AâB-Intersite-Dependent Magnetic Order and Electronic Structure of

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A-B Inter-Site Dependent Magnetic Order and Electronic Structure in LaMnNiMnO : A First-Principles Study 3

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Min Liu, Cui-E Hu, Cai Cheng, and Xiang-Rong Chen J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b10591 • Publication Date (Web): 12 Jan 2018 Downloaded from http://pubs.acs.org on January 12, 2018

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The Journal of Physical Chemistry

A-B Inter-Site Dependent Magnetic Order and Electronic Structure in LaMn3Ni2Mn2O12: A First-Principles Study Min Liu,†,‡ Cui-E Hu,¶ Cai Cheng,†,‡ and Xiangrong Chen∗,† †Institute of Atomic and Molecular Physics, College of Physical Science and Technology, Sichuan University, Chengdu 610064, China ‡Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China ¶College of Physics and Electronic Engineering, Chongqing Normal University, Chongqing 400047, China E-mail: [email protected]

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Abstract LaMn3 Ni2 Mn2 O12 , an recently synthesized A- and B-site ordered quadruple perovskite, was discovered to show a nontrivial orthogonal spin ordering in the B and B′ -sites, which is the first example of orthogonal spin ordering in quadruple perovskites reported in experiment. To understand this novel spin order, we perform comprehensive first-principles calculations to investigate the magnetic properties and electronic structure in LaMn3 Ni2 Mn2 O12 . The B-site ordered quadruple perovskite La2 NiMnO6 and A-site ordered quadruple perovskite LaMn3 Al4 O12 are also investigated. Our calculations show that both LaMn3 Al4 O12 and LaMn3 Ni2 Mn2 O12 have a collinear G-type AFM order in the A′ -site Mn3+ spins. For La2 NiMnO6 , B-site Ni2+ and B′ -site Mn4+ present a collinear FM order. However, the Ni2+ and Mn4+ sites in LaMn3 Ni2 Mn2 O12 show a 90◦ canted spin alignment, leading to a noncollinear FM spin order consistent with experimental observations. Our calculated magnetic interactions further demonstrate that the A′ -site Mn3+ spins play a crucial role in determining the novel spin structure of the B and B′ -sites.

Introduction In the manganese oxides with perovskite structure, its spin, charge, and orbital degrees of freedom are usually strongly coupled, so that a variety of magnetic and electronic phases appear in response to external conditions such as temperature, chemical doping, magnetic field, and pressure. 1–4 For example the ferromagnetic ground state of the metallic La1−x Srx MnO3 is due to the so-called double-exchange interaction between the Mn3+ and the Mn4+ ions which display a rich variety of competing spin, charge, and orbital orderings. 5–7 The stability of these orderings is attributed to the anisotropic d-orbitals of transition metal, their degeneracy splitting under the perovskite structure, and interactions between spins and charges. The Goodenough-Kanamori-Anderson (GKA) rules, 8–10 which provide a general frame for describing the strength and sign of the magnetic interaction, are widely employed to ex2


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plain various orderings found in experiment. In addition, the Jahn-Teller (JT) effect drives the polarization of dx2 y2 or dz2 orbitals in the MnO6 octahedron. In this respect, the key parameters are the Mn-O-Mn bond angle, and the Mn-O bond length. It follows that the stabilization of magnetic, charge, and orbital orderings requires a well-defined pattern of cooperative buckling and distortion of the octahedral network. Based on the GKA rules, for the B-site ordered double perovskite La2 NiMnO6 (LNMO) with Ni2+ (t62g e2g ) and Mn4+ (t32g e0g ) electronic configurations, a FM interaction is predicted between the half-filled eg orbitals of Ni2+ and the empty eg orbitals of Mn4+ via a straight Ni2+ -O-Mn4+ pathway. According to experiment results, LNMO presents a collinear FM alignment with TC ≈ 280 K, 11,12 which is in well agreement with the GKA prediction. It has received considerable attention, not only because of the possibility of combining multiple electronic properties (ferromagnetism, magnetoresistance, magneto capacitance, and semiconductivity) in this material, but also because of the expectation that a fundamental understanding of the Ni2+ -O-Mn4+ electronic interaction would provide new guidelines for designing multiple property materials. It suggests that if Ni2+ and Mn4+ ions can be introduced in an ordered fashion into the B-site in AMn3 B4 O12 with a rocksalt-type manner, the long-range FM behavior may be still possible to realize for a larger-size A-site cation like La3+ . But among manganese containing perovskite oxides in A-site ordered perovskites AA′3 B4 O12 -type phases with Mn at the A′ -site, since the MnO4 square planes have different orientations as shown in Fig. 1, the ordering pattern is expected to be much more complex, such as YMn3 Al4 O12 . It has been experimentally confirmed that YMn3 Al4 O12 undergoes an AFM transition at 34 K, 13,14 while the ground-state spin structure has not been clarified yet. However, with Mn at both the A- and the B-site perovskites, they present a wide variety of interesting physical properties such as colossal magnetoresistance, huge dielectric constant, charge disproportionation, intermetallic charge transfer, negative thermal expansion and multiferroicity etc. 15–21 Two independent long-range antiferromagnetic (AFM) phase transitions usually arise from the Mn ions in AMn3 Mn4 O12 . 22–24 For example LaMn3 Mn4 O12 ,

