Impact of spin-orbit coupling on the magnetism of Sr₃MIrO₆ (M = Ni, Co)

Abstract

Iridates are of current great interest for their entangled spin-orbital state and possibly exotic properties. In this work, using density functional calculations, we have demonstrated that the hexagonal spin-chain materials Sr₃MIrO₆ (M = Ni, Co) are an iridate system in which the spin-orbit coupling (SOC) tunes the magnetic and electronic properties. The significant SOC alters the orbital state, the exchange pathway and thus the magnetic structure. This work clarifies the nature and the origin of the intra-chain antiferromagnetism of Sr₃MIrO₆ and well accounts for the most recent experiments.

Introduction

Charge, spin and orbital states are often coupled in 3d transition-metal oxides due to their multiple degrees of freedom and electron correlation. These states are closely related to diverse material properties and functionalities, e.g., charge ordering, orbital ordering, spin-state and magnetic transition, metal-insulator transition, superconductivity, colossal magnetoresistance and multiferroicity. It is therefore very important to study those charge-spin-orbital states and their fascinating coupling for modeling and understanding of the abundant properties. This has formed a research stream in condensed matter physics over past decades, see e.g., a short review by Dagotto¹. Very recently, research interest has been extended to 5d transition-metal oxides, which probably possess a significant spin-orbit coupling and provide an avenue to novel magnetic and electronic properties due to an entangled spin-orbital state.

In this respect, iridates are a representative example^{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17}. An octahedrally coordinated iridium ion normally has a large t_2g-e_g crystal-field splitting due to the delocalized character of its 5d electrons. The resultant low-spin state with only a t_2g occupation makes an open shell Irⁿ⁺ ion (e.g., for n = 4) behave effectively like p electrons (with an effective orbital momentum ). As a result, an intrinsic strong spin-orbit coupling (SOC) splits the t_2g levels into a lower quartet and a higher doublet. Then, for an Ir⁴⁺ constituent oxide, the half-filled doublet may form, due to a moderate electron correlation, a novel Mott insulating state^2,3. It has been proposed that such a spin-orbital entangled state can bring about exotic properties, e.g., correlated topological insulator^4,5, superconductivity⁶, Kitaev model^7,8, Weyl semimetal⁹ and unusual magnetism¹⁰.

In this Report, we have studied the 3d–5d transition-metal hybrid material Sr₃MIrO₆ (M = Ni, Co) and find that the SOC has a significant impact on its magnetism by tuning its spin-orbital states and the Ir-M inter-orbital interactions. This system has a general chemical formula A₃MM'O₆ (A = Ca, Sr; M = 3d transition metal, M' = 3d, 4d, 5d transition metal) and displays an in-plane hexagonal structure and out-of-plane spin chains, see Fig. 1. Those quasi one-dimensional spin chains each consist of alternating face-sharing MO₆ trigonal prisms and M'O₆ octahedra. This system drew a lot of attention in the past decade^{18,19,20,21,22,23,24,25,26,27}, because of its intriguing step-wise magnetization, significant Ising-like magnetism, thermoelectricity and multiferroicity. Sr₃NiIrO₆ and Sr₃CoIrO₆ also possess fascinating magnetism^28,29,30,31. Owing to their complex temperature-dependent magnetic transitions, a standing issue is the nature of their dominant intrachain magnetism: either an intrachain ferromagnetic (FM) exchange^28,29 or an antiferromagnetic (AF) coupling^30,31 was proposed in previous studies. Moreover, the origin of the magnetism remains elusive. Therefore, Sr₃NiIrO₆ and Sr₃CoIrO₆ call for a prompt study to clarify the nature and origin of their intriguing intrachain magnetism. As seen below, we make a comparative study for Sr₃NiIrO₆ and Sr₃CoIrO₆, by carrying out a systematic set of electronic structure calculations. Our results consistently explain the experimental observations and settle the standing issue. In particular, we find that the SOC of the Ir⁴⁺ ion plays an essential role in determining the intrachain AF structure of Sr₃MIrO₆ (M = Ni, Co) by tuning the crystal-field level sequence and altering the Ir-M inter-orbital interactions. Therefore, Sr₃MIrO₆ is added to the iridate category which highlights the significance of the SOC.

Figure 1

Crystal structure plot of Sr₃MIrO₆ (a) projected onto the ab plane and (b) in a perspective view.

It has a hexagonal ab plane and quasi one-dimensional MIrO₆ spin chains extending along the c-axis, in which the IrO₆ octahedra and MO₆ trigonal prisms are alternating.

