# Case Studies

Here we examine some case-studies that demonstrate how the Magnetic Materials Discovery App can be used for materials discovery and prediction.

All examples here can found in the associated paper:

Court, C. J. & Cole, J. M. "Magnetic and Superconducting Phase Diagrams and Transition Temperatures Predicted Using Text-mining and Machine Learning", NPJ Comput Mater 6, 1, 1-9 (2020) https://doi.org/10.1038/s41524-020-0287-8

# 1. Perovskite Oxides

The perovskite-type oxides are inorganic compounds with general formula $$ABO_3$$ where $A$ is a large 12-coordinated cation and $B$ is a smaller 6-coordinated cation.

The generic perovskite structure is cubic; however, this form is rarely found due to structural deformation. These deformations cause the perovskites to exhibit a wide variety of interesting and useful properties including ferroelectricity, piezoelectricity, superconductivity and magnetism. As such, perovskite materials are found in a vast number of applications.

The properties and phase diagrams of the common perovskite series have been widely reported, and as such, these materials make great candidates as case studies to validate the performance of our database and phase-diagram phase transition prediction toolkit.

Magnetism in perovskites arises through the incorporation of paramagnetic cations. Commonly, cationic species are lanthanides or transition metals, who have partially filled d and f orbitals. Through the crystal-field interaction, local-coordination environments determine the orbital energy levels and hence the spin state and magnetic moment of the cation. The large dependence of the magnetic properties on the crystal-field interaction leads to a huge variation in magnetic state with temperature and composition. With only minor changes in doping concentration of the A-site and B-site cations, the material undergoes transitions between multiple different magnetic phases. A classic example of this is the $La_{1-x}Sr_xMnO_3$ system which displays a bulk metallic ferromagnetic phase and 4 different antiferromagnetic phases with varying $Sr$ composition.

Using our toolkit and the excellent visualisation tools provided by Plot.ly the auto-generated phase diagram below clearly distinguishes the ferromagnetic phase ($0.1 \leq x \leq 0.6$) and the antiferromagnetic phases ($x \leq 0.1$ and $x \geq 0.5$).

# 2. Antiferromangetic Perovskites

Antiferromagnetic interactions in perovskites originate from the superexchange mechanism. This is defined as an indirect exchange interaction between non-neighbouring magnetic cations mediated by a non-magnetic anion; an example of this is shown below.

In the orthorhombic perovskite structure, which displays $180^\circ$ superexchange, the geometry favours antiferromagnetic alignment, and thus the orthorhombic perovskites typically demonstrate a clear Néel transition. Such examples include the rare-earth orthochromite series $LNCrO_3$ where $LN$ is a lanthanide ion. The theory of superexchange indicates that the strength of the antiferromagnetic interaction, and hence the Néel temperature of the material, depends on the degree of overlap between the cations and their mediating anion. This in turn indicates that the transition temperature should be highly dependent on the properties of the $LN$ cation.

Panels (a) and (b) in the figure below show the auto-generated and reported Néel phase for the $LNCrO_3$ series. As predicted, these show a strong linear dependence of the Néel temperature on the ionic radius of the $LN$ cation.

We note that the auto-generated diagram is missing its Ce-member, $CeCrO_3$. Making use of the available auto-generated data and machine-learning techniques, we can attempt to make a prediction of the Néel temperature, and compare this to the r eported value of ~260 K.

Using the prediction and feature selection methods of the phase-transition toolkit outlined in the Methodology, we predicted the Néel temperature of $CeCrO_3$ to be in the range of 250-270 K, very close to the manually reported experimental value of 260 K. The optimal features for the model were determined to be the ionic radius and Pauling electronegativity of the $LN$ ion, both of which can be directly related to the theory of superexchange that depends primarily on the overlap of the $LN$ and $O$ orbitals.

# 3. Rare-earth Manganites

Other antiferromagnetic perovskite oxides include the rare-earth manganite series $LNMnO_3$. The auto-generated Néel temperature as function of ionic radius of the LN ion is given in panel (c) above. Again, the auto-generated temperatures match closely to the values from manually curated experimental reports. However, in contrast to the orthochromites, the relationship between Néel temperature and ionic radius is non-linear for $LNMnO3$ compounds. This non-linearity results from a structural phase transition within this series. For $LN = Ho, Er, Yb, Lu$ the compounds typically crystallise in a stable hexagonal structure.

In hexagonal perovskites the linkage between the cations can be either $180^\circ$ or $90^\circ$, yielding very different superexchange mechanisms to those of orthorhombic $LNMnO3$ compounds with $LN = La, Pr, Nd, Sm, Eu, Gd, Tb$; thus their different dependence on the Néel temperature. These orthorhombic $LNMnO3$ compounds have similar structures and magnetic behaviour to rare-earth orthochromites, given their common crystal system.

# 4. Rare-earth Ferropnictide Superconductors

The ferropnictides are a series of recently discovered iron-based superconductors formed from layers of iron and a pnictide material (see the inset in the figure below). The first measurement of superconductivity in these materials was reported in 2008, where a critical temperature of 26 K was discovered in $LaFeAsO_{0.89}F_{0.11}$: