Exploration of D022-Type Al3TM(TM = Sc, Ti, V, Zr, Nb, Hf, Ta): Elastic Anisotropy, Electronic Structures, Work Function and Experimental Design

The structural properties, elastic anisotropy, electronic structures and work function of D022-type Al3TM (TM = Sc, Ti, V, Y, Zr, Nb, La, Hf, Ta) are studied using the first-principles calculations. The results indicate that the obtained formation enthalpy and cohesive energy of these compounds are in accordance with the other calculated values. It is found that the Al3Zr is the most thermodynamic stable compound. The mechanical property indexes, such as elastic constants, bulk modulus, shear modulus, Young’s modulus, Poisson’s ratio, and Vickers hardness are systematically explored. Moreover, the calculated universal anisotropic index, percent anisotropy and shear anisotropic factors of D022-type Al3TM are analyzed carefully. It demonstrates that the shear modulus anisotropy of Al3La is the strongest, while that of Al3Ta is the weakest. In particular, the density of states at Fermi level is not zero, suggesting that these phases have metal properties and electrical conductivity. More importantly, the mechanisms of correlation between hardness and Young’s modulus are further explained by the work function. Finally, the experimental design proves that D022-Al3Ta has an excellent strengthening effect.


Introduction
With the increasing demand of aerospace and automotive industry for structural material properties, aluminum rich compounds containing transition metal (TM, i.e., Sc, Ti, V, Y, Zr, Nb, La, Hf, Ta) elements have attracted extensive attention [1][2][3]. Among them, trialuminides (Al 3 TM) are the most potential candidate compounds, mainly because it can meet a variety of advantages, such as a high melting point, good thermal conductivity, low temperature damage resistance, strong creep resistance, and high specific strength. Furthermore, most of these intermetallics have different crystal structures of L1 2 , D0 19 , D0 22 , or D0 23 [4][5][6][7][8][9][10][11]. Usually, the Al 3 TM series of trialuminides can crystallize into the cubic L1 2 (space group Pm3m) and tetragonal D0 22 (space group I4/mmm) crystal structures. The L1 2 structure has better ductility due to its higher symmetry and more slip systems. However, it is generally believed that the lower symmetry of D0 22 structure is the main cause of poor ductility. Many attempts had been made to convert D0 22 into L1 2 , which can make the aluminide have good ductility [12][13][14][15]. However, the D0 22 structure has a good strengthening effect in the recent design of high entropy alloy. Hereinto, one of the critical challenges is to explore the internal mechanism of D0 22 -type trialuminides.
Previous studies have made a lot of efforts to reveal the properties of these intermetallic compounds. Recently, Jahnatek M et al. [16] investigated the interatomic bonds and the

Structural Properties and Stability
Firstly, the structural stability of these D0 22 -type compounds was investigated. The crystal structure of the binary compound D0 22 -Al 3 TM, where TM is Sc, Ti, V, Y, Zr, Nb, La, Hf, Ta, is displayed in Figure 1. The D0 22 -type compounds are tetragonal crystal with a space group of I4/mmm. Except for Al element, the composition of these compound elements is mainly composed of transition elements. Although these transition elements form the same crystal structure with aluminum, they have a greater impact on the properties of the compound. Being able to understand their internal mechanisms is very important for the design of alloys. In order to determine the phase stability of Al3TM (TM = Sc, Ti, V, Y, Zr, Nb, La, Hf, Ta) intermetallics, the formation enthalpy and cohesive energy of these compounds are calculated by following equations [31,32]: In order to determine the phase stability of Al 3 TM (TM = Sc, Ti, V, Y, Zr, Nb, La, Hf, Ta) intermetallics, the formation enthalpy and cohesive energy of these compounds are calculated by following equations [31,32]: where ∆H is the formation enthalpy of Al 3 TM compound. E tot is the total energy of Al 3 TM phase. E A solid and E B solid are the energy of Al and TM atom, respectively. In addition, E coh is the cohesive energy of Al 3 TM compound. E A atom and E B atom is the energy of Al and TM free atom, respectively. The x and y are the number of Al and TM atom in the D0 22 -Al 3 TM crystal structure, respectively. The calculated results of this work are noted in Table 1 and compared with other previous values [13,21,32,33]. The order of the compounds is arranged according to the atomic number of the TM atom in the periodic table of the elements. After analysis, it is evidence that the obtained results are basically coincident with the other calculated values, which indicates the reliability and good self-consistency of the proposed method.  [13] 8.312 [13] 3.773 [21] 8.325 [21] 118.510 [21] −0.277 [ [34] 8.598 [34] −0.318 [34] The first principle calculation is carried out at the ground state of 0 K and 0 Pa. The formation enthalpy of the compound is negative, indicating that the formation of the compound is an exothermic process. The more negative the formation enthalpy, the more stable the compound is. It can be seen from Table 1 that the formation enthalpy of these nine D0 22 -type compounds is less than zero, meaning that these compounds are thermodynamically stable in the ground state.
