Disorder-induced domain wall velocity shift at high fields in perpendicularly magnetized thin films

Michele Voto, Luis Lopez-Diaz, Luis Torres, and Simone Moretti

Phys. Rev. B 94, 174438 – Published 23 November 2016

Abstract

Domain wall dynamics in a perpendicularly magnetized system is studied by means of micromagnetic simulations in which disorder is introduced as a dispersion of both the easy-axis orientation and the anisotropy constant over regions reproducing a granular structure of the material. High field dynamics show a linear velocity-field relationship and an additional grain size dependent velocity shift, weakly dependent on both applied field and intrinsic Gilbert's damping parameter. We find the origin of this velocity shift in the nonhomogeneous in-plane effective field generated by the tilting of anisotropy easy axis introduced by disorder. We show that a one-dimensional analytical approach cannot predict the observed velocities and we augment it with the additional dissipation of energy arising from internal domain wall dynamics triggered by disorder. This way we prove that the main cause of higher velocity is the ability of the domain wall to irradiate energy into the domains, acquired with a precise feature of disorder.

Received 2 August 2016
Revised 14 October 2016

DOI:https://doi.org/10.1103/PhysRevB.94.174438

Physics Subject Headings (PhySH)

Domain walls Magnetic anisotropy

Random & disordered media Thin films

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Michele Voto^*, Luis Lopez-Diaz, Luis Torres, and Simone Moretti

Departamento de Física Aplicada, Universidad de Salamanca, Plaza de la Merced s/n, 37008 Salamanca, Spain

^*michele.voto@usal.es

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Issue

Vol. 94, Iss. 17 — 1 November 2016

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Images

Figure 1
(a) Schematic of the system under study: Bloch domain wall separating up and down domains. The external field is applied out-of-plane, favoring the expansion of the red domain. A typical Voronoi tessellation for a grain distribution is shown on the $x − y$ plane together with the polar $(η)$ and azimuthal $(ζ$ , inset) angles of a tilted anisotropy easy-axis $\hat{u}$ . (b) Distribution of the in-plane components of $\hat{u}$ for a sample with grain size $d = 30$ nm. A Gaussian distribution of standard deviation $σ = 0.05$ and mean 0 is used to perturb the $x$ and $y$ components at each grain. (c) Corresponding distribution of the polar $(η)$ and azimuthal $(ζ)$ angles of the easy axis for the same sample.
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Figure 2
(a) DW velocity as function of applied field from micromagnetic simulations with Gilbert's damping $α = 0.015$ and various grain diameters, together with simulations with no disorder and 1D model analytical calculations (green line). (b)–(d) DW velocity at high field for various values of $α$ and a fixed grain size, showing the shift in the zero field intercept with introduction of disorder. Dashed lines are linear fittings to the 1D linear precessional regime.
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Figure 3
(a) DW velocity versus field from micromagnetic simulations with Gilbert's damping $α = 0.015$ of a perfect sample (blue circles) and a sample with disorder and grain size $d = 10$ nm; continuous lines are 1D model velocity calculations with a specified tilting $η$ of easy axis; spades $(♠)$ symbols show DW velocity for simulations where easy axis has been kept uniform $\hat{u} = \hat{z}$ ; clubs $(♣)$ symbols show DW velocity for simulations where anisotropy constant $k_{u}$ has been kept uniform. (b) DW velocity versus field from simulations with $α = 0.05$ and various grain size. Continuous lines are 1D model velocity calculations for various $η$ .
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Figure 4
(a) Time evolution of absolute topological charge variation calculated using Eq. (7) for a sample with no disorder (blue line) and another with grain size $d = 15$ nm (red line), extracted from simulations with $α = 0.015, μ_{0} H = 140 mT$ and shown over a time window of 20 ns. (b) Time evolution of the total dissipated power calculated using Eq. (2) for the same simulations as in (a), displayed over the same time window. It appears clear the correlation between peaks in dissipated energy and change in topological charge in the case with disorder. The dotted line marks the average dissipation rate ${\bar{P}}_{d} = 〈 P_{d} - P_{0} 〉$ . (c) and (d) Snapshots of dissipated power and magnetization around the DW of the perfect sample at the instant of time indicated in (b) by the blue circle. (e) and (f) Snapshots showing the same features for the disordered sample at the instant of time indicated in (b) by the red circle. Hot-cold color gradient represents the amount of local dissipated power in logarithmic scale, in-plane magnetization angle is represented by a color as in the color wheel.
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Figure 5
(a) DW velocity versus applied field from simulations with $α = 0.015$ without disorder (blue circles) and with different grain sizes $(d = 5$ nm red diamonds, $d = 15$ nm magenta hexagons, $d = 30$ nm gray down-triangles). Cyan dots show DW velocity computed from 1D model (11) with a realistic $η = 4 . 5^{\circ}$ plus dissipation contribution extracted from simulations. Same results for $α = 0.03$ and $α = 0.05$ are shown in (b) and (c), respectively, where the dissipation contribution is the one extracted from the simulations in (a) with $α = 0.015$ .
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