Introduction

Heusler alloys have long attracted attention due to their remarkable multifunctional properties and their suitability for diverse applications, including spintronics, optoelectronics, thermoelectric, and magnetic shape memory devices 1. Since their discovery by Fritz Heusler in 1903, these compounds have remained a fascinating class of intermetallic materials with unique and tunable properties 2. In recent decades, ferromagnetic Heusler alloys have emerged as particularly promising candidates for spin-based and magneto-electronic technologies due to their high Curie temperature (\(T_{\textrm{C}}\)), robust ferromagnetism, and high spin-polarization (P) 3,4,5,6. The spin polarization \(P\) is given by:

$$\begin{aligned} P = \frac{N^{\uparrow }(E_F) - N^{\downarrow }(E_F)}{N^{\uparrow }(E_F) + N^{\downarrow }(E_F)} \times 100\%, \end{aligned}$$
(1)

where \(N^{\uparrow }(E_F)\) and \(N^{\downarrow }(E_F)\) are the densities of states (DOSs) at the Fermi level (\(E_F\)) with up and down spins, respectively. A finite spin polarization arises from an imbalance between these two spin channels, while a perfect half-metallic system exhibits \(P = 100\%\) due to the complete absence of one spin channel, and the other spin channel is present at \(E_{\textrm{F}}\). Such half-metallic ferromagnets (HMFs) are ideal for generating highly spin-polarized currents, making them essential building blocks for spintronic devices such as magnetic tunnel junctions and spin valves 7,8.

Although a wide range of materials have been theoretically predicted and experimentally confirmed to display half-metallicity, such as transition-metal oxides (e.g., CrO\(_2\) 9, Fe\(_3\)O\(_4\) 10), manganites (e.g., (La,Sr)MnO\(_3\) 11), and diluted magnetic semiconductors (e.g., (Ga,Mn)As 12), their practical applicability is often limited. These materials usually suffer from low \(T_{\textrm{C}}\) near room temperature (RT) and phase instability, especially in oxides, where complex stoichiometry can lead to multiple competing phases during growth. In contrast, Heusler alloys stand out as robust half-metallic systems with \(T_{\textrm{C}}\) often exceeding 800–1000 K, stable ordered phases, and chemical tunability, making them superior for RT spintronic applications 13,14,15,16.

Among these, Heusler alloys have demonstrated excellent spin-dependent transport properties even at room temperature, primarily due to their high spin polarization (P) and high Curie temperature (\(T_{\textrm{C}}\)) 17,18,19. Structurally, Heusler alloys are commonly classified into three principal categories: half-Heusler (XYZ, C1\(_b\)-type, space group: F\(\bar{4}\)3m (No. 216)), full-Heusler (X\(_2\)YZ, L2\(_1\)-type, Fm\(\bar{3}\)m (No. 225)), and equiatomic quaternary Heusler (XX\(^{\prime }\)YZ, Y-type, F\(\bar{4}\)3m (No. 216)) alloys. Here, X, X\(^{\prime }\), and Y are typically transition metals, while Z represents a main-group element 20. The vast number of possible elemental combinations and crystal configurations within this family allows systematic tuning of structural, magnetic, and electronic behavior, making Heusler alloys a uniquely versatile platform for spintronic and multifunctional material design.

It is well-known that Heusler alloys are very sensitive to structural disorder 21, which results in the formation of minority spin states in the half-metallic gap, thereby reducing P, degrading the half-metallic nature, and lowering device efficiency. Therefore, it is essential to overcome this issue. In recent years, equiatomic quaternary Heusler alloys (EQHAs) have attracted increasing attention due to their enhanced chemical tunability, ordered atomic arrangements, and reduced structural disorder compared to conventional ternary Heusler systems. EQHAs are derived from the full-Heusler structure (X\(_2\)YZ) by replacing one of the X atoms with a different transition metal (X\(^{\prime }\)) 22. These compounds crystallize in a Y-type structure (LiMgPdSn prototype) with four interpenetrating face-centered cubic (fcc) sublattices. The presence of distinct atomic species at each sublattice site helps suppress antisite disorder, a major drawback in half- and full-Heusler alloys, which often leads to reduced spin polarization and degraded transport performance. Consequently, EQHAs offer an excellent platform for realizing half-metallicity with high structural order, large Seebeck coefficients, and low carrier scattering, making them ideal for both spintronic and thermoelectric applications. Experimentally, several EQHAs such as CoFeCrAl, CoFeMnSi, and CoFeCrGa have shown large magnetoresistance, high Seebeck coefficients, and robust Curie temperatures well above RT 23,24,25, highlighting their technological promise. In addition to these systems, Cr-based quaternary Heusler alloys have also gained considerable attention, with several studies reporting their structural stability, half-metallicity, high Curie temperatures, and promising spintronic and thermoelectric characteristics 26,27,28. These findings further highlight the growing interest in Cr-based EQHAs and reinforce their potential for multifunctional applications.

Despite some progress in Co- or Cr- and Fe-based EQHAs, Ti- and Ni-based quaternary systems remain largely unexplored, particularly those containing Cr and Mn, leaving gaps in our understanding regarding spin polarization, magnetic, electronic, transport properties, and dynamics (phonons). In this context, we investigate a new series of Ti- and Ni-based EQHAs, namely XMnCrZ (X = Ti, Ni; Z = Sb, Sn), which have not yet been reported either experimentally or theoretically. These compositions were strategically selected to combine Mn and Cr with the contrasting electronic configurations of Ti and Ni as well as the heavy main-group elements Sb and Sn, which are known to enhance spin-orbit coupling and thermoelectric performance. In particular, TiNiSn and related systems promise thermoelectric properties 29,30, whereas NiMnSb displays spin-dependent transport properties 31. However, their equiatomic quaternary counterparts remain unstudied.

