn atomic species, and iii) longer-ranged $J_{4}$ and $J_{5}$
exchange interactions, not considered in the literature, thereby improves our
description with respect to previous models.

In tables Tab. 1 and Tab. 2 we list all the magnetic parameters estimated for
$U=1$ eV and $U=2$ eV.

(meV) |  $J^{\mathrm{iso}}_{1}$ |  $J^{\mathrm{iso}}_{2}$ |  $J^{\mathrm{iso}}_{3}$ |  $J^{\mathrm{iso}}_{4}$ |  $J^{\mathrm{iso}}_{5}$ |  $A_{c}^{\mbox{\scriptsize Mn${}_{(1)}$}}$ |  $A_{c}^{\mbox{\scriptsize Mn${}_{(2)}$}}$ |
---|---|---|---|---|---|---|---|---
$U=1$ |  26.06 |  3.87 |  9.25 |  0.84 |  2.57 |  0.33 |  1.14 |
$U=2$ |  20.68 |  2.33 |  6.39 |  0.52 |  1.61 |  0.13 |  1.02 |
Table 1: Estimated isotropic exchange interactions and single-ion anisotropies of MST using two different values of $U$. Within our definition of the Heisenberg model, a positive energy corresponds to an antiferromagnetic interaction. Fulfilling the symmetry properties of the system, SIA can be parametrized by a unique coupling constant $A_{c}=\mathcal{A}_{zz}-\mathcal{A}_{xx}$. All values are in units of meV, assuming classical spins of length $|S|=1$. (meV) |  $J^{S}_{1xx}$ |  $J^{S}_{1zz}$ |  $D_{1z}$ |  $J^{S}_{2xx}$ |  $J^{S}_{2yy}$ |  $J^{S}_{2zz}$ |  $J^{S}_{2xz}$ |
---|---|---|---|---|---|---|---|---
$U=1$ |  -0.02 |  0.04 |  0.66 |  -0.05 |  0.01 |  0.04 |  0.04 |
$U=2$ |  -0.03 |  0.06 |  0.49 |  -0.05 |  0.02 |  0.04 |  0.04 |
Table 2: Estimated anisotropic exchange interactions of MST using two
different values of $U$. Symmetry imposes $J_{1xx}=J_{1yy}$, while the
cartesian components of tensor $J^{S}_{2}$ are given in a local reference
frame with the $y$ axis perpendicular to the Mn(1)-Mn(1) bond and the $z$ axis
parallel to the crystallographic $c$ vector. The Dzyaloshinskii vector
component is defined as $D_{1z}=(J_{1xy}-J_{1yx})/2$. All values are in units
of meV, assuming classical spins of length $|S|=1$.

Consistent with previous studies[9, 13], all (isotropic) interactions are
found to be antiferromagnetic, denoting a non-negligible magnetic frustration.
On the other hand, our calculations also reveal sizeable longer-range
interactions between Mn(1) magnetic moments within the honeycomb layers, with
the third-nearest-neighbor exchange $J^{\mathrm{iso}}_{5}$ being of the same
order of magnitude as $J_{2}^{\mathrm{iso}}$. This is likely due to both the
presence of Si-Si dimers, which are located at the center of the Mn(1)
hexagons where they can efficiently mediate the magnetic interactions, as well
as to ligand contributions [40]. While the (antiferromagnetic) Mn(1)-Mn(1)
interactions within the honeycomb layer do lead to magnetic frustration of the
system, they are much smaller than the (antiferromagnetic) Mn(1)-Mn(2)
exchange coupling, so that a ferrimagnetic configuration with antiparallel
Mn(1) and Mn(2) is expected to be the lowest energy state.

