Magnetism and hybrid improper ferroelectricity in LaMO

Cite this:Phys. Chem. Chem. Phys., 2019,21, 20132

Magnetism and hybrid improper ferroelectricity in LaMO

₃

/YMO

₃

superlattices

Pengxia Zhou,†*^abShuaihua Lu,†^aChuanfu Li,^bChonggui Zhong, *^a Zhiyun Zhao,^aLihua Qu,^aYi Min,^aZhengchao Dong,^abNa Zhang^band Jun-Ming Liu ^b

Using first-principles calculations, we investigate the structural, electronic, and magnetic properties of perovskite LaMO3/YMO3superlattices (M = Cr, Mn, Co and Ni). It is found that ferroelectricity can emerge in LaMO3/YMO3superlattices (M = Cr, Mn, Co), allowing them to be promising multiferroic candidates, while no ferroelectricity is found in the LaNiO3/YNiO3superlattice. The electronic structure calculations indicate that the LaCrO3/YCrO3, LaMnO3/YMnO3, and LaCoO3/YCoO3 superlattices are insulators, and their magnetic ground states exhibit G-type antiferromagnetic (AFM), A-type AFM, and G-type AFM order, respectively, while the LaNiO3/YNiO3superlattice is however a half-metallic ferromagnet. The electronic structure and magnetic ground state are discussed, based on the projected density of states data and Heisenberg model, respectively, and the magnetic phase transition temperature is evaluated based on mean-field theory. In the meantime, the spontaneous ferroelectric polarization of the LaMO3/YMO3

superlattices (M = Cr, Mn, Co) is determined respectively using the Born effective charge model and Berry phase method, and their hybrid improper ferroelectric character is predicted, with the net polarization mainly from the different displacements of the LaO layers and YO layers along theb-axis. It is suggested that alternative multiferroic materials can be obtained by properly designing superlattices that consist of two non-polar magnetic materials but exhibit tunable magnetic ground states and transition temperature and hybrid improper ferroelectricity.

1 Introduction

Multiferroic materials have unique physical properties and a wide range of potential applications. The coexisting charge dipole ordering and magnetic ordering in multiferroics allow the possible cross-control of the magnetic and electrical properties, which promises an impact on data storage technologies.^1,2 However, usually the formation of charge dipoles needs empty d-orbitals (the so-called d⁰ rule), while the magnetic moment needs partially occupied d-orbitals. Such a contradiction makes multiferroic materials not common in nature.³

In the past decade, many routes have been developed to generate ferroelectric polarization in magnetic materials.^4,5For example, by introducing elements with 6s²lone pair electrons (e.g.Bi³⁺),^6,7proper ferroelectricity with large polarization can be obtained by violating the d⁰ rule. Besides, some special magnetic order, e.g. cycloidal spin order, can also generate

improper ferroelectricity, although the polarization in these systems is typically small (e.g.o1.0mC cm²).^8,9In addition to these two branches, the geometric ferroelectric family does not exclude magnetism, such as hexagonal YMnO₃,¹⁰ where the condensation of multiple non-polar phonon modes leads to improper ferroelectricity.

Recently, a new kind of geometric ferroelectricity, i.e. the so-called hybrid improper ferroelectricity (HIF), has been predicted^11,12and then experimentally verified in some Ruddle–

Popper structural A3B2O7 (e.g. Ca3Ti2O7).^13–15 Besides A3B2O7, HIF can also be obtained in some perovskite (ABO3)n/(A⁰BO3)n

(n: odd number) or (ABO3)n/(AB⁰O3)n (n: even number) super- lattices.^16–18Indeed, the PbTiO3/SrTiO3superlattice was the first example proposed for such HIF.¹⁹In detail, HIF arises from the coupling between the combination of rotation and tilting of oxygen octahedra.¹¹ The as-generated polarization is moderate (typically41mC cm²) and the ferroelectric transition tempera- tures are usually higher than room temperature since the rota- tion and tilting of oxygen octahedra in most oxides can occur at high temperatures.²⁰

Following this way, the 1 + 1 LaMO3/YMO3 superlattices (M = Cr, Mn, Co, and Ni) are systematically investigated using density functional theory (DFT) in this work. For the A site

aSchool of Science, Nantong University, Nantong, 226007, China.