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a C-type antiferromagnetic ordering originating from the B-site Mn3+ sublattice is found to occur at TN ≈ 78 K, 23 markedly differs from the A-type antiferromagnetic structure of LaMnO3 . 25 In addition, TN ≈ 21 K, a second AFM transition is found for the square coordination of A′ -site Mn3+ ions. 23 However, the long-range FM behavior has never been found as yet in the family of AMn3+ 3 B4 O12 . A new oxide LaMn3 Ni2 Mn2 O12 (LMNMO), which crystallizes in an AA′3 B2 B′2 O12 -type A-site and B-site ordered quadruple perovskite structure, was synthesized in experiment recently. 26 A G-type antiferromagnetic ordering originating from the A′ -site Mn3+ sublattice was found to occur at TN ≈ 46 K. And experiment demonstrated that the spin coupling between the B-site Ni2+ and B′ -site Mn4+ sublattices show an orthogonally ordered spin alignment, leading to a ferromagnetic order near TC ≈ 34 K, which is the first example of orthogonal spin ordering in quadruple perovskites reported in experiment. 26 However, the origin of the novel spin oder in LMNMO is yet to be revealed. The effect of structures and compositions on the spin and charge order of this complex quadruple perovskite is also intriguing to investigate which could provide valuable insights into novel electronic and magnetic properties of quadruple perovskites. In this work, we perform density functional calculations to study the magnetic properties and electronic structure of LMNMO and two related quadruple perovskites, a B-site ordered double perovskite LNMO and an A-site ordered perovskite LaMn3 Al4 O12 (LMAO). Our calculations show that the A′ -site Mn3+ spin of both LMAO and LMNMO have a collinear G-type AFM structure. The B-site Ni2+ and B′ -site Mn4+ in LNMO present a collinear FM alignment. However, in LMNMO, they show a 90◦ canted spin alignment, leading to a noncollinear FM spin order which is consistent with experimental observations. 26 In particular, we investigate the magnetic interactions between different magnetic sublattices, e.g. the magnetic interactions between A′ -B, A′ -B′ , A′ -A′ , BB′ , B-B, and B′ -B′ sites. Our computed magnetic interactions indicates that the competing exchange interactions among the three different magnetic ions (A′ -site Mn3+ , B-site Ni2+ , and B′ -site Mn4+ ) are responsible for the unique spin structure in LMNMO.

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Methods Electronic Structure Calculations The electronic structure were analyzed by density functional theory (DFT) calculations using the full potential linearized augmented plane-wave (LAPW) method, 27 with the augmented plane-wave plus local orbitals implementation 28 both in the WIEN2k code 29 and the non-collinear WIENNCM code. 30 To include the strong correlations in the transition-metal elements, we took the generalized gradient approximation-Perdew, Burke and Ernzerhof (GGA-PBE) exchange correlation potential 31 with different effective Coulomb repulsion U eff test for Mn and Ni in the GGA+U calculations with a method of self-interaction correction introduced by Anisimov. 32 Finally, we took UMn =3 eV, UNi =5 eV for the calculations. The Muffin-tin radii (RMT ) are 2.50 a.u. for La, 1.90 a.u. for Mn, 2.00 a.u. for Ni, 1.80 a.u. for Al and 1.60 a.u. for O, respectively. The maximum modulus for the reciprocal vectors Kmax was chosen such that RMT *Kmax = 8.0. The lattice parameters and ionic positions derived from experiment results were used to perform the structural and ionic relaxations. In the calculations of magnetic exchange interactions, the same structure are used for different magnetic configurations as we find the effect of magnetic configurations on the structure is negligible. The Brillouin-zone samplings were checked by directly increasing the density of the k-point meshes until convergence was reached, that is, until the changes in calculated properties were insignificant. Finally, 1000 k-points meshes in the Brillouin zone were used.

Magnetic exchange interaction calculations The magnetic stability and exchange interaction are computed from the comparison of total energy in different magnetic states. We adopt an effective Heisenberg model to model the magnetic exchange interactions in the three materials we investigated. 33,34 The equation is

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given by: H = E0 −

∑

Jij Si · Sj

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(1)

ij

Where E0 is the total energy of nonmagnetic ground-state. Jij are the exchange constants for exchange interactions between spin Si and Sj at sites i and j, respectively. A positive (negative) Jij represents FM (AFM) coupling of the two spins. The total energy differences between different magnetic states are attributed to the spin-spin interaction terms, Jij Si · Sj , and Jij are the key parameters that determine the ground state magnetic order of materials. To determine the value of Jij , we choose a set of magnetic states. We compute the total energy and Si for each magnetic state using DFT calculations. We then obtain a set of linear equations in terms of E0 and Jij according to Eq. 1. The value of Jij could be determined by solving the linear equations if the number of magnetic states is larger than the number of parameters Jij .

Results and Discussion Structural properties As the peculiar magnetic properties of LNMO, LMAO and LMNMO, we first have compared their crystal structures. The crystal structure symmetry changes from monoclinic (P21 /n) to cubic (Im3 ) and cubic (Pn3) as shown in Fig. 1. Because of such geometrical differences, they exhibit different ordering patterns of the Mn-3d states. So it is very important to investigate the geometrical differences of them. LNMO, having the general structure of a double ordered perovskite (A2 BB′ O6 ), is distorted from the ideal double perovskite, and the distortion changes as the temperature varies. The structure of LNMO is rhombohedral (R3) at high temperature and transforms to monoclinic (P21 /n) at low temperature, with these two structures coexisting over a wide temperature range. 35,36 Here we only focus on the monoclinic (P21 /n) structure of LNMO. The LNMO monoclinic in experiment (β = 89.9091◦ ,

