# Magnetism in iron alloys: thermodynamics, kinetics and defects

Tilmann Hickel, Sergiy Divinski, Sergei Starikov, Frédéric Soisson, Christian Mény, Omka Hedge, Marius Gerlitz, Gladyce Magnifouet, Anton Schneider, Cyrille Barreteau, et al.

► To cite this version:

Tilmann Hickel, Sergiy Divinski, Sergei Starikov, Frédéric Soisson, Christian Mény, et al., Magnetism in iron alloys: thermodynamics, kinetics and defects. International Journal of Materials Research, 2025, pp.1. 10.1515/ijmr-2023-0225. cea-05115472

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Tilmann Hickel*, Sergiy Divinski, Sergei Starikov, Frédéric Soisson, Christian Mény, Omkar Hegde, Marius Gerlitz, Gladice Magnifouet, Anton Schneider, Cyrille Barreteau, Isabelle Mirebeau, Van Truong Tran, Georg Daniel Förster, Alexis Front, Aleksei Egorov, Gerhard Wilde, Hakim Amara, Thomas Hammerschmidt, Matous Mrovec, Véronique Pierron-Bohnes, Ralf Drautz and Chu-Chun Fu

# Magnetism in iron alloys: methodological advances for thermodynamics, defects, and kinetics

https://doi.org/10.1515/ijmr-2023-0225

Received July 16, 2023; accepted March 17, 2025;

published online June 16, 2025

**Abstract:** Steels are among the technologically and economically most relevant materials. Key innovations in important sectors of human society such as mobility, energy and safety, are currently based on alloying of Fe with other transition-metal elements such as Mn, Cr, or Co. Due to strong impacts and conceptual challenges related to magnetism, however, the fundamental understanding and the ability to computationally design these steels in high-throughput approaches lags behind other classes of alloys. In this article, we will provide a substantial review of the role of magnetism, magnetic excitations and transformations for alloy thermodynamics, point defects, interfaces and kinetics. This will be achieved by combining insights from different methods: Ab initio simulations have the advantage that the magnetic ground state is intrinsic part of the electronic minimization. Due to the coarsening of the many-electron structures and

therewith magnetic interactions, tight-binding methods can handle larger system sizes. Effective interaction models provide the freedom to exploit more sophisticated magnetic interactions. The performance of these methods in terms of magnetic properties of Fe alloys will be evaluated by providing state-of-the-art results for their sensitivity to magnetism. Furthermore, dedicated experiments will be discussed to complete the understanding of magnetic effects in Fe alloys and to validate the modeling strategy.

**Keywords:** magnetic excitations; defects; diffusion; density functional theory; tight-binding; effective interaction models

## Acronyms

1nn first nearest neighbor.

2nn second nearest neighbor.

ACE atomic cluster expansion.

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<footnote>*Corresponding author: Tilmann Hickel, Department of Computational Materials Design, Max-Planck-Institute for Sustainable Materials, 40237 Düsseldorf, Germany; and BAM Federal Institute for Materials Research and Testing, 12489, Berlin, Germany, E-mail: tilmann.hickel@bam.de. https://orcid.org/0000-0003-0698-4891</footnote>

<footnote>Sergiy Divinski, Marius Gerlitz and Gerhard Wilde, Institute of Materials Physics, Westfälische Wilhelms-University of Münster, Münster, Germany, E-mail: divin@uni-muenster.de (S. Divinski), mariusgerlitz@gmx.de (M. Gerlitz), gwilde@uni-muenster.de (G. Wilde)</footnote>

<footnote>Sergei Starikov, Aleksei Egorov, Thomas Hammerschmidt, Matous Mrovec and Ralf Drautz, ICAMS, Ruhr-Universität Bochum, 44780, Bochum, Germany, E-mail: sergei.starikov@icams.ruhr-uni-bochum.de (S. Starikov), aleksei.egorov@rub.de (A. Egorov), thomas.hammerschmidt@rub.de (T. Hammerschmidt), matous.mrovec@icams.ruhr-uni-bochum.de (M. Mrovec), ralf.drautz@rub.de (R. Drautz)</footnote>

<footnote>Frédéric Soisson, Anton Schneider, Van Truong Tran and Chu-Chun Fu, Université Paris-Saclay, CEA, Service de recherche en Corrosion et Comportement des Matériaux, SRMP, 91191, Gif-sur-Yvette, France, E-mail: frederic.soisson@cea.fr (F. Soisson), anton.schneider3393@gmail.com (A. Schneider), vantruongtran.nanophys@gmail.com (V.T. Tran), chuchun.fu@cea.fr (C.-C. Fu)</footnote>

<footnote>Christian Mény, Gladice Magnifouet and Véronique Pierron-Bohnes, Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), Université de Strasbourg, 67034, Strasbourg, Cedex 2, France, E-mail: christian.meny@cnrs.fr (C. Mény), gladice.magnifouet@ipcms.unistra.fr (G. Magnifouet), vero@ipcms.unistra.fr (V. Pierron-Bohnes)</footnote>

<footnote>Omkar Hegde, Department of Computational Materials Design, Max-Planck-Institute for Sustainable Materials, 40237, Düsseldorf, Germany, E-mail: omkaarg@gmail.com</footnote>

<footnote>Cyrille Barreteau, Université Paris-Saclay, CEA, CNRS, SPEC, 91191, Gif-sur-Yvette, France, E-mail: cyrille.barreteau@cea.fr</footnote>

<footnote>Isabelle Mirebeau, Laboratoire Léon Brillouin, CEA-CNRS UMR 12, Centre de Saclay, F-91191, Gif-sur-Yvette, France, E-mail: isabelle.mirebeau@cea.fr</footnote>

<footnote>Georg Daniel Förster and Alexis Front, Laboratoire d'Etude des Microstructures, Université Paris-Saclay, ONERA, CNRS, F-92322, Châtillon, France, E-mail: daniel.foerster2@gmail.com (G.D. Förster), alexisfront@protonmail.com (A. Front)</footnote>

<footnote>Hakim Amara, Laboratoire d'Etude des Microstructures, Université Paris-Saclay, ONERA, CNRS, F-92322, Châtillon, France; and Laboratoire Matériaux et Phénomènes Quantiques (MPQ), Université de Paris Cité, F-75013, Paris, France 3ICMMO, E-mail: hakim.amara@onera.fr</footnote>
<table><tr><td>ADP</td><td>angular-dependent potential.</td></tr><tr><td>AFD</td><td>anti-ferromagnetic double-layer.</td></tr><tr><td>AFM</td><td>anti-ferromagnetic.</td></tr><tr><td>ASA</td><td>atomic sphere approximation.</td></tr><tr><td>bcc</td><td>body-centred cubic.</td></tr><tr><td>BOP</td><td>bond-order potential.</td></tr><tr><td>CPA</td><td>coherent potential approximation.</td></tr><tr><td>DFT</td><td>density functional theory.</td></tr><tr><td>DLM</td><td>disordered local moment.</td></tr><tr><td>DMFT</td><td>dynamical mean-field theory.</td></tr><tr><td>DNS</td><td>diffuse neutron scattering.</td></tr><tr><td>DOS</td><td>density of states.</td></tr><tr><td>EAM</td><td>embedded-atom method.</td></tr><tr><td>EIM</td><td>effective interaction model.</td></tr><tr><td>fcc</td><td>face-centred cubic.</td></tr><tr><td>FM</td><td>ferromagnetic.</td></tr><tr><td>FMA</td><td>fourth moment approximation.</td></tr><tr><td>FNR</td><td>Ferromagnetic Nuclear Resonance.</td></tr><tr><td>GB</td><td>grain boundary.</td></tr><tr><td>GGA</td><td>generalized gradient approximation.</td></tr><tr><td>GMS</td><td>ground magnetic state.</td></tr><tr><td>hcp</td><td>hexagonal closed-packed.</td></tr><tr><td>HF</td><td>hyperfine field.</td></tr><tr><td>KKR</td><td>Korringa-Kohn-Rostoker.</td></tr><tr><td>kMC</td><td>kinetic Monte Carlo.</td></tr><tr><td>LDA</td><td>Local Density Approximation.</td></tr><tr><td>LMTO</td><td>linear muffin-tin orbital.</td></tr><tr><td>LRO</td><td>long-range order.</td></tr><tr><td>MC</td><td>Monte Carlo.</td></tr><tr><td>MD</td><td>molecular dynamics.</td></tr><tr><td>mEAM</td><td>magnetic EAM.</td></tr><tr><td>MF</td><td>mean-field.</td></tr><tr><td>ML</td><td>machine learning.</td></tr><tr><td>MSRO</td><td>magentic short-range order.</td></tr><tr><td>MTP</td><td>moment tensor potential.</td></tr><tr><td>NEB</td><td>nudged elastic band.</td></tr><tr><td>NMR</td><td>Nuclear Magnetic Resonance.</td></tr><tr><td>NNP</td><td>neural network potential.</td></tr><tr><td>NOTB</td><td>non-orthogonal tight binding.</td></tr><tr><td>OTB</td><td>orthogonal tight binding.</td></tr><tr><td>PAW</td><td>projector augmented-wave.</td></tr><tr><td>PM</td><td>paramagnetic.</td></tr><tr><td>rMC</td><td>rescaled Monte Carlo.</td></tr><tr><td>RPA</td><td>random-phase approximation.</td></tr><tr><td>SD</td><td>spin dynamics.</td></tr><tr><td>SQS</td><td>special quasirandom structure.</td></tr><tr><td>SSA</td><td>spin-space average.</td></tr><tr><td>TB</td><td>tight binding.</td></tr><tr><td>TRIP</td><td>transformation induced plasticity.</td></tr></table>

# 1 Introduction

## 1.1 From fundamentals of magnetism to methodological challenges

Magnetism has been known since ancient times. One can only imagine the mystical quality it must have had for sailors that

relied on magnetism to navigate the seas a thousand years ago. Today we know that magnetism is a companion of charge. Moving charge creates a magnetic field, a consequence of frame invariance under Lorentz transformation of special relativity as described by the Maxwell equations. Magnetic phenomena observed in solids are largely derived from the spin of the electron. Electrons in an atom favour a parallel alignment of their spins that results in a magnetic moment. When atoms are brought in contact, their orbitals overlap, thereby increasing the kinetic energy of the electrons. Because of Pauli's exclusion principle, the kinetic energy of electrons that form a magnetic moment with more up-spin than down-spin states is larger than a non-magnetic arrangement with equal distribution of up and down spin states. In the Stoner picture of magnetism, the magnetic state of a solid may therefore be viewed as a compromise between minimizing the kinetic or band energy of the electrons, which favours a non-magnetic state, and maximizing the atomic exchange energy, which favours parallel spin alignment. For example, while the 3d elements Cr, Mn, Fe, Co and Ni are magnetic, the overlap of the d-states in the isoelectronic 4d and 5d elements quenches magnetism and makes these elements non-magnetic.

An intricate dependence of magnetism on band filling means that elements with an approximately half-full d-valence (Cr,Mn) prefer anti-parallel alignment of the atomic magnetic moments, while towards a full d-band (Co,Ni) the magnetic moments order in parallel. Iron is in between the two limiting cases and depending on crystal structure sustains different arrangements of the magnetic moments. Iron stands out for other reasons. Among the 3d elements, the magnetic contribution to the cohesive energy is largest in iron by far. Magnetism depends sensitively on crystal structure and by undergoing a phase transition from the non-magnetic to the magnetic state, the body-centred cubic (bcc) lattice gains significantly more energy than the hexagonal closed-packed (hcp) or face-centred cubic (fcc) lattices. Therefore, the ground state of iron is bcc and not hcp, as found for isoelectronic ruthenium and osmium.

The magnetic energy of iron depends on volume and temperature. At small volumes d-band overlap increases and quenches magnetism. At increased temperature the atomic magnetic moments fluctuate, which also lowers the magnetic energy and contributes to the stabilization of close-packed phases at elevated temperatures. It is thus clear that the phase diagram of iron cannot be understood without accounting for magnetism. The same holds for iron alloys: magnetism contributes to make steel one of the most versatile classes of materials. Other properties, like diffusion, are equally affected by magnetism in iron and its alloys.

The complex interplay between electronic and atomic structure in magnetic materials cannot be captured by
thermodynamic models that focus on state variables. There-
fore magnetic materials, and in particular iron and its alloys,
are only poorly described by state-of-the-art thermodynamic
models and databases. A thermodynamic and kinetic analysis
of iron has to incorporate the essential features of magnetism.
As magnetism depends sensitively on the atomic structure,
atomic scale modelling is critical for leveraging fundamental
theory of magnetism to applications for designing alloys of
iron and steels. Our review of magnetism in iron alloys
therefore starts from the electronic structure and has its main
focus on atomic scale modelling, from which thermodynamic
and kinetic properties of iron alloys are obtained.

We highlight three different electronic structure methods
that have been developed independently and collaboratively<sup>a</sup>
by the groups of the co-authors over the last decade and that
all have their advantages and disadvantages when capturing
the magnetic challenges of iron and its alloys:

(1) Density functional theory (DFT) is nowadays an estab-
lished approach for predictive materials science. Due to
the fundamental coupling of the spin and the charge of
electrons, the determination of the magnetic ground
state is an intrinsic part of the selfconsistent electronic
minimization within (spin-polarized) DFT. The challenge
of DFT is, however, to capture the long-range magnetic
interactions in small supercells and to handle magneti-
cally excited states at finite temperatures.

(2) Within tight binding (TB) approaches a coarsening of the many-electrons structure is achieved by approximating its energetics by effective terms for the superposition of wave functions of isolated atoms located at the atomic sites. This simplification allows one to overcome the finite-size limitations of DFT. However, the explicit magnetic interaction energy is in TB approaches often limited to a single parameter for capturing the Stoner interaction. Furthermore, it needs to be clarified to what extend the restriction to d-electrons is sufficient for the magnetism of iron alloys.

(3) An effective interaction model (EIM) provides the freedom to introduce more sophisticated magnetic interactions than the TB approach, while keeping the advantage of being computationally much cheaper than DFT. The disadvantage is the ambiguity in the choice of the model and the decoupling of magnetic and chemical interactions in the Hamiltonian. Since its derivation is not possible from first principles, the EIM depends on the fitting to the results of more fundamental approaches such as DFT.

1.2 From methodological achievements to
practical applications

In our review we discuss recent advances and current
results achieved with these methods in terms of magnetism
in Fe and its alloys. To this end, we closely relate theory,
simulation and experiment for thermodynamic and kinetic
properties of selected binary Fe-based materials that have a
practical relevance in engineering.

Of course, steels form the most abundant class of
Fe-based materials that are important for structural appli-
cations. They owe their versatility to a large number of
alloying elements that are in combination with sophisticated
heat treatments and other processing steps responsible for
the formation of different microstructures. Empirical
approaches need to test various annealing or tempering
steps to obtain desired compositional homogeneity, micro-
structure, precipitate or secondary phase formation in these
materials. More efficient, simulation-guided strategies for
the development of new steels with specific properties,
therefore, require a detailed understanding of the impact of
magnetism on the thermodynamics and kinetics of steels.

Steels derived from Fe–Mn alloys exhibit high strength
due to sophisticated deformation mechanisms. If the Mn
content is in the range of 18–25 %, a stabilization of the fcc
crystal structure along with a change in the magnetic state is
observed. It adapts to a deformation through a diffusionless
(martensitic) transformation to a hcp or bcc crystal struc-
ture, the so-called transformation induced plasticity (TRIP).
A reduction of the Mn-content to 10–15 % reduces some
technological problems with high-Mn steels, while substan-
tial amounts of retained austenite still provide access to the
TRIP mechanism. However, one of the challenges of these
materials is the sensitivity to grain boundary embrittlement
in the bcc phase caused by Mn segregation. The segregation
kinetics as well as the resulting decohesion strength of grain
boundaries is again strongly dependent on the magnetic
state.

Much of the current interest in Fe–Cr is driven by its
tolerance to radiation damage at a composition of about 12 %
Cr. At low Cr concentrations Fe–Cr shows a tendency to
mixing that changes to de-mixing and phase separation at
about 12 % Cr. The mixing tendency at low Cr concentrations
is due to magnetism. Computations without magnetism find
a strong driving force for phase separation. Therefore, a
fundamental challenge in Fe–Cr alloys is the control of the
kinetics of coherent decomposition into an Fe-rich phase (a)
and a Cr-rich phase (a'). The precipitation kinetics includes
nucleation-and-growth or spinodal decomposition regimes,
depending on composition and temperature. In both cases,
the ferromagnetic (FM)/paramagnetic (PM) transition affects

<footnote>a The collaborative work has been performed within the DFG-ANR
project MAGIKID from 2016 to 2021: https://gepris.dfg.de/gepris/projekt/
316673557?language=en</footnote>
the thermodynamics, the diffusion and the microstructural evolution of the alloy.

Fe–Co is the base material for many permanent magnets. At equiatomic composition, the material undergoes an order-disorder transition between an ordered bcc-based B2 phase and a disordered bcc A2 phase at 730 °C, while it remains ferromagnetic up to the transition to the γ-phase at 980 °C. The diffusion coefficients change significantly at the chemical order-disorder transition temperature. This behavior needs to be compared and contrasted to other dilute Fe alloys (here: dilute Fe–Cr and Fe–Mn), where the diffusion coefficient is affected by magnetic disordering.

In this article we discuss recent relevant results for all these materials by comparing theory, simulation and experiment. The focus will be on the thermodynamics of binary Fe alloys, the properties of point as well as extended defects, and on diffusion phenomena. After providing first a brief overview of theoretical methodologies, the subsequent chapters address these topics, respectively. In the final chapter, we summarize the present state-of-the-art and outline potential topics for future research.

# 2 Computational methods for magnetic materials

## 2.1 Density functional theory

DFT represents a compromise between an accurate quantum-mechanical description of magnetic material properties and the ability to calculate chemically and structurally complex materials. Within this approach, the local magnetic moments are derived from the local spin polarization of the magnetic moment density. In this way, the description of magnetically ordered structures at the ground state is directly accessible and often in good agreement with experiment. The prediction of magnetically excited states is, however, much more challenging.¹

### 2.1.1 Spin constraints

A pre-requisite for handling magnetic disorder in DFT codes is the implementation of spin constraints, for which several versions are available in the literature. They have in common that the atomic moment $M^a$ of an atom a needs to be defined. Within the projector augmented-wave (PAW) formalism, which is an established transformation of the all-electron wave function, it is given as

$$M^a = \sum_{ij a} \chi_a D_{ij}^{aa} \Omega_{ij} \quad (1)$$

Here, $\chi = \pm 1$ is the spin sign, $D_{ij}^{aa}$ is the one-center density matrix for atom a with spin $\alpha$, and $\Omega_{ij}$ is an integral over the product of partial waves $\phi_i(\mathbf{r})$ and $\phi_j(\mathbf{r})$ within a cut-off radius $r^{cut}$, and has the dimension of volume.

Based on this definition, most DFT codes have implemented the spin constraint via a penalty contribution to the expression of the total energy. For example, Ma et al.² have suggested a version of a constraining magnetic field $\mathbf{b}^a(\mathbf{r})$ on an atom a given by

$$\mathbf{b}^a(\mathbf{r}) = -\lambda \left( \frac{\mathbf{M}^a}{|\mathbf{M}^a|} - \mathbf{e}^a \right) F^a(\mathbf{r} - \mathbf{r}^a) \quad (2)$$

Here, $F^a$ corresponds to a cutoff sphere for the atom, $\mathbf{e}^a$ is the unit vector of the expected direction of the magnetic moment, and $\lambda$ is a numerical scaling parameter. A slightly different approach is the implementation of the spin constraints via the standard Lagrange formalism as suggested by Hegde et al.³ Here, the goal are the stationary points of the functional

$$\mathcal{L} = F^{el} [\psi_{\{n\sigma\}}, f_{\{n\sigma\}}] - \sum_{a} v_{a} (M^{a} [\psi_{\{n\sigma\}}, f_{\{n\sigma\}}] - M^{a, target})$$

where $M^{a, target}$ are the target moments, and $v_a$ the corresponding constraints that enter the equation as Lagrangian multipliers. $F^{el}$ refers to the electronic free energy functional. The stationary points are found by minimizing the functional with respect to Lagrangian parameters, i.e.,

$$\frac{\delta \mathcal{L}}{\delta \langle \psi_n |} = 0, \quad \frac{\partial \mathcal{L}}{\partial f_n} = 0, \quad \text{and} \quad \frac{\partial \mathcal{L}}{\partial v_a} = 0 \quad (3)$$

In this way the Kohn-Sham equation for the electronic degrees of freedom and the spin constraint are treated on the same footing. The implementation of this scheme is numerically efficient and avoids energy penalties in the Hamiltonian.

### 2.1.2 Paramagnetic disorder

One of the possible concepts for handling magnetic disorder is the disordered local moment (DLM) approach.⁴,⁵ It is often based on the Korringa-Kohn-Rostoker (KKR) version of the coherent potential approximation (CPA), which is an effective medium theory and suffers from the underlying single-site approximation when carrying out ensemble averages. The DLM approach is based on adiabatic decoupling of magnetic and electronic degrees of freedom, of which the first is assumed to be much slower than the latter. Furthermore, a complete magnetic disorder is assumed, i.e., possible short-range order effects are ignored. One of the achieved results is an estimate of the Curie temperature of Fe to be
1250 K⁵ and a reduction of the magnitude of the magnetic moment above this temperature to 85 % of the value at the ground state.⁴

A formally equivalent approach is the spin-wave method.⁶ It performs an average over total energies of planar spin spirals, but with the possibility to use a special point technique⁷ for the Brillouin zone integration. It can be extended to take magnetic short-range order effects into consideration and has been applied to calculate defect-formation energies, elastic constants, and phonon spectra.⁶

An alternative is the consideration of supercells that mimic the magnetic disorder. The concept of a special quasirandom structure (SQS)⁸ is often used to represent collinear magnetic disorder of spin-up and spin-down atoms,⁹,¹⁰ a justified approximation for complete disorder and neglected spin-orbit coupling.⁵ A supercell approach for non-collinearly disordered magnetic configurations²,¹¹ is computational more demanding. It has been demonstrated for DFT codes with plane-wave basis sets and gives access to ensemble averages beyond the single-site CPA approach.¹¹

The consideration of paramagnetic materials in terms of disordered spin configurations has the disadvantage that it describes the adiabatic limit of slowly evolving spins in the dynamic microstructure. This is certainly not correct for the atomic relaxation of atoms next to defects, and might also be an unjustified approximation for lattice vibrations and diffusion processes. Therefore, the concept of a spin-space average (SSA) has been developed,¹² which defines forces as

$$F_{SSA}^{a} = \sum_{K=1,N} p_{K} F_{K}^{a} \quad (4)$$

where for a set of N magnetic configurations $F_K^a$ is the Hellmann-Feynman force on atom a and $p_K$ the Boltzmann weight. The latter is $p_K = \exp[-E_K^{BO}/k_B T]/Z$, where $E_K^{BO}$ is the magnetic Born-Oppenheimer energy for the configuration K, $k_B$ is the Boltzmann constant and Z is the magnetic partition sum. In the fully paramagnetic limit of $T \to \infty$ all configurations have the same weight $p_K = 1/N$.