The present work aims to fill this gap by conducting a comprehensive first-principles investigation of the structural, electronic, magnetic, mechanical, vibrational, and thermoelectric properties of the XMnCrZ (X = Ti, Ni; Z = Sb, Sn) quaternary Heusler alloys. Using Density Functional Theory (DFT) 32 and Density Functional Perturbation Theory (DFPT) 33, we systematically explore their ground-state properties and stability, while Boltzmann transport theory is employed to estimate their thermoelectric performance. Special attention is given to the half-metallic behavior, spin polarization, and Seebeck coefficients that determine their potential for multifunctional device applications.

Computational method

The electronic structure and physical properties of XMnCrZ (X = Ti, Ni; Z = Sb, Sn) Heusler alloys were systematically investigated using Density Functional Theory (DFT) within the Full-Potential Linearized Augmented Plane Wave (FLAPW) framework, as implemented in the FLEUR code 34. Structural optimization, magnetic analysis, and calculations of electronic, elastic, and thermoelectric properties were performed using the Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional 35 within the framework of the Generalized Gradient Approximation (GGA). To assess the sensitivity of the results to the choice of exchange–correlation treatment, additional magnetic calculations were carried out using the PBE0 hybrid 36, SCAN meta-GGA 37, and DFT+\(U\) approach 38 functionals for evaluating the magnetic properties. Meanwhile, the electronic band structures were determined using PBE, SCAN and DFT+U, ensuring a more accurate description of both the band dispersion and correlation effects.

For the basis set, a plane-wave cutoff parameter of \(\textit{R}_{MT}K_{\text {max}} = 9\) was employed, where \(\textit{R}_{MT}\) denotes the smallest muffin-tin radius and \(K_{\text {max}} = 4.5\) corresponds to the maximum value of the reciprocal lattice vector. For the \(XMnCrZ\) (\(X = \textrm{Ti}, \textrm{Ni};\ Z = \textrm{Sn}, \textrm{Sb}\)) quaternary Heusler alloys, the selected muffin-tin radii (\(\textit{R}_{MT}\), in atomic units) were: Ti = 2.1, Ni = 2.0, Mn = 2.0, Cr = 2.0, Sn = 2.4, and Sb = 2.4. Inside the muffin-tin spheres, the valence wavefunctions were expanded using spherical harmonics up to an angular momentum quantum number of l\(_{max}\) = 10. Brillouin zone integrations were carried out using 72 k-points in the irreducible wedge, corresponding to a \(12 \times 12 \times 12\) Monkhorst–Pack mesh 39. The band structure is produced along high-symmetry points using 400 k-points. Self-consistent field (SCF) calculations were considered converged when the charge density distance between successive iterations was below \(10^{-5}\) meV/a.u.\(^{3}\)

Phonon dispersion spectra were computed to examine the dynamical stability of the compounds using the plane-wave pseudopotential method implemented in the Quantum Espresso package 40. To model the interaction between ionic cores and valence electrons, norm-conserving pseudopotentials 41 were utilized along with a \(4 \times 4 \times 4\) q-point mesh.

Thermoelectric transport properties were evaluated using the semi-classical Boltzmann theory with the BoltzTraP2 code 42. To assess the suitability of XMnCrZ alloys for thermoelectric applications, we systematically examined their key properties, including the Seebeck coefficient (S), electrical conductivity (\(\sigma /\tau\)), and electrical thermal conductivity (\(\kappa _{e}/\tau\)) 43, by using the BoltzTraP2 code, which employs the semi-classical Boltzmann transport theory within the framework of the relaxation time approximation 44.

Elastic constants (C\(_{ij}\)) and related thermophysical parameters were computed using the thermo_pw package within Quantum Espresso, employing the Voigt–Reuss–Hill averaging scheme 45 to provide reliable estimates of bulk, shear, and Young’s moduli.

Post-processing and visualization of the computed results were carried out using Python-based analysis scripts, along with the VESTA 46 and XCrySDen 47 software packages.

Results and discussions

Structural and mechanical properties

Understanding the stability and mechanical performance of XMnCrZ (X = Ti, Ni; Z = Sb, Sn) alloys is crucial for evaluating their suitability in practical applications. In the present study, we begin by examining the equilibrium structural parameters of XMnCrZ alloys through first-principles calculations, followed by an evaluation of their mechanical properties. Structural optimization was carried out to determine the equilibrium lattice parameters, atomic positions, and total energies, which were subsequently used to assess thermodynamic stability. The optimized crystal structures also served as the basis for calculating elastic constants, enabling verification of their mechanical robustness via Born–Huang stability criteria.

EQHAs can be described as four interpenetrating fcc sublattices with Wyckoff positions 4a (0.0, 0.0, 0.0), 4b (0.5, 0.5, 0.5), 4c (0.25, 0.25, 0.25), and 4d (0.75, 0.75, 0.75) 48. When the Z atoms occupy the position 4a, the three remaining atoms X, X\('\) = Mn, and Y = Cr yield only three independent, energetically nondegenerate configurations out of six possible permutations, because interchanging atoms in the 4c and 4d positions produces energetically equivalent structures. These three distinct configurations, referred to as \(\alpha\), \(\beta\), and \(\gamma\), are each characterized by specific Wyckoff positions 49 and are considered in the present study to examine the structural stability of XMnCrZ (X = Ti, Ni; Z = Sb, Sn). Table 1 summarizes the atomic occupations used in our DFT calculations, and Figure 1 illustrates the corresponding unit cell structures.

