Rare-Earth-Tuned Evolution from d- to f-Orbital Dominance and Giant Anomalous Hall Effect in Topological RGaGe (R = Ce, Pr, Nd) Semimetals

This study establishes the noncentrosymmetric rare-earth germanides RGaGe (R = Ce, Pr, Nd) as a tunable platform for exploring the evolution from d- to f-orbital dominated topology and demonstrates their giant intrinsic anomalous Hall effect driven by robust Weyl semimetallic states coupled with magnetic order.

The Gist

Imagine a group of tiny, magical building blocks called atoms. Usually, these atoms arrange themselves in neat, symmetrical patterns, like a perfectly balanced seesaw. But in a special family of materials called RGaGe (made from rare-earth metals like Cerium, Praseum, and Neodymium mixed with Gallium and Germanium), the atoms arrange themselves in a way that breaks this balance. They are "lopsided," or what scientists call noncentrosymmetric.

Think of this lopsided structure like a spiral staircase that only goes up, never down. This unique shape is the key to unlocking some very strange and powerful behaviors in electricity and magnetism.

Here is what the researchers discovered about these materials, explained simply:

1. The Magnetic "One-Way Street"

These materials are magnets, but they are very picky about which way they point.

• The Analogy: Imagine a crowd of people holding compasses. In most magnets, the compasses might point in all directions or flip easily. In RGaGe, the compasses are glued to a specific track. They strongly prefer to point "up and down" (along the vertical axis of the crystal) rather than "side to side."
• The Discovery: When the researchers cooled these crystals down, the atoms lined up in a specific pattern: they acted like a unified team pointing up (ferromagnetic) vertically, but behaved like a tug-of-war team pointing in opposite directions horizontally (antiferromagnetic-like). This "one-way street" behavior is called strong magnetic anisotropy.

2. The "Giant" Electrical Shortcut (The Anomalous Hall Effect)

Usually, when electricity flows through a wire, it goes straight. If you put a magnet near it, the electricity might curve slightly. This is the "Hall Effect."

• The Analogy: Imagine driving a car on a highway. Normally, you drive straight. If you hit a strong crosswind (magnetism), you might drift a little. But in these RGaGe materials, the road itself is twisted like a rollercoaster. Even without a strong external wind, the car (electrons) is forced to swerve wildly to the side just because of the road's shape and the car's own internal engine (magnetism).
• The Discovery: The researchers found that these materials create a massive sideways electrical current (called the Anomalous Hall Effect). It was so strong that in one version (PrGaGe), it was nearly 1.3 times stronger than in similar, well-known materials (RAlGe). It's like finding a shortcut that is significantly faster than the highway everyone else uses.

3. The "Ghost" Particles (Weyl Semimetals)

Why is the electricity swerving so much? The researchers found that the electrons in these materials aren't just normal electrons; they are behaving like Weyl fermions.

• The Analogy: Think of normal electrons as cars driving on a flat road. Weyl fermions are like cars driving on a mountain pass where the road twists into a knot. At the very center of this knot, the road splits and rejoins in a way that creates a "portal."
• The Discovery: Because the crystal structure is lopsided, it creates these "portals" (called Weyl points) right where the electrons are moving. These portals act like traffic directors, forcing the electrons to take a specific, curved path, which creates that giant electrical shortcut.

4. The "Orbital Evolution" (Changing the Engine)

The researchers looked at three different versions of this material: one with Cerium (Ce), one with Praseum (Pr), and one with Neodymium (Nd). They noticed a fascinating change as they moved from one to the next.

• The Analogy: Imagine three cars that look identical on the outside.
  • The Cerium and Praseum cars are powered by a standard d-engine (like a reliable V6).
  • The Neodymium car, however, has been upgraded with a powerful f-engine (like a high-tech electric hybrid).
• The Discovery: As they moved from Cerium to Neodymium, the "engine" powering the electrons changed. In the first two, the electrons were dominated by d-orbitals (a specific type of electron cloud). In the Neodymium version, the f-orbitals (a more complex, inner electron cloud) took over. This shift changed how the electrons interacted with the magnetic fields, creating a "tunable" system where you can adjust the material's properties just by swapping the rare-earth ingredient.

5. The "Ghost" That Lingers

One of the most surprising findings was that this giant electrical shortcut didn't disappear when the material stopped being magnetic.

• The Analogy: Usually, if you turn off the engine of a car, it stops moving. But in these materials, even when the "magnetic engine" cooled down and stopped aligning (above the magnetic ordering temperature), the "twisted road" (the topological structure) remained.
• The Discovery: The giant electrical effect persisted even when the material was warm and no longer magnetic. This proves that the effect comes from the shape of the road itself (the electronic structure), not just the magnetism. It's a built-in feature of the material's geometry.

Summary

The paper describes a new family of materials that act like magnetic, lopsided rollercoasters for electricity. By swapping different rare-earth ingredients, the scientists can tune the "engine" of the electrons from one type to another. These materials create a massive, natural shortcut for electricity that is driven by the unique, twisted shape of their atomic structure, offering a new playground for understanding how magnetism and quantum physics work together.