We speak about SSA relaxation,³ if the atoms are relaxed using the forces as given in (4). In order to reduce the number of necessary configurations, it is often convenient to symmetrize these forces according to the point group symmetry of the given atom. Still, the numerical challenge of the approach is the fact that the averaged forces cannot be determined within a single DFT calculation, but instead require the communication between an external software tool with N individual DFT calculations.

## 2.2 Tight-binding methods

### 2.2.1 Coarsening the electronic structure

To study large systems, a coarsening of the electronic-structure description towards approximate and computationally efficient models that include magnetic contributions is required. In this context, the TB framework is perfectly suited since it provides an explicit treatment of electronic interactions so that the description of magnetism is naturally contained within the model. As TB is a parameterised quantum-based representation, the involved functional forms and the level of approximations may differ. Indeed, there are many variants of magnetic TB to describe transition metals and their alloys that differ in the choice of the atomic basis (e.g., a complete *spd* basis¹³ or only a minimal *d* basis¹⁴,¹⁵ for pure transition metals) and the treatment of self-consistency (e.g., local charge neutrality). Two primary types of TB models are orthogonal tight binding (OTB) models, which neglect the overlap matrix,¹⁶,¹⁷ and non-orthogonal tight binding (NOTB) models, which consider the orbitals to remain non-orthogonal on different atomic sites.¹⁸

The magnetic term is derived using the Stoner Hamiltonian by taking into account the local exchange fields in the band energy.¹⁹ As a result, the degeneracy between the two spin directions is suppressed by a potential that generates a band separation between the up and down spins. In the case of collinear magnetism, the two populations are subtracted leading to a spin moment $M_i$ on an atom i given by $M_i = N_{i\uparrow} - N_{i\downarrow}$, with $N_{i\uparrow}$ and $N_{i\downarrow}$ being the number of electrons in the majority and minority spin bands, respectively. In this approach, the exchange potential is described by an effective magnetic field $I M_i/2$ and local magnetic on-site levels:

$$e_i^\sigma = e_i^0 \pm \frac{I}{2} M_i \quad (5)$$

where I is the Stoner exchange integral. The sign is chosen if the spin $\sigma$ is parallel or antiparallel to the direction of the local magnetic field. In practice, the levels as defined in Equation (5) must be determined self-consistently. The critical task in the construction of all TB models is the determination of model parameters. For instance, a *d*-OTB model requires parametrization of the $dd\sigma$, $dd\pi$, and $dd\delta$ hopping integrals as functions of the interatomic distance, the choice of the number of valence electrons, and the value of the Stoner parameter *I*. Since all these quantities have a clear physical meaning, the fitting procedure can remain rigorous and the model retains transparency and a high degree of transferability.
Moreover, there have been developments of TB-based interatomic potentials to enable simulations of larger systems without applying periodic boundary conditions and/or to perform simulations at finite temperatures. In the case of non-magnetic systems, the total energy (with respect to the energy of free atoms) can be written as a sum of an attractive contribution, which describes the formation of an energy band when atoms are put together, and of a phenomenological repulsive term. For magnetic systems, the contribution of the exchange potential to the total energy is given by $-Im_i^2/4$ after summing over the whole electron population and considering the double counting states.²⁰⁻²²

The standard TB approach consists of performing a full diagonalization of the Hamiltonian to obtain the exact band energy contribution. However, due to the cubic scaling of the diagonalization, the computational time increases rapidly for larger systems, which limits its use in large-scale Monte Carlo (MC) or molecular dynamics (MD) simulations. In addition, in configurations with broken symmetries where atoms are shifted away from their perfect bulk positions a significant number of iterations may be necessary to achieve the magnetic convergence.²³ The expensive diagonalization can be circumvented by the moment²⁴ or recursion methods.²⁵,²⁶ By limiting the numerical calculations to the first moments of the local density of states (DOS) only, a significant speed up can be achieved.²⁷ As a result, the are various TB models in the literature, ranging from very accurate spd-NOTB parametrizations employing exact diagonalizations to the simplest d-OTB models evaluated either via the fourth moment approximation (FMA) or the bond-order potential (BOP) formalism.

### 2.2.2 Bond order potentials

The TB bond model²⁸ can be seen as coarse-grained description of the electronic structure that is obtained by first-order expansion of the DFT functional in terms of the electron density.²⁹,³⁰ The TB models can be further coarse-grained to BOPs¹⁵,³¹,³² by making use of the recursion method for solving the Schrödinger equation.³³ In this way the BOPs replace the computational bottleneck of TB calculations, i.e., the diagonalization of the Hamiltonian in reciprocal space, by a linear-scaling real-space calculation of an approximate electronic DOS for the underlying TB problem (see, e.g., References 32,34 for computational details). The approximate solution at the TB/BOP level is represented by the moments of the DOS that provide a direct link between the local atomic structure and the local electronic structure.³⁵,³⁶ The resulting BOP methodology provides a robust

and physically transparent model of the interatomic interaction for significantly reduced numerical effort.

The TB/BOP models contain adjustable parameters that have been successfully tuned to describe the structural stability of crystalline phases and defects of several transition metals. The treatment of Fe and its alloys at the TB/BOP level requires to account for the effects of magnetism that are usually included by a Stoner model³⁷ (cf. Equation (5)). The resulting contribution of the magnetic energy arises within the self-consistency cycle of the onsite levels of the TB/BOP Hamiltonian. The BOP formalism can treat collinear arrangements of atomic spins²²,³⁸ as well as non-collinear arrangements.³⁹ The existing BOP models for Fe²²,³⁸,⁴⁰,⁴¹ and Fe–Co alloys⁴² provide a reliable description of the respective ground-states and defects³⁸,⁴³ as well as the magnetic⁴⁴ and structural⁴⁵ phase transitions of Fe, as outlined in the upcoming chapters.

## 2.3 Effective interaction models

### 2.3.1 Heisenberg-like models for spin dynamics

To handle effective magnetic interactions between atoms, different classes of model Hamiltonians are discussed in materials science (see⁴⁶ for a recent review). If the magnetic moments of individual electrons are considered, then the Hubbard⁴⁷ (cf. Equation (9)) and the Anderson⁴⁸ models are established approaches. In 3d-transition elements like Fe and its alloys, the magnetism is determined by the set of itinerant electrons that are strongly localized to an atom. Mathematical approximations like the limit of large coupling constants in the Hubbard model will then give rise to the notion of average local moment operators. The corresponding model Hamiltonian is the Heisenberg model, which can be formulated in a classical or a quantum-mechanical version:

$$ \mathcal{H}^{\text{mag}} = -\sum_{i,j \neq i} J_{ij} (R_{ij}) \mathbf{M}_i \cdot \mathbf{M}_j = -\sum_{ij} \tilde{J}_{ij} \mathbf{S}_i \cdot \mathbf{S}_j \quad (6) $$

Here $\mathbf{M}_i$ represents the classical magnetic moment of the i-th atom and $\mathbf{S}_i$ denotes a quantum-mechanical spin operator. The $J_{ij}(R_{ij})$ is the distance-dependent magnetic exchange parameter, which can be derived from DFT calculations by fitting the energies of different magnetic configurations. More rigorously, a second-order perturbation theory approach can be applied, where the $J_{ij}$ parameters are taken as the second derivative of the total energy of a magnetic system with respect to the relative orientations of the moments $\mathbf{M}_i$.⁴⁶ Alternatively, it is possible to calculate the energy of spin-wave excitations for certain wave vectors in DFT
and perform a Fourier transformation in order achieve the real-space $J_{ij}$ representation.⁴⁹

Given the Hamiltonian (6), the spin dynamics (SD) of the magnetic system can be determined by the Landau-Lifshitz-Gilbert equation,⁵⁰,⁵¹ where the effective field acting on the other magnetic moments is given by the derivative $-\partial H/\partial \mathbf{M}_i$. This approach gives rise to magnetisation curves and Curie temperatures. In more advanced studies the dynamical coupling of spin and lattice excitations and its impact on the properties of materials has been investigated.⁵² To this end, the MD snapshots give rise to the distances $R_{ij}$ that determine the exchange parameters $J_{ij}$ entering the SD, whereas the magnetic snapshots determine the atomic forces relevant for the MD time step.

### 2.3.2 Lattice based effective interaction models

The temperature evolution of phase stability, equilibrium point defect concentration and atomic diffusion coefficients can be determined by lattice-based effective interaction models⁵³ combined with either equilibrium or kinetic MC simulations. Such an on-lattice kinetic Monte Carlo (kMC) approach is particularly powerful for the simulation of processes (e.g., precipitation, phase ordering) driven by thermal activated events such as atomic diffusion.⁵⁴⁻⁵⁷ If the EIM contains explicitely both chemical and magnetic variables, it also allows to gain insights into the effects of magnetochemical interplay behind the thermodynamic and kinetic properties. The EIM parameters are obtained via fitting on DFT data.

In the present study, the discussion of EIMs refers to the following Hamiltonian, where magnetic interactions are described via a generalized Heisenberg approach, allowing both longitudinal and transversal spin fluctuations. In addition, a pairwise “chemical” (or nonmagnetic) interaction term is included. In the application of these EIMs in kMC simulations, both the atomic and the magnetic variables can be followed. For the latter, a quantum-statistic treatment of spin is necessary for a correct prediction of the magnetic entropy at low temperatures. Therefore, the Bose-Einstein statistics in the spin-MC simulations of Reference 58 up to the Curie point is adapted, following the quasi-harmonic approach of Reference 59.

$$ H = \sum_{i}^{N} A_{i}^{(S)} M_{i}^{2} + B_{i}^{(S)} M_{i}^{4} + \sum_{i}^{N} \sum_{n}^{P} \sum_{j}^{Z_{n}} J_{ij}^{(n)} \mathbf{M}_{i} \cdot \mathbf{M}_{j} + \sum_{i}^{N} \sum_{n}^{P} \sum_{j}^{Z_{n}} V_{ij}^{(n)} \sigma_{i} \cdot \sigma_{j} \quad (7) $$

Here, the third term is the lattice based version of the Heisenberg model (6) with $N$ the total number of atoms, $P$ is

the maximum range of interactions in terms of neighbor shells, $Z_n$ is the coordination number of the $n$-shell. $V_{ij}^{(n)}$ and $J_{ij}^{(n)}$ represent respectively the chemical pair-interaction and the magnetic exchange-coupling parameters between sites $i$ and $j$, as $n$-shell neighbors. The interaction parameters $V_{ij}^{(n)}$ and $J_{ij}^{(n)}$ depend on the chemical species of atoms $i$ and $j$, and can be functions of the local chemical environment. $\sigma_i$ is the occupation variable of the $i$-th site. It is 1 if there is an atom on the site, and 0 if there is a vacancy. $A_i^{(S)}$ and $B_i^{(S)}$ are the magnetic on-site parameters of the $i$-th atom, where "S" indicates the distance between the site and a vacancy. These onsite parameters for the first and second nearest neighbors of a vacancy differ from the values for a bulk atom. This review discusses parameterization of such model Hamiltonians for bcc Fe-Cu and Fe-Mn systems. For bcc Fe-Co alloys, the Hamiltonian just consists of a sum of two terms: a classical Heisenberg and a chemical pair interaction term, where the atomic magnetic moments are functions of the local chemical environment. All the corresponding parameter values, as well as a description of the parameterization procedure are given in References 58,60–62.

## 2.4 Spin dynamics based on classical potential

Classical simulation of a magnetic material can be conveniently divided into two components: (1) modeling of the ground magnetic state (GMS) with classical interatomic potentials and (2) an additional spin contribution describing the temperature-induced magnetic excitation.

### 2.4.1 Classical interatomic potentials

The description of the interatomic interaction in Fe using classical potentials has the evident limitation that the existence of the various magnetic states and the complex phase diagram can hardly be captured by these simple models. Nevertheless, they may still provide valuable insights into the common features of diffusion processes in pure Fe and Fe-based alloys provided that the relevant thermodynamic and vibrational properties of defects are accurately reproduced. A large number of interatomic potentials has been reported for atomistic simulation of Fe and its alloys.⁶³⁻⁷² Some of these models attempt to include a rudimentary description of magnetic interactions.⁷³,⁷⁴ To illustrate an applicability of such classical approaches, we present here results obtained using three different classical potentials:
- a widely used embedded-atom method (EAM) potential for Fe developed by Ackland et al.,⁷⁵ which was later extended for the binary Fe–Cr system;⁷⁶

- an EAM-based potential for pure Fe developed by Chiesa et al.⁶⁹ (version CS3-33) which incorporates an approximate description of magnetic contributions using an additional embedding term (magnetic EAM (mEAM));

- an angular-dependent potential (ADP) for Fe–Cr developed by Starikov et al.⁷⁷,⁷⁸ based entirely on DFT calculated data using a machine-learning-like force-matching method.⁷⁹,⁸⁰ The idea of the force-matching method is to adjust the potential functions to optimally reproduce DFT energies, forces and stresses for an optimally selected set of reference structures.

These potentials illustrate the different types of classical approaches to model magnetic Fe systems. The EAM potential may be considered as the simplest model that describes Fe over a wide range of temperatures, but without distinguishing between the FM and PM states. The ADP parametrization aims to capture accurately the properties of Fe in the GMS where magnetic excitations are negligible. The mEAM potential includes analytic functions to mimic the magnetic description of FM bcc Fe. MD and static simulation using ADP and mEAM models should be considered as the modeling of iron in the GMS.

## 2.4.2 Classical spin dynamics

Qualitative description of the temperature-induced magnetic excitation, can be achieved by the combination of MD based on the ADP potential with classical SD.⁷⁴,⁸¹,⁸² For instance, such coupled MD+SD simulations are nowadays possible using the LAMMPS code.⁸² The atomic energy for such coupled systems can be written as a sum of three contributions:

$$U_i = U_{\text{clas}} - \sum_{j \neq i} J_{ij} (R_{ij}) \mathbf{M}_i \cdot \mathbf{M}_j + \sum_{j \neq i} J_{ij} (R_{ij}) M_i M_j \quad (8)$$

where $U_{clas}$ is represented by a classical potential, fitted to the GMS. $J_{ij}(R_{ij})$ is the magnetic exchange paramater as defined in Equation (6), and $M$ is an atomic moment vector as defined by the Landau–Lifshitz–Gilbert equation.⁵⁰ The last term in Equation (8) compensates for the magnetic contribution in $U_{clas}$. The spin contributions in Equation (8) describing the magnetic excitations become significant only at elevated temperatures (approximately at $T > 0.5T_c$ for Fe) since at low temperatures $M_i M_j/M_i M_j \approx 1$ and $U_i \approx U_{clas}$. Here, we present several results of MD-SD simulations of $\alpha$-Fe in FM/PM states to illustrate some fundamental features of such an approach.

## 2.5 Alternative simulation methods

The above four classes of methods are the focus of this paper and their performance to describe magnetism in Fe-based materials will be discussed in the coming chapters. It goes without saying that the present study cannot cover the entire range of methods that could also be applied to these topics. In the following, however, some alternatives are briefly mentioned.

### 2.5.1 DMFT and many-body theories

Since magnetism is a quantum-mechanical effect that is determined by the electronic structure of the materials, methods that can accurately handle electronic correlation effects are particularly well suited to predict the impact of magnetism on materials properties. One example is the random-phase approximation (RPA),⁸³ which is briefly addressed in Section 3.1.2. Dynamical mean-field theory (DMFT) belongs to the most advanced electronic-structure methods employed in materials science.⁸⁴⁻⁸⁶ It connects the solution of a lattice model that is solved with mean-field concepts to an impurity model that can be accurately solved, e.g., by quantum Monte-Carlo methods. The most fundamental lattice model used in DMFT approaches is the Hubbard model:⁴⁷

$$\mathcal{H} = t \sum_{\langle i,j \rangle} \sum_{\sigma} c_{i\sigma}^{\dagger} c_{j\sigma} + U \sum_{i} \hat{n}_{i\uparrow} \hat{n}_{i\downarrow} \quad (9)$$

in which $t$ describes the hopping integral between two neighboring sites $\langle i, j \rangle$ and $U$ the Coulomb interaction between two electrons of opposite spins at the same site $i$. The operators $c_{i\sigma}^{\dagger}$, $c_{j\sigma}$, and $\hat{n}_{i\sigma}$ are the creation, annihilation and occupation number of an electron, respectively. The corresponding impurity model used for the mapping is typically the Anderson model,⁴⁸ which contains the Hubbard $U$ interaction only at a single site and a hybridization between this impurity level and an otherwise uncorrelated energy band. Several extensions of this model exist, including the coupling of the lattice Hamiltonian to DFT to capture realistic band structures.

In order to take magnetic disorder into account, a common approach is the combination of DMFT with the CPA, e.g., implemented within the KKR method (cf. Section 2.1.2). The method has, for instance, been used to describe structural $\gamma - \varepsilon$ phase transitions in Fe–Mn alloys including their dependence on the Mn content⁸⁷ as well as the dependence of the Curie temperature on the Mn content.⁸⁸ The advantage of these methods is that electronic and magnetic excitations and fluctuations are handled on the same footing, i.e., magnetic disorder is self-consistently taken into account.
However, even the extension to the more advanced Local Density Approximation (LDA)+DMFT approach, has several shortcomings for Fe-based alloys. One is the substantial overestimation of the Curie temperature (1900 K instead of 1043 K);⁸⁹ another one is the missing access to an accurate description of forces; and a third one is the huge computational effort, which makes the handling of structurally and chemically complex alloys not feasible.

### 2.5.2 Machine learning potentials

In the past 15 years, a new class of machine learning (ML) potentials has emerged and quickly became tremendously successful. Most ML parametrizations aim to provide numerically efficient surrogate models of the atomic interactions that are parameterized based on large, typically DFT-computed, datasets. The ML potentials employ advanced interpolation algorithms so that they achieve errors with respect to the reference data of only a few meV/atom while being orders of magnitude faster than the reference DFT computations. Therefore, they promise simulations of thermodynamic and kinetic material properties at near DFT-accuracy. A common disadvantage is very poor extrapolation and hence transferability to configurations not included in the training dataset.

The extension of ML potentials to include magnetic degrees of freedom is at present an active field of research. First publications based on a neural network potential (NNP)⁹⁰ and a moment tensor potential (MTP)⁹¹ are available. The extension of the atomic cluster expansion (ACE)⁹²,⁹³ to include magnetism allows for a fully non-collinear treatment of magnetic moments.⁹⁴,⁹⁵

## 2.6 Experimental methods

In the present manuscript, dedicated experimental results are added to complete the understanding of magnetic effects in Fe alloys and to validate the modeling strategy. Hence, the upcoming selection of experimental methods is limited to those methods, for which results are shown in the upcoming chapters, and does not claim to be exhaustive.

### 2.6.1 Ferromagnetic Nuclear Resonance

Ferromagnetic Nuclear Resonance (FNR) refers to Nuclear Magnetic Resonance (NMR) when implemented for studying ferromagnetic materials (it is also sometime called zero field NMR or Internal Field NMR).⁹⁶ The NMR measurements shown in the present study have been performed at 4 K by

frequency sweepings with a homemade state of the art automated broadband pulsed spectrometer with phase sensitive detection.

The direct output of FNR is the nuclear spin resonance frequency that is proportional (through the nucleus gyro-magnetic ratio) to the magnetic field on the probed nucleus site. The main contribution to the field experienced by the nuclei is the hyperfine field (HF) that results from the spontaneous magnetisation of the sample.⁹⁷ Although no unique relationship can be defined between the HF and the magnetisation of the probed atom, it is believed that the trends revealed by the FNR spectra are reflecting the trends of the magnetisation carried by the probed atom.

Because of the small natural abundance of Fe atoms with non-zero nuclear spin nuclei ($^{57}$Fe), Fe is not a very efficient element to perform FNR experiments. However, if Fe constitutes the matrix of the samples it is possible to perform Fe FNR measurements. Co and Mn are very efficient elements for FNR and experiments can be easily carried out even if the concentration of these elements is very small. The small gyromagnetic ratio of the $^{53}$Cr nuclei does not make it possible to perform FNR measurements. Indeed, it would request a very large hyperfine field to make its resonance frequency larger than the lower limit of the bandwidth of typical FNR spectrometer (20–700 MHz).

### 2.6.2 Diffuse neutron scattering

Diffuse neutron scattering (DNS) is the best method to get information on atomic and magnetic properties in alloys because the neutron interacts with both the nuclei and the magnetic moments in matter. The atomic and magnetic distributions give rise to the diffuse scattering, a modulation of the background between the Bragg peaks related to their Fourier transform. To separate atomic and magnetic contributions, the best way is to combine the measurements with polarised and unpolarised incident beams, and to perform them using different external magnetic fields.

The experimental details for the DNS results presented in this paper can be found for example in References 98,99. Unpolarized neutron measurements were performed in zero magnetic field and with a magnetic field saturating the sample. Similar measurements were also performed using polarized neutrons in a saturating field, keeping the incident neutron polarization either parallel or anti-parallel to the field. Combining these measurements allows one to obtain the nuclear and magnetic cross sections separately, as well as the nuclear-magnetic interference term. The presented DNS experiments were performed at 8 K on the G6.1 diffractometer of the Orphée reactor in Saclay, equipped with a multidetector, with an incident neutron wavelength
of 4.73 Å. A vertical magnetic field of 1.5 T could saturate the sample magnetization in the direction perpendicular to the scattering plane. The details on measurements and analysis can be found in.¹⁰⁰

### 2.6.3 Transmission microscopy for visualization of magnetic domains

A basic understanding of magneto-chemical coupling, especially at defects, like grain boundaries or dislocations, requires direct visualization of local magnetic moments. Reliable three-dimensional measurements of the local magnetic moments with atomic resolution were recently shown to be practicable using electron magnetic chiral dichroism.¹⁰¹ The measurements are both, site specific and element specific and provide a high spatial resolution.¹⁰² Even in 3D systems, the magnetic fields can be visualized using tomographic imaging with about 80 nm resolution.¹⁰³ The local magnetic moments can be determined using polarized neutron diffuse scattering,¹⁰⁰ see previous subsection.

