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Spin polarization gives rise to Cu precipitation in Fe-matrix Wen-xiong Song and Shi-jin Zhao* This article tries to uncover the physical reason of Cu precipitation from an Fe matrix at the electronic level. The general rule is obtained that the more bonds among Cu atoms, the more stable the system is. It was shown that Cu would precipitate from the matrix with Fe spin-polarization but not without spinpolarization. The partial density of states (PDOS) analysis illustrated that the d states of Fe near the Fermi level potentially have strong interaction with other atoms, but Cu d states below the Fermi level lack this potential, which results in weak covalent d orbital interaction between Fe and Cu. Furthermore, the
Received 18th December 2013, Accepted 22nd January 2014
charge density difference also confirmed the weaker bond between Fe and Cu with spin-polarization compared to without spin-polarization, due to the decreased charge between them. In addition, the
DOI: 10.1039/c3cp55319a
{110} interface energy between Fe and Cu, estimated by the ‘‘dangling bond’’, is 676.3 mJ m2, which
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agrees with the DFT calculation, 414.2 mJ m2. Finally, this study also revealed that Ni atoms can reduce the ‘‘dangling bond’’ when it locates at the interface and separates Fe and Cu.
1. Introduction Copper precipitation plays an important role in the hardening of high-strength low-carbon steels1–4 and results in the radiation-embrittlement of operating reactor pressure vessels (RPV) steel.5–7 It is assumed to act as obstacles to dislocation motion. Cu precipitation is studied experimentally1–3,5–15 with many characterization techniques, such as 3D atom probes (3D-AP),12,16,17 high-resolution transmission electron microscopy (HRTEM),18 small-angle neutron scattering (SANS)19 and positron annihilation techniques.9,20–22 Many simulation methods are also used to research Cu precipitation, mainly using first principles23–32 at an electronic level, kinetic Monte Carlo (KMC)33–36 and molecular dynamics (MD)3,37 at the atomic level, and also others.38–40 The mechanism of impurity precipitation has been studied intensively. Isheim and Fine et al.1 studied the local composition of small and coherent Cu-rich precipitates in an a-Fe matrix using 3D-AP. They observed that the precipitates with an average diameter of 2.5 0.3 nm, contain 33 1 at% Fe and were enriched in Al (0.5 0.1 at%); meanwhile, nickel and Mn are significantly segregated at the a-Fe matrix/precipitate heterophase interfaces. Miller et al.5 utilized the 3D-AP technique to conduct the microstructural characterization of pressure vessel steels and understand the embrittlement of these materials during neutron irradiation, finding that there is a complex pattern of segregation of various solutes (including P, Ni, Mn, or Mo) to the grain Institute of Materials Science, Shanghai University, Shanghai 200072, China. E-mail: [email protected]; Fax: +86-21-56331480; Tel: +86-21-56331480
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boundaries. Seko et al.31,32 calculated the Cu clusters’ free energies using the DFT method, in which the critical nucleation number and estimated activation energy are 13 atoms and 0.67 eV, respectively. Medvedeva4 showed that cobalt is rejected from the core of the Cu particle and segregated in the interfacial region, increasing the cleavage fracture energies and cleavage stress of the Fe/Co/Cu interface. Reith et al.25,26 found that including the vibrational free energy via the vibrational entropy is very important for explaining the rather wide miscibility range of mixing Cu to bcc Fe; and the vibrational free energy is absolutely vital for the correct first-principles prediction of Cu solubility in the bcc Fe matrix. Choi et al.29 studied the effect of the nucleated Cu phase on magnetic properties and electronic structures in bcc Fe. Shim et al.3 presented a study on the strengthening effect of nanosized Cu precipitates in bcc Fe by MD simulations. Vincent and Becquart et al.35 wrote a critical review of AKMC simulations about the precipitation in the FeCu system. Recently, the method was also used to study the stability and mobility of copper-vacancy clusters in Fe.34 In addition, many researchers38,41,42 explained the Cu nucleation pathway of a coherent precipitate, some deeming that Cu precipitation is pure while others believing that nuclei has a lower Cu content than pure Cu. To summarise, many experiments and simulations12,16,17,20–22,43 focused on the process of Cu precipitation from the Fe matrix; what’s more, calculations3,27,28 were carried out to construct the energy criterion of Cu precipitating from the Fe matrix. There are few articles that attempt to discover the physical essence of Cu precipitation by analyzing the orbital interaction between Fe and Cu at an electronic level. However, this paper may provide the reason why Cu precipitates from the Fe matrix
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at this level, which is a further explanation for the energy criterion. The magnetism or spin polarization effect on Cu precipitation, which is thought to be the root cause, is systematically discussed here. There are already some similar studies in which carbon segregates in nickel44 and also hydrogen in nickel,45,46 a phenomenon called ‘‘magnetoexpulsion’’. Therefore, one important purpose of this paper is to systematically research the spin polarization which results in Cu precipitation in an Fe matrix, focused at the electronic level. This study is very promising in a wide range of applications, as the precipitation phase is a key determinant of a material’s properties. In addition, the analytical method can be also suited for other non-magnetic elements precipitating from the magnetism matrix. Spin polarization is the natural property of a-Fe. Many theories have already been proposed to predict or explain magnetism and ferromagnetism in physics and chemistry. Among them, the Stoner mechanism47,48 is popularly admitted and the orbital origin of magnetism49,50 mechanism is also useful to explain it. In the Stoner mechanism, accepted by most solid-state physicists, the d states of the elements Fe and Ni are split into the full majority a-spin and the partially filled minority b-spin. Concisely speaking, the Stoner criterion is IDOS(eF) 4 1 where I is a measure of the strength of the exchange interaction in the metal and DOS(eF) is the density of states at the Fermi level eF. A decade ago, Landrum and Dronskowski, as a chemical view, used the concepts of bonding and electronic shielding to account for the orbital origins of magnetism. In this theory, the critical valence electron concentrations in Fe and Ni result in local electronic instabilities due to the population of the antibonding states at the Fermi level eF. Removal of these antibonding states from the vicinity of eF is the origin of ferromagnetism, causing a distortion of electrons. It results an inequivalence of the a and b spin sub-lattices and leads to energy level shifts, repopulation of the spin sublattices, and a redistribution of charge density. In summary, it is a fact that the d states of the element Fe will split, but Cu will be not, which results in Cu precipitation. In the next section of this contribution, the computational details are shown. In the third section, the binding energy was calculated with or without spin polarization; the partial density of states (PDOS) was shown to describe the orbital interaction; spin density and charge density differences were used to illustrate the charge redistribution after spin polarization and to try to capture the bonding character between the impurity and matrix; the effect of Ni on Cu precipitation was also discussed here. Finally, conclusions were drawn.
2. Methodology 2.1
QM method
We carried out non-spin polarization and spin polarization calculations. The Kohn–Sham equations were solved using the Vienna Ab Initio Simulation package (VASP).51–54 The valence electron and core interactions were described using the projector augmented wave (PAW) method.55 Electron exchange and
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correlation were described using the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) functional with a kinetic energy cutoff of 360 eV and a plane wave basis.56 The relaxations were assumed to have converged when the energy in two consecutive ionic relaxation steps differed by less than 100 meV. Integrations in reciprocal space were made according to the Monkhorst–Pack scheme, and a reasonably converged grid spacing and numbers of k-points were used, such as a 4 4 4 super-cell of 64 unit-cells with 128 atoms (MP 5 5 5) and a 3 3 3 super-cell of 27 unit-cells with 54 atoms (MP 7 7 7). Quasi-Newton (variable metric) and conjugate-gradient algorithms were used to relax the ions into their instantaneous ground state. In order to express their bonding character, the crystal orbital Hamilton population (COHP)57 was calculated by the SIESTA package58 with numerical atomic orbital basis sets59 and Troullier–Martins norm-conserving pseudopotentials.60 The exchange–correlation functional utilized is GGA-PBE61 and the double-z plus polarization (DZP) basis set was employed. The cutoff for the realspace grid is set as 400 Ry and the tolerance in maximum difference between the input and output density matrix is 0.0001. The method of COHP is in many ways analogous to crystal orbital overlap population (COOP).62 The difference is that COOP is a Mulliken overlap population weighted density-of-states, while COHP instead is a Hamiltonian population weighted density of state. By doing so, the band structure energy is partitioned into bonding, nonbonding and antibonding contributions. The bonding state for the COHP has a negative sign (compared to COOP’s positive sign), due to its unit in energy. Energy, PDOS, spin density and charge density difference were calculated by the VASP package, while PDOS and COHP were calculated by SIESTA. We also calculated the binding energy with SIESTA to verify the accuracy of the results, and we obtained similar results to VASP; both predicted Cu precipitation from the Fe matrix.
3. Results and discussion In order to study the effect of spin-polarization on Cu precipitation, different calculations were carried out. Firstly, it was illustrated that the Cu solute tends to gather in the matrix under the effect of spin-polarization, as demonstrated by calculation of the solute–solute binding energy, but scatter without spin-polarization. Then, for the sake of a systematic study of the effect of spinpolarization on binding energy, we calculated sequentially many electronic properties, including the PDOS, COHP, spin density and charge density difference. Finally, we also discussed the effect of Ni on Cu precipitation. 3.1
Binding energy
Binding energies were calculated to illustrate the solubility of Cu impurities in the Fe matrix at dilute concentration, defined as the following (formula (1)), and also can be found elsewhere,35 E b(n) XY (A) = E [(N 1) A + 1 X] + E [(N 1) A + 1Y ] E [NA] E [(N 2) A + 1 X + 1Y ]
(1)
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Table 1
4 3 3 3 3
4 3 3 3 3
Binding energies relative to bcc-Fe calculated by VASP or SIESTA packages
4a 3a 3b 3c 3d
E b(1) CuCu (Fe)/eV
E b(2) CuCu (Fe)/eV
E b(1) (Fe)/eV NiNi
E b(2) (Fe)/eV NiNi
E b(1) CuNi (Fe)/eV
E b(2) CuNi (Fe)/eV
0.22 0.21 0.08 0.11 0.03
0.04 0.04 0.03 0.04 0.03
0.02 0.00 0.03 0.11 0.07
0.00 0.00 0.00 0.02 0.01
0.13 0.14 0.05 0.01 0.01
0.03 0.04 0.01 0.02 0.02
a Calculated by VASP package with spin-polarization. b Calculated by VASP package without both spin-polarization and relaxation. c Calculated by SIESTA package with spin-polarization and without relaxation. d Calculated by SIESTA package without both spin-polarization and relaxation.
where E represents the energy of the related system, [(N 1)A + 1X] represents the system that the A matrix with N atoms is replaced by one X atom, [(N 2)A + 1X + 1Y] represents that the A matrix with N atoms is replaced by one X atom and one Y atom and two of them with the nth nearest-neighbor. The binding energy describes whether it is easy or difficult for the two-atom cluster to dissolve in the matrix (A) with isolated status. A positive value means that the cluster will be stable without considering entropy and temperature; negative or around zero value represents the opposite result. Before these calculations, equilibrium lattice constants were optimized. The lattice constant of bcc Fe optimized by VASP and SIESTA was 2.834 Å or 2.98 Å, respectively, close to the experimental value of 2.87 Å.63 Bcc Cu and Ni are considered as impurities to replace Fe atoms. Table 1 shows the binding energy calculated by the VASP or SIESTA packages with difference spin-polarization and relaxation conditions. In order to test the size effect, 3 3 3 and 4 4 4 super-cells were calculated initially, and the results indicated that the smaller one was large enough. Then, all of the calculations were carried out later with a b(2) 3 3 3 super-cell. The binding energies E b(1) CuCu (Fe) and E CuCu (Fe) are positive with spin-polarization by both the VASP and SIESTA packages, although with a small difference. So, it can be demonstrated that Cu atoms tend to gather under spinpolarization. In contrast, values of the binding energy E b(1) (Fe) NiNi and E b(2) (Fe) are negative or around zero, no matter what spinNiNi polarization condition and package was used. Besides, the b(2) binding energy E b(1) CuNi (Fe) and E CuNi (Fe) are closer to zero, b(1) b(2) compared with E CuCu (Fe) and E CuCu (Fe), respectively, indicating that the Cu-rich cluster is more stable when it contains Ni atoms at the interface between Fe and Cu. 3.2
PDOS and COHP
PDOS and COHP curves were drawn to explain the opposite values of binding energy with different spin-polarization conditions, as shown in Fig. 1–3. PDOS curves will be used to illustrate the orbital interaction and COHP curves will be used to characterize the system stability.49,50 All the calculations were carried out by SIESTA and VASP packages without structure relaxation, containing two first nearest neighbor (1nn) impurity atoms (Fe52Cu2, Fe52Ni2, Fe52CuNi). Fig. 1 mainly exhibits PDOS of Cu, Fe (matrix) and Fe (nearest neighbor to Cu), as well as COHP of Fe–Fe, Fe–Cu and Cu–Cu bonds in the Fe52Cu2-1nn system considering the spin-polarization (Fig. 1(a) and (b)), or not (Fig. 1(c) and (d)). In the COHP curve of Fig. 1(d), it is notable that the Fermi level
