Berry curvature and field-induced intrinsic anomalous Hall effect in an antiferromagnet FeTe

This theoretical study demonstrates that the antiferromagnet FeTe exhibits a large, temperature- and field-sensitive intrinsic anomalous Hall effect driven by Berry curvature, establishing it as a promising platform for exploring quantum transport in low-dimensional correlated systems.

The Gist

Imagine you are trying to drive a car through a city. Usually, if you turn the steering wheel, the car goes in that direction. But in the world of tiny particles (electrons) inside certain magnetic materials, things get weird. Sometimes, even if you push the car straight, it drifts sideways. This sideways drift is called the Hall Effect.

In most magnets, this drift is caused by the magnet itself pulling the car off course. But in Antiferromagnets (a special type of magnet where the tiny internal magnets cancel each other out, like a tug-of-war where both teams are equally strong), there is no net pull. So, how can the car drift?

This paper explores a mysterious material called FeTe (Iron Telluride) and explains how we can make it drift sideways using a magnetic field, even though it's an antiferromagnet. Here is the story, broken down with simple analogies.

1. The Material: A Tug-of-War City

Think of FeTe as a city built on a grid. In this city, every building has a tiny "compass" (a magnetic spin) on its roof.

• Normal Magnets: All compasses point North.
• Antiferromagnets (FeTe): The compasses are arranged in a pattern where neighbors point in opposite directions (North, South, North, South). Because they cancel each other out, the city looks "magnetically neutral" from the outside. It's like a tug-of-war where the rope doesn't move.

Usually, because the compasses cancel out, you can't get a "sideways drift" (the Anomalous Hall Effect) in these materials. But the authors of this paper discovered a trick.

2. The Secret Ingredient: The "Berry Curvature"

To understand the drift, imagine the electrons aren't just driving on a flat road. Instead, they are driving on a hilly, warped landscape made of invisible energy.

In physics, this warped landscape is called Berry Curvature.

• The Analogy: Imagine driving a car on a smooth, flat road. If you hit a bump (a magnetic field), the car might bounce. But imagine driving on a curved ramp or a spiral slide. Even if you steer straight, the shape of the road forces you to turn.
• In FeTe, the electrons are driving on these invisible, warped ramps. The "shape" of the road is determined by the arrangement of the magnetic compasses and the material's atomic structure. This shape creates a "force" that pushes the electrons sideways, creating a voltage.

3. The Magic Switch: Turning on the Magnetic Field

Here is the cool part: In FeTe, this warped landscape is hidden when the material is cold and quiet. The "roads" are flat, and there is no sideways drift.

But, if you apply an external magnetic field (like bringing a giant magnet close to the city), something magical happens:

1. The Tug-of-War Tilts: The external magnet slightly tilts the compasses in the city. They don't all point North, but they lean a little bit.
2. The Road Warps: This slight tilt changes the shape of the invisible energy landscape. Suddenly, the flat roads turn into spiral slides.
3. The Drift Begins: Now, when electricity flows through the material, the electrons get pushed sideways by the shape of the road, creating a strong Anomalous Hall Effect.

4. The Temperature Twist: A Shape-Shifting Road

The researchers found that the "road" is incredibly sensitive to temperature.

• Hot (Above 60 Kelvin): The compasses are jittery and random. The road is flat. No drift.
• Cold (Below 60 Kelvin): The compasses lock into their tug-of-war pattern.
• The Surprise: When they applied the magnetic field at different cold temperatures, the "sideways drift" didn't just get stronger; it flipped direction.
  • At one temperature, the electrons drifted Left.
  • At a slightly lower temperature, the "road" warped the other way, and they drifted Right.

It's like driving on a road that changes from a left-turning spiral to a right-turning spiral just because you turned the thermostat down a few degrees.

5. Why Does This Matter?

This discovery is a big deal for two reasons:

1. New Electronics (Spintronics): We want to build computers that use the "spin" of electrons instead of just their charge. These are faster and use less energy. FeTe shows us that we can control these electron flows using tiny magnetic fields, even in materials that don't seem magnetic at all.
2. Topological Physics: It proves that the "shape" of the energy landscape (topology) is just as important as the material itself. By understanding how to warp these roads, scientists can design new materials that act like super-highways for electricity.

Summary

The paper tells us that FeTe is a "shape-shifting" material.

• Normally, it's a neutral antiferromagnet with no magnetic drift.
• But, if you cool it down and apply a magnetic field, you warp the invisible roads the electrons travel on.
• This warping creates a powerful sideways push (the Hall Effect) that can even flip directions depending on the exact temperature.

It's like finding a hidden switch in a tug-of-war game that, when flipped, suddenly turns the rope into a spinning slide, sending everyone in a new direction. This opens the door to building smarter, faster, and more efficient electronic devices.

Technical Summary

1. Problem Statement

The paper addresses the challenge of understanding and generating the Anomalous Hall Effect (AHE) in collinear antiferromagnets (AFM).

• The Challenge: Traditional AHE requires broken time-reversal symmetry ($T$) and net magnetization. In collinear AFMs, the spin sublattices are compensated (net magnetization $\approx 0$), and combined symmetries (like $P \otimes T$, where $P$ is spatial inversion) often protect band degeneracies, suppressing Berry curvature and resulting in zero intrinsic AHE.
• The Gap: While non-collinear AFMs (e.g., Mn$_3$Sn) exhibit large AHE due to non-zero scalar spin chirality, the mechanism for generating a robust, intrinsic AHE in collinear AFMs, particularly under external fields, remains less explored.
• The Material: FeTe (Iron Telluride) is a semimetallic van der Waals (vdW) antiferromagnet with bicollinear AFM order. It possesses frustration and is a candidate for low-dimensional spintronics, but its intrinsic Hall response mechanisms were not fully understood theoretically.

