# Unraveling Magnetic Anisotropy Energy in Ferromagnetic Monolayer on
Ferroelectric ABO3 via DFT and Machine Learning

Dameul Jeong  Department of Physics, Kyung Hee University, Seoul, 02447, Korea
Seoung-Hun Kang<EMAIL_ADDRESS>Materials Science and Technology
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, United States of
America  Young-Kyun Kwon<EMAIL_ADDRESS>Department of Physics, Kyung Hee
University, Seoul, 02447, Korea  Department of Information Display and
Research Institute for Basic Sciences, Kyung Hee University, Seoul, 02447,
Korea

(May 2022)

###### Abstract

Spin-based devices have attracted attention as an alternative to CMOS-based
technology. However, one of the challenges in spintronics devices is reducing
the spin-switching energy in ferromagnetic (FM) materials. To address this, we
considered ferroelectric (FE) materials, which may affect the magnetic
properties of FM materials. We explored various oxide perovskites ABO3 as FE
materials, onto which a Fe monolayer was placed as the FM material. We
evaluated the magnetic anisotropy energy (MAE) of the Fe monolayer while
varying the polarization of ABO3. Our analysis showed that the MAE depends on
the magnetic dipole moment induced in the FE material at the interface between
the FE and FM materials due to structural modifications. Machine learning
techniques were also employed to identify universal behaviors of the MAE in
the presence of FE layers, confirming the importance of magnetic moments near
the interface in explaining the dependence of the MAE on FE materials.

The development of complementary metal–oxide–semiconductors (CMOS) has been
driven by the need for small and fast electronic devices. While CMOS has been
very successful, its development has stagnated at sizes of 5 nm or less due to
the quantum tunneling effect that impedes the smooth movement of electrons in
the circuit Wang (2006). Despite intensive efforts to overcome this obstacle,
finding a fundamental solution remains a challenge Ratnesh et al. (2021);
Samal et al. (2020); Prasad (2019). To resolve such limitations of CMOS,
spintronics is emerging as an alternative Wolf et al. (2001); Žutić et al.
(2004); Fert (2008). Spintronics, which uses the spin of electrons to control
devices, has enabled the commercialization of spin-transfer torque (STT)
magnetoresistive random access memory (MRAM) with near-zero standby and zero
leakage power Wolf et al. (2010); Salehi et al. (2017). However, the energy
required to record and maintain magnetic information in STT-MRAM is higher
than that of conventional dynamic random access memory (DRAM) as well as
static random access memory (SRAM) Nozaki et al. (2019). Therefore, research
is being actively conducted to find ways to reduce such high energy
consumption Gupta and Bhargava (2019); Eswar et al. (2021); Cai et al. (2021).

Intel developed a spin-based logic device known as magnetoelectric spin-orbit
(MESO) several years ago, which outperforms traditional CMOS devices by a
factor of 10 to 30. MESO utilizes two physical principles: spin-to-charge
conversion through topological and robust spin-orbit coupling (SOC) materials
and magnetization switching using the magnetoelectric effect of multiferroic
materials such as La-doped BiFeO3 Manipatruni et al. (2019). Despite its
potential benefits, BiFeO3 has not been widely used in practical applications
due to reliability issues and high costs Yang et al. (2015). Furthermore,
there are only a few reports of multiferroic materials that exhibit both
strong ferromagnetism and ferroelectricity at room temperature, which limits
their commercial viability Mundy et al. (2016). Therefore, new materials and
magnetic switching approaches are necessary to develop novel spintronic
devices. One possible strategy is to manipulate the magnetic anisotropy energy
(MAE) of ferromagnetic (FM) materials. This involves lowering the energy
required to flip a spin, which reduces the write energy requirement. Certain
ferroelectric (FE) materials, such as BaTiO3 and HfO2, have been shown to
modulate the MAE of FM materials through their spontaneous polarization Duan
et al. (2006, 2008); Lukashev et al. (2012); Radaelli et al. (2014); Vermeulen
et al. (2019).

