Signatures of two ferromagnetic states and goniopolarity in LaCrGe3 in the Hall effect

This study utilizes Hall effect measurements to confirm the existence of two distinct ferromagnetic phases and the phenomenon of goniopolarity in LaCrGe3, while also reporting a large intrinsic anomalous Hall conductivity that highlights the material's potential for future electronic applications.

The Gist

Imagine a material called LaCrGe3 (let's call it "L-C-G" for short) as a bustling, microscopic city. In this city, tiny particles called electrons are the citizens, and they are constantly moving around. Usually, in magnetic materials, these electrons all agree to march in the same direction, creating a strong magnetic field. This is called being "ferromagnetic."

But L-C-G is a bit of a rebel. It doesn't just have one way of being magnetic; it has two distinct "moods" or phases, and the scientists in this paper figured out how to spot them using a clever trick.

Here is the story of their discovery, broken down into simple concepts:

1. The Two Magnetic Moods (The "Two-Phase" Mystery)

Think of L-C-G like a person who has two different personalities depending on the temperature.

• Personality A (FM1): At a certain temperature (around 70 Kelvin, which is very cold), the electrons settle into one specific pattern.
• Personality B (FM2): As it gets even colder, the electrons suddenly shift into a different pattern.

For a long time, scientists suspected these two moods existed, but it was hard to prove because the change is subtle. It's like trying to tell if a person is slightly annoyed versus slightly angry just by looking at them from far away.

The Detective Work:
The researchers used the Hall Effect as their magnifying glass. Imagine you are driving a car (the electrons) down a straight road. If you turn on a strong wind (a magnetic field), the car gets pushed to the side.

• In a normal material, the car gets pushed a predictable amount.
• In L-C-G, the researchers watched how hard the car was pushed as they slowly cooled the road down.
• The Clue: They saw the "push" (Hall resistivity) spike and then dip exactly at the temperature where the material switches from Personality A to Personality B. It was like seeing the car suddenly swerve twice in a row, proving there were two distinct driving modes happening.

2. The "Goniopolarity" Trick (The Shape-Shifting City)

This is the most fascinating part of the paper. Usually, in a city, if the traffic flows East, it's because the roads are built for Eastward travel. If you look North, the traffic might be slower, but it's still the same type of traffic.

But in L-C-G, the "roads" (the electronic structure) are shaped so strangely that the direction of the traffic changes depending on which way you look.

• The Analogy: Imagine a hill that looks like a saddle (like a horse saddle).
  • If you roll a ball forward, it rolls down the hill (like an electron).
  • If you roll a ball sideways, it rolls up the hill (like a hole, or a missing electron).
• The Result: When the scientists applied a magnetic field pointing "up" (out of the crystal), the current acted like it was made of positive charges. But when they applied the field pointing "sideways" (along the crystal), the current acted like it was made of negative charges.
• This phenomenon is called Goniopolarity. It's as if the material is a chameleon that changes its electrical identity based on the angle of the light hitting it.

3. Why Does This Matter?

Why should we care about a material that has two magnetic moods and changes its electrical personality?

• Better Electronics: Modern computers and phones rely on controlling the flow of electricity and magnetism. A material that can switch between different magnetic states or change its charge type based on direction is like a Swiss Army knife for engineers. It could lead to new types of memory storage or sensors that are much more efficient.
• Understanding the Unseen: The fact that the electrons behave this way tells us that the "roads" inside the material (the Fermi surface) are incredibly complex and twisted. Understanding this helps physicists predict how other exotic materials might behave.

Summary

In short, this paper is about finding a hidden "double life" in a magnetic material.

1. They proved the material has two different magnetic states by watching how electricity reacts to a magnetic field as it cools down.
2. They discovered the material is directionally tricky: it acts like a positive conductor in one direction and a negative conductor in another, a rare trait called goniopolarity.

It's a bit like discovering a city where the traffic laws change depending on which street you are driving on, and the citizens change their uniforms depending on the time of day. This makes LaCrGe3 a very exciting candidate for the next generation of high-tech devices.

Technical Summary

1. Problem Statement

The itinerant ferromagnet LaCrGe$_3$ is a critical system for studying quantum critical phenomena and unconventional magnetic states. Previous studies have proposed the existence of two distinct ferromagnetic (FM) phases (FM1 and FM2) separated by a transition temperature ($T_x \approx 70$ K) below the Curie temperature ($T_C \approx 85$ K). However, experimental evidence for these two phases has been conflicting:

• Some techniques (resistivity derivatives, magnetization, muon spin relaxation) suggest two phases.
• Others (neutron diffraction, electron spin resonance, thermodynamic measurements) have failed to reveal clear signatures of two distinct FM states.
• The nature of the magnetic ordering and the electronic transport properties in the paramagnetic (PM) phase remain incompletely understood, particularly regarding carrier polarity anisotropy.

The study aims to resolve this ambiguity by employing Hall effect measurements, which are highly sensitive to changes in electronic band structure and spin-dependent scattering, to definitively identify the two FM phases and investigate the electronic transport mechanisms in the PM phase.

