- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Thermodynamic ground and Berry-phase origin of the planar spin Hall effect

Haolin Pan1,2, Zheng Liu1, Dazhi Hou1,2,*, Yang Gao1,2,3,†, and Qian Niu1,2,3

  • 1Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, China
  • 2Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China
  • 3Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China

  • *dazhi@ustc.edu.cn
  • †ygao87@ustc.edu.cn
PDF

Phys. Rev. Research 6, L012034 – Published 21 February, 2024

DOI: https://doi.org/10.1103/PhysRevResearch.6.L012034

Abstract

The thermodynamic ground of the spin transport is elaborated, demonstrating the Onsager's reciprocal relation. The latter is also confirmed by deriving the spin Hall effect and its inverse using the semiclassical theory and the linear response theory. Importantly, the intrinsic contribution to the spin conductivity that satisfies the Onsager's relation has only Hall-type component and hence does not dissipate heat, in analogy to the anomalous Hall effect and fulfilling the thermodynamic requirement. Based on the expression of the spin Hall conductivity, the Berry-phase origin is elaborated for the recent-discovered planar spin Hall effect which features current-induced spin polarization within the plane of the Hall deflection. We unravel a spin-repulsion vector governing the planar spin Hall effect, providing a transparent criteria for identifying intrinsic planar spin Hall materials. Finite spin-repulsion vector is found permitted in 13 crystalline point groups, revealing a big number of unexplored planar spin Hall systems. Our result can be used for the quantitative calculation of the planar spin Hall effect, facilitating further material hunting and experimental investigation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (49)

