Theoretical Investigation of Pressure Dependent Structural and Elastic Properties of MnSi Compound

MnSi is a metallic compound that exhibits a structural phase transition as a function of pressure. At ambient conditions, MnSi has a cubic structure such as NaCl type crystal structure. However, at high pressures, it undergoes a structural phase transition to a CsCl type crystal structure. The pressure-dependent structural phase transition in MnSi has been studied using various experimental and theoretical techniques. In the present work, we have used an efficient inter-ionic potential approach to predict pressure dependent structural phase change and associated volume collapse in MnSi. Therein, the potential includes the long-range Coulomb, van der Waals (vdW) interaction, and the short-range repulsive interaction up to second neighbour ions. It has been demonstrated that the Hafemeister and Flygare approach successfully evaluates the equation of state for change in volume with respect to applied pressure. We have identified a structural phase transition from B1(NaCl) type structure to B2 (CsCl) type structure in this compound. The estimated value of phase transition pressure (Pt) is 42 GPa, which is consistent with the available reported data. The identified first order phase transformation showed a volume collapse of about 10% in the vicinity of transition. In addition, we have also investigated second order of elastic constants for MnSi compound.


Introduction
In response to the thermodynamical variables, transition metal monosilicides have been subjected to significant research for the purpose of determining their unique thermoelectric capabilities, superconductivity, structural phase transitions as well as their resistance to oxidation [1][2][3][4][5][6][7][8][9][10].In particular, researchers have done in-depth investigations on the high-pressure transformations of transition metal monosilicides.In order to acquire a deeper comprehension of the physical and chemical features of these monosilicides when subjected to extreme pressurized conditions [5].Manganese silicide (MnSi) is an example of an interesting binary compound that has been the subject of widespread research in the fields of condensed matter physics and materials science due to its high-pressure structural phase transition and other properties [6][7].For instance, Guo et al. [7] used first-principles computations and the CALYPSO structure seeking method to produce a complete analysis of the highpressure structural development of MnSi.In their investigation of the structure and phase stability of the transition metal MnSi compound at high pressure, they discovered a structural phase transition from e-FeSi type to CsCl type, both of which were shown to be mechanically and dynamically stable.
The magnetic behavior of MnSi is one of its most remarkable characteristics [8][9][10][11][12].As a function of pressure, magnetic moments and exchange splitting have been studied long back [8].Corti et al. examined spin dynamics in the MnSi compound [9].As mentioned above, MnSi has attracted as a potential material for spintronics applications, such as magnetic memory and data storage, due to its intriguing magnetic phases.It's interesting to note that one of the most important properties of MnSi is its ability to superconduct [13].Even though MnSi is not a superconductor at atmospheric pressure, it is possible to make it in such a way that it will demonstrate superconductivity.Numerous studies [13,14] have been carried out on this topic in order to gain a deeper comprehension of the fundamental properties of superconductivity and the applications that may be made of it.
In addition, MnSi has been investigated for the thermal and electrical transport capabilities it possesses [15], in addition to the magnetic and superconducting properties it possesses.In contrast to its relatively low electrical conductivity, which has motivated studies aimed at improving its electrical properties for future usage in electronic devices, its high thermal conductivity makes it a promising contender for applications relating to thermal management.Several studies [16][17][18] have been prompted by the fact that its comparatively low electrical conductivity has prompted these studies.For example, monitoring the evolution of the Raman spectrum as a function of the laser intensity is used to explore the thermal stability of MnSi as well as the phase change of the material [19].This allows for the identification of three distinct compositions: MnSi, MnSiO3, and Mn5Si3.After that, a proposal for the included oxidation reaction is made, and it is confirmed by carrying out thermogravimetric and x-ray diffraction examination.
It has also been observed in a prior work that the temperature dependency of the electrical resistivity deviates from that predicted by the traditional Fermi liquid approach as a result of the expansion of the magnetic susceptibility as Tc moves towards 0 K, together with an increment in the amplitude and intensity of quasiparticle interactions at the Fermi level [20].Further, it can be understood that on the magnetic transition on the application of pressure by Koyama et al. [21].In point of fact, Yamada et al. [22] performed an in-depth investigation into the magnetic variation that occurred within the pressure range of 0-1.7 GPa.
As was seen in the above paragraphs, this compound exhibits a number of phenomena that are associated with condensed matter physics.A comprehensive investigation into the pressuredependent structural phase transition and elastic properties of this compound will be a crucial step towards the further realization of this material for its technological applications.In order to accurately forecast the pressure-dependent structural phase transition and elastic properties in MnSi and the related volume collapse, we made use of an effective inter-ionic potential method in this paper.

