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Quadrupole formation and coupling to magnetic and structural degrees of freedom in the $5d^1$ double perovskites Ba $_2$ MgReO $_6$ and Ba $_2$ NaOsO $_6$

This study utilizes first-principles calculations to reveal that while both Ba $_2$ MgReO $_6$ and Ba $_2$ NaOsO $_6$ exhibit spontaneous quadrupolar order and spin-orbit-mediated magnetic canting, only the Re-compound's strong Jahn-Teller coupling successfully stabilizes the observed antiferroic structural order, leaving the magnetic ground state of the Os-compound partially unexplained.

Original authors: Francesco Martinelli, Claude Ederer

Published 2026-02-27

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CC BY 4.0

Original authors: Francesco Martinelli, Claude Ederer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have two identical-looking Lego castles, built from the same number of bricks, arranged in the exact same pattern. They are so similar that if you looked at them from far away, you'd think they were twins. These are the two materials in this study: Ba₂MgReO₆ and Ba₂NaOsO₆.

Both are made of heavy metals (Rhenium and Osmium) that have a very specific, tricky electronic personality. They are "5d¹" materials, which is a fancy way of saying they have one lonely electron hanging out in a specific orbital room. Because these atoms are so heavy, they have a superpower called Spin-Orbit Coupling. Think of this as a very strong magnetic glue that ties the electron's spin (its tiny internal compass) to its orbital shape (the shape of the room it lives in).

The scientists wanted to figure out why these two "twins" behave so differently when they get cold.

The Mystery: The "Wobbly" Electron

In these materials, that single electron isn't just sitting still. It wants to arrange itself in a specific pattern to save energy. This pattern is called a quadrupole.

The Analogy: Imagine a balloon. If you squeeze it from the top and bottom, it becomes an egg shape. If you squeeze it from the sides, it becomes a different egg shape. The electron is like that balloon, trying to find the perfect "squeeze" (shape) to be most comfortable.
The Goal: The researchers wanted to see if the electrons in these two castles would spontaneously decide to squeeze themselves into a specific shape (order) on their own, even before the atoms themselves move.

The Discovery: Twins with Different Personalities

The scientists used a powerful computer simulation (like a virtual microscope) to watch what happens.

1. The "Willing" Twin (Ba₂MgReO₆):
In the first castle, the electrons are very eager to organize. Even when the atoms are perfectly still in a cube shape, the electrons start to line up their "balloon squeezes" in a specific pattern.

The Result: When the material gets cold, the atoms actually move to match the electrons. The oxygen atoms around the metal shift slightly (a Jahn-Teller distortion). It's like the whole castle rearranging its furniture to match the new shape of the electron balloons.
The Outcome: This creates a perfect, stable structure that matches what experimentalists see in the real world. The electrons and the atoms are in a happy, synchronized dance.

2. The "Reluctant" Twin (Ba₂NaOsO₆):
In the second castle, the electrons are a bit more lazy or indecisive. They want to organize, but the urge is much weaker.

The Result: When the scientists tried to force the atoms to move to match the electrons, the atoms refused to budge much. The "Jahn-Teller distortion" (the furniture rearrangement) didn't happen strongly.
The Mystery: Here is the plot twist. Experiments show that in this second castle, the magnetic compasses (the spins) do tilt weirdly (a phenomenon called canting). But the scientists' computer said, "Wait, if the atoms don't move to support the electron pattern, the spins shouldn't tilt like that."
The Conflict: The computer says the electrons are too weak to force the atoms to move, yet the real-world experiments show the spins are definitely tilted. The scientists are left scratching their heads, wondering what invisible force is making the spins tilt in this second material.

The Connection: The Magnetic Dance

The paper also discovered a fascinating link between the shape of the electron balloon and the direction of the magnetic compass.

The Analogy: Imagine a dancer (the electron) holding a partner (the magnetic spin).
The Mechanism: If the dancer changes their pose (the electron quadrupole changes shape), they physically pull the partner's arm, forcing the partner to turn.
The Finding: The scientists showed that the "tilting" of the magnetic spins (the canting) is directly caused by the electrons trying to arrange themselves in that alternating pattern. In the first castle, the atoms move to help this happen. In the second castle, the atoms don't move enough, so the computer can't explain why the spins are tilting so much.

