# First-principles study of structural, electronic and magnetic properties at
the (0001)Cr2O3-(111)Pt interface

Marlies Reher  Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093
Zürich, Switzerland  Nicola A. Spaldin  Materials Theory, ETH Zürich,
Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland  Sophie F. Weber
Materials Theory, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich,
Switzerland

###### Abstract

We perform first-principles density functional calculations to elucidate
structural, electronic and magnetic properties at the interface of
(0001)Cr2O3-(111)Pt bilayers. This investigation is motivated by the fact
that, despite the promise of Cr2O3-Pt heterostructures in a variety of
antiferromagnetic spintronic applications, many key structural, electronic,
and magnetic properties at the Cr2O3-Pt interface are poorly understood. We
first analyze all inequivalent lateral interface alignments to determine the
lowest energy interfacial structure. For all lateral alignments including the
lowest-energy one, we observe an accumulation of electrons at the interface
between Cr2O3 and Pt. We find an unexpected reversal of the magnetic moments
of the interface Cr ions in the presence of Pt compared to surface Cr moments
in vacuum-terminated (0001)Cr2O3. We also find that the heterostructure
exhibits a magnetic proximity effect in the first three Pt layers at the
interface with Cr2O3, providing a mechanism by which the anomalous Hall effect
can occur in (0001)Cr2O3-(111)Pt bilayers. Our results provide the basis for a
more nuanced interpretation of magnetotransport experiments on
(0001)Cr2O3-(111)Pt bilayers and should inform future development of improved
antiferromagnetic spintronic devices based on the Cr2O3-Pt material system.

††preprint: APS/123-QED

##  I Introduction

Antiferromagnetic spintronic devices offer several advantages over their
ferromagnetic counterparts. Due to their zero net magnetization,
antiferromagnets are robust against external magnetic field perturbations,
rendering them stable against data erasure [1, 2, 3]. Furthermore, the absence
of a magnetic stray field for monodomain antiferromagnets allows for
crosstalk-free miniaturization of bits and consequently a higher storage
density [1, 4, 5]. Antiferromagnets exhibit ultrafast switching of the Néel
vector, with rates in the range of THz as opposed to the GHz limit of
ferromagnets [6]. Finally, antiferromagnets are more ubiquitous than
ferromagnets. Therefore, the exploitation of antiferromagnets in spintronic
devices vastly expands the suite of candidate materials [7].
Despite their appealing properties, a major roadblock to the widespread use of
antiferromagnets in practical applications is the inherent challenge of Néel
vector switching and readout arising from the vanishing bulk magnetization [1,
8]. Metallic antiferromagnets with broken time-reversal, for example CuMnAs
and Mn_2Au, can be switched via electrical current pulses [3, 9]. The
underlying microscopic mechanism in this case is proposed to be a relativistic
spin-orbit torque. On the other hand, insulating antiferromagnets with broken
time-reversal symmetry can exhibit a magnetoelectric effect (that is, an
induced bulk magnetization in response to an applied electric field, or
conversely an induced bulk electric polarization in response to an applied
magnetic field). The Néel vector in magnetoelectric antiferromagnets can be
switched via a combination of magnetic fields and electric fields [10]
(instead of a current) leading to lower energy consumption. In addition, it
has been shown that a rigorous symmetry correspondence exists between surface
magnetization (a symmetry-allowed dipole moment per unit area) and bulk
magnetoelectricity, such that antiferromagnets that have a finite
magnetoelectric response for an electric field along some direction
$\mathbf{r}$ also have finite surface magnetization on a surface whose normal
is parallel to $\mathbf{r}$ [11, 12]. Since the sign of this surface
magnetization couples to the sign of the bulk Néel vector, the finite surface
magnetization provides a direct method of readout [10, 13, 11, 12].
Cr2O3 in particular is an extensively studied magnetoelectric antiferromagnet
with an unusually high magnetic ordering temperature of $\approx 300$
$\mathrm{K}$, making it ideal for device applications [14, 15, 16, 17, 18,
19]. The roughness-robust, uncompensated surface magnetization of the
(0001)Cr2O3 surface, which has the same symmetry origin as the linear
magnetoelectric response for an electric field along the [0001] direction, has
been discussed theoretically [11, 12, 20] as well as experimentally [21, 22,
23, 13]. An adjacent layer of Pt can in fact be used to read out the sign of
surface magnetization on (0001)Cr2O3 via measuring the transverse Hall voltage
of Pt in response to a current applied parallel to the interface [24].
While prototype antiferromagnetic spintronic devices based on Cr2O3-Pt
heterostructures have been built [25], the electronic and magnetic
configuration at the interface of Cr2O3 and Pt, as well as the detailed origin
of the Hall voltage in Cr2O3-Pt heterostructures are not well understood.
One effect which almost certainly contributes to the Hall voltage and its
dependence on the sign of surface magnetization is the so called “Spin Hall
magnetoresistance” (SMR), in which an applied in-plane electric current first
generates a spin current in Pt perpendicular to the interface via the Spin
Hall effect. Then, the transverse voltage in Pt generated due to the Inverse
Spin Hall effect will have opposite sign depending on the sign of the Cr2O3
surface magnetization (and correspondingly, the sign of the Néel vector),
which the spin current interacts with [26, 27].
Another possible contributing factor is the anomalous Hall affect, or AHE. The
AHE, in which an electric current develops transverse to an applied current,
is symmetry-forbidden in both bulk Cr2O3 and nonmagnetic bulk fcc Pt. Note
that Cr2O3 in fact has broken time-reversal symmetry in the bulk, which is the
minimal symmetry breaking required for the AHE, but in this case the AHE must
still vanish due the product of inversion and time-reversal symmetry being
preserved, although each symmetry individually is broken. However, the AHE has
been reported to be present in Cr2O3-Pt heterostructures [24, 28, 29, 30]. A
finite Pt magnetization induced in a (0001)Cr2O3-(111)Pt heterostructure via a
magnetic proximity effect [31, 24] would break time-reversal symmetry in Pt,
thus allowing for a nonvanishing AHE. Because the AHE and the above-described
SMR, which does not require spin polarization of the Pt, both generate the
same electric response, it is difficult to disentangle the contribution of the
two effects experimentally. Detailed ab-initio investigations of the magnitude
of induced magnetization in Pt, as well as the penetration depth, in
(0001)Cr2O3-(111)Pt heterostructures can help to reveal how much the AHE
contributes to the Hall conductivity in Pt in magnetotransport experiments
and/or spintronic devices.
Although broadly accepted that Cr2O3 affects the electronic, and possibly
magnetic (for example, spin-polarization of Pt via magnetic proximity to
Cr2O3) properties of Pt, there has been little to no investigation of how the
magnetic and electronic properties of Cr2O3 are altered due to proximity to
Pt. Elucidating whether Pt alters (0001)Cr2O3 surface magnetization is
important for establishing the degree to which the magnetic properties at
Cr2O3 vacuum- and heavy metal-terminated surfaces can be directly compared.
Based on these motivations, in this work we perform a detailed ab-initio
investigation of the structural, electronic and magnetic properties at the
(0001)Cr2O3-(111)Pt interface. In Sec. III.1 we calculate the energies of all
distinct high-symmetry lateral alignments of (0001)Cr2O3 and (111)Pt to
determine the most energetically favorable lattice matching for the
unreconstructed (0001)Cr2O3-(111)Pt interface. In Sec. III.2, we analyze the
charge density distribution and layer-projected density of states for the
(0001)Cr2O3-(111)Pt heterostructure with the lowest energy interface. We find
a significant charge redistribution due to the interaction of Cr2O3 and Pt,
specifically, a substantial electron accumulation in the interstitial region
between the Cr and Pt atoms closest to the interface. Finally, in Sec. III.3,
we shed light on the magnetic properties of both Cr2O3 and Pt near their
interface, in particular, how the presence of Pt affects the Cr2O3
antiferromagnetic order (Subsec. III.3.1), and how the proximity of boundary
magnetization in Cr2O3 affects the spin polarization of Pt (Subsec. III.3.2).

