Direct detection and stochastic analysis on thermally activated domain-wall depinning events in micropatterned Nd-Fe-B hot-deformed magnets

Abstract

Although the magnetization reversal process in the permanent magnets has long been studied, it has not been fully revealed. The recent progress of the computational science realizes the atomistic calculations for the thermally activated magnetization reversal process in permanent magnets. In contrast, the experimental study on the magnetization reversal has remain to be the macroscopic evaluation. In this study, Nd-Fe-B hot-deformed magnets are micropatterned, and a staircase-like magnetization curve corresponding to the elemental domain-wall depinning event at each grain boundary is successfully observed. Each elemental domain-wall depinning event fluctuates with respect to thermal activation, and the stochastic analysis based on the Néel-Arrhenius model gives the energy barrier parameters H₀ and E₀ of each depinning event, which are the intrinsic domain-wall depinning field and energy barrier height, respectively. Three types of Nd-Fe-B hot-deformed magnets with different coercivities are adopted for this stochastic analysis. As a result, E₀ exhibits very little dependence on H₀, and its slope becomes steeper for the lower-coercivity magnet. The stochastic micromagnetics simulation based on the Landau-Lifshitz-Gilbert equation for the two-grain model with various inter-grain exchange coupling reproduces the experimentally observed relationship between E₀ and H₀. Moreover, the behavior for the lower-coercivity magnet can be explained by assuming the presence of a low magnetic anisotropy layer on the grain surface.

Introduction

High-performance permanent magnets are one of the key materials for low-energy consuming and/or low-environmental cost technologies, such as electric vehicles. Therefore, high-performance magnets have been extensively studied during the last decade in terms of many points, such as elemental strategy [1,2], microstructure engineering [3], [4], [5], [6], [7], [8], new material exploration [9], [10], [11], and coercivity mechanism [12], [13], [14], [15].

Although the coercivity mechanism in the permanent magnets has been a very classical issue since the Brown's paradox was proposed in 1945 [16], it remains under investigation. Traditionally, the coercivity mechanism was mainly discussed by means of micromagnetics theory [17], [18], [19], [20]. However, most of them have failed to explain the actual coercivity values of permanent magnets. In the micromagnetics theory, an axially symmetric magnet body or an infinite slab is assumed, and then the nucleation is defined as the starting point of magnetization deflection under the continuum approximation. On the other hand, because the actual magnetization at a finite temperature largely fluctuates due to thermal activation, the nucleation takes place as the growth of a reversed domain embryo by overcoming the energy barrier. This picture of nucleation is quite different from that assumed in the micromagnetics theory. Recently, the energy barrier was computationally evaluated from the energy landscape calculation with the energy minimizing path method [21,22]. Moreover, very recently, the thermally activated nucleation process of a reversed domain embryo in a Nd₂Fe₁₄B magnet was successfully computed using the atomistic simulation method based on the first-principles calculations [23]. Thus, a single event of thermally activated magnetization reversal in permanent magnets have been rigorously calculated.

In an actual bulk permanent magnet, however, numerous nucleations and/or domain-wall depinnings simultaneously take place. Therefore, it is very difficult to directly compare the experimental results with the theoretical calculations. To overcome this problem, it is essential to measure the elemental magnetization reversal events in a permanent magnet. One example method in which the elemental magnetization reversal events can be measured is the Barkhausen noise measurement [24]. However, the Barkhausen noise is the assembly of randomly occurring elemental magnetization reversal events. Therefore, it is impossible to stochastically analyze each elemental magnetization reversal event using this measurement.

To tackle this problem, we attempted to reduce the number of randomly occurring elemental magnetization reversal events by micropatterning a Nd-Fe-B hot-deformed magnet [25]. Nd-Fe-B hot-deformed magnets are known as one of the heavy rare-earth free high-performance magnets [26] and have already been practically adopted for the traction motors of hybrid vehicles. One of the features of Nd-Fe-B hot-deformed magnets is their unique microstructure consisting of ultra-fine platelet shaped Nd₂Fe₁₄B main phase (MP) grains enveloped by the ~3nm thin Nd-rich grain boundary (GB) phase, as schematically shown in Fig. 1(a). Representative scanning electron microscopy (SEM) images along the c-plane and c-axis are shown in Fig. 1(b) and (c), respectively. The MP grains pile along the c-axis like a brick wall. This microstructure is a key factor for the micropatterning. In the conventional Nd-Fe-B sintered magnets, which consist of much larger polygonal grains with several micrometers in diameter, the magnetic hardness easily gets deteriorated due to the presence of a surface damaged layer [27]. On the other hand, the effect of the surface damaged layer to the coercivity is negligibly small for the Nd-Fe-B hot-deformed magnet owing to its unique microstructure [25].

The magnetic domain-wall in the Nd-Fe-B hot-deformed magnet propagates first along the c-axis and gets pinned at the GB phase [28]. The domain-wall depinning was proposed to be the dominant magnetization reversal process [29]. A thin cross-shape micropatterned Nd-Fe-B hot-deformed magnet with the thickness of 5 μm and the width of 13 μm was fabricated [25]. By using the anomalous Hall effect (AHE), the magnetization signal of the very small cross-center region can be sensitively detected. As a result, a staircase-like magnetization curve was successfully observed, indicating that each staircase step corresponds to an elemental step of domain-wall depinning at the grain boundary. The magnetization curve, however, varies its staircase-like pattern in every measurement. This fact indicates that the sequence of the elemental domain-wall depinning changes every time, and the stochastic analysis for the elemental domain-wall depinning is difficult in this size range of the micropatterned Nd-Fe-B hot-deformed magnet.

In this study, we have further extended the micropatterning of the Nd-Fe-B hot-deformed magnet into the sub-micron size region. Consequently, the good repeatability of the staircase step on the magnetization curve is realized. Thus, stochastic analysis can be implemented for the thermally activated elemental domain-wall depinning events. Three types of Nd-Fe-B hot-deformed magnets with different coercivities are adopted for this measurement. As a result, different energy barrier properties for these three magnets are found reflecting the different GB phase condition. The physical origin of the different energy barrier properties is discussed by comparing them with the elaborated micromagnetics simulations. Moreover, it is also concluded that the energy barrier parameters are considered as good measures for intergrain exchange and MP grain surface magnetic anisotropy, which are the information hardly accessed usually.