Soft Magnetic Nanocrystalline Alloys: Materials and Models

Desired Properties

The dependence of coercivity on crystal size exhibits two different behavioral regimes. The traditional method of decreasing coercivity in soft magnetic materials has historically been to increase grain size because this decreases the amount of grain boundaries that act as pinning centers for the domain walls. By increasing grain size from 100 nm to 100 μm, it is possible to decrease Hc from ~10,000 A/m to values on the order of 1 A/m. The counter-example of this approach appeared with the emergence of amorphous and nanocrystalline materials. By decreasing grain size to values on the order of 10 nm or even making the solid amorphous, coercivity decreases to reach values comparable to those of very large crystals. Taking into account that an amorphous structure is, by definition, full of defects, the previous argument of decreasing coercivity with decreasing concentration of defects no longer holds true. This characteristic behavior can be explained with the Random Anisotropy Model (RAM) of Alben, Becker, and Chi3 and has been applied to nanocrystalline alloys by Herzer.4

Random Anisotropy Model: An Explanation Based on Analogies

A formal explanation of the RAM remains elusive for researchers more interested in material properties than in the development of physical models. However, understanding the physics behind the model can help in designing materials with the desired performance. We use several analogies taken from everyday life to grasp the concept of RAM. The message here is that the existence of very different length scales in the sample is responsible for the vanishing effective anisotropy of the amorphous and nanocrystalline alloys, which gives rise to extremely low coercivity values.

The first example takes us on a walk to a traditional monument found in many old cities, like Sevilla. Many of these ancient cities have patios with a floor made of numerous pebbles. When walking on top of these floors, those wearing high heels will typically have more difficulty walking than those wearing flat-soled shoes. The most reasonable explanation for this circumstance is that high heels have a typical size comparable to the pebbles and the gaps between them so they detect the discrete maximum and minimum heights of the floor. Flat shoes average the irregularities and allow us to walk more steadily because the heel is much longer than the pebbles and gaps.

Another example uses two refrigerator magnets with the same characteristics (ideally, two identical ones). Magnets of this type are fabricated using polymer-bonded ferromagnetic particles with a domain structure consisting of a set of stripes pointing outward and inward on one side, with the other side containing closure domains. The side with the closure domains is typically decorated and will not stick to the fridge. When one magnet is placed directly on top of the other with the same orientation (decorated sides facing the exterior), the magnets stick together. However when one magnet is slid perpendicular to the stripe domain, the upper magnet jumps as it moves along the attractive and repulsive regions with a periodicity corresponding to twice the domain width. As both the upper and lower magnets have the same characteristic length, i.e., the width of the domains, the system samples energy maxima and minima and they are detected macroscopically. This can be felt in our hands when trying to move the magnets, often producing a sound due to the abrupt attaching and detaching of the flexible laminae. However, if we rotate the upper magnet 90° with respect to the lower magnet, in a perpendicular orientation of the stripe domains of both laminae, the movement in any direction becomes smooth. It is not because the local maxima and minima have disappeared. There are still regions where equal poles face each other and therefore repel, as well as areas in which opposite poles lie on top of each other, creating attraction. The difference with respect to the previous situation is that any position of the upper magnet samples the same amount of attractive and repelling areas. The periodicity of the lower magnet remains the width of the domains, while the periodicity of the upper magnet turns to the length of the magnet. As one is much larger than the other, we cannot macroscopically detect the local energy difference and the magnets displace smoothly, similar to the flat-soled shoes on the pebbled patio.

A Simple Model for Nanocrystalline Alloys

Nanocrystalline alloys also can be modeled in a simple fashion as an ensemble of single domain nanoparticles (shown as squares in Figure 2A) with a typical grain size of 10 nm and with randomly oriented easy axes. Even if Fe-based nanocrystals have cubic magnetocrystalline anisotropy, we consider uniaxial anisotropy in the current model. We will show that the effective anisotropy is almost averaged to zero, thus this assumption of symmetry does not affect our conclusion. Consider the case in which the domain size (connected to the magnetic correlation length and marked as a thick white square in Figure 2B) is comparable to the structural correlation length (the grain size in Figure 2A). Assuming we have a domain in which the magnetization is oriented in the horizontal direction inside the white square, we color code the anisotropy energy of the different crystals from blue (minimal anisotropy energy) to red (maximal anisotropy energy). It is easy to see in the figure that it is not equivalent to displace this domain in all directions, since in some cases we gain blue squares (when moving to the right), while we don’t notice a remarkable difference in red or blue when moving up because there is anisotropy in the displacement of the domain. When we progressively decrease grain size while keeping the domain size constant, differences in energy will become gradually smaller when we move the domain in any direction, becoming negligible below a certain grain size. Each grain has its own anisotropy energy, but the system as a whole does not present anisotropy. In order to perform this simulation more quantitatively, we create different grids using different ratios of grain size to domain size and place the correlated domain at all positions in the grid, calculating the total value of energy in each position and the difference between maximal and minimal energy values for each of the grids. When we plot the difference in energy as a function of the number of particles inside each domain, N, we clearly observe a decreasing trend that can be fit to a 1 /√Ν law. Taking into account, for this model, anisotropy is a function of the difference in energy for each position of the domain inside the grid, we observe that the effective anisotropy, K*, corresponds to K* = K /√Ν, where K is the anisotropy of the crystals. This is the same result formally obtained from the RAM and originates from the difference between the magnetic correlation length and the structural correlation length. As a consequence of this discussion, when particle size decreases noticeably below the exchange correlation length, the effective anisotropy decreases toward zero and makes coercivity practically null.

This simplistic explanation, while keeping the essence of the phenomenology, was improved by also considering nanocrystalline alloys are not only formed by the nanocrystals, but they are two-phase systems in which the nanocrystals are embedded in a residual amorphous matrix, which is also ferromagnetic. More sophisticated models for multiphase materials have been developed5,6 that also include the contributions of magnetoelastic energy.