Technical Summary

Technical Summary: Rare-Earth-Tuned Evolution from d- to f-Orbital Dominance and Giant Anomalous Hall Effect in Topological RGaGe (R = Ce, Pr, Nd) Semimetals

Problem and Context
The interplay between strong magnetism, electronic correlations, and topological band structures remains a central theme in quantum materials research. While the noncentrosymmetric rare-earth aluminosilicon/germanide family ($RAlX$, where $R$ is a rare earth and $X$ is Si or Ge) has been established as a fertile platform for investigating Weyl semimetal (WSM) states and anomalous transport, the systematic modulation of these properties via chemical substitution within this structural family is less developed. Specifically, the evolution of orbital character (from $d$- to $f$-orbital dominance) and its impact on the interplay between topology and magnetism in the germanide analogs ($RGaGe$) requires further exploration to advance the fundamental understanding of correlated topological phases.

Methodology
The study employs a combined experimental and theoretical approach to investigate single crystals of the $RGaGe$ series ($R = \text{Ce, Pr, Nd}$).

• Synthesis and Characterization: High-quality single crystals were grown using the self-flux method with a mixed Ga-In flux. Structural quality was verified via single-crystal X-ray diffraction (SXRD).
• Transport and Magnetic Measurements: Comprehensive magnetotransport measurements (resistivity, Hall effect) and magnetic characterization (DC susceptibility, isothermal magnetization) were conducted over a temperature range of 2–300 K under various magnetic fields.
• Theoretical Analysis: First-principles calculations based on density functional theory (DFT) were performed using the Vienna ab initio simulation package (VASP) with the generalized gradient approximation and a Hubbard $U$ parameter ($U=5$ eV) applied to rare-earth $f$-orbitals. The intrinsic anomalous Hall conductivity (AHC) was computed using Wannier functions and the WannierTools package.

Key Results

1. Crystal Structure and Metallic Behavior: The synthesized $RGaGe$ compounds crystallize in a noncentrosymmetric tetragonal structure (space group $I4_1md$). All members exhibit metallic behavior down to 2 K, with distinct anomalies in resistivity corresponding to magnetic ordering temperatures ($T_C$) of 6.74 K (Ce), 17.9 K (Pr), and 9.72 K (Nd).
2. Magnetic Anisotropy and Ground States: The materials display pronounced uniaxial magnetocrystalline anisotropy. Magnetic susceptibility measurements reveal a ferromagnetic (FM) transition with moments aligned along the crystallographic $c$-axis, while the $ab$-plane exhibits antiferromagnetic (AFM)-like behavior. The magnetization ratio $\chi_c/\chi_{ab}$ increases from $\sim 10$ in the paramagnetic regime to $\sim 100$ at 2 K, driven by the synergistic effect of strong spin-orbit coupling (SOC) and the crystalline electric field (CEF).
3. Giant Anomalous Hall Effect (AHE): The $RGaGe$ family exhibits a significantly enhanced intrinsic anomalous Hall conductivity (AHC) compared to their $RAlGe$ counterparts. At 2 K, the maximum experimental AHC values are 446, 948, and 670 $\Omega^{-1}\cdot\text{cm}^{-1}$ for Ce, Pr, and Nd, respectively. Notably, PrGaGe reaches 948 $\Omega^{-1}\cdot\text{cm}^{-1}$, approximately 1.3 times larger than the previously reported value for PrAlGe.
4. Orbital Evolution: First-principles calculations reveal a distinct evolution in the electronic structure near the Fermi level. While CeGaGe and PrGaGe are dominated by $d$-orbital contributions, NdGaGe shows significant involvement of $f$-orbitals. This signals a progressive transition from $d$-orbital to $f$-orbital dominated topology across the series.
5. Topological Origin: The large AHC is attributed to a robust Weyl semimetallic state arising from inversion symmetry breaking. Weyl points located near the Fermi level couple strongly to the magnetic order. Crucially, the AHC persists well above the magnetic ordering temperature (e.g., remaining finite at 30 K in PrGaGe), confirming its intrinsic electronic origin rather than a purely magnetic one.

Significance and Claims
The authors establish the $RGaGe$ system as a tunable platform for systematically extending the $RAlGe$-related family. The primary contributions of this work are:

• Enhanced Transport: Demonstrating that substituting Ga for Al in the $RAlX$ framework yields materials with substantially larger anomalous Hall responses.
• Orbital Tunability: Revealing a rare-earth-dependent evolution from $d$-orbital to $f$-orbital dominance near the Fermi level, which modulates the Berry curvature and topological transport properties.
• Robust Topology: Confirming that the topological band structure (Weyl points) and the resulting AHE are intrinsic features that coexist with and are modulated by magnetism, persisting even in the absence of long-range magnetic order.

These findings advance the understanding of the interplay between localized $f$-electron physics and topological band properties, offering a pathway to realize correlated topological phases through orbital engineering.