Fig. 1
Fig. 1
Full size image

Three structural configurations of XMnCrZ (X = Ti, Ni; Z = Sb, Sn) quaternary Heusler alloys based on different Wyckoff occupations for (a) \(\alpha\), (b) \(\beta\), and (c) \(\gamma\) phases.

Table 1 Atomic occupations of the \(\alpha\), \(\beta\), and \(\gamma\) phases of equiatomic quaternary Heusler alloys used in the DFT calculations.
Full size table

To identify the ground-state phase of each XMnCrZ (X = Ti, Ni; Z = Sb, Sn) alloy, total-energy optimization was performed for the \(\alpha\), \(\beta\), and \(\gamma\) structural configurations. For every phase, we further evaluated four distinct spin arrangements among the transition-metal atoms (Ti/Ni, Mn, and Cr), together with a non-magnetic configuration (NM), in order to establish the energetically preferred magnetic state. From these calculations, the most stable magnetic and structural configuration for each alloy was selected for detailed analysis.

The optimized equilibrium lattice parameters were then obtained by fitting the total energy–lattice parameter data using the third order Birch–Murnaghan equation of state, expressed as 50:

$$\begin{aligned} E(V) = E_{0} + \frac{9V_{0}B}{16} \left\{ \left[ \left( \frac{V_{0}}{V} \right) ^{\tfrac{2}{3}} - 1 \right] ^{3} B' + \left[ \left( \frac{V_{0}}{V} \right) ^{\tfrac{2}{3}} - 1 \right] ^{2} \left[ 6 - 4 \left( \frac{V_{0}}{V} \right) ^{\tfrac{2}{3}} \right] \right\} , \end{aligned}$$
(2)

where E(V) is the total energy at a given volume V, \(E_{0}\) is the minimum energy at the equilibrium volume \(V_{0}\), B is the bulk modulus, and \(B'\) is its pressure derivative. The corresponding fitted curves and equilibrium parameters are presented in Figure 2, which summarizes the optimized energetics of all studied alloys.

As observed from the optimized energy profiles, the Ni-based alloys exhibit distinct phase preferences. For the NiMnCrSn compound, the \(\gamma\) phase emerges as the most stable configuration, corresponding to the magnetic arrangement Ni \(\uparrow\), Mn \(\uparrow\), and Cr \(\downarrow\). In contrast, the NiMnCrSb alloy stabilizes in the \(\alpha\) phase (magnetic ordering \(\uparrow\)–\(\downarrow\)–\(\uparrow\)). However, the energy difference between the \(\alpha\) and \(\gamma\) phases in this alloy is extremely small, with the \(\gamma\) phase lying only about 0.31 mRy higher in energy. For the Ti-based systems, both TiMnCrSn and TiMnCrSb alloys favor the \(\beta\) phase as the ground-state structure. In each case, the lowest-energy configuration corresponds to the magnetic ordering \(\downarrow\)–\(\uparrow\)–\(\uparrow\), indicating a consistently stable spin arrangement across the Ti-containing compounds.

Overall, all the alloys investigated in this work are found to stabilize in ferrimagnetic ground states(FIM). The optimized equilibrium lattice parameters derived from the Birch–Murnaghan equation of state are summarized in Table 2 below.

Fig. 2
Fig. 2
Full size image

Variation of total energy as a function of lattice parameter for (a) NiMnCrSn (b) NiMnCrSb (c) TiMnCrSn (d) TiMnCrSb in three distinct configurations.

Table 2 Equilibrium lattice parameter \(a_{0}\), ground state energy \(E_{0}\), energy difference \(\Delta E\), bulk modulus B, and its pressure derivative \(B'\) for FIM and NM states of XMnCrZ (X = Ti, Ni; Z = Sb, Sn) alloys.
Full size table

In addition to stability, we computed the elastic constants (\(C_{11}\), \(C_{12}\), and \(C_{44}\)) to evaluate the alloy’s response to mechanical deformation. Elastic constants are essential for understanding and analyzing these mechanical properties. The mechanical stability in conventional elasticity is governed by the Born-Huang criteria  51 as follows:

$$\begin{aligned} C_{11}> 0, \quad C_{44}> 0, \quad C_{11} - C_{12}> 0, \quad C_{11} + 2C_{12} > 0, \quad C_{12}< \frac{C_{11} + 2C_{12}}{3} < C_{11} \end{aligned}$$
(3)

These conditions must be fulfilled to guarantee that the material can resist small deformations without undergoing mechanical instability. The computed values for the XMnCrZ (X= Ti, Ni, Z= Sb, Sn) alloys at 0 K, zero pressure, and zero strain are listed in Table 3. The Voigt-Reuss-Hill scheme was adopted to evaluate the bulk modulus (B), shear modulus (G), Young’s modulus (E), Poisson’s ratio (\(\nu\)), and the Zener anisotropic factor (A) from the given mathematical formulae 52:

$$\begin{aligned} G = \frac{C_{11} - C_{12} + 3C_{44}}{5} \end{aligned}$$
(4)
$$\begin{aligned} B = \frac{C_{11} + 2C_{12}}{3} \end{aligned}$$
(5)
$$\begin{aligned} E = \frac{9BG}{3B + G} \end{aligned}$$
(6)
$$\begin{aligned} \nu = \frac{3B - 2G}{2(3B + G)} \end{aligned}$$
(7)
$$\begin{aligned} A = \frac{2C_{44}}{C_{11} - C_{12}} \end{aligned}$$
(8)

The Young’s modulus of a material is related to its resistance to shape deformation, whereas the bulk modulus provides insight into the volume change against external pressure. A higher B value indicates that the material is less compressible and more resistant to volume change under pressure, while a larger value of Young’s modulus describes the structural rigidity under uniaxial tension or compression 53. Table 3 indicates that TiMnCrSn has the highest values of both modulus of elasticity and bulk modulus among all studied alloys, suggesting it is the stiffest and hardest. This trend is also consistent with the results obtained earlier from the Birch-Murnaghan equation of state fitting, where TiMnCrSn exhibited the highest bulk modulus of 178.3 GPa.

