Global magnetic phase diagram and multiple unconventional magnets in NiAs-type compounds

This study establishes a global magnetic phase diagram for NiAs-type compounds using classical modeling and DFT calculations to identify new $g$-wave altermagnetic and $f$-wave odd-parity magnetic states, revealing that interlayer coupling drives a unique mixed-parity state in CrSe and CrTe$_{1-x}$Se$_x$ and offering strategies to engineer unconventional magnets via doping or strain.

The Gist

Imagine a giant, three-dimensional dance floor made of atoms. In this specific dance floor, known as the NiAs-type structure, the atoms are arranged in a triangular pattern, like a honeycomb that has been stretched into a stack of pancakes.

For a long time, scientists knew about two main types of "dancers" (magnetic states) on this floor:

1. The Ferromagnets: Everyone spins in the same direction, like a crowd doing the wave.
2. The Antiferromagnets: Neighbors spin in opposite directions, canceling each other out so the whole room feels "neutral."

But recently, scientists discovered some very weird, "unconventional" dancers. These are the Altermagnets (AM) and Odd-Parity Magnets (OPM). They are tricky because they look neutral from a distance (no net magnetism), but if you look closely at how they spin in momentum space (a fancy way of describing their energy and movement), they have a hidden, complex pattern. Think of them as dancers who look like they are standing still, but their internal rhythm is actually a complex, spinning jazz solo.

The Big Map

The authors of this paper wanted to find all the possible dance moves this specific atomic floor could do. They didn't just guess; they built a global magnetic phase diagram.

Think of this diagram as a weather map for magnets. Just as a weather map tells you where it's sunny, rainy, or snowy based on temperature and pressure, this map tells you which magnetic "dance" will happen based on how strongly the atoms talk to their neighbors.

They used two tools to draw this map:

1. A Simple Model (The Heisenberg Model): Imagine the atoms are little magnets connected by invisible springs. The authors tweaked the strength of these springs (called $J_1, J_2, J_3$) to see what happens.
2. Supercomputer Simulations (DFT): They ran complex math on a computer to see exactly how the electrons behave in real materials like Chromium Selenide (CrSe) or Manganese Telluride (MnTe).

The New Discoveries

On their "weather map," they found four new types of magnetic weather:

• Two "Even-Parity" Storms (g-wave AM): These are like the known Altermagnets (found in CrSb and MnTe). They have a specific symmetry, like a four-leaf clover pattern.
• Two "Odd-Parity" Storms (f-wave OPM): These are the new, rare finds. They have a different, more complex symmetry, like a three-leaf clover or a flower with an odd number of petals. These are the "Odd-Parity Magnets" (OPM) that are hard to find in nature.

The "Umbrella" Surprise

The most exciting discovery is a mixed state. The authors found that under certain conditions, the atoms don't just pick one dance move; they do a hybrid.

Imagine an umbrella.

• The handle of the umbrella represents the "Even-Parity" dance (flat on the floor).
• The ribs of the umbrella represent the "Odd-Parity" dance (sticking up).
• When the atoms form an umbrella-like structure, they are doing both dances at once.

The paper claims that materials like Chromium Selenide (CrSe) and a mix of Chromium Telluride and Selenium (CrTe$_{1-x}$Se$_x$) naturally form this "umbrella" shape. They are mostly doing the "Odd-Parity" dance, but with a tiny bit of the "Even-Parity" dance mixed in. This creates a unique "mixed-parity" state that hasn't been seen clearly before.

The Secret Ingredient: The "Third Neighbor"

Why does this happen? The authors point to a specific invisible spring called $J_3$ (the interaction between atoms that are two steps away, not just neighbors).

Think of it like a game of tug-of-war.

• Usually, the immediate neighbors ($J_1$) decide the game.
• But in this system, the "second-neighbor" spring ($J_3$) is surprisingly strong. It pulls the system in a different direction, creating a fierce competition between the normal magnetic states and these weird, unconventional ones.

Because this spring is so sensitive, the authors show that you can change the "weather" just by tweaking the material:

• Chemical Doping: Swapping a few atoms (like replacing Tellurium with Selenium) changes the tension on the springs.
• Strain: Squeezing or stretching the material (changing the size of the dance floor) also changes the springs.

By doing this, they showed you can force a material to switch from a normal magnet to one of these exotic "Odd-Parity" or "Mixed-Parity" states.

Summary

In short, this paper draws a complete map of the magnetic possibilities for a specific family of materials. It proves that these materials are a playground for exotic magnetism. They found new types of magnetic order (f-wave OPMs) and showed that nature can easily mix them together into a hybrid "umbrella" state. This gives scientists a recipe book: if you want to build a specific type of exotic magnet, just adjust the "springs" (strain or doping) in these NiAs-type compounds, and the map tells you exactly what you will get.

