First-principles predictions of structural and magnetic phase stability in irradiated α-Fe

Abstract

We here use density functional theory and the creation-relaxation algorithm to investigate the appearance of polymorphism in α-Fe, driven by irradiation-induced microstructural changes. Local constriction leads to magnetic instability and provides excess energy required for structural phase transformation. Under extreme conditions, α-Fe undergoes local transformations into icosahedral C15 Laves phase with highly close-packed stacking and internal short-range ferromagnetic ordering, antiparallel to the bulk magnetisation. Analysing local magnetic moments and atomic volumes, in conjunction with the magneto-volume relations of different Fe structures, suggests two other alternatives for local phase transformation under irradiation conditions: the double-layer antiferromagnetic γ-Fe and non-magnetic ϵ-Fe.

GRAPHICAL ABSTRACT

IMPACT STATEMENT

This work predicts the formation of icosahedral C15 Laves phase structure with short-range ferromagnetic order in irradiated α-Fe using dynamic first-principles calculations. This finding suggests potential detection through advanced magnetic-based transmission electron microscopy techniques, with implications for understanding radiation damage in structural materials.

KEYWORDS:

• Density functional theory
• short-range ferromagnetic ordering
• C15 Laves phase
• polymorphism

1. Introduction

Iron undergoes several structural and magnetic phase transformations activated by temperature and/or pressure changes [1]. At ambient pressure, iron exhibits two primary crystal structures. In the ground state, the most stable form of iron is body-centred cubic (bcc) α-phase, and it demonstrates ferromagnetic (FM) ordering up to its Curie temperature of 1041 K [2]. By increasing temperature to 1184 K and at standard pressure, the α-phase transforms into the second form, i.e. the face-centred cubic (fcc) γ-phase, and it stabilises up to 1665 K. Within its stability range, the γ-phase exhibits paramagnetism, although theoretical studies reveal that its ground state is antiferromagnetic (AF) [3–5] at low temperature, far below its stability range. This AF ground state has been shown through low-temperature Mössbauer experiment [6] conducted on γ-phase precipitates in Cu [7] and CuAl [8], as well as on γ-phase thin films grown on Cu₃Au [9], confirming the predictions derived from first-principles calculations [4]. Furthermore, subjecting Fe films to cryogenic self-ion irradiation has been observed to bring about alterations in both structure and magnetism [10,11].

Additionally, according to first-principles predictions [5], γ-Fe not only exhibits the conventional AF ordering with alternating layers of up and down spin arrangements but also features an alternative AF state known as the double-layer antiferromagnetic (AFD) state. In this AFD state, there is a doubling of intra-layer coupling of short-range FM order, resulting in twice the number of parallel-aligned spins among nearest neighbours compared to the regular AF structure, as suggested by Herper et al. [5]. The AFD state demonstrates a higher µ of approximately 38% and a slightly larger equilibrium volume of about 3% compared to the regular AF state, aligning more closely with experimental observations of the γ-phase [5,7]. The short-range FM coupling inherent in the AFD state plays a crucial role in explaining the anti-Invar effect in iron, providing additional experimental support for the existence of the AFD state in nature [5].

Under high pressures, iron can also adopt a hexagonal close-packed (hcp) ϵ-phase as a third form [12]. The experimental measurements of Mössbauer experiments on ϵ-Fe prove that no magnetic ordering occurs at low temperatures (T = 0.03 K) and pressures up to 21 GPa [13].

The existence of such multi-crystallographic states is a common phenomenon among elements and is known as polymorphism [14]. Therefore, investigating the polymorphism of iron under extreme conditions, particularly understanding the factors that stabilise its various forms under irradiation, is crucial for fundamental physics, and this letter addresses this inquiry from multiple perspectives. Theoretical investigations through comprehensive total energy calculations, including band energy analysis and examination of the density of states (DOS), offer valuable insights into the structural stability and magnetic phase of irradiated systems. By examining the total energy landscape and electronic structure, these calculations provide a basis for predicting the stable crystal structures and elucidating the magnetic behavior of the system under irradiation.

2. Methodology

In this letter, we applied electronic structure calculations in the framework of density functional theory (DFT) in order to illustrate the potential occurrence of polymorphism in an irradiated α-Fe. Drawing from our recent DFT-driven creation-relaxation algorithm (CRA) calculations [15], we demonstrated that, in low-energy and low-temperature irradiation conditions, α-Fe undergoes a transformation into locally formed icosahedral Laves phase structures, specifically the C15-type cluster. Furthermore, we observed that the constituent atoms in these C15-type clusters were in antiparallel alignment with the host bcc lattice atoms while locally exhibiting spontaneous short-range FM characteristics. Our supplementary analysis in this study suggests alternative possibilities, including the possible formation of the non-magnetic (NM) ϵ-phase and AFD γ-phase, induced by irradiation in α-Fe.