Similarly, analysis of the shear modulus is another significant parameter that can be derived from the elastic constants. The modulus of rigidity measures a material’s resistance to plastic deformation 54. Based on our calculations, the following sequence represents the resistance to irreversible shape changes: TiMnCrSn \(\;>\;\) TiMnCrSb \(\;>\;\) NiMnCrSb \(\;>\;\) NiMnCrSn. This result also confirms the high stiffness and ability to maintain structural integrity under torsional or sliding loads of the TiMnCrSn alloy.

Further insights were obtained from ductility indicators. The statement of Pettifor 55 refers to the fact that Cauchy pressure (\(C_p\)) can be evaluated by the \(C_{12} - C_{44}\) relation. From the Cauchy pressure, one can predict the nature of atomic bonding along with the ductile and brittle behavior of a material 56. Positive Cauchy pressure \(C_p\) suggests metallic bonding in all alloys. Additionally, both Pugh’s ratio (\(B/G > 1.75\)) and Poisson’s ratio (\(\nu > 0.26\)) indicate a ductile nature. Those with \(B/G < 1.75\) and \(\nu < 0.26\) are brittle in nature. The Zener anisotropic factor (A) describes the degree of elastic anisotropy in a crystal. It represents how differently a material behaves mechanically along different crystallographic directions. If \(A = 1\), then the crystal is elastically isotropic 57. The higher values of A reflect that crystals are highly anisotropic. The A values of the present alloys significantly deviate from unity, revealing pronounced elastic anisotropy.

Table 3 Elastic and thermodynamic properties of XMnCrZ (X = Ti and Ni, Z = Sb and Sn) Heusler alloys. The elastic constants \(C_{11}\), \(C_{12}\), and \(C_{44}\), along with the bulk modulus B, shear modulus G, and Young’s modulus E, are expressed in GPa. Additional parameters include the Pugh’s ratio (B/G), Cauchy pressure (\(C_P\)), Poisson’s ratio (\(\nu\)), anisotropy factor (A), and Debye temperature (\(\theta _D\) in K).
Full size table
Fig. 3
Fig. 3
Full size image

(a) Young’s modulus, (b) Shear modulus, and (c) Poisson’s ratio: 3D surfaces (left) and 2D projections (right) illustrating elastic anisotropy.

When a material is under uniaxial tensile stress, it elongates along the direction of loading and undergoes deformation 58. To visualize this anisotropy, we employed an open-source software ELATE 59 to plot 2D and 3D directional maps of Young’s modulus, shear modulus, and Poisson’s ratio along the xy, yz, and xz planes for the TiMnCrSb alloy, as illustrated in Figure 3. Moreover, the three-dimensional projections of these planes are provided for better visualization. For isotropic materials, these plots appear as perfect spheres (3D) or circles (2D), indicating uniform mechanical behavior in all directions. However, deviations from these ideal shapes reflect elastic anisotropy 60. The 3D surface for Young’s modulus, shown in green, along with its 2D projections on the xy, xz, and yz planes, reveals pronounced petal-like distortions. These indicate a strong directional dependence in stiffness. For the shear modulus, the outer blue and inner green surfaces represent the maximum and minimum values, reflecting directional variation in shear resistance. Poisson’s ratio plot similarly reveals orientation-dependent transverse strain. Overall, the clear deviation from spherical/circular symmetry across all three elastic tensors highlights the strong directional dependence of elastic properties, confirming the anisotropic mechanical response of these alloys.

Phonon dispersion

To further evaluate the dynamical stability of the investigated alloys, phonon dispersion curves (PDC) and phonon density of states (PhDOS) were calculated using DFPT. These were examined along high-symmetry directions in the Brillouin zone for XMnCrZ (\(X =\) Ti, Ni; \(Z =\) Sb, Sn) alloys, portrayed in Figure 4. Since each unit cell contains four atoms, a total of 12 vibrational branches were obtained. The first three modes in the low-frequency band (LFB) region represent the acoustic modes, while the remaining nine are optical modes.

The eigenfrequency of the acoustic phonon modes carries information about the dynamical stability 61. As observed in Fig. 4, all Ti-based alloys exhibit real and positive phonon frequencies throughout the Brillouin zone, confirming the absence of negative (imaginary) frequencies and thereby validating their dynamical stability. Moreover, a noticeable gap exists between the acoustic and optical branches, particularly in TiMnCrSb, where the maximum acoustic phonon frequency reaches slightly above 100 cm\(^{-1}\), while TiMnCrSn extends up to 171 cm\(^{-1}\). The well-separated optical modes and the existence of this acoustic–optical gap further support the stable lattice dynamics of the Ti-based systems. The absence of significant gaps within the optical region also indicates that the constituent atoms possess comparable masses, consistent with Sb and Sn being relatively heavy main-group elements.

In contrast, the Ni-based alloys (NiMnCrSb and NiMnCrSn) show the presence of negative frequencies in their phonon spectra, indicating dynamical instability at 0 K. Additionally, these compounds lack a distinct gap between the acoustic and optical branches, which can lead to strong acoustic-optical phonon interactions. Such interactions typically enhance phonon scattering and lead to reduced lattice thermal conductivity 62, a feature that could be favorable for thermoelectric applications but detrimental for structural stability.

