Strength and length-scale of the interaction between domain walls and pinning disorder in thin ferromagnetic films

We explore the magnetic-field-driven motion of domain walls with different chiralities in thin ferromagnetic films made of Pt/Co/Pt, Au/Co/Pt, and Pt/Co/Au. From the analysis of domain wall dynamics, we extract parameters characterizing the interaction between domain walls and weak pinning disorder of the films. The variations of domain wall structure, controlled by an in-plane field, are found to modify the characteristic length-scale of pinning in strong correlation with the domain wall width, whatever its chirality and the interaction strength between domain walls and pinning defects. These findings should be also relevant for a wide variety of elastic interfaces moving in weak pinning disordered media.

The controlled motion of magnetic textures such as chiral domain walls (DWs) [1] or skyrmions resulting from the Dzyaloshinskii-Moriya interaction (DMI) [2] are at the basis of potential applications of spintronic devices [3]. However, magnetic textures are very sensitive to weak pinning due to ubiquitous inhomogeneities of magnetic materials, which strongly reduces their velocity and produces stochastic universal behaviors [4,5]. Despite numerous recent studies [6][7][8][9][10][11] focusing on the dynamics of pinned chiral magnetic textures, their interactions with weak pinning disorder is far from being understood.
DWs are well known to present universal behaviors [12][13][14] similar to those encountered by interfaces in a wide variety of other physical systems. Those behaviors can be described by minimal statistical-physics models [15][16][17] as an interplay between interface elasticity, weak pinning, thermal activation, and a driving force f . The dynamical regimes of interfaces strongly depend on the relative magnitude of the drive f compared to a depinning threshold force f d . In the creep regime (f < f d ), the velocity follows an Arrhenius law v ∼ e −∆E/(k B T ) , where the barrier height presents an asymptotic power law behavior ∆E ∼ f −µ with the force f close to zero. At and just above the depinning threshold, the velocity presents a power law variation with the temperature The critical exponents µ, ψ, and β are universal (i.e. material and temperature independent). Their values characterize the universality class of the motion and reflects the dimension of the interface and embedding medium, the range of elasticity and the interaction with pinning defects. In ultrathin films with perpendicular anisotropy, a perpendicular magnetic field H can serve as an isotropic driving force (H ∝ f ). A large majority of experimental studies on DW dynamics reported in the literature [18] is compatible with the theoretical predictions (µ = 1/4, ψ = 0.15, and β = 0.25) for short range (random bond) interactions between pinning dis-order and DWs. However, the minimal models ignore the exact structure of interfaces and the characteristic length-scale of pinning is a parameter chosen arbitrarily [17,19]. Since the seminal work of Lemerle et al. [12] on the creep motion, basic issues such as the length-scale and the strength of interaction between DW and defects in magnetic materials remain open.
Recent experiments on the creep motion of chiral DWs have evidenced the correlations between the DW magnetic texture and its dynamics. In thin films with perpendicular anisotropy, the DMI results in an in-plane effective magnetic field H DMI pointing in the direction perpendicular to the DW. The DMI field combined with an in-plane field H x can be used to adjust the in-plane component of magnetization direction in the DW and to control the DW magnetic structure. The recent observation of asymmetric expansion of initially circular domains [6] has lead to precise investigations of the variation of DW width, energy and stiffness [8][9][10][11] with the direction and magnitude of the in-plane and DMI fields and to re-examine more generally the creep motion of chiral DWs [11]. Rather accurate descriptions of the shape of the velocity curves versus in-plane field are now obtained [10,11]. The proposed models discuss the variations of creep barrier height ∆E with the DW energy [6][7][8][9][10][11]. Surprisingly, the interaction between the DW and random pinning disorder is assumed to be independent of DW's magnetic structure.
Here, we evidence a strong correlation between the variation of DW width controlled by an in-plane field and the characteristic length of pinning, in films with three different chiralities. Our argument is organized as follows. We first extract the material and in-plane field dependent pinning parameters controlling DW dynamics, from the self-consistent analysis proposed in Ref. [13,14]. We then numerically compute the variation with in-plane and DMI fields of the DW energy and width. The latter are compared to the variation of pinning range and strength deduced from the pinning parameters via scaling arXiv:2005.08862v1 [cond-mat.mes-hall] 18 May 2020 relations [18].
Experimental techniques. The samples are Pt/Co/Pt, Pt/Co/Au, and Au/Co/Pt films (with thicknesses of 5 nm for Pt and Au and 0.9 nm for Co) with perpendicular magnetic anisotropy, which have been grown by e-beam evaporation in ultra high vacuum on Si(001)/SiO 2 (100 nm)/Ta(5 nm) templates. They present (111) oriented crystallites with a typical grain size of 15 nm [20]. The micromagnetic parameters characterizing the films are detailed in Table I. The DW displacement was observed by polar Kerr microscopy. The in-plane magnetic field H x controlling the magnetization direction in the DW was generated by two large coils supplied with DC current. The out-of-plane field H used to move DW was produced by a small coil (diameter ≈ 1 mm) placed in the close vicinity of the films and supplied by pulses of duration ∆t between 1 µs and 10 ms. The explored range of µ 0 H x (±187 mT) and µ 0 H(0 − 120 mT) was limited by the nucleation of multiple domains, which impede measurement of DW displacement. The DW velocity v corresponds to the ratio ∆x/∆t, where ∆x was exclusively measured in the direction of H x (see the inset of Fig. 1).
Domain wall dynamics. The velocity curves obtained for the three films and different values of µ 0 H x are reported in Fig. 1. For their analysis, we used the self-consistent description of the creep and depinning regimes [13,14,18,21,22]: is the energy barrier of the creep regime and the depinning law is the asymptotic velocity behavior in the limit of negligible contribution of thermal fluctuations [14]. In Eq. 1, µ = 1/4, β = 0.25, and ψ = 0.15 are universal critical exponents and x 0 = 0.65 a universal constant [14] characterizing the quenched Edwards Wilkinson universality class [14,15,23] and are therefore fixed. The nonuniversal parameters are the characteristic height of effective pinning barrier k B T d and coordinates of the depinning threshold (corresponding to ∆E → 0) H d and v(H d ). Those three parameters depend on the film magnetic and pinning properties, and external parameters as the in-plane field, and the temperature [18]. As it can be observed in Fig. 1 a-c, all the fits of Eqs. 1, obtained with only three fitting parameters, present a good agreement with the velocity curves (see Refs. [18] for details on the fitting procedure). This finding is compatible with the results reported for a large variety of other magnetic materials [13,14,18,21,22] and confirms that chiral and non-chiral DWs follow very similar universal behaviors as predicted in Ref. 11 for the creep regime.
Pinning at the microscopic scale. Having discussed universal behaviors, we can now focus on the obtained non-universal parameters (see Fig. 1 d-e). (Here, we do not discuss the values of v(H d ) since it is partly determined by the contribution of DW dynamics in the flow regime at the threshold H d [18].) Surprisingly, the values of H d and T d vary with the in-plane magnetic field H x . This strongly suggests that the pinning properties of DWs depend on with their magnetic texture, in contrast to the usual assumption found in the literature [6,[8][9][10][11]. In order to understand the variations of H d and T d , we shall discuss the DW pinning at the microscopic scale and, more precisely, to compare the effect of the in-plane field on the DW energy and width and on the strength and length-scale of pinning (see Fig. 2).
To predict accurately the variation of DW energy and structure with in-plane and DMI fields, we have decided not to use simplified descriptions of DWs [6,[8][9][10][11]. We have calculated the magnetization profile M (x) of a DW whose plane is ⊥ H x , from numerical micromagnetic calculations (MuMax3 [24]), using the parameters reported in Table I. In addition, an analytical model leading to very similar results has been developed [25]. For the DW width, we use both the geometrical Hubert [26] ∆ H and dynamic Thiele [27,28]  In order to discuss the DW pinning at the microscopic scale with in-plane field H x , we use standard scaling arguments [12,19]. The free energy of a DW segment of length L, deformed over a distance u [11,12] can be written as δF (L, u) = δF elas (L, u) − δF pin (L, ξ) − δF z (L, u), where δF elas = σtu 2 /L is the elastic energy produced by the increase of DW length and, δF z = 2µ 0 M s HtLu is the gain of Zeeman energy due to magnetization reversal. The DW stiffness σ is, at this stage, assumed to be given by the DW surfacic energy. For a weak disorder producing fluctuations of DW energy, the pin-  A good agreement with the data is obtained up to the cross-over between the depinning and flow regimes [14], which can be observed at µ0Hx = 0 mT for Pt/Co/Au (µ0H > 70 mT) and Pt/Co/Pt (µ0H > 90 mT). The curves (d) and (e) correspond to the variation with µ0Hx of the depinning field (H d ) and temperature (T d ) deduced from the fit of Eqs. 1. Insert of c. Displacement of DW in the Au/Co/Pt film produced by a magnetic field pulse (µ0H = 3 mT, ∆t = 1 µs) for µ0Hx = 48 mT.
