Covalency and vibronic couplings make a nonmagnetic j=3/2 ion magnetic

Abstract

For 4d¹ and 5d¹ spin–orbit-coupled electron configurations, the notion of nonmagnetic j=3/2 quartet ground state discussed in classical textbooks is at odds with the observed variety of magnetic properties. Here we throw fresh light on the electronic structure of 4d¹ and 5d¹ ions in molybdenum- and osmium-based double-perovskite systems and reveal different kinds of on-site many-body physics in the two families of compounds: although the sizable magnetic moments and g-factors measured experimentally are due to both metal d–ligand p hybridisation and dynamic Jahn–Teller interactions for 4d electrons, it is essentially d−p covalency for the 5d¹ configuration. These results highlight the subtle interplay of spin–orbit interactions, covalency and electron–lattice couplings as the major factor in deciding the nature of the magnetic ground states of 4d and 5d quantum materials. Cation charge imbalance in the double-perovskite structure is further shown to allow a fine tuning of the gap between the t_2g and e_g levels, an effect of much potential in the context of orbital engineering in oxide electronics.

Introduction

A defining feature of d-electron systems is the presence of sizable electron correlations, also referred to as Mott–Hubbard physics. The latter has been traditionally associated with first-series (3d) transition-metal (TM) oxides. But recently one more ingredient entered the TM oxide ‘Mottness’ paradigm—large spin–orbit couplings (SOCs) in 4d and 5d quantum materials.^1,2 It turns out that for specific t_2g-shell electron configurations, a strong SOC can effectively augment the effect of Hubbard correlations:¹ although the 4d and 5d orbitals are relatively extended objects and the Coulomb repulsive interactions are weakened as compared with the more compact 3d states, the spin–orbit-induced level splittings can become large enough to break apart the ‘nonrelativistic’ t_2g bands into sets of well-separated, significantly narrower subbands for which even a modest Hubbard U acting on the respective Wannier orbitals can then open up a finite Mott–Hubbard-like gap.¹ On top of that, SOC additionally reshuffles the intersite superexchange.³ The surprisingly large anisotropic magnetic interactions that come into play via the strong SOCs in iridates,^3–8 e.g., are responsible for an exotic assortment of novel magnetic ground states and excitations.^2,3,9

For large t_2g−e_g splittings, the spin–orbit-coupled $t_{2g}^1$ and $t_{2g}^5$ electron configurations can be in first approximation viewed as ‘complementary’: in the simplest picture, the d-shell manifold can be shrunk to the set of j=1/2 and j=3/2 relativistic levels, with a j=3/2 ground state for the TM $t_{2g}^1$ configuration and a j=1/2 ground state for $t_{2g}^5$.^10–12 Although strongly spin–orbit-coupled $t_{2g}^5$ oxides and halides—iridates, rhodates and ruthenates, in particular—have generated substantial experimental and theoretical investigations in recent years, much of the properties of 5d and 4d $t_{2g}^1$ systems remain to large extent unexplored.

From textbook arguments,^10–12 the $t_{2g}^1$ j=3/2 quadruplet should be characterised by a vanishing magnetic moment in cubic symmetry, due to perfect cancellation of the spin and angular momentum contributions. But this assertion leaves unexplained the wide variety of magnetic properties recently found in 4d¹ and 5d¹ cubic oxide compounds.^13–20 Ba₂YMoO₆, e.g., develops no magnetic order despite a Curie–Weiss temperature of ≈−200 K (refs 13,14) and features complex magnetic dynamics that persists down to the mK range, possibly due to either a valence-bond-glass¹⁵ or spin-liquid¹⁶ ground state. Also, Ba₂NaOsO₆ displays an antiferromagnetic Curie–Weiss temperature^18,19 but orders ferromagnetically below 7 K (ref. 20), whereas Ba₂LiOsO₆ is a spin-flop antiferromagnet.²⁰

