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Thermal conductivity of CdCr $_{2}$ Se $_{4}$ ferromagnet at low temperatures: role of grain boundaries and porosity

This study experimentally confirms the T $^{3/2}$ specific heat and T $^{2}$ thermal conductivity scaling of magnons in the ferromagnetic insulator CdCr $_{2}$ Se $_{4}$ , while revealing that grain boundary scattering limits magnon transport more severely than phonon transport, resulting in a phonon-dominated thermal conductivity with an anomalous T $^{2.3}$ temperature dependence.

Original authors: Jiří Hejtmánek, Kyo-Hoon Ahn, Zdeněk Jirák, Petr Levinský, Jiří Navrátil, Sandy Al Bacha, Emmanuel Guilmeau, Karel Knížek

Published 2026-03-03

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CC BY 4.0

Original authors: Jiří Hejtmánek, Kyo-Hoon Ahn, Zdeněk Jirák, Petr Levinský, Jiří Navrátil, Sandy Al Bacha, Emmanuel Guilmeau, Karel Knížek

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

The Big Picture: A Heat Highway in a Magnetic Crystal

Imagine you have a tiny, magical city made of atoms. This city is called CdCr₂Se₄. It's a special kind of rock that acts like a magnet (a ferromagnet) but doesn't conduct electricity like a metal.

Scientists want to understand how heat travels through this city. Heat moves in two main ways here:

The "Vibrations" (Phonons): Imagine the atoms in the city are like people dancing. When they dance faster, they pass energy to their neighbors. These dancing waves are called phonons.
The "Spin Waves" (Magnons): Because this city is magnetic, the atoms also have tiny internal compasses (spins) that point in the same direction. When these compasses wobble together in a wave, they carry heat too. These are called magnons.

The goal of this research was to figure out exactly how much heat is carried by the dancing atoms versus the wobbly compasses, and what happens when the city isn't perfectly built (like having potholes or broken walls).

The Two Versions of the City: Porous vs. Dense

The researchers built two versions of this atomic city to see how the structure affects heat travel:

The "Sponge" City (Porous Sample): This version was made with lots of tiny air pockets (17% empty space). It's like a sponge. The "streets" between the buildings (grains) are full of gaps.
The "Skyscraper" City (Dense Sample): This version was crushed together under immense pressure (using a technique called Spark Plasma Sintering). It's almost solid, with very few air pockets (only 5% empty space).

The Result: The dense city conducted heat four times better than the sponge city. This makes sense: if you remove the potholes and gaps, the heat can flow much faster.

The Mystery of the "Short-Sighted" Compasses

Here is where things get really interesting and unexpected.

Usually, scientists think that magnons (the magnetic compass waves) are like ghosts. They can easily slip through cracks and walls because they are so small and fast. Phonons (the atomic vibrations) are like heavy trucks; they get stuck easily if there are cracks in the road.

However, this paper found the opposite!

The Magnons (Compasses): In this material, the magnetic waves got stuck almost immediately. They couldn't even cross from one "building" (grain) to the next. It's as if the compasses are short-sighted and refuse to walk through the "neighborhood boundaries" (grain boundaries). They only travel a tiny distance (about 0.25 micrometers) before hitting a wall and bouncing back.
The Phonons (Trucks): The heat-carrying vibrations were much better at navigating the city. They could travel all the way across the buildings (about 1.4 micrometers) before getting stuck.

Why?
The paper suggests that the "roads" between the buildings aren't just empty space; they are messy and slightly misaligned. Because the magnetic forces in this material are extremely sensitive to the exact distance between atoms, even a tiny bump in the road confuses the magnetic compasses and stops them dead. The atomic vibrations (phonons) are a bit more robust and can handle the bumps better.

The "Magic Trick" of the Magnetic Field

The researchers used a giant magnet (13 Tesla, which is incredibly strong) to test their theory.

