The structures, stability, mechanical properties and electronic properties of X (X=Nb, Ge, Ni) doped Al₂Cu: a first-principle study

The formation energy of the reaction is equal to the total energy of the reaction product minus the total energy of the reactant. It can measure the degree of difficulty in the formation of intermetallic compounds, determine whether a stable structure can be formed. When the formation energy is negative, the smaller the formation energy, the easier the formation of intermetallic compounds. The formula for the reaction of X doped Al₂Cu are as follows:

$$\begin{eqnarray*}&&\bigcirc\!\!\!\!\!{1} \,{{\rm{Al}}}_{64}{{\rm{Cu}}}_{32}+{\rm{X}}\to {{\rm{Al}}}_{63}{{\rm{Cu}}}_{32}{\rm{X}}+{\rm{Al}}\end{eqnarray*}$$

$$\begin{eqnarray*}&&\bigcirc\!\!\!\!\!{2} \,{{\rm{Al}}}_{64}{{\rm{Cu}}}_{32}+{\rm{X}}\to {{\rm{Al}}}_{64}{{\rm{Cu}}}_{31}{\rm{X}}+{\rm{Cu}}\end{eqnarray*}$$

It can be seen from figure 2 that the except for the positive value of Al₆₄Cu₃₂Ge, the formation energies of other structures are negative, indicating that all structures can exist stably except Al₆₄Cu₃₂Ge. For stable structures, the formation energies in accordance with the order from low to high is Al₆₃Cu₃₂Nb < Al₆₄Cu₃₁Nb < Al₆₄Cu₃₁Ni < Al₆₃Cu₃₂Ni < Al₆₃Cu₃₂Ge. So, Al₆₃Cu₃₂Nb is easiest to form, and Al₆₃Cu₃₂Ge is most difficult to form. Moreover, However, for the three elements, the formation energies of Nb-doped solid solution are the smallest, whereas that of Ge-doped solid solution are the largest, regardless of where X replaces Al or Cu. This indicates that Al-Cu-Nb doped with Nb are the easiest to form, while Al-Cu-Ge compounds doped with Ge are the most difficult to form.

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The cohesive energy of a crystal is the energy released by free atoms combine into a crystal. In other words, the work done by the outside world when a crystal resolved into single atoms. The structural stability of the crystal is closely related to the cohesive energy, the greater the absolute value of cohesive energy, the more stable the crystal formed. The formula of cohesive energy is given by reference [41].

$$\begin{eqnarray*}&&{E}_{coh}=\displaystyle \frac{1}{a+b+c}\left({E}_{tot}-a{E}_{atom}^{Al}-b{E}_{atom}^{Cu}-c{E}_{atom}^{X}\right)\end{eqnarray*}$$

Where ${E}_{atom}^{Al},$ ${E}_{atom}^{Cu},$ ${E}_{atom}^{X}$ are the energies of the free atoms of Al, Cu and X, respectively. The results are shown in figure 3. Observing all the cohesive energies, the absolute values are in turn: Al₆₃Cu₃₂Nb > Al₆₄Cu₃₁Nb > Al₆₄Cu₃₁Ni > Al₆₃Cu₃₂Ni > Al₆₃Cu₃₂Ge, indicating that the structure of Al₆₃Cu₃₂Nb is the most stable, and the structure of Al₆₃Cu₃₂Ge is the most unstable. For the three elements, the absolute value of binding energies after Nb doping are the largest, while that after Ge doping are the smallest. This indicates that the Nb is more stable in the form of Al-Cu-X, whereas Ge is just the contrary. Al-Cu-Nb system is the most stable, while Al-Cu-Ge system is the most instable. This corresponds to the case of formation energies.

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The elastic constant is an important index to reflect the external stress of a crystal. In order to test the mechanical stability of the cell after doping, we calculated the elastic constants of the six structures, as shown in table 2. The mechanical stability criterion of tetragonal system are as follows: [42, 43].

$$\begin{eqnarray*}&&\begin{array}{l}{{\rm{C}}}_{11}\gt 0,{{\rm{C}}}_{33}\gt 0,{{\rm{C}}}_{44}\gt 0,{{\rm{C}}}_{66}\gt 0,{{\rm{C}}}_{11}+{{\rm{C}}}_{33}-2{{\rm{C}}}_{13}\gt 0,{{\rm{C}}}_{11}-{{\rm{C}}}_{12}\gt 0,\\ \,2\left({{\rm{C}}}_{11}+{{\rm{C}}}_{12}\right)+{{\rm{C}}}_{33}+4{{\rm{C}}}_{13}\gt 0.\end{array}\end{eqnarray*}$$

