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Domain wall dynamics in cubic magnetostrictive materials subject to Rashba effect and nonlinear dissipation

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Abstract

This work focuses on the analytical investigation of domain wall motion occurring along the major axis of a thin magnetostrictive nanostrip perfectly arranged on the top of a thick piezoelectric actuator. The motion is driven by magnetic fields, spin-polarized currents, and spin–orbit torque effects and takes place in cubic magnetostrictive materials characterized by a nonlinear dissipation. The main aim is to describe how magnetoelasticity, Rashba field, dry-friction, chemical composition, and crystal symmetry affect the steady and precessional dynamics of magnetic domain walls. In detail, it is here analytically inspected how the key features (threshold, breakdown, domain wall mobility, and propagation direction) can be effectively manipulated by the above contributions. Finally, the theoretical results are numerically illustrated for realistic materials, revealing a satisfying qualitative agreement with experimental observations.

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Abbreviations

DW:

Domain wall

MSL:

Magnetostrictive layer

PEL:

Piezoelectric layer

ELLG:

Extended Landau–Lifshitz–Gilbert equation

STT:

Spin-transfer torque

SOT:

Spin–orbit torque

WB:

Walker breakdown

References

  1. Eerenstein, W., Mathur, N.D., Scott, J.F.: Multiferroic and magnetoelectric materials. Nature 442(7104), 759–765 (2006)

    Google Scholar 

  2. Vaz, C.A., Hoffman, J., Ahn, C.H., Ramesh, R.: Magnetoelectric coupling effects in multiferroic complex oxide composite structures. Adv. Mater. 22(26–27), 2900–2918 (2010)

    Google Scholar 

  3. Lei, N., Devolder, T., Agnus, G., Aubert, P., Daniel, L., Kim, J.-V., Zhao, W., Trypiniotis, T., Cowburn, R.P., Chappert, C., Ravelosona, D., Lecoeur, P.: Strain-controlled magnetic domain wall propagation in hybrid piezoelectric/ferromagnetic structures. Nat. Commun. 4, 1378 (2013)

    Google Scholar 

  4. Mathurin, T., Giordano, S., Dusch, Y., Tiercelin, N., Pernod, P., Preobrazhensky, V.: Stress-mediated magnetoelectric control of ferromagnetic domain wall position in multiferroic heterostructures. Appl. Phys. Lett. 108(8), 082401 (2016)

    Google Scholar 

  5. Zighem, F., Faurie, D., Mercone, S., Belmeguenai, M., Haddadi, H.: Voltage-induced strain control of the magnetic anisotropy in a Ni thin film on flexible substrate. J. Appl. Phys. 114(7), 073902 (2013)

    Google Scholar 

  6. Balinskiy, M., Chavez, A.C., Barra, A., Chiang, H., Carman, G.P., Khitun, A.: Magnetoelectric spin wave modulator based on synthetic multiferroic structure. Sci. Rep. 8(1), 1–10 (2018)

    Google Scholar 

  7. Allwood, D.A., Xiong, G., Faulkner, C.C., Atkinson, D., Petit, D., Cowburn, R.P.: Magnetic domain-wall logic. Science 309(5741), 1688–1692 (2005)

    Google Scholar 

  8. Parkin, S.S., Hayashi, M., Thomas, L.: Magnetic domain-wall racetrack memory. Science 320(5873), 190–194 (2008)

    Google Scholar 

  9. Hayashi, M., Thomas, L., Moriya, R., Rettner, C., Parkin, S.S.: Current-controlled magnetic domain-wall nanowire shift register. Science 320(5873), 209–211 (2008)

    Google Scholar 

  10. Depassier, M.C.: Speed of field-driven domain walls in nanowires with large transverse magnetic anisotropy. Europhys. Lett. 111(2), 27005 (2015)

    Google Scholar 

  11. Hu, J.-M., Yang, T., Momeni, K., Cheng, X., Chen, L., Lei, S., Trolier-McKinstry, S., Gopalan, V., Carman, G.P., Nan, C.-W., Chen, L.-Q.: Fast magnetic domain-wall motion in a ring-shaped nanowire driven by a voltage. Nano Lett. 16(4), 2341–2348 (2016)

    Google Scholar 

  12. De Ranieri, E., et al.: Piezoelectric control of the mobility of a domain wall driven by adiabatic and non-adiabatic torques. Nat. Mater. 12, 808–814 (2013)

    Google Scholar 

  13. Consolo, G., Valenti, G.: Analytical solution of the strain-controlled magnetic domain wall motion in bilayer piezoelectric/magnetostrictive nanostructures. J. Appl. Phys. 121(4), 043903 (2017)

    Google Scholar 

  14. Consolo, G., Valenti, G.: Magnetic domain wall motion in nanoscale multiferroic devices under the combined action of magnetostriction, Rashba effect and dry-friction dissipation. Atti della Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche, Matematiche e Naturali 96(S1), 3 (2018)

    MathSciNet  Google Scholar 

  15. Consolo, G., Federico, S., Valenti, G.: Strain-mediated propagation of magnetic domain-walls in cubic magnetostrictive materials. Ricerche di Matematica 70(1), 81–97 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Consolo, G.: Modeling magnetic domain-wall evolution in trilayers with structural inversion asymmetry. Ricerche di Matematica 67(2), 1001–1015 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Zhang, J.X., Chen, L.Q.: Phase-field microelasticity theory and micromagnetic simulations of domain structures in giant magnetostrictive materials. Acta Mater. 53, 2845–2855 (2005)

    Google Scholar 

  18. Dwivedi, S., Singh, Y.P., Consolo, G.: On the statics and dynamics of transverse domain walls in bilayer piezoelectric-magnetostrictive nanostructures. Appl. Math. Model. 83, 13–29 (2020)

    MathSciNet  MATH  Google Scholar 

  19. Weiler, M., Brandlmaier, A., Geprags, S., Althammer, M., Opel, M., Bihler, C., Huebl, H., Brandt, M.S., Gross, R., Goennenwein, S.T.B.: Voltage controlled inversion of magnetic anisotropy in a ferromagnetic thin film at room temperature. N. J. Phys. 11(1), 013021 (2011)

    Google Scholar 

  20. Zhu, B., Lo, C.C.H., Lee, S.J., Jiles, D.C.: Micromagnetic modeling of the effects of stress on magnetic properties. J. Appl. Phys. 89(11), 7009–7011 (2001)

    Google Scholar 

  21. Hubert, A., Schafer, R.: Magnetic domains: the analysis of magnetic microstructures. Springer Science and Business Media (2008)

    Google Scholar 

  22. Chikazumi, S., Graham, C.D.: Physics of Ferromagnetism. Oxford University Press, Oxford (2009)

    Google Scholar 

  23. Cullity, B.D., Graham, C.D.: Introduction to Magnetic Materials. Wiley, New York (2009)

    Google Scholar 

  24. Consolo, G., Federico, S., Valenti, G.: Magnetostriction in transversely isotropic hexagonal crystals. Phys. Rev. B 101(1), 014405 (2020). (24)

    Google Scholar 

  25. Clark, A.E., et al.: Extraordinary magnetoelasticity and lattice softening in bcc Fe-Ga alloys. J. Appl. Phys. 93, 8621–8623 (2003)

    Google Scholar 

  26. Wuttig, M., Dai, L., Cullen, J.R.: Elasticity and magnetoelasticity of Fe-Ga solid solutions. Appl. Phys. Lett. 80, 1135–1137 (2002)

    Google Scholar 

  27. Rafique, S., et al.: Magnetic anisotropy of FeGa alloys. J. Appl. Phys. 95, 6939–6941 (2004)

    Google Scholar 

  28. Gopman, D.B., et al.: Static and dynamic magnetic properties of sputtered Fe-Ga thin films. IEEE Trans. Magn. 53, 1–4 (2017)

    Google Scholar 

  29. Goussev, A., Lund, R.G., Robbins, J.M., Slastikov, V., Sonnenberg, C.: Domain wall motion in magnetic nanowires: an asymptotic approach. Proc. R. Soc. A 469(2160), 20130308 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Agarwal, S., Carbou, G., Labbè, S., Prieur, C.: Control of a network of magnetic ellipsoidal samples. Math. Control Relat. Fields 1(2), 129–147 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Dubey, S., and Dwivedi, S.: On controllability of a two-dimensional network of ferromagnetic ellipsoidal samples. Differ. Equ. Dyn. Syst. D. 1–21, (2018)

  32. Dwivedi, S., Dubey, S.: Field-driven magnetization reversal in a three-dimensional network of ferromagnetic ellipsoidal samples. Rendiconti del Circolo Matematico di Palermo Series 2 69(2), 497–519 (2020)

    MathSciNet  MATH  Google Scholar 

  33. Schryer, N.L., Walker, L.R.: The motion of \(180^{\circ }\) domain walls in uniform dc magnetic fields. J. Appl. Phys. 45(12), 5406–5421 (1974)

    Google Scholar 

  34. Dwivedi, S.: On the dynamics of transverse domain walls in biaxial magnetic nanostrips with crystallographic defects. AIP Conf. Proc. 1975(1), 030028 (2018)

    Google Scholar 

  35. Dwivedi, S., Dubey, S., Singh, Y.P.: On the statics of transverse domain walls in ferromagnetic nanostrips. Iran. J. Sci. Technol. Trans. A. Sci. 44(3), 717–724 (2020)

    MathSciNet  Google Scholar 

  36. Consolo, G., Currò, C., Martinez, E., Valenti, G.: Mathematical modeling and numerical simulation of domain wall motion in magnetic nanostrips with crystallographic defects. Appl. Math. Model. 36(10), 4876–4886 (2012)

    MathSciNet  MATH  Google Scholar 

  37. Mougin, A., Cormier, M., Adam, J.P., Metaxas, P.J., Ferré, J.: Domain wall mobility, stability and Walker breakdown in magnetic nanowires. Europhys. Lett. 78(5), 57007 (2007)

    Google Scholar 

  38. Consolo, G., Curro, C., Valenti, G.: Curved domain walls dynamics driven by magnetic field and electric current in hard ferromagnets. Appl. Math. Model. 38, 1001–1010 (2014)

    MathSciNet  MATH  Google Scholar 

  39. Consolo, G., Valenti, G.: Traveling wave solutions of the one-dimensional extended Landau-Lifshitz-Gilbert equation with nonlinear dry and viscous dissipations. Acta Appl. Math. 122, 141–152 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Landau, L.D., Lifshitz, E.M.: On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion 8, 153–169 (1935)

    MATH  Google Scholar 

  41. Gilbert, T.L.: A Lagrangian formulation of the gyromagnetic equation of the magnetization field. Phys. Rev. 100, 1243 (1955)

    Google Scholar 

  42. Dwivedi, S., Dubey, S.: On Dynamics of Current-Induced Static Wall Profiles in Ferromagnetic Nanowires Governed by the Rashba Field. Int. J. Appl. Comput. Math. 3(1), 27–42 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Thiaville, A., Nakatani, Y., Miltat, J., Suzuki, Y.: Micromagnetic understanding of current-driven domain wall motion in patterned nanowires. Europhys. Lett. 69, 990–996 (2005)

    Google Scholar 

  44. Miron, I.M., Moore, T., Szambolics, H., et al.: Fast current-induced domain-wall motion controlled by the Rashba effect. Nat. Mater. 10(6), 419–423 (2011)

    Google Scholar 

  45. Liu, L., Moriyama, T., Ralph, D.C., Buhrman, R.A.: Spin-torque ferromagnetic resonance induced by the spin Hall effect. Phys. Rev. Lett. 106(3), 036601 (2011)

