Accelerated ANE in Magnetic Thermoelectrics

• ANE acceleration is a strategy to enhance the anomalous Nernst effect by engineering Berry curvature hotspots and leveraging magnon-drag mechanisms.
• Intrinsic contributions from Berry curvature and extrinsic magnon-induced spin transfer collectively boost ANE coefficients in various magnetic materials.
• Experimental advances in doping, strain engineering, and device design have achieved significant enhancements, enabling efficient energy harvesting and spin-caloritronic applications.

ANE acceleration refers to strategies for enhancing the anomalous Nernst effect (ANE)—a transverse thermoelectric phenomenon wherein a temperature gradient in a magnetic material induces an orthogonal electric field via spin–orbit- and Berry-phase-driven electronic structure effects. Maximizing the ANE is of central interest for magnetic energy harvesting, spin-caloritronic devices, and integrated thermoelectric converters, where large ANE coefficients (both thermopower $S_{xy}$ and thermoelectric conductivity $\alpha_{xy}$) enable efficient conversion of waste heat to electrical power. Contemporary research encompasses both intrinsic mechanisms (Berry curvature of the electronic bands near the Fermi energy) and extrinsic mechanisms (notably magnon-drag effects) in a range of magnetic materials, including ferromagnets, compensated magnets, and altermagnets.

1. Theoretical Foundations of ANE and Its Enhancement

The intrinsic contribution to the anomalous Nernst conductivity $\alpha_{xy}$ is encoded by Berry curvature hotspots near the Fermi level. In metals with nonzero Berry curvature, such as Heusler alloys, ferromagnetic thin films, and collinear altermagnets, the ANE can be written as: $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$ where $\Omega^z_n(\mathbf{k})$ is the Berry curvature, $\epsilon_n(\mathbf{k})$ the band energies, $\mu$ the chemical potential, and $s(E)$ is an entropy kernel peaked within $k_B T$ of $\mu$ (Han et al., 2024). This formalism underlines that rapid ANE acceleration is possible when Berry curvature “hotspots” closely coincide with the Fermi energy.

Alongside the intrinsic term, recent studies (e.g., in MnBi) demonstrate a giant extrinsic ANE originating from magnon-induced spin angular momentum transfer to conduction electrons. This advective magnon term also depends strongly on material parameters—especially spin–orbit coupling (SOC), magnon–electron coupling strength, and magnetic order topology (He et al., 2020).

2. Berry Curvature Engineering and Electronic Structure Design

A principal avenue for ANE acceleration is the deliberate positioning of Berry curvature hotspots at the Fermi level. In collinear altermagnets such as Mn$\alpha_{xy}$0Si$\alpha_{xy}$1, an alternating spin-splitting of 3$\alpha_{xy}$2-derived bands—gapped by SOC—produces sharply peaked Berry curvature regions. First-principles calculations identify a strong sensitivity of $\alpha_{xy}$3 to small shifts of $\alpha_{xy}$4, with even a $\alpha_{xy}$5 upward shift leading to a sixfold enhancement of $\alpha_{xy}$6 and $\alpha_{xy}$7 (Han et al., 2024). This is experimentally realized by controlled Mn doping in Mn$\alpha_{xy}$8Si$\alpha_{xy}$9.

In ferromagnetic thin films (e.g., Mn$\alpha_{xy}$0Ge$\alpha_{xy}$1C$\alpha_{xy}$2), interstitial C expands the lattice, modifies the $\alpha_{xy}$3-band density of states, and drives avoided crossings near $\alpha_{xy}$4. These changes amplify intrinsic $\alpha_{xy}$5 and the overall Hall angle, causing S$\alpha_{xy}$6 (the anomalous Nernst coefficient) to increase by a factor of three at $\alpha_{xy}$7 versus undoped films (Kraft et al., 2020).

Design principles emerging from these results include:

• Chemical substitution or strain engineering to align Fermi level with topological band features.
• Selection of nontrivial topological materials (e.g., with Weyl or Dirac points) to maximize Berry curvature.
• Alloying/doping with heavy elements to enhance SOC without degrading magnetic ordering.

3. Extrinsic Magnon Contributions: Advective Mechanisms

In certain metallic ferromagnets, the ANE is dramatically enhanced via extrinsic magnon-drag mechanisms. Under a thermal gradient, magnon flow in the bulk generates a transverse spin current; in the presence of strong SOC, this is converted to a charge voltage by the inverse spin Hall effect (ISHE): $\alpha_{xy}$8 where $\alpha_{xy}$9 reflects the spin Hall angle, $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$0 the magnon–electron coupling, $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$1 and $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$2 magnon velocity and population, and $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$3 the magnon–electron scattering time (He et al., 2020).

