Nonequivalent Spin Exchanges of the Hexagonal Spin Lattice

Jul 23, 2018 - Nonequivalent Spin Exchanges of the Hexagonal Spin Lattice Affecting the Low-Temperature Magnetic Properties of RInO3 (R = Gd, Tb, Dy):...
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Nonequivalent Spin Exchanges of the Hexagonal Spin Lattice Affecting the Low-Temperature Magnetic Properties of RInO3 (R = Gd, Tb, Dy): Importance of Spin−Orbit Coupling for Spin Exchanges between Rare-Earth Cations with Nonzero Orbital Moments Elijah E. Gordon,† Xiyue Cheng,‡ Jaewook Kim,§ Sang-Wook Cheong,§ Shuiquan Deng,*,‡ and Myung-Hwan Whangbo*,†,‡,∥ Downloaded via KAOHSIUNG MEDICAL UNIV on July 23, 2018 at 20:36:09 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

†

Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695, United States State Key Laboratory of Structural Chemistry, Fujian Institute of Research on the Structure of Matter (FJIRSM), Chinese Academy of Sciences (CAS), Fuzhou 350002, China § Rutgers Center for Emergent Materials and Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, United States ∥ State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, China ‡

S Supporting Information *

ABSTRACT: Rare-earth indium oxides RInO3 (R = Gd, Tb, Dy) consist of spin-frustrated hexagonal spin lattices made up of rare-earth ions R3+, where R3+ = Gd3+ (f7, L = 0), Tb3+ (f8, L = 3), and Dy3+ (f9, L = 5). We carried out DFT calculations for RInO3, including on-site repulsion U with/without spin− orbit coupling (SOC), to explore if their low-temperature magnetic properties are related to the two nonequivalent nearest-neighbor (NN) spin exchanges of their hexagonal spin lattices. Our DFT + U + SOC calculations predict that the orbital moments of the Tb3+ and Dy3+ ions are smaller than their free-ion values by ∼2μB while the Tb3+ spins have an inplane magnetic anisotropy, in agreement with the experiments. This suggests that the f orbitals of each R3+ (R = Tb, Dy) ion are engaged, though weakly, in bonding with the surrounding ligand atoms. The magnetic properties of GdInO3 with the zero orbital moment are adequately described by the spin exchanges extracted by DFT + U calculations. In describing the magnetic properties of TbInO3 and DyInO3 with nonzero orbital moments, however, the spin exchanges extracted by DFT + U + SOC calculations are necessary. The spin exchanges of RInO3 (R = Gd, Tb, Dy) are dominated by the two NN spin exchanges J1 and J2 of their hexagonal spin lattice, in which the honeycomb lattice of J2 forms spin-frustrated (J1, J1, J2) triangles. The J2/J1 ratios are calculated to be ∼3, ∼1.7, and ∼1 for GdInO3, TbInO3, and DyInO3, respectively. This suggests that the antiferromagnetic (AFM) ordering of GdInO3 below 1.8 K is most likely an AFM ordering of its honeycomb spin lattice and that TbInO3 would exhibit low-temperature magnetic properties similar to those of GdInO3 while DyInO3 would not.

1. INTRODUCTION Rare-earth indium oxides RInO3, crystallizing in the hexagonal space group P63cm, are made up of InO5 trigonal bipyramids containing In3+ ions and axially compressed RO6 octahedra containing rare-earth ions R3+.1 The InO5 trigonal bipyramids form a hexagonal layer by sharing their equatorial corners (Figure 1a) and so do the RO6 octahedra by sharing their edges (Figure 1b). RInO3 has two nonequivalent rare-earth sites, R(1) and R(2), in a 2:1 ratio. The R(1) atoms make honeycomb layers, with the R(2) atoms occupying the center of each hexagon of R(1) atoms. The InO5 layers alternate with the RO6 layers along the c direction by sharing their O corners. A given RO6 layer shares its O corners with its adjacent InO5 layers, lying above or below. For each R(1)O6 this corner sharing occurs with the lower-lying InO5 layer, while for each © XXXX American Chemical Society

