Observation of the In-plane Anomalous Hall Effect induced by Octupole in Magnetization Space

This paper challenges conventional theories by demonstrating that in-plane magnetization induces an Anomalous Hall Effect in cubic ferromagnets like iron and nickel, a phenomenon driven by a previously overlooked octupole moment in the anomalous Hall conductivity that promises to revolutionize magnetic sensor design.

The Gist

Imagine you have a compass, but instead of pointing North, it reacts to electricity flowing through a metal. This is the Anomalous Hall Effect (AHE). For a long time, scientists thought this "compass" only worked when the magnetic force (magnetization) was pointing straight up or down, like a flagpole sticking out of a table. If the magnetism was lying flat on the table (in-plane), the compass was supposed to be blind.

This paper says: "Not so fast." The researchers discovered that in common metals like Iron and Nickel, this compass can actually see the flat, in-plane magnetism. They found a way to make the "flagpole" effect work even when the magnet is lying down.

Here is how they did it, using some simple analogies:

1. The Old Rule: The Perfectly Aligned Arrow

Usually, when electricity flows through a magnet, the resulting voltage (the signal) points in the exact same direction as the magnet's internal force.

• The Analogy: Imagine a perfectly synchronized dance. If the magnet (the dancer) moves North, the electrical signal (the partner) also moves North. If the magnet lies flat on the floor, the signal lies flat too. Because of this perfect alignment, if you try to measure a signal coming out of the floor (which is what we usually look for), you get nothing when the magnet is flat.

2. The New Discovery: The "Octupole" Twist

The researchers found that in these metals, there is a hidden, complex rule that breaks this perfect synchronization. They call this hidden rule an "Octupole."

• The Analogy: Imagine the magnet is a dancer, but instead of just moving in a straight line, they have a secret, complex spin.
  • In the old view, the dancer moves North, and the partner moves North.
  • With this new "Octupole" twist, if the dancer moves in a specific direction (like a diagonal), the partner doesn't just follow; they get pushed slightly to the side.
  • The Result: Even though the magnet is lying flat on the table, this "twist" pushes the electrical signal slightly up into the air. Suddenly, the "flat" magnet creates a "vertical" signal that we can finally detect!

3. The Experiment: Testing the Theory

The team tested this on two very common materials: Iron and Nickel.

• They made thin films of these metals and set up a specific orientation (like tilting the metal at a specific angle).
• They ran electricity through the metal and applied a magnetic field that lay flat on the surface.
• The Outcome: Just like the theory predicted, they saw a voltage signal appearing perpendicular to the flat magnetism.
  • When they aligned the magnetic field with one specific direction on the metal, the "twist" happened, and they saw the signal.
  • When they rotated the field to a different direction, the "twist" canceled out, and the signal disappeared.
• They also checked a different type of Iron film (Fe 001) and found no signal, proving that this effect depends entirely on the specific crystal shape of the metal, just as their math predicted.

4. Why This Matters (According to the Paper)

The paper claims this is a major shift in understanding.

• Breaking the Rules: For decades, theories said this "in-plane" signal was impossible in these common, symmetrical metals. This paper proves that theory wrong by finding the hidden "Octupole" mechanism.
• A New Tool: This discovery means we can now detect flat magnetism in common metals without needing complex, special-shaped devices.
• Future Possibilities: The authors suggest that because this "Octupole" effect exists in the mathematical structure of magnetism, it might also explain similar "flat" effects in other areas, like thermal electricity (heat turning into electricity), though they didn't test those specifically in this study.

In short: The researchers found a hidden "twist" in the physics of Iron and Nickel that allows them to detect flat magnetism, a feat previously thought impossible. They didn't just find a new material; they found a new way to look at old, common materials.

Technical Summary

Technical Summary: Observation of the In-plane Anomalous Hall Effect Induced by Octupole in Magnetization Space

Problem Statement
The Anomalous Hall Effect (AHE) is a fundamental phenomenon in condensed matter physics, widely utilized for probing magnetism in nanoscale devices due to its high sensitivity and ease of integration. However, a critical limitation exists: conventional AHE is exclusively sensitive to out-of-plane magnetization. In ferromagnetic films, where in-plane magnetization is prevalent, the AHE fails to discern the magnetization direction, necessitating complex device architectures or indirect methodologies. While theoretical studies suggested that in-plane AHE could occur in systems with singular mirror symmetry or strong crystalline anisotropy, the scarcity of materials satisfying both ferromagnetism and reduced symmetry has hindered experimental realization. Furthermore, existing theories for cubic crystal systems (such as iron and nickel) generally forbid in-plane AHE, as the anomalous Hall conductivity vector ($\sigma_{AHE}$) is assumed to align strictly parallel to the magnetization ($M$).

