Explanations on how MAE factors into a crucial permanent magnet property, coercivity, and how we can use calculated MAE values to get a good feel for which candidates have permanent magnet potential.
Coercivity, as I understand it, is incredibly difficult to evaluate computationally. I would hate to call it impossible. This coercivity value , , measures how resistant a magnet is to demagnetization, and it’s one of the critical parameters that separates high-performance magnets like NdFeB from less capable alternatives.
Today, measuring coercivity requires synthesizing high-quality samples, carefully controlling microstructure, and running magnetic hysteresis experiments.
Our goal with this post is to talk through another property, Magneto-crystalline Anisotropy Energy (MAE), and how we can use this property to separate promising permanent magnet candidates from the deluge of mostly soft magnets we're finding with the tree search:
Detailing our open experimentation with SakanaAI's Treequest algorithm, AB-MCTS, and its potential applicability in rare-earth free permanent magnet discovery.
At its core, MAE is the energy difference between aligning a crystal’s magnetization along different crystallographic directions. In density functional theory (DFT), we compute total energies with spin–orbit coupling for various orientations:
This microscopic quantity can be converted into the anisotropy constant
Once we have
is the saturation magnetization (A/m).
is the permeability of free space (also called the magnetic constant).
Physically,
So in this anisotropy field equation:
has units of J/m³ (energy density).
Dividing by (Tesla × A/m = J/m³) gives a field in A/m.
This field represents the theoretical maximum coercivity a material could achieve if it were a perfect, defect-free single domain.
Building some intuition for the Easy Axis vs. Hard Axis:
Every crystal has directions where magnetization “wants” to point. The easy axis is the orientation where magnetization naturally aligns, minimizing energy. The hard axis is the orientation where magnetization resists pointing, because it costs extra energy (the MAE) to hold it there.
The larger this energy difference, the more strongly magnetization is locked into its easy axis, and the greater the potential coercivity becomes. However, real coercivity is always lower than the anisotropy-derived "ceiling". Grain boundaries and defects provide nucleation sites for reversal, domain walls move under smaller fields than , and temperature effects reduce anisotropy strength.
So while MAE sets the idealized maximum, real magnets usually reach only a fraction of it.
Having said this, MAE is invaluable. By computing it, we can, screen new systems without synthesis, identify candidates with strong anisotropy before committing to experimental validation, and ultimately narrow down the search space in the race for rare-earth-free permanent magnets.
It’s similar to how we use saturation magnetization () as the theoretical upper bound for field strength:
: “If every spin lined up, this is the maximum magnetization.”
MAE: “If every grain were perfect, this is the maximum coercivity.”
Both serve as useful, idealized ceilings that we can’t quite reach, but they still provide crucial direction.
Computationally evaluating MAE isn’t a replacement for measuring coercivity, but it’s the best shortcut we have right now.
More to come as always.
It’s cool to see that these boosted tree/ random forest models can be performant even with these more complex properties. It also further reinforces your work with the Curie temperature predictor!
True, though I think there's a unique challenge that we don't really have to worry about in classic data science and predictive modeling where predictions could be coming from structural similarity but not actually modeling the physics properly. There needs to be very careful holdouts and validation across chemical families.
This is just to say it's pretty easy to have a model look better than it is because of the nature of "correlated" training/test examples.
This is actually a common challenge in developing permanent magnets, including rare-earth-free ones.
Here's why this disconnect can occur:
MAE is an intrinsic property that represents the energy barrier for rotating magnetization from an easy axis to a hard axis in a perfect single crystal. It sets the theoretical upper limit for coercivity.
Coercivity is an extrinsic property that depends heavily on microstructure and real-world imperfections. The actual coercivity is typically much lower than what the MAE would predict because:
Grain boundaries and defects - These act as nucleation sites for reverse magnetic domains, allowing magnetization reversal at much lower fields than the theoretical limit.
Grain size effects - If grains are too large, domain walls can form and move easily. If they're too small (below the single-domain limit), thermal fluctuations can destabilize magnetization.
Grain alignment - Randomly oriented grains mean the effective anisotropy is averaged out, reducing coercivity even if individual grains have high MAE.
Surface/interface effects - Surface defects, oxidation, or poor magnetic isolation between grains can dramatically reduce coercivity.
Brown's paradox - This famous observation notes that real permanent magnets typically achieve only 15-30% of their theoretical maximum coercivity based on anisotropy field.
For example, L1₀-ordered FeNi has very high calculated MAE, but achieving high coercivity in practice has been extremely challenging due to difficulties in creating the proper ordered structure and microstructure. Similarly, some MnAl and MnBi phases show high MAE but struggle with coercivity due to phase stability and microstructural control issues.
The key to developing good permanent magnets is not just finding materials with high MAE, but also engineering the microstructure to better utilize that intrinsic anisotropy.
This really helps clarify things. So it's possible that even if we get a good candidate with good MAE, the material still might not be a great permanent magnet because coercivity is more what matters in the real world. It would be good to know how often a material might have high MAE and low coercivity, and on average how large the drop-off is.
This is the paper I was talking about that mentioned MAE as a powerful predictor to coercivity.
Increased demand for high-performance permanent magnets in the electric vehicle and wind turbine industries has prompted the search for cost-effective alternatives. Nevertheless, the discovery of new magnetic materials with the desired intrinsic and extrinsic permanent magnet properties presents a significant challenge. Traditional Density Functional Theory (DFT) accurately predicts intrinsic permanent magnet properties such as magnetic moments, magneto-crystalline anisotropy constants, and exchange interactions. However, it cannot compute extrinsic macroscopic properties, such as coercivity (Hc), which are influenced by factors like microscopic defects and internal grain structures. Although micromagnetic simulation helps compute Hc, it overestimates the values almost by an order of magnitude due to Brown’s paradox. To circumvent these limitations, we employ Machine Learning (ML) methods in an extensive database obtained from experiments, DFT calculations, and micromagnetic modeling. Our novel ML approach is computationally much faster than the micromagnetic simulation program, the mumax3. We successfully utilize it to predict Hc values for materials like cerium-doped Nd2Fe14B, and subsequently compare the predicted values with experimental results. Remarkably, our ML model accurately identifies uniaxial magnetic anisotropy as the primary contributor to Hc. With DFT calculations, we predict the Nd-site dependent magnetic anisotropy behavior in Nd2Fe14B, confirming 4f-site planar and 4g-site uniaxial to crystalline c-direction in good agreement with experiment. The Green’s function atomic sphere approximation calculated a Curie temperature (TC) for Nd2Fe14B that also agrees well with experiment. Paper by Churna Bhandari, Gavin N. Nop, Jonathan D.H. Smith, Durga Paudyal
The MAE and realized coercivity gap can been seen in these experimental results from Jiro Kitagawa’s work in characterizing as-cast and minimally post-processed FeCoNiPd and FeCoNiPt: https://arxiv.org/abs/2209.14506
Where the MAE we saw for equiatomic FeCoNiPt is high, but the “as cast” measured coercivity is almost nonexistent.