Differential phase contrast as a contrast mechanism can be used in conjunction with scanning transmission electron microscopy.¹⁰⁴ Local mapping of magnetic domains is performed here using a four-segmented detector of the on-axis bright filed/dark field detector.¹⁰⁵ Lorentz microscopy is also often used for understanding the physical and electromagnetic structures of magnetic nano-assemblies.¹⁰⁶

### 2.6.4 Calorimetric measurements

Reliable experimental measurements in magnetic materials and materials containing magnetic additions are generally challenging. The heat capacity is one of the fundamental thermodynamic quantities, which allows to experimentally quantify enthalpy, entropy and the Gibbs energy and thus enables quantitative thermodynamic modelling and its validation. The relevant experimental techniques are divided into low- and high-temperature measurements.¹⁰⁷ The low-temperature heat capacity is typically measured using a two $\tau$ method¹⁰⁸ by fitting the characteristic time constants for heating and cooling stages after application of a heat pulse. The measurement steps are repeated several times and an average value of the heat capacity is recorded. At higher temperatures, differential scanning calorimetry is used, often in combination with sapphire as standard material and it provides reasonably accuracy.¹⁰⁹ Applying an external magnetic field, the magnetic contributions to heat capacity can be determined, too.

### 2.6.5 Tracer diffusion

Reliable measurements of diffusion properties of magnetic materials are imperative for prediction of long-term response and optimization of thermo-mechanical treatments. Radiotracer diffusion is a unique and powerful method to access (and/or resolve) both bulk and short-circuit (grain boundary, dislocation, triple junction, etc.) contributions.¹¹⁰ Typically, a tiny amount of suitable radioisotope of a given element is applied to the sample surface and is allowed to diffuse as a result of heat treatment at the given temperature for the given time. Afterwards, the sample is sectioned parallel to the sample surface and the section's radioactivity (which is proportional to the diffusing element concentration) is measured by nuclear counting techniques.¹¹⁰ The method reveals highest sensitivity at the cost of a complete loss of lateral resolution.

Secondary-ion mass spectroscopy¹¹¹,¹¹² or atom probe tomography¹¹³⁻¹¹⁶ are used to trace the diffusing atom distributions with sub-micron or even nano-meter resolutions, respectively. Non-radioactive isotopes are used, e.g., the highly-enriched <sup>64</sup>Ni isotope with its abundance of about 0.9 % in natural nickel or suitable solute elements. In this way, one may directly image grain-boundary or triple-junction diffusion¹¹⁴,¹¹⁵ in polycrystalline materials, but with a reduced chemical resolution and natural limitations with respect to measurements of self-diffusion.¹¹⁷

## 3 Thermodynamics

As already outlined in Section 1, Fe alloys are characterized by a large variety of phases that determine their structural as well as functional properties. While the hcp $\varepsilon$ structure is stabilized by pressures above 12 GPa, a unique sequence of structural phase transformations from bcc to fcc and back to bcc is observed as a function of temperature. This specific behaviour is consequence of the magnetic nature of Fe. Only the magnetic order stabilizes the more open bcc $\alpha$ structure in the low-temperature regime, whereas in the magnetically disordered PM regime the bcc $\delta$ structure can only be stabilized against the fcc $\gamma$ structure at temperatures close to the melting point. It is therefore clear that the phase stability of Fe alloys critically depends on magnetic excitations.

This is a major challenge for the theoretical predictions of thermodynamic phase stabilities and properties of Fe-based alloys.¹ Even in empirical approaches for computational thermodynamics, such as Calphad, the treatment of the magnetic degrees of freedom is considerably less developed than for other entropy contributions.⁴⁶ Here, the early approach of Inden,¹¹⁸ which handles magnetism like a
chemical order-disorder transition, is still the state of the art. As a consequence, the λ-shaped discontinuity in the heat capacity of the second-order phase transformation is fitted below and above the Curie temperature by two different logarithmic functions. Similarly, within DFT the treatment of the magnetic entropy contribution to the Helmholtz free energy is much less clear as compared to the consideration of vibrational and electronic entropy contributions. In the following some of these challenges are addressed and the performance of the methods introduced in the previous chapters in this context is highlighted.

# 3.1 Elementary thermodynamic properties of Fe

## 3.1.1 Experimental determination of heat capacity

The heat capacity is a fundamental thermodynamic property, the knowledge of which helps to understand various thermodynamic physical properties of solids and to facilitate technological developments (cf. Section 2.6.4). Furthermore, it is used for benchmarking DFT-informed calculations (see, e.g., Reference 119). In magnetic materials, like α-Fe, the magnetic excitations could fully be analysed using heat capacity measurements with and without external magnetic field. As heat capacity measurements of dilute binary Fe alloys are rare, we provide in this paper unpublished measurements for the Fe–Mn system and use the repetition of pure Fe results as a benchmark for the quality of the measurements.

The measured low-temperature heat capacity of pure Fe (purity 99.99 wt.%) is shown in Figure 1a and compared with other literature data. A very good agreement of the results is seen, the minor deviations might be induced by residual impurities present in different materials. In the temperature interval from 2 to 400 K, the low-temperature heat capacities of Fe with Mn impurities turned out to be similar to that of pure Fe, Figure 1b. Some small deviations are seen at the temperatures of 20–40 K.

To go further, the magnetic entropy contribution to the measured heat capacity can be determined repeating the same measurements with an applied external magnetic field that yields magnetic saturation. The measured specific heat capacity curves are typically fitted using cubic β splines and the entropy, S(T), is calculated by integrating the specific heat capacity as follows:

$$S(T) = \int_{T_0}^{T} \frac{C_p(T')}{T'} dT' \qquad (10)$$

Having determined the entropy contributions without, S₀, and with applied magnetic field, Smag, the magnetic entropy is estimated as ΔSmag = Smag – S₀. Our preliminary results indicate, however, relatively small magnetic contributions to the total heat capacities of pure Fe and Fe–Mn alloys for temperatures below 400 K.

## 3.1.2 DFT calculations of Cₚ, phonon and magnon spectra

The heat capacity measurements shown in Figure 1a and b form a perfect benchmark to assess the quality of DFT approaches to determine temperature dependent free energies.¹²³ These combine electronic, vibrational, and magnetic effects as well as structural defects in an integrated approach. Established methods for the consideration of quasiharmonic lattice vibrations and electronic excitations based on Fermi smearing can also successfully be applied to Fe⁸³ and its alloys.

Most challenging is, however, to achieve accurate results for the magnetic entropy of magnetic materials, even for pure Fe. Several approaches have been suggested to describe transversal and longitudinal fluctuations with DFT-based approaches, some of which are summarized in Figure 2 and are described in more detail in Reference 1. Most of them are based on a mapping of magnetic interactions on the Heisenberg model in Equation (6). This requires the determination of the magnetic exchange integrals Jij between localized spins at lattice sites i and j, which can be the result of a fitting to a large number of DFT calculations for well-defined spin configurations. As mentioned in Section 2.3.1, an alternative is the calculation and subsequent Fourier transformation of the magnon excitation spectrum. This has the additional advantage that a direct comparison with experimentally measurable energies becomes possible. In Figure 3 such a comparison is provided for several methods, including DFT, the linear muffin-tin orbital (LMTO) method with the atomic sphere approximation (ASA) and TB-LMTO methods. While a detailed discussion of these results can be found in,⁸³ we note here that the vicinity of the Γ point is decisive for the spin-wave stiffness and most of the methods provide a reasonable accuracy in this region. The differences between the two lattice constants shown in Figure 3 are more important than those between the LDA and GGA for the same lattice constant, for example.

In Figure 2, one of the employed approximations is the mean-field (MF) solution of the (quantum-mechanical version of the) Heisenberg Hamiltonian (6). In this case, the individual (quantum-mechanical) spins do not interact with one another, but only with an effective field, SᵢSⱼ → Bₐₗₐₗᵢⱼ, where Bₐₗ = 2J₀⟨Sz⟩, J₀ = 1/N∑ijJij, and ⟨Sz⟩ is the
<figure><img src="image_5.png" /><figcaption>**Figure 1:** Heat capacity of pure Fe (a) and of Fe alloyed with small amounts of Mn (b), measured using a Physical Property Measurement System PPMS 6000 by Quantum Design. For pure Fe, the present results (open squares) are compared with the data of Kelley¹²⁰ (triangles down), Bendick et al.¹²¹ (circles) and Wallace et al.¹²² (triangles up).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 2:** Comparison of different methods (MF, RPA, cMC-classical MC and rMC-quantum MC, see text) to determine the magnetic contribution to the temperature dependence of the heat capacity of bcc Fe. All results contain the vibrational (vib) and electronic (el) entropy contributions. The vertical dashed line indicates the structural transition temperature (1,184 K). Adapted from Ref. [1].</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 3:** Calculated magnon spectra using planar spin spirals for two lattice constants, $a = 2.82$ Å (dashed lines) and $a = 2.92$ Å (solid lines), within GGA (black lines) and LDA (blue lines). For comparison the spectra obtained using a small cone angle and TB-LMTO employing GGA (red line) and TB-LMTO+ASA within LDA (orange lines). Experimental spectra have been measured at 10 K (open squares, ¹²⁴) and 300 K (Fe(12-at.% Si), open circles, ¹²⁵). Adapted from Ref. [83].</figcaption></figure>

magnetization. Though physical trends can be reproduced by MF,⁸³ the quantitative prediction of the Curie temperature is usually too high. The RPA is another analytic approach, which is based on a decoupling scheme of higher order Green's functions.⁸³ While for the magnetization and the Curie temperature it is sufficient to consider the transversal Green's function, $\langle \langle \mathbf{S}_i^\dagger ; \mathbf{S}_j^- \rangle \rangle$, the access to $\langle \mathcal{H}^{\text{mag}} \rangle$ and eventually the magnetic free energy, $F^{\text{mag}}$, also requires the determination of the longitudinal Green's function, $\langle \langle \mathbf{S}_i^z ; \mathbf{S}_j^z \rangle \rangle$. The magnetic moment can be related to the spin quantum number via $M = g\mu_B S$, where $g \approx 2$ denotes the

Landé factor. A corresponding RPA theory for spin quantum numbers beyond $S = 1/2$ has been proposed in.⁸³ Since the magnetic moment in itinerant magnets $M$ can be continuous, an interpolation between quantum solutions for half-integer spin quantum-numbers is required to apply such an analytical approach.¹²⁶

We further learn from Figure 2 that a treatment of the Heisenberg model with classical methods (e.g., MC) is insufficient, because quantum effects have a strong impact. To overcome this problem, an analytical approach has been developed. An alternative are MC simulations that either allow to consider quantum corrections to classical MC, for
which the so-called rescaled Monte Carlo (rMC) is a promising example,¹²⁷ or rely on quantum MC simulations of effective Hamiltonians.¹²⁸ The latter approach has also been applied to Cr¹²⁶ and to compounds such as cementite.¹²⁹ However, even for these methods substantial development is still required in order to apply them to disordered alloys.

It is an additional complication for the computation of phase stabilities that the assumption of an adiabatic decoupling of entropy contributions brakes down at elevated temperatures. In particular, we realized the importance of a magnetically driven modification of lattice vibrations not only in the paramagnetic limit,¹²,⁴⁶ but already close to the ferro-to paramagnetic transition.¹³⁰ The SSA technique provides a way to calculate the forces that enter the dynamical matrix in the PM limit. We have combine these results with the forces obtained from fully FM calculations by using a coupling parameter α, which is determined by quantum Monte-Carlo simulations for an effective nearest-neighbor Heisenberg model. In this way phonon spectra have been obtained that are in perfect agreement with neutron-scattering experiments for the full range of temperatures, at which the measurements have been performed (Figure 4).

<figure><img src="image_8.png" /><figcaption>**Figure 4:** Experimental (grey) and computed phonon density of states (left) and phonon dispersion (right) for bcc Fe at three different temperatures. The experimental data (symbols) are obtained from neutron scattering. The DFT results are performed with the SSA approach. The light-gray shading indicates the range between the PM limit (mixing parameter α = 0) and the fully FM solution (α = 1). Adapted from Ref. [130].</figcaption></figure>

### 3.1.3 TB calculations of magnetism, phonon spectra and phase stability

There exist several TB models for pure Fe²¹,¹³¹⁻¹³⁵ that differ mainly in the type and extent of the basis (orthogonal or non-orthogonal, *spd* or *d* only), the employed functional forms, and their parameterizations (cf. Section 2.2.1). The *d*-band orthogonal models²¹,¹³¹⁻¹³³ are computationally efficient and can describe reliably the stability of various magnetic Fe phases. However, the lack of explicit treatment of the *s*-electrons typically leads to overestimation of the atomic magnetic moments. The *spd*-NOTB models developed by Barreteau et al.²³,¹³⁴,¹³⁵ give excellent agreement with DFT even for magnetic properties including non-collinear spin effects, but they are computationally more expensive and can be applied to limited system sizes only. Examples of the electronic DOS from the *d*-band OTB and BOP models¹³¹ are shown in Figure 5. These figures demonstrate that the essential features of the electronic structure are correctly captured for both FM bcc and FM fcc Fe phases even by the simplest TB models coupled with the Stoner model. The DOS for the FM bcc phase has a characteristic bimodal shape while for the FM fcc phase it is more homogeneous. In both magnetic phases the spin-up and spin-down channels are shifted with respect to each other according to the Stoner criterion. The physical origin of TB models also leads to an appropriate variation of the magnetic moment as a function of volume. As shown in Figure 6, where BOP and TB-FMA calculations are presented, the magnetic moment correctly decreases with decreasing volume for all considered Fe phases (here bcc and fcc) and eventually vanishes.

Another great success of TB models is the correct prediction of phase stability, in particular, the stabilization of the FM bcc ground state over the close packed fcc and hcp phases due to magnetism. A comparison of energy-volume curves from the BOP and TB-FMA models is presented in Figure 6. The TB models are even sufficiently robust to predict the *α* (bcc) – *γ* (fcc) – *δ* (bcc) structural phase transitions of Fe with temperature in very good agreement with experiment.⁴⁵ It can thus be seen that a simplified TB model can be greatly beneficial to perform finite temperature simulations of large systems. In addition, most models are also able to reproduce with a good accuracy elastic and vibration properties (e.g., see phonon spectra in Figure 7) as well as energies, structures and local magnetic variations of various crystal defects, such as vacancies, interstitials, dislocations, surfaces, grain boundaries.³⁸,¹³⁶
<figure><img src="image_9.png" /><figcaption>**Figure 5:** Electronic DOS from d-OTB (grey area) and BOP (red line)¹³¹ for (a) FM-bcc Fe and (b) FM-fcc Fe.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 6:** Relative stabilities of different Fe bulk phases and dependencies of magnetic moment for bulk phases calculated by BOP (a) and TB-FMA (b).</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 7:** Phonon band-structure of (a) FM bcc Fe and (b) FM fcc Fe from *d*-OTB (black symbols) and BOP (red lines).</figcaption></figure>

### 3.1.4 Classical interatomic potentials: $C_p$, phonon spectra and thermal expansion

Large-scale MD (or coupled MD-SD) simulations with interatomic potentials enable to take into account directly the influence of temperature on the thermodynamic properties. As discussed for the TB models, the accuracy of a potential plays a fundamental role in such modelling. For a comparison of the

methods, the phonon spectrum by means of the small displacement method as implemented in the PHONOPY software package¹³⁷ is considered as a first test for three classical potentials. Figure 8 shows the phonon curves for bcc Fe. The obtained results reveal that all considered potentials describe the atomic vibrations in appropriate manner.

As mentioned above, the magnetic disordering at high temperatures can be taken into account in classical atomistic

<figure><img src="image_12.png" /><figcaption>**Figure 8:** Phonon spectra and DOS of bcc Fe predicted by three classical interatomic potentials. The black dots and the black line correspond to experimental data.¹³⁸</figcaption></figure>
simulations by the combination of MD and SD approaches. However, it should be noted that the magnetic subsystem is still described in a classical way, which leads to a good qualitative but not fully quantitative representation of the properties over a wide range of temperatures. Figure 9 illustrates the dependence of the total magnetic moment on the temperature obtained by the MD-SD simulations. As expected, the total magnetic moment decreases with temperature but its change is more gradual in comparison with the step-like trend seen in the experiments. Nevertheless, the Curie temperature for the simulated system may be estimated at about 900 K which is close to the experimental value of 1,043 K.

As mentioned in the DFT Section 3.1.2, another indicative feature of the magnetic phase transition is the dependence of the heat capacity ($C_p$) on temperature. The MD-SD results are shown in Figure 9. As expected, the classical MD simulation without the SD contribution predicts almost no temperature dependence of the heat capacity as the magnetic excitations are absent. The heat capacity calculated in the MD-SD simulation at low temperature is significantly overestimated because of the classical description of the magnetic subsystem (similar to cMC results in Figure 2). The value of $C_p$ at $T < 600$ K should be less than 30 J/K·mole¹²² due to the quantum suppression of the magnetic excitation. Nevertheless, the temperature dependence $C_p(T)$ shows the characteristic $\lambda$ peak near the Curie temperature similar to experimental data, although less peaked and shifted to lower temperature.

Lastly, Figure 10 shows the temperature dependencies of the thermal expansion coefficients calculated for the different models together with the measured values.¹³⁹ The comparison with the measured values reveals that the ADP potential reproduces the dependence better than the other models. The incorporation of magnetic excitations via the coupled MD-SD simulation leads to an almost constant shift of the classical prediction. This is most likely related to the absence of quantum suppression of the magnetic excitations as in the case of the heat capacity.

<figure><img src="image_13.png" /><figcaption>Figure 10: Calculated thermal expansion coefficient for different atomistic models in comparison with experimental data.¹³⁹ The arrow indicates the SD contribution.</figcaption></figure>

## 3.2 Mixing enthalpy of binary Fe alloys

### 3.2.1 DFT results for mixing enthalpies

Mixing enthalpies of bcc Fe–Cr, Fe–Mn and Fe–Co systems at 0 K have been determined in various DFT calculations.⁶²,¹⁴⁰⁻¹⁴⁷ They provide information on the mixing (or unmixing) tendency of the alloys, and about the relative stability of ordered phases and solid solutions. The mixing enthalpy of an Fe-X system (X = Cr, Mn or Co) is defined with the following expression:

$$H^{\text{mix}}(\text{Fe} - \text{X}) = \frac{H^{\text{tot}}(n\text{Fe} + p\text{X}) - nH(\text{Fe}) - pH(\text{X})}{n + p} \quad (11)$$

where $H^{\text{tot}}(n\text{Fe} + p\text{X})$ is the enthalpy of an Fe-X system, $H(Fe)$ is the enthalpy per atom of pure bcc Fe in its magnetic ground state (the FM state) and $H(X)$ is the enthalpy per atom of the pure bcc X system (here we consider the anti-

<figure><img src="image_14.png" /><figcaption>Figure 9: (left) Temperature dependence of the total magnetic moment: the black dashed line is the experimental data; the blue dots are the results using MD-SD simulations with the ADP potential. (right) Temperature dependence of the specific heat: the black dots represent experimental data;¹²² the blue dashed line is the result of a MD simulation only and the blue dash dot dot line of the MD-SD simulation.</figcaption></figure>
ferromagnetic (AFM) state, the anti-ferromagnetic double-layer (AFD) state, and the FM state for Cr, Mn and Co, respectively). In addition to some most relevant ordered structures, the SQS method,⁸ with negligible short-range orders, is adopted to mimic random solid solutions. SQS can also be used to handle magnetic disorder (cf. Section 2.1.2) and is required for rather small simulation cells with periodic boundary conditions in DFT calculations. The obtained values are shown in Figure 11.

Fe–Cr alloys exhibit an unmixing tendency (positive mixing enthalpy) for most compositions, except at very low Cr concentrations (below approximately 10%-Cr) where they show slightly negative mixing enthalpies. The latter can be explained by a stabilization of isolated Cr solutes in a FM-Fe lattice¹⁸,²³,¹⁴¹ due to strong Fe–Cr anti-ferromagnetism with an induced moment on Cr. The magnitude of the Cr magnetic moment in bcc Fe is around $1 \mu_B$ larger than its value in pure AFM bcc Cr. For a SQS system with a higher Cr concentration, it is unavoidable that some Cr atoms get close to each other. Therefore, the ferromagnetism between Fe atoms, the anti-ferromagnetism between Cr atoms, and the tendency of Fe and Cr moments to be antiparallel (especially for the 1nn and 2nn Fe–Cr pairs) cannot be simultaneously satisfied. Such a magnetic frustration contributes to the unmixing behaviour. Note that in the very dilute Fe region, the magnetic frustration also promotes a clustering of Fe atoms in the AFM-Cr matrix.¹⁴⁸

Regarding Fe–Mn alloys, it is well known that the bcc phase is thermodynamically stable only up to a few percent of Mn, while for most compositions there is an $\alpha + \gamma$ dual-phase domain. However, in order to have a thorough understanding of the compositional dependence of energetic and magnetic behaviors of bcc Fe–Mn alloys, it is interesting and relevant to perform a systematic study for all

concentrations. As can be seen in Figure 11, the obtained mixing energies are positive for all the concentrations (although smaller than the Fe–Cr values in the concentrated region), which reveals an unmixing tendency. Moreover, the shape follows well the regular solution model (described with $H^{mix} = A \chi_{Fe} \chi_{Mn}$) with $A = 192$ meV (18.5 kJ/mol). This is in good agreement with previous theoretical and experimental results, as detailed in Ref. [143] and references therein.

Concerning the magnetic properties, we look especially at bcc Fe–Mn alloys in the Fe-rich domain, up to 50 % of Mn. The magnetization in the SQS solid solutions shows a linear decrease with increasing Mn concentration, in very good agreement with the Slater-Pauling curve.¹⁴³ This behaviour is also consistent with available experimental studies, showing a linear decrease up to 11 % of Mn before a bcc-fcc phase transition.¹⁴⁹ This can be overall explained by a magnetic-dilution scenario. On one hand, as expected, the Fe moment remains close to 2.2 $\mu_B$ in any configuration of any composition, showing a very small perturbation of the bcc-Fe moment due to the presence of Mn atoms. On the other hand, the Mn moment shows smaller magnitudes. Due to the half-filled 3d-band of Mn, its magnetism is very sensitive to the local chemical environment.¹⁴³ Depending on the local configuration, the magnetic moment of Mn can be either antiparallel (AFM-Mn) or parallel (FM-Mn) to the Fe moments.

