Site-Preference, Electronic, Magnetic, and Half-Metal Properties of Full-Heusler Sc 2 VGe and a Discussion on the Uniform Strain and Tetragonal Deformation E ﬀ ects

: A hypothetical full-Heusler alloy, Sc 2 VGe, was analyzed, and the comparison between the XA and L2 1 structures of this alloy was studied based on ﬁrst-principles calculations. We found that the L2 1 -type structure was more stable than the XA one. Further, the electronic structures of both types of structure were also investigated based on the calculated band structures. Results show that the physical nature of L2 1 -type Sc 2 VGe is metallic; however, XA-type Sc 2 VGe is a half-metal (HM) with 100% spin polarization. When XA-type Sc 2 VGe is at its equilibrium lattice parameter, its total magnetic moment is 3 µ B , and its total magnetism is mainly attributed to the V atom. The e ﬀ ects of uniform strain and tetragonal lattice distortion on the electronic structures and half-metallic states of XA-type Sc 2 VGe were also studied. All the aforementioned results indicate that XA-type Sc 2 VGe would be an ideal candidate for spintronics studies, such as spin generation and injection.

In recent years, HH compounds with HM properties have been widely reported [21][22][23]. Some examplesare as follows: in 2011, Chen et al. [24] found that GeKCa and SnKCa exhibit HM properties, and that these alloys have large HM band gap values of 0.28 eV and 0.27 eV, respectively. In 2012, Yao et al. [25] found that CoCrP and CoCrAs have HM properties with HM band gap values of 0.46 eV and 0.50 eV, respectively. It was also found that, in terms of lattice distortion, the HM properties of these alloys can be maintained in the range of −4.8% to 6.6% and −7.7% to 4.5%, respectively. The discovery of the presence of HM properties in HH alloys has led to the availability of more options for spintronics materials [26,27].
A series of FH compounds with HM properties were also reported by researchers [28]. For example, Kogachi et al. [29] studied the electronic and magnetic properties of Co 2 MnZ (Z = Si, Ge, Sn). Liu et al. [30] investigated the electronic structures of Mn 2 CoZ (Z = Al, Si, Ge, Sn, Sb) in detail and found two mechanisms to induce the band gap for minority spin states near the Fermi level; Wang et al. [31] studied the electronic and magnetic properties of FH alloy Zr 2 CoZ (Z = Al, Ga, In, Si, Ge, Sn, Pb, Sb) and found that the half-metallicities are robust against lattice distortion; Wang et al. also studied the site preferences of the Titanium-based [32] and Hf 2 V-based [33] FH alloys, and found that most of these alloys are likely to form the L2 1 structure instead of the XA structure. Thus, the traditional site-preference rule (SPR) may not be suitable for all FH alloys, such as X 2 YZ, where X is a low-valent transition metal element, such as, Ti, Zr, Sc, and Hf.
There are also some reports about the scandium-based (SB) FH alloys. In 2013, Zhang et al. [34] studied a series of SB FH compounds and found that some SB compounds with the XA structure can exhibit nontrivial topological band ordering. In 2017, Li et al. [35] studied the thermoelectric characteristics of FH Sc 2 FeSi and Sc 2 FeGe and found maximum power factors of 48.77 × 10 14 µW cm −1 K −2 s −1 and 47.11 × 10 14 µW cm −1 K −2 s −1 for Sc 2 FeSi and Sc 2 FeGe, respectively.
In this study, we will focus on a new FH alloy, Sc 2 VGe, and perform a complete first-principle study of the site-preference, electronic, magnetic, and half-metallic properties of this material. Further, phase stability in terms of the calculated formation and cohesive energies is also explained. Moreover, the conduction band minimum (CBM), valence band maximum (VBM), band gap and half-metallic band gap, total and atomic magnetic moments, and electronic band structures as functions of the lattice parameter and c/a ratio will be discussed in detail.

Calculation Methods
In this study, the plane-wave pseudopotential method within CASTEP [36], which uses density functional theory, was used to calculate the physical properties of the material. The generalized gradient approximation (GGA) [37] was used in the scheme of Perdew-Burke-Ernzerh of (PBE) [38] to deal with the exchange and correlation functions between electrons. The cutoff energy for plane waves was set to 450 eV, and the convergence was set to 5 × 10 −7 eV using 12 × 12 × 12 mesh points grid. The above parameters ensure the accuracy of the calculated results based on the references [39]. Similar methods to investigate the electronic structures of Heusler alloys can be found in [30][31][32][33].