It is shown in table 2 that all structures satisfy the criterion. This shows that all structures are stable at the mechanical level. The C₁₁, C₂₂ and C₃₃ of these six structures (Al₆₄Cu_32, Al₆₃Cu₃₂Nb, Al₆₄Cu₃₁Nb, Al₆₃Cu₃₂Ge, Al₆₃Cu₃₂Ni, Al₆₄Cu₃₁Ni) are larger than other elastic constants, which indicates that the ability of resisting deformation under normal stress is stronger than the ability of resisting deformation under shearing stress in these six structures.

We calculated the bulk modulus (K), shear modulus (G), Young's modulus (E), the ratio of K/G and Poisson's ratio (u) for Al₆₄Cu_32, Al₆₃Cu₃₂Nb, Al₆₄Cu₃₁Nb, Al₆₃Cu₃₂Ge, Al₆₃Cu₃₂Ni, Al₆₄Cu₃₁Ni. For K, G, E, the Voigt, Reuss, and Hill (VRH) approximations are used for estimation, and the related formulas are presented by reference [44–46]. The results of elastic modulus are shown in figure 4.

$$\begin{eqnarray*}&&\begin{array}{l}{K}_{VRH}=\displaystyle \frac{{K}_{V}+{K}_{R}}{2},\,{G}_{VRH}=\displaystyle \frac{{G}_{V}+{G}_{R}}{2},{E}_{VRH}=\displaystyle \frac{9{K}_{VRH}}{1+3{K}_{VRH}/{G}_{VRH}}\\ {K}_{V,R}=\displaystyle \frac{{C}_{11}+2{C}_{12}}{3},\,{G}_{V}=\displaystyle \frac{{C}_{11}-{C}_{12}+3{C}_{44}}{5},{G}_{R}=\displaystyle \frac{5({C}_{11}-{C}_{12})\times {C}_{44}}{{C}_{44}+3({C}_{11}-{C}_{12})}\end{array}\end{eqnarray*}$$

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The Young's modulus (E) reflects the stiffness of the materials. The smaller the Young's modulus, the smaller the stiffness of the material, the better the ductility. From figure 4, we know that the values of Young's modulus (E) from low to high is Al₆₄Cu₃₂ < Al₆₄Cu₃₁Nb < Al₆₃Cu₃₂Nb < Al₆₃Cu₃₂Ni < Al₆₃Cu₃₂Ge < Al₆₄Cu₃₁Ni, which shows the stiffness of Al₆₄Cu₃₂ is the smallest, the stiffness of Al₆₄Cu₃₁Ni is the largest, Al₆₄Cu₃₂ has the best ductility. With the addition of Nb, Ge and Ni, the stiffness of Al₆₄Cu₃₂ became larger, and the ability to resist the elastic deformation are stronger. It can be seen from figure 4 that doping does not significantly change the bulk modulus of Al₆₄Cu₃₂, but affects the shear modulus, which leads to a significant change in Young's modulus. The ratio of bulk modulus (K) and shear modulus (G) can be used to evaluate the brittleness and ductility of crystals. When K/G < 1.75, the materials are brittle, conversely the materials behave as toughness. The values of K/G are shown in table 3. As shown in table 3, the values of Al₆₄Cu₃₂, Al₆₃Cu₃₂Nb and Al₆₄Cu₃₁Nb are greater than 1.75, so they are ductile materials, and the K/G values of Al₆₃Cu₃₂Ge, Al₆₃Cu₃₂Ni, Al₆₄Cu₃₁Ni are less than 1.75, so Al₆₃Cu₃₂Ge, Al₆₃Cu₃₂Ni, Al₆₄Cu₃₁Ni are brittle. This indicates that the doping of Ge and Ni causes a transition from plasticity to brittleness in Al₆₄Cu₃₂. Poisson's ratio (u) is used to evaluate the shear stability of the structure. The bigger the value of Poisson's ratio, the better the ductility of the material. The results of Poisson's ratio (u) are also shown in table 3. From table 3, we observed that the Poisson's ratios of Al₆₄Cu₃₂ are the largest, indicated the ductility are the best, followed are Doping of Nb, Ge and Ni, which are consistent with the results of ductility obtained by Young's modulus (E).