The Bottom Line

This paper is a story about two chemical twins.

Twin A is a team player: The electrons organize, the atoms move to help, and everything works perfectly. The computer model matches the real world.
Twin B is a bit of a mystery: The electrons are less organized, the atoms don't move, yet the magnetic spins still do something strange.

The scientists have successfully explained the "perfect" behavior of the first twin but have hit a wall with the second. They have shown us how the electron shapes and magnetic spins are linked, but for the second material, there is still a missing piece of the puzzle that future research needs to find.

In short: They figured out why one heavy-metal crystal behaves exactly as predicted, but the other one is still keeping a secret about why its magnetic spins are tilting.

1. Problem Statement

The paper investigates the complex interplay between charge, magnetic, and structural degrees of freedom in two isostructural and isoelectronic 5d $^1$ double perovskites: Ba $_2$ MgReO $_6$ and Ba $_2$ NaOsO $_6$ . Both materials contain magnetic cations (Re $^{6+}$ and Os $^{7+}$ ) with a nominal $5d^1$ configuration, resulting in a four-fold degenerate $J_{eff}=3/2$ ground state due to strong spin-orbit coupling (SOC) and octahedral crystal fields.

Despite their chemical similarity, these materials exhibit distinct low-temperature behaviors:

Ba $_2$ MgReO $_6$ : Undergoes two phase transitions. At $T_q = 33$ K, it exhibits a symmetry lowering from cubic $Fm\bar{3}m$ to tetragonal $P4_2/mnm$ , attributed to antiferroic (AF) ordering of $x^2-y^2$ -type quadrupoles coupled with Jahn-Teller (JT) distortions. At $T_m = 18$ K, it enters a non-collinear magnetic state with canted moments.
Ba $_2$ NaOsO $_6$ : Shows a similar canted magnetic order below 6.8 K but lacks any detectable long-range structural distortion or global symmetry breaking above $T_m$ . However, local symmetry breaking (indicated by NMR splitting) persists, suggesting a dynamical JT scenario.

The central problem is to explain why these two nearly identical systems behave differently regarding the stabilization of long-range quadrupolar order and structural distortions, and to elucidate the microscopic mechanisms coupling quadrupoles, magnetic dipoles, and lattice distortions.

2. Methodology

The authors employed first-principles electronic structure calculations based on Density Functional Theory (DFT) with the following key features:

Framework: DFT+U within the Projector Augmented Wave (PAW) framework using the VASP code. The PBE functional was used for exchange-correlation, with an effective Hubbard parameter $U_{eff} = 1.8$ eV (chosen for consistency with previous literature, though linear response calculations suggested $\sim 1.3$ eV).
Spin-Orbit Coupling (SOC): Included in all calculations, as it is critical for $5d$ systems.
Constraint Methods:
- Quadrupole Constraints: Used Lagrange multipliers to constrain specific components of the local charge quadrupole moments ( $Q_{xy}, Q_{x^2-y^2}$ , etc.) to map the energy landscape as a function of quadrupole magnitude.
- Magnetic Constraints: Used penalty terms to constrain the direction of local magnetic moments to simulate canting angles ( $\theta$ ) and paramagnetic states (using random moment orientations in supercells).
Structural Relaxation: Performed full structural relaxations starting from experimental low-temperature structures to determine the stability of JT distortions (characterized by Van Vleck modes $Q_1-Q_6$ ).
Paramagnetic Simulation: Modeled the paramagnetic phase using $2\times2\times2$ supercells with random magnetic moment orientations to average out dipolar effects and isolate quadrupolar tendencies.

3. Key Contributions and Results

A. Spontaneous Quadrupolar Order in the Cubic Phase

Even in the absence of magnetic order and structural distortion (perfect cubic $Fm\bar{3}m$ ), both materials exhibit a tendency toward spontaneous quadrupolar order.