It is clear from  3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 La, Al 3 Hf, Al 3 Ta, respectively. Generally speaking, the large cohesive energy of compounds can only show that the free atoms of the two elements release more energy when they bond. If the energy needed to break the combination of the two elements is also large, the compound is still relatively unstable. However, the thermodynamic stability of compounds is affected by their enthalpy of formation. To some extent, the lower the enthalpy of formation, the easier the compounds are to form and the higher the thermodynamic stability [35,36]. The results show that the value of formation enthalpy of these D0 22 -type compounds is −0.366, −0.396, −0.283, −0.295, −0.464, −0.419, −0.134, −0.406, −0.322 eV/atom for Al 3 Sc, Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 La, Al 3 Hf, Al 3 Ta, respectively. The formation enthalpy of Al 3 Zr is the lowest, which suggests that Al 3 Zr alloy has the strongest formation ability and the most stable thermodynamics, while Al 3 La is on the contrary. Therefore, the alloying ability of the nine D0 22 -type compounds from strong to weak can be arranged as Al 3 Zr > Al 3 Nb > Al 3 Hf > Al 3 Ti > Al 3 Sc > Al 3 Ta > Al 3 Y > Al 3 V > Al 3 La, as showed in Figure 2. Considering all these compounds together, it is found that the Al 3 Zr is the most thermodynamic stable compound.
ing, the large cohesive energy of compounds can only show that the free atoms of the two elements release more energy when they bond. If the energy needed to break the combination of the two elements is also large, the compound is still relatively unstable. However, the thermodynamic stability of compounds is affected by their enthalpy of formation. To some extent, the lower the enthalpy of formation, the easier the compounds are to form and the higher the thermodynamic stability [35,36]. The results show that the value of formation enthalpy of these D022-type compounds is −0.366, −0.396, −0.283, −0.295, −0.464, −0.419, −0.134, −0.406, −0.322 eV/atom for Al3Sc, Al3Ti, Al3V, Al3Y, Al3Zr, Al3Nb, Al3La, Al3Hf, Al3Ta, respectively. The formation enthalpy of Al3Zr is the lowest, which suggests that Al3Zr alloy has the strongest formation ability and the most stable thermodynamics, while Al3La is on the contrary. Therefore, the alloying ability of the nine D022type compounds from strong to weak can be arranged as Al3Zr > Al3Nb > Al3Hf > Al3Ti > Al3Sc > Al3Ta > Al3Y > Al3V > Al3La, as showed in Figure 2. Considering all these compounds together, it is found that the Al3Zr is the most thermodynamic stable compound.

Mechanical Stability, Elastic Properties and Moduli
As well known, elastic constant is a very significant index to characterize the properties of compounds. The elastic constant is the index of material elasticity, which is related to the stress-strain relationship in the anisotropic medium. To some extent, the elastic constant also indicates the influence of crystal dynamics on mechanical behavior. The stressstrain method is used to calculate the elastic constants in the present calculation process and the results can be listed in Table 2. Furthermore, for the D022-type crystal, tetragonal

Mechanical Stability, Elastic Properties and Moduli
As well known, elastic constant is a very significant index to characterize the properties of compounds. The elastic constant is the index of material elasticity, which is related to the stress-strain relationship in the anisotropic medium. To some extent, the elastic constant also indicates the influence of crystal dynamics on mechanical behavior. The stress-strain method is used to calculate the elastic constants in the present calculation process and the results can be listed in Table 2. Furthermore, for the D0 22 -type crystal, tetragonal phase (C 11 , C 33 , C 44 , C 66 , C 12 , and C 13 ) [37], the elastic constants can be restricted by the following Formula (3): C 11 > 0, C 33 > 0, C 44 > 0, C 66 > 0 (C 11 − C 12 ) > 0, (C 11 + C 33 − 2C 13 ) > 0, [2(C 11 + C 12 ) + C 33 + 4C 13 ] > 0.
