Ab initio Calculation to Predict Possible Non ...

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Journal of the Physical Society of Japan Vol. 71, No. 1, January, 2002, pp. 141–143 #2002 The Physical Society of Japan

Ab initio Calculation to Predict Possible Non-Equilibrium Solid Phases in an Immiscible Y–Nb System L. T. K ONG, J. B. L IU and B. X. LIU* Laboratory of Advanced Materials, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Also at Laboratory of Solid-State Microstructure, Nanjing University, Nanjing 210008, China (Received September 3, 2001)

In the equilibrium immiscible Y–Nb system, the total energies of the possible structures for Y3 Nb and YNb3 non-equilibrium phases were calculated as a function of their lattice constant(s), under the frame work of the Vienna ab initio simulation package (VASP) and the calculated results predicted the relative stability of the Y3 Nb and YNb3 phases crystallizing in four possible simple structures, i.e. A15, D019 , L12 and L60 structures, respectively. Experimentally, a fcc Y3 Nb non-equilibrium phase was indeed obtained by ion beam mixing and its lattice constant determined by diﬀraction analysis was in agreement with the calculated value. KEYWORDS: Y–Nb system, non-equilibrium phase, phase stability, ab initio DOI: 10.1143/JPSJ.71.141

1.

Introduction

2.

Since early 1980s, a scheme named solid-state reaction (SSR) of multilayers has been introduced to obtain nonequilibrium solid phases with either an amorphous or crystalline structure in the binary metal systems. 1) Meanwhile, another powerful method, i.e. ion beam mixing (IBM) of multilayers, has also been introduced 2) and various nonequilibrium solid phases have been acquired in many systems with diverse characteristics. 3) It is noted that a good deal of non-equilibrium crystalline phases obtained by SSR and/or IBM in some binary metal systems have similar alloy compositions near A3 B, where A and B stand for the two constituent transition metals, respectively and that these phases prone to crystallize in a similar structure (hcp or fcc). Such a new category of A3 B non-equilibrium solid phases of similar crystalline structure raises a challenging issue for theoretical pursuing to reveal the underlying physics of its formation. In this respect, some researchers have studied the stability of the non-equilibrium phases based on thermodynamics of solids. For instance, based on Miedema’s semiquantitative model, Alonso and Gallego attempted to calculate the free energy diagram of a binary metal system for predicting the formation of non-equilibrium phases. 4) To the present authors’ view, to clarify the underlying physics concerning the formation as well as the stability of the nonequilibrium phases at a depth of electronic structure, it is necessary to pursue ﬁrst-principles calculation. We therefore conducted, in the present work, the ﬁrst-principles calculation to predict the possible non-equilibrium states of A3 B type in a representative Y–Nb system, which is essentially equilibrium immiscible and characterized with a positive heat of formation of þ44 kJ/mol. In fact, there is no equilibrium Y–Nb alloy phase over the whole composition range in the Y–Nb system. Besides, the calculation results were compared with those from IBM experiments obtained previously by the authors’ group. 5)

Method of ab initio Calculation

It is known that there are about 20 possible diﬀerent structures for the A3 B or AB3 phases, i.e., A15, Ae , D02 , D03 , D09 , D011 , D018 , D019 , D020 , D021 , D022 , D023 , D024 , D0a , D0b , D0c , D0d , L12 , L1a , and L60 et al. The choosing of the possible structures in our ab initio calculations was based on the following considerations. First, in the above 20 possible structures, some are very complicated, such as the D02 and D021 structures containing more than 16 atoms per unit cell. For the L1a structure, it can also be regarded as a complicated one, as it has 32 atoms per unit cell, 6,7) or 8 atoms per unit cell according to Pearson notation. These phases probably require highly suﬃcient kinetic conditions to nucleate and grow, e.g. requiring relatively high temperature to enhance the atomic mobility and long time to enable the atoms to organize themselves into a speciﬁc atomic conﬁguration. In other words, if such kinetic conditions are not available, these phases are hardly to be formed. 8) Second, the non-equilibrium crystalline phases have so far been obtained by IBM and/or SSR were only of simple structures like hcp and fcc, probably because of the restricted kinetic conditions available in both SSR and IBM processes. 9) Consequently, in our ab initio calculation, the relatively simple A15, D019 , L60 and L12 structures of A3 B and AB3 types were selected in calculation for investigate their relative stability in the equilibrium immiscible Y–Nb system. The calculations are based on the Vienna ab initio simulation package (VASP). 10–12) In the program the exchange and correlation items are described by the specially deﬁned functions proposed by Perdew and Zunger, 13) in which the nonlocal corrections in the form of the generalized-gradient approximation (GGA) of Perdew and Wang 14) are added for improving the computation. The calculations are performed in a plane-wave basis, using fully nonlocal Vanderbilt-type ultrasoft pseudopotentials to describe the electron-ion interaction, 15) which allows the use of a moderate cutoﬀ for the construction of the plane-wave basis for the transition metals. The integration in the Brillouin zone is done on some special k points determined according to the Monckhorst–Pack scheme, 16) and in the

