Anomalous Nernst Effect (ANE) in Magnetic Materials

• ANE is a transverse thermoelectric phenomenon in magnetic materials that converts a temperature gradient into a voltage perpendicular to both magnetization and thermal flow.
• Recent research highlights large ANE signals in diverse systems like ferromagnets, noncollinear antiferromagnets, and Weyl semimetals, expanding its applications in spin caloritronics and nanoscale imaging.
• Engineering strategies such as doping, strain, and interface tuning allow precise control of Berry curvature and ANE magnitude, facilitating advances in energy harvesting and ultrafast spin technologies.

The anomalous Nernst effect (ANE) is a transverse thermoelectric phenomenon observed in magnetic materials, in which the application of a temperature gradient (∇T) generates an electromotive force perpendicular to both the magnetization (M) and the gradient. While traditionally associated with ferromagnets, recent research has established the ANE as a generic probe of Berry curvature at the Fermi level, with large signals observed in noncollinear antiferromagnets, compensated ferrimagnets, magnetic Weyl semimetals, and even collinear altermagnets. The ANE is of central importance in spin caloritronics, energy harvesting, ultrafast spintronics, and nanoscale magnetic imaging due to its compatibility with planar device architectures, its sensitivity to band-topological features, and its ability to operate without external magnetic fields.

1. Theoretical Foundation and Formalism

The ANE is described phenomenologically by the relation

$$\mathbf{E}_{\rm ANE} = S_{\rm ANE} (\nabla T \times \mathbf{M})$$

where $\mathbf{E}_{\rm ANE}$ is the induced electric field, $S_{\rm ANE}$ the anomalous Nernst coefficient, $\nabla T$ the applied thermal gradient, and $\mathbf{M}$ the (local) magnetization. In tensor notation, the effect is encoded in the off-diagonal thermoelectric tensor components $\alpha_{xy}$, yielding a transverse current density under $\nabla_xT$ as $j_y = -\alpha_{yx} \nabla_xT$.

The microscopic mechanism, for the intrinsic contribution, is rooted in the Berry curvature $\Omega_n(\mathbf{k})$ of electronic bands,

$$\alpha_{xy} = -\frac{1}{T} \sum_n \int_{\text{BZ}} \frac{d^3k}{(2\pi)^3} \Omega_n^z(\mathbf{k}) s\big(\varepsilon_n(\mathbf{k})\big)$$

where $\mathbf{E}_{\rm ANE}$0 is the entropy density per state and $\mathbf{E}_{\rm ANE}$1 indexes bands. At low temperature, the Mott relation links the transverse thermoelectric to the energy derivative of the anomalous Hall conductivity:

$\mathbf{E}_{\rm ANE}$2

This formalism undergirds the vast majority of recent quantitative theory and simulation of ANE in both topological and conventional magnetic materials (Pan et al., 2021, Ikhlas et al., 2017, Yang et al., 2018, Zhou et al., 2019).

2. Material Classes and Symmetry Considerations

The ANE requires both time-reversal symmetry breaking and significant spin–orbit coupling (SOC), but does not necessarily require net magnetization. Material systems exhibiting large or tunable ANE include:

• Ferromagnets and Ferrimagnets: The conventional context, but recent work shows the ANE magnitude can far exceed predictions based on $\mathbf{E}_{\rm ANE}$3 scaling if Berry curvature is sharply peaked at the Fermi level (e.g., magnetic Weyl semimetals Co$\mathbf{E}_{\rm ANE}$4Sn$\mathbf{E}_{\rm ANE}$5S$\mathbf{E}_{\rm ANE}$6 (Yang et al., 2018), UCo$\mathbf{E}_{\rm ANE}$7Ru$\mathbf{E}_{\rm ANE}$8Al (Asaba et al., 2021), Co$\mathbf{E}_{\rm ANE}$9MnAl$S_{\rm ANE}$0Si$S_{\rm ANE}$1 Heuslers (Sakuraba et al., 2018)).
• Noncollinear Antiferromagnets: In Mn$S_{\rm ANE}$2Sn (Ikhlas et al., 2017) and Mn$S_{\rm ANE}$3NiN (Zhou et al., 2019), a chiral triangular spin structure enables a large zero-field ANE, circumventing the vanishing Berry curvature of conventional collinear compensated AFMs.
• Canted and Collinear Antiferromagnets: Canted structures (YbMnBi$S_{\rm ANE}$4 (Pan et al., 2021)) or altermagnets (Mn$S_{\rm ANE}$5Si$S_{\rm ANE}$6 (Han et al., 2024)) can support an ANE via symmetry-allowed Berry curvature even with vanishing net magnetization.
• Compensated Ferrimagnets and Amorphous Alloys: As demonstrated in Co$S_{\rm ANE}$7Gd$S_{\rm ANE}$8 (Liu et al., 2022), TbCo (Odagiri et al., 2024), and GdFe (Kurokawa et al., 2021), a substantial ANE persists even at magnetic compensation; the sign is controlled by the dominant transition-metal sublattice and its associated Berry curvature.

