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The electronic effect of N impurity in an 〈100〉 edge dislocation core system in α-iron
Computational Materials Science 22 (2001) 144±150
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The electronic eect of N impurity in an h1 0 0i edge dislocation core system in a-iron Yuan Niu a, Shan-Ying Wang b,*, Dong-Liang Zhao a, Chong-Yu Wang a,b,c a
Central Iron and Steel Research Institute, Beijing 100081, China Physical Department, Tsinghua University, Beijing 100084, China International Center for Materials Physics, Academia Sinica, Shenyang 110016, China b
c
Received 2 January 2001; received in revised form 30 March 2001; accepted 18 April 2001
Abstract First-principles methods are employed to investigate the structure relaxation and the electronic structure of a N impurity in an h1 0 0i edge dislocation core (DC) system in a-iron. A 96-atom cluster model is used to simulate the local environment of N impurity in the edge dislocation. By use of the DMol method, we obtained an optimized atomic con®guration for the system by calculating the forces on N impurity and its neighboring Fe atoms, and by minimizing the total energy of the cluster model. The optimization results show that the N impurity moves away from compression region to a stable position in the dilated region. By use of the discrete variational (DV) method, we calculated energetic parameters (structural energy and interatomic energy) and charge distribution. From these results, one can ®nd that N impurity has a strong interaction with its adjacent Fe atoms in the DC system. Moreover, notable charge redistribution between the N impurity and Fe atoms indicates the formation of N impurity±Fe dislocation complex, which implies the trapping eect of DC on N impurity. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 81.05.Bx; 71.20.Be; 61.72.Lk Keywords: Structure relaxation; Electronic structure; Edge dislocation core system; First-principles method; a-iron
1. Introduction Dislocation is one of the important structural defects in the steel materials. Its geometrical feature and motion behavior aect greatly the mechanical properties of the materials. The classical elastic theory has been well developed and can
*
Corresponding author. E-mail address: [email protected] Wang).
(S.-Y.
describe correctly the dislocation behavior to some extent. However, due to the limitation of the continuum model (the presence of continuum singularity), the classical elastic theory fails to describe the behavior of dislocation core (DC) system, especially the interaction between impurity and DC system. The radius of an eective DC is about 1.25±1.65 Burgers vectors [1]. It can be expected that in such a local system, the electronic eect may be important in determining the interaction between impurity and DC. So the investigation of the electronic structure of DC is essential
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 8 3 - 5
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
and may give us more insight into the correlation between the microscopic structure and macroscopic properties. Actually, the DC problems in steel materials are very complicated and to our knowledge, there are little experimental investigations on them at either atomic level or electronic level up to now. Nevertheless, people still have made many theoretical eorts to explore the DC problem by means of atomistic simulation and by some semi-empirical electronic structure methods [2±10]. Dehosson [11] investigated the localized states of an h1 0 0i edge dislocation in airon based on the multiple-scattering model method. He determined the variation in electronic structure of the lattice near the core region of defect in a-iron, and the variation of hydrostatic pressure with an edge dislocation producing a rearrangement of the conduction electrons. In his research, the cluster model contains only nine Fe atoms and the structure distortion induced by the impurities cannot be determined. Masuda et al [12] studied the core structures and the core energies of 1=2h1 1 1if1 1 0g and 1=2h1 1 1if1 1 2g edge dislocations in a bcc transition metal by use of a tight-binding type electronic theory. They showed that the electronic eect on DC is important and the Peierls stress of the screw dislocation is approximately 10 times larger than that of the edge dislocation. More recently, using the semi-empirical atom superposition and electron delocalization molecular orbital theory, Gesari et al. [13] have studied the interaction of hydrogen with a-iron. They simulated the adsorption process of hydrogen on 1 1 2 and (1 1 0) faces in an edge dislocation. They found that the H±Fe interaction is stronger near the center of the dislocation. By use of the same method, Moro et al. [14] have studied the interaction of hydrogen with c-iron. In their study, the in¯uence of the stacking fault zone was considered. They found that H occupies nearly octahedral site on the stacking fault plane. So far, there are little ®rst-principles studies on the electronic structure of DC system in a-iron. In this paper, we report our ®rst-principles study on the electronic eect of a N impurity in an h1 0 0i edge DC system in a-iron.
