UDK 669.715:544.225 Original scientific article/Izvirni znanstveni članek ISSN 1580-2949 MTAEC9, 47(1)65(2013) STRUCTURAL STABILITY AND ELECTRONIC PROPERTIES OF AlCu3, AlCu2Zr AND AlZr3 STABILNOST STRUKTURE IN ELEKTRONSKE LASTNOSTI AlCu3, AlCu2Zr IN AlZr3 Rong Cheng, Xiao-Yu Wu Shenzhen Key Laboratory of Advanced Manufacturing Technology for Mold & Die, College of Mechatronics and Control Engineering, Shenzhen University, Shenzhen 518060, P. R. China songxp12345@yeah.net Prejem rokopisa — received: 2012-07-23; sprejem za objavo - accepted for publication: 2012-08-28 First-principles calculations were performed to study the alloying stability and electronic structure of the Al-based intermetallic compounds AlCu3, AlCu2Zr and AlZr3. The results show that the lattice parameters obtained after the full relaxation of the crystalline cells are consistent with the experimental data, and these intermetallics have a strong alloying ability and structural stability due to their negative formation energies and their cohesive energies. A further analysis revealed that the single-crystal elastic constants at zero-pressure satisfy the requirements for the mechanical stability of cubic crystals. The calculations on Poisson's ratio show that AlCu3 is much more anisotropic than the other two intermetallics. In addition, calculations on the densities of states indicate that the valence bonds of these intermetallics are attributed to the valence electrons of Cu 3d states for the AlCu3, Cu 3d and Zr 4d states for AlCu2Zr, and Al 3s, Zr 5s and 4d states for AlZr3, respectively. In particular, the electronic structure of the AlZr3 shows the strongest hybridization. Keywords: AlCu3, AlCu2Zr, first-principles, electronic structure Narejeni so bili prvi načelni izračuni stabilnosti legiranja in elektronske strukture aluminijevih intermetalnih zlitin (AlCu3, AlCu2Zr in AlZr3). Rezultati kažejo, da se mrežni parametri po polni relaksaciji kristalnih celic ujemajo z eksperimentalnimi podatki in da imajo te intermetalne zlitine veliko sposobnost legiranja ter stabilno strukturo zaradi negativne tvorbene energije in kohezivnih energij. Nadaljnje analize so pokazale, da elastična konstanta monokristala pri ničelnem tlaku ustreza zahtevam mehanske stabilnosti kubičnega kristala. Izračun Poissonovega količnika pokaže, da je AlCu3 bolj anizotropen kot drugi dve intermetalni zlitini. Dodatno izračun gostote stanj pokaže, da so valenčne vezi teh treh intermetalnih zlitin vezane na valenčne elektrone Cu 3d-stanja za AlCu3, Cu 3d in Zr 4d-stanja za AlCu2Zr, ter Al 3s, Zr 5s in 4d-stanja za AlZr3, posebno elektronska struktura AlZr3 pa kaže najmočnejšo hibridizacijo. Ključne besede: AlCu3, AlCu2Zr, načelen izračun, elektronska struktura 1 INTRODUCTION Intermetallics involving aluminum and transition metals (TMs) are known to have a high resistance to oxidation and corrosion, elevated-temperature strength, relatively low density, and high melting points, which make them desirable candidates for high-temperature structural applications12. In particular, zirconium can effectively enhance the mechanical strength of the alloys when copper and zinc elements exist in aluminum and Al-based alloys3. Adding Zr to the Al-Mg alloys can effectively remove or reduce hydrogen, grain refinement, pinholes, porosity and hot cracking tendency, and so improve the mechanical properties4. Many investigations have focused on the constituent binary systems, such as Al-Cu, Al-Zr, and Cu-Zr 5-10; however, there has been a lack of systematic theoretical and experimental investigations for binary and ternary systems, especially for ternary alloy systems. In recent years, first-principles calculations based on the density-functional theory have become an important tool for the accurate study of the crystalline and electronic structures and mechanical properties of solids11. In the present study, we report on a systematic investigation of the structural, elastic and electronic properties of Al-based alloys (AlCu3, AlZr3 and AlCu2Zr) using first-principles calculations, and the results are discussed in comparison with the available experimental data and other theoretical results. 