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eF passes through the antibonding area of all the bonds, indicating the system is unstable; besides, Cu Fermi level lies near the end of the antibonding section and there is little tendency for spin polarization because of the almost completely filled d orbits. However, in order to lessen the antibonding area the d states of Fe shifts in energy, with the a spin sub-lattice decreasing to fill completely the d block; meanwhile, the b spin increases to maintain a constant number of electrons in the system as a whole, as shown in Fig. 1(a) and (b). Besides, when analyzed by the Stoner mechanism, the IDOS(eF) of Fe is larger than Cu, assuming that both of the I values are the same, indicating that the PDOS of Fe will be split and result in magnetism, but Cu will be not split thus with no magnetism. From the above, both the chemical bonding approach and the Stoner mechanism illustrate that the d states of Fe will be split but Cu will not be. Similar to the frontier molecular orbital theory, the states near Fermi level are important to orbital interaction in that the occupied state is analogous to the highest occupied molecular orbital (HOMO) and the unoccupied state is analogous to the lowest unoccupied molecular orbital (LUMO). Then, one atom’s unoccupied states interact with other atom’s occupied states, and a bond is formed between the two atoms. According to this view, the interaction of the d states between Fe and Cu is stronger in Fig. 1(b) than in Fig. 1(a), because the Cu d states are too far below the Fermi level after spin polarization. So, the weak d-state interaction of Cu may be the essential reason for Cu precipitation. It has been confirmed64 that the width of the DOS curve is directly proportional to the strength of the interatomic interactions in a crystal; and the smaller the curve’s slope, the stronger the interaction. Therefore, the PDOS curves of Cu in the Fe52Cu2-1nn system (Fig. 1) show that the orbital interaction without spin-polarization between Cu and Fe is stronger, due to the wider energy interval ([12, 7] eV) and a smaller curve slope near the top of the d state in Fig. 1(c), than that under spin-polarization (a-spin [12.5, 8.5] eV, b-spin [12, 9] eV in Fig. 1(a)). In Fig. 1, the shapes of the PDOS calculated by the two packages are similar and the conclusions obtained are the same, although VASP calculated them more finely. The reasons for this phenomenon can be concluded as follows: (1) the hybridization of a-spin d states between Fe and Cu becomes weaker after spin polarization, mainly because both of their a-spin d states are almost completely filled, although they largely overlap with each other; (2) another reason for the weaker hybridization of b-spin d states between
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Fig. 1 PDOS of Fe and Cu atoms and COHP of Fe–Fe, Fe–Cu and Cu–Cu bonds under spin-polarization (a), (b) and (e), or not (c), (d) and (f) in Fe52Cu2-1nn system; (a), (b), (c) and (d) are calculated by SIESTA, while (e) and (f) are calculated by VASP package.
Fe and Cu is that the b-spin of the Cu d states is almost fully filled and there is a large energy interval gap between them after spin polarization, even resulting in a narrower energy interval of Cu b-spin d states compared to a-spin states. All the reasons mentioned above can be used to explain why the positive or negative E b(1) CuCu (Fe) values occur under different spin-polarization conditions, and the positive value illustrates Cu precipitation. So, the bond between Fe and Cu can be considered as a ‘‘dangling bond’’. In addition, the related PDOS and COHP curves were also drawn in the systems of Fe52Ni2-1nn (Fig. 2) and Fe52CuNi-1nn (Fig. 3) under spin-polarization conditions. In the Fe52Ni2-1nn system, there is a little shift between the a-spin and b-spin in the PDOS of Ni, because Ni has 8 electrons filled the d state, 2 electrons more than the d state of Fe. In spite of the large a-spin overlap between Ni and Fe, the interaction between them is still weak because of their fully filled d states. However, the b-spin orbital interaction between Fe and Ni is strong since the Fermi level passes through both of their b-spin d states. Like the previous Fe52Cu2-1nn and Fe52Ni2-1nn systems, Fig. 3 illustrates that the a-spin d states of Fe, Cu, and Ni can hybridize with each other. On the other side, the difference of b-spin energy interval between Fe and Cu is larger than that between Cu and Ni, resulting in a stronger interaction between Cu and Ni than that between Fe and Cu. Then, Ni can reduce the ‘‘dangling bond’’
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between Fe and Cu when it locates at the ‘‘interface’’ between them. 3.3
Spin density and charge density difference
In order to understand further the process of the d state shift, spin density, calculated by quantity r(spin) = r(up) r(down), is shown in Fig. 4. It is found that spin density gradually decreases from the nucleus to outside. Additionally, spin density diminishes successively for Fe, Ni and Cu, which agrees with their a- and b-spin energy shifting under spin-polarization. As a result, spin polarization leads to the rearrangement of electron density. The larger the spin density, the greater the rearrangement. The charge density difference, plotted by the quantity Dr = r(magnetic) r(nonmagnetic), is given in Fig. 5. It can be seen that Fe shows an obvious rearrangement of the electron density, an increase in the dz2 and dx2y2 orbitals and a decrease in other regions (Fig. 5(b)); while Ni shows a small rearrangement of the electron density, a decrease in dz2 and dx2y2 orbitals and an increase in other regions (Fig. 5(c)). Further analysis of the electron number of elemental Fe in d orbitals reveals that the electron number in dxy, dyz and dxy orbitals is the same and it is larger than that in dz2 and dx2y2 orbitals. So, bcc-Fe shows octahedral crystal field splitting, indicating that the charge effect of the next nearest neighbor atoms is the key to field splitting; however, Ni shows tetrahedral
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Fig. 2 The PDOS of Fe and Ni atoms, COHP of Fe–Fe, FeNi and NiNi bonds under spin-polarization, Fe52Ni2-1nn system; (a) and (b) are calculated by SIESTA, while (c) and (d) are calculated by VASP package.