E-mail: chgzhong@ntu.edu.cn, ntzhoupx@ntu.edu.cn

bLaboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

†These authors contributed equally to this work.

Received 29th June 2019, Accepted 23rd August 2019 DOI: 10.1039/c9cp03675j

rsc.li/pccp

PAPER

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elements, La and Y are chosen due to the larger difference in ionic radius between them, which may lead to relatively large distortion and thus large polarization. Here the transition metal element M is magnetic, making these systems promising multiferroic candidates if HIF can be achieved. The tuning of B site element M leads to multiple effects. First, the local magnetic moment can be tuned. Second, the magnetic order can be tuned. Third, the size of M can affect the lattice distortions,i.e.

the octahedral rotation and tilting, which directly tune the HIF.

Fourth, the electronic structure can be tuned. Therefore, these systems may exhibit diversified physical properties.

We have investigated the multiferroic properties of four LaMO3/YMO3 superlattices (M = Cr, Mn, Co and Ni) from several aspects. First, the lattice structures of the parent bulk materials in the superlattice form have been investigated by calculating the tolerance factor. Second, the magnetic ground state and phase transition temperature (Curie temperatureTC

for ferromagnetic (FM) materials or Ne´el temperature TN for antiferromagnetic (AFM) materials) have been investigated.

Third, the electric structures of the four superlattices have been studied by discussing the density of states (DOS) and projected DOS (PDOS). Finally, the ferroelectric polarization of these superlattices has been calculated and discussed.

2 Methods

First principles DFT calculations were performed using the Viennaab initioSimulation Package (VASP).^21,22The Perdew–

Burke–Ernzerhof (PBE) form of the generalized gradient approxi- mation (GGA)²³ was chosen for the exchange–correlation potential. Considering the strong Coulomb interactions for d- and f- electrons of transition metal ions, the Hubbard-type Uand Jhave been added to the La f orbitals and M d orbitals (except for the Cr element),²⁴ as summarized in Table 1. The plane-wave energy cutoff was set as 500 eV. The Monkhorst–Pack k-point mesh was 7 75 for all the superlattices. Both the lattice constants and atomic positions were fully relaxed until the Hellman–Feynman force on each atom was below 0.01 eV Ź.

In the calculations, we first check whether the A-AFM order is the magnetic ground state of orthorhombic LaMnO3.²⁷Then, given this ground magnetic structure, half of the La atoms are replaced by Y atoms, and the Mn atoms are replaced by M (M = Cr, Mn, Co, Ni) atoms, as shown in Fig. 1, where the minimum unit cell of an orthorhombic perovskite is adopted. This unit can be regarded as the 1 + 1 superlattice and the stacking of La and Y layers is along the c-axis. We optimize the lattice and atom positions with finiteUand different magnetic states to reach the ground structure. Here, ferromagnetic (FM), A-type antiferromagnetic (A-AFM), C-type antiferromagnetic (C-AFM),

and G-type antiferromagnetic (G-AFM) order are considered in the structure calculation of superlattice LaMO3/YMO3(M = Cr, Mn, Co, Ni). After accessing the magnetic ground state struc- ture, the electronic structure and polarization are calculated.

The ferroelectric polarization was calculated using the Born effective charge method and Berry phase method. The Born effective charges were estimated based on density functional perturbation theory (DFPT).^30,31The magnetic transition tem- peratureT_C(orT_N) was estimated with the energy of different magnetic order by using the mean field method.