145
2. Computation details In order to get an atomic con®guration of an h1 0 0i edge DC system for the ®rst-principles calculation, we ®rst simulated an h1 0 0i edge dislocation in a-iron by molecular dynamics (MD) method with the Finnis±Sinclair potential [15,16], which is a semi-empirical N-body potential suitable for transition metals. The atomic con®guration of the dislocation system for MD simulation was determined according to the elastic displacement ®eld theory, where the Burger's vector and dislocation line are along [1 0 0] direction and [0 0 1] direction, respectively, and the slip plane is (0 1 0). After a full relaxation under the periodic condition along [0 0 1] direction and the ®xed boundary conditions along [0 1 0] and [1 0 0] directions, respectively, we obtained an atomic con®guration with symmetry close to C2v for the h1 0 0i edge dislocation system. The result is quite similar to that obtained by Gehlen [1]. Then for ®rst-principles calculation, we constructed a cluster model of DC from the MD results (named by initial DC system), which consists of 96 Fe atoms as illustrated in Fig. 1. Some atoms in the cluster model are labeled with numbers for the purpose of discussing the results clearly. The cluster model contains ®ve layers with a stacking sequence ABABA along the [0 0 1] direction. We can see an interstice enclosed by eight Fe atoms labeled from Fe1 to Fe8 in the initial DC system. A N atom is introduced into the interstice in order to study the interaction of N impurity with the h1 0 0i edge DC system. For the dierent size and shape between the N impurity and the interstice in the DC, it may be expected that N impurity could result in a large local distortion of neighboring Fe sites. In order to obtain the equilibrium atomic con®guration, DMol method [17,18] which is capable of calculating energy gradient for each atom was used to optimize the local structure. In the present DMol relaxation calculation, considering the short-range eect of N impurity, we allow only the N atom and its 13 neighboring Fe atoms from Fe1 to Fe13 to move around by the forces on them. We obtained an optimized geometry structure and the binding energy (de®ned as the dierence of the total energy
146
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
atomic units) in the one-electron-like wave function equations is expressed as: H [010] direction
Fe11
Fe13
Fe5(6)
Fe12
Fe7(8)
Fe1 N
Fe2
slip plane
Fe3(4) Fe9
Fe10 Fe14(15)
[100] direction Fig. 1. The atomic con®gurations in initial DC system and in N-doped DC system. The triangle and square denote the Fe atoms on stacking plane A and B, respectively, along the direction [0 0 1] of dislocation line. The triangle and square with and without ``+'' symbol denote the Fe atom in N-doped DC system, respectively. A solid circle symbolizes N atom. The dashed line is guidance for the clear sight of the slip plane (0 1 0). One can see clearly that Fe1±Fe13 atoms shift away by DMol relaxation.
between the bonding atom system and the free atom system) for the h1 0 0i DC system within the tolerances of 0.005 Ry/a.u and 0.005 a.u for energy gradient and atomic displacement, respectively. Here, we name the DMol relaxation result on the N impurity in h1 0 0i DC system as N-doped DC system. The double numerical polarized bases as well as the Vosko±Wilk±Nusair [19] local exchange-correlation functional with Becke±Perdew±Wang general gradient approximation correction [20] are used for the sake of giving more reliable relaxation results. Based on the atomic con®guration obtained by the DMol relaxation calculation, we calculated the energy parameters including structural energy and interatomic energy, and charge distribution by the discrete variational (DV) method [21±26]. As is well known, DV method which is a ®rst-principles numerical method for solving the local-densityfunctional equations, has been successfully used to study the electronic structure of metals, alloys and intermetallic compounds [25±29]. In the DV method, the Hamiltonian quantity (in Hartree
r2 =2 VCoul VXC ;
where VCoul are the Coulomb potentials of electron±nucleus and electron±electron, and VXC is the exchange-correlation potential. The one-electronlike wave functions ui are expanded as the linear combinations of atomic orbitals /i , namely P un i Cni /i . The equations are solved self-consistently and the solutions can be obtained till the linear combination coecients of the two neighboring calculation steps are changeless. In this work, an embedded potential produced by several hundreds of environment atoms is added in the Hamiltonian quantity, which includes the contributions of the Coulomb potential and exchangecorrelation potential from the environment atoms. In the self-consistent calculation, the funnel potential is added in the numerical base calculation and the frozen-core approximation is adopted. The Coulomb potential is calculated in a selfconsistent charge approximation and about 1000 integration points per atom are used.
3. Results and discussion 3.1. Atomic con®guration We know that the long-range MD relaxation is sometimes insucient in dealing with the problem of local system. In order to avoid the possible insuciency by the MD relaxation and to clarify the signi®cance of the local distortion induced by impurity, we performed both a DMol total energy calculation on the initial DC system, and a DMol relaxation calculation on the initial DC system similar to that of the N-doped DC system (the result named by clean DC system). The main results for the three systems are summarized in Table 1. We discuss the results on the initial DC system and the clean DC system ®rst. We can see the pronounced modi®cation on the interatomic distance in clean DC system by DMol relaxation. Comparing to the MD results, DMol relaxation
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
147
Table 1 for the selected atomic pairs in initial DC system, clean DC system and N-doped DC system The calculated interatomic distances (in A) dFe1±Fe2 dFe3±Fe4 dFe5±Fe6 dFe5±Fe7 dFe3±Fe14 dFe9±Fe10 dFe11±Fe12 Eb a a
Initial DC
Clean DC
N-doped DC
4.20 2.87 2.87 2.35 3.08 3.02 5.26 )432.41
3.61 2.49 3.22 2.59 3.05 2.94 5.20 )433.50
3.42 3.17 3.20 2.62 2.51 3.16 5.24 )441.84
Eb is the binding energy of the system (in eV).
gave out the increased interatomic distance in the section above the slip plane and the decreased interatomic distance under the slip plane. In the stacking layers A and B, the distance between the Fe1 and Fe2 atoms (denoted by dFe1±Fe2 ) reduces by 14.1% and dFe9±Fe10 reduces by 2.6%, while dFe5±Fe7 expands by 10.2%, dFe3±Fe14 and dFe11±Fe12 change little. Along the direction of dislocation line, dFe3±Fe4 reduces by 13.2% but dFe5±Fe6 expands by 12.2%. The dierence of the binding energies between the initial DC system and the clean DC system are small, only 1.09 eV. It appears that for the study of the edge dislocation system in a-iron, the correction on atomic bond length made by ®rst-principles relaxation calculation may be more essential. Similar experiences can also be found in the studies on Fe grain boundary by Tang et al. [30]. We now turn to the discussions on the N-doped DC system. Comparing with the clean DC system, but dFe9±Fe10 increases by dFe1±Fe2 reduces by 0:19 A 0:22 A, while dFe5±Fe6 change little, only 0:02 A. In sequence of distance, the interatomic distances for the selected Fe±N atomic pairs are: dN±Fe1 is dN±Fe3 is 1:83 A, dN±Fe11 is 2:55 A and 1:80 A, dN±Fe5 is 3:16 A, respectively. It is very interesting that in the interstice, N impurity prefers to stay at a stable position in the dilated region. As a result, Fe3 and Fe4 atoms in the dilated region are pu by N impurity along [0 1 0] shed down for 0:54 A direction, and the distance between them increases The binding energy of Nnotably, about 0:68 A. doped DC system is 8.34 eV lower than that of clean DC system, which imply that the N-doped DC system is more stable. These analyses show
that N impurity could induce a remarkable distortion of the environment atomic sites in edge DC system in a-iron. 3.2. Energy analysis In order to study the interaction of N impurity with the DC system, we have calculated two energetic quantities: the structural energy in realspace Green Function method [31] and the interatomic energy [32,33]. Structural energy El of atom l is expressed as: Z EF El nl EE dE; 1