R(2)O6 it occurs with the upper-lying InO5 layer (Figure 1c). Note that three adjacent InO5 trigonal bipyramids share their axial corners with a RO6 octahedron such that the common equatorial corner of the three InO5 trigonal bipyramids caps one face of the RO6 to form a RO7 polyhedra. In every RO6 layer, the R(2)O6 octahedra are capped from above, but the R(1)O6 octahedra are capped from below (Figure 1c). Since there are twice as many R(1) as R(2) atoms, the resulting structure of RInO3 becomes polar. This polar structure is identical in energy to the alternative one in which the R(2)O6 octahedra are capped from below and the R(1)O6 octahedra, from above. Thus, the polarity of RInO6 is reversed by Received: May 9, 2018

A

DOI: 10.1021/acs.inorgchem.8b01274 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry

how they affect the magnetic anisotropies of the R3+ ions and the spin exchanges between adjacent R3+ ions. In the present work we evaluate the orbital moments and magnetic anisotropies of the R3+ ions of RInO3 (R = Gd, Tb, Dy) by DFT + U + SOC calculations and extract the spin exchanges between adjacent R3+ ions by the energy-mapping method7 using both DFT + U + SOC and DFT + U calculations. It is shown that the Tb3+ and Dy3+ ions have orbital moments (μL) smaller than their free-ion values by ∼2μB. The spin exchanges extracted by DFT + U calculations are adequate in describing the magnetic properties of GdInO3 with zero orbital moment, but those extracted by DFT + U + SOC calculations are necessary in describing the magnetic properties of TbInO3 and DyInO3 with nonzero orbital moments. The low-temperature magnetic properties of oxides RInO3 are governed by the J2/J1 ratio of their spin-frustrated hexagonal spin exchanges.

Figure 1. Essential features of the crystal structure of RInO3: (a) A layer made up of corner-sharing InO5 trigonal bipyramids, where the blue spheres represent the In atoms. (b) A layer made up of edgesharing RO6 octahedra, where the red and pink spheres represent the R(1) and R(2) atoms, respectively. (c) The corner-sharing pattern between a layer of corner-sharing InO5 trigonal bipyramids and a layer of edge-sharing RO6 octahedra. (d) A hexagonal layer of R(1)3+ and R(2)3+ ions, where the numbers 1 and 2 refer to the spinexchange paths J1 and J2, respectively. (e) Two adjacent layers of R(1)3+ and R(2)3+ ions, where the numbers 3 and 4 refer to the spinexchange paths J3 and J4, respectively.

2. COMPUTATIONAL DETAILS Spin-polarized DFT calculations were carried out using the Vienna ab initio simulation package8,9 by employing the projector-augmented wave method10,11 with the PBE exchange−correlation functionals,12 the plane-wave cutoff energy was set at 520 eV with a set of 3 × 3 × 2 k points for sampling the irreducible Brillouin zone, and the threshold of selfconsistent-field energy was 10−6 eV. The electron correlation associated with the 4f states of R3+ was taken into consideration using the DFT + U method with Ueff = U − J = 5−7 eV.13 Thus, our study of RInO3 was carried out by using DFT + U + SOC calculations.14 Unless otherwise mentioned, our DFT + U + SOC calculations were carried out with spins oriented along the c direction (hereafter referred to as the ||c direction). Calculations with spins oriented along the a direction (hereafter referred to the ⊥c direction) lead to a convergence problem in that the orbital moment (μL) obtained differs from that found for the ||c calculations and varies from calculation to calculation. The plots of the density of states (DOS) calculated for the various states of GdInO3, TbInO3, and DyInO3 are presented in Figure S1−S3, respectively.

