# Magnetism in Two-Dimensional Ilmenenes:
Intrinsic Order and Strong Anisotropy

R.H Aguilera-del-Toro1-2∗, M. Arruabarrena2, A. Leonardo1-3, A. Ayuela1-2
1Donostia International Physics Center (DIPC), 20018 Donostia, Spain  2Centro
de Física de Materiales - Materials Physics Center (CFM-MPC), 20018 Donostia,
Spain  3EHU Quantum Center, Universidad del País Vasco/Euskal Herriko
Unibertsitatea UPV/EHU, Leioa, Spain<EMAIL_ADDRESS>
###### Abstract

Iron ilmenene is a new two-dimensional material that has recently been
exfoliated from the naturally-occurring iron titanate found in ilmenite ore, a
material that is abundant on earth surface. In this work, we theoretically
investigate the structural, electronic and magnetic properties of 2D
transition-metal-based ilmenene-like titanates. The study of magnetic order
reveals that these ilmenenes usually present intrinsic antiferromagnetic
coupling between the 3d magnetic metals decorating both sides of the Ti-O
layer. Furthermore, the ilmenenes based on late 3d brass metals, such as
CuTiO3 and ZnTiO3, become ferromagnetic and spin compensated, respectively.
Our calculations including spin-orbit coupling reveal that the magnetic
ilmenenes have large magnetocrystalline anisotropy energies when the 3d shell
departs from being either filled or half-filled, with their spin orientation
being out-of-plane for elements below half-filling of 3d states and in-plane
above. These interesting magnetic properties of ilmenenes make them useful for
future spintronic applications because they could be synthesized as already
realized in the iron case.

  * October 2022

## 1\. Introduction

Technology for synthesizing two-dimensional materials has greatly improved in
recent years. Since the synthesis of graphene[1, 2], a large number of
extensive systems only a few atoms thick have been obtained. The study of
these 2D materials has brought new physical phenomena with countless
applications into play, like their magnetic properties[3, 4]. Obtaining
magnetism in 2D isotropic crystals is forbidden in the Heisenberg model as
explained by the Mermin-Wagner theorem[5]: the magnon dispersion is reduced
with respect to their 3D counterparts, and it has an abrupt onset, which
translates into low thermal agitation and a collapse of the spin order.
However, 2D systems with uniaxial magnetic anisotropy are able to withstand
thermal agitation, allowing magnetic states in mono and multilayer materials.

In the past decades, the study of 2D magnetic properties has been performed on
epitaxially grown thin films, in which phenomena such as oscillating exchange
coupling[6, 7], giant magnetoresistance[8, 9] and Hall effect[10, 11] have
been observed. Nevertheless, the study of the intrinsic magnetic properties of
these 2D systems is novel, and most of the materials that have been
synthesized are magnetic van der Waals crystals[3, 4]. In the last five years,
non-van der Waals two-dimensional materials have also been synthesized, mainly
by exfoliating naturally occurring ores. By liquid exfoliation of natural iron
ore hematite ($\alpha$-Fe2O3), Balan et al.[12] synthesized a new 2D material
which called hematene. Contrary to its antiferromagnetic bulk, hematene
presents ferromagnetic order. Similarly, other promising material for 2D
magnets is ilmenene [13], which has been synthesized using liquid phase
exfoliation from titanate ore ilmenite (FeTiO3). Motivated by the synthesis of
iron ilmenene, other similar 2D materials could be exfoliated from the bulk
systems with their 3D counterpart ilmenite-like structures.

The aim of the present work is then to characterize the ilmenene-like
titanates (TMTiO3, TM = V to Zn) to set a first theoretical basis for this
exciting family of materials. Within the framework of the density functional
theory, we systematically analyze the crystalline structure of these
compounds, finding that most of the ilmenenes exhibit triangular symmetry for
TMs on both sides of a Ti-Ti hexagonal graphene-like sublattice. In the
chromium and copper ilmenenes, we find structural distortions of Jahn-Teller
origin. Our electronic structure calculations reveal that most of these
compounds are magnetic semiconductors which have TM layers
antiferromagnetically coupled. The calculations including the spin-orbit
interaction show a strong magnetic anisotropy, with the magnetization being
oriented out-of-plane (in-plane) below (above) half-filling of the TM
electronic 3d shell. The presence of out-of-plane anisotropy in some of these
compounds suggests potential applications for spintronics in thin layers and
2D materials.