It is noticeable that the elastic constants calculated in Table 2 conform to the mechanical stability criterion. According to Equation (3), these D0 22 -type compounds are mechanically stable at 0 K. The elastic constants C 11 , C 22 , and C 33 mean the compressibility of the crystal structure along the a-axis, b-axis, and c-axis, respectively. In the tetragonal system, C 11 and C 22 have the same value. Similarly, the value of C 44 in the tetragonal system is the same as that of C 55 . The value of C 44 indicates the ability to resist shear strain in (100) or (010) plane, while the value of C 66 represents the ability to resist shear strain in (001) plane. In view of Al 3 Sc, Al 3 Y and Al 3 La compounds, the calculated value of C 33 is less than that of C 11 , which proves that the a-axis has greater compression resistance than the c-axis. However, for other compounds, such as Al 3 Ti, Al 3 V, Al 3 Zr, Al 3 Nb, Al 3 Hf, Al 3 Ta, the elastic constants of C 33 are higher than that of C 11 , suggesting that the c-axis has greater compression resistance than the a-axis, as illustrated in Figure 3. In particular, it was found that Al 3 Ta has the highest resistance along the a-axis, b-axis, and c-axis. Besides, the values of C 11 and C 33 are larger than that of C 44 and C 66 . It means that these D0 22 -type phases are highly deformation resistant under uniaxial stress along the a-and c-axis.  [13] 60.00 [13] 42.00 [13] 158.00 [13] 63.00 [13] 93.00 [ [13] 84.00 [13] 49.00 [13] 216.00 [13] 94.00 [13] 122.00 [ [13] 77.00 [13] 47.00 [13] 258.00 [13] 104.00 [13] 129.00 [13] 220.87 [21] 92.69 [21] 45.26 [21] 256.95 [21] 98.57 [21] 130. 25   On the other hand, the bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (σ) of Al3TM crystal can be calculated by Viogt-Reuss-Hill (VRH) approximation method. Using the following formulas [38], the calculated values are shown in Table 3.
where the subscript symbols H denotes the modulus values obtained by the Hill approximation. The subscript V and R mean the modulus values obtained by the Voigt and Reuss approximation methods, respectively. Furthermore, the Voigt approximation limits the maximum of elastic modulus and the Reuss approximation is considered to be the minimum of elastic modulus. The Hill approximation uses the average value of Voigt and Reuss to express the elastic constants of materials.  On the other hand, the bulk modulus (B), shear modulus (G), Young's modulus (E) and Poisson's ratio (σ) of Al 3 TM crystal can be calculated by Viogt-Reuss-Hill (VRH) approximation method. Using the following formulas [38], the calculated values are shown in Table 3.
where the subscript symbols H denotes the modulus values obtained by the Hill approximation. The subscript V and R mean the modulus values obtained by the Voigt and Reuss approximation methods, respectively. Furthermore, the Voigt approximation limits the maximum of elastic modulus and the Reuss approximation is considered to be the minimum of elastic modulus. The Hill approximation uses the average value of Voigt and Reuss to express the elastic constants of materials. It is noted that the calculated values of elastic moduli are summarized in the Table 3. Obviously, the bulk modulus of these D0 22 -type compounds decreased in order: Al 3 Ta > Al 3 Nb > Al 3 V > Al 3 Hf > Al 3 Ti > Al 3 Zr > Al 3 Sc > Al 3 Y > Al 3 La, as shown in Figure 4. The bulk modulus of crystal reflects the resistance of crystal under water pressure. At the microscopic level, the bulk modulus of the crystal is determined by the strength of the chemical bond. The larger the bulk modulus of the crystal, the stronger the chemical bond strength and the stronger the compression resistance. The calculated results show that Al 3 Ta has the strongest resistance of compression. Oppositely, Al 3 La has the weakest resistance of compression. It is noted that the calculated values of elastic moduli are summarized in the Table 3. Obviously, the bulk modulus of these D022-type compounds decreased in order: Al3Ta > Al3Nb > Al3V > Al3Hf > Al3Ti > Al3Zr > Al3Sc > Al3Y > Al3La, as shown in Figure 4. The bulk modulus of crystal reflects the resistance of crystal under water pressure. At the microscopic level, the bulk modulus of the crystal is determined by the strength of the chemical bond. The larger the bulk modulus of the crystal, the stronger the chemical bond strength and the stronger the compression resistance. The calculated results show that Al3Ta has the strongest resistance of compression. Oppositely, Al3La has the weakest resistance of compression. Generally, shear modulus refers to the ability of a material to resist shear strain. The higher the shear modulus, the stronger the rigidity of the material. As listed in Table 3, Al3Ta and Al3La have the largest shear modulus (108.19 GPa) and smallest one (22.41 GPa), respectively. At the same time, Young's modulus is a term of material mechanics, which is used to express the deformation resistance of solid materials. The rigidity of the material can be reflected by the value of Young's modulus. That is to say, the greater the Generally, shear modulus refers to the ability of a material to resist shear strain. The higher the shear modulus, the stronger the rigidity of the material. As listed in Table 3, Al 3 Ta and Al 3 La have the largest shear modulus (108.19 GPa) and smallest one (22.41 GPa), respectively. At the same time, Young's modulus is a term of material mechanics, which is used to express the deformation resistance of solid materials. The rigidity of the material can be reflected by the value of Young's modulus. That is to say, the greater the Young's modulus is, the greater the rigidity of the material is, and the harder it is to deform. It can be seen that Al 3 Ta and Al 3 La have the largest Young's modulus (254.17 GPa) and smallest one (59.62 GPa). It is evident that Al 3 Ta has the greater rigidity and is difficult to deform.
Furthermore, the brittleness and ductility of the materials can be judged by Poisson's ratio. This is due to Poisson's ratio being the ratio of transverse strain to longitudinal strain when the material is deformed under tension or compression. A value of 0.26 is taken as the critical value of brittle and plastic separation. When the value of Poisson's ratio is greater than 0.26, it can be considered as ductile material. On the contrary, it can be determined as brittle material.
