Relativistic and nonrelativistic spin splitting above and below the Fermi level in a $g$-wave altermagnet

This study combines first-principles calculations, symmetry analysis, and dual spin-resolved spectroscopies (spin-ARPES and spin-ARRES) to experimentally map and distinguish relativistic and nonrelativistic spin splittings in the altermagnet CoNb$_4$Se$_8$ across both occupied and unoccupied states, thereby establishing a prototype for exploring spin-group phenomena in layered antiferromagnets.

The Gist

Imagine a bustling city where the traffic rules are different depending on which side of the street you are on. In most cities (standard magnets), cars (electrons) are either all going one way or the other, creating a net flow. In some cities (antiferromagnets), for every car going north, there is a matching car going south right next to it, so the net flow is zero. Usually, this means the traffic is perfectly balanced and indistinguishable.

But in this new city, CoNb4Se8, the rules are stranger. Even though the total traffic flow is zero (no net magnetism), the cars on the "spin-up" side are driving on a different lane than the cars on the "spin-down" side. They are separated by invisible walls that only exist in specific directions. This is the discovery of a "g-wave altermagnet."

Here is a breakdown of the paper's findings using simple analogies:

1. The Mystery of the "Ghost" Traffic Split

For decades, scientists believed that if a material had no net magnetism (like a perfect tug-of-war where both sides pull equally), the electrons inside would be identical twins—indistinguishable from each other.

However, a new theory suggested that in certain crystals, the electrons could still split into two distinct groups (spin-up and spin-down) based purely on the crystal's shape and symmetry, without needing the heavy "relativistic" forces usually required to do this. This is called Non-Relativistic Spin Splitting (NRSS).

• The Analogy: Imagine a dance floor. In a normal ballroom, everyone dances in pairs. In this new "altermagnet" ballroom, the dancers are split into two groups: those wearing red shirts and those wearing blue shirts. Even though the room is perfectly balanced (equal numbers of red and blue), the red dancers are forced to dance on the left side of the room, and the blue dancers on the right. The rule isn't a magnetic pull; it's the architecture of the room itself.

2. The Challenge: Seeing the Invisible

The problem is that this "traffic split" is very subtle. It's like trying to see if two identical twins are wearing slightly different shoes while they are running at high speed.

• The Obstacle: Most tools can only see the "occupied" dance floor (where electrons currently are). They can't see the "unoccupied" floor (where electrons could go but aren't right now).
• The Limitation: Previous tools were like a camera with a wide lens; they could see the whole room but couldn't zoom in on specific dancers, or they got confused by other effects (like the "relativistic" split caused by heavy atoms).

3. The New Toolkit: Two Cameras, One City

To solve this, the researchers used two different "cameras" to map the entire city, both the occupied and unoccupied zones:

• Camera 1: Spin-ARPES (The Occupied Floor): This is a standard high-tech camera that takes pictures of the electrons currently dancing. It confirmed that the "red" and "blue" dancers are indeed separated in specific directions.
• Camera 2: Spin-ARRES (The Unoccupied Floor): This is the new invention in the paper. Imagine a radar that shoots electrons into the material to see where they could land. It's like shining a flashlight into a dark room to see the furniture that isn't lit up yet. This allowed them to see the spin-splitting in the "empty" energy levels, proving the effect exists everywhere, not just where the electrons are currently sitting.

4. The "G-Wave" Pattern

The researchers found that the separation of red and blue dancers follows a specific pattern called a "g-wave."

• The Analogy: If you look at the dance floor from above, the separation isn't just a simple line down the middle (like a d-wave). It has four distinct "nodal planes" (invisible walls where the red and blue dancers mix and become identical). It looks like a four-leaf clover pattern. If you rotate the room by 60 degrees, the pattern flips: the red dancers become blue, and blue become red. This flipping is the signature of the "g-wave" altermagnet.

5. Distinguishing the "Real" Split from the "Fake" Split

There was a danger of confusion. Sometimes, heavy atoms in the crystal can cause a "fake" split (Relativistic Spin Splitting) that looks similar but behaves differently.

• The Test: The researchers used temperature as a test.
  • The Real Split (NRSS): When they heated the material above a certain temperature (the Néel temperature), the magnetic order broke, and the "red vs. blue" separation disappeared. The dancers mixed back together. This proved it was a genuine magnetic phenomenon.
  • The Fake Split (RSS): The relativistic split, caused by the heavy atoms, stayed even when the material was hot.
• The Result: By watching what happened when the material got hot, they could clearly separate the "new" g-wave effect from the "old" relativistic effects.

