The $Sc_{2}$CoSi Heusler has the Fm$3^{-}$m space group^[31-32] by 6.28Å lattice constant,^[22] and also the $Sc_{2}$CoSi films are grown in [001] direction including the 18 atomic layers by 20 {\AA} vacuum. The surfaces are terminated to four forms of Sc-Si, Sc-Co, Co-Si, and Sc-Sc atomic layers, which all surface structures are relaxed by miniposition command and their thermodynamic stability are discussed in the following. In the mentioned structures, Brich-Mournaghan equations are used to optimize the unit cell volumes to obtain the E-V curves which are shown in Fig. 1 and results are listed in Table 1.

In Fig. 2(a), the band structure curves of the $Sc_{2}$CoSi bulk composition have been shown in both majority and minority spins because the band structure gives us very important information about the electronic structure of matter, such as metallic, semiconducting, magnetic and HM and etc. behaviors. As can be seen, $Sc_{2}$CoSi bulk is a semiconductor in spin up with a direct band gap 0.54 eV, and it has a metallic behavior in the spin down. In the spin up, the maximum valance level is located at the Γ symmetric point while the minimum conducting band is placed along Γ→X, so it implies the p-type semiconductor with an indirect band gap. Another point, the valance levels at the Γ point have a steeper slope than the conduction ones for both spins, so the holes mobility is far more than the electron mobility. In the following, band structure of the Sc-Si termination [001] is plotted in Fig. 2(b) for two spins of up and down which comparison between the two spin modes reveals the semiconductor conditions for down, and metallic behavior for up.

It is important to note that the majority carriers for the bulk state belong to the spin down, but for Sc-Si [001], they are on the spin up. In addition, in the film phase, the band gap is converted into a direct gap for the spin down at the Γ point. It should also be mentioned that the curve slope of the energy levels in the region of maximum valence at the Γ point is declined regarding to its bulk state, and the flip spin gap is almost zero in the bulk and thin film [001].

Figure 3(a) shows the electron density of states (DOS) curves of the $Sc_{2}$CoSi bulk in two spins up and down. As it can be seen, this composition has a magnetic behavior with full spin polarization (100%) by zero spin flip gap implying to perfect HM behavior. In the following, by comparing the partial density of states of the atoms forming this composition, the highest contribution at Fermi level and the conduction region belongs to the half-filled d orbitals of Sc, and the biggest share below Fermi level and the valence region belongs to the cobalt atom states while the largest contribution of the magnetic anisotropy belongs to the d orbitals of Sc atom. In the following, the density of states curves of the $Sc_{2}$CoSi thin films at two Sc-Co and Sc-Si terminations [001] are shown in Figs. 3(b) and 3(c). Clearly, the Sc-Si termination has 100 % spin polarization at Fermi level while the spin polarization for the Sc-Co termination is about 40\%. Therefore, magnetic anisotropy is much less in the Sc-Co termination and shows metallic behavior, but one of the challenges related to the Heusler films is the absence of or decrease in the 100 % spin polarization at the Fermi level.^[33-34]

In Figs. 3(d), 3(e), 3(f), the partial densities of states for the Sc-Si termination are plotted for different layers so that the contributions of the Sc electron states in the surface and subsurface layers are much larger than those of the middle layers at Fermi level. Moreover, the surface effects of the Sc atoms lead to the magnetic anisotropy and splitting the states in the minority spin, although the contribution of magnetic anisotropy of Si atoms is very small. More interestingly, the electron states of valance band for the Si atoms of the middle and surface layers have almost isotropic behavior in the energy range of ($-$1 eV to $-$5 eV) and are completely non-magnetic. After that, the electron states of Co atoms for the middle and the subsurface layers are compared, and consequently, there is a little magnetic anisotropy in the valence region for middle Co atoms while the anisotropy of subsurface Co atoms is greater. In this case, the Co atoms of the mentioned layers have the magnetic asymmetry as the electron states of these layers demonstrate a polarization of 100%, but at higher energies and the conduction region, the anisotropy effects are greatly reduced.