When the value of Poisson's ratio is less than 0.26, it can be judged as brittle material. Otherwise, it can be judged as ductile material [39,40]. As for these D0 22 -type compounds, the Poisson's ratios are 0.185, 0.173, 0.175, 0.211, 0.176, 0.172, 0.183, and 0.175 for Al 3 Sc, Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 Hf, and Al 3 Ta respectively, while the Poisson's ratio of Al 3 La is 0.330. Therefore, it is noted that Al 3 La shows toughness and the other eight compounds exhibit brittle property. Similarly, brittleness and ductility of compounds can be predicted by the ratio of bulk modulus to shear modulus [41,42]. Based on the Pugh standard, the material presents brittleness when the value of B/G is less than 1.75, otherwise the material exhibit toughness. As can be shown in Table 3 that the B/G of Al 3 Sc, Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 Hf, and Al 3 Ta is less than 1.75, indicating that these eight D0 22 -type compounds are brittle. On the contrary, the B/G value of Al 3 La is greater than 1.75, which shows toughness.
Ultimately, it is reported that the hardness of a compound is directly related to its shear modulus and Young's modulus. At present, although the accurate relationship between hardness and elastic modulus has not been determined, the larger elastic modulus can represent the higher hardness. For the nine compounds studied, their Vickers hardness (H V ) of Al 3 TM-type compounds can be forecasted by the following empirical formula [43]: As presented in Table 3, the maximum Vickers hardness of Al 3 TM is Al 3 Ta, which is 21.94 GPa. The minimum Vickers hardness is Al 3 La and its value is 1.02 GPa. Similar trends indicate that Al 3 Ta and Al 3 La have the largest Young's modulus and smallest one, as indicated in Figure 4. It is also confirmed that Young's modulus has a decisive effect on the hardness of the compound.

Mechanical Anisotropy
It is very important to study the anisotropy of D0 22 -type compounds since this index has influence on the macroscopic mechanical properties of the alloy. In the present calculation, the universal anisotropic index (A U ), the percent anisotropy index (A B and A G ) and the shear anisotropic factors (A 1 , A 2 and A 3 ) are estimated by the following expressions [44]: The generated universal anisotropic index (A U ), percent anisotropy (A B and A G ) and shear anisotropic factors (A 1 , A 2 and A 3 ) of these D0 22 -type compounds are exhibited in Table 4. The anisotropy of D0 22 -type compounds can be directly reflected by the value of A U . In case of A U is equal to zero, the crystal is isotropic. The larger the value of A U , the stronger the anisotropy, and vice versa. It is noted that the values of A U of Al 3   When the values of A B and A G are not equal to zero, it shows that the bulk modulus and shear modulus of the crystal are anisotropic. The values of A B and A G correspond to the anisotropy. It reveals that Al 3 La has the largest A B value 2.839, indicating that the bulk modulus anisotropy of Al 3 La is the strongest. Then, the bulk modulus anisotropy of Al 3 Y, Al 3 Sc, Al 3 Zr, Al 3 Nb, Al 3 Ti, Al 3 Hf, Al 3 V, and Al 3 Ta decrease gradually. Evidently, The A G value of Al 3 La is the highest and that of Al 3 Ta is the lowest, indicating that Al 3 La and Al 3 Ta have the strongest and weakest shear modulus anisotropy, respectively. Furthermore, the shear factors A 1 , A 2 , and A 3 can be used to represent anisotropy on (100), (010), and (001) planes. When the values of A 1 , A 2 , and A 3 are 1, the crystal is isotropic. According to the calculated results, all D0 22 -type compounds have different degrees of elastic anisotropy, as noted in the Table 4. Among these D0 22 -type phases, the A 1 (A 2 ) values of Al 3 La deviate most severely from 1, which imply that Al 3 La exhibits the strongest shear anisotropy in (100) and (010) planes. Conversely, the values of A 1 (A 2 ) of Al 3 Ta have the least deviation among these D0 22 -type compounds, which mean that the Al 3 Ta exists the lowest shear anisotropy in the (100) plane and (010) plane. The A 3 value of Al 3 La has the most deviation from 1, hinting that the Al 3 La shows the highest shear anisotropy in the (001) plane. These D0 22 -type phases all have a wide deviation from 1 in the (001) plane.