Experimental data from single-crystal MnBi demonstrates two-orders-of-magnitude excess in $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$4 over intrinsic predictions (up to $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$5 and $$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$6), clearly indicating a dominant magnon-induced mechanism. The design levers for maximizing this advective term are:

• High SOC (via heavy element incorporation)
• Maximized magnon density (by lowering spin-wave stiffness, optimizing temperature, and maintaining moderate fields)
• Enhanced magnon–electron interaction (via exchange doping or controlled disorder)

4. Experimental Realizations and Material Optimization

Tables of representative strategies and enhancements:

System	Tuning Parameter	Enhancement Factor (Approx.)
Mn$$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$7Ge$$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$8C$$\alpha_{xy}(\mu, T) = k_B \sum_n \int_{BZ} \frac{d^3k}{(2\pi)^3} \Omega^z_n(\mathbf{k})\, s[\epsilon_n(\mathbf{k})-\mu]$$9 thin films	C doping ($\Omega^z_n(\mathbf{k})$0)	$\Omega^z_n(\mathbf{k})$1 in $\Omega^z_n(\mathbf{k})$2
Mn$\Omega^z_n(\mathbf{k})$3Si$\Omega^z_n(\mathbf{k})$4 thin films	Mn excess ($\Omega^z_n(\mathbf{k})$5)	$\Omega^z_n(\mathbf{k})$6 in $\Omega^z_n(\mathbf{k})$7/$\Omega^z_n(\mathbf{k})$8
MnBi single crystal	Magnon drag (SOC)	$\Omega^z_n(\mathbf{k})$9 vs. theory

Controlled experiments consistently highlight the utility of:

• Perpendicular heat-flow geometry (film on low-thermal-conductivity substrate) to maximize $\epsilon_n(\mathbf{k})$0
• Thin films and high-quality interfaces to confine and control heat and charge flows
• Maintaining moderate resistivity to preserve a large Hall angle without sacrificing conductivity
• Ensuring high Curie or Néel temperature for device stability at operational $\epsilon_n(\mathbf{k})$1

In altermagnets, stabilization of the collinear order and crystalline epitaxy sustain the alternating spin-splitting and thereby the Berry curvature hotspots crucial for robust, nonvolatile ANE (Han et al., 2024).

5. Temperature Dependence and Magnetic Order Effects

The scaling of $\epsilon_n(\mathbf{k})$2 with temperature is critically dependent on the type of magnetic order. In Mn$\epsilon_n(\mathbf{k})$3Si$\epsilon_n(\mathbf{k})$4, with a collinear Néel vector, the scaling follows $\epsilon_n(\mathbf{k})$5 with $\epsilon_n(\mathbf{k})$6, distinct from non-collinear cases such as Mn$\epsilon_n(\mathbf{k})$7Sn ($\epsilon_n(\mathbf{k})$8). The peak in $\epsilon_n(\mathbf{k})$9 is sharply tied to the onset of antiferromagnetic order: the ANE rises rapidly below the Néel temperature, peaks at intermediate $\mu$0, and decays slowly at low $\mu$1 (Han et al., 2024). This is contrasted with ferromagnets, where extrinsic contributions peak well below $\mu$2, displaying monotonic decrease with increasing $\mu$3 (He et al., 2020, Kraft et al., 2020).

Such behaviors are attributed to different regimes of thermal fluctuation of the Néel vector and the presence or absence of net magnetization. In collinear altermagnets, field-free, non-volatile ANE is possible, with the sign of $\mu$4 controlled by the Néel vector, independent of net magnetization.

6. Practical Guidelines for ANE Acceleration

Synthesizing the findings across materials systems:

• Fermi Level Tuning: Fine control (doping, gating) to align $\mu$5 with calculated Berry curvature peaks.
• Crystalline and Strain Engineering: High-quality, epitaxial growth and strain application to target optimal $\mu$6/$\mu$7 and band structure.
• Material Selection/Alloying: Use of heavy elements for SOC, selection for high ordering temperature, and attention to nontrivial band topology.
• Magnon Population and Coupling: Enhance at moderate $\mu$8 ($\mu$9–$s(E)$0), maximize magnon–electron interactions, and utilize magnetic fields to align domains without gapping low-energy magnon modes.
• Device Integration: Leverage perpendicular geometries and optimal layer stacks for maximal $s(E)$1 and readout efficiency.

These principles yield transverse thermoelectric responses substantially exceeding those dictated by band structure alone, supporting device concepts in integrated energy harvesting, high-frequency thermoelectric sensors, and spintronic logic (He et al., 2020, Kraft et al., 2020, Han et al., 2024).

7. Outlook and Open Challenges

Despite significant progress, the quantitative model–experiment agreement, especially in highly doped or extrinsically dominated systems, remains incomplete—pointing to unidentified electronic and magnonic scattering mechanisms (Kraft et al., 2020). Further, extension to new classes of compensated magnets and the development of robust, field-free ANE devices require ongoing advancement in band structure modeling, thin film synthesis, and multi-modal doping strategies. The intersection of Berry curvature engineering, magnonics, and device nanofabrication defines the next frontier for ANE acceleration in condensed matter and applied spin-caloritronics.