    Google Scholar 

  46. Manchon, A., Koo, H.C., Nitta, J., Frolov, S.M., Duine, R.A.: New perspectives for Rashba spin-orbit coupling. Nat. Mater. 14(9), 871–882 (2015)

    Google Scholar 

  47. Pylypovskyi, O.V., Sheka, D.D., Kravchuk, V.P., Yershov, K.V., Makarov, D., Gaididei, Y.: Rashba torque driven domain wall motion in magnetic helices. Sci. Rep. 6(1), 1–11 (2016)

    Google Scholar 

  48. Wang, X., Manchon, A.: Diffusive spin dynamics in ferromagnetic thin films with a Rashba interaction. Phys. Rev. Lett. 108(11), 117201 (2012)

    Google Scholar 

  49. Xu, Y., Yang, Y., Yao, K., Xu, B., Wu, Y.: Self-current induced spin-orbit torque in FeMn/Pt multilayers. Sci. Rep. 6(1), 1–11 (2016)

    Google Scholar 

  50. Shahu, C.K., Dwivedi, S., Dubey, S.: Curved domain walls in the ferromagnetic nanostructures with Rashba and nonlinear dissipative effects. Appl. Math. Comput. 420, 126894 (2022)

    MathSciNet  MATH  Google Scholar 

  51. Martinez, E.: The influence of the Rashba field on the current-induced domain wall dynamics: a full micromagnetic analysis, including surface roughness and thermal effects. J. Appl. Phys. 111(7), 07D302 (2012)

    Google Scholar 

  52. He, P.B., Zhou, Z.D., Wang, R.X., Li, Z.D., Cai, M.Q., Pan, A.L.: Stability analysis of current-driven domain wall in the presence of spin Hall effect. J. Appl. Phys. 114(9), 093912 (2013)

    Google Scholar 

  53. Martinez, E., Emori, S., Beach, G.S.: Current-driven domain wall motion along high perpendicular anisotropy multilayers: The role of the Rashba field, the spin Hall effect, and the Dzyaloshinskii-Moriya interaction. Appl. Phys. Lett. 103(7), 072406 (2013)

    Google Scholar 

  54. Seo, S.M., Kim, K.W., Ryu, J., Lee, H.W., Lee, K.J.: Current-induced motion of a transverse magnetic domain wall in the presence of spin Hall effect. Appl. Phys. Lett. 101(2), 022405 (2012)

    Google Scholar 

  55. Ryu, J., Lee, K.J., Lee, H.W.: Current-driven domain wall motion with spin Hall effect: reduction of threshold current density. Appl. Phys. Lett. 102(17), 172404 (2013)

    Google Scholar 

  56. Martinez, E., Finocchio, G.: Domain wall dynamics in asymmetric stacks: the roles of Rashba field and the spin Hall effect. IEEE Trans. Magn. 49(7), 3105–3108 (2013)

    Google Scholar 

  57. Osborn, J.A.: Demagnetizing factors of the general ellipsoid. Phys. Rev. 67(11–12), 351 (1945)

    Google Scholar 

  58. Carbou, G.: Stability of static walls for a three-dimensional model of ferromagnetic material. Journal de mathématiques pures et appliquées 93(2), 183–203 (2010)

    MathSciNet  MATH  Google Scholar 

  59. Dwivedi, S., Dubey, S.: On the stability of steady-states of a two-dimensional system of ferromagnetic nanowires. J. Appl. Anal. 23(2), 89–100 (2017)

    MathSciNet  MATH  Google Scholar 

  60. Dwivedi, S., Dubey, S.: On the stability of static domain wall profiles in ferromagnetic thin film. Res. Math. Sci. 6(1), 2 (2019)

    MathSciNet  MATH  Google Scholar 

  61. Federico, S., Consolo, G., Valenti, G.: Tensor representation of magnetostriction for all crystal classes. Math. Mech. Solids 24(9), 2814–2843 (2019). (37)

