Vibronic interaction as main reason of magnetic ordering in insulating manganites R1–xAxMnO3

The model of orbitally dependent magnetic structure of charge ordered insulated manganites is proposed. The model is semi-phenomenological. It allows using a few parameters to describe possible magnetic structures of compounds. The experimental crystal structure of compounds also could be taken into account. The compounds LaMnO3, La1/2Ca1/2MnO3, La1/3Ca2/3MnO3, BiMnO3 are considered.


Introduction
The manganites compounds are interesting because of strong correlation between crystal, charge, orbital, and magnetic subsystems.The general formula of compounds under consideration is R 1-x A x MnO 3 , where R 3+ =La 3+ or Bi 3+ , A 2+ =Ca 2+ , x=0, 1/2, 2/3.A field of correlation is a sublattice of Mn 3+ ions.Mn 3+ ion in octahedral oxygen neighborhood is a Jahn-Teller (JT) ion.The orbital states of Mn 3+ ions form a regular structure, called orbital ordering (OO).In the case of non-isovalent doping, the charge carriers could be localized on Mn ions.This effect leads to charge ordering (CO).It means the regular structure of Mn 3+ /Mn 4+ ions.The magnetic structures (MS) of manganies are various [1][2][3][4][5][6].There are A-type, E-type, CE-type, and F-type structures.OO is dependent upon crystal structure and charge distribution [7][8][9][10][11].The interrelation between orbital and magnetic subsystems is also discussed [8][9][10][11].In OO description, a choice of driving mechanism of cooperative ordering is discussed.The ab initio calculations require a complicated account of correlations [12,13].The most popular Kugel-Khomskii model [8] usually is used within the approximation of comparable vibronic and superexchange interaction.Nevertheless, the main vibronic term is estimated as ~1 eV [14], and the exchange parameters are ~1 meV [9,10].The rich set of MS makes difficult to create a uniform model.The main rules for estimation of the signs and relative values of superexchange (SE) parameters are Goodenough-Kanamory rules [1].These rules use OO for MS explanation.The absence of precise values makes the MS description impossible for magnetically frustrated compounds such as La 1/3 Ca 2/3 MnO 3 or BiMnO 3 .For SE parameters description, the ab initio calculations [12], Kugel-Khomskii model [8], orbitally-dependent phenomenological model [4,9,10,17], next-nearest neighbor model [4,7,15], double-exchange model [7,11] are applied.The ab initio calculations require preliminary assumption of MS [12,13].The doubleexchange model is not applicable in suggestion of CO.We use previously proposed orbitally-dependent SE model [9,10] to find possible MS for different orbital mixing within the framework of crystal symmetry.The aim of the paper is the description of orbital-magnetic correlation in insulating magnanites including frustrated cases.

Orbital state of Mn 3+ ion
The ground state of Mn 3+ ion's orbital subsystem with total spin S=2 in undistorted octahedral ligand neighborhood is 5 E multiplet.This state is twofold degenerated, thus some distortions of the neighborhood lead to removing of degeneration.We can use e-type orbital operators acting on many-electron eigenfunctions of 5 E (|,|) in a linear combination as effective Hamiltonian where operators in matrix form are .0 1 The  1 and  2 coefficients are mainly phenomenological and could be described by linear vibronic interaction, non-linear vibronic interaction or crystal field models [9,10].Taking into account the effective interaction (1), the ground state function is considered as a linear combination of 5 E eigenfunctions: where 1 where "+" is for  2 <0, and visa versa.For magnetic properties prediction, the mean values of X-operators are needed.
In the crystal, there is a sublattice of Mn ions.The orbital wave functions and orbital operators must be indexed by Mn 3+ ion number.Thus, there are two sets of orbital interaction characteristics for n th Mn 3+ ion: 1)  1,n and  2,n ; 2) ρ n , Θ n .For orbital state wave function ( 6) only one of characteristics-Θ n -is sufficient.