Results

Crystal-field levels

We first carry out spin-restricted LDA calculations to estimate the crystal field splitting. The calculated DOS (density of states) results for Sr₃NiIrO₆ are shown in Fig. 2. The O 2p valence bands lie in between −7 and −2 eV. They have a significant covalency with the delocalized Ir 5d orbital and bring about the Ir 5d bonding state around −6 eV. A relatively weak Ni-O hybridization yields the Ni 3d bonding state in between −2 and −4 eV. Both the Ir 5d and Ni 3d antibonding states lie above −2 eV. For the Ir ion, its local octahedral coordination but a trigonal crystal field in the global coordinate system split the otherwise t_2g triplet into the doublet and the a_1g singlet both of the concern. The e_g doublet is far above them by 3 eV and is out of the concern. The a_1g singlet can be written as 3z² − r², as the z-axis of the hexagonal lattice is along the [111] direction of the local IrO₆ octahedra. Moreover, the doublet can be expressed as and , when the y-axis is set along the [] direction of the local IrO₆ octahedra (then the x-axis is uniquely defined and the xy is in the hexagonal ab plane). By integrating the DOS and determining the center of gravity for each eigen orbital within the antibonding energy range, we find that the doublet is lower than the a_1g singlet by 0.21 eV. Actually, the a_1g-e'_g level splitting is an important issue. The trigonal distortion of the IrO₆ octahedron¹⁹, e'_g-e_g mixing³² and long-ranged crystal field due to the lattice anisotropy³³ all contribute to the splitting. All these effects are properly included in the LDA calculation. Although the contribution of the lattice anisotropy is often quite complicate³³, the a_1g-e'_g level ordering can be understood, as the elongation of the IrO₆ octahedron along the [111] direction (the resultant O-Ir-O bond angle deviating from the ideal 90° by 5.4°²⁸) raises a_1g with respect to e'_g and the e'_g-e_g mixing pushes e'_g downwards³². For the Ni ion, its trigonal prismatic coordination produces the crystal-field level sequence of the Ni 3d electrons as 3z² − r²/ xy, x² − y²/xz, yz (0/0.57/0.76 eV). Furthermore, there are inter-site interactions between the Ir 5d and Ni 3d orbitals, see Fig. 2. The Ir a_1g 3z² − r²and Ni 3z² − r² electrons have a lobe pointing to each other and have a ddσ hybridization. The Ir orbital has, via it's xz or yz component, a ddπ hybridization with the Ni xz/yz; and via it's xy or x² − y² component, a weak ddδ hybridization with the Ni xy/x² − y². Note that those crystal-field level sequences and the Ir-Ni inter-orbital hybridizations are crucial for understanding of the spin-orbital state and the intrachain magnetism in Sr₃MIrO₆, as seen below.

Figure 2

Partial density of states (DOS) of Sr₃NiIrO₆ in the nonmagnetic state calculated by LDA.

The octahedral Ir ion has a common large t_2g − e_g crystal field splitting of more than 3 eV; and in a trigonal crystal field (elongation of the IrO₆ octahedron along the local [111] direction, i.e., the z-axis of the hexagonal lattice), the t_2g splits further into a lower doublet and a higher a_1g singlet. The trigonal prismatic coordination produces the Ni 3d crystal-field level sequence (from low to high) as 3z² − r²/xy, x² − y²/xz, yz.

Spin polarization and electron correlation

Then we perform spin-polarized LSDA calculations. We start with a FM or an AF state with the Ni²⁺ spin = 1 and Ir⁴⁺ spin = 1/2, but both calculations converge to a same FM metallic solution (not shown here). It has a total spin moment of 2.82 μ_B/fu, consisting of the local spin moments of 1.46 μ_B/Ni²⁺, 0.54 μ_B/Ir⁴⁺ and 0.11 μ_B/O. The Ni²⁺ ion has the electronic configuration and the Ir⁴⁺ has a single t_2g hole mostly on the a_1g orbital [i.e., ]. The Ir-O and Ni-O covalencies bring about an appreciable spin moment of 0.11 μ_B on each oxygen. As the a_1g orbital is a higher crystal-field level than the , the a_1g hole state allows a direct a_1g electron hopping from Ni²⁺ to Ir⁴⁺ and this prompts the FM metallic solution.