The $J^{\mathrm{iso}}_{4},J^{\mathrm{iso}}_{5}$ exchange interactions had not
previously been estimated in Refs. [9, 13], where total energies of different
magnetic configurations defined within the crystallographic unit cell were
mapped onto a classical Heisenberg model. We confirmed that such an approach
cannot provide information on $J^{\mathrm{iso}}_{4}$, while
$J^{\mathrm{iso}}_{5}$ interactions give rise to a spurious contribution to
the nearest-neighbor Mn(1)-Mn(1) exchange that, within this total-energy
mapping procedure, would correspond to
$J^{\mathrm{iso}}_{2}+J^{\mathrm{iso}}_{5}$. Taking into account the
differences between the four-state and the total-energy mapping procedures,
our results for
$J^{\mathrm{iso}}_{1},(J^{\mathrm{iso}}_{2}+J^{\mathrm{iso}}_{5}),J^{\mathrm{iso}}_{3}$
are in excellent agreement with previously reported estimates[13]. On the
other hand, both SIA and anisotropic exchange interactions support the
experimentally observed easy-plane magnetic anisotropy here.

The magnetic anisotropy energy per Mn ion, defined as the energy difference
$E_{\mbox{\scriptsize MAE}}=E_{\perp}-E_{\parallel}$ of a ferrimagnetic
configuration perpendicular or parallel to the honeycomb ($ab$) layer, can be
expressed from our model in Eq. 4 as:

$\displaystyle E_{\mbox{\scriptsize MAE}}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left(2A_{c}^{\mbox{\scriptsize
Mn${}_{(1)}$}}+A_{c}^{\mbox{\scriptsize
Mn${}_{(2)}$}}\right)+\frac{2}{3}\left(J_{1zz}-J_{1xx}\right)$ (5)
$\displaystyle+\left[J_{2zz}-\frac{1}{2}\left(J_{2xx}+J_{2yy}\right)\right].$

Using the parameters listed in Tables 1 and 2, we find a MAE of $\sim 0.70$
and $\sim 0.54$ meV/Mn for $U=1$ eV and $U=2$ eV, respectively. The dominant
contribution to the MAE is provided by the Mn single-ion anisotropies. Here,
the inequivalency of the Mn atoms is clearly reflected via distinct SIAs: both
display an easy-plane character with the hard axis parallel to the crystal $c$
axis, but the SIA of Mn(1) is one order of magnitude smaller than that of
Mn(2). Interestingly, we found that roughly 15% of the magnetic anisotropy
energy is contributed by anisotropic exchange interactions. In agreement with
previous studies [9, 13], we find that the dominant Mn(1)-Mn(2)
antiferromagnetic interactions favour a ferrimagnetic ground state comprising
ferromagnetic Mn(1) and Mn(2) layers antiferromagnetically aligned.

We estimate $T_{\mathrm{C}}$ from our calculated exchange couplings using
Monte Carlo simulations (Fig. 5). The transition temperature extracted from
the temperature dependence of the calculated specific heat per spin, $C_{v}$,
is estimated to be $T_{c}\simeq 89$ K and $T_{c}\simeq 118$ K from our
calculations taking $U=2$ eV and $U=1$ eV, respectively. These are in good
qualitative agreement with the experimentally determined $T_{C}=75-79$ K,
further supporting the identified range of $U=1-2$ eV as appropriate for MST,
and highlighting the crucial role of magnetic anisotropy, which leads to a
significantly reduced $T_{\mathrm{C}}$ in our calculations in comparison to an
isotropic model. Our findings from analysis of the calculated specific heat
are further validated by direct extraction of the ferrimagnetic order
parameter (Fig. 5(b), see Methods) which, together with the abrupt change of
its relative susceptibility, shows the ferrimagnetic state onset. Finally, we
note that the dependence of the ferromagnetic order parameter with temperature
(Fig. 5(c), see Methods) indicates the magnetic easy-plane nature of the
system, further validating our discussions above.

##  IV Conclusions

Our results demonstrate an integrated approach to predicting and understanding
the interactions governing long-range magnetic order in the layered
ferrimagnet Mn3Si2Te6. Our approach allows characterising the importance of
covalency from ligand-metal hybridisation, which we find to be significant in
this system, weakening the electronic correlations. Using these spectroscopic
results to constrain our calculations, we demonstrate how combined first-
principles and Monte Carlo methods can be used to accurately predict exchange
interactions and magnetic anisotropies, elucidating their role in stabilising
MST’s novel ferrimagnetic order and predicting its ordering temperature within
$\sim\\!15$ K of the experimental value. Our results reveal a key role of
covalency in ruling these properties in MST, which we expect to be key to
understanding magnetism across the emerging class of 2D and layered magnets.