Computational Method
The effective interionic interaction potential (EIoIP) model that was developed and used herein to analyze MnSi compound at ambient as well as high pressure.In a crystalline lattice, the potential energy that exists between two ions is referred to as the effective interionic interaction potential.This potential is influenced by a number of different parameters, some of which are the size and charge of the ions, the distance between them, and the dielectric constant or polarizability of the medium in which they are situated.This model is particularly helpful for understanding the nature of the atomic interactions that occur within these compounds.In most cases, a change in the structural characteristics of a solid state that is brought about by the application of hydrostatic pressures may be explained using lattice models within the thermodynamic limit.
In order to undertake a study on the thermodynamic properties of semiconducting compounds, it is necessary to first establish an effective interionic potential in such a way that it is able to take into account the way in which the distance between atoms changes as pressure is applied.As a result, in order for a crystal to have the interionic potential of a crystal, there must be a drop in the volume of the unit cell whenever there is an increase in the static pressure of the crystal.The following is an assumption made for the sake of this discussion on the concept: inside the crystal, there is a tiny change in force constant; the short-range interactions are still effective up to the next neighbour ions; and the harmonic elastic forces bind the atoms or ions together without causing any internal strains.If this is the case, the effective interionic interaction potential can be described as follows: Here, the first term represents the long-range Coulomb, the second term indicating short-range repulsive energies, whereas, the third and fourth terms denote the van der Waals interactions, respectively.Further, the symbols: cij and dij are the coefficients of van der Waals (vdW) and βij are the Pauling coefficients expressed as   = 1 + (   /  ) + (   /  ) with Zi(Zj) as the valence and ni(nj) represents the number of electrons in the outermost, respectively, in the equation (1).Moreover, the Coulomb screening effect is explained by the term Zme, as well as b and  which is the short-range parameters.Consequently, only three parameters determine the effective interionic potential: modified ionic charge (Zm), range () and hardness parameters (b) as calculated from crystal properties [23].
It is well knowledge that an isolated phase can only be considered stable if the free energy associated with it is at its lowest possible value under the given thermodynamic conditions.When there is a change in the temperature, pressure, or any other factor that is acting on the systems, there is also a change in the smooth and continuous flow of the free energy.It is said that a phase transition has taken place when there have been changes in the structural details of the phase that have been triggered by differences in the free energy.U is the internal energy, which at 0 K corresponds to the cohesive energy.S is the vibrational entropy at absolute temperature T, pressure P, and volume V.It should come as no surprise that the stability of a specific structure is determined by the minimum of Gibb's free energy.The Gibbs's free energies GB1(r) = UB1(r) + 2Pr 3 for NaCl (B1) phase and GB2(r') = UB2(r') + [8/3√3]Pr' 3 for CsCl (B2) phase become equal at the phase-transition pressure P and at zero temperature i. e., G (= GB1 -GB2).Within this context, UB1 and UB2 stand for the cohesive energies of the B1 and B2 phases, respectively, and are The variable r represents the nearest-neighbor distance in the NaCl structure, whereas r' represents the same distance in the CsCl structure.It is essential to remember that r and r' will have distinct values because the two crystal structures have different lattice constants.The structure of NaCl is a facecentered cubic lattice, whereas the structure of CsCl is a body-centered cubic lattice.Therefore, the nearest-neighbor distances between the two structures will differ.
The variables r and r' denote nearest-neighbor (nn) separations that correspond to the NaCl and CsCl phases, respectively.
In the above equations, we can define b and ρ being the short-range parameters.In materials science and condensed matter physics, the study of second-order elastic constants (SOEC) is an essential area of research.The SOECs, which include the C11, C12, and C44 constants, provide information about a material's elastic properties, which are essential for comprehending its mechanical behavior.
The vibrations of atoms in a material are minimized at 0 Kelvin (or absolute zero), allowing for a more precise determination of the SOECs.The study of these constants at 0 Kelvin can shed light on the nature of the interatomic forces that hold the material together, such as covalent, metallic, and van der Waals interactions.Calculating the SOECs requires intricate mathematical modelling and simulations, the details of which can be found in relevant literature [24].Overall, the study of SOECs at 0 K is a crucial instrument for comprehending the fundamental properties of materials and creating novel materials with the desired mechanical properties.The expressions for the elastic constants can be obtained for B1 structure phase, it follows (5) Similarly for CsCl structured crystal (B2 phase), following determines the expressions for the elastic constants.It follows Here, the coefficient A1, A2, B1 and B2 are defined as the short-range parameters as discussed elsewhere [24] in terms of following