Here we carry out a detailed ab initio investigation of the Mo⁵⁺ 4d¹ and Os⁷⁺ 5d¹ relativistic electronic structure in the double-perovskite compounds Ba₂YMoO₆, Ba₂LiOsO₆ and Ba₂NaOsO₆. In addition to providing reliable results for the energy scale of the d-level splittings, t_2g−e_g and induced by SOC within the $t_{2g}^1$ manifold, we analyse the role of TM d−O p orbital mixing plus the strength of electron–lattice couplings. It is found that strong metal d−O p hybridisation generates a finite magnetic moment even for perfectly cubic environment around the TM site, providing ab initio support to the phenomenological covalency factor introduced in this context by Stevens.²¹ The TM d¹ magnetic moment is further enhanced by tetragonal distortions, against which the octahedral oxygen cage is unstable. According to our results, such Jahn–Teller (JT) effects are particularly strong for the Mo 4d¹ ions in Ba₂YMoO₆. Although additional investigations are needed for clarifying the role of intersite cooperative couplings,^22,23 our calculations emphasise the high sensitivity of the effective magnetic moments to both metal–ligand covalency effects and local JT physics. The material dependence for the ratio among the strengths of the spin–orbit interaction, the JT coupling parameter and the effective covalency factor that we compute here provide a solid basis for future studies addressing the role of intersite interactions on the double-perovskite fcc lattice.

Results

Quantum chemistry calculations were first performed to resolve the essential features of the electronic structure of the cubic lattice configuration, without accounting for electron–lattice couplings (see Materials and methods for computational details and Figure 1 for a sketch of a three-dimensional double-perovskite crystal). Results for the splitting between the Mo⁵⁺ $t_{2g}^1$ j=3/2 and j=1/2 spin–orbit states, Δ_3/2→1/2, are provided in Table 1 at two levels of approximation, i.e., multiconfiguration complete-active-space self-consistent field (CASSCF) and multireference configuration interaction (MRCI) with single and double excitations on top of the CASSCF wave function.²⁴ Knowing the splitting Δ_3/2→1/2, the strength of the SOC parameter can be easily derived as $\lambda =\frac{2}{3}{\Delta }_{\mathrm{3/2}\to \mathrm{1/2}}^{\mathrm{CASSCF}}$.¹¹ The resulting λ of 89 meV is somewhat smaller than earlier estimates of 99 meV for Mo⁵⁺ impurities in SrTiO₃.²⁵ A most interesting finding, however, is that despite the cubic environment the quantum chemistry calculations yield a nonvanishing magnetic moment and a finite g-factor. This obviously does not fit the nonmagnetic j=3/2 quartet ground state assumed to arise in standard textbooks on crystal-field theory^11,12 from exact cancellation between the spin and the orbital moments.

Figure 1

Sketch of the atomic positions in a cubic double-perovskite compound, Ba₂BB′O₆. B stands here for Y, Na or Li (site at the centre of a dark-blue octahedron); B′ is either Mo or Os (site at the centre of a light-green octahedron). O ions are shown as small red spheres, whereas the Ba sites are the larger blue spheres.

Table 1 Mo⁵⁺ 4d-shell splittings (j=3/2 to j=1/2 and t_2g−e_g) and ‘static’ g_|| factor in cubic Ba₂YMoO₆

At a qualitative level, it has been argued by Stevens²¹ that finite g-factor values can in fact occur for j=3/2 ions due to TM-O covalency on the TM O₆ octahedron. For better insight into the nature of such effects, we therefore performed a simple numerical experiment in which the six ligands coordinating the reference Mo⁵⁺ 4d¹ ion are replaced by −2 point charges with no atomic basis functions. In that additional set of computations the magnetic moment and the g-factor do vanish, in agreement with the purely ionic picture of Kotani, Abragam and Bleaney.^10,11 This shows that one tuning knob for switching magnetism on is indeed the TM 4d−O 2p orbital hybridisation. The latter is strong for high ionisation states such as Mo⁵⁺ (as the tails of the 4d-like valence orbitals indicate in the case the nearest-neighbor ligands are provided with atomic basis sets, see Figure 2a), gives rise to partial quenching of the orbital moment and makes that the exact cancellation between the spin and the orbital moments no longer holds.

