Universal temporal characteristics and vanishing of multifractality in Barkhausen avalanches

G. Z. dos Santos Lima, G. Corso, M. A. Correa, R. L. Sommer, P. Ch. Ivanov, and F. Bohn

Phys. Rev. E 96, 022159 – Published 31 August 2017

Abstract

Barkhausen effect in ferromagnetic materials provides an excellent area for investigating scaling phenomena found in disordered systems exhibiting crackling noise. The critical dynamics is characterized by random pulses or avalanches with scale-invariant properties, power-law distributions, and universal features. However, the traditional Barkhausen avalanches statistics may not be sufficient to fully characterize the complex temporal correlation of the magnetic domain walls dynamics. Here we focus on the multifractal scenario to quantify the temporal scaling characteristics of Barkhausen avalanches in polycrystalline and amorphous ferromagnetic films with thicknesses from 50 to 1000 nm. We show that the multifractal properties are dependent on film thickness, although they seem to be insensitive to the structural character of the materials. Moreover, we observe for the first time the vanishing of the multifractality in the domain walls dynamics. As the thickness is reduced, the multifractal behavior gives place to a monofractal one over the entire range of time scales. This reorganization in the temporal scaling characteristics of Barkhausen avalanches is understood as a universal restructuring associated to the dimensional crossover, from three- to two-dimensional magnetization dynamics.

Received 17 March 2017
Revised 28 June 2017

DOI:https://doi.org/10.1103/PhysRevE.96.022159

Physics Subject Headings (PhySH)

Critical exponents Critical phenomena

Statistical Physics & Thermodynamics

Authors & Affiliations

G. Z. dos Santos Lima^1,2,3, G. Corso², M. A. Correa⁴, R. L. Sommer⁵, P. Ch. Ivanov^3,6,7, and F. Bohn^4,*

¹Escola de Ciências e Tecnologia, Universidade Federal do Rio Grande do Norte, 59078-970 Natal, RN, Brazil
²Departamento de Biofísica e Farmacologia, Universidade Federal do Rio Grande do Norte, 59078-970 Natal, RN, Brazil
³Keck Laboratory for Network Physiology, Department of Physics, Boston University, Boston, Massachusetts 02215, USA
⁴Departamento de Física, Universidade Federal do Rio Grande do Norte, 59078-900 Natal, RN, Brazil
⁵Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, Urca, 22290-180 Rio de Janeiro, RJ, Brazil
⁶Harvard Medical School and Division of Sleep Medicine, Brigham and Women Hospital, Boston, Massachusetts 02115, USA
⁷Institute of Solid State Physics, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria

^*felipebohn@fisica.ufrn.br

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Vol. 96, Iss. 2 — August 2017

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Figure 1
Representative Barkhausen noise in ferromagnetic films and its dependence with thickness. In the left side, schematic representation of the dimensionality of the films with thickness. It is important to point out that films thicker than 100 nm present three-dimensional magnetic behavior, while films thinner than 100 nm are in the two-dimensional regime. In the right, experimental Barkhausen time series, measured for the smallest driving magnetic field frequency, 50 mHz, obtained for amorphous FeSiB films with selected thicknesses. Similar behavior with thickness is verified for the polycrystalline NiFe films. Barkhausen noise in different time windows of observation (insets in the top panel) reveals statistical self-similarity of avalanches at smaller scales, indicating an underlying scale-invariant temporal organization.
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Figure 2
Dimensionality and transition in the temporal scaling characteristics of Barkhausen avalanches. Barkhausen avalanches have different temporal characteristics in three- and two-dimensional films. A peculiar behavior of the temporal scaling characteristics is observed at thickness of 100 nm, i.e., $α = 1.0$ , characterizing an imminent transition. (a) Log-log plot of the fluctuation function $F (s)$ as a function of the time scale $s$ , obtained from the DFA applied to experimental Barkhausen noise time series measured with driving magnetic field frequency of 50 mHz in amorphous FeSiB films with the selected thicknesses of 1000, 100, and 50 nm. Each panel in (a) shows ten $F (s)$ curves, from different time series, indicating consistent scaling behavior. To guide the eyes, the black solid lines are power laws with the slope $α$ . (b) Dependence of the crossover time scale $t_{x}$ with thickness for amorphous FeSiB (blue triangles) and polycrystalline NiFe (blue circles) films. In the inset, $t_{x}$ as a function of the film thickness in the log-log scale, indicating a power-law scaling behavior. (c) Dependence of the scaling exponents $α$ and $α_{LR}$ with thickness for the amorphous FeSiB and polycrystalline NiFe films. Here, $α$ (blue filled symbols) represents the correlation behavior at scales shorter than $t_{x}$ , while $α_{LR}$ (red open symbols) is obtained over long-range scales [for instance, above 3 ms, as marked by the fitting line in the top panel of (a)]. In particular, for each film, the analyses are obtained from 200 experimental Barkhausen time series. The error bars are estimated using the standard deviation.
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Figure 3
Multifractal behavior and vanishing of the multifractality in Barkhausen avalanches. Multifractal behavior of Barkhausen avalanches is strongly dependent on the dimensionality. As the thickness is reduced from 100 to 50 nm, the multifractality of the DW dynamics vanishes, and the multifractal behavior gives place to a monofractal one over the entire range of time scales. (a) Log-log plot of the fluctuation function $F_{q} (s)$ as a function of the time scale $s$ for several moments $q$ , ranging from $q = - 4.0$ to $q = + 4.0$ , obtained from the MF-DFA applied to experimental Barkhausen noise time series measured with driving magnetic field frequency of 50 mHz in the amorphous FeSiB films with the selected thicknesses of 1000 and 100 nm. In particular, these scaling curves represent the analysis of a single Barkhausen noise time series, and the scaling curve for $q = 2$ , corresponding to the DFA fluctuation function $F (s)$ , and $t_{x}$ , the crossover time scale, are the very same previously presented in Fig. 2. To guide the eyes, the black solid lines are power laws with slope $α$ . The temporal dynamics of Barkhausen avalanches is characterized by two temporal scaling regimes: fractal range $Δ s_{F}$ and multifractal range $Δ s_{MF}$ , which are separated at the crossover time scale $t_{x}$ . (b) Range of time scales $Δ s_{MF}$ , in units of decades, in which the Barkhausen avalanches exhibit multifractal behavior, as a function of the thickness. (c) The width $Δ α$ of the multifractal spectrum $f (α)$ as a function of film thickness. The insets show the multifractal spectrum $f (α)$ of five representative Barkhausen noise time series for the amorphous FeSiB films with thicknesses of 100 and 1000 nm. In particular, for the 50-nm-thick film, $Δ s_{MF}$ and $Δ α$ are not shown in (b) and (c), since the multifractality vanishes for thickness just below 100 nm. In particular, for each film, the analyses are obtained from 200 experimental Barkhausen time series. The error bars are estimated using the standard deviation.
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