  1. J. E. Hirsch, Spin Hall effect, Phys. Rev. Lett. 83, 1834 (1999).
  2. P. Song, C. H. Hsu, G. Vignale, M. Zhao, J. Liu, Y. Deng, W. Fu, Y. Liu, Y. Zhang, H. Lin, V. M. Pereira, and K. P. Loh, Coexistence of large conventional and planar spin Hall effect with long spin diffusion length in a low-symmetry semimetal at room temperature, Nat. Mater. 19, 292 (2020).
  3. Q. Xie, W. Lin, S. Sarkar, X. Shu, S. Chen, L. Liu, T. Zhao, C. Zhou, H. Wang, J. Zhou, S. Gradecak, and J. Chen, Field-free magnetization switching induced by the unconventional spin-orbit torque from WTe2, APL Mater. 9, 051114 (2021).
  4. L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A. Buhrman, Current-induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin Hall effect, Phys. Rev. Lett. 109, 096602 (2012).
  5. D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A. Buhrman, J. Park, and D. C. Ralph, Control of spin-orbit torques through crystal symmetry in WTe2/ferromagnet bilayers, Nat. Phys. 13, 300 (2017).
  6. T. Nan, C. X. Quintela, J. Irwin, G. Gurung, D. F. Shao, J. Gibbons, N. Campbell, K. Song, S. Y. Choi, L. Guo, R. D. Johnson, P. Manuel, R. V. Chopdekar, I. Hallsteinsen, T. Tybell, P. J. Ryan, J. W. Kim, Y. Choi, P. G. Radaelli, D. C. Ralph et al., Controlling spin current polarization through non-collinear antiferromagnetism, Nat. Commun. 11, 4671 (2020).
  7. L. Liu, C. Zhou, X. Shu, C. Li, T. Zhao, W. Lin, J. Deng, Q. Xie, S. Chen, J. Zhou, R. Guo, H. Wang, J. Yu, S. Shi, P. Yang, S. Pennycook, A. Manchon, and J. Chen, Symmetry-dependent field-free switching of perpendicular magnetization, Nat. Nanotechnol. 16, 277 (2021).
  8. J. Zhou, X. Shu, Y. Liu, X. Wang, W. Lin, S. Chen, L. Liu, Q. Xie, T. Hong, P. Yang, B. Yan, X. Han, and J. Chen, Magnetic asymmetry induced anomalous spin-orbit torque in IrMn, Phys. Rev. B 101, 184403 (2020).
  9. Y. Liu, Y. Liu, M. Chen, S. Srivastava, P. He, K. L. Teo, T. Phung, S. H. Yang, and H. Yang, Current-induced out-of-plane spin accumulation on the (001) surface of the IrMn3 antiferromagnet, Phys. Rev. Appl. 12, 064046 (2019).
  10. T. C. Chuang, D. Qu, S. Y. Huang, and S. F. Lee, Magnetization-dependent spin Hall effect in a perpendicular magnetized film, Phys. Rev. Res. 2, 032053(R) (2020).
  11. X. Chen, S. Shi, G. Shi, X. Fan, C. Song, X. Zhou, H. Bai, L. Liao, Y. Zhou, H. Zhang, A. Li, Y. Chen, X. Han, S. Jiang, Z. Zhu, H. Wu, X. Wang, D. Xue, H. Yang, and F. Pan, Observation of the antiferromagnetic spin Hall effect, Nat. Mater. 20, 800 (2021).
  12. R. Galceran, B. Tian, J. Li, F. Bonell, M. Jamet, C. Vergnaud, A. Marty, J. H. García, J. F. Sierra, M. V. Costache, S. Roche, S. O. Valenzuela, A. Manchon, X. Zhang, and U. Schwingenschlögl, Control of spin-charge conversion in van der Waals heterostructures, APL Mater. 9, 100901 (2021).
  13. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Universal intrinsic spin Hall effect, Phys. Rev. Lett. 92, 126603 (2004).
  14. M. Gradhand, D. V. Fedorov, F. Pientka, P. Zahn, I. Mertig, and B. L. Györffy, First-principle calculations of the berry curvature of bloch states for charge and spin transport of electrons, J. Phys.: Condens. Matter 24, 213202 (2012).
  15. J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87, 1213 (2015).
  16. M. Seemann, D. Ködderitzsch, S. Wimmer, and H. Ebert, Symmetry-imposed shape of linear response tensors, Phys. Rev. B 92, 155138 (2015).
  17. Q. F. Sun, X. C. Xie, and J. Wang, Persistent spin current in nanodevices and definition of the spin current, Phys. Rev. B 77, 035327 (2008).
  18. R. S. Akzyanov and A. L. Rakhmanov, Bulk and surface spin conductivity in topological insulators with hexagonal warping, Phys. Rev. B 99, 045436 (2019).
  19. C. Xiao and Q. Niu, Unified bulk semiclassical theory for intrinsic thermal transport and magnetization currents, Phys. Rev. B 101, 235430 (2020).
  20. A. Roy, M. H. D. Guimarães, and J. Sławińska, Unconventional spin Hall effects in nonmagnetic solids, Phys. Rev. Mater. 6, 045004 (2022).
  21. Y. Zhang, Q. Xu, K. Koepernik, R. Rezaev, O. Janson, J. Železný, T. Jungwirth, C. Felser, J. van den Brink, and Y. Sun, Different types of spin currents in the comprehensive materials database of nonmagnetic spin Hall effect, npj Comput. Mater. 7, 167 (2021).
  22. J. H. Cullen, P. Bhalla, E. Marcellina, A. R. Hamilton, and D. Culcer, Generating a topological anomalous Hall effect in a nonmagnetic conductor: An in-plane magnetic field as a direct probe of the berry curvature, Phys. Rev. Lett. 126, 256601 (2021).
  23. R. B. Atencia, Q. Niu, and D. Culcer, Semiclassical response of disordered conductors: Extrinsic carrier velocity and spin and field-corrected collision integral, Phys. Rev. Res. 4, 013001 (2022).
