---
title: Berry-Curvature-Driven AEE in Magnetic Materials
url: https://www.emergentmind.com/topics/berry-curvature-driven-anomalous-ettingshausen-effect
type: topic
---

# Berry-Curvature-Driven AEE in Magnetic Materials

Berry-curvature-driven anomalous Ettingshausen effect is the intrinsic component of the anomalous Ettingshausen effect (AEE) in which a transverse heat current in a magnetic material is rooted in the Berry curvature of occupied bands near the Fermi level. In the experimentally established case of the Heusler alloy \(\mathrm{Co}_2\mathrm{MnGa}\), a micron-sized on-chip cooler exhibits a record anomalous Ettingshausen coefficient of \(-2.1\) mV at room temperature, and \(44\%\) of the total AEE is attributed to the intrinsic, Berry-curvature-related contribution, linking non-trivial band topology directly to transverse thermoelectric cooling performance [2403.13598].

## 1. Definition and transport coefficients

The anomalous Ettingshausen effect is a transverse thermoelectric effect in magnetic materials, analogous to the anomalous Hall and anomalous Nernst effects. When a charge current \(\mathbf{J}_c\) flows through a ferromagnetic material with magnetization \(\mathbf{M}\), a heat current is generated perpendicular to both \(\mathbf{J}_c\) and \(\mathbf{M}\). In the notation used for \(\mathrm{Co}_2\mathrm{MnGa}\),  
\[
\mathbf{J}_{q,\mathrm{AEE}}=\Pi_{\mathrm{AEE}}\left(\mathbf{J}_c\times \mathbf{m}\right),
\]
where \(\mathbf{m}\) is the unit vector of magnetization and \(\Pi_{\mathrm{AEE}}\) is the anomalous Ettingshausen coefficient [2403.13598].

The total AEE separates into intrinsic and extrinsic parts,
\[
\Pi_{\mathrm{AEE}}=\Pi_1+\Pi_2=(S_1+S_2)T=(\rho_{xx}\alpha_{xy}+\rho_{xy}\alpha_{xx})T.
\]
Here, \(\Pi_1\) and \(S_1\) denote the intrinsic, Berry-curvature-related contribution, while \(\Pi_2\) and \(S_2\) denote the extrinsic, impurity-scattering-related contribution; \(\alpha_{xy}\) is the transverse thermoelectric conductivity, and \(\rho_{xx}\) and \(\rho_{xy}\) are the longitudinal and transverse resistivities [2403.13598]. The anomalous Nernst effect (ANE) is the Onsager counterpart of the AEE, so discussions of Berry-curvature-driven ANE and AEE are closely coupled throughout the literature.

A common source of confusion is the relation to the classical Ettingshausen effect. The classical effect is conventionally framed through Lorentz-force-driven transverse thermomagnetic transport in an external magnetic field, whereas the anomalous effect is intrinsic to the material’s electronic structure and is present in magnetic systems because transverse charge and heat currents can be generated by Berry curvature without requiring an external magnetic field in the same way [2403.13598].

## 2. Microscopic origin in Berry curvature

The intrinsic contribution to AEE is rooted in the Berry curvature, which acts as a “magnetic field” in momentum space and gives rise to an anomalous velocity perpendicular to the applied electric field. In the oxide-interface literature, the semiclassical velocity is written as
\[
\mathbf{v}(\mathbf{k})=\frac{1}{\hbar}\frac{\partial E(\mathbf{k})}{\partial \mathbf{k}}-\frac{e}{\hbar}\mathbf{\mathcal E}\times \mathbf{b}(\mathbf{k}),
\]
where \(\mathbf{b}(\mathbf{k})\) is the Berry curvature. The intrinsic anomalous Hall conductivity is correspondingly expressed as
\[
\sigma_{xy}=-\frac{e^2}{\hbar}\int_{\mathrm{BZ}}\frac{d^3k}{(2\pi)^3}f(E_n(\mathbf{k}))\,b_z^n(\mathbf{k}),
\]
showing that the sign and magnitude depend sensitively on the occupation of bands with nontrivial Chern number near avoided crossings [1810.05619].

For anomalous thermoelectric transport in spin-orbit-coupled noncentrosymmetric metals, the anomalous conductivity and anomalous Peltier tensor are written as
\[
\sigma^a_{yz}=\frac{e^2}{\hbar}\sum_\lambda\int\frac{d^3\mathbf{k}}{(2\pi)^3}f^\lambda_{\mathbf{k}}\Omega^\lambda_x,\qquad
\alpha^a_{yz}=-\frac{k_B e}{\hbar}\sum_\lambda\int\frac{d^3\mathbf{k}}{(2\pi)^3}s^\lambda_{\mathbf{k}}\Omega^\lambda_x,
\]
with \(s^\lambda_{\mathbf{k}}\) the entropy density in momentum space. In this formulation, Berry curvature is the intrinsic source of transverse Hall and thermoelectric transport, and the AEE follows once these anomalous off-diagonal coefficients are converted into a temperature gradient or heat current under the appropriate boundary conditions [2508.13147].

