# A Data-Efficient Approach for Long-Term Human Motion Prediction Using Maps
of Dynamics

Yufei Zhu1, Andrey Rudenko2, Tomasz Piotr Kucner3, Achim J. Lilienthal1,4,
Martin Magnusson1 1AASS MRO lab, Örebro University, Sweden
<EMAIL_ADDRESS>Corporate Research, Robert Bosch GmbH, Stuttgart,
Germany<EMAIL_ADDRESS>University, Finland4Chair “Perception
for Intelligent Systems”, TU Munich, GermanyThis work has received funding
from the European Union’s Horizon 2020 research and innovation programme under
grant agreement No 101017274 (DARKO).

###### Abstract

Human motion prediction is essential for the safe and smooth operation of
mobile service robots and intelligent vehicles around people. Commonly used
neural network-based approaches often require large amounts of complete
trajectories to represent motion dynamics in complex semantically-rich spaces.
This requirement may complicate deployment of physical systems in new
environments, especially when the data is being collected online from onboard
sensors. In this paper we explore a data-efficient alternative using _maps of
dynamics_ (MoD) to represent place-dependent multi-modal spatial motion
patterns, learned from prior observations. Our approach can perform efficient
human motion prediction in the long-term perspective of up to 60 seconds. We
quantitatively evaluate its accuracy with limited amount of training data in
comparison to an LSTM-based baseline, and qualitatively show that the
predicted trajectories reflect the natural semantic properties of the
environment, e.g. the locations of short- and long-term goals, navigation in
narrow passages, around obstacles, etc.

##  I INTRODUCTION

Long-term human motion prediction (LHMP) is important for autonomous robots
and vehicles to operate safely in populated environments [1]. Accurately
predicting the future trajectories of people in their surroundings over
extended time periods is essential for enhancing motion planning, tracking,
automated driving, human-robot interaction, intelligent safety monitoring and
surveillance.

Human motion is complex and may be influenced by several hard-to-model
factors, including social rules and norms, personal preferences, and subtle
cues in the environment that are not represented in geometric maps. To address
these challenges, popular neural network approaches learn motion dynamics
directly from data, with many recent studies developing models based on LSTMs
[2], GANs [3], CNNs [4], CVAEs [5] and transformers [6]. Most of these
approaches focus on learning to predict stochastic interactions between
diverse moving agents in the short-term perspective in scenarios where the
effect of the environment topology and semantics is minimal.

When predicting long-term human motion in complex, large-scale environments,
the influence of the surrounding space (e.g. passages, stairs, entrances,
various objects and semantically-meaningful areas) on human motion goes beyond
what is contained in the current state of the moving person or the observed
interactions. This impact of the environment has to be modelled explicitly,
for instance by informing the prediction method with a semantic map [7, 8, 9].
Another effective approach to address this challenge is to use _maps of
dynamics_ (MoDs). MoDs [10] are maps that encode spatial or spatio-temporal
motion patterns as a feature of the environment. MoD-informed long-term human
motion prediction (MoD-LHMP) approaches are particularly suited to predict
motion in the long-term perspective, where the environment effects become
critical for making accurate predictions. MoDs efficiently encode the
stochastic local motion patterns over the entire map, informing the predictor
in areas which may have no influence on the immediate decisions of the walking
people, but become critical in the long-term perspective.

As a proof of concept for MoD-LHMP, we propose to build CLiFF MoDs [11] from
training data and use them to bias a constant velocity motion prediction
method, generating stochastic trajectory predictions for up to
$60\text{\,}\mathrm{s}$ into the future.

Figure 1: Maps of dynamics provide an efficient and lightweight encoding of
sparse and incomplete velocity data to characterize the motion flows in the
environment. We propose a method to predict long-term multi-modal human motion
using data-efficient CLiFF maps [11]. Left: trajectories from the ATC dataset
used for training. Right: CLiFF map.

One crucial advantage of the MoD-LHMP approach is its data efficiency. Prior
art neural network-based approaches often require large amounts of data for
training, and their performance can significantly degrade in absence thereof.
Typically, these approaches also need complete sequences of tracked positions
for training. The proposed MoD-LHMP approach, on the other hand, allows
encoding human motion from sparse and incomplete data, requiring only observed
velocities in discrete locations and interpolating the missing motion in
between. This property is relevant, for instance, when the deployed robot
collects the data in an online fashion from on-board sensors and with a
limited field of view.

In this work, we evaluate the efficiency of MoD-based motion encoding for
making accurate long-term predictions. In our experiments we sample few
trajectories from the ATC dataset and use them to build a CLiFF map and train
LSTM-based baselines. We then compare these methods using the ADE/FDE
prediction accuracy metrics. Furthermore, we qualitatively demonstrate that
the CLiFF-LHMP approach has the ability to predict human motion in complex
environments over very long time horizons, implicitly inferring common goal
points and correctly predicting trajectories that follow the complex topology
of the environment, e.g. navigating around corners or obstacles or passing
through narrow passages such as doors.

##  II METHOD

###  II-A Maps of Dynamics

In the proposed approach for human motion prediction, we exploit Maps of
Dynamics (MoD) which encode human dynamics as a feature of the environment. By
using velocity observations, human dynamics can be represented through flow
models. In this work, we employ Circular-Linear Flow Field map (CLiFF-map)
[11] to represent the flow of human motion. CLiFF-map represents local flow
patterns as a multi-modal, continuous joint distribution of speed and
orientation. As the orientation of velocity is a circular variable, and
magnitude of velocity is a linear variable, CLiFF-map associates a semi-
wrapped Gaussian mixture model (SWGMM) with each location, describing flow
patterns around the given location, see Fig. 1. By using SWGMM, CLiFF-map is
able to properly address multimodality in the data, thereby enhancing its
capability to predict uncertain long-term human motion. A CLiFF-map represents
motion patterns based on local observations and estimates the likelihood of
motion at a given query location. As it can be built from incomplete or
spatially sparse data, CLiFF-map efficiently captures human motion patterns
without requiring large amounts of data or complete trajectories. This
characteristic makes CLiFF-LHMP a data-efficient approach for predicting human
motion.

###  II-B Motion Prediction

We frame the task of predicting a person’s future trajectory as using a short
observed trajectory to infer a sequence of future states. The length of the
observation history is $O_{s}\in\mathbb{R}^{+}$ $\text{\,}\mathrm{s}$,
equivalent to an integer $O_{p}>0$ observation time steps. With the current
time-step denoted as the integer $t_{0}\geq 0$, the sequence of observed
states is $\mathcal{H}=\langle s_{t_{0}-1},...,s_{t_{0}-O_{p}}\rangle$, where
$s_{t}$ is the state of a person at time-step $t$. A state is represented by
2D Cartesian coordinates $(x,y)$, speed $\rho$ and orientation $\theta$:
$s=(x,y,\rho,\theta)$.

From the observed sequence $\mathcal{H}$, we derive the observed speed
$\rho_{\mathrm{obs}}$ and orientation $\theta_{\mathrm{obs}}$ at time-step
$t_{0}$. Then the current state becomes
$s_{t_{0}}=(x_{t_{0}},y_{t_{0}},\rho_{\mathrm{obs}},\theta_{\mathrm{obs}})$.
The values of $\rho_{\mathrm{obs}}$ and $\theta_{\mathrm{obs}}$ are calculated
as a weighted sum of the finite differences in the observed states, as in the
recent ATLAS benchmark [12], such that
$\rho_{\mathrm{obs}}=\sum_{t=1}^{O_{p}}v_{t_{0}-t}g(t)$ and
$\theta_{\mathrm{obs}}=\sum_{t=1}^{O_{p}}\theta_{t_{0}-t}g(t)$, where
$g(t)=(\sigma\sqrt{2\pi}e^{\frac{1}{2}(\frac{t}{\sigma})^{2}})^{-1}$.

Given the current state $s_{t_{0}}$, the goal is to estimate a sequence of
future states. Future states are predicted for a given horizon
$T_{s}\in\mathbb{R}^{+}$ $\text{\,}\mathrm{s}$. $T_{s}$ is equivalent to
$T_{p}>0$ prediction time steps assuming the constant time interval $\Delta t$
between two predictions. Thus, the prediction horizon is $T_{s}=T_{p}\Delta
t$. The future sequence is then denoted as $\mathcal{T}=\langle
s_{t_{0}+1},s_{t_{0}+2},...,s_{t_{0}+T_{p}}\rangle$.

