Spin Wave Spectra in Pseudoperovskite Manganites with Superexchange Interaction Competition

In compounds La1/3Ca2/3MnO3 and BiMnO3, a calculation of spin-wave dispersion dependences along pseudoperovskite directions of reciprocal space is made. The model includes orbitally dependent superexchange interaction and single-ion anisotropy. Considered compounds present competing exchange interactions within magnetic unit cell because of orbital or charge-orbital ordering. The nearest neighbor superexchange interaction is taken into account. The magnetic structure and dispersion dependences of spin waves frequencies are calculated within the framework of regular multi-sublattice model. The peculiarities of frustrated magnetic spin-wave spectra are found. It is shown, that crossing and splitting of spin-waves branches in non-symmetric points of magnetic Brillouin zone is a common feature of the spectra due to exchange competition. The band structure of Г-point spectra for both compounds are predicted.


Introduction
The Jahn-Teller magnets with manganese sublattice are widely investigated as compounds with lattice, charge, orbital and magnetic subsystems interaction. The main application of CMR properties in these crystals awoke a vivid interest to some fundamental features including low-dimensional and frustrated magnetic structures.
The pseudoperovskite R 1-x A x MnO 3 manganites (where R 3+ is a rare earth ion, A 2+ is an alkaline earth ion, and x is a doping level) are distorted to orthorhombic or nearly orthorhombic crystal structure [1][2][3]. If the R 3+ position is occupied by bismuth ions, the crystal structure distortions are mostly monoclinic without doping. The Mn 3+ magnetic sublattice is orbitally degenerated in the non-distorted ideal crystal. The Jahn-Teller effect removes degeneration, thus the complicated superexchange interaction takes place [4]. BiMnO 3 has more complicated crystal structure of monoclinic symmetry, which is sharply different comparing with rare-earth undoped or doped manganites. It causes unusual orbital structure [5]. In BiMnO 3 compound and in doped rare-earth manganites at some doping levels, the competition of superexchange parameters is possible.
The main problem of investigation is that there are no studies of spin-wave spectra on multi-sublattice magnets with exchange competition. Two kinds of theoretical papers: geometrical frustration in triangular [6,7], honeycomb, or kagome lattice and exchange competition of next and other neighbors [8,9] on square lattice. Manganites under consideration are 3D magnets which have competing exchange interaction between nearest magnetic neighbors due to peculiar orbital ordering.
The aim of the study is to find specific features of low-temperature spin-wave spectra due to non-geometric frustration. The compounds of different symmetry and different manganese sublattices-BiMnO 3 and La 1/3 Ca 2/3 MnO 3 -have competing nearest-neighbor superexchange interactions within a pseudocubic cell. The current investigation uses these crystals as examples.

Methods
The ordering of orbital states in manganites is described by angle parameters Θ n . They are used in linear combination of 5 E ground-state eigenfunctions � ⟩ n , � ⟩ n for nth Mn 3+ ion. The wave functions due to Jahn-Teller (JT) effect are [10] where sign " ± " mainly depends upon e-type symmetrized distortions of local oxygen coordination [10]. This characteristic strongly affects the magnetic subsystem.
To describe the magnetic properties and particularly spin-wave spectra, the following Hamiltonian is used: where S i , S j are total manganese spins, and orbital structure parameters Θ i , Θ j are given by Eq. (1) for trivalent manganese ions, J ij (Θ i , Θ j ) are exchange parameters in i-j pair, D , i (Θ i ) is a single-ion anisotropy constant for ith Mn 3+ ion. The orbitally dependent superexchange interaction parameters of Eq. (2) in general form are as follows [10]: where parameters of interacting pair configuration are φ ij (the bond angle of Mn-O-Mn) and r ij (the average Mn-O bond length in i-j pair), J 0,k and F k,ij (Θ i , Θ j ) factors depend upon the charge and orbital states of interacting ions in the pair.
The second interaction of Hamiltonian (2) is a single-ion anisotropy on trivalent manganese ions: where D i = -3PcosΘ i , E i = -√3PsinΘ i , and P = -0.1 meV, S iz , S ix , S iz are total spin projections on local oxygen octahedron axes of each Mn 3+ ion.