Here, we report our first-principles and machine learning investigation on the
MAE of Fe monolayers in various FE ABO3 materials, comprising alkaline (A=Ca,
Sr, or Ba) and group 4 transition metal (B=Ti, Zr, or Hf) elements. Our study
disclosed unusual factors affecting the MAE of Fe monolayers leading to
valuable insights that can aid in identifying and improving suitable FE and FM
materials for spintronics applications. Specifically, we found that the MAE of
Fe monolayers depend not only on the spontaneous polarization of ABO3 but also
on the magnetic dipole moment induced in the FE material near the interface
between FE and FM materials due to the structural strain and the proximity
effect. Furthermore, we employed a machine learning approach to examine the
universal behavior of FM MAE in the presence of FE layers, and our findings
demonstrate that the induced magnetic moment near the interface is indeed a
crucial feature that can explain the FE-dependent behavior of FM MAE.

We carried out ab initio calculations based on density functional theory (DFT)
Hohenberg and Kohn (1964); Kohn and Sham (1965) as implemented in the Vienna
ab initio simulation package (VASP) Kresse and Hafner (1993); Kresse and
Furthmüller (1996) with projector augmented wave potentials Blöchl (1994). For
the exchange-correlation (XC) functional, we employed the Ceperley-Alder (CA)
form Ceperley and Alder (1980) within the local density approximation (LDA).
The energy cutoff was set to 500 eV for all our calculations. Structural
configurations relaxed until the Hellmann-Feynman force on every atom was
lower than 0.01 eV/Å. The Brillouin zone was sampled using a $10\times
10\times 10$ and a $12\times 12\times 1$ $\Gamma$-centered $k$-grids for the
bulk and slab structures. We considered the spin-orbit interaction and the
Berry phase approach Spaldin (2012) to evaluate the MAE and the spontaneous
polarization, respectively.

To decipher the universal behaviors of the Magnetic Anisotropy Energy (MAE) of
the Fe monolayer on ABO3, we applied the Sure Independence Screening and
Sparsifying Operator (SISSO) approach Ouyang et al. (2018). This method
prompted us to consider a vast feature space encompassing a wide variety of
potential attributes that reflect the essential properties of our Fe monolayer
on the ABO3 slab model. During this process, we put many mathematical
operations to work, simple ones such as addition, subtraction, multiplication,
and division, and more complex functions like exponentials and trigonometric
functions. We put these operations into play between the features,
systematically and recursively creating a wealth of meaningful combinations in
our system. This thorough, step-by-step approach brought us to six unique
features, each with a complexity level of three. We selected a subset of
10,000 features from the abundant collection that we felt deserved a closer
look. This in-depth exploration led us to discover an optimized descriptor
that beautifully outlines the universal behaviors of the MAE of the Fe
monolayer across a variety of ABO3 compounds.

Figure 1: The Atomic structures of ABO3 and Fe on ABO3 with porlarization. (a)
Three ABO3 bulk perovskite structures show different polarization states. The
Equilibrium structure of bulk ABO3 is tetragonal in which spontaneous
polarization is induced, shown in the top and bottom configurations with the
opposite polarization directions, whereas the center one is the central
symmetric structure without the spontaneous polarization. (b) Slab model of
the Fe monolayer on seven-layer ABO3. The lower four layers of the ABO3
substrate were fixed, and the upper three layers were fully relaxed to
describe the bulk and surface effect, respectively. Especially, we named two
nonequivalent oxygen atoms near the interface as OI and OII corresponding to
those at the top ABO3 surface and below the top B atom, respectively.