Input: $\mathcal{H}$, $x_{t_{0}}$, $y_{t_{0}}$

Output: $\mathcal{T}$

1 $\mathcal{T}=\\{\\}$

2$\rho_{\mathrm{obs}},\theta_{\mathrm{obs}}\leftarrow$
getObservedVelocity($\mathcal{H}$)

3$s_{t_{0}}=(x_{t_{0}},y_{t_{0}},\rho_{\mathrm{obs}},\theta_{\mathrm{obs}})$

4for  _ $t=t_{0}+1$, …, $t_{0}+T_{p}$_ do

5

6 $x_{t},y_{t}\leftarrow$ getNewPosition($s_{t\textendash 1}$)

7 $\theta_{s}$ $\leftarrow$ sampleVelocityFromCLiFFmap($x_{t},y_{t}$)

8 ($\rho_{t}$, $\theta_{t}$) $\leftarrow$ predictVelocity($\theta_{s}$,
$\rho_{t\textendash 1}$, $\theta_{t\textendash 1}$)

9 $s_{t}\leftarrow(x_{t},y_{t},\rho_{t},\theta_{t})$

10 $\mathcal{T}\leftarrow\mathcal{T}\cup s_{t}$

11return $\mathcal{T}$

Algorithm 1 CLiFF-LHMP

The CLiFF-LHMP algorithm is presented in Alg. 1. With the input of a CLiFF-map
and past states of a person, the algorithm predicts a sequence of future
states. To estimate $\mathcal{T}$, for each prediction time step, we sample a
velocity from the CLiFF-map at the current position ($x_{t}$, $y_{t}$) to bias
the prediction with the learned motion patterns represented by the CLiFF-map.
To sample a velocity at a given location $(x,y)$, we first get the SWGMMs
$\Xi_{\mathrm{near}}$ whose distances to $(x,y)$ are less than $r_{s}$, where
$r_{s}$ is the sampling radius. After getting the sampled velocity, the
velocity ($\rho_{t}$, $\theta_{t}$) is predicted by assuming that a person
will continue walking with the same speed as in the last time step,
$\rho_{t}=\rho_{t-1}$, and biasing the direction of motion with the sampled
orientation $\theta_{s}$ as:

$\begin{gathered}\theta_{t}=\theta_{t-1}+(\theta_{s}-\theta_{t-1})\cdot
K(\theta_{s}-\theta_{t-1}),\\\ \end{gathered}$ (1)

where $K(\cdot)$ is a kernel function that defines the degree of impact of the
CLiFF-map. We use a Gaussian kernel with a parameter $\beta$ that represents
the kernel width:

$K(x)=e^{-\beta\left\|x\right\|^{2}}.\\\ $ (2)

With kernel $K$, we scale the CLiFF-map term by the difference between the
velocity sampled from the CLiFF-map and the current velocity according to a
constant velocity model (CVM). The sampled velocity is trusted less if it
deviates more from the current velocity. A larger $\beta$ value makes the
method behave more like a CVM, and a smaller $\beta$ makes it more closely
follow the CLiFF-map.

##  III Evaluation

In this section, we evaluate the data efficiency and accuracy of the proposed
CLiFF-LHMP approach and compare it to the LSTM-based human motion prediction
methods. Vanilla LSTM [13] is used as the baseline representative of LSTM-
based methods.

###  III-A Implementation Details

We evaluate the prediction performance using the ATC dataset [14], which
contains trajectories recorded in a shopping mall in Japan. The dataset covers
a large indoor environment with a total area of around
$900\text{\,}{\mathrm{m}}^{2}$. The ATC dataset consists of 92 days in total.
Given the immense length of the ATC dataset for each recording day, a subset
covering the first four days can be considered representative. We use the
subset in the experiments, with the first day (Oct.24) for training, and the
remaining 3 days for testing. Both the LSTM and CLiFF-LHMP approaches are
trained with same data and evaluated with same data to ensure a fair
comparison.

In ATC dataset, the original detection rate is $30\text{\,}\mathrm{Hz}$. We
downsample the data to $2.5\text{\,}\mathrm{Hz}$ to align with
$0.4\text{\,}\mathrm{s}$ observation time interval, as commonly used in human
motion prediction. For each trajectory, we take $3.2\text{\,}\mathrm{s}$ (the
first 8 positions) as the observation history and use the remaining trajectory
(up to the maximum prediction horizon) as the prediction ground truth. Instead
of using a fixed prediction horizon, we explore a wider range of values
$T_{s}$ up to a maximum value in our evaluation. The maximum prediction
horizon is determined based on the tracking duration distribution of the
dataset. We use the 90th percentile value, which is $60\text{\,}\mathrm{s}$,
as maximum prediction horizon for experiments of ATC dataset. As LSTM-based
approaches require complete trajectories for training, we use for all compared
approaches trajectories equal or longer than $60\text{\,}\mathrm{s}$ for both
training and testing.

Given the area and tracking duration in the ATC dataset, when evaluating
CLiFF-LHMP, we set prediction time step $\Delta t$ to $1\text{\,}\mathrm{s}$,
CLiFF-map resolution to $1\text{\,}\mathrm{m}$, sampling radius $r_{s}$ to
$1\text{\,}\mathrm{m}$ and kernel parameter $\beta$ to 1. For training vanilla
LSTM model, we set the dimension of hidden state of the LSTM model set to 128
and the learning rate set to 0.003.

For the evaluation of the predictive performance we use the _Average_ and
_Final Displacement Errors_ (ADE and FDE) metrics. ADE describes the error
between points on the predicted trajectories and the respective ground truth
at the same time step. FDE describes the error at the last prediction time
step.

We stop predicting when the sample reaches an area outside of the MoD, in case
of the CLiFF map, i.e. when no SWGMMs are available within the radius $r_{s}$
around the sampled location. Predicted trajectories that end before $T_{s}$
will only be included in the ADE/FDE evaluation up to the last predicted
point. When predicting for each ground truth trajectory, the prediction
horizon $T_{s}$ is set either equal to its length or $60\text{\,}\mathrm{s}$
for longer trajectories.

###  III-B Experiments and Results

####  III-B1 Efficiency of motion prediction with limited data

To evaluate the data efficiency of the CLiFF-LHMP method, we ran a series of
experiments with varying amount of training data [100, 200, …, 1000
trajectories]. The training data were randomly selected multiple times. Once
selected, we fed the same data to train the CLiFF-map and the LSTM model, and
the evaluation metrics were averaged from all the runs.

Figure 2 shows the ADE and FDE results for CLiFF-LHMP and vanilla LSTM for
prediction horizon of $60\text{\,}\mathrm{s}$, with the number of training
trajectories ranging from 100 to 1000. CLiFF-LHMP consistently outperforms
LSTM when predicting long-term human motion in these cases. When more than 200
training trajectories are used, the standard deviation of ADE and FDE of
CLiFF-LHMP is also lower than for LSTM. While the performance of LSTM drops
substantially for smaller training data sets, especially when training with
fewer than 200 trajectories, CLiFF-LHMP has a stable performance even with as
few as 100 training trajectories. When decreasing the training dataset size
from 1000 to 100 trajectories, the error merely increases 4% in ADE and 1% in
FDE for CLiFF-LHMP, while for LSTM the ADE increases by 35% and the FDE by
27%. Figure 3 shows a comparison on different prediction horizons from
$10\text{\,}\mathrm{s}$ to $60\text{\,}\mathrm{s}$ for three sizes of the
training dataset (200, 600, 1000 trajectories). When the prediction horizon
increases, CLiFF-LHMP becomes slightly more sensitive to the amount of
training data.

Figure 2: ADE/FDE of CLiFF-LHMP and LSTM in the ATC dataset, using different
amounts of trajectories (100–1000) as training data. The prediction horizon is
$60\text{\,}\mathrm{s}$. The shade represents one std. dev.

Figure 3: ADE/FDE of CLiFF-LHMP in the ATC dataset with training dataset of
200, 600, 1000 trajectories and with prediction horizon
10–$60\text{\,}\mathrm{s}$, and $12\text{\,}\mathrm{s}$.