Having a closer look on the compositional dependence of the Mn magnetism,¹⁴³ the average Mn moment is antiparallel to Fe moments at the lowest concentrations, followed by a rapid decrease of the moment magnitude with increasing Mn concentration. This finally leads to a transition from AFM to FM Fe–Mn coupling tendency at approximately 6 % Mn (in SQS systems). After that, the average Mn magnetic moment converges to a value around $0.8 \mu_B$, parallel to Fe moments. This constant value of $0.8 \mu_B$ results from the presence of an approximately constant fraction of 75 % FM–Mn and 25 % AFM-Mn atoms in each SQS configuration, where both types of Mn present an average moment magnitude of around $1.6 \mu_B$ (that is, $1.6 \times 0.75 - 1.6 \times 0.25 = 0.8$).

At variance with the Fe–Cr and Fe–Mn alloys, the alloying of Fe with Co (having a higher 3d-band occupation than Fe) in a bcc lattice shows a mixing tendency for the whole range of compositions (see Figure 11). In both ordered and disordered structures, the Fe and Co magnetic moments tend to be parallel to each other. The moment magnitude of Co only varies slightly with the change of Co content, which is consistent with experimental observations. On the other hand and different from the Fe–Cr and Fe–Mn systems, the moment of Fe is strongly affected by the presence of Co solutes. In the SQSs, the average moment of Fe atoms increases rapidly from around $2.2 \mu_B$ to $2.5 \mu_B$ in the range from 0 to 20 %-Co Co, then it

<figure><img src="image_15.png" /><figcaption>**Figure 11:** Mixing enthalpy for bcc Fe–Cr, Fe–Mn and Fe–Co alloys from DFT calculations.</figcaption></figure>
varies slowly up to a saturated value of 2.6 $\mu_B$ and remains almost constant above 50 %-Co. In the ordered B2 phase, the magnetic moment of Fe increases with Co concentration and reaches a maximum at 50 %-Co before decreasing. It is worth noting that this stoichiometric B2 phase, showing the highest magnetization, is also the most stable phase of bcc Fe–Co at low temperatures.¹⁵⁰

### 3.2.2 TB and classical potential results for Fe–Cr mixing enthalpies

As previously highlighted, the mixing enthalpy anomaly in the Fe–Cr binary system results from the competition between ferro-and antiferromagnetism in Fe and Cr.<sup>18,141,151</sup> Despite its subtlety already discussed within DFT approaches, the TB models are able to capture the anomaly and provide a correct description of bulk as well as defective Fe–Cr systems.<sup>18,152,153</sup> As shown in Figure 12, the dependence of the mixing enthalpy is reproduced qualitatively well by different TB models for the whole range of concentrations, but the best description is obtained for the NOTB approximation. Apart from the stability of bulk phases, the developed model also provides a robust description of the energetics of small Cr clusters in Fe matrix as well as various crystal defects in the Fe–Cr systems.

Even though the classical atomistic models are not capable of distinguishing between the different magnetic orderings, a correct (at least qualitatively) description of the anomalous dependence of $H^{mix}$ on Cr concentration is desired also in classical simulations. Interestingly, both the EAM and the ADP interatomic potentials developed for the binary Fe–Cr system are indeed able to reproduce this feature. Figure 12 shows the calculated dependencies of $H^{mix}$ on the alloy composition. The reproduction of this unusual feature with classical potentials is possible due to the intrinsic many-body character of these two potentials.

### 3.2.3 EIM based MC simulations for mixing energies

We have determined mixing energies of bcc Fe–Cr, Fe–Mn and Fe–Co random alloys via MC simulations using the DFT parameterized EIMs. We have adopted the SQS method to represent the random solutions (Figure 13a–c). For each concentration and temperature, we performed spin-MC simulations with fixed atomic configurations. The obtained lowest-temperature values are in good agreement with the 0 K DFT results (Figure 11<sup>60,62,154</sup>).

The main advantage of the MC approach is to extend the DFT calculations and to study the mixing energies at finite temperatures. Figure 13a shows the Fe–Cr mixing energies at 10 K and 2,000 K. At 10 K, the system is found at the magnetic ground state and it is paramagnetic at 2,000 K for all compositions. The low-temperature curve is asymmetric due to the magnetism-driven anomaly at the low-Cr limit, as described in Section 3.2.1. As expected, this anomaly disappears with increasing temperature, making the high-temperature curve symmetric at the paramagnetic state. Interestingly, the height of both mixing-energy curves appears to be similar.<sup>154</sup> When comparing this result with literature data, the recent Calphad study by Xiong et al. concludes that the maximum of mixing enthalpies $H^{mix}$ near 50 % Cr decreases by only 0.015 eV from 0 to 1529 K,<sup>155</sup> while the difference is approximately 0.03 eV between the ideal FM and PM states according to DFT-DLM.<sup>156</sup> This effect is not reproduced by the present EIM, but it may be overestimated by DLM.

Regarding Fe–Mn, the temperature evolution of the bcc Fe–Mn mixing energy is closely related to the magnetization (see Figure 13b). Indeed, for any given concentration, two regimes can be clearly identified: below the magnetic transition temperature (see Figure 14), the mixing energy curve remains quite similar to the 0 K limit, while beyond the magnetic transition temperature, it converges to a high

<figure><img src="image_16.png" /><figcaption>**Figure 12:** Enthalpy of mixing of the Fe–Cr system as predicted by DFT, *d*-OTB, BOP and *spd*-NOTB (left) and EAM and ADP interatomic potentials (right) compared with DFT<sup>142</sup> (both).</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Figure 13:** Mixing energies of bcc (a) Fe–Cr, (b) Fe–Mn and (c) Fe–Co alloys, for various temperatures, as predicted by EIM + MC simulations.</figcaption></figure>

temperature paramagnetic limit. It can also be noticed that at both high and low temperature limits, that is, as soon as for all the concentrations the temperature is located either below or above the magnetic transition temperatures, the mixing-energy curve is symmetric. However, for the intermediate temperatures an asymmetry appears between the Mn-rich and the Fe-rich domains. Also, the Mn-rich mixing energies decrease faster with increasing temperature than the Fe-rich values. This asymmetry is consistent with the above-mentioned two-regime behavior, considering the difference between the Fe-rich and Mn-rich magnetic transition temperatures. Indeed, we predict the magnetic transition temperature to decrease with Mn concentration, in agreement with experimental data (see Figure 14).

We notice that the paramagnetic limit (2,000 K) of the mixing energy curve is very similar to the non-magnetic contribution of the 0 K mixing energy,⁶⁰ indicating that the mixing between Fe and Mn atoms has a negligible impact on the average magnitude of their respective magnetic moments. Overall, the present results suggest that spin dis-ordering favors the mixing of Fe and Mn, while spin ordering favors the phase separation tendency. It is worth mentioning that our results are in qualitative agreement with a previous Calphad prediction,¹⁶¹ in which the thermodynamic

<figure><img src="image_18.png" /><figcaption>**Figure 14:** Curie temperature versus Mn concentration of Fe–Mn random solutions: a comparison between EIM + MC simulation results and various experimental data.¹⁵⁷–¹⁶⁰</figcaption></figure>

parameters lead to a fully positive mixing energy of bcc Fe–Mn alloys which decreases with temperature.

In Figure 13c, the mixing energies of the disordered Fe–Co structures are presented for several temperatures: namely low (1 K), intermediate (700 K) and high (1,000 K, 1,500 K) temperatures. For all the studied temperatures, the mixing energies remain negative. The mixing energy curve
presents a minimum around 40 % Co at 1 K and this minimum slightly shifts to a lower concentration of Co when temperature increases. More specifically, the mixing energies decrease slightly when the temperature increases from 1 K to 700 K. At higher temperatures, the variation is more visible as the mixing energies drop faster when the temperature varies from 700 K to 1,000 K. Interestingly, the trend is reversed as the mixing energies increase with temperature from 1,000 K to 1,500 K in the paramagnetic regime.

These temperature-dependent variations can be understood in terms of distinct Curie temperatures for systems of different compositions. As an example, we consider the 50%-Co system: We look at the change of total energy of the systems in the calculation of the mixing energy (pure Fe, pure Co, and Fe-50% Co A2) with respect to their respective 1 K value.⁶² At low temperatures, the change of energy in all the three systems is almost the same, it leads to the almost unchanged mixing energy. In the range of 700 K – 1,100 K, the change in the energies of pure Fe and pure Co is faster than that in the Fe-Co system, due to the strong magnetic disordering near the magnetic phase transition in the pure systems (1,065 K for Fe and 1,077 K for Co). As a result, the mixing enthalpy decreases. Above 1,100 K, the energies of the pure Fe and Co systems tend to saturate, while the one of the Fe-Co grows up more rapidly as this system goes close to its Curie point and thus the mixing energy increases.

## 3.3 Magnetic and chemical transitions in Fe and Fe alloys

### 3.3.1 TB determination of Curie temperature in iron

The numerical efficiency of the TB models makes it possible to consider relatively large system sizes and to perform finite temperature simulations. A good illustration is their use to determine the Curie temperature ($T_C$) as crucial quantity in magnetic systems. Remarkably, it turns out that interatomic potentials based on reliable TB models are well suited for determining the Curie temperature. As can be seen in Figure 15a, BOP gives an estimate of the $T_C$ of iron around 1,200 K in good agreement with the experimental value of 1,043 K.⁴⁴ Moreover, it is possible to go further by characterizing the evolution of the local magnetic moment distribution as a function of the temperature. This can be done with a TB-FMA model where the calculations are sufficiently fast for obtaining average quantities accurately with extensive statistical sampling.¹⁶²,¹⁶³ As an example (see Figure 15b), the case of fcc Fe is considered where the Curie temperature calculated within the TB-FMA model is close to

structural phase transformation.¹⁶³ To go deeper, we consider the distribution of the local magnetic moments in the paramagnetic and ferromagnetic regimes. At low temperature, the magnetic moment distribution is characterized by a narrow Gaussian distribution centered around 2.30 $\mu_B$. When the temperature is increased, the Gaussian distributions become a little wider due to thermal fluctuations. Above the Curie temperature, the magnetic moment distribution is wider and its amplitude is lower, as expected in the PM state. Interestingly, the TB-FMA interatomic potential coupled with specific MC magnetic moment tests proves to be particularly effective in obtaining randomly distributed collinear magnetic moments in a straightforward manner. This is a direct result of the simulations without specific hypothesis in the TB model.

### 3.3.2 EIM predicted phase transitions in bcc Fe–Mn and Fe–Co

On-lattice MC simulations using the developed EIM allows us to predict various thermodynamic properties of the binary bcc Fe alloys. In the following, we present two examples: the concentration-dependent Curie temperature in Fe–Mn,⁶⁰ and the magnetic and chemical phase boundaries in Fe–Co.⁶²

First, we investigated the Mn concentration dependence of $T_C$ in dilute Fe–Mn alloys, and compared the results with experimental data. At each Mn concentration, the Curie temperature was determined as the inflection point of the magnetization-versus-temperature ($M(T)$) curve.¹⁶⁴ The obtained values are shown in Figure 14. As can be seen, $T_C$ decreases with Mn concentration with a slope of approximately 10 K per Mn at.%, in excellent agreement with most literature data. Indeed, various experimental works have shown that $T_C$ tends to decrease in the dilute limit linearly with Mn concentration, at a rate of approximately 10 K per Mn at.%,¹⁵⁷⁻¹⁶⁰ as shown in Figure 14.

For intermediate compositions, up to 20 at.% of Mn, some discrepancy in experimental data appears. While there is still a decrease of 10 K per Mn at.% in the case of Paduani et al.,¹⁵⁷ Yamauchi et al.¹⁴⁹ found a larger decreasing slope of around 43 K per Mn at.%. One possible explanation for the deviation of Yamauchi et al. is that their magnetic measurements are biased by the use of cold-rolling on the samples in order to stabilize the body-centered cubic phase,¹⁴⁹,¹⁶⁵ which is not the case in the other experimental studies. A recent Calphad assessment¹⁶¹ also assumes such a decrease of $T_C$ with Mn concentration at a rate of approximately 10 K per Mn at.%, as shown in Figure 14.

Finally, magnetic and chemical phase diagram of bcc Fe–Co systems are considered. In this case, Curie
<figure><img src="image_19.png" /><figcaption>**Figure 15:** Performance of interatomic potentials based on reliable TB models. (a) Temperature-dependent contribution of magnetism to the specific heat of bcc Fe from simulations with a magnetic BOP with a resulting Curie temperature of approximately 1,200 K. (Reprinted from Reference 44). (b) Curie temperature of fcc Fe from TB-FMA calculations. (c) Local magnetic moment of 500 configurations at different temperatures: below $T_c$ (top) and above $T_c$ (bottom). (Reprinted from Reference 162).</figcaption></figure>

temperatures were determined to be respectively 1,316 K and 1,461 K for the 50%-Co A2 and B2 structures. Systematic calculations were also performed for other Co concentrations. The results allow us to predict the magnetic phase diagram of bcc Fe–Co alloys (Figure 16). The experimental data extracted from References 166–168 were also added for a comparison. However, these experimental Curie temperatures may not always come from direct measurements, but obtained by data extrapolations.¹⁶⁸ They roughly coincide with the bcc → fcc transition temperatures for a large range of Co concentrations. As can be seen, the Curie points obtained by our model via MC simulations are in good agreement with the experimental data, with the highest discrepancy not exceeding 100 K, which is within a reasonable margin of error using an effective model. The obtained result is also compatible with the equilibrium phase diagram, in which the bcc (B2 and A2) phases present a FM state while the fcc phase presents a PM state. Our predicted $T_c$ is found to increase significantly, from around 1060 K to above

<figure><img src="image_20.png" /><figcaption>**Figure 16:** Curie temperature and B2-A2 transition temperature in bcc Fe–Co alloys: a comparison between EIM + MC simulation results and various experimental data.¹⁶⁶–¹⁶⁹</figcaption></figure>

1220 K, in the range from 0 to 20 % Co, then it reaches the maximum value of about 1340 K at 40 % Co before decreasing to about 1200 K for 70 % Co.
According to the equilibrium phase diagram, the ordered-B2 and the disordered-A2 phases, both FM, are separated by a second-order transition.¹⁶⁹ We have then determined the phase boundary between these two phases, via a determination of the temperature-dependence of the chemical long-range order (LRO) at each Co concentration.⁶² To this end, coupled spin-atom MC simulations were performed.

It is worth mentioning that the present model, like most on-lattice EIMs, does not include the contribution of lattice vibrations. Nevertheless, our model still provides a satisfactory prediction of the B2-A2 transition temperature. This is because in Fe-Co systems, the vibrational entropies of the B2 and the A2 structures are found to be very similar. For instance, 3.25 $k_B$, per atom for the B2 and 3.31 $k_B$ for the A2 phase, at 50 % Co. Indeed, for this transition, the effect of the lattice vibrations is negligible compared to that of magnetic excitations.⁶² Together with the predicted Curie temperatures, the present model, although rather simple, is able to predict simultaneously very satisfactory magnetic and chemical phase diagrams for a broad range of temperature and alloy concentration (Figure 16).

# 4 Point and extended defects

The thermodynamic bulk phase diagrams (cf. Section 3) can be used as a roadmap, in which state variables such as temperature and composition are used to realise phase mixtures for materials with desirable structural or functional properties. Similarly fundamental concepts of lattice defects are important for the design of structural materials and can be used to control mechanical performance. The relevant defects include vacancies, solutes and segregated atoms, dislocations, grain and phase boundaries, as well as surfaces, including those at cracks and pores. While the impact of magnetism for the bulk thermodynamics is already an established concept, this is much less the case for the defects. In the following chapter, the mechanisms are discussed for selected examples and insights into this topic are introduced. The focus on point defects prepares the considerations on diffusion (cf. Section 5). In the case of extended defects, the magneto-structural coupling is more pronounced and will be therefore highlighted as well.

## 4.1 Impact of solutes on magnetism

### 4.1.1 Experimental FNR results in diluted Fe alloys versus DFT magnetic moments

Simulated results on the impact of Cr, Mn and Co on the magnetism in Fe alloys have already been discussed in

Section 3.2.1 and Section 3.2.3 in order to interpret the mixing enthalpies provided there. Here, we add experimental insights into the influence of solutes on the neighboring Fe magnetic moments, which have not yet been published elsewhere. To this end, FNR experiments⁹⁶ have been performed on diluted FeX (with X = Co, Cr, Mn) alloys. The magnetic moments induced by the Fe matrix on the impurities have also been probed when possible (for Co and Mn).

Following a procedure established for alloys with Co matrices,¹⁷⁰,¹⁷¹ FNR spectra for FeCo diluted samples have been obtained. The Fe FNR spectra (left panel in Figure 17a) are dominated by a line at 46.7 MHz that corresponds to the pure bulk Fe resonance frequency, i.e., with “0” Co first nearest neighbor (1nn) atoms. Increasing the Co content from 0.1 to 4 % the high frequency contributions increase in intensity. Only one resolved contribution is observed in the high frequency range at 47.2 MHz (labelled “1”) and is attributed to the presence of one Co atom among the 1nn Fe atoms. In contrary to¹⁷⁰ no resolved FNR contribution corresponding to 2 or more Co atoms among the 1nn Fe atoms can be identified. This might be due to the similar distance of the six second nearest neighbor (2nn) and the eight 1nn atoms of Fe in the bcc structure, which can result in a large variety of local HF signals and the large distribution of resonance frequencies in the unresolved tail. The analysis of the spectra suggests that when Co atoms are situated in the 1nn environment of Fe atoms they always increase the magnetic moment of the Fe atoms. This is consistent with the FeCo macroscopic magnetisation measurements that show an increase of the samples magnetic moments up to 20 % of Co content.¹⁷² DFT-generalized gradient approximation (GGA) calculations also indicate an increase of Fe magnetic moment of around 5 % if located at a 1nn site of a Co atom (Section 3.2.1).¹⁵⁰

The Co spectra in the right panel of Figure 17a are dominated by the line at 288.7 MHz corresponding to single Co atoms diluted into the Fe matrix (labelled as “1”). With increasing Co content from 0.1 % to 4 % satellite lines increase in intensity and alternatively on each side of the dominant line. From the evolution of the intensity of the satellites with the Co concentration, the contributions labelled “2”, “3” and “4” can be attributed to Co pairs, triplets and small clusters of 4 Co atoms, respectively. The observations suggest that the Co magnetic moments oscillate significantly (several %) with the clustering of the Co atoms.

The FNR spectra recorded for the FeMn samples shown in Figure 17b again contain a main line attributed to pure Fe. Increasing the Mn content, satellite lines increase in intensity at lower frequencies around 44 MHz (1 Mn atom) and 41 MHz (2 Mn atoms). In opposition to the FeCr sample, lines 1 and 2 seem to be composed of a single contribution and no sign of distinct contributions originating from Mn atoms as
<figure><img src="image_21.png" /><figcaption>**Figure 17:** FNR spectra of magnetic elements and diluted alloys. (a) Fe (left) and Co (right) FNR spectra of FeCo diluted alloys. (b) Fe (left) and Mn (right) FNR spectra of FeMn diluted alloys.</figcaption></figure>

1nn or 2nn can be observed. The Mn FNR spectra (Figure 17b right panel) show a main line at 240 MHz attributed to isolated Mn atoms in Fe (label 1). Increasing the Mn concentration, low frequency lines increase in intensity (label 2 and 3) that are attributed to Mn pairs and triplets respectively. These FNR spectra suggest that each Mn atom in the vicinity of an Fe atom decreases the magnetic moment of Fe and a clustering of Mn atoms also decreases their magnetic moments. This finding is in contradiction to some experimental and theoretical (Section 3.2.1)¹⁴³ reports that in the studied concentration range the Mn atoms could show a ferromagnetic as well as an antiferromagnetic coupling with the Fe atoms, which should result in two different sets of lines.

To study the sign of the magnetic coupling between the Fe and Mn atoms a Mn FNR spectrum of the sample containing 2 % Mn under an external static field (Figure 18b) is used. The comparison with the FNR spectra of the FeMn 2 at.% sample recorded in zero field shows that the under field spectrum does shift as a whole towards lower frequencies. This means

that compared to the direction of the applied field the HF points in the opposite direction, and hence the magnetic moment points (due to general rules for transition metals⁹⁷) in the same direction. This suggests that the magnetic moment of Mn atoms observed in these samples are ferromagnetically coupled to the magnetic moment of the Fe atoms. If comparing the present experimental data with DFT-GGA results (Section 3.2.1),¹⁴³ the latter also show a decrease of Fe moment (of around 2 %) due to the presence of a 1nn or a 2nn Mn atom that presents a spin parallel to the Fe moments. However, as already mentioned above, DFT predicts that the FM-Mn is slightly higher in energy (0.065 eV) than the AFM state of Mn. But, this relative energy is very sensitive to the local chemical environment of the Mn atom.¹⁴³

As seen in Figure 18a the Fe FNR spectrum for the FeCr 2 at.% is again dominated by a contribution close to 46.7 MHz attributed to Fe atoms with “0” Cr 1nn. The vicinity of Cr atoms strongly decreases the HF of Fe atoms as observed by the lines around 43 MHz, suggesting a decrease of the Fe
<figure><img src="image_22.png" /><figcaption>Figure 18: FNR spectra of Fe-based alloys. (a) Fe FNR spectrum of a FeCr 2 at.% alloy. (b) Comparison of zero field and in static field FNR spectra of the FeMn 2 at.% sample.</figcaption></figure>

magnetic moments. The contributions at 43 MHz are actually constituted by two lines, one at 43.8 MHz and one at 42.5 MHz. Since they are close in frequency position, they are both attributed to one Cr atom among the Fe neighbours (labelled 1 and 1' in Figure 18a).

Since the probability of a Cr atom in one of the eight 1nn sites will be a little bit larger than one in the six 2nn sites, it is tempting to attribute the slightly more intense line at 43.8 MHz to Cr atoms among the eight 1nn, though this imposes that the 2nn Cr atoms have a stronger impact on the Fe HF field. Interestingly, this result is in qualitative agreement with the DFT-GGA data, that is, a 2nn Cr atom decreases the Fe moment by around 1%,¹⁰⁰ while 1nn Cr has a similar impact, but the decrease is negligible.