Perovskite is a mineral with the same crystal form as CaTiO3, a structure
derived from the ABX3 configuration. Its unit cell is a cube of space group
$Pm\bar{3}m~{}(221)$ with cations A and B at the vertices and body centers,
respectively, and anion X at the face centers, as illustrated in Figure 1(a).
The cation A acts as a fixed shell, so its chemical and physical properties
are of relatively little importance, but the displacement of the B and X
atoms, which breaks the centrosymmetry transforming the cube into a
tetrahedral structure, causes spontaneous polarization. This displacement,
which governs the FE properties of perovskites, can be controlled by changing
the electronic configuration of the cation B through geometric modification of
the BX6 octahedron Tilley (2016). In this study, only divalent (Group 2: Ca,
Sr, Ba, Ra) and tetravalent (Group 4: Ti, Zr, Hf) elements were considered as
cations A and B, respectively, and only oxygen as anion X.

To investigate how the FE spontaneous polarization affects the MAE of a Fe
monolayer, a slab model was constructed as shown in Figure 1(b). To minimize
the lattice mismatch between ABO3 and the Fe monolayer, we chose the (001)
plane of the bcc structure of Fe as the monolayer because it matches very well
with the (001) plane of ABO3 and thus forms a square lattice. We calculated
the lattice mismatch between ABO3 and the Fe monolayer using the Fe bulk
lattice constant of 2.75 and obtained the following results: -2.27%, 2.65%,
2.65%, -0.30%, 4.75%, 4.30%, 1.73%, 6.63%, 5.84%, for CaTiO3 (CTO), CaZrO3
(CZO), CaHfO3 (CHO), SrTiO3 (STO), SrZrO3 (SZO), SrHfO3 (SHO), BaTiO3 (BTO),
BaZrO3 (BZO), and BaHfO3 (BHO), respectively. We applied a strain only to the
Fe monolayer to compensate for these mismatches. The top three layers of the
ABO3 substrate with Fe monolayers were allowed to fully relax to obtain an
equilibrium interfacial structure, while the bottom four layers were fixed to
maintain the bulk structure that requires spontaneous polarization. The
direction of relative displacement of the BO6 octahedra within the ABO3 cube
determines the direction of polarization, which becomes negative when the
octahedron is displaced toward the Fe monolayer, and vice versa.

Figure 2: The relative MAE $\Delta\eta_{\mathrm{Fe}}$, defined in Eq. 2, of
the Fe monolayer on various ABO3 (A = Ca, Sr, or Ba; B = Ti, Zr, or Hf)
substrates. Since the lattice mismatch between the Fe monolayer and ABO3
substrate was released by adjusting the size of the Fe monolayer, the ABO3
substrate determines the strain of the Fe monolayer. Divalent elements Ca, Sr,
and Ba corresponding to the cation A are represented by symbols of solid
squares, crosses, and solid circles, whereas tetravalent ones Ti, Zr, and Hf
for the cation B are done by colors of blue, red, and yellow, respectively.
For each given B atom, $\Delta\eta_{\mathrm{Fe}}$ values for three different A
atoms were fitted with a quadratic function, which was plotted with a dashed
line with the color corresponding to the B atom. It is clear that the trends
of $\Delta\eta_{\mathrm{Fe}}$ are classified by the tetravalent elements. (b)
$\Delta\eta_{\mathrm{rel}}$, defined in Eq. 3, as a function of relative
polarization $\xi$ of six ABO3 substrates. The other three ABO3 substrates are
not shown here because their spontaneous polarizations are zero. While
$\Delta\eta_{\mathrm{rel}}$ of SZO (red cross), SHO (yellow cross), and BTO
(blue circle) show almost linear behaviors, those of CTO (blue square), CZO
(red square), and CHO (yellow square) do unexpected upturns as $\xi$
decreases.

