Exploration of Structural, Thermodynamic, Magnetic, Mechanical, and Dynamical Properties of Martensite Fe₃Pt Alloys: A Density Functional Theory Study

Abstract

The current study explored the martensite structures of Fe₃Pt alloys, specifically $Cmmm$-Fe₃Pt, $P{6}_3/mmc$-Fe₃Pt, $P4/mmm$-Fe₃Pt, and $R\overline{3}m$-Fe₃Pt, aiming to provide a comprehensive understanding of the mechanisms that govern their physical and chemical properties. We have focused on their structural, thermodynamical, magnetic, electronic, mechanical, and dynamical characteristics, utilizing the density functional theory (DFT) technique. Our study revealed that in addition to the previously reported austenitic cubic $Pm\overline{3}m$-Fe₃Pt and martensite tetragonal $I4/mmm$-Fe₃Pt with L1₂ structure, there exist additional Fe₃Pt phases that exhibit excellent structural, thermodynamic, magnetic, and mechanical properties. The calculated enthalpies of formation were found to be negative and less than −0.39 eV in all the structures considered, indicating thermodynamic stability and formation under experimental synthetic conditions. Moreover, the computed magnetic moments are in the range 2.94 to 3.04 ${\mathsf{\mu }}_{\mathrm{B}}$, which is relatively comparable to 3.24 ${\mathsf{\mu }}_{\mathrm{B}}$ of the widely reported $Pm\overline{3}m$-Fe₃Pt alloy. The analysis of the electronic structure also revealed strong magnetism due to the presence of asymmetry in the spin-up and -down states of the density of states (DOS) plots. To determine the mechanical response of Fe₃Pt structures under loading conditions, we computed the independent elastic constants, macroscopic properties, and stress–strain relationship under hydrostatic stress. All four phases were studied, but the hypothetical $P{6}_3/mmc$-Fe₃Pt showed excellent mechanical stability at ambient conditions and exceptional hardness and resistance to compression in the elastic region 0% ≤ strain ≤ 10%. This evidence is provided by satisfying the Born necessary stability conditions, large bulk modulus, and a strong linear relationship fit ($R^2$) of greater than 0.94. Moreover, the phonon dispersion curves revealed dynamical stability for $Cmmm$-Fe₃Pt and $R\overline{3}m$-Fe₃Pt, and metastability for $P4/mmm$-Fe₃Pt, while the hypothetical $P{6}_3/mmc$-Fe₃Pt is unstable.

1. Introduction

The exploration of bimetallic Fe-Pt alloys has long garnered significant attention due to their intriguing properties and promising applications in various fields such as ultra-high density data storage, spintronics, catalysis, and nanotechnology [1,2,3,4]. As indicated by their binary stability phase diagram [5], there have been three well-known and widely reported structures of Fe-Pt alloys which are chemically ordered. These include the tetragonal L1₀ $P4/mmm$-FePt phase which exhibits ferromagnetic ordering, and the face-centered cubic (FCC) phases FePt₃ and Fe₃Pt, both crystallizing in the $Pm\overline{3}m$ space group with L1₂-type crystal symmetry. The latter two phases support both antiferromagnetic and ferromagnetic magnetic ordering [6]. These alloys have been previously investigated both experimentally and computationally and have been found to exhibit excellent thermodynamic, vibrational, and mechanical stability. Moreover, they possess desirable magnetic properties due to large magnetic moments, large uniaxial magnetocrystalline anisotropy (MCA) energies originating from spin–orbit interactions and asymmetry of spin-up and -down in the electronic density of states [7,8,9]. Variations in crystallization and chemical composition significantly influence their magnetic behavior, electronic structure, and lattice dynamics. For instance, although both Fe and Pt contribute to the magnetic moments, Fe contributes approximately eight times more than Pt, thereby making the magnetic behavior predominantly dependent on the Fe content [10]. The $P4/mmm$-FePt phase crystallizes at temperatures below 1570 K in equiatomic Fe:Pt concentration, while the FCC structures crystallize around 25% and 75% Fe contents at temperatures below 1120 K and 1620 K, respectively [11]. The tetragonal L1₀ FePt shows one of the highest MCA energy ranging from $6\times {10}^6 \mathrm{J}/{\mathrm{m}}^3$ to $9 \times {10}^6 \mathrm{J}/{\mathrm{m}}^3$ along the magnetically easy $c$ axis [001], making it a promising candidate to overcome superparamagnetic limit, which is the loss data due to thermally activated fluctuations of magnetization [12,13]. It consists of alternating stacked atomic layers of Fe and Pt atoms along [100] direction similar to the AuCu structure, leading to a tetragonal distortion along the $c$-axis. As per the previous communication, the cubic Fe₃Pt and FePt₃ alloys exhibit the highest MCA energies of $1.1\times {10}^6 \mathrm{J}/{\mathrm{m}}^3$ (0.375 meV) and $1.5\times {10}^5 \mathrm{J}/{\mathrm{m}}^3$ (0.055 meV) along the [100] easy axis direction [12]. In the Fe-rich Fe₃Pt alloy, Fe atoms occupy the corner lattice positions, while Pt atoms are located at face-centered positions. Conversely, in the Pt-rich FePt₃ alloy, Pt atoms occupy the corner positions, and Fe atoms are situated at the face-centered positions.

In addition, the L1₂ face-centered cubic (austenitic) Fe₃Pt alloy is reported to exhibit strong magnetoelastic coupling and undergoes a thermoelastic martensitic transformation from the ordered L1₂ phase into low-temperature martensitic structures, which inherently depend on the degree of order of the L1₂ structure [14,15]. One such martensite structure is the body-centered tetragonal (BCT) Fe₃Pt phase, which crystallizes in the I4/mmm space group [16,17]. The equilibrium cell parameters of the I4/mmm bulk structure as produced by X-ray diffraction are $a=5.73 \mathrm{Å}$ and $c=6.34 \mathrm{Å}$ [18]. It was found that the spin-up Fe $d$ states in the density of states (DOS) in this martensite structure are situated below the Fermi level, while the spin-down states overlaps above the Fermi, indicating a pronounced spin asymmetry. Consequently, this spin polarization results in relatively large magnetic moments of 2.61 ${\mu }_B$ and 2.82 ${\mu }_B$ for Fe1 and Fe2 atoms, respectively. Moreover, Pt atoms were also found to exhibit a finite magnetic moment of 0.45 ${\mu }_B$, suggesting a degree of spin polarization induced by hybridization with neighboring Fe atoms [16]. Our previous communication further reported magnetic moments of 2.744 ${\mu }_B$ for Fe and 0.45 ${\mu }_B$ Pt along with the MCA energy of 1.364 meV along the [001] easy axis [12], which are higher than those of the austenitic Fe₃Pt.

The structural models of other Fe₃Pt martensite phases have long been proposed; however, their properties remain unstudied [19]. According to the Materials Explorer within the Materials Project database [20,21], three additional martensitic phases of Fe₃Pt are predicted to exist whose structural, thermodynamic, magnetic and mechanical properties are not yet fully explored. These correspond to the P4/mmm, $Cmmm$, and R3m space groups as shown in Figure 1. The $Cmmm$–Fe₃Pt phase crystallizes at approximately 55 K in an orthorhombic lattice characterized by primitive lattice parameters $\left(a=3.732 \mathrm{Å}, b=3.714 \mathrm{Å}, \mathrm{c}=3.750 \mathrm{Å}\right)$ as determined by XRD. The structure comprises two crystallographically distinct Fe sites. This orthorhombic structure is an intermediate phase, appearing between face-centered tetragonal (FCT1) ($c/a<1$) and FCT2 ($c/a>1$) [15]. The system considered in this current work is a conventional $c$-centered (centering doubles the number of lattice points along one direction) orthorhombic Fe₃Pt which is equivalent, but twice the volume and number of atoms (eight) as the primitive (four atoms) reported in the literature. In the first site, Fe atoms adopt a two-coordinate geometry, bonded to two Fe and two Pt atoms, whereas the second Fe site exhibits eightfold coordination with neighboring Fe atoms. The Pt atoms display a four-coordinate geometry, each bonded to four Fe atoms, completing the orthorhombic network. The $P4/mmm$–Fe₃Pt phase exhibits a USi₂-type structure and crystallizes in a body-centered tetragonal (BCT) lattice. This phase forms through a thermoelastic transformation. During the formation, the tetragonality ratio $c/a$ shifts from 1 to 0.79 at the transformation temperature, signifying a discontinuous lattice distortion characteristic of a first-order martensitic transformation [15,17]. The structure contains two inequivalent Fe sites. Each Fe atom is coordinated by eight Fe and four Pt atoms, forming distorted FeFe₈Pt₄ cuboctahedra. These polyhedra interconnect through corner-, edge-, and face-sharing with neighboring FeFe₈Pt₄ and PtFe₁₂ units, forming a highly integrated three-dimensional framework. The Pt atoms are twelvefold coordinated by Fe atoms, generating PtFe₁₂ cuboctahedra that are similarly interconnected through shared corners, edges, and faces with surrounding FeFe₈Pt₄ polyhedra. The $R\overline{3}m$–Fe₃Pt phase crystallizes in a rhombohedral-centered hexagonal or trigonal lattice. Each Fe atom is surrounded by nine Fe and three Pt atoms, forming distorted FeFe₉Pt₃ cuboctahedra that share corners, edges, and faces with adjacent FeFe₉Pt₃ and PtFe₆Pt₆ polyhedra. The Pt atoms are coordinated by six Fe and six Pt atoms, creating PtFe₆Pt₆ cuboctahedra that link through corner-, edge-, and face-sharing, producing a robust trigonal framework. Lastly, the $P{6}_3/mmc$-Mn₃Ir structure served as a prototype for Fe₃Pt, since Fe and Mn as well as Pt and Ir are chemically analogous neighboring transition metals with comparable atomic radii and electronic configurations. Such substitution preserves the symmetry and coordination environment, enabling reliable DFT modeling of synthetical Fe–Pt system within the same crystallographic framework. This is the beta Cu₃Ti- and Mn₃Ga-type structure which crystallizes in the hexagonal lattice with eight atoms in a primitive cell [22,23]. Fe is bonded to four equivalent Pt atoms to form a mixture of distorted corner, edge, and face-sharing FePt₄ cuboctahedra. Pt is bonded to twelve equivalent Fe atoms to form a mixture of corner and face-sharing PtFe₁₂ cuboctahedra.

The magnetic, thermodynamic, mechanical, and dynamical properties of these four martensite phases are yet to be systematically explored either experimentally or theoretically, which limits a comprehensive understanding of the structure–property relationships and the potential functional applicability in advanced magnetic and electronic technologies. Experimental exploration of these alloys may be challenging due to the complexities associated with detecting such phases arising from dynamical effects on diffraction reflections [16]. However, the development of density functional theory (DFT)-based first-principles methods enables theoretical exploration and optimization of L1₂ bimetallic alloys by accurately calculating the interatomic forces acting on the nuclei [24,25,26]. The current study focuses on the structural, thermodynamic, magnetic, electronic, mechanical, and dynamical properties of the four martensite Fe₃Pt bimetallic alloys at 0 K to assess their stability and deduce their suitability as magnetic materials. We have employed the density functional theory (DFT) quantum mechanical method embedded in the CASTEP code to perform first-principles quantum mechanical calculations on Fe₃Pt alloys. Our computational findings revealed that three martensite Fe₃Pt alloys exhibit excellent thermodynamic and mechanical stability and magnetic moments comparable to those of the extensively studied Fe-Pt alloys.