Results and Discussion
The crystal structure of BiMnO 3 is monoclinic with C2/c space symmetry group [5]. Mn 3+ ions belong to positions 2e and 2d. The orbital structure of manganese sublattice is determined in paper [11] and is shown in Fig. 1. In the terms of orbital mixing angles, it is described by relations: The Eq. (5) relation is changed for Θ d by 2π period staying the same as in paper [11]. The superexchange interaction is determined by Eqs. (3), (4). The superexchange and single-ion anisotropy parameters of Eq. (5) used for calculations are: The parameters of crystal structure used for estimation are taken from experimental paper [5].
As one can see in Fig. 2a, the ferromagnetic (FM) bonds form a zigzag-like stripes, coupled with antiferromagnetic (AFM) interaction within the pseudoperovskite y p z p plane. But considering the shift of the stripe in the neighbor plane and the FM character of J x , there is the competition of superexchange interactions. The resulting magnetic structure is FM. The competition is important with account of slight difference between positive and negative values of superexchange in y p z p plane is crucial for the magnetic ordering and is dependent upon Θ d parameter of orbital ordering [11]. The magnetic ordering is quasi-one-dimensional because the relation |J x /J y,z | is about value 5.
The calculation of the magnetic structure is made within the framework of 16-sublattice model. The magnetic cell includes two lines along y p direction in two neighboring y p z p planes (see Fig. 2a). The direction of the magnetic moment is approximately along monoclinic c axis and contradicts with the experiment [5], which found it to be directed along b-axis. The discrepancy may be caused by necessity of Dzyaloshinskii-Moriya (DM) antisymmetric exchange interaction ij i × j account. The vectors d ij -parameters of DM interaction (DMI)-are not estimated yet. The LaMnO 3 parameters found in experiment [12] are not suitable in this case because of strongly distorted structure of BiMnO 3 and its orbital structure (6), which has another symmetry. In the paper [13], the second-and third-neighbors Fig. 1 The orbital structure of BiMnO 3 [11]. The Mn 3+ ions are drawn as electronic densities, oxygen and bismuth ions are omitted 1 3 Spin Wave Spectra in Pseudoperovskite Manganites with… superexchange interactions are proposed, but the nearest-neighbor's DMI is not explicitly described. As certain magnetic moment direction does not affect spinexcitations dispersion and there are no determined parameters of interaction, the DMI term is not included in the spin-Hamiltonian (2) for BiMnO 3 .
The crystal structure of La 1/3 Ca 2/3 MnO 3 is orthorhombic with Pnma space symmetry group with enlarged unit cell [3]. At low temperatures, it is a charge ordered compound with two manganese sublattices of Mn 3+ and Mn 4+ ions. The orbital structure of Mn 3+ sublattice is determined earlier [10] and is shown in Fig. 3. In the terms of orbital mixing angles, it is described by relations: The superexchange interaction is determined by Eqs.  [3].
The calculation of the magnetic structure is made within the framework of 24-sublattice model. The magnetic cell includes four Mn 3+ ions and eight Mn 4+ ions in each of two neighboring ac planes (one plane is drawn in Fig. 4a). The magnetic structure is markable non-collinear and consists of strongly-coupled FM trimers Mn 4+ -Mn 3+ -Mn 4+ along x p and y p pseudoperovskite axes, stacked along orthorhombic c-axis. AFM exchange between trimers leads to competition of interactions [10,14]. The magnetic structure is drawn in Fig. 4.   [10,14] In the compounds under consideration, the role of orbital ordering in magnetic competition is crucial. It causes non-geometrical magnetic frustration with account of nearest-neighbors superexchange interaction only. A calculation of spin-wave dispersion dependences along pseudoperovskite directions of reciprocal space is made. The peculiarities of frustrated magnetic spin-wave spectra are found. The crossing and splitting of spin-waves branches in non-symmetric points of magnetic Brillouin The low-frequency spin-wave spectrum along pseudoperovskite reciprocalspace direction [1 0 0] p for BiMnO 3 drawn in Fig. 5b is similar to charge-ordered La 1/2 Ca 1/2 MnO 3 , but one can note non-symmetric features due to exchange competition. In [0 1 0] p (see Fig. 5c) direction there is a mode without dispersion due to quasi-low-dimensional character of the magnetic structure. The direction of magnetic moment does not affect the dispersion dependence. Because of approximately collinear ferromagnetic ordering, a supposed spectrum should be reduced to one clear branch and some branches with weak intensity.
The branches of spin-wave dispersion curves are crossing in non-symmetric points of Brillouin zone. It is a common feature for compounds under consideration and classical case of triangular magnet [6,7].
In paper [12], the importance of antisymmetric DM exchange term is emphasized in orthorhombic manganites and the detailed study of experimental ESR spectra in La 0.95 Sr 0.05 MnO 3 was carried out. The values of DM and anisotropic constants were obtained for T > T N . For Mn 3+ -Mn 3+ pairs DMI account in La 1/3 Ca 2/3 MnO 3 , the results of [12] could be used. But DMI of other types of superexchange pairs were not investigated. Nevertheless, we have shown that for LaMnO 3 with DMI parameters of [12], the DM interaction account gives only quantitative changes in AFMR spectrum dependences [17].
In the case of frustrated compounds, the forms of dependences Figs. 5 and 6 are mostly determined by superexchange interaction. The DMI cannot change the general symmetry magnetic ordering, it gives a contribution in spin canting and shifts spin-wave spectrum's branches, particularly, in Г-point.

Conclusions
The theoretical study of Jahn-Teller compounds BiMnO 3 and La 1/3 Ca 2/3 MnO 3 with superexchange competition is carried out. The superexchange dependence orbital and charge orderings are crucial for the magnetic structure forming. The model does not consider second and third magnetic neighbors exchange. The spin wave spectra of low-temperature phase are predicted. The dispersion curves demonstrate similar features as in case of triangular magnet.
Author Contributions L.E. Gonchar as a single author has made calculation, wrote the manuscript text and prepared all Figures.
Funding No funding was received to assist with the preparation of this manuscript.