switching the capping pattern along the c direction. A computational study on RInO3 (R = Nd, Sm, Gd, Dy, Er) showed that these oxides are ferroelectric,2 and this prediction was experimentally confirmed for GdInO3 and DyInO3.2 In RInO3 (R = Gd, Tb, Dy), only the rare-earth cations R3+ are magnetic. These magnetic ions form hexagonal spin lattices (Figure 1d) in which the R(2)3+ ions lie slightly above the plane of the R(1)3+ ions (Figure 1e), and there are two such hexagonal lattices per unit cell. Given the triangular arrangements of the NN R3+ ions (Figure 1d,e), the spin-exchange interactions are spin-frustrated when the NN spin exchanges are all AFM.3 The magnetic susceptibilities of GdInO3,4 TbInO3,5 and DyInO35 follow the Curie−Weiss law with negative Curie−Weiss temperatures (θCW), suggesting that the dominant NN spin exchanges are most likely AFM. The spin frustrations expected for GdInO3,4 TbInO3,6 and DyInO36 were examined. By symmetry, the hexagonal spin lattice of RInO3 has two different NN spin exchanges, J1 and J2 (Figure 1d), and hence is not identical to the ideal hexagonal spin lattice in which J1 = J2. Depending on the relative strengths of J1 and J2, therefore, the magnetic properties of RInO3 at low temperature can deviate from those expected for the ideal hexagonal spin lattice. GdInO3 has been found to undergo an AFM ordering, with a sharp increase in the specific heat and a precipitous drop in the inverse magnetic susceptibility below 1.8 K.5 It is of interest to know if these observations arise from the difference in J1 and J2. Magnetization measurements for RInO3 (R3+ = Tb3+, Dy3+) at 0.55 K up to 65 T show the total moments (μT) of 6.9μB and 8.1μB for the Tb3+ and Dy3+ ions, respectively.6 Thus, the Tb3+ and Dy3+ ions have orbital moments (μL) of 0.9μB and 3.1μB, respectively, given their spin moments (μS) of 6μB and 5μB, respectively. Though smaller than expected for free ions, the observed orbital moments of Tb3+ and Dy3+ ions are still substantial (i.e., 0.9μB vs 3μB for Tb3+ and 3.1μB vs 5μB for Dy3+). The nonzero orbital moments of the R3+ ions result from their SOC. Two important issues associated with nonzero orbital moments of the R3+ ions are

3. UNQUENCHED ORBITAL MOMENTS OF TB3+ AND DY3+ For the ferromagnetic (FM) spin arrangement of the R3+ spins with a ||c spin orientation, our DFT + U + SOC calculations for RInO3 (R = Gd, Tb, Dy) using Ueff = 5, 6, and 7 eV show the μS and μL of the R(1)3+ and R(2)3+ ions listed in Table 1a and S1a. The two different rare-earth ions, R(1)3+ and R(2)3+, have Table 1. Spin Moments μS, Orbital Moments μL, and Magnetic Anisotropies E(⊥c) − E(||c) Obtained for the R(1)3+ and R(2)3+ Ions of RInO3 (R = Gd, Tb, Dy) Determined by DFT + U + SOC Calculations with Ueff = 7 eV for Their FM Statesa Gd Tb Dy

a

B

site

μS

μL

E(||) − E(⊥)

Gd(1) Gd(2) Tb(1) Tb(2) Dy(1) Dy(2)

7.10 7.02 6.16 6.07 5.08 5.04

0.03 0.03 1.00 0.99 2.75 2.74

−0.07 −7.33 ?

The μS and μL are in μB, and the E(⊥c) − E(||c) are in meV/R. DOI: 10.1021/acs.inorgchem.8b01274 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry nearly the same μS and μL values. μL ≈ 0 for Gd3+ (f7), which is as expected because the free-ion L value of Gd3+ (f7) is zero. The free-ion μL values of Tb3+ and Dy3+ are 3μB and 5μB, respectively, while their calculated μL values are close to +1μB and +3μB, respectively. These calculated μL values are in agreement with the experimental values (i.e., 0.9μB and 3.1μB for Tb3+ and Dy3+, respectively).6 In our analysis for the spin exchanges of RInO3 (R = Tb, Dy) using eq 2, we use μL values of +1μB and +3μB (and hence the μT values of 7 and 8) for Tb3+ and Dy3+, respectively (see below). We briefly comment on how the orbital moments of the Tb3+(f8) and Dy3+(f9) ions are reduced from their free-ion values by examining the PDOS plots calculated for the Lz = ±3, ±2, ±1, and 0 components of their up-spin and down-spin f states. The PDOS plots calculated for GdInO3, TbInO3, and DyInO3 are presented in Figures 2−4, respectively. The f states

Figure 4. PDOS plots of Lz components ±3, ±2, ±1, and 0 calculated for the f states of the Dy(1)3+ and Dy(2)3+ ions in the ferromagnetic state of DyInO3 by the DFT + U + SOC calculations with Ueff = 7 eV.

each fully occupied, as expected. The remaining two down-spin f electrons are described as a linear combination of the Lz = +3, +2, and 0 states, leading to μL ≈ + 2.75μB for Dy3+(f9).