On the other hand, the three-dimensional (3D) image has the characteristics of clear hierarchy and visual intuition, which can show the anisotropic characteristics of these D0 22type compounds more clearly and vividly. In this study, bulk modulus, shear modulus, and Young's modulus of different D0 22 -type compounds are drawn with spherical coordinates. The relationship between bulk modulus, shear modulus, Young's modulus, and different directions can be realized by the following formulas [45,46]: where, l 1 = sinθcosφ, l 2 = sinθsinφ, l 3 = cosθ, l 1 , l 2 and l 3 the directional cosines, S ij the elastic compliance constants. If the system is isotropic, the three-dimensional directional correlation is spherical. The deviation of spherical shape suggests the degree of anisotropy. It is clear from the 3D stereoscopic pictures of bulk modulus, shear modulus, and Young's modulus that D0 22 -type compounds with the same crystal structures and different compositions reveal different degree anisotropies. It can also be seen from Figure 5 that the bulk modulus of Al 3 La shows the strongest anisotropy. This result is consistent with the maximum anisotropy index A B of Al 3 La calculated in Table 4. As proved in Figure 6, the shear modulus anisotropy of Al 3 La is the strongest, while that of Al 3 Ta is the weakest. This phenomenon is consistent with the results obtained by the anisotropy index A G in Table 4. Finally, the anisotropic properties of Young's modulus of these D0 22 -type compounds can be arranged as Al 3 La > Al 3 Zr > Al 3 Hf > Al 3 Ti > Al 3 Y > Al 3 Sc > Al 3 V > Al 3 Nb > Al 3 Ta, as noted in Figure 7. terials 2021, 14, x FOR PEER REVIEW weakest. This phenomenon is consistent with the results obtained by the anisotr AG in Table 4. Finally, the anisotropic properties of Young's modulus of these compounds can be arranged as Al3La > Al3Zr > Al3Hf > Al3Ti > Al3Y > Al3Sc > Al3 > Al3Ta, as noted in Figure 7.      In particular, the projecting images of bulk modulus, shear modulus, and Young's modulus can display the anisotropy details of the compounds more clearly, which are given in Figure 8. It should be noted that the projection features of (100) and (010) crystal planes are the same, so only the projections of (001), (010), and (110) plane are listed here. Obviously, the bulk modulus of these compounds shows greater anisotropic on the (010) and (100) planes, but it exhibits isotropy on the (001) plane. The degree of anisotropy of the same compound on (010) and (100) is the same, suggesting that the order of anisotropy is Al 3 La > Al 3 Y > Al 3 Sc > Al 3 Zr > Al 3 Nb > Al 3 Ti > Al 3 Hf > Al 3 V > Al 3 Ta. Additionally, the shear moduli of these D0 22 -type compounds are anisotropic on all three planes. Especially, the 3D diagram morphology of Al 3 La shear modulus anisotropy is different from that of the other eight compounds. These eight compounds have the similar anisotropy on the (001) crystal plane, while the anisotropy degree of Al 3 Ta displays the lowest. Furthermore, among these eight compounds, the anisotropy of Al 3 Ta in three crystal faces (001), (010), and (100) is smaller. Similarly, the Young's moduli of the nine compounds are anisotropic in three planes. The degree of anisotropy of Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 Hf, and Al 3 Ta is smaller on (100) and (010) crystal planes, while that of Al 3 Sc and Al 3 La is larger on (100) and (010) crystal planes. However, it is interesting to note that the anisotropy of Young's modulus is in the opposite direction to that of the shear modulus. Obviously, the Young's modulus of Al 3 Ta is less anisotropic in all three planes, while the anisotropy of Al 3 La is greater.
Materials 2021, 14, x FOR PEER REVIEW 13 of In particular, the projecting images of bulk modulus, shear modulus, and Young modulus can display the anisotropy details of the compounds more clearly, which a given in Figure 8. It should be noted that the projection features of (100) and (010) cryst planes are the same, so only the projections of (001), (010), and (110) plane are listed her Obviously, the bulk modulus of these compounds shows greater anisotropic on the (01 and (100) planes, but it exhibits isotropy on the (001) plane. The degree of anisotropy the same compound on (010) and (100) is the same, suggesting that the order of anisotrop is Al3La > Al3Y > Al3Sc > Al3Zr > Al3Nb > Al3Ti > Al3Hf > Al3V > Al3Ta. Additionally, th shear moduli of these D022-type compounds are anisotropic on all three planes. Especiall the 3D diagram morphology of Al3La shear modulus anisotropy is different from that the other eight compounds. These eight compounds have the similar anisotropy on th (001) crystal plane, while the anisotropy degree of Al3Ta displays the lowest. Furthermor among these eight compounds, the anisotropy of Al3Ta in three crystal faces (001), (010 and (100) is smaller. Similarly, the Young's moduli of the nine compounds are anisotrop in three planes. The degree of anisotropy of Al3Ti, Al3V, Al3Y, Al3Zr, Al3Nb, Al3Hf, an Al3Ta is smaller on (100) and (010) crystal planes, while that of Al3Sc and Al3La is larg on (100) and (010) crystal planes. However, it is interesting to note that the anisotropy Young's modulus is in the opposite direction to that of the shear modulus. Obviously, th Young's modulus of Al3Ta is less anisotropic in all three planes, while the anisotropy Al3La is greater.  Strangely, from Figures 5-8, the anisotropic appearances of Al 3 La in bulk modulus, shear modulus, and Young's modulus is quite different from those of the other eight compounds. As discussed earlier, Al 3 La has higher enthalpy of formation, lower modulus values, unique toughness, and smallest Vickers hardness. Thus, it is speculated that D0 22 -type Al 3 La may not exist.
