Article pubs.acs.org/cm
Analysis of the Difference between the Pyroxenes LiFeSi2O6 and LiFeGe2O6 in Their Spin Order, Spin Orientation, and Ferrotoroidal Order Changhoon Lee,†,‡ Jinhee Kang,† Jisook Hong,‡ Ji Hoon Shim,*,‡,§ and Myung-Hwan Whangbo*,† †
Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695, United States Department of Chemistry, Pohang University of Science and Technology, Pohang 790-784, Korea § Division of Advanced Nuclear Engineering, Pohang University of Science and Technology, Pohang 790-784, Korea ‡
S Supporting Information *
ABSTRACT: The pyroxenes LiFeSi2O6 and LiFeGe2O6 are isostructural and isoelectronic, but they differ in their spin order, spin orientation and ferrotoroidal order. The reasons for these differences were probed by density functional theory electronic structure calculations and magnetic dipole−dipole (MDD) interaction energy calculations. The ferromagnetic Fe-zigzag chains of LiFeSi2O6 arise from the antiferromagnetic interchain spin exchanges and the antiferromagnetic Fe-zigzag chains of LiFeGe2O6 from the antiferromagnetic intrachain spin exchange. The preferred spin orientations of LiFeGe2O6 and LiFeSi2O6 are not caused by spin−orbit coupling but by MDD interactions. LiFeSi2O6 undergoes a ferrotoroidal order because it has spin exchange rings made up of an even number of antiferromagnetic spin exchanges paths with comparable strengths. A ferrotoroidal order is not expected for LiFeGe2O6 because it has no such spin exchange rings.
1. INTRODUCTION The pyroxenes with the general formula AFeSi2O6 (A = Li, Na)1−10 and its germanium analogues AFeGe2O63,11−13 have received much attention in recent years due largely to their exotic magnetic properties. NaFeSi2O6 becomes ferroelectric in a magnetically ordered state below ∼6 K, whereas LiFeSi2O6 becomes ferroelectric only when a magnetic field is applied.4 NaFeGe2O6 becomes ferroelectric in a magnetically ordered state,13 but NaFeSi2O6 exhibits a ferrotoroidal order.8 LiFeSi2O6 undergoes a ferrotoroidal order1 as well as a threedimensional (3D) antiferromagnetic (AFM) order,3 whereas LiFeGe2O6 exhibits only a 3D AFM order. Thus, these pyroxenes exhibit subtle differences in their magnetic properties, which are triggered by merely replacing their nonmagnetic ions with other nonmagnetic ions within the same family. This reflects the fact that magnetic properties are sensitive to small changes in the bond lengths and bond angles around magnetic ions.14,15 Ferroelectricity in a system occurs when it loses inversion symmetry.16 A spiral spin order, which occurs to reduce the extent of spin frustration17 in a collinear spin arrangement, removes inversion symmetry thereby leading to a magnetic-order-induced ferroelectricity.18,19 Thus, the observed 3D AFM order in LiFeSi2O6 and LiFeGe2O63 implies that they do not possess strong spin frustration, whereas the magneticorder-induced ferroelectricity in NaFeSi2O6 and NaFeGe2O6 implies that they most likely possess a spiral spin order due to the presence of spin frustration. The magnetic-field-driven ferroelectricity in LiFeSi2O6 indicates that its 3D AFM order is destroyed by an external magnetic field, and the resulting ordered magnetic structure has no inversion symmetry. It also © 2014 American Chemical Society
implies that the driving force toward a 3D AFM order is not strong in LiFeSi2O6. Compared with ferroelectricity, the phenomenon of ferrotoroidal order is rather rare, and its chemical picture is lacking. In a localized spin system, a ferrotoroidal order is pictured as arising from spin rings made up of an even number of spin moments arranged in a head-totail arrangement,20 which leads to a toroidal moment perpendicular to the rings. From the viewpoint of spin exchanges, the occurrence of such spin rings is possible only if there are spin exchange rings made up of an even number of AFM exchange paths with comparable strengths. Then, a ferrotoroidal order would not occur in a magnetic system with strong one-dimensional (1D) chain character. We test these implications by analyzing the difference in the magnetic properties of LiFeSi2O6 and LiFeGe2O6. The pyroxenes LiFeSi2O6 and LiFeGe2O6 crystallize in a monoclinic space group P21/c at room temperature and do not undergo any structural phase transition down to very low temperatures (1.4 and 5 K, respectively).3 The structural building blocks of LiFeSi2O6 (M = Si, Ge) are FeO6 octahedra and MO4 (M = Si, Ge) tetrahedra. The FeO6 octahedra share their cis edges to form FeO4 zigzag chains (Figure 1a,b) along the c-direction, and the MO4 tetrahedra share their corners to form MO3 zigzag chains (Figure 1c) along the c-direction. The FeO4 zigzag chains share their corners with the MO3 zigzag chains such that each FeO4 chain is surrounded by six MO3 Received: January 14, 2014 Revised: January 31, 2014 Published: February 3, 2014 1745