4. SPIN ORIENTATION The magnetic anisotropy of the R3+ ions of RInO3 was examined by determining the relative energies of the spin orientations along the c direction (||c) and the ||a direction (hereafter referred to as ⊥c) moment orientations in the ferromagnetic (FM) state of RInO3 by DFT + U + SOC calculations. Our calculations for RInO3 with ||c spin orientation always converge to the states with μL mentioned in the previous section. In contrast, it was difficult to obtain the states with the same μL values for DyInO3 when calculations were carried out with ⊥c spin orientation. The results of our calculations are summarized in Tables 1b and S1b. For the Gd3+ ions of GdInO3, the ||c and the ⊥c orientations are nearly the same in energy. This is expected because μL ≈ 0 for the Gd3+ (f7) ion so that its SOC, which determines the preferred spin orientation, is zero. The Tb3+ ions of TbInO3 strongly prefer the ⊥c orientation to the ||c spin orientation (by ∼7 meV/Tb). This result is consistent with the experimental findings based on a single crystal sample of TbInO3; its magnetic susceptibility is considerably greater when the probe magnetic field is along the ⊥c than along the ||c orientation.5 For DyInO3 with the ⊥c spin orientation of the Dy3+ ions, our calculations did not converge to a state of μL close to 2.75μB (i.e., the value found for the ||c spin orientation). In the DFT + U + SOC calculations, the energy of a state depends sensitively on the value of μL; the smaller the μL, the lower the energy becomes. (A lower orbital moment means that the system has a stronger orbital quenching; i.e., it uses orbitals more extensively to lower its energy.) Consequently, it was not possible to estimate the magnetic anisotropy of the Dy3+ ions in DyInO3. We now examine the preferred spin orientations of the Tb3+ and Dy3+ ions in RInO3 using the qualitative rule7,15,16 derived from perturbation theory based on the SOC, Ĥ = λŜ L̂ . This rule allows one to predict the preferred spin orientation for a magnetic ion, located at a certain coordinate site forming a polyhedron with the surrounding ligands, by considering the difference in the magnetic quantum numbers, |ΔLz|, of the HOMO and LUMO of the polyhedron. (For RInO3 the z direction is the ||c direction, i.e., the 3-fold rational axis of each RO7 polyhedron containing the magnetic ion R3+.) If the

Figure 2. PDOS plots of Lz components ±3, ±2, ±1, and 0 calculated for the f states of the Gd(1)3+ and Gd(2)3+ ions in the ferromagnetic state of GdInO3 by DFT + U + SOC calculations with Ueff = 7 eV.

Figure 3. PDOS plots of Lz components ±3, ±2, ±1, and 0 calculated for the f states of the Tb(1)3+ and Tb(2)3+ ions in the ferromagnetic state of TbInO3 by DFT + U + SOC calculations with Ueff = 7 eV.

of R3+(1) are slightly lower in energy than those of R3+(2). The PDOS plots of GdInO3 (Figure 2) reveal that the up-spin Lz states of ±3, ±2, ±1, and 0 of Gd3+(f7) are each fully occupied while the corresponding down-spin Lz states are fully unoccupied, leading to μL ≈ 0 as expected. The PDOS plots of TbInO3 (Figure 3) show that the seven up-spin Lz states (i.e., ±3, ±2, ±1, and 0) of Tb3+(f8) are each fully occupied, as expected. The one down-spin f electron is described as a linear combination of Lz = +3 and 0 states, leading to μL ≈ +1μB for Tb3+(f8). The PDOS plots of DyInO3 (Figure 4) reveal that the up-spin Lz states of ±3, ±2, ±1 and 0 states of Dy3+(f9) are C

DOI: 10.1021/acs.inorgchem.8b01274 Inorg. Chem. XXXX, XXX, XXX−XXX

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Inorganic Chemistry

energies of its ordered spin states. To probe this question, we extract the spin exchanges J1−J4 of RInO3 (R = Gd, Tb, Dy) by both DFT + U and DFT + U + SOC calculations. In extracting the spin exchanges from the DFT + U + SOC calculations, one may use eqs 1 and 2 with Q = S. In the LS coupling scheme, the SOC is described by λS⃗ L⃗ ,15,16 which leads to the total angular momentum Ji⃗ = S⃗ i + L⃗ i in terms of the spin and orbital angular momenta S⃗ i and L⃗ i, respectively. The SOC constant λ is negative (positive) for the ion with more (less) than a half-filled shell, so the total angular momenta of Tb3+(f8) and Dy3+ (f9) are given by J = S + L. The magnetic energy spectrum for a system made up of rare-earth cations R3+ can also be described by the model Hamiltonian