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the space group P21 /n) with the lattice constants a=5.5064 Å, b=5.4558 Å, c=7.7315 Å. The structure is shown as Fig. 1(a). Owing to the rotation and tilting of the NiO6 and MnO6 octahedra, the Mn-O-Ni bond angle ranges from 160.32◦ to 162.26◦ , which is shown in Fig. 1(b). The lattice constants a, b and c are systematically overestimated by about 1%, as commonly seen when the GGA functional is used for the exchange-correlation energy. The optimized structural parameters (such as Mn-O bond length and Mn-O-Ni bond angle) within GGA+U show good agreement with the experimental ones proving the reliability of our calculation scheme as tabulated in Table 1. Hence, we adopted the optimized crystal structures in our calculations. In order to get the ground state of LNMO, we calculate four magnetic states, namely, ferromagnetic (FM), A-type antiferromagnetic (A-AFM), C-type antiferromagnetic (C-AFM) and G-type antiferromagnetic (G-AFM) four spin configurations [see Fig. 2(a)] using GGA+U method. All the calculations are based on the P21 /n symmetry. FM state always is the ground state (shown in Table 2). It is in accordance with the experimental result. For LMAO, we model its crystal structure based on one of its sister compound YMn3 Al4 O12 13,14 (YMAO) as experimental structure data of LMAO is not available. Since the role of both La and Y atoms in the compound is to provide 3 electrons and to stabilize the MnO6 octahedron, it is reasonable to assume that they share a similar structure, although the different radius of La and Y ions might lead to minor difference in the structure. We obtain LMAO structure by replacing Y atom in YMAO with La atom, and then fully relaxing the structure to take into account the structure changes due to the replacement of ions, as shown in Fig. 1(c). LMAO crystallizes in a cubic Im3 structure with Mn3+ ions occupy the A′ -sites. The Mn3+ ions locate at the D4h local symmetry, which is no longer JT active. The relax result shows lattice parameter a=7.2924 Å, which is a little larger than that of YMAO. This may be because the ionic radius of La is larger than Y. And the ionic positions are La (0, 0, 0), Mn (0, 0.5, 0.5), Al (0.25, 0.25, 0.25), O (0, 0.1821, 0.3045). In LMAO, the MnO4 square planes determine the ordering of the Mn-3d orbital states, since the square planes

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have different orientations as shown in Fig. 1(c), the ordering pattern is expected to be much more complex than the LNMO case. From Fig. 1(d), it shows the Al-O-Al bond angle is only 141.60◦ , which is much smaller than Mn-O-Ni bond angles in LNMO. It indicates AlO6 has a smaller distortion than MnO6 and NiO6 . Based on these optimized structure parameters, the magnetic stability is derived from the comparison of total energy, which is calculated by imposing FM, A-AFM, B-AFM, and G-AFM four spin configurations [see Fig. 2(b)]. Note that a system with A-AFM spin order has Pmmm symmetry that is lower than the original symmetry of the crystal (Im3). For the sake of accurate comparison of total energies, the calculations are always done with Pmmm symmetry even for the FM phase. The trend of the magnetic stability ∆E (shown in Table 2) is in reasonable agreement with the experiment. Through the total energy calculations, G-AFM magnetic structure is always found to be the ground states. For LMNMO, A-site and B-site ordered quadruple perovskite structure, a unique feature of this specially ordered perovskite is that three different atomic sites (A′ , B, and B′ -sites) can all accommodate magnetic transition metals [see Fig. 1(e)]. It is cubic structure with the group Pn3, with the lattice constants a=7.36720 Å. In general, for the ordered AA′3 B2 B′2 O12 perovskite, the A-site substitution with smaller-size transition metal significantly decreases the average ionic radius for this atomic site. The BO6 and B′ O6 octahedra thus become heavily tilting (typically, B-O-B′ ≈ 140◦ ) and the A′ -site transition metal forms square planar coordinated A′ O4 units. As seen in Fig. 1(f), in LMNMO, the Mn4+ -O-Ni bond angle is 138.46◦ , which is much smaller than that of LNMO. And Mn3+ -O-Mn4+ angle is 113.26◦ and Mn3+ -O-Ni is 109.97◦ . Unlike the structure of LMAO, the B-site nonmagnetic Al3+ is in place of B-site magnetic Ni2+ and B′ -site magnetic Mn4+ . As a consequence, multiple magnetic and electrical interactions may occur among A′ -site Mn3+ , B-site Ni2+ , and B′ -site Mn4+ . It is also expected that the strong coupling among these magnetic sublattices have important impacts on further enhancing the spin or charge ordering temperature. As it is the first example that shows orthogonal ferromagnetic spin ordering in the B/B′ -sites assisted by

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A-site antiferromagnetic ordering, it is interesting to investigate the magnetic interaction of different magnetic sublattices, such as the magnetic interaction of A′ -B, A′ -B′ , A′ -A′ , B-B′ , B-B, B′ -B′ sites.

Figure 1: Structural comparison of LNMO, LMAO and LMNMO, the crystal structure symmetry changes from monoclinic(P21/n) to cubic (Im3) and cubic(Pn3) symmetry. The red, blue, green, yellow, and cyan spheres denote Ni, Mn4+ , Mn3+ , La, and O atoms, respectively.

Magnetic Properties in La2 NiMnO6 For LNMO, in order to calculate the magnetic interaction between the Ni2+ and Mn4+ ions, we choose four magnetic states as shown in Fig. 2(a). All these calculations are done in one single unit cell. The total energy results are listed in Table 2. It is found that FM state is always the ground state with a local spin moment of 1.679 µB /Ni2+ , 2.875 µB /Mn4+ and 9



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0.030 µB /O2− within GGA+U (see Table 1). We have also test different values of U eff for Ni and Mn, it is the same that FM state is still the ground state and the magnetic moments are always enhancing with the increasing of U eff . Since exchange coupling between nearest neighbor are dominant, we only take into account exchange couplings between the nearest neighboring Ni2+ -Mn4+ , Ni2+ -Ni2+ , and Mn4+ -Mn4+ , which are labeled, for convenience, as J1 , J2 , and J3 respectively. According to Eq. 1, we can write down one equation for each magnetic states in terms of total energy E, E0 and J. We give the equations of four magnetic states we selected by E(FM) = E0 − 6J1 S1 S2 − 6J2 S12 − 6J3 S22 E(A − AFM) = E0 − 2J1 S1 S2 + 2J2 S12 + 2J3 S22 E(C − AFM) = E0 + 2J1 S1 S2 +

