Doping position estimation for FeRh-based alloys

Abstract

FeRh-based alloys have attracted significant attention due to their magnetic phase transition and significant magnetocaloric effects. These properties position them as promising candidates for fundamental research and practical applications, including magnetic cooling and targeted drug delivery. The study of FeRh alloys, particularly those where Rhodium or Iron atoms are substituted with other transition metals, is crucial as certain substitutions preserve the alloy’s magnetocaloric properties. However, even within a specific structural type and without considering competing phases, determining which atom (Fe or Rh) is replaced upon introducing a third element remains unclear. This paper addresses this ambiguity through ab initio calculations. We propose an approach to predict whether a dopant will replace Fe or Rh, offering insights into the electronic and structural factors influencing the substitution. Additionally, we present a dataset of ab initio calculations on doped FeRh alloys, which will support future data-driven modeling efforts. Our findings not only advance the understanding of FeRh-based alloys but also contribute to the design of novel materials for experimental and industrial applications.

Introduction

Currently, FeRh-based alloys are generating considerable interest within the scientific community^1–10. These alloys have recently attracted the attention of researchers because of their first-order magnetic phase transition occurring at temperatures close to those of the human body, along with their magnetic and crystalline structure properties and significant magnetocaloric effect. These properties mark FeRh-based alloys as promising materials for practical applications in fields such as magnetic cooling, magnetic recording technology, and targeted drug delivery^11–16.

A number of recent works^17–29 have investigated doped FeRh alloys. They demonstrate that doping FeRh alloys with Cu, Pd, and Ni in low concentrations (from approximately 1–5%) increases the refrigerant capacity of the resulting ternary alloy. However, doping these alloys may introduce specific defects where the doping occurs alongside vacancy defect. In this setting, two vacancy defects (one among Fe atoms and one among Rh atoms) occur in the crystal structure. The doping atom then substitutes one of two vacancies. Naturally, the doping atom favors one of two possible substitutions, and it is crucial to understand the underlying energetics of this process as it directly influences the properties of doped solutions³⁰. From the computational point of view, comparing the per-atom energies of these two cases is straightforward, as the atomic composition stays constant no matter which substitution is favored by the doping atom. However, this issue has been only partially addressed in existing literature^20–27, with recent studies indicating the absence of a comprehensive theory to predict which atom in the alloy is substituted by the doping element³¹.

From the computational perspective, the thermodynamic properties of defects and favorability of specific substitutions can be efficiently evaluated for binary compounds^32,33. However, implementation of such convex-hull-based approaches still needs to be improved for ternary and more chemically complex systems due to the absence of complete lists of competing phases³⁴. It was recently proposed^28,29 to study the magnetocaloric properties of FeRh alloys as well as its doped variant FeRh_1−xPd_x using ab initio methods. Another work³⁵ focuses on the magnetic and electronic properties of FePd alloy doped with Rh. However, these studies either consider pure FeRh alloys without doping elements or consider a single doping element such as Pd. Moreover, all these papers mainly focus on the magnetic properties of alloys.

In this work, we propose a unique computational scheme based on ab initio methods to study the energetics and thermodynamic properties of doped FeRh alloys where doping occurs together with a vacancy defect. We provide calculations for the full set of possible configurations for 16 different dopants and dopant concentration levels from 1.5 to 5%. Additionally, this work proposes a novel method for determining whether a dopant is more likely to substitute Rh or Fe. Specifically, we hypothesize that the favorability of substitution does not depend on dopant concentration or its position in the lattice and can be determined by considering specific dopant atom features such as ionic radius, electronegativity, and maximum valence. We demonstrate that a decision tree that operates in a three-dimensional feature space induced by the features mentioned above trained on the collected ab initio data perfectly separates all the dopants (see Fig. 4).

Fig. 4.

A visualization of the classification decision tree. A leaf node contains the Gini impurity value, the total amount of samples from data $\mathbf{X}$ that ended up in the leaf, class values for these samples (i.e., $\text{values}=[0,5]$ stands for 0 samples from class “0” and 5 samples from class “1”), and the class name of the majority of samples in the leaf. A non-leaf node additionally contains a rule by which to partition the samples.

Finally, we compose a dataset comprising geometry optimization trajectories for doped FeRh alloys with a vacancy defect. This dataset covers various dopant concentration levels and different dopant and vacancy positions. The dataset can be utilized in future work to train data-driven models such as Graph Neural Networks^36–42.