##  II Computational methods

We use density functional theory (DFT) employing the Vienna ab-initio
simulation package (VASP) [32, 33] with projector-augmented-wave (PAW)
pseudopotentials [33, 34]. The local spin-density approximation (LSDA) [35,
36, 37, 38, 39] exchange-correlation functional is combined with a Hubbard U
correction (LSDA+U) [40] on the Cr $d$ states in order to approximately
account for their localized nature. We use the rotationally invariant method
of Dudarev et al. [41] in which a single effective parameter
$\mathrm{U}_{\mathrm{eff}}$ is adjusted, as opposed to separately specifying a
Hubbard U and a Hund J. We set $\mathrm{U}_{\mathrm{eff}}=4$ $\mathrm{eV}$
based on experience from previous studies [42, 12]. The LSDA functional and
the pseudopotentials $\mathrm{Cr_{sv}}$, $\mathrm{O}$, and $\mathrm{Pt_{pv}}$
with valence electrons of $\mathrm{Cr_{sv}}$:$3s^{2}3p^{6}3d^{5}4s^{1}$;
$\mathrm{O}$:$2s^{2}2p^{4}$; and $\mathrm{Pt_{pv}}$:$5p^{6}5d^{9}6s^{1}$ are
chosen based on good agreement of calculated and experimental lattice
parameters, see appendix. Based on our convergence tests, we choose a 7x7x1
Gamma-centered k-point grid and a cutoff energy of 800 eV.
For the calculations described in the main text, we use a slab structure with
six layers of Cr-terminated (0001)Cr2O3, six layers of (111)Pt, and 20 Å
vacuum, see Fig. 1, unless specified otherwise. Pt forms a (111) surface on
(0001)Cr2O3 as shown in several X-ray diffraction studies [43, 44, 30, 45, 46,
47, 48]. The structure files for Cr2O3 and Pt are obtained from the Materials
Project database [49] and transformed into a slab structure using the Atomic
Simulation Environment (ASE) in python [50, 51]. (111)Pt with a lattice
parameter of a=4.79 Å is strained to (0001)Cr2O3 with a lattice parameter of
a=4.92 Å, yielding a lattice mismatch of $\approx 3\%$. We fix the lattice
parameters and relax the internal coordinates (while constraining the
structure to maintain the original symmetry) until forces on all atoms are
less than $0.01$ $\mathrm{eV}/\mathrm{\AA}$. During these relaxations, we fix
the Cr magnetic moments to the collinear “up down up down” antiferromagnetic
order with the Néel vector directed along $[0001]$ (Fig. 1), which is known to
be the ground-state order for both bulk and vacuum-terminated (0001)Cr2O3 [42,
20]. Given the strong spin-orbit coupling of heavy metals such as Pt, we
include spin-orbit coupling self-consistently for the heterostructure
relaxations, as well as all subsequent calculations described in Secs. III.1,
III.2 and III.3.

Figure 1: The Cr2O3-Pt heterostructure consisting of six Cr2O3 layers, six Pt
layers and 20 Å of vacuum. The antiferromagnetic structure is illustrated
through the corresponding magnetic moments (green arrows). The numbers on the
Cr atoms show the labels used in this work. Cr12 is the interfacial Cr atom.

##  III Results

###  III.1 Structural properties

First, we determine the most energetically favorable lateral alignment of the
(0001)Cr2O3-(111)Pt interface. An analogous ab-initio investigation of
interface energetics in the isostructural heterostructure (0001)Fe2O3-(111)Pt
was performed in Ref. [52]. For (0001)Cr2O3-(111)Pt bilayers on the other
hand, the one prior DFT calculation [30] in the literature which we are aware
of did not compare different lateral alignments.
Possible adsorption sites for adatoms on the (111)Pt surface are the so-called
bridge, top, hollow fcc, and hollow hcp site [53] (Fig. 2). The difference
between hollow fcc and hollow hcp is the presence of either a void or an atom
in the second Pt layer beneath/above the adsorption site in the first Pt
layer. We use this terminology to describe the lateral alignment of the
interfacial Cr ion of (0001)Cr2O3 relative to the (111)Pt surface in the
Cr2O3-Pt heterostructure (Fig. 2). Two possibilities arise for the hollow fcc
and hcp sites: the Pt atoms of the first Pt layer can be positioned laterally
above or between the O atoms of Cr2O3 resulting in the four configurations
hollow fcc (Pt between O), hollow hcp (Pt between O), hollow fcc (Pt on O),
and hollow hcp (Pt on O) in addition to the bridge and top configuration.

Figure 2: Terminology to describe the lateral alignment of the
(0001)Cr2O3-(111)Pt interface. Left: Top view of (111)Pt surface, right: side
view of hollow hcp (Pt on O) lateral alignment.

We performed DFT calculations for these six lateral alignments to calculate
the relaxed structure and determine the lowest energy interface configuration.
Note that we were not able to converge calculations using the bridge lateral
alignment. The authors of Ref. 52 mentioned the same issue for their study of
(0001)Fe2O3-(111)Pt heterostructures, so we suspect that the bridge alignment
is highly unstable. Tab. 1 gives the relative energies for the five remaining
lateral alignments in $\mathrm{eV}$ per 6-layer-Cr2O3-6-layer-Pt slab (48
atoms total) used in the calculations. Note that the atomic positions for each
lateral alignment are fully relaxed as described in Sec. II using the
antiferromagnetic Cr2O3 order shown in Fig. 1. The effect of magnetic order
will be studied later. The hollow hcp (Pt on O) lateral alignment exhibits the
lowest energy followed by the hollow fcc (Pt on O) lateral alignment. The
other lateral alignments exhibit significantly higher energies. Thus, in
addition to the clear energetic preference for Cr to lie between the Pt atoms
in the interfacial layer, rather than directly below (corresponding to the top
lateral alignment), it is also energetically favorable for Pt to be positioned
on top of the oxygen atoms.

Table 1: Energies of different lateral alignments relative to the hollow hcp (Pt on O) lowest-energy lateral alignment. Lateral alignment | Energy difference [eV]
---|---
Top | +0.161
Hollow hcp (Pt between O) | +0.158
Hollow fcc (Pt between O) | +0.113
Hollow fcc (Pt on O) | +0.026
Hollow hcp (Pt on O) | 0.00

The energy differences of the order of 0.01 to 0.1 eV per unit cell (Tab.1)
are significantly larger than the resolution of our calculations
($\approx$10-6 eV), thus we can be confident that the hollow hcp (Pt on O)
lateral alignment is the most energetically stable at 0 $\mathrm{K}$. Thus,
for the remainder of the main text we focus on the electronic and magnetic
properties of (0001)Cr2O3-(111)Pt heterostructures with the hollow hcp (Pt on
O) alignment (the results for other lateral alignments, given in the appendix,
are qualitatively similar).
Note that at room temperature (298 K), the thermal energy $k_{B}$T = 0.0257
$\mathrm{eV}$ is close to the energy difference between hollow hcp (Pt on O)
and hollow fcc (Pt on O). Thus, at the growth temperature it is possible that
the two lateral alignments coexist, or hollow fcc (Pt on O) could even
dominate. However, at lower temperatures closer to the 0-kelvin DFT limit, we
expect only the hollow hcp (Pt on O) alignment to be stable.
Before moving on to the electronic properties, we stress that in this work, we
focus on pristine, unreconstructed interfaces and have not considered possible
surface reconstructions or sources of disorder such as vacancies or
interstitials. Taking such disorder into account will be an important followup
study.

###  III.2 Electronic properties

We now analyze the electronic properties of the (0001)Cr2O3-(111)Pt interface
in the lowest-energy hollow hcp (Pt on O) lateral alignment. Specifically, we
analyze the layer-projected density of states and calculate the real-space
distribution of electronic charge using the charge density difference method.
The charge density difference $\Delta\rho$ between the charge density of the
Cr2O3-Pt heterostructure $\rho_{\mathrm{Cr_{2}O_{3}-Pt}}$ and the charge
density of a Cr2O3 slab $\rho_{\mathrm{Cr_{2}O_{3}}}$ and a Pt slab
$\rho_{\mathrm{Pt}}$ can be expressed as

$\Delta\rho=\rho_{\mathrm{Cr_{2}O_{3}-Pt}}-\rho_{\mathrm{Cr_{2}O_{3}}}-\rho_{\mathrm{Pt}}.$
(1)

$\Delta\rho$ captures the changes in electron density, relative to the summed
densities of isolated Cr2O3 and Pt structures, due to the mutual interaction
at their interface, and is visualized with the structure visualization
software VESTA in Fig. 3. While no charge transfer is visible for layers 1 to
5 in Cr2O3, we observe a charge density difference in all Pt layers as well as
in the layer of Cr2O3 at the Cr2O3-Pt interface. The Pt atoms exhibit a slight
depletion of electrons throughout all Pt layers, marked in blue in Fig. 3.
Electrons accumulate between the Cr12 interface ion and the first Pt layer.
Note that this accumulation of electrons, depicted in yellow, is present for
all lateral alignments, with only the shape of the electron accumulation
changing, see appendix.