The Trick: When they turned on the strong magnet, it forced all the magnetic compasses to line up perfectly and stop wiggling.
The Effect: Since the compasses stopped wiggling, they stopped carrying heat. The researchers could then measure exactly how much heat was left being carried only by the dancing atoms (phonons).
The Discovery: They confirmed that at very low temperatures (near absolute zero), the magnetic waves (magnons) were responsible for a huge chunk of the heat (about 87% of the specific heat capacity!). But because they couldn't cross the neighborhood boundaries, they weren't very good at transporting heat across the whole sample.

The Temperature Dance

The paper also looked at how heat moves at different temperatures:

At Super Cold Temps (Near 0 K): The heat flow follows a specific mathematical rhythm (like a song). The magnetic part sings a song that goes up as $T^2$ , and the atomic part usually goes up as $T^3$ .
The Twist: The atomic part in this material didn't sing the perfect $T^3$ song. It sang a slightly off-key $T^{2.3}$ song.
The Reason: The "potholes" and rough edges of the city (grain boundaries) were scattering the heat waves in a weird way. It's like trying to run through a crowd where people are moving randomly; you don't just run in a straight line, you zigzag, slowing you down more than expected.

The Takeaway

This paper teaches us that in certain magnetic materials, structure is everything.

Even though the material is a solid block, the tiny boundaries between the microscopic crystals act like massive walls for magnetic heat carriers, but only like small speed bumps for atomic heat carriers. It turns the usual rules of physics upside down: the "ghosts" (magnons) are the ones getting stuck, while the "trucks" (phonons) are the ones doing the heavy lifting of moving heat through the material.

This discovery is crucial for designing better materials for electronics and thermoelectric devices, where controlling how heat moves is the key to efficiency.

1. Problem Statement

The low-temperature thermal properties of ferromagnetic insulators are theoretically well-understood (involving contributions from phonons and magnons), but experimentally difficult to isolate. In most materials, magnon contributions are masked by:

Itinerant electron excitations in metallic systems.
Nuclear specific heat (Schottky anomalies) in rare-earth systems.
Extrinsic factors like impurities, glassy phases, or sample morphology.

The authors sought to resolve this by studying CdCr2Se4, a ferromagnetic insulator ( $T_C \approx 130$ K) that lacks itinerant electrons, has no significant nuclear magnetic moments, and can be synthesized in high purity. The specific challenge addressed was quantifying the distinct roles of magnons (spin waves) and phonons (lattice vibrations) in specific heat and thermal conductivity, and understanding how grain boundaries and porosity in polycrystalline ceramics affect these transport mechanisms.

2. Methodology

Sample Preparation: Two types of ceramic CdCr2Se4 samples were synthesized to vary microstructure:
1. Porous Sample: Hot-pressed at 763 K (50 MPa), resulting in ~17% porosity and ~83% relative density.
2. Dense Sample: Spark Plasma Sintered (SPS) at 773 K (300 MPa), followed by annealing in Se atmosphere, resulting in ~5% porosity and >95% density.
- Both samples exhibited grain sizes of ~1 $\mu$ m.
Characterization:
- Magnetic: SQUID magnetometry to confirm ferromagnetic ordering and homogeneity.
- Thermal: Physical Property Measurement System (PPMS) to measure specific heat ( $C_p$ ) and thermal conductivity ( $\kappa$ ) from 0.35 K to 300 K.
- Magnetic Field Dependence: Measurements were taken under magnetic fields up to 13 T to suppress magnon contributions (via Zeeman gap), allowing for the isolation of phonon-only contributions.
Theoretical Framework: Data were analyzed using kinetic gas theory, Debye models for phonons, and quadratic dispersion models for magnons ( $E = Dk^2$ ).