2J2 S12

+

(2)

2J3 S22

E(G − AFM) = E0 + 6J1 S1 S2 − 6J2 S12 − 6J3 S22 Here, we used S1 = 1 and S2 = 3/2 for the B-site Ni2+ and B′ -site Mn4+ , respectively. The sign of each interaction terms is determined by spin directions of two sites connected by J, and the prefactor is give by the number of equivalent interaction terms. From the above equations, we could only get the magnetic interaction of Ni2+ -Mn4+ J1 directly. The G-AFM state lies at a much higher energy than the FM ground state by 207.908 meV/f.u. (equal to 12J1 S1 S2 ). This allows us to estimate the Ni2+ -Mn4+ exchange energy (J1 S1 S2 ) to be about 17.326 meV, which shows the FM magnetic interaction in agreement with the experiment. However, the J1 S1 S2 can also be estimated from the energy difference between the A-AFM and C-AFM states (equal to −4J1 S1 S2 ), to be about -19.316 meV, which shows the AFM magnetic interaction in opposite with experiment. So we take the FM magnetic interaction of J1 S1 S2 . Now we probe the possible magnetic coupling within both the fcc Ni2+ and Mn4+ sublattices, whose values, however, cannot be extracted from the energy differences of those magnetic states listed in Table 2. In order to estimate the magnetic interaction

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of Ni2+ -Ni2+ J2 and Mn4+ -Mn4+ J3 , the artificial La2 NiGeO6 and La2 GeMnO6 are used (both assuming the same structural parameters as LNMO, and moreover, 0.69 Å/Ni2+ , 0.53 Å/Mn4+ , 0.73 Å/Ge2+ , 0.53 Å/Ge4+ are all similar size. 37 We calculate the FM state and the layered AFM state (FM ab planes and an AFM stacking along the c axis) for both the artificial La2 NiGeO6 and La2 GeMnO6 . For La2 NiGeO6 , the calculations show that the layered AFM state is more stable than the FM state by 12.824 meV/f.u. (equal to −8J2 S12 ), which allows us to estimate the AFM Ni2+ -Ni2+ magnetic interaction J2 S12 to be about 1.603 meV in GGA+U (UNi = 5 eV). In La2 NiGeO6 calculations, the magnetic moment of Ni2+ is 1.726 µB , which are almost same in LNMO. For La2 GeMnO6 , the GGA+U calculations (UMn = 3 eV) show that the layered AFM state is also more stable than the FM state. However, the magnetic moment of Mn4+ is 3.742 µB , which are almost deviate from the moments in LNMO. So it is the unreliable result for this artificial La2 GeMnO6 . We can get the magnetic interaction of J3 S22 from the above equations with the value of J1 S1 S2 and J2 S12 , which is 7.725 meV within GGA+U . There exist a strong geometrical magnetic frustration. Both values of J1 S1 S2 and J3 S22 have an overwhelmed trend than J2 S12 , which make the FM magnetic structure of LNMO much more stable. Hence, from the value of magnetic interaction, it is obvious that the FM state of LNMO is the most stable one. The ferromagnetic transition temperatures (TC ) is calculated within the mean-field approximation as follows: 33,34 ⟨Siz ⟩ =

Si (Si + 1) ∑ Jij ⟨Sjz ⟩ 3kB T j

(3)

The TC is given by the largest eigenvalue of the matrix Θij = Si (Si + 1) Jij /3kB . Here kB is the Boltzmann constant. We obtain the TC =378 K for LNMO, which is a little higher than the experimental one (280 K).

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(a)

J+ J1 J*

FM

A-AFM

C-AFM

G-AFM

A-AFM

B-AFM

G-AFM

(b)

J,

JJ FM

Figure 2: Magnetic configurations of the LNMO (a) and LMAO (b). The pink, blue, and green arrows denote the magnetic moment directions of Ni, Mn4+ , and Mn3+ , respectively.

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Magnetic exchange interactions in LaMn3 Al4 O12 For LMAO, based on these optimized structure parameters, the magnetic stability is derived from comparison of total energy. Four spin configurations are also calculated [see Fig. 2(b)]. In LMAO, only one magnetic ion Mn3+ , so it is very important to investigate the magnetic interactions. All these calculations are done in one single unit cell. However, it is found that G-AFM state is always the ground state with a local spin moment of 3.648 µB /Mn3+ and 0.033 µB /O2− within GGA+U , UM n = 3 eV (see Table 1). We have mapped the total energy of the above magnetic configurations into an effective Heisenberg model. Here, we take into account the nearest neighbor (NN) magnetic interaction J4 , the next-nearest neighbor (NNN) magnetic interaction J5 and the third-nearest neighbor (3-NN) magnetic interaction J6 of Mn3+ at A′ -site [see Fig. 2(b)]. Hence, for one f.u., the equations can be written: E(FM) = E0 − 6J4 S32 − 12J5 S32 − 12J6 S32 E(B − AFM) = E0 + 2J4 S32 + 4J5 S32 − 4J6 S32 E(A − AFM) = E0 −

2J4 S32

+

4J5 S32

+

(4)

12J6 S32

E(G − AFM) = E0 + 6J4 S32 − 12J5 S32 + 12J6 S32 Here, we used S3 =2 for the A′ -site Mn3+ . The antiferromagnetic transition temperatures (TN ) is calculated within the mean-field approximation as follows: 33,34

TN =

2S (S + 1) (2J4 − 4J5 + 4J6 ) 3kB

(5)

The calculated TN is 108 K. The result of total energy difference between FM and AFM spin configurations are presented in Table 2. We also test different Coulomb effects. It is found that the magnetic moment of Mn3+ at A′ -site increases as the Coulomb effect U increases, which means the correlation effect has a great influence on the Mn3+ . And the exchange interaction constants are shown that all the magnetic interaction constants present AFM

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interaction, indicting the G-AFM is the ground state. In addition, it is found that all the magnetic interaction constants increase as the Coulomb effect U increases. And J4 is about 10 times larger than J5 , 100 times larger than J6 , indicating the nearest neighbor magnetic interaction J4 plays the major role in the G-AFM magnetic state.