Figure 3: Left: Calculated charge density difference for the hollow hcp (Pt on
O) lateral alignment (0001)Cr2O3-(111)Pt heterostructure. Yellow represents an
accumulation of electrons, while cyan represents a depletion of electrons. The
isosurface value is set to 0.006
$\frac{e}{${\mathrm{\SIUnitSymbolAngstrom}}^{3}$}$. Right: Layer-projected
density of states, with the layers defined in the left-hand structure. Insets
in the center zoom in on properties near the interface.

The accumulation of electrons at the interface indicates a metallic behavior
that is also reflected in the layer-projected density of states, shown on the
right in Fig. 3. While layers 1 to 5 in Cr2O3 are insulating with a pronounced
band gap, the interface layer (layer 6) shows a finite density of states at
the Fermi energy and no clear bandgap. These effects are also robust across
different alignments, see appendix.
The substantial charge accumulation between the Cr12 ions closest to the
interface and the Pt directly above already provides a qualitative hint that
the properties of Cr moments at the (001)Cr2O3-(111)Pt interface may be
altered relative to bulk or vacuum-terminated Cr2O3. This is especially
evident in the Cr2O3 magnetism, as we will see in the next section.

###  III.3 Magnetic properties

Next, we calculate the magnetic properties of the (0001)Cr2O3-(111)Pt
interface to investigate whether the surface magnetization of Cr2O3 is
affected by Pt, and whether Pt is affected by Cr2O3.

####  III.3.1 Effect of Pt on Cr2O3 – Flipped magnetic moment

First, we determine how the presence of Pt in the (0001)Cr2O3-(111)Pt
heterostructure affects the (0001)Cr2O3 surface magnetization, in particular
the monolayer of Cr12 moments closest to the vacuum or Pt interface. By
revealing how the presence of the heavy metal modifies Cr2O3 magnetic
properties at the interface compared to the vacuum-terminated surface, our
investigation should indicate how directly magnetotransport measurements
exploiting electrical readout in heavy metals can be compared with alternative
experimental surface magnetization probes, such as nitrogen vacancy
magnetometry.[54, 55].
We use constrained magnetic DFT as implemented by Ma and Dudarev [56] to
rotate the Cr12 interface magnetic moment in steps of 45∘ from the bulk
orientation, in which it is antiferromagnetically aligned with its nearest
neighbor Cr11, to the opposite orientation. We then calculate the
corresponding energies to check whether canting of the surface magnetization
is energetically favorable. Without Pt (vacuum termination), the energy is
lowest at 0∘, that is, when the Cr2O3 slab retains its bulk antiferromagnetic
ordering (Fig. 4). This is consistent with our prior DFT studies using vacuum-
terminated (0001)Cr2O3 [12] as well as ab-initio studies by other authors
[57].
In stark contrast, for the (0001)Cr2O3-(111)Pt heterostructure, the energy is
lowest when the magnetic moment of Cr12 is rotated by 180∘ (Fig. 4). This has
the surprising implication that, at least for pristine, unreconstructed
(0001)Cr2O3-(111)Pt heterostructures, the interface magnetic moment of Cr12
flips with respect to its direction in bulk and vacuum-terminated (0001)Cr2O3,
disrupting the antiferromagnetic order at the interface and leading to a net
magnetic moment in the Cr2O3 slab. We also confirm that the flipped Cr
interface magnetic moment is energetically favorable for the four additional
lateral alignments we explored in Sec. III.1, see appendix. Thus, we can
conclude that the reversal of (0001)Cr2O3 magnetization at the interface is a
generic consequence of Cr2O3-Pt interactions, and is robust to the microscopic
details of different lateral alignments. This is a key result of this work.

Figure 4: Top: Energies calculated at different rotation angles (with $\theta$
defined in the bottom figure) of the Cr12 interface magnetic moment relative
to the energy at 0 degrees. 0 degrees corresponds to the antiferromagnetic
structure of Cr2O3, while 180 degrees corresponds to Cr12 reversed (pointing
downwards in this case). Bottom: Exchange interaction coupling parameters,
described further in the text, for both bulk-like (Cr7) and surface/interface
(Cr12) magnetic sites.

Exchange interactions: Next, we calculate the isotropic Heisenberg spin
exchange interactions for Cr12 moments in the (0001)Cr2O3-(111)Pt
heterostructure and compare them to those in bulk Cr2O3 in order to
rationalize the flipping of the interface moments. We take the convention with
the Heisenberg contribution to the energy given by

$H_{\mathrm{Heis}}=\sum_{\left<i,j\right>}J_{ij}\mathbf{\hat{e}}_{i}\cdot\mathbf{\hat{e}}_{j},$
(2)

where $J_{ij}$ is the Heisenberg exchange coupling parameter between magnetic
moments on sites $i$ and $j$, and $\mathbf{\hat{e}}_{i}$ is the unit vector
parallel to the magnetic moment of the Cr ion at site $i$. Note that the
convention of Eq. 2 absorbs the value of spin into the exchange constant, and
thus when comparing this work’s $J_{ij}$ values to literature values where the
Hamiltonian is written as
$H_{\mathrm{Heis}}=\sum_{\left<i,j\right>}J_{ij}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$,
such as Ref. 42, it is necessary to divide by $S^{2}$ with $S$ being the spin
of the Cr^3+ ion. Also, with the convention of Eq. 2, positive values of
$J_{ij}$ indicate preferred antiferromagnetic alignment for moments $i$ and
$j$, whereas negative values indicate preferred ferromagnetic alignment. We
use the four-state total energy method [58] whereby the exchange coupling
parameter $J_{ij}$ of the magnetic moments on atoms $i$ and $j$ is calculated
by constraining the magnetic moments of atoms $i$ and $j$ to all four possible
collinear configurations $\uparrow\uparrow$, $\downarrow\downarrow$,
$\downarrow\uparrow$, and $\uparrow\downarrow$ while all other magnetic
moments remain the same. Then