3. Key Contributions

Unambiguous Separation of Contributions: The study successfully demonstrated that in CdCr2Se4, the low-temperature specific heat follows $C_p \sim T^{3/2}$ (magnon-dominated) and $C_p \sim T^3$ (phonon-dominated) regimes, which are clearly distinguishable due to the absence of masking effects.
Discrepancy in Mean Free Paths: A major finding is the counter-intuitive result that magnons have a significantly shorter mean free path than phonons in polycrystalline CdCr2Se4, despite phonons usually being more susceptible to boundary scattering.
Mechanism of Scattering: The paper proposes that grain boundaries act as a "defective medium" with reduced spin-wave stiffness, causing strong reflection of magnons, whereas phonons (specifically long-wavelength ones) can traverse boundaries more effectively.
Porosity Effects: The study quantifies how porosity acts as a multiplicative factor reducing thermal conductivity without altering the fundamental scattering mechanisms of the intrinsic material.

4. Key Results

Specific Heat Capacity ( $C_p$ )

Magnon vs. Lattice Ratio: At 2 K, the ratio of magnon to lattice specific heat is 87:13.
Parameters: The data fits the model $C_{tot} = \beta T^3 + \delta T^{3/2}$ $C_{t o t} = β T^{3} + δ T^{3/2}$ .
- Debye temperature ( $\theta_D$ ): 237 K.
- Spin-wave stiffness ( $D$ ): 33.5 meVÅ² (consistent with previous resonance data).
Field Effect: Applying a 13 T field suppresses the magnon contribution, leaving a pure $T^3$ phonon dependence.

Thermal Conductivity ( $\kappa$ )

Temperature Dependence:
- Phonons: Instead of the standard Casimir limit prediction of $\kappa_{ph} \sim T^3$ , the experimental data shows $\kappa_{ph} \sim T^{2.3}$ (porous) and $\sim T^{2.5}$ (dense). This indicates frequency-dependent scattering (likely due to grain boundary strain or dislocations) rather than simple boundary reflection.
- Magnons: The magnon contribution follows the predicted $\kappa_m \sim T^2$ dependence.
Mean Free Path Analysis (at 2 K):
- Phonons: The estimated mean free path ( $l_{ph}$ ) is ~1.4 $\mu$ m, which is larger than the grain size (~1 $\mu$ m), suggesting long-wavelength phonons traverse boundaries efficiently.
- Magnons: The estimated mean free path ( $l_m$ ) is ~0.25 $\mu$ m, which is four times smaller than the grain size.
- Conclusion: Magnons are heavily scattered by grain boundaries (approx. 75% reflection), likely due to local variations in interatomic distances altering the exchange interactions (competition between AFM direct exchange and FM superexchange).
Porosity Impact: The dense sample (5% porosity) showed a thermal conductivity 4 times higher than the porous sample (17% porosity). The ratio of magnon to phonon contribution remained constant, suggesting porosity acts as a geometric scaling factor.
High-Temperature Anomaly: Above the phonon peak (~50 K), thermal conductivity drops steeply near $T_C$ (130 K) and follows a $T^{-0.5}$ trend at higher temperatures, attributed to strong coupling between lattice dynamics and spin fluctuations.

5. Significance

Fundamental Physics: This work provides one of the clearest experimental validations of the $T^{3/2}$ magnon specific heat and $T^2$ magnon thermal conductivity in a ferromagnetic insulator, free from electronic or nuclear interference.
Microstructure Engineering: It challenges the assumption that grain boundaries affect all heat carriers equally. The finding that magnons are more sensitive to grain boundaries than phonons in this system is crucial for designing spintronic or thermomagnetic devices.
Scattering Mechanisms: The study highlights that in polycrystalline magnetic insulators, the "defective" nature of grain boundaries (atomic displacements) can drastically alter magnetic exchange paths, leading to strong magnon scattering. This suggests that nanostructuring could be an effective tool to suppress thermal conductivity in thermoelectric applications by targeting specific heat carriers.
Material Design: The identification of CdCr2Se4 as an ideal model system for studying spin-lattice coupling offers a benchmark for future theoretical and experimental studies on magnetic insulators.