Figure 2

(a) Mo 4d t_2g charge distribution as obtained by CASSCF calculations. The tails at the nearest-neighbor O sites have substantial weight. (b) TM t_2g splittings in cubic (left) and tetragonal (right) symmetry; δ₁=0 and δ₂=Δ_3/2→1/2 for cubic octahedra. (c) Ground-state energy as function of z axis tetragonal distortion, MRCI results both with and without spin–orbit coupling. (d) Variation of the Mo 4d¹ magnetic moment (μ_||) and g-factor (g_||) with the amount of z axis tetragonal distortion, MRCI results including spin–orbit interactions.

We find that this effect is even stronger for the formally 7+ Os ion in Ba₂LiOsO₆ and Ba₂NaOsO₆. As shown in Table 2, g-factors as large as 0.4 are computed in this case. The quantum chemistry results also allow us to estimate the strength of the effective Os⁷⁺ 5d¹ SOC constant, with $\lambda =\frac{2}{3}{\Delta }_{\mathrm{3/2}\to \mathrm{1/2}}^{\mathrm{CASSCF}}\mathrm{=387}\mathrm{meV}$, lower than λ=468 meV in tetravalent 5d⁵ iridates.²⁶

Table 2 Os⁷⁺ 5d-shell splittings (j=3/2 to j=1/2 and t_2g−e_g) and ‘static’ g_|| factors in cubic Ba₂LiOsO₆ and Ba₂NaOsO₆

As the $t_{2g}^1$ electron configuration is susceptible to JT effects, we carried out further investigations on the stability of an ideal TM O₆ octahedron against tetragonal (z axis) distortions. A total-energy profile for specified geometric configurations is provided in Figure 2c for an embedded MoO₆ octahedron. It is seen that the minimum corresponds to ~3% tetragonal compression, as compared with the cubic octahedron of the Fm3m crystalline structure.¹⁴ As expected, the magnetic moment rapidly increases in the presence of distortions, as illustrated in Table 3 and Figure 2d.

Table 3 Mo⁵⁺ $t_{2g}^1$ electronic structure with ‘static’ tetragonal squeezing of the reference MoO₆ octahedron

Depending on further details related to the strength of the intersite couplings among ‘JT centres’, static deformations away from cubic symmetry may be realised in some systems, as observed for example in the Re⁶⁺ 5d¹ double-perovskite Sr₂MgReO₆ (ref. 27) and rare-earth molibdates.^28,29 If the local JT couplings and intersite interactions are relatively weak, one may be left on the other hand in a dynamic JT regime, as earlier pointed out for the particular $t_{2g}^1$ configuration by, e.g., Kahn and Kettle.³⁰ The relevant vibrational modes that couple to the ²T_2g$\left(t_{2g}^1\right)$ electronic term are those of E_g symmetry ((3z²−r²)- and (x²−y²)-like). From the quantum chemistry calculations, we find that the potential-energy well is significantly shallower for these normal coordinates, as compared with z axis-only compression. The value we computed for the Mo⁵⁺ ion in Ba₂YMoO₆, ≈40 meV, is comparable to the estimate made in the 1970s for Mo⁵⁺ t_2g¹ impurity ions within the SrTiO₃ matrix, ≈60 meV.²⁵ For the osmates, the depth of this potential well is much reduced, with E_JT values in the range of 10–15 meV by spin–orbit MRCI calculations (Table 4).

Table 4 TM g_|| factors using the Kahn–Kettle vibronic model³⁰ and ab initio estimates for λ, k_cov and E_JT

The vibronic model of Kahn and Kettle³⁰ provides specific expressions for the g-factors. In particular, g_|| can be parametrised as³⁰

$$\begin{array}{}\text{(1)}& g_{\parallel }\mathrm{=2}\left(1-k_{\mathrm{cov}}{k}_{\mathrm{vib}}\right),\end{array}$$

where k_cov is Stevens’ covalency factor²¹ and

$$\begin{array}{}\text{(2)}& k_{\mathrm{vib}}=\exp \left[-x\mathrm{/(1}+\rho )\right].\end{array}$$