HOMO and LUMO have the same spin, the preferred spin orientation is the ||z direction when |ΔLz| = 0 but the ⊥z direction when |ΔLz| = 1.14−16 In TbInO3, the HOMO is dominated by the (Lz = 3)↓ state while the LUMO is dominated by the (Lz = 2)↓ state for both Tb(1)3+ and Tb(2)3+ (Figure 3). Therefore, |ΔLz| = 1 so that the preferred spin orientation is predicted to be the ⊥z direction for both Tb(1)3+ and Tb(2)3+, in agreement with experiment. In DyInO3, the HOMO is dominated by the (Lz = 3)↓ state for both Dy(1)3+ and Dy(2)3+, as is the LUMO (Figure 4). Thus, |ΔLz| = 0 so that the preferred spin orientation is the ||z direction for both Dy(1)3+ and Dy(2)3+.

5. SPIN EXCHANGES For the spin-exchange paths in RInO3, we consider the two intralayer exchanges J1 and J2 (Figure 1d) as well as the two interlayer exchanges J3 and J4 (Figure 1e). The geometrical parameters associated with J1−J4 are summarized in Table S2. For magnetic transition-metal cations, μT ≈ μS because their orbital moments are typically quenched (μL ≈ 0). (The only exception is found for the cations with uniaxial magnetism for which μL ≠ 0.15,17) Thus, for most magnetic systems made up of transition-metal cations, their magnetic energy spectra are well described by the spin Hamiltonian

H = −∑ Jij Ji ⃗ Jj ⃗

and eq 2 with Q = J. In using eq 3, we do not employ the freeion L values of R3+ but their observed L values so that J = S + L = 3.5 for GdInO3, 4 for TbInO3, and 5.5 for DyInO3. The relative energies of the five ordered spin states and the values of J1−J4 determined from the DFT + U calculations are summarized in Tables S3 and S4, respectively, and those from the DFT + U + SOC calculations, in Tables S5 and S6, respectively. For GdInO3, the J1−J4 values from the DFT + U + SOC calculations are similar to those from the DFT + U calculations. This is understandable because L = 0 for Gd3+. For the DFT + U + SOC calculations on TbInO3 and DyInO3 with L ≠ 0 ions Tb3+ and Dy3+, respectively, the J1−J4 values obtained by using eq 1 are similar to those obtained by using eq 2. This reflects the fact that Ji⃗ Jj⃗ ∝ S⃗ iS⃗ j for a system of identical rare-earth cations. The spin exchanges J1−J4 obtained from the DFT + U + SOC calculations show that interlayer exchanges J3 and J4 are negligible compared with intralayer exchanges J1 and J2. To test the reliability of the J1−J4 values extracted, we calculate the relative energies of the two additional ordered spin states (AF5 and AF6 in Figure S5) using spin exchanges J1−J4 and compare them with those determined by the DFT + U and DFT + U + SOC calculations (Table S7). For GdInO3 for which L = 0, the relative energies of the states obtained from the DFT + U and DFT + U + SOC calculations agree well with those estimated by using the calculated J1−J4 values (Table S7). For TbInO3 and DyInO3 with L ≠ 0 rare-earth cations, this is also the case for the DFT + U + SOC calculations (Table S7b) but not for the DFT + U calculations (Table S7a). We further test the reliability of the calculated J1− J4 values by estimating the Curie−Weiss temperatures of RInO3, θCW, in terms of the J1−J4 using the mean-field theory approximation.18 Since RInO3 has the R(1)3+ and R(2)3+ ions in a 2:1 ratio, we approximate the θCW value by the weighted average of the values expected from the R(1)3+ and R(2)3+ ions. Thus,