Overall, phonon analysis reveals a clear distinction between Ti- and Ni-based quaternary Heusler alloys: the Ti-based compounds are dynamically stable, satisfying the criteria of no negative frequencies and a clear acoustic–optical gap, whereas the Ni-based counterparts exhibit dynamical instabilities that compromise their structural integrity.

Fig. 4
Fig. 4
Full size image

Phonon dispersion relations and phonon density of states (PhDOS) for (a) NiMnCrSn, (b) NiMnCrSb, (c) TiMnCrSn, and (d) TiMnCrSb quaternary Heusler alloys.

Electronic properties

Following the structural and stability (mechanical and dynamical) assessments, the electronic properties were investigated to gain deeper insight into the nature of bonding in the XMnCrZ alloys. Spin-polarized band structures, calculated using both the PBE and SCAN exchange–correlation functionals, are presented in Figure. 5 along the high-symmetry path X–K–\(\Gamma\)–L–W–X–\(\Gamma\) in the Brillouin zone. The electronic band structure reveals a distinct spin splitting between the majority- and minority-spin channels in the vicinity of the Fermi level (\(E_{\textrm{F}}\)), clearly indicating the ferrimagnetic nature of all the investigated alloys. This spin-resolved behavior plays a crucial role in determining the electronic transport characteristics and underscores the importance of magnetic ordering in these quaternary Heusler systems.

Among the investigated alloys, TiMnCrSb distinctly exhibits half-metallic behavior, characterized by a metallic nature in the spin-down channel and a semiconducting nature in the spin-up channel. The spin-up channel shows an energy bandgap of 0.54 eV within the PBE exchange–correlation framework, while the SCAN functional, known for its improved treatment of electronic localization and more accurate bandgap predictions, increases this gap to 1.12 eV. For TiMnCrSn, a trace amount of electronic states is observed at the Fermi level in both spin channels, indicating a slight deviation from ideal half-metallicity. In contrast, the Ni-based counterparts (NiMnCrZ ( Z= Sn, Sb) and display metallic character, with several bands crossing the Fermi level in both spin channels. This metallic nature is consistently confirmed by both PBE and SCAN calculations, confirming the intrinsic metallic nature of the Ni-containing alloys.

Fig. 5
Fig. 5
Full size image

Spin-polarized band structures of XMnCrZ (X = Ti, Ni; Z = Sb, Sn) quaternary Heusler alloys. Panels (a–b) show TiMnCrSb, (c–d) TiMnCrSn, (e–f) NiMnCrSb, and (g–h) NiMnCrSn, with spin-up (left) and spin-down (right) channels.

To further examine the role of electron correlation in shaping the electronic structure, particularly the \(d\)-orbital hybridization and the origin of half-metallicity, DFT+U calculations were performed using the PBE exchange-correlation functional for TiMnCrSb and for the remaining alloys (See Supplimentary sec. 1.0). A Hubbard parameter of \(U = 2.5\) eV was applied to both Mn and Cr atoms to account for on-site Coulomb interactions in their localized \(3d\) states. The spin-resolved band structure of TiMnCrSb obtained from DFT+\(U\) is shown in Figure. 6, while the results for the other alloys are provided in the Supplementary Section. As expected, the inclusion of \(U\) widens the band gap in the spin-up channel, increasing it by approximately 0.67 eV compared with the PBE value in TiMnCrSb alloy. This enhancement primarily originates from the downward shift of the \(t_{2g}\) manifold to around \(-0.443\) eV along the \(\Gamma\) path, with dominant contributions from Mn-\(1d\) and Cr-\(3d\) states. The \(e_{g}\) manifold (\(0.23\)eV), also formed by Mn and Cr \(d\)-orbitals, is similarly affected. The projected band structures of this alloy can be found in the Supplementary Sec. 2.0

The physical origin of the half-metallic gap in TiMnCrSb can be understood from the combined effects of crystal-field splitting, exchange splitting, and strong d–d / d–p hybridizations among the transition-metal atoms 63,64. In fact, a larger exchange integral enhances the exchange splitting and consequently widens the gap. In this alloy, Ti and Mn occupy the tetrahedral sites, while Cr resides in the octahedral site, resulting in distinct crystal-field environments and a clear separation between the \(t_{2g}\) and \(e_g\) manifolds of Mn and Cr. The strong hybridization between Mn-1d, Cr-3d, and Sb-p states further redistributes the electronic states around the Fermi level, giving rise to the characteristic gap in the spin-up channel. Moreover, the sizeable exchange splitting between the \(t_{2g}\) and \(e_g\) states plays a crucial role in pushing the majority- and minority-spin bands apart, consistent with the mechanism reported for other half-metallic Heusler alloys 65. The inclusion of the Hubbard correction enhances the localization of Mn and Cr \(d\)-electrons, strengthening the crystal-field driven splitting and shifting the hybridized bands, thereby reinforcing the half-metallic nature of this alloy 66.

Fig. 6
Fig. 6
Full size image

Spin-polarized band structure of TiMnCrSb alloy using DFT+\(U\): (a) spin-up and (b) spin-down electronic bands.

To gain a deeper understanding of the electronic properties of the investigated XMnCrZ (X = Ti and Ni, Z = Sb and Sn) alloys, the total and partial density of states (TDOS and PDOS) around the Fermi level were analyzed using the GGA method, as illustrated in Figure. 7. For improved readability, we present only the TDOS in the main text, while the detailed PDOS for each alloy is made available in the Supplementary Sec. 3.0.

