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Static properties and current-induced dynamics of pinned 90∘ magnetic domain walls under applied fields: An analytic approach

Pavel Baláž, Sampo J. Hämäläinen, and Sebastiaan van Dijken
Phys. Rev. B 98, 064417 – Published 21 August 2018
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Article Text
  • INTRODUCTION
  • MODEL
  • 90∘ DOMAIN WALL
  • CURRENT-INDUCED DOMAIN WALL DYNAMICS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Magnetic domain walls are pinned strongly by abrupt changes in magnetic anisotropy. When driven into oscillation by a spin-polarized current, locally pinned domain walls can be exploited as tunable sources of short-wavelength spin waves. Here, we develop an analytical framework and discrete Heisenberg model to describe the static and dynamic properties of pinned domain walls with a head-to-tail magnetic structure. We focus on magnetic domain walls that are pinned by 90∘ rotations of uniaxial magnetic anisotropy. Our model captures the domain wall response to a spin-transfer torque that is exerted by an electric current. Model predictions of the domain wall resonance frequency and its evolution with magnetic anisotropy strength and external magnetic field are compared to micromagnetic simulations.

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    • Received 19 February 2018
    • Revised 4 May 2018

    DOI:https://doi.org/10.1103/PhysRevB.98.064417

    ©2018 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Domain wallsFerroelectric domainsFerromagnetismMagnetic domainsMicromagnetismSpin currentSpin waves
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Pavel Baláž1,2,*, Sampo J. Hämäläinen3, and Sebastiaan van Dijken3

    • 1Charles University, Faculty of Mathematics and Physics, Department of Condensed Matter Physics, Ke Karlovu 5, CZ 121 16 Prague, Czech Republic
    • 2IT4Innovations Center, VSB Technical University of Ostrava, 17. listopadu 15, CZ 708 33 Ostrava-Poruba, Czech Republic
    • 3NanoSpin, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, FI-00076 Aalto, Finland

    • *balaz@karlov.mff.cuni.cz

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    Vol. 98, Iss. 6 — 1 August 2018

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    Images

    • Figure 1
      Figure 1

      Collective coordinates of a head-to-tail 90∘ magnetic DW. The red arrows point in the direction of local magnetization. The left (L) and right (R) parts of the magnetic layer differ in the direction of uniaxial magnetic anisotropy axis. The red solid line indicates the abrupt anisotropy boundary, and the black dashed line marks the DW center. The displacement of the DW center from the anisotropy boundary is given by coordinate q while the DW tilting angle from the film plane is given by ψ. The black arrow indicates the direction of applied magnetic field (Happ).

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    • Figure 2
      Figure 2

      Spherical coordinate system used in the analytical model.

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    • Figure 3
      Figure 3

      Domain wall profiles obtained from Heisenberg model simulations for different values of applied magnetic field μ0Happ. In the simulations, N=2000, a=0.5nm, Ms=1.5×106A/m, A=2.1×10−11J/m, Ku=2.5×104J/m3, and D⊥=0.1Du. The open circles indicate zero-field solutions of the analytic model [Eq. (19)].

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    • Figure 4
      Figure 4

      (a) Angle ζ and (b) p=λ′/λ as a function of applied magnetic field, Happ, for various values of Ku. The others parameters in the calculations are the same as in Fig. 3.

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    • Figure 5
      Figure 5

      Head-to-tail DW profiles under the influence of an electric current for (a) in-plane, Sz, and (b) out-of-plane, Sx, spin coordinates as obtained from Heisenberg model simulations. The inset of (a) shows the displacement of the DW center from the anisotropy boundary at y=0. In the calculations α=0.15, P=0.5, β=0.4, and |I|=1012A/m2. The other parameters are the same as in Fig. 3.

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    • Figure 6
      Figure 6

      Comparison of the linearized analytical model and Heisenberg model simulations for α=0.15, P=0.5, β=0.4, and I=1012A/m2. The other parameters are the same as in Fig. 3. (a) DW displacement, (b) DW tilting angle, (c) DW width. In (c), the DW width obtained from Heisenberg model simulations is compared to Eq. (18).

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    • Figure 7
      Figure 7

      (a) DW resonance frequency calculated for Happ=0. The line is calculated using the 1D analytical model and the open diamonds are obtained from micromagnetic simulations. (b) Field dependence of the DW resonance frequency calculated using the 1D model (lines) for various values of Ku and extracted from micromagnetic simulations (solid symbols) using Ku=1.0×104J/m3 (pentagons) and 5.0×104J/m3 (triangles).

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    • Figure 8
      Figure 8

      Micromagnetic simulations of the DW profile during current-induced magnetization dynamics. Panels (a) and (b) show results for different magnetic bias fields along the y axis. The solid black lines depict DW profiles in equilibrium (zero current). The dashed red and dotted blue lines show snapshots of the displaced and distorted DW during current-driven oscillations. The anisotropy constant in the simulations is Ku=105J/m3.

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