*

Corresponding author. Fax: +86-10-62771160. E-mail: [email protected] 141

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L. T. KONG et al.

present work a mesh of 11 11 11 k-points was adopted. 3.

Results and Discussion

For the possible Y3 Nb phase, the total energies were calculated as a function of the lattice constant(s) for the four selected structures, i.e. D019 , L12 , A15, and L60 , respectively. The c=a ratios of the D019 and L60 structures were optimized. Figure 1 exhibits the four calculated curves showing the correlation of the total energy versus the average atomic volume. One sees from the ﬁgure that the increasing order of the relative stability of the possible Y3 Nb phase is A15, L60 , L12 , and D019 . The A15 structure, which is cubic, has the highest minimum total energy. The L60 structure has a much lower energy than the A15 structure. While the D019 and L12 structures have similar energies and are both lower than the A15 and L60 structures. The D019 structure has a lower minimum total energy than L12 by about 0.03 eV per atom, and the cohesive energy diﬀerence between L12 and D019 structures is therefore less than 0.03 eV/atom. Table I lists the calculated minimum to energies and other properties for the four possible Y3 Nb structures. It can be seen that the diﬀerence of the atomic volume between the D019 and L12 structures is less than 0.4%, reﬂecting the well-known fact that the fcc (L12 ) and hcp (D019 ) structures are very much alike. Because of the similarity of the fcc and hcp structures in their atomic conﬁgurations, it is quite reasonable that the calculated total energies of the fcc and hcp phases are very close. Similar calculations were also conducted for the possible YNb3 phase and the four calculated curves are displayed in Fig. 2, showing the correlation of the total energy versus the average atomic volume. The calculated cohesive properties

Average Fig. 1. The calculated total energy vs average atomic volume for Y3 Nb non-equilibrium phases with diﬀerent structures.

Table I. The calculated cohesive properties (lattice constant a and c=a, atomic volume V, minimum total energy Emin ) of Y3 Nb in the D019 , L12 , A15 and L60 structures. D019 a (A)

3.37

c=a 3 /atom) V (A

1.59

Emin (eV/atom)

L12

A15

L60

4.75

6.01

4.98

26.89

26.79

27.14

26.65

7:09

7:06

6:89

7:02

0.86

Fig. 2. The calculated total energy vs average atomic volume for YNb3 non-equilibrium phases with diﬀerent structures.

Table II. The calculated cohesive properties (lattice constant a and c=a, atomic volume V, minimum total energy Emin ) of YNb3 in the D019 , L12 , A15 and L60 structures. D019 a (A)

6.16

c=a 3 /atom) V (A

1.64

Emin (eV/atom)

L12

A15

L60

4.35

5.44

4.55

23.56

23.39

23.27

23.27

8:71

8:73

8:83

8:77

0.88

for the four structures are listed in Table II. One sees that the increasing order of the relative stability for the possible YNb3 phase is D019 , L12 , L60 , and A15. The four calculated structures of the YNb3 phase could be divided into three groups. The D019 and the L12 structures in the ﬁrst group have a minimum total energy diﬀerence of about 0.02 eV/ atom. The L60 structure has a little lower minimum total energy than the D019 and L12 structures, belonging to the second group. For the third group, the A15 structure has a lower minimum total energy than the L60 structure as well as the lowest minimum total energy among the four calculated structures, suggesting that the A15 structure may be more stable. In comparing the energetic states of the possible structures, two mechanical mixtures at the concentrations of Y3 Nb and YNb3 , respectively, are set as reference states. It turned out that at the concentration of Y75 Nb25 , the total energy of the mixture was 7:35 eV/atom, which is much lower than those of the four calculated structures of Y3 Nb and that the total energy of the mixture at the concentration of Y25 Nb75 was 9:23 eV/atom, which is also lower than that of the A15 structure by about 0.40 eV/atom. Apparently, these calculated results conﬁrm the metastability of the predicted Y3 Nb and YNb3 phases. It is of interest to compare the above calculation results with the experimental results obtained previously by the authors’ group. 5) First, at the Y-rich side, a fcc nonequilibrium phase was indeed obtained by IBM of the Y–Nb multilayers with an overall composition of Y77 Nb23 , which is quite close to Y3 Nb. In experiments, the lattice constant of the fcc phase was determined through diﬀraction analysis to be aexp fcc ¼ 4:87 A. As mentioned above, the calculated lattice cal for the L12 (fcc) structure and the constant is afcc ¼ 4:75 A