Our contributions are twofold:

• We introduce a method to determine the favorability of dopant substituting Rh or Fe atom upon introduction into the alloy, verified through extensive ab initio calculations.

• We create a large dataset comprising geometry optimization trajectories for FeRh alloys with various dopants in different spatial configurations to facilitate the training of data-driven models.

Methods and results

This section describes how we compiled the dataset using ab initio calculations. Then, we propose a method to determine whether the dopant atom is more likely to substitute Fe or Rh from the ab initio calculations. Lastly, we show that the favorability of substitution is independent of the dopant concentration or its position in the lattice and propose a model that relies on the dopant’s atomic radius, electronegativity, and valency.

Supercells and dopants

In this part, we describe the part of the configuration space that will be studied. We start with a standard Fe–Rh cubic lattice structure. The primitive cell of this alloy is defined with orthogonal lattice vectors equal to $\left(l_0,\approx ,2.99,\mathring{\text{A}}\right)$ in which one Fe atom and one Rh atom are placed with coordinates (0, 0, 0) and (0.5, 0.5, 0.5), respectively, in the lattice basis. We study doped FeRh alloys where doping occurs alongside a vacancy defect. To achieve that, we remove one atom of Fe and one atom of Rh from the lattice structure and then substitute one of the vacancies with a dopant atom. This procedure ensures that the solution’s atomic composition does not depend on the substitution type (whether the atom of Rh or the atom of Fe is substituted), which allows for the straightforward comparison of the per-atom energy.

To calculate the concentration of the dopant, we use $c=\frac{1}{N}$, where N is the total amount of atoms in a lattice structure excluding the vacancies. This work considers dopant concentration levels in the alloy of 1% to 5%. To obtain the desired concentration percentages, we expanded the original $(l_0,l_0,l_0)$ configuration by periodically repeating our initial primitive cell in space. We assume that there is only a single dopant atom and a single vacancy in each resulting configuration. For example, to obtain a dopant concentration of approximately 5% percent, we repeat the original lattice three times along one dimension and two times along two other dimensions. We call the resulting lattice $(3l_0,2l_0,2l_0)$ Small Supercell (SS). To get lower percentages of dopant concentration, we perform an analogous procedure and get Medium Supercell (MS) and Large Supercell (LS). The details on these configurations can be found in Table 1. Our work investigates the following $M=16$ dopants: Ir, Co, Cr, Ru, Pd, Pt, Mn, Ni, Te, Ti, Ta, Re, Nb, Os, Sn, Sb.

Table 1.

Atomic composition of considered supercells.

Supercell type	Total atoms (N)	Fe atoms	Rh atoms	Dopant atoms	Dopant concentration (c) (%)	Lattice type
Small supercell (SS)	23	11	11	1	4.35	$(3l_0,2l_0,2l_0)$
Medium supercell (MS)	35	17	17	1	2.86	$(3l_0,3l_0,2l_0)$
Large supercell (LS)	53	26	26	1	1.89	$(3l_0,3l_0,3l_0)$

Note that the formation of interstitial solid solutions is unlikely for these transition metals since the atomic radii of these elements differ by not more than 5% to the atomic radius of Fe and Rh. Thus, according to the assumptions of Hume-Rothery rules⁴³, all of them form substitutional solid solutions and occupy sites in the nodes of the crystalline lattice of FeRh. As indicated by previously published experimental studies, the first eight chosen components are elements that maximally preserve the significant magneto-thermal properties of binary alloys^22,27,31.

Non-equivalent configurations

Generally, there are three different scenarios in double substitutions: we can either substitute two Fe atoms (Fe–Fe type), two Rh atoms (Rh–Rh type), or both Fe and Rh atoms (Fe–Rh type). For the reasons described in Section “Determining the replacement type”, we only consider Fe–Rh type substitutions in this work. Our goal is to study the thermodynamic properties of each symmetrically inequivalent realization of the structure, which can correspond to fixed atomic occupancies and, consequently, hinder applications of the virtual crystal⁴⁴ and coherent potential approximations⁴⁵.