    MathSciNet  MATH  Google Scholar 

  62. Baňas, L.U.: A numerical method for the Landau-Lifshitz equation with magnetostriction. Math. Methods Appl. Sci. 28(16), 1939–1954 (2005)

    MathSciNet  MATH  Google Scholar 

  63. Liang, C.Y., Keller, S.M., Sepulveda, A.E., Bur, A., Sun, W.Y., Wetzlar, K., Carman, G.P.: Modeling of magnetoelastic nanostructures with a fully coupled mechanical-micromagnetic model. Nanotechnology 25(43), 435701 (2014)

    Google Scholar 

  64. Shu, Y.C., Lin, M.P., Wu, K.C.: Micromagnetic modeling of magnetostrictive materials under intrinsic stress. Mech. Mater. 36, 975–997 (2004)

    Google Scholar 

  65. Mudivarthi, C., Datta, S., Atulasimha, J., Evans, P.G., Dapino, M.J., Flatau, A.B.: Anisotropy of constrained magnetostrictive materials. J. Magn. Magn. Mater. 322(20), 3028–3034 (2010)

    Google Scholar 

  66. Mballa-Mballa, F.S., et al.: Micromagnetic modeling of magneto-mechanical behavior. IEEE Trans. Magn. 50, 1–4 (2014)

    Google Scholar 

  67. Yang, J.: An Introduction to the Theory of Piezoelectricity. Springer, New York (2005)

    MATH  Google Scholar 

  68. Consolo, G., and Valenti, G.: Optimized voltage-induced control of magnetic domain-wall propagation in hybrid piezoelectric/magnetostrictive devices. In: Actuators (Vol. 10, No. 6, p. 134). MDPI (2021)

  69. Baltensperger, W., Helman, J.S.: A model that gives rise to effective dry friction in micromagnetics. J. Appl. Phys. 73, 6516–6518 (1993)

    Google Scholar 

  70. Chen, D.-X., Pardo, E., Sanchez, A.: Demagnetizing factors of rectangular prisms and ellipsoids. IEEE Trans. Magn. 38(4), 1742–1752 (2002)

    Google Scholar 

  71. Cayssol, F., Menéndez, J.L., Ravelosona, D., Chappert, C., Jamet, J.P., Ferré, J., Bernas, H.: Enhancing domain wall motion in magnetic wires by ion irradiation. Appl. Phys. Lett. 86, 022503.1-022503.3 (2005)

    Google Scholar 

  72. Shepley, P.M., Rushforth, A.W., Wang, M., Burnell, G., Moore, T.A.: Modification of perpendicular magnetic anisotropy and domain wall velocity in Pt/Co/Pt by voltage-induced strain. Sci. Rep. 5(1), 1–5 (2015)

    Google Scholar 

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Acknowledgements

S. Dwivedi would like to thank the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India, and the National Institute of Technology Andhra Pradesh, for the financial support through Projects CRG/2019/003101 and NITAP/SD-G/15/2020, respectively, and G. Consolo gratefully acknowledges support from INdAM-GNFM and from Italian MIUR through project PRIN2017 “Multiscale phenomena in Continuum Mechanics: singular limits, off-equilibrium and transitions” (project number: 2017YBKNCE).

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Authors and Affiliations

  1. Department of Mathematics, School of Sciences, National Institute of Technology Andhra Pradesh, Tadepalligudem, 534 101, India

    Sumit Maity, Sarabindu Dolui & Sharad Dwivedi

  2. Department of Mathematics, Computer, Physical and Earth Sciences, University of Messina, 98166, Messina, Italy

    Giancarlo Consolo

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  1. Sumit Maity
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Maity, S., Dolui, S., Dwivedi, S. et al. Domain wall dynamics in cubic magnetostrictive materials subject to Rashba effect and nonlinear dissipation. Z. Angew. Math. Phys. 74, 23 (2023). https://doi.org/10.1007/s00033-022-01911-9

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