Density functional theory study of the magnetic moment of solute Mn in bcc Fe

. An unexplained discrepancy exists between the experimentally measured and theoretically calculated magnetic moments of Mn in  -Fe. In this study, we use density functional theory to suggest that this discrepancy is likely due to the local strain environment of a Mn atom in the Fe structure. The ferromagnetic coupling, found by experiment, was shown to be metastable and could be stabilised by a 2% hydrostatic compressive strain. The effects of Mn concentration, vacancies and interstitial defects on the magnetic moment of Mn are also discussed. It was found that the ground state, anti-ferromagnetic (AFM) coupling of Mn to Fe requires long range tensile relaxations of the neighbouring atoms along <111> which is hindered in the presence of other Mn atoms. Vacancies and Fe interstitial defects stabilise the AFM coupling but are not expected to have a large effect on the average measured magnetic moment. manner with an average moment of ~0.06  B and median of ~0.70  B . By providing a statistically significant dataset, we show that the previous theory of a uniform switch from AFM to FM between


I. INTRODUCTION
Steels are ubiquitous in technological applications due to the abundance and low cost of Fe and its highly desirable mechanical and corrosion properties with alloying additions. Manganese is second only to C in its importance and use in steels. Historically, Mn has been included as both a minor and major alloying addition. The former to increase workability by supressing FeS formation and latter to increase ductility, through twinning induced plasticity, in the fairly recently developed, so-called TWIP steels. The state-of-the art theoretical description of Mn in Fe has a large impact on our understanding of phenomena such as solute clustering 1 and vacancy-solute clustering 2 , which compromise the structural integrity of the steels during operation. The former occurs in the ferritic phase of duplex steels as a result of thermal ageing (573 -773 K for >1000 h) [3][4][5] and in low-alloy steels [6][7][8] resulting from longterm (> 1 year) elevated temperature (~550 K); the latter, occurs due to neutron irradiation damage, which is of interest to life extension of nuclear fission reactors and for fast neutron damage of steels to be used in future fusion reactors. It is therefore important for atomic scale processes such as binding, substitution and migration to be understood at a fundamental level, to be used in high order methods and analyses [9][10][11][12][13] to model these phenomena in industrial settings.
Mn has long been regarded as one of the most troublesome transition metals in terms of predicting its magnetic behaviour and electronic interaction with other elements 14 . In its elemental ground state, it adopts a 58 atom cubic unit cell of spacegroup 4 ̅ 3 , and exhibits a non-collinear (NC) magnetic structure, whereby atoms located in 2a, 8c and two 24g sites (in Wyckoff notation) exhibit different spin vectors 15,16 . This magnetic structure undergoes a transition to paramagnetic (PM) structure at the Néel temperature of ~95 K 17 . In contrast, Fe is one of the most well studied transition metals [18][19][20] . In its ground-state, Fe adopts a ferromagnetic (FM) BCC structure 21 , which it maintains until its Curie temperature at ~1043 K 22 .
With minor alloying additions of Mn to Fe the crystal structure will remain BCC as a solid solution. The solubility limit is reached at ~3 at. % at ~600 K 23 , at equilibrium; however, increased additions are reported to be stabilised using cold working 24 . The magnetic structure of Fe-Mn alloys is highly variable and is thought to be affected by many different factors; namely, the Mn concentration 25 , the local atomic environment 26 , lattice parameter 14 and temperature 24 . This behaviour is observed to a much lesser extent for Fe-Cu, Fe-Ni and Fe-Si solid solutions 27 . It is therefore no surprise that discrepancies exist between experimental observations and theoretical frameworks. In the current study, we use density functional theory (DFT) to study the effect of concentration, local environment, strain and point defects on the magnetic moment and stability of Mn in α-Fe.