2 COMPUTATIONAL METHOD All the calculations were performed using the Vienna ab-initio Simulation Package (VASP)1213 based on the density-functional theory (DFT)14. The exchange and correlation energy was treated within the generalized gradient approximation of Perdew-Wang 91 version (GGA-PW91)15. The interaction between the valence electrons and the ions was described by using potentials generated with Blochl's projector augmented wave (PAW) method16. The PAW potential used for Al treats the 3s, 3p states as valence states, and the other electron-ion interaction was described by the 3d, 4s valence states for Cu, 5s, 4d, 5p valence states for Zr. A plane-wave energy cut-off was set at 450 eV for the AlCu3 and AlCu2Zr, and at 350 eV for the AlZr3. Brillouin Zone integrations were performed using the Monkhorst-Pack17 k-point meshes, e.g., the k-point meshes for AlCu3, AlCuaZr and AlZrs were 15x15x15, 9x9x9 and 13x13x13 for optimizing the geometry and calculating the elastic constants, and 25x25x25, 19x19x19and 23x23x23 for calculations of the density of states (DOS) at the equilibrium volume, respectively. Optimizations of the structural parameters (atomic positions and the lattice constants) for each system were performed using the conjugate gradient method, and the coordinates of the internal atoms were allowed to relax until the total forces on each ion were less than 0.01 eV/(10-1 nm). The total energy and density of states (DOS) calculations were performed with the linear tetrahedron method using Blochl corrections18. In order to avoid wrap-around errors, all the calculations were performed using the "accurate" setting within VASP. 3 RESULTS AND DISCUSSION where Etot is the total energy of the compound at the equilibrium lattice constant, and E A atom ' EB EC are the energies of the isolated atoms A, B and C in the freedom states. Na, Nb and Nc refer to the numbers of A, B and C atoms in each unit cell. The energies of the isolated Al, Cu and Zr atoms are -0.276 eV, -0.254 eV and -2.054 eV, respectively. The cohesive energies (Ecoh) per atom of all the crystal or primitive cells are calculated from Eq. (1), and the results of the calculations are listed in Table 2. From the calculated values we find that the cohesive energy of AlZr3 is 2.237 eV and 1.413 eV per atom lower than that of AlCu3 and AlCu2Zr, respectively. Therefore, of the three phases, the AlZr3 phase has the highest structural stability, followed by AlCu2Zr and finally the AlCu3. This means that for the AlZr3, AlCu2Zr, and AlCu3 alloys the structural stability is higher with increasing amounts of Zr in the crystal. 3.1 Equilibrium properties The AlCu3 and AlZr3 alloys have the simple cubic CusAu (L12 type, space group Pm-3m) structure19,20. The AlCu2Zr alloy is a partially ordered Cu2MnAl-type fcc structure with the Fm-3m space group21. Firstly, these crystal structures were optimized with a relaxation of the cell shape and the atomic positions. The equilibrium volume V0, bulk modulus B0 and the pressure derivation of the bulk modulus B'0 of the AlCu3, AlCu2Zr and AlZr3 were determined by fitting the total energy calculated at different lattice-constant values to a Birch-Murnaghan equation of state22. The results of the first-principles calculations are listed in Table 1. From Table 1 it is clear that the results of our calculations