## 2\. Model Details

In this work, we study the structural, electronic and magnetic properties of
ilmenene-type materials using the projector augmented wave method (PAW)
implemented in the Vienna Ab-initio Software Package (VASP)[14, 15]. For the
exchange and correlation potential we use the Perdew-Burke-Ernzerhof form of
the generalized gradient approximation (GGA), with the formulation of
Dudarev[16] for GGA+U. The Hubbard U parameters for each element are chosen
based on the available literature on transition metal oxides[17], and are
shown in Table S1 of the Supplementary Information (SI) with the considered
valence states. A test calculation of the partial density of states (PDOS) of
cobalt titanate using the HSE06 hybrid functional shows that including a U
parameter in the oxygen p-orbitals is needed to correctly describe the
electronic structure. This finding is in agreement with previous
investigations concerning titanium oxide and hematite[18, 19, 20, 21, 22, 23].
All calculations are performed with a well-converged plane-wave cutoff energy
of 800 eV, a gamma-centered 4x4x1 Monkhorst-Pack k-point mesh, and a Fermi
smearing of 20 meV. Atomic coordinates are relaxed until forces in all
directions were smaller than 0.5 meV/Å. An energy convergence criterion of
10-7 eV is used. Further tests using even a larger plane wave cutoff up to
1000 eV and 6x6x1 k-point mesh do not modify the presented results.
Differences in local charges and local magnetic moments are univocally
analyzed using the Bader method[24, 25]. By including the spin-orbit coupling,
additional tests are performed to converge the magnetocrystalline anisotropy
energy with respect to the Brillouin Zone sampling.

The d-metal ilmenene sheets are obtained by cutting their respective bulk
titanates (TMTiO3, TM = V to Zn) in the hexagonal [001] direction. For the
iron ilmenene FeTiO3, transmission electron microscopy measurements confirm
the 2D structure in this direction[13]. The layer structure is shown in Fig.
1. For all the compounds under analysis, two different layer-ilmenenes are
tested: titanium and transition metal terminated ilmenene layers. This work
focuses on the TM ended layers because they are found to be more stable for
all the materials under analysis.

## 3\. Results and discussion

###  3.1 Structural properties

The structure of the TM ilmenenes is graphically depicted from two viewpoints
in Fig. 1. After structural relaxations, we find that most of the compounds
keep the input symmetry, except for chromium and copper ilmenenes, which show
structural deformations of the perfect lattice due to the Jahn-Teller effect
(see Fig. 1(b)). The orange area in panel (a) denotes the chemical cell that
reproduces the crystalline structure of the ilmenenes when periodically
repeated. We calculate magnetic configurations using a larger 2x2x1 magnetic
cell. For chromium and copper titanates, due to structural distortions, the
unit cell and the magnetic cell coincide. In the distorted compounds, the two
in-plane lattice vectors become slightly different: for the chromium case
a=10.57Å and b=10.00 Å, and for the copper case a=10.44 Å and b=10.21 Å. Note
that these ilmenenes become anisotropic. Overimposed on the global lattice
distortions, the inner atomic distortions become more noticeable. For
instance, the largest-smallest distances are 6.0 Å- 3.9 Å and 5.4 Å- 4.8 Å for
Cr and Cu ilmenenes, respectively. The large 1D anisotropy of these two Janh-
Teller distorted ilmenenes is shown by the stripes of the green and violet
areas in Fig. 1 (b). A summary of the interatomic distances of the TM
titanates is shown as a function of the elements across the period in the
periodic table in Fig. 1. We find that the horizontal distance between
transition metals in the same layer $l_{TM-TM}$ increases from V to Mn, then
decreases until Ni, and finally increases for the brass metals Cu and Zn. The
height distance between metals in different layers $h_{TM-TM}$ generally
decreases. The distances involving Ti in the Ti-Ti and Ti-O bonds remain
nearly constant, while the TM-O distances decrease. In essence, TM-ilmenenes
can be thought of as a solid network formed by Ti and O atoms on which the TM
atoms attach on both sides. For the Cr and Cu ilmenenes showing structural
distortions, the TM-TM distances have two values because the triangular
symmetry is broken, and the Ti-O distances are split into several values as
shown by the extra marks in the lower panel of Fig. 1.