The details of the ab initio calculation for this study can be found in [15]. Moreover, for obtaining the equation of states and energy minimisation of the cubic structures like α-Fe and γ-Fe, we employed a conventional lattice structure with 11 × 11 × 11 k-mesh sampling, while close-packed structures (i.e. ϵ-Fe and hypothetical icosahedral C15 structures) and body-centred tetragonal¹ AFD γ-Fe were treated with slightly smaller k-meshes (11 × 11 × 9), particularly in the c-axis directions. Other simulation criteria, e.g. cut-off energy of 350 eV and 0.1 meV energy minimisation convergence, remained consistent with the full volume relaxation calculations in [15].

3. Results and discussion

As shown in Figure 1, the total energies relative to the ground state energy of α-Fe are plotted as a function of the atomic volume (Ω_a) for various structures and magnetic states. Furthermore, we incorporated the average irradiation-induced energy (δE) accumulated in the damaged structure, i.e. 97.5 meV as determined in our recent DFT-CRA calculations [15] for an irradiation dose of 0.35 displacements per atom (dpa). This inclusion facilitates a comprehensive comparison with equation-of-state curves and allows for the prediction of potential phase transformations in damaged α-Fe. The calculated δE corresponds to 1131 K according to the energy-temperature equation (E = k_B.T), a magnitude sufficient to induce possible phase transformations, particularly those from FM α-phase to AFD γ-phase or to NM ϵ-phase.

Figure 1. Equations of state representing distinct iron structures with diverse magnetic ordering. The arrow denotes the energy shift observed in the damaged α-Fe structure up to an irradiation dose of 0.35 dpa. To interpret the colour references in this figure legend, the reader is referred to the web version of the article.

Referring to Figure 1, the formation of new structures with diverse magnetic states in Fe strongly depends on the Ω_a and packing factor of the constituent atoms, as noted elsewhere [5,16–18]. Specifically, the energetics of Fe indicate that for the lowest possible Ω_a of 10.2 ų/atom, the energetically most stable structure is NM ϵ-phase, as also experimentally reported under high pressure [13]. Whereas for the greater Ω_a of about 12.2 ų/atom, the two metastable states are either the high-spin (HS)² FM γ-phase or the hypothetical FM C15 Laves phase structure according to energetics of iron from our DFT calculations.

It is worth noting that the C15 Laves phase should be considered as a short-range ordered structure according to topological short-range order theory [12], rather than a long-range lattice structure. That is, within an irradiated structure, the irradiation-induced isolated self-interstitial atoms initially accumulate to form short-range order icosahedral clusters [15], as also demonstrated in Figure 2.

Figure 2. The formation of the short-range FM C15 Laves phase structure (blue atoms) among locally close-packed structure and bcc lattice atoms (red atoms) extracted from DFT-CRA calculation. The orange clouds, enveloping atoms with spin-down orientation (blue arrows), represent the three-dimensional spin density isosurfaces of atoms underwent spin flip or quench. Red arrows symbolize the spin-up orientations, while the white and black elliptical lines represent the logarithmic positive and negative drawing contours, respectively. For colour references, the reader is referred to the web version of the article.

Therefore, for small interstitial clusters, there is necessarily no need to construct long-range lattice structure. Instead, the energetically most favorable configuration for the aggregation of self-interstitial atoms is expected to be the three-dimensional icosahedral C15 structure, among other possible cluster configurations, as previously demonstrated by Marinica et al. [19]. Additionally, in complete agreement with other first-principles studies [18,19], our DFT-CRA calculations also revealed energy relaxation of up to 6 eV once the C15-type structures (e.g. clusters of triangular or hexagonal di-interstitial rings) are formed, compared to the conditions with the same number of isolated self-interstitial defects. It is noteworthy that further increasing the size of the atomic icosahedral cluster leads to strong distortions and accompanying strains, therefore, the icosahedral structure will eventually, as it grows, become unstable and transform into other forms of extended defects such as 1/2〈111〉 or 〈100〉 dislocation loops in damaged α-Fe [20,21].

Figure 2 shows an example of a locally formed short-range FM C15 cluster (blue atoms) obtained from our first-principles DFT-CRA calculations at an irradiation dose of 0.22 dpa, accompanied by respective atomic site indices. As seen, although the constituent atoms of the imperfect C15 cluster antiferromagnetically align with host atoms of the α-phase (red atoms), they are internally coupled with a short-range FM ordering. This short-range FM ordering is visualized through the three-dimensional spin density map isosurfaces (orange clouds) surrounding the C15-type atoms among other close-packed or bcc lattice atoms. The C15-type atoms undergo reductions in their Ω_a, leading to increased atomic forces and pressure. Consequently, they exhibit spin flip or quench and stabilise at an average µ of −0.7 µ_B, suggests a potential for detection using advanced magnetic-induced mapping technique coupled with transmission electron microscopy [22].