The above LSDA metallic solution contradicts the experimental insulating behavior. In order to probe the electron correlation effect, we now carry out LSDA + U calculations. The electron correlation stabilizes the Ni²⁺ S = 1 state and the calculated spin moment of Ni²⁺ is enhanced to 1.68 μ_B. As seen in Fig. 3(a), the single Ir⁴⁺ t_2g hole now fully occupies the down-spin a_1g orbital. The Ir⁴⁺ spin moment is also increased to 0.64 μ_B. Owing to the correlation driven d-electron localization, the induced 2p spin moment on oxygen via the Ir-O and Ni-O hybridizations is reduced to 0.08 μ_B. Apparently, electron correlations make the Ir 5d and Ni 3d spin-orbital states fully polarized, thus giving an insulating solution. The tiny band gap is within the Ir t_2g shell and it is due to a relatively weak electron correlation of delocalized 5d electrons. As Ir⁴⁺ has the single t_2g hole on the down-spin a_1g (3z² − r²) orbital, the down-spin 3z² − r² electron of Ni²⁺ can hop forth and back. In order to maximize the local Hund exchange on the virtual Ni³⁺ ion in the excited intermediate state, the hopping 3z² − r² electron should be in the minority-spin channel, see Fig. 4 (b). Then this exchange mechanism gives the FM coupling between Ir⁴⁺ S = 1/2 and Ni²⁺ S = 1. Indeed, our LSDA + U calculation gives a total maximal spin moment of 3 μ_B/fu for this FM insulating state. Here we note that the Ir⁴⁺ a_1g (3z² − r²) orbital is orthogonal to the Ni²⁺ xz/yz and xy/x² − y² orbitals and hence there is no hopping between them to account for the magnetism. Moreover, the Ir⁴⁺e_g empty bands are too high (see Fig. 2) to be relevant for the magnetic coupling. Therefore, with the crystal-field level diagrams depicted in Figs. 4(a) and 4(b), the electron correlation and inter-orbital hybridization give rise to the FM insulating solution.

Figure 3

The partial DOS of the Ir-5d a_1g and orbitals in Sr₃NiIrO₆ in (a) the FM state calculated by LSDA + U and in (b) the AF state by LSDA + U + SOC.

Taking the Ni²⁺ S = 1 as a reference, the Ir⁴⁺ ion has a down-spin a_1g empty state in (a) but an up-spin empty state (i.e., a complex orbital with l_z = 1) in (b).

Figure 4

Schematic level diagrams of the Ir⁴⁺ 5d and Ni²⁺ 3d orbitals.

The down-spin 3z² − r²electron mediates a FM coupling via a ddσ hybridization [(a) and (b)]. The up-spin xz/yz electrons mediate an AF coupling via a ddπ hybridization [(b) and (c)], in which the SOC splits the doublet.

Spin-orbit coupling

As 5d transition metals have an intrinsic strong SOC and particularly iridates are a representative example in this respect, now we are motivated to study the SOC effect by doing LSDA + U + SOC calculations. It is interesting to note that now we get an AF insulating solution with a small band gap of 0.15 eV, see Fig. 3(b). Particularly, this solution has the Ir⁴⁺ single t_2g hole on the orbital, in sharp contrast to the a_1g hole state in the above LSDA + U FM insulating solution. The Ni²⁺ retains its configuration state and has a spin moment of 1.69 μ_B. Owing to the small crystal-field splitting of 0.19 eV between (xy, x² − y²) and (xz,yz), a finite mixing between them due to the Ni SOC gives also a small orbital moment of 0.21 μ_B on Ni²⁺. The Ir⁴⁺ ion has now a spin moment of −0.44 μ_B. Moreover, the doublet can form a complex orbital with l_z = ±1. Then in the up-spin channel, the SOC lowers l_z = −1 state and the l_z = 1 state is pushed above the Fermi level by SOC and moderate electron correlation, determining the modest band gap and giving an orbital moment of −0.51 μ_B on Ir⁴⁺. Owing to the significant Ir-O covalency, both the spin and orbital moments are reduced from their respective ideal unit value. In the AF state of Ni²⁺ S = 1 and Ir⁴⁺ S = −1/2, the induced magnetic moment on oxygen gets tiny (only about 0.01 μ_B).