Magnetic Properties in LaMn3 Ni2 Mn2 O12 In contrast to the magnetism with the perovskite B-sites that has been extensively and comprehensively studied for decades, magnetic interactions between the cations on the perovskite A (or A′ )-sites still requires investigation. An understanding of them would also be necessary for the study of more complex materials containing magnetic transition metals both at the A′ -sites and B-sites, such as CaMn7 O12 that shows magnetoelectric coupling 38 and BiCu3 Mn4 O12 that is magnetoresistive. 39 For LMNMO, as a G-type antiferromagnetic order originating from the A′ -site Mn3+ sublattice is found to occur at TN ≈ 46 K. However, a non-collinear ferromagnetic order of the B-site Ni2+ and B′ -site Mn4+ sublattices is found near TC ≈ 34 K. This is an amazing phenomenon for LMNMO. As it shows a net FM component originating from the orthogonal spin ordering in the B and B′ -sites (Ni2+ and Mn4+ ). In this spin structure model, the calculated total moment is 7.070 µB /f.u. causing by the 90◦ canted Ni2+ and Mn4+ , which is close to the experimental one (6.600 µB /f.u.). The calculated magnetic moments are 3.560 µB , 2.830 µB , 1.700 µB for Mn3+ , Mn4+ and Ni2+ by GGA+U within non-collinear magnetic calculation From the above discussion, without the A′ -site Mn3+ effect, it is found the ferromagnetic transition temperature is TC ≈ 280 K for LNMO, which is much higher than TC ≈ 34 K for LMNMO. The only difference is the ferromagnetic order, which is collinear for LNMO but an orthogonally ordered spin alignment for LMNMO. If we neglect the B′ -site Ni2+ effect, it is estimated that the antiferromagtic transition temperature is TN ≈ 108 K in LMAO, which is much lower than TN ≈ 46 K for LMNMO. So it is very interesting to investigate the magnetic interaction of LMNMO to find the mechanism for this complex 14


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FM

FiM1

FiM2

FiM3

FiM4

FiM5

FiM6

FiM7

FiM8

Figure 3: The magnetic configurations of the perovskite LMNMO, the magnetic moment directions of A′ -site Mn3+ , B-site Ni2+ , and B′ -site Mn4+ are denoted by green, red, and blue arrows, respectively.

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magnetic structure. In order to get a deep insight into the complex magnetic exchange interactions, the magnetic stability is derived from comparison of total energy of LMNMO, which is calculated with nine different collinear magnetic configurations: FM, FiM1, FiM2, FiM3, FiM4, FiM5, FiM6, FiM7, FiM8 (see Fig. 3) within GGA+U calculations. We have calculated the strength of exchange interactions J1 -J8 between Mn3+ , Ni2+ and Mn4+ spins, with the exchange pathways shown in Fig. 4(a). All these calculations are also done on one single unit cell with the same symmetry structure. The calculated result of total energy difference among these spin configurations are presented in Table 2. We have mapped the total energy of the above magnetic configurations into the effective Heisenberg model. For simplicity, we have expressed Jij in terms of J1 -J8 for different pairs of Mn-Ni, Mn-Mn and Ni-Ni interactions. Here, we used, S1 = 1, S2 = 3/2 and S3 = 2 for the B-site Ni2+ and B′ -site Mn4+ and A′ -site Mn3+ , respectively. The equations corresponding to the above 9 magnetic configurations are listed as the set of Eqs. 6 in one formula. E(FM) = E0 − 12J1 S1 S2 − 12J2 S12 − 12J3 S22 − 6J4 S32 − 12J5 S32 − 12J6 S32 − 12J7 S1 S3 − 12J8 S2 S3 E(FiM1) = E0 + 12J1 S1 S2 − 12J2 S12 − 12J3 S22 − 6J4 S32 − 12J5 S32 − 12J6 S32 − 12J7 S1 S3 + 12J8 S2 S3 E(FiM2) = E0 + 12J1 S1 S2 − 12J2 S12 − 12J3 S22 − 2J4 S32 + 4J5 S32 + 12J6 S32 E(FiM3) = E0 + 12J1 S1 S2 − 12J2 S12 − 12J3 S22 + 2J4 S32 + 4J5 S32 − 4J6 S32 E(FiM4) = E0 − 4J1 S1 S2 + 4J2 S12 + 4J3 S22 − 6J4 S32 − 12J5 S32 − 12J6 S32 E(FiM5) = E0 − 12J1 S1 S2 − 12J2 S12 − 12J3 S22 − 2J4 S32 + 4J5 S32 + 12J6 S32 E(FiM6) = E0 − 12J1 S1 S2 − 12J2 S12 − 12J3 S22 + 6J4 S32 − 12J5 S32 + 12J6 S32 E(FiM7) = E0 − 12J1 S1 S2 − 12J2 S12 − 12J3 S22 − 6J4 S32 − 12J5 S32 − 12J6 S32 + 12J7 S1 S3 + 12J8 S2 S3 E(FiM8) = E0 + 4J1 S1 S2 − 12J3 S22 − 6J4 S32 − 12J5 S32 − 12J6 S32 − 12J8 S2 S3 (6) The calculated total energy difference between 9 different spin configurations and spin exchange interactions of J1 -J8 are listed in Table 3. Here, we use the effective spin exchange constants Jij eff = Jij Si · Sj to compare the relative strengths of the spin exchange inter16


