Planning a magnet property prediction model

We're starting to be able to generate some decent, better permanent magnet candidates. The next step in a end-to-end pipeline would be to quickly calculate the full set of relevant magnet properties so we can further estimate their viability.

Ideally, we can come up with a model that predicts a number of features, like:

• Herfindahl–Hirschman index (HHI): a measure of market concentration, typically used in economics to quantify how monopolized an industry is. Here, it is repurposed to assess the concentration of critical elements in a given material.

• Magnetic Density: the total magnetization per unit volume or mass.

• Saturation Magnetization ($M_s$): the maximum magnetization a material can reach when fully magnetized under an external magnetic field.

• Coercivity ($H_c$): the resistance of a magnet to being demagnetized by an external field.

• Energy Product ($BH_{\max}$): the maximum energy density a magnet can store.

  • $$BH_{\max} = \frac{1}{4} M_s H_c$$

• Anisotropy energy (Magnetocrystalline Anisotropy Energy, MAE): refers to the energy required to rotate the magnetization away from its preferred direction (easy axis). Higher anisotropy energy means that the magnetization is more stable, leading to higher coercivity ($H_c$).

• Curie temperature: is the temperature above which a ferromagnetic material loses its permanent magnetism and becomes paramagnetic.

• Electrical Resistivity: how well the material resists electrical currents. In high-speed rotating machines (e.g., electric motors), conductive magnets can suffer from eddy current losses, which cause heating and efficiency loss.

A good permanent magnet should have high saturation magnetization ($M_s$), high coercivity ($H_c$), and a high energy product ($BH_{\max}$).

Summary table of key properties

We don't necessarily need to look at all of these properties, but for sure we should have a grasp on $BH_{\max}$.

Micromagnetism

Asking Claude about datasets and other ways to understand a materials permanent magnet potential, I came across micromagnetism. It's a step above the quantum mechanical level of theory. @will @mahyar check this out.

There is an open source, GPU-accelerated library for micromagnetic simulations: https://mumax.github.io/

Learn more about micromagnetism here: http://micromagnetics.org/micromagnetism/

Different models have been proposed in order to approximately describe ferromagnetic materials on a macroscopic scale. Depending on the simplifications introduced by a particular model it is able to describe the system accurately only under certain assumptions and on a certain length scale. Table 1 gives an overview of established models for the description of ferromagnets on different length scales.

For the description of ferromagnetism on the micron scale the theory of micromagnetism has proved to be a reliable tool. In contrast to domain theory it is able to resolve the inner structure of domain walls. On the other hand the micromagnetic equations can be solved numerically for relatively large system compared to atomistic approaches.

Limits of the Micromagnetic Model

Bloch Points

The central assumption in micromagnetics is the homogeneous saturation magnetization $M_s$ . This assumption is justified by the fact that ferromagnetic materials are subject to exchange coupling which leads to locally almost perfectly aligned magnetic moments.

However, certain magnetic processes involve the creation of magnetic singularities called Bloch points. At these Bloch points the magnetization changes rapidly in space, which is inconsistent with the basic assumption of a homogeneous $M_s$ in micromagnetics. Despite this fact it was shown in Thiaville et al. (2003) that micromagnetic simulations involving the creation of Bloch points are able to describe the corresponding processes in accordance with experiments, although the energy density at the Bloch point is underestimated.

Temperature

Another example for the violation of the micromagnetic assumption of a locally homogeneous magnetization is given by thermal effects. Thermal effects are most naturally reflected by local perturbation of magnetic moments. Perturbation of a single magnetic moment, however, obviously breaks the homogeneity of the magnetization. In the framework of classical micromagnetics a possible approach for consideration of finite temperature is the reduction of the saturation magnetization according to a mean-field approximation, see Kittel and McEuen (1996) . Another approach is to add a fluctuating field to the effective field, which converts the Landau-Lifshitz-Gilbert equation into a stochastic differential equation, see Lyberatos, Berkov, and Chantrell (1993) . Both of these techniques do not account for local changes in the saturation magnetization and as a result both methods fail to describe the magnetization dynamics correctly when approaching the Curie temperature. This deficiency is overcome by the Landau-Lifshitz-Bloch equation, which extends the Landau-Lifshitz-Gilbert equation not only by a fluctuating field, but also by a term that allows the change of the magnetization modulus Garanin (1997).

It's possible we don't need a machine learning model to predict these properties. Instead, we could use simulations to calculate them (ab initio or a higher-level of theory). This would also save us the pain of having to curate the dataset.

For example, see how we could use MuMax3 to estimate some of the properties: