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    Modeling and analysis of second-harmonic magnetoresistance with arbitrary spin polarization and unconventional spin-orbit torques

    Fangye Lin1,2, Tianyi Zhang3, Xinjian Zhou1,2, Guoyi Shi4, Zhifeng Zhu5, Zichen Zhang1, Yue Zhang1,6, Caihua Wan3,*, Hyunsoo Yang4 et al.

    Weisheng Zhao1,2,† and Shuyuan Shi1,2,‡

    • 1Fert Beijing Institute, MIIT Key Laboratory of Spintronics, School of Integrated Circuit Science and Engineering, Beihang University, Beijing 100191, People’s Republic of China
    • 2National Key Laboratory of Spintronics, Hangzhou International Innovation Institute, Beihang University, Hangzhou 311115, People’s Republic of China
    • 3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
    • 4Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore
    • 5School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, People’s Republic of China
    • 6Anhui Province Key Laboratory of Spintronic Chip Research and Manufacturing, Hefei Innovation Research Institute, Beihang University, Hefei 230013, People’s Republic of China

    • *Contact author: wancaihua@iphy.ac.cn
    • †Contact author: wszhao@buaa.edu.cn
    • ‡Contact author: smeshis@buaa.edu.cn
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    Phys. Rev. Applied 25, 024066 – Published 20 February, 2026

    DOI: https://doi.org/10.1103/sr8x-26ls

    Abstract

    The second-harmonic method is an important approach for studying the traditional spin-orbit torques (SOTs). In recent years, research on unconventional SOTs has provided a new approach for zero-field switching of the perpendicular magnetization and thus has received extensive attention. In this work, we theoretically extend the second-harmonic method to include the unconventional SOTs. For the four common magnetic configurations, we have derived the analytical relationship between the second-harmonic Hall (longitudinal) resistance and the (un)conventional SOT coefficients. We have comprehensively taken into account the effects of the anomalous Hall effect, the planar Hall effect, anisotropic magnetoresistance, the spin Hall magnetoresistance effect, the anomalous Nernst effect, the planar Nernst effect, and thermal fluctuations. The fitting results obtained from the numerical simulation verify the correctness of the analytical formula. This work provides a theoretical framework for measuring unconventional SOTs using the second-harmonic method, which is conducive to the development of new all-electrical spintronic devices.

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    Phys. Rev. Applied 25, 024066– Published 20 February, 2026
    • Received 28 October 2025
    • Accepted 23 January 2026
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