### 4.1.2 Experiments: local magnetic moments from neutron diffuse scattering

In FeX alloys (with X = Co or Cr), the solute X has an effect on the magnetic moment of its neighbor atoms. Polarized neutron diffuse scattering and DFT gave access to this physical value in chemically disordered (random distribution of Fe and X atoms on the bcc lattice) Fe₁⁻ₓCrₓ alloys and Fe₁⁻ₓCoₓ alloys (x ranging between 2 at.% and 20 at.%). Due to the closeness of the two first shells, it is impossible to determine them separately in the first two shells. We found (Figure 19a) that Cr decreases the moments of Fe (the parameter gᵢ is the change of μFe for a Fe atom with a Cr atom in the ith neighbor shell) and increases the moment amplitude of Cr in the first 2 shells (the parameter hᵢ is the change of μCr for a Cr atom with a Cr atom in the ith neighbor shell), whereas (Figure 19b) Co increases the moment of Fe and slightly that of Co. The experimental values of φ₁₂, the pondered average of both effects φᵢ = (1 - x)gᵢ + xhᵢ, is in good

agreement with the DFT values in FeCr (Figure 19a). In FeCo (Figure 19b), the sign and the order of magnitude of φ₁₂ are in agreement, but there is a factor of at least 2 between experiment and DFT φ₁₂ values. Co increases the Fe moment in its 2nn shells and has a small effect on the Co atoms in the same shells. These results are in agreement with the FNR results in Section 4.1.1.

Magnetization measurements on the same samples in the same ordered state were combined with the neutron results in order to obtain the average moments on Fe and X, that agrees well with the calculated values¹⁰⁰,¹⁵⁰ and FNR from Section 4.1.1.

## 4.2 Vacancy formation in pure iron and its alloys

### 4.2.1 DFT: vacancy formation energies in Fe and Fe-Mn

Besides the above mentioned experiments, the impact of magnetism on diffusion in Fe-based materials has been measured by several groups (cf. Section 5.2). This is strongly related to the formation energy of vacancies in these materials. From a DFT point of view, the comparison of calculated vacancy formation energies with experimental values also serves as a chance to assess the treatment of magnetic configurations and atomic relaxations next to defects.

The vacancy concentration at a given temperature can be expressed as

$$c_v(T) = \exp\left[-\frac{G_v^f(T)}{k_B T}\right] = \exp\left[\frac{S_v^f(T)}{k_B}\right]\exp\left[-\frac{H_v^f(T)}{k_B T}\right] \quad (12)$$

where H_v^f and S_v^f are the vacancy formation enthalpy and entropy, respectively. At zero pressure, the vacancy formation
<figure><img src="image_23.png" /><figcaption>**Figure 19:** Fe<sub>1-x</sub>Cr<sub>x</sub> alloys (a) and Fe<sub>1-x</sub>Co<sub>x</sub> alloys (b): Variation of MSRO parameter with x: measured using polarized neutron diffuse scattering φ<sub>12</sub> (black open squares) and calculated by DFT, averaged for the first two neighbor shells: g<sub>12</sub> (red bullets), h<sub>12</sub> (blue squares), and φ<sub>12</sub> (green diamonds). In panel (a), h<sub>12</sub> has been divided by 10. (a: Figure from Ref. [100] DFT results were obtained using the method described in Section 2.1).</figcaption></figure>

enthalpy is equal to the formation energy, $H'_{v} = E'_{v}$. Table 1 summarizes the vacancy formation energies in pure Fe and with dilute Mn concentrations reported in the literature, while the corresponding migration energies are discussed in Section 5.2. The different DFT result for the case of FM saturation are in good agreement, where small deviations are also influenced by inconsistent choices of the lattice constant.

The paramagnetic DFT result of 1.62 eV in the upper part of the table is obtained by the SSA relaxation method. In this method, an SQS structure is used to mimic a collinear PM disorder in a given supercell. As described in Section 2.1.2 the forces acting on the atoms are obtained by an average over all magnetic configurations in the SQS. The result is in excellent agreement with one of the available experimental values.<sup>176</sup> It is compared with different relaxation schemes, as also visualized in Figures 26 and 27. As expected, the energy is higher in a PM calculation, in which no relaxation is considered. The SSA relaxed energy is also lower as compared to a calculation, in which the relaxation is performed in the ferromagnetic state,<sup>174,175</sup> confirming experimental insights from tracer diffusion that “relaxation of the lattice around a vacancy is larger in the PM state than in the FM state”.<sup>180</sup> Remarkable is, however, the observation that a relaxation of individual spin configurations and a subsequent average of the vacancy formation energy yields a value that is much lower than within the SSA relaxed scheme. On the other hand, there is a good agreement of the SSA approach with an adiabatic averaging of positions obtained from a set of calculations with changing magnetic configurations.<sup>173</sup> For comparison, the results of other approaches given in the literature such as, e.g., the spin-wave method,<sup>181</sup> are also provided.

Table 1 also contains values for the vacancy formation energy next to a Mn atom. It turns out that Mn has a stronger

impact in the FM as compared to the PM state. This is also reflected by the first nearest-neighbour Mn-vacancy binding energies, which is –0.15 eV in the FM state and –0.10 eV in the PM state, where the negative sign refers to an attractive interaction. The second nearest neighbour Mn-vacancy interaction for both the FM and the PM states are weak and their respective binding energies are –0.06 eV and –0.03 eV.

Early measurements of the vacancy formation energy in α-Fe resulted in similar values for FM and PM states, about 1.40 eV.<sup>178</sup> Carefully performed experiments in nominally pure (4N) Fe for the ferromagnetic state predict 1.68 eV as the formation energy of vacancies.<sup>179</sup> However, Table 1 indicates a difference between the experimental results and theoretical data, especially those provided by DFT calculations, for the ferromagnetic state; the experiments predict smaller values by about 20–30 %. Residual impurities, especially interstitial elements like C or N, may account, at least partially, for the difference in view of their strong binding to vacancies. For example, a value of –0.65 eV is predicted for vacancy–interstitial C pairs.<sup>182</sup> For further details, see the corresponding discussion of vacancy diffusion in ferromagnetic α-Fe, Section 5.2.1.

### 4.2.2 TB/BOP and classical potentials: vacancy formation and concentration in Fe

The vacancy formation energies in FM bcc Fe predicted by various TB/BOP parametrizations and classical interatomic potentials are given in Table 1. In most cases, they are underestimated in comparison with the available DFT values. Nevertheless, all TB/BOP models correctly show an increase of magnetic moments on the 1nn atoms surrounding the vacancy, albeit the OTB parametrizations only marginally.
<table><caption>**Table 1:** A comparison of vacancy formation energies in bcc Fe and in dilute Fe₅₃Mn₁ obtained by experiment and different levels of theory for the FM and PM magnetic configurations.</caption><thead><tr><th>Vacancy formation energy E<sub>v</sub><sup>f</sup> (eV)</th><th>FM</th><th>PM</th></tr></thead><tbody><tr><td colspan="3"><strong>DFT: bcc Fe</strong></td></tr><tr><td>Hegde et al.<sup>3</sup></td><td></td><td></td></tr><tr><td>unrelaxed, a = 2.83 Å</td><td></td><td>1.94</td></tr><tr><td>SSA relaxed, a = 2.83 (2.90) Å</td><td></td><td>1.62 (1.96)</td></tr><tr><td>FM relaxed</td><td>2.15</td><td>1.71</td></tr><tr><td>individually relaxed</td><td></td><td>0.99</td></tr><tr><td>Gambino et al.<sup>173</sup> (a = 2.84 Å)</td><td>2.20</td><td>1.61</td></tr><tr><td>Ruban et al.<sup>140</sup> (a = 2.90 Å)</td><td></td><td>1.77</td></tr><tr><td>Sandberg et al.<sup>174</sup></td><td>2.15</td><td>1.54</td></tr><tr><td>Ding et al.<sup>175</sup> (a = 2.90 Å)</td><td>2.13</td><td>1.98</td></tr><tr><td colspan="3"><strong>EIM: bcc Fe</strong></td></tr><tr><td>Schneider et al.<sup>60</sup></td><td>2.20</td><td>1.99</td></tr><tr><td colspan="3"><strong>TB: bcc Fe</strong></td></tr><tr><td>Paxton et al.<sup>18</sup></td><td>1.57</td><td></td></tr><tr><td>Madsen et al.<sup>40</sup></td><td>1.83</td><td></td></tr><tr><td>Barreteau et al.<sup>13</sup></td><td>1.92</td><td></td></tr><tr><td colspan="3"><strong>BOP: bcc Fe</strong></td></tr><tr><td>Mrovec et al.<sup>131</sup></td><td>1.67</td><td></td></tr><tr><td>Ford et al.<sup>22</sup></td><td>2.02</td><td></td></tr><tr><td>Ladines et al.<sup>41</sup></td><td>2.03</td><td></td></tr><tr><td colspan="3"><strong>Classical potentials: bcc Fe (nonmagnetic)</strong></td></tr><tr><td>EAM<sup>75</sup></td><td>1.72</td><td></td></tr><tr><td>mEAM<sup>69</sup></td><td>1.95</td><td></td></tr><tr><td>ADP<sup>77</sup></td><td>2.05</td><td></td></tr><tr><td colspan="3"><strong>Experiments: bcc Fe</strong></td></tr><tr><td>Matter et al.<sup>176</sup></td><td></td><td>1.60</td></tr><tr><td>de Schepper et al.<sup>177</sup></td><td></td><td>1.79</td></tr><tr><td>Kim & Buyers<sup>178</sup></td><td>1.40</td><td>1.40</td></tr><tr><td>Seeger<sup>179</sup></td><td>1.68</td><td></td></tr><tr><td colspan="3"><strong>DFT: vacancy next to an isolated Mn atom</strong></td></tr><tr><td>Hegde et al.<sup>61</sup></td><td></td><td></td></tr><tr><td>a = 2.86 Å</td><td>2.07</td><td></td></tr><tr><td>SSA relaxed, a = 2.90 Å</td><td></td><td>1.86</td></tr></tbody></table>

Using Equation (12), the formation energy enters the equilibrium vacancy concentration, $c_v$. However, magnetic excitations and anharmonic effects lead to a nonlinear temperature dependence of the vacancy formation energy and non-Arrhenius behavior of the vacancy concentration. To explore the dependency of $E_v^f$ on $T$, we performed extensive finite-temperature MD+SD simulations using the NVT ensemble (at each temperature the volumes

corresponded to zero pressure). The simulation cells contained 2000 atoms for the bulk case and 1999 atoms for the system with a single vacancy. Details of the methodology can be found in Reference 183. The temperature dependence of $E_v^f$ found from the MD+SD simulation with the ADP potential is plotted in Figure 20 together with results obtained using the EIM model.

The knowledge of $E_v^f$ enables us to compute the Gibbs free energy $G_v^f(T)$ of vacancy formation by numerical integration of the Gibbs-Helmholtz equation

$$ \frac{G_v^f(T)}{T} = \frac{E_v^0 - T_0 S_v^0}{T_0} - \int_{T_0}^{T} \frac{E_v^f(T')}{T'^2} dT' $$

$$ G_v^f(T) = E_v^0 \frac{T}{T_0} - T S_v^0 - T \int_{T_0}^{T} \frac{E_v^f(T')}{T'^2} dT' \quad (13) $$

where $T_0$ is a reference temperature (chosen as 300 K), and $E_v^0$ and $S_v^0$ are the vacancy formation energy and entropy, respectively, at $T_0$ and zero pressure. The predicted temperature variations of $G_v^f(T)$ are displayed in Figure 20. In addition, by using the thermodynamic relation $TdS/dT = dE/dT$ and the calculated phonon entropy at room temperature, the temperature dependence of the vacancy formation entropy $S_v^f$ is shown. It is important to note that the obtained temperature variation of $E_v^f$ is caused mostly by magnetic excitation as the classical MD simulation without SD predicts almost temperature independent values of $E_v$ and $S_v^f$ (heating up to 1,000 K increases the energy by 0.05 eV.) As it can be seen in Figure 20, the EIM model predicts even more pronounced non-Arrhenius behaviour of $c_v(T)$ than the MD+SD simulation.

The interatomic potentials for the binary Fe–Cr system enable to investigate also atomic diffusion in a dilute alloy (e.g., at low Cr concentrations), namely, how is Fe diffusion affected by the solute Cr atoms. The impact of alloying on the vacancy concentration is characterized by the solute binding energy, $E_b$, with the vacancy. We used the EAM and ADP potentials to calculate this energy for a Cr-vacancy pair in a dilute Fe–Cr alloy. The predicted values are close to –0.04 eV for both potentials, while DFT calculation predicts $E_b \sim 0.05$ eV for the Cr-vacancy pair.<sup>184</sup> Despite the opposite sign of $E_b$, the absolute binding energies are very small and hence $c_v$ near a Cr atom will likely be only slightly different from $c_v$ in pure Fe.

### 4.2.3 TB/BOP: point defects in Fe–Co

In FeCo alloys, point defects are crucial for phase stability and ordering. At 1:1 (stoichiometric) composition, FeCo forms an ordered B2 (CsCl) phase below 1,000 K<sup>185,186</sup> and
<figure><img src="image_24.png" /><figcaption>**Figure 20:** Simulation results obtained with MD+SD (red lines) and EIM (blue lines) methods, where the vertical arrows indicate the Curie temperatures for both models. The subfigures contain (a) the formation energy, (b) the formation entropy, and (c) the Gibbs free energy of a single vacancy; the thin dash-dotted lines indicate the temperature dependence of the energy without magnetic disordering: $E_v^0 - TS_v^0$. (d) The calculated concentration of vacancies; the thin dashed lines show the dependence $c_v(T)$ in the PM state.</figcaption></figure>

transformsto disordered bcc above this temperature.¹⁸⁷ This low order-disorder transition temperature (e.g., NiAl and FeAl with B2 structure preserve the order above 1,500 K) indicates that the exchange of Fe and Co atoms on the bcc lattice is comparably easy. This is not surprising as these neighboring elements in the periodic table have alike atomic sizes and similar electronic structures of bcc Fe and bcc Co.¹⁸⁸ For this reason, we can conclude that the antisite defects (i.e., a Fe atom on the Co sublattice or vice versa) are the dominating defects in stoichiometric B2 FeCo. Table 2 summarizes the formation energies of vacancies and exchange defect (sum of Fe and Co antisite defects) estimated from experiments¹⁸⁹ and obtained by DFT,¹⁹⁰⁻¹⁹² BOP,⁴² and EAM/MEAM.¹⁹³⁻¹⁹⁵

The different calculations consistently indicate that the exchange defect is significantly lower in formation energy than vacancies and, therefore, the dominating defect in stoichiometric B2 FeCo. For the non-stoichiometric case, Fe antisites are the structural defects for Fe-rich compositions

and Co antisites for Co-rich. The formation energies for the non-stoichiometric case calculated with DFT are close to those for the stoichiometric case.¹⁹⁰ The small but positive formation energy of exchange defects prevents the spontaneous exchange of Fe and Co atoms and a transformation to disordered bcc. We note that the DFT and BOP calculations in Table 2 were performed, allowing spinpolarization to

<table><caption>**Table 2:** Vacancy formation energy on the Fe sublattice, $E_v^f(Fe)$, and the Co sublattice, $E_v^f(Co)$, and exchange defect $E_{ex(0)}$ in B2 FeCo obtained by experiment (a: not chemically resolved data of positron annihilation¹⁸⁹), DFT (b,¹⁹⁰ c,¹⁹¹ h¹⁹²), EAM/MEAM (d,¹⁹³ e,¹⁹³ f¹⁹⁴) and BOP (g⁴²) calculations.</caption><thead><tr><th></th><th>Experiment</th><th>DFT</th><th>EAM, MEAM</th><th>BOP</th></tr></thead><tbody><tr><td>E<sub>v</sub><sup>f</sup>(Fe)</td><td>1.45<sup>a</sup></td><td>1.65<sup>b</sup>, 1.78<sup>c</sup>, 1.35<sup>h</sup></td><td>1.65<sup>d</sup>, 1.69<sup>e</sup>, 1.7<sup>f</sup></td><td>1.77<sup>g</sup></td></tr><tr><td>E<sub>v</sub><sup>f</sup>(Co)</td><td>1.45<sup>a</sup></td><td>1.68<sup>b</sup>, 1.78<sup>c</sup>, 1.32<sup>h</sup></td><td>1.65<sup>d</sup>, 1.88<sup>e</sup>, 1.7<sup>f</sup></td><td>2.36<sup>g</sup></td></tr><tr><td>E<sub>ex(0)</sub></td><td>–</td><td>0.30<sup>b</sup>, 0.34<sup>c</sup></td><td>0.25<sup>d</sup>, 0.16<sup>e</sup>, 0.29<sup>f</sup></td><td>0.36<sup>g</sup></td></tr></tbody></table>
include the effects of magnetism. In non-spinpolarized DFT calculations, the formation energies of antisite defects take small negative values, leading to a spontaneous exchange of Fe and Co atoms and transformation to disordered bcc.¹⁹⁰ In this sense, magnetism stabilizes the ordered arrangement of Fe and Co atoms in the B2 phase.

### 4.2.4 EIM: vacancy formation and vacancy-solute binding versus temperature

Temperature evolution of vacancy-solute interaction properties, such as the magnetic free energy of binding (accounting for the magnetic entropy), can be determined via MC simulations coupled with EIMs. The vacancy-solute binding energy at a 1nn distance dictates the vacancy concentration next to a solute atom, which is especially important for the determination of solute diffusion coefficients via a vacancy mechanism.¹⁹⁶,¹⁹⁷

Using the on-lattice MC approach, we compute the binding energy as the difference of the magnetic free energy of formation of a vacancy at a 1nn site of the solute and in a pure Fe system. The vacancy formation energies are determined by evaluating the corresponding equilibrium vacancy concentrations as a function of temperature. A detailed description of the numerical method employed can be found in References 58,60.

The left panel of Figure 21 shows the vacancy formation magnetic free energy in pure Fe and at 1nn sites of a solute (Cu or Mn) in Fe, as functions of temperature. Concerning the pure Fe case, the vacancy formation magnetic free energy obtained in the low and high temperature regimes (respectively 2.20 and 1.99 eV in FM and PM magnetic states) is in agreement with previous experimental and DFT data from literature (cf. Table 1), ranging from 2.00 to 2.24 eV in the FM state and from 1.54 to 1.98 eV in the PM state.¹⁷⁴,¹⁷⁵,¹⁷⁷,¹⁹⁸,¹⁹⁹ We note that the vacancy formation magnetic free energy in the PM state shows very scattered results in the literature. The values are extremely sensitive to the computational details, while the various studies are very consistent concerning the FM state.⁶¹,¹⁷⁴,¹⁷⁵,¹⁷⁷,¹⁹⁸,¹⁹⁹ At low temperatures the formation free energy at 1nn sites of a Mn (resp. Cu) solute is approximately 0.14 eV (resp. lower than the value in pure Fe), which is consistent with the FM binding obtained via DFT calculations, that is, between 0.15 eV and 0.17 eV for vacancy-Mn,⁶⁰,⁶¹,²⁰⁰ and 0.24 eV for a vacancy-Cu pair.⁵⁸

As the temperature increases, the formation energy difference or the binding energies decrease towards approximately zero in the fully paramagnetic regime, for both Mn and Cu cases (right panel of Figure 21). This solute-vacancy binding behaviour indicates that the magnetic

disorder is able to erase the chemical-interaction effects in the very dilute Fe alloys. Since we have observed an identical behavior in the case of two solutes that exhibit quite different electronic and magnetic properties, it appears to be a general behaviour in bcc Fe systems, independent of the chemical nature of the solute. This dominance of magnetic-disorder over chemical effects is fully coherent with a significant increase of the solute kinetic correlation factor (indicating the approach to a random walk) from the FM to the PM state, as discussed below, in the diffusion section.

## 4.3 Grain boundaries

### 4.3.1 Thermodynamics and defects of grain boundaries

Grain boundaries are important defects in crystalline materials that contribute to fundamental relationships between physical properties, microstructure and composition, especially for advanced technological applications. GBs represent often (especially in metallic materials) pathways of short-circuit diffusion²⁰¹ and hence can tremendously influence the microstructure evolution. In addition, they usually present preferable sites for segregation of solute atoms as well as intrinsic point defects such as vacancies and interstitial atoms. Hence, the atomic structure and the phase stability of GBs needs to be closely analysed as their finite temperature properties could be fundamentally different from that of bulk materials.²⁰²⁻²⁰⁵

This analysis can be performed systematically in terms of defect phase diagrams, a concept which was recently advanced as a materials design strategy.²⁰⁶ Using a density-based model,²⁰⁷ i.e., considering a GB as a defect with an excess free volume or reduced density, multi-component GB phase diagrams can even be computed based on available bulk thermodynamic data.⁴⁵ The magnetic contributions represent further degrees of freedom which become especially important for magnetically sensitive materials. For example, the adsorption and spinodal decomposition theoretically predicted in Fe–Mn alloys²⁰⁵,²⁰⁷,²⁰⁸ have their origin in magnetic ordering, and these phenomena result in nucleation and phase transitions at GBs. A GB spinodal decomposition was also observed experimentally or Fe–Mn²⁰⁸,²⁰⁹ or Ni–Cr-based²¹⁰ alloys, proofing the relevance of magnetism.

The structure, segregation and phase stability of GBs is not only determined by macroscopic degrees of freedom. In fact, GB defects like, e.g., disconnections or steps²¹¹,²¹² have to be taken into account. As an example, recent investigations of the GB structure-property relationships in a Ni-base 602CA coarse-grained alloy revealed a co-existence of
<figure><img src="image_25.png" /><figcaption>**Figure 21:** Magnetic free energy versus temperature (in K) of (left): vacancy formation in pure Fe (blue), at a 1nn site of a Cu (orange) or a Mn (green) solute, and (right): binding between a solute and a 1nn-vacancy, as predicted by EIM + MC simulations.</figcaption></figure>

several families of high-angle GBs²¹⁰ distinguished by precipitation and segregation levels. For example, irregular (hackle) GBs were found to be covered by platelet-like carbides, whereas straight high-angle GBs contained globular carbides. Moreover, alternating Cr/Ni layers of nanometer widths were observed between the globular carbides featuring spinodal-like decomposition.²¹⁰ The structure characteristics were found to be correlated with distinct values of the GB diffusion coefficients for these two GB families.²¹⁰ Elastic fields were found to affect significantly the diffusion properties of GBs. The observed mechanochemical coupling, i.e., the coupling between interface structure defects, local chemistry, stress/strain fields and precipitation, contributes to the formation of metastable complexions resulting in time-dependent GB diffusion coefficients.²¹⁰ A contribution of magnetic degrees of freedom, i.e., the magneto-mechano-chemical coupling, into the interrelation of the GB structure, particle shapes and element segregation is in a focus of current investigations.