A critical insight is the pivotal role of symmetry: noncollinear spin arrangements or altermagnetic band structures may induce finite Berry curvature and hence ANE even in the absence of macroscopic M (Han et al., 2024, Ikhlas et al., 2017).

3. Microscopic Origins: Berry Curvature and Extrinsic Mechanisms

While the intrinsic ANE is dominated by Berry curvature at or near the Fermi surface, extrinsic mechanisms such as skew scattering and side-jump also contribute, especially in classic 3$S_{\rm ANE}$9 perovskite ferromagnets La$\nabla T$0Na$\nabla T$1MnO$\nabla T$2 (Ghosh et al., 2018). The magnitude, sign, and temperature dependence of the ANE can thus be tailored via electronic structure engineering:

• Weyl Nodes, Nodal Lines, and Flat Bands: Materials with Fermi-energy–proximate Weyl points or topological nodal lines (e.g., Co$\nabla T$3Sn$\nabla T$4S$\nabla T$5, Mn$\nabla T$6NiN, Fe$\nabla T$7Ga) generate sharp Berry curvature hot spots, greatly enhancing the ANE (Yang et al., 2018, Zhou et al., 2019, Stejskal et al., 2023).
• Band Engineering: Doping, strain, and alloying shift the Fermi level with respect to Berry curvature features, enabling sign and magnitude control of $\nabla T$8 (e.g., Fe$\nabla T$9Ga under strain and doping, Mn$\mathbf{M}$0Si$\mathbf{M}$1 via Mn content) (Stejskal et al., 2023, Han et al., 2024).
• Interfaces and Multilayers: In Ni/Pt and Pt/Fe multilayers, interfacial electronic structure modification can boost $\mathbf{M}$2 and the ANE far above bulk values, independent of proximity magnetism (Seki et al., 2020, Uchida et al., 2015).

4. Experimental Observation and Quantification

Experimental quantification of ANE requires precise control and measurement of temperature gradients and detection of transverse voltages:

Methodology	Key Features	Representative Systems
Heater-based gradients	Steady-state, macroscale $\mathbf{M}$3; broad applicability	Co, Ni/Pt, Heuslers
Laser/AFM-induced gradients	Localized, intense, fast-modulated $\mathbf{M}$4; micro/nanoscale, time-resolved	Co thin films, near-field imaging
Strain tuning	Control of magnetic phase and Berry curvature via epitaxy or piezoactuators	Mn$\mathbf{M}$5SnN, Fe$\mathbf{M}$6Ga, Mn$\mathbf{M}$7NiN
Terahertz emission	Ultrafast detection of ANE-driven currents (sub-ps) via THz radiation	Fe, Co films (Feng et al., 2023)

Finite-element modeling is essential for extracting accurate $\mathbf{M}$8 values under inhomogeneous or nanoscale temperature profiles (Mochizuki et al., 28 Jan 2025, Pandey et al., 2024). ANE-based imaging achieves sub-100 nm spatial resolution using near-field laser excitation (Pandey et al., 2024).