Corresponding to Figure 8, the directions. The reason is that the C 33 value of Al 3 Sc and Al 3 La is smaller than C 11 and C 22 , which leads to greater compressibility in c-axis direction.

Electronic Structures
The compounds in present work are all D0 22 -type compounds that are bound with transition metals. Because of their orientation-related bonding properties, it is of great significance to analyze the chemical bonding properties and electronic structures of these D0 22 -type compounds. In Figure 9, we calculated the total density of states (TDOS) and partial density of states (PDOS) of D0 22 -Al 3 TM compounds. The vertical dashed line at zero energy shows the Fermi level (E F ). Obviously, the density of states at Fermi level is not zero, suggesting that these phases have metal properties and electrical conductivity. For Al 3 Sc, Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 La, Al 3 Hf, and Al 3 Ta phases, there is a peak near the Fermi level of the TDOS curves, implying their good conductivity.
It is obvious that the TDOS at the Fermi level for Al 3 Sc, Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 La, Al 3 Hf and Al 3 Ta are 3.43, 4.25, 1.99, 3.59, 3.85, 1.30, 4.65, 3.71, and 1.08 electrons/eV, respectively. The TDOS curve in Figure 9 also shows the pseudogap near E F , which is due to the electron transfer to the low energy region. The lower the position of E F in the gap, the more stable the structure of intermetallics is. Thus, Al 3 Ta can be considered as the most stable compound. When the pseudogap is larger, it means stronger bond strength and higher deformation resistance. Compared with Al 3 V, Al 3 Nb, and Al 3 Ta in Figure 9, it can be noted that the pseudogap of Al 3 Ta has the widest gap, meaning that the covalent bond strength of Al 3 Ta is stronger than that of Al 3 V and Al 3 Nb. This explains why Al 3 Ta has the highest hardness. In view of nine compounds, the shapes of TDOS for Al 3 Sc, Al 3 Ti, Al 3 Y, Al 3 Zr, Al 3 La, and Al 3 Hf are similar, which demonstrate that their chemical bonds are similar. The TDOS curves of Al 3 V, Al 3 Nb, and Al 3 Ta are alike, which is to say that the V-shaped tip intersects the dotted line of Fermi level. It is obvious that the TDOS at the Fermi level for Al3Sc, Al3Ti, Al3V, Al3 Al3Nb, Al3La, Al3Hf and Al3Ta are 3.43, 4.25, 1.99, 3.59, 3.85, 1.30, 4.65, 3.71, and trons/eV, respectively. The TDOS curve in Figure 9 also shows the pseudogap which is due to the electron transfer to the low energy region. The lower the po EF in the gap, the more stable the structure of intermetallics is. Thus, Al3Ta can b ered as the most stable compound. When the pseudogap is larger, it means stron strength and higher deformation resistance. Compared with Al3V, Al3Nb, and Figure 9, it can be noted that the pseudogap of Al3Ta has the widest gap, meanin covalent bond strength of Al3Ta is stronger than that of Al3V and Al3Nb. This why Al3Ta has the highest hardness. In view of nine compounds, the shapes of T Al3Sc, Al3Ti, Al3Y, Al3Zr, Al3La, and Al3Hf are similar, which demonstrate that th ical bonds are similar. The TDOS curves of Al3V, Al3Nb, and Al3Ta are alike, w say that the V-shaped tip intersects the dotted line of Fermi level. Meanwhile, in order to explore the chemical bonds and charge transfer in all D0 22 -type compounds, the charge density difference on the (010) basal plane of these compounds were considered. In Figure 10, the distribution state of charge density difference of Al 3 Sc, Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 La, Al 3 Hf, and Al 3 Ta can be seen clearly. The values of charge density difference map are plotted from −0.04 to 0.04 e/Å 3 . Besides, the red and blue mean separately the aggregation and reduction of electrons.

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indicating that the Al-Al bond in the compound has covalent bond characteristics. Furthermore, a larger degree of localization of electrons reflects a stronger bond. Consequently, the electron concentration between the Al and TM (V, Nb and Ta) atoms in Al3V, Al3Nb and Al3Ta compounds is higher, which leads to the conclusion that Al-TM (V, Nb and Ta) bonds are stronger. On the contrary, it can be suggested that the bonds of Al-La and La-La are the weakest, which is because the less the localization degree of electrons, the weaker the bonds are.