In Table 1, the magnetic moments of the films are listed accompany with comparison to the bulk state. Significantly, the magnetic moment of Sc-Co [001] film is below 1 μB while the magnetic moment of the Sc-Si [001] film is the large number of 3 μB, where the magnetic moment of the bulk is $-$1 μB, the negative sign indicates the weak magnetic responsiveness of the bulk state, and the positive sign and the large number of 3 μB for the Sc-Si [001] film indicates high responsiveness and appropriate sensitivity to the external magnetic fields. As a result, this compound is a good candidate for GMR and TMR applications.

The optical parameters give us very good information about the electronic structure and their optical properties, which are extractable by Kramers-Kronig equations. The matter response to the incident light in the real space is shown by the real part of the dielectric function, and for the energy space, it is the imaginary part of the dielectric function, which both can be extracted from the following equations:^[35]

The real part of the dielectric function is shown in Fig. 4(a) for the $Sc_{2}$CoSi bulk. The static value of Re ${\varepsilon }({\omega })$ has a very large amount indicating the strong metallic behavior at lower energies, but the sign of Re ${\varepsilon }({\omega })$ becomes negative in the regions of 0.35 eV to 0.65 eV (infrared region) and above 3.7 eV (UV region) meaning that the light does not pass through. Most significantly, due the peaks in the infrared and visible regions, this composition is a good candidate for optical applications in the optoelectronic industry and the optical electrodes of the solar cells.^[36-37] After that, Re ${\varepsilon }({\omega })$ is zero at energies of 0.3 eV, 0.65 eV, 2.45 eV, 3.65 eV, and 16.3 eV, which can be a sign of Plasmon oscillations or the light refraction in matter, in the following, the Re ${\varepsilon }({\omega })$ curves have been illustrated for the [001] film with Sc-Si termination in two directions of $x$ and $z$. In general, except for differences in the static value of Re ${\varepsilon }({\omega })$ and in the shape of the Re ${\varepsilon }({\omega })$ curve in the visible region and the UV edge, the Re ${\varepsilon }({\omega })$ graphs are relatively similar in the two directions. In both directions, the static value of the Re ${\varepsilon }({\omega })$ is still a large number confirming the metallicity, but this metallic behavior at low energies along $x$ is larger than $z$ ones. Compared with the bulk Re ${\varepsilon }({\omega })$ curve, it is clear that the metallicity has decreased which, of course, was expected because the number of atoms in $z$ direction is reduced. Moreover, the film optical response rate is decreased with a steep slope so that it reaches a minimum value at about 0.3 eV. Importantly, the light responsiveness, Re ${\varepsilon }({\omega })$, has a positive value from zero energy to 3.4 eV for $x$ direction and 3.75 eV for $z$ one, which is in contrary to its bulk state that was negative in the infrared region. Therefore, this thin film has a high optical sensitivity in the infrared, visible and even UV edge regions, which may be due to the presence of bonding arms from Sc and Si atoms on the film surface. Another significant point is that the sign of the dielectric function is negative at other energies in the UV region up to 15 eV in both directions, so light does not pass through and the thin film is dark. The Re ${\varepsilon }({\omega })$ value is zero at energies of 3.3 eV and 13.3 eV for $x$ direction and 3.8 eV and 12.9 eV for $z$ direction, and this can also be another positive feature, which the light energy reduction is only occurred at the higher energies, not in the visible and infrared regions. The imaginary part of the dielectric function of the $Sc_{2}$CoSi structure is plotted in Fig. 4(b) for the bulk phase such that the peaks of the Im ${\varepsilon }({\omega })$ represent optical transitions from filled to empty levels. There is a very large peak in zero energy confirming again the metallic behavior, because with the minimum incident light, excited weak electrons transit from filled levels to empty ones. Furthermore, the Im ${\varepsilon }({\omega })$ curve experiences a significant decrease occurring at the energy of 0.88 eV corresponding to the zero value of Re ${\varepsilon }({\omega })$ because wherever there is Plasmon oscillations, there would be electron transitions and the matter response to the light vanishes completely. In addition, there is a large peak at 1.75 eV (in the visible area), however, the intensity of the peaks is reduced exponentially, so that the later peaks at the energies of 3.71 eV, 5.56 eV, and 7.8 eV are less and less.