Superexchange interaction
The exchange interaction of two Mn ions in octahedral neighborhood could be described by general expression where SE operators are dependent upon orbital operators (2) of interacting ions: Let us review the different kinds of orbitally-dependent 180 0 -SE interaction in insulating perovskite manganties.
Finding the mean values of orbital operators on the orbital state wave functions (6) of interacting ions and taking into account the Mn-O-Mn bond configuration [9,10], one could write the SE parameters in pseudoperovskite axes as a function of orbital angles ( i ,  j ), Mn-O-Mn bond angle  ij and mean Mn-O distance r ij .1) Mn 3+ -Mn 3+ (x, y, z): 3) Mn 4+ -Mn 4+ : . cos .Thus, we can determine the main component of magnetic interaction.

Results and discussion
Let us apply the theory above to certain manganite crystals.This application was performed sometimes earlier in previous papers [9,10,17], but using another characteristics of OO, i.e.  angle instead of .It helps to remove the equivalence with MS description used in Kugel-Khomskii model [8] 4.1 LaMnO 3 The space symmetry of the compound is Pnma [1].The OO of LaMnO 3 (Fig. 1) is caused by vibronic interaction of Mn 3+ ion with local distortions of oxygen octahedra.These distortions are described by symmetrized coordinates Q ,n and Q ,n .The coefficients of effective Hamiltonian (1) are: where -|V e |Q ,n =0.11 eV, -|V e |Q ,n =0.34 eV.All Mn 3+ ions are divided into four sublattices due to symmetry considerations (see Fig. 1).The correlation between angles is: The SE interaction is of type 1.The SE dependences upon Θ angle are drawn on Fig. 2. In orthorhombic Pnma notations, J b =J z , J ac =J x, y .Fig. 2. Orbitally-dependent SE interaction.Arrows denote LaMnO 3 orbital states of Mn 3+ ions.Letters A and G denote MS types.
As one can see at Fig. 2, the SE interaction allows only two types of AFM structure -A and G.The experimentally determined MS of LaMnO 3 is A-type AFM structure [1].Some OO allow one-dimensional MS (=52 0 , 149 0 , 211 0 , 308 0 ).The G-type MS is not observed in manganites.

La 1/2 Ca 1/2 MnO 3
The space symmetry of the compound is P2 1 /m [5].At low temperature, there is CO state of the compound.The CO and OO are presented on Fig. 3. Fig. 3. CO and OO of La 1/2 Ca 1/2 MnO 3 .Mn 3+ ions as mixed singleelectron e g -functions and Mn 4+ ions as spheres are drawn.Ions of 2a and 2b positions are numbered The OO of Mn 3+ sublattice of La 1/2 Ca 1/2 MnO 3 is caused mainly by linear vibronic interaction of Mn 3+ ions with local distortions of oxygen octahedra.The coefficients of vibronic Hamiltonian (1) of Mn 3+ ions have the same form (10) as for LaMnO 3 .The coefficients of effective Hamiltonian (1) are: The wave functions of the Mn 3+ sublattice is of the same form as in Eqs.(11).The OO is described by correlation: . 56 2 , , The SE interaction is of types 1-3.The second type of SE interaction defines planar interaction (J x , J y ), and 1, 3 types are interplanar (J z ).The dependences of SE parameters upon  angle are drawn on Fig. 4. In monoclinic P2 1 /m notations, J b =J z , J ac =J x, y .SE parameter of Mn 4+ -Mn 4+ interaction is J=15 K [9].The space symmetry of the compound is Pnma [21] that is tripled compared with LaMnO 3 .At low temperature, the CO state is also proposed [17,18].The CO and OO are presented on Fig. 5.The OO of Mn 3+ sublattice of La 1/3 Ca 2/3 MnO 3 is caused mainly by vibronic interaction of Mn 3+ ion with local distortions of oxygen octahedra.The coefficients of vibronic Hamiltonian (1) of Mn 3+ ions have the same form (11) as for LaMnO 3 .They must be slightly corrected, because it helps describing MS [17].
The wave function of the Mn 3+ sublattice is of the same form as in Eq. ( 11).The OO is described by correlation: The SE interaction is of types 1-3.The SE parameters dependent upon  angle are the same as in La 1/2 Ca 1/2 MnO 3 and are drawn on Fig. 4. SE parameter of Mn 4+ -Mn 4+ interplanar interaction is J b =15 K [9], and J ac =15 K [17] in the ac-plane.Taking into account the frustration of MS in Mn 3+ -Mn 4+ or Mn 4+ -Mn 4+ pairs within the ac-plane, one can suppose three possible types of MS.The magnetic moments in two structures form FM trimers along elongated O-Mn 3+ -O bond.They are differently located relative to each other.The structure 1 (Fig 6a) has Mn 3+ -Mn 4+ FM ordered bond with AFM SE interaction.It does not double the crystal unit cell.On Fig. 6b, the FM Mn 4+ -Mn 3+ -Mn 4+ trimers are ordered as stripes, which are perpendicular to each other.Fig. 6b structure has doubled crystal unit cell along c-axis.

