Loss of altermagnetic order and smooth restoration of Kramers' spin degeneracy with increasing temperature in CrSb and MnTe

This study demonstrates that while local magnetic moments persist above the Néel temperature in the altermagnets CrSb and MnTe, Kramers' spin degeneracy is smoothly restored as thermal disorder increases, with the transition occurring well below $T_\mathrm{N}$ in metallic CrSb due to electronic smearing but only near or above $T_\mathrm{N}$ in semiconducting MnTe where the band gap remains stable.

The Gist

Imagine you have a dance floor filled with pairs of dancers. In a normal crowd, everyone moves randomly. But in a special kind of magnetic material called an altermagnet, the dancers are arranged in a very specific, rigid pattern.

This paper is about two specific dance floors: one made of CrSb (a metal) and one made of MnTe (a semiconductor). The researchers wanted to know what happens to the electronic "dance" when you heat these materials up, causing the dancers to get jittery and lose their perfect formation.

Here is the story of what they found, broken down into simple concepts:

1. The Special Dance: Altermagnetism

First, let's understand the starting point.

• The Setup: In these materials, the electrons (the dancers) are split into two teams: "Spin Up" and "Spin Down."
• The Twist: Usually, if you have equal numbers of Up and Down dancers, the whole floor looks neutral (like a standard magnet). But in altermagnets, even though the total number of Up and Down dancers is equal, they are arranged in a way that creates a "spin-split" effect.
• The Analogy: Imagine a checkerboard where all the black squares are dancing one way and all the white squares are dancing the opposite way. Even though the total energy is balanced, the pattern creates a unique "spin traffic jam" that allows for cool electronic tricks, like conducting electricity with a specific spin direction. This is the "altermagnetic order."

2. The Heat Test: What happens when it gets hot?

The researchers asked: What happens when we turn up the heat?
Normally, when you heat a magnet, the dancers get so jittery that they forget the pattern entirely, and the material becomes "paramagnetic" (just a random mess).

The Big Surprise:
The researchers found that even when the material gets hot enough to be "paramagnetic" (above the critical temperature, $T_N$), the dancers don't actually stop dancing.

• The Metaphor: Imagine a chaotic mosh pit. Even though everyone is moving randomly, if you zoom in on just one person, they are still spinning wildly in place. They haven't lost their "local spin"; they've just lost the group choreography.
• The Finding: In both CrSb and MnTe, the individual atoms (Cr and Mn) keep their strong magnetic "spins" even when the material is hot. They are just pointing in random directions.

3. The Two Different Reactions

Here is where the two materials act differently, like two different types of crowds reacting to a loud noise:

Case A: The Metal (CrSb) - The "Smoothie" Effect

• The Situation: CrSb is a metal, meaning its electrons are free to roam everywhere, like a crowd of people running through a hallway.
• The Heat Effect: As soon as the temperature rises, the random spinning of the atoms starts to "smear" the electronic paths.
• The Analogy: Imagine trying to run through a hallway where people are suddenly spinning in circles randomly. You can't run in a straight line anymore; you get bumped and slowed down. The clear, sharp "spin-split" pattern (the altermagnetism) gets washed out and turned into a blurry smoothie.
• The Result: The special electronic properties disappear at temperatures well below the point where the material is technically "magnetically dead." The "spin traffic" gets too messy to use for technology.

Case B: The Semiconductor (MnTe) - The "Fence" Effect

• The Situation: MnTe is a semiconductor. Its electrons are stuck in specific lanes (energy bands) with a gap in the middle, like cars on a highway with a wide, empty ditch in the center.
• The Heat Effect: Even when the atoms start spinning randomly, the "ditch" (the band gap) stays open. The electrons can't jump across it easily.
• The Analogy: Imagine the dancers are spinning wildly, but they are all trapped inside their own little cages. They can spin, but they can't bump into each other enough to break the wall between the lanes.
• The Result: The special "spin-split" pattern stays visible for much longer. The material keeps its unique electronic structure until the temperature gets very close to the point where the spins finally give up completely.