ning term can be written δF pin = f pin √ nξLξ, where f pin and ξ are the characteristic force and range of the DW-defects interaction, respectively and, n is the density of pinning defects per unit surface area. Here we go beyond the conventional relation ξ ∝ b (≈ 1/ √ n 2 ) used throughout in the literature [12]: we reintroduce a clear distinction between the characteristic length-scale of the pinning disorder b, which is fixed by the material inhomogeneities, and the range ξ of the DW-defects interaction, which may vary with the DW structure [29] as shown in the following discussion. For a short segment length (L < L c ), the DW is too rigid to follow the random pinning potential (δF elas (L, u) > δF pin ). The segment is collectively pinned by a set of pinning defects. The characteristic collective pinning lengthscale can be deduced from δF elas (L c , ξ) ∼ δF pin (L c , ξ), which reads L c ∼ (ξ/n) 1/3 (σt/f pin ) 2/3 . For L > L c , the pinning energy is larger than the elastic energy: a DW can be seen as a set of rigid segments of length L c , whose orientation follows the random pinning landscape. In order to depin a rigid segment, the magnetic field H has to reach a threshold field H d . The latter can be determined from δF pin (L c , ξ) ∼ δF z (L c , ξ), which leads to H d ∼ nξ/L c f pin /(2µ 0 M s t). Then, assuming [18] that the pinning barrier height is given by k B T d ∼ δF pin (L c , ξ), we obtain the scaling relations: which relate the characteristic range ξ and force f pin of the DW-defect interaction, to the measured depinning field H d (µ 0 H x ) and temperature T d (µ 0 H x ) (see Fig. 1 d-e) and the predicted DW surface energy σ(µ 0 H x ) (see Fig. 2 b). The variation with in-plane field of ξ and f pin , determined from the measurements of H d , T d (see Fig. 1 d-e), the predictions for σ(µ 0 H x ) (see Fig. 2 b) and Eqs 2-3, are shown in Fig. 2. For a comparison with the predicted variations of the DW width, the values of ξ were rescaled with a constant factor, which is the only free parameter. As it can be observed in Fig. 2 a, the variations with in-plane field of the pinning range ξ(µ 0 H x ) present strong correlations with the predictions for both ∆ H and ∆ T : the agreement is very good for the Pt/Co/Au except for µ 0 H x > 100 mT and it is even better for the Pt/Co/Pt and Au/Co/Pt films. The DMI field seems to essentially modify the value of µ 0 H x at which a minimum of DW width (and pinning range) is observed and no chiral contribution of pinning can be evidenced. Those observations suggest that the range of the DW-defect interaction is close to the DW width. As a direct consequence, the characteristic length-scale of pinning defects (b ≈ 1/ √ n 2 ) should be close to or smaller than the DW width [29], which rules out the approximation considering DW as one dimensional line [12,30] interacting with remote defects (∆ H and ∆ T b). Let us now discuss the pinning force f pin , whose vari-ations with in-plane field is reported in Fig.2 b. Here also, the only free parameter is the rescaling factor. As it can be observed, the pinning force follows rather well the variations of σ(H x ). This is expected since the weak pinning of DWs results from fluctuations of DW energy, whose amplitude should decrease as σ decreases. An additional contribution is the increase of DW width, which (for b < ∆) is predicted to reduce the strength of pinning interaction [31,32].
In conclusion, variations of the range and the strength of pinning following modifications of the wall structure have been evidenced. This effect, observed on magnetic domain walls with chiral structure submitted to large hard-axis magnetic fields, should be relevant for elastic interfaces moving in weak pinning disordered media [16,17] in a wide variety of others systems.
We wish to thank J. Sampaio for careful reading of the manuscript.