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actions between Mn3+ , Ni2+ and Mn4+ ions with GGA+U . The exchange interactions B(Ni2+ )-B′ (Mn4+ ) J1 eff, B(Ni2+ )-B(Ni2+ ) J2 eff and B′ (Mn4+ )-B′ (Mn4+ ) J3 eff are 6.530 meV, 2.150 meV and 1.350 meV, respectively. It is obvious that all of the values are FM coupling. The effective exchange constants J4 eff, J5 eff and J6 eff, corresponding to the NN, NNN and 3-NN exchange interactions between A′ (Mn3+ )-A′ (Mn3+ ) spins, are -5.360 meV, -0.007 meV and -0.240 meV, respectively. According to these values, it is clearly that J4 eff and J6 eff overwhelm J5 eff interaction and all favor the AFM ordering. These calculated results all are consistent with the experiments. It is obvious that LNMO and LMNMO have the same trend of magnetic interaction for B-site and B′ -site [see Fig. 4(b)]. However, it is seen that LMAO and LMNMO has different NNN and 3-NN magnetic interactions for A′ -site. It may indicate that the inter-site exchange interactions have the important effect on LMNMO. As the inter-site exchange interactions like the A′ -B inter site spin coupling as reported in YMn3 Al4 O12 and LaMn3 Cr4 O12 are negligible. 13,40 However, the significant A′ (Mn3+ )B(Ni2+ ) J7 eff (-3.500 meV) and A′ (Mn3+ )-B′ (Mn4+ ) J8 eff (4.770 meV) exchange interactions are presented in our LMNMO. Moreover, the interaction energies of J7 eff and J8 eff are comparable to those of the NN or NNN J4 eff and J1 eff-J3 eff, strongly suggesting that the A′ -site magnetic ions play an important role for the spin ordering of the B-site and B′ -site magnetic ions in LMNMO. The inter-site exchange interactions J7 eff is FM while J8 eff is AFM, indicating there being a competition between these two interactions. These comparable FM and AFM exchange interactions J1 eff (FM), J7 eff (FM) and J8 eff (AFM) occurring in the A′ -site Mn3+ , B-site Ni2+ and B′ -site Mn4+ spins can form a geometrical magnetic triangle. We attribute the highly non-collinear magnetic structure at the B and B′ -sublattices to be the result of the competition in this magnetic triangle, which is also responsible for the reduced spin moments obtained in the neutron refinements

17



(a)

(b)

Magnetic Exchange coupling (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 37

18 12 6 0

LNMO

LMAO

LMNMO

J1eff J2eff J3eff J4eff J5 eff X10 J6eff X10 J7eff J8eff

Figure 4: (a) The exchange pathway of spin order for the A′ , B and B′ -site of the LMNMO, (b) the effective magnetic interaction of LNMO, LMAO and LMNMO calculated by GGA+U .

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Electronic structure To further understand these specific spin interactions among the three different perovskites, electronic structure calculations are performed based on the spin model. Fig. S1 (see Supporting Information) provides the total and site-projected density of states (DOS) for ferromagnetic LNMO with the optimized structure by the GGA+U method. The total DOS shows clearly the insulating nature with a gap size about 1.470 eV, which is in agreement with the experimental value 1.400 eV. 41 It has been seen from the total DOS that Mn4+ and Ni2+ ions determine the electronic structure. The octahedral surrounding of Mn and Ni atoms splits the Mn-d and Ni-d manifolds into t2g and eg levels. Fig. 5 shows the siteprojected density of states in order to compare these compounds. Since the projected DOS in ferromagnetic LNMO is spin dependent, we show the site projected DOS for both spin-up and spin-down channels. It can be seen that Ni-t2g and Ni-eg levels are found almost in valence band. In fact, from the spin site-projected density of states, the t2g and eg levels of both Mn and Ni ions can be clearly identified. In the spin-up channel, the Ni-t2g and Ni-eg levels are found in the energy range -4 eV to Fermi energy and show a significant mixing with Mn-d states and O-p states. In the spin-down channel, the Ni-t2g bands are located between -4 eV and -1 eV showing a strong hybridization with O-p bands, while Ni-eg states are located at 3 eV above the Fermi level. This corresponds to the nominal valence of Ni2+ (t62g e2g ). In the spin-up channel the Mn-t2g bands are almost filled, which are localized between Ni-t2g and Ni-eg bands. However, the Mn-eg bands located above the Fermi energy are empty, and are separated by a gap of 3 eV from the Mn-t2g bands. In the spin-down channel, both Mn-t2g and Mn-eg bands are located above Fermi level. It indicates that the valence of Mn is nominally 4+ (t32g e0g ), which agree with the experimental results. 42,43 The occupation of the Ni-d states and Mn-d states suggests again the nominal valence states of Ni and Mn ions to be 2+ and 4+, respectively. The energy splitting between t2g and eg states are about 7 eV for Mn and about 2-4 eV for Ni. For both Mn and Ni, the splitting energies are close to the exchange splitting energy, so their spin states are not the high-spin (HS) states but 19