We find that in comparison with their bulk counterparts, TM ilmenenes are
compacted along the c direction, so that a two-dimensional hexagonal layer of
titanium ions is formed similar to graphene. The Ti-Ti sublattice in ilmenenes
remains almost constant with the distances varying within 3%. 111However, the
chromium titanate layer does not show the fully compacting behavior to a flat
titanium sheet (see Fig. 1(b)). This compression for the iron ilmenene has a
minimum height about 2.91 Å, in agreement with another work [13]. The
theoretical value of 2.59 Å for the interplanar space corresponding to the
($11\bar{2}0$) and ($\bar{2}110$) lattice spacing compares well with the
experimental one ($\sim$ 2.53 Å) [13], which is also supporting our
calculations since the interatomic distances depend on the chosen U values.
For cobalt titanates, a layer thickness of 4.03 Å has been reported[26] in
bulk, which decreases to 2.95 Å in the ilmenene layer, while the horizontal
distance between cobalt atoms in the same layer expands slightly from 5.06 Å
in bulk to 5.17 Å in the layer. The significant changes in the height
distances are due to the layer being decorated with half of the TM atoms than
in the corresponding bulk, in order to keep the stoichiometry. It seems that
the ilmenene layers depart largely from those found in ilmenites and not only
their structural properties but their magnetic ones are thus deserving a
thoroughly study.

###  3.2 Electronic properties: magnetic semiconductors

We now focus on the electronic structure of ilmenenes and find that they show
a gap opening that is linked to their stability. The calculated electronic
band gaps range between 1.8 and 4 eV, as displayed in Fig. 2 (a), with values
that are typical for semiconductors. The gaps are increasing when going from V
to Mn, and they are oscillating for Fe, Co and Ni. In general, the values are
smaller than the gap for TiO2 in the rutile bulk phase ($\sim$ 3.2 eV). Note
that the Mn and the Co ilmenenes remain within the same order of the TiO2 gap
values. These trends follow the filling of 3d TM electronic levels as
discussed below.

The TM atoms in the ilmenene compounds show local magnetic moments, which are
calculated using the Bader method, and are shown in panel (b) of Fig. 2. The
calculated values are close to the ones expected by applying the Hund rules to
isolated ions. Furthermore, the local magnetic moment as a function of atomic
number follows a Slater-Pauling-like rule, increasing from vanadium to
manganese, then decreasing until it vanishes for zinc. Thus, the 3d ilmenenes
can be taken as magnetic semiconductors.

The band structures and the PDOS projected onto the transition metal atoms are
collected in the SI. From their analysis, an electronic filling model for each
TM ilmenene is presented in Fig. 2(b). We find that the orbitals can be
classified into three groups: (i) out-of-plane $d_{z^{2}}$, (ii) in-plane
$d_{x^{2}-y^{2}}$ and $d_{xy}$, and (iii) mixing in- and out- of plane
contributions for $d_{xz}$ and $d_{yz}$. The $d_{x^{2}-y^{2}}$ and $d_{xy}$
orbitals are highly hybridized, as the $d_{xz}$ and $d_{yz}$ orbitals are.
Because the in-plane degenerated energy levels of chromium and copper
titanates are being partially occupied, they are held responsible for the
Jahn-Teller-like distortions in these structures, where a splitting of these
otherwise degenerated orbitals is observed under distortions.

All these electronic trends can be better understood following the orbital
model in real space for each case, as shown in Fig. 2 (c). Below half-filling,
we show how the magnetic moment increases in correlation with the gap values.
For next elements, above half-filling, the Fe case becomes similar to the Ni
one which has a weaker spin distribution. In between, we find the Co ilmenene
with a large in-plane spin distribution has also an out-of-plane component
that cannot be neglected, an extra component that is related to the complexity
added to this case when trying to write down a single spin hamiltonian. For
brass metals, the local magnetization is nearly screened having a spin
compensated cloud around the brass metal centers. This model is going to be
interesting in the discussion of magnetic anisotropies below. For instance, we
note that the spin distribution is not anisotropic for the case of Cr (Cu)
having one-electron less below a half-filled (filled) 3d shell.

###  3.3 Magnetic ordering

We next consider the magnetic behavior of the ilmenene-type materials, and
calculate the total energies of the magnetic configurations shown
schematically in Fig. 3.

We find that, with the exception of ferromagnetic copper and spin-compensated
zinc titanates, the ilmenenes show antiferromagnetism between the 2D TM layers
in the so-called AFM-1 configuration.