Methodology and Theoretical Framework
The authors challenge this conventional understanding by proposing a multipolar expansion of the anomalous Hall conductivity in magnetization space. They express $\sigma_{AHE}$ as a series of multipoles dependent on the unit magnetization vector $\hat{M}$:
$$\sigma^i_{AHE} = p_{ij}\hat{M}_j + \frac{1}{15}o_{ijkl}\hat{M}_j\hat{M}_k\hat{M}_l$$
where the first term represents the dipole contribution and the second term represents the octupole contribution.

• Dipole Term: In cubic crystals (e.g., Fe, Ni), the dipole coefficient reduces to a scalar, forcing $\sigma_{AHE}$ to be parallel to $M$, which precludes in-plane AHE.
• Octupole Term: The authors identify the rank-4 octupole tensor ($o_{ijkl}$) as the origin of in-plane AHE. In cubic lattices, this term introduces a component $\beta \hat{M}^3_i$ that causes a misalignment between $\sigma_{AHE}$ and $M$. This misalignment generates a finite anomalous Hall conductivity vector component ($\sigma^\perp_{AHE}$) normal to $M$, inducing an in-plane Hall voltage even when the electric field ($E$) is parallel to $M$.

Experimentally, the study utilizes epitaxially grown thin films of Fe(103) and Ni(111) on MgO substrates. The researchers employed Hall bar geometries to measure transverse resistivity ($\rho_{xy}$) under various in-plane magnetic field orientations. Key experimental strategies included:

1. Field Orientation Dependence: Sweeping in-plane magnetic fields along specific crystallographic directions (e.g., Fe[30$\bar{1}$] vs. Fe[0$\bar{1}$0]) to test the symmetry-dependent generation of $\sigma^\perp_{AHE}$.
2. High-Field Verification: Applying fields up to 90,000 Oe to distinguish the in-plane AHE signal from artifacts caused by out-of-plane magnetization components (which would decay with field saturation).
3. Angular Rotation: Rotating the in-plane magnetic field to analyze the harmonic dependence of the Hall signal, separating the odd-parity in-plane AHE from the even-parity Planar Hall Effect (PHE) using Fourier analysis.

Key Results

1. Observation in Iron: In Fe(103) films, a clear anomalous Hall signal was observed when the magnetic field was aligned along Fe[30$\bar{1}$], but the signal vanished when aligned along Fe[0$\bar{1}$0]. This behavior contradicts the expectation of isotropic AHE in cubic systems and confirms the presence of an in-plane AHE. The signal persisted at high fields (90,000 Oe), ruling out contributions from tilted out-of-plane magnetization.
2. Observation in Nickel: Similar results were obtained in Ni(111) films. An antisymmetric Hall component (indicative of AHE) was detected when $H \parallel$ Ni[2$\bar{1}$$\bar{1}$] but was absent for $H \parallel$ Ni[10$\bar{1}$].
3. Symmetry Analysis: The angular dependence of the in-plane AHE signal in Fe(103) followed a $\sin^3\theta$ pattern, while Ni(111) exhibited a three-fold symmetry ($\cos(3\theta)$). These experimental curves matched the theoretical predictions derived from the octupole term in Eq. (2) with high precision.
4. Absence in Fe(001): Control experiments on Fe(001) films showed no in-plane AHE, consistent with the theoretical prediction that the octupole contribution vanishes for this specific orientation in cubic lattices.
5. Quantitative Values: The measured in-plane AHE conductivity in Fe was approximately $-34.5 \, \Omega^{-1}\text{cm}^{-1}$, which, while smaller than the linear AHE ($1122 \, \Omega^{-1}\text{cm}^{-1}$), is comparable to the largest AHE values found in ambient-temperature 2D ferromagnets.

Significance and Claims
The paper claims to have realized the in-plane AHE in common ferromagnets (Fe and Ni) by exploiting the previously neglected octupole of the anomalous Hall conductivity in magnetization space. The significance of this work is threefold:

• Theoretical Breakthrough: It identifies the AHE octupole as a fundamental origin of in-plane AHE, challenging the conventional view that cubic ferromagnets cannot support this effect.
• Paradigm Shift: It highlights the importance of geometric quantities (like Berry curvature) distributed within the magnetization space, shifting focus beyond real and momentum spaces.
• Technological Potential: The discovery offers a new physical mechanism for detecting in-plane magnetization, promising to revolutionize the design of magnetic devices and sensors by enabling direct AHE-based readout of in-plane magnetic states without complex architectures.

The authors further anticipate that this octupole mechanism may be ubiquitous across various Bravais lattices and could extend to other transport phenomena, such as the anomalous Nernst and Ettinghausen effects, constituting a novel classification of physical phenomena in condensed matter physics.