where nl E is the local density of states of atom l, and EF is the Fermi energy of the system. It was used to de®ne the impurity formation energy in treating defect system [34]. An impurity at its stable atomic site usually displays a low structural energy. Thus structural energy can re¯ect the interaction of the impurity with the environment in host. Interatomic energy Elm describes the interaction strength between atom l and m. It is de®ned as XX Elm Nn anal anbm Halbm ; n
ab
where Nn is the electron occupation number in molecular orbital un , anal h/al jun i, and Halbm is the Hamiltonian matrix element connecting the atomic orbital /bm of atom m to the atomic orbital /al of atom l. Interatomic energy diers from the usual bond order parameter by that Hamiltonian
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Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
matrix and overlap matrix are used, respectively, for the two quantities' de®nitions. Interatomic energy was successfully applied to the studies on the Ni3 Al grain boundary [27±29]. Usually the lower the interatomic energy, the stronger the interaction is. In this work, the structural energy and the interatomic energy are calculated using the data obtained by DV method. The structural energies for some interesting atoms are listed in Table 2. In the N-doped DC system, N impurity has the lowest structural energy about )8.90 eV, which shows that N impurity stays at a site of energy valley. The result re¯ects the trapping eect of DC on N impurity. In addition, the structural energies of Fe1 and Fe3 atoms display noticeable changes compared to those in the clean DC system, which implies the strong interactions of N impurity with Fe1 and Fe3 atoms. The interactions re¯ect the pinning eect of N impurity on the dislocation. In Table 3, the interatomic energies of some typical atomic pairs in clean DC system and Ndoped DC system are presented. We can see that when N impurity enters into the DC system, the bonding strength between its neighboring Fe atoms is weakened. Comparing to the clean DC
Table 2 The structural energies (in eV) of the selected atoms in clean DC system and N-doped DC system N Fe1 Fe3 Fe5 Fe9
Clean DC
N-doped DC
)2.83 )2.76 )2.98 )2.77
)8.90 )2.72 )2.95 )2.97 )2.76
Table 3 The interatomic energies (in eV) of the typical atomic pairs in clean DC system and N-doped DC system Atomic pair
Clean DC
N-doped DC
DEa
Fe1±Fe3 Fe3±Fe4 Fe3±Fe5 Fe3±Fe9 Fe5±Fe6 Fe5±Fe7 N±Fe1 N±Fe3 N±Fe5 N±Fe9
)2.13 )1.83 )0.83 )1.70 )0.33 )1.66
)0.35 0.25 )0.15 )1.47 )0.32 )1.73 )4.50 )3.95 0.17 0.28
1.75 2.08 0.68 0.23 0.01 )0.07
a
DE is the dierence of the interatomic energy of atomic pair in N-doped DC system and clean DC system.
Table 4 The electron occupation number n n0 in valence orbital of the selected atoms in N-doped DC system (clean DC system)a Clean cluster n0 N
Fe1
Fe3
Fe9
a
2s 2p Q
N-doped cluster n
Dn
1.68 3.87 5.55
DQ
+0.55
3d 4s 4p Q
6.34 0.76 0.87 7.97
6.37 0.62 0.72 7.71
+0.03 )0.14 )0.15
3d 4s 4p Q
6.35 0.78 0.91 8.04
6.34 0.69 0.82 7.85
)0.01 )0.09 )0.09
3d 4s 4p Q
6.34 0.75 0.80 7.89
6.36 0.72 0.80 7.88
+0.02 )0.03 )0.00
)0.26
)0.19
)0.01
Q is the summation of the electron occupation number in valence orbitals. For Fe atom, Dn n Q N-doped DC Q clean DC. For N impurity, DQ Q N-doped DC Q free atom.
