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The spontaneous Nernst coefficient of ferromagnets from the interplay of electron scattering and Berry curvature

This paper employs a Boltzmann transport approach to demonstrate that the spontaneous Nernst coefficient in ferromagnetic metals is inversely proportional to the scattering time and directly linked to itinerant orbital angular momentum, providing a theoretical framework and practical guidelines for maximizing this effect through band structure engineering.

Original authors: Vittorio Basso, Adriano Di Pietro, Alessandro Sola

Published 2026-06-29

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CC BY 4.0

Original authors: Vittorio Basso, Adriano Di Pietro, Alessandro Sola

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a busy highway inside a metal. Electrons are the cars zooming along this road. Usually, if you heat one end of the metal, the cars just drift toward the cold end, creating a simple flow. But in certain magnetic metals (ferromagnets), something magical happens: the cars don't just move forward; they start swerving sideways, creating a "side current" even without any external magnetic field pushing them. This sideways movement caused by heat is called the Spontaneous Nernst Effect.

This paper is like a detective story trying to figure out exactly why these cars swerve and, more importantly, how to build a better highway to make the swerving as strong as possible.

Here is the breakdown of their findings using simple analogies:

1. The "Traffic Jam" Surprise

In normal physics, you usually think that for traffic to flow smoothly and efficiently, you want a clear road with no obstacles. You want the cars (electrons) to zoom freely without hitting anything.

However, the authors discovered a stark contrast for these magnetic metals. To get the strongest sideways swerving (the Nernst effect), you actually want more traffic jams (more scattering).

The Analogy: Imagine a dance floor. If everyone is gliding perfectly across the floor without bumping into anyone, they don't spin. But if the floor is crowded and people are bumping into each other (scattering), they start spinning and moving sideways more vigorously.
The Finding: The paper claims that efficient materials for this effect need to be "messy" with short distances between collisions, which is the opposite of what you need for standard electrical wires.

2. The Invisible "Curved Road" (Berry Curvature)

Why do the electrons swerve in the first place? The paper explains that the electrons aren't just moving on a flat, straight road. Because of the magnetic nature of the material, the "road" itself is curved in a hidden dimension called Berry Curvature.

The Analogy: Think of a car driving on a flat map. If the map is actually a globe (curved), the car will naturally drift off course even if the driver is steering straight. The electrons are "feeling" a curvature in their energy landscape that forces them to turn sideways.
The Connection: The authors link this sideways turn to something called Orbital Angular Momentum. Imagine the electrons aren't just driving; they are also spinning like tops. The paper connects the strength of the sideways swerve directly to how much these "spinning tops" are spinning as they move through the metal's energy bands.

3. The "Two-Lane Highway" Model

To predict how different metals (like Iron, Cobalt, and Nickel) behave, the authors built a simplified model. They imagine the electrons traveling in two separate lanes: one for "majority" spins (cars going one way) and one for "minority" spins (cars going the other way).

The Finding: By looking at how full these lanes are (electron filling), they could predict whether the metal would swerve left or right.
- Iron (Fe): The lanes are filled in a way that creates a strong negative swerve.
- Cobalt (Co) and Nickel (Ni): The filling changes, flipping the direction of the swerve.
The Result: Their simple model matched real-world experiments surprisingly well, correctly predicting the direction and size of the effect for these common metals.

4. How to Build a Better "Swerving Machine"

The ultimate goal of the paper is to find a recipe to make this effect stronger, which could help in making better heat sensors or energy harvesters.

The Recipe: The authors suggest tweaking the "traffic" by mixing metals. They specifically looked at mixing Nickel and Copper.
The Prediction: By adding Copper to Nickel, you create a "disordered" highway (more scattering, which helps the effect) and you also shift the "filling level" of the electron lanes to a sweet spot where the swerving is maximized.
The Outcome: Their model predicts that a specific mix of Nickel and Copper could be about 400 times more efficient at this effect than pure Nickel.

Summary

The paper argues that to harness the spontaneous Nernst effect (turning heat into sideways electricity in magnets), we shouldn't look for the cleanest, most perfect metals. Instead, we should look for materials with:

Strong scattering (a bit of "traffic" or disorder).
Specific electron filling (adjusting how full the energy lanes are).
Strong orbital spin (electrons that are "spinning" as they move).

By mixing metals like Nickel and Copper, we might be able to engineer materials that are much better at harvesting heat energy than anything we have today.

Technical Summary: The Spontaneous Nernst Coefficient of Ferromagnets from the Interplay of Electron Scattering and Berry Curvature

Problem Statement
The spontaneous (or anomalous) Nernst effect in metallic ferromagnets is a promising mechanism for heat harvesting, flux sensing, and active cooling without external magnetic fields. Despite the existence of candidate materials such as MnBi, Fe $_3$ Ga, and various transition metal-rare earth compounds, the figure of merit ($zT$) for spontaneous Nernst devices remains orders of magnitude lower than that of ordinary Nernst materials or longitudinal thermoelectrics. A primary obstacle to optimization is the lack of a clear theoretical framework connecting the spontaneous Nernst coefficient ( $\varepsilon^s_N$ ) to microscopic material properties, particularly the role of electron scattering and the specific contributions of Bloch band topology (Berry curvature). Unlike the ordinary Nernst effect, where efficiency is maximized by minimizing scattering, the dependence of the spontaneous effect on scattering mechanisms has been less understood, hindering the design of optimized active materials.