For the TiMnCrSb alloy, the density of states confirms the half-metallic nature suggested by the band structure. Specifically, a distinct energy gap appears in the majority-spin channel near \(E_{\textrm{F}}\), while the minority-spin channel exhibits metallic behavior due to the presence of electronic states at \(E_{\textrm{F}}\). This spin asymmetry results in complete spin polarization, a characteristic feature of half-metals. The total DOS reveals that all atomic species contribute to the minority-spin states at \(E_{\textrm{F}}\), with Cr and Mn atoms playing the dominant role. The partial DOS further highlights the crystal field splitting of the Mn and Cr 3d orbitals into \(t_{2g}\) and \(e_{g}\) components. This splitting, driven by the surrounding crystal field and Coulomb interactions, leads to partial localization of the d electrons and the formation of the energy gap in the majority-spin channel 67. A similar electronic structure is observed in the TiMnCrSn alloy, supporting its classification as a half-metallic ferrimagnet. However, the Ni-based counterparts, NiMnCrSb and NiMnCrSn, display notably different behavior. In these alloys, both majority- and minority-spin channels have states at \(E_{\textrm{F}}\), indicating metallic character rather than half-metallicity. This is consistent with the overlapping valence and conduction bands observed in their spin-resolved band structures and density of states.

Fig. 7
Fig. 7
Full size image

Total density of states for Ni- and Ti-based Heusler alloys: (a) NiMnCrSb, (b) NiMnCrSn, (c) TiMnCrSb, (d) TiMnCrSn.

Table 4 Spin polarization of Ti- and Ni-based quaternary Heusler alloys.
Full size table

Magnetic properties

The magnetic characteristics of Heusler alloys are crucial in determining their suitability for spintronic applications. In this study, we evaluated the magnetic stability of XMnCrZ (X = Ti, Ni; Z = Sb, Sn) alloys by analyzing their spin polarization and total magnetic moments. For Ti-based alloys, TiMnCrSb exhibits complete spin polarization (see Table 4), confirming its half-metallic character, while TiMnCrSn shows a slightly reduced value (\(P \approx 92.9\%\)), still making it an excellent candidate for spintronic devices. In contrast, both NiMnCrSb and NiMnCrSn display finite density of states at the Fermi level in both spin channels, resulting in metallic behavior and the absence of half-metallicity.

The total magnetic moments were further examined using the Slater–Pauling rule 68, expressed as \(M_{t} = Z_{t} - 24\), where \(M_{t}\) is the total magnetic moment (in \(\mu _{B}\)) per formula unit and \(Z_{t}\) is the total number of valence electrons. For TiMnCrSb (\(Z_{t} = 22\)) and TiMnCrSn (\(Z_{t} = 21\)), the expected total magnetic moments are \(2.0~\mu _{B}\) and \(3.0~\mu _{B}\), respectively. Our DFT calculations show excellent agreement with these predictions, confirming that the Ti-based alloys strictly obey the Slater–Pauling rule. In contrast, the Ni-based alloys deviate from this behavior, a trend commonly observed in several Heusler systems and often attributed to strong \(d\)–\(d\) hybridization effects 69 or the involvement of transition-metal elements, which modify the electronic structure and reduce the applicability of simple electron-counting models 70.

Since magnetic exchange interactions and spin polarization in Heusler alloys are highly sensitive to the treatment of electronic correlations, the use of different exchange–correlation functionals was implemented to ensure the reliability and accuracy of the predicted magnetic properties. The atom-resolved magnetic moments, summarized in Table 5, show that Mn and Cr atoms dominate the total magnetization in all compounds, while Ti/Ni and Sb/Sn atoms contribute only weakly, often exhibiting antiparallel alignment in the Ni-based systems. The negligible interstitial magnetic moments further confirm that the magnetization is highly localized around atomic sites. Additionally, both GGA and hybrid PBE0 functionals yield nearly identical magnetic moments, reinforcing the reliability of the results.

Importantly, the SCAN meta-GGA functional predicts magnetic moments that generally fall between those obtained from PBE and DFT+U, providing a balanced treatment of electron localization without requiring empirical parameters. SCAN slightly enhances the Mn and Cr moments in the Ni-based alloys while moderately adjusting the Ti-site compensation in the Ti-based systems. In contrast, DFT+U produces the largest Mn and Cr magnetic moments due to its explicit correction for on-site Coulomb interactions, which strengthens electron localization on d states 71,72.

In summary, Ti-based alloys (TiMnCrSb and TiMnCrSn) exhibit robust half-metallicity, high spin polarization, and adherence to the Slater–Pauling rule, making them strong candidates for spintronic devices. Conversely, the Ni-based alloys show metallic character and deviations from the Slater–Pauling rule, primarily due to complex electronic hybridization effects.

Table 5 Magnetic Moments (\(\mu _B\)) of Atoms in TiMnCrZ and NiMnCrZ (\(Z =\) Sb, Sn) Alloys using Different Functionals.
Full size table

Thermoelectric properties

Developing high-efficiency thermoelectric materials remains a critical challenge for both industrial applications and fundamental research. Thermoelectric materials are capable of converting waste heat into energy through the Seebeck effect. The Seebeck effect refers to the generation of an electric potential across the material in response to the temperature gradient. The thermoelectric properties of Heusler alloys were investigated to evaluate their potential for energy conversion applications 73. To assess the suitability of XMnCrZ alloys for thermoelectric applications, we systematically examined their key properties, including Seebeck coefficient (S),electrical conductivity (\(\sigma /\tau\)), electrical thermal conductivity (\(\kappa _{e}/\tau\)), and power factor (\(PF = S^{2}\sigma /\tau\)) using the BoltzTraP2 code within the semi-classical Boltzmann framework 74.