the intermediate-spin or the low spin (LS) states. The oxygen DOS is included to show the profound mixing effect between d-states of transition metals and p-states of oxygen from -6 eV to Ef . The calculated total and site-projected density of states (DOS) of the antiferromagnetic LMAO with the optimized volume by the GGA+U method are shown in Figure S2 . The obtained energy gap is about 2.6 eV, which is in consistent with the insulating behavior of YMn3 Al4 O12 . 13,14 The occupations are t32g e1g for Mn, indicating valence states of Mn3+ . The Mn-3d and O-2p orbitals are largely near the Fermi energy. The energy splitting between t2g and eg states are about 0.5 eV for Mn3+ (see Fig. 5). It is close to the exchange splitting energy, and so the spin states is the high-spin (HS) states but not the intermediate-spin or the low spin (LS) states. The oxygen DOS is included to show the profound mixing effect between d-states of transition metals and p-states of oxygen from -6 eV to Ef . However, the orbitals of La or Al have negligible contributions to the bands near the Fermi energy. This implies that the Al orbitals do not hybridize with Mn-3d or O-2p orbitals and they do not contribute to the magnetic interactions between the Mn-3d spins. When the B-site contains non-magnetic ions, we can observe A′ -A′ magnetic interaction. The A′ -site Cu-Cu interaction in CaCu3 B4 O12 (B = Ge and Sn) is ferromagnetic because of the direct exchange interaction between the nearest neighboring Cu spin, while that in CaCu3 Ti4 O12 is antiferromagnetic primarily. As the orbital hybridization of the non-magnetic Ti4+ ions at the B-site mediates the antiferromagnetic interaction between the Cu2+ spins through Cu-O-Ti-O-Cu paths. 44 In LMAO, it differs markedly from the antiferromagnetism in CaCu3 Ti4 O12 , Mn-O-Mn superexchange interaction does not seem to be responsible for the antiferromagnetism of LMAO, because the Mn-O bond lengths (1.96 Å) in the Mn-O-Mn paths is too long to mediate such interaction and because the Mn-O-Mn bond angle (101◦ ) is far from the 180◦ expected to induce antiferromagnetic interaction according to the Kanamori-Goodenough rule. Instead, Mn-Mn direct exchange interaction appears to play a major role in the antiferromagnetism of LMAO unlike the major role for Cu-Cu direct

20


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exchange interaction in the ferromagnetism of CaCu3 Sn4 O12 . In YMn3 Al4 O12 , the half-filled dz2 and dxy orbitals of the nearest neighboring Mn ions are directed toward each other. 13,14 Thus the overlap of those orbitals can produce antiferromagnetic direct exchange interaction between the Mn spins. We think this is the same mechanism for LMAO. Electronic structure calculations further reveal that the Al orbitals do not contribute to the magnetic interaction between the Mn3+ spins. The antiferromagnetic interaction in this material is attributed to the nearest Mn-Mn direct exchange interaction. LMNMO crystallizes in an AA′3 B2 B′2 O12 -type A-site and B-site ordered quadruple perovskite structure. A G-type antiferromagnetic ordering originating from the A′ -site Mn3+ sublattice was found in experiment to occur at TN ≈ 46 K. However, the spin coupling between the B-site Ni2+ and B′ -site Mn4+ sublattices shows an orthogonally ordered spin alignment, leading to a ferromagnetic order near TC ≈ 34 K. 26 Hence, we first investigate the electronic structure of collinear magnetic calculations, in order to find A-site, B-site and B′ -site effect. We calculated the electronic structure of LMNMO with the optimized volume by the GGA+U method. The calculated density of states (DOS) and band structure are shown in Fig. S3. The 3d states of A′ -site Mn, B-site Ni and B′ -site Mn cations show very large overlap, indicating that the inter-site exchange interactions between A′ - and B/B′ -sites are comparable to the A′ -A′ and B/B′ -B/B′ exchange interactions. We also perform GGA+U calculations on LMNMO with a non-collinear magnetic structure that reported in low-temperature NPD experiments. The calculated band structure and density of states (DOS), shown in Fig. S4, demonstrate a wide indirect energy gap about 1.5 eV, indicating the insulating behavior of LMNMO as expected from the high spin Ni2+ and Mn4+ states at the corner-sharing B/B′ O6 octahedron. In addition, the DOS of the A′ -site Mn3+ , B-site Ni2+ and B′ -site Mn4+ are considerably overlapped, indicating that the inter-site exchange interactions between A′ - and B/B′ -sites are comparable to the A′ -A′ and B/B′ -B/B′ exchange interactions. It helps to explain the strong spin interactions among these magnetic ions. By comparison, in YMn3 Al4 O12 13 and LaMn3 Cr4 O12 , 40 the overlap of

21


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

DOS (states/eV) DOS (states/eV)DOS (states/eV atom) ).u.f Ve/setatDOS s( SO(states/eV D f.u.)


44 3 22 1 -6 0 00 1 -6 2 3 4

Mn4+-t2g LNMO 4+-eg Mn Energy (eV) spin

-4

-2

2

4

-4

-2 0 2 spin Energy (eV)

4

Ni-eg Ni-t2g

0

Page 22 of 37

O-p

4 3

O Mn-t2g Mn-eg

LMAO

Mn3+-t2g Mn3+-eg

Ni-t2g Ni-eg Mn-t2g Mn-eg O O Mn-eg Mn-t2g Ni-eg Ni-t2g

2 1 0 66 -6 5 44 3 22 1 00 -6 -6

-4

-2

0

2

4

LMNMO Energy (eV)

-4 -4

-2 -2

00 22 Energy (eV) Energy (eV)

Ni-t2g Ni-eg Mn4-t2g Mn4-eg Mn3-t2g Mn3-eg O

44

Figure 5: The spin- and site-projected partial DOS for ferromagnetic LNMO, and the siteprojected partial DOS for antiferromagnetic LMAO and LMNMO with non-collinear magnetic structure.

22


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DOS between the A′ - and B-site ions is negligible.