crystal field splitting due to the opposite d orbital energy order to Fe, indicating that the charge effect of the nearest neighbor atoms is the key to field splitting. Besides, the electron number tends to be uniform after spin polarization. Therefore, the electron number increases in Fe dz2 orbitals after spinpolarization while it decreases in Ni dz2 orbitals, as shown in Fig. 5(b) and (c). After the Fe atoms are replaced by Cu and Ni impurities, electron transfer also occurs among them to different degrees. Bader charge analysis65 describes the charge movement such that Cu and Ni obtain 0.31 and 0.36 electron per atom, respectively. This is in large part due to the fact that the Fe d states show larger exchange splitting compared to Ni.49 Cu has an almost unfilled 4p orbital to obtain electrons from Fe. Thus, Ni has an obvious charge density difference in Fig. 5(a) compared with Fig. 5(c). The region with a positive charge density difference compared Ni is the closest to Fe, indicating strong
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Ni bonding with Fe. The region between Ni and Cu is also slightly positive, indicating partial bonding between them, due to the Cu d states being almost completely filled. We find that the region between Fe and Cu is more negative than the region between Fe atoms, indicating the weakest bonding between them. The above-mentioned illustrates that the electron transfer among different orbitals can be used to describe their properties of bonding, although the d-state shift is the key factor affecting the strength of d orbital interaction between Cu and other atoms. 3.4
Effect of Ni
From the above analysis, it can be shown that Ni has an important effect on Cu precipitation. Because the Fe–Cu bond is unstable due to the weak b-spin orbital interaction, that the bond between Fe and Cu is considered as a similar ‘‘dangling bond’’, Fe–Cu bonds tend to break and form more Fe–Fe and
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Fig. 3 The PDOS of Fe, Cu and Ni atoms, COHP of FeFe, FeCu, FeNi and CuNi bonds under spin-polarization, Fe52CuNi-1nn system; (a) and (b) are calculated by SIESTA, while (c) and (d) are calculated by VASP package.
Cu–Cu bonds. The energy difference of those bonds can be considered as an ‘‘interface energy’’, as shown in formula (2). When many Fe–Cu bonds arrange in a plane, the interface is formed between Fe and Cu atoms and the interface energy can be simply calculated based on formula (2) by counting Fe–Cu
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bonds. In other words, Cu precipitation from the Fe-matrix is also caused by the ‘‘interface energy’’ of Fe–Cu bonds. 1 sFeCu ¼ eFeCu ðeFeFe þ eCuCu Þ 2
(2)
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Fig. 4 Spin density of the (1% 10) plane sketched in Fig. 5(e).
where e represents the bond energy. So, the interface energy s is approximate to the binding energy shown in Table 1. In the Fe–Cu (110) interface, each Cu atom has two first- and two secondnearest neighbor Fe atoms. Here, the bond energy difference is 0.24 eV while the related area is 5.678 A2. Thus, the (110) interface
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energy of Fe–Cu is 676.3 mJ m2, which is similar to that from DFT calculation, 414.2 mJ m2 obtained from VASP. Above all, regarding the effect of Ni on Cu precipitating from the Fe-matrix, it can be concluded that Ni can enhance the Cu nucleation density but reduce the growth velocity of Cu-rich clusters. b-Spin energy intervals of both Cu and Fe can overlap with Ni, so both of them can hybridize with the b-spin d states of Ni. Then, the Cu–Ni and Fe–Ni bonds are stable in the Fe-matrix, as shown in Table 1. The interface energies of sFe–Ni and sCu–Ni are both negative, illustrating that they are liable to mix, and not to precipitate to form an interface between Fe and Ni or Cu and Ni. When Cu precipitates from the Fe matrix, Ni can be regarded as a ‘‘protective shell’’ to avoid the contact of Cu with Fe. This is why a Cu core with an Ni shell structure was observed in related experiments.1,11 Simulation43 reveals that the addition of Ni notably increases Cu nucleation density and reduces the size of the clusters. It is easy to understand that Ni can enhance Cu nucleation because Ni prefers contact with Cu to reduce the interface energy (or the ‘‘dangling bond’’) and the critical nucleation number.43 Because E b(1) (Fe) is not a posiNiNi tive value, there is no driving force to make Ni gather, which reduces the chance of two Ni-shell Cu clusters forming a bigger cluster when two ‘‘protective shell’’ Cu clusters get close.
4. Conclusions We confirmed that orbital interaction is the root-cause of Cu precipitation. From the above analysis, the d state interaction between Fe and Cu is weak when considering spin-polarization,
Fig. 5 Charge density difference upon spin polarization, red color indicates a positive value and blue color indicates a negative value: (a) charge density difference of the (1% 10) plane sketched in (e); (b) charge density difference of Fe (1% 10) plane sketched in (d); (c) charge density difference of Ni (1% 10) plane sketched in (d); (d) body-centered cubic (bcc) unit cell where the slope is the (1% 10) plane; (e) 3 3 3 super-cell of Fe element replaced by Cu and Ni impurities as the nearest neighbor, the gray atom is Ni, blue is Cu and brown is Fe, and the slope is the (1% 10) plane.
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because the Cu d states are too far below the Fermi level. On the other hand, the d states near the Fermi level of the Fe atom have strong interaction with other Fe atoms. Therefore, the bond between Fe and Cu is similar to a ‘‘dangling bond’’, which mainly induces Cu precipitation. The charge density difference also illustrates the weaker bond between Fe and Cu after spinpolarization because of the decreased charge between them. The ‘‘dangling bond’’ is also the origin of interface energy between Fe and Cu. The {110} interface energy calculated by the ‘‘dangling bond’’ concept is 676.3 mJ m2, which agrees with the DFT calculation 414.2 mJ m2 obtained by the VASP package. In truth, Ni can reduce the ‘‘dangling bond’’ between Fe and Cu, which finally enhances Cu nucleation density but reduces the growth velocity of Cu-rich clusters. Finally, there is a general rule that the more bonds are formed among Cu atoms and the more Ni atoms locate at the interface between Fe and Cu, the more stable is the system in enthalpy.
Acknowledgements The authors thank Dr Li-Ming Liu, San-Yi Ma, Yun Xue, Li-Chun Xu, Yao-Ping Xie and Yu-Bin Si for great help. This work is financially supported by National Natural Science Foundation of China (Grant Nos. 50931003, 51301102), and 085 project at Shanghai University. High performance computing resources are provided by the Ziqiang Supercomputer Center at Shanghai University.
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Spin polarization gives rise to Cu precipitation in Fe-matrix Wen-xiong Song and Shi-jin Zhao* This article tries to uncover the physical reason of Cu precipitation from an Fe matrix at the electronic level. The general rule is obtained that the more bonds among Cu atoms, the more stable the system is. It was shown that Cu would precipitate from the matrix with Fe spin-polarization but not without spinpolarization. The partial density of states (PDOS) analysis illustrated that the d states of Fe near the Fermi level potentially have strong interaction with other atoms, but Cu d states below the Fermi level lack this potential, which results in weak covalent d orbital interaction between Fe and Cu. Furthermore, the
Received 18th December 2013, Accepted 22nd January 2014
charge density difference also confirmed the weaker bond between Fe and Cu with spin-polarization compared to without spin-polarization, due to the decreased charge between them. In addition, the
DOI: 10.1039/c3cp55319a
{110} interface energy between Fe and Cu, estimated by the ‘‘dangling bond’’, is 676.3 mJ m2, which
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agrees with the DFT calculation, 414.2 mJ m2. Finally, this study also revealed that Ni atoms can reduce the ‘‘dangling bond’’ when it locates at the interface and separates Fe and Cu.
1. Introduction Copper precipitation plays an important role in the hardening of high-strength low-carbon steels1–4 and results in the radiation-embrittlement of operating reactor pressure vessels (RPV) steel.5–7 It is assumed to act as obstacles to dislocation motion. Cu precipitation is studied experimentally1–3,5–15 with many characterization techniques, such as 3D atom probes (3D-AP),12,16,17 high-resolution transmission electron microscopy (HRTEM),18 small-angle neutron scattering (SANS)19 and positron annihilation techniques.9,20–22 Many simulation methods are also used to research Cu precipitation, mainly using first principles23–32 at an electronic level, kinetic Monte Carlo (KMC)33–36 and molecular dynamics (MD)3,37 at the atomic level, and also others.38–40 The mechanism of impurity precipitation has been studied intensively. Isheim and Fine et al.1 studied the local composition of small and coherent Cu-rich precipitates in an a-Fe matrix using 3D-AP. They observed that the precipitates with an average diameter of 2.5 0.3 nm, contain 33 1 at% Fe and were enriched in Al (0.5 0.1 at%); meanwhile, nickel and Mn are significantly segregated at the a-Fe matrix/precipitate heterophase interfaces. Miller et al.5 utilized the 3D-AP technique to conduct the microstructural characterization of pressure vessel steels and understand the embrittlement of these materials during neutron irradiation, finding that there is a complex pattern of segregation of various solutes (including P, Ni, Mn, or Mo) to the grain Institute of Materials Science, Shanghai University, Shanghai 200072, China. E-mail: [email protected]; Fax: +86-21-56331480; Tel: +86-21-56331480
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boundaries. Seko et al.31,32 calculated the Cu clusters’ free energies using the DFT method, in which the critical nucleation number and estimated activation energy are 13 atoms and 0.67 eV, respectively. Medvedeva4 showed that cobalt is rejected from the core of the Cu particle and segregated in the interfacial region, increasing the cleavage fracture energies and cleavage stress of the Fe/Co/Cu interface. Reith et al.25,26 found that including the vibrational free energy via the vibrational entropy is very important for explaining the rather wide miscibility range of mixing Cu to bcc Fe; and the vibrational free energy is absolutely vital for the correct first-principles prediction of Cu solubility in the bcc Fe matrix. Choi et al.29 studied the effect of the nucleated Cu phase on magnetic properties and electronic structures in bcc Fe. Shim et al.3 presented a study on the strengthening effect of nanosized Cu precipitates in bcc Fe by MD simulations. Vincent and Becquart et al.35 wrote a critical review of AKMC simulations about the precipitation in the FeCu system. Recently, the method was also used to study the stability and mobility of copper-vacancy clusters in Fe.34 In addition, many researchers38,41,42 explained the Cu nucleation pathway of a coherent precipitate, some deeming that Cu precipitation is pure while others believing that nuclei has a lower Cu content than pure Cu. To summarise, many experiments and simulations12,16,17,20–22,43 focused on the process of Cu precipitation from the Fe matrix; what’s more, calculations3,27,28 were carried out to construct the energy criterion of Cu precipitating from the Fe matrix. There are few articles that attempt to discover the physical essence of Cu precipitation by analyzing the orbital interaction between Fe and Cu at an electronic level. However, this paper may provide the reason why Cu precipitates from the Fe matrix
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at this level, which is a further explanation for the energy criterion. The magnetism or spin polarization effect on Cu precipitation, which is thought to be the root cause, is systematically discussed here. There are already some similar studies in which carbon segregates in nickel44 and also hydrogen in nickel,45,46 a phenomenon called ‘‘magnetoexpulsion’’. Therefore, one important purpose of this paper is to systematically research the spin polarization which results in Cu precipitation in an Fe matrix, focused at the electronic level. This study is very promising in a wide range of applications, as the precipitation phase is a key determinant of a material’s properties. In addition, the analytical method can be also suited for other non-magnetic elements precipitating from the magnetism matrix. Spin polarization is the natural property of a-Fe. Many theories have already been proposed to predict or explain magnetism and ferromagnetism in physics and chemistry. Among them, the Stoner mechanism47,48 is popularly admitted and the orbital origin of magnetism49,50 mechanism is also useful to explain it. In the Stoner mechanism, accepted by most solid-state physicists, the d states of the elements Fe and Ni are split into the full majority a-spin and the partially filled minority b-spin. Concisely speaking, the Stoner criterion is IDOS(eF) 4 1 where I is a measure of the strength of the exchange interaction in the metal and DOS(eF) is the density of states at the Fermi level eF. A decade ago, Landrum and Dronskowski, as a chemical view, used the concepts of bonding and electronic shielding to account for the orbital origins of magnetism. In this theory, the critical valence electron concentrations in Fe and Ni result in local electronic instabilities due to the population of the antibonding states at the Fermi level eF. Removal of these antibonding states from the vicinity of eF is the origin of ferromagnetism, causing a distortion of electrons. It results an inequivalence of the a and b spin sub-lattices and leads to energy level shifts, repopulation of the spin sublattices, and a redistribution of charge density. In summary, it is a fact that the d states of the element Fe will split, but Cu will be not, which results in Cu precipitation. In the next section of this contribution, the computational details are shown. In the third section, the binding energy was calculated with or without spin polarization; the partial density of states (PDOS) was shown to describe the orbital interaction; spin density and charge density differences were used to illustrate the charge redistribution after spin polarization and to try to capture the bonding character between the impurity and matrix; the effect of Ni on Cu precipitation was also discussed here. Finally, conclusions were drawn.
2. Methodology 2.1
QM method
We carried out non-spin polarization and spin polarization calculations. The Kohn–Sham equations were solved using the Vienna Ab Initio Simulation package (VASP).51–54 The valence electron and core interactions were described using the projector augmented wave (PAW) method.55 Electron exchange and
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correlation were described using the Perdew–Burke–Ernzerhof (PBE) generalized gradient approximation (GGA) functional with a kinetic energy cutoff of 360 eV and a plane wave basis.56 The relaxations were assumed to have converged when the energy in two consecutive ionic relaxation steps differed by less than 100 meV. Integrations in reciprocal space were made according to the Monkhorst–Pack scheme, and a reasonably converged grid spacing and numbers of k-points were used, such as a 4 4 4 super-cell of 64 unit-cells with 128 atoms (MP 5 5 5) and a 3 3 3 super-cell of 27 unit-cells with 54 atoms (MP 7 7 7). Quasi-Newton (variable metric) and conjugate-gradient algorithms were used to relax the ions into their instantaneous ground state. In order to express their bonding character, the crystal orbital Hamilton population (COHP)57 was calculated by the SIESTA package58 with numerical atomic orbital basis sets59 and Troullier–Martins norm-conserving pseudopotentials.60 The exchange–correlation functional utilized is GGA-PBE61 and the double-z plus polarization (DZP) basis set was employed. The cutoff for the realspace grid is set as 400 Ry and the tolerance in maximum difference between the input and output density matrix is 0.0001. The method of COHP is in many ways analogous to crystal orbital overlap population (COOP).62 The difference is that COOP is a Mulliken overlap population weighted density-of-states, while COHP instead is a Hamiltonian population weighted density of state. By doing so, the band structure energy is partitioned into bonding, nonbonding and antibonding contributions. The bonding state for the COHP has a negative sign (compared to COOP’s positive sign), due to its unit in energy. Energy, PDOS, spin density and charge density difference were calculated by the VASP package, while PDOS and COHP were calculated by SIESTA. We also calculated the binding energy with SIESTA to verify the accuracy of the results, and we obtained similar results to VASP; both predicted Cu precipitation from the Fe matrix.
3. Results and discussion In order to study the effect of spin-polarization on Cu precipitation, different calculations were carried out. Firstly, it was illustrated that the Cu solute tends to gather in the matrix under the effect of spin-polarization, as demonstrated by calculation of the solute–solute binding energy, but scatter without spin-polarization. Then, for the sake of a systematic study of the effect of spinpolarization on binding energy, we calculated sequentially many electronic properties, including the PDOS, COHP, spin density and charge density difference. Finally, we also discussed the effect of Ni on Cu precipitation. 3.1
Binding energy
Binding energies were calculated to illustrate the solubility of Cu impurities in the Fe matrix at dilute concentration, defined as the following (formula (1)), and also can be found elsewhere,35 E b(n) XY (A) = E [(N 1) A + 1 X] + E [(N 1) A + 1Y ] E [NA] E [(N 2) A + 1 X + 1Y ]
(1)
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Table 1
4 3 3 3 3
4 3 3 3 3
Binding energies relative to bcc-Fe calculated by VASP or SIESTA packages
4a 3a 3b 3c 3d
E b(1) CuCu (Fe)/eV
E b(2) CuCu (Fe)/eV
E b(1) (Fe)/eV NiNi
E b(2) (Fe)/eV NiNi
E b(1) CuNi (Fe)/eV
E b(2) CuNi (Fe)/eV
0.22 0.21 0.08 0.11 0.03
0.04 0.04 0.03 0.04 0.03
0.02 0.00 0.03 0.11 0.07
0.00 0.00 0.00 0.02 0.01
0.13 0.14 0.05 0.01 0.01
0.03 0.04 0.01 0.02 0.02
a Calculated by VASP package with spin-polarization. b Calculated by VASP package without both spin-polarization and relaxation. c Calculated by SIESTA package with spin-polarization and without relaxation. d Calculated by SIESTA package without both spin-polarization and relaxation.
where E represents the energy of the related system, [(N 1)A + 1X] represents the system that the A matrix with N atoms is replaced by one X atom, [(N 2)A + 1X + 1Y] represents that the A matrix with N atoms is replaced by one X atom and one Y atom and two of them with the nth nearest-neighbor. The binding energy describes whether it is easy or difficult for the two-atom cluster to dissolve in the matrix (A) with isolated status. A positive value means that the cluster will be stable without considering entropy and temperature; negative or around zero value represents the opposite result. Before these calculations, equilibrium lattice constants were optimized. The lattice constant of bcc Fe optimized by VASP and SIESTA was 2.834 Å or 2.98 Å, respectively, close to the experimental value of 2.87 Å.63 Bcc Cu and Ni are considered as impurities to replace Fe atoms. Table 1 shows the binding energy calculated by the VASP or SIESTA packages with difference spin-polarization and relaxation conditions. In order to test the size effect, 3 3 3 and 4 4 4 super-cells were calculated initially, and the results indicated that the smaller one was large enough. Then, all of the calculations were carried out later with a b(2) 3 3 3 super-cell. The binding energies E b(1) CuCu (Fe) and E CuCu (Fe) are positive with spin-polarization by both the VASP and SIESTA packages, although with a small difference. So, it can be demonstrated that Cu atoms tend to gather under spinpolarization. In contrast, values of the binding energy E b(1) (Fe) NiNi and E b(2) (Fe) are negative or around zero, no matter what spinNiNi polarization condition and package was used. Besides, the b(2) binding energy E b(1) CuNi (Fe) and E CuNi (Fe) are closer to zero, b(1) b(2) compared with E CuCu (Fe) and E CuCu (Fe), respectively, indicating that the Cu-rich cluster is more stable when it contains Ni atoms at the interface between Fe and Cu. 3.2
PDOS and COHP
PDOS and COHP curves were drawn to explain the opposite values of binding energy with different spin-polarization conditions, as shown in Fig. 1–3. PDOS curves will be used to illustrate the orbital interaction and COHP curves will be used to characterize the system stability.49,50 All the calculations were carried out by SIESTA and VASP packages without structure relaxation, containing two first nearest neighbor (1nn) impurity atoms (Fe52Cu2, Fe52Ni2, Fe52CuNi). Fig. 1 mainly exhibits PDOS of Cu, Fe (matrix) and Fe (nearest neighbor to Cu), as well as COHP of Fe–Fe, Fe–Cu and Cu–Cu bonds in the Fe52Cu2-1nn system considering the spin-polarization (Fig. 1(a) and (b)), or not (Fig. 1(c) and (d)). In the COHP curve of Fig. 1(d), it is notable that the Fermi level
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eF passes through the antibonding area of all the bonds, indicating the system is unstable; besides, Cu Fermi level lies near the end of the antibonding section and there is little tendency for spin polarization because of the almost completely filled d orbits. However, in order to lessen the antibonding area the d states of Fe shifts in energy, with the a spin sub-lattice decreasing to fill completely the d block; meanwhile, the b spin increases to maintain a constant number of electrons in the system as a whole, as shown in Fig. 1(a) and (b). Besides, when analyzed by the Stoner mechanism, the IDOS(eF) of Fe is larger than Cu, assuming that both of the I values are the same, indicating that the PDOS of Fe will be split and result in magnetism, but Cu will be not split thus with no magnetism. From the above, both the chemical bonding approach and the Stoner mechanism illustrate that the d states of Fe will be split but Cu will not be. Similar to the frontier molecular orbital theory, the states near Fermi level are important to orbital interaction in that the occupied state is analogous to the highest occupied molecular orbital (HOMO) and the unoccupied state is analogous to the lowest unoccupied molecular orbital (LUMO). Then, one atom’s unoccupied states interact with other atom’s occupied states, and a bond is formed between the two atoms. According to this view, the interaction of the d states between Fe and Cu is stronger in Fig. 1(b) than in Fig. 1(a), because the Cu d states are too far below the Fermi level after spin polarization. So, the weak d-state interaction of Cu may be the essential reason for Cu precipitation. It has been confirmed64 that the width of the DOS curve is directly proportional to the strength of the interatomic interactions in a crystal; and the smaller the curve’s slope, the stronger the interaction. Therefore, the PDOS curves of Cu in the Fe52Cu2-1nn system (Fig. 1) show that the orbital interaction without spin-polarization between Cu and Fe is stronger, due to the wider energy interval ([12, 7] eV) and a smaller curve slope near the top of the d state in Fig. 1(c), than that under spin-polarization (a-spin [12.5, 8.5] eV, b-spin [12, 9] eV in Fig. 1(a)). In Fig. 1, the shapes of the PDOS calculated by the two packages are similar and the conclusions obtained are the same, although VASP calculated them more finely. The reasons for this phenomenon can be concluded as follows: (1) the hybridization of a-spin d states between Fe and Cu becomes weaker after spin polarization, mainly because both of their a-spin d states are almost completely filled, although they largely overlap with each other; (2) another reason for the weaker hybridization of b-spin d states between
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Fig. 1 PDOS of Fe and Cu atoms and COHP of Fe–Fe, Fe–Cu and Cu–Cu bonds under spin-polarization (a), (b) and (e), or not (c), (d) and (f) in Fe52Cu2-1nn system; (a), (b), (c) and (d) are calculated by SIESTA, while (e) and (f) are calculated by VASP package.
Fe and Cu is that the b-spin of the Cu d states is almost fully filled and there is a large energy interval gap between them after spin polarization, even resulting in a narrower energy interval of Cu b-spin d states compared to a-spin states. All the reasons mentioned above can be used to explain why the positive or negative E b(1) CuCu (Fe) values occur under different spin-polarization conditions, and the positive value illustrates Cu precipitation. So, the bond between Fe and Cu can be considered as a ‘‘dangling bond’’. In addition, the related PDOS and COHP curves were also drawn in the systems of Fe52Ni2-1nn (Fig. 2) and Fe52CuNi-1nn (Fig. 3) under spin-polarization conditions. In the Fe52Ni2-1nn system, there is a little shift between the a-spin and b-spin in the PDOS of Ni, because Ni has 8 electrons filled the d state, 2 electrons more than the d state of Fe. In spite of the large a-spin overlap between Ni and Fe, the interaction between them is still weak because of their fully filled d states. However, the b-spin orbital interaction between Fe and Ni is strong since the Fermi level passes through both of their b-spin d states. Like the previous Fe52Cu2-1nn and Fe52Ni2-1nn systems, Fig. 3 illustrates that the a-spin d states of Fe, Cu, and Ni can hybridize with each other. On the other side, the difference of b-spin energy interval between Fe and Cu is larger than that between Cu and Ni, resulting in a stronger interaction between Cu and Ni than that between Fe and Cu. Then, Ni can reduce the ‘‘dangling bond’’
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between Fe and Cu when it locates at the ‘‘interface’’ between them. 3.3
Spin density and charge density difference
In order to understand further the process of the d state shift, spin density, calculated by quantity r(spin) = r(up) r(down), is shown in Fig. 4. It is found that spin density gradually decreases from the nucleus to outside. Additionally, spin density diminishes successively for Fe, Ni and Cu, which agrees with their a- and b-spin energy shifting under spin-polarization. As a result, spin polarization leads to the rearrangement of electron density. The larger the spin density, the greater the rearrangement. The charge density difference, plotted by the quantity Dr = r(magnetic) r(nonmagnetic), is given in Fig. 5. It can be seen that Fe shows an obvious rearrangement of the electron density, an increase in the dz2 and dx2y2 orbitals and a decrease in other regions (Fig. 5(b)); while Ni shows a small rearrangement of the electron density, a decrease in dz2 and dx2y2 orbitals and an increase in other regions (Fig. 5(c)). Further analysis of the electron number of elemental Fe in d orbitals reveals that the electron number in dxy, dyz and dxy orbitals is the same and it is larger than that in dz2 and dx2y2 orbitals. So, bcc-Fe shows octahedral crystal field splitting, indicating that the charge effect of the next nearest neighbor atoms is the key to field splitting; however, Ni shows tetrahedral
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Fig. 2 The PDOS of Fe and Ni atoms, COHP of Fe–Fe, FeNi and NiNi bonds under spin-polarization, Fe52Ni2-1nn system; (a) and (b) are calculated by SIESTA, while (c) and (d) are calculated by VASP package.
crystal field splitting due to the opposite d orbital energy order to Fe, indicating that the charge effect of the nearest neighbor atoms is the key to field splitting. Besides, the electron number tends to be uniform after spin polarization. Therefore, the electron number increases in Fe dz2 orbitals after spinpolarization while it decreases in Ni dz2 orbitals, as shown in Fig. 5(b) and (c). After the Fe atoms are replaced by Cu and Ni impurities, electron transfer also occurs among them to different degrees. Bader charge analysis65 describes the charge movement such that Cu and Ni obtain 0.31 and 0.36 electron per atom, respectively. This is in large part due to the fact that the Fe d states show larger exchange splitting compared to Ni.49 Cu has an almost unfilled 4p orbital to obtain electrons from Fe. Thus, Ni has an obvious charge density difference in Fig. 5(a) compared with Fig. 5(c). The region with a positive charge density difference compared Ni is the closest to Fe, indicating strong
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Ni bonding with Fe. The region between Ni and Cu is also slightly positive, indicating partial bonding between them, due to the Cu d states being almost completely filled. We find that the region between Fe and Cu is more negative than the region between Fe atoms, indicating the weakest bonding between them. The above-mentioned illustrates that the electron transfer among different orbitals can be used to describe their properties of bonding, although the d-state shift is the key factor affecting the strength of d orbital interaction between Cu and other atoms. 3.4
Effect of Ni
From the above analysis, it can be shown that Ni has an important effect on Cu precipitation. Because the Fe–Cu bond is unstable due to the weak b-spin orbital interaction, that the bond between Fe and Cu is considered as a similar ‘‘dangling bond’’, Fe–Cu bonds tend to break and form more Fe–Fe and
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Fig. 3 The PDOS of Fe, Cu and Ni atoms, COHP of FeFe, FeCu, FeNi and CuNi bonds under spin-polarization, Fe52CuNi-1nn system; (a) and (b) are calculated by SIESTA, while (c) and (d) are calculated by VASP package.
Cu–Cu bonds. The energy difference of those bonds can be considered as an ‘‘interface energy’’, as shown in formula (2). When many Fe–Cu bonds arrange in a plane, the interface is formed between Fe and Cu atoms and the interface energy can be simply calculated based on formula (2) by counting Fe–Cu
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bonds. In other words, Cu precipitation from the Fe-matrix is also caused by the ‘‘interface energy’’ of Fe–Cu bonds. 1 sFeCu ¼ eFeCu ðeFeFe þ eCuCu Þ 2
(2)
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Fig. 4 Spin density of the (1% 10) plane sketched in Fig. 5(e).
where e represents the bond energy. So, the interface energy s is approximate to the binding energy shown in Table 1. In the Fe–Cu (110) interface, each Cu atom has two first- and two secondnearest neighbor Fe atoms. Here, the bond energy difference is 0.24 eV while the related area is 5.678 A2. Thus, the (110) interface
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energy of Fe–Cu is 676.3 mJ m2, which is similar to that from DFT calculation, 414.2 mJ m2 obtained from VASP. Above all, regarding the effect of Ni on Cu precipitating from the Fe-matrix, it can be concluded that Ni can enhance the Cu nucleation density but reduce the growth velocity of Cu-rich clusters. b-Spin energy intervals of both Cu and Fe can overlap with Ni, so both of them can hybridize with the b-spin d states of Ni. Then, the Cu–Ni and Fe–Ni bonds are stable in the Fe-matrix, as shown in Table 1. The interface energies of sFe–Ni and sCu–Ni are both negative, illustrating that they are liable to mix, and not to precipitate to form an interface between Fe and Ni or Cu and Ni. When Cu precipitates from the Fe matrix, Ni can be regarded as a ‘‘protective shell’’ to avoid the contact of Cu with Fe. This is why a Cu core with an Ni shell structure was observed in related experiments.1,11 Simulation43 reveals that the addition of Ni notably increases Cu nucleation density and reduces the size of the clusters. It is easy to understand that Ni can enhance Cu nucleation because Ni prefers contact with Cu to reduce the interface energy (or the ‘‘dangling bond’’) and the critical nucleation number.43 Because E b(1) (Fe) is not a posiNiNi tive value, there is no driving force to make Ni gather, which reduces the chance of two Ni-shell Cu clusters forming a bigger cluster when two ‘‘protective shell’’ Cu clusters get close.
4. Conclusions We confirmed that orbital interaction is the root-cause of Cu precipitation. From the above analysis, the d state interaction between Fe and Cu is weak when considering spin-polarization,
Fig. 5 Charge density difference upon spin polarization, red color indicates a positive value and blue color indicates a negative value: (a) charge density difference of the (1% 10) plane sketched in (e); (b) charge density difference of Fe (1% 10) plane sketched in (d); (c) charge density difference of Ni (1% 10) plane sketched in (d); (d) body-centered cubic (bcc) unit cell where the slope is the (1% 10) plane; (e) 3 3 3 super-cell of Fe element replaced by Cu and Ni impurities as the nearest neighbor, the gray atom is Ni, blue is Cu and brown is Fe, and the slope is the (1% 10) plane.
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because the Cu d states are too far below the Fermi level. On the other hand, the d states near the Fermi level of the Fe atom have strong interaction with other Fe atoms. Therefore, the bond between Fe and Cu is similar to a ‘‘dangling bond’’, which mainly induces Cu precipitation. The charge density difference also illustrates the weaker bond between Fe and Cu after spinpolarization because of the decreased charge between them. The ‘‘dangling bond’’ is also the origin of interface energy between Fe and Cu. The {110} interface energy calculated by the ‘‘dangling bond’’ concept is 676.3 mJ m2, which agrees with the DFT calculation 414.2 mJ m2 obtained by the VASP package. In truth, Ni can reduce the ‘‘dangling bond’’ between Fe and Cu, which finally enhances Cu nucleation density but reduces the growth velocity of Cu-rich clusters. Finally, there is a general rule that the more bonds are formed among Cu atoms and the more Ni atoms locate at the interface between Fe and Cu, the more stable is the system in enthalpy.
Acknowledgements The authors thank Dr Li-Ming Liu, San-Yi Ma, Yun Xue, Li-Chun Xu, Yao-Ping Xie and Yu-Bin Si for great help. This work is financially supported by National Natural Science Foundation of China (Grant Nos. 50931003, 51301102), and 085 project at Shanghai University. High performance computing resources are provided by the Ziqiang Supercomputer Center at Shanghai University.
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