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Electronic Mechanisms on the Iso-Structural Phase Transition in Fe-Mn Solid Solution

Experiments overall suggested that dilute solid solution of manganese in body-centered cubic iron transforms from antiferromagnetic coupling into a ferromagnetic coupling at about 2 at.% Mn. Despite long-term theoretical studies, this phase transition is poorly understood, and the transition mechanism is still open. Based on DFT calculations with dense k-point meshes, we reveal that this"iso-structural"phase transition (IPT) occurs at 1.85 at.% Mn, originating from the shifting of 3d eg level of Mn across the Fermi level and consequent intra-atomic electron transfer within 3d states of Mn. The IPT involves a sudden change of the bulk modulus accompanied by a small yet detectable change of the lattice constant, an inversion of magnetic coupling between solute Mn and Fe matrix, and a change in bonding strength between Mn and the first-nearest neighboring Fe atoms. Our interpretation of this IPT plays an enlightening role in understanding similar IPTs in other materials.

In condensed matter physics, iso-structural phase transition (IPT) is particularly remarkable because of its unusual nature. For example, cerium (Ce) transforms from γ to α phase at ~8 kBar and ambient temperature with a volume collapse of ~17% and an electric-resistance drop of ~40% [1]. Experiments showed that IPTs in some materials are associated with intriguing phenomena, such as negative thermal expansion, giant magneto-elastic coupling, and large piezoelectric response, indicating potential technological applications [2][3][4]. Great efforts have been made in studying the IPT mechanisms for more than a half-century. Some studies concentrated on the role of phonon dynamics in these IPTs [5,6], while others concentrated on electronic origin. It was found that the electronic valence transition of rare-earth elements leads to the IPTs in YbGaGe [2], YbInCu 4 [7], Eu(Pd 1-x Au x ) 2 Si 2 [8] and SmS [9]. However, it has not been fully understood about the electronic origin of some other prototypical IPTs, including the γ→α Ce transition and the insulator→metal transition of V 2 O 3 [10]. This is because experimental measurements were obviously inconsistent with the valence transition model and meanwhile could not offer detailed information to interpret the electronic process during these IPTs [11,12]. From a theoretical viewpoint, a reasonable description of these IPTs needs accurately calculating correlation effects by treating the many-body N-electron configuration [13][14][15], which is beyond density-functional theory (DFT). The electronic density of states (DOS) of Ce in either phase calculated by DFT did not reproduce the famous three-peak structure of experimental spectra [12,16]. The mechanism of the insulator→metal transition of V 2 O 3 has been under debate so far [17,18]. Furthermore, current miscellaneous DFT methods, as well as merged methods with DFT, cannot produce two solutions (phases) for these materials and carry out transitions between them within a single available computational framework.
On the other hand, experiments suggested that a phase transition in body-centered cubic (bcc) Fe-Mn alloy with increasing Mn concentration [19]. Neutron diffraction measurements yield the μ Mn values of -0.82, 0, and 0.6 μ B , respectively at 0.79, 1.5, and 2.0 at.% Mn at room temperature [20][21][22]. At 2.0 at.% Mn and higher concentrations, all experiments presented positive μ Mn values [19,23,24]. Long-term efforts have been made in theoretical studying the variation of μ Mn with Mn concentration in bcc Fe-Mn alloy. Calculations of linear-muffin-tin-orbital (LMTO) Green's function method showed that ferromagnetic (FM, μ Mn >0) coupling and antiferromagnetic (AFM, μ Mn <0) coupling states of Mn with bcc Fe matrix are both stable, and suggested that fluctuations between the two opposite states are responsible for the discrepancies in experimental μ Mn data [25]. Recent DFT calculations using Perdew-Burke-Ernzerhof (PBE) functional indicated that the AFM coupling phase is the ground state, while the FM coupling phase, which may be referred to as the excited state, is in a rather flat local minimum in the energy landscape about 0.07 eV per Mn atom higher than the ground state [19,26]. Further calculations within special quasirandom structure model showed that the proportion of FM-Mn solutes is very small at dilute solution limit, but increases with increasing Mn concentration, which finally makes the average μ Mn turn from negative to positive at ~6 at. % Mn [19]. These results provide a detailed scenario for the μ Mn inversion as a function of Mn content and also an explanation for the much scattered theoretical μ Mn results [27]. Two other theoretical studies predicted that the inversion of μ Mn occurs respectively at ~13 at.% and ~1.5 at.% Mn in disordered bcc Fe-Mn alloy [28,29].

Within our reach, this phase transition might be firstly proposed by Mirzoev et al. based on
Tight-Binding LMTO calculations [29]. Generally speaking, it is still known little about the phase transition, especially about the underlying mechanism, which motivates this study.
From early theoretical studies, Ce, Mn, and Fe are all at the cross-over between electron bonding behavior and magnetic moment formation [30]. For these metals, the electronic delocalization-localization transition can be induced by small perturbations, which ultimately modifies their properties. In this sense, similar IPT might occur in a "simpler" metallic system only with 3d narrow bands, e.g., the bcc Fe-Mn solid solution, so that the mechanism can be studied more easily. In this letter, the phase transition from AFM coupling to FM coupling between Mn and Fe matrix is demonstrated to occur at 1.85 at.% Mn from DFT calculations using a dense k-point mesh. This phase transition is confirmed to be an "iso-structural" one, involving a shift of localized state, electron transfer, change of bonding strength, and sudden jump of the bulk modulus. Our calculations present a detailed view of the electronic process in this IPT, which is essential for further understanding of the complex interplay between electronic and lattice degrees of freedom. Besides, we find an excited state in FM coupling order at 1.39 at.% Mn [31]. This letter is arranged as follows. The variation of elastic and structural properties with solute content of seven Fe-X (X=3d transition metals) solid solutions are examined. Investigations on electronic structure and magnetism are made to explain the anomalous behavior in Fe-Mn alloy. An IPT is confirmed, and its mechanism is suggested. Enthalpy calculations are performed to address why only positive magnetic moments of Mn are detected in experiments at 1.85 at.% Mn and higher concentrations.
Discussions on the excited state at 1.39 at.% Mn, high-pressure effect and Mn-cluster effect on this IPT are presented in the end.
All ab initio calculations are performed by the projector augmented wave method within DFT, as implemented in the Vienna ab initio simulation package (VASP) [32,33].
Exchange-correlation functionals are parameterized by Perdew and Wang (PW91) within the generalized gradient approximation [34], while the interpolation of the correlation part is done by Vosko-Wilk-Nusair (VWN) method [35]. The pseudopotential that treats semi core 3p 6 as valence electrons is used for Mn, while the standard is used for all the other 3d elements.
Detailed DFT settings can be found in Refs. [31,36]. Six different solute concentrations from 0.4 to 2.08 at.%, are considered by six different supercells that each has one solute besides Fe atoms. The dimensions of these supercells and the corresponding k-point meshes are listed in under zero strain, is defined in Ref. [38].
As long-established, the elastic moduli of bcc Fe based solid solutions are linear functions of solute content, especially at the dilute solution limit [39]. For example, the calculated bulk modulus difference, ΔB values of six Fe-X (X=Ti, V, Cr, Co, Ni, and Cu) alloys decrease linearly with increasing solute content [31]. However, we find an exceptional case in Fe-Mn alloy. As shown by the filled circle in FIG. 1(a) [29]. These anomalous phenomena in Fe-Mn alloy may correlate with the dual character of the 3d 5 electrons of Mn [30], which is unique among these 3d solutes.
The underlying mechanisms behind these anomalous phenomena might be discovered by investigating the evolution of electronic structure with Mn concentration. Firstly, we examine the evolution of electron localization function (ELF) [40]   Towards a further understanding of this IPT, we examine the evolution of differential electron density ρ diff in (110) plane of these solutes. Herein, ρ diff is calculated by the electron density at a certain solute content subtracting that at the next lower solute content than the former, and a preprocessing is performed before the calculation [31]. There is only a slight change (within

5(a) and (b).