5. Quantitative Trends and Optimization Principles

The magnitude of ANE in notable systems spans several orders:

Material/System	Reported $\mathbf{M}$9 ($\alpha_{xy}$0V/K)	$\alpha_{xy}$1 (A/K·m)	Remarks
UCo$\alpha_{xy}$2Ru$\alpha_{xy}$3Al (Asaba et al., 2021)	23	21	Colossal ANE, 5f systems, %%%%54$\mathbf{E}_{\rm ANE}$055%%%% Weyl nodes
Co$\alpha_{xy}$6MnAl$\alpha_{xy}$7Si$\alpha_{xy}$8 (Sakuraba et al., 2018)	6.2	—	Giant ANE in B2/L2$\alpha_{xy}$9-Heuslers, Berry curvature peak
Co$\nabla_xT$0Sn$\nabla_xT$1S$\nabla_xT$2 (Yang et al., 2018)	5	10	Weyl semimetal, large $\nabla_xT$3
YbMnBi$\nabla_xT$4 (Pan et al., 2021)	6	10	Canted AFM, extremely low M
Mn$\nabla_xT$5NiN (AFM) (Zhou et al., 2019)	0.5 (in films)	1.8	Largest in AFMs; symmetry-protected
Amorphous Tb$\nabla_xT$6(Fe$\nabla_xT$7Co$\nabla_xT$8)$\nabla_xT$9 (Imaeda et al., 19 Dec 2025)	1.8	0.8	Combinatorial maximum via direct + indirect ANE terms
Gd$j_y = -\alpha_{yx} \nabla_xT$0Fe$j_y = -\alpha_{yx} \nabla_xT$1 glass (Kurokawa et al., 2021)	2.13	—	Flexible, amorphous, near-compensation
Co$j_y = -\alpha_{yx} \nabla_xT$2Gd$j_y = -\alpha_{yx} \nabla_xT$3 films (Liu et al., 2022)	0.13–0.15 (near compensation)	—	ANE polarity governed by Co sublattice
Classic 3d FM (bulk Ni)	0.2–0.6	0.1–1	Reference for enhancement

Strategies for maximizing ANE focus on maximizing the derivative $j_y = -\alpha_{yx} \nabla_xT$4 at $j_y = -\alpha_{yx} \nabla_xT$5 (steep Berry curvature slopes) and optimizing phase, band filling, and disorder to maintain transport.

6. Nanoscale and Ultrafast Applications

The ANE serves as a detection principle in several advanced device architectures:

• Planar Thermopiles: The transverse geometry enables modules with laterally arranged elements, offering full heat-source coverage and higher packing density without the need for stray-field mitigation, particularly using antiferromagnetic or compensated materials (Ikhlas et al., 2017, Zhou et al., 2019).
• Heat-Flux Sensors/Harvesters: Zero-field operation and large coercive force in materials such as amorphous TbCo and GdFe films facilitate robust, self-powered devices (Odagiri et al., 2024, Kurokawa et al., 2021).
• Near-field Magnetothermal Imaging: Laser-induced ANE voltages provide quantitative, spatially-resolved detection of magnetic textures in both ferromagnets and antiferromagnets, with sub-100 nm spatial resolution (Pandey et al., 2024).
• Ultrafast Spin Caloritronics: Femtosecond laser pulses drive picosecond ANE currents, emitting terahertz radiation and enabling interrogation of ultrafast magnetization dynamics (Feng et al., 2023).

7. Future Directions and Open Questions

Ongoing and future research is focused on:

• Altermagnets and Compensated Systems: Confirming and optimizing ANE in collinear, compensated systems where symmetry breaking is exclusively encoded in band topology, as recently shown in Mn$j_y = -\alpha_{yx} \nabla_xT$6Si$j_y = -\alpha_{yx} \nabla_xT$7 (Han et al., 2024).
• Fermi Level and Band Structure Engineering: Systematic control of doping, strain, and interface design to position Berry curvature hotspots at $j_y = -\alpha_{yx} \nabla_xT$8, maximize $j_y = -\alpha_{yx} \nabla_xT$9, and realize giant or sign-reversible ANE (Stejskal et al., 2023, Imaeda et al., 19 Dec 2025).
• Disorder and Amorphous Effects: Determining the extent to which short-range order and transition-metal content in amorphous alloys determines Berry curvature and ANE, as exemplified by compositional engineering in Tb-Fe-Co (Imaeda et al., 19 Dec 2025).
• Scalability and Device Integration: Implementing robust, substrate-flexible, and zero-field devices with high $\Omega_n(\mathbf{k})$0 for thermoelectric and spintronic applications, including integration with complementary effects (e.g., spin Seebeck).

The ANE is thus a canonical example of macroscopic observable directly determined by quantum topological properties and is increasingly central in the design of next-generation thermoelectric, spintronic, and magneto-optical technologies.