In the following, the imaginary part of the dielectric function is plotted for the Sc-Si [001] termination for two mentioned directions. Although anisotropy is seen for optical behavior in the infrared area, the overall behavior is similar at higher energies. In this regard, the static value of Im ${\varepsilon }({\omega })$ in $x$ direction is a large number experiencing a sharp drop at about 0.7 eV while the static value of the dielectric function for $z$ direction is much smaller about 1/3 times of the value for $x$ one. Correspondingly, a sharp Dirac peak occurs at about 0.27 eV, and it then falls rapidly after 0.88 eV. In $x$ direction, there is a main peak in the visible range at the energy of 1.7 eV, but there are three other peaks in $z$ direction at 1.7 eV, 3.7 eV and 5.4 eV, which the peaks intensities constantly decrease. Comparing these two curves, if the light enters along $x$ direction, most of the light energy will be used for the electron transition, while in $z$ axis, a smaller portion of the light energy is spent on these transitions and a greater part of the light passes through. Due to the incident light and the electron excitation and transition, electrons lead to the optical conduction that have the chance of crossing Fermi level.

The optical conductivity of the $Sc_{2}$CoSi bulk phase has been depicted in Fig. 4(c), and as it appears, there is a large peak at lower energies indicating a drastic decline at about 0.8 eV. This peak is a strong verification of the metallic behavior, but significant conduction peaks are occurred in the visible and UV regions (energies of 2.2 eV, 3.7 eV, 5.5 eV and 8 eV). Afterward, a decrease in conductivity can be seen so that the conductivity becomes zero after 25 eV. The steep conduction valleys also occur at the same energies that Plasmon oscillations or refraction are observed. The optical conductivity of Sc-Si [001] film is shown for two directions of $x$ and $z$ in Fig. 4(c). In the $x$ direction, more metallic behavior can be detected especially in zero energy, and the optical conductivity is more uniform. Moreover, the reduction in the optical conductivity is much less than the bulk one and even along $z$ direction.

Figure 4(d) indicates the absorption graph of the bulk phase rising as a Gaussian curve so that in the visible region of 1.7 eV experiences a steep absorption slope. Additionally, high absorption is observed at the UV edge (3.6 eV) to reach the maximum light absorption in energy range between 6.2 eV and 10 eV, but the light absorption drops with a mild slope afterward to reach about zero at 25 eV. In the Sc-Si termination, similar to the bulk state, absorption experiences a pungent elevation in the visible and the UV edge region, and a small valley can be seen in the peak point at 7.5 eV for both directions owing to the refraction occurred in this energy area. The ELOSS curve in Fig. 4(e) shows the degree of light loss in the matter, and those peaks of the ELOSS spectrum that match the zero points of Re ${\varepsilon }({\omega })$ represent the Plasmon oscillations. As can be seen from the Re ${\varepsilon }({\omega })$ curve of the bulk phase, there is a sharp and large peak in the mentioned region, so the large Plasmon oscillations occur at these energies such that the energy of the plasma oscillations is 20.4 eV. Then, the ELOSS curve for the Sc-Si [001] film is plotted for two directions of $x$ and $z$. A point that should be considered is that the peaks of the mentioned curves show a red shift, and their main peak has shifted to about 14.5 eV. Accordingly, with respect to zero values of Re ${\varepsilon }({\omega })$ at this energy, the Plasmon oscillation has occurred in the thin film. One of the important optical coefficients derived from the dielectric function is the refractive index extracted from the following equation:^[35]