2. Methodology

• Sample Synthesis: High-quality single crystals of LaCrGe$_3$ were grown using the self-flux method with an excess of Germanium. The crystals exhibited a hexagonal rod shape.
• Characterization:
  • Structural: X-ray diffraction (XRD) and Laue diffraction confirmed the hexagonal $P6_3/mmc$ structure and crystal orientation.
  • Magnetic/Transport: Measurements were performed using a Quantum Design Dynacool system (VSM and ETO options). Temperature-dependent magnetization (ZFC, FCC, FCW) and resistivity were recorded.
  • Hall Effect: Transverse resistivity ($\rho_{ij}$) was measured under varying magnetic fields ($B$) and temperatures ($T$). Data was symmetrized/antisymmetrized to isolate longitudinal and Hall components.
  • Seebeck Effect: Thermovoltage measurements were conducted to determine carrier polarity without external magnetic fields.
• Theoretical Calculations: Density Functional Theory (DFT) calculations were performed using VASP (LDA and GGA functionals) including Spin-Orbit Coupling (SOC) to compute electronic band structures and Fermi surfaces.

3. Key Contributions and Results

A. Identification of Two Ferromagnetic Phases via Hall Effect

The study provides robust evidence for two FM phases (FM1 and FM2) through continuous temperature-dependent Hall resistivity ($\rho_{yx}$) measurements at fixed low magnetic fields:

• Anomalies at Transition Points: The Hall resistivity shows distinct features at $T_C$ (PM-FM1 transition) and $T_x$ (FM1-FM2 transition).
• Remanent Properties: The remanent Hall resistivity ($\rho_{yx}|_{Sat}$) and the coercive field ($B_c$) both exhibit a maximum near $T_x$ ($\approx 73$ K).
• Hall Coefficient ($R_0$): The ordinary Hall coefficient shows a minimum and a sharp change in slope at the FM1-FM2 boundary.
• Mechanism: These anomalies are attributed to domain wall pinning and depinning. The transition between FM1 and FM2 involves a change in domain wall width and dynamics, leading to a "pinned" state at low temperatures and another pinned state just below $T_C$, with a depinning region in between that enhances resistivity.
• Conclusion: The Hall effect serves as a more sensitive probe than standard resistivity or magnetization for distinguishing these phases, as the electronic band structure reconstruction at $T_x$ strongly affects carrier scattering.

B. Anomalous Hall Effect (AHE) Analysis

• Magnitude: A large anomalous Hall conductivity ($\sigma_{xy}^A$) of 1160 $\Omega^{-1}$cm$^{-1}$ was observed at 2 K when the field was applied along the magnetic easy axis ($c$-axis).
• Origin: The temperature independence of $\sigma_{xy}^A$ at low temperatures and its scaling behavior indicate that the AHE is dominated by intrinsic mechanisms (Berry curvature) arising from band crossings near the Fermi energy, rather than extrinsic skew scattering.
• Anisotropy: The anomalous Hall angle ($\theta_{AH}$) peaks at $T_x$, reaching $\sim 7.5\%$ for $B \parallel c$, further highlighting the significance of the phase transition.

C. Discovery of Goniopolarity in the Paramagnetic Phase

In the paramagnetic phase ($T > T_C$), the material exhibits goniopolarity (direction-dependent conduction polarity):

• Hall Effect:
  • With $B \parallel c$ (out-of-plane), the Hall slope is positive, indicating hole-like carriers.
  • With $B \parallel b$ (in-plane), the Hall slope is negative, indicating electron-like carriers.
• Seebeck Effect: Consistent with the Hall data, the Seebeck coefficient ($S$) is positive (p-type) for in-plane measurements above 108 K but negative (n-type) for out-of-plane measurements below 257 K.
• Theoretical Support: DFT calculations reveal a highly anisotropic, multi-sheeted Fermi surface. The curvature of the bands near the Fermi energy changes sign depending on the crystallographic direction (e.g., $\Gamma$-A vs. $\Gamma$-M-K). This geometric anisotropy causes the effective mass sign to flip with direction, resulting in opposite carrier polarities.

4. Significance

• Resolution of Magnetic Controversy: The paper definitively confirms the existence of two FM phases in LaCrGe$_3$ using Hall transport signatures, resolving previous inconsistencies in the literature. It links these phases to complex domain wall dynamics rather than simple spin reorientation.
• Goniopolarity Demonstration: It identifies LaCrGe$_3$ as a rare goniopolar material where the majority carrier type (electron vs. hole) is determined by the measurement direction, even in the absence of a magnetic field (in the PM phase). This is driven by the unique geometry of the Fermi surface.
• Device Potential: The coexistence of unconventional magnetic phases, large intrinsic anomalous Hall effects, and goniopolar transport makes LaCrGe$_3$ a promising candidate for novel electronic and thermoelectric devices, particularly those exploiting transverse thermoelectric effects and spintronic applications.
• Methodological Insight: The study underscores the Hall effect as a superior tool for probing subtle magnetic phase transitions and Fermi surface reconstructions in itinerant magnets compared to traditional magnetization or resistivity measurements.