II. METHODOLOGY
A plane-wave density functional theory method was used, as implemented in the Vienna Ab initio Simulation Package (VASP) 28 . The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 29 , based on the Generalised Gradient Approximation (GGA), is used with the projector augmented wave (PAW) method 30 . Pseudopotentials provided with the VASP distribution were used, where 14 electrons (3p 6 3d 7 4s 1 ) for Fe and 13 electrons (3p 6 3d 5 4s 2 ) for Mn are considered as valence electrons.
The k-points, energy cut-off and lattice parameters the perfect lattice of both elements (α-Fe and α-Mn) were converged. It was determined that a real space k-point density of 0.03 Å -3 and energy cutoff of 500 eV provided accurate results and were kept consistent for all calculations (resulting in 6×6×6, 4×4×4 and 2×2×2 k-point grids for the 54, 128 and 250 atom supercells). Constant pressure, full relaxations with Methfessel-Paxton 31 smearing (with a width of 0.1 eV) were used to obtain the groundstate configurations. The tetrahedron smearing method with Blöchl corrections 32 was used on fixed dimension/volume calculations to generate the electronic density of states (DOS). Spin polarisation effects were also included; non-collinear and collinear theory was used for the α-Mn structure. The electronic energy and ionic relaxation convergence criteria were set to 10 -6 and 10 -4 eV, respectively, for all calculations.
Supercells of 54 (3×3×3), 128 (4×4×4) and 250 (5×5×5) Fe atoms were relaxed at constant pressure. To understand the behaviour of a dilute concentration of Mn in the -Fe lattice, a single Mn was substituted for an Fe atom in each of the supercell sizes. For higher concentrations of Mn and Mn-Mn interactions within the Fe matrix, a Mn content of 4.69 at. % was achieved by pseudo-randomly selecting six lattice sites for Mn atom substitution in the 128 atom supercell. As this method is stochastic in nature, this was repeated to produce 20 unique supercells (120 Mn atoms in total) to achieve a good statistical understanding of the system. Cohesive, binding, substitutional, vacancy and interstitial formation energies were calculated following standard procedures, outlined in the supplementary material.
Linear elastic theory, as implemented within the ANETO framework 33 for calculating the dipole tensor using the strain method 34,35 , was used calculate the relaxation volume and interaction energies for Mn in BCC ( Fe in the. In accordance with past literature, and the observed equilibrium behaviour of pure Mn, the -Mn phase is the most energetically favourable at 0 K 42 . This is then followed by  and  phases, which are temperature stabilised. The different known magnetic arrangements of -Mn were simulated, again predicting the behaviour as found in past literature 14,16 . The magnetic moments of each atomic site can be found in Table S1 (supplementary material). The overestimation of the cohesive enthalpies compared to the experimental values can be attributed to the approximation of the exchange-correlation functional 36,43 .  45 and PAW method and that there is a large discrepancy between the magnetic moment. This was attributed to the inability of the USPP method to correctly represent the semi-core d electrons. In this section we investigate the effect of magnetic moment of Mn on Esub. It is known that the magnetic moment of Mn is particularly difficult to converge in -Fe as a shallow energy landscape exists between local FM and AFM states 46 (where all magnetic states are henceforth described, locally, in reference to the Fe matrix). Within this study it was found that the AFM state of Mn is the ground state in -Fe and that it is also possible to model the FM state in 128 and 250 atom supercells, provided a sufficiently high initial magnetic moment is set prior to energy minimisation, see  These results demonstrate that it is possible to converge to both the FM and AFM state of Mn when its initial magnetic moment is 4 or ≤ 2, respectively, for supercell sizes ≥128 atoms. The corresponding 6 substitutional energies highlight the importance of the resultant magnetic moment of Mn to its stability in the -Fe matrix. When it is in the FM state the Esub becomes less favourable by 0.10 eV and 0.05 eV for the 4×4×4 and 5×5×5 supercell, respectively. Since it is predicted that the AFM state is the more stable of the two, it is recommended that it is used as the reference state for the calculation of binding energy calculations.