As we can see, the peaks in the downward branch of t 2g DOS shrink while the upward e g peak shifts in the low-energy direction to be below E F level from 1.39 to 1.85 at.% Mn. Our results of the shrink of t 2g peak and the simultaneously increased density of e g state at E F level are similar to the experimental observations of the shortening of the peak at around -2 eV and the simultaneously height-increasing of the peak at E F level in the photoemission spectra during the γ→α Ce transition [12]. These DOS curves perfectly correspond to the differential electron density observations; for example, the shift of the upward e g peak from above E F level to below corresponds to the electron transfer at 1.  Interestingly, according to the band structure theory suggested by Goodenough [41], it is the 3d electron transfer that gives rise to the inversion of magnetic coupling order and the In summary, our DFT calculations using dense k-point meshes describe two distinct solutions     Figure S1 presents Mn species in the PW91 pseudopotential files provided with the VASP distribution, which induces very small effect of Pulay stress [2]. In our case, the Pulay stress arising from the incompleteness of plane wave basis set leads to a constant shift of the output external pressure, so that it is added by 4.5 kBar as a correction. Furthermore, in this work, the calculated lattice constant and bulk modulus are respectively added by 0.0022 Å and -1.85 GPa to eliminate the effect of Pulay stress. In our relaxing calculations, the adopted k-point meshes are so dense that the output stresses are adequate for accurately calculating elastic moduli [3].
Noticeably, in the Brillouin-zone sampling, a Γ-centered k-mesh with even subdivision k-point mesh, as used in Ref. [5]. In the relaxing and the static calculations, the first order Methfessel-Paxton smearing method and the tetrahedron smearing method with Blӧchl corrections are respectively employed as in Ref. [5]. We adopt the force convergence criterion in Ref. [5], 0.01 eV/Å, in relaxing calculations. However, we employ the PW91 functional and the pseudopotentials that treat 3d 7 4s 1 and 3p 6 3d 5 4s 2 as valence electrons respectively for Fe and Mn in our calculations, in contrast with the PBE functional and the pseudopotentials that treat 3p 6 3d 7 4s 1 and 3p 6 3d 5 4s 2 as valence electrons respectively for Fe and Mn in Ref. [5].
We obtain a ground state with the magnetic moment of Mn (μ Mn ) of -1.803 μ B and the lattice constant of 2.8355 Å, and also an exited state with the μ Mn of 0.355 μ B and the lattice constant of 2.8318 Å. The μ Mn in the ground state is very similar to that in Ref. [5], -1.89 μ B , and that in Ref. [6], -1.91 μ B . The μ Mn in the excite state is very similar to that in Ref. [6], 0.321 μ B .
The DFT calculations in Ref. [6] employ an energy cutoff of 400 eV and a k-point mesh equivalent to a bcc cubic unit cell with a 16*16*16 shifted grid. Reference [6] uses the PBE functional and the pseudopotentials that treat 3d 7 4s 1 and 3d 5 4s 2 as valence electrons respectively for Fe and Mn. The total energy of the exited state is around 0.053 eV higher than that of the ground state, which is very similar to those in Refs. [5,6] FIG. S2). These results point out that there are equivalent probabilities for solute Mn to be FM coupling and to be AFM coupling (μ Mn~0 ) with Fe matrix at 1.85 and 2.08 at.% Mn. Figure 3( Preprocessing for plotting the differential electronic density ρ diff of (110) plane. Before calculating ρ diff , we subtract the superposition of the isolated atomic electron density from the electron density of Fe . FM and AFM are abbreviated for ferromagnetic and antiferromagnetic, respectively. ΔH and ΔV are respectively the enthalpy and the volume differences between the ab initio results of FM phase and the fitting results of AFM phase. The AFM phase is simulated by magnetic-moment constraining method implemented in VASP. The third-order Birch-Murnaghan equation [10] is fitted to these ab initio energy and volume values of AFM phase. The fitting results of bulk modulus (B) and volume at zero pressure, and the derivative of B relative to pressure are respectively 179.79 GPa and 615.47 Å 3 , and 3.27. The difference between the ab initio energies and the fitting results is less than 0.007 eV.