Berry curvature can also enter through the phase-space measure. In the \(C_4K\) analysis of Hall transport, the phase-space density becomes
\[
D(\mathbf{k})=1+\frac{e}{\hbar}\mathbf{B}\cdot\mathbf{\Omega}(\mathbf{k}),
\]
and this produces a Berry-curvature-induced Hall term distinct from the conventional Hall conductivity. This broadens the microscopic picture of Berry-curvature-driven transverse transport beyond anomalous velocity alone [2504.08882].

## 3. Experimental benchmark: \(\mathrm{Co}_2\mathrm{MnGa}\)

\(\mathrm{Co}_2\mathrm{MnGa}\) is described as a magnetic Weyl semimetal/Heusler compound whose nontrivial topological band structure hosts Weyl nodes and large Berry curvature near the Fermi level. This amplifies intrinsic thermoelectric transport coefficients and produces giant anomalous Nernst and Ettingshausen effects. The paper further states that \(\mathrm{Co}_2\mathrm{MnGa}\) shows a strong departure from trivial scaling with magnetization seen in conventional ferromagnets, indicating that its topological character is essential to the size of the response [2403.13598].

The experimental demonstration uses micron-scale U-shaped \(\mathrm{Co}_2\mathrm{MnGa}\) devices together with scanning thermal microscopy and magnetotransport measurements. The reported room-temperature anomalous Ettingshausen coefficient is \(\Pi_{\mathrm{AEE}}=-2.1\) mV, and the transverse thermoelectric conductivity is \(\alpha_{xy}=-2.14~\mathrm{A~(Km)}^{-1}\). Using the decomposition
\[
\Pi_{\mathrm{AEE}}=(\rho_{xx}\alpha_{xy}+\rho_{xy}\alpha_{xx})T,
\]
the work quantifies the Berry-curvature-related intrinsic contribution as \(44\%\) of the total AEE. The paper emphasizes that this is a record for the highest \(\Pi_{\mathrm{AEE}}\) at room temperature and identifies the Berry curvature contribution as essential for explaining the enormous observed \(\Pi_{\mathrm{AEE}}\) [2403.13598].

In application terms, these measurements connect a band-topological quantity to device-level cooling. The transverse nature of the effect allows localized temperature control, and the large intrinsic component implies that the response is not limited to impurity-scattering optimization alone. In this sense, \(\mathrm{Co}_2\mathrm{MnGa}\) functions as a benchmark material for Berry-curvature-driven AEE-based spot cooling [2403.13598].

## 4. Symmetry, cancellation, and distinction from Lorentz-force transport

The Berry-curvature-driven anomalous Ettingshausen effect is symmetry constrained. In spin-orbit-coupled noncentrosymmetric metals, the unperturbed state preserves time-reversal symmetry, and the two relevant Fermi surfaces carry equal and opposite Berry curvature fluxes. In the time-reversal-symmetric state, Berry-curvature contributions cancel, so there is no anomalous Hall or anomalous Ettingshausen effect. Applying a real magnetic field breaks time-reversal symmetry, makes the two bands asymmetric in energy, and prevents exact cancellation in the band integrals, thereby unlocking the anomalous response [2508.13147].

This intrinsic anomalous contribution is qualitatively distinct from the conventional Lorentz-force-driven Ettingshausen effect. In the strain-induced SOC-NCM analysis, the Lorentz-force-driven contribution is odd in \(B\), linear at weak fields, and strongly dependent on scattering. By contrast, the Berry-curvature-driven anomalous Ettingshausen effect is described as even in \(B\), sometimes non-monotonic, and governed by the Berry curvature and orbital magnetic moment structure; its angular patterns include \(\xi_x\sim \sin(2\gamma)\), \(\xi_y\sim \cos\gamma\), and \(\xi_z\sim \sin^2\gamma\) for the geometries considered [2508.13147].

A related symmetry setting appears in \(C_4K\) materials, where \(C_4\) and time reversal \(K\) are broken individually but their combination is preserved. In that case the anomalous Hall effect is forbidden because \(\int_{\mathrm{BZ}}\Omega_z=0\), yet the Berry-curvature phase-space correction
\[
\sigma_H^B=\tau\frac{e^3}{\hbar}\int\frac{d^2\mathbf{k}}{(2\pi)^2}v_xv_y\Omega_z\frac{\partial f_0}{\partial\varepsilon}
\]
is symmetry allowed. The paper argues that thermomagnetic analogs, including Ettingshausen and Nernst responses, should inherit the same symmetry structure and exhibit a square-root onset at an altermagnetic transition, \(\sigma_H^B(T\to T_c^-)\propto \sqrt{T_c-T}\) [2504.08882]. This suggests that Berry-curvature-driven AEE is not restricted to ferromagnetic anomalous transport in the narrow sense, but extends to broader symmetry-enabled thermomagnetic phenomena.