Conclusions In summary, we performed first-principles calculations on the structural, magnetic and electrical properties in B-site ordered quadruple perovskite LNMO, A-site ordered quadruple perovskite LMAO, and A- and B-site ordered quadruple perovskite LMNMO. We found that the angle of Mn4+ -O-Ni in LNMO and LMNMO show a significant difference, which may be responsible for the different TC of the two materials. Our results from magnetic properties calculations show that the LNMO and LMNMO have the same trend of magnetic interaction for B-site and B′ -site. However, LMAO and LMNMO have different NNN and 3-NN magnetic interactions for A′ -site. Both A′ (Mn3+ )-B(Ni2+ ) and A′ (Mn3+ )-B′ (Mn4+ ) pathways show considerable superexchange interactions comparable with those of A′ -A′ , BB, and B′ -B′ pathways. And the inter-site exchange interactions of A′ -site Mn3+ , B-site Ni2+ and B′ -site Mn4+ spins can form a geometrical magnetic triangle. We attribute the highly noncollinear magnetic structure at the B and B′ -sublattices to be a result of the competition in this magnetic triangle, which is also responsible for the reduced spin moments obtained in the neutron refinements. Our results indicate that the inter-site exchange interactions play an essential role in the novel magnetic order in LMNMO.

Supporting Information Available Supporting Information. Site-projected partial density of states and electronic band structures for La2 NiMnO6 , LaMn3 Al4 O12 , and LaMn3 Ni2 Mn2 O12 .

23



Acknowledgement This work was supported by the Science Challenge Project (Grant No. TZ2016001) and the NSAF (Grant No. U1430117).

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(44) Toyoda, M.; Yamauchi, K.; Oguchi, T. Ab initio study of magnetic coupling in CaCu3 B4 O12 (B=Ti, Ge, Zr, and Sn). Phys. Rev. B 2013, 87, 224430.

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Graphical TOC Entry Magnetic Exchange coupling (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 37

18

Ni

J7

J1

12 3+

Mn

J3

J2

J4

J8

J5

J6

4+

Mn

6

J1

J2 J3 J4 J5 J6 J7

0

La 2NiMnO6

LaMn3Al4 O12

LaMn3 Ni2 Mn2 O12

30


J8

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Table 1: Calculated structural parameters within GGA+U at the optimized crystal structure (lattice constant, Mn-O bond length and bond angle ϕMn3+ −O−Mn3+ ), the local moments on Mn3+ , Mn4+ , Ni2+ and O2− ions (mMn3+ , mMn4+ , mNi2+ , mO2− ) and the corresponding experimental data are also shown for comparison. LNMO a (Å) b (Å) c (Å) β (deg) dMn3+ −O (Å) dMn4+ −O (Å) dNi2+ −O (Å) ϕMn3+ −O−Ni2+ (deg) ϕMn3+ −O−Mn4+ (deg) ϕMn4+ −O−Ni2+ (deg) mMn4+ (µB ) mMn3+ (µB ) mNi2+ (µB ) mO2− (µB )

Cal. 5.5219 5.5181 7.7848 89.836

Exp. 5.5064 5.4558 7.7315 89.909

1.934 2.013

1.914 2.020

157.199 2.875

160.321 3.000

1.679 0.030

1.900

LMAO Cal. 7.2924

LMNMO Cal. Exp. 7.3672 7.3569

90.000 1.931

90.000 1.909 1.908 1.916 106.98 113.87 139.18 2.830 3.560 1.70 0.057

3.648 0.033

90.000 1.913 1.905 2.028 107.90 113.20 138.40 0.440 2.940 0.65

Table 2: The relative total energies difference (meV/f.u.) between different spin configurations for LNMO, LMAO and LMNMO and the calculated TC /TN by GGA+U method. UMn =3 eV and UNi =5 eV are adopted in the calculations. LNMO LMAO LMNMO LMNMO

FM 0 FM 60.340 FM 110.170 FiM5 85.737

A-AFM 141.248 A-AFM 36.750 FiM1 651.680 FiM6 0

C-AFM 63.985 B-AFM 17.250 FiM2 398.780 FiM7 170.732

31


G-AFM 207.908 G-AFM 0 FiM3 363.445 FiM8 251.461

TC (K) 378 TN (K) 108 FiM4 356.809 TC (K) 56


Page 32 of 37

Table 3: The calculated spin exchange interactions of J1 -J8 for LNMO, LMAO and LMNMO by GGA+U method, respectively. LNMO LMAO LMNMO LMNMO

J1 11.550 J4 1.244 J1 4.350 J5 -0.002

J2 -0.712 J5 0.048 J2 2.150 J6 -0.0598

32


J3 7.724 J6 0.006 J3 0.060 J7 -1.750

J4 -1.340 J8 1.590

Page 33 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42



The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


Page 34 of 37

Page 35 of 37 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56


FM

FiM3

FiM1

FiM4

FiM2

FiM5


FiM6

FiM7

FiM8

(a)

(b)

Magnetic Exchange coupling (meV)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44


18 12 6 0

LNMO

LMAO

LMNMO


Page 36 of 37

J1eff J2eff J3eff J4eff J5 eff X10 J6eff X10 J7eff J8eff

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47


DOS (states/eV)

Page 37 of 37

44 3 22 1 -6 0 00 1 -6 2 3 4

Mn4+-t2g LNMO Mn4+-eg spin

-4

-2

0

2

4

-4

-2

0

2

4

spin

Ni-eg Ni-t2g

Mn-t2g Mn-eg Mn-eg Mn-t2g

O-p

4 3

LMAO

Mn3+-t2g Mn3+-eg

Mn-t2g Mn-eg

2 1 0 66 -6 5 44 3 22 1 00 -6 -6

-4

-2

0

2

4

LMNMO

Mn4-t2g Mn4-eg Mn3-t2g Mn3-eg -4 -4

-2 -2

00 22 Energy (eV) ACS Paragon Plus Environment

44

AâB-Intersite-Dependent Magnetic Order and Electronic Structure of

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