The parameters x and ρ are defined as³⁰ $x=3E_{\mathrm{JT}}\mathrm{/2}\overline{h}\omega$ and $\rho =3\lambda \mathrm{/2}\overline{h}\omega$, where $\overline{h}\omega$ is the E_g-mode vibrational energy and g_⊥=0 by symmetry.^11,21,30 Recent infrared transmission spectra indicate that $\overline{h}\omega \approx 560$ cm⁻¹≈70 meV for the bond-stretching phonons.^17,31 The effective parameter k_cov we can easily evaluate from the static g_|| values obtained in the MRCI spin–orbit treatment (Tables 1 and 2) if vibronic interactions are neglected (k_vib=1 for ‘frozen’ cubic octahedra), with $k_{\mathrm{cov}}\equiv 1-g_{\parallel }^{\mathrm{MRCI}}\mathrm{/2}$. This yields covalency reduction factors of 0.90 for Ba₂YMoO₆ and 0.80 for the osmates.

Estimates for g_|| are provided in Table 4, using the Kahn–Kettle vibronic model and the quantum chemistry results for λ, k_cov and E_JT. It is seen that a large ρ/x ratio (i.e., large λ/E_JT) makes that g_|| is generated mostly through covalency effects in the osmates, with minor contributions from vibronic couplings. On the other hand, the small ρ/x ratio in Ba₂YMoO₆ gives rise to a strong enhancement of g_|| through vibronic effects, with a factor of nearly 4 between (1−k_covk_vib) and (1−k_cov). This way, the interesting situation arises that the TM magnetic moment is mainly due to vibronic effects in Ba₂YMoO₆ and predominantly to strong covalency in Ba₂LiOsO₆ and Ba₂NaOsO₆.

Discussion

Experimentally, the measured magnetic moments are indeed significantly smaller in Ba₂LiOsO₆ and Ba₂NaOsO₆ (refs 19,20) as compared with Ba₂YMoO₆ (refs 13,14,17,25). With regard to the estimates we make here for g_||, possible sources of errors concern the accuracy of the calculated E_JT when using the experimental crystal structure as reference and correlation and polarisation effects beyond a single TM O₆ octahedron.^32,33 The latter effects would only increase E_JT. With respect to the former aspect, it is known that by advanced quantum chemistry calculations, the lattice constants of TM oxides can be computed with deviations of <0.5% from the measured values,³² which implies rather small corrections to E_JT. Interestingly, recent findings of additional phonon modes at low temperatures¹⁷ indicate static distortions of the MoO₆ octahedra in Ba₂YMoO₆ and indeed a rather large E_JT. More detailed investigations on this matter are left for future work. Valuable experimental data that can be directly compared with our calculations would be the results of electron spin resonance measurements of the g-factors.

It is also worth pointing out that using the Kahn–Kettle model even a E_JT of 75 meV, five to seven times larger than the values computed by MRCI for the osmates (Table 4), still yields a rather moderate g_|| factor of 0.65 for the Os 5d¹ ion. Such g_|| factors of 0.4–0.6 compare quite well with the low-temperature magnetic moment derived from magnetisation and muon spin relaxation measurements on Ba₂NaOsO₆, ≈0.2 μ_B.^19,20 For the Mo 4d¹ ion in Ba₂YMoO₆, the computed g_|| factor is much more sensitive to variations of E_JT—increasing E_JT from, e.g., 40–200 meV enhances g_|| of Equation (1) from ≈0.6 to ≈1.6.

One other remarkable prediction of Kahn and Kettle³⁰ is that the splitting of the j=3/2 and j=1/2 states is increased through vibronic couplings, by a factor

$$\begin{array}{}\text{(3)}& \gamma =1+x\frac{3+\rho }{\mathrm{3(}{\rho }^2-\mathrm{1)}}.\end{array}$$

This effect turns out to be small in the osmates, given the small x and large ρ in those compounds. But we compute a strong modification of the j=3/2 to j=1/2 excitation energy for Ba₂YMoO₆, from ~0.13 eV in the absence of vibronic interactions (Table 1) to ≈0.20 eV with JT effects included (E_JT=40 meV). Experimentally, the situation can be clarified by direct resonant inelastic X-ray scattering (RIXS) measurements on Ba₂YMoO₆. High-resolution RIXS measurements could also address the occurrence of static distortions at low temperatures, suggested for Ba₂YMoO₆ on the basis of extra phonon modes in the low-T infrared transmission spectra¹⁷ and for Ba₂NaOsO₆ from the integrated entropy through the magnetic phase transition at ~7 K.¹⁹ According to the MRCI data in Table 3, a reduction by 0.5–1.5% of the interatomic distances on one set of O–Mo–O links already gives a splitting of 20–50 meV of the low-lying spin–orbit states. Splittings of this size should be accessible with last-generation RIXS apparatus.