Competition of L2 1 and XA Structurein Full-Heusler Sc 2 VGe
It is well known that there are four atomic sites for FH alloys, i.e., A (0, 0, 0), B (0.25, 0.25, 0.25), C (0.5, 0.5, 0.5), and D (0.75, 0.75, 0.75). When atomic occupancy is X (A)-X (B)-Y (C)-Z (D), the XA structure is formed; on the other hand, when atomic occupancy is X (A)-Y (B)-X (C)-Z (D), the L2 1 structure is formed [40]. For Sc 2 VGe compound, the two structures are shown in Figure 1. According to the traditional SPR [40][41][42][43][44][45], the V atom has more valence electrons than the Sc atom and tends to occupy the C position, the two Sc atoms occupy A and B positions, respectively, and the Ge atom occupies the D position, thus, preferring to form the XA structure. However, we should point out that traditional SPR may not be suitable for all FH alloys. Therefore, for both the types, we focused on the relationship between their total energies during FM state and lattice parameters in this study. From Figure 2 and Table 1, we see that L2 1 type has a lower total energy; therefore, L2 1 -type structure is considered more stable than the XA type. Interestingly, the calculated results are contrary to that of SPR, that is, here, the most stable structure of Heusler Sc 2 VGe alloy is found to be the L2 1 type instead of XA type. For the XA-type Sc 2 VGe, the calculated magnetic moment is 3 µ B and it conforms well to the Slater-Pauling rule: M t = Z t − 18 [46,47].   We further calculated the band structures of the two types of Sc2VGe compound. It can be noted from Figure 3 that XA-type Sc2VGe (Figure 3a) is a HM with a direct band gap (at point G). Furthermore, the spin-up channel of Sc2VGe exhibits semiconducting property and the spin-down channel of it shows a metallic behavior. Based on the obtained band structures of XA-type Sc2VGe, we can see that 100% spin polarization [48,49] occurs near the Fermi level. On the other hand, L21-type Sc2VGe (Figure 3b) shows metallic behavior. In short, the band structures for both the spin channels in the Fermi level overlap with each other and reflect a metallic behavior.    We further calculated the band structures of the two types of Sc2VGe compound. It can be noted from Figure 3 that XA-type Sc2VGe ( Figure 3a) is a HM with a direct band gap (at point G). Furthermore, the spin-up channel of Sc2VGe exhibits semiconducting property and the spin-down channel of it shows a metallic behavior. Based on the obtained band structures of XA-type Sc2VGe, we can see that 100% spin polarization [48,49] occurs near the Fermi level. On the other hand, L21-type Sc2VGe (Figure 3b) shows metallic behavior. In short, the band structures for both the spin channels in the Fermi level overlap with each other and reflect a metallic behavior.  We further calculated the band structures of the two types of Sc 2 VGe compound. It can be noted from Figure 3 that XA-type Sc 2 VGe ( Figure 3a) is a HM with a direct band gap (at point G). Furthermore, the spin-up channel of Sc 2 VGe exhibits semiconducting property and the spin-down channel of it shows a metallic behavior. Based on the obtained band structures of XA-type Sc 2 VGe, we can see that 100% spin polarization [48,49] occurs near the Fermi level. On the other hand, L2 1 -type Sc 2 VGe ( Figure 3b) shows metallic behavior. In short, the band structures for both the spin channels in the Fermi level overlap with each other and reflect a metallic behavior.

Thermal Stability of XA-Type Sc2VGe
To determine the stability of XA-type Sc2VGe compound, the cohesive energy (Ec) was calculated. The Ec per unit cell can be expressed using the following formula [50]: where is the total energy of the Sc2VGe compound, , , and are the energies of the isolated atoms Sc, V, and Ge, respectively. The calculated Ecof Sc2VGe compound is 20.62 eV, indicating that the chemical bonding of Sc2VGe compound is firm.
In addition, formation energy (Ef) is another way to describe the stability of crystals. We use the following formula to characterize Ef per unit cell of Sc2VGe [50]: where is the same as mentioned above, , , and are energies of Sc, V, and Ge bulks, respectively. The calculated Ef of Sc2VGe compound is −3.64 eV, which is a negative value theoretically indicating that Sc2VGe compound is thermally stable.
Based on the results of Ec and Ef, it can be said that the XA-type Sc2VGe compound is found to be stable in terms of theory. We hope this material can be experimentally synthesized in the near future.