$J_{ij}=\frac{E(\uparrow\uparrow)+E(\downarrow\downarrow)-E(\downarrow\uparrow)-E(\uparrow\downarrow)}{4N},$
(3)

with $N$ being the coupling degeneracy (the number of symmetry-equivalent
nearest neighbors corresponding to the $J_{ij}$ of interest) and $E$ the total
DFT energy for each $(ij)$ configuration [42, 58].
There are five relevant exchange interaction coupling parameters $J_{1}$ to
$J_{5}$ for bulk Cr2O3,with $J_{n}$ denoting the $n$th nearest neighbor.
Couplings beyond $J_{5}$ can be safely neglected [42]. At the Cr-terminated
surface of Cr2O3, the dominating, antiferromagnetic $J_{1}$ and $J_{2}$
interactions are cut off and only $J_{3,surf}$ to $J_{5,surf}$ are retained,
as illustrated in Fig. 4. As a result, Cr12 in vacuum-terminated (0001)Cr2O3
is only weakly coupled to other Cr moments, and consequently the surface
magnetization is effectively paramagnetic at room temperature, even though the
bulk Néel vector is magnetically ordered [12].
We first calculate the Heisenberg exchange parameters for a Cr site in the
center of the Cr2O3 slab (specifically Cr7, Fig. 4) to see whether Pt affects
the magnetic structure far from the interface. The values (we show only
$J_{1}$ and $J_{2}$ in Tab. 2 as these are the dominant couplings in bulk) are
very close to our previous findings in bulk Cr2O3 see Tab. 2. Consequently, we
can be confident that the heterostructure we use in our DFT calculations is
sufficiently thick to reproduce bulk-like behavior of Cr2O3 in the middle of
the slab (this is also reflected in the layer-projected density of states in
Fig. 3 in Sec. III.2).
In contrast, the surface exchange parameters for the interface Cr12 are
strongly affected by the presence of Pt. The most significant change with
respect to the values in a relaxed vacuum-terminated slab of (0001)Cr2O3 (Tab.
2, Ref. [12]) is for $J_{3,surf}$, which increases by more than a factor of
$20$ in the heterostructure and becomes ferromagnetic. This indicates a
preference for Cr12 and the Cr11 just below to align parallel, consistent with
the observed energy lowering in Fig. 4 for flipping Cr12 with respect to the
antiferromagnetic order. $J_{4,surf}$ ($J_{5,surf}$) on the other hand favor
ferromagnetic (antiferromagnetic) alignment between Cr12 and Cr10 (Cr 9), both
of which favor the normal orientation $\theta=0$ for Cr12. However, by
calculating the contribution to the Heisenberg energy if Cr12 is flipped,
based on Eq. 2 and the coupling degeneracies in Tab. 2, we obtain

$H_{\mathrm{Heis}}^{\mathrm{Cr12,flipped}}=3J_{3,surf}-3J_{4,surf}+J_{5,surf}\\\
=-2.2\mathrm{meV}.$ (4)

Thus, the flipping of the top layer next to the (0001)Cr2O3-(111)Pt interfaces
is overall energetically favorable due to the strong ferromagnetic
$J_{3,surf}$ in the presence of Pt 111We must mention that the rigorous
quantitative accuracy of $J_{ij}$ as obtained from Eq. 3 relies on the
magnetic and electric properties of all atoms other than $i$ and $j$ remaining
unchanged for the calculations. If this is the case, then
$E(\theta=0)-E(\theta=180^{\circ})$, which as shown in Fig. 4 is about 7
$\mathrm{meV}$, should be precisely equal to
$H_{\mathrm{Heis}}^{\mathrm{Cr12},\theta=0}-H_{\mathrm{Heis}}^{\mathrm{Cr12},\theta=180^{\circ}}$.
However, using the values in Tab.2, we find that the difference between
flipped and antiferromagnetic configurations of Cr12 is 4.4 $\mathrm{meV}$.
The slight difference comes from the fact that in some of the configurations
used in the 4-state method, specifically $E(\downarrow\uparrow)$, the Pt
magnetization reverses (discussed in Sec. III.3.2) with respect to its
orientation in the three other magnetic configurations. Thus, our $J$ values
include a small contribution due to Cr2O3-Pt interactions which does not
cancel. However, the _relative_ values of $J_{3,surf}$, $J_{4,surf}$, and
$J_{5,surf}$, which are what determines the energetic favorability of the
flipped Cr2O3 configuration in the presence of Pt, should be reliable..
Note however, that the magnitude $\sim 2$ $\mathrm{meV}$ of the effective
coupling for $Cr12$ in the (0001)Cr2O3-(111)Pt bilayer is very similar to the
magnitude of effective coupling for Cr12 with vacuum termination[12]. Thus, we
would expect Cr12 to also be effectively paramagnetic near the bulk Néel
temperature in the presence of Pt. The key difference is that at low
temperature where Cr12 is ordered, the interaction with Pt flips the effective
coupling, and hence the ground-state direction of Cr12, with respect to the
vacuum-terminated case.

Table 2: Exchange parameters $J_{n}$ in bulk Cr2O3, the (0001)Cr2O3-(111)Pt bilayer with hollow hcp (Pt on O) lateral alignment, and a relaxed vacuum-terminated (0001)Cr2O3 slab as calculated in Ref. [12]. $N$ indicates the coupling degeneracy. Negative values of $J$ indicate ferromagnetic couplings, and positive values refer to antiferromagnetic couplings. (Note that in Ref. [12] we used a different convention for the Heisenberg Hamiltonian such that all reported values were divided by two and multiplied by minus one relative to this table.) |  | bulk Cr2O3 [12] |  Cr2O3-Pt |  Cr2O3-vac. [12]
---|---|---|---|---
$J_{n}$ | $N$ |  $J$ [meV] |  $J$ [meV] |  $J$ [meV]
$J_{1}$ | 1 | 20.92 | 21.10 | $-$
$J_{2}$ | 3 | 15.76 | 15.98 | $-$
$J_{3,surf}$ | 3 | $-$ | -10.50 | 0.30
$J_{4,surf}$ | 3 | $-$ | -9.01 | -8.88
$J_{5,surf}$ | 1 | $-$ | 2.20 | 0.78