compare very favorably with the experimental data. This shows that the used parameters are reasonable. It is known that the stability of a crystal structure is correlated to its cohesive energy23, which is often defined as the work that is needed when the crystal is decomposed into single atoms. Hence, the lower the cohesive energy is, the more stable the crystal structure is23. In the present study, the cohesive energies (Ecoh) of the AlCu3, AlCuZr and AlZr3 crystal cells can be calculated by: E T = coh ( e - N Ea - N EB - N EC ) V-^tot A atom Jv Blatom JVC^atom/ N A + N B + N C (1) Table 2: Total energy Etot, cohesive energy Ecoh and formation energy AH of AlCu3, AlCu2Zr and AlZr3 Tabela 2: Celotna energija Etot, kohezivna energija Ecoh in tvorbena energija AH za AlCu3, AlCu2Zr in AlZr3 Compound Etot /eV per atom Ecoh /eV per atom AH/eV per atom AlCu3 -3.897 -3.637 -0.177 AlCu2Zr -5.261 -4.551 -0.359 AlZr3 -7.574 -5.964 -0.307 In order to compare the alloying abilities of the present compounds, we calculate the formation energy AH, which can be given by: ahabc - " ( e - N EA - N EB - N EC ) V-^tot A solid B solid ^C^soM^ A v solid_ N A + N B + N C (2) are the energies per atom of the where EAl d, EBl d, EC,. solid ' solid ' soli pure constituents A, B and C in the solid states, respectively. And the other variables are as defined for Eq. (1). If the formation energy is negative, the formation of a compound from its elements is usually an exothermic process. Furthermore, the lower the formation energy is, the stronger the alloying ability is, and the more stable the crystal structure is23. The calculated energies of Al, Cu and Zr in their respective crystals are -3.696 eV, -3.728 eV, -8.457 eV. The calculated results of these compounds are also listed in Table 2. It is clear that all Table 1: Calculated and experimental lattice parameters a (nm), equilibrium volume Vo (nm3), bulk modulus Bo (GPa) and the pressure derivation of the bulk modulus B o for AlCu3, AlCu2Zr, AlZr3 Tabela 1: Izračunani in eksperimentalno določeni mrežni parametri a (nm), ravnotežni volumen Vo (nm3), modul pri stiskanju Bo (GPa) in izpeljava modula iz tlaka B o za AlCu3, AlCu2Zr, AlZr3 AlCu3 AlCu2Zr AlZr3 Present. Expt. Present. Expt. Present. Expt. a/nm 0.3693 0.3607 19 0.6256 0.6216 21 0.4381 0.4392 20 Vo/nm3 50.358 • 10-3 - 244.805 • 10-3 240.210 • 10-3 21 84.110 • 10-3 84.700 • 10-3 20 Bo/GPa 131.010 - 128.600 - 100.800 101.47 B'o 4.47 - 4.280 - 3.48 3.33 7 the AH is negative, which means that the structure of these compounds can exist and be stable. A further comparison and analysis showed that the alloying abilities of AlCu2Zr were much stronger than AlCu3 and AlZr3. It should be noticed that the alloying ability of AlZr3 was higher than that of the AlCu3 alloy. 3.2 Elastic properties The density-functional theory has become a powerful tool for investigating the elastic properties of materials (in the limit of zero temperature and in the absence of zero-point motion). For a given crystal it is possible to calculate the complete set of elastic constants by applying small strains to the equilibrium unit cell and determining the corresponding variations in the total energy. The necessary number of strains is imposed by the crystal symmetry24. For a material with cubic symmetry, there are only three independent elastic constants, Cn, C11 and C11. The strain tensor is given by: f ô = àn ô 21 VÔ 31 '33 7 (3) In the present study we applied three kinds of strains (50, C11 > 0, C44 > 0, (C11 + 2 C12) > 0. This shows that AlCu3, AlCu2Zr and AlZr3 have a stable structure. The average bulk modulus is identical to the single-crystal bulk modulus, i.e., B = (C11 + 2 Ci2)/3. Interestingly, we noted that the bulk modulus calculated from the