### 4.3.2 Experiment: GB energy in α-Fe

Experimentally, it is extremely involved to measure absolute values of GB energies, $γ_{gb}$. The intensively used thermal grooving method provides relative values of $γ_{gb}$ as a ratio $γ_s/γ_{gb}$, where $γ_s$ is the surface energy.²¹³ Roth²¹⁴ used a combination of the zero-creep technique and the thermal grooving method to determine $γ_{gb}$ for δ-Fe as 0.985 J/m². Significantly smaller values of about 0.47 J/m² were obtained by Price et al.²¹⁵ using the same technique (Table 3). Values of about 0.8–0.9 J/m² were also reported for δ-Fe at about 1,700 K by Vlack²¹⁹ and Hondros.²²⁰ To our knowledge, there are no reports of directly measured $γ_{gb}$ values for α-Fe yet.

Using Gupta's representation²²¹ of a semi-empirical formalism of Borisov et al.,²¹⁶ one may alternatively estimate the GB energy, $γ_{gb}$, via a combination of GB self-diffusion, $D_{gb}$, and volume self-diffusion, $D_v$, measurements

$$γ_{gb} = \frac{k_B T}{2a^2} \ln \frac{D_{gb}}{D_v} \qquad (14)$$

Here, *a* is the lattice constant of the material and the GB width is taken as 2*a*. This expression has been derived by Guiraldenq²²² in a slightly modified way from Borisov's original approach and was suggested to be useful also for GB diffusion in concentrated alloys. The applicability of Borisov's approach for at least simple metals and dilute alloys was demonstrated by Prokoshkina et al.²²³ Recently, the approach was extended to multi-principal element alloys²²⁴,²²⁵ and its universality has been tested for both fcc²²⁶ and bcc²²⁷ alloys.

Using the literature data on volume¹⁸⁰,²²⁸ and GB self-diffusion²¹⁷,²¹⁸ (cf. Section 5.5.1), one can estimate the GB

<table><caption>**Table 3:** GB energies estimated from GB diffusion data in α-Fe using a semi-empirical approach of Borisov et al.²¹⁶ and available sets of GB energies measured by a direct technique for δ-Fe (T = 1,753 K) for comparison.</caption><thead><tr><th rowspan="3">T (K)</th><th colspan="4">Grain boundary energies (J/m²)</th></tr><tr><th rowspan="2">Iijuma et al.<sup>217</sup></th><th colspan="2">Divinski et al.<sup>218</sup></th><th rowspan="2">Roth<sup>214</sup></th></tr><tr><th>Pure Fe</th><th>Less pure Fe</th></tr></thead><tbody><tr><td>766</td><td>1.47</td><td>1.18</td><td>1.07</td><td></td></tr><tr><td>905</td><td>1.47</td><td>1.13</td><td>1.09</td><td></td></tr><tr><td>1,005</td><td>1.58</td><td>1.06</td><td>1.05</td><td></td></tr><tr><td>1,092</td><td>1.60</td><td>0.98</td><td>0.98</td><td></td></tr><tr><td>1,753</td><td></td><td></td><td></td><td>0.985</td></tr><tr><td></td><td></td><td></td><td></td><td>0.47</td></tr></tbody></table>
<table><caption>**Table 4:** GB energies of the Σ5(310) GB for Fe and FeMn in the FM and PM states. For FeMn, a Fe atom at the GB is replaced by a Mn atom.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Grain boundary energies (J/m²)</th></tr><tr><th>Ferromagnetic</th><th>Paramagnetic</th></tr></thead><tbody><tr><td>Fe</td><td>1.57</td><td>0.96</td></tr><tr><td>FeMn</td><td>1.50</td><td>0.94</td></tr></tbody></table>

energy in nominally pure α-Fe and the results are given in Table 3. These data can be compared with the calculations presented in Section 4.3.3, namely 1.57 J/m² and 0.96 J/m² for fully FM and PM states, respectively, see Table 4. At least qualitatively, the GB energy estimated using the data of Divinski et al.²¹⁸ for pure α-Fe show trends similar to the current predictions. Most importantly, the GB energy decreases with increasing temperature as it has to be for a nominally pure metal. Furthermore, Table 3 suggests a reasonable extrapolation of the GB energies for α-Fe for both pure and less pure Fe materials to the high temperatures of direct measurements of δ-Fe. An alternating trend would be predicted when the GB diffusion data of Iijima et al.²¹⁷ are used; the increase of the GB energy with temperature indicates an impurity contamination of the Fe material used for the corresponding GB diffusion measurements in Reference 217. Moreover, from the data set of Iijima et al.²¹⁷ the GB energies in the PM state are predicted to be higher than those in the FM state, in obvious contradiction to the predictions from DFT-informed calculations, Sections 4.3.3 and 5.5.1.

### 4.3.3 Theoretical prediction of GB energetics in pure Fe

There exist numerous atomistic studies investigating the structures and energetics of GBs in metals. Figure 22 shows a comparison of theoretical values of GB energies (at 0 K) predicted by various atomistic methods for bcc Fe. The DFT calculations are typically performed with relatively small periodic supercells containing two identical symmetric tilt GBs. The interatomic potentials enable to investigate GB properties at finite temperatures including even diffusion processes.²³⁴ The results in Figure 22 demonstrate that BOP and ADP agree quantitatively very well with DFT, while the simpler EAM potential underestimates the GB energies. It is worth mentioning that for the Σ5(210) GB there exists a rather broad range of DFT results. This is likely related to several possible metastable configurations that exist for this GB. The larger GB energies reported in References 232,233 probably correspond not to the ground state but to a metastable structure of higher energy.

<figure><img src="image_26.png" /><figcaption>**Figure 22:** Comparison of GB energies computed with different methods. The literature DFT data are taken from DFT¹ – Ref.,²²⁹ DFT² – Ref.,²³⁰ DFT³ – Ref.,²³¹ DFT⁴ – Ref. ²³² and DFT⁵ – Ref.²³³</figcaption></figure>

While classical potentials do not contain any explicit treatment of magnetism, DFT can provide information about GBs for different magnetic configurations. Table 4 shows the formation energies for the Σ5(310) GB for Fe and FeMn in the FM and PM (SSA relaxed) states. A substantial decrease of formation energy in the PM state compared to the FM state agrees with the experimental findings discussed above. In addition, the segregation of Mn at the GB reduces the formation energy in both magnetic states.

### 4.3.4 Segregation of Mn to GBs

Mn segregation energies are calculated as the difference between the GB formation energies with Mn and without Mn. Figure 23 shows the segregation energies of a single Mn atom at different layers from the Σ5[001](310) GB.²³⁵,²³⁶ Here, the negative segregation energy indicates favorable segregation. Clearly, the FM case shows a more favorable segregation that is substantially different than the PM (SSA relaxed) case. Moreover, the layer next to the GB is the most favored segregation site in the FM state while the GB layer is the most favored site in the PM state. A second Mn atom is also most favorable in the second layer, and shows an even larger segregation energy in the FM state (–0.59 eV). In the PM state, the trend is similar to the case of a single Mn atom. The large differences in the FM and the PM segregation energies underlie the significant impact of magnetic states on the chemical effects in the vicinity of the GB.

### 4.3.5 Segregation of Cr to GBs

The segregation of Cr atoms to GBs in stainless steel may strongly affect the GB cohesion as well as local Cr concentration profiles. However, the knowledge of Cr segregation energies at Fe GBs is rather limited. DFT calculations of Wachowicz²³² predict favorable segregation of Cr at two
<figure><img src="image_27.png" /><figcaption>**Figure 23:** Segregation energies of single Mn atoms at different layers from the Σ5[001](310) GB for the FM and the PM state.</figcaption></figure>

high-angle tilt GBs (Σ3(111) and Σ5(210)) in Fe. These results were confirmed recently for additional GBs by Yang et al.²³⁷ and our own DFT studies.²³⁸ Eich and Schmitz²³⁹ presented a systematic theoretical study of GB segregation in Fe–Cr alloys using their EAM potential. They predicted a small segregation tendency on the Fe-rich side but a strong segregation of Fe to GBs on the Cr-rich side. We used the ADP parametrization for the Fe–Cr system⁷⁸ to calculate Cr segregation energies for two tilt GBs. Representative simulation results are summarized in Table 5 where the most favorable segregation sites are listed only. The theoretical segregation energies indicate that Cr atoms exhibit a moderate tendency to enrich the GB.

# 5 Diffusion

## 5.1 Overview about present challenges related to magnetism

A spectacular illustration of the effect of magnetism on diffusion in iron-based alloys is the relative acceleration of diffusion in ferromagnetic matrix when approaching the transition temperature from the FM to the PM state. This is

<table><caption>**Table 5:** Computed segregation energies (in eV) of Cr at two high-angle tilt GBs in bcc Fe.</caption><thead><tr><th>Model</th><th>Σ5(210)</th><th>Σ13(510)</th></tr></thead><tbody><tr><td>EAM<sup>239</sup></td><td>-0.35</td><td>-0.29</td></tr><tr><td>ADP<sup>78</sup></td><td>-0.27</td><td>-0.38</td></tr><tr><td>DFT<sup>232</sup></td><td>-0.25</td><td></td></tr><tr><td>DFT<sup>238</sup></td><td>-0.16</td><td>-0.12</td></tr></tbody></table>

manifested by a characteristic kink on the Arrhenius plot of self-diffusion and impurity diffusion coefficients (see Figures 25 and 33 below). Several empirical models have been proposed to describe this effect, the best known being Ruch's diffusion model²⁴⁰ (see Equation (19) below). These models fit well the existing experimental data in pure iron, but do not predict the amplitude of the acceleration in alloys – or the respective contributions to vacancy formation and migration energies (enthalpies).

In metals and metallic alloys, diffusion of substitutional elements occurs by vacancy jumps and is therefore controlled by the vacancy concentration and vacancy jump frequencies. An evaluation of these properties from first principles requires the calculation of formation and migration Gibbs free energies.

For example, the self-diffusion coefficient of iron is:²⁴¹

$$D_{\text{Fe}}^{\text{Fe*}} = f_0 c_v^{\text{eq}} D_v \quad (15)$$

where *a* is the lattice parameter, $f_0 ≈ 0.727$ the correlation factor, $c_v^{\text{eq}}$ the equilibrium vacancy concentration given by Equation (12) and $D_v$ the vacancy diffusion coefficient:

$$D_v = a^2 \Gamma_0 = a^2 \nu_0 \exp\left(-\frac{G_v^m}{k_B T}\right) \quad (16)$$

where $\nu_0$ is an attempt frequency and $G_v^m$ is the migration Gibbs free energies of a vacancy in pure iron. $\Gamma_0$ is the vacancy jump frequency in pure iron (which include vibrational, magnetic and electronic entropy contributions).

Similarly, accounting for different types of vacancy jumps in a vicinity of a solute atom according to Le Claire's model,²⁴² the impurity diffusion coefficient of an element *X* is given by:

$$D_{\text{Fe}}^{\text{X*}} = a^2 f_2 c_v^{\text{eq}} \frac{\Gamma_4}{\Gamma_3} \quad (17)$$

where $\Gamma_2$ is the frequency of the X–V exchange (V denotes a vacancy), $\Gamma_3$ and $\Gamma_4$ the Fe–V exchange frequencies that lead to the dissociation and association of the X–V pairs. The correlation factor $f_2$ is itself a function of various jump frequencies (the number of which depends on the range of interaction between the solute and the vacancy, see Figure 24).

Alternatively, one can perform a sufficient number of vacancy jumps in a system of finite size during a time *t*, using MD or kMC simulations, and directly deduce the diffusion coefficients of the species *i* (*i* = V, Fe*, X*, etc.) from the measurements of its mean-square displacement $\langle R_i^2 \rangle$:²⁴¹

$$D_{\text{Fe}}^{i} = \frac{\langle R_{i}^{2} \rangle}{6t} \quad (18)$$
<figure><img src="image_28.png" /><figcaption>**Figure 24:** The diffusion vacancy jump frequencies around a solute X (gray) and its neighboring Fe atoms (white). The arrows indicate the direction of the vacancy jumps; the circled numbers give the distance to the solute atom (adopted from Reference 54).</figcaption></figure>

As such simulations are performed with a vacancy concentration $c_v^{sim}$ (usually corresponding to one vacancy in the simulation box) that differs from the actual equilibrium vacancy concentration $c_v^{eq}$, the resulting diffusion coefficients must be divided by $c_v^{sim}/c_v^{eq}$.

The calculation of formation and migration energies or enthalpies using standard DFT methods is straightforward at 0 K, in a perfectly FM state for α-Fe. The task is more challenging in the PM state at high temperature, and in partially ordered magnetic states around the magnetic order-disorder transition temperature. The estimation of the magnetic entropy is also challenging.

The vacancy formation is discussed in Section 4.2.1. For the vacancy migration, three complementary approaches, based on the methods presented in Chap. 2, have been developed and are presented in the following sections.

- SSA: a direct calculation of the migration barrier by DFT calculations can be performed, including the relaxation of atomic positions around the vacancy, in the PM state (by averaging the atomic forces over magnetically disordered snapshots). Then Eqs. (15)–(17) can be used to compute the vacancy and atomic diffusion coefficients. Ruch’s model is used for the interpolation between the FM and PM states.⁶¹

- EIM: vacancy jumps and spin relaxations are performed by MC simulations at any given temperature, and the diffusion coefficients deduced from the mean-square displacements using Equation (18).⁵⁸

- TB and classical interatomic potentials: vacancy jumps are performed by MD simulations and the diffusion coefficients deduced for the mean-square displacements

using Equation (18). With the ADP potential, MD simulations can be coupled with spin dynamics (MD+SD) to model the effects of magnetic excitations. The classical MD simulations can be also used for evaluating the GB diffusion coefficients.

This section addresses the diffusion properties starting with the simplest case: self-diffusion in α-iron. We then consider impurity diffusion coefficients in α-iron, with special attention to the case of Mn, whose behavior is atypical (the effect of the magnetic transition is much less pronounced than for other elements). Finally, we present some results for a few concentrated alloys and for the diffusion at GBs.

## 5.2 Self-diffusion in α-Fe

### 5.2.1 Experimental methods and results

Tracer self-diffusion in α-Fe has been measured extensively in the past, especially with respect to the impact of magnetic transition on the diffusion behavior. Already in pioneering works of Borg and Birchenall²⁴³ or Buffington et al.,²⁴⁴ a strongly non-linear Arrhenius temperature dependence of Fe self-diffusion has been established and discussed. The most reliable data measured for extended temperature intervals by Iijima et al.¹⁸⁰ and by Lübbenhusen and Mehrer²²⁸ are shown in Figure 25. Both mechanical sectioning and ion-beam sputtering techniques were used that allowed to cover a large range of the diffusion coefficients from about $10^{-23}$ m²/s to $10^{-15}$ m²/s, in total over seven orders of magnitude.

While the two experimental data sets agree perfectly for the FM region, the results of Lübbenhusen and Mehrer²²⁸ are systematically (but not significantly, less than by a factor of two) larger than those of Iijima et al.¹⁸⁰ in the PM region, Figure 25. This fact may probably be explained by the impact of dislocation diffusion on the results of mechanical grinding by Lübbenhusen and Mehrer.²²⁸ For this reason, the results of Iijima et al.¹⁸⁰ for the PM region are considered as more reliable.

As it was mentioned above, several semi-empirical models have been suggested to account for the impact of magnetic ordering on diffusion in α-Fe, e.g., by Ruch et al.,²⁴⁰ Hettich et al.,²⁴⁵ Kucera,²⁴⁶ or Braun and Feller-Kniepmeier.²⁴⁷ All these models could be traced back to the seminal work of Girifalco on the impact of chemical ordering on diffusion in alloys.²⁴⁸ From an experimental point of view only, it is impossible to decide which model is most appropriate for describing the data, since all fits result in similar residuals by a regression analysis.²²⁸,²⁴⁹
According to the Ruch model,²⁴⁰ for self and impurity diffusion, the temperature dependence of the tracer diffusion coefficient, $D_{\text{Fe}}^{\text{Fe}^*}$, can be approximated by the dependence

$$D_{\text{Fe}}^{\text{Fe}^*} = D_0^p \exp \left[ - \frac{Q_p (1 + \alpha M^2)}{k_B T} \right] \quad (19)$$

Here $D_0^p$ and $Q_p$ are the prefactor and the activation enthalpy of diffusion in the paramagnetic state, respectively and M is the relative magnetization (order parameter). The parameter $\alpha$ expresses the impact of the magnetic transition on the activation enthalpy of diffusion of the given element. The values of $\alpha$ are summarized in Table 6. The experimental data, see Figure 25, were used to quantify the relative magnetization as a function of temperature.

The activation enthalpy of self-diffusion is composed of vacancy formation and migration enthalpies, see Equation (15). The vacancy formation enthalpy, $H_v^f$, has already been discussed in Section 4.2.1. Experimentally, it can be determined by a number of techniques, such as differential dilatometry,²⁵⁵,²⁵⁶ resistivity²⁵⁷,²⁵⁸ or positron annihilation,²⁵⁹ just to mention a few of them. According to the positron annihilation study,²⁵⁹ the vacancy formation energy in α-Fe is almost the same in ferro- and para-magnetic regions and it equals to about (1.4 ± 0.1) eV. Seeger¹⁷⁹ estimated the vacancy formation, $H_v^f$, and migration, $H_v^m$, energies as $H_v^f = (1.68 ± 0.07)$ eV and $H_v^m = (1.23 ± 0.11)$ eV. The migration energy of vacancies in α-Fe has been deduced from the growth of voids during electron irradiation as well²⁶⁰,²⁶¹ and the value of $H_v^m = (1.3^{+0.1}_{-0.3})$ eV has been obtained.

Since self-diffusion in α-iron occurs by the vacancy mechanism, the diffusion activation energy is the sum of the vacancy formation and migration energies. Then in its magnetically fully ordered state, the activation energy of self-diffusion is predicted to be $H_v^f + H_v^m = (2.91 ± 0.04)$ eV in good agreement with the tracer diffusion data.¹⁸⁰,²²⁸

DFT-informed calculations in FM Fe, see following section, result in a similar self-diffusion energy, but are partitioned as the sum of a larger formation energy, $H_v^f \approx 2.2$ eV, and a smaller migration energy, $H_v^m \approx 0.7$ eV.⁶⁰ The difference comes probably from the fact that in many experiments, impurities like C are strongly bounded to vacancies, slow down their migration and increase their concentrations. Very recent experimental observations support the smaller values of $H_v^m$, too.²⁶²

<table><caption>**Table 6:** Arrhenius parameters (the pre-factor $D_0^p$, the activation enthalpy $Q_p$ and the constant $\alpha$) for diffusion of different elements in α-Fe. $Q^f = Q^p(1 + \alpha)$ is the activation enthalpy of diffusion deeply in the FM state.</caption><thead><tr><th>Element</th><th>D<sub>0</sub><sup>P</sup><br>m<sup>2</sup>/s</th><th>Q<sup>P</sup><br>eV</th><th>α</th><th>Q<sup>P</sup>(1 + α)<br>eV</th><th>Reference</th></tr></thead><tbody><tr><td>Fe</td><td>2.76 × 10<sup>-4</sup></td><td>2.597</td><td>0.156 ± 0.003</td><td>3.003</td><td>180</td></tr><tr><td>Co</td><td>9.9 × 10<sup>-5</sup></td><td>2.500</td><td>0.19</td><td>2.969</td><td>250</td></tr><tr><td>Ni</td><td>4.5 × 10<sup>-5</sup></td><td>2.731</td><td>0.08</td><td>2.950</td><td>251</td></tr><tr><td>Nb</td><td>1.4 × 10<sup>-1</sup></td><td>3.106</td><td>0.061</td><td>3.296</td><td>249</td></tr><tr><td>Zn</td><td>6 × 10<sup>-2</sup></td><td>2.722</td><td>0.118</td><td>3.043</td><td>252</td></tr><tr><td>Cr</td><td>3.73 × 10<sup>-3</sup></td><td>2.771</td><td>0.133 ± 0.005</td><td>3.140</td><td>253</td></tr><tr><td>Cu</td><td>3 × 10<sup>-2</sup></td><td>2.905</td><td>0.073</td><td>3.118</td><td>254</td></tr><tr><td>Mn</td><td>2.9 × 10<sup>-2</sup></td><td>2.931</td><td>0.0</td><td>2.931</td><td>61</td></tr></tbody></table>

<figure><img src="image_29.png" /><figcaption>**Figure 25:** Experimentally measured Fe self-diffusion in α-Fe across the Curie temperature, $T_c$. The most reliable results of Iijima et al.¹⁸⁰ and Lübbenhusen and Mehrer²²⁸ are presented. The solid line represents a fit according to the Ruch model,²⁴⁰ Equation (19).</figcaption></figure>
## 5.2.2 SSA approximation

As discussed in Section 2.1.2, the SSA relaxation scheme³ assumes the adiabatic limit for the PM state, where the magnetic degrees of freedom are much faster compared to the atomic degrees of freedom at high temperatures. Similar to the discussion of the vacancy formation energy in Section 4.2.1, the vacancy activation energies obtained from such a relaxation scheme are compared with other approximate schemes that are commonly employed in literature in Figure 26.

Here, the activation energies of unrelaxed scheme (magnetic configurations frozen in the perfect bcc geometry), FM relaxed scheme (magnetic configurations frozen in the geometry obtained from the atomic relaxation in the FM state), and the individually relaxed scheme (atoms in each magnetic configuration relaxed according their instantaneous forces) are significantly different from the exact SSA relaxation scheme. Moreover, the inclusion of volume expansion is shown to be important for evaluating vacancy activation energy. The atomic displacements obtained from different relaxation schemes are compared in Figure 27. These results emphasize that the importance of proper relaxation scheme and the inclusion of volume expansion for defect energies in the PM state.

The self-diffusion coefficients are computed from the FM and the PM activation energies from the Ruch's model,²⁴⁰ as shown in Figure 28. For a valid comparison with experiments, experimental lattice parameters of 2.86 Å for the FM state and 2.90 Å for the PM state have been considered. These results are in good agreement with the experimental literature.