Results and Discussion
To determine the stability of structures with different volumes, force constants are important parameters.Two crystal structures relatively stable each other always required a very precise prediction.There are two aspects that influence the behavior of every crystal structure when we applied pressure: its size and its shape.Changes occurred in nearest-neighbor distances, which are the first indicator affecting overlaps and bandwidths.The second is the changes in symmetry between the atoms, which affects hybridization and bond-repulsion.For the calculation of material parameters, initially we obtained the vdW coefficients using the Slater-Kirlwood variational approach [25] and the estimated values of these coefficients are represented in Table 1.
We were able to calculate the values of three material-dependent characteristics with the assistance of the reported lattice constant (a) [6] and the bulk modulus (BT) [6].These material-dependent parameters are modified ionic charge, hardness, and range, and their values are shown in Table 2.It is important to mention that the pressure-dependent ionic radii of respective ions of both structural phases have been determined by making use of the free parameters as estimated above.This indicates that the characteristics that were accessible to us were utilised in the calculation of the ionic radii of the ions in the crystal lattice of both of the structural phases.Here, the pressure-dependent ionic radii refer to the modifications that occur in the radii of the ions as a result of an increase in pressure.As the pressure is increased, there is a possibility that the size of the ions will alter as a result of the compression of the crystal lattice.When investigating the effects of pressure on the properties of MnSi at varying pressures, it is crucial to take into account the pressure dependency of ionic radii.
In the next step, we then minimised the Gibb's free energy GB1(r) and GB2(r') for the equilibrium interatomic distance (r) and (r') for each and every value of the pressure that is being applied.The calculations have been carried until the point where the self-consistent technique that determined the total energy of the system to be at its lowest possible value.Following that, a stable structure with the minimum Gibbs free energy was For the MnSi compound, the Gibb's free energy difference, denoted by the equation ∆ =  2 ( ′ ) −  1 (), was displayed in Fig. 1 as a function of pressure (Pt), utilizing the effective interionic interaction potential that was mentioned in earlier section.The point at which the Gibb's free energy or enthalpies of the two competing phases become equal is referred to as the phase transition pressure, abbreviated as (Pt).Because the Gibb's free energy for the MnSi compound in crystal phase B1 is lower than that of the B2 phase below the phase transition pressure (Pt), this phase is more thermodynamically and mechanically stable than the B2 phase, which is unstable.This can be seen clearly from the Gibb's free energy curve.As seen in figure 1, the Gibb's free energy for the B2 phase decreases more quickly than it does for the B1 phase as the pressure continues to increase beyond Pt.This makes the B2 phase more stable than the B1 phase.In Table 2, we have reported the calculated values of Pt as 42 GPa.This result that we estimated based on our theoretical calculations is in perfect accord with the results that were presented previously [6].
Next, we have proceeded to assess the relative volumes associated with different compressions based on the Murnaghan equation of state [26].
assume that V0 (B0) represents the cell volume (bulk modulus) at ambient conditions, whereas B' is the pressure derivative of the bulk modulus.Volume compression rate (∆V(P)/V(0) is defined as (V0-Vp)/V0 where (V0 is the volume of test compound under zero pressure, Vp is the volume of test compound under specific pressure).We are able to estimate the volume discontinuity at the transition pressure by employing such a methodology as the phase diagram.Figure 2 of the phase diagrams for MnSi, which can be found here, demonstrates that there is an abrupt reduction in volume at the transition pressure (Pt).A phase transition with such a nature is an example of a transition of the first order.Table 2 contains a tabulation of the calculated magnitude of the volume collapse that was determined from the phase diagram.The value that was derived for volume collapse is comparable to previous values that have been reported [6].
Following the equations from reference [24], the dynamical matrix was calculated for cubic symmetry in order to obtain the second order elastic constants (SOECs) C11, C12, and C44 for the MnSi compound in both of its phases.The change in second order elastic constants (SOECs) for the MnSi compound as a function of the pressure that was applied is depicted in Fig 3 .Under hydrostatic pressure P, the value of second-order elastic constants Cij can be seen with respect to the finite strain.The role of the long-range interactions like Coulomb, the short-range interactions like overlap repulsion that applied up to the second neighbor ions, and the van der Waals interactions are all extremely important.

Conclusions
MnSi is a metallic compound that exhibits a structural phase transition as a function of pressure.Hafemeister and Flygare approach evaluates the equation of state for change in volume with respect to applied pressure.The estimated value of phase transition pressure (Pt) is 42 GPa, which is consistent with the available reported data.In the vicinity of the transition, the first order phase transformation exhibited a 10% volume decrease, as well as discontinuity in second order of elastic constants.

Figure 1 .
Figure 1.Indicates the relationship between the estimated Gibbs free energy difference (ΔG) and pressure (P) for MnSi compound.

Figure 2 .
Figure 2. Illustrates the relationship between the dimensionless volume V(P)/V(0) and pressure for MnSi compound.

Figure 3 .
Figure 3. Variation of second order elastic constants (SOEC) for MnSi with pressure

Table 1 :
van der Waals coefficients of MnSi compound.(cij in units of 10 -60 erg cm6and dij in unit of 10 -76 erg cm 8 ).C and D are the overall van der Waals coefficients.

Table 2 :
Model parameters, calculated transition pressures and volume collapse for MnSi compound.