Crystal electric field excitations and spin dynamics in a spin-orbit coupled distorted honeycomb magnet BiErGeO$_5$

This study investigates the magnetic properties and crystal electric field scheme of the spin-orbit coupled distorted honeycomb magnet BiErGeO$_5$, revealing short-range antiferromagnetic correlations, long-range order at 0.4 K, and persistent slow spin fluctuations below the transition temperature through a combination of magnetization, heat capacity, muon spin relaxation, and inelastic neutron scattering experiments.

The Gist

Imagine a bustling city made entirely of tiny, spinning magnets. In most cities, these magnets just spin randomly or line up neatly in rows. But in the material studied in this paper, BiErGeO₅, the magnets are trapped in a very specific, tricky neighborhood that forces them to behave in a strange, quantum way.

Here is the story of that neighborhood, explained simply.

1. The Neighborhood: A Wobbly Honeycomb

The "city" in this material is built on a honeycomb lattice (like a beehive), but it's not a perfect hexagon. It's distorted, like a honeycomb that got squished.

• The Residents: The main characters are Erbium (Er) ions. Think of them as tiny, heavy-duty tops (spinning magnets) that are also deeply connected to the "fabric" of the material (spin-orbit coupling).
• The Trap: These tops are stuck in a cage made of oxygen atoms. This cage isn't symmetrical; it's lopsided. This lopsidedness is called the Crystal Electric Field (CEF).

2. The Energy Levels: A Staircase with Weird Steps

Because the cage is lopsided, the Erbium tops can't spin in just any direction. They are forced to choose specific "energy levels," like standing on specific steps of a staircase.

• The Ground Floor: At very low temperatures, the tops settle on the bottom step (the ground state).
• The Excitations: If you give them a little energy (like a gentle nudge), they can jump to higher steps. The scientists used a giant "flashlight" (neutrons) to see these jumps. They found eight distinct steps (energy levels) the tops could jump to.
• The Metaphor: Imagine a piano. Usually, you can play any note. But in this material, the piano only has 8 specific keys that work, and the distance between the keys is uneven. The scientists mapped out exactly where these keys are.

3. The Temperature Drama: What Happens When it Gets Cold?

The researchers cooled this material down to near absolute zero (colder than outer space!) to see what the magnets did.

• The "Short-Range" Flirtation (1.4 K): As it cooled, the magnets started to notice each other. They began to align slightly, but only with their immediate neighbors. It was like a crowd of people whispering to the person right next to them, but not organizing the whole room. This created a "broad hump" in the data.
• The "Long-Range" Order (0.4 K): At 0.4 Kelvin, the whole city finally agreed on a direction. They formed a long-range order (a solid magnetic state). Usually, when magnets line up, they freeze solid and stop moving.
• The Twist: Here is the surprise. Even though they "ordered" (lined up), they didn't freeze.

4. The Ghost in the Machine: Muon Spin Relaxation

To check if the magnets were truly frozen, the scientists used Muons (tiny, short-lived particles) as spies. They shot these spies into the material and watched how they spun.

• The Expectation: If the magnets were frozen solid, the muon spies would see a static, unchanging magnetic field and spin in a predictable, rhythmic pattern (like a clock ticking).
• The Reality: The muons saw no rhythm. Instead, they saw a constant, slow "jitter."
• The Analogy: Imagine a crowd of people who have all agreed to stand in a line (ordered). In a normal frozen state, they would stand perfectly still. But in this material, even though they are in a line, they are constantly shuffling their feet, wiggling, and swapping places very slowly. The "line" exists, but the people inside it are still dancing.

5. Why is this Special?

The scientists compared this material to a cousin made with Ytterbium (Yb) instead of Erbium.

• The Ytterbion Cousin: In the Yb version, the magnets were so frustrated they couldn't decide on a line at all. They stayed in a chaotic, disordered state (a "quantum spin liquid").
• The Erbium Hero: By swapping Ytterbium for Erbium, the material finally decided to form a line (order). BUT, because of the unique shape of the Erbium ion and its "lopsided cage," it kept its quantum jitter even while ordered.

