Measuring the Hall effect in hysteretic materials

Measurement of the Hall effect is a ubiquitous probe for materials discovery, characterization, and metrology. Inherent to the Hall measurement geometry, the measured signal is often contaminated by unwanted contributions, so the data must be processed to isolate the Hall response. The standard approach invokes Onsager-Casimir reciprocity and antisymmetrizes the raw signal about zero applied magnetic field. In hysteretic materials this becomes nontrivial, since Onsager-Casimir relations apply only to microscopically reversible states. Incorrect antisymmetrization can lead to artifacts that mimic anomalous or topological Hall signatures. The situation is especially subtle when hysteresis loops are not centered at zero applied field, as in exchange-biased systems. A practical reference for generically extracting the Hall response in hysteretic materials is lacking. Here, using Co$_3$Sn$_2$S$_2$ as a bulk single-crystal model that can be prepared with or without exchange-biased hysteresis, we demonstrate two procedures that can be used to extract the Hall effect: (1) reverse-magnetic-field reciprocity and (2) antisymmetrization with respect to applied field. We then measure the Hall effect on CeCoGe$_3$, a noncentrosymmetric antiferromagnet which can be prepared to have asymmetric magnetization and magnetoresistance, and demonstrate how improper processing can generate artificial anomalous Hall signals. These methods are generic and can be applied to any conductor.