Figure 4: Heatmap of KL divergence of three pairs of CLiFF-map snapshots built
with different amount of training data. Left: KL divergence of between CLiFF-
maps trained with 1000 and 100 trajectories, Middle left: 200 and 100
trajectories, Middle right: 600 and 500 trajectories, and Right: 1000 and 900
trajectories. The color scale indicates the magnitude of KL divergence, with
warmer colors indicating higher values, meaning larger differences between the
CLiFF-map pair. When the size of training dataset increases, the impact on the
CLiFF-map model becomes less significant, showing that the sensitivity of the
CLiFF-map to the amount of training data decreases.

Figure 5: Predictions in ATC with $T_{s}=60$ $\text{\,}\mathrm{s}$. Red lines
show the ground truth trajectory and green line show the observed tracklet.
Prediction trajectories of CLiFF-LHMP and LSTM approaches are shown in blue
and orange, respectively. When the trajectory predicted by LSTM is unfeasible
by crossing the walls, CLiFF-LHMP make predictions along the corridor.

####  III-B2 Efficiency of motion representation

To compare the quality of the underlying CLiFF-map itself, trained with
different amounts of data, we compute the Kullback-Leibler (KL) divergence
[15] between the distributions represented in the CLiFF-maps. The KL
divergence results are shown as heatmaps in Figure 4. CLiFF-map associates a
Gaussian Mixture Model to each location, and we use a KL divergence heatmap to
visualize the changes between two different CLiFF-maps. The first image in
Figure 4 shows the changes of CLiFF-maps built with 100 and 1000 trajectories,
respectively. It is evident that as the number of training trajectories
increases, the primary alterations in the CLiFF-map occur predominantly along
the boundary regions. Moreover, in highly constrained environments, such as
the eastern corridor of the ATC map, the velocity distributions exhibit
comparatively minimal variations. The other three figures in Figure 4 shows
the sensitivity of CLiFF-map to the input data. When the number of training
data increases from 900 to 1000 (see the fourth image in Figure 4), the CLiFF-
map changes less than when the number of training data increases from 100 to
200 (see the second image in Figure 4). This shows that the CLiFF-map can
capture major human motion patterns already with small amounts of training
data.

####  III-B3 Descriptive power of compact motion representation models

Figure 5 shows qualitative examples of predicted trajectories using Maps of
Dynamics in the long-term perspective. As no explicit knowledge is given about
obstacle layout, LSTM predicts unfeasible trajectory which crosses the walls.
In contrast, by exploiting learned motion patterns encoded in the CLiFF-map,
our method predicts realistic trajectories that follow the complex topology of
the environment e.g. navigating around corners or obstacles or passing through
narrow passages such as doors, stairs (in the top part of the map) and exits
(in the left part).

##  IV CONCLUSIONS

In this paper, we present the idea to exploit _Maps of Dynamics_ (MoDs) for
long-term human motion prediction. As a proof of concept for MoD-LHMP, we
propose CLiFF-LHMP. Our method uses the CLiFF-map, a specific MoD that
probabilistically represents human motion patterns within a velocity field.
Our approach involves sampling velocities from the CLiFF-map to bias constant
velocity predictions, generating stochastic trajectory predictions for up to
$60\text{\,}\mathrm{s}$ into the future. We evaluate CLiFF-LHMP using the ATC
dataset, with a vanilla LSTM as the baseline approach. The experiments
highlight the data efficiency advantage of our method. CLiFF-LHMP accuracy is
only affected to a minor degree when using less than 200 trajectories as
training data, while LSTM requires about three times as many trajectories to
approach its optimal performance. The results also demonstrate that our
approach consistently outperforms the LSTM method at the long prediction
horizon of $60\text{\,}\mathrm{s}$. By exploiting learned motion patterns
encoded in the CLiFF-map, our method implicitly accounts for the obstacle
layouts and predicts trajectories that follow the complex topology of the
environment.

The current implementation of MoD-LHMP uses spatial motion patterns that are
built offline based on past observations. In the future we plan to extend the
approach to online life-long learning, enabling live updates based on the
motion observations. One future direction is the evaluation of additional
types of MoDs for long-term human motion prediction, including those capturing
temporally-conditioned motion patterns. Another future direction is to learn
MoDs online for life-long learning enabling updates based on live motion
observations. Additionally, in future work, we aim to formally describe and
analyze the MoD-LHMP methodology, include further datasets [16, 17] in the
evaluation.

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# Structural and magnetic investigation of the interfaces of Fe3O4/MgO(001)
with and without NiO interlayer

Tobias Pohlmann<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Notkestr. 85, 22607 Hamburg, Germany  Department of Physics, Osnabrück
University, 49076 Osnabrück, Germany  Florian Bertram  Deutsches Elektronen-
Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany  Jannis Thien
Department of Physics, Osnabrück University, 49076 Osnabrück, Germany  Jari
Rodewald  Department of Physics, Osnabrück University, 49076 Osnabrück,
Germany  Kevin Ruwisch  Department of Physics, Osnabrück University, 49076
Osnabrück, Germany  Timo Kuschel  Center for Spinelectronic Materials and
Devices, Department of Physics, Bielefeld University, Universitätsstr. 25,
33615 Bielefeld, Germany  Eugen Weschke  Helmholtz-Zentrum Berlin für
Materialien und Energie, Wilhelm-Conrad-Röntgen-Campus BESSY II, Albert-
Einstein-Strasse 15, 12489 Berlin, Germany  Karsten Küpper  Department of
Physics, Osnabrück University, 49076 Osnabrück, Germany  Joachim Wollschläger
Department of Physics, Osnabrück University, 49076 Osnabrück, Germany

###### Abstract

We present an investigation on the structural and magnetic properties of the
interfaces of $\mathrm{Fe_{3}O_{4}/MgO(001)}$ and
$\mathrm{Fe_{3}O_{4}/NiO/MgO(001)}$ by extracting cation-selective
magnetooptical depth profiles by means of x-ray magnetic reflectivity (XRMR)
in combination with charge-transfer multiplet simulations of x-ray magnetic
circular dichroism (XMCD) data. For $\mathrm{Fe_{3}O_{4}/MgO(001)}$, the
magnetooptical depth profiles at the $\mathrm{Fe^{2+}_{oct}}$ and the
$\mathrm{Fe^{3+}_{oct}}$ resonant energies follow exactly the structural
profile, while the magnetooptical depth profile at the
$\mathrm{Fe^{3+}_{tet}}$ resonance is offset by $3.2\pm 1.3\,$Å from the
interface, consistent with a B-site interface termination of
$\mathrm{Fe_{3}O_{4}}$ with fully intact magnetic order. In contrast, for
$\mathrm{Fe_{3}O_{4}/NiO(001)}$, the magnetooptical depth profiles at the
$\mathrm{Fe^{2+}_{oct}}$ and the $\mathrm{Ni^{2+}}$ resonances agree with the
structural profile, but the interface positions of the magnetooptical depth
profiles at the $\mathrm{Fe^{3+}_{oct}}$ and the $\mathrm{Fe^{3+}_{tet}}$
resonances are laterally shifted by $3.3\pm 1.4\,$Å and $2.7\pm 0.9\,$Å,
respectively, not consistent with a magnetically ordered stoichiometric
interface. This may be related to an intermixed $\mathrm{(Ni,Fe)O}$ layer at
the interface. The magnetooptical depth profiles at the Ni $L_{3}$ edge reveal
uncompensated magnetic moments throughout the NiO film.

###### pacs:

Valid PACS appear here

##  I Introduction

Magnetite ($\mathrm{Fe_{3}O_{4}}$) is a half-metallic ferrimagnet in the
inverse spinel structure. This structure consists of a cubic close-packed
oxygen lattice whose interstitial sites are populated by three different iron
species: 1/2 of the octahedral B-sites are occupied randomly by divalent
$\mathrm{Fe^{2+}_{oct}}$ and trivalent $\mathrm{Fe^{3+}_{oct}}$ cations, and
1/8 of the tetrahedral A-sites are occupied by trivalent
$\mathrm{Fe^{3+}_{tet}}$ cations. The two octahedrally coordinated species
$\mathrm{Fe^{2+}_{oct}}$ and $\mathrm{Fe^{3+}_{oct}}$ are ferromagnetically
coupled by double exchange, while the $\mathrm{Fe^{3+}_{oct}}$ and
$\mathrm{Fe^{3+}_{tet}}$ cations couple antiferromagnetically via
superexchange. Therefore, the magnetic moments of the $\mathrm{Fe^{3+}}$
cations compensate each other and the resulting macroscopic moment of
$\mathrm{Fe_{3}O_{4}}$ of $4.07\,\mathrm{\frac{\mu_{B}}{f.u.}}$ is determined
by the magnetic moments of the $\mathrm{Fe^{2+}_{oct}}$ cations Weiss and
Forrer (1929).