Having established the interface between the Fe monolayer and ABO3 substrate,
we first studied the MAE of the Fe monolayer on the unpolarized ABO3
structures with the centrosymmetric structure shown in Figure 1(a). The MAE
$\eta$ is the energy required to change the direction of magnetization from
the out-of-plane direction to the in-plane direction in the magnetic thin film
material, defined as

$\eta=E_{\mathrm{tot}}(S_{\rightarrow})-E_{\mathrm{tot}}(S_{\uparrow}),$ (1)

where $E_{\mathrm{tot}}(S_{\rightarrow})$ and $E_{\mathrm{tot}}(S_{\uparrow})$
are the total energies for the systems with the in-plane ($S_{\rightarrow}$)
and out-of-plane ($S_{\uparrow}$) spin configurations, respectively. A
positive (negative) value of the MAE means that the spin prefers the out-of-
plane (in-plane) direction. The MAE arises from the SOC effect, where the
orbitals of atoms affected by the crystal field interact with spins Daalderop
et al. (1990). To explore the effects of FE substrates alone on the MAE of the
Fe monolayer, excluding the strain effect, we evaluated
$\Delta\eta_{\mathrm{Fe}}$ defined as

$\Delta\eta_{\mathrm{Fe}}=\eta_{\mathrm{Fe/ABO_{3}}}-\eta_{\mathrm{Fe^{free}_{s}}},$
(2)

where $\eta_{\mathrm{X}}$ is the MAE of a system X. Here, X is either Fe/ABO3
or Fe$\mathrm{{}^{free}_{s}}$, which represents a Fe/ABO3 slab system or a
freestanding strained Fe monolayer with the same lattice constant determined
by ABO3, respectively. As shown in Figure 2(a), the evaluated
$\Delta\eta_{\mathrm{Fe}}$ ranges from 0.4 to 0.9 meV/atom on various ABO3
substrates indicating the out-of-plane spin direction is favored. It is also
shown that for a given B element of ABO3 substrate, the heavier the A element,
the higher the MAE. Since the lattice mismatch between the Fe monolayer and
ABO3 substrate was relieved by adjusting the lattice constant of the former as
mentioned above, the size of the A atom is directly related to the strain of
the Fe monolayer given as the $x$-axis. The MAE data were well fitted to a
quadratic function of the strain of the Fe monolayer for each B element, as
shown in Figure 2(a). To explore the inherent effect of the spontaneous
polarization of ABO3 substrate alone on the MAE of the Fe monolayer, we also
evaluated the relative MAE $\Delta\eta_{\mathrm{rel}}$ using

$\Delta{\eta}_{rel}={\eta(\xi)-\eta(0)},$ (3)

where $\xi$ represents the normalized atomic displacement corresponding to the
relative polarization ($P$) of each ABO3 with respect to its spontaneous
polarization ($P_{\mathrm{s}}$), or $\xi=P/P_{\mathrm{s}}$. For example,
$\eta(0)$ denotes the MAE of the Fe monolayer placed on the corresponding
centrosymmetric ($\xi=0$) ABO$3$ substrate with zero polarization. Figure 2(b)
presents our evaluated $\Delta\eta_{\mathrm{rel}}$ as a function of $\xi$. It
is noted that negative values in Figure 2(b) do not indicate that the spins on
the Fe monolayer prefer the in-plane direction but rather demonstrate the
change in MAE with respect to their unpolarized cases, in which the Fe
monolayer still favors the out-of-plane spins. In this figure, we did not
include three ABO3 materials, STO, BZO, and BHO, since they are not FE
indicating zero spontaneous polarization. Our results demonstrate that the MAE
of the Fe monolayer is linearly dependent on $\xi$ on SZO, SHO, and BTO, as
naturally expected that the MAE would be affected by the electric field
induced by the spontaneous polarization of the substrate. However, when the Fe
monolayer is placed on the other three substrates containing Ca element for
the A-site, we observed unusual upturns in $\Delta\eta_{\mathrm{rel}}$ as
$\xi$ decreases.

Figure 3: The relative MAE $\Delta\eta_{\mathrm{rel}}$ and the relative
magnetic moments $\Delta\mu_{\mathrm{O_{II}}}$ induced on OII, which is
denoted in Figure 1(b), of (a) Fe/CTO, (b) Fe/CZO, and (c) Fe/CHO, as a
function of $\xi$. These relative quantities were evaluated with respect to
those of each of the corresponding centrosymmetric systems. The upturns of
$\Delta\eta_{\mathrm{rel}}$ as $\xi$ decreases, observed in these cases, are
quite strongly correlated with $\Delta\mu_{\mathrm{O_{II}}}$.