H = −∑ Jij Si⃗ Sj⃗ (1)

i>j

where Jij is the spin-exchange constant associated with spins S⃗ i and S⃗ j at sites i and j, respectively. The values of the spin exchanges Jij chosen for a given magnetic system (e.g., Jij = J1 − J4 in RInO3) are extracted by performing an energy-mapping analysis based on DFT electronic structure calculations.7 In this analysis, one carries out DFT calculations for a certain number of ordered spin states of the magnetic system to determine their relative energies and equates them to the corresponding energies expressed in terms of the spinexchange constants. To extract the values of the four spinexchange constants J1−J4 of RInO3, we consider the five ordered spin states (FM, AF1, AF2, AF3, and AF4) shown in Figure S4. In terms of the spin Hamiltonian, eq 1, the energies per unit cell of the five ordered spin states are given by FM: (+ 12J1 + 6J2 + 2J3 + 4J4 )Q 2 AF1: ( AF2: ( +6J1

+ 6J2 − 2J3 + 4J4 )Q 2 − 2J3

)Q 2

AF3: ( −12J1 + 6J2 + 2J3 + 4J4 )Q 2 AF4: (

+ 6J2 + 2J3 − 4J4 )Q 2

(3)

i>j

(2)

where Q = S when eq 1 is used. The values of J1−J4 can be determined by mapping the relative energies of the five states from either the DFT + U or DFT + U + SOC calculation onto the corresponding relative energies from the spin Hamiltonian, eq 1. The energy-mapping analysis based on the DFT + U calculations provides adequate descriptions for magnetic systems of 3D transition-metal ions. However, this approach may be inaccurate in determining the spin exchanges of a rareearth compound because the DFT + U calculations do not include SOC and hence may not provide accurate relative

θCW =

Q (Q + 1) (12J1 + 6J2 + 2J3 + 4J4 ) 9

(4)

where Q = S or J. For polycrystalline samples of RInO3, the experimental θCW values are found to be −50 K4 and −9.7 K5 for R = Gd, −21 K5 for R = Tb, and −10 K5 for R = Dy. For GdInO3 with the L = 0 rare-earth cation, both DFT + U and DFT + U + SOC calculations predict similar θCW values (i.e., close to −22 K using Ueff = 7 eV) (Table S8a,b). For polycrystalline samples of TbInO3 and DyInO3 with L ≠ 0 D

DOI: 10.1021/acs.inorgchem.8b01274 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry rare-earth cations, this is also the case for DFT + U + SOC calculations (namely, −19 K for TbInO3 and −13 K for DyInO3 using Ueff = 7 eV) (Table S8b), while predictions from DFT + U calculations deviate strongly from the experimental values (Table S8a).

frustration in DyInO3 is stronger than those of GdInO3 and TbInO3. At low temperature, then, it would be difficult for the honeycomb lattices of the Dy(1)3+ ions to undergo AFM ordering. Thus, DyInO3 would not exhibit the magnetic properties found for GdInO3 at low temperature.

6. EFFECT OF NONEQUIVALENT INTRALAYER SPIN EXCHANGES We now discuss the important implications of the spin exchanges obtained for RInO3 from the DFT + U + SOC calculations on their magnetic properties at low temperature by considering the two dominant exchanges J1 and J2. These intralayer exchanges are both AFM and form spin-frustrated (J1, J1, J2) triangles so that the hexagonal spin lattices are spinfrustrated. However, the nature of this spin frustration depends on the ratio of J2/J1. The values of J1−J4 calculated for RInO3 (R = Gd, Tb, Dy) from our DFT + U + SOC calculations are summarized in Table 2. The J2 paths form the honeycomb

7. CONCLUDING REMARKS As expected, the magnetic properties of GdInO3 are equally well described by the DFT + U and DFT + U + SOC calculations because L = 0 for Gd3+. The μL of the Tb3+ and Dy3+ ions are smaller in magnitude than those of their free-ion values by ∼2μB, and the Tb3+ spin prefers the ⊥c orientation. The magnetic properties of TbInO3 and DyInO3 are well explained by the spin exchanges J1−J4 extracted from DFT + U + SOC calculations but not by those extracted from DFT + U calculations, showing the need to include SOC in evaluating the spin exchanges between rare-earth cations with nonzero L by energy-mapping analysis. The spin exchanges of RInO3 (R = Gd, Tb, Dy) are dominated by the intralayer exchanges J1 and J2, which are both AFM and form the honeycomb lattices of J2 with spin-frustrated (J1, J1, J2) triangles. For GdInO3 (J2/J1 ≈ 3), the honeycomb spin lattices of J2 undergo an AFM ordering at low temperature with a spin-gap opening. Similar AFM ordering is expected for TbInO3 (J2/J1 ≈ 1.7) but not for DyInO3 (J2/J1 ≈ 1).