Due to these magnetic properties, magnetite is a long-standing candidate to
contribute to all-oxide thin-film spintronic devices, as a source for spin-
polarized currents Moussy (2013); Coey and Chien (2003); Bibes and Bartheélemy
(2007); Žutić  _et al._ (2004); Moyer  _et al._ (2015); Marnitz  _et al._
(2015). These kinds of devices utilize the fact that many metal oxides with
varying electronic and magnetic properties grow in spinel or rock-salt
structures, such as the conducting ferrimagnets $\mathrm{Fe_{3}O_{4}}$ and
$\gamma$-$\mathrm{Fe_{2}O_{3}}$, the insulating ferrimagnets
$\mathrm{NiFe_{2}O_{4}}$ and $\mathrm{CoFe_{2}O_{4}}$, the insulating
antiferromagnets NiO, CoO and FeO, or the insulating diamagnets MgO and
$\mathrm{MgAl_{2}O_{4}}$, which all share a cubic close-packed oxygen lattice
with very similar lattice constants Moussy (2013). This allows epitaxial
growth of film stacks with a large variety of spin electronic functionality
but with little strain and thus supposedly well-matching interfaces.

However, a drawback of this concept is that the structural similarity of these
metal oxides also means that undesired modifications at their interfaces are
difficult to detect, such as interdiffusion of Mg Kim-Ngan  _et al._ (2009),
Ni Farrow  _et al._ (2000); Kuschel  _et al._ (2016) or Co Rodewald  _et al._
(2019) into $\mathrm{Fe_{3}O_{4}}$ films, or the transformation of the
different iron oxides into each other Bertram  _et al._ (2011, 2012). All-
oxide spintronic devices with $\mathrm{Fe_{3}O_{4}}$ electrodes did indeed not
prove to be very successful yet; their shortcomings were speculated to stem
from magnetic dead layers at the substrate interface van der Heijden  _et al._
(1996); van der Zaag  _et al._ (2000) or other interface effects Marnitz  _et
al._ (2015).

Of particular interest have been the interfaces between Fe3O4 films and the
tunnel barrier material MgO Marnitz  _et al._ (2015); van der Zaag  _et al._
(2000); Kado (2008), as well as the interface between Fe3O4 and
antiferromagnetic films, e.g. NiO, exhibiting exchange bias Keller  _et al._
(2002); Gatel  _et al._ (2005); Krug  _et al._ (2008); Kuepper  _et al._
(2016). This effect can cause a shift of the coercive fields of the
ferrimagnetic Fe3O4 film and be used to pin its magnetization state.

In this study, we investigate the structural and magnetic properties of the
$\mathrm{Fe_{3}O_{4}/NiO}$ and $\mathrm{Fe_{3}O_{4}/MgO}$ interfaces. We grow
$\mathrm{Fe_{3}O_{4}}$ single layers and $\mathrm{Fe_{3}O_{4}/NiO}$ bilayers
on MgO(001) by reactive molecular beam epitaxy (RMBE) and investigate the
distribution and magnetic order of the three cations $\mathrm{Fe^{2+}_{oct}}$,
$\mathrm{Fe^{3+}_{tet}}$ and $\mathrm{Fe^{3+}_{oct}}$ of
$\mathrm{Fe_{3}O_{4}}$ and of the $\mathrm{Ni^{2+}}$ cations of NiO by x-ray
resonant magnetic reflectivity (XRMR) combined with charge-transfer multiplet
analysis of x-ray magnetic circular dichroism (XMCD) spectra. We find that the
magnetic structure of $\mathrm{Fe_{3}O_{4}}$ on MgO is intact down to the
interface, while on the $\mathrm{Fe_{3}O_{4}/NiO}$ interface the data
indicates a disturbed magnetic order. XRMR data on the Ni $L_{3}$ edge
indicate uncompensated magnetic moments in the antiferromagnet NiO.

##  II Experimental details

The deposition and characterization methods of the samples followed the ones
presented in Refs. Kuepper  _et al._ (2016); Kuschel  _et al._ (2016). We
prepared $\mathrm{Fe_{3}O_{4}/MgO(001)}$ and
$\mathrm{Fe_{3}O_{4}/NiO/MgO(001)}$ samples in a multichamber ultra-high-
vacuum system with a base pressure of $p_{0}<1\times 10^{-8}\,$mbar. Before
deposition, the MgO(001) substrates were annealed at $400^{\circ}$C in an
oxygen atmosphere of $1\times 10^{-4}\,$mbar for 1 hour. Our films were grown
by RMBE. For the NiO, we deposited nickel in an oxygen pressure of $1\times
10^{-5}\,$mbar, and for the $\mathrm{Fe_{3}O_{4}}$, we deposited iron in an
oxygen pressure of $5\times 10^{-6}\,$mbar. We limited the substrate
temperature to $250^{\circ}$C in order to avoid interdiffusion of Mg into the
films Kim-Ngan  _et al._ (2009). After growth, the electronic structure of the
samples were characterized in situ by x-ray photoelectron spectroscopy (XPS)
using a Phoibos HSA 150 hemispherical analyzer and an Al K$\alpha$ anode, and
their surface structure by low-energy electron diffraction (LEED). The Fe $2p$
XPS spectra show the $\mathrm{Fe^{2+}}$ and the $\mathrm{Fe^{3+}}$ features
typical for $\mathrm{Fe_{3}O_{4}}$, and the LEED patterns confirm the
characteristic ($\sqrt{2}\times\sqrt{2}$)R$45^{\circ}$ surface structure of
$\mathrm{Fe_{3}O_{4}}$ Mariotto  _et al._ (2002); Bliem  _et al._ (2014) (both
not shown here).

$\mathrm{Fe_{3}O_{4}/MgO(001)}$ and $\mathrm{Fe_{3}O_{4}/NiO/MgO(001)}$
samples were transported under ambient conditions to BESSY II for x-ray
absorption spectroscopy (XAS), XMCD, x-ray reflectivity (XRR) and XRMR on the
XUV diffractometer at beamline UE46_PGM-1 Weschke and Schierle (2018). The
samples were placed between two permanent magnets in a magnetic field of
$200\,$mT at room temperature. The x-rays had a degree of 90% circular
polarization.

All XAS and XMCD spectra were recorded in total electron yield (TEY) mode with
an incidence glancing angle of $30^{\circ}$. XRR and XRMR curves were obtained
by $\theta$-$2\theta$ scans in the range $2\theta=0^{\circ}-140^{\circ}$ at
selected resonant photon energies with both right and left circularly
polarized x-rays. The structural properties of the samples (thickness $d$,
roughness $\sigma$) obtained by XRR at an off-resonant energy ($1000\,$eV, cf.
Fig. 1) are summarized in Tab. 1.

In order to obtain magnetic information with higher depth sensitivity, we
measured XAS and XMCD in total fluorescence yield mode (TFY) and reflection
mode on a different $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ sample at Diamond Light
Source (DLS), on the RASOR diffractometer of beamline I10. The sample was
placed in a similar magnet setup, again in a magnetic field of $200\,$mT and
at room temperature. Here, the x-rays had a degree of circular polarization of
99%. The structural properties of this sample can be found in Tab. 1, too.