####  III.3.2 Effect of Cr2O3 on Pt – Magnetic proximity effect

Finally, we briefly investigate the influence of antiferromagnetic Cr2O3 on
Pt. A spin polarization of Pt induced via proximity to Cr2O3, that is a
magnetic proximity effect is a-priori likely, considering that Pt is close to
the Stoner criterion. Our DFT calculations reveal a sizeable magnetization of
the order of $\pm$ 0.01 $\mu_{B}$ per atom in the first three Pt atomic layers
at the interface with Cr2O3 (Fig. 5). Although Pt is strained to Cr2O3,
magnetization caused by strain can be excluded, as a calculation of a Pt slab
strained to match the Cr2O3 lattice constant gives zero magnetization in Pt
without the presence of Cr2O3. From the fourth layer onward, the magnetization
becomes negligible, dropping down to the order of $\pm$ 0.001 $\mu_{B}$. We
find that the Pt magnetic polarization is confined to the first three layers
at the interface in all other lateral alignments as well, see appendix. Thus,
based on our calculations, Pt in (0001)Cr2O3-(111)Pt heterostructures exhibits
a clear and localized magnetic proximity effect (MPE).
Moreover, the magnetization in Pt is inherently coupled to the
antiferromagnetic domain state of Cr2O3. When the domain state of Cr2O3 is
switched (Domain A $\rightarrow$ Domain B), the magnetization values in Pt
have the same magnitude but the opposite sign (Fig. 5). The coupling of Pt to
the Cr interface magnetic moment, Cr12, is antiferromagnetic for all three Pt
layers with significant spin polarization with, interestingly, the maximum
spin polarization occurring in the second Pt layer from the Cr2O3-Pt
interface, rather than in the Pt layer directly adjacent to Cr2O3.
Lastly, related to the prior section III.3.1 in which we found that for a
given domain, the interface Cr12 moments flip with respect to their standard
antiferromagnetic order, we also investigate the effect on spin polarization
in Pt when Cr12 has the flipped $(\theta=180^{\circ})$, rather than
antiferromagnetic $(\theta=0^{\circ})$ orientation. Importantly for
experimental implications, from the dark blue lines in Fig. 5 we see that
flipping the Cr12 magnetic moment into its lowest energy configuration has a
similar effect on Pt as switching the domain state of Cr2O3. In addition, the
Cr2O3-Pt interface with flipped Cr interface magnetic moment has an enhanced
magnetic proximity effect in the first two Pt layers compared to the Cr2O3-Pt
interface with antiferromagnetic order in Cr2O3.

Figure 5: Magnetization in Pt for the two different domain states in Cr2O3 for
the antiferromagnetic configuration and the configuration with Cr12 flipped
interface magnetic moment. Layer 1 is located towards Cr2O3 and layer 6
towards vacuum. A magnetic proximity effect is present in the first three Pt
layers and changes sign when the Cr12 magnetic moment is flipped.

##  IV Discussion and Conclusions

In summary, we have carried out an in-depth computational exploration of the
interface energetics, and the electronic and magnetic properties of
(0001)Cr2O3-(111)Pt heterostructures.

Our analysis of all possible pristine lateral alignments shows that interface
stackings (both fcc and hcp) with Pt directly on top of O are both very close
in energy to each other, as well as being substantially lower in energy than
all other lateral alignments. This finding indicates that the position of Pt
relative to O is more relevant in the heterostructure energetics than whether
the alignment is hollow hcp or hollow fcc. We also stress again, that the
experimental relevance of our findings on lateral alignment stability must be
explored further in followup studies by taking into account likely
possibilities of surface reconstruction and other disorder.
Our calculations for the lowest-energy hollow hcp (Pt on O) alignment indicate
substantial electronic and magnetic effects due to Cr2O3-Pt interactions in
the layers closest to the interface. Beyond the interface region of one Cr2O3
layer and three Pt layers, both Pt and Cr2O3 are bulklike.
One of our most potentially consequential findings is that a reversed Cr
interface magnetic moment in (0001)Cr2O3-(111)Pt heterostructures is
energetically more favorable than the usual antiferromagnetic structure of
Cr2O3. This is reflected also in our calculation of the Heisenberg coupling
constants for the interface Cr, in particular, a strongly ferromagnetic
$J_{3,surf}$ which favors a flipping of the interface Cr compared to its
alignment in the bulk or with vacuum termination. The microscopic mechanism of
this flipping needs to be investigated further. Qualitatively, it is
presumably tied to the observed accumulation of charge density between Cr and
Pt interface moments.
Our finding of the flipping of interface Cr for (0001)Cr2O3-(111)Pt
heterostructures suggests that heavy metals cannot be assumed to be passive
readout elements for surface magnetic properties, but rather, their presence
can drastically alter the magnetism at the surface. For (0001)Cr2O3-(111)Pt
heterostructures in particular, our findings (Fig. 5) reveal that a given
antiferromagnetic domain with the vacuum-terminated (0001)Cr2O3 surface moment
orientation produces a qualitatively identical response in Pt as the opposite
(0001)Cr2O3 domain with the flipped interface moment orientation. Thus, it is
important to verify whether for example established field-cooling procedures
are actually switching bulk Cr2O3 domains, or simply switching the interface
Cr with a fixed bulk domain.
Additionally, we find a substantial magnetization $\sim 0.01\mu_{B}$ in the
first three layers of Pt induced via Cr2O3. This calculated magnetic proximity
effect is consistent with experiments utilizing the magneto-optical Kerr
effect (MOKE) measurements [31], though it must be emphasized that spurious
signals in Kerr and soft x-ray measurements can arise due to dirty samples and
be confused with magnetic signals from a pure sample. In contrast, other
measurements using x-ray magnetic circular dichrosim (XMCD) do not find a
significant MPE in (0001)Cr2O3-(111)Pt heterostructures [30]. We hope our work
motivates experimentalists to look more closely into the possible spin
polarization of Pt, which via breaking of time-reversal symmetry would allow
the AHE in Pt on (0001)Cr2O3.
If an MPE in Pt for (0001)Cr2O3-(111)Pt exists, an interesting experiment to
test whether it contributes substantial at low temperatures to the transverse
voltage signal is to compare Pt electrical resistance at depths within and
beyond three layers from the interface. Based on the apparent penetration
depth of the MPE, the electrical response within the first three layers of Pt
should contain an additional component due to the AHE induced by the
magnetized Pt. This will disappear in deeper regions of the Pt as the MPE
falls off.
To conclude, our first-principles investigation of structural, electronic and
magnetic properties of (0001)Cr2O3-(111)Pt heterostructures can help with
interpretation of magnetotransport experiments in assessing the sign of the
Néel vector, as well as provide guidance in the future design of AFM
spintronic devices. We also hope that our work inspires further investigation,
both theoretical and experimental, of how the interactions of
antiferromagnetic and heavy metal materials can lead to unexpected electric
and magnetic phenomena at their interfaces.