Ferroic vs. Antiferroic: The ferroic (FQ) $xy$-type order is energetically favored over the antiferroic (AFQ) $x^2-y^2$ order in the cubic phase for both materials.
Material Difference: The energy gain for FQ-$xy$ order is slightly larger in Ba $_2$ MgReO $_6$ ( $\sim 3$ meV/f.u.) than in Ba $_2$ NaOsO $_6$ ( $\sim 2$ meV/f.u.). Ba $_2$ NaOsO $_6$ shows no energy minimum for AFQ- $x^2-y^2$ order in the cubic phase, whereas Ba $_2$ MgReO $_6$ shows a very shallow minimum.
Conclusion: The intrinsic electronic tendency favors ferroic order; the experimentally observed antiferroic order in Ba $_2$ MgReO $_6$ requires coupling to structural distortions.

B. Coupling Between Charge Quadrupoles and Magnetic Dipoles

The study reveals an intimate coupling mediated by strong SOC:

Canting Mechanism: Imposing an AFQ- $x^2-y^2$ order (alternating rotation of charge anisotropy) forces the local magnetic moments to cant away from the global magnetization direction ( $[110]$ ).
Orbital vs. Spin: The local orbital moments follow the rotation of the quadrupole almost rigidly. The spin moments follow the orbital moments but with a smaller canting angle due to competition between SOC (aligning spin and orbital) and inter-site exchange (favoring collinear alignment).
Reciprocity: Conversely, constraining the spin moments to a canted angle induces a corresponding AFQ- $x^2-y^2$ component. This confirms that the observed magnetic canting is intrinsically linked to the emergence of antiferroic quadrupolar order.

C. Role of Structural Jahn-Teller Distortions

This is the critical differentiator between the two materials:

Ba $_2$ MgReO $_6$ : When structural relaxation is allowed, the coupling between the electronic quadrupoles and the lattice stabilizes the AFQ- $x^2-y^2$ order and a corresponding AF-Q2 Jahn-Teller distortion. This results in excellent quantitative agreement with experimental data regarding the distortion amplitude, canting angles ( $\sim 15^\circ$ calculated vs. $\sim 40^\circ$ estimated experimentally), and net magnetization.
Ba $_2$ NaOsO $_6$ : The coupling to JT modes is significantly weaker. Structural relaxation fails to stabilize the AFQ- $x^2-y^2$ $x^{2} - y^{2}$ order or the AF-Q2 distortion. Instead, the system relaxes to a state with a small ferroic trigonal distortion and collinear ferromagnetic order.
- Note: Even with a higher $U_{eff}$ (3.4 eV) and constrained spins, the AF-Q2 distortion in Ba $_2$ NaOsO $_6$ is unstable or very weak compared to the Re compound.

4. Significance and Remaining Discrepancies

Significance:

The work successfully elucidates the mechanism stabilizing the hidden order in Ba $_2$ MgReO $_6$ , confirming that the interplay between AF quadrupoles and JT distortions is essential.
It demonstrates that the "canted" magnetic state is a direct consequence of the antiferroic quadrupolar order, driven by SOC.
It provides a systematic comparison showing that subtle differences in electron-lattice coupling strength can dictate whether a material exhibits long-range structural symmetry breaking or only local/dynamical effects.

Discrepancy in Ba $_2$ NaOsO $_6$ :

The calculations predict that without a stable AFQ- $x^2-y^2$ order, there should be no canting of magnetic moments.
However, experiments (NMR) indicate a strong canting ( $\theta \approx 67^\circ$ ) in Ba $_2$ NaOsO $_6$ despite the absence of long-range structural distortion.
Conclusion: The authors conclude that while their model explains the Re-compound perfectly, the origin of the magnetic canting in the Os-compound remains elusive. It suggests that either the canting arises from a mechanism not captured by the current static DFT+U approach (e.g., dynamic fluctuations, different correlation effects, or a different type of multipolar order) or that the experimental interpretation of the NMR data requires re-evaluation in the context of the calculated weak JT coupling.

In summary, the paper establishes that Ba $_2$ MgReO $_6$ is a textbook example of coupled quadrupolar, magnetic, and structural order stabilized by strong JT coupling, while Ba $_2$ NaOsO $_6$ represents a more complex case where weak JT coupling prevents long-range structural symmetry breaking, leaving the origin of its magnetic canting an open question for future research.