Uncovering critical temperature dependence in Heusler magnets via explicit machine learning

TL;DR

This work addresses the problem of understanding how the magnetic transition temperature $T_c$ in collinear Heusler magnets depends on composition and magnetic moments. It introduces an interpretable explicit machine learning framework, hierarchical dependence extraction (HDE), applied to the JuHemd database with targets $T_c^{\rm exp}$ and $T_c^{\rm calc}$ and descriptors including chemical proportions $d_{\rm Y}$, their products, and element-resolved magnetization amplitudes $M_{\rm Z}$ and products $M_{\rm Z}M_{\rm Z'}$, yielding explicit, interpretable expressions for $T_c$. The main findings show that $T_c$ is dominated by Fe, Co, and Mn proportions, and that HDE achieves accuracy comparable to other ML methods, while enabling the construction of an explicit order parameter: an element-resolved $\mu$ for $T_c^{\rm calc}$ and a total magnetization-based form that scales as $T_c^{\rm est}=9681\,M^{1.18}$. This work provides physical insight into the mechanisms controlling $T_c$ in Heusler magnets and demonstrates a pathway for materials design using interpretable, physics-informed ML frameworks.

Abstract

We employ interpretable explicit machine learning to analyze the material dependence of the magnetic transition temperature $T_c$ in ferromagnetic and ferrimagnetic Heusler compounds. For around 200 compounds, we consider both experimental $T_c$ and calculated $T_c$ using \textit{ab initio} determination of magnetic interactions together with a Monte-Carlo solution. We use the hierarchical dependence extraction (HDE) procedure [Morée and Arita, Phys. Rev. B 110, 014502 (2024)] to extract the dependencies of $T_c$ on chemical proportions and magnetic moments from the main order to the higher order, and construct an explicit expression of $T_c$ from these dependencies. The main results are: (a) $T_c$ is mainly controlled by the proportions of Fe, Co, and Mn, and increases with these proportions, consistent with previous machine learning analyses of ferromagnetic materials. (b) The HDE describes $T_c$ with an accuracy that is comparable to that of other machine learning procedures. (c) The HDE expression of $T_c$ can be interpreted as a generalized order parameter that increases with increasing magnetization amplitude, in qualitative agreement with various theories of phase transitions. These results strengthen our understanding of the material dependence of $T_c$ in collinear Heusler magnets and motivate the further use of HDE in material design.

Figures (7)

• Figure 1: Crystal structure of Heusler compounds. The general chemical formula is denoted as X$_1$X$_2$YZ, where X$_1$, X$_2$, Y, and Z are either magnetic elements with an open $3d$ shell (from Ti to Ni) or nonmagnetic elements. In the unit cell (delimited by the gray lines), the atoms are ordered as Y$-$X$_1-$Z$-$X$_2$. Atoms within the unit cell are marked by gray diamonds. We show the primitive vectors that span the unit cell, ${\bf a} = [0 \ a/2 \ a/2]$, ${\bf b} = [a/2 \ 0 \ a/2]$, and ${\bf c} = [a/2 \ a/2 \ 0]$ in Cartesian coordinates, where $a$ is the lattice constant.
• Figure 2: Summary of the dependencies that are analyzed in this work. We start from the chemical proportions $d_{\rm Y}$ and the products $d_{\rm Y}d_{\rm Z}$, which are immediately deduced from the chemical formula. (Y is any chemical element, and Z is a magnetic element between Ti and Ni.) First, we analyze the dependence of experimental and calculated magnetic transition temperatures $T_c^{\rm exp}$ and $T_c^{\rm calc}$ on the $d_{\rm Y}$ and $d_{\rm Y}d_{\rm Z}$. Second, we analyze the dependence of $T_{c}^{\rm calc}$ on the element-resolved magnetization amplitudes $M_{\rm Z}$ and their products $M_{\rm Z}M_{\rm Z'}$, which are obtained by postprocessing the ab initio calculation results. (Z and Z' are magnetic elements between Ti and Ni.) Third, we analyze the dependence of $M_{\rm Z}$ on the $d_{\rm Y}$ and $d_{\rm Z}d_{\rm Y}$.
• Figure 3: Dependence of experimental and calculated magnetic transition temperatures $y=T_c^{\rm exp}$ (left column) and $y=T_{c}^{\rm calc}$ (right column) on the products of chemical proportions $x=\{ d_{\rm Z}d_{\rm Y}\}$ in the HDE. Upper panels: Isosurfaces of the dependence of $y^{\rm HDE}_{(3)}$ on the $d_{\rm Y}$. Middle and lower panels: Dependence of $y$ on $y^{\rm HDE}_{\rm (3)}$ and $y^{\rm HDE}_{\rm (50)}$. The values of $R^{2}_{(g)}$ are shown in each panel.
• Figure 4: Dependence of calculated magnetic transition temperature $y=T_c^{\rm exp}$ on the element-resolved magnetization amplitudes and their products, i.e. $x=\{M_{\rm Z}, M_{\rm Z}M_{\rm Z'}\}$, in the HDE. The left (right) column shows the dependence of $y$ on $y^{\rm HDE}_{(g)}$ for $\mu_{\rm thr} = 0 \mu_{\rm B}$ ($0.5 \mu_{\rm B}$), at $g=3$ (upper panels) and $g=10$ (lower panels). The values of $R^{2}_{(g)}$ are shown in each panel.
• Figure 5: Main-order dependencies of $M_{\rm Fe}$ and $M_{\rm Co}$ on $d_{\rm Fe}$, $d_{\rm Co}$, and $d_{\rm Mn}$ in Eqs. \ref{['eq:mfemfe']} and \ref{['eq:mcomco']}.