Measuring the Hall effect in hysteretic materials

TL;DR

The paper tackles the problem of reliably extracting the intrinsic Hall response in hysteretic materials, where history-dependent magnetization can contaminate the measurement. It introduces two practical extraction routes—reverse-magnetic-field reciprocity and antisymmetrization with respect to the applied field—and validates them on Co$_3$Sn$_2$S$_2$ and CeCoGe$_3$, outlining a decision framework for centered and non-centered hysteresis. The work demonstrates that improper antisymmetrization can create artifacts mimicking anomalous or topological Hall effects, while the mirror-method based on time-reversed states yields artifact-resistant results and doubles data efficiency. Collectively, the methods are generalizable to a broad class of conductors and offer a robust workflow for artifact-free Hall analysis in magnetic and non-m magnetic systems alike.

Abstract

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.

Figures (10)

• Figure 1: Schematics of (a) topological Hall effect -like and (b) anomalous Hall effect - like artifacts that can arise in measuring the magnetic field $H$ dependent Hall resistance $R_{yx}^{odd}$. The blue lines symbolize data with no artifact, and the red dashed lines symbolize the signal contaminated with experimental artifacts. The (a) -like artifact shows up as an unexpected feature at finite applied magnetic field $H$, while the (b) -like artifact gives finite $R_{yx}^{odd}$ at $H$= 0.
• Figure 2: Isothermal magnetization ($M$) (blue,left axis) and longitudinal resistivity ($\rho_{xx}$) (pink,right axis) measured on Co$_3$Sn$_2$S$_2$ at $T$ = 5 K after cooling the sample from 300 K using the (a) zero-field cooled () protocol, (b) positive field cool protocol with $\mu_0H_{FC}~=~+1$ T and (c) negative field cool protocol with $\mu_0H_{FC}~=~-1$ T. For the protocol in (a), the sample was cooled in zero field from $T_{max}=300$ K to $T_{meas}=5$ K, then the magnetic field was swept $0 \rightarrow +1~\text{T} \rightarrow -1~\text{T} \rightarrow +1~\text{T}$. For the positive field-cool protocol in (b), the sample was cooled from $T_{max}$ to $T_{meas}$ in $\mu_0 H_{FC}=+1$ T, and the field was swept $+1~\text{T} \rightarrow -1~\text{T} \rightarrow +1~\text{T}$. For the negative field-cool protocol in (c), the sample was cooled from $T_{max}$ to $T_{meas}$ in $\mu_0 H_{FC}=-1$ T, and the field was swept $-1~\text{T} \rightarrow +1~\text{T} \rightarrow -1~\text{T}$. Q1-Q4 label the sequential quarter-segments of the field sweep; for the (a) loop these correspond to $+1~\text{T}\rightarrow 0$, $0\rightarrow -1~\text{T}$, $-1~\text{T}\rightarrow 0$, and $0\rightarrow +1~\text{T}$, respectively. Q1$'$-Q4$'$ denote the analogous segments for the (b) positive field-cool protocol, while Q1$"$-Q4$"$ in (c) sequentially correspond to $-1~\text{T}\rightarrow 0$, $0\rightarrow 1~\text{T}$, $1~\text{T}\rightarrow 0$, and $0\rightarrow -1~\text{T}$ (c) for the negative field-cool protocol. All measurements were performed with the applied magnetic field $H \parallel c$ while the transport measurements were performed with the current $I\parallel ab$.