## Acknowledgments

We thank Denys Makarov and Oleksandr Pylypovskyi for useful discussions. This
work was funded by the ERC under the European Union’s Horizon 2020 research
and innovation programme with grant No. 810451, and by ETH Zürich.
Computational resources were provided by ETH Zürich’s EULER cluster.

## Appendix

##  Appendix A DFT Details: Assessment of functionals

We relaxed the lattice parameters of bulk Pt and bulk Cr2O3 with different DFT
functionals and compared them to experimental values to determine the
functional that best describes the Cr2O3-Pt heterostructure. Tables 3 and 4
give the values of the lattice parameters of Cr2O3 and Pt obtained with the
local (spin) density approximation (L(S)DA) and the Perdew-Burke-Ernzerhof
(PBE) functional. A Hubbard U correction is applied on the Cr 3d orbitals in
both cases, as described in Sec. II. We used the VASP Cr_sv and O PAW
pseudopotentials as established in previous studies [12], and compared the
standard pseudopotential (Pt) and extended pseudopotential (Pt_pv) for
platinum for the LDA and the PBE functional. The extended pseudopotentials
ending with _pv and _sv additionally take into account semi-core p and s
electrons respectively as valence electrons in addition to the usual outer
shell electrons.

Table 3: Relaxed lattice parameters of Cr2O3 computationally obtained with LDA and PBE functionals compared to the experimental value determined in Ref. [60]. $\Delta$a and $\Delta$c describe the relative difference between the computational and experimental lattice parameters. | a [Å] |  $\Delta$a | c [Å] |  $\Delta$c
---|---|---|---|---
LSDA | 4.9156 | –0.84 % | 13.5186 | –0.54 %
PBE | 5.0340 | +1.55 % | 13.8026 | +1.55 %
Experiment [60] | 4.9572 | – | 13.5917 | –
Table 4: Relaxed lattice parameters of Pt computationally obtained with LDA and PBE functionals in combination with Pt and Pt_pv PAW pseudopotentials compared to the experimental value determined in Ref. [61]. $\Delta$a describes the relative difference between the computational and experimental lattice parameters. | a [Å] |  $\Delta$a
---|---|---
LDA (Pt) | 3.8326 | –2.3 %
LDA (Pt_pv) | 3.9096 | –0.4 %
PBE (Pt) | 3.9137 | –0.3 %
PBE (Pt_pv) | 3.9771 | +1.4%
Experiment [61] | 3.9236 | –

The results in Tables 3 and 4 show that the LDA functional with the extended
Pt_pv pseudopotential yields lattice parameters closest to the experimental
lattice parameters. Based on these results, the L(S)DA functional (+U
correction) as well as Cr_sv, O, and Pt_pv pseudopotentials are selected to
model the Cr2O3-Pt heterostructure.

##  Appendix B Electronic properties - More lateral alignments

While Sec. III.2 focuses on the electronic properties of the lowest-energy
lateral alignment, here we present the electronic properties of the other
lateral alignments. We find an electron accumulation in the Cr2O3 interface
layer for all lateral alignments, with the detailed shape of the electron
accumulation depending on the lateral alignment, as shown in Fig. 6. The
hollow fcc and hollow hcp configurations have a triangular-shaped accumulation
of electrons, with the triangle corners pointing towards the Pt atoms. The
hollow fcc (Pt between O) and hollow hcp (Pt between O) lateral alignment show
similar accumulations to each other; likewise the hollow fcc (Pt on O) and
hollow hcp (Pt on O) alignments are similar. Consequently, the relative O-Pt
position plays a larger role in the shape of the electron accumulation than
whether the lateral alignment is of fcc or hcp type. The electron accumulation
in the ‘top’ lateral alignment occurs in a doughnut-like shape between the Pt
atom above and the Cr12 ion below. The shape of the electron accumulation
stays similar upon variation of the isosurface value for all different lateral
alignments. We note that the layer-projected density of states also shows a
finite DOS in the Cr2O3 interface layer for all lateral alignments, consistent
with the observed electron accumulations in Fig. 6.

Overall, the electron accumulations appear in proximity to the Cr12 ion
irrespective of whether the Cr12 magnetic moment is flipped or not, and their
shape is controlled by the platinum interface lateral alignment.

