The critical Barkhausen avalanches in thin random-field ferromagnets with an open boundary

The interplay between the critical fluctuations and the sample geometry is investigated numerically using thin random-field ferromagnets exhibiting the field-driven magnetisation reversal on the hysteresis loop. The system is studied along the theoretical critical line in the plane of random-field disorder and thickness. The thickness is varied to consider samples of various geometry between a two-dimensional plane and a complete three-dimensional lattice with an open boundary in the direction of the growing thickness. We perform a multi-fractal analysis of the Barkhausen noise signals and scaling of the critical avalanches of the domain wall motion. Our results reveal that, for sufficiently small thickness, the sample geometry profoundly affects the dynamics by modifying the spectral segments that represent small fluctuations and promoting the time-scale dependent multi-fractality. Meanwhile, the avalanche distributions display two distinct power-law regions, in contrast to those in the two-dimensional limit, and the average avalanche shapes are asymmetric. With increasing thickness, the scaling characteristics and the multi-fractal spectrum in thicker samples gradually approach the hysteresis loop criticality in three-dimensional systems. Thin ferromagnetic films are growing in importance technologically, and our results illustrate some new features of the domain wall dynamics induced by magnetisation reversal in these systems.

1 Critical exponents for the entire range of thicknesses To determine the critical exponents in Table SI-I, τ 1 , α 1 and τ 2 , α 2 , representing two slopes of the avalanche size distribution P (S, l) and the duration distribution P (T, l) at different thickness l, we fit the whole distribution using the non-linear weighted least squares method. This gives the best unbiased estimator of the fitting parameters' values provided that the weight of each point is taken as to reciprocal of the variance of that point. We the function (Eq. (4) in Model and Methods): where B is the bending size and [5] The simulated data for the avalanche size distributions P (S, l) and P (T, l) both in the hysteresis loop centre and the loop-integrated distributions at different thickness l, see Fig.4 in the paper, are all fitted in this way; a good initial guess for each fitting parameter is very important here, and a lot of attention was dedicated to finding it. The representative fits are illustrated in Fig. SI-1.

Table SI-I of all exponents
The resulting sets of scaling exponents for the entire range of thicknesses are summarised in Table SI-I. The values of the critical exponents τ 1 and τ 2 for the size distribution and α 1 and α 2 for the duration distribution are shown, estimated from the double power-law and cut-off fits of both loop-integrated and loop-centred avalanche distributions. The corresponding data are shown in the left and the middle column of Figure 4 of the main paper. Empty fields in the

Comparisons with theoretical predictions
An analytical form of the distribution of the avalanche sizes with a bump, similar as in our simulations, has been derived by functional renormalization group theory for the interface depinning dynamics, see [1,2] and other references in the main text. Correctly, the avalanche size distribution reads [1] where the exponents τ and δ as well as the parameters A, B, C are estimated from the -expansion. Precisely, from the one-loop expansion, these parameters are determined [1] (we are quoting only the case applicable to RF model): In the RG theory, the parameter ≡ d c − d, where d c = 4 for the interface depinning model. Therefore, for comparisons of these theoretical curves with the data in 3-dimensional systems, we set = 1, while = 2 for the 2-dimensional case.
Here, < S > is the average size of the avalanche according to that distribution, while S m is a fitting parameter [2]. Note that in this theoretical approach, no expression was given for the related distribution of avalanche duration.

Fits for the distribution of avalanche sizes
It should be stressed that only the avalanches in the central part of the hysteresis loop can have an extended domain wall, such that the interface dynamics can be dominant. By setting = 1 in the above expressions, we attempt to fit our simulated data for the avalanche size distribution in HLC for large thickness, i.e., l = L = 256; the fit is shown in the top-right panel in  (1), the scaling exponent τ varies with the thickness. Moreover, the exponent k = 3.95 that fits the cut-off differs from the theoretical one 1/2. Apart from higher-order contributions to the expression (3), these differences call for a new RG theory for systems with finite thicknesses, which will also account for the scaling forms that involve additional scaling variable l/L [6]. , the second part of the distribution is fitted using = 2, i.e., 2-dimensional system, while the first part has the slope of a 3-dimensional system. The data for thick sample l = L = 256 (right) are fitted using the parameters for = 1, i.e., 3-dimensional system.

Shape of avalanches
The RG theory also predicts the asymmetric shape of the avalanches with the exponential term, see [4] and the -expansion for the asymmetry parameter A d as Notice that in the empirical expression [3] in Eq. (5) in the main paper, the asymmetric term represents the first order expansion of the above exponential expression with the asymmetry parameter a ≡ A d . In [5], by performing an analysis of many samples with different thicknesses and the avalanche duration T up to 2048, the corresponding values of the parameter a are found to be negative and dependent on the actual thickness in agreement with the RG predictions (4), see Table SI-II for details. It is interesting to notice that, by taking d c = 4, the expression (4) gives the value a=-0.084 for d = 3, while a=-0.168 for d = 2; the latter value is close to the numerical one for large avalanche durations in the thin sample.

Distributions away from the critical line
To add to the discussion in the main paper, here we present the simulation results for the situations that typically occur in empirical investigations, that is away from the critical line.

Fixed disorder-varied thickness
This situation often occurs in experiments; it represents a horizontal cut through the phase diagram, cf. Fig. SI-5.

Fixed thickness-varied disorder
This situation represents a vertical cut through the disorder-thickness phase diagram. In the laboratory experiments, it assumes samples of a different composition, as in the case of thin films, cf. . SI-5. Theoretically, below the critical disorder line, the interface depinning occurs, and the distributions dominate with the spanning avalanches. Whereas, above the critical line, the cut-offs are limiting the power-law distributions, with a systematically decreasing cut-off length. The nature of the accompanying Barkhausen noise is changed accordingly, as shown in Fig.8 in the paper.

Comparisons with experimental results
In Fig. SI-5, we plot the disorder-thickness phase diagram using the relative scales (see the figure caption), that enable to plot several experimental results from the literature relative to the studied critical disorder line. , where x = 81, and a horizontal line is set at the equivalent theoretical critical disorder. Then all other thicknesses from different measurements are placed relative to that point. Different measurements obtained in the alloy of the same composition belong to the same line, while the points representing a varied composition at the fixed thickness belong to a vertical line, where we can not assess the actual disorder but only plot the theoretical lower and upper limits. Similar reasoning are applied to assess relative disorder line for other samples, in particular, amorphous Fe 75 Si 15 B 10 and Fe 73.5 Si 22.5−x Cu 1 Nb 3 B x ; here, the exponents measured in Ref. [15] systematically remain in the range left/above the theoretical critical line, suggesting a stronger disorder in these amorphous alloys. In the legend, we indicate the theoretical exponents that correspond (within error bars) to the experimental values in the respective materials: (τ 2 , α 2 ) referring to the theoretical second slope observable in thin materials, while (τ 1 , α 1 ) are the exponents of the first slope, which dominates in the distributions computed for the thicker samples, see Methods in the main text for the exact definition. The abbreviation "HLC" indicate that these exponents are observed by considering strictly central segments of the hysteresis loop and "INT" indicates integration over an extended segment of the loop (or loop branch). H c in the legend indicates that these experiments were done by imposing the field close to the coercive value (the corresponding symbols are empty), in contrast to field sweeping with a given rate as it is standardly done in other experiments. The corresponding range of the theoretical exponents are highlighted in the Table SI-I as (τ 2 , α 2 ) INT-bold, (τ 1 , α 1 ) HLC-italic, and (τ 2 , α 2 ) HLC-italic-bold fonts.