J. Phys. Soc. Jpn., Vol. 71, No. 1, January, 2002

diﬀerence between the experimental and calculated values is less than 2.5%. Moreover, the predicted possible fcc YNb3 non-equilibrium phase also appeared in the Nb-rich Y–Nb multilayered ﬁlms upon room temperature IBM, conﬁrming the relevance of the calculation. It should be noted that in experiments, the diﬀraction analysis only conﬁrmed a fcc structure for the Y3 Nb phase and did not give information for lattice occupation for the Y and Nb atoms. Whereas in the present calculation, it is assumed an exactly ordered Y3 Nb fcc structure, as required by the VASP code. Nonetheless, the IBM results did conﬁrm the existence of such a non-equilibrium state located near a stoichiometry of Y3 Nb with a structure of fcc-type in the equilibrium immiscible Y–Nb system. One may question that at the Nb-rich side, the calculated minimum total energies of the A15 and L60 structures are lower than those of the L12 and D019 , yet they did not appear in the previous IBM experiments, instead, some fcc nonequilibrium phases were obtained. From a structural point of view, as the A15 and L60 structures are relatively complicated in comparison to those of L12 and D019 , they probably require highly suﬃcient kinetic condition to nucleate and grow. While in the process of IBM, the kinetic condition available for growing a solid crystalline phase is very restricted during the relaxation period immediately after atomic collision triggered by ion irradiation and therefore allows forming only those simple structured solid phases. 9) Consequently, it is the kinetic constraint that prevents the A15 and L60 structures from growing in the IBM process. Another interesting question is that why did the D019 structure, on the Y-rich side, not show out so far in IBM experiments, though its structure is a simple one having a lower energy than L12 . This issue certainly deserves further investigation to see if the D019 structure could be obtained by some other non-equilibrium processing methods.

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Conclusion

In summary, employing the Vienna ab initio simulation package (VASP), ab initio calculation results are quite relevant in predicting the existence of some non-equilibrium states in the equilibrium immiscible Y–Nb system, and in experiments, some of the corresponding solid phases, such as a fcc Y3 Nb was indeed obtained by ion beam mixing, which is known as a powerful means for synthesizing new and non-equilibrium materials. Acknowledgement The authors gratefully acknowledge the ﬁnancial aid from the National Natural Science Foundation of China, the Ministry of Science and Technology of China through a Grant No. G2000067207 and the Administration of Tsinghua University. 1) R. B. Schwarz and W. L. Johnson: Phys. Rev. Lett. 51 (1983) 415. 2) B. X. Liu, W. L. Johnson, M. A. Nicolet and S. S. Lau: Appl. Phys. Lett. 42 (1983) 45. 3) B. X. Liu and O. Jin: Phys. Status Solidi A 161 (1997) 3. 4) F. R. de Boer, R. Boom, W. C. M. Mattens, A. R. Miedema and A. K. Niessen: Cohesion in Metals: Transition Metal Alloys (North-Holland, Amsterdam, 1989). 5) Y. G. Chen and B. X. Liu: Acta Mater. 47 (1999) 1389. 6) Y. C. Tang: J. Appl. Crystallogr. 4 (1951) 377. 7) A. Schneider and U. Esch: Z. Elektrochem. 50 (1944) 290. 8) B. X. Liu, L. J. Huang, K. Tao, C. H. Shang and H. D. Li: Phys. Rev. Lett. 59 (1987) 745. 9) B. X. Liu and O. Jin: Phys. Status Solidi A 116 (1997) 3. 10) G. Kress and J. Hafner: Phys. Rev. B 47 (1993) 558. 11) G. Kress and J. Furthmuller: Comput. Mater. Sci. 6 (1996) 15. 12) G. Kress and J. Furthmuller: Phys. Rev. B 54 (1996) 11169. 13) J. Perdew and A. Zunger: Phys. Rev. B 23 (1981) 5048. 14) J. Perdew and Y. Wang: Phys. Rev. B 45 (1992) 13244. 15) D. Vanderbilt: Phys. Rev. B 41 (1990) 7892. 16) H. J. Monkhorst and J. D. Pack: Phys. Rev. B 13 (1976) 5188.