Uncommon clustering in dilute Ti-Fe alloys

We present the results of ab initio modeling of structure of dilute Ti-Fe, a typical representative of quenched Ti-based transition-metal alloys. We have demonstrated that beyond the solubility limit this alloy cannot be described in common terms of substitutional and interstitial alloys. Instead, very stable local clusters are formed in both low-temperature hcp and high-temperature bcc phases of alloys, with almost identical local structures. This gives an example of geometrically frustrated state and explains unusual concentration behavior of M\"ossbauer spectra discovered long ago for this system.

the most stable intermetallic phase TiFe with CsCl structure and therefore just substitution of Ti by Fe is not optimal energetically. The assumption [10] is quite unusual for crystalline metallic alloys.
Electronic structure calculations show that Fe as a substitutional impurity should be magnetic [11][12][13], which seems to agree with experiment [12] for a very small concentration of Fe, before the solubility limit 3 10   c x @RT, but clearly contradicts experimental data for c x x  [3]. Only interstitial position of Fe was considered as an alternative in electronic structure calculations up to now [12,13]. Here we present detailed first-principle calculations which confirm the hypothesis [9,10] on clustering of Fe in quenched Ti-Fe alloys with a local structure of the clusters that is different from those dictated by crystal lattice structure of the matrix. This means that even for so chemically simple and non-exotic alloys as Ti-Fe the conventional separation of metallic alloys into two classes, substitutional and interstitial, is not enough and much more complicated local structural state can be formed. More specifically, this state can be described as a formation of complexes of more or less substitutional Fe impurities with interstitial Ti atoms.

2.Methods
The modeling was performed by density functional theory (DFT) in the pseudopotential code SIESTA [14], as was done in our previous work on a similar subject [15]. All calculations were performed using the generalized gradient approximation (GGA-PBE) with spin-polarization [16]. The ion cores were described by norm-conserving pseudo-potentials [17] and the wave functions are expanded with a double-ζ plus polarization basis of localized orbitals for iron and titanium. Full optimization of the atomic positions was performed, and the forces and the total energy were calculated with the accuracy of 0.04 eV/Å and 1 meV, respectively. For the modeling of all configurations the 3×3×3 supercell of 54 titanium atoms in hcp and bcc structures was used.
To check the effect of supercell size, larger supercells with 96 titanium atoms in hcp and 128 atoms in bcc configuration were used. All the calculations were carried out with an energy mesh cut-off of 300 Ry and a k-point mesh of 6×6×4 (3×3×2 for larger supercell) and 6×6×6 (3×3×3 for larger supercell) in the Monkhorst-Pack scheme [17] for α and β phases, respectively. For the plots of DOS the k-point mesh was increased up to 8×8×6 and 8×8×8, respectively. Formation energy of considered configurations was calculated by the standard formula: Eform = EnFe+mTi -(nEFe+mETi), where EnFe+mTi is the total energy of the supercell contain m atoms of Ti and n atoms of Fe, EFetotal energy per atom of iron in α-Fe, and ETi is the total energy per titanium atom in the corresponding phase (α or β).

Computational results
The first step of our calculations is the check of energetically preferable positions of iron impurities in α-titanium at various concentrations. We examine four possible configurations: (i) quasi-random distribution of substitutional iron impurities, (ii) aggregation of substitutional iron impurities with further formation of clusters of substitute atoms (nFe(S)) where n is the number of iron atoms in supercell), (iii) interstitial iron impurities with further formation of cluster of substitutional iron impurities around interstitial one (nFe(S)+Fe(I)), (iv) formation of clusters of substitutional iron impurities around atom of titanium in interstitial void (nFe(S) + Ti(I), see Fig.   1a,b). We examined all these configurations; further we discuss the results only for the structures with the lowest formation energies.

α-phase
The computational results ( Fig. 2) allow us to suggest energetically optimal configurations of iron impurities dependent on their concentration. First of all, iron is more soluble in β-Ti than in α-Ti, in agreement with the previous computations [11] and the well-known fact that Fe is βstabilizer [1]. In α-Ti at the lowest studied concentrations (x<0.06) formation of single substitutional impurity (~0.48 eV/Fe) and pairs of substitutional iron atoms is more energetically favorable than other types of defects (curves 2,3 in Fig. 2a). Formation of the single and double defects does not provide visible changes in lattice parameters of the system. For concentration of Fe in α-phase roughly between 4 and 10 at% a quasi-random distribution of substitutional impurities is not energetically favorable; note however that α-β transition happens at the average concentration Fe near 4at% at room temperature [1,5,9]. Thus, we can consider the concentration of 4 at% as the lowest margin of dilution. In the samples of α-Ti with concentrations of Fe impurities 6-12 at% the most energetically favorable configuration is the cluster of substitutional impurities around interstitial titanium atom which formation energies are of the order of 0.1~0.2 eV/Fe atom lower than for other configurations (see Fig. 2a). Note that the formation energy of interstitial Ti-atom without substitutional Fe-impurity in its vicinity is rather high (~2.1 eV/Ti), thus, the formation of suggestion [10] that geometric factors play a crucial role in the structure of Ti-based alloys above the solubility limit. Thus, we can describe nFe(s)+Ti(I) clusters as the smallest nuclei of distorted TiFe phase. At equilibrium, the system is a two-phase, with coexisting TiFe and α-Ti phases [1].
In the quenched state, there is nanoscale inhomogeneity of the type described above. In general, this inhomogeneous state can be stabilized by misfit strains [19]. Microscopic description of these phenomena requires calculations for specific materials, and this is what we have done for Ti-Fe.
Noticeable experimental result is the absence of local magnetic moments on Fe in the concentration range under consideration [3,12,13]. From all four studied models, only nFe(s)+Ti(I) clusters have zero magnetic moments, other three studied configurations provide an appearance of magnetic moments on iron impurities of the order of 2.2 μB/Fe. The other experimental fact is the formation of pseudo-gap in electronic structure in diluted Fe-doped titanium [2][3][4][5][6]. The calculated electronic structures for nFe(s)+Ti(I) clusters (Fig. 3a) do demonstrate similarity of the electronic structure of iron and Ti(I) with electronic structure of B2-FeTi (Fig. 3c). Note that the difference in electronic structure of Ti atom inside the iron cluster is rather different from the electronic structure of other Ti-atoms in the system. Due to rather low concentration of the impurities, the electronic structure of the whole system (total DOS) is mainly determined by the contribution from titanium atoms (Fig. 3a,b). The formation of the clusters decreases the density of states at the Fermi level, which can be related to the experimentally observed pseudo-gap formation in these systems [2,[4][5][6]. Therefore, we can say that all structural, magnetic and electronic properties of nFe(s)+Ti(I) clusters are in a qualitative agreement with the experimental results.