Total and Partial Density of States of XA-Type Sc2VGe
To analyze the contribution of each atom to the energy bands, we calculated the total density of states (TDOS) and the partial density of states (PDOS) for each atom. It can be seen from Figure 4 that the low range of energy states (lower than −2 eV) of TDOS are mainly due to the contribution of the atoms of the main group element Ge, such as the peak in the energies between −4 eV and −5 eV and peak in the energies between −3 eV and −2 eV. There are two obvious peaks in spin-up channels near the Fermi level that range from −1 eV to 0 eV, and the main TDOS in this range comes from the contributions of V atom. Further, Sc1 and Sc2 atoms also contribute a small part to the energy states from −1 eV to 0 eV. Near the Fermi level, the TDOS of the spin-up channel is zero and has a large energy gap, whereas the spin-down channel is not zero, which mainly results from the hybridization among Sc1, Sc2, and V atoms. From the DOS, similar to the calculated band structures, one can see that half-metallic property with full spin polarization is found in XA-type Sc2VGe. Moreover, from the TDOS, we can see that the energy gap in the spin-up direction is generated by four peaks, two peaks below, and two peaks above the Fermi level. As discussed previously, the two peaks below the Fermi level are mainly derived from the d orbital of V atoms. The two peaks above the Fermi level, as can be clearly seen, are mainly derived from the d orbital of Sc atoms. Moreover, the hybridization of d-d orbitals between V and Sc atoms plays an important role, which cannot be ignored, in the formation of energy gap in the spin-up channel.

Thermal Stability of XA-Type Sc 2 VGe
To determine the stability of XA-type Sc 2 VGe compound, the cohesive energy (E c ) was calculated. The E c per unit cell can be expressed using the following formula [50]: where E Sc 2 VGe total is the total energy of the Sc 2 VGe compound, E iso Sc , E iso V , and E iso Ge are the energies of the isolated atoms Sc, V, and Ge, respectively. The calculated E c of Sc 2 VGe compound is 20.62 eV, indicating that the chemical bonding of Sc 2 VGe compound is firm.
In addition, formation energy (E f ) is another way to describe the stability of crystals. We use the following formula to characterize E f per unit cell of Sc 2 VGe [50]: where E Sc 2 VGe total is the same as mentioned above, E Sc bulk , E V bulk , and E Ge bulk are energies of Sc, V, and Ge bulks, respectively. The calculated E f of Sc 2 VGe compound is −3.64 eV, which is a negative value theoretically indicating that Sc 2 VGe compound is thermally stable.
Based on the results of E c and E f , it can be said that the XA-type Sc 2 VGe compound is found to be stable in terms of theory. We hope this material can be experimentally synthesized in the near future.

Total and Partial Density of States of XA-Type Sc 2 VGe
To analyze the contribution of each atom to the energy bands, we calculated the total density of states (TDOS) and the partial density of states (PDOS) for each atom. It can be seen from Figure 4 that the low range of energy states (lower than −2 eV) of TDOS are mainly due to the contribution of the atoms of the main group element Ge, such as the peak in the energies between −4 eV and −5 eV and peak in the energies between −3 eV and −2 eV. There are two obvious peaks in spin-up channels near the Fermi level that range from −1 eV to 0 eV, and the main TDOS in this range comes from the contributions of V atom. Further, Sc 1 and Sc 2 atoms also contribute a small part to the energy states from −1 eV to 0 eV. Near the Fermi level, the TDOS of the spin-up channel is zero and has a large energy gap, whereas the spin-down channel is not zero, which mainly results from the hybridization among Sc 1 , Sc 2 , and V atoms. From the DOS, similar to the calculated band structures, one can see that half-metallic property with full spin polarization is found in XA-type Sc 2 VGe. Moreover, from the TDOS, we can see that the energy gap in the spin-up direction is generated by four peaks, two peaks below, and two peaks above the Fermi level. As discussed previously, the two peaks below the Fermi level are mainly derived from the d orbital of V atoms. The two peaks above the Fermi level, as can be clearly seen, are mainly derived from the d orbital of Sc atoms. Moreover, the hybridization of d-d orbitals between V and Sc atoms plays an important role, which cannot be ignored, in the formation of energy gap in the spin-up channel.