The transport coefficients were calculated both at the intrinsic Fermi level (Fig. 8) and over a range of chemical potentials (Fig. 9) by employing the constant relaxation time approximation (CRTA) 75. This enables us to determine how shifts in the Fermi level influence the Seebeck coefficient, electrical conductivity, and power factor. By analyzing this energy-dependent behavior, we illustrate how suitable tuning of the Fermi level through doping or carrier adjustment can enhance the thermoelectric performance of the XMnCrZ alloys. Given these considerations, it is important to acknowledge the inherent limitations of the CRTA. Because BoltzTraP employs a constant relaxation time rather than momentum- or energy-dependent scattering rates, the resulting transport coefficients should be interpreted with caution in a quantitative sense. As highlighted by Anooja et al. 76, incorporating realistic scattering mechanisms is essential for achieving fully quantitative accuracy. Nevertheless, the CRTA remains useful for analyzing relative trends and comparative transport behavior across materials.

As shown in Figure 8, the Seebeck coefficient exhibits a noticeable temperature dependence in the range of 300–800 K at the Fermi level. The Ti-based alloys display negative Seebeck values across the entire temperature range, whereas the Ni-based counterparts show positive values. Among all compounds, TiMnCrSn attains the largest negative Seebeck coefficient, reaching –30.53 µV/K at room temperature, which confirms its dominant n-type behavior with electrons as the primary charge carriers. In contrast, both NiMnCrSb and NiMnCrSn show p-type conduction, with only a modest increase in the Seebeck coefficient at higher temperatures.

Understanding the interplay between electronic and thermal transport properties is essential for optimizing thermoelectric performance. In this context, analyzing the temperature-dependent behavior of electrical conductivity and electronic thermal conductivity provides valuable insights into the underlying carrier transport mechanisms 77. The \(\kappa _{e}/\tau\) increases with temperature for all alloys due to the thermal excitation of charge carriers. The parallel variation of \(\sigma /\tau\) and \(\kappa _{e}/\tau\) supports the Wiedemann–Franz law 78, which states that electronic thermal conductivity is proportional to electrical conductivity. Comparatively, the Ni-based alloys show higher electrical conductivity values and, proportionally, higher electronic thermal conductivity as well, primarily due to their metallic character, which is confirmed by the band structure calculations, resulting in larger carrier concentration and higher mobility near the Fermi level. This inherently lower \(\kappa _e\) in Ti-based alloys is advantageous for thermoelectric performance, as it minimizes electronic heat loss and supports higher efficiency. It is worth noting that the total thermal conductivity (\(\kappa\)) consists of both electronic (\(\kappa _e\)) and lattice (\(\kappa _L\)) contributions. Since the lattice part arises from phonon-mediated heat transport, which is beyond the present computational framework, we report here only the electronic contribution. For completeness, the lattice thermal conductivity was estimated semi-empirically using the widely employed Slack equation 79, and the details can be found in the Supplementary Sec. 4.0.

In thermoelectric materials, the power factor (PF) measures how effectively a material can produce electrical power from a temperature gradient, with a higher PF indicating better thermoelectric performance 80. Among the investigated systems, TiMnCrSn exhibits the highest power factor at the Fermi level, increasing steadily with temperature and reaching its peak near 800 K. TiMnCrSb follows a similar trend with slightly lower magnitude, while both NiMnCrSn and NiMnCrSb show relatively smaller PF values with limited change over temperature. These results indicate that Ti-based alloys are more suitable for energy harvesting applications, primarily due to their more favorable electronic structures and enhanced carrier mobility, which together optimize the electrical response at elevated temperatures.

Fig. 8
Fig. 8
Full size image

Thermoelectric properties at the Fermi level for the investigated quaternary Heusler alloys: (a) Seebeck coefficient, (b) electrical conductivity, (c) power factor, and (d) electronic thermal conductivity.

Next, we investigated the thermoelectric properties when the Fermi level is shifted to specific energy values (\(E'\)), as depicted in Figure 9. Upon fine adjustment, the Seebeck coefficient of TiMnCrSb drastically increases, reaching 441.47 µV/K at room temperature, clearly indicating that this alloy exhibits the most promising thermoelectric performance among the systems studied. For the other alloys, the Seebeck coefficient also shows a noticeable enhancement with this energy shift. Similarly, the power factor values for all alloys increase significantly under these conditions, emphasizing the critical role of material engineering and doping in optimizing thermoelectric performance. To explore these effects in more detail, the variation of the thermoelectric properties over a range of chemical potentials is provided in the Supplementary Section, which illustrates how manipulation of the electronic structure and energy levels can be strategically employed to enhance the thermoelectric response.

Fig. 9
Fig. 9
Full size image

Calculated thermoelectric properties of XMnCrZ alloys. For each temperature, the Seebeck coefficient (S) is scanned over the full range of chemical potentials, and the maximum S value is selected (a). (b) The electrical conductivity, (c) power factor, and (d) electronic thermal conductivity shown here correspond to the same chemical potential at which the maximum S occurs.

Thermodynamic properties

In addition to thermoelectric performance, understanding the thermodynamic properties of materials is crucial for evaluating their thermal stability and suitability for high-temperature applications. Thermodynamic analysis provides fundamental insights into lattice dynamics, interatomic interactions, and the overall thermal behavior of materials. In this study, key parameters such as specific heat capacity (\(C_v\)), entropy (S), vibrational free energy (F), and internal energy (E) are computed using the quasi-harmonic Debye model 81. These properties are essential for characterizing the response of the alloys with temperature variation and for assessing their potential under realistic operating conditions.