The magnetic configurations in Fig. 3 are used to fit the Heisenberg
Hamiltonian

$H=-\sum_{ij}J_{ij}\mathbf{\tilde{S}_{i}}\cdot\mathbf{\tilde{S}_{j}}$ (1)

where the $\tilde{S}_{i}$ is the pseudospin of each isolated atomic species
(e.g: $\tilde{S}=3/2$ for Co), and the $J_{i}$ with $i=1,2$ are the inter-
layer ($i=1$) and intra-layer ($i=2$) magnetic couplings schematically
depicted in 3. The magnetic couplings are computed from the energy differences
$\Delta E_{1}=E_{FM}-E_{AFM_{1}}$, $\Delta E_{2}=E_{AFM_{2}}-E_{AFM_{1}}$.

The fitted Ji values are shown in Fig. 3 (b). The inter-layer coupling is the
leading interaction because J1 is an order of magnitude larger than J2. The
inter-layer coupling J1 is negative for most of the ilmenenes, that indicates
a preference for antiferromagnetism. The intra-layer coupling J2 is positive
in all cases, and the compounds favor intralayer ferromagnetism. Since the
coordination of the TM atoms on each layer side changes from being hexagonal
in bulk to triangular in the layer, the structural differences between bulk
and two dimensional titanates mentioned above play an important role in the
magnetic ordering. It is noteworthy that the manganese ilmenene even favors
ferromagnetism within layers in contrast with its antiferromagnetic bulk
counterpart.

###  3.4 Magnetic anisotropy

We last study the magnetic anisotropy of d-metal ilmenenes. The spin-orbit
interaction couples the spin magnetic moment to the crystal lattice, which
means that some spin orientations are more stable than others. The
magnetocrystalline anisotropy energy (MAE) is defined as the energetic
difference between two magnetic configurations with different spin
orientations. We calculate the total energy of the ilmenenes for a number of
spin orientations, and find that, anisotropy-wise, ilmenenes can be classified
in two groups: out-of-plane ilmenenes, in which the spin magnetic moment is
aligned in the c-axis, and in-plane ilmenenes, with the spin aligned in any of
the directions along TM-Ti bonds projected in the plane (see Fig. 4). Due to
symmetry around TM atoms, there are six such equivalent directions in the ab
plane. The vanadium, chromium and manganese titanates in addition to copper
ones have an out-of-plane anisotropy, and the magnetization in iron, cobalt
and nickel titanates is oriented in-plane. These MAE trends agree with the
electronic filling model, in which below half-filling TMs the electrons near
the Fermi level are those that include an out-of-plane component, and for
iron, cobalt and nickel titanates they are mainly in-plane.

The obtained MAE values range between several orders of magnitude from 10-2
meV up to tenths of meVs because the spin-orbit interaction varies greatly for
the different ilmenenes. The chromium, iron and cobalt titanates have a strong
MAE around 5 meV, and in the case of the vanadium and nickel layers, this
value is still large about 0.6 and 0.13 meV, respectively. The manganese
titanate has the smallest anisotropy ($\sim$ 0.04 meV) because the orbital
magnetic moment is nearly fully quenched. This finding summed to having the
energetic difference between the magnetic AFM-1 and AFM-2 configurations of
the manganese titanate being very small (around 0.25 meV) points to non-
collinear magnetic configurations, a topic that might merit future
experimental investigation on this specific layer, but being beyond the actual
scope of the paper. Furthermore, the iron ilmenene is also a particular 2D
layer because it shows a strong in-plane anisotropy, contrary to the ilmenite
bulk with a non-negligible out-of-plane component.

We observe a noticeable trend for the magnetic anisotropy. Below half-filling,
the V and Cr ilmenenes exhibit out-of-plane anisotropy, while the half-filled
manganese ilmenene has a very small MAE. Above half-filling, most the
compounds have an in-plane anisotropy. In fact, these trends in anisotropy
follow the levels depicted in Fig. 2 (c). For V and Cr ilmenenes, it shows
that the spin density is perpendicular to the plane. The spin density in
manganese ilmenene is isotropic, and this correlates with its low MAE. Above
half-filling, the spin density of Fe, Co and Ni ilmenenes lies in-plane, in
agreement with their in-plane MAE. The case of Cu ilmenene shows out-of-plane
anisotropy even if the spin density lies in-plane, a finding that can be
explained by the out-of-plane changes in the orbitals when the spin-orbit
coupling term is included with the distortions.