dx.doi.org/10.1021/cm5001413 | Chem. Mater. 2014, 26, 1745−1750
Chemistry of Materials
Article
dashed interchain interactions, respectively.) It should be noticed from Figure 2a that each Fe-zigzag chain is surrounded by four Fe-zigzag chains, with two solid and two dashed interchain interactions. The Fe-zigzag chains along the (a+b) direction form layers parallel to the (a+b, c) plane (see the red rectangular box in Figure 2a), in which there occur only the solid interchain interactions. Similarly, the Fe-zigzag chains along the (a−b) direction form layers parallel to the (a−b, c) plane (see the green rectangular box in Figure 2a), in which there occur only the dashed interchain interactions. A view along the b-direction of those Fe-zigzag chains contained in the blue rectangular box of Figure 2a is presented in Figure 2b, where the shaded chains lie higher in the b-axis height than the unshaded ones and the solid and dashed lines indicate the solid and dashed interchain interactions, respectively. Using this layer representation, the observed magnetic structure of LiFeSi2O6 can be described in Figure 2c and that of LiFeGe2O6 in Figure 2d.3 In LiFeSi2O6, the Fe3+ ions form ferromagnetically coupled Fe-zigzag chains, and these ferromagnetic (FM) chains are antiferromagnetically coupled to form layers parallel to the (a +b, c) plane as well as layers parallel to the (a−b, c) plane (Figure 2c). In LiFeGe2O6, however, the Fe3+ ions form antiferromagnetically coupled Fe-zigzag chains. These AFM chains are ferromagnetically coupled along the (a+b) direction to form layers parallel to the (a+b, c) plane, but are antiferromagnetically coupled along the (a−b) direction to form layers parallel to the (a−b, c) plane (Figure 2d). In addition, LiFeSi2O6 and LiFeGe2O6 differ in their spin orientation, which is along the c-direction in LiFeSi2O6, but perpendicular to the c-direction (in the ac plane, i.e., approximately along the a-direction) in LiFeGe2O6. So far, only a few theoretical studies on the pyroxenes have been reported. Siegel et al. carried out density functional theory (DFT) calculations to examine the changes in the crystal structure of LiFeSi2O6 induced by the phase transition between the high-temperature C2/c and low-temperature P21/c phase.2 Jodlauk et al. estimated three spin exchange parameters (namely, J3, J4, J5 of Figure 2b) of LiFeSi2O6 by DFT calculations to find that they lead to spin-frustrated triangles, and hence suggested the possibility that a spiral spin structure induced by spin frustration is responsible for the ferroelectric behavior of LiFeSi2O6.4a In addition, Streltsov and Khomskii,4b who also considered the three spin exchange interactions, argued that the intrachain spin exchange in the Fe-zigzag chains of LiFeSi2O6 should be AFM. However, as shown later by Redhammer et al.,3 the magnetic structure of LiFeSi2O6 is not spiral but collinear, and the spins in each Fe-zigzag chain are ferromagnetically coupled. Thus, in describing the magnetic properties of the pyroxenes, considering only three spin exchange interactions is inadequate. The present work explores the reasons why the magnetic structures of LiFeSi2O6 and LiFeGe2O6 are different in their ordered magnetic states. To explain the observed spin order of the pyroxenes LiFeM2O6 (M = Si, Ge), we evaluate their spin exchange parameters J1−J6 defined in Figure 2b by performing energy-mapping analysis based on DFT calculations. 14,15 These spin exchange parameters allow us to discuss if spin exchange rings needed for a ferrotoroidal order can be formed in LiFeM2O6. To account for the observed spin orientations of LiFeM2O6 (M = Si, Ge), we determine the magneto-crystalline (MCA) energies for their Fe3+ ions by performing DFT calculations including spin−orbit coupling (SOC) and also by performing magnetic dipole−dipole (MDD) interaction energy calculations.21
Figure 1. (a) FeO4 zigzag chain along the c-direction resulting from cis edge-sharing FeO6 octahedra in LiFeM2O6 (M = Si, Ge). (b) A Fezigzag chain representing the FeO4 zigzag chain. (c) A MO3 zigzag chain along the c-direction resulting from the corner-sharing MO4 tetrahedra. (d) A perspective view of the crystal structure of LiFeM2O6 viewed approximately along the c-direction. Red circle = Fe, gray circle = M, cyan circle = Li, white circle = O.