4. Why Does This Matter?

The researchers discovered that the "Kramers spin degeneracy" (a fancy way of saying the two spin states are identical) is smoothly restored as the heat increases.

• In the cold: The spins are perfectly ordered, creating a unique, split pattern.
• In the heat: The spins get messy. The split pattern blurs and eventually disappears, making the two spin states look the same again (degenerate).

The Takeaway for Technology:
If you want to build a new type of computer chip (spintronics) using these materials, you have to be careful about temperature.

• If you use CrSb, you have to keep it very cool, or the "spin magic" will turn into a blurry mess before you even reach the "magnetic death" temperature.
• If you use MnTe, it's more robust; it keeps its special properties up to higher temperatures, making it a better candidate for real-world devices.

Summary

This paper is a report on how two special magnetic materials behave when they get hot. They found that even when the "group dance" breaks down, the individual dancers keep spinning. However, in the metal (CrSb), this chaos ruins the electronic structure quickly, while in the semiconductor (MnTe), the structure holds up much better. This helps scientists decide which material is best for building future spin-based electronics.

Technical Summary

1. Problem Statement

Altermagnetism is a newly identified class of magnetic order characterized by zero net magnetization (like antiferromagnets) but with spin-split electronic bands across large portions of the Brillouin zone (like ferromagnets). While the ground-state properties of altermagnets (specifically CrSb and MnTe) have been experimentally and theoretically established, a critical gap exists in understanding how these materials behave at finite temperatures.

Standard theoretical approaches often treat the paramagnetic state (above the Néel temperature, $T_N$) as a completely non-magnetic state. The authors argue this is unphysical for materials containing mid-to-late 3d elements, as it fails to account for persistent local magnetic moments that fluctuate thermally. Consequently, existing models cannot accurately predict:

• The persistence of local moments above $T_N$.
• The mechanism and temperature dependence of the restoration of Kramers' spin degeneracy.
• The impact of thermal spin fluctuations on electronic transport and band structure broadening.

2. Methodology

The authors employ a Disordered Local Moment (DLM) picture combined with ab initio density functional theory (DFT) calculations to model the electronic structure of CrSb and MnTe across a range of temperatures.

• Theoretical Framework:

  • DLM Picture: Instead of assuming a non-magnetic state, the authors model the paramagnetic phase as a statistical ensemble of thermally fluctuating local magnetic moments. The orientation of these moments is described by a probability distribution $P(\hat{e}_i)$, which becomes uniform (isotropic) in the high-temperature limit.
  • Coherent Potential Approximation (CPA): To handle the disorder of local moment orientations, the authors use the CPA within the KKR (Korringa-Kohn-Rostoker) multiple scattering theory framework. This allows for the calculation of the average Green's function for the disordered system.
  • Order Parameter ($m$): A magnetic order parameter $m$ is defined to interpolate linearly between the perfectly ordered altermagnetic state ($m=1$) and the fully disordered paramagnetic state ($m=0$).

• Computational Details:

  • Code: All-electron Hutsepot code implementing the KKR formulation of DFT.
  • Exchange-Correlation Functionals:
    • CrSb (Metallic): Local Spin-Density Approximation (LSDA).
    • MnTe (Semiconducting): Local Self-Interaction Correction (LSIC) applied to LSDA to correctly capture strong electronic correlations and the band gap.
  • Structural Models: Both materials crystallize in the hexagonal NiAs structure (Space Group $P6_3/mmc$). The calculations utilize experimentally determined lattice constants.
  • Symmetry Analysis: The study explicitly tracks symmetry changes, particularly the restoration of Kramers degeneracy and the behavior of non-symmorphic symmetries (glide mirrors and screw rotations) in the paramagnetic limit.