Also of interest is an experimental confirmation of the unusually large t_2g−e_g gap we predict in the double-perovskite heptavalent osmates, ≳6 eV (Table 2). According to the results of additional computations we carried out, the source of this exceptional d-level splitting is the stabilisation of the Os t_2g states due to the large effective charge (formally 7+) at the nearest-neighbor Os sites. The latter are situated on the axes along which the lobes of the t_2g orbitals are oriented; in contrast, the lobes of the e_g functions point towards the monovalent species (Li¹⁺ or Na¹⁺). For example, test CASSCF calculations in which the size of the point charges placed at the 12 Os and 6 alkaline-ion nearest-neighbor sites are modified from the formal ionic values 7+ and 1+ (12×7+6×1=90) to 5+ and 5+ (12×5+6×5=90) show a reduction of about 2 eV of the t_2g−e_g level splitting. Similar effects, with relative shifts and even inversion of the d-electron energy levels due to charge imbalance at nearby cation sites, were recently evidenced in Sr₂RhO₄ and Sr₂IrO₄,^8,26 the rare-earth 227 iridates R₂Ir₂O₇ (ref. 34) and Cd₂Os₂O₇.³⁵ The mechanism has not been thoroughly explored so far experimentally but seems to hold much potential in the context of orbital engineering in TM compounds.

To summarise, it is well known that nominal orbital degeneracy gives rise in 3d TM oxides to subtle couplings between the electronic and lattice degrees of freedom and very rich physics. Here we resolve the effect of electron–lattice interactions on the magnetic properties of heavier, 4d and 5d TM ions with a formally degenerate t_2g¹ electron configuration in the double-perovskite materials Ba₂YMoO₆, Ba₂LiOsO₆ and Ba₂NaOsO₆. In particular, using advanced quantum chemistry electronic-structure calculations, we reconcile the notion of a nonmagnetic spin–orbit-coupled t_2g¹ j=3/2 ground state put forward by Kotani, Abragam, Bleaney and others^10–12 with the variety of magnetic properties recently observed in 4d¹ and 5d¹ double perovskites. Our analysis shows that the sizable magnetic moments and g-factors found experimentally are due to strong TM d–ligand p hybridisation and dynamic JT effects, providing new perspectives on the interplay between metal–ligand interactions and SOCs in TM oxides. It also highlights the proper theoretical frame for addressing the remarkably rich magnetic properties of d¹ double perovskites^{2,15,16,19,20} in particular. Over the past two decades, vibronic couplings have unjustifiably received low attention in the case of these intriguing materials.

Materials and methods

All ab initio calculations were carried out with the quantum chemistry package Molpro.³⁶ Crystallographic data as derived in ref. 14 for Ba₂YMoO₆ and in ref. 18 for Ba₂LiOsO₆ and Ba₂NaOsO₆ were employed.

We used effective core potentials, valence basis functions of triple-zeta quality and two f polarisation functions for the reference Mo/Os ions^37,38 for which the d-shell excitations are explicitly computed. All-electron triple-zeta basis sets supplemented with two d polarisation functions³⁹ were applied for each of the six adjacent O ligands. The eight Ba nearest neighbours were in each case modelled by Ba²⁺ ‘total-ion’ pseudopotentials supplemented with a single s function.⁴⁰ For Ba₂YMoO₆, the six nearby Y sites were described by effective core potentials and valence basis functions of double-zeta quality.³⁷ In Ba₂LiOsO₆ and Ba₂NaOsO₆, we employed total-ion pseudopotentials for the six nearest Li and Na cations and sets of one s and one p functions.⁴¹ The farther solid-state surroundings enter the quantum chemistry calculations at the level of a Madelung ionic potential. How the complexity and accuracy of quantum chemistry calculations for an infinite solid can be systematically increased is addressed in, e.g., refs 32,33,42,43.