Why Does This Matter?

This discovery is a big deal for the future of technology:

• Faster Computers: Because these materials have no net magnetism, they don't create magnetic interference with each other. This means we could pack computer chips much closer together without them messing up each other's signals.
• Brain-like Computing: The ability to control these "spin textures" could lead to neuromorphic computing (computers that work like human brains), which is much more energy-efficient than today's silicon chips.
• New Physics: It opens the door to "unconventional superconductivity" (electricity with zero resistance) in new types of materials.

In Summary:
The team discovered a new type of magnetic material where electrons are naturally sorted into two lanes based on the crystal's shape, not just magnetism. They proved this by using a new "radar" technique to see both the occupied and empty electron zones, confirming that this "g-wave" pattern is real, robust, and distinct from other known effects. It's like finding a new law of physics that allows for a perfectly balanced tug-of-war where the teams are still clearly separated.

Technical Summary

1. Problem Statement

The field of spintronics has recently identified a new class of magnetic materials called altermagnets. Unlike conventional antiferromagnets (AFMs) which have doubly degenerate electronic bands, altermagnets exhibit nonrelativistic spin splitting (NRSS) driven by crystal symmetries (specifically spin-group symmetries) rather than spin-orbit coupling (SOC). This results in momentum-dependent spin splitting even in collinear AFMs with zero net magnetization.

Despite theoretical predictions, experimental verification of NRSS faces significant hurdles:

• Sample Quality: Many candidate materials suffer from domain formation, mosaicity, or degradation, obscuring intrinsic electronic signatures.
• Distinguishing Mechanisms: It is difficult to distinguish NRSS from relativistic spin splitting (RSS) (caused by SOC and inversion symmetry breaking) using standard techniques.
• Energy Range Limitations: Most techniques (like spin-ARPES) only probe occupied states near the Fermi level ($E_F$), whereas NRSS signatures often extend far above and below $E_F$.
• Indirect Evidence: Many existing methods (e.g., Anomalous Hall Effect) require coexisting ferromagnetism or external fields, which contradicts the pure AFM nature of altermagnets.

The paper aims to definitively identify and map both NRSS and RSS in a specific candidate material, CoNb$_4$Se$_8$, across a wide energy range to establish a prototype for g-wave altermagnetism.

2. Methodology

The authors employed a multifaceted approach combining advanced theoretical modeling with two complementary spin-resolved spectroscopies:

• Theoretical Framework:

  • Symmetry Analysis: Used group theory to identify the spin point group ($26/2m2m1m$) and predict nodal planes where spin splitting vanishes.
  • Density Functional Theory (DFT): Performed calculations to map the electronic structure, magnetic moments, and spin textures.
  • Tight-Binding Model: Developed a minimal model incorporating crystal field effects and antiferromagnetic exchange to isolate the microscopic mechanism of NRSS (crystal field + Zeeman splitting) versus RSS (SOC + inversion symmetry breaking).

• Experimental Techniques:

  • Spin-ARPES (Angle-Resolved Photoemission Spectroscopy): Used to probe occupied electronic states below the Fermi level. Measurements were conducted at 13 K (below the Néel temperature, $T_N$) and 200 K (above $T_N$) to track temperature dependence.
  • Spin-ARRES (Spin- and Angle-Resolved Electron Reflection Spectroscopy): A newly developed technique used to probe unoccupied electronic states (up to 11 eV above $E_F$). This method uses spin-polarized low-energy electrons and measures reflectivity contrast, offering higher spatial resolution ($\sim$1 $\mu$m) and interaction cross-sections compared to inverse photoemission.

• Material: High-quality single crystals of CoNb$_4$Se$_8$, a 1/4-intercalated transition metal dichalcogenide (TMD) with a hexagonal structure ($P6_3/mmc$) and collinear AFM order.

