Exploring the thermomagnetic behavior of Co2TiZ (Z=Al, Si, Ga, Ge, and Sn) alloys: a computational study

A comprehensive computational exploration of the structural and thermomagnetic properties of Co2TiZ (Z = Al, Si, Ga, Ge, and Sn) Heusler alloys are conducted utilizing both density functional theory (DFT) and Monte Carlo simulations (MC). Our calculations revealed that the XA prototype consistently exhibited larger lattice parameters than the L21 structure. Furthermore, the investigation of exchange parameters uncovered distinct differences between the L21 and XA prototypes. The L21 structures consistently exhibited stronger Co-Co interactions, while the XA prototypes showcased more pronounced Co-Ti interactions. The calculated Curie temperatures (Tc) varied between the L21 and XA prototypes, highlighting the significance of atomic arrangement. The calculated critical temperature Tc of Co2TiAl exhibited variation depending on the structural prototype, and it is determined to be equal to 131K for the L21 prototype, while in the XA structure, it increases significantly to 248K. The higher Tc indicates improved thermal stability, expanding the material’s operational range and making it suitable for applications that require magnetic functionality at higher temperatures.


Introduction
Full Heusler alloys have been the subject of extensive research due to their unique properties and potential applications in spintronics [1][2][3][4][5][6][7].These intermetallic compounds, typically represented by the formula X 2 YZ are characterized by their high degree of order and half-metallic nature, which refers to metallic behavior in one spin channel and semiconducting behavior in the other.This dual nature positions them as ideal candidates for integration into spintronic devices [8][9][10][11].Among the full Heusler alloys, the cobalt-based full Heusler alloys are known for their high Curie temperatures and robust half-metallicity, which make them promising candidates for high-temperature spintronic applications.Recently, cobalt-based alloys, such as Co 2 TiZ (Z = Al, Si, Ga, Ge, and Sn), have attracted significant attention [12][13][14][15][16].It has been established that these alloys have adopted the conventional L2 1 prototype structure as their ground state [17].In addition, the majority exhibit perfect halfmetallic behavior with 100% spin polarization, a characteristic that makes them ideal for spintronic applications [13,[18][19][20][21][22].However, Co 2 TiGa deviates from this trend, demonstrating near half-metallicity due to a slightly reduced spin polarization [23].In addition, despite the promising properties of Co 2 TiZ alloys, their practical application in spintronics is constrained by relatively low Curie temperatures calculated by previous experimental investigations (ranging from 130K to 380K) in the L2 1 structure.Exploring alternative structures like XA could potentially yield higher Curie temperatures [14,16,17,[24][25][26][27][28][29].
Despite the extensive research on Co 2 TiZ (Z = Al, Si, Ga, Ge, and Sn) full Heusler alloys, their thermomagnetic properties, particularly under varying structural conditions, remain not fully explored.In this context, the novel application of Monte Carlo simulations presents an opportunity for a more comprehensive exploration of these properties.Therefore, by utilizing a combination of density functional theory (DFT) calculations and Monte Carlo simulations, we systematically investigate the thermomagnetic properties of these alloys under both L2 1 and XA prototypes.Expanding upon traditional approaches, we have systematically explored and contrasted the L2 1 and XA prototypes, uncovering nuanced distinctions in their structural and magnetic properties.Our findings pave the way for a more informed selection of atomic arrangements in future alloy designs, emphasizing the potential of these prototypes in advancing the field of material science.

Calculations details
Our calculations are carried out using density functional theory (DFT) implemented via the Vienna ab-initio Simulation Package (VASP) [30,31] and spin-polarized relativistic Korringa-Kohn-Rostoker (SPR-KKR) [32] codes.Specifically, VASP is used for volume optimization, accurately representing our Heusler alloys' structures in their respective L2 1 and XA configurations.The projector augmented wave (PAW) [31,33,34] method is selected to describe the electron-ion interaction, a choice that combines accuracy and computational efficiency.The exchange-correlation energy is approximated within the framework of the Generalized Gradient Approximation (GGA) as proposed by Perdew-Burke-Ernzerhof (PBE) [35,36].The cut-off energy of the plane-wave basis set is chosen at 350 eV to ensure the precision of our calculations.The Brillouin zone integration was conducted using a 16 16 16 ´´Monkhorst-Pack k-points grid.The convergence criterion of the total energy of the system is set to be 10 −7 eV.In addition, the effect of Coulomb interaction (U) is taken into account to improve the localization of the 3d orbital using approach of Dudarev [37].This approach is described by U U J eff =where U is coulomb repulsion and J is the Hund exchange parameter.The values of U for Co and Ti are 3.2, and 0.72 eV, respectively, while J is fixed to 0.1 eV for both elements [38, 39].The calculations of Heisenberg exchange interaction parameters are carried out using a technique proposed by Liechtenstein as implemented in the SPRKKR (spin-polarized relativistic Korringa-Kohn-Rostoker) code.These calculations are performed using a grid of 57 3 k-points in the irreducible part of the Brillouin zone, with a total of 4495 k-points.The spin-polarized scalar-relativistic (SP-SREL) Hamiltonian is used with an energy convergence criterion of 10 −7 Ry.Furthermore, we will extend our study to investigate the temperature-dependent magnetic properties using Metropolis algorithm [40] in framework of Monte Carlo simulation (MC).These calculations are carried out using Uppsala atomistic spin dynamics (UppASD) code [41] and execute using a 24 24 24 ´´supercell with periodic boundary conditions.Also, a total of 5×10 4 Monte Carlo steps were implemented for the purpose of equilibrating the system under investigation.Subsequent to this equilibration process, an equivalent number of steps for each spin configuration has been applied.The calculations of thermomagnetic properties are discussed in detail in our previous works [42][43][44][45].

