Synchrotron x-ray diffraction and DFT study of non-centrosymmetric EuRhGe3 under high pressure

This study combines synchrotron X-ray diffraction and DFT calculations to investigate the high-pressure structural behavior of non-centrosymmetric EuRhGe3, revealing a smooth volume contraction up to 35 GPa without phase transitions, anisotropic lattice compression, and a deviation between experimental and theoretical volumes at higher pressures attributed to non-integer Eu valence.

The Gist

Imagine a tiny, intricate crystal made of Europium, Rhodium, and Germanium. Think of this crystal as a microscopic, three-dimensional scaffolding or a Lego structure. The scientists in this paper wanted to see what happens to this structure when you squeeze it incredibly hard, like putting it in a giant, high-tech vice.

Here is the story of their experiment, broken down simply:

The Setup: A High-Stakes Squeeze

The researchers took a crystal called EuRhGe3. This isn't just any crystal; it has a special, "lopsided" shape (scientists call this non-centrosymmetric), which gives it interesting magnetic properties.

To test it, they didn't use a regular vice. They used a Diamond Anvil Cell. Imagine two tiny diamonds (the hardest material on Earth) pressing against each other. The crystal is crushed between them, surrounded by helium gas to keep the pressure even, like a tiny, high-pressure submarine. They squeezed it until the pressure was 35,000 times the atmospheric pressure we feel at sea level.

The Main Discovery: A Smooth Squeeze, Not a Snap

Usually, when you squeeze things too hard, they snap, break, or suddenly change their shape (a "phase transition"). Think of it like a sponge that suddenly turns into a rock.

However, this crystal was surprisingly resilient.

• No Breaking: Even under that massive pressure, the crystal didn't break or change its fundamental shape. It kept its original "Lego pattern" all the way up to the limit.
• Getting Smaller: Instead of snapping, it just got smaller and smaller, like a stress ball being squeezed. The whole unit shrank smoothly.

The Twist: One Side Shrinks Faster

Here is where it gets interesting. The crystal isn't a perfect cube; it's a bit like a tall, thin box.

• When squeezed, the width (the a-axis) shrank much faster than the height (the c-axis).
• Imagine a tall, skinny soda can. If you squeeze it, the sides might buckle inward quickly, but the top and bottom stay relatively rigid for a while. That's what happened here. The crystal got "squatter" as the pressure increased.

The Mystery of the "Valence" (The Invisible Weight)

There is a hidden character in this story: the Europium atom.

• At normal pressure, Europium acts like it has a "charge" of about +2 (let's call it Eu2+).
• As the pressure increased, the scientists noticed the Europium atoms started acting a bit more like they had a charge of +3 (Eu3+).
• Why does this matter? An atom with a +3 charge is physically smaller than one with a +2 charge (about 10% smaller).

The scientists used a super-computer (DFT calculations) to predict how the crystal should shrink.

• Below 13 GPa: The computer prediction matched the real experiment perfectly. The crystal shrank exactly as the math said it would.
• Above 13 GPa: The real crystal started shrinking faster than the computer predicted.
• The Explanation: The computer assumed the Europium atoms stayed the same size (like Eu2+). But in reality, the atoms were getting smaller (turning into Eu3+). Because the atoms themselves were shrinking, the whole crystal got smaller than the computer thought it would. It's like if you were predicting how much a suitcase would shrink if you packed it tighter, but you forgot that the clothes inside were also shrinking!

The "Goldilocks" Comparison

The paper compares this crystal to its cousins, EuCoGe3 and EuNiGe3.

• These cousins behave very similarly: they also get squished without breaking, and their Europium atoms slowly change their "charge" without ever fully turning into the smaller version.
• This is different from other similar crystals (called Eu122 systems) which often snap into a completely new shape and change their charge drastically at lower pressures. Our crystal is the "Goldilocks" of the group—it changes slowly and smoothly, never making a sudden jump.

The Bottom Line

The scientists squeezed a magnetic crystal to extreme limits and found that:

1. It is incredibly tough and doesn't change its shape or break, even under 35 GPa of pressure.
2. It squishes unevenly (width shrinks faster than height).
3. The reason it gets smaller than computer models predict at high pressure is that the Europium atoms inside are slowly changing their internal size, a subtle shift that the computer models didn't fully account for.

In short, this crystal is a master of adaptation, shrinking gracefully under pressure without ever losing its identity.