To examine more possible magnetic and/or structural phase transformation in the damaged α-Fe, we plotted the distribution of the local µ of all 1024 constituent atoms relative to respective local Voronoi polyhedral cell volumes (i.e. Ω_a) in Figure 3, corresponding to the irradiation dose of 0.22 dpa. Furthermore, to provide a thorough understanding of the potential irradiation-induced magnetic and structural phase changes, the magneto-volume relations (Figure 3(a)) and energetics (Figure 3(b)) of iron have been incorporated with the distribution of moment-volume (µ – Ω_a) data points of damaged lattice atoms.

Figure 3. Distribution of local µ of 1024 Fe atoms in relation to their local Ω_a at 0.22 dpa. The µ – Ω_a data points are compared with (a) magneto-volume relations in different iron structures and (b) energetics of iron in various magnetic states with respect to the α-Fe ground state energy. The black-dashed line separates LS/LV and HS/HV regions, emphasizing deviations in local µ and Ω_a of the atoms in the damaged α-Fe compared to the undamaged bcc structure. The blue-dotted line on the right panel shows the equilibrium atomic volume of NM/AF hcp structure, while yellow-shaded regions depicts volume expansion (ΔV) in irradiated Fe. *In the left panel (a), the magneto-volume curve for the C15-FM structure is multiplied by −1 for better comparison with the irradiated α-Fe, indicating antiparallel spin orientation with respect to the host bcc lattice atoms. For colour references, the reader is referred to the web version of the article.

In addition, to interpreter the correlations between atomic volume, magnetic moment and potential phase transformation in defected α-Fe, the plots are partitioned by vertical dashed lines in Figure 3(a) and (b), highlighting the high-spin/high-volume (HS/HV) and low-spin/low-volume (LS/LV) regions. These regions are referenced against the Ω_a = 11.37 ų/atom and local µ = 2.2 µ_B of an iron atom in a typical pristine α-Fe lattice. The yellow-shaded regions in Figure 3 also represent the predicted irradiation-induced volume expansion (ΔV) of about 2.2%, which has been reported in our recent study [15].

As seen in Figure 1, the increase in total energy (δE = 97.5 meV/atom) of the irradiated Fe is sufficiently high to thermodynamically initiate the FM α-phase to AFD γ-phase or NM/FM ϵ-phase transformations. Significantly, the irradiated α-Fe undergoes an average reduction in total magnetisation (δM) by −0.043 µ_B/atom within the expanded volume of ΔV, as illustrated in Figure 3(a) and reported in [15]. This reduction intersects with the magneto-volume relation of the AFD γ-phase, emphasizing the potential transformation from α to γ induced by irradiation.

Furthermore, within the HS/HV region depicted in Figure 3(a), the distribution of µ – Ω_a data aligns with the magneto-volume relation of the bcc structure. This observation suggests that, under irradiation conditions, the majority of the system retains its crystal structure, albeit with the formation and clustering of vacancies, as also demonstrated experimentally under cryogenic irradiation conditions [10]. Consequently, the local magnetic moments near vacant sites experience slight increases, asymptotically approaching the moment of a free Fe atom. In contrast, the LS/LV region exhibits a notable departure in the local moments from the magneto-volume relation of the bcc structure to those of highly close-packed structures such as FM-C15, AFD fcc, and even NM/AF hcp. To further substantiate this conclusion, a detailed examination involves comparing the equation of states of those aforementioned structures are compared with the µ – Ω_a data distribution, as indicated in Figure 3(b).

As shown in Figure 3(a) and (b), the DFT-CRA model causes a rearrangement of atoms in the LS/LV region, resulting in a localized reduction in Ω_a. This leads to the formation of diverse stress fields around displaced atoms, consequently quenching their local moments. The spontaneous spins of atoms experiencing reductions in interatomic spacing are distributed between 1 and −1 µ_B, indicative of a potential AF ordering with a high packing density. Notably, certain atoms lose their local spins, resulting in their magnetic moments being distributed along or close to the horizontal line of µ = 0. This behavior signifies a highly close-packed non-magnetic configuration similar to NM ϵ-Fe, as one can realize from Figure 3(a).

The formation of NM or even AF ϵ-Fe, as suggested by our DFT-CRA calculations, aligns seamlessly with the energetic considerations for NM and AF ϵ-Fe, as illustrated in Figure 3(b). Indeed, the clustering of µ – Ω_a data points around a vertical blue-dotted line, representing the ground state of the NM/AF hcp phase, reinforces the evidence for the influence of this close-packed structure in irradiated bcc Fe.