<figure><img src="image_30.png" /><figcaption>**Figure 26:** Comparison of vacancy activation energies in bcc Fe for different relaxation schemes, which consist of formation energies (lower parts) and migration energies (upper parts). Calculations are performed at the equilibrium lattice constant (blue bars) and at a high temperature lattice constant close to the Curie temperature (orange bars).</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>**Figure 27:** Displacements of atoms in the nearest-neighbor shell of the vacancy in the vacancy state (blue bars and right inset) and in the transition state (yellow/green/orange bars and left inset). The SSA relaxation scheme has been compared with other approximate relaxation schemes.</figcaption></figure>

<figure><img src="image_32.png" /><figcaption>**Figure 28:** Arrhenius diagram for the self-diffusion coefficients in α-Fe: comparison of the SSA, EIM + MC and MD + SD models with the experiments of Iijima et al.¹⁸⁰</figcaption></figure>

## 5.2.3 EIM results

An alternative way to determine diffusion properties in Fe-based systems is based on the DFT-parameterized EIM coupled with on-lattice MC simulations. As outlined in Section 2.3, we need to construct EIMs to describe the magnetism and the energetics of Fe atoms in the bulk, near a vacancy, and at a saddle point site for a vacancy-atom exchange. Details of the model parameters and of the MC setup can be found in Reference 58. As mentioned in References, 59,199,263 a quantum treatment of spins is necessary for a correct prediction of the magnetic entropy at low temperatures. We therefore adopted the Bose-Einstein statistics in our spin-MC simulations up to the Curie point, following the
quasi-harmonic approach of Reference 59. The self-diffusion coefficients are obtained by directly simulating the tracer diffusion experiments with MC simulations.⁵⁸ We compute the mean square displacement of the tracers and we determine the physical time at each temperature using Equation (18).⁵⁸ Note that the lattice-vibration effects (vacancy formation entropy and attempt frequency) are not intrinsically accounted for in the present EIM-MC setup but calculated separately by DFT in the FM state. The magnon-phonon effects are not considered.

Via the same MC simulations, we also determined the magnetic free energy of vacancy migration. Adding the latter to the magnetic free energy of vacancy formation, as described previously, we obtained the activation free energy for Fe tracer diffusion. The values are shown in Figure 29. Regarding the activation free energies, we reproduce closely the DFT value in the low-temperature limit (2.90 eV). The asymptotic PM value is evaluated at 2,000 K (2.46 eV). These energies are in good agreement with previous DFT and available experiments data (2.63–3.10 eV and 2.48–2.92 eV for FM and PM states, respectively¹⁷⁵,¹⁷⁷).

As shown in Figure 28, this approach predicts the self-diffusion coefficients versus temperature in excellent agreement with experimental data.¹⁸⁰,²⁴⁴,²⁴⁵,²⁵⁴,²⁶⁴–²⁷¹ Especially, the sudden deviation from Arrhenius law near the Curie temperature is consistently predicted without any additional assumption. The change of slope (activation

energy) between the FM and the PM regimes is also successfully predicted. It is worth mentioning that if we adopt the Boltzmann (instead of the Bose–Einstein) distribution at low T in the spin-MC, the local acceleration of diffusion around the Curie point is significantly underestimated.⁵⁸

At variance with the direct-DFT (DLM or SSA) approaches, where an additional interpolation model is required to describe diffusion properties at intermediate temperatures, the current approach provides an equal description for all temperatures regardless of the magnetic state. The crucial issue is, of course, an accurate parameterization of the EIMs. On the other hand, it allows to reach time scales of several orders of magnitudes longer than the spin-lattice dynamics simulations. Therefore, it is also promising to address more complex kinetic processes than the atomic diffusion, such as phase ordering, precipitation or segregation.

### 5.2.4 TB and classical interatomic potentials

The way how the self-diffusion coefficient may be determined using TB and classical potentials is via the product of the equilibrium vacancy concentration, $c_v^{eq}$, and a single-vacancy diffusivity, $D_v$ (see Equation (15)), which depend on the vacancy formation and migration free energies, respectively. For 0 K conditions, these can be computed easily in the same manner as in the first principles calculations (and also validated against those), but one can additionally carry out MD simulations to obtain the free energies and their temperature dependencies.

The vacancy formation energy at 0 K calculated with various TB models and interatomic potentials have been presented in Table 1 (Section 4.2.1). The vacancy migration energy, $E_v^m$, at zero temperature can be evaluated using the nudged elastic band (NEB) method. Figure 30 shows the calculated energy profiles associated with the nearest-neighbor vacancy jump along the 〈111〉 direction in bcc Fe. All classical potentials predict $E_v^m$, determined as the maximum of the curves, rather close to the DFT result (~0.7 eV), with errors of about 0.1 eV. However, the shape of the curves appears to be double-humped in contrast to the DFT result. Practically, such deviation of the migration barrier shape observed for the potentials should result only in an error in the pre-factor of $D_v$ while the order of magnitude of the diffusion coefficient is unaffected.²⁷² The right panel of Figure 30 contains the results calculated using three TB models. Two of the models give a correct shape but overestimate the barrier significantly, while for the OTB parametrization of Madsen et al.¹³³ the diffusing Fe atom flips its magnetic moment in the middle of the path, which

<figure><img src="image_33.png" /><figcaption>**Figure 29:** Activation magnetic free energy for self-diffusion in bcc iron, from EIM simulations. The predicted Curie temperature (1050 K) is indicated by a vertical dashed line.</figcaption></figure>
<figure><img src="image_34.png" /><figcaption>**Figure 30:** Minimum energy path associated with the migration of a vacancy in bcc Fe along the (111) direction; (a) results obtained using three interatomic potentials, (b) results computed using OTB models.</figcaption></figure>

leads to a discontinuous lowering of the energy. These results illustrate that it is challenging for the approximate models to capture the energetics of vacancy migration quantitatively correct.

The high efficiency of interatomic potentials enables to study the diffusion of a single vacancy directly by MD simulations. We present here results obtained for supercells containing 1999 Fe atoms (i.e., 2000 atoms arranged in a bcc lattice with a single vacancy). Each calculation was performed in the NVE ensemble, with the volume adjusted to obtain zero pressure. The single-vacancy diffusivity, $D_v$, was extracted from the mean-square displacements of all atoms in the simulated system, cf. Equation (18). MD time-length of 2 ns was found sufficient for a reliable determination of $D_v$ at high temperatures, while much longer times (of up to 10 ns) were necessary to achieve a sufficient number of jump events at lower temperatures. The simulation results are shown in Figure 31. The slopes of the obtained dependencies $D_v(T)$ in Arrhenius coordinates are close to the $E_v^m$ computed using NEB. In addition to the classical MD simulations with three interatomic potentials, we performed also the coupled MD-SD simulations (using the ADP potential only). The comparison of diffusion coefficients obtained in MD and MD-SD simulations indicates only a weak and constant effect of the magnetic excitations on the vacancy mobility in the present SD model. Figure 31 also contains the prediction of the EIM model which shows a strong non-Arrhenius behavior. The large variations in the obtained results ($D_v(T)$ slopes as well as absolute values) again demonstrate that rather small differences in the fundamental quantities (formation and migration energies, phonon spectra) may lead to diffusivities varying by several orders of magnitude.

Finally, the temperature dependence of the self-diffusion coefficient, $D_{Fe}^{Fe*}$, obtained via MD-SD as the product of $c_v^{eq}(T)$ and $D_v(T)$, is included with those of other methods in Figure 28. At low temperature, where magnetic excitations are negligible, the prediction is close to the experimental data, but it does not exhibit the characteristic change close to $T_c$ despite the qualitatively correct temperature dependence of $G_v^f$ at high temperature (see above). Hence, a satisfactory representation of self-diffusion in the PM state by interatomic potentials requires further work.

<figure><img src="image_35.png" /><figcaption>**Figure 31:** Dependence of single vacancy diffusivity on temperature in bcc Fe as predicted by three classical interatomic potentials using MD and coupled MD-SD simulations; the plot includes also results of the EIM model for comparison.</figcaption></figure>
## 5.3 Solute diffusion in α-Fe (dilute case)

### 5.3.1 Experimental results

Recently, diffusion of Mn in α-Fe has been measured using a combination of mechanical grinding and ion-beam sputtering techniques.⁶¹ The results are presented in Figure 32 and compared with the previous data which were measured over significantly narrower temperature intervals mainly in the PM state of α-Fe.

The diffusion coefficients of Mn measured across the Curie temperature follow almost a single Arrhenius line substantiating only a marginal, if any impact of the magnetic transition on Mn diffusion. If the diffusion coefficients of Mn would be fitted separately for the PM and FM regions, a difference of about 10 kJ/mol (0.1 eV/at) might be indicated, though this value is within the experimental uncertainties. By applying the Ruch model,²⁴⁰ Equation (19), the value of α was estimated to be 0.056 ± 0.024.⁶¹

In Figure 33 the diffusion behavior of Mn is compared to that of other solutes. While the magnetic ordering is seen to slow down diffusion of other investigated solutes, Mn is an exception for solutes with a strong magnetic moment.⁶¹ In Table 6 the available diffusion data are summarized.

### 5.3.2 SSA: Mn diffusion

To understand the Mn diffusion profile as observed in the tracer diffusion experiments discussed above, the migration

<figure><img src="image_36.png" /><figcaption>**Figure 32:** Arrhenius diagram for the impurity of Mn in α-Fe measured using two sectioning methods (circles⁶¹) in comparison to the previous data of Kirkaldy et al. (dashed-dotted line)²⁷³ and Nohara and Hirano (dashed line).²⁷⁴ The black solid line gives the iron self-diffusion coefficient by Iijima et al.¹⁸⁰</figcaption></figure>

<figure><img src="image_37.png" /><figcaption>**Figure 33:** Experimental Arrhenius diagram for solute diffusion of Co,²⁵⁰ Ni,²⁵¹ Cr,²⁵³ Cu,²⁵⁴ Zn,²⁵² Nb²⁴⁹ and Mn⁶¹ in α-Fe across the Curie temperature in comparison to that of Fe self-diffusion.¹⁸⁰</figcaption></figure>

barriers for different jumps are calculated from DFT for both the FM and the PM states and are listed in Table 7.

The SSA relaxation method has been employed to calculate migration barriers in the PM state. These migration barriers are further used as input for Le Claire's model²⁴² to evaluate the ratios of Mn diffusion coefficients to self-diffusion coefficients as shown in Figure 35. The values are in good quantitative agreement with both the experimental and EIM results. Further, using these ratios and the self-diffusion coefficients discussed in 5.2.3, we obtain Mn diffusion coefficients as shown in Figure 34.

### 5.3.3 EIM: Mn and Cu diffusion

Figure 36 provides a comparison of the tracer diffusion coefficients for Fe, Cu and Mn in α-iron, as resulted from the

<table><caption>**Table 7:** The migration barriers for the nearest neighbor jumps of the vacancy in both FM and PM states obtained with DFT. The numbering of the migration energies $E^{(i)}$ corresponds to that for the jumps frequencies $\Gamma_i$, which are shown in Figure 24: $E^{(0)}$ is the migration energy in pure Fe, $E^{(2)}$ is the migration energy of Mn solute in Fe, $E^{(3)}$ is the migration energy of Fe jump that dissociates Mn and vacancy, and finally $E^{(4)}$ is the migration energy of Fe jump that associates Mn and vacancy.</caption><thead><tr><th></th><th colspan="2">Migration energies (eV)</th></tr><tr><th></th><th>FM</th><th>PM</th></tr></thead><tbody><tr><td>$E^{(0)}$</td><td>0.73</td><td>0.43</td></tr><tr><td>$E^{(2)}$</td><td>0.46</td><td>0.41</td></tr><tr><td>$E^{(3)}$</td><td>0.64</td><td>0.42</td></tr><tr><td>$E^{(4)}$</td><td>0.58</td><td>0.36</td></tr></tbody></table>
<figure><img src="image_38.png" /><figcaption>**Figure 34:** Arrhenius diagram for Mn diffusion in α-Fe: comparison between the SSA and EIM models, and the experiments. Figure taken from Reference 61.</figcaption></figure>

<figure><img src="image_39.png" /><figcaption>**Figure 35:** Ratio of Mn diffusion to Fe self-diffusion coefficients, as a function of temperature: comparison between the SSA and EIM models, and experiments. Figure taken from Reference 61.</figcaption></figure>

EIM + MC approach.⁵⁸,⁶¹ The simulation results are compared with the experimental results of Iijima et al.¹⁸⁰ and Lübbenhusen & Mehrer²²⁸ (Fe), Hegde et al.⁶¹ (Mn) and Salje et al.²⁵⁴ (Cu).

In order to simulate solute diffusion in iron, we adopted the same methodology as for the self-diffusion. We parameterized an EIM to describe interactions between Fe, the solute and a vacancy, all at bcc lattice positions, and another EIM with an atom (Fe or the solute) at a saddle-point site. We

employed DFT data on solute-vacancy binding energies and atom-vacancy exchange barriers, with various spin configurations. The numerical parameters of the various EIMs are given in Reference 58,60.

From the results (Figure 36), we note a kink of a different size at the Curie point, in the Arrhenius plot of the different elements, which is proportional to the respective ΔQ (difference of activation energy between the FM and the PM regimes). The ΔQ is the largest for Fe (0.46 eV), followed by Cu (0.22 eV), and smallest for Mn (0.11 eV). This hierarchy is in full agreement with available experimental data. In addition, it is interesting to note that the difference of activation energy (the slope of the Arrhenius curve) between the different solutes is significantly larger in the FM state (up to 0.48 eV) than in the PM state (up to 0.13 eV).

Figure 36 also shows that the kinetic correlation factor $f_2$, Equation (17), for Cu and Mn diffusion increases with temperature up to an asymptotic limit of 0.73, which is the $f_0$ value in pure bcc Fe. To clarify the role of magnetic disorder on $f_2$, we have performed similar MC simulations for Cu and Mn diffusion, but imposing a perfect FM order for all atomic-MC temperatures. The results show that the kinetic correlation factor of solute diffusion increases much more slowly in the absence of magnetic disorder. This observation, together with the smaller difference between the activation free energies at the PM than at the FM state, suggest a dominance of the magnetic-disorder over the chemical-interaction effect in the very dilute Fe-X systems. Indeed, the magnetic free energy of binding between a vacancy and a solute atom (both Mn and Cu) practically vanishes in the high-temperature limit.

### 5.3.4 Comparison between different solutes

As can be observed experimentally (Figure 33), Fe and any solute in bcc-iron exhibit a rather similar activation energy (Q) in the PM regime, while their Q in the FM state can be significantly different. Therefore, the size of the kink and the difference between the FM and PM activation energies (ΔQ) of the distinct solutes are mainly dictated by the Q in the FM state. To better understand this, we have determined, by means of DFT, the solute-vacancy binding energy and the solute-vacancy exchange barrier (two important ingredients of the activation energy) in α-Fe for all the 3d elements.

It is well known that several properties of the magnetic 3d elements (Cr, Mn, Fe, Co, Ni) present a deviation from the expected parabolic behaviour as a function of the d-band filling.²⁷⁵ We have therefore computed the binding energy (Figure 37a) and the exchange energy barrier (Figure 37b) under two different conditions: (i) either imposing the solute magnetic moment to zero while relaxing the moment of all
<figure><img src="image_40.png" /><figcaption>**Figure 36:** Tracer diffusion coefficients for Fe, Mn and Cu in iron from EIM + MC and experiments (left), and the corresponding kinetic correlation factors (right). For the latter, the values are also determined with constraining the system to a FM state, for a comparison. The experimental results of Iijima et al.<sup>180</sup> and Lübbenhusen & Mehrer<sup>228</sup> (Fe), Hegde et al.<sup>61</sup> (Mn) and Salje et al.<sup>254</sup> (Cu) are shown.</figcaption></figure>

other atoms, or (ii) relaxing the magnetic moment of all the atoms. In the former case, we observe the expected nearly-parabolic behaviour, while we note some deviation from the overal trend for certain magnetic elements, especially in the case of Mn and Co. For Mn, the very strong deviation leads to an increase of vacancy-Mn binding energy and a decrease of vacancy-Mn exchange barrier, both of around 0.2 eV. Such a decrease of the activation energy (Q) of around 0.4 eV in the FM regime, makes it approaching the value of Q in the PM state. Therefore, the resulting "kink" in the diffusion coefficients is significantly reduced for Mn in α-iron, as shown both experimentally and by DFT-based simulations.<sup>61</sup> For Co, the deviation leads to an opposite effect, i.e., an increase of the Q in the FM regime, and therefore a larger kink at the Curie point.

These deviations can be rationalized by comparing the energy differences between the system with and without imposing the solute moment to zero, with the three local environments for the solute. We see from Figure 38a that, at the saddle-point position, they follow the solute moment magnitude (Figure 38b), in the bulk and next to the vacancy, except for Mn, and for Co in a less extent.

For Mn, the deviations are due to the strong local-environment dependence of these energy differences shown in Figure 38a, with the smallest ΔE for a Mn solute in the bulk, and then next to the vacancy. As a peculiarity of Mn, DFT calculations predicted the presence of two energy minima for a single Mn solute in bcc iron. The AFM-Mn (with a large Mn moment anti-parallel to the Fe moments) is the ground state, and the FM-Mn state, showing a small Mn moment,

<figure><img src="image_41.png" /><figcaption>**Figure 37:** Solute-vacancy interaction in α Fe. (a) Solute-vacancy binding energy and (b) exchange barrier from DFT. We show the values with fully relaxed magnetic moments and with the solute moment constrained to zero.</figcaption></figure>
<figure><img src="image_42.png" /><figcaption>**Figure 38:** Diffusion energetics of solutes in α Fe. (a) Energy difference between the cases with the solute-moment relaxed and constrained to zero. (b) Relaxed magnetic moment magnitude of the solute (in μ<sub>B</sub>). Data are shown for the solute in the bulk, next to a vacancy, or in the saddle-point position.</figcaption></figure>

parallel to the Fe moments. The energy differences for Mn shown in Figure 38a, follow the same trend of the energy difference between the AFM-Mn and the FM-Mn states. In the bulk, there is only 0.07 eV of energy difference between AFM-Mn and FM-Mn. The FM-Mn configuration is energetically less favorable next to a vacancy (with an energy difference between FM-Mn and AFM-Mn of 0.28 eV), and such energy difference is even larger for a Mn at the saddle-point position (0.53 eV).

### 5.3.5 Impact of external magnetic field on solute diffusion in α-Fe

The magnetic state was shown above to influence self- and solute diffusion in, e.g., α-Fe resulting in characteristic slow down of the diffusion rate with FM ordering. The ferromagnetic ordering near the Curie temperature can be influenced by an applied external magnetic field. Thus, one may expect an impact of the magnetic field on element diffusion in α-Fe near the Curie temperature.

Indeed, application of external magnetic fields was found to affect strongly solute diffusion in the ferromagnetic materials changing the diffusion coefficients by 20–70 %.²⁷⁶⁻²⁸⁰ In the case of pulsed magnetic field, diffusion of solutes becomes to be generally retarded, although a controversial enhancement of Sn diffusion at the frequency *f* = 13 Hz is seen, Figure 39. In Figure 39 the diffusion coefficients of Sn in α-Fe measured in a pulsed magnetic field of the intensity *H* = 0.5 T and the repetition frequency *f* normalized on the corresponding value without filed, *H* = 0, are shown. In experiments, the measured diffusion enhancement/retardation might be triggered by magnetic field-induced changes of the jump barriers or attempt frequencies, on one hand, as well as by contributions from pipe diffusion.

Two different methods have been applied for the measurements of the diffusion coefficients, X–Ra²⁷⁶ and

conventional Secondary Ion Mass Spectroscopy (SIMS) analysis (for a critical discussion of the existing methods for the experimental determination of the tracer diffusion coefficients see e.g. a recent overview¹¹⁰). The mechanism is still not properly understood, especially in the view of a relatively low amplitude of the applied magnetic fields, often below 0.7 T.²⁷⁷,²⁷⁹ A rigorous theoretical description and modelling of the influence of external magnetic field on self- and solute diffusion has to be still elaborated.

## 5.4 Diffusion in concentrated alloys

There are a few experimental investigations of diffusion in concentrated bcc iron-based alloys: Borg and Lai in Fe–Si,²⁸¹ Raghunathan and Sharma in Fe–Al and Fe–Si,²⁸² Braun and Feller-Kniepmeier in Fe–Cr,²⁵³ Jonsson in Fe–Cr–Ni,²⁸³ some of them dealing with the effect of the magnetic transition. Braun and Feller-Kniepmeier, for example, have shown that the interdiffusion coefficient in Fe–Cr alloys displays a complex evolution with the Cr concentration, due to the effect of the magnetic transition, but also to the proximity of the *α* – *α'* miscibility gap.

### 5.4.1 Diffusion in Fe–Mn alloys

Diffusion of both, Mn and Fe, was recently measured in a series of bcc Fe–Mn alloys with up to 1.9 wt.% Mn in the temperature interval from 838 K to 1,120 K in order to address the impact of magnetic transition on both self- and solute diffusion.²⁸⁴ The results are presented in Figure 40a and b for temperatures of 1,120 K and 838 K, respectively (filled symbols) and compared with the predictions of the EIM-KMC simulations (open symbols).²⁸⁴

While the magnetic transition was found to have a marginal impact on Fe enhancement factors, Mn diffusion in
<figure><img src="image_43.png" /><figcaption>**Figure 39:** A relative value of the diffusion coefficient of Sn in α-Fe as a function of the pulse frequency of the applied magnetic field of 0.5 T at three investigated temperatures.²⁷⁶ The measurements were performed using X-ray technique (open symbols)²⁷⁶ or SIMS profiling (filled symbols).</figcaption></figure>

paramagnetic Fe–Mn alloys was observed to be significantly enhanced with increased Mn content, Figure 40a. On the other hand, Mn diffusion changes only slightly with increasing Mn content in the ferromagnetic Fe–Mn alloys, see the data at 838 K in Figure 40b.