Additionally, we obtain different resulting conformations by varying the locations from which Fe and Rh atoms are removed. Note that many double substitutions are equivalent to each other when translations and reflections are considered. To find all possible unique double substitutions, we employ Supercell software⁴⁶ to obtain all non-equivalent pairs of atoms. We have two options for placing the dopant for any such pair, so the total number of double substitutions $K=2\ast [\text{total number of non-equivalent pairs}]$. We illustrate non-equivalent double substitutions in Fig. 1. There are $K_{\mathit{SS}}=4$ unique double substitutions for small supercell, $K_{\mathit{MS}}=6$ unique double substitutions for medium supercell, and $K_{\mathit{LS}}=8$ unique double substitutions for large supercell. The total number of double substitutions is thus $(K_{\mathit{SS}}+K_{\mathit{MS}}+K_{\mathit{LS}})\ast M=288$.

Fig. 1.

3D representations of all possible non-equivalent double substitutions. Red atoms correspond to Fe; blue atoms correspond to Rh. The dotted bounding box represents the volume spanned by lattice vectors. Yellow atoms indicate pairs of atoms that are first removed and then successively replaced by the dopant atom.

Overall database

To complete the database, we extend it with additional structures. First, we include the baseline supercells without any substitutions, thus adding three additional conformations to the database. Second, we include the so-called “single substitutions,” in which a single Fe (Rh) atom is replaced with a dopant. The total number of single substitutions for all considered supercells is $2\ast 3(\text{number of supercells})\ast M=96$. Additionally, it is interesting to consider the case when some of the Fe and Rh atoms are interchanged with each other. Since this procedure could be done for each combinatorially non-equivalent case, thus we increase our database by $\frac{1}{2}(K_{\mathit{SS}}+K_{\mathit{MS}}+K_{\mathit{LS}})=9$ entities. Our dataset consists of $288+3+96+9=396$ initial configurations. To get the final dataset, we obtain geometry optimization trajectories for all initial configurations (see Section “Determining the replacement type”).

Determining the replacement type

In this section, we propose a method to determine whether the dopant atom is more likely to substitute Fe or Rh in case of the double vacancy defect through ab initio calculations (see Section “Additional details” for the details on calculations). For a given dopant type D and supercell type SC, we divide the set of all non-equivalent double substitutions into two subsets $S{C}_{\mathit{Fe}}(D)$ and $S{C}_{\mathit{Rh}}(D)$, where an atom of Iron and atom of Rh is substituted by a dopant atom, correspondingly. Next, for each configuration c, we calculate the per-atom energy $\text{E}(c)$ in the relaxed state. To get the relaxed structure and the potential energy for a substitution, we perform geometry relaxation using Vienna Ab initio Simulation Package(VASP)^47–49 (see Section “Additional details”). We simultaneously perform volume relaxation to account for different sizes of dopant atoms. Finally, we compare the minimum energies of two types: $\underset{c\in S{C}_{\mathit{Fe}}(D)}{min}\text{E}(c)$ and $\underset{c\in S{C}_{\mathit{Rh}}(D)}{min}\text{E}(c)$. The configuration with lower energy indicates a more desirable substitution type for the dopant. Note that the direct comparison of energies for two configurations is only possible for structures with the same atomic composition. This is the main reason we only consider double substitutions with one atom of each Fe and Rh removed. Such alloys can be denoted as ${\text{Fe}}_{x-1}{\text{Rh}}_{x-1}{\text{D}}_1$, where x is the total number of Fe(Rh) atoms in a supercell, and D denotes a dopant.

To illustrate the preferred replacement types for various dopants, we calculate the difference in minimal total energies after the relaxation between the cases where the dopant atom replaces either Fe or Rh: $\mathrm{\Delta }{E}^{D,SC}=\underset{c\in S{C}_{\mathit{Rh}}(D)}{min}\text{E}(c)-\underset{c\in S{C}_{\mathit{Fe}}(D)}{min}\text{E}(c)$, where D stands for a dopant type, and SC stands for the supercell type. The results are summarized in Fig. 2. We color entries in Fig. 2 depending on the $\mathrm{\Delta }{E}^{D,SC}$ sign. If $\mathrm{\Delta }{E}^{D,SC}<0$ (replacing the atom of Fe is more energetically preferable for the dopant atom), the color is set to purple; if $\mathrm{\Delta }{E}^{D,SC}>0$ (replacing the atom of Rh is more energetically preferable for the dopant atom), the color is set to blue. Qualitatively, we can observe that Cr, Mn, Te, Ti, Ta, Re, Nb, Sn, and Sb tend to replace Fe, while Ir, Co, Ru, Pd, Pt, Ni, and Os are more likely to substitute Rh. Moreover, for each dopant, we observe that the concentration determined by the supercell type does not influence (with precision up to 1.25 meV) the preferability of the substitution.