Structural properties
The structural properties of a material are pivotal for understanding its overall performance and behavior.These are the characteristics that determine how a material responds to temperature changes.Within the framework of full Heusler alloys, two prominent crystallographic structures, denoted as L2 1 and XA prototypes, as shown in figure 1, are commonly employed.These prototypes, characterized by their distinctive atomic arrangements, result in distinct electronic configurations, ultimately affecting the magnetism of the alloy due to the relationship between electron configuration and magnetic properties.Determining lattice parameters and total energy is a vital part of our investigation into the structural properties of the L2 1 and XA prototypes in full Heusler alloys.As depicted in figure 2(a), (a) comprehensive calculation of the lattice parameters for both L2 1 and XA structures is performed, and a clear correlation between the atomic size and the unit cell size can be observed.In both L2 1 and XA structures, the trend of increasing lattice parameters with the atomic size of the Z element is evident.The smallest element, Si, corresponds to the smallest lattice parameter, while the largest element, Sn, leads to the largest lattice parameter in both structures.This highlights the influence of atomic radii on the dimensions of the unit cell.The calculated lattice parameters of the studied Heusler alloys correspond closely with the experimentally observed values.For the Co 2 TiAl alloy, the experimental measurements for the lattice parameter are reported as 5.848 and 5.837 [14,17].Also, the experimental lattice parameter of Co 2 TiSi is observed as 5.74 and 5.733 [16,17].In the case of Co 2 TiGa, the experimentally derived lattice parameters are 5.848 and 5.857 [17,46], while for Co 2 TiGe, the values are 5.831 and 5.82 [17,47].Lastly, for Co 2 TiSn, the experimental lattice parameters have been reported as 6.07 and 6.072 [17,48].A comprehensive analysis and further discussion regarding the lattice parameters are reported in the appendix.As illustrated in figure 2(b), the calculated total energy aligns well with the experimental observations, reinforcing the theoretical understanding that the L2 1 structure exhibits higher stability than the XA prototype.The lower total energy of the L2 1 structure implies a more energetically favored configuration, indicating that under typical conditions, the Co 2 TiZ (Z = Al, Si, Ga, Ge, and Sn) alloys will preferentially adopt this prototype.