The refractive index of the $Sc_{2}$CoSi in the bulk phase is shown in Fig. 4(f). Correspondingly, due to its metallic nature, the static refractive index is a large number and an intense decline can be seen, as a result, most of the incident light energy passes through or reflects. Furthermore, there is a peak in the visible region, so the light speed in matter is as high as possible. The refractive index decreases so that it exhibits semiconductor behavior in the range of 5 eV to 12 eV, since then, the refractive index is almost zero and becomes a transparent material. Along the $x$ axis of the [001] film, an exponential decrease can be detected after a sharp decline of the refractive index in the infrared region, and the refractive index peak also has a red shift. In the same way, along $z$ axis, a fading sawtooth exponential curve can be observed where the highest rate of the light refraction is again occurred in the visible area for both directions. Another important optical parameter in the field of the light transition through material is the imaginary part of the refractive index or the extinction coefficient, which by the definition, is the amount of loss occurring in the light intensity for penetration in matter. For the bulk state, the most loss has occurred at infrared energies, which it was also expected because it has a metallic nature. Even though there is a keen falloff in the extinction coefficient at about 0.8 eV, this is the area where the sign of Re ${\varepsilon }({\omega })$ was negative. Afterwards, there is a peak in the visible area and two other peaks in the UV region which, of course, this reduction rate is insignificant compared to the lower energies. In order that the reduction rate in the absorption amplitude is practically minimized at the higher energies, more light passes through the matter. Hereupon, the extinction coefficient of the mentioned film for two directions indicates that the extinction coefficient in the infrared area is more than the other energies as same as the bulk state; of course, it is still less than the bulk one. Comparing the two directions, the extinction coefficient value along the $z$ axis is approximately half of the one along the $x$ axis. Additionally, there are two peaks in the visible region with this difference that the reduction of the extinction coefficient is smooth in $x$ direction, but sawtooth in $z$ direction. In general, it is possible to say from the concepts of the optical section that the Sc-Si [001] thin films has a red shift and its optical sensitivity has been shifted to the visible area and the UV edge, which can have very useful applications in the optoelectronic devices.^[36]

In this section, the relative surface stabilities are studied from the thermodynamic point of view; including Sc-Si, Sc-Co, Co-Si, and Sc-Sc terminations. For this purpose, the surface free energies of these terminations are studied as a function of the atomic chemical potential in the framework of thermodynamic calculations. The surface free energy is calculated as following^[37] in which $N{_{i}}$ and ${\mu }{_{i}}$ are the number and the chemical potentials of the $i$ atom, $G$ is Gibbs free energy for the surface, $2A$ is the total area of the supercell surface, and Γ is the surface free energy per unit area:

In order to achieve the phase diagram, the various surface free energies are compared in a permissible chemical potential region to find a stable level with the lowest surface free energy. According to the fact that the created termination is considered in the thermodynamic equilibrium with the bulk core layers, the appropriate equilibrium conditions can be written as follow:

which $g{_{\rm bulk}}$ is the bulk Gibbs free energy. By considering the dynamic contribution of Gibbs free energy at lower temperatures, the total energy of the DFT calculations is only considered to obtain these energies.

By comparing the calculated values of Γ for different surfaces, the most stable surfaces can be obtained in different chemical potential values. By determining the most stable surface for each coordinate couple, ${\mu }{_{\rm Sc}}$ and ${\mu }{_{\rm Co}}$, the two-dimensional phase diagram is plotted on the coordinates of these two chemical potentials which the surface free energy is obtained by calculating ${\mu }{_{\rm Si}}$ from Eq. (2) in the terms of ${\mu }{_{\rm Sc}}$ and ${\mu }{_{\rm Co}}$. In this regard, the highest value of ${\mu }{_{\rm Sc}} ({\mu }{_{\rm Co}})$ is related to the structure of Sc (Co), hence, the permissible zone in the phase diagram is limited by the bulk Gibbs free energies of Sc and Co. The maximum values of ${\mu }{_{\rm Sc}} and {\mu }{_{\rm Co}}$ are obtained from the relationships of ${\mu }{_{\rm Sc}}$ $=$ $g{_{\rm Sc}}$ and ${\mu }{_{\rm Co}}$ $=$ $g{_{\rm Co}}$. Here, if the chemical potential of Sc and Co atoms is low, the escape probability of these atoms from the bulk structure increases; ergo, the permissible area will be limited to the minimum values of these two chemical potentials (according to the relationship ${\mu }{_{\rm Sc}}$ $=$ $g{_{\rm CoSi}}$ and {\mu }{_{\rm Co}} $=$ $g_{{\rm Sc}_2{\rm Si}})$. According to the obtained results, all for terminations of Sc-Si, Sc-Co, Co-Si, and Sc-Sc are accessible within the permissible area, and the location that these lines cross each other represents the stability zone for the bulk structure. The phase diagram is shown in Fig. 5, and the title of each region indicates the termination with the least surface energy; correspondingly the Sc-Co and Sc-Si terminations are more stable than the other surfaces by having the lowest $E{_{c}}$ value. Furthermore, it is observed that by increasing the chemical potential of cobalt and scandium, the concentration of these two increases in the surface. The major area within the permissible area is also appertained to the Sc-Sc and Sc-Co surfaces, and these two terminations are the most stable surfaces.