BESSY II UE46$\\_$PGM-1

DLS I10

| $\mathrm{Fe_{3}O_{4}/MgO}$ | $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ | $\mathrm{Fe_{3}O_{4}/NiO/MgO}$
---|---|---|---
$d_{\mathrm{Fe_{3}O_{4}}}$ |  $25.2\pm 0.3\,$nm |  $9.3\pm 0.1\,$nm |  $17.6\pm 0.1\,$nm
$d_{\mathrm{NiO}}$ | – |  $4.3\pm 0.1\,$nm |  $27.3\pm 0.2\,$nm
$\sigma_{\mathrm{Fe_{3}O_{4}}}$ |  $3.3\pm 0.5\,$Å |  $3.2\pm 0.5\,$Å |  $2.0\pm 0.6\,$Å
$\sigma_{\mathrm{NiO}}$ | – |  $3.0\pm 0.2\,$Å |  $4.2\pm 0.8\,$Å
$\sigma_{\mathrm{MgO}}$ |  $3.5\pm 0.5\,$Å |  $2.7\pm 0.5\,$Å |  $2.5\pm 0.3\,$Å

Table 1:  Film thicknesses $d_{i}$ and rms roughnesses $\sigma_{i}$ of the
three investigated samples, obtained from off-resonant XRR measurements
recorded at 1000 eV. Corresponding data and fits for the UE46$\\_$PGM-1
samples are shown in Fig. 1. Figure 1:  XRR measurements of the two samples
recorded at the three resonant energies of the Fe L3 XMCD spectrum of Fe3O4
(708.4 eV, 709.5 eV, 710.2 eV), the energy of the Ni L3 maximum (853.6 eV) and
at an off-resonant energy (1000 eV).

##  III Data analysis

###  III.1 XMCD

To obtain cation- and lattice-site-selective magnetooptical depth profiles,
first the XMCD spectra have to be analyzed. Figures 2(a) and (b) show
exemplarily XAS and XMCD spectra, respectively, of the Fe $L_{2,3}$ edges of
the $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ sample. Charge-transfer multiplet
calculations of the three Fe cations of $\mathrm{Fe_{3}O_{4}}$ using the Thole
code van der Laan (1997) with assistance of CTM4XAS de Groot (2005); Stavitski
and de Groot (2010) provide the three individual XAS and XMCD spectra shown
below the data. For these calculations, we used Ref. Kuepper  _et al._ (2016)
as a starting point: we assumed the three-cation model with crystal field
energies of $\mathrm{10Dq^{oct}}=1.0\,$eV in octahedral and
$\mathrm{10Dq^{tet}}=-0.6\,$eV in tetrahedral coordination. The splittings
between the initial and final charge-transfer states were chosen as
$\Delta_{\mathrm{init}}=6\,$eV and $\Delta_{\mathrm{final}}=5\,$eV, and for
the exchange splitting, $g\cdot\mu_{\mathrm{B}}=12\pm 1\,$meV was used. The
multiplet states resulting from these calculations were compared to the
experimental data by assuming a Gaussian instrumental broadening of $0.25\,$eV
and a Lorentzian lifetime broadening of $0.3\,$eV at $L_{3}$ and $0.6\,$eV at
$L_{2}$. Adding the three individual cation spectra with a 1:1:1 ratio, as
expected for Fe3O4, results in a total XAS and a total XMCD spectrum (orange
lines in Figs. 2(a),(b), respectively), which fit both the XAS and the XMCD
data well. The multiplet analysis reveals that at those energies for which the
XMCD spectrum has its extrema ($708.4\,$eV, $709.5\,$eV, $710.2\,$eV), most of
the XMCD signal originates from one dominant cation species Pohlmann  _et al._
(2020). The individual contributions of each cation species to the XMCD
spectrum at these three energies can be found in Tab. 2. Therefore, XRMR
measurements on those energies are mostly sensitive to one specific cation
species. This allows to disentangle the contributions of the individual
cations to the magnetooptical depth profiles.

Energy | $\mathrm{Fe^{2+}_{oct}}$ | $\mathrm{Fe^{3+}_{tet}}$ | $\mathrm{Fe^{3+}_{oct}}$
---|---|---|---
$708.4\,$eV | $70\pm 5\%$ | $-8\pm 3\%$ | $22\pm 5\%$
$709.5\,$eV | $19\pm 3\%$ | $-63\pm 3\%$ | $18\pm 3\%$
$710.2\,$eV | $4\pm 2\%$ | $-25\pm 8\%$ | $71\pm 10\%$

Table 2:  Contributions of the three cation species to the extrema in the XMCD
spectrum in Fig. 2(b) as obtained by the multiplet analysis.

Since the TEY mode has a probing depth of about $3\,$nm in
$\mathrm{Fe_{3}O_{4}}$ Gomes  _et al._ (2014), the TEY signal from the buried
NiO film was strongly attenuated at the Ni $L_{3}$ edge. For a clearer signal,
we brought a $\mathrm{18\,nm~{}Fe_{3}O_{4}/27\,nm~{}NiO/MgO}$ sample to
beamline I10 of DLS, and measured XAS and XMCD in TFY mode at a fixed incident
angle of $30^{\circ}$, whose probing depth is only limited by the x-ray
attenuation length in $\mathrm{Fe_{3}O_{4}}$ of about $80\,$nm Henke  _et al._
(1993) (cf. Figs. 3(a),(b)). Simultaneously, we measured the reflected
intensity in order to obtain an estimate of the magnetooptic effects in
reflection (cf. Figs. 3(c),(d)).

Figure 2: (a) XAS and (b) XMCD spectrum at the Fe L2,3 edges for the
Fe3O4/NiO/MgO sample, taken at 200 mT external magnetic field, at room
temperature and in TEY mode. A step function was subtracted from the XAS
spectrum. Black circles are data points; green, red and blue spectra are
multiplet calculations for the three cation species of Fe3O4 assuming an ideal
1:1:1 stoichiometry; the orange line is their sum. The cation spectra are
offset for better visibility.

###  III.2 XRMR

The XRMR data were recorded by measuring XRR curves at resonant photon
energies $E_{i}$ with extrema in the XMCD signal (maximum at $708.4\,$eV,
minimum at $709.5\,$eV, maximum at $710.2\,$eV, see Fig. 2(b)) with both left
and right circularly polarized x-rays. Resonant ’non-dichroic’ XRR curves were
obtained by averaging the signals $I^{\sigma}$ from both helicities
($\sigma=$right/left)

$I=(I^{\mathrm{right}}+I^{\mathrm{left}})/2$ (1)

and the XRMR asymmetry ratios by subtracting and normalizing them:

$\Delta
I=\frac{I^{\mathrm{right}}-I^{\mathrm{left}}}{I^{\mathrm{right}}+I^{\mathrm{left}}}.$
(2)

These curves were then fitted with the Zak matrix formalism using the software
ReMagX Macke and Goering (2014) to determine the depth profiles of the complex
refractive index $n(z)$

$n(z)=1-\delta(z)+i\beta(z)$ (3)

along the film height $z$. The optical dispersion $\delta$ and the optical
absorption $\beta$ can be split into non-magnetic components $\delta_{0}$,
$\beta_{0}$ and magnetooptical components $\Delta\delta$, $\Delta\beta$. In
the case of an in-plane magnetic field longitudinal to the x-ray beam as
applied here, they can be written as Macke and Goering (2014)

$\displaystyle\delta(z)=\delta_{0}(z)\mp\Delta\delta(z)\cdot\mathrm{cos}(\theta)$
(4)
$\displaystyle\beta(z)=\beta_{0}(z)\pm\Delta\beta(z)\cdot\mathrm{cos}(\theta)\
\ \ ,$ (5)

for which the magnetooptical contributions depend on the x-ray incidence
glancing angle $\theta$ and their sign on the helicity of the x-rays. The
optical absorption $\beta_{0}$ is proportional to the XAS signal, while the
magnetooptical absorption $\Delta\beta$ is proportional to the XMCD signal.
Thus, $\Delta\beta(z)$ is a measure of the magnetization along the film depth.
A detailed review of the XRMR method and the software is given in Ref. Macke
and Goering (2014), and a conclusive recipe for fitting XRMR data can be found
in Refs. Kuschel  _et al._ (2015); Klewe  _et al._ (2016); Krieft  _et al._
(2020).

Figure 3:  (a) XAS and (b) XMCD spectra in TFY mode at the Ni L3 edge for a 18
nm Fe3O4/27 nm NiO/MgO sample. (c) Energy scans of the reflected intensity for
left and right circularly polarized x-rays. (d) Difference of the curves in
(c). The green circles indicate the energy at which the XRMR curves were taken
(853.6 eV). The XAS data are normalized to the maximum of the XAS spectrum
XASleft+XASright. All data are recorded at a fixed incidence glancing angle of
30∘.

