Using MAE to Filter Permanent Magnet Candidates From Soft Magnets

Coercivity, as I understand it, is incredibly difficult to evaluate computationally. I would hate to call it impossible. This coercivity value , $H_c$, 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:

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:

$$\text{MAE} = E_{\text{hard axis}} - E_{\text{easy axis}}$$This microscopic quantity can be converted into the anisotropy constant $K_u$ by normalizing per unit volume:

$$K_u = \frac{\text{MAE}}{V}$$Once we have $K_u$, we can estimate the anisotropy field:

$$H_A = \frac{2K_u}{\mu_0 M_s}$$

• $M_s$ is the saturation magnetization (A/m).

• $\mu_0$ is the permeability of free space (also called the magnetic constant).

$$\mu_0 = 4\pi \times 10^{-7} \ \text{H/m} \quad (\text{Henries per meter})$$Physically, $\mu_0$ provides the conversion between magnetic field strength $H$ (A/m) and magnetic flux density $B$ (Tesla):

$$B = \mu_0 (H + M)$$So in this anisotropy field equation:

• $K_u$ has units of J/m³ (energy density).

• Dividing by $\mu_0 M_s$ (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 $H_A$, 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 ($B_{sat}$) as the theoretical upper bound for field strength:

• $B_{sat}$: “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. @mmoderwell can add some color here, but although its our best shortcut, it is still extremely challenging to compute. One of the open research items we have is finding similar adjacent properties or characteristics that help us filter for promising permanent magnet candidates, while being simpler to evaluate than Matt's current MAE computation.

More to come as always.