n0 , and DQ
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
149
(a)
13
[ 010] direction
11
1
(b)
13
12
2 N
9
N 3
4
14
15
10
[100] direction
[001] direction
Fig. 2. The charge density dierences for two planes: (a) the center stacking plane A along the [0 0 1] direction containing N, Fe1 and Fe2 atoms; (b) a (1 0 0) plane containing N, Fe3 and Fe4 atoms. The circled numbers correspond to those in Fig. 1. The contour 3 . Solid lines and dashed lines indicate the gain and the loss of electrons, respectively. spacings are 0:001 e=A
system, the interatomic energies between the Fe atoms which are adjacent to N impurity such as Fe1 ±Fe3 and Fe3 ±Fe5 in the N-doped DC system decrease by about 17.8% and 18.1%, respectively. In the direction of the dislocation line, the sign of interatomic energy between Fe3 and Fe4 changes from minus to plus, which implies the strongly weakened interaction between Fe3 and Fe4 atoms because of their increased interatomic distance induced by the N impurity. In addition, N displays strong interactions with its adjacent Fe atoms. The interatomic energies of atomic pairs of N±Fe1 and N±Fe3 are )4.50 and )3.95 eV, respectively. The above results indicate that N impurity could attract the adjacent Fe atoms in DC, and conversely DC could exert the trapping eect on N impurity. 3.3. Charge distribution Naturally, the charge transfers among the impurity and the host Fe atoms play an important
role in the formation of atom bond. Table 4 lists the electron occupation number in the valence orbitals of N impurity and its neighboring Fe atoms obtained by Mulliken population [35]. It can be seen that N impurity obtains 0.55 electrons from the adjacent Fe atoms and assigns them mainly to its 2p valence orbital. The adjacent Fe1 , Fe3 and Fe11 atoms lose 0.26, 0.19 and 0.01 electrons, respectively, and parts of the electrons from the 4s and 4p orbitals are transferred to N impurity. One can also obtain directly an insight into the bonding character by charge density distribution. Figs. 2(a) and (b) illustrate the charge density dierences on the center stacking plane A which contains N, Fe1 and Fe2 atoms, and on a (0 1 0) plane which contains N, Fe3 and Fe4 atoms, respectively. The charge density dierences are obtained by subtracting the charge density of the clean DC system and that of a free N atom from that of N-doped DC system. Thus the impurity-
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Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
induced charge redistribution in the DC system can be seen clearly. We can see that a remarkable number of electrons accumulate near the N impurity, and there are strong charge correlation curves connecting the N impurity with its adjacent Fe atoms from Fe1 to Fe4. The results indicate that the covalent-like bonds are formed between the N impurity and Fe atoms from Fe1 to Fe4. The feature of the charge density distribution re¯ects again that the N impurity has a strong interaction with its adjacent Fe atoms, which agrees well with the analyses on interatomic energy results. These results imply the nature of quantummechanical eect of Cottrell atmosphere in DC system. 4. Conclusion We have studied the electronic eect of N impurity in an h1 0 0i edge DC system in a-iron by use of ®rst-principles methods. We found that the N impurity prefers to stay at a stable position in the dilated region in the edge DC system. The calculated energetic parameters indicate that the N impurity has a strong interaction with its adjacent Fe atoms. There are charge transfers from the host Fe atom to the N impurity. The charge density distribution shows that the N impurity forms covalent-like bond with its adjacent Fe atoms. These results suggest the formation of N impurity±Fe edge dislocation complex, which implies the trapping eect of DC on N impurity. Acknowledgements The authors are grateful to Dr. Hongbo Liu for his helpful work in the MD simulation of the dislocation. This work was supported by the National Natural Science Foundation of China (59971041) and 973 National Pandeng Project (G2000067102).
References [1] P.C. Gehlen, A.R. Rosen®eld, G.T. Hahn, J. Appl. Phys. 39 (1968) 5246. [2] K. Masuda, A. Sato, Philos. Mag. B 37 (1978) 531. [3] A. Sato, K. Masuda, Philos. Mag. B 43 (1981) 1. [4] K. Masuda, A. Sato, J. Phys. Soc. Jpn. 50 (1981) 569. [5] D.J. Bacon, J.W. Martin, Philos. Mag. A 43 (1981) 901. [6] Y. Minonishi, S. Ishioka, M. Koiwa, S. Morozumi, M. Yamaguchi, Philos. Mag. A 43 (1981) 1017. [7] Y. Minonishi, S. Ishioka, M. Koiwa, S. Morozumi, M. Yamaguchi, Philos. Mag. A 44 (1981) 1225. [8] D.J. Bacon, M.H. Liang, Philos. Mag. A 53 (1986) 163. [9] M.H. Liang, D.J. Bacon, Philos. Mag. A 53 (1986) 181. [10] M.H. Liang, D.J. Bacon, Philos. Mag. A 53 (1986) 205. [11] J.Th.M. Dehosson, Int. J. Quant. Chem. 18 (1980) 575. [12] K. Masuda, K. Kobayashi, A. Sato, T. Mori, Philos. Mag. B 43 (1981) 19. [13] S. Gesari, B. Irigoyen, A. Juan, J. Phys. D 31 (1998) 2179. [14] L. Moro, E. Roviglione, D. Hermida, A. Juan, J. Phys. D 31 (1998) 893. [15] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 50 (1984) 45. [16] M.W. Finnis, J.E. Sinclair, Philos. Mag. A 53 (1986) 161. [17] B. Delley, J. Chem. Phys. 92 (1990) 508. [18] B. Delley, J. Chem. Phys. 94 (1991) 7245. [19] S.J. Vosko, L. Wilk, M. Nusair, Can. J. Phys. 58 (1980) 1200. [20] P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13244. [21] D.E. Ellis, G.S. Painter, Phys. Rev. B 2 (1970) 2887. [22] E.J. Maerends, D.E. Ellis, P. Ros, Chem. Phys. 2 (1973) 41. [23] F.W. Averill, D.E. Ellis, J. Chem. Phys. 59 (1973) 6412. [24] B. Delley, D.E. Ellis, A.J. Freeman, E.J. Baerends, D. Post, Phys. Rev. B 27 (1983) 2132. [25] D.E. Ellis, G.A. Benesh, E. Byrom, Phys. Rev. B 16 (1977) 3308. [26] D. Guenzburger, D.E. Ellis, Phys. Rev. B 45 (1992) 285. [27] F.H. Wang, C.Y. Wang, J. Phys.: Condens. Matter 9 (1997) 4499. [28] F.H. Wang, C.Y. Wang, J.L. Yang, J. Phys. Condens. Matter 8 (1996) 5527. [29] F.H. Wang, C.Y. Wang, Phys. Rev. B 57 (1998) 289. [30] S.P. Tang, A.J. Freeman, Phys. Rev. B 50 (1994) 1. [31] V. Heine, Solid State Phys. 35 (1980) 1. [32] C.Y. Wang, C.Y. Dn, Y.P. Zeng, L. Gao, F.S. Liu, in: Proceedings of the International Workshop on Physics of Materials, Shenyang, B5-1, 1989. [33] C.Y. Wang, Defect and Diusion Forum 125±126 (1995) 79. [34] C.Y. Wang, D.L. Zhao, Mater. Res. Soc. Sym. Proc. 571 (1994) 318. [35] R.S. Mulliken, J. Chem. Phys. 23 (1955) 1833.
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The electronic eect of N impurity in an h1 0 0i edge dislocation core system in a-iron Yuan Niu a, Shan-Ying Wang b,*, Dong-Liang Zhao a, Chong-Yu Wang a,b,c a
Central Iron and Steel Research Institute, Beijing 100081, China Physical Department, Tsinghua University, Beijing 100084, China International Center for Materials Physics, Academia Sinica, Shenyang 110016, China b
c
Received 2 January 2001; received in revised form 30 March 2001; accepted 18 April 2001
Abstract First-principles methods are employed to investigate the structure relaxation and the electronic structure of a N impurity in an h1 0 0i edge dislocation core (DC) system in a-iron. A 96-atom cluster model is used to simulate the local environment of N impurity in the edge dislocation. By use of the DMol method, we obtained an optimized atomic con®guration for the system by calculating the forces on N impurity and its neighboring Fe atoms, and by minimizing the total energy of the cluster model. The optimization results show that the N impurity moves away from compression region to a stable position in the dilated region. By use of the discrete variational (DV) method, we calculated energetic parameters (structural energy and interatomic energy) and charge distribution. From these results, one can ®nd that N impurity has a strong interaction with its adjacent Fe atoms in the DC system. Moreover, notable charge redistribution between the N impurity and Fe atoms indicates the formation of N impurity±Fe dislocation complex, which implies the trapping eect of DC on N impurity. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 81.05.Bx; 71.20.Be; 61.72.Lk Keywords: Structure relaxation; Electronic structure; Edge dislocation core system; First-principles method; a-iron
1. Introduction Dislocation is one of the important structural defects in the steel materials. Its geometrical feature and motion behavior aect greatly the mechanical properties of the materials. The classical elastic theory has been well developed and can
*
Corresponding author. E-mail address: [email protected] Wang).
(S.-Y.
describe correctly the dislocation behavior to some extent. However, due to the limitation of the continuum model (the presence of continuum singularity), the classical elastic theory fails to describe the behavior of dislocation core (DC) system, especially the interaction between impurity and DC system. The radius of an eective DC is about 1.25±1.65 Burgers vectors [1]. It can be expected that in such a local system, the electronic eect may be important in determining the interaction between impurity and DC. So the investigation of the electronic structure of DC is essential
0927-0256/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 1 8 3 - 5
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
and may give us more insight into the correlation between the microscopic structure and macroscopic properties. Actually, the DC problems in steel materials are very complicated and to our knowledge, there are little experimental investigations on them at either atomic level or electronic level up to now. Nevertheless, people still have made many theoretical eorts to explore the DC problem by means of atomistic simulation and by some semi-empirical electronic structure methods [2±10]. Dehosson [11] investigated the localized states of an h1 0 0i edge dislocation in airon based on the multiple-scattering model method. He determined the variation in electronic structure of the lattice near the core region of defect in a-iron, and the variation of hydrostatic pressure with an edge dislocation producing a rearrangement of the conduction electrons. In his research, the cluster model contains only nine Fe atoms and the structure distortion induced by the impurities cannot be determined. Masuda et al [12] studied the core structures and the core energies of 1=2h1 1 1if1 1 0g and 1=2h1 1 1if1 1 2g edge dislocations in a bcc transition metal by use of a tight-binding type electronic theory. They showed that the electronic eect on DC is important and the Peierls stress of the screw dislocation is approximately 10 times larger than that of the edge dislocation. More recently, using the semi-empirical atom superposition and electron delocalization molecular orbital theory, Gesari et al. [13] have studied the interaction of hydrogen with a-iron. They simulated the adsorption process of hydrogen on 1 1 2 and (1 1 0) faces in an edge dislocation. They found that the H±Fe interaction is stronger near the center of the dislocation. By use of the same method, Moro et al. [14] have studied the interaction of hydrogen with c-iron. In their study, the in¯uence of the stacking fault zone was considered. They found that H occupies nearly octahedral site on the stacking fault plane. So far, there are little ®rst-principles studies on the electronic structure of DC system in a-iron. In this paper, we report our ®rst-principles study on the electronic eect of a N impurity in an h1 0 0i edge DC system in a-iron.