##  IV Results

Figure 4:  XRMR data (open circles) and fits (solid lines) from (a) the
Fe3O4/MgO and (b) the Fe3O4/NiO/MgO sample, recorded at the three resonant
energies of the Fe XMCD L3 edge and the Ni L3 edge. The magnetooptical depth
profile models corresponding to the fits in (a) are displayed in Fig. 5(b),
and the interface regions of the magnetooptical depth profiles corresponding
to the fits in (b) are shown in Fig. 5(c).

Figure 3(a) shows the XAS spectra at the Ni $L_{3}$ edge of the
$\mathrm{18\,nm~{}Fe_{3}O_{4}/27\,nm~{}NiO/MgO}$ sample recorded with left and
right circularly polarized x-rays in TFY mode, and Fig. 3(b) shows their
difference. The XMCD signal is about 3% of the XAS maximum. In reflection, the
magnetooptical effects become even more apparent. Note that the data are
recorded at fixed incidence angle of $\theta=30^{\circ}$. Therefore the
scattering vector $q_{z}$ changes from $0.426\,\mathrm{\r{A}^{-1}}$ to
$0.440\,\mathrm{\r{A}^{-1}}$ along with the energy in the range displayed in
Figs. 3(c),(d). At the Ni $L_{3}$ resonant energy of $853.6\,$eV, the
magnetooptically induced change of the reflected intensity ranges up to about
8% demonstrating a strong magnetooptical signal from NiO. This energy was
chosen for the XRMR measurements at the Ni $L_{3}$ resonance and is indicated
by the green circles in Fig. 3.

The data and fits of the resonant XRR measurements can be found in Fig. 1
alongside with the off-resonant XRR curves recorded with $1\,$keV photons.
Figures 4(a) and (b) show the XRMR data for the $\mathrm{Fe_{3}O_{4}/MgO}$ and
the $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ samples at the three Fe $L_{3}$ resonant
energies $708.4\,$eV, $709.5\,$eV and $710.2\,$eV, and the Ni $L_{3}$ resonant
energy $853.6\,$eV, together with their respective fits, which describe the
data very well. The fits in Fig. 4(b) were obtained from the magnetooptical
depth profile models which are displayed in Fig. 5(a) for all four resonant
energies of the $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ sample. The grey line
represents the optical absorption $\beta_{\text{off-res}}$ obtained from the
off-resonant XRR measurement, and represents the structural depth profiles of
the sample. It has three plateaus corresponding to the MgO substrate, the NiO
film and finally the $\mathrm{Fe_{3}O_{4}}$ film, as illustrated by the sketch
on top of the panel. The filled areas are the magnetooptical depth profiles
$\Delta\beta(z)$ at the resonant energies of the three iron cation species and
the Ni $L_{3}$ energy, obtained from the XRMR asymmetry ratios.

One notable feature of these magnetooptical depth profiles is found at the
surface of the $\mathrm{Fe_{3}O_{4}}$ films. Here, a thin layer of enhanced
magnetooptical absorption is observed at $709.5\,$eV and $710.2\,$eV, both for
the $\mathrm{Fe_{3}O_{4}/MgO(001)}$ and the
$\mathrm{Fe_{3}O_{4}/NiO/MgO(001)}$ samples. This is likely related to a
modification of the cation stoichiometry at the $\mathrm{Fe_{3}O_{4}(001)}$
surface. This effect is discussed in detail in Ref. Pohlmann  _et al._ (2020)
for $\mathrm{Fe_{3}O_{4}/MgO(001)}$, and it is interesting to note that it
occurs on the $\mathrm{Fe_{3}O_{4}/NiO/MgO(001)}$ sample, too, as exemplified
by Fig. 5(a). However, it is not the subject of the current study, which
focuses on the $\mathrm{Fe_{3}O_{4}/MgO}$ and $\mathrm{Fe_{3}O_{4}/NiO}$
interfaces.

Figure 5:  (a) $\Delta\beta$(z) depth profiles at the three Fe L3 resonant
energies and the Ni L3 energy of the Fe3O4/NiO/MgO sample, corresponding to
the fits in Fig. 4(b). (b) Close-up of the interface region of the Fe3O4/MgO
film, obtained from the fits in Fig. 4(a). (c) Close-up of the Fe3O4/NiO
interface region of the magnetooptical depth profiles in (a). The grey lines
indicate the optical absorption profiles $\beta_{\text{off-res}}$(z) obtained
by off-resonant XRR fits. Dashed lines highlight the interface positions of
the magnetooptical depth profiles.

Therefore, Fig. 5(b) focuses upon the interface region of the magnetooptical
depth profiles of the $\mathrm{Fe_{3}O_{4}/MgO}$ sample, according to the fits
in Fig. 4(a). Both the interfaces of the magnetooptical depth profile at
$708.4\,$eV (green) and of the one at $710.2\,$eV (blue) are collocated with
the structural interface (grey line) at $z=0\,$Å. However, the interface of
the magnetooptical depth profile at $709.5\,$eV (red) is shifted by a distance
$\Delta z_{\mathrm{709.5\,eV}}=3.2\pm 1.3\,$Å away from the interface into the
Fe3O4 film. The roughnesses of magnetooptical depth profiles at both
$\mathrm{Fe^{3+}}$ resonances follow the structural depth profile. In
contrast, the roughness of the magnetooptical depth profile recorded at the
$\text{Fe}^{2+}_{\text{oct}}$ energy, $\sigma_{\mathrm{708.4\,eV}}=4.4\pm
0.2\,$Å, is slightly larger than the structural roughness
$\sigma_{\mathrm{substrate}}=3.5\pm 0.5\,$Å.

Figure 6:  (a) Illustration of the stacking order at a B-terminated
Fe3O4/MgO(001) interface. The blue and red shaded areas indicate octahedral
and tetrahedral layers, respectively. Arrows depict magnetic moments. (b)
Simulated cation profiles at the interface of a Fe3O4 film with a roughness of
$\sigma$ = 3.5 Å, in scale with (a). (c) Expected magnetooptical depth
profiles at the three resonant energies of the Fe L3 edge, stemming from the
cation profiles in (b), and assuming the magnetooptical absorption
contributions of the individual cation spectra in Fig. 2(b).

For the $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ film, the results are slightly
different. Figure 5(c) shows the $\mathrm{Fe_{3}O_{4}/NiO}$ interface region
of the $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ sample. The XRMR data at the Ni $L_{3}$
edge can be well fitted with a homogeneous magnetization profile throughout
the NiO film. The interfaces of the magnetooptical depth profiles at the Ni
$L_{3}$ edge and at the $\mathrm{Fe^{2+}_{oct}}$-related resonance at
$708.4\,$eV are collocated with the structural interface, indicating intact
structural and magnetic order for both species. Notably, both their
roughnesses are slightly higher, $\sigma_{\mathrm{708.4\,eV}}=6.3\pm 0.7\,$Å,
$\sigma_{\mathrm{853.6\,eV}}=5.0\pm 0.7\,$Å, compared to the structural
roughness $\sigma_{\mathrm{Fe_{3}O_{4}/NiO}}=3.0\pm 0.2\,$Å.

In contrast to the magnetooptical depth profile at $708.4\,$eV, which directly
follows the structural profile, the profiles at $709.5\,$eV and $710.2\,$eV
are rising with shifts of $\Delta z_{\mathrm{709.5\,eV}}=2.6\pm 0.9\,$Å and
$\Delta z_{\mathrm{710.2\,eV}}=3.3\pm 1.4\,$Å, respectively, apart from the
interface, pointing to a lack of magnetooptical absorption at resonant
energies of both $\mathrm{Fe^{3+}_{oct}}$ and $\mathrm{Fe^{3+}_{tet}}$ at the
interface.

