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Description
Persistent spin textures give rise to robust spin states, unlocking wide-range opportunities for quantum computing, data storage, and advanced spintronic technologies [1]. In this work, we explore the potential for realizing a persistent spin texture in monolayer phosphorene - a promising spintronic material with high carrier mobility, tunable semiconducting band gap and weak spin-orbit coupling [2,3].
Using first-principles calculations, we investigate the effects of strain and external transverse electric field on spin anisotropy in phosphorene bands. Strain is varied from -5% to +5% along zigzag and armchair directions across different electric field strengths. We find that while strain applied along the armchair direction has a minor effect on spin physics, strain along the zigzag direction leads to qualitative changes in spin anisotropy. In particular, for 1.2% strain in the zigzag direction, a persistent spin texture, with spins polarized in the x-direction, appears in the conduction band of phosphorene, even for arbitrarily small electric fields. In effect, spin relaxation triggered by the Dyakonov-Perel [4] spin relaxation mechanism is effectively deactivated for S$_x$ spins. The analysis of intrinsic spin-orbit coupling by the spin-mixing parameter b$^2$ [5] reveals that tensile strain has a minor impact of up to 4%, with the anisotropy ratio reaching 2. For 5%, the anisotropy sharply rises to 100, driven by band anticrossing and evolving with doping. In zero-electric field case with preserved inversion symmetry, the Elliott-Yafet mechanism dominates the spin relaxation process.
Our findings highlight that tuning the strain in the phosphorene enables an excellent platform for manipulating the spin texture or spin anisotropy, simultaneously enhancing the spin lifetime [6].
Acknowledgement
The authors acknowledge financial support from NCBR under the V4-Japan project BGapEng V4-JAPAN/2/46/BGapEng/2022 and ICM UW, within grant no. G83-27 and Slovak Academy of Sciences project IMPULZ IM-2021-42.
References
[1] L. L. Tao and E. Y. Tsymbal, Persistent spin texture enforced by symmetry, Nat. Commun. 9, 2763 (2018).
[2] L. Cording, J. Liu, J. Y. Tan, et al., Highly anisotropic spin transport in ultrathin black phosphorus, Nat. Mater. 23, 479 (2024).
[3] H. Liu, A. T. Neal, Z. Zhu, Z. Luo, X. Xu, D. Tománek, and P. D. Ye, Phosphorene: An unexplored 2D semiconductor with a high hole mobility, ACS Nano 8, 4033 (2014).
[4] M. I. Dyakonov and V. I. Perel, Spin relaxation of conduction electrons in noncentrosymmetric semiconductors, Sov. Phys. Solid State 13, 3023 (1971).
[5] B. Zimmermann, P. Mavropoulos, S. Heers, N. H. Long, S. Blügel, and Y. Mokrousov, Anisotropy of spin relaxation in metals, Phys. Rev. Lett. 109, 236603 (2012).
[6] P. Jureczko, M. Milivojević, and M. Kurpas, Strain-tuning of spin anisotropy in single-layer phosphorene: Insights from Elliott-Yafet and Dyakonov-Perel spin relaxation rates, arXiv:2501.05911 (2025).
