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Ab Initio Characterization of Magnetoelectric Coupling in Fe/BaTiO_{3}, Fe/SrTiO_{3}, Co/BaTiO_{3} and Co/SrTiO_{3} Heterostructures

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## Abstract

**:**

_{3}and SrTiO

_{3}that exhibit magnetism, ferroelectric polarization and piezoelectric effects. Within the structures under investigation, magnetic moments can be tuned by an external electric field via the ferroelectric dipoles. We investigate the effect of magnetoelectric coupling by means of ab initio spin-polarized and spin–orbit calculations. We study the structural, electronic and magnetic properties of heterostructures, and show that electrostriction can reduce the magnitude of the magnetization vector of a ferromagnet. This approach can become the basis for controlling the properties of one of the ferromagnetic layers of a superconducting spin valve, and thus the superconducting properties of the valve.

## 1. Introduction

_{3}) demonstrate an ideal crystallinity and possibilities of epitaxial growth. In addition, such materials have some other outstanding properties, which positively affect the performance of devices based on them. In particular, they exhibit good multiferroic properties, a magnetoelectric (ME) coupling effect and reverse tunability of electronic and magnetic properties. The conjugate magnetic and electric fields can be used to control the respective order parameter with cross-coupling. For example, switching the ferromagnetic (FM) order using an electric field promises a significant impact on the development of next-generation devices. Magnetic tunnel junctions (MTJs) are of particular interest to the experimental and theoretical community due to their promising applications in magnetic random-access memory (MRAM). Multiferroic materials are also suitable for spin-filter purposes [13,14].

_{c}> ∂T

_{c}, where ∆Tc is the magnitude of the effect of the superconducting spin valve and ∂Tc is the width of the superconducting transition. At present, the control of a superconducting current under the action of an external magnetic field in the design of a superconducting spin valve has reached its maximum efficiency. To progress further, it is necessary to study new heterostructures with other switching capabilities. One such approach is the study of structures based on piezoelectric substrates. This offers the possibility of controlling the superconducting current in a superconducting spin valve using an electric field.

_{3}(BTO), Fe/SrTiO

_{3}(STO), Co/BaTiO

_{3}and Co/SrTiO

_{3}were studied using density functional theory (DFT) calculations. The choice of components was motivated by the fact that Fe and BaTiO

_{3}are two “classical” ferroic materials with well-known bulk properties. In addition, fcc Fe and BaTiO

_{3}(Figure 1a,b) have similar cubic structures, which permits the experimental layer-by-layer epitaxial growth of Fe = BaTiO

_{3}multilayers without significant misfit dislocations, as well as simplifying the simulation of the heterostructures on a computer. Furthermore, since the Fe/BTO heterostructure has already been studied in depth [15,16,17,18,19,20], in this work, we present a comparison with heterostructures based on similar compounds, namely, ferromagnetic Co and quantum paraelectric SrTiO

_{3}(potential ferroelectric), in which the quantum fluctuations suppress the phase transition from the paraelectric to the ferroelectric state [21]. Duan et al. [16] showed that changing the direction of polarization could change the magnetization inside a ferromagnetic film. Thus, a change in direction of the magnetization could be used in devices for switching magnetization and superconductivity by an electric field in a superconducting spin valve. Here, we show that isotropic striction affects the magnitude of the magnetic moments. To study the striction effect, compression and tension were applied isotropically within the x and y axes.

## 2. Material and Method Details

^{−5}eV. The Brillouin zones were separated using Monkhorst–Pack grids [28,29,30], including 5 × 5 × 1 k-points for Fe/BaTiO

_{3}, Fe/SrTiO

_{3}, Co/BaTiO

_{3}and Co/SrTiO

_{3}film heterostructures (Figure 1c), with a Gaussian smearing of 0.05 eV. To take into account strong correlations between the electrons of the d-shells, the calculations were performed within the GGA + U method using a simplified approach proposed by Dudarev et al. [31], which takes into account only the difference between the Coulomb screening parameter U and the Stoner exchange parameter J (U

_{eff}= U − J). In addition, a correct exchange is necessary for both the correct electronic and magnetic structure and correct geometry [32]. We applied additional local correlations U

_{eff}equal to 4.4 eV and 4.6 eV for the Ti 3d and Fe 3d orbitals [33], respectively. To simplify the calculations, the following procedure was used: the geometry optimization was performed within the spin-polarized approach, whereas electronic and magnetic properties were extracted within the spin–orbit calculations [34].

## 3. Results and Discussions

_{B}, respectively. These values, together with the cell parameters, coincide well with previously published experimental values and theoretical predictions [10,26]. For our calculations, we used fcc cells of ferromagnetic materials (Figure 1b, note, that Co and Fe have the same structure). As for the BTO and STO chosen in this work, they have a perovskite cell with Ti surrounded by an oxygen octahedral (Figure 1a). The ferroelectric BTO in the tetragonal phase has a Ti–O displacement of 0.13 Å (the displacement direction is shown by an arrow in Figure 1a), which corresponds to 31 μC/cm

^{2}, while in STO, there are no displacements in volume.

_{BTO}− a

_{Fe})/a

_{BTO}, where a is a lattice parameter. To compose heterostructures, the components were brought into contact with an expansion of ferromagnetic slabs, but without any rotation. Alongside that, the BTO and STO were considered substrates, so the lattice parameters of the constructed heterostructures were kept frozen during the optimization procedure to the bulk values of the BTO and STO.

_{2}interface with O atoms over Fe/Co atoms (1) or Ti over Co atoms; (2) as well as a Ba(Sr)O interface with Ba(Sr) atoms; (3) and O over Co atoms and O in quadruple voids of the fcc Co(001) plane (4). Figure 1c shows the most stable interface configuration corresponding to the first type. This conclusion is consistent with the previous works [19,35]. Further discussion is presented for the most stable geometry.

#### 3.1. Spin-Polarized Calculations: Structural and Magnetic Properties

_{2}layers are associated with polarization. Similar plots were observed in [36], where the incensement of rumpling was observed close to the interfaces. More smooth curves would be observed with increased BTO(STO) slabs. It should be noted that the middle three layers were frozen in our computation scheme, and because of that, there are zero displacements. The other layers undergo complex distortions, with the highest out-of-plane shifts for interfacial atomic layers. The biggest magnitude corresponds to the BTO/Co heterostructure.

_{B}. For BTO/Fe, the values of the magnetic moments at the interface are slightly higher than those in [12]. In addition, the magnitudes of the Ti moments were also summarized in Figure 3b. We found that only interfacial titanium atoms receive magnetic moments, and the magnitude of magnetization agrees well with the values published in [10] for the BTO/Fe case. The highest magnitudes of ~0.4–0.5 µ

_{B}correspond to the heterostructures with Fe. Alongside that, the Ti magnetic moments have directions opposite to that of the Fe moments which are toward the ferroelectric slab.

_{B}per Co atom, which is slightly higher than the value in the bulk cobalt. In contrast to the case with iron, no rumpling was observed.

#### 3.2. Spin–Orbit Calculations

#### 3.3. Density of State Calculations

#### 3.4. Striction Calculations

_{B}for the initial structure to ~2.8 µ

_{B}for the structure with 15% compression. Additionally, we found that the difference between the interfacial and bulk Fe increases as the striction increases. In particular, structures with significant compression demonstrated the occurrence of maximum at the interfaces and in the middle of the ferromagnetic slab, while the minimum corresponded to the second layer from the interface.

## 4. Discussion and Conclusions

_{3}, Fe/SrTiO

_{3}, Co/BaTiO

_{3}and Co/SrTiO

_{3}model superlattice systems using the density functional theory approach. It was found that the structural properties of all considered heterointerfaces are similar: the structural optimization performed taking into account the magnetic nature of the materials led to the distortions associated with the movement of Ti and Ba(Sr) atoms from theirs bulk positions. The resulting magnitudes of out-of-oxygen plane displacements are the biggest at the interfacial layer due to the proximity of the ferromagnet. This is true for all investigated structures. On the contrary, the layer-dependent magnitude of displacement is different for all, and it depends both on the type of ferroelectric and ferromagnet used in the heterostructure. Taken together, these distortions lead to the appearance of polarization directed from the interface toward the ferroelectric substrate.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) The unit cells of BaTiO

_{3}at the tetragonal phase (SrTiO

_{3}has similar perovskite structure) and (

**b**) fcc Fe(Co); and (

**c**) the heterostructures Ba(Sr)TiO

_{3}/Fe(Co) used for simulations with arrows indicating the displacement of Ti from the center of oxygen octahedra and corresponding ferroelectric polarization in BTO.

**Figure 2.**The displacement of Ti atoms out of oxygen planes in TiO

_{2}layers within the substrate (BTO or STO). The middle three atomic layers were frozen. Layers 1 and 11 correspond to the interfacial layers of the BTO (STO) in various heterostructures.

**Figure 3.**Average magnetic moments calculated within the spin-polarized approach for heterostructures with seven atomic layers of ferromagnet. (

**a**) The magnetic moment values calculated per Fe/Co ion for different layers of the ferromagnetic film. (

**b**) The magnetic moment values calculated per Ti ion for different layers of the BTO (STO). Layer 1 and layer 7 correspond to the interfacial layers within the ferromagnetic slab.

**Figure 4.**Average magnetic moments calculated taking into account the spin–orbit interaction. Magnetic moments were calculated per Fe (Co) ion in the ferromagnetic layers of the heterostructures. The results are shown for the z Cartesian direction only, whereas the magnetic moments in other directions are negligible (in accordance with Figure 1). Layer 1 and layer 7 correspond to the interface layers, whereas layer 4 is the middle layer of the ferromagnetic slab.

**Figure 5.**Atom-resolved density of states plots calculated for the BTO/Fe (

**a**) and BTO/Co (

**b**) heterostructures. Majority and minority spin DOSs are shown in the upper and lower panels, respectively. Zero at the energy scale corresponds to the Fermi level. Ti states are shown by the black curve, O by the green, Fe by the red and Co by the blue.

**Figure 6.**Density of states plots calculated for the STO/Fe (

**a**) and STO/Co (

**b**) heterostructures. Majority and minority spin DOSs are shown in the upper and lower panels, respectively. Zero at the energy scale corresponds to the Fermi level. Ti states are shown by the black curve, O by the green, Fe by the red and Co by the blue.

**Figure 7.**Magnetic moments of the Fe atoms calculated for the BTO/Fe heterostructure within the plane striction. Each curve corresponds to the in-plane a = b lattice parameters.

SrTiO_{3} | BaTiO_{3} | |

Co | 9.4% | 11.8% |

Fe | 6.5% | 8.9% |

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**MDPI and ACS Style**

Piyanzina, I.; Evseev, K.; Kamashev, A.; Mamin, R.
Ab Initio Characterization of Magnetoelectric Coupling in Fe/BaTiO_{3}, Fe/SrTiO_{3}, Co/BaTiO_{3} and Co/SrTiO_{3} Heterostructures. *Magnetism* **2023**, *3*, 215-225.
https://doi.org/10.3390/magnetism3030017

**AMA Style**

Piyanzina I, Evseev K, Kamashev A, Mamin R.
Ab Initio Characterization of Magnetoelectric Coupling in Fe/BaTiO_{3}, Fe/SrTiO_{3}, Co/BaTiO_{3} and Co/SrTiO_{3} Heterostructures. *Magnetism*. 2023; 3(3):215-225.
https://doi.org/10.3390/magnetism3030017

**Chicago/Turabian Style**

Piyanzina, Irina, Kirill Evseev, Andrey Kamashev, and Rinat Mamin.
2023. "Ab Initio Characterization of Magnetoelectric Coupling in Fe/BaTiO_{3}, Fe/SrTiO_{3}, Co/BaTiO_{3} and Co/SrTiO_{3} Heterostructures" *Magnetism* 3, no. 3: 215-225.
https://doi.org/10.3390/magnetism3030017