We next bring the magnetic results on magnetic ilmenenes into contact with
experimental facts. Note that Fe ilmenenes can be obtained by exfoliation and
deposited on substrates for some experimental setups [13, 12]. Then, further
measurements on the magnetic properties of the Fe ilmenenes can be today
performed in line with our theoretical results. Furthermore, we have studied
Co ilmenenes that could be interesting as they can behave in a similar way to
study magnon physics in two dimensions as their bulk counterpart is already
being used [27, 28, 26]. In fact, Cr ilmenenes seem to become key as they are
candidates to show out-of-plane anisotropy when in the form of ultrathin
layers. These findings are suggesting the future use of TM ilmenenes in
spintronics devices for injecting magnons and study magnetism in exfoliated 2D
magnetic layers.

## 4\. Conclusion

In this work, we have analyzed the structural, electronic and magnetic
properties of TM ilmenene-like systems. Our calculations reveal that most of
the materials under analysis present a triangular crystalline structure for
TMs, with an ironed compression of the internal titanium ion layer with
respect to bulk. The chromium and copper ilmenenes exhibit notable structural
distortions, which seem to have a Jahn-Teller origin. All the compounds
studied have been found to be magnetic semiconductors with band gaps in the
range between 1.7 and 4 eV. The magnetic ground state is mainly
antiferromagnetic between layers, and ferromagnetic and spin-compensated for
CuTiO3 and ZnTiO3, respectively. Furthermore, spin-orbit calculations revealed
that TM ilmenenes can be divided into two groups: out-of-plane ilmenenes, for
less than half-filling of the 3d bands, and in-plane ilmenenes, above half-
filling of these levels, with the spin aligned in the TM-titanium directions
projected in the hexagonal plane. We believe that this family of materials
paves the way to other types of promising 2D candidates with potential
applications in the field of spintronics.

## 5\. Data availability statement

All data that support the findings of this study are included within the
article (and any supplementary files).

## 6\. Acknowledgement

This work has been supported by the Spanish Ministry of Science and Innovation
with PID2019-105488GB-I00 and PCI2019-103657. We acknowledge financial support
by the European Commission from the NRG-STORAGE project (GA 870114). The
Basque Government supported this work through Project No. IT-1569-22. M.A was
supported by the Spanish Ministry of Science and Innovation through the FPI
PhD Fellowship BES-2017-079677. R.H.A-T acknowledge the postdoctoral contract
from the Donostia International Physics center.

## 7\. References

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Figure 1:  Magnetic unit cell for transition metal ended ilmenene-like
systems: (a) symmetric for most ilmenenes, and (b) distorted for chromium
titanate CrTiO3. The color code of the atoms is as follows: TM (Cr) atoms in
blue (purple), titanium in cyan, and oxygen in red. The orange area in panel
(a) represents the smaller chemical cell. For chromium and copper titanates,
the chemical and magnetic cells coincide. Calculated interatomic distances:
(c) horizontal TM-TM distance $l_{TM-TM}$, (d) layer height $h_{TM-TM}$ and
Ti-Ti distances, and (e) Ti-O and TM-O distances. For the chromium and copper
ilmenenes we present an average of the distances in their distorted
structures.  Figure 2:  (a) Electronic band gaps for TM ilmenenes. Vertical
lines separate different regions with TM below half-filling, TM above half-
filling, and brass metals. (b) Calculated local magnetic moment around
transition metal atoms. For each compound, the electronic filling model of the
ground state is also shown. Red levels represent the in-plane
$d_{x^{2}-y^{2}}$ and $d_{xy}$ orbitals; green, the out-of-plane $d_{z^{2}}$
orbital; and blue, the $d_{xz}$ and $d_{yz}$ orbitals with in- and out-plane
components. (c) Orbital models of the local magnetization in 3d TMs and brass
metals within ilmenenes. Blue and green regions denote spin polarized regions;
grey, spin compensated ones.  Figure 3:  (a) Magnetic ordering configurations
of TM ilmenenes: “FM” Ferromagnetic, “AFM-1” antiferromagnetic by layers,
“AFM-2” and “AFM-3” antiferromagnetic. (b) Couplings between both layer sides
(J1), and within the same side layer of TM ions (J2). Figure 4:
Magnetocrystalline anisotropy energy of the transition metal ilmenenes. In
blue, compounds with out-of-plane anisotropy; in red, with in-plane magnetic
moments. For the case of manganese titanate, the magnitude was enhanced by a
factor of 3 to make it visible in the shown range.