values of the elastic constants is in good agreement with the one obtained through fitting to the Birch-Murnaghan equation of state (B0), giving a consistent estimation of the compressibility for these com-pounds26. In order to further validate our results, the elastic modulus, such as the shear modulus G(GPa), Young's modulus E(GPa), Poisson's ratio v and anisotropy constant A for a polycrystalline material were also calculated with the single-crystal elastic constants Cj, all of these elastic moduli are shown in Table 4. In the present study we adopted Hershey's averaging method27, which has been known to give the most accurate relation between single-crystal and polycrystalline values for a cubic lattice28. According to this method, G is obtained by solving the following equation: 5C + 4C C (7C - 4C ) 3 Jl, 11 T-l-l, 12 44 ^ 11 12^ 8 G - 8 G3 + G - C 44(C 11 C12)(C 11 + C12) 8 (5) = 0 The calculated shear moduli G for AlZr3 are the largest, while the quantities for AlCu2Zr are less than for AlCu3. Pugh29 found that the ratio of the bulk modulus to the shear modulus (B/G) of polycrystalline phases can predict the brittle and ductile behavior of the materials. A high and low value of B/G are associated with ductility and brittleness, respectively. The critical value which separates ductility from brittleness is about 1.75. From B/G calculated in Table 4 we can see that all the B/G ratios are larger than 1.75. Therefore, AlCu3, AlCu2Zr and AlZr3 have good ductility. In contrast, the biggest B/G ratio for AlCu2Zr indicates that AlCu2Zr is of very good ductility in these three Al-based alloys. AlCu3 has an intermediate ductility, while AlZr3 has the worst ductility. Besides B/G, the Young's modulus E and the Poisson's ratio v are important for technological and engineering applications. The Young's modulus is used to provide a measure of the stiffness of the solid, i.e., the larger the value of E, the stiffer the material24. According to Hershey's averaging method, the Young's modulus is defined as: E = 9GB/3(B+G). Based on the calculated results, we find that AlZr3 has a Young's modulus that is 18.806 GPa and 24.663 GPa larger than AlCu3 and AlCu2Zr, respectively. This indicates that the AlZr3 phase has the highest stiffness, followed by AlCu3 and finally the AlCu2Zr. In addition, the Poisson's ratio v has also Table 4: Calculated elastic constants (GPa) and elastic modulus (bulk modulus B (GPa), shear modulus G (GPa), Young's modulus E (GPa), Poisson's ratio v and anisotropy constant A) of AlCu3, AlCu2Zr and AlZr3 Tabela 4: Izračunane konstante elastičnosti (GPa) in elastični moduli (modul pri stiskanju B (GPa), strižni moduli G (GPa), Youngovi moduli E (GPa), Poissonov količnik v in konstante anizotropije A) za AlCu3, AlCu2Zr in AlZr3 Compound c11 c12 c44 B G B/G E v A reference AlCu3 150.707 120.565 81.880 130.612 43.593 2.996 117.686 0.350 1.887 this study 176.000 117.400 92.400 136.900 49.600 132.800 0.340 5 AlCu2Zr 157.504 115.305 62.685 129.371 41.237 3.137 111.829 0.356 1.528 this study AlZr3 148.653 79.387 70.834 102.476 53.400 1.919 136.492 0.278 1.487 this study 163.800 79.300 86.500 107.670 6 been used to measure the shear stability of the lattice, which usually ranges from -1 to 0.5. The greater the value of the Poisson's ratio v, the better the plasticity of 25 20 15 10 5 0 0.16 0.12 0.08 0.04 120 90 60 30 1.0 0.5 0.0 105 70 35 0 20 15 10 5 0 42 ■ 0 0.6 0.4 0.2 0.0 6. « i ii 1 I 1 I 1 I 1 I 1 (a) " -^-1- —1—'— Iks ' 1 ' 1 -Cu:s 1 - =S5 ft A ■ i...... Energy (eV) J -AlCUjZr ft ■ -w ™ l& Jl - -cuTot 1mfV ■ - Zr Tot J \ (b) ---- - A irw- ------ c" 1 ■ 1 ^ A. 1 1 ■ ........ - -Zr:s - - Zr:p - -Zr:d -1-1-1 ~ I-i . 1 —.