Study of the Nonlinear Dependence of Anomalous Hall Conductivity on Magnetization in Weak Itinerant Ferromagnet ZrZn2

TL;DR

This study investigates the nonlinear dependence of anomalous Hall conductivity on magnetization in ZrZn₂, revealing deviations from linearity at small magnetic moments and providing insights into the underlying physics.

Contribution

It provides direct calculations of AHE in ZrZn₂ showing the breakdown of linear relation at small magnetizations, extending understanding beyond previous models.

Findings

Linear relation holds as M approaches zero.

Sign reversal of AHE occurs at small magnetic moments (~0.4 μB/Zr).

Deviations from linearity are observed at low magnetization.

Abstract

As opposed to the ordinary Hall effect, the anomalous Hall effect (AHE) remained unexplained for decades, and, amazingly, some misconceptions have survived even now, in particular, the claim that AHE is linearly related to the net magnetization. Karplus and Luttinger provided a quantum-mechanical explanation of AHE by explicitly including the SOC and the Berry curvature of electronic bands. They did address the question of linearity, but only in the relatively uncommon limit of the exchange coupling smaller than SOC. Now the linear relation in traditional ferromagnets is understood as a domain population effect: both AHE and magnetization are independently proportional to the domain disbalance. In this connection, it is interesting to check to what extent this relation will hold in {\em single-domain} itinerant ferromagnet, the closest case to that analyzed by Karplus and Luttinger? We answer this question by direct calculations, using the Karplus-Luttinger formula, of AHE in a prototypical itinerant ferromagnet, ZrZn$_2$. We show that in the zero-magnetization limit, $M\rightarrow 0$, the linear relation hold, but at rather small moments of $\sim 0.4\ \mu_B$/Zr breaks down completely and even flips the sign.