Thermomagnetic properties
Thermomagnetic properties involve changes in magnetic properties due to temperature variations.Understanding thermomagnetic properties is crucial as it provides insights into the stability of a material's magnetic state over a range of temperatures.The exchange interaction is the quantum mechanical phenomenon that describes the magnetic interaction between neighboring spins in a material.Calculating the exchange interaction parameters is essential for understanding the magnetic properties of materials.Figure 3 shows the Computed exchange interaction parameters for Co 2 TiAl and Co 2 TiGa Heusler alloys with both L2 1 and XA prototypes.These compounds have the same number of valence electrons, which leads to similarities in their electronic configurations.In the case of the L2 1 prototype, we observe that the interaction between cobalt atoms (Co-Co) manifests as the strongest exchange interaction.Specifically, the nearest neighbor interaction in Co 2 TiAl is approximately 3.5 meV, indicating a significant magnetic coupling between the adjacent cobalt atoms.
Similarly, for Co 2 TiGa, the Co-Co exchange interaction's strength is higher, at approximately 4.   Co atoms are noticeably larger than those of the Ti atoms, which suggests a stronger influence on the overall magnetism of the alloy.A similar trend in exchange interactions is observed for the XA prototype as in the L2 1 structure.The Co-Co exchange interaction remains prominent but with a key distinction.This interaction exhibits antiferromagnetic characteristics in the XA prototype, marking a significant departure from the behavior seen in the L2 1 structure.This antiferromagnetic interaction in the Co-Co pairing under the XA configuration suggests a more complex magnetic behavior in these Heusler alloys.In the context of the XA prototype of Co 2 TiAl and Co 2 TiGa full Heusler alloys, the observed antiferromagnetic (AFM) behavior between second nearest neighbors of Co 1 atoms is attributed to super exchange interactions mediated by the nonmagnetic, either Al or Ga, atoms existing within the structure.These atoms, due to their non-magnetic nature, are hypothesized to play a pivotal role in the mediation of super exchange interactions.The path connecting Co and non-magnetic atoms, if conducive to suitable orbital overlap and electron hopping, is likely to facilitate super exchange mechanisms, thus inducing AFM coupling between Co atoms.This occurs despite the typically ferromagnetic nature of direct Co-Co interactions.The mechanism's efficiency hinges on the electronic configuration of the non-magnetic atoms, where the hybridization of non-magnetic s and p orbitals with the d orbitals of Co atoms is crucial for enabling the super exchange interaction.Further discussion on the super exchange mechanism can be found in [49] In the case of Co 2 TiSi, Co 2 TiGe, and Co 2 TiSn compounds with the L2 1 structure, a trend similar to that observed for Co 2 TiAl and Co 2 TiGa is noted, as shown in figure 4. The interaction between cobalt atoms (Co-Co) emerges as the most significant, with approximately 8 meV, indicating its dominance in influencing the magnetic behavior of these alloys.In the XA structure, an intriguing change is observed in the Co-Ti exchange interaction.Compared to the L2 1 prototype, the Co-Ti interaction in the XA structure becomes notably stronger, with the exchange parameter value exceeding one meV.On the other hand, the Co-Co exchange interaction in the XA structure appears to weaken compared to that in the L2 1 structure.For instance, in Co 2 TiSn, this interaction drops to approximately 0.5 meV.
The thermomagnetic properties of a material, which depict the variation in magnetic behaviors as a function of temperature, are of immense scientific and technological significance.These properties can reveal the critical temperatures at which phase transitions occur, such as from ferromagnetic to paramagnetic states.Additionally, understanding how magnetism varies with temperature can aid in predicting the material's performance under different thermal conditions and offer valuable insights into the potential uses and limitations of materials.The C ( ) At T , C the thermal energy becomes sufficient to randomize the orientations of the magnetic moments.As a result, the spontaneous magnetization of the material goes to zero, where the material transitions from a ferromagnetic to a paramagnetic state.On the other hand, the magnetic susceptibility-temperature ( T c -) curve is often more accurate in predicting the Curie temperature compared to the magnetization-temperature due to the inherent nature of the properties they depict.Near the T C in T ccurve, the susceptibility often exhibits a sharp peak which is generally very discernible, making T ccurves advantageous for accurately determining T .
C In contrast, the decay in M T -curve is often more gradual and lacks the sharp feature observed in T ccurves, making the exact point of T C more challenging to pinpoint accurately.The calculated Curie temperatures of Co 2 TiAl and Co 2 TiGa using Monte Carlo simulations yield 131K and 156K, respectively, as shown in figure 5.These findings align well with the range observed experimentally.The experimental data shows the T C of Co 2 TiAl to be within the range of 125K to 138K [14,17,24,25,27] and that of Co 2 TiGa to range from 124K to 130K [17,25,28].The T C of Co 2 TiAl and Co 2 TiGa fall within the same range due to their similar atomic and electronic structures and the similarity of their exchange interaction parameters.Switching from the L2 1 to the XA prototype can change the calculated Curie temperatures.For Co 2 TiGa, the calculated Curie temperature remains in the same range when using the XA prototype.This suggests that the XA and L2 1 prototypes lead to similar exchange interactions in this compound, possibly due to similar degrees of orbital overlap in the two structures.The increase in the total magnetic moment at zero temperature may indicate a higher degree of alignment of the magnetic moments in the structure, leading to a stronger net magnetization.In contrast, the much higher calculated Curie temperature for Co 2 TiAl in the XA structure suggests a significant difference in the exchange interactions compared to the L2 1 structure.The higher T C indicates stronger ferromagnetic exchange interactions, which might be due to different atomic arrangements leading to a more favorable overlap of atomic orbitals in the XA prototype.The increase in the T C significantly bolsters the performance and stability of a ferromagnetic material.With a higher T , C the ferromagnetic material exhibits improved stability and expands the operational range of the material.The calculated Curie temperatures for Co 2 TiSi, Co 2 TiGe, and Co 2 TiSn, as shown in figure 6, are 366 K, 358 K, and 362 K.These results show remarkable consistency with existing experimental data.For instance, the observed Curie temperatures for Co 2 TiSi reported in the literature range around 375 to 380 K [16,24], closely mirroring our calculated value.Similarly, our calculation for Co 2 TiGe aligns with the reported experimental value of 386 K [17].Furthermore, the calculated Curie temperature for Co 2 TiSn converges to a mean of the experimental range of 355 K to 364 K [17,29].The higher Curie temperatures observed for Co 2 TiSi, Co 2 TiGe, and Co 2 TiSn compared to Co 2 TiAl and Co 2 TiGa can be attributed to these compounds' strength and type of magnetic exchange interactions.The observed decrease in both the Curie temperature and magnetic moment when transitioning from the L2 1 to the XA prototype for Co 2 TiSi, Co 2 TiGe, and Co 2 TiSn is likely attributable to changes in the atomic arrangement and the subsequent alterations in electronic structure.A reduction in Curie temperature suggests that the ferromagnetic exchange interactions become weaker in the XA prototype for these compounds, possibly due to reduced overlap of the magnetic atomic orbitals, leading to decreased magnetic moment as well.A drastic decrease in the Curie temperature from the L2 1 to XA prototype for Co 2 TiSn, down to 81K, suggests a significantly weakened ferromagnetic exchange interaction within the XA structure for this compound.