- ' 1 -4 -2 0 Energy (eV) -AIZr3 Tot -Al Tot -Zr Tot Xh Arv -1--1-1-1--1---1-r-- ^ \ ^pva/W. ——■—1—^— Zr's ' ' -Zr:p W 1 1 Azx Energy (eV) Figure 1: The total and partial density of states (DOS) of: a) AlCu3 crystal cell, b) AlCu2Zr crystal cell, c) AlZr3 crystal cell. The vertical dot line indicates the Fermi level. Slika 1: Skupna in parcialna gostota stanj (DOS) za: a) kristalno celico AlCu3, b) kristalno celico AlCu2Zr, c) kristalno celico AlZr3. Navpična pikčasta linija prikazuje Fermijev nivo. the materials. So we can see that AlCu3, AlCu2Zr and AlZr3 have a better plasticity. The elastic anisotropy of the crystals has an important application in engineering materials since it is highly correlated with the possibility of inducing micro-cracks24,30. For cubic symmetric structures31, the elastic anisotropy is defined as A = (2C44 + Ci2)/Cn. For a completely isotropic material the value of will be 1, while values smaller or bigger than 1 measuring the degree of elastic anisotropy24. Interestingly, we note that the values of A (Table 4) do not deviate far from unity, suggesting that the present cubic-structure alloys also do not deviate far from being isotropic. The calculated results also indicate that AlCu3 is much more anisotropic than the other two alloys. 3.3 Density of states For a beter understanding of the electronic characteristic and structural stability, the total density of states (DOS) for AlCu3, AlCuZrand AlZr3 were calculated, as shown in Figure 1, as well as the partial density of states (PDOS) of Al, Cu and Zr atoms in these Al-based intermetallic compounds. Figure 1 has evidence for the metallic character of these considered AlCu3, AlCu2Zr and AlZr3 structures because of the finite DOS at the Fermi level. With regard to the total density of states curve of AlCu3, it is clear from Figure 1a that the whole valence band of AlCu3 is located between -7 eV and 9 eV, which is dominated by Cu 3d states and a small contribution from the 3s and 3p states of Al. The valence band of AlZr3 (Figure 1c) can be divided into three areas. The first area is dominated by the valence electron numbers of Al 3 s and Zr 4d states are mostly located between -7 eV and -5 eV, the second by the Zr 5s and 4d states located between -4 eV and -3 eV, and the third by Zr 4d states located between -2.8 eV and 3.0 eV. Both below and above the Fermi level, the hybridization between the Al-p states and Zr-d states is strong. Due to the strong hybridization (or covalent interaction) the entire DOS can be divided into bonding and anti-bonding regions, and that a pseudogap resides in between. The characteristic pseudogap around the Fermi level indicates the presence of the directional covalent bonding. The Fermi level located at a valley in the bonding region implies the system has a pronounced stability. It is also generally considered that the formation of covalent bonding would enhance the strength of the material in comparison with the pure metallic bonding32. According to the covalent approach, the guiding principle is to maximize the bonding. Therefore, for a series of compounds having the same structure, the greater the occupancy in the bonding region the higher the stability33. It is indeed seen that the structural stability increases from AlCu3 to AlZr3. For AlCu2Zr (see Figure 1b) it is clear that the main bonding peaks between -6 eV and -2 eV are predominantly derived from the Cu 3d orbits, while the main bonding peaks between the Fermi level and 3 eV predominantly derived from the Zr 4d orbits. It should be noted that the phase stability of intermetallics depends on the location of the Fermi level and the value of the DOS at the Fermi level, i.e. N(Ef) 34 35. A lower N(EF) corresponds to a more stable structure. The value of the total DOS at the Fermi level is 3.64 states per eV for AlZr3, and the value of the total DOS at the Fermi level is 5.74 states per eV for AlCuZr. Therefore, AlZr3 has a more stable structure in these three Al-based intermetallics. This is in accordance with the calculation of cohesive energy. 