BiMnO 3
The space symmetry of the compound is C2/c [6] that is unusual for manganites.The possible OO is presented on Fig.
8 [10].The OO of BiMnO 3 is caused not only by vibronic interaction of Mn 3+ ion with local distortions of oxygen octahedra [10].The non-local and non-linear vibronic terms could be taken into account as parameters: where n is Mn 3+ sublattice number (n=1e, 2e, 1d, 2d).  .These values are inconsistent with found MS.However, the account of non-linear vibronic terms drastically changes the orbital states of d-position Mn 3+ ions [10].The superexchange interaction is of 1 type (Mn 3+ -Mn 3+ ).As opposed to LaMnO 3 , there are six different values of SE interaction.The diagram of possible MS is drawn on Fig. 9. OO, caused by linear vibronic interaction (dark arrows on Fig. 9), leads to MS frustration in different directions (y p or z p ).The possible structures are: G-type AFM without frustration, F-type FM structure, C-type AFM with FM chains approximately along x p direction, A 1 -type AFM with FM planes parallel to x p y p -planes, A 2 -type AFM with FM planes parallel to x p z p -planes.The regions of orbital mixing angles supporting the experimental F-type MS approximately are:  e 300 0 , 90≤ d 160.

Conclusions
The uniform approach to description of orbital and magnetic structure of insulating manganite crystals is proposed.The model of orbital structure uses 2 parameters per Mn 3+ position.For magnetic structure description -only 1 parameter per JT position is needed.
The support by Act 211 Government of the Russian Federation, contract № 02.A03.21.0006 is acknowledged.

Fig. 1 .
Fig.1.OO of LaMnO 3 .Mn 3+ ions as mixed single-electron e g -functions are drawn.Here and on other figures, oxygen and lanthanum ions are omitted.Numbers denote Mn 3+ ions of 4a position.

Fig. 4 .
Fig. 4. Orbitally-dependent SE interaction.Arrows denote two states of Mn 3+ sublattice in La 1/2 Ca 1/2 MnO 3 .Letters A, G, and CE denote MS types.The magnetic cell is described by doubled crystal cell.The changes of OO may cause the change in the MS.The experimental MS of La 1/2 Ca 1/2 MnO 3 is CE structure [5].It consists of FM zigzag chains along a axis with AFM order between chains.The A-type of MS is the nearest structure on the diagram.

Fig. 6 .
Fig. 6.Frustrated MS of 1 and 2 types of La 1/3 Ca 2/3 MnO 3 in one of ac-planes.In neighbor plane the spin moments are directed opposite.The experimental investigation [18] has determined MS of type 2. However, the local oxygen octahedral distortions account in OO lead to type 1 structure.Taking into account possible terms  ,n ,  ,n in vibronic interaction (15),  may change slightly.On Fig. 7, the region of mixing angle is drawn as wide arrows.The possible values of  angles supportingthe experimental MS are ≤305 0 , ≥55 0 The additional parts of Hamiltonian's coefficients could be estimated as | ,n |, | ,n |~0.01 eV/Å.The edge of the region is established

Fig. 7 .
Fig. 7. Orbitally-dependent energy of possible MS of La 1/3 Ca 2/3 MnO 3 .The white and colored stripes mean the regions of orbital-mixing angles corresponding to different MS.

Fig. 9 .
Fig. 9. Phase diagram of possible MS depending upon orbital mixing angles for C2/c space group.Black arrows denote OO of Mn 3+ sublattices in BiMnO 3 due to local linear vibronic interaction [10].Red arrow shows possible orbital state for experimental [6] MS.A 1 , A 2 , C, G, F denote types of MS.