Technical Summary

Technical Summary: Synchrotron X-ray Diffraction and DFT Study of Non-Centrosymmetric EuRhGe3 under High Pressure

Problem and Context
Rare-earth intermetallic compounds with non-centrosymmetric BaNiSn3-type structures (EuTX3) exhibit rich magnetic properties and pressure-induced phenomena, including superconductivity. While pressure-induced valence transitions and structural collapses are well-documented in the isostructural Eu122-systems (ThCr2Si2-type), where Eu2+ transitions to Eu3+ accompanied by a significant volume collapse, the behavior of the EuTGe3 series remains less understood. Specifically, it is unclear whether the gradual pressure-induced increase in Eu valence observed in EuRhGe3 (from ~2.1 to ~2.4) correlates with specific structural changes or if the crystal lattice remains stable. Previous studies on isostructural compounds EuCoGe3 and EuNiGe3 showed stable antiferromagnetic ordering and continuous volume contraction without symmetry changes, but a systematic structural study of EuRhGe3 to elucidate the relationship between lattice evolution and Eu valence fluctuations was lacking.

Methodology
The study employed a combined experimental and theoretical approach:

• Experimental: High-pressure synchrotron powder X-ray diffraction (XRD) was performed on EuRhGe3 at room temperature up to 35 GPa using the PSICHE beamline at SOLEIL. Samples were prepared by grinding single crystals into powder and loaded into a membrane diamond anvil cell (DAC) with a helium pressure-transmitting medium and gold as a pressure reference. Diffraction data were analyzed using Rietveld refinement (Profex software) to extract lattice parameters and unit cell volumes.
• Theoretical: Density Functional Theory (DFT) calculations were conducted using the Quantum ESPRESSO package. The study utilized the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional and included a Hubbard U correction (DFT+U) with U = 3.8 eV to account for antiferromagnetic ordering on Eu ions. Calculations were performed assuming a static Eu2+ valence state across the pressure range to compare against experimental data.
• Analysis: The experimental and theoretical unit cell volumes were fitted to the third-order Birch-Murnaghan equation of state (EOS) to determine the bulk modulus ($B_0$) and its pressure derivative ($B'_0$).

Key Results

1. Structural Stability: EuRhGe3 maintains its non-centrosymmetric tetragonal symmetry ($I4mm$) throughout the entire pressure range up to 35 GPa. No structural phase transitions were observed.
2. Anisotropic Compression: The unit cell exhibits anisotropic compression. The lattice parameter $a$ contracts more significantly than $c$ under pressure. This behavior is consistent with isostructural compounds EuCoGe3 and EuNiGe3.
3. Axial Ratio Evolution: The axial ratio ($c/a$) shows a change in slope around 13 GPa, dividing the pressure evolution into low-pressure (LP, <13 GPa) and high-pressure (HP, >13 GPa) regions. The rate of increase for $c/a$ decreases from 0.003 GPa$^{-1}$ in the LP region to 0.001 GPa$^{-1}$ in the HP region. The authors note that unlike in Eu122-systems, this slope change does not coincide with a discontinuous structural transition or a first-order valence transition.
4. Equation of State: The bulk modulus ($B_0$) and its pressure derivative ($B'_0$) were determined to be 73 (1) GPa and 5.5 (2), respectively. These values are comparable to those of EuCoGe3 (75.6 GPa) and EuNiGe3 (79 GPa).
5. DFT vs. Experiment:
   • Low Pressure (<13 GPa): Experimental lattice parameters and volumes show good agreement with DFT calculations.
   • High Pressure (>13 GPa): A deviation emerges where experimental unit cell volumes are smaller than DFT-predicted values. The authors attribute this discrepancy to the pressure-induced increase in the mean Eu valence (approaching 2.4), which reduces the ionic radius. Since the DFT calculations assumed a fixed Eu2+ valence, they could not capture this volume reduction associated with valence fluctuation.

Significance and Claims
The paper establishes that EuRhGe3 undergoes a smooth, continuous contraction of its unit cell volume under high pressure without undergoing a structural phase transition, despite the continuous evolution of the Eu valence state. The study highlights that the change in the compressibility of the $c$-axis (reflected in the $c/a$ slope change) occurs independently of the valence transition dynamics observed in Eu122-systems.

The primary significance lies in clarifying the structural response of the BaNiSn3-type EuTGe3 series to pressure. The authors demonstrate that while the Eu valence increases under pressure, it does not trigger a catastrophic volume collapse or symmetry breaking as seen in ThCr2Si2-type compounds. Furthermore, the divergence between experimental and DFT results at high pressures serves as an indicator of the limitations of static valence models in DFT when applied to systems with significant valence fluctuations, reinforcing the need to consider valence changes when modeling the high-pressure behavior of these intermetallics.