Magnetic moment and stability
When assessing the local density of states (LDOS) of Mn in -Fe, the FM and AFM spin-states are distinct, see Fig. 1. The difference in states is observed as the shift in the anti-bonding peak of the majority spin channel, from above the Fermi level for the AFM, to below the Fermi level in the FM magnetic coupling of Mn. The occupation of the anti-bonding peak explains the relative stability difference between the two states 47 .

Calculated vs experimental magnetic moment
In this section we compare the DFT calculated magnetic moments in the current study to the past experimental studies that used diffuse neutron scattering. For the former, the Mn concentration was varied by using the three different supercell sizes 250, 128 and 54 atoms correspond to 0.40, 0.78 and 1.85 at. %, respectively. A higher concentration of 4.69 at. % (above the expected solubility limit) was obtained by randomly populating the 128 atom supercell with 6 Mn atoms, repeated in 20 supercells; the average of all the magnetic moments were taken and the standard error is plotted as error bars. Figure 2 shows that there is little agreement among the experimental data and that the ground state theoretical values, obtained in this work, over-predict the magnitude of the magnetic moment below 5 at. %. However, it should be emphasised that the solubility of Mn in -Fe is at maximum ~3 at % at 600 K 23 . In the studies that surpass this solubility, Nakai et al. 24 reportedly stabilises the -phase by cold working and the study by Radhakrishna et al. 26 does not provide such information. Formation of secondary phases such as -(Mn,Fe) is likely to occur at equilibrium for Mn concentrations >3 at. %.
Nevertheless, the results in the current study are in better agreement with experimental findings for Mn in excess concentrations. In the following sections, the possible factors that lead to the discrepancies between experiment and theoretical calculations and how these differences can be captured through a mechanistic understanding of the Fe-Mn system are explored.

Atomic relaxations
By taking the difference between the atomic coordinates of the relaxed and unrelaxed supercells, the vectors corresponding to the atomic relaxations upon substitution of Fe with Mn can be calculated.  For the smaller supercells (54 and 128 atoms) the long-range movement is more constrained and may explain the slight variance in magnetic moment and absence of FM moment for the 54 atom case. Figure 4 shows the elastic self-interaction energy 48,49 . It can be seen that the supercells are well converged by 128 atoms. lattice is much more localised than the AFM, where the former has self-interaction energies three orders of magnitude smaller than the latter. These results suggest that the increased stability of the AFM state over FM state of Mn is due to electronic rather than elastic effects.

Mn concentration
The interaction between neighbouring Mn species in -Fe is not well understood. Past theoretical studies that report the binding energies between two Mn atoms are not in accord 27,44,50 . Within this study, a stochastic behaviour was found in the resultant magnetic moments of the Mn. When two Mn atoms were placed in the 54, 128 and 250 atom supercells, in different nn positions, each adopted FM or AFM moments with varying magnitudes and no clear trend (reported in Figure S1 of the supplementary material). The binding energies between the two Mn atoms varied by ±0.05 eV between supercells and did not correlate with their magnetic moment. Figure 6 reports the binding energies calculated for the 250 atom supercell, and provides a comparison with past literature. The calculation of the binding energy was done using the AFM and FM reference state, which results in a significant difference (0.14 eV) in energy. When comparing to the work of Olsson et al. 27 , the average binding energies calculated from the FM Mn in -Fe reference state are somewhat in agreement. However, the FM state is metastable therefore the AFM state should be taken as reference. This offset does not explain the discrepancy with respect to the work of Vincent et al. 44 . It is more likely that the difference stems from the use of USPPs and other theoretical parameters.
When considering the difference in atomic relaxations for AFM and FM isolated Mn atoms, it is perhaps no surprise that there is a large variation in magnetic moment between nearest neighbour Mn with varying supercell size. If Mn does indeed prefer to exhibit AFM behaviour, then a more complex atomic relaxation is required. However, at certain geometries of nearest neighbour Mn and without sufficient long-range movement allowed, it will be more favourable for AFM and FM Mn atoms to coexist.    53 ; it is expected that these volumes are of the m length scale (although not explicitly stated) in references from Section III(B) (2). Therefore, the overall average magnetic moment is an important result for comparison to literature.