Technical Summary

Technical Summary: Global Magnetic Phase Diagram and Multiple Unconventional Magnets in NiAs-type Compounds

Problem Statement
The discovery of altermagnetism (AM) has revitalized interest in unconventional antiferromagnets (AFM) that exhibit momentum-dependent spin splitting without net magnetization. While even-parity AMs (e.g., $d$-wave RuO$_2$, $g$-wave CrSb) and odd-parity magnets (OPM) have been theoretically proposed, a comprehensive understanding of the magnetic landscape in NiAs-type compounds remains incomplete. Specifically, there is a lack of a global magnetic phase diagram to guide the search for OPMs and potential mixed-parity states within this family. Furthermore, the conditions under which even-parity AMs and odd-parity OPMs might coexist or mix in a natural magnetic structure have not been systematically explored.

Methodology
The authors employ a dual approach combining a classical $J_1$-$J_2$-$J_3$ Heisenberg model and Density Functional Theory (DFT) calculations:

1. Classical Model: A minimal Heisenberg model is constructed considering interlayer nearest-neighbor ($J_1$) and next-nearest-neighbor ($J_3$) couplings, and intralayer nearest-neighbor ($J_2$) coupling. The model analyzes total energies for various magnetic configurations, including collinear AFM, ferromagnetic (FM), stripe, zigzag, and noncollinear (NCL) structures. The analysis extends to include intralayer next-nearest-neighbor coupling ($J_4$) in supplementary calculations to refine phase boundaries.
2. DFT Calculations: First-principles calculations are performed using the Vienna Ab-initio Simulation Package (VASP) with the GGA-PBE functional. Magnetic exchange couplings ($J_i$) are extracted via the total energy difference method. The study focuses on representative NiAs-type compounds: CrSb, MnTe, CrSe, CrTe, CrTe$_{1-x}$Se$_x$, and hexagonal FeTe.
3. Symmetry Analysis: Spin space groups are utilized to classify magnetic states based on parity in momentum space ($s(k) = \pm s(-k)$). Band structures and Fermi surfaces are analyzed to identify spin-splitting characteristics.

Key Contributions and Results

• Global Magnetic Phase Diagram: The authors construct a $J_2/|J_1|$ vs. $J_3/|J_1|$ phase diagram for both AFM ($J_1 > 0$) and FM ($J_1 < 0$) interlayer couplings. This diagram reveals a rich landscape of ground states, identifying two $g$-wave AMs and two $f$-wave OPMs as stable phases.
• Identification of New Unconventional Magnets:
  • $g$-wave AMs: Confirmed as the ground state for CrSb and MnTe (labeled $g$-AM1 and $g$-AM2).
  • $f$-wave OPMs: Identified two distinct $f$-wave OPM states ($f$-OPM1 and $f$-OPM2) characterized by odd-parity spin textures ($s_z(k) = -s_z(-k)$).
• Mixed-Parity States: A significant finding is the natural emergence of a mixed-parity state in an umbrella-like noncollinear magnetic structure. This state arises when out-of-plane magnetic moments break the symmetry required for pure OPMs, creating a superposition of an $f$-wave OPM and a $g$-wave AM.
  • CrSe: DFT calculations indicate CrSe resides in this mixed-parity state, dominated by the $f$-OPM1 component with a small $g$-AM1 contribution (canted angle $\theta \sim 2^\circ$).
  • CrTe$_{1-x}$Se$_x$: Chemical doping drives CrTe (which is FM) into the $f$-OPM1 state for $x \geq 0.2$. The resulting CrTe$_{0.8}$Se$_{0.2}$ also exhibits a mixed-parity state with a dominant $f$-wave component.
• Role of Interlayer Coupling ($J_3$): The study highlights that the interlayer next-nearest-neighbor coupling $J_3$ is crucial. Despite being smaller in magnitude than $J_1$, its multiplicity allows it to strongly compete with and stabilize unconventional states. For instance, a ferromagnetic $J_3$ can stabilize an $f$-OPM1 state even when $J_1$ or $J_2$ are antiferromagnetic, leading to strong competition between FM and OPM phases.
• Phase Transitions via Strain and Doping: The paper demonstrates that phase transitions between conventional and unconventional magnets can be induced by:
  • Chemical Doping: Replacing Te with Se in CrTe shifts the system from FM to $f$-OPM1.
  • Strain: Applying in-plane biaxial strain to CrSe and FeTe drives transitions between FM, $f$-OPM, and $g$-AM states. For example, compressive strain on FeTe stabilizes the $f$-OPM2 state.
• FeTe: The hexagonal FeTe is found to be near a phase boundary between FM and $f$-OPM2. While the ground state is a distorted Zigzag-2 AFM, compressive strain can drive it into the $f$-OPM2 state, exhibiting $f$-wave spin splitting.

Significance
The paper provides a systematic framework for designing and searching for unconventional magnets in NiAs-type compounds. By establishing a global phase diagram, it offers a guide for realizing even-parity (AM), odd-parity (OPM), and mixed-parity magnetic states. The identification of mixed-parity states in umbrella-like structures expands the theoretical classification of magnetic order. Furthermore, the demonstration that chemical doping and strain can tune these materials between conventional and unconventional phases suggests a feasible route for engineering materials with specific spin-splitting properties. The metallic nature of the predicted candidates (CrSe, CrTe$_{1-x}$Se$_x$, FeTe) is noted as beneficial for constructing heterostructures with superconductors to explore topological superconducting states via the proximity effect.