We now turn our attention to the abrupt upturns in $\Delta\eta_{\mathrm{rel}}$
observed in three cases (Fe/CTO, Fe/CZO, and Fe/CHO) under the negative
spontaneous polarization or $\xi<0$, as shown in Figure 2(b). It is clear that
there must be something other than the induced electric field from spontaneous
polarization causing this sudden rise. We found that the MAEs observed in
these three cases are governed more strongly by interface effects, such as
induced interfacial magnetic dipole moment. As shown in Figure 3,
$\Delta\mu_{\mathrm{O_{II}}}$, the relative magnetic moment induced in the
oxygen atom OII, located just below the topmost B atom of ABO3 shown in Figure
1(b), also exhibits a very similar sharp rise. Note that in the other three
cases (Fe/SZO, Fe/SHO, and Fe/BTO), where MAE does not show a sharp rise,
$\Delta\mu_{\mathrm{O_{II}}}$ remains nearly zero over the entire range of
$\xi$.

Building on these findings, our analysis reveals two distinct factors
contributing to the MAE of the Fe monolayer in the ABO3 systems impacted by
spontaneous polarization. The first factor, the ”interface effect,” is
influenced by atomic displacements caused by ferroelectricity. As the
spontaneous polarization $\xi$ becomes more negative, the separation between
the Fe and B atoms increases. This displacement subsequently brings the
O${}_{\text{II}}$ atom, as depicted in Figure 1(b), closer to the B atom. The
outcome is the induction of a magnetic moment in O${}_{\text{II}}$, resulting
in an upswing in the MAE following an initial decrease. Simultaneously, the
second factor is facilitated by the electric field induced by spontaneous
polarization itself. It is particularly prominent in the three cases (Fe/CTO,
Fe/CZO, and Fe/CHO), as seen in Figure 3. In these cases, the movement of the
B atom, coupled with a reduced magnetic moment at the interface, triggers a
magnetic moment in O${}_{\text{II}}$, which aligns with the abrupt upturns
observed in the MAE of the Fe monolayer (See Supplementary Note S1). Our
results underscore the crucial role played by changes in the magnetic moment
of interface atoms due to spontaneous polarization. These changes
significantly influence the MAE of the Fe monolayer, thus demonstrating the
multi-dimensional effects of spontaneous polarization in these complex
material systems.

Figure 4: Machine learning results of the relative MAE
$\Delta\eta_{\mathrm{rel}}$ for the Fe monolayer on various ABO3 (A = Ca, Sr,
or Ba; B = Ti, Zr, or Hf) substrates and test sets with RaBO3(B= Ti, Zr, Hf).
(a) Relationship between MAE and descriptors. The descriptor trained for the
Fe monolayer on the ABO3 layer (A = Ca, Sr, Ba; B = Ti, Zr, Hf) were $\alpha$,
while $\Delta\eta_{rel}$ was used for machine learning properties. The solid
black circle is a training set with R2=0.933, and the empty red circle is the
validation set with R2=0.898. The Blue rhombus point is the test set with
R2=0.865. (b) Plotted $\Delta\eta$ of Fe monolayer on the RaBO3 on top of
Figure 2(a). Ra closely follows the trend of the training set.