Conclusion
In this comprehensive computational study, we have examined the structural and thermomagnetic properties of Co 2 TiZ (Z = Al, Si, Ga, Ge, and Sn) using density functional theory and Monte Carlo simulations, highlighting the significant effects of atomic arrangements on the magnetism of these Heusler alloys.The calculated lattice parameters for both XA and L2 1 structural prototypes corroborated well with available experimental data.Furthermore, it was observed that the XA prototype consistently displayed a larger lattice parameter compared to the L2 1 structure.The calculated exchange interaction parameters revealed distinct trends between the L2 1 and XA prototypes.The L2 1 structures consistently demonstrated stronger Co-Co interactions, while the XA prototypes exhibited more robust Co-Ti interactions, underlining the influence of differing atomic arrangements on the magnetic interactions within these Heusler alloys.Moreover, our calculations of the Curie temperatures revealed notable differences between the L2 1 and XA prototypes.While the L2 1 structure typically resulted in lower Curie temperatures for Co 2 TiAl and Co 2 TiGa, and higher temperatures for Co 2 TiSi, Co 2 TiGe, and Co 2 TiSn.Notably, the Curie temperature of Co 2 TiAl in the XA prototype increased to 248K from 131K in the L2 1 structure, highlighting a substantial improvement in thermal stability.This increase in Curie temperature suggests that these materials can maintain magnetic functionality over a wider temperature range, extending their utility in high-temperature environments.This work seeks to identify potential applications of the investigated alloys within the fields of spintronics, high-density data storage, and magnetic cooling systems.

ORCID iDs
Bilal Aladerah https:/ /orcid.org/0000-0002-2835-7904Maen Gharaibeh https:/ /orcid.org/0000-0003-2797-7746Abdalla Obeidat https:/ /orcid.org/0000-0002-8504-0583 5 meV.Comparatively, the Co-Ti and Ti-Ti exchange interactions within the L21 structure are negligible when set against the pronounced Co-Co interaction.Despite the presence of these atomic pairs, their contribution to the overall magnetic behavior of Co 2 TiAl and Co 2 TiGa alloys is relatively minimal.The relative insignificance of the Co-Ti and Ti-Ti exchange interactions can be attributed to the comparatively low magnetic moments of the Ti atoms.Specifically, for Co 2 TiAl, the calculated magnetic moments for Co and Ti are 0.54 B m and −0.105 , B m respectively, and for Co 2 TiGa, these values are 0.947 B m for Co and −0.024 B m for Ti.The magnetic moments of

Figure 2 .
Figure 2. Comparative analysis of (a) lattice parameters and (b) total energy for L21 and XA prototypes of Co2TiZ (Z = Al, Si, Ga, Ge, and Sn) Heusler alloys.

Figure 3 .
Figure 3. Exchange interaction parameters for Co 2 TiAl and Co 2 TiGa Heusler alloys with both L2 1 and XA prototypes.

Figure 5 .
Figure 5. Calculated M T and T ccurves for Co 2 TiAl and Co 2 TiGa Heusler alloys with both L2 1 and XA prototypes.

Figure 6 .
Figure 6.Calculated M T and T ccurves for Co 2 TiSi, Co 2 TiGe and Co 2 TiSn Heusler alloys with both L2 1 and XA prototypes.

Table A1 .
Calculated lattice parameters for both prototypes of Co 2 TiZ (Z = Al, Si, Ga, Ge, and Sn) alloys alongside with available reported theoretical and experimental values.