Figure 7:  Simulated magnetooptical depth profiles for three models of the Fe3O4/MgO interface. (a),(d),(g) Illustrations of the stacking orders of Fe3O4/MgO interfaces (a) with a single non-magnetic Fe1-$\delta$O interlayer, (d) with a single magnetic Fe1-$\delta$O interlayer, (g) with an additional $\text{Fe}^{3+}_{\text{oct}}$ cation in the interface layer, following the model of Chang et al Chang  _et al._ (2016). (b),(e),(h) Cation depth profiles corresponding to the illustrations in (a),(d),(g), respectively. (c),(f),(i) Magnetooptical profiles resulting from the cation profiles in (b),(e),(h), respectively. Distances of the rising edges of the magnetooptical profiles from the structural interface can be found in Tab. 3. | $\mathrm{Fe_{3}O_{4}/MgO}$ | $\mathrm{Fe_{3}O_{4}/NiO/MgO}$ | B-term. |  $\mathrm{Fe_{1-\delta}}$ (NM) |  $\mathrm{Fe_{1-\delta}}$ (FM) | Chang
---|---|---|---|---|---|---
$\Delta z_{708.4\,\mathrm{eV}}$ (Å) | 0 | 0 | 0 | 1.7 | 0 | 0
$\Delta z_{709.5\,\mathrm{eV}}$ (Å) | $3.2\pm 1.3$ | $2.6\pm 0.9$ | 2.5 | 4.1 | 5.5 | 2.5
$\Delta z_{710.2\,\mathrm{eV}}$ (Å) | 0 | $3.3\pm 1.4$ | 0 | 1.7 | 1.1 | -0.5
Table 3:  Distances of the rising edges of the magnetooptical profiles from
the structural interface at the three resonant energies. Considered here are
the results from the two investigated samples as well as the discussed
interface models: the B-terminated model (cf. Fig. 6), the non-magnetic (NM)
$\mathrm{Fe_{1-\delta}}$ interface layer (cf. Figs. 7(a)-(c)), the
ferromagnetic (FM) $\mathrm{Fe_{1-\delta}}$ interface layer (cf. Figs.
7(d)-(f)) and the model proposed by Chang et al. Chang  _et al._ (2016) (cf.
Figs. 7(g)-(i)).

##  V Discussion

For the Fe3O4/MgO sample, it is shown in Fig. 5(b) that the magnetooptical
depth profile recorded at a photon energy of $709.5\,$eV is displaced from the
interface into the Fe3O4 film by a shift $\Delta z_{\mathrm{709.5\,eV}}$. From
the quality of the fits, we can determine this shift to the range $\Delta
z_{\mathrm{709.5\,eV}}=3.2\pm 1.3$Å. Since this resonance is governed by the
tetrahedrally coordinated Fe ions, this result is consistent with a
B-terminated interface having octahedrally coordinated $\mathrm{Fe^{2+}}$ and
$\mathrm{Fe^{3+}}$-cations in the Fe3O4 interface layer. Figure 6(a) shows the
ideal stacking order at a B-terminated $\mathrm{Fe_{3}O_{4}/MgO(001)}$
interface. The oxygen lattice of the substrate continues as the oxygen lattice
of the film. In [001] direction, $\mathrm{Fe_{3}O_{4}}$ can be described as a
stack of subsequent B layers consisting of $\mathrm{O^{2-}}$ anions as well as
$\mathrm{Fe^{2+}_{oct}}$ and $\mathrm{Fe^{3+}_{oct}}$ cations, and A layers
containing $\mathrm{Fe^{3+}_{tet}}$ cations with a distance of $1.05\,$Å
between them. The stacking of B- and A-layers is depicted as blue and red
shaded areas, respectively, in Fig. 6(a) with the interface layer being a B
layer (B-termination). Simulated cation depth profiles following this model
are shown in Fig. 6(b). The atomically sharp distributions are smeared out
using an interface roughness of $\sigma=3.5\,$Å corresponding to the
experimentally determined roughness of the $\mathrm{Fe_{3}O_{4}/MgO}$
interface. The rising edge of the $\mathrm{Fe^{3+}_{tet}}$ depth profile is
shifted by $\Delta z_{\mathrm{tet}}=1.05\,$Å from the interfaces of the
$\mathrm{Fe^{2+}_{oct}}$ and $\mathrm{Fe^{3+}_{oct}}$ profiles into the Fe3O4
film. Because of the overlap of the individual cation spectra (cf. Fig. 2(b)),
the expected magnetooptical depth profiles at the different resonant energies
do not follow this behavior exactly.

Taking into account the magnetooptical contributions as derived from the
multiplet calculations of each cation at each of the three energies, the
expected magnetooptical depth profiles of a B-terminated
$\mathrm{Fe_{3}O_{4}/MgO(001)}$ interface can be calculated. They are shown in
Fig. 6(c). The expected shift of the magnetooptical depth profile at
$709.5\,$eV is $\Delta z_{\mathrm{709.5\,eV}}=2.5\,$Å, consistent with the
experimental result of $\Delta z_{\mathrm{709.5\,eV}}=3.2\pm 1.3\,$Å.
Therefore, the magnetooptical depth profiles indicate a B-terminated
$\mathrm{Fe_{3}O_{4}/MgO(001)}$ interface with no interlayer, and evidently,
also no magnetic dead layer.

In this scenario, the magnetic order of all three sublattices has bulk
properties down to the interface. The simulations of the B-terminated
interface also predict that the apparent roughness
$\sigma_{\mathrm{708.4\,eV}}$ of the magnetooptical depth profiles at
$708.4\,$eV appears to be about $0.5\,$Å larger than the structural profile
$\sigma_{\mathrm{substrate}}$. This offers an explanation for the slight
mismatch of these two roughnesses observed in the experiment. However, while
the discrepancy between the model distance and the experimental distance is
well within the error range, it is still substantial enough to make a
discussion of alternative models worthwhile.

Both the $\mathrm{Fe_{3}O_{4}/MgO}$ and the $\mathrm{Fe_{3}O_{4}/NiO}$
interfaces have been studied by various methods. Spintronic devices require
interfaces that are structurally, but especially also magnetically sharp.
Therefore, focus has been laid on the possible presence of interlayers and
intermixing at the interfaces. In the case of $\mathrm{Fe_{3}O_{4}}$ directly
grown on a substrate, the formation of FeO interlayers has been reported on
both metal and oxide substrates Gota  _et al._ (1999); Karunamuni  _et al._
(1999); Schlueter  _et al._ (2011), and particularly on MgO(001) for films
deposited at room temperature Bertram  _et al._ (2012). The possibility of a
single atomic $\mathrm{Fe_{1-\delta}O}$ interlayer is discussed in the
following. The corresponding magnetooptical depth profiles are calculated in
Figs. 7(a)-(f) for two scenarios.

Therefore, the first reasonable scenario is that the $\mathrm{Fe_{1-\delta}O}$
interlayer forms a magnetic dead layer at the interface, presented in Figs.
7(a)-(c). Figure 7(a) shows an illustration of the stacking order, Fig. 7(b)
the cation depth profiles simulated with a roughness of $\sigma=3.5\,$Å, and
Fig. 7(c) the resulting magnetooptical depth profiles.
$\mathrm{Fe_{1-\delta}O}$ is paramagnetic at room temperature. Due to the
magnetically dead $\mathrm{Fe_{1-\delta}O}$ layer, the rising edges of the
magnetooptical depth profiles at both 708.4 eV and 710.2 eV are shifted about
2 Å into the Fe3O4 film compared to the structural interface (cf. Tab. 3), not
consistent with the observed profiles in Fig. 5(b).

The second scenario assumes that the very thin FeO layer magnetically couples
to $\mathrm{Fe_{3}O_{4}}$ and the magnetic order of its
$\text{Fe}^{2+}_{\text{oct}}$ sublattice is extended into the FeO layer. The
resulting magnetooptical depth profiles show non-monotonic behavior and differ
even more from the observed ones (cf. Figs. 7(d)-(f) and Tab. 3).

Our data therefore do not indicate any magnetically dead layers, which had
been considered to be the cause of the magnetization reduction in
$\mathrm{Fe_{3}O_{4}}$ ultrathin films van der Heijden  _et al._ (1996); Zhou
_et al._ (2006); Gomes  _et al._ (2014); Chang  _et al._ (2016), nor
interlayers of ferromagnetic FeO. Since it has already been shown that the
magnetooptical depth profiles can be explained without an interlayer, it is
unlikely that a FeO interlayer of more than a single atomic layer is present.