β-phase
Mössbauer spectra [9] demonstrate that local Fe-Ti structures with similar characteristics exist in both αand β-Ti phases, with quadrupolar splitting and isomer shift being almost concentration independent (except a close vicinity of the structural transition point [9]). Importantly, both these phases, α and β, at room temperature and below are obtained by quenching of the high-temperature phase (β-Ti).
In contrast to hcp α-Ti, in bcc β-Ti the interstitial voids are too small to allow appearance interstitial Fe impurities. Therefore, we start out modeling from the pair of impurities and consider only three types of configurations: (i) quasi-random distribution of iron atoms, (ii) clusters of substitutional iron impurities, and (iii) aggregation of iron impurities in the vicinity of one titanium atom (Figs. 1с,d) which further could be transform to nFe(s)+Ti(I) clusters in α-Ti (see below).
Formation energy of the single impurity in β-Ti is -1.96 eV, that is, corresponds to instability of the host system. Incorporation of the single iron impurity does not change lattice parameters of the supercell. At the same time, at x = 0.04 -0.05 quasi-random distribution of impurities turns out to be the most favorable (see Fig. 2b). Note that experimentally [9] in this region, contrary to large and smaller Fe concentration, the Mössbauer spectra cannot be described as a single, quadruplesplit doublet and demonstrate a broad distribution of observable Fe positions. In this respect, our results also seem to agree with the experiment. For the case of 4 at% Fe (two iron impurities per 54 atomic supercell) the formation of Fe-Ti-Fe clusters is about 0.4 eV/Fe less energetically favorable and formation of Fe-Fe neighbor's pairs is 0.8 eV less preferable than the random configuration.
We do not show the results for x < 0.04 since bcc phase does not exist at these concentrations. Due to these features of Fe-Fe interaction the increase of impurity concentration from 4 to 15% makes the formation of iron clusters around titanium atom (Fig. 1d) the most energetically favorable. For higher concentration of iron the random distribution of impurities makes appearance of Fe-Fe pairs unavoidable and aggregation of impurities around one Ti atom turns out to be the best choice to prevent the presence of unfavorable Fe-Fe pairs. This explains peculiarities of the Mössbauer spectra near the transition point, with a broad distribution of observable iron positions [8]. For higher (in α-phase) and lower (in β-phase) concentrations there is an unique favorable atomic configuration. Now we will describe this configuration in more detail.
Fe-Ti distances in the energetically favorable clusters in β-Ti are in the range of 2.59~2.61 Å and angles close to 70°, like in α-phase (see Fig. 1b,d), that suggests a tendency to formation of very stable local clusters of distorted TiFe phase in both hosts. It turns out that only these clusters in β-Ti among all studied ones are nonmagnetic (other considered configurations provide appearance of magnetic moments about 2.4 μB/Fe). Electronic structure of clusters with iron surrounding Ti atom (Fig. 3b) is similar to the electronic structure of nFe(s)+Ti(I) clusters in αphase and electronic structure of the Ti-atoms inside the cluster is similar to that in B2-FeTi. There is a clear tendency to the formation of pseudo-gap in the density of states near the Fermi energy.

Discussion and conclusions
Based on the presented results we can speculate about possible mechanism of survival of Fe clusters during the bcc-hcp transformation when cooling from high temperatures. As commonly accepted, the bcc-hcp transformation in Ti during the cooling is realized by phonon mechanism ensuring the reconstruction of bcc lattice under the Burgers scheme [20]. This scheme involves two Based on the results of our modeling one can conclude that quenched Ti-Fe alloys above the solubility limit can be considered for 0.06 < x < 0.13 as neither substitutional nor interstitial alloys. Instead, a formation of local clusters with close to optimal Ti-Fe interatomic distances takes place. This explains mysterious concentration evolution of the Mössbauer spectra [9] of these alloys and supports a hypothesis [10] on the formation of locally symmetry broken structure. At the same time, we specify this hypothesis and suggest a model for real structural state of these very common and practically important alloys.
MIK acknowledges a financial support by NWO via Spinoza Prize. The work was supported by Act 211 Government of the Russian Federation, contract No. 02.A03.21.0006. DWB acknowledge support from the Ministry of Science and Higher Education of the Russian Federation, Project № 3.7372.2017/8.9      (1) and distances (2) in cluster of six iron atoms around titanium center as function of lattice distortion of during hcp-bcc transition along the Burgers path [18].