Modeling of a magnetic field sensor based on spin Hall magnetoresistance

TL;DR

The paper addresses the limitations of tunneling magnetoresistance sensors by proposing Spin Hall Magnetoresistance (SMR) in a Wheatstone-bridge configuration and develops a multiphysics model that couples SMR/AMR, Spin-Orbit Torque (SOT), current distribution via Fuchs-Sondheimer, and magnetization dynamics including domain-wall motion. The model combines a uniform Stoner-Wohlfarth approach with a truncated astroid to capture domain statistics, and validates predictions against Pt/FeCoB and Ta/FeCoB bilayers, enabling design guidance for reduced power and increased sensitivity. Key findings show that the bridge output can be linearized by SOT biasing and that material choice (Pt vs Ta) tunes the trade-off between electrical performance and magnetic softness; the framework yields practical guidelines for geometry, layer thickness, and material properties. Overall, the work provides a pathway to optimize SMR-based sensors beyond traditional MR devices, with Ta-based devices offering improved linearity in the tested range and the modeling framework guiding future enhancements including orbital torque concepts.

Abstract

Next-generation spintronic sensors aim to overcome the limitations of traditional tunneling-magnetoresistance (TMR) devices, such as complex manufacturing, high $1/f$ noise, and significant offsets. This work presents a comprehensive modeling and experimental validation of a magnetic field sensor based on Spin Hall Magnetoresistance (SMR) in a Wheatstone bridge configuration. Utilizing a multiphysics approach, we simulate the interplay between SMR, Anisotropic Magnetoresistance (AMR), and Spin-Orbit Torque (SOT) using a Stoner-Wohlfarth model complemented by a Fuchs-Sondheimer analysis of current distribution. To account for the presence of magnetic domains, we incorporate a modified Stoner-Wohlfarth framework that considers non-uniform magnetization and domain wall motion through a "truncated astroid" approach, allowing for a statistical distribution of single-domain particles. The model is validated against experimental measurements of Pt/$\text{Fe}_{60}\text{Co}_{20}\text{B}_{20}$ and Ta/$\text{Fe}_{60}\text{Co}_{20}\text{B}_{20}$ bilayers patterned into Hall bars and Wheatstone bridges. The model provides critical design guidelines for optimizing material properties, layer thickness, and device layout to minimize power consumption and maximize sensitivity in SMR-based sensing applications.

Figures (11)

• Figure 1: Spin current $J_\mathrm{s}$ injection in a bilayer system composed of a heavy metal (HM) and a ferromagnet (FM) due to spin Hall effect in presence of an electrical current $J_\mathrm{e}$.
• Figure 2: The coefficient $A$ as a function of $\Delta R_{\mathrm{SMR}}$. The magnitude of $A$ depends on the ratio between $R_{\mathrm{HM}}$ and $R_{\mathrm{FM}}$. Here a fixed $R_{\mathrm{FM}}$=5000$\ohm$ is used with a fixed typical value $\Delta R_{\mathrm{AMR}}$ of 4% of $R_{\mathrm{FM}}$. The values of $\Delta R_{\mathrm{HM}}$ are shown up to a maximum of 5% of $R_{\mathrm{HM}}$ assuming the SMR to be of the same order as the AMR.
• Figure 3: Sketch of a Wheatstone bridge resistance configuration.
• Figure 4: Truncated astroid (orange threshold) representing the modified Stoner-Wohlfarth model of a single "particle" in the ensemble of statisically distributed ones, characterized by an easy axis with anisotropy constant $K$ forming an angle $\varphi$ with $\mathbf{x}$. Two $\textbf{H}$ paths not involving (olive) or involving (orange) the threshold corresponding to a local Barkhausen jump are shown. Note the difference between the two Gibbs free energy double-well profiles corresponding to the point where $\textbf{H}$ crosses the classical (thick olive) or truncated (thick orange) threshold.
• Figure 5: The two sensor configurations which were modeled, pattern p45 at 45° (left) and pattern p0 parallel (right) with respect to the applied magnetic field $H$. $K$ indicates the direction of an eventually present magnetic anisotropy.