## Acknowledgments:

We gratefully acknowledge support from The Leverhulme Trust via Grant No.
RL-2016-006 and the European Research Council (through the QUESTDO project,
714193). P.B. and S.P. acknowledge financial support from the Italian Ministry
for Research and Education through PRIN-2017 projects ‘Tuning and
understanding Quantum phases in 2D materials—Quantum 2D’ (IT-MIUR grant No.
2017Z8TS5B) and ‘TWEET: Towards ferroelectricity in two dimensions’ (IT-MIUR
grant No. 2017YCTB59), respectively. MCH, DM and GB acknowledge financial
support by the UK Engineering and Physical Sciences Research Council through
grant EP/T005963/1. We thank the Elettra synchrotron for access to the APE-HE
beamline under proposal number 20195300. We thank Diamond Light Source for
beamtime on the I10 beamline under proposal number MM28727-1. The research
leading to this result has been supported by the project CALIPSOplus under
Grant Agreement 730872 from the EU Framework Programme for Research and
Innovation HORIZON 2020. G.V., V.P., D.D. and F.M. acknowledge financial
support from the Nanoscience Foundry and Fine Analysis (NFFA-MUR Italy
Progetti Internazionali) project (www.trieste.NFFA.eu). For the purpose of
open access, the authors have applied a Creative Commons Attribution (CC BY)
licence to any Author Accepted Manuscript version arising. The research data
supporting this publication can be accessed at [[DOI TO BE INSERTED]].