3 Results and discussion

A Structures

The generation of HIF in a superlattice requires proper lattice distortion modes and magnitudes of octahedra. One of the most common space groups of perovskites, i.e. Pbnm, allows antiferroelectric polarization due to octahedral rotation and tilting.³²Thus, it is possible that ferroelectricity emerges in the superlattice consisting of two perovskite components with the Pbnmstructure because of the different magnitudes of octahedra distortion.

For perovskites, the stable structure is usually determined by the tolerance factor, as defined ast = (rA+rO)/(rB +rO)2^1/2,³³ whererA,rB, and rOare the radii of A, B, and O, respectively.

Perovskite materials favour thePbnmstructure if 0.75oto0.9.³⁴ The ions’ radii involved in the present work and corresponding tolerance factors are shown in Table 2. Here, for the 1 + 1 super- lattice constitution, the tolerance factor of the LaMO₃/YMO₃super- lattice is the average value of the parent materials LaMO₃ and YMO₃,i.e.,t₃= (t₁+t₂)/2.

Table 1 The values of coefficientsUandJ(in unit of eV) for La f electrons and M d electrons

La^{3+ 25,26} Mn^{3+ 27} Co^{3+ 28} Ni^{3+ 29}

U 11 2.7 6 7

J 0.68 1 0.8 0.65

Fig. 1 Schematic crystal structures of LaMO3/YMO3superlattices (M = Cr, Mn, Co and Ni) and the atom number assignment. Here, LaMO3and YMO3

are placed alternatively along thec-axis,i.e. the so-called 1 + 1 type superlattice.

Table 2 The ions’ radii (R, in units of Å) and tolerance factors of bulk LaMO3and YMO3

M La³⁺ Y³⁺ O² Cr³⁺ Mn³⁺ Co³⁺ Ni³⁺

R³⁵ 1.032 0.9 1.35 0.615 0.645 0.61 0.56 t₁(LaMO₃) — — — 0.857 0.844 0.859 0.882 t₂(YMO₃) — — — 0.810 0.798 0.812 0.833 t₃(LaMO₃/YMO₃) — — — 0.834 0.821 0.836 0.858

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Since all the tolerance factor values of LaMO3and YMO3are between 0.75 and 0.9, the constituting bulk perovskite materials can form the Pbnmstructure, part of which are consistent with previous results^36–41 and are conducive to the formation of HIF.

In the following, we will start from the ‘fakePbnm’ (AMO3/A⁰MO3, AaA⁰) structure and perform structural relaxation.

The optimized structure of the LaMO3/YMO3 superlattice becomes the Pmc21 group, which is lowered from the Pbnm group. ThePmc2₁group is polar, with the polar axis along the b-axis (to be discussed later). The lattice constants and volumes are listed in Table 3. The volume of LaMO₃/YMO₃ decreases when M changes from Mn to Co and Ni atoms (except Cr becauseU was not considered, whereUcan make the lattice constant larger than the actual value), which is reasonable considering the different radius of the M ions.

B Magnetism

To explore the magnetic properties, the four possible magnetic states were calculated. By comparing their energies (see Fig. 2), it is found that the ground state of the LaMO3/YMO3 superlattice prefers G-AFM order for M = Cr and Co, and A-AFM order for M = Mn, while the FM state is preferred for M = Ni. It is worth mentioning that there is a fifth type of linear magnetic order in a cubic structure,i.e., E-AFM.⁴²E-AFM can result in exchange stric- tion and thus can induce electric polarization.⁴³We calculate the energies of the E-type AFM state in the four kinds of superlattices.

The results suggest that the E-AFM energies are 4.795 eV, 1.221 eV,

2.908 eV and 3.792 eV, higher than the ground state one for the LaCrO3/YCrO3, LaMnO3/YMnO3, LaCoO3/YCoO3and LaNiO3/YNiO3

superlattices respectively (including 20 atoms). Thus, the E-AFM magnetic order neither behaves as the ground state nor induces polarization. The magnetic order of each superlattice can be under- stood simply from the exchange coupling constants.