The Big Takeaway

This paper tells us that order doesn't always mean stillness.
In the world of quantum magnets, you can have a material where the magnets are perfectly lined up (ordered), yet they are still constantly fluctuating and dancing (dynamic). This happens because the "cage" holding the magnets is so weirdly shaped that it forces them to keep moving, even at the coldest temperatures imaginable.

It's like a traffic jam where every car is stuck in a line, but the drivers are all still revving their engines and shifting gears, creating a unique, humming energy that defies the usual rules of physics. This discovery helps scientists understand how to build new materials for future quantum computers, where controlling these "wiggly" magnets is key.

Technical Summary

Here is a detailed technical summary of the paper "Crystal electric field excitations and spin dynamics in a spin-orbit coupled distorted honeycomb magnet BiErGeO5".

1. Problem Statement and Motivation

Rare-earth magnets with strong spin-orbit coupling (SOC) and crystal electric field (CEF) effects are a primary platform for exploring unconventional quantum phenomena, such as quantum spin liquids (QSL) and anisotropic magnetic ordering. While Yb-based honeycomb magnets (e.g., BiYbGeO5) have been extensively studied and often exhibit disordered ground states, the behavior of Er-based analogues remains less understood.

The specific problem addressed is determining how substituting the Yb$^{3+}$ ion with the Er$^{3+}$ ion in the distorted honeycomb lattice of BiYbGeO5 alters the magnetic ground state. Er$^{3+}$ ions possess a $J=15/2$ ground state and a different CEF environment compared to Yb$^{3+}$ ($J=7/2$), potentially leading to distinct anisotropic exchange interactions and magnetic ordering. The study aims to characterize the CEF scheme, magnetic interactions, and low-temperature spin dynamics of BiErGeO5 to understand the interplay between CEF-driven anisotropy, reduced dimensionality, and magnetic ordering.

2. Methodology

The researchers employed a multi-faceted experimental approach on polycrystalline samples of BiErGeO5 and its non-magnetic analogue, BiYGeO5:

• Synthesis & Structural Characterization: Samples were synthesized via solid-state reaction. Phase purity and crystal structure were confirmed using Powder X-ray Diffraction (XRD) with Rietveld refinement, identifying an orthorhombic $Pbca$ space group with a quasi-2D distorted honeycomb lattice of Er$^{3+}$ ions.
• Thermodynamic Measurements:
  • Magnetization ($M$): Measured using a SQUID magnetometer down to 0.4 K under various magnetic fields.
  • Heat Capacity ($C_p$): Measured down to 100 mK using relaxation techniques. The magnetic contribution ($C_{mag}$) was isolated by subtracting the lattice contribution derived from the non-magnetic BiYGeO5 analogue.
• Inelastic Neutron Scattering (INS): Performed at the ISIS Neutron and Muon Source (MARI spectrometer) using zero-field conditions. Data were collected at 5, 50, and 100 K with incident neutron energies of 10, 25, 80, and 120 meV. Phonon contributions were subtracted using data from BiYGeO5 to isolate CEF excitations.
• Muon Spin Relaxation ($\mu$SR): Conducted at the Swiss Muon Source (S$\mu$S) under zero-field (ZF) and longitudinal-field (LF) conditions down to 30 mK. This technique probed local magnetic fields and spin dynamics on a timescale of $10^{-11}$to$10^{-5}$ s.
• Theoretical Modeling:
  • CEF Analysis: The INS spectra were fitted using a Stevens operator Hamiltonian ($H_{CEF}$) to extract crystal field parameters and energy levels.
  • Thermodynamic Simulation: The extracted CEF parameters were used to calculate theoretical magnetic susceptibility, magnetization, and heat capacity.
  • Exchange Modeling: A spin-1/2 XXZ model was used to simulate low-temperature thermodynamic data to estimate exchange coupling constants ($J_{xy}$ and $J_z$).