3. Key Contributions

• Validation of Local Moments: The study demonstrates that sizeable local magnetic moments persist on Cr and Mn atoms even well above the Néel temperature, necessitating a spin-polarized description of the electronic structure in the paramagnetic regime.
• Mechanism of Degeneracy Restoration: The paper elucidates how Kramers' spin degeneracy is "smoothly restored" on average as thermal disorder increases, contrasting this with the abrupt loss of order in simple mean-field models.
• Material-Specific Responses: The authors reveal a fundamental difference in how metallic vs. semiconducting altermagnets respond to thermal disorder:
  • CrSb (Metallic): Experiences heavy smearing of electronic states near the Fermi energy ($E_F$) at temperatures well below $T_N$, leading to a rapid loss of altermagnetic signatures.
  • MnTe (Semiconducting): The band gap remains largely intact despite magnetic disorder, with spin degeneracy only fully restored near or above $T_N$.
• Quantitative Prediction of $T_N$: The authors calculate Néel temperatures ($T_N$) and the temperature dependence of the order parameter $m(T)$, providing a benchmark for experimental validation.

4. Key Results

Electronic Structure Evolution

• Ground State ($T=0$): Both materials exhibit clear non-relativistic spin splitting (altermagnetism) along specific high-symmetry paths (e.g., $-\Gamma'-M'$). CrSb shows spin-splitting at the Fermi surface, while MnTe exhibits a band gap (indirect 0.6 eV, direct 1.2 eV).
• Paramagnetic State (DLM):
  • Local Moments: Calculated moments remain large ($2.58 \mu_B$ for Cr in CrSb; $4.63 \mu_B$ for Mn in MnTe), confirming they are "good" local moments.
  • Band Smearing: Magnetic disorder induces significant broadening (smearing) of the Bloch spectral function (BSF). In CrSb, this smearing is severe around $E_F$, effectively washing out the Fermi surface features and destroying the distinct altermagnetic band splitting at temperatures significantly lower than $T_N$.
  • MnTe Stability: In contrast, the band gap of MnTe is robust against magnetic disorder. The spin splitting vanishes, but the insulating nature is preserved until $T \approx T_N$.

Symmetry and Degeneracy

• Kramers Degeneracy: In the altermagnetic state, Kramers degeneracy is broken due to specific crystal symmetries combined with time-reversal symmetry breaking. In the paramagnetic state, the ensemble average restores Kramers degeneracy.
• Symmetry Analysis: The authors analyze the $k_z = 0$ and $k_z = \pi/c$ planes. In the paramagnetic state, the combination of global $PT$ symmetry and non-symmorphic glide mirror symmetries leads to a four-fold degeneracy on the $k_z = \pi/c$ plane, whereas the $k_z = 0$ plane retains only two-fold degeneracy.

Transition Temperatures and Order Parameter

• Calculated $T_N$:
  • CrSb: $T_N \approx 920$ K (with Onsager correction), compared to the experimental 705 K.
  • MnTe: $T_N \approx 160$ K, compared to the experimental 306 K.
  • Note: The underestimation for MnTe is attributed to the strong localization of 3d states in the LSIC-LSDA model. Using pure LSDA for MnTe incorrectly predicts a metallic state and an overestimated $T_N$ (709 K).
• Order Parameter $m(T)$: The order parameter decays more rapidly with temperature for CrSb than for MnTe. This correlates with the observation that CrSb loses its altermagnetic transport signatures at lower temperatures relative to $T_N$.

5. Significance and Implications

• Spintronics Applications: The results suggest that the attractive spin transport properties of altermagnets (e.g., large anomalous Hall effects, spin-splitting torque) are highly temperature-dependent. For metallic altermagnets like CrSb, these effects may degrade significantly below $T_N$ due to disorder-induced band smearing, whereas semiconducting altermagnets like MnTe may retain functional properties closer to $T_N$.
• Theoretical Modeling: The study establishes that modeling altermagnets requires going beyond the "non-magnetic" approximation. The DLM approach is essential for interpreting experimental spectroscopy (ARPES) and transport data, as the "paramagnetic" state is actually a state of dynamic local moments with distinct electronic signatures.
• Future Directions: The work lays the foundation for fully relativistic studies to investigate the interplay between spin fluctuations, spin-orbit coupling, and transport properties in these emerging functional materials.

In summary, this paper provides a rigorous first-principles framework for understanding the thermal evolution of altermagnets, highlighting the critical role of persistent local moments and the distinct thermal responses of metallic versus semiconducting altermagnetic materials.