Specific heat capacity determines the thermal stability and energy storage capability of alloys, which is necessary to understand their high-temperature behavior. The enhancement of lattice vibrations with elevated temperature directly elevates the material’s specific heat capacity. The variation of \(C_v\) with respect to temperature is shown in Figure 10 for the four alloys under investigation. The specific heat at constant volume increases with temperature up to 667 K for TiMnCrSb, 631 K for TiMnCrSn, 243 K for NiMnCrSb, and 358 K for NiMnCrSn. Beyond these temperatures, \(C_v\) tends to saturation, approaching the Dulong–Petit limit. At very low temperatures, the heat capacity (\(C_v\)) follows a \(T^3\) dependence, as predicted by Debye’s theory of lattice vibrations 82,83. The maximum recorded \(C_v\) values are 98.43 J/mol-K for TiMnCrSb, 97.13 J/mol-K for TiMnCrSn, 99.78 J/mol-K for NiMnCrSb, and 98.43 J/mol-K for NiMnCrSn.

Fig. 10
Fig. 10
Full size image

Variation of the specific heat capacity (Cv) as a function of temperature for the XMnCrZ (X=Ti, Ni and Z = Sb and Sn) alloys.

By considering lattice vibrations, the temperature variation of vibrational free energy (F) can be related to the following equation 84:

$$\begin{aligned} F = 3nN k_{B} T \int _{0}^{\infty } \ln \left[ 2 \sinh \left( \frac{\hbar \omega }{2 k_{B} T} \right) \right] g(\omega ) \, d\omega , \end{aligned}$$
(9)

where n is the number of atoms per formula unit, \(k_{B}\) is Boltzmann’s constant, T is the absolute temperature, \(\hbar\) is the reduced Planck’s constant, \(\omega\) is the phonon frequency, and \(g(\omega ) \, d\omega\) describes the phonon density of states. For the alloys, the lattice vibration variation with respect to temperature is The vibrational energy as a function of temperature is plotted in Figure 11. This figure clearly highlights the presence of vibrational energy even at zero kelvin. This zero-point energy indicates that lattice vibrations persist at absolute zero temperature in all these alloys, a direct consequence of the Heisenberg uncertainty principle 85. The variation of vibrational energy for our alloys follows the sequence: TiMnCrSn > TiMnCrSb > NiMnCrSn > NiMnCrSb. This behavior arises because atoms with smaller masses tend to vibrate at higher frequencies.

Fig. 11
Fig. 11
Full size image

Variation of the vibrational free energy as a function of temperature for the XMnCrZ (X=Ti, Ni and Z = Sb and Sn) alloys.

Along with vibrational free energy, the temperature dependence of internal energy E and entropy S can be expressed using the following equations 86:

$$E = \frac{3nN\hbar }{2} \int _{0}^{\infty } \omega \coth \left( \frac{\hbar \omega }{2k_{B}T}\right) g(\omega ) \, d\omega$$
$$S = 3nNk_{B} \int _{0}^{\infty } \left[ \frac{\hbar \omega }{2k_{B}T} \coth \left( \frac{\hbar \omega }{2k_{B}T}\right) - \ln \left( 2\sinh \left( \frac{\hbar \omega }{2k_{B}T}\right) \right) \right] g(\omega ) \, d\omega$$

Here, the internal energy E accounts for the thermal excitation of phonons, while the entropy S reflects the configurational disorder associated with lattice vibrations. Other parameters in the equations retain their usual physical meanings. Lattice vibrations, being the dominant contributor to internal energy in solids, significantly influence thermodynamic behavior. Moreover, entropy plays a central role in the Gibbs free energy expression \(G = H - TS\), which governs phase stability 87. The temperature-dependent variation of these thermodynamic quantities is depicted in the following plots (Fig. 12). As temperature increases, internal energy exhibits a near-linear growth, while entropy shows a logarithmic increase due to enhanced atomic motion 88.

Fig. 12
Fig. 12
Full size image

Variation of (a) internal energy and (b) entropy with respect to temperature for the XMnCrZ (X=Ti, Ni and Z = Sb and Sn) alloys.

Conclusion

In summary, we performed a systematic first-principles investigation of Ti- and Ni-based equiatomic quaternary Heusler alloys XMnCrZ (X = Ti, Ni; Z = Sb, Sn). The elastic and mechanical analysis revealed that all compounds are mechanically stable and ductile, with TiMnCrSn showing the highest stiffness among the investigated systems. Phonon dispersion confirmed the dynamical stability of Ti-based alloys, while Ni-based alloys exhibited soft modes, indicating potential instability without external stabilization.

Electronic and magnetic properties showed a clear contrast between Ti- and Ni-based systems. TiMnCrSb and TiMnCrSn were identified as half-metallic ferromagnets with nearly complete spin polarization and magnetic moments following the Slater–Pauling rule, making them suitable for spintronic applications. In contrast, Ni-based alloys exhibited metallic behavior and deviations from the Slater–Pauling rule due to complex hybridization effects. The use of different exchange–correlation functionals, including DFT+U, ensured reliable predictions of the magnetic and electronic structures.

Thermoelectric analysis identified TiMnCrSb as the most promising multifunctional alloy, combining half-metallicity, strong ferrimagnetic ordering, and excellent transport properties. NiMnCrSb also exhibited good thermoelectric performance at elevated temperatures. Thermodynamic modeling confirmed that all alloys remain stable across a wide temperature range, with entropy and specific heat following expected temperature dependencies.

Altogether, TiMnCrSb emerges as the most promising multifunctional alloy among the studied systems, combining half-metallicity, strong ferrimagnetic ordering, and favorable thermoelectric behavior. The present study not only expands the understanding of Ti- and Ni-based quaternary Heusler alloys but also provides useful guidance for future experimental synthesis and device integration in spintronics, thermoelectrics, and high-temperature functional materials.