• Figure 3: Generic time ($t$) histories of magnetic field ($H$) and temperature ($T$) for (a, top axis) the zero-field-cooled () protocol and (b, top axis) the positive and negative field-cooled (FC) protocols. For Co$_3$Sn$_2$S$_2$ with applied field $H \parallel c$, $\mu_0 H_{max}=1$ T, $T_{max}=300$ K, and $T_{meas}=5$ K, the protocol yields centered hysteresis loops whereas FC protocols yield non-centered loops. Q labels indicate the measurement order of the field-sweep segments: for , Q0 denotes $0 \rightarrow +1~\text{T}$, followed by Q1--Q4: $+1~\text{T} \rightarrow 0$, $0 \rightarrow -1~\text{T}$, $-1~\text{T} \rightarrow 0$, and $0 \rightarrow +1~\text{T}$. For the positive FC protocol, Q1$'$--Q4$'$ use the same segment definitions as Q1--Q4 but without Q0. For the negative FC protocol, Q1$"$--Q4$"$ sequentially label $-1~\text{T} \rightarrow 0$, $0 \rightarrow +1~\text{T}$, $+1~\text{T} \rightarrow 0$, and $0 \rightarrow -1~\text{T}$. Corresponding histories of the measured resistances $R_m$, labeled m1 (blue) and m2 (red), used to extract the Hall resistivity $\rho_{yx}^{odd}$ of Co$_3$Sn$_2$S$_2$ are shown for (a, bottom axis) the protocol and (b, bottom axis) the positive and negative FC protocols. The measurements were performed with current $I \parallel ab$. Shown for a Van der Pauw geometry (c), m1 and m2 correspond to $R_{m1}=R_{12,34}$ and $R_{m2}=R_{34,12}$, where the first index pair denotes the current source and drain contacts and the second index pair denotes the high and low voltage contacts. (c) Schematic comparison of reverse-magnetic-field reciprocity () and antisymmetrization methods for extracting the Hall effect. The method uses m1 and the $I$--$V$-rotated measurement m2 to generate a pair of measurements equivalent to reversing the polarity of $B_z$ resulting in time-reversed states (TRS), while antisymmetrization uses m1 and m1$'$ measured with the same contact geometry at equal and opposite $H$ to measure TRS. Depending on whether the hysteresis loops are centered or non-centered, one or two hysteresis loops are required to access $H$-dependent TRS; the pairing used in (a,b) is encoded according to the legend in (c). $\rho_{yx}^{odd}$ obtained from $R_m$ in (a,b) for (d) the protocol, (e) the positive FC protocol with $\mu_0 H_{FC}=+1$ T, and (f) the negative FC protocol with $\mu_0 H_{FC}=-1$ T. Yellow curves in (d--f) are obtained using , while gray curves are obtained using antisymmetrization. Thick lines denote sweeps from $+1$ T$\rightarrow -1$ T, and thin lines denote $-1$ T$\rightarrow +1$ T. The inset of (e) highlights the coercive-field region; the maroon data are an independent measurement using the antisymmetrization method.
• Figure 4: A decision tree that can be used to minimize experimental artifacts in Hall effect measurements.
• Figure 5: (a) Isothermal magnetization ($M$) and (b) longitudinal resistivity ($\rho_{xx}$) measured at temperature $T$ = 3 K after cooling the sample from 100 K in $\mu_0H_{FC}~=~+4$ T using the positive field cool procedure. Q1'-Q4' in (a) label the sequential quarter-segments of the field sweep; $+4~\text{T}\rightarrow 0$, $0\rightarrow -4~\text{T}$, $-4~\text{T}\rightarrow 0$, and $0\rightarrow +4~\text{T}$, respectively. (c) $M$ and (d) $\rho_{xx}$ after field-cooling the sample in $\mu_0H_{FC}~=~-4$ T using the negative field cool procedure. Q1$"$-Q4$"$ in (c) label the sequential quarter-segments of the field sweep; $-4~\text{T}\rightarrow 0$, $0\rightarrow 4~\text{T}$, $4~\text{T}\rightarrow 0$, and $0\rightarrow -4~\text{T}$ (c) for the negative field-cool protocol. The main panels show data from $\mu_0H = \pm1$ T, while the insets show the same data from $\mu_0H = \pm4$ T. Light lines in all panels correspond to data measured from $+4~\text{T}\rightarrow-4~\text{T}$ while dark lines correspond to $-4~\text{T}\rightarrow+4~\text{T}$. All measurements were performed with the applied magnetic field $H \parallel c$ while the transport measurements were performed with the current $I\parallel ab$. Anomalies corresponding to metamagnetic transitions for increasing $H$ and $H>0$ or decreasing H and $H<0$ are labeled as $H_{c1}$, $H_{c2}$, $H_{c3}$ and $-H_{c1}$, $-H_{c2}$, $-H_{c3}$, respectively.