3. Key Results

A. Crystal Structure and CEF Scheme

• Structure: BiErGeO5 features a quasi-2D distorted honeycomb lattice of Er$^{3+}$ ions in the $ac$-plane, separated by non-magnetic Bi$^{3+}$ and Ge$^{4+}$ ions along the $b$-axis. The Er-O bonds are anisotropic ($\sim$3.46 Å and $\sim$3.59 Å).
• CEF Excitations: INS spectra revealed eight CEF doublets (Kramers doublets) for the Er$^{3+}$ ion. The ground state is a Kramers doublet.
• CEF Parameters: The fitted CEF Hamiltonian yielded an anisotropic g-factor ratio of $g_{xy}/g_z \approx 1.38$, indicating an easy-plane anisotropy (where the magnetic moment prefers the plane perpendicular to the crystallographic $b$-axis).

B. Magnetic Ordering and Thermodynamics

• Long-Range Order (LRO): Heat capacity data shows a sharp $\lambda$-like anomaly at $T_N \approx 0.4$ K, signaling the onset of antiferromagnetic (AFM) long-range order.
• Short-Range Correlations: A broad maximum in magnetic susceptibility and heat capacity appears around $T_{SR} \approx 1.4$ K, indicating short-range AFM correlations preceding the LRO.
• Exchange Anisotropy: Analysis of the Curie-Weiss law and XXZ modeling yielded exchange constants of $J_{xy} \approx 2.96$ K and $J_z \approx 1.56$ K, confirming significant exchange anisotropy ($J_{xy} > J_z$) consistent with the CEF ground state.
• Field Dependence: The magnetic LRO is suppressed by fields above 0.5 T. The heat capacity shows complex field-dependent features, including a new broad maximum at low fields attributed to Zeeman splitting of the ground state doublet.

C. Spin Dynamics ($\mu$SR)

• Absence of Static Order Signatures: Despite the thermodynamic evidence for LRO at 0.4 K, zero-field $\mu$SR spectra do not show coherent oscillations or the characteristic static 1/3 tail expected for a conventional static magnetic order down to 30 mK.
• Relaxation Behavior: The relaxation rates ($\lambda$) display a plateau below $T_N$ and follow an Orbach-type activated behavior at higher temperatures (involving excited CEF levels).
• Slow Fluctuations: Longitudinal-field measurements up to 1.5 T show only weak decoupling of the muon spin relaxation. This indicates that the relaxation is driven by slow, persistent spin fluctuations rather than static internal fields, even deep within the ordered phase.

4. Key Contributions

1. Identification of CEF Scheme: The study provides a complete CEF level scheme for BiErGeO5, identifying eight doublets and quantifying the strong anisotropy ($g_{xy} > g_z$) inherent to the distorted honeycomb lattice.
2. Anisotropic Exchange Quantification: It successfully extracts anisotropic exchange parameters ($J_{xy} \approx 2.96$ K, $J_z \approx 1.56$ K) that explain the thermodynamic behavior and the easy-plane nature of the magnet.
3. Unconventional Ordered State: The paper demonstrates that BiErGeO5 enters a magnetic long-range ordered state that coexists with slow spin fluctuations. This "dynamic" ordered state is distinct from conventional static antiferromagnets and is evidenced by the lack of static signatures in $\mu$SR despite clear thermodynamic ordering.
4. Rare-Earth Substitution Effect: By comparing with BiYbGeO5 (which has a disordered ground state), the study highlights that the specific choice of rare-earth ion (Er vs. Yb) critically determines the ground state, driven by differences in CEF wavefunctions and resulting anisotropic exchange.

5. Significance

This work establishes BiErGeO5 as a prototypical anisotropic spin-orbit coupled honeycomb magnet. It reveals that strong CEF-driven anisotropy and reduced dimensionality can lead to unconventional low-temperature magnetic behavior where long-range order and quantum fluctuations coexist. This challenges the simple dichotomy between ordered and disordered states in frustrated magnets. The findings are crucial for the broader search for Kitaev materials and quantum spin liquids, suggesting that even in systems with apparent long-range order, significant dynamic correlations may persist, potentially hosting exotic excitations or fragmented magnetic moments. The study underscores the necessity of analyzing CEF schemes to predict and understand the magnetic ground states of rare-earth based quantum magnets.