We employed the Sure Independence Screening and Sparsifying Operator (SISSO)
scheme for machine learning to pinpoint the key parameters needed to represent
$\Delta\eta_{rel}$ accurately. Training the model on a 30-set that excludes
unpolarized ABO3, we evaluated various features, like atomic bonds of
interface atoms, ABO3’s lattice constant, and individual atom magnetic moments
for the Fe monolayer atop the ABO3 layer (where A = Ca, Sr, Ba; B = Ti, Zr,
Hf), in the presence of spontaneous polarization. The ensuing machine-learning
model offered the descriptors as follows:

${\alpha}=(\mu_{B}+\mu_{Fe})({\lambda_{1}}\mu_{O_{\text{II}}}\mu_{Fe}+{\lambda_{2}}a^{6})+{\gamma}$
(4)

In this equation, $\alpha$ signifies the descriptor, $\lambda_{1,2}$ represent
coefficients (with respective values of $1.814\times 10^{3}$ and $4.312\times
10^{-4}$), and $\gamma$ is a constant ($1.951\times 10^{-2}$). The lattice
constant is denoted by $a$, while $\mu_{B}$, $\mu_{Fe}$, and
$\mu_{O_{\text{II}}}$ symbolize the magnetic moments of the B site, Fe, and
O${}_{\text{II}}$ atoms, respectively. Our two-dimensional descriptor with a
complexity of three operators utilizes $\alpha$ to express $\Delta\eta_{rel}$,
not solely based on the magnetic moments of Fe and B atoms, but also
integrating the magnetic moment of O${}_{\text{II}}$ and the lattice constant.
Figure 4(a) shows that our descriptor $\alpha$ displays high accuracy
(R2=0.933; RMSE=0.035) in predicting $\Delta\eta_{rel}$ for the training set
of Fe monolayers on the ABO3 layer, indicated by solid black circles.
Remarkably, our descriptor exhibits accurate prediction values for untrained
data (R2=0.898; RMSE=0.050), represented by empty red circles. Furthermore, it
performs well for additional ABO3 types, like RaTiO3 (not part of the training
set) with an R2 value of 0.865 and RMSE=0.063, denoted by blue rhombi. As
shown in Figure 4(b), RaBO3 (B = Ti, Zr, Hf) closely adheres to the trend of
the training dataset demonstrated in Figure 2(b). However, for MgBO3 (B = Ti,
Zr, Hf), the descriptor’s prediction of $\Delta\eta_{rel}$ is less accurate,
primarily attributed to the A site element’s lower atomic weight compared to
other ABO3 compounds. Refer to Supplementary Note S2 for comprehensive
details. Thus, our machine learning findings are beneficial for predicting MAE
values for Fe monolayers on diverse ABO3 compounds.

In summary, our study aims to control magnetic anisotropy energy (MAE) in
ferromagnetic (FM) materials for low-power spin-based devices by exploring the
use of ferroelectric (FE) spontaneous polarization. By examining ABO3
materials, which can transform into FE, we observed variations in the MAE of a
Fe monolayer based on the A and B elements of ABO3. We discovered a decreasing
trend followed by a jump in the MAE due to the interface effect between the Fe
monolayer and ABO3, which was unexpected. Using machine learning, we
identified descriptors for physical quantities impacting the MAE and
determined that the magnetic moments of interface atoms were the most
efficient features. Our machine-learning model demonstrated high accuracy,
with an R-square of 0.933 and a root-mean-square error of 0.035. We validated
the model by testing it with Ra as one of the A elements and comparing the
results to a graph constructed from training elements. Our study highlights
how selecting A and B elements in the perovskite structure can affect the
magnetic anisotropy energy (MAE) of ferromagnetic (FM) materials, with certain
B elements having a more significant impact. It also reveals that the jump in
MAE results from the interface effect between the Fe monolayer and ABO3 rather
than the electric field effect due to ABO3 polarization. The machine learning
model developed in this research can help identify the most compelling
features for controlling the MAE, such as the magnetic moments of interface
atoms. Overall, these findings could enhance the development of efficient
spin-based devices and have broad potential applications in materials science.

###### Acknowledgements.

This research was supported by the Korea goverment (MSIT) through the National
Research Foundation of Korea (NRF-2022R1A2C1005505 and
NRF-2022M3F3A2A01073562) and Institute for Information & Communications
Technology Planning & Evaluation (IITP) (2021-0-01580). This work used the
resources of the KISTI Supercomputing Center (KSC-2022-CRE-0062 and
KSC-2022-CRE-0379).

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