Discussion

The above results indicate an interesting evolution of the intrachain magnetic structure, from the LSDA + U FM state to the LSDA + U + SOC AF state. It is ascribed to the SOC tuning orbital state of the Ir⁴⁺ ion. Although the a_1g is a higher crystal-field level than the by 0.21eV, the significant SOC of the Ir⁴⁺ ion (being about 0.5 eV) can well split the doublet and eventually places the upper branch above the a_1g (Fig. 4(c)). As a result, the single t_2g hole of the Ir⁴⁺ ion lies in the state. Then, the up-spin xz/yz electrons of the Ni²⁺ ion can hop, forth and back, to the up-spin empty state (i.e., the up-spin l_z = 1 branch), giving rise to the AF coupling via the ddπ hybridization (Figs. 4(b) and 4(c)). Actually, using the orbital states depicted in Figs. 4(b) and 4(c), we also calculated the FM state. Our LSDA + U + SOC calculations find that the FM state is indeed less stable than the AF ground state by 67 meV/fu. Corresponding LSDA + U + SOC calculations, with Hund exchange parameter J = 0.9 eV for Ni 3d and J = 0.4 eV for Ir 5d, find that the AF ground state is more stable than the FM state by 63 meV/fu. In addition, the spin (orbital) magnetic moment changes within 0.1 (0.05) μ_B. Therefore, here the J parameter has an insignificant influence on the calculated results. As the SOC is intrinsic in iridates, the AF ground state is deemed reliable from the LSDA + U + SOC calculations, but the FM state seems fictitious from the LSDA + U calculations without inclusion of the SOC. Indeed, the AF ground state agrees with the most recent experiment³⁰. Therefore, we can conclude that it is the significant SOC of the Ir⁴⁺ ion which tunes the spin-orbital states and Ir-Ni inter-orbital interactions and hence determines the AF structure of Sr₃NiIrO₆.

Now we turn to Sr₃CoIrO₆. As this material has practically the same crystal structure as Sr₃NiIrO₆^28,30, both systems have many common features in the electronic and magnetic structures. The a_1g singlet of the Ir⁴⁺ ion is higher than the doublet in the crystal-field level diagram. Moreover, the high-spin Co²⁺ ion has the same crystal-field level sequence as Ni²⁺, but it now has one hole on the xy/x² − y² doublet (compared with Ni²⁺, see Fig. 4(b)). Our LSDA + U calculations give a FM metallic solution due to the 3z² − r² electron hopping and the 3/4 filled xy/x² − y² bands. Apparently, this solution contradicts the experimental AF insulating behavior³⁰.

However, when we include SOC by doing LSDA + U + SOC calculations, we have obtained the correct AF insulating solution (see Fig. 5) in good agreement with the experiment³⁰. The high-spin Co²⁺ ion (S = 3/2) has a spin moment of 2.66 μ_B. In its down-spin channel, the xy/x² − y² doublet form the complex orbitals (x² − y²) ± ixy with l_z = ±2 and the Co²⁺ SOC lowers the l_z = 2 state but lifts the l_z = −2 state. The electron correlation places the former at −1.5 eV and the latter at 2 eV. As a result, the Co²⁺ ion has also a huge orbital moment of 1.71 μ_B. In total, the Co²⁺ ion has the magnetic moment of 4.37 μ_B and it is firmly aligned, due to the SOC, along the hexagonal c-axis (i.e., a significant Ising-like spin system). Moreover, the Ir 5d states are almost the same as in Sr₃NiIrO₆: the Ir⁴⁺ SOC places the single t_2g hole on the doublet; the SOC and moderate electron correlation determine the small insulating gap within the Ir⁴⁺ t_2g shell, see Fig. 5. The Ir⁴⁺ ion has the spin (orbital) moment of −0.39 (−0.47) μ_B and in total −0.86 μ_B. Using these spin-orbital states, our LSDA + U + SOC calculations find that the AF ground state is more stable than the FM state by 122 meV/fu, which changes little to 130 meV/fu when including Hund exchange J = 0.9 eV for Co 3d and J = 0.4 eV for Ir 5d.

Figure 5

The partial DOS of the Ir 5d and Co 3d eigen orbitals in AF Sr₃CoIrO₆ calculated by LSDA + U + SOC.

The high-spin Co²⁺ ion (S = 3/2) has the orbitals 3z² − r² (red curve) and occupied (green curve) in its down-spin channel. The Ir⁴⁺ (S = −1/2) has the empty state (blue curve) in its up-spin channel.