Fig. 2.

Absolute energy differences $|\mathrm{\Delta }{E}^{D,SC}|$ per atom in relaxed states for different dopants and supercell types. All the values are in milielectronvolts (meV). The red and blue colors indicate double substitutions where $E^{D,SC}<0$ and $E^{D,SC}>0$ respectively.

Rules for prediction

In this work, we are concerned with substitutional solid solutions of three metals. There is a basic set of rules⁵⁰ that indicates when an element could dissolve in a metal, forming a substitutional solid solution. This set of rules operates with three numerical features of atoms: empirically measured covalent radius (also known as Slater radius⁵¹), electronegativity by Pauling scale⁵², and maximum valency. However, this set of rules was initially derived for two-element alloys, whereas our work studies three-element alloys. Moreover, these rules do not provide an intuition on this manuscript’s central question: whether the dopant atom is more likely to substitute Fe or Rh in case of the double vacancy defect in the alloy.

We hypothesize that the favorability of the substitution does not depend on the concentration of the dopant or its position in the lattice. We propose using the aforementioned numerical features to infer rules to determine the favorability directly from ab initio data using classification decision trees⁵³. To train a classification decision tree, we must first map dopant atoms to vectors in the feature space defined by atomic radius, electronegativity, and valency of the atom. This procedure results in a feature matrix $\mathbf{X}\in {\mathbf{R}}^{M\times 3}$, which is visualized in Fig. 3.

Fig. 3.

Barplots of atomic radii, electronegativities, and maximum valencies of dopants. Dopants replacing Fe, according to the proposed computational method, are colored red, while elements replacing Rh are colored blue. Dashed lines indicate the atomic radii, electronegativities, and maximum valencies of Fe and Rh.

To define the targets for classification, we use ab initio calculations described in Section “Determining the replacement type” and presented in Fig. 2. We say that an element belongs to class “0” (replaces Fe) if it replaces Fe in the majority of concentrations (see Fig. 2). Analogously, we say that an element belongs to class “1” (replaces Rh) if it replaces Rh in the majority of concentrations. This results in a target vector $\mathbf{y}\in {\{0,1\}}^M$.

To infer the classification rules, we use the DecisionTreeClassifier from the sklean package⁵⁴ and fit it using $\mathbf{X}$ as the feature matrix and $\mathbf{y}$ as the target vector. We specifically limit the maximum depth of the tree to 3 to avoid complex rules. The resulting tree is visualized in Fig. 4. Every leaf node is a terminal node with a certain class (“replaces Fe” or “replaces Rh”) assigned to it. The class of the leaf node is determined during the tree fitting as the class that the majority of the elements in the leaf have. The leaf nodes also contain the Gini impurity⁵³ value that indicates how homogeneous the objects in the leaf are in terms of their target value. The Gini impurity of 0 means that all objects in the leaf belong to the same class. Note that the Gini impurity equals 0 in all leaf nodes, so the decision tree perfectly separates all dopants.

Every non-leaf node contains a rule consisting of a feature and a threshold. For example, the root node contains the following rule: If the dopant’s electronegativity is larger than 2.15, the dopant replaces Rh (the right child of the root node is the leaf node with class “1” assigned to it); otherwise, continue with the rule in the left child. Note that all non-leaf nodes contain rules with unique features, which indicates that all three features are essential to classify all the dopants perfectly.