2. Methodology

The authors employed a realistic spin-fermion model coupled with Density Functional Theory (DFT) to investigate the electronic structure and transport properties of FeTe.

• Model Construction:
  • Electronic Structure: Derived from DFT calculations (using VASP) on the experimental structure at 80 K. A Wannier tight-binding model was constructed projecting onto Fe $d$-states to capture the itinerant electrons.
  • Magnetic Model: Localized Fe moments were modeled using a Heisenberg Hamiltonian ($H_S$) with exchange integrals ($J_1$ to $J_4$) derived from DFT total energy differences of various magnetic configurations. The bicollinear AFM state was identified as the ground state.
  • Coupling: A spin-fermion Hamiltonian ($H_{sf} = H_d + H_S + H_{d-S}$) was formed, where $H_{d-S}$ represents the exchange coupling between itinerant $d$-electrons and localized spins.
• Simulation Conditions:
  • Mean-Field Approximation: Used to solve for the magnetic state at finite temperatures ($T$) and under applied magnetic fields ($B$).
  • Symmetry Breaking: An external magnetic field ($B$) was applied perpendicular to the plane to break the $P \otimes T$ symmetry, lifting band degeneracies.
  • Transport Calculation: The Berry curvature ($\Omega_{xy}$) and Anomalous Hall Conductivity (AHC, $\sigma_{xy}$) were calculated by integrating the Berry curvature over the Brillouin zone, weighted by the Fermi-Dirac distribution.
  • Parameters: Calculations were performed across a range of temperatures (above and below the Néel temperature $T_N \approx 60$ K), magnetic fields (up to 20 Tesla), and Fermi level shifts (simulating doping).

3. Key Contributions

• Theoretical Framework: Established a microscopic framework linking the evolution of bicollinear AFM order under magnetic fields to the generation of Berry curvature in a collinear AFM.
• Mechanism Identification: Demonstrated that the AHE in FeTe is intrinsic and driven by Berry curvature, rather than extrinsic scattering mechanisms, which are suppressed in compensated AFMs.
• Symmetry Analysis: Clarified how an external magnetic field breaks the $P \otimes T$ symmetry (present below $T_N$ without a field) and $T$ symmetry (present above $T_N$), thereby opening gaps at band crossings and inducing non-zero Berry curvature.
• Lifshitz Transition Insight: Identified a specific mechanism for the sign reversal of the Hall conductivity involving a Lifshitz transition (disappearance of hole pockets) near $0.68 T_N$.

4. Key Results

• Field-Induced AHE:
  • Without a magnetic field, FeTe exhibits zero AHC due to symmetry protection (Kramers degeneracy above $T_N$; $P \otimes T$ degeneracy below $T_N$).
  • Applying a magnetic field ($B=10$ T) breaks these symmetries, inducing a large, intrinsic AHC.
• Temperature Dependence & Sign Reversal:
  • Above $T_N$: AHC is large and positive.
  • Below $T_N$: AHC rapidly switches to negative values within a narrow temperature range.
  • Peak Behavior: A pronounced negative peak in AHC occurs at $T \approx 0.68 T_N$. This is attributed to a Lifshitz transition where small hole pockets (contributing positive AHC) vanish, leaving electron pockets with strong positive Berry curvature that dominate the integral to produce a net negative AHC.
• Field Dependence:
  • The AHC shows a nearly linear dependence on the magnetic field at most temperatures.
  • Near $T = 0.68 T_N$, the dependence becomes nonlinear, reflecting the complex evolution of the Berry curvature and spin canting near the Fermi level.
• Doping Sensitivity:
  • The sign crossover temperature is highly sensitive to the Fermi level position ($\Delta E_F$). Hole doping ($\Delta E_F < 0$) significantly lowers the crossover temperature.
• Ordinary Hall Effect (OHE):
  • The calculated OHE shows a monotonic temperature dependence with a kink at $T_N$, consistent with experimental trends, though the theoretical magnitude differs slightly, suggesting the need to account for temperature-dependent carrier mobility.
• Magnitude: The calculated field-induced AHC reaches several hundred S/cm, comparable to or exceeding values reported for Dirac semimetals.

5. Significance

• Prototypical Platform: FeTe is established as a prototypical platform where magnetism and topology cooperate to produce robust intrinsic Hall responses in collinear AFMs.
• Spintronics Potential: The ability to tune the Hall effect (magnitude and sign) via temperature and magnetic field in a low-power, zero-net-moment material makes FeTe a promising candidate for ultrafast, low-power spintronic logic devices.
• General Applicability: The theoretical framework developed here can be extended to other correlated van der Waals magnets, offering a general route to explore Berry-curvature-driven transport in low-dimensional systems.
• Experimental Validation: The results provide a microscopic explanation for recent experimental observations of field-induced AHE in FeTe, distinguishing intrinsic topological effects from ordinary Lorentz force contributions.

In summary, this work demonstrates that even in compensated collinear antiferromagnets, external fields can unlock large topological transport phenomena, bridging the gap between magnetic order and quantum topology.