NMR study on equilateral triangular lattice antiferromagnet Ba2La2CoTe2O12

This study utilizes $^{139}$La-NMR to characterize the magnetic phase diagram of the $S=1/2$ equilateral triangular-lattice antiferromagnet Ba$_2$La$_2$CoTe$_2$O$_{12}$, revealing a zero-field 120$^\circ$ ordered state below 3.26 K and a field-induced sequence of transitions involving up-up-down and triangular coplanar phases above 3 T.

The Gist

Imagine a tiny, magical dance floor made of triangles. On this floor, there are dancers (atoms) who are constantly spinning. In most dance floors, the dancers can easily agree on who spins which way. But on this specific floor, the geometry is tricky: it's an equilateral triangle, and the dancers are "frustrated." They want to spin in opposite directions to their neighbors, but on a triangle, you can't have everyone be opposite to everyone else at the same time. It's like a three-way tug-of-war where no one can win.

This scientific paper is about a specific material, Ba₂La₂CoTe₂O₁₂, which acts like this frustrated dance floor. The researchers used a special technique called NMR (Nuclear Magnetic Resonance) to listen in on the "heartbeat" of these dancers to see how they behave when it gets very cold or when they are pushed by a magnetic field.

Here is the story of what they found, broken down simply:

1. The Setup: The Triangular Dance Floor

The material is a "triangular antiferromagnet."

• The Dancers: The Cobalt (Co) ions are the main dancers. They have a spin of 1/2, which is like a tiny magnet pointing up or down.
• The Audience: The Lanthanum (La) ions are like the audience sitting in the middle of the triangle, watching the dancers. The researchers used the La ions as microphones to "hear" what the Co dancers were doing.
• The Goal: They wanted to see how the dancers arrange themselves when the temperature drops or when a strong magnetic field (a "wind" pushing them) is applied.

2. The First Act: The 120° Spin Structure (No Wind)

When the material is cold but has no magnetic wind blowing on it, the dancers settle into a calm, organized pattern.

• Imagine three dancers on a triangle. To be fair, they don't point straight up or down. Instead, they point 120 degrees apart from each other, like the hands of a clock at 12, 4, and 8.
• This is called the 120° structure. It's a perfect, balanced dance where everyone is happy with the compromise.
• The Discovery: At 3.26 Kelvin (which is incredibly cold, just a few degrees above absolute zero), the dancers suddenly lock into this formation. The researchers saw this happen because the "heartbeat" (relaxation rate) of the audience (La) went crazy, signaling a big change.

3. The Second Act: The "Up-Up-Down" Formation (A Gentle Wind)

Now, imagine turning on a magnetic fan (applying a magnetic field).

• When the wind gets strong (above 3 Tesla), the dancers can't keep the 120° balance anymore.
• They rearrange into a new pattern called Up-Up-Down (uud). Two dancers point with the wind, and one points against it.
• This is a very famous state in physics because it creates a "magnetization plateau." It's like the dancers hit a speed bump where the total spin of the group stays stuck at exactly one-third of the maximum possible, no matter how much you push the wind.
• The researchers confirmed this by seeing the "heartbeat" of the audience change exactly when the specific heat measurements (another way to measure energy) showed a peak.

4. The Third Act: The Mystery Shift (Stronger Wind)

Here is where the paper gets really interesting.

• The researchers thought the "Up-Up-Down" dance would last forever as they got colder.
• The Surprise: At a specific field strength (5.4 Tesla), they noticed something weird happening to the "noise" (linewidth) of the audience's signal. As they got colder, the signal didn't just get louder; it suddenly got quieter (narrower).
• The Explanation: Why would the signal get quieter if the dancers are spinning harder?
  • Think of the audience member (La) sitting in the middle. In the "Up-Up-Down" phase, the three dancers around them are pushing the audience in different directions, creating a messy, strong net force.
  • But at this specific point, the dancers switch to a Triangular Coplanar phase. In this new dance, they all lie flat on the floor (coplanar) and arrange themselves so that their pushes cancel each other out perfectly in the direction the audience is listening.
  • It's like three people pushing a car: if they push from three different angles that cancel out, the car doesn't move. The "noise" of the push disappears.
  • This explains why the signal got narrower: the net magnetic force on the audience dropped because the dancers found a new, more efficient way to cancel each other out.

5. The Takeaway

This paper is a detective story about how atoms rearrange themselves in a frustrated triangle.

• Without wind: They dance in a 120° circle.
• With medium wind: They do the "Up-Up-Down" dance (the famous 1/3 plateau).
• With strong wind and very cold temps: They switch to a "flat, cancelling" dance (Triangular Coplanar).

The researchers used the "heartbeat" of the Lanthanum atoms to prove that the material doesn't just stay in one state; it goes through a series of transformations, changing its dance routine as the conditions change. This helps scientists understand how quantum mechanics works in complex, crowded systems, which could one day help us build better computers or new types of magnetic materials.