Effect of strain
In the past, it has been demonstrated that the lattice parameter of -Mn has a large effect on its magnetic structure 14 . To-date this effect has not been investigated in the Fe-Mn system, however pioneering work by Cable et al. has explored this effect in Ni-and Co-Mn alloys 54 . Two methods of straining the environment around the Mn atom were used in this study, both using 128 atom supercells: Applying tension and compression (1) hydrostatically and fixing the volume and shape of the supercell while allowing internal relaxation of the atoms and (2) locally in the 1 st nn Fe, fixing supercell shape and atomic positions of atoms. The latter performed to investigate the influence of only the 1 st nn.
At the dilute limit, the effect of strain on the magnetic moment of Mn is clear; compression (negative strain) induces the FM moment and tension induces the AFM moment, see Fig. 8(A). The results from the hydrostatic method show a transition from AFM to FM moment between 0 and 2% strain. For the locally strained case, the transition occurs at higher compressive strains and to a lesser degree. This discrepancy is considered to be the result of the influence of the neighbouring atoms, other than the 1 st nn, which remain fixed. The difference in internal energy is calculated as: where and are the total internal energy of the strained and ground state supercell, respectively, normalised to the number of displaced atoms (nd). The results follow the expected parabolic distribution where deviation from the ground state configuration leads to less favourable energies ( Fig. 8(B)).  56 and Nakai et al. 24 . However, caution must be used when making direct comparisons to experiment. This is because the approximation of the exchange-correlation functional used by DFT introduces discrepancies to both the lattice parameter and magnitude of magnetic moment 57 . Therefore, this result should be taken qualitatively and an experimental relation between strain and magnetic moment is required to establish a quantitative relationship.

Effect of vacancies
Vacancies are ubiquitous in crystalline systems and vary in concentration depending on the condition and processing of the sample 58 . Vacancy-solute clusters are a documented phenomenon 59,60 and the binding of solute atoms to vacancies in -Fe is important for understanding solute partitioning, diffusivity and precipitation [61][62][63] . In Figure 9(A-C), the resultant magnetic moment, binding energy and vacancy formation energy of Mn substitution are shown as a function of distance from a vacancy from 1 st  6 th nn. The magnitudes and trend of the binding energies are in accordance with past theoretical findings 27,44,64 . It is predicted that it is favourable for Mn to bind to an Fe vacancy, with a relatively longrange attraction compared to other 3d transitional metals 27,65,66  These findings follow the trend identified within this study i.e. the FM moment is stabilised by compressive and AFM by tensile lattice strain. At equilibrium, without a vacancy, the 2 nd nn polyhedron volume containing Mn is calculated to be 45.60 Å 3 , which corresponds to the ground-state magnetic moment of 2 B. When only using 1 st nn polyhedron volumes no correlation was found.
From an applied perspective the equilibrium vacancy concentration is quite small in Fe ≤10 -6 below the melting temperature 2,67 . However, when subject to displacive radiation damage these concentrations can become non-trivial. It is possible that the increased stability when Mn is under local tensile strain, in the presence of a vacancy, is the basis for solute-vacancy stabilisation seen experimentally 2 or hypothesised to occur due to nucleation of solutes from defects in RPV steels 6 .