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Figure 1:  Ferrimagnetism in Mn3Si2Te6. (a) Crystal structure of MST,
indicating the ferrimagnetic alignments of Mn moments within the (001) easy
$ab$-plane. (b) Side- and Top-views showing the atomic arrangements together
with the exchange interactions $J_{i}$ between the Mn atoms considered in this
work. (c) $M$ vs. $H$ magnetization curves measured at $T=2$ K for $H$
$\parallel$ $c$ and $H$ $\perp$ $c$ geometries. (d) Temperature-dependent dc
magnetic susceptibility ($\chi$) as measured in zero-field cooled (ZFC)
warming mode in a small field of 10 mT applied along the crystal directions
parallel and perpendicular to the $c$-axis. (e) Planar resistivity
($\rho_{xx}$) measured in zero field. Figure 2:  Spectroscopic investigation
of the magnetic order in grazing incidence geometry. _Top graph:_ Mn $L_{2,3}$
XAS measured at 10 K with an applied field of $1.4$ T for the two opposite
x-ray helicities: circular right (CR, red) and circular left (CL, blue). The
yellow curve shows the Mn $L_{2,3}$ XAS summed over the CR and CL spectra
after the step-edge background (grey) subtraction. _Bottom graph:_ x-ray
magnetic circular dichroism (XMCD) obtained from the difference between the
two absorption spectra. The black arrow denotes the pre-peak feature, which is
a fingerprint for Mn hybridisation with the ligand. _Inset:_ Temperature-
dependent XMCD asymmetry measured at remanence (i.e., in zero applied field as
a function of increasing temperature) after first field-cooling the sample.
The magnetic susceptibility curve obtained from bulk measurements (green
curve) are also shown. The dashed-dotted vertical line marks the magnetic
ordering temperature. Figure 3:  Orbital hybridisation in the Mn3Si2Te6
valence band. (a) Angle integrated, resonant photoemission (ResPES) of the
valence band measured scanning the excitation energy across the Mn $L_{3}$
absorption edge. Coloured circles on the XAS spectra (_left panel_) mark the
photon energies at which the valence band photoemission measurements were
performed (_right panel_). The energy is referenced to the top of the valence
band (EVB) and the VB spectra are vertically offset for clarity. (b) The
Mn-(_left_) and Te-(_right_) partial density of states (P-DOS) as determined
by DFT calculations as a function of $U$ and compared to the experimental
data. The Mn partial density of states was experimentally evaluated as the
difference between the on- and off-resonance photoemission spectra (measured
at $h\nu=641.4$ eV and $h\nu=635.5$ eV, respectively) while the Te partial DOS
was taken as the VB measured just before the Mn absorption edge (i.e., at
$h\nu=635.5$ eV). Best agreement between the experimental and calculated data
(see, e.g., dashed line as a guide to the eye) is found for the Mn partial DOS
calculated at $U=1-2$ eV. Figure 4:  Electronic structure of Mn3Si2Te6.
Calculated band dispersions along high symmetry directions, projected on the
(a) Mn(1) and (b) Mn(2) $d$-orbitals and (c) the Te $p$-orbitals. (d) shows
the corresponding partial DOS. Calculations were performed with $U=1$ eV.
ARPES spectra acquired along the $\bar{\Gamma}$ \- $\bar{\text{M}}$ at $58.5$
eV photon energy for both (e) horizontal (LH) and (f) vertical (LV) linearly
polarized light, which show a strong dichroism particularly for the Te-derived
states. (g) A magnified view of (e) with enhanced contrast, better
highlighting the upper Te-bands. (h) Bulk states projected on the (001)
surface Brillouin zone plane. (i) Bulk and surface-projected Brillouin zone
marking the relevant high-symmetry points. Figure 5:  Monte Carlo simulations.
(a) Specific heat of the full model (dark symbols) with parameters obtained
from DFT+$U$ calculations for $U=1,2$, compared with the specific heat
calculated on a simplified model with isotropic
$J_{1}^{\mathrm{iso}},(J_{2}^{\mathrm{iso}}+J_{5}^{\mathrm{iso}}),J_{3}^{\mathrm{iso}}$
(light symbols). The stronger magnetic frustration of the full model reduces
the critical temperature $T_{\mathrm{C}}$, signalled by the peak in $C_{v}$,
by roughly 20% with respect to the simplified model. (b) Temperature evolution
of the ferrimagnetic order parameter and its associated susceptibility which
displays a sharp peak at the transition (only $U=2$ eV results are shown; the
same trend was observed with the other set of parameters) and signalling the
transition to a ferrimagnetic state. (c) The in-plane
$F_{\parallel}=\sqrt{(}F_{x}^{2}+F_{y}^{2})$ and out-of-plane
$F_{\perp}=|F_{z}|$ components of the ferromagnetic order parameter are shown
as a function of temperature, confirming the easy-plane character of the
magnetically ordered phase.

##  V Supplementary Information:

Supplementary Figure S6:  X-ray Diffraction pattern of Mn3Si2Te6 single
crystal. The pattern shows the (00 _l_) reflections. The inset shows typical
Mn3Si2Te6 single crystals. Supplementary Figure S7:  (a) Experimental
geometries for X-ray Magnetic Circular Dichroism (XMCD) measurements. In
grazing incidence geometry (_left_) circularly polarized light impinges on the
sample at $20^{\circ}$ from the sample surface (i.e., the $ab$-plane), while
the normal incidence geometry is achieved when the photon beam is aligned
perpendicularly to the $ab$-plane. In both geometries, the magnetic field is
collinear with the incident photon beam and the signal is acquired in total
electron yield (TEY) detection by measuring the drain current of the sample.
(b) XMCD asymmetry signal measured in the two different geometries with $-1.4$
T applied field and the sample temperature kept at $10$ K. The XMCD signal in
grazing incidence being approximately twice larger than for the normal
geometry identifies $ab$-plane as the easy magnetization plane and it is in
good agreement with the bulk magnetic properties measured by SQUID (Fig.1(c),
(d) in the main text). (c) Remanent XMCD spectra series of the Mn $L_{2,3}$
absorption edge measured in the Mn3Si2Te6 $ab$-plane at different temperatures
across the ferrimagnetic transition (i.e., between $10$ K and $90$ K) showing
the onsetting of the long range magnetic order below $75$ K. Spectra are
offset along the vertical axis for clarity.