4 CONCLUSIONS In summary, using the first-principles method we have calculated the alloying stability, the electronic structure, and the mechanical properties of AlCu3, AlCuZr and AlZr3. These intermetallics have a strong alloying ability and structural stability due to the negative formation energies and the cohesive energies. In particular, AlCu3 is much more anisotropic than the other two intermetallics. The valence bonds of these inter-metallics are attributed to the valence electrons of the Cu 3d states for AlCu3, Cu 3d and Zr 4d states for AlCu2Zr, and Al 3s, Zr 5s and 4d states for AlZr3, respectively, and the electronic structure of the AlZr3 shows the strongest hybridization, leading to the worst ductility. Acknowledgements This work was financially supported by the National Natural Science Foundation of China (No. 51175348). The authors are also grateful to colleagues for their important contribution to the work. 5 REFERENCES I G. Sauthoff, In: J. H. Westbrook, R. L. Fleischer, editors, Inter-metallic compounds, Wiley, New York 1994, 991 2R. W. Cahn, Intermetallics, 6 (1998), 563 3 P. K. Rajagopalan, I. G. Sharma, T. S. Krishnan, J. Alloys Compd., 285 (1999), 212 4 P. Wonwook, Mater Design, 17 (1996), 85 5W. Zhou, L. J. Liu, B. L. Li, Q. G. Song, P. Wu, J. Electron Mater., 38 (2009), 356 6E. Clouet, J. M. Sanchez, Phys. Rev. B, 65 (2002), 094105 7 G. Ghosh, M. Asta, Acta Mater., 53 (2005), 3225 8 G. Ghosh, Acta Mater., 55 (2007), 3347 9W. J. Ma, Y. R. Wang, B. C. Wei, Y. F. Sun, Trans Nonferrous Met. Soc. China, 17 (2007), 929 10 S. Pauly, J. Das, N. Mattern, D. H. Kim, J. Eckert, Intermetallics, 17 (2009), 453 II H. Baltache, R. Khenata, M. Sahnoun, M. Driz, B. Abbar, B. Bouhafs, Physica B, 344 (2004), 334 12 G. Kresse, J. Hafner, Phys. Rev. B, 49 (1994), 14251 13 G. Kresse, J. Furthmüller, Phys. Rev. B, 54 (1996), 11169 14W. Kohn, L. J. Sham, Phys. Rev., 140 (1965), 1133 15 J. P. Perdw, Y. Wang, Phys. Rev. B, 45 (1992), 13244 16 P. E. Blöchl, Phys. Rev. B, 50 (1994), 17953 17 H. J. Monkhorst, J. D. Pack, Phys. Rev. B, 13 (1976), 5188 18 P. E. Blöchl, O. Jepsen, O. K. Andersen, Phys. Rev. B, 49 (1994), 16223 19 M. Draissia, M. Y. Debili, N. Boukhris, M. Zadam, S. Lallouche, Copper, 10 (2007), 65 20 W. J. Meng, J. Jr Faber, P. R. Okamoto, L. E. Rehn, B. J. Kestel, R. L. Hitterman, J. Appl. Phys., 67 (1990), 1312 21 R. Meyer zu Reckendorf, P. C. Schmidt, A. Weiss, Z. Phys. Chem. N F, 163 (1989),103 22 F. Birch, J. Geophys. Res., 83 (1978), 1257 23 V. I. Zubov, N. P. Tretiakov, J. N. Teixeira Rabelo, Phys. Lett. A, 194 (1994), 223 24 M. Mattesini, R. Ahuja, B. Johansson, Phys. Rev. B, 68 (2003), 184108 25 W. Y. Yu, N. Wang, X. B. Xiao, B. Y. Tang, L. M. Peng, W. J. Ding, Solid State Sciences, 11 (2009), 1400 26 B. Y. Tang, N. Wang, W. Y. Yu, X. Q. Zeng, W. J. Ding, Acta Mater., 56 (2008), 3353 27 H. M. Ledbetter, J. Appl. Phys., 44 (1973), 1451 28 A. Taga, L. Vitos, B. Johansson, Grimvall G., Phys. Rev. B, 71 (2005), 14201 29 S. F. Pugh, Philos. Mag., 45 (1954), 823 30 V. Tvergaard, J. W. Hutchinson, J. Am. Ceram. Soc., 71 (1988), 157 31 B. B. Karki, L. Stixrude, S. J. Clark, M. C. Warren, G. J. Ackland, J. Crain, Am. Miner., 82 (1997), 51 32 P. Chen, D. L. Li, J. X. Yi, W. Li, B. Y. Tang, L. M. Peng et al., Solid State Sciences, 11 (2009), 156 33 J. H. Xu, W. Lin, A. J. Freeman, Phys. Rev. B, 48 (1993), 4276 34 J. H. Xu, T. Oguchi, A. J. Freeman, Phys. Rev. B, 36 (1987), 4186 35 T. Hong, T. J. Watson-Yang, A. J. Freeman, T. Oguchi, J. H. Xu, Phys. Rev. B, 41 (1990), 12462