Effect of the <110> Fe interstitial
Unlike the majority of BCC metals, the most stable self-interstitial configuration in -Fe is the <110> dumbbell 68 . Although Mn has been shown to also have a strong binding to the <111> Fe dumbbell interstitial, it is expected that the <110> will be the most prevalent interstitial in -Fe 27 . For this reason, only the <110> orientation was considered with Mn at three positions around the Fe interstitial: Mixed (M), Compressive (C) and Tensile (T), see Fig. 10. Within this study these denotations are arbitrary, however, to remain consistent to past literature where the names are given in reference to their respective Voronoi polyhedra volumes 12 , we continue their use. From our current work, it is identified that the 2 nd nn displacement has a non-negligible effect on the magnetic moment of Mn (Section B(5)), therefore a Voronoi analysis is not expected to correlate with the resultant magnetic moment of Mn. In agreement with past work, the binding energies between Mn and an Fe <110> interstitial were favourable in all three sites, see Fig. 11. A difference in binding energy between FM and AFM Mn reference state was found to be 0.08 and 0.06 eV for 4×4×4 and 5×5×5 supercells, respectively.
Interestingly, the relative magnetic moments do not follow the same trend as seen with the vacancy defects. Analysis of the 2 nd nn polyhedra volumes again display a negative correlation with the magnetic moments, as seen with the vacancies. In the absence of Mn, the magnetic moment of the two Fe atoms in the <110> dumbbell configuration, exhibit a weak AFM coupling to the matrix 0.27(2) B. Further, the 1 st nn Fe atoms to the dumbbell display a significant deviation in magnitude of their magnetic moment from the matrix.
The 4 Fe atoms in the geometrically equivalent C and T sites yield magnetic moments of ~1.72(7) and ~2.42(1) B, respectively, which vary only slightly with Mn substitution. It is likely that the introduction of a self-interstitial atom sufficiently changes the atomic and magnetic structure so that the magnitudes of the 2 nd nn polyhedron volume vs magnetic moment relationship, determined for the undefective and vacated cases, no longer holds true. Nevertheless, the coupling of Mn to Fe is consistently AFM.

IV. CONCLUSIONS
• The discrepancy in magnetic moment of Mn in -Fe, that exists between theory and experiment, is likely due to susceptibility of the magnetic moment to change with strain. Here, it was found that a hydrostatic compressive strain of 2% (0.06 Å) stabilised the FM state of Mn atoms (relative to Fe). Strains imposed by quenching, cold working or thermal expansion are expected to dominate over the effect due to Mn concentration, vacancies and Fe interstitial atoms. Therefore, it is extremely important to provide experimental details pertaining to the lattice parameter and sample conditions in future assessment of the magnetic moment of Mn in Fe.
• In the dilute case, it is possible to simulate both FM and AFM states of Mn, where the former is metastable, by initialising the spin state on Mn to 4 and ≤ 2 B, respectively. The relaxations of the Fe atoms around the FM and AFM Mn differ significantly. A long range tensile relaxation in the <111> and compression in the <100> and <201> occur for the AFM case and strictly compressive relaxations for the FM. The AFM state is the ground state for dilute Mn defects in -Fe should be used as a reference state to calculate the binding energies of defect complexes in future work.
• At higher Mn concentrations, Mn exhibits AFM and FM spin states in a stochastic manner with an average moment of ~0.06 B and median of ~0.70 B. By providing a statistically significant dataset, we show that the previous theory of a uniform switch from AFM to FM between 2 -3 at. % Mn, is an artefact of inadequate statistical sampling. This randomness is likely due to a combination of shallow energy landscape between AFM and FM spin-states and the inability for relaxations of the surrounding atoms to occur for specific geometries and concentrations of Mn, as to allow for exclusively one state to exist.
• A vacancy and <110> Fe interstitial were both found to stabilise the AFM moment of Mn. The atomic relaxations due to the defects dominate over those around the Mn substitution.
Polyhedron analysis of the atoms within the 2 nd nn to Mn show a negative correlation between volume and magnetic moment with varying nn distance to the vacancy/interstitial. This result suggests that the relationship between magnetic moment and local strain still exists when in the presence of a vacancy or interstitial.