Figure 6: Charge density differences at the interface between Cr12 and the
first Pt layer for (0001)Cr2O3-(111)Pt heterostructures with different lateral
alignments. Yellow represents an accumulation of electrons, while cyan
represents a depletion of electrons The isosurface value in VESTA is set to
0.006 $\frac{e}{${\mathrm{\SIUnitSymbolAngstrom}}^{3}$}$.

##  Appendix C Magnetic properties - Thicker slabs and supercell

In the main text, Cr2O3-Pt heterostructures with six Cr2O3 layers are
evaluated. In this section, the effect of an increasing number of Cr2O3 layers
is analyzed.

The resulting energy differences between antiferromagnetic order and magnetic
order with Cr12 flipped magnetic moment of four slab structures with different
numbers of Cr2O3 layers are listed in Table 5. The flipped Cr12 magnetic
moment is energetically favorable for all four of them. The energy difference
is similar for all four, and importantly there is no trend in increasing or
decreasing energy difference that would suggest an additional thickness-
dependent contribution to the energy.

Table 5: Energy difference $\Delta$E between the flipped and antiferromagnetic Cr12 configurations for different slabs of the lowest energy lateral alignment (hollow hcp (Pt on O)) with four Pt layers. No. Cr2O3 layers | Energy difference $\Delta$E [meV]
---|---
6 | -18.3
7 | -15.0
9 | -16.1
12 | -19.1

In addition, we analyzed the Cr interface magnetic moments in a 2x1x1
supercell to check whether a combination of flipped and AFM interface
$\mathrm{Cr}$ atoms is lower in energy than flipping all $\mathrm{Cr}$
moments. We found that the 2x1x1 supercell has lowest energy for both top Cr12
flipped, intermediate energy for one Cr12 flipped and one with the AFM
configuration, and highest energy for both primitive magnetic cells having the
bulk AFM order for the Cr12 moments. Thus, it seems that lateral spin
rearrangements of Cr at the (0001)Cr2O3-(111)Pt which lead to a mixed flipped-
AFM interface do not lower the energy compared to a uniformly flipped Cr12
layer. Moreover, the energy difference between both Cr flipped and both Cr in
the bulk AFM configuration for the 2x1x1 supercell is almost exactly twice as
large as the energy difference for the single magnetic unit cell used in the
main text. This implies that no other energy contributions (for example, a
non-negligible magnetic exchange between adjacent Cr12 moments mediated via
RKKY interactions) which were absent in the primitive magnetic cell arise in
the supercell containing two Cr12 moments. Thus, interactions between
neighboring surface Cr moments in the heterostructure, like the case of the
vacuum-terminated structure, can be safely neglected, and the observed
flipping of Cr surface moments for the single magnetic unit cell should occur
for all Cr12 moments at an experimental (0001)Cr2O3-(111)Pt interface.

##  Appendix D Magnetic properties - Further analysis of MPE

Section III.3.2 described the observation of a magnetic proximity effect (MPE)
for a (0001)Cr2O3-(111)Pt heterostructure with six Pt layers in a hollow hcp
(Pt on O) lateral alignment. This section analyzes the MPE of the other
lateral alignments as well as its dependence on the thickness of Pt.

Fig. 7 and Fig. 8 show the calculated magnetic moment per Pt atom as a
function of the Pt layer number for different lateral alignments. For all
cases, the Pt magnetization points along the [0001] hexagonal direction,
perpendicular to the (0001)Cr2O3-(111)Pt interface, with negligible in-plane
components. We see that the MPE is present for all lateral alignments, and in
all cases occurs in the first three Pt layers adjacent to Cr2O3. The hollow
fcc (Pt on O) magnetization values (purple) and hollow hcp (Pt on O)
magnetization values (brown) have the same dependence on distance from the
interface; likewise, the hollow fcc (Pt between O) values (green) and hollow
hcp (Pt between O) values (red) have the same dependence. This suggests that
the relative alignment of Pt and O affects the MPE more than whether the
alignment is of fcc or hcp type. This is analogous to the behavior of the
electron accumulation described in Sec. B.

Figure 7: Magnetization of Pt atoms in the hexagonal [0001] direction as a
function of Pt layer for all different lateral alignments with
antiferromagnetic order in Cr2O3. A magnetic proximity effect is pronounced in
the first three Pt layers. Figure 8: Magnetization of Pt atoms in the
hexagonal [0001] direction as a function of Pt layer number for different
lateral alignments. The Cr2O3 slab has the Cr12 magnetic moment flipped, in
agreement with our calculated ground state for the heterostructure. A magnetic
proximity effect is pronounced in the first three Pt layers.

In the main text a heterostructure with six Pt layers was analyzed. Here, we
vary the number of Pt layers to see whether the MPE is affected, in particular
to determine whether the thickness of the penetration depth of the MPE depends
on the thickness of Pt. Fig. 9 and Fig. 10 show the magnetization in Pt for up
to eight Pt layers for a heterostructure with antiferromagnetic Cr2O3 and a
heterostructure with Cr2O3 with flipped Cr12 magnetic moment, respectively.
Regardless of the magnetic order in Cr2O3 and regardless of the thickness of
Pt, the MPE is pronounced only in the first three Pt layers adjacent to Cr2O3.
As described in Sec. III.3, Pt is coupled antiferromagnetically to the Cr12
interface magnetic moment and the flipped Cr12 magnetic moment increases the
MPE, which explains the change in sign as well as the difference in magnitude
between Fig. 9 and Fig. 10.

Figure 9: Magnetization of Pt atoms in the hexagonal [0001] direction as a
function of Pt layer number for different numbers of Pt layer thicknesses with
antiferromagnetic order in Cr2O3. A magnetic proximity effect is pronounced in
the first three Pt layers. Figure 10: Magnetization of Pt atoms in the
hexagonal [0001] direction as a function of Pt layer number for different Pt
layer thicknesses with flipped Cr12 magnetic moment in Cr2O3. A magnetic
proximity effect is pronounced in the first three Pt layers.

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