Another interesting model, which was proposed by Chang et al. for the growth
dynamics of $\mathrm{Fe_{3}O_{4}}$ Chang  _et al._ (2016) shall also briefly
be mentioned here. This model suggests the first interface B-layer to contain
one additional $\mathrm{Fe^{3+}_{oct}}$ cation per unit formula
[$(\mathrm{Fe^{2+}_{oct}})_{2}(\mathrm{Fe^{3+}_{oct}})_{3}\mathrm{O_{8}}$
instead of
$(\mathrm{Fe^{2+}_{oct}})_{2}(\mathrm{Fe^{3+}_{oct}})_{2}\mathrm{O_{8}}$]. An
illustration can be seen in Fig. 7(g). Both the cation depth profiles and the
simulated magnetooptical depth profiles resulting from this model, presented
in Figs. 7(h),(i), hardly differ from the B-terminated interface shown in
Figs. 6(b),(c). Although XRMR would in principle be an ideal method to test
this model, the lacking spatial resolution in our experiments can neither
confirm nor reject a faint phenomenon like an additional
$\mathrm{Fe^{3+}_{oct}}$ cation in the interface layer.

For the Fe3O4/NiO/MgO(001) sample, a noteworthy finding is the dichroic signal
of the NiO film. Bulk NiO is an antiferromagnet at room temperature and should
not show any circular dichroism. However, as demonstrated in Fig. 3, we
clearly observe magnetooptical effects in both TFY mode and in reflection at
the Ni $L_{3}$ edge. Interestingly, we can exclude the XMCD signal to stem
from uncompensated surface spins, since the magnetooptical depth profiles
clearly show a homogeneous magnetization of the entire film. Size-effects of
the magnetic properties of NiO, including ferromagnetic behaviour at room
temperature, have been frequently reported before, mostly for NiO
nanoparticles Kodama  _et al._ (1997); Winkler  _et al._ (2005); Li  _et al._
(2006); Ravikumar  _et al._ (2015); Rinaldi-Montes  _et al._ (2016). For
$\mathrm{Fe_{3}O_{4}/NiO}$ ultrathin films, a spin-flop coupling of NiO to the
$\mathrm{Fe_{3}O_{4}(001)}$ interface has been reported Krug  _et al._ (2008).
In that case, the antiferromagnetic order of NiO aligns perpendicular to the
magnetization of $\mathrm{Fe_{3}O_{4}}$, but with a small canting of the
$\mathrm{Ni^{2+}}$ moments, resulting in a magnetization component parallel to
the ferrimagnet. This reaction of NiO to outer magnetic fields has also been
confirmed by spin Hall magnetoresistance measurements Hoogeboom  _et al._
(2017); Fischer  _et al._ (2018), and can explain the presence of the observed
XMCD signal.

The interface of $\mathrm{Fe_{3}O_{4}/NiO}$ has mostly been discussed
regarding the presence of a $\mathrm{NiFe_{2}O_{4}}$ interlayer. In reports by
Gatel et al. Gatel  _et al._ (2005) and Pilard et al. Pilard  _et al._ (2007),
high-resolution transmission electron microscopy (HRTEM) images show generally
sharp interfaces between the rock salt structure of NiO and the spinel
structure of $\mathrm{Fe_{3}O_{4}}$. However, this only partly solves the
question, because the lack of chemical and magnetic contrast cannot exclude,
for instance, the formation of a rock salt $\mathrm{(Fe,Ni)O}$ phase or a
spinel $\mathrm{NiFe_{2}O_{4}}$ phase. Gatel et al. adressed this issue by
performing both HRTEM and electron energy loss spectroscopy (EELS) Gatel  _et
al._ (2005). For a $\mathrm{NiO/Fe_{3}O_{4}/MgO(001)}$ sample, for which the
NiO film was deposited at a substrate temperature of $700^{\circ}$C, indeed an
intermediate $\mathrm{NiFe_{2}O_{4}}$ phase was observed. For a
$\mathrm{Fe_{3}O_{4}/NiO/MgO(001)}$ stack, which had the Fe3O4 film deposited
at $400^{\circ}$C, the interface appeared to be chemically sharp with at most
minor interdiffusion Gatel  _et al._ (2005). The $\mathrm{NiFe_{2}O_{4}}$
phase they observe in the $\mathrm{NiO/Fe_{3}O_{4}/MgO(001)}$ stack is likely
caused by thermal interdiffusion due to the high deposition temperature
Kuschel  _et al._ (2016).

A formation of a well-ordered $\mathrm{NiFe_{2}O_{4}}$ interlayer does not
match our observations. $\mathrm{NiFe_{2}O_{4}}$ crystallizes in the same
inverse spinel structure as Fe3O4, but with $\mathrm{Ni_{oct}^{2+}}$ cations
instead of $\mathrm{Fe_{oct}^{2+}}$ sharing the B-sites with
$\text{Fe}^{3+}_{\text{oct}}$. Analogous to Fe3O4, the magnetic moments of the
$\mathrm{Ni_{oct}^{2+}}$ and $\text{Fe}^{3+}_{\text{oct}}$ cations on the
B-sites align antiferromagnetically to the magnetic moments of the
$\text{Fe}^{3+}_{\text{tet}}$ cations on the A-sites. For the magnetooptical
depth profiles, this would imply a decrease of $\mathrm{Fe^{2+}_{oct}}$
cations close to the interface and thus of the magnetooptical absorption at
$708.4\,$eV, while the magnetooptical absorption depth profiles for
$709.5\,$eV and $710.2\,$eV should stay constant in a $\mathrm{NiFe_{2}O_{4}}$
layer. Instead, we observe a reduction of the magnetooptical absorption at
$709.5\,$eV and $710.2\,$eV as compared to the case of the
$\mathrm{Fe_{3}O_{4}/MgO}$ interface. This behavior, together with the
increased roughnesses of the magnetooptical depth profiles at the
$\mathrm{Fe^{2+}_{oct}}$ and the $\mathrm{Ni^{2+}}$ energies, might indicate a
slight interdiffusion of $\mathrm{Fe^{2+}_{oct}}$ into the rock salt structure
of $\mathrm{NiO}$. However, this effect may not extend farther than a single
atomic layer. Notably, the ferromagnetic coupling between the
$\text{Fe}^{2+}_{\text{oct}}$ cations is retained down to the interface
regardless of the intermixing.

##  VI Conclusion

We have prepared ultrathin $\mathrm{Fe_{3}O_{4}/MgO(001)}$ and
$\mathrm{Fe_{3}O_{4}/NiO/MgO(001)}$ films by RMBE and performed XMCD and XRMR
measurements to extract magnetooptical depth profiles for the individual
cation species $\mathrm{Fe^{2+}_{oct}}$, $\mathrm{Fe^{3+}_{tet}}$ and
$\mathrm{Fe^{3+}_{oct}}$ as well as for $\mathrm{Ni^{2+}_{oct}}$. These
magnetooptical depth profiles show that for $\mathrm{Fe_{3}O_{4}/MgO(001)}$,
the magnetic order of all three cation species is stable for the entire film
with no interlayer or magnetic dead layer at the interface. For
$\mathrm{Fe_{3}O_{4}/NiO}$ films, we observe a magnetooptical absorption at
the Ni $L_{3}$ edge in the NiO film corresponding to uncompensated magnetic
moments throughout the entire NiO film. The magnetooptical profiles of the
iron cations reveal an intact magnetic order for the
$\text{Fe}^{2+}_{\text{oct}}$ cation species down to the interface, while the
magnetooptical depth profiles at the $\mathrm{Fe^{3+}_{oct}}$ and the
$\mathrm{Fe^{3+}_{tet}}$ resonances are shifted about $3\,$Å into the Fe3O4
film, possibly indicating a single intermixed layer containing both
$\mathrm{Fe^{2+}}$ and $\mathrm{Ni^{2+}}$ cations.

##  VII  Acknowledgements

Financial support from the Bundesministerium für Bildung und Forschung (FKZ
05K16MP1) is gratefully acknowledged. We are also grateful for the kind
support from the Deutsche Forschungsgemeinschaft (DFG under No. KU2321/6-1,
and No. WO533/20-1). The authors would like to thank Diamond Light Source for
beamtime (proposal SI19173-1), and the staff of beamline I10 for assistance
with data collection. We thank HZB for the allocation of synchrotron beamtime
at beamline $\mathrm{UE46\\_PGM}$-1 (181/06266ST/R) where we recorded the XRMR
and XMCD measurements.

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