Effect of Uniform Strain on XA-Type Sc2VGe
Uniform strain is an important way to regulate the band structures, i.e., the physics nature of alloys. In this section, we will discuss the effect of uniform strain on the band structures of XA-type Sc2VGe alloy. Firstly, we aim to study the physical nature transition of XA-type Sc2VGe as the lattice parameter changes from 5.80 Å to 7.20 Å. The results are shown in Figure 5. When the lattice parameter of XA-type Sc2VGe is smaller than 6.16 Å, it is considered to be a magnetic metal, as shown in Figure 5a. When the lattice parameter is between 6.16 Å and 6.54 Å (see Figure 5b), XA-type Sc2VGe is a HM material with an indirect band gap in the spin-up channel. When the lattice parameter is between 6.54 Å and 6.69 Å, half-metallic behavior with direct band gap can be found in the XA-type Sc2VGe, as exhibited in Figure 5c,d. When the lattice parameter is larger than 6.69 Å, the HM property of this alloy breaks and a metallic property appears instead.

Effect of Uniform Strain on XA-Type Sc 2 VGe
Uniform strain is an important way to regulate the band structures, i.e., the physics nature of alloys. In this section, we will discuss the effect of uniform strain on the band structures of XA-type Sc 2 VGe alloy. Firstly, we aim to study the physical nature transition of XA-type Sc 2 VGe as the lattice parameter changes from 5.80 Å to 7.20 Å. The results are shown in Figure 5. When the lattice parameter of XA-type Sc 2 VGe is smaller than 6.16 Å, it is considered to be a magnetic metal, as shown in Figure 5a. When the lattice parameter is between 6.16 Å and 6.54 Å (see Figure 5b), XA-type Sc 2 VGe is a HM material with an indirect band gap in the spin-up channel. When the lattice parameter is between 6.54 Å and 6.69 Å, half-metallic behavior with direct band gap can be found in the XA-type Sc 2 VGe, as exhibited in Figure 5c,d. When the lattice parameter is larger than 6.69 Å, the HM property of this alloy breaks and a metallic property appears instead. Figure 6a shows the calculated CBM, VBM, band gap, and half-metallic band gap in the spin-up channel at different lattice parameters for XA-type Sc 2 VGe. One can see that the half-metallic band gap gradually increases from 6.2 Å, and then gradually decreases after reaching a maximum value at 6.6 Å, and finally disappears at approximately 6.7 Å. More importantly, the maximum half-metallic band gap at around 6.6 Å is about 0.2 eV. Such a large value ensures that the half-metallic property of this material is not affected by external factors. On the other hand, we can see that the band gap in the spin-up direction of this material almost remains constant over the lattice range of 6.2 Å to 6.4 Å. When the lattice parameter is in the range of 6.4 Å to 6.7 Å, the band gap is significantly reduced.  Figure 6a shows the calculated CBM, VBM, band gap, and half-metallic band gap in the spin-up channel at different lattice parameters for XA-type Sc2VGe. One can see that the half-metallic band gap gradually increases from 6.2 Å, and then gradually decreases after reaching a maximum value at 6.6 Å, and finally disappears at approximately 6.7 Å. More importantly, the maximum half-metallic band gap at around 6.6 Å is about 0.2 eV. Such a large value ensures that the half-metallic property of this material is not affected by external factors. On the other hand, we can see that the band gap in the spin-up direction of this material almost remains constant over the lattice range of 6.2 Å to 6.4 Å. When the lattice parameter is in the range of 6.4 Å to 6.7 Å, the band gap is significantly reduced. The total and atomic magnetic moments under uniform strain were also studied. The results are shown in Figure 6b. We can see that the total magnetic moment of XA-type Sc2VGe hardly changes as the lattice parameter changes. For atomic magnetic moments, as the lattice parameters increase, the atomic magnetic moment of V gradually increases, while the atomic magnetic moments of other atoms gradually decrease. The total and atomic magnetic moments under uniform strain were also studied. The results are shown in Figure 6b. We can see that the total magnetic moment of XA-type Sc 2 VGe hardly changes as the lattice parameter changes. For atomic magnetic moments, as the lattice parameters increase, the atomic magnetic moment of V gradually increases, while the atomic magnetic moments of other atoms gradually decrease.