  24. D. Culcer, J. Sinova, N. A. Sinitsyn, T. Jungwirth, A. H. MacDonald, and Q. Niu, Semiclassical spin transport in spin-orbit-coupled bands, Phys. Rev. Lett. 93, 046602 (2004).
  25. Q. F. Sun and X. C. Xie, Definition of the spin current: The angular spin current and its physical consequences, Phys. Rev. B 72, 245305 (2005).
  26. J. Shi, P. Zhang, D. Xiao, and Q. Niu, Proper definition of spin current in spin-orbit coupled systems, Phys. Rev. Lett. 96, 076604 (2006).
  27. N. Nagaosa, Spin currents in semiconductors, metals, and insulators, J. Phys. Soc. Jpn. 77, 031010 (2008).
  28. N. Sugimoto, S. Onoda, S. Murakami, and N. Nagaosa, Spin Hall effect of a conserved current: Conditions for a nonzero spin Hall current, Phys. Rev. B 73, 113305 (2006).
  29. T. W. Chen, C. M. Huang, and G. Y. Guo, Conserved spin and orbital angular momentum Hall current in a two-dimensional electron system with Rashba and Dresselhaus spin-orbit coupling, Phys. Rev. B 73, 235309 (2006).
  30. P. Zhang, Z. Wang, J. Shi, D. Xiao, and Q. Niu, Theory of conserved spin current and its application to a two-dimensional hole gas, Phys. Rev. B 77, 075304 (2008).
  31. T. W. Chen and G. Y. Guo, Torque and conventional spin Hall currents in two-dimensional spin-orbit coupled systems: Universal relation and hyperselection rule, Phys. Rev. B 79, 125301 (2009).
  32. G. Liu, P. Zhang, Z. Wang, and S. S. Li, Spin Hall effect on the kagome lattice with Rashba spin-orbit interaction, Phys. Rev. B 79, 035323 (2009).
  33. A. Rückriegel and P. Kopietz, Spin currents, spin torques, and the concept of spin superfluidity, Phys. Rev. B 95, 104436 (2017).
  34. R. J. H. Wesselink, K. Gupta, Z. Yuan, and P. J. Kelly, Calculating spin transport properties from first principles: Spin currents, Phys. Rev. B 99, 144409 (2019).
  35. L. Dong, C. Xiao, B. Xiong, and Q. Niu, Berry phase effects in dipole density and the Mott relation, Phys. Rev. Lett. 124, 066601 (2020).
  36. S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics (Dover, New York, 1962).
  37. J. Jackson, Classical Electrodynamics (Wiley, New York, 1998).
  38. D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Rev. Mod. Phys. 82, 1959 (2010).
  39. Y. Zhao, Y. Gao, and D. Xiao, Electric polarization in inhomogeneous crystals, Phys. Rev. B 104, 144203 (2021).
  40. See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevResearch.6.L012034 for discussions of the entropy production rate, the Onsager's reciprocal relation, the definition and physical meaning of the total spin current, the charge current in response to the Zeeman field gradient using the semiclassical and linear response theory, the total spin current in response to the electric field, the non-Abelian spin conductivity, the spin conductivity in a model system with time reversal and inversion symmetry, and first-principles calculation of the spin conductivity in VAu4.
  41. L. D. Landau and E. M. Lifshitz, Statistical Physics (Butterworth-Heinemann, Oxford, 1980).
  42. D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Berry-phase effect in anomalous thermoelectric transport, Phys. Rev. Lett. 97, 026603 (2006).
  43. Y. Gao, D. Vanderbilt, and D. Xiao, Microscopic theory of spin toroidization in periodic crystals, Phys. Rev. B 97, 134423 (2018).
  44. Y. Zhang, Y. Sun, H. Yang, J. Železný, S. P. P. Parkin, C. Felser, and B. Yan, Strong anisotropic anomalous Hall effect and spin Hall effect in the chiral antiferromagnetic compounds Mn3X (X=Ge, Sn, Ga, Ir, Rh, and Pt), Phys. Rev. B 95, 075128 (2017).
  45. X. Q. Sun, S. C. Zhang, and Z. Wang, Helical spin order from topological Dirac and Weyl semimetals, Phys. Rev. Lett. 115, 076802 (2015).
  46. R. González-Hernández, L. Šmejkal, K. Výborný, Y. Yahagi, J. Sinova, T. Jungwirth, and J. Železný, Efficient electrical spin splitter based on nonrelativistic collinear antiferromagnetism, Phys. Rev. Lett. 126, 127701 (2021).
  47. A. Bose, N. J. Schreiber, R. Jain, D. F. Shao, H. P. Nair, J. Sun, X. S. Zhang, D. A. Muller, E. Y. Tsymbal, D. G. Schlom, and D. C. Ralph, Tilted spin current generated by the collinear antiferromagnet ruthenium dioxide, Nat. Electronics 5, 267 (2022).
  48. C. Xiao and Q. Niu, Conserved current of nonconserved quantities, Phys. Rev. B 104, L241411 (2021).
  49. R. Raimondi, P. Schwab, C. Gorini, and G. Vignale, Spin-orbit interaction in a two-dimensional electron gas: A SU(2) formulation, Ann. Phys. (Berlin) 524, 153162 (2012).

Outline

Information

Phys. Rev. Research 6, L012034– Published 21 February, 2024
  • Received 4 October 2021
  • Accepted 25 January 2024
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation