Investigation of the thermodynamic, structural, electronic, mechanical and phonon properties of D0c Ru-based intermetallic alloys: an ab-initio study

The crystal structures of X₃Ru alloys (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn) belong to the body centered tetragonal D0c phase with the I4/mcm space group. Within the unit cell, the Ru atoms are positioned at the Wyckoff site 4a (0, 0, 0.25), while X atoms are located at 4b (0, 0.5, 0.25) and 8 h (0.25, 0.75, 0) positions, as illustrated in figure 1.

To determine the equilibrium ground state properties of X₃Ru alloys, structural optimization was executed under zero temperature-pressure conditions. The results for the equilibrium lattice constants, volume, heats of formation, magnetic moments, and density are presented in table 1. Notably, an observed trend in X₃Ru (X = Sc-Cu) reveals a consistent decrease in calculated lattice parameters (hence volume) across the 3d series, aligning with the trends in atomic radii of the 3d elements across the series. Table 2 shows the calculated and experimental lattice parameters for all 3d-transition metal elements such as Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu and Zn. We note that the calculated and experimental data agree.

The heat of formation (ΔH_f ) for an alloy is defined as the energy required to either form or break chemical bonds and is calculated as:

$$\begin{eqnarray}&&{\rm{\Delta }}{H}_{f}=\left[\displaystyle \frac{E\left({A}_{x}{B}_{y}\right)-[xE(A)+yE(B)]}{x+y}\right]\end{eqnarray} \tag{ 1 }$$

where, E(A_x B_y ), E(A), and E(B) are the calculated equilibrium total energies of the alloy system AB and that of the individual elemental species A and B, with atomic concentrations x and y respectively.

A negative value for ΔH_f  indicates chemical stability, while a positive value of ΔH_f indicates instability. Table 1 displays the calculated heat of formation for X₃Ru alloys. The calculated heat of formation for Cr₃Ru is observed to be comparable to the previous theoretical value. The slight discrepancy can be primarily attributed to the use of different exchange correlation functionals. In the previous study, the GGA functional was employed, whereas in this investigation, the GGA + U (U = 2.5 eV) was used. Notably, Sc₃Ru, Ti₃Ru, V₃Ru, Mn₃Ru, and Zn₃Ru exhibit negative heat of formation, signifying thermodynamic stability. Therefore, these alloy systems can readily be synthesized experimentally under equilibrium conditions. Considering that the stability of intermetallic alloys is intertwined with their magnetic moments through orbital hybridization, we also computed the magnetic moments of X₃Ru alloys. It was observed that all X₃Ru alloys are magnetic except Cu₃Ru and Zn₃Ru. Additionally, it is noteworthy that most of the stable systems display significant magnetic moments, with Mn₃Ru having the highest magnetic moment.

The density of materials is a vital tool used to characterize its use in lightweight applications such as in aerospace, and is calculated as:

$$\begin{eqnarray}&&{\rho }^{cal}=\left[\displaystyle \frac{{M}_{W}\,\ast \,N}{Vol\,\ast \,{A}_{0}}\right]\end{eqnarray} \tag{ 2 }$$

where $Vol$ represents the volume of a unit cell, ${M}_{W}$ is the average molecular weight of the elements in the unit cell, $N$ is the total number of atoms and ${A}_{0}$ is the Avogadro's number (6.022 × 10²³). Table 1 gives the calculated densities of D0c X₃Ru alloys. Importantly, it is observed that the densities of the most stable alloys, namely Sc₃Ru (4.54 g/cm³), Ti₃Ru (5.86 g/cm³), V₃Ru (6.31 g cm⁻³), and Mn₃Ru (6.57 g cm⁻³), are comparable to that of L1₂-Ni₃Al (6.14 g cm⁻³) [12] commonly employed in the aerospace industry. Therefore, these structures are promising candidates for high-temperature lightweight structural applications.

Research on D0c X₃Ru alloys is relatively new and as such there is limited data of these materials systems in literature. Our calculated lattice constant of Cr₃Ru—which is the prototypical alloy in this class of materials agrees with literature. To further validate our theoretical model, our calculated lattice parameters for the individual elements (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu) are agreement with experimental values (table 2).