For the CASSCF calculations of the d-shell splittings, we used active spaces of either three (t_2g) or five (t_2g plus e_g) orbitals. The CASSCF optimisations were carried out for an average of either the ²T_2g ($t_{2g}^1$) or ²T_2g($t_{2g}^1$)+²E_2g($e_g^1$) eigenfunctions of the scalar relativistic Hamiltonian. All O 2p and Mo/Os 4d/5d electrons on the reference TM O₆ octahedron were correlated in the MRCI treatment. The latter was performed with single and double substitutions with respect to the CASSCF reference, as described in refs 44,45. The spin–orbit treatment was carried out according to the procedure described in ref. 46.

The g-factors were computed following a scheme proposed by Bolvin⁴⁷ and Vancoillie.⁴⁸ For a Kramers-doublet ground state $\{\mathit{\psi },\overline{\mathit{\psi }}\}$, the Abragam–Bleaney tensor¹¹ G=gg^T can be expressed in matrix form as

$$\begin{array}{}\text{(4)}& \begin{array}{ccc}\hfill G_{kl}& \hfill =\hfill & 2\sum _{u,v=\psi ,\overline{\psi }}〈u|{\hat{L}}_k+g_e{\hat{S}}_k|v〉〈v|{\hat{L}}_l+g_e{\hat{S}}_l|u〉\hfill \\ \hfill & \hfill =\hfill & \sum _{m=x,y,z}\left({\Lambda }_{km}+g_e{\sum }_{km}\right)\left({\Lambda }_{lm}+g_e{\sum }_{lm}\right),\hfill \end{array}\end{array}$$

where g_e is the free-electron g-factor and

$$\begin{array}{}\text{(5)}& \begin{array}{ccc}\hfill {\Lambda }_{kx}& \hfill =\hfill & 2Re\left[〈\overline{\psi }\left|{\hat{L}}_k\right|\psi 〉\right],{\sum }_{kx}\mathrm{=2}Re\left[〈\overline{\psi }\left|{\hat{S}}_k\right|\psi 〉\right],\hfill \\ \hfill {\Lambda }_{ky}& \hfill =\hfill & 2Im\left[〈\overline{\psi }\left|{\hat{L}}_k\right|\psi 〉\right],{\sum }_{ky}\mathrm{=2}Im\left[〈\overline{\psi }\left|{\hat{S}}_k\right|\psi 〉\right],\hfill \\ \hfill {\Lambda }_{kz}& \hfill =\hfill & 2\left[〈\psi \left|{\hat{L}}_k\right|\psi 〉\right],{\sum }_{kz}\mathrm{=2}\left[〈\psi \left|{\hat{S}}_k\right|\psi 〉\right].\hfill \end{array}\end{array}$$

The matrix elements of $\hat{L}$ were extracted from the Molpro outputs, whereas the matrix elements of $\hat{S}$ were derived using the conventional expressions for the generalised Pauli matrices:

$$\begin{array}{}\text{(6)}& \begin{array}{ccc}\hfill {\left({\hat{S}}_z\right)}_{MM\prime }& \hfill =\hfill & M{\delta }_{MM\prime },\hfill \\ \hfill {\left({\hat{S}}_x\right)}_{MM\prime }& \hfill =\hfill & \frac{1}{2}\sqrt{(S+M)(S-M+\mathrm{1)}}{\delta }_{M-\mathrm{1,}M\prime }\hfill \\ \hfill & \hfill \hfill & +\frac{1}{2}\sqrt{(S-M)(S+M+\mathrm{1)}}{\delta }_{M+\mathrm{1,}M\prime },\hfill \\ \hfill {\left({\hat{S}}_y\right)}_{MM\prime }& \hfill =\hfill & -\frac{i}{2}\sqrt{(S+M)(S-M+\mathrm{1)}}{\delta }_{M-\mathrm{1,}M\prime }\hfill \\ \hfill & \hfill \hfill & +\frac{i}{2}\sqrt{(S-M)(S+M+\mathrm{1)}}{\delta }_{M+\mathrm{1,}M\prime }.\hfill \end{array}\end{array}$$

The g-factors were calculated as the positive square roots of the three eigenvalues of G.