Work Function
In order to further understand the properties of D022-type compounds in this study, some surface work functions were calculated. Theoretically, the work function is the energy barrier used to move electrons from the surface of solid compounds to the free space, as noted by the following expression [47,48]: where φ is work function, and symbol Vvac means the electrostatic potential of the vacuum region near the surface. EF corresponds to the Fermi energy of the slab. The schematic diagram of the electronic movement is presented in Figure 11. When a compound has a higher electron work function, it takes more energy to change its electron state. Therefore, there are greater obstacles to improving the state or properties of compounds such as mechanical and elastic properties. The higher electron work function of the compound indicates that it has stronger atomic bond [49,50]. In present work, the work function value of Al3Sc, Al3Ti, Al3V, Al3Y, Al3Zr, Al3Nb, Al3La, Al3Hf, and Al3Ta is 3.838, 4.305, 4.206, 3.754, Figure 10. The electron density difference of Al 3 TM compounds.
As can be seen from the Figure 10, there are electron deletions around both Al and TM atoms (blue), and electron aggregation between Al and TM atoms (red). This suggests that the electrons are aggregated into a covalent Al-TM bond between the atoms Al and TM. Similarly, there is a large amount of charge accumulation between Al and Al atoms, indicating that the Al-Al bond in the compound has covalent bond characteristics. Furthermore, a larger degree of localization of electrons reflects a stronger bond. Consequently, the electron concentration between the Al and TM (V, Nb and Ta) atoms in Al 3 V, Al 3 Nb and Al 3 Ta compounds is higher, which leads to the conclusion that Al-TM (V, Nb and Ta) bonds are stronger. On the contrary, it can be suggested that the bonds of Al-La and La-La are the weakest, which is because the less the localization degree of electrons, the weaker the bonds are.

Work Function
In order to further understand the properties of D0 22 -type compounds in this study, some surface work functions were calculated. Theoretically, the work function is the energy barrier used to move electrons from the surface of solid compounds to the free space, as noted by the following expression [47,48]: where ϕ is work function, and symbol V vac means the electrostatic potential of the vacuum region near the surface. E F corresponds to the Fermi energy of the slab. The schematic diagram of the electronic movement is presented in Figure 11. When a compound has a higher electron work function, it takes more energy to change its electron state. Therefore, there are greater obstacles to improving the state or properties of compounds such as mechanical and elastic properties. The higher electron work function of the compound indicates that it has stronger atomic bond [49,50]. In present work, the work function value of Al 3  3.953, 4.100, 3.291, 4.079, and 4.078 eV on (100) plane, respectively. It is found that the value of Al3La is the smallest. Therefore, the Young's modulus and Vickers hardness of Al3La are the lowest, which is consistent with the previous calculated results. Figure 11. The schematic diagram of the electronic movement related to work function. Figure 12a shows the SEM and XRD images of representative CoCrFeAlTa0.16 HEA at 50 h annealing state. It is obvious that the alloy exhibits the crystal structure of FCC. The grain distribution in the image is clearly angular. The typical stress-strain curves of the solid solution and different heat treatment states can be seen in Figure 12b. It can be observed that the strength and toughness of the sample is better when it is treated at 700 °C/50 h. However, the strength and toughness of the samples were reduced when they were treated for 700 °C/100 h. It is possible that the grain size will be further enlarged as the solution time increases. In fact, the specimen annealed for 50 h will have D022-type Al3Ta phase, resulting in its tensile strength up to 1120 MPa and elongation up to 26%. This result fully reflects the strengthening effect of Al3Ta phase.  Figure 12a shows the SEM and XRD images of representative CoCrFeAlTa0.16 HEA at 50 h annealing state. It is obvious that the alloy exhibits the crystal structure of FCC. The grain distribution in the image is clearly angular. The typical stress-strain curves of the solid solution and different heat treatment states can be seen in Figure 12b. It can be observed that the strength and toughness of the sample is better when it is treated at 700 • C/50 h. However, the strength and toughness of the samples were reduced when they were treated for 700 • C/100 h. It is possible that the grain size will be further enlarged as the solution time increases. In fact, the specimen annealed for 50 h will have D0 22 -type Al 3 Ta phase, resulting in its tensile strength up to 1120 MPa and elongation up to 26%. This result fully reflects the strengthening effect of Al 3 Ta phase. In order to more clearly characterize the microstructure of D022, Figure 13 exhibits the TEM structure of Al3Ta phase. According to the image, the Al3Ta phase is needle like and has a certain orientation. It presents a crystallographic relationship with the matrix, [001]D022//[001]Matrix. The diffraction pattern in Figure 13b clearly illustrates this relationship. It can see that plane (2 2 0) of the matrix corresponds to plane (110) of D022-type Al3Ta In order to more clearly characterize the microstructure of D0 22 , Figure 13 exhibits the TEM structure of Al 3 Ta phase. According to the image, the Al 3 Ta phase is needle like and has a certain orientation. It presents a crystallographic relationship with the matrix, [001] D022 //[001] Matrix . The diffraction pattern in Figure 13b clearly illustrates this relationship. It can see that plane (220) of the matrix corresponds to plane (110) of D0 22 -type Al 3 Ta phase. Moreover, stereographic projection of the orientation relationship denotes the crystal plane correspondence between the two phases more clearly, as shown in Figure 13c,d. This orientation relationship further indicates that the formation of Al 3 Ta contributes to the improvement of its strength and toughness. Therefore, the reliability of the simulation is further confirmed by the experiments, which has important reference values for the design and application of this material. In order to more clearly characterize the microstructure of D022, Figure 13 exhibits the TEM structure of Al3Ta phase. According to the image, the Al3Ta phase is needle like and has a certain orientation. It presents a crystallographic relationship with the matrix, [001]D022//[001]Matrix. The diffraction pattern in Figure 13b clearly illustrates this relationship. It can see that plane (2 2 0) of the matrix corresponds to plane (110) of D022-type Al3Ta phase. Moreover, stereographic projection of the orientation relationship denotes the crystal plane correspondence between the two phases more clearly, as shown in Figure 13c,d. This orientation relationship further indicates that the formation of Al3Ta contributes to the improvement of its strength and toughness. Therefore, the reliability of the simulation is further confirmed by the experiments, which has important reference values for the design and application of this material.
