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Scaling properties of a ferromagnetic thin film model at the depinning transition

In this paper, we perform a detailed study of the scaling properties of a ferromagnetic thin film model. Recently, interest has increased in the scaling properties of the magnetic domain wall (MDW) motion in disordered media when an external driving field is present. We consider a (1+1)-dimensional model, based on evolution rules, able to describe the MDW avalanches. The global interface width of this model shows Family-Vicsek scaling with roughness exponent $ζ\simeq 1.585$ and growth exponent $β\simeq 0.975$. In contrast, this model shows scaling anomalies in the interface local properties characteristic of other systems with depinning transition of the MDW, e.g. quenched Edwards-Wilkinson (QEW) equation and random-field Ising model (RFIM) with driving. We show that, at the depinning transition, the saturated average velocity $v_\mathrm{sat}\sim f^θ$ vanished very slowly (with $θ\simeq 0.037$) when the reduced force $f=p/p_\mathrm{c}-1\to 0^{+}$. The simulation results show that this model verifies all accepted scaling relations which relate the global exponents and the correlation length (or time) exponents, valid in systems with depinning transition. Using the interface tilting method, we show that the model, close to the depinning transition, exhibits a nonlinearity similar to the one included in the Kardar-Parisi-Zhang (KPZ) equation. The nonlinear coefficient $λ\sim f^{-ϕ}$ with $ϕ\simeq -1.118$, which implies that $λ\to 0$ as the depinning transition is approached, a similar qualitatively behaviour to the driven RFIM. We conclude this work by discussing the main features of the model and the prospects opened by it.