chains (Figure 1d) with the Li+ ions located in the channels formed by MO3 and FeO4 zigzag chains. In both LiFeSi2O6 and LiFeGe2O6, the Fe atoms are present as high-spin Fe3+ ions (d5, S = 5/2), and both compounds undergo a long-range antiferromagnetic (AFM) order below ∼20 K; the magnetic susceptibility χ(T) of LiFeSi2O6 (LiFeGe2O6) exhibits a sharp maximum at 20.4 K (24.4 K) with the Curie−Weiss temperature θcw ≈ −25.4 K (−78.6 K).3 However, LiFeSi2O6 and LiFeGe2O6 are strikingly different in their spin order and spin orientation.3 To illustrate these differences, it is convenient to consider a view of the Fe-zigzag chains approximately along the c-direction (Figure 2a). There occur two types of interchain interactions indicated by solid and dashed lines, along the (a +b) and (a−b) directions, respectively. (For convenience of our discussion, they will be hereafter referred to as the solid and
Figure 2. (a) View of the Fe-zigzag chains of LiFeM2O6 (M = Si, Ge) approximately along the c-direction. (b) A view along the b-direction of those Fe-zigzag chains contained in the blue rectangular box of Figure 2a. The shaded chains lie higher in the b-axis height than the unshaded chains while the solid and dashed lines indicate the solid and dashed interchain interactions, respectively. The numbers 1−6 represent the spin exchange paths J1−J6, respectively. (c, d) The observed magnetic structures of LiFeSi2O6 and LiFeGe2O6. The gray rectangular boxes are given to indicate that the Fe-zigzag chain lie higher in the b-axis height than the chains without gray rectangular boxes. 1746
dx.doi.org/10.1021/cm5001413 | Chem. Mater. 2014, 26, 1745−1750
Chemistry of Materials
Article
2. COMPUTATIONAL DETAILS
Table 1. Some Geometrical Parameters Associated with the Spin Exchange Paths J1−J6 of LiFeSi2O6 and LiFeGe2O6a
In our density functional calculations for LiFeM2O6 (M = Si, Ge), we employed the frozen-core projector augmented wave method22,23 encoded in the Vienna ab initio simulation package (VASP),24 the local density approximation (LDA) with the Perdew−Zunger parametrization25 of the correlation energy of a homogeneous electron gas calculated by Ceperley and Alder,26 the plane-wave-cutoff energy of 450 eV and a set of 80 k-points in the irreducible Brillouin zone. To examine the effect of electron correlation in the Fe 3d states, the LDA plus on-site repulsion U (LDA+U) method of Dudarev et al.27 was used with the effective Ueff = U − J values of 4, 6, and 8 eV on the Fe atom. These are typical Ueff values used to confirm trends in the calculated spin exchanges.28 We also investigate the spin-orientations of LiFeSi2O6 and LiFeGe2O6 by performing LDA+U calculations with spin orbit coupling (SOC) effect included and also by carrying out MDD interaction energy calculations.21
J1 (SSE) J2 (SSE) J3 (SSE) J4 (SSE) J5 (SE)
3. SPIN EXCHANGE INTERACTIONS AND THEIR IMPLICATIONS The spin exchange paths J1−J6 of LiFeM2O6 (M = Si, Ge) considered in our study are specified in Figure 3. The intrachain
J6 (SSE)
∠Fe−O−Fe Fe···Fe O···O
LiFeSi2O6
LiFeGe2O6
6.555 2.758 6.441 2.694 5.357 2.758 (×2) 5.280 2.694 (×2) 3.173 1.969, 2.196 2.120, 2.094 99.1, 97.7 5.277 2.808
6.731 2.923 6.517 2.979 5.546 2.979 (×2) 5.422 2.923 (×2) 3.126 2.126, 2.026 2.137, 2.044 100.5, 101.5 5.317 2.814
a
The bond distances and angles are given in units of Å and degrees, respectively. SE and SSE refer to the Fe−O−Fe super exchange and the Fe−O···O−Fe super-superexchange, respectively.
where N refers to the unpaired spins per spin site (in the present case, N = 5).28 The six numbers n1−n6 defining the energy of each spin state are given in the parentheses (n1, n2, n3, n4, n5, n6) in Figure S1 of the Supporting Information. Thus, by mapping the relative energies of the seven spin ordered states (see Table S2 of the Supporting Information) calculated from the LDA+U calculations onto the corresponding energies expected from the total spin exchange energies, we finally obtain the values of the spin exchanges J1−J6 summarized in Table 2. In terms of these spin exchange parameters, we calculate the Curie−Weiss temperature θCW of LiFeM2O6 (M = Si, Ge), which in the mean field theory29 is written as
Figure 3. Structural units representing the spin exchange paths J1−J6.
Table 2. Values of the Spin Exchanges J (in kBK) and the Curie−Weiss Temperatures θCW (in K) of LiFeM2O6 (M = Si, Ge) Obtained from LDA+U Calculations with Effective Ueff (in eV)
nearest-neighbor spin exchange J5 is a Fe−O−Fe superexchange (SE), and the remaining spin exchanges are all Fe− O···O−Fe supersuperexchanges (SSEs).14,15 The geometrical parameters associated with these spin exchange paths are summarized in Table 1 (More details about the geometrical parameters are given in Table S1 of the Supporting Information.) To evaluate the values of J1−J6, we first determine the relative energies of the seven ordered spin states depicted in Figure S1 of the Supporting Information on the basis of LDA+U calculations27 with effective Ueff = U − J = 4, 6, and 8 eV on the Fe atom. The relative energies of these states per formula unit (FU) obtained from our calculations are summarized in Table S1 of the Supporting Information. In terms of the spin Hamiltonian
(a) LiFeSi2O6 U
eff
J1 J2 J3 J4 J5 J6 θCW
Ĥ = −∑ Jij Sî ·Sĵ i