Understanding the density of states (DOS) is crucial for clarifying the link between the stability of a material and its magnetic properties [46]. The partial density of states (PDOS) for X₃Ru is shown in figure 2, showing the contribution arising from the s, p and d orbitals in the unit cell. Significantly, there is an electronic overlap between the valence and conduction bands near the Fermi energy level, suggesting metallic conductivity in these alloys. In Cu₃Ru and Zn₃Ru, the spin up and spin down bands are symmetric, hence zero spin polarization. This symmetry leads to the elimination of the magnetic moment linked to electronic spin, explaining the absence or minimal calculated magnetic moments (table 1). Conversely, the density of states in Sc₃Ru, Ti₃Ru, V₃Ru, Cr₃Ru, Mn₃Ru, Fe₃Ru, and Co₃Ru indicates spin polarization, confirming the existence of magnetic moments in these systems. The existence of magnetic moments in these alloys makes them potentially viable for applications in spintronics and spin injection. These findings align with our earlier research in similar material systems [32, 47–49].

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Notably, around the Fermi energy, a prominent feature is the presence of a valley referred to as the pseudo-gap in both the spin-up and spin-down bands, which indicates covalent bonding [50, 51]. The existence of the pseudo-gap arises from strong hybridization in X-3d and Ru-3d states, effectively separating the bonding states from the anti-bonding states.

Mechanical stability serves as a metric to gauge a material's strength and is employed for characterizing the structural stability and deformation of a system under external load [52]. This stability is defined in terms of elastic constants C_ij , Bulk Modulus (B), Shear Modulus (G), and Young's modulus (E), providing insights into a material's hardness and ductility. For tetragonal crystals, mechanical stability criteria at zero pressure are defined by the following set of equations [53];

$$\begin{eqnarray}&&\begin{array}{c}C11\gt 0,C33\gt 0,C44\gt 0,C66\gt 0,C11-C12\gt 0,\\ \left(C33+C11-2C13\right)\gt 0,\left(2C11+2C12+C33+4C13\right)\gt 0\end{array}\end{eqnarray} \tag{ 3 }$$

In tetragonal X₃Ru (X = Sc-Zn) alloys, the six independent elastic constants C₁₁ , C₃₃ , C₁₂ , C₄₄ , C₁₃ , and C₆₆ are outlined in table 3. It is noteworthy that the elastic constants of V₃Ru, Ni₃Ru, Co₃Ru, Cu₃Ru, Mn₃Ru, Cr₃Ru, and Zn₃Ru meet the Bohr mechanical stability criteria (C_ij > 0), signifying their mechanical stability. In contrast, Sc₃Ru, Ti₃Ru, and Fe₃Ru structures have C₄₄ and C₆₆ values below zero, indicating mechanical instability.

The melting temperature ($Tm$) of a material depends on its mechanical properties, and it follows a linear relationship with its elastic constants [54–57]. For tetragonal structures, the melting point is given by equation (4):

$$\begin{eqnarray}&&Tm=354K+\left(\displaystyle \frac{450K}{Mbar}\right)* \left[\displaystyle \frac{1}{3}\left(2{C}_{11}+{C}_{33}\right)\right]\pm 300K\end{eqnarray} \tag{ 4 }$$

In table 3, we noticed a rising pattern in melting temperatures from Sc to Zn alloys, with Fe₃Ru exhibiting the highest melting temperature. Moreover, the melting temperatures of Fe₃Ru, Co₃Ru, and Ni₃Ru exceed those of the presently employed L1₂ Ni₃Al (1691 K [12] and 1668 K [58]). As a result, these particular alloys present themselves as promising candidates for high-temperature structural conditions.

The Bulk Modulus (B) is a measure for a material's resistance to compression, and its magnitude is influenced by the crystal structure of the material. Materials with high compressibility exhibit large values of bulk modulus, while low bulk modulus values indicate materials with low compressibility. In the context of tetragonal structures, the expression for B is:

$$\begin{eqnarray}&&{B}_{V}=\displaystyle \frac{1}{9}\left((2{C}_{11}+{C}_{12})+{C}_{33}+4{C}_{13}\right)\end{eqnarray} \tag{ 5 }$$

and the Reuss bounds are:

$$\begin{eqnarray}&&{B}_{R}=\displaystyle \frac{{C}^{2}}{M}\end{eqnarray} \tag{ 6 }$$

where,

$$\begin{eqnarray}&&M={C}_{11}+{C}_{12}+2{C}_{33}-4{C}_{13}\end{eqnarray} \tag{ 7 }$$

with B_R , B = B_H and B_V being the Bulk modulus for Reuss, Voigt and Hill approximations [59]. The shear modulus (G) of a material describes its response to shear stress and is a measure of a material's stiffness. For tetragonal structures, G is expressed as

$$\begin{eqnarray}&&{G}_{V}=\displaystyle \frac{1}{30}(M+3{C}_{11}-3{C}_{12}+12{C}_{44}+6{C}_{66})\end{eqnarray} \tag{ 8 }$$

where, $M={C}_{11}+{C}_{12}+2{C}_{33}-4{C}_{13}$

$$\begin{eqnarray}&&{G}_{R}=15{\left[\left(\displaystyle \frac{18{B}_{V}}{{C}^{2}}\right)+\left(\displaystyle \frac{6}{{C}_{11}-{C}_{12}}\right)+\left(\displaystyle \frac{6}{{C}_{44}}\right)+\left(\displaystyle \frac{3}{{C}_{66}}\right)\right]}^{-1}\end{eqnarray} \tag{ 9 }$$

where, ${C}^{2}=({C}_{11}+{C}_{12}){C}_{33}-2{C}_{13}^{2}$

where the G_R and G_V are the Reuss and Voigt bounds [60], Young's modulus and Poison's ratio ʋ are independent of the type of a material's crystal structure and are given by:

$$\begin{eqnarray}&&E=\left(\displaystyle \frac{9{B}_{X}{G}_{X}}{3{B}_{X}+{G}_{X}}\right)\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\upsilon =\left(\displaystyle \frac{3{B}_{X}-2{G}_{X}}{2({B}_{X}+{G}_{X})}\right)\end{eqnarray} \tag{ 11 }$$

where, X = Voigt, Reuss and Hill approximations.

From the calculated elastic constants provided in table 3, the bulk modulus (B), shear modulus (G), and Young's modulus (E) are computed and presented in table 4. Across all the X₃Ru alloys, it is noteworthy that B_H,R consistently surpasses G_H,R, indicating that the principal parameter governing the stability of base-centered tetragonal X₃Ru alloys is the shear modulus [61]. On the other hand, Young's modulus characterizes a material's strain response to uniaxial stress, reflecting its stiffness. Higher values of Young's modulus are associated with stiffer materials, whereas lower values indicate less stiffness. Table 3 reveals that Ni₃Ru stands out as the stiffest, while Ti₃Ru is the least stiff.

We find that the elastic moduli across the 3d-series (X) is generally lower in the early 3d-elements and higher in the middle 3d-element, with Fe₃Ru, Co₃Ru and Ni₃Ru having the largest B and G values. Interestingly, we therefore conclude that, high elastic moduli in X₃Ru alloys correspond to the high melting temperature (figure 3). Further, the bulk, shear, young's moduli have a monotonic relationship with the melting temperature. Therefore, increased stiffness, hardness, and compression resistance in these structures explains the higher melting temperatures correspond. These findings align with the trends previously observed by Popoola et al [12].

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Applying Pugh's criterion [62], the ratio of bulk to shear modulus (B_H/G_H) is employed to predict the ductility/brittleness of a material, with the critical value set at 1.75. Ductile behavior is predicted if B_H/G_H is greater than 1.75; otherwise, the material is deemed brittle. Notably, Sc₃Ru Mn₃Ru, Fe₃Ru, Co₃Ru, Ni₃Ru, Cu₃Ru, and Zn₃Ru exhibit ductile behavior, while Ti₃Ru, V₃Ru, and Cr₃Ru display brittleness. Additionally, the ductility/brittleness of materials can be assessed using Poisson's ratio, with the threshold value set at 0.33 [63]. Employing the Poisson's ratio criteria, it is observed that all X₃Ru alloys are ductile except V₃Ru and Cr₃Ru, aligning with the results from Pugh's criterion, as shown in the figure 4.

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In evaluating the crystal's stability against shear deformation, Poisson's ratio has been considered [64, 65]. A higher Poisson's ratio indicates better plasticity in crystals, and it also serves as a characterization of the bonding forces within crystals [66]. The central forces in a solid typically fall within the range of 0.25 ≤ υ ≤ 0.5. The computed Poisson's ratios (υ) for X₃Ru range from 0.14 to 0.54. All the structures exhibit a Poisson's ratio within the range of 0.25 ≤ υ ≤ 0.5, confirming that the interatomic forces in these structures are central, except for Ti₃Ru, V₃Ru and Cr₃Ru.

Hardness is a critical attribute in high-temperature environments, particularly in aerospace applications. It defines a material's resistance to localized plastic deformation caused by either mechanical indentation or abrasion. Hardness, while not a fundamental property, relies on tensile strength, yield strength, and the modulus of elasticity. It is determined through the Vickers hardness equation (12) [67]

$$\begin{eqnarray}&&{H}_{V}=2{\left({K}^{2}G\right)}^{0.585}-3\end{eqnarray} \tag{ 12 }$$

where K represents the ratio of shear modulus to bulk modulus (G_H/B_H), and H denotes the Hill approximation. The calculated hardness values for tetragonal X₃Ru structures are presented in table 3. It is evident that V₃Ru exhibits the highest hardness, while Sc₃Ru has the lowest. Notably, the hardness for Ti₃Ru is undefined due to the negative square root in G_H/B_H, indicating instability associated with a phase change in this structure. This type of instability is also observed in ferro-elastic phase transformations [68], as explained by Landau theory [69], where two local minima form in a strain energy function.

Understanding the anisotropic behavior of a material is crucial in engineering science and crystal physics, as it is closely related to micro-cracks in materials. Calculating elastic anisotropy provides valuable information about a material, including details about micro-cracks, phase transformation, precipitation, and dislocation dynamics [70]. For tetragonal alloys, elastic anisotropy can be expressed through three elastic factors: shear anisotropy factors (such as A₁ , A₂ and A₃ ) for different crystallographic planes, the universal anisotropic index (A^U ), and the percentage in compression (A_B ) anisotropy as given below:

$$\begin{eqnarray}&&\begin{array}{c}A1=\displaystyle \frac{2C66}{C11-C12},\,A2=A3=\displaystyle \frac{4C44}{C11+C33-2C13}\\ {A}^{U}=\,5\displaystyle \frac{GV}{GR}+\displaystyle \frac{BV}{BR}-6\,{\rm{and}}\,AB=\displaystyle \frac{BV-BR}{BV+BR}\,\ast \,100 \% \end{array}\end{eqnarray} \tag{ 13 }$$

where A₁ , A₂ and A₃ represent the shear anisotropic factors for the (001), (010) and (100) shear planes. In the case of locally isotropic structural materials, the shear anisotropy factors A₁ , A₂ and A₃ are expected to be equal to one. Additionally, the universal anisotropic index A^U should be zero in such materials. Any deviations from one or zero, indicates the degree of elastic anisotropy [71, 72]. The B_V , B_R , G_V and G_R represents the Voigt and Reuss approximation for bulk modulus (B) and shear modulus (G). The maximum value of (A_B ) is 100%, which corresponds to maximum anisotropy [73, 74] while the minimum is zero which corresponds to an isotropic material.

Table 5 summarizes the calculated values of A_1, A₂, A^U andA_B in the X₃Ru alloys. It is evident that A₁ values in the X₃Ru alloys are smaller than one, indicating elastically anisotropic structures. On the other hand, the A₂ values are larger than one (except in Ti₃Ru and Fe₃Ru) suggesting that they are slightly isotropic. Regarding A^U , it is observed that V₃Ru and Ni₃Ru and Zn₃Ru exhibit a small degree of universal elastic isotropic behavior as they are close to unity. The (A_B ) results clearly indicate that X₃Ru (X = V, Mn, Fe, Co, Ni, Cu, and Zn) structures are close to zero, suggesting they are slightly isotropic, while Sc₃Ru (0.0) is isotropic, indicating that these structures are predicted to have nearly identical values of A_B in all x, y and z directions, attributing to external forces.

Phonons play a vital role in the dynamical behavior and thermal conductivities, which are critical aspects in the research advancements of novel materials. The phonon dispersion curves provide insights into dynamic stability or instability of a material by showcasing positive (real branches) and negative (imaginary branches) phonon frequencies. Positive phonon frequency modes signify dynamic stability in a material, while negative frequency modes indicate dynamic instability in a compound [75, 76]. Figure 5 shows the dispersion curves of D0c X₃Ru (X = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) at plotted along the highest symmetry k-points of the Brillouin zone. The phonon dispersion curves of Cr₃Ru, Fe₃Ru, Ni₃Ru, Cu₃Ru, and Zn₃Ru are dynamically stable due to the absence of imaginary frequencies. Conversely, the phonon dispersion plots of Ti₃Ru, V₃Ru, Co₃Ru, and Mn₃Ru exhibit negative phonon modes, indicating dynamic instability, which can be drawback in the application of these materials where dynamic stability is critical.

In summary, by considering the thermodynamic, mechanical and dynamic stability of the X₃Ru alloys presented in this study, our findings suggest that: (1) D0c X₃Ru (X = Sc, Ti, V, Cr, Mn, and Zn) alloys exhibit negative heat of formation, hence are thermodynamical stable (2) X₃Ru (X = V, Mn, Co, Ni, Cu, and Zn) alloys are mechanically stable, with Co₃Ru and Ni₃Ru exhibiting elevated melting temperatures compared to the currently utilized alloy L1₂ Ni₃Al (1691 °C) [12] (3) X₃Ru (X = Cr, Fe, Ni, Cu, and Zn) are dynamically stable. Therefore, Zn₃Ru meets all three stability criteria, which could give a slight advantage for structural applications over the other systems examined in this work.

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