Technical Summary

1. Problem Statement and Scientific Context

The paper addresses the complex magnetic behavior of Ba$_2$La$_2$CoTe$_2$O$_{12}$, an $S=1/2$ quasi-two-dimensional antiferromagnet on an equilateral triangular lattice. This system is of significant interest due to:

• Geometrical Frustration: The triangular arrangement of spins often leads to exotic ground states.
• 1/3 Magnetization Plateau: Previous bulk measurements (specific heat, magnetization) indicated the existence of a 1/3-magnetization plateau between 8.7 and 15.2 T at 1.3 K, associated with a collinear up-up-down (uud) spin structure.
• Successive Phase Transitions: Under magnetic fields above 3 T, the system exhibits two distinct phase transitions ($T_{N1}$ and $T_{N2}$). While $T_{N1}$ marks the transition from the paramagnetic (PM) phase to the ordered state, the nature of the transition at $T_{N2}$ (from the uud phase to a triangular coplanar phase) required microscopic verification.
• Gap in Knowledge: While bulk properties were known, the microscopic evolution of the spin structure and the local magnetic environment across these transitions remained unclear.

2. Methodology

The authors employed $^{139}$La Nuclear Magnetic Resonance (NMR) as a microscopic probe to investigate the local magnetic environment.

• Sample: Powder sample of Ba$_2$La$_2$CoTe$_2$O$_{12}$.
• Conditions: Measurements were conducted at low temperatures (down to 1.8 K) under magnetic fields ranging from 4 to 9 T.
• Probe Site: The La$^{3+}$ ion is located 1.2 Å above the center of the equilateral triangle formed by three Co$^{2+}$ ions. This geometry allows the NMR signal to probe the local magnetization averaged over the triangle via hyperfine and dipole-dipole interactions.
• Key Measurements:
  1. NMR Spectra: To determine the linewidth and Knight shift.
  2. Spin-Lattice Relaxation Rate ($1/T_1$): Measured using the saturation-recovery method to probe spin dynamics and critical fluctuations.
  3. Data Analysis: Recovery curves were fitted using a stretched exponential function to account for sample inhomogeneity.

3. Key Contributions and Results

A. Confirmation of Magnetic Ordering ($T_{N1}$)

• Critical Divergence: The relaxation rate $1/T_1$ exhibited a critical divergence at $T_{N1}$ (approx. 3.26 K in zero field, shifting with field). This confirms the onset of long-range magnetic order and characterizes the transition from the paramagnetic phase to the ordered state as a second-order phase transition.
• Agreement with Bulk Data: The observed transition temperatures and fields align with previous specific heat measurements, validating the NMR data.

B. Distinct Behavior of Linewidth at $T_{N2}$

The most significant finding concerns the behavior of the NMR linewidth under different magnetic fields:

• At 7.5 T: The linewidth increased below $T_{N1}$ and remained constant down to 1.8 K. This indicates the system remains in the uud phase throughout this temperature range.
• At 5.4 T: The linewidth increased below $T_{N1}$ but exhibited an anomalous decrease below $T_{N2}$ (approx. 3.1 K).
  • Interpretation: Since neutron diffraction shows the ordered moment magnitude increases monotonically with cooling, the linewidth reduction cannot be due to a loss of magnetic order. Instead, it is attributed to a change in spin structure.
  • Mechanism: The reduction is caused by the geometrical cancellation of ordered moments within the triangle when the system transitions from the collinear uud phase to the triangular coplanar phase. In the coplanar phase, the net magnetic field at the La site (surrounded by three Co spins) is smaller than in the uud phase.

C. Absence of Critical Slowing Down at $T_{N2}$

• Unlike $T_{N1}$, the relaxation rate $1/T_1$ showed no critical divergence at $T_{N2}$.
• Significance: This confirms that the transition at $T_{N2}$ occurs between two statically ordered phases (uud $\to$ coplanar) rather than involving critical fluctuations associated with the onset of order.

D. Phase Diagram Refinement

The study successfully mapped the field-temperature phase diagram, confirming the sequence of phases:

1. Paramagnetic Phase ($T > T_{N1}$)
2. uud Phase ($T_{N2} < T < T_{N1}$)
3. Triangular Coplanar Phase ($T < T_{N2}$ at lower fields)

4. Significance and Conclusion

• Microscopic Validation: This work provides the first microscopic evidence (via NMR) confirming the transition from the uud phase to the triangular coplanar phase in Ba$_2$La$_2$CoTe$_2$O$_{12}$, a transition previously only hypothesized based on bulk thermodynamic data.
• Mechanism of Plateau Formation: The results clarify the stability regions of the 1/3-magnetization plateau, showing that the plateau (uud phase) is stable at higher fields (e.g., 7.5 T) but gives way to a coplanar state at lower fields (e.g., 5.4 T) upon cooling.
• Methodological Insight: The study demonstrates the utility of NMR linewidth analysis in distinguishing between changes in magnetic order magnitude and changes in spin geometry (cancellation effects) in frustrated systems.

Future Work: The authors note that further measurements in a wider field range and at lower temperatures are necessary to fully quantify the linewidth reduction and to investigate spin excitation gaps or magnon properties via $1/T_1$ behavior at very low temperatures.