As mentioned earlier, the first-principles prediction of irradiation-induced phase transformation from bcc to NM hcp under increased pressure is expected and experimentally demonstrated by x-ray magnetic circular dichroism spectroscopy [23,24]. This transformation is attributed to a reduction in interatomic distance, leading to an overlap between the 3d-state and 4s-state bands and inducing the delocalization of 3d electrons by lowering the DOS at Fermi energy (E_F) and weakening the condition for FM ordering [25,26].

Referring to Figure 3, an intriguing observation within the constrained atomic volume range (9.5 < Ω_a < 10.5) reveals that C15-type atoms (depicted by data points enclosed within red circles) distinctly exhibit FM ordering. Clearly, these FM-ordered C15-type atoms demonstrate antiferromagnetic alignment with the host bcc lattice atoms (depicted as black circles).

A more in-depth analysis of the partial projected DOS (PDOS) for the atoms forming the C15 cluster (as shown in Figure 2) before and after irradiation unveils a charge transfer phenomenon. Specifically, as depicted in Figure 4, there is a charge transfer from majority states below the E_F into unoccupied minority states above it when the atom is moved into a constrained environment. As mentioned, by comparing the partial integrated DOS (IDOS), this charge transfer results in a reduction of the DOS at E_F by 26%, weakening the FM ordering, manifested by the upward shift of E_F to higher energy levels by approximately 1.3 eV (as illustrated in Figure 4(a)). Qualitatively, the transferred charge is schematically represented by the shaded area in the partial PDOS of the atoms in the undamaged structure (Figure 4(a)), which later form the C15 cluster following irradiation (Figure 4(b)).

Figure 4 The comparison of partial PDOS and IDOS of C15-type atoms before and after irradiation: (a) illustration of transferred charge from majority into minority state before irradiation and (b) established partial DOS of atoms forming the imperfect C15 Laves phase structure after irradiation. The vertical dashed lines show the Fermi energy levels, set at E_F = 0.

4. Conclusions

In summary, our investigation, employing electronic structure calculations within the DFT framework, sheds light on the potential polymorphic transformations occurring in irradiated α-Fe. The extensive DFT-driven CRA simulations for modeling low-energy, low-temperature irradiation without thermally activated diffusion, unveil a noteworthy transformation leading to the emergence of locally formed icosahedral Laves phase structures, specifically in the form of C15-type clusters. Within this localized structure, constituent atoms exhibit antiferromagnetic orientation in relation to the host bcc lattice atoms, while displaying spontaneous short-range ferromagnetic characteristics. This dualistic magnetic behavior adds a layer of complexity to the understanding of the structural and magnetic responses of α-Fe under irradiation. Moreover, our study elucidates the impact of irradiation-induced pressure on the system, revealing the possibility of NM ϵ-phase formation with a highly close-packed structure. Concurrently, the emergence of the AFD γ-phase is anticipated from the average increase in total energy and decrease in magnetisation of the irradiated α-Fe, respectively.

The transformations observed in the LS/LV regions result from a reduction in atomic volume, leading to the delocalization of 3d electrons at the Fermi level due to the overlapping of 3d and 4s bands. Consequently, the distinctive characteristics of the irradiation-induced polymorphism, such as spin–flip or quenching magnetisation traits, stabilise the local magnetic moment of the C15 cluster at approximately −0.7 µ_B. This suggests a remarkable potential for experimentally detecting C15 Laves phase structures using advanced magnetic-induced mapping techniques when coupled with transmission electron microscopy.

Acknowledgments

Financial support was provided by the Swedish Foundation for Strategic Research (SSF) under grant number ARC19-0043 (SUNRISE centre) and the Swedish Research Council (VR) for projects 2017-06458 and 2019-04156. The work was also affiliated with the EUROfusion Consortium, funded by the European Union under Grant Agreement No 101052200. Additional funding was received from the Euratom Research and Training Programme under grant agreements No 755039 (M4F project) and No 740415 (IL TROVATORE project) during 2019–2020. The authors acknowledge computational resources from the Swedish National Infrastructure for Computing (SNIC; PDC at KTH and NSC at LiU) and CINECA HPC Center in Italy. The views expressed are solely those of the authors and do not necessarily represent the views of the European Union or the European Commission, which cannot be held responsible for them.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by European Union: [Grant Number 101052200]; Swedish Foundation for Strategic Research (SSF): [Grant Number ARC19-0043]; SUNRISE centre, Swedish Research Council (VR): [Grant Number 2017-06458].

Notes

1 We selected the body-centred tetragonal cell with a ratio of c/a = 2√2 to represent the AFD structure, comprising two consecutive layers with up- and down-spin orientations. This choice, as described by Herper et al. [5], describes the short-range order FM within an antiferromagnetic ordering [5].

2 Depending on the Ω_a, γ-Fe exhibits two FM states. In small atomic volume, it has an unstable low-spin (LS) state with respect to AF states. The second form is associated with a high-spin (HS) state, which gains stability over AF state at expanded volumes [5].