3. Key Contributions

1. First Complete Momentum-Resolved Mapping: The study provides the first experimental mapping of both relativistic and nonrelativistic spin splitting in a single system across occupied and unoccupied states.
2. Introduction of Spin-ARRES: Demonstrates the utility of the newly developed spin-ARRES technique for probing unoccupied spin-polarized bands, overcoming the energy limitations of spin-ARPES.
3. Mechanism Disentanglement: Successfully distinguishes NRSS from RSS based on:
   • Nodal Planes: NRSS vanishes at specific symmetry-enforced nodal planes (e.g., $k_z=0$ and specific glide planes), while RSS follows different symmetry constraints (e.g., vanishing at $\Gamma-M$ paths but splitting at $K$ points).
   • Temperature Dependence: NRSS collapses above the Néel temperature ($T_N$), whereas RSS persists.
4. Prototype Identification: Establishes CoNb$_4$Se$_8$ as a prototypical g-wave altermagnet (characterized by 4 nodal surfaces), distinct from previously studied d-wave altermagnets like MnTe.

4. Key Results

A. Crystal Structure and Magnetic Order

• CoNb$_4$Se$_8$ crystallizes in a centrosymmetric hexagonal structure with collinear AFM order.
• The two magnetic sublattices are related by 12 symmetry operations, including 6-fold screw axes and glide reflections.
• DFT confirms the AFM ground state is energetically favored over the ferromagnetic state by 60 meV/formula unit.

B. Nonrelativistic Spin Splitting (NRSS)

• Mechanism: Driven by the coupling of crystal field splitting and Zeeman splitting in the absence of strong SOC.
• Observation:
  • Occupied States (ARPES): Spin-split bands were observed along the $M''-\Gamma'-M'$ path. The spin polarization flips sign upon 6-fold rotation or crossing nodal planes.
  • Unoccupied States (ARRES): Similar alternating spin textures were observed 11 eV above $E_F$.
  • Temperature Dependence: The spin splitting along $M''-\Gamma'-M'$ disappears above $T_N$ (168 K), confirming its magnetic origin.
  • Nodal Planes: Splitting is zero at the $\Gamma$ point and along specific symmetry lines, consistent with g-wave symmetry (4 nodal planes).

C. Relativistic Spin Splitting (RSS)

• Mechanism: Driven by SOC and local inversion symmetry breaking at the surface termination (Rashba-type effect).
• Observation:
  • Occupied States (ARPES): Significant spin splitting was observed at the $K$ and $K'$ valleys (surrounding the Brillouin zone corners).
  • Unoccupied States (ARRES): Alternating spin textures were observed at $K$ valleys 11 eV above $E_F$.
  • Temperature Dependence: The spin splitting at the $K$ valleys persists above $T_N$, confirming it is a relativistic effect independent of magnetic order.
  • Symmetry: Splitting is present at $K$ points but vanishes along the $\Gamma-M$ path, distinct from the NRSS pattern.

D. Comparison and Distinction

• NRSS vs. RSS: The study clearly separates the two phenomena. NRSS is tied to the magnetic sublattice symmetry and vanishes at $T > T_N$; RSS is tied to surface inversion symmetry breaking and persists at $T > T_N$.
• Spatial Resolution: The high spatial resolution of ARRES ($\sim$1 $\mu$m) suggests that the relatively weak spin polarization measured in ARPES ($\sim$5%) might be due to averaging over altermagnetic domains smaller than the ARPES beam spot ($\sim$100 $\mu$m).

5. Significance

• Fundamental Physics: This work provides the first "smoking gun" experimental evidence for g-wave altermagnetism, validating the theoretical framework of spin-group symmetries. It clarifies the coexistence and distinction between relativistic and nonrelativistic spin splitting mechanisms.
• Technological Impact: By demonstrating that NRSS exists in both occupied and unoccupied states, the paper opens new avenues for engineering spin-based functionalities. Potential applications include:
  • Neuromorphic Computing: Utilizing the distinct domain structures and switching behaviors.
  • Unconventional Superconductivity: Exploring how altermagnetic order affects Cooper pair symmetry in TMDs.
  • Spintronics: Developing devices that utilize momentum-dependent spin textures without net magnetization.
• Methodological Advancement: The successful integration of spin-ARRES with spin-ARPES sets a new standard for characterizing the full electronic structure of quantum materials, particularly for probing unoccupied states where many exotic phenomena reside.

In conclusion, this paper establishes CoNb$_4$Se$_8$ as a robust platform for studying altermagnetism and demonstrates a comprehensive experimental toolkit capable of disentangling complex spin textures in quantum materials.