2. Computational Approach

The present study utilized the Cambridge Serial Total Energy Package (CASTEP) version 2020 [27] which employs the plane-wave density functional theory (DFT) plane-wave pseudopotential method to perform first-principles calculations on Fe₃Pt alloys. The Generalized Gradient Approximation (GGA) and Perdew–Burke–Ernzerhof (PBE) exchange–correlation functional were preferred since they account for inhomogeneities in the electron distribution [28,29]. To determine the minimum optimal plane-wave cut-off energies that converge the total energy, single-point energy calculations were performed for all the systems at several increasing cut-off values without relaxing the atomic positions and cell parameters. This was conducted until the total energy between successive steps was below 1 meV. The cut-off energies 400 eV, 300 eV, 400 eV, and 300 eV were, respectively, found sufficiently to converge the total energies of $Cmmm$, $P{6}_3/mmc$, $P4/mmm,$ and $R\overline{3}m$ systems. Moreover, the customized Monkhorst–Pack [30] k-points grid size of 9 × 7 × 4, 4 × 4 × 6, 5 × 5 × 6, and 8 × 8 × 1 for self-consistent field (SCF) calculations were applied for Brillouin zone sampling. Before subsequent calculations of electronic structure and elastic properties, full geometry optimization was performed to obtain the arrangement of atoms and cell parameters corresponding to the minimum total energy, i.e., the ground-state structures. Both cell volume and shape were allowed to change until the final energy between two iterations was less than 5.7 meV. Moreover, to ensure that electronic structure is self-consistent before updating forces and atoms reach equilibrium positions, the total energy convergence and maximum ionic force tolerances of $0.2\times {10}^{-4} \mathrm{e}\mathrm{V}/\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}$ and $0.5\times {10}^{-1} \mathrm{e}\mathrm{V}/\mathrm{Å}$ were used. To allow separation of majority and minority spin projections, the spin density was treated by the collinear spin polarization using formal spin as initial. The low-memory Brayden–Fletcher–Goldfarb–Shanno (LBFGS) algorithm [31] that scales linearly in memory and performs better for large systems was preferred for cell optimization over the normal BFGS [32] that scales as a square of the system size for cell. Furthermore, the LBFGS algorithm employs a general sparse preconditioner that enhances the efficiency of geometry optimization, achieving up to a twofold speed increase for small systems and up to tenfold for large systems [33]. Subsequent computation of density of states and elastic constants were conducted using the equilibrium structures. Calculation of the elastic properties was performed using the Elastic Constants task with a maximum strain amplitude of 0.003, which is adequately low for the material to stay in the linear elasticity. The calculation of electronic band structure and density of states was conducted with a band energy tolerance of ${10}^{-5}$ eV. For phonon dispersion curves computation, the finite displacement method was preferred over the linear response, as the latter does not support spin–orbit coupling and is restricted to collinear spin polarization.

3. Results and Discussion

3.1. Structural Properties

The equilibrium cell parameters, densities, and enthalpies of formation of the four martensite Fe₃Pt structures as shown in

Table 1. Calculated equilibrium cell parameters, densities, and enthalpies of formation and density of Fe₃Pt alloys.

Parameter	$\mathit{P}\mathit{m}\overline{3}\mathit{m}$	$\mathit{C}\mathit{m}\mathit{m}\mathit{m}$	$\mathit{P}{6}_3/\mathit{m}\mathit{m}\mathit{c}$	$\mathit{P}4/\mathit{m}\mathit{m}\mathit{m}$	$\mathit{R}\overline{3}\mathit{m}$
a (Å)	3.756 (3.73) [34]	4.089	5.262	3.827	8.550
b (Å)	-	4.092	-	-	-
c (Å)	-	6.197	4.685	3.593	-
V (Å3)	53.000	103.694	112.34	52.620	54.330
$∆H_{f }\left(\mathrm{e}\mathrm{V}\right)$	−1.564	−1.564	−3.128	−1.564	−1.564
$∆H_f \left(\mathrm{e}\mathrm{V}/\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\right)$	−0.391	−0.391	−0.391	−0.391	−0.391
$\rho \left(\mathrm{g}/{\mathrm{c}\mathrm{m}}^3\right)$	11.361	11.614	10.720	11.440	11.082

were obtained by performing full geometry optimization. The experimental volume for the orthorhombic $Cmmm$-Fe₃Pt and lattice parameters of the previously reported austenitic $Pm\overline{3}m$-Fe₃Pt are given in parenthesis to validate DFT calculations against existing experimental data. The calculated lattice parameters of $Pm\overline{3}m$-Fe₃Pt are more than 99% in agreement with experimental data [34] and previous calculations [8], while the optimized volume of $Cmmm$-Fe₃Pt in 96% in agreement with the XRD results [15]. These agreements indicate that the employed approach correctly predicts the ground state properties of Fe₃Pt alloys. The conventional hexagonal $P{6}_3/mmc$-Fe₃Pt structure shows the largest unit cell volume and smallest density corresponding to weaker interatomic interactions. The enthalpies of formation ($∆H_f$) per atom were computed according to Equation (1):

$$∆H_f={\displaystyle \frac{E_{{Fe}_{3}Pt}-\left(3E_{Fe}+E_{Pt}\right)}{4}}$$

(1)

where $E_{{Fe}_{3}Pt}$ is the total ground state energy of Fe₃Pt formula unit, while $E_{Fe}$ and $E_{Pt}$ are the elemental energies of Fe and Pt atoms. Negative enthalpies of formation indicate the release of energy when forming a compound (exothermic reaction) and thus thermodynamic stability of the compounds in relation to its constituent elements, i.e., $E_{compound}<E_{elements}$. Our DFT calculations predict that all the four martensite Fe₃Pt alloys are thermodynamically stable due to negative enthalpies of formation. Expectedly, the normalized enthalpy of formation values per atom of austenitic $Pm\overline{3}m$ and martensite $Cmmm$, $P{6}_3/mmc$, $P4/mmm,$ and $R\overline{3}m$ are similar since they have the same stoichiometry and elemental reference state. This indicates that, on a per-atom basis, all structural variants are energetically degenerate. The differences observed in the total formation enthalpy (e.g., −1.564 eV vs. −3.128 eV) arise solely from variations in the number of atoms per unit cell.

3.2. Magnetic Properties

Table 2. Calculated magnetic moments, Hirshfield spins, and charges of Fe₃Pt alloys. $\uparrow$ and $\downarrow$ represents spin up and down, respectively.

$\mathbf{Magnetic} \mathbf{Moments} \left({\mathit{\mu }}_{\mathit{B}}\right)$
	$\mathit{C}\mathit{m}\mathit{m}\mathit{m}$	$\mathit{P}{6}_3/\mathit{m}\mathit{m}\mathit{c}$	$\mathit{P}4\mathit{m}\mathit{m}\mathit{m}$	$\mathit{R}\overline{3}\mathit{m}$
${Fe}_1\uparrow$	5.35	5.33	5.29	5.34
${Fe}_1\downarrow$	2.38	2.41	2.47	2.46
${Fe}_1\uparrow -{Fe}_1\downarrow$ (${\mathsf{\mu }}_{\mathrm{B}}$)	2.98	2.92	2.83	2.87
${Fe}_2\uparrow$	5.18	-	5.26	5.29
${Fe}_2\downarrow$	2.74	-	2.50	2.64
${Fe}_2\uparrow -{Fe}_2\downarrow \left({\mathsf{\mu }}_{\mathrm{B}}\right)$	2.43	-	2.75	2.65
$Pt\uparrow$	16.46	16.45	16.43	16.34
$Pt\downarrow$	16.16	16.34	16.29	16.13
$Pt\uparrow -Pt\downarrow$ (${\mathsf{\mu }}_{\mathrm{B}}$)	0.30	0.12	0.15	0.21
Total (${\mathsf{\mu }}_{\mathrm{B}}/\mathrm{f}.\mathrm{u}.)$	3.01	3.04	2.94	2.97
Hirshfeld Analysis (${\mathsf{\mu }}_{\mathbf{B}}$)
${Fe}_1$	2.83	2.81	2.72	2.82
${Fe}_2$	2.45	-	2.68	2.61
$Pt$	0.57	0.45	0.44	0.35
Charge (e)
${Fe}_1$	0.27	0.26	0.24	0.20
${Fe}_2$	0.08	-	0.24	0.07
$Pt$	−0.62	−0.79	−0.72	−0.46

presents the calculated spin polarized magnetic moments and charge distribution characteristics of the four considered Fe₃Pt crystal structures derived from Mulliken atomic population analysis. The magnetic moments are calculated as the difference between spin-up and spin-down ($\uparrow -\downarrow$) charge densities. Moreover, the average of Fe1 and Fe2 were taken when quantifying the total spin moment per formula unit. The calculated total moments for $Cmmm$, $P{6}_3/mmc$, $P4/mmm,$ and $R\overline{3}m$ are 3.01 ${\mathsf{\mu }}_{\mathrm{B}}$, 3.04 ${\mathsf{\mu }}_{\mathrm{B}}$, 2.94 ${\mathsf{\mu }}_{\mathrm{B}}$, and 2.97 ${\mathsf{\mu }}_{\mathrm{B}}$, respectively. These values are comparable with those reported for other Fe-Pt alloys and bimetallic systems of similar composition [10,35,36]. For all space groups, the Fe atoms exhibit substantial local magnetic moments compared to Pt, indicative of strong ferromagnetic ordering within the Fe sublattice and large spin–orbit effects from Pt. Moreover, in all the structures, the spin-up ($\uparrow$) for both Fe1 and Fe2 are more densely than the spin-down ($\downarrow$), revealing pronounced spin asymmetry in the electronic density of states and confirming strong magnetic polarization within the system.

Our DFT calculations reveal that the orthorhombic $Cmmm$ space group possesses the largest Fe1 magnetic moment (2.98 ${\mathsf{\mu }}_{\mathrm{B}}$), followed by the hexagonal $P{6}_3/mmc$ ($2.92 {\mathsf{\mu }}_{\mathrm{B}}$), while the trigonal $R\overline{3}m$ and tetragonal $P4/mmm$ display relatively decreased moment values of 2.87 ${\mathsf{\mu }}_{\mathrm{B}}$ and 2.83 ${\mathsf{\mu }}_{\mathrm{B}}$, respectively. This may be attributed to the increase in restrictions on independent lattice parameters and angles, i.e., symmetry from orthorhombic to tetragonal, which slightly decreases the exchange splitting, consistent with increased Fe–Pt orbital hybridization. The smaller variation in Fe2 moments indicates comparable local environments across the structures, except for the $P{6}_3/mmc$ phase, where Fe occupies only one unique crystallographic site due to higher symmetry. The Pt atoms exhibit relatively small induced magnetic moments, ranging from 0.12 ${\mathsf{\mu }}_{\mathrm{B}}$ to 0.30 ${\mathsf{\mu }}_{\mathrm{B}}$, arising from Fe–Pt hybridization and spin polarization of Pt 5d orbitals. $Cmmm$-Fe₃Pt shows the largest Pt moment (0.30 ${\mathsf{\mu }}_{\mathrm{B}}$), whereas the $P{6}_3/mmc$ phase displays the smallest (0.12 ${\mathsf{\mu }}_{\mathrm{B}}$).

The Hirshfeld partitioning of electron density into atomic contributions further confirms the dominance of Fe atoms in the overall magnetic behavior. The Fe1 moments range from 2.72 ${\mathsf{\mu }}_{\mathrm{B}}$ to 2.83 ${\mathsf{\mu }}_{\mathrm{B}}$, while Fe2 moments vary between 2.45 ${\mathsf{\mu }}_{\mathrm{B}}$ and 2.68 ${\mathsf{\mu }}_{\mathrm{B}}.$ In contrast, Pt contributes only 0.35–0.57 ${\mathsf{\mu }}_{\mathrm{B}}$, consistent with the induced magnetization mechanism. The $Cmmm$ structure again exhibits the highest overall Fe magnetization, while the $P4/mmm$ phase shows a marginally reduced value, reflecting differences in local coordination and exchange interactions.

Hirshfeld charge analysis indicates notable charge redistribution between Fe and Pt atoms. Fe atoms possess positive charges (+0.07 to +0.27 e), signifying electron donation, while Pt atoms carry negative charges (−0.46 to −0.79 e), confirming their role as electron acceptors. The $P{6}_3/mmc$ phase shows the highest charge transfer (−0.79 e), suggesting stronger Fe–Pt bonding and more effective hybridization.

3.3. Electronic Structure

Figure 2 presents the electronic total and orbital-projected spin polarized density of states (DOS) for the martensitic Fe₃Pt, with the Fermi level set as the origin of the energy scale. As anticipated, all the investigated Fe₃Pt alloys display metallic behavior, as indicated by the finite density of states at the Fermi level arising from the overlap of valence and conduction bands, most prominently in the spin-down states. In all total DOS plots, the spin-up states are predominantly located below the Fermi level, whereas the spin-down states extend into the conduction band above the Fermi level, reflecting a pronounced spin asymmetry which indicates spin polarization and gives rise to substantial magnetic moments and strong ferromagnetic character consistent with previous experimental findings on Fe₃Pt thin films [37]. Notably, the $P{6}_3/mmc$ phase exhibits a further strong asymmetry (enhanced spin polarization) in the valence states between −4 eV and −2 eV, consistent with its highest calculated magnetic moment. Shallow pseudo-gaps exist near the Fermi level, indicating enhanced Fe-Pt hybridization and electronic stability for all phases.

The orbital-projected DOS shows a dominant contribution from Fe 3d states in the total plots, with minor contribution from the Pt 5d states hybridizing with Fe d-bands. Moreover, there exist strong spin asymmetry in Fe states and weaker in Pt. This strong exchange splitting reflects localized Fe moments stabilized by the low structural symmetry. Furthermore, the strong spin polarization arises from the Fe 3d orbitals, while Pt-5d orbitals show small but non-zero induced moments, contributing slightly to the total magnetization.

The electronic properties of these systems were further analyzed using the band structure along the high symmetry direction in the Brillouin zone as presented in Figure 3. All four structures exhibit metallic characteristics, as evidenced by the continuous overlap of energy bands at the Fermi level and the finite DOS at $E_F$ and transitioning of electron wave highest to lowest energy levels through the Brillouin zone [38]. Moreover, there exists a clear spin splitting: red and blue bands diverge near the Fermi level. Interestingly, the $R\overline{3}m$ phase shows almost similar spin-up (red) and spin-down (blue) bands, indicating nearly degenerate spins.

3.4. Mechanical Properties

Table 3. Calculated independent elastic constants and macroscopic properties of Fe₃Pt alloys.

Elastic Constants (GPa)
Structure	${\mathit{C}}_{11}$	${\mathit{C}}_{22}$	${\mathit{C}}_{33}$	${\mathit{C}}_{44}$	${\mathit{C}}_{55}$	${\mathit{C}}_{66}$	${\mathit{C}}_{12}$	${\mathit{C}}_{13}$	${\mathit{C}}_{14}$	${\mathit{C}}_{23}$
$Cmmm$	300.43	298.95	224.53	120.12	119.83	36.27	92.71	161.85	-	160.68
$P{6}_3/mmc$	249.31	-	22.23	−302.21	-	-	92.47	41.74	-	-
$P4/mmm$	198.88	-	166.50	46.35	-	12.11	141.24	190.46	-	-
$R\overline{3}m$	239.92	-	152.93	48.48	-	-	125.14	88.28	5.36	-
Macroscopic Properties (GPa)
	B	G		E	$\mathit{\sigma }$	$\mathit{B}/\mathit{G}$	${\mathit{\theta }}_{\mathit{D}}$ (K)		${\mathit{\upsilon }}_{\mathit{m}}$$(\mathit{m}/\mathit{s}$)	$\mathit{A}$
$Cmmm$	183.77	70.85		188.34	0.33	2.59	343.02		2711.57	1.97
$P{6}_3/mmc$	57.87	−25.51		74.88	0.76	−2.27	-		-	−14.01
$P4/mmm$	179.09	38.75		108.44	0.40	4.62	262.7		2082.99	−2.79
$R\overline{3}m$	131.99	52.15		138.23	0.33	2.53	306.5		2430.37	0.24

shows the calculated independent elastic constants ($C_{ij}$) and derived macroscopic properties of the orthorhombic, hexagonal, tetragonal, and trigonal Fe₃Pt alloys at 0 K. CASTEP uses the generalized Hooke’s law shown in Equation (2) to fit the linear relationship between stress and strain components in the elastic regime to compute the independent elastic constants.

$${\sigma }_i=\sum _j{C}_{ij}{\epsilon }_j$$

(2)

where ${\sigma }_i$ and ${\epsilon }_j$ are the Voigt notation components of the stress and strain tensor, respectively. Calculation and elastic constants and macroscopic mechanical properties are crucial in understanding the intrinsic mechanical stability, bonding nature, and deformation behavior of solid-state materials. For any solid-state crystal lattice to be considered mechanically stable, certain stability conditions as postulated by Born must be satisfied [39].

Hexagonal crystals possess five independent elastic constants, while the tetragonal contain six due to the added $C_{66}=\left(C_{11}-C_{12}\right)/2$. Since the Laue classes of the hexagonal and tetragonal crystal lattice have similar elastic matric, their Born necessary stability conditions are also similar and are given as [40]:

$$C_{11}>\left|C_{12}\right|; {2C}_{13}^2<C_{33}\left(C_{11}+C_{12}\right); C_{44}>0; C_{66}>0$$

(3)

The condition $C_{66}>0$ is redundant for hexagonal crystal lattices. The trigonal or rhombohedral lattice has six independent elastic constants and like the hexagonal one, $C_{66}=\left(C_{11}-C_{12}\right)/2,$ it is dependent. The following four conditions are sufficient for stability:

$$C_{11}>\left|C_{12}\right|; C_{44}>0; C_{13}^2<{\displaystyle \frac{1}{2}}{C}_{33}\left(C_{11}+C_{12}\right); C_{14}^2<{\displaystyle \frac{1}{2}}{C}_{44}\left(C_{11}-C_{12}\right)=C_{44}{C}_{66}$$

(4)

The orthorhombic crystal lattice with lower symmetry contains a larger number (nine) of independent elastic constants and the Born stability conditions are

$$\begin{array}{c}{C}_{11}>0; C_{11}{C}_{22}>C_{12}^2; C_{11}{C}_{22}{C}_{33}+2C_{12}{C}_{13}{C}_{23}-C_{11}{C}_{23}^2-C_{22}{C}_{13}^2-C_{33}{C}_{12}^2>0; \\ C_{44}>0; C_{55}>0; C_{66}>0\end{array}$$

(5)

The independent elastic constants of the orthorhombic $Cmmm$, tetragonal $P4/mmm,$ and trigonal $R\overline{3}m$ structures are positive and obey all the necessary Born stability conditions, which indicates mechanical stability. In contrast, the $P{6}_3/mmc$ phase exhibits anomalous elastic constants, due to a negative $C_{44}$ value which violates the Born stability criteria. Such behavior signifies mechanical instability, suggesting that this hypothetical hexagonal structure cannot exist under ambient conditions and would spontaneously transform into a lower-symmetry, more stable phase with space group $mmm$ [11,41]. The elastic stiffness coefficients $C_{11}$, $C_{22}$, and $C_{33}$ for the $Cmmm$-Fe₃Pt are relatively high, above 224.53 GPa, which signifies a strong resistance to axial deformation along all crystallographic directions. Furthermore, the off-diagonal $C_{12}=92.71 \mathrm{G}\mathrm{P}\mathrm{a}$ and $C_{13}=161.85 \mathrm{G}\mathrm{P}\mathrm{a}$ are also large, indicating considerable interplanar coupling, while the comparatively smaller shear constant ($C_{66}=$ 36.270 GPa) reveals moderate shear anisotropy. The $P4/mmm$-Fe₃Pt structure shows moderately high values of $C_{11}$ and $C_{33}$, indicating fair axial stiffness. However, its small $C_{66}$ value (12.11 GPa) reflects weak shear resistance, implying significant anisotropy in its mechanical response. The $R\overline{3}m$-Fe₃Pt phase displays intermediate stiffness with $C_{11}=239.92 \mathrm{G}\mathrm{P}\mathrm{a}$ and $C_{33}=152.93 \mathrm{G}\mathrm{P}\mathrm{a}$, while the relatively small off-diagonal $C_{14}$ value (5.36 GPa) indicates limited shear coupling.

The corresponding macroscopic mechanical properties representing a polycrystalline aggregate using Voigt–Reuss–Hill (VRH) averaging schemes were computed from the independent elastic constants to determined deformation characteristics [42].

Table 4. Mathematical expressions and descriptions of the macroscopic mechanical properties of the polycrystalline material.

Property	Expression	Description
Bulk modulus (B)	$B_V={\displaystyle \frac{1}{9}}\left(C_{11}+C_{22}+C_{33}\right)+{\displaystyle \frac{2}{9}}\left(C_{12}+C_{13}+C_{23}\right)$ $B_R={\left[S_{11}+S_{22}+S_{33}+2\left(S_{12}+S_{13}+S_{23}\right)\right]}^{-1}$ $B={\displaystyle \frac{B_V+B_R}{2}}$	Resistance to uniform volume compression
Shear modulus (G)	$G_V={\displaystyle \frac{1}{15}}\left(C_{11}+C_{22}+C_{33}-C_{12}-C_{13}-C_{23}\right)+{\displaystyle \frac{1}{5}}\left(C_{44}+C_{55}+C_{66}\right)$ $G_R=15{\left[4\left(S_{11}+S_{22}+S_{33}-S_{12}-S_{13}-S_{23}\right)+3\left(S_{44}+S_{55}+S_{66}\right)\right]}^{-1}$ $G={\displaystyle \frac{G_V+G_R}{2}}$	Resistance to shape change
Young’s modulus (E)	$E={\displaystyle \frac{9BG}{3B+G}}$	Stiffness
Poisson’s ratio (σ)	$\sigma ={\displaystyle \frac{3B-2G}{6B+2G}}$	Ductility and brittleness
Pugh ratio	${\displaystyle \frac{B}{G}}$	Ductility
Anisotropy factor (A)	$A={\displaystyle \frac{2C_{44}}{C_{11}-C_{12}}}$	Elastic anisotropy
Longitudinal and transverse bulk velocities	${\upsilon }_m={\left[{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{1}{{\upsilon }_L^3}}+{\displaystyle \frac{2}{{\upsilon }_T^3}}\right)\right]}^{-1/3}$	Mechanical stiffness
Debye temperature	${\mathsf{\Theta }}_D={\displaystyle \frac{\hslash }{k_B}}{\left(6{\pi }^2{\displaystyle \frac{N}{V}}\right)}^{1/3}{\upsilon }_m$	Lattice stiffness

$S_{ij}$ represents the compliance tensor, the inverse of $C_{ij}$.

summarizes the expressions and descriptions of the calculated macroscopic properties.

The $Cmmm$ Pphase shows large bulk modulus $B = 183.77 \mathrm{G}\mathrm{P}\mathrm{a}$ and shear modulus $G = 70.85 \mathrm{G}\mathrm{P}\mathrm{a},$ further confirming its robustness and greater resistance to compression, while the relatively high Young’s modulus $E = 188.34 \mathrm{G}\mathrm{P}\mathrm{a}$ indicates good stiffness and metallic bonding characteristics. Moreover, the Pugh ($B/G= 2.59$) ratio greater than the critical value 1.5 of ductility and brittleness and Poisson’s ratio ($\sigma =0.33$) classifies this phase as ductile, while the positive anisotropy factor (A = 1.97) that diverges from unity suggests moderate elastic anisotropy.

The calculated Poisson’s ratio ranges from 0.33 to 0.40, except for $P{6}_3/mmc,$ and is within the error limit (0.32 $\pm$ 0.09) and similar to $\sigma =1/3$ recorded in most alloys [11,43]. Consistent with the negative $C_{44}$, the hexagonal $P{6}_3/mmc$ shows a negative shear modulus and Pugh ration. Its Poisson ration is 0.76 which is beyond the metallic limit (~0.5), indicating that this phase exhibits extreme lateral expansion when compressed (or contraction when stretched), very high ductility, low shear resistance, and strong anisotropic bonding or magnetostrictive behavior. This phase likely reflects magnetoelastic softening, not purely mechanical elasticity. The bulk modulus (179.09 GPa) of the $P4/mmm$ phase is relatively larger than the shear (38.75 GPa), leading a larger Pugh ratio, confirming a ductile but less rigid nature relative to the orthorhombic phase. The moderate Poisson’s ratio (0.40) also supports a metallic and ductile bonding character. Similar behavior is observed for the $R\overline{3}m$.

To further analyze the mechanical properties of the martensite Fe₃Pt alloys, we plotted stress vs. strain profiles which represent the elastic response of crystal systems under applied strain, from which elastic constants and mechanical stability are evaluated. Figure 4 illustrates the calculated stress vs. strain relationships for the $Cmmm$, $P{6}_3/mmc$, $P4/mmm$, and $R\overline{3}m$ phases of Fe₃Pt. The $Cmmm$, $P4/mmm,$ and $R\overline{3}m$ structures exhibit smooth and well-defined linear elastic regions, 0% ≤ strain ≤ 10% (red line), with strong correlations between the fitted and computed data ($R^2=0.980, 0.939$ and 0.970, respectively).

This excellent agreement confirms excellent elastic behavior, numerical accuracy, mechanical stability with the elastic regime, and validates the reliability of the derived elastic constants. Conversely, the hypothetical $P{6}_3/mmc$-Fe₃Pt phase displays a noticeable deviation from linearity even at small strains, reflected by a relatively low $R^2=0.860$. The irregular stress response suggests internal atomic relaxations or numerical instabilities during deformation, which is consistent with the observed negative $C_{44}$ elastic constant and shear modulus values. This behavior indicates that although predicted to be thermodynamically stable, the hypothetical $P{6}_3/mmc$–Fe₃Pt phase is mechanically unstable and prone to soft-mode distortions under elastic perturbation.

Moreover, beyond the elastic region, all phases undergo a pronounced softening characterized by a broad pressure minimum between ~0.08 and 0.12 strain, reflecting structural relaxation or lattice rearrangements that allow deformation with minimal additional stress. At larger strains, the curves gradually recover, showing an increase in pressure consistent with hardening or densification as the lattice resists further deformation. Collectively, these results highlight that while all Fe₃Pt polymorphs follow similar overall deformation stages, elastic response, softening, and high-strain recovery, the stiffness and nature of the softening region are strongly dependent on crystallographic symmetry, with the $Cmmm$ and $R\overline{3}m$ phases being the most mechanically rigid and $P{6}_3/mmc$ the most accommodating under strain. Therefore, the combination of stress vs. strain analysis and correlation fitting provides a robust means of assessing the elastic consistency and mechanical resilience of crystal systems.

3.5. Phonon Dispersion Curves

Figure 5 illustrates the calculated phonon dispersion relations of Fe₃Pt alloys in four crystallographic structures: (a) orthorhombic $Cmmm$, (b) hexagonal $P{6}_3/mmc$, (c) tetragonal $P4/mmm$, and (d) trigonal $R\overline{3}m$ plotted along their respective high-symmetry directions in the Brillouin zone. The phonon spectra were obtained to evaluate the dynamical stability and vibrational properties of the different structural configurations.

The phonon dispersion curves of the orthorhombic $Cmmm$ and trigonal $R\overline{3}m$ show no imaginary modes with negative frequencies below the zero-frequency line, indicating dynamical stability at 0 K within the harmonic approximation. The imaginary modes observed at finite $q (G\to Z)$ for $R\overline{3}m$ indicate a finite-wavelength dynamical instability and not the collapse of the entire structure. The absence of imaginary modes signifies that the atomic arrangements in these configurations correspond to local minima on the potential energy surface and do not exhibit any tendency toward structural distortion. In contrast, the hexagonal $P{6}_3/mmc$ and tetragonal $P4/mmm$ phases display imaginary frequencies along certain high-symmetry paths, revealing dynamical instability. This instability is especially pronounced in the $P{6}_3/mmc$ structure, where extensive imaginary modes appear along all examined high-symmetry directions. The dynamical behavior is fully consistent with the elastic properties obtained for this phase as shown in Table 3. In particular, the negative elastic constant $C_{44}$ and shear modulus unambiguously indicates a mechanical instability associated with shear deformation. A negative $C_{44}$ implies that the crystal releases, rather than stores, elastic energy under shear strain, a phenomenon that manifests microscopically as soft transverse acoustic branches adopting imaginary frequencies. This correspondence is clearly reflected in the spectrum, where several transverse-like modes exhibit strong softening and cross into the imaginary region. For the $P4/mmm$ phase, imaginary modes are confined mainly to the region between the $M$ and $Z$ points, indicating that its instability is present but comparatively less severe. These modes can be assigned to a negative anisotropy factor (A = −2.79), which indicates a significant deviation from isotropic elastic behavior. In practice, this magnitude of anisotropy manifests as direction-dependent phonon behavior: acoustic branch slopes (and hence sound velocities) differ markedly between high-symmetry directions and formerly degenerate optical branches split and shift unevenly. In the $P4/mmm$ spectrum, this is reflected by localized softening that is stronger along specific directions. The small, localized imaginary dips observed along particular paths therefore appear as directional soft modes rather than a uniform lattice collapse; this is the microscopic vibrational fingerprint of elastic anisotropy. The phonon bandwidths differ among the four phases, reflecting variations in bonding stiffness and lattice dynamics. The $P{6}_3/mmc$ structure (Figure 5b) exhibits the widest phonon dispersion range, stretching between −13 THz and 15 THz, followed by the $Cmmm$ (−1–9.24 THz, Figure 5a) and $P4/mmm$ (2.2–9.6 THz, Figure 5c) phases, while the $R\overline{3}m$ phase (−1.2–7.1 THz, Figure 5d) displays the narrowest frequency range. This hierarchy implies that the interatomic force constants are strongest in the $P{6}_3/mmc$ structure and weakest in the $R\overline{3}m$ phase, indicating progressively softer lattice vibrations from $P{6}_3/mmc$ to $R\overline{3}m$.

Moreover, the spectra show that the orthorhombic $Cmmm$ and trigonal $R\overline{3}m$ phases exhibit well-behaved, non-anomalous acoustic slopes at $G$, indicating well-defined longitudinal and transverse sound velocities and consequently mean acoustic velocity (${\upsilon }_m$) and implying relatively high characteristic phonon velocities and Debye temperatures (${\mathsf{\Theta }}_D$) as shown in Table 3. By contrast, the hexagonal $P{6}_3/mmc$ phase shows extensive imaginary modes and therefore lacks meaningful long-wavelength elastic phonons in the harmonic limit; extraction of sound speeds or a Debye temperature for $P{6}_3/mmc$ from the present 0 K phonons is not physically meaningful. A physically meaningful Debye temperature requires a sensible mean sound velocity, which in turn requires positive (real) transverse and longitudinal sound speeds. These come from the shear modulus and elastic constants like $C_{44}$. Hence, ${\mathsf{\Theta }}_D$ and ${\upsilon }_m$ cannot be computed meaningfully for the $P{6}_3/mmc$ alloy. The tetragonal $P4/mmm$ phase occupies an intermediate position: the acoustic branches are intact along some directions but are softened (and locally imaginary) near $M-Z$, indicating strongly direction-dependent (anisotropic) sound velocities and a reduced Debye temperature along the softened directions.

4. Conclusions

Density functional theory calculation on the martensite $Cmmm$, $P{6}_3/mmc$, $P4/mmm,$ and $R\overline{3}m$ Fe₃Pt alloys were successfully carried out to determine the structural, thermodynamic, magnetic, electronic and mechanical properties. The negative enthalpies of formation showed that all the structures are thermodynamically stable. The substantial magnetic moments ranging from 2.94 ${\mathsf{\mu }}_{\mathrm{B}}$ to 3.04 ${\mathsf{\mu }}_{\mathrm{B}}$ and strong asymmetry of the spin-up and -down in the density of states plots suggest strong magnetism and ferromagnetism behavior of the Fe₃Pt alloys. Calculations of independent and macroscopic elastic constant predicted mechanical stability on the $Cmmm$, $P4/mmm,$ and $R\overline{3}m$ phases. However, the hypothetical $P{6}_3/mmc$ phase possesses negative $C_{44}$ and shear modulus values, suggesting that it cannot exist under ambient conditions and would transform into a lower-symmetry and more stable phase. Computations of phonon dispersion spectra corroborated the mechanical properties as they revealed dynamical stability on $Cmmm$ and $R\overline{3}m$ phases, metastability on $P4/mmm,$ and instability on $P{6}_3/mmc$. The observations collectively demonstrate that in addition to the $I4/mmm$-Fe₃Pt alloy, there exist other martensite $Cmmm$, $P4/mmm,$ and $R\overline{3}m$ phases which are thermodynamically, mechanically, and dynamically stable, highly magnetic, strongly ferromagnetic, ductile, and stiff, consistent with their martensitic character.