Predicting the Curie temperature of ferromagnets using machine learning

The magnetic properties of a material are determined by a subtle balance between the various interactions at play, a fact that makes the design of new magnets a daunting task. High-throughput electronic structure theory may help to explore the vast chemical space available and offers a design tool to the experimental synthesis. This method efficiently predicts the elementary magnetic properties of a compound and its thermodynamical stability, but it is blind to information concerning the magnetic critical temperature. Here we introduce a range of machine-learning models to predict the Curie temperature $T_{\mathrm{C}}$ of ferromagnets. The models are constructed by using experimental data for about 2500 known magnets and consider the chemical composition of a compound as the only feature determining $T_{\mathrm{C}}$. Thus we are able to establish a one-to-one relation between the chemical composition and the critical temperature. We show that the best model can predict $T_{\mathrm{C}}$'s with an accuracy of about 50 K. Most importantly our model is able to make predictions in regions of the chemical space, where only a small fraction of the data was considered for training. Furthermore, it is able to assess the uncertainty of such predictions. This is demonstrated by tracing the $T_{\mathrm{C}}$ of binary intermetallic alloys along their composition space and for the Al-Co-Fe ternary system.

Figure 1

Histogram of the $T_{\mathrm{C}}$'s of about 2,000 known ferromagnets. The median value of the distribution is 227 K. The insert shows the relative elemental abundance, in logarithmic scale, for the ferromagnets included in the dataset (the logarithm of the number of compounds containing a particular element). The most frequent magnetic element is Co followed by Fe and Gd.

Figure 2

RF model for the $T_{\mathrm{C}}$ of ferromagnets. In the left-hand side panel, we compare the experimental and predicted $T_{\mathrm{C}}$ for the test set. The $R^2$ coefficient of the model is 0.87 and the mean absolute value is 57 K. The upper left panel shows the distribution of absolute errors, $|T_{\mathrm{C}}^{\exp }-T_{\mathrm{C}}^{\mathrm{predic}}|$, while the lower one displays the distribution of the relative errors for compounds with $T_{\mathrm{C}}>300$ K. The orange lines indicate the best fit respectively to an exponential and a Gaussian distribution.

Figure 3

$T_{\mathrm{C}}$ prediction as a function of composition for three binary transition-metal systems, namely Co-Mn, Fe-Ni, and Ni-Rh. Data are presented as a function of the atomic fraction of one of the two species. The blue line traces the ML prediction, black crosses (green dots) are experimental points included (not included) in the training set. The experimental results are taken from references [40, 41, 42]. The light-blue shadowed area corresponds to the range of predicted $T_{\mathrm{C}}$'s of subsets of the trees, namely it indicates the uncertainty of the ML model. In the middle panel the red dots correspond to the $T_{\mathrm{C}}$ associated to the ${\mathrm{FeNi}}_3$ intermetallic phase, while the green ones correspond to random Ni-Fe alloys.

Figure 4

$T_{\mathrm{C}}$ prediction as a function of composition for the ternary system Al-Co-Fe. Data are presented as a function of the atomic fraction of the three species and the $T_{\mathrm{C}}$ is expressed as a heat map. The figure also introduces a detailed analysis of the three relevant binary phase diagrams, where the blue line traces the ML prediction, black crosses (green dots) are experimental points included (not included) in the training set. The light-blue shadowed area in the binary plots corresponds to the range of predicted $T_{\mathrm{C}}$'s, namely it indicates the uncertainty of the ML model. The solid square (circles) included in the ternary $T_{\mathrm{C}}$ diagram are for experimental data included (not included) in the training set, with the color code describing the $T_{\mathrm{C}}$. Numbers correspond to three known stoichiometric phases: (1) ${\mathrm{Co}}_2\mathrm{FeAl},T_{\mathrm{C}}=1000$ K, (2) ${\mathrm{Fe}}_2\mathrm{CoAl},T_{\mathrm{C}}>873$ K, and (3) ${\mathrm{Fe}}_3\mathrm{Al},T_{\mathrm{C}}=573$ K. The plot was partially made using python ternary [45].