The experimental findings are supported by theoretical results, which are analyzed in terms of the various constituents of the tracer diffusion coefficients.²⁸⁴ The experimentally observed diffusion behaviour (filled symbols) is first in overall good agreement with results of kMC simulations using the same EIM as in Reference 61 (open symbols), Figure 40. The simulation results show a more visible

composition dependency of the Mn than the Fe diffusion coefficients.²⁸⁴ These results also allow us to gain insight into the distinct behaviour of the equilibrium vacancy concentration, the effective jump frequencies, the correlation factor and the solute-vacancy binding energy. For instance, above the Curie temperature (1,120 K), we note a larger Mn-vacancy binding energy with increasing Mn concentration, which is the main reason of the increased Mn diffusion coefficients with respect to the infinitely diluted case. On the other side, below the Curie temperature (838 K), we note a decrease of the correlation factor and an increase of the solute migration energy with a larger Mn content, making the overall decrease of the Mn diffusion coefficients comparable to the infinitely diluted case.²⁸⁴

As mentioned in previous subsections, there is a negligible impact of the magnetic transition on the diffusion of Mn in bcc-Fe, reflected by a very small “kink” in the Arrhenius plot around the Curie point. Interestingly, our study suggests an increase of the size of such “kink” with increasing Mn content. For instance, it becomes similar to the one of Fe diffusion with 2 % of Mn.

### 5.4.2 Diffusion in Fe–Ni alloys

The role of magnetic disorder was evidenced via DFT-informed EIM-kMC simulations of diffusion in PM fcc Fe–Ni alloys for the whole range of composition, revealing the diffusion-behavior difference between the paramagnetic and the magnetic ground states.²⁸⁵ Generally, predicting atomic diffusion in concentrated magnetic systems is challenging due to thermal magnetic effects and complex magneto-chemical interplay. In Reference, 285 an efficient approach is proposed employing kMC feded by ab initio parametrized models. The weak composition dependence of

<figure><img src="image_44.png" /><figcaption>**Figure 40:** Mn (red symbols) and Fe (black symbols) diffusion in dilute bcc Fe–Mn alloys at 1120 K (a) and 838 K (b). The experimental data (filled symbols) are compared with the predictions of DFT-informed calculations (open symbols). The dashed lines represent the experimentally established trends.</figcaption></figure>
diffusion coefficients are successfully predicted and explained as a compensation of distinct contributions of their constituents, namely the equilibrium vacancy concentration and the effective jump frequency.

Interestingly, a much stronger composition dependence of the diffusion properties is predicted if the 0-K magnetic configurations are imposed instead of the PM state. This effect persists well beyond the dilute limit, regardless of the alloy composition. This results suggests that the magnetic disorder (in the PM state) tends to reduce the chemical-interaction effects. The different diffusion properties of ordered and disordered magnetic states substantiate that the magnetic ground state is not a proper descriptor of diffusion in the paramagnetic concentrated alloys.

We also show that the atom-vacancy short-range ordering is one of the main components of the tracer diffusion coefficient. It provides a generalization of the vacancy-solute binding term as part of the diffusion-coefficients expression, well known for extremely dilute alloys.²⁸⁵

### 5.4.3 High-throughput determination of concentration-dependent tracer diffusion coefficients: Fe–Ga as a case system

Tracer diffusion in concentrated alloys can conveniently be measured using conventional radioisotopes, as it was done, e.g., for the Fe–Mn alloys. However, a number of important elements, like Al, Si or Ga, do not have such radioisotopes.ᵇ A recently developed augmented tracer-interdiffusion couple approach²⁸⁶ could be applied to determine the tracer diffusion coefficients of such elements in binary alloys (including magnetic ones), if the tracer diffusivities of another component could be measured and a proper thermodynamic description of the alloy is available. Recently this method has been applied to the Fe–Ga system,²⁸⁷ a system with a strong composition dependence of the diffusion coefficients.

Tracer-($^{59}$Fe) and inter-diffusion were simultaneously measured in three diffusion couples Fe/Fe-16Ga, Fe/Fe-24Ga and Fe-16Ga/Fe-24Ga at 1,143 K in the PM region (Ga content in at.%). The results obtained for the couples with different end-members turned out to be in an excellent agreement with each other for the overlapping composition intervals. The influence of the molar volume on the measured tracer- and inter-diffusion coefficients was found to be marginal for the particular case of small deviations from Vegard's law of mixing.²⁸⁷ Using thermodynamic calculations, the Ga tracer

diffusion coefficients and the vacancy wind factor were determined via the Darken-Manning relation for the composition range of 0–24 at.% Ga.

The combination of (radio-) tracer source and diffusion couple experimental techniques was shown to allow a reliable determination of the composition dependence of tracer diffusion coefficients of Fe along the whole diffusion path in the PM state, black solid line in Figure 41. Strong composition dependencies of the inter-diffusion coefficients were established, too, blue dashed line in Figure 41. Furthermore, using a thermodynamic assessment,²⁸⁷ the tracer diffusion coefficients of Ga were determined, red dashed-dotted line in Figure 41.

The reliability of the method was checked measuring directly the tracer diffusion coefficients of Fe in end-members of the diffusion couples, black squares in Figure 41. Furthermore, the tracer diffusion coefficients of Ga were determined at selected concentrations (corresponding to the positions of the Kirkendall planes) from the measured ratio of the intrinsic diffusivities, red circles in Figure 41. The whole set of diffusion data indicates a strong requirement of further improvement of the thermodynamic description of the Fe–Ga alloys.

The elaborated augmented tracer-interdiffusion couple approach can be applied for magnetic materials and will allow to quantify separately the impacts of magnetism on atomic jumps and alloy thermodynamics in a high-throughput manner.

<figure><img src="image_45.png" /><figcaption>**Figure 41:** Diffusion in concentrated Fe-Ga alloys at T ≈ 1143 K: comparison between the tracer ($D^{Fe}$, $D^{Ga}$) and interdiffusion (D) coefficients in Fe–Ga alloys.</figcaption></figure>

<footnote>b In fact, such radioisotopes could be either short-living like $^{31}$Si (half-time 2.6 h) or very expensive, e.g. $^{26}$Al (extremely long half-time of about 7.4 × 10$^5$ years, correspondingly low activation probability).</footnote>
## 5.5 Diffusion in vicinity of extended defects

### 5.5.1 Impact of magnetic transition on grain boundary diffusion

Grain boundary self-diffusion of Fe in α-Fe was measured in a number of studies, however, the results remain controversial. Whereas no impact of the magnetic transition has been substantiated by Divinski et al.²¹⁸ for Fe self-diffusion in α-Fe of two different purity levels, a strong impact of the magnetic transition on Fe grain boundary diffusion has been reported by Iijima et al.,²¹⁷ see Figure 42a. The difference may be related to the experimental techniques used for the corresponding investigations, i.e., magnetron sputtering by Iijima et al.²¹⁷ versus mechanical grinding by Divinski et al.²¹⁸ Dislocation pipe diffusion, carefully discriminated from the GB diffusion contribution by Divinski et al.,²¹⁸ may have affected the results of Iijima et al.²¹⁷ due to an increased dislocation density in the samples. Note that Divinski et al.²¹⁸ measured Fe GB self-diffusion in Fe materials of two purity levels. The results occurred to be consistent with general trends, i.e., the purer the matrix is the faster is the GB diffusion and the lower is the corresponding activation energy, Figure 42a. Furthermore, the GB energies estimated using the data of Divinski et al.²¹⁸ agree qualitatively with the DFT-informed predictions, while an opposite trend is seen with the data of Iijima et al.,²¹⁷ see Section 4.3.2 and Table 3. Thus, the problem of the impact of magnetic transition on the GB self-diffusion rates remains experimentally unresolved.

One has to indicate that it was the so-called double product $P, P = δ × D_{gb}$, which was measured experimentally for self-diffusion at relatively high temperatures (above about 750 K for the data shown in Figure 42a) in the B-type kinetics after Harrison's classification.²⁰¹ Here $δ$ is the GB width and $D_{gb}$ is the GB diffusion coefficient. The value of $δ$ has been measured to be about 0.5 nm in pure metals and alloys.²²³,²²⁵,²⁸⁹

At lower temperatures, i.e. below about 750 K for the data shown in Figure 42a, the diffusion conditions corresponded to the C-type regime and the GB diffusion coefficients were measured directly. The results obtained by Iijima et al.²¹⁷ are presented in Figure 42a by circles. For a reliable comparison, the measured GB diffusion coefficients are multiplied by the GB width taken as 0.5 nm and shown in Figure 42a. One recognizes that the value $δ = 0.5$ nm is at least a reasonable estimate of the GB width in bcc Fe.²¹⁷

The data on solute GB diffusion in α-Fe reported so far do not reveal any significant impact of the magnetic transition on the kinetics of atomic transport along the interfaces.²⁴⁹ As an example, the data on Nb diffusion in α-Fe measured exclusively in the B-type kinetic regime are shown in Figure 42b, triangles.

Recently,²⁸⁸ GB diffusion of Mn in α-Fe has been investigated and the results indicate an absence of any measurable impact of the magnetic transition on solute (Mn) GB diffusion rates, Figure 42b. The so-called triple product $P, P = s × δ × D_{gb}$, was measured for solute GB diffusion at relatively high temperatures in the B-type kinetics after Harrison's classification.²⁰¹ Here $s$ is the segregation factor of

<figure><img src="image_46.png" /><figcaption>**Figure 42:** Grain boundary diffusion in α-Fe. (a) Grain boundary self-diffusion in α-Fe as reported by Iijima et al.²¹⁷ (black squares and circles) and by Divinski et al.²¹⁸ (Fe of a higher purity, triangles up, and Fe of a lower purity, triangles down). (b) Grain boundary diffusion of Mn²⁸⁸ (red squares and circles) and Nb²⁴⁹ (Fe of higher purity, triangles up, and Fe of lower purity, triangles down) in α-Fe. The circles represent the diffusion coefficients measured in the C-type kinetic regime multiplied by the grain boundary width, $δ$, taken as 0.5 nm. The estimation of the segregation factor for Mn atoms at about 800 K is indicated. The vertical dotted and dashed lines indicate the $a/γ, T_{a/γ}$, and Curie, $T_c$, temperatures, respectively.</figcaption></figure>
the solute (which is just unity in the case of self-diffusion). In Figure 42b, the corresponding results for Mn diffusion are shown by squares (turned out to be measured at temperatures above about 800 K).

At lower temperatures, $T < 800$ K, the diffusion conditions corresponded to the C-type regime and the GB diffusion coefficients were measured directly. For an adequate comparison of the B- and C-type type diffusion data, the GB diffusion coefficients of Mn measured in the C-type kinetic regime are multiplied by the GB width taken as 0.5 nm. The comparison provides an estimate of the Mn segregation factor, $s = P/(\delta D_{gb})$, and it is about 100 at 800 K. This value is in surprisingly good agreement with the theoretical predictions for Mn segregation at $\Sigma 5$ GBs in α-Fe, Section 4.3.4, if the Mn segregation factor is estimated as $s = \exp(-H_s/RT)$. Here $H_s$ is the segregation energy ($H_s < 0$ in the case of segregation).

Furthermore, the DFT-informed calculations allow to shed light on an apparent increase of the segregation factor determined as $s = P/(\delta D_{gb})$ with increasing temperature above 800 K. In the ferromagnetic state, the Mn atoms segregate to second-layer positions with respect to the GB core, whereas segregation directly to the GB core in the paramagnetic state (with a significantly lower segregation energy) dominates. The barriers for atomic jumps increase with the distance from the GB core that explains the enhancement of the triple product values $P$ measured in the paramagnetic state and near the Curie temperature. Still, the controversy is not fully resolved and the case of Mn GB diffusion and segregation in α-Fe needs further analysis. We might conclude that the magnetic transition affects GB diffusion of Mn influencing the segregation characteristics. The impact on the corresponding kinetic barriers has still to be elucidated.

### 5.5.2 Classical potentials: GB diffusion

Classical MD simulations provide the only means to directly investigate such a complex diffusion phenomenon as GB self-diffusion. Recent large-scale MD simulations,$^{77,234}$ albeit without any explicit magnetic degrees of freedom, revealed that the GB self-diffusion in bcc Fe is mostly governed by self-interstitial atoms, in contrast to the bulk diffusion where the vacancy mechanism is dominant. The main reason of such unusual diffusion mechanism is the extremely low formation energy of self-interstitial atom inside GBs.$^{234}$ Figure 43 shows representative results using the ADP potential together with an estimation of the semi-empirical Borisov's approach$^{216}$ (see Section 4.3.2) and experimental data. The simulations predict a significantly faster diffusion along GBs

<figure><img src="image_47.png" /><figcaption>**Figure 43:** Bulk and GB self-diffusion in bcc Fe predicted by classical MD simulations using the ADP potential and the experimental data.$^{180,217,218,228}$ The solid blue line corresponds to GB self-diffusion coefficient calculated with the Borisov's semi-empirical approach.</figcaption></figure>

in excellent agreement with experiments. However, the influence of magnetism still needs to be examined.

# 6 Conclusions and outlook

## 6.1 Summary and conclusions of the review

The key factors that make a rational design approach for Fe-based materials more challenging than for other materials are magnetism and magnetic excitations. Magnetism is a quantum property of materials and its impact on thermodynamic, kinetic or mechanical properties is far from obvious. With this article we review the present knowledge about this impact by focusing on three Fe-based binary alloys, Fe–Cr, Fe–Mn and Fe–Co. While magnetism plays a critical role in all these systems, they represent different materials properties and allow us to discuss different simulation aspects related to magnetism.

As outlined in Section 2, several classes of methods are considered. A particular focus is on DFT, since it is not only the “work horse” of modern materials modelling, but also forms the basis for all other modelling methods introduced in this review. We introduce tight binding, bond order potentials, classical spin-dependent potentials, effective interaction models, and give a brief outlook on alternative methods such as magnetic machine learning potentials. From the experimental point of view, we focus on measurements of thermodynamic, kinetic and structural properties in magnetic materials.

Section 3 starts with a discussion of thermodynamic properties and heat capacities of iron and its alloys. This enables, among other properties, to study magnetic and chemical phase transitions and the Curie temperature. Experimentally, the contribution of magnetic entropy to
heat capacity is evaluated by comparing measurements with and without an external magnetic field that yields magnetic saturation. On the theoretical side, the magnetic entropy prediction is quite challenging, in particular for the DFT-based approaches. In addition, an exact or approximate quantum treatment of the spin variables is necessary when employed for example in MC simulations. A significant phonon-magnon coupling, even in the case of pure Fe, is evidenced by DFT-SSA calculations. A similar investigation could also be performed for Fe alloys.

The mixing enthalpy is a key thermodynamic property, providing information on the ordering or unmixing tendency of an alloy. We have computed the mixing enthalpy of bcc Fe–Cr, Fe–Mn and Fe–Co alloys employing various approaches. In particular, DFT predicts the atypical sign change of the 0 K mixing enthalpy in Fe–Cr consistent with experimental observations. The semi-empirical models, TB, BOP and EIM, which include magnetic variables explicitly, are also able to capture this feature that is caused by a magnetic frustration occurring at low temperatures. This anomaly disappears in the PM state, as revealed by EIM-MC calculations. The latter approach also predicts a reduction of the absolute value of the mixing enthalpy for Fe–Mn and Fe–Co alloys in the high-temperature PM state, suggesting a reduction of bonding interactions due to magnetic disorder. Classical potentials, being parameterized on 0 K DFT data, can also reproduce the magnetically driven anomaly in the Fe–Cr mixing behaviour. However, they can only provide the mixing enthalpy at high temperatures correctly if they are combined with SD simulations.

An important goal of our studies is the prediction of magnetic and chemical phase boundaries. The Curie point in pure Fe or Co is determined via TB, BOP, empirical potentials + SD, and EIM in reasonable agreement with experimental data. The EIM–MC simulations also provide the composition evolution of the Curie temperature for bcc Fe–Cr, Fe–Mn and Fe–Co alloys. Together with the determination of the chemical phase boundary, the complete magnetochemical phase diagram can be obtained, as in the example presented for bcc Fe–Co. By employing approaches that include both chemical and magnetic variables explicitly, the interplay between magnetic and chemical phase stability can be elucidated as well. A promising next step is the analysis of potential effects of magneto-chemo-vibrational coupling on the phase stability of the alloys.

In Section 4, we focus on point and extended defects. We present dedicated experiments to investigate the magnetism of Cr, Mn and Mo solutes in Fe, as well as their impact on the magnetic state of the host material. Our DFT predictions are consistent with the experimental data in most cases, except

for the magnetism of Mn. Further studies are required to solve the apparent discrepancy.

An interesting comparison between various theoretical approaches, including DFT, EIM, TB, BOP and empirical potentials, can be obtained by analyzing thermodynamic properties of a mono-vacancy in iron. The DFT-SSA method can only give the vacancy formation energy in the limiting FM and PM phases while EIM-MC simulations provide a continuous determination of vacancy formation free energies versus temperature, including the effect of magnetic excitations and transitions. Although the effect of lattice vibrations can be estimated through the EIM parameters, the anharmonicity and magnon-phonon coupling effects are still missing. By applying TB, BOP or empirical potentials in MD + SD simulations, the vacancy formation free energy versus temperature containing both magnetic and lattice excitations, thermal expansion and their coupling could be predicted. However, parameterizations of these approximate models are usually not sufficiently accurate to achieve quantitative predictions. Including DFT-SSA data for the fitting of the EIM and of the models for coupled MD + SD simulations could improve their accuracy.

The second part of Section 4 is devoted to grain boundaries and stacking faults. After reviewing existing experimental results, the GB structure, energetics and GB segregation are discussed. All theoretical studies reveal an important impact of magnetic entropy on defect properties. For instance, we find a significant reduction of chemical interactions due to magnetic disorder. For example, the Mn segregation to Fe GBs is shown to be largely driven by magnetic interactions. Mn segregation energies and occupancies in the PM state are much smaller than those in the FM state. In general, magnetism is identified as a key descriptor for GB thermodynamics in Fe–Mn alloys, with a complex interplay of magnetism, local structure and chemistry. Experimental measurements of Mn GB diffusion in bcc Fe revealed a strong segregation of Mn at lower temperatures in the FM state, in full accord with the DFT-informed predictions.

In Section 5, we turn to kinetic properties of Fe-based alloys, in particular diffusion. After a short outline of models and challenges, the first focus is given to diffusion in pure Fe, motivated by a large amount of experimental results. This analysis is further extended to diffusion of solute atoms, including Cr, Mn and Cu. The SSA method provides, in principle, accurate migration barriers in FM and model PM systems, as the barriers are directly calculated via DFT. However, some approximations are still involved. For instance for the PM state, it is assumed that the characteristic time of spin fluctuations is much shorter than the time for atomic relaxations, for both stable and saddle-point
configurations. This assumption still needs a further confirmation. Besides, an empirical interpolation scheme is needed for intermediate temperatures. It is also not obvious how to extend this method to address diffusion in concentrated alloys, as the number of distinct barriers to be considered becomes very large. In this case, the coarse-grained EIM and TB/classical potential may be more efficient approaches.

The presently applied spin-atomic kMC with the EIMs and the MD-SD simulations using classical potentials provide a continuous and consistent description of the diffusion coefficients as a function of temperature and across the magnetic transition. The on-lattice kMC is more efficient for the treatment of thermally-activated events such as the vacancy-mediated diffusion. However, the lattice-vibration effects are currently only implicitly taken into account and the magnon-phonon coupling is neglected. These approximations are naturally taken care of in MD-SD. Hence, the MD-SD approach is more suitable for the study of diffusion in systems involving strong lattice deformation, for instance GB diffusion. Of course, the accuracy of the results depends strongly on the accuracy of the parameterized EIMs and interatomic potentials. Our current study seems to suggest that it is particularly challenging to achieve accurate empirical potentials for diffusion in Fe alloys.

Similar to the GBs, the migration barriers and activation energies for distinct atom-vacancy exchanges in the PM state are strongly reduced compared to the magnetic ground state. At the same time, an increase of the kinetic correlation factor upon magnetic disordering is observed. A reduced 'kink' in the diffusion-coefficient temperature evolution for Mn, compared to the case of Fe or other 3d solutes, is predicted. This result is shown to be in excellent agreement with the experimental data, and can be explained by the smaller difference between the FM and PM activation enthalpies for Mn diffusion compared to those for self-diffusion of Fe. Our results show a good agreement between SSA and EIM Fe self- and Mn solute tracer diffusion coefficients in bcc-Fe, that is also in line with experimental data. The EIM-kMC approach is employed to investigate these diffusion properties as a function of Mn content, up to 2%-Mn. The outcome is fully consistent with the experimental tracer diffusion data, evidencing the close interplay between the Curie transition and the alloy composition. Beyond diffusion phenomena in single crystals, we show a promising agreement on the GB self-diffusion coefficients between experimental polycrystalline data and MD simulation results that consistently confirm a faster GB diffusion with respect to the bulk diffusion in bcc Fe.

## 6.2 Outlook

The atomistic modelling approaches discussed here allow accurate predictions of the impact of magnetic excitations and magnetic transitions on thermodynamic and kinetic properties in three classes of Fe alloys. Some of the more advanced methods also consider the interplay between chemical interaction, magnetism and lattice vibrations. While a lot of progress has been made recently, this study reveals at the same time challenges to be overcome in future research.

On the methodological side, the efficiency of the approaches needs to be improved continuously in order to go beyond binary materials. On the one hand, workflows that automatize magnetically constrained DFT computations and enable efficient parameterization and validation of accurate magnetic interaction models need to be developed. On the other hand, atomistic simulation frameworks (such as, e.g., pyiron<sup>290)</sup> for the fast sampling of atomic positional and magnetic degrees of freedom need to be available in order that computations of thermodynamics and phase diagrams, kinetics, diffusion and segregation, as well as extended defects like grain boundaries and dislocations and their interactions become feasible. Furthermore, multi-component surrogate models of the atomic interaction that provide an accurate representation of the DFT reference need to be developed. Ideally, these models do not rely on a lattice, but capture the full complexity of a magnetic atomic bond. Modern machine-learning potentials, as discussed in Section 2.5.2, take now magnetism into account and give hope to expect numerous new insights.

With improved efficiency, several extensions of the physical concepts discussed here can be addressed. This includes the consideration of vibrational degrees of freedom as well as phonon-magnon-chemical coupling in the primarily magnetic EIMs discussed here. Furthermore, a combination of lattice-based and off-lattice MC approaches would allow one to consider the change of structural configurations due to a chemo-magnetic coupling. The simultaneous consideration of these multiple degrees of freedom forms in particular a major challenge for the simulation of precipitation dynamics. On the one hand, one has to resolve sophisticated details in the electronic structure that drive magnetic interactions and magnetic excitations. On the other hand, large scale simulations are required to capture the formation of precipitates, their interfaces and interactions with the matrix. The same applies to the role of magnetism for planar defects such as grain boundaries. As the example of chemo-magnetic coupling in Fe–Mn alloys