It is noted that the thermodynamic properties of the nine D022-type compounds are stable. The alloying ability of the nine D022-type compounds from strong to weak can be arranged as Al3Zr > Al3Nb > Al3Hf > Al3Ti > Al3Sc > Al3Ta > Al3Y > Al3V > Al3La. The order of the bulk modulus of these D022-type compounds is Al3Ta > Al3Nb > Al3V > Al3Hf > Al3Ti > Al3Zr > Al3Sc > Al3Y > Al3La. Specifically, Al3Ta and Al3La have the largest shear modulus (108.19 GPa) and smallest one (22.41 GPa) separately. It can be found that Al3Ta and Al3La have the largest Young's modulus (254.17 GPa) and smallest one (59.62 GPa). It is

Conclusions
Overall, first principles calculations have been performed on elastic anisotropy, electronic structures, and work function of D0 22 -type Al 3 TM (TM = Sc, Ti, V, Y, Zr, Nb, La, Hf, Ta), including Al 3 Sc, Al 3 Ti, Al 3 V, Al 3 Y, Al 3 Zr, Al 3 Nb, Al 3 La, Al 3 Hf, and Al 3 Ta, respectively. The obtained results agree with the existing theoretical values.
It is noted that the thermodynamic properties of the nine D0 22 -type compounds are stable. The alloying ability of the nine D0 22 -type compounds from strong to weak can be arranged as Al 3 Zr > Al 3 Nb > Al 3 Hf > Al 3 Ti > Al 3 Sc > Al 3 Ta > Al 3 Y > Al 3 V > Al 3 La. The order of the bulk modulus of these D0 22 -type compounds is Al 3 Ta > Al 3 Nb > Al 3 V > Al 3 Hf > Al 3 Ti > Al 3 Zr > Al 3 Sc > Al 3 Y > Al 3 La. Specifically, Al 3 Ta and Al 3 La have the largest shear modulus (108.19 GPa) and smallest one (22.41 GPa) separately. It can be found that Al 3 Ta and Al 3 La have the largest Young's modulus (254.17 GPa) and smallest one (59.62 GPa). It is evident that Al 3 Ta has the higher hardness and is not easy to deform. Furthermore, the universal anisotropy of these D0 22 -type compounds can be listed as Al 3 La > Al 3 Zr > Al 3 Hf > Al 3 Ti > Al 3 Y > Al 3 Sc > Al 3 V > Al 3 Nb > Al 3 Ta. The Al 3 La and Al 3 Ta have the highest and lowest A G values, demonstrating that the shear modulus anisotropy of Al 3 La is the strongest, while that of Al 3 Ta is the weakest. The A 3 value of Al 3 La has the most deviation from 1, indicating that the Al 3 La shows the highest shear anisotropy in the (001) plane. Except for Al 3 Sc and Al 3 La, the Young's modulus of other compounds in [001] direction is greater than that in [100] and [010] directions. From the standpoint of anisotropy, it is speculated that D0 22 -type Al 3 La may not exist.
It can be suggested that the density of states at Fermi level is not zero, suggesting that these phases have metal properties and electrical conductivity. Obviously, the pseudogap of Al 3 Ta has the widest width, meaning that the covalent bond strength of Al 3 Ta is stronger than that of Al 3 V and Al 3 Nb. The electron concentration between the Al and TM (V, Nb and Ta) atoms in Al 3 V, Al 3 Nb and Al 3 Ta compounds is higher, which shows that Al-TM (V, Nb and Ta) bonds are stronger. Ultimately, the work function of Al 3 La is the smallest on (100), demonstrating that the Young's modulus and Vickers hardness of Al 3 La are the lowest.
Based on the calculated results, a kind of D0 22 reinforced alloy was designed, which proves that this phase has an excellent strengthening effect. The matrix is strengthened by