Cracks in materials are an important cause of material failure. Cracks expand and merge slowly through the initial micro-cracks. Ultimately, the strength of the material decreases markedly, even breaks. This has seriously affected the active life of materials and will bring incalculable consequences to the project. For this reason, we have calculated the elastic anisotropy (A_U) of the materials, which can give us an understanding of the cause of induce micro-cracks in materials. The value of A_U indicates the extent of the anisotropy. When A_U is 0, the material is isotropic, and the more the A_U deviates from the 0, the higher the degree of anisotropy of the materials. The formula for A_U are given by references [47, 48].

$$\begin{eqnarray*}&&{A}_{U}=5\displaystyle \frac{{G}_{V}}{{G}_{R}}+\displaystyle \frac{{K}_{V}}{{K}_{R}}-6\end{eqnarray*}$$

Table 3 shows the values of the anisotropy of the five structures. It can be seen that all structures are anisotropic. The value of Al₆₄Cu₃₂ is the largest, the degree of anisotropy is the highest, the value of Al₆₃Cu₃₂Ge is minimum, indicated the degree of anisotropy is lowest. The results show the periodicity, density and physicochemical properties of the Al₆₄Cu₃₂ are most different from each other in the different directions. Under the same conditions, cracking caused by external factors is the easiest. But Al₆₃Cu₃₂Ge is just the opposite of Al₆₄Cu₃₂.

In most cases, the cracks initiate from the surface of the materials. The difficulty of surface cracking directly affect the occurrence of micro-cracks. In order to determine the surface where cracks are most likely to occur, the analysis of shear anisotropy of materials is an effective method. The shear anisotropy is an important criterion for judging the formation of micro-cracks on surface. The shear anisotropic factor reflects the degree of anisotropy of the bonding between atoms in different planes, which explains the reason of the cracking on the surface of the materials. For the {100} shear planes between 〈011〉 and 〈010〉 directions, the {010} shear planes between 〈101〉 and 〈001〉 directions, and the {001} shear planes between 〈110〉 and 〈010〉 directions, the shear anisotropic factors are given by reference [49].

$$\begin{eqnarray*}&&{A}_{1}=\displaystyle \frac{4{C}_{44}}{{C}_{11}+C{}_{33}-2{C}_{13}},{A}_{2}=\displaystyle \frac{4{C}_{55}}{{C}_{22}+{C}_{33}-2{C}_{23}},{A}_{3}=\displaystyle \frac{4{C}_{66}}{{C}_{11}+{C}_{22}-2{C}_{12}}\end{eqnarray*}$$

For isotropic crystals, the values of A₁, A₂, and A₃ must be 1, while any values less than or greater than 1 are the measures of elastic anisotropy, and the more the deviation between the value and 1, the higher the degree of anisotropy [50]. The results of A₁, A₂, and A₃ are shown in figure 5.

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As shown in figure 5 the six structures are all anisotropic on three planes. The anisotropic values of A₂ and A₃ are generally larger than those of A₁, implying that the degree of anisotropy in the {010} and {001} surface are higher than that in {100}. On the {001} plane, Al₆₄Cu₃₁Ni has the highest anisotropy, which means that the distribution of the charge on {001} plane has the highest anisotropy, and most easily induce the germination of micro-cracks. The value of Al₆₃Cu₃₂Ge in {001} is closest to 1, and the degree of anisotropy is the lowest in this plane, indicated it is most difficult to germinate micro-cracks. On the {100} and {010} planes, The values of A₁ and A₂ of the six structures are similar, which indicates that their anisotropic degrees on these two surfaces are little difference. Al₆₄Cu₃₂, Al₆₄Cu₃₁Nb, Al₆₃Cu₃₂Ni and Al₆₄Cu₃₁Ni have the closest A₁ and A₂ values. Overall analysis, the anisotropic value of Al₆₃Cu₃₂Ge is closest to 1, followed by Al₆₃Cu₃₂Nb. No matter what element is doped, the anisotropic value of Al₆₄Cu₃₂ will be closer to 1, and the anisotropy of Al₆₄Cu₃₂ will be reduced, except for the A₃ of Al₆₄Cu₃₁Ni. In summary, the addition of Nb, Ge and Ni can greatly reduce the anisotropy of the {100}, {010} and the whole anisotropy, especially Nb and Ge replaced Al sites are more obvious.