Methodology
The authors employ the Boltzmann transport approach, explicitly incorporating the Berry curvature of Bloch bands as a source of transverse current density. The derivation treats the transverse current arising from Berry curvature as a Fermi surface property, distinct from the conventional group velocity contribution.

Key methodological steps include:

Transport Formalism: The authors derive the transverse current density by combining the Boltzmann equation with the anomalous velocity term induced by Berry curvature ( $\Omega_n$ ). They demonstrate that the spontaneous transverse conductivity ( $\sigma_{s,xy}$ ) can be expressed as a Fermi surface integral involving the Berry connection.
Scattering Dependence: By applying Mott's expression for thermoelectric conductivity, the authors analyze the energy dependence of the transport coefficients. They explicitly contrast the scattering dependence of the ordinary Nernst effect (proportional to scattering time $\tau_c$ ) with the spontaneous effect.
Connection to Orbital Angular Momentum: The study establishes a direct link between the spontaneous transverse conductivity and the density of itinerant orbital angular momentum ( $L_b$ ) arising from Bloch bands, distinguishing it from the atomic orbital contribution ( $L_a$ ).
Modeling: To evaluate thermoelectric coefficients and predict trends, the authors construct a rigid two-band tight-binding model for 3d transition metal ferromagnets. This model incorporates spin-orbit coupling, exchange splitting, and a semi-circular density of states (DOS) approximation to simulate the electronic structure of Fe, Co, and Ni.

Key Contributions and Results

Inverse Scattering Dependence: A central finding is that the spontaneous Nernst coefficient ( $\varepsilon^s_N$ ) is proportional to the inverse of the scattering time constant ( $\tau_c^{-1}$ ). This implies that, contrary to ordinary Nernst materials which require long mean free paths, efficient spontaneous Nernst materials should exhibit relatively strong electron scattering. This result arises because the spontaneous transverse conductivity ( $\sigma_{s,xy}$ ) is an intrinsic, scattering-independent property, while the longitudinal conductivity ( $\sigma_e$ ) scales with $\tau_c$ . Consequently, the Nernst coefficient $\varepsilon^s_N \propto \sigma_{s,xy}/\sigma_e$ scales as $\tau_c^{-1}$ .
Orbital Angular Momentum Connection: The authors demonstrate that the strength and sign of the spontaneous Nernst coefficient are directly connected to the itinerant contribution to the orbital angular momentum density ( $L_b$ ) derived from the Bloch bands. They propose a "third Hund's rule" for Bloch states to predict the placement of large $L_a$ and $L_b$ states within the energy bands of ferromagnets.
Sign and Magnitude Prediction: Using the two-band rigid model, the authors successfully reproduce the signs and orders of magnitude of experimental thermoelectric coefficients for Fe, Co, and Ni. The model explains the sign changes in thermopower and Nernst coefficients as a function of electron filling, attributing them to the interplay between majority and minority spin bands and the energy derivative of the orbital momentum density.
Optimization Strategy: The model suggests that maximizing the spontaneous Nernst effect involves tailoring the electronic band structure to position the Fermi level near the complete filling or emptying of the d-bands (e.g., $N_d \approx 0$ or $N_d \approx 4$ ). In these regions, the contributions from the spontaneous Hall conductivity and the intrinsic Nernst conductivity act constructively.
Case Study (Ni-Cu Alloys): Applying the model to Ni $_{1-x}$ Cu $_x$ alloys, the authors predict a significant improvement in $zT$ (reaching $\sim 1.32 \times 10^{-3}$ at $x \approx 0.53$ ) compared to pure Ni. This improvement is attributed to two factors: optimized band structure contributions (constructive interference of coefficients) and enhanced scattering due to the disordered alloy structure.

Significance and Claims
The paper claims to provide a unified theoretical framework that clarifies the microscopic origin of the spontaneous Nernst effect, resolving the confusion regarding its dependence on scattering. By identifying the inverse relationship with scattering time, the authors offer a counter-intuitive but practical guideline for material optimization: rather than seeking ultra-pure crystals, one should look for magnetic materials with strong scattering mechanisms combined with specific band structure features.

The authors modestly state that while their rigid band model is a simplification, it successfully captures the essential physics governing the signs and magnitudes of the relevant coefficients. They propose that the derived "recipes" for maximizing the effect—specifically through electronic band structure tailoring and the utilization of alloy disorder to enhance scattering—provide a concrete path forward for developing high-performance spontaneous Nernst materials. The work suggests that future extensions could replace the rigid band model with more accurate DFT calculations to refine these predictions for other classes of magnetic metals.

1. The "Traffic Jam" Surprise

2. The Invisible "Curved Road" (Berry Curvature)

3. The "Two-Lane Highway" Model

4. How to Build a Better "Swerving Machine"

Summary

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