Universal pinning energy barrier for driven domain walls in thin ferromagnetic films

We report a comparative study of magnetic field driven domain wall motion in thin films made of different magnetic materials for a wide range of field and temperature. The full thermally activated creep motion, observed below the depinning threshold, is shown to be described by a unique universal energy barrier function. Our findings should be relevant for other systems whose dynamics can be modeled by elastic interfaces moving on disordered energy landscapes.

Domain walls are at the basis of future applications of high-density memories in ferromagnetic materials [1]. In this type of memories data bits are built up of magnetic domains with opposite magnetization varying in size and/or in position, and hence working memories implies the displacement of domain walls. Noteworthy, even weak random pinning due to local defects or inhomogeneities in the host materials is known to have a strong effect on domain walls [2]. Pinning tends to stabilize domain wall positions [3], introduces stochasticity [4], induces domain wall roughness and dramatically modifies the driven dynamics at small field and current [5][6][7]. A fundamental understanding on how weak disorder affects the dynamics of domain walls is thus critical for applications. This question is also particularly relevant on a wider context since pinning dependent motion of elastic interfaces is observed in a large variety of other systems such as ferroelectric materials [8], contact lines in wetting [9], crack propagation [10], and earthquake models [11]. In all those systems, the competition between the interface elasticity and pinning leads to rich and complex thermally activated motion over effective energy barriers (see Fig. 1) described by universal law [2,12,13]. Although remarkable efforts have been made in the last decades, a quantitative description of the thermally activate regime of slow motion, so-called creep regime, has remained elusive.
An essential starting point to seize the universal character of the pinning dependent motion is the zero temperature behavior (see Fig. 1A). For an elastic line driven by a force f , a depinning threshold f d separates a zero velocity regime for f < f d from a finite velocity regime for f > f d . A finite temperature value T results in a thermally activated subthreshold creep motion with the velocity following an Arrhenius law v ∼ exp(−∆E/k B T ). For a near zero driving force (f → 0), the motion of an elastic line requires to overcome a universal divergent energy barrier presenting a power law variation ∆E ∼ f −µ . The universal creep exponent µ presents a good agreement [5,7,[14][15][16][17][18][19] with the predicted value (µ = 1/4), for an elastic line moving in an uncorrelated disordered potential [20]. A more stringent test of universality demonstrating a compatibility between universal scaling exponents and dimensionality [20] was performed in ferromagnetic Pt/Co/Pt ultrathin films [5,14] but was not yet reproduced for other ferromagnetic materials or other physical systems. Moreover, In this work, we address the question of universality from a study of magnetic field driven creep motion in magnetic films with perpendicular anisotropy. In magnets, the elastic interfaces are magnetic domain walls and the force f is proportional to the applied field H (see Fig. 1B). The field-induced domain wall dynamics is studied in a singlecrystalline (Ga 0.95 ,Mn 0.05 )(As 0.9 ,P 0.1 ) semi-conductor [25] . 2B). This non-universal regime is controlled by material dependent microscopic dynamical structure of domain walls [24].
where k B T d is the characteristic pinning energy scale, and k B the Boltzmann constant.
Note that a similar empirical law was proposed in various theoretical works [31,32]. It is easy to see that Eq. 2 yields the creep law ∆E ∼ k B T d (H/H d ) −µ for H → 0 and a linear vanishing of the energy barrier (∆E → 0) for H → H d [31]. Each velocity-field characteristic was fitted by Eqs. 1 and 2 (see [33]) in order to determine the parameters H d , v(H d , T ), and T d . As reported in the supplementary material [33], the obtained values are material and temperature dependent reflecting the microscopical origin of pinning [5,18].
However a discussion on this non-trivial subject goes beyond the scope of this letter. As it can be observed in Fig.2  [21] whose energy scale was adjusted to experimental data (see [33]). The dashed line corresponds to the linear variation of the energy barrier close to the depinning field (H = H d ).
Inset: Universal barrier presented in semi-log scale showing a good quantitative agreement with Eq. 2 over more than three orders of magnitude.
the creep regime can be very well described by a unique (reduced) barrier function of the (reduced) temperature and field. Furthermore, it compares fairly well with the results obtained by using numerical simulations [21] of a minimal model for a one-dimensional elastic line in a two-dimensional disordered medium (see Fig. 3). As this model does not take into account the properties of a specific system, the barrier function of the creep regime is expected to be relevant for a larger variety of systems other than ferromagnets.
In conclusion, we provide evidence of the universal character of the whole thermally activated subthreshold creep motion in magnetic thin films. In this dynamical regime, the magnetic domain wall motion is shown to be controlled by a unique universal reduced energy barrier function. The compatibility of this universal law with the predictions of a minimal model strongly suggests our results to be relevant to understand the creep dynamics in other systems than magnetic thin films whose emergent properties are also controlled by the competition between quenched disorder and the elasticity of a driven fluctuating string.