Power efficiency of Hall-like devices: comparison between reciprocal and anti-reciprocal Onsager relations

TL;DR

This work analyzes the stationary transport in Hall-like devices driven by a lateral load, comparing Planar Hall effect (PHE) and Anomalous Hall effect (AHE) within a second-law–driven variational framework. By minimizing dissipation under global constraints and including electrostatic screening, it derives current and edge-charge distributions and the power delivered to a load circuit for arbitrary load resistance, recovering known limits such as the perfect Hall bar ($α\to0$) and the Corbino disk ($α\to∞$). The key finding is that the transverse Hall current distributions are identical for AHE and PHE to leading order, while longitudinal currents and dissipated power differ only at second order in the Hall angle $Θ$, with both systems exhibiting maximum power transfer at $α=1$. Overall, the second-law perspective unifies the macroscopic dissipative behavior of these two Onsager classes and provides predictive, analytic insight into edge charging and load-driven power in Hall-like devices across materials supporting an effective magnetic field.

Abstract

Two well-known Hall-like effects are occurring in ferromagnets: the Anomalous Hall effect and the Planar Hall effect. The former is analogous to the classical Hall effect and is defined by the Onsager reciprocity relation of the second kind (antisymmetric conductivity matrix), while the latter is defined by the Onsager reciprocity relation of the first kind (symmetric conductivity matrix). The difference is fundamental, as it is based on time-invariance symmetry breaking at the microscopic scale. We study the Hall current generated in both cases, together with the power that can be extracted from the edges of Hall device. The expressions of the distribution of the electric currents, the distribution of electric carriers, and the power efficiencies (i.e. the power that can be injected into a load circuit) are derived at stationary regime from a variational method based on the second law of thermodynamics. It is shown that the distribution of the transverse Hall-current is identical in both cases but the longitudinal current and the power dissipated differ at the second order in the Hall angle.

Figures (3)

• Figure 1: Left: sketch of the Hall bar contacted to a load circuit (resistance $R_\ell$) that preserves the translation invariance along $x$. The sign of the electric-charge accumulation is indicated by red (-) and blue (+) colors (see the calculated profile in Fig.3). Right: Integration loop ABCDA defined on one of the $yz$ vertical planes. A magnetic vector $\vec{m}$ is present but it is not represented in the picture (see definition of the Hall angles in the text).
• Figure 2: Power efficiency $P_{lat}/P_0$ for both AHE (red lines) and PHE (blue dashed line). Left: Efficiency as a function of the ratio $\alpha = R/R_{\ell}$ for values of $\Theta$ ranging from 0 to 0.5 with increments of 0.05. Right: Efficiency as a function of the Hall angle $\Theta$.
• Figure 3: The profiles of the charge accumulations $\delta n(y)/\nu$ for both AHE (blue) and PHA (red) for different values of the Hall angle $\Theta$ at the maximum $\alpha =1$ (i.e., $R=R_{\ell}$) and for $\nu = 0.1n_0$. Left: over the whole sample. Right: close to the right boundary. In both panels, the angle $\Theta$ range from $0$ to $0.5$ by increments of $0.05$.