Table 2. Intralayer Spin Exchanges J1 and J2 (in kBK) as Well as Interlayer Spin Exchanges J3 and J4 (in kBK) of RInO3 (R = Gd, Tb, Dy) Obtained by Using Eq 2a GdInO3

TbInO3

DyInO3

J1

J2

J2/J1

J3

J4

−0.403 −0.271 −0.207 −0.664 −0.338 −0.199 −0.812 −0.485 −0.295

−1.327 −0.818 −0.548 −1.117 −0.580 −0.315 −0.810 −0.630 −0.326

3.29 3.02 2.65 1.68 1.72 1.58 1.00 1.30 1.11

0.012 0.012 0.012 0.016 0.018 0.007 −0.084 −0.015 0.004

−0.133 0.002 −0.003 −0.035 −0.019 −0.009 −0.139 −0.056 −0.050

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.inorgchem.8b01274. The calculated spin and orbital moments of RInO3, the geometrical parameters associated with the spinexchange paths J1−J4 of RInO3, the relative energies calculated for the five ordered spin states of RInO3, the calculated J1−J4 values of RInO3, the relative energies calculated for two additional ordered spin states of RInO 3 , the Curie−Weiss temperatures of RInO 3 estimated from the calculated J1−J4 values, the PDOS plots of RInO3 as well as the spin arrangements of the five plus two additional ordered spin states of RInO3 (PDF)

a For each compound, the first, second, and third rows of numbers refer to the results of DFT + U + SOC calculations with Ueff = 7, 6, and 5 eV, respectively.

lattices (Figure 1d,e), with the J1 paths forming the spinfrustrated (J2, J1, J1) triangles. For GdInO3, the J2/J1 ratio is close to ∼3, implying that the honeycomb spin lattices of the Gd(1)3+ ions would undergo an AFM ordering with a concomitant spin gap opening at low temperature where the thermal energy is not high enough to break the AFM ordering in the honeycomb lattice of Gd(1)3+ ions. In such a case, the magnetic susceptibility of GdInO3 is contributed only by the magnetism arising from the “isolated” Gd(2)3+ ions, the spins of which are no longer spin-frustrated. Therefore, the magnetic susceptibility of GdInO3 below 1.8 K would become greater than its value above 1.8 K where spin frustration exists. This explains why the inverse magnetic susceptibility curve (χ−1 vs T) of GdInO3 shows a sharp drop below 1.8 K4 and why the specific heat curve (Cp/T vs T) shows a sharp upturn below 1.8 K.4 Namely, the AFM phase transition of GdInO3 seen below 1.8 K is identified as an AFM ordering of the honeycomb lattices made up of the J2 bonds. For TbInO3, the J2/J1 ratio (∼1.7) is considerably greater than 1 (Table 2). At low temperature, therefore, it is possible that the honeycomb lattices of the Tb(1)3+ ions in GdInO3 undergo an AFM ordering, with a spin gap opening and the magnetic susceptibility contributed only by the Tb(2)3+ ions. Then, the low-temperature magnetic properties of TbInO3 would exhibit features resembling those of GdInO3. For DyInO3, the J2/J1 ratio is close to ∼1 (Table 2) so that the extent of spin

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Shuiquan Deng: 0000-0001-5538-6738 Myung-Hwan Whangbo: 0000-0002-2220-1124 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS M.-H.W. thanks J. L. Musfeldt for invaluable discussions in the early stage of this work and the High Performance Computing services of NCSU for computing resources. The work at Rutgers University was supported by the DOE under grant no. DOE: DE-FG02-07ER46382. The work at Fujian Institute of Research on the Structure of Matter, Chinese Academy of Sciences was financially supported by the National Natural E

DOI: 10.1021/acs.inorgchem.8b01274 Inorg. Chem. XXXX, XXX, XXX−XXX

Article

Inorganic Chemistry Science Foundation of China (21703251), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB20000000), the National Key Research and Development Program of China (2016YFB0701001), the 973 Program of China (2014CB932101), and the 100 Talents Program of CAS and Fujian Province.

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DOI: 10.1021/acs.inorgchem.8b01274 Inorg. Chem. XXXX, XXX, XXX−XXX