## 5. Routes to engineering the effect

One route is direct topological band-structure design in magnetic Weyl semimetals. In \(\mathrm{Co}_2\mathrm{MnGa}\), Weyl nodes and large Berry curvature near the Fermi level are identified as the origin of the giant intrinsic thermoelectric coefficients, and the large AEE is explicitly tied to these nontrivial band-topological features [2403.13598].

A second route is interface engineering. In \(\mathrm{SrRuO}_3\) heterostructures, Berry curvature is manipulated by choosing \(\mathrm{SrIrO}_3\) or \(\mathrm{SrTiO}_3\) as adjacent layers, which changes orbital hybridization, magnetization, and lattice strain at atomically sharp interfaces. The anomalous Hall resistance reverses sign between symmetric \(\mathrm{SIO/SRO/SIO}\) and \(\mathrm{STO/SRO/STO}\) stacks, and asymmetric \(\mathrm{STO/SRO/SIO}\) structures show two coexisting anomalous Hall channels. The paper states that an immediate implication is that manipulation of Berry curvature at oxide interfaces could extend to the anomalous Ettingshausen effect and related thermoelectric phenomena because the transverse effects depend on the same Berry-curvature physics as their electrical analogs [1810.05619].

A third route is orbital design in low-symmetry crystals. In systems with effective \(L=1\) orbital degrees of freedom, acentric crystal fields and orbital Rashba coupling generate Berry-curvature hot-spots and singular pinch points, which naturally yield giant Berry-curvature dipoles and large nonlinear transport responses in time-reversal-symmetric conditions. The same work explicitly identifies nonlinear Hall, Ettingshausen, and Nernst effects as Berry-curvature-induced responses in such settings [2301.04548]. This suggests that Berry-curvature-driven AEE can be engineered not only through Weyl nodes and magnetism, but also through orbital multiplicity and symmetry lowering.

A fourth route is magnetization-orientation control. In \(\mathrm{DyCo}_5\), density functional theory and Kubo calculations show that spin-orbit-coupling-induced avoided crossings of Co \(3d\) bands create strongly peaked Berry-curvature hot spots, and that \(\alpha_{xy}\) exhibits an about two orders of magnitude contrast in the spin reorientation transition window for \(\mathbf{M}\parallel c\) versus \(\mathbf{M}\perp c\). Using \(S_{\mathrm{ANE}}\) and \(\Pi_{\mathrm{AEE}}=T S_{\mathrm{ANE}}\), the work demonstrates robust orientation-controlled switching under a fixed in-plane bias current [2512.14335].

## 6. Nonlinear generalizations and device implications

Although the canonical AEE is a linear transverse thermoelectric effect, recent work extends Berry-curvature control to higher-order moments. In systems with broken time-reversal symmetry where Berry-curvature monopole and dipole moments vanish by symmetry, the Berry-curvature quadrupole becomes the leading source of nonlinear anomalous thermal Hall and Nernst coefficients. These responses are cubic in the temperature gradient and scale as \(\tau^2\), and the paper describes them as of the Ettingshausen type [2401.08155]. A plausible implication is that future AEE research will increasingly classify materials not only by the magnitude of their Berry curvature, but also by the symmetry-allowed multipole structure of that curvature.

The device relevance is already explicit in the linear regime. For \(\mathrm{Co}_2\mathrm{MnGa}\), the transverse geometry is presented as a new approach for on-chip cooling, with strong localization of AEE-induced temperature gradients that is described as essential for microelectronic, quantum, healthcare, and sensing devices. Because charge and heat currents are decoupled by geometry, the platform is framed as surpassing classical thermoelectrics for micron-scale spot cooling [2403.13598].

Device concepts can also incorporate autonomous control. In \(\mathrm{DyCo}_5\), the combination of AEE with a temperature-driven spin reorientation transition provides a sensor-free, self-regulating thermal switch: as temperature passes through the spin reorientation window, the Berry-curvature hot-spot structure changes, \(\alpha_{xy}\) changes by about two orders of magnitude, and the AEE-derived heat flow switches under fixed current bias without external sensors or feedback electronics [2512.14335].

Methodologically, the broader Berry-curvature literature reinforces that anomalous transport can be used to access band geometry directly. Wave-packet dynamics have been used to map Berry curvature from anomalous displacement under pumping, establishing that semiclassical anomalous transport is not merely a phenomenological signature but also a probe of local geometric structure [1609.09412]. This supports the interpretation of the Berry-curvature-driven anomalous Ettingshausen effect as both a transport phenomenon and a diagnostic of momentum-space geometry.

Source: https://www.emergentmind.com/topics/berry-curvature-driven-anomalous-ettingshausen-effect