Additional details

Spin-polarized total energy DFT calculations and total energy relaxations were executed using the Vienna Ab initio Simulation Package(VASP)^47–49. We used a fully automatic generation scheme of the Bloch vectors with $R_k=20\mathring{\text{A}}$ where $R_k$ defines subdivisions $N_i$ along the reciprocal lattice vectors ${\mathbf{b}}_{\mathbf{i}}$ in the following way: $N_i=int(max(1,R_k|{\mathbf{b}}_{\mathbf{i}}|+0.5))$. The energy cutoff for the plane-wave basis set was set at 600 eV. For self-consistent field conditions, the criteria for breaking the cycle were that the total energy change and the band structure energy change between two steps are smaller than ${10}^{-6}$ eV. Gaussian smearing was used for electronic occupancies with a smearing of 0.04 eV. All calculations were considered spin-polarized ones with initial magnetic moments equal to $2{\mu }_b$ for each atom. Ionic relaxation was executed via the conjugate gradient algorithm until the norms of all the forces acting on the atoms were smaller than ${10}^{-2}$ eV/Å. We used the default blocked-Davidson-iteration scheme. Finally, the Projector Augmented-Wave (PAW)⁵⁵ pseudopotentials were used to reproduce the atomic core effects in the electronic density of the valence electrons. In the standard mode, VASP performs a fully relativistic calculation for the core electrons and treats valence electrons in a scalar relativistic approximation.

Table 2 presents the averages and standard deviations of energies and maximum forces for each system, along with the number of structures analyzed. These properties characterize the systems at the final stage of ionic optimization. Notably, the average maximum force meets the criterion for halting the ionic relaxation, as the forces are smaller than ${10}^{-2}$ eV/Å.

Table 2.

Averages and standard deviations of energies per atom (eV) and maximum forces (eV/Å) for each system, alongside the number of analyzed structures at the final stage of ionic optimization.

System	Avg. Energy	Std. Dev. Energy	Avg. Max Force	Std. Dev. Force	Num. of Struc.
SS baseline	$-$ 7.7799	0.0000	0.000387	0.000000	1
SS Fe–Rh exchange	$-$ 7.7641	0.0008	0.007956	0.000455	2
SS single substitution	$-$ 7.7708	0.1210	0.007397	0.002083	32
SS double substitution (1)	$-$ 7.7350	0.1219	0.008203	0.001394	32
SS double substitution (2)	$-$ 7.7286	0.1246	0.008292	0.001186	32
MS baseline	$-$ 7.7799	0.0000	0.000566	0.000000	1
MS Fe–Rh exchange	$-$ 7.7723	0.0012	0.007211	0.001703	3
MS single substitution	$-$ 7.7712	0.0792	0.007937	0.001402	32
MS double substitution (1)	$-$ 7.7498	0.0804	0.008385	0.001317	32
MS double substitution (2)	$-$ 7.7460	0.0836	0.008477	0.001366	32
MS double substitution (3)	$-$ 7.7468	0.0838	0.008522	0.001234	32
LS baseline	$-$ 7.7799	0.0000	0.000455	0.000000	1
LS Fe–Rh exchange	$-$ 7.7750	0.0009	0.008095	0.000913	4
LS single substitution	$-$ 7.7727	0.0537	0.007196	0.002112	32
LS double substitution (1)	$-$ 7.7610	0.0532	0.008620	0.001066	32
LS double substitution (2)	$-$ 7.7550	0.0552	0.008697	0.000999	32
LS double substitution (3)	$-$ 7.7564	0.0552	0.008616	0.001097	32
LS double substitution (4)	$-$ 7.7552	0.0547	0.007517	0.001771	32

Conclusions

In this study, we focused on FeRh-based alloys and their doped variants. We explored the issue of determining whether a dopant replaces a Rh or Fe atom. To address this, we collected a substantial dataset of ab initio calculations and developed a decision-tree-based algorithm that accurately categorizes the data. We hypothesize that the favorability of substitution depends only on the dopant’s atom type, which agrees with the computation data up to 1.25 meV. Furthermore, we compiled a comprehensive dataset of geometry optimization trajectories for FeRh-based alloys to aid in training data-driven models. We consider the exploration of the configurational space and properties of FeRh-based alloys using neural networks to be a promising direction for future research.

Author contributions

Conceptualization: E.R., K.K., R.E., V.Z., A.K. Methodology: E.R., K.K., R.E., V.Z., A.K. Software: E.R., K.K., A.T. Validation: K.K., V.Z., N.P., R.G. Investigation: E.R., K.K., V.Z., N.P., R.G. Data Curation: E.R., K.K. Writing—Original Draft: E.R., K.K., V.Z., N.P., R.G., A.K. Writing - Review and Editing: E.R, K.K., A.T., R.E., A.K. Visualization: E.R. Supervision: A.K. Project administration: A.K.

Data availability

The full data obtained and used within the manuscript is available free of charge at https://github.com/AI4DD/DOPED.

Competing interests

The authors declare no competing interests.