The effective exchange coupling constants between the nearest neighbour M spins can be obtained by mapping the DFT energies to the classical Heisenberg model, expressed as:

E_FM¼E₀þ4 2Jð _abþJ_cÞ E_AFM¼E₀þ4 2Jð _abJ_cÞ E_C-AM¼E₀4 2Jð _abJ_cÞ E_G-AM¼E₀4 2Jð _abþJ_cÞ

(1)

whereEFM,EA-AFM,EC-AFM, andEG-AFMdenote the energies of the FM, A-AFM, C-AFM, and G-AFM states, respectively. In eqn (1), all the spins are normalized to unit 1, despite the concrete magnetic moments of the M species being different.J_abandJ_c are the nearest in-plane and out-of-plane exchange coupling constants respectively, and E₀ is the non-magnetic contribu- tion. The values ofJ_abandJ_cfor different LaMO₃/YMO₃super- lattices are displayed in Table 3. Negative and positive signs of Jabimply that the two in-plane nearest-neighbour M ions favour FM and AFM coupling, respectively, and theJcsign is the same asJabexcept when the magnetic order is between the two out-of- plane M ions. From Table 3, the signs of Jab and Jc are consistent with the ground magnetic state.

The magnetic transition temperature is the point where the system changes from paramagnetic to magnetic order. Here, based on the calculations of different magnetic state energies and exchange coupling constants, the magnetic transition temperature is estimated using mean field theory:

T¼ JP

iaj

S* iS^*_j

3k_B (2)

wherekB= 1.3810²³J K¹is the Boltzmann constant. With the values of Jgiven in Table 2 and eqn (2), we obtained the transition temperaturesTN= 211.31 K, 28.05 K, and 955.54 K, Table 3 The optimized lattice constants (in units of Å), volumes (in units of ų) and the out-of-plane M–O–M angleyout(1) and in-plane M–O–M angle yin(1) for LaMO3/YMO3superlattices. Here, the data are taken for the lowest energy magnetic state structure (see the text). The effective nearest neighbour exchange coupling constants are calculated based on the different magnetic state energies. The magnetic moment per M atom and total magnetic moment are also shown

LaCrO₃/YCrO₃ LaMnO₃/YMnO₃ LaCoO₃/YCoO₃ LaNiO₃/YNiO₃

Lattice constants a 5.414 5.452 5.364 5.397

b 5.577 5.968 5.621 5.565

c 7.761 7.593 7.924 7.717

Volume 234.324 247.063 238.893 231.778

y_out 146.770, 159.023 141.287, 154.425 145.749, 158.412 146.395, 160.565

y_in 150.803, 152.870 146.788, 148.796 148.940, 151.226 151.537, 153.445

Magnetic ground state G-AFM A-AFM G-AFM FM

M(total) (m_B) 2.653 (0.0) 3.664 (0.0) 2.983 (0.0) 1.461(4.232)

J_ab 9.205 3.121 36.348 28.181

J_c 8.941 2.612 50.982 34.236

Fig. 2 The energies of superlattice LaMO₃/YMO₃(M = Cr, Mn, Co and Ni) in FM, A-AFM, C-AFM and G-AFM magnetic states; here, for each super- lattice, the ground state energy is regarded as the zero energy, and the other three state energies are relative values to the ground state one.

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and TC = 699.97 K for the LaCrO3/YCrO3, LaMnO3/YMnO3, LaCoO3/YCoO3, and LaNiO3/YNiO3superlattices, respectively.

C Electronic structures

To clarify the electronic structures of the LaMO3/YMO3super- lattices, the atom-projected density of states (DOS) is calculated, as shown in Fig. 3. The magnetic ground states are adopted in these calculations. According to Fig. 3, the DOS near the Fermi level for each case is mainly contributed by O 2p and M 3d orbitals. The DOS distribution shows that the LaMO3/YMO3(M = Cr, Mn, Co) superlattices are insulators while LaNiO3/YNiO3is a half-metal material. p–d hybridization can also be evidenced.

D Polarization

As stated in the introduction, the 1 + 1 LaMO3/YMO3 super- lattice may generate HIF, although neither LaMO3 nor YMO3is polar electrically. To check this property, the Born Effective Charge (BEC) method is used to evaluate the polarization of the LaMO3/ YMO3superlattices. The as-evaluated polarizationPBECis given by:

P_BEC¼ e O

X

i

Z_iu_i (3)

whereeis the electron charge,Ois the unit cell volume, andu_iis the displacement of ioniin the polar structure with respect to the high symmetric non-polar structure.Z* is the BEC calculated by density functional perturbation theory (DFPT). The polarization values of the first three superlattices are listed in Table 4. For the LaNiO3/YNiO3 superlattice, no polarization is generated in the common sense since it exhibits an FM metal electronic structure.

Furthermore, the more strict Berry phase method is employed to check the polarization, as compared in Table 4.

To analyze the emergent polarization, the contribution of each ion is displayed in Fig. 4. Namely, for each ion, the production of

nominal charges (2 for O, +3 for La and Y, and +3 for M) and displacements along theb-axis with respect to the high symmetry structure are calculated. The index of the twenty ions in the super- lattice can be found in Fig. 1. It is clear that the contributions of the MO2layers (including M, O5 to O12) are relatively small. The LaO layer (La, O1, O4) and YO layer (Y, O2, O3) contribute oppositely and the effective displacement of the YO layer is larger than that of the LaO layer, which can be reasonably understood due to the lighter weight of Y. Thus, the net polarization of all the superlattices mainly arises from the uncompensated displacements of the LaO layer and the YO layer. In addition, the p–d orbital hybridization DOS of each LaMO3/YMO3superlattice (M = Cr, Mn, Co) may also contribute to the polarization (see Fig. 3). Although the LaNiO3/YNiO3superlattice is a metal, one finds that the product of nominal charge with ion displacement is also similar to the results of first three super- lattices. It is suggested that there also exists ‘‘fake polarization’’ due to the distortion of the octahedron.

4 Conclusion

In conclusion, our calculations predict that the LaMO₃/YMO₃ (M = Cr, Mn and Co) superlattices are multiferroic materials according to first principles calculations. Although the magnetic ground state and magnetic transition temperature are different for these systems, the appearance of hybrid improper ferroelectricity is predicted. The ferroelectric polarization is contributed by not only the ion displacements but also the p–d orbital hybridization. The Fig. 3 The density of states (DOS) and projected DOS (PDOS) of the

ground states in these LaMO₃/YMO₃superlattices. (a) G-AFM LaCrO₃/ YCrO₃, (b) A-AFM LaMnO₃/YMnO₃, (c) G-AFM LaCoO₃/YCoO₃, and (d) FM LaNiO₃/YNiO₃. The Fermi energy is set as 0 eV.

Table 4 The calculated polarization of these superlattices (units:mC cm²).

Only theb-axis component is nonzero

M Cr Mn Co Ni

PBEC 7.865 4.248 5.542 —

P_BP 7.558 4.569 4.8945 —

Fig. 4 The polarization value of each atom along the b-axis in each superlattice. Here, numbers 1 to 12 denote the oxygen atoms, numbers 13 to 16 denote the magnetic atoms M (Cr, Mn, Co or Ni), numbers 17 and 18 denote the La atoms and numbers 19 and 20 denote the Y atoms. (a) LaCrO₃/ YCrO₃; (b) LaMnO₃/YMnO₃; (c) LaCoO₃/YCoO₃; and (d) LaNiO₃/YNiO₃.

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calculation on the electronic structure and atom displacements for the LaNiO3/YNiO3superlattice shows a polar lattice structure but a half-metal FM state. For all these cases, the ferroelectricity mainly comes from the structural distortion.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was financially supported by the national Natural Science Foundation of China (Grant No. 11604164, 11504093) and Nantong University Natural Science Foundation (Grant No. 15B16 and 15ZY14), and computations were supported by Nantong University Computation Center.

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