By looking at Figs. 4(b) and 4(c) and now having also one hole on the down-spin xy/x² − y² doublet for Co²⁺, we find that the net spin = 1 from xz/yz and the net spin = 1/2 from xy/x² − y² both contribute to the AF exchange with the net Ir spin = −1/2 from the . The former is via a ddπ hybridization as in Sr₃NiIrO₆ and the latter a weaker ddδ one (which is missing in Sr₃NiIrO₆). This, together with the spin values (Co²⁺ S = 3/2 vs Ni²⁺ S = 1), qualitatively accounts for a higher stability of the AF ground state over the FM state in Sr₃CoIrO₆ than in Sr₃NiIrO₆, i.e., 122 vs 67 meV/fu. In addition, the ddδ hybridization results in the smaller spin and orbital moments of the Ir⁴⁺ ion in Sr₃CoIrO₆ (0.39 + 0.47 μ_B) than in Sr₃NiIrO₆ (0.44 + 0.51 μ_B). Note that all these results qualitatively explain the (slightly) higher intrachain AF transition temperature of 90 K in Sr₃CoIrO₆ than 85 K in Sr₃NiIrO₆³⁰. Moreover, the calculated magnetic moments of the significant Ising type, 4.37 μ_B/Co²⁺ and −0.86 μ_B/Ir⁴⁺, also agree reasonably well with the experimental ones of 3.6 and −0.6 μB in Sr₃CoIrO₆³⁰.

In summary, using density functional calculations including spin-orbit coupling (SOC) and electron correlation, we have demonstrated that the SOC of the Ir⁴⁺ ion plays an essential role in determining the antiferromagnetism of the hexagonal spin-chain system Sr₃MIrO₆ (M = Ni, Co) by tuning the crystal-field level sequence and altering the Ir-M inter-orbital interactions. The SOC splits the doublet of the octahedral Ir⁴⁺ ion () in a trigonal crystal field and the single t_2g hole resides on the upper branch and gives rise to the antiferromagnetic superexchange. In absence of the SOC, however, the single t_2g hole would occupy the a_1g singlet instead, which would mediate an unreal ferromagnetic exchange due to a direct a_1g hopping along the Ir-M chain. We also find that the Ni²⁺ and Co²⁺ ions are both in a high-spin state and moreover the Co²⁺ ion carries a huge orbital moment. This work well accounts for the most recent experiments and magnifies again the significance of the SOC in iridates.

Methods

We have carried out density functional calculations, using the full-potential augmented plane wave plus local orbital code (Wien2k)³⁴. We use the structural data of Sr₃NiIrO₆ measured by neutron diffraction at 10 K²⁸ and of Sr₃CoIrO₆ at 4 K³⁰. They have practically the same crystal structure: the Ir-O bondlength of 2.01 Å, the M-O 2.18 Å and the M-Ir 2.78 Å for M = Ni and Co; the small deviation of the O-Ir-O bond angle (from the ideal 90°) due to a small elongated trigonal distortion of the IrO₆ octahedron, being 5.4° for M = Ni and 5.2° for M = Co. The muffin-tin sphere radii are chosen to be 2.8, 2.2, 2.2 and 1.5 Bohr for Sr, Ni/Co and Ir and O atoms, respectively. The plane-wave cut-off energy of 16 Ry is set for the interstitial wave functions and 5 × 5 × 5 k mesh for integration over the rhombohedral Brillouin zone. Using 7 × 7 × 7 k mesh (more than a doubling) gives practically the same results, with the total energy converging within 2 meV/fu. We employ the local spin density approximation³⁵ plus Hubbard U (LSDA + U) method³⁶ to describe the electron correlation of the M 3d and Ir 5d electrons, with the LSDA + U double counting correction made in a fully atomic limit³⁶. The typical values, effective U = 2, 4 and 5 eV (U_eff = U − J with Hund exchange parameter J being set at zero) are used for the Ir 5d, Co 3d and Ni 3d states, respectively. Note that our key results –the coupled spin-orbital state and the magnetic ground state –are independent of the tested U values (1–3 eV for Ir 5d and 3–7 eV for Co/Ni 3d). To account for (near) degeneracy of the Ir 5d orbitals (and of M 3d orbitals as well), the SOC is included by the second-variational method with scalar relativistic wave functions. In order to probe diverse possible spin-orbital states and magnetic structures, we excess them in our calculations by setting their respective occupation number matrix and thus orbitally dependent potentials and then do self-consistent calculations including a full electronic relaxation. (Otherwise, some states of the concern or even the ground state cannot be achieved.) An advantage of this procedure is such that we can reliably determine the magnetic ground state by a direct comparison of the different states^19,21.