The effect of Tetragonal Lattice Distortion on XA-Type Sc 2 VGe
The effect of tetragonal distortion on the electronic structures of XA-type Sc 2 VGe was studied. Firstly, we studied the physical transition during a change in the c/a ratio when in the range of 0.75~1.25. As shown in Figure 7, when the range of c/a ratio is between 0.80 to 1.22 (see Figure 7b-e), the compound shows the HM property. On the other hand, it is a metal when the c/a ratio is lower than 0.80 (see Figure 7a) or higher than 1.22 (see Figure 7f). As shown in Figure 8a, the largest HM gap appears when c/a=1. As the c/a ratio increases (or decreases), the VBM in the spin-up channel gradually decreases, and therefore, the HM band gap decreases. The band gap does not change much when the c/a ratio is in the range of 0.85 to 1.15. The reason behind this can be understood from Figure 8a.In this interval, when c/a increases or As shown in Figure 8a, the largest HM gap appears when c/a=1. As the c/a ratio increases (or decreases), the VBM in the spin-up channel gradually decreases, and therefore, the HM band gap decreases. The band gap does not change much when the c/a ratio is in the range of 0.85 to 1.15. The reason behind this can be understood from Figure 8a. In this interval, when c/a increases or decreases, although the VBM always decreases, the CBM shows an increasing trend, such that the overall band gap remains almost unchanged.

Conclusion
In this study, we focused on FH alloy Sc2VGe, and showed a complete first-principle study on the site-preference, electronic, magnetic, and half-metallic properties of this material. The main results are as follows: (i) The site-preference of FH alloy Sc2VGe was examined, and results showed that the L21 type is more stable than the XA type. We further calculated the electronic structures of both types of Sc2VGe and found that the XA-type alloy was an excellent half-metallic material, whereas the L21-type alloy was a magnetic metal. XA-type Sc2VGe can intrinsically provide single spin channel electrons, and therefore this material can be used for pure spin generation and injection. (ii) When XA-type Sc2VGe is at its equilibrium lattice parameter, its total magnetic moment is 3 μ , which is in accordance with the well-known Slater-Pauling rule, and the main contribution to the total magnetism came from V atoms. (iii) The effects of uniform strain and tetragonal lattice distortion on the electronic structures of XA-type Sc2VGe were also studied. We found that the half-metallic state can be maintained in a large area of the lattice parameter and the c/a ratio, indicating that XA-type Sc2VGe is a robust half-metallic material. (iv) The formation energy and cohesive energy were calculated and results showed that this alloy has extensive scope for use in experiments.
(v) The half-metallic band gap and the band gap in the spin-up channel as a function of the lattice parameter and the c/a ratio were taken into consideration for XA-type Sc2VGe, and we found that the maximum half-metallic band gap around 6.6 Å was approximately 0.2 eV. Such a large value ensures that the half-metallic property of this material is not affected by external factors. (vi) All the aforementioned results indicate that XA-type Sc2VGe would be an ideal candidate in spintronics.  Finally, we study the effect of tetragonal distortion of XA-type Sc 2 VGe on its magnetic property. It can be seen from Figure 8b that the magnetic moments of unit cell and each atom vary slightly when the c/a ratio ranges from 0.75 to 1.25, which show that the magnetism of the material is quite stable and has a strong resistance to tetragonal lattice distortion.

Conclusions
In this study, we focused on FH alloy Sc 2 VGe, and showed a complete first-principle study on the site-preference, electronic, magnetic, and half-metallic properties of this material. The main results are as follows: (i) The site-preference of FH alloy Sc 2 VGe was examined, and results showed that the L2 1 type is more stable than the XA type. We further calculated the electronic structures of both types of Sc 2 VGe and found that the XA-type alloy was an excellent half-metallic material, whereas the L2 1 -type alloy was a magnetic metal. XA-type Sc 2 VGe can intrinsically provide single spin channel electrons, and therefore this material can be used for pure spin generation and injection. (ii) When XA-type Sc 2 VGe is at its equilibrium lattice parameter, its total magnetic moment is 3 µ B , which is in accordance with the well-known Slater-Pauling rule, and the main contribution to the total magnetism came from V atoms. (iii) The effects of uniform strain and tetragonal lattice distortion on the electronic structures of XA-type Sc 2 VGe were also studied. We found that the half-metallic state can be maintained in a large area of the lattice parameter and the c/a ratio, indicating that XA-type Sc 2 VGe is a robust half-metallic material. (iv) The formation energy and cohesive energy were calculated and results showed that this alloy has extensive scope for use in experiments. (v) The half-metallic band gap and the band gap in the spin-up channel as a function of the lattice parameter and the c/a ratio were taken into consideration for XA-type Sc 2 VGe, and we found that the maximum half-metallic band gap around 6.6 Å was approximately 0.2 eV. Such a large value ensures that the half-metallic property of this material is not affected by external factors.
(vi) All the aforementioned results indicate that XA-type Sc 2 VGe would be an ideal candidate in spintronics.