145
2. Computation details In order to get an atomic con®guration of an h1 0 0i edge DC system for the ®rst-principles calculation, we ®rst simulated an h1 0 0i edge dislocation in a-iron by molecular dynamics (MD) method with the Finnis±Sinclair potential [15,16], which is a semi-empirical N-body potential suitable for transition metals. The atomic con®guration of the dislocation system for MD simulation was determined according to the elastic displacement ®eld theory, where the Burger's vector and dislocation line are along [1 0 0] direction and [0 0 1] direction, respectively, and the slip plane is (0 1 0). After a full relaxation under the periodic condition along [0 0 1] direction and the ®xed boundary conditions along [0 1 0] and [1 0 0] directions, respectively, we obtained an atomic con®guration with symmetry close to C2v for the h1 0 0i edge dislocation system. The result is quite similar to that obtained by Gehlen [1]. Then for ®rst-principles calculation, we constructed a cluster model of DC from the MD results (named by initial DC system), which consists of 96 Fe atoms as illustrated in Fig. 1. Some atoms in the cluster model are labeled with numbers for the purpose of discussing the results clearly. The cluster model contains ®ve layers with a stacking sequence ABABA along the [0 0 1] direction. We can see an interstice enclosed by eight Fe atoms labeled from Fe1 to Fe8 in the initial DC system. A N atom is introduced into the interstice in order to study the interaction of N impurity with the h1 0 0i edge DC system. For the dierent size and shape between the N impurity and the interstice in the DC, it may be expected that N impurity could result in a large local distortion of neighboring Fe sites. In order to obtain the equilibrium atomic con®guration, DMol method [17,18] which is capable of calculating energy gradient for each atom was used to optimize the local structure. In the present DMol relaxation calculation, considering the short-range eect of N impurity, we allow only the N atom and its 13 neighboring Fe atoms from Fe1 to Fe13 to move around by the forces on them. We obtained an optimized geometry structure and the binding energy (de®ned as the dierence of the total energy
146
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
atomic units) in the one-electron-like wave function equations is expressed as: H [010] direction
Fe11
Fe13
Fe5(6)
Fe12
Fe7(8)
Fe1 N
Fe2
slip plane
Fe3(4) Fe9
Fe10 Fe14(15)
[100] direction Fig. 1. The atomic con®gurations in initial DC system and in N-doped DC system. The triangle and square denote the Fe atoms on stacking plane A and B, respectively, along the direction [0 0 1] of dislocation line. The triangle and square with and without ``+'' symbol denote the Fe atom in N-doped DC system, respectively. A solid circle symbolizes N atom. The dashed line is guidance for the clear sight of the slip plane (0 1 0). One can see clearly that Fe1±Fe13 atoms shift away by DMol relaxation.
between the bonding atom system and the free atom system) for the h1 0 0i DC system within the tolerances of 0.005 Ry/a.u and 0.005 a.u for energy gradient and atomic displacement, respectively. Here, we name the DMol relaxation result on the N impurity in h1 0 0i DC system as N-doped DC system. The double numerical polarized bases as well as the Vosko±Wilk±Nusair [19] local exchange-correlation functional with Becke±Perdew±Wang general gradient approximation correction [20] are used for the sake of giving more reliable relaxation results. Based on the atomic con®guration obtained by the DMol relaxation calculation, we calculated the energy parameters including structural energy and interatomic energy, and charge distribution by the discrete variational (DV) method [21±26]. As is well known, DV method which is a ®rst-principles numerical method for solving the local-densityfunctional equations, has been successfully used to study the electronic structure of metals, alloys and intermetallic compounds [25±29]. In the DV method, the Hamiltonian quantity (in Hartree
r2 =2 VCoul VXC ;
where VCoul are the Coulomb potentials of electron±nucleus and electron±electron, and VXC is the exchange-correlation potential. The one-electronlike wave functions ui are expanded as the linear combinations of atomic orbitals /i , namely P un i Cni /i . The equations are solved self-consistently and the solutions can be obtained till the linear combination coecients of the two neighboring calculation steps are changeless. In this work, an embedded potential produced by several hundreds of environment atoms is added in the Hamiltonian quantity, which includes the contributions of the Coulomb potential and exchangecorrelation potential from the environment atoms. In the self-consistent calculation, the funnel potential is added in the numerical base calculation and the frozen-core approximation is adopted. The Coulomb potential is calculated in a selfconsistent charge approximation and about 1000 integration points per atom are used.
3. Results and discussion 3.1. Atomic con®guration We know that the long-range MD relaxation is sometimes insucient in dealing with the problem of local system. In order to avoid the possible insuciency by the MD relaxation and to clarify the signi®cance of the local distortion induced by impurity, we performed both a DMol total energy calculation on the initial DC system, and a DMol relaxation calculation on the initial DC system similar to that of the N-doped DC system (the result named by clean DC system). The main results for the three systems are summarized in Table 1. We discuss the results on the initial DC system and the clean DC system ®rst. We can see the pronounced modi®cation on the interatomic distance in clean DC system by DMol relaxation. Comparing to the MD results, DMol relaxation
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
147
Table 1 for the selected atomic pairs in initial DC system, clean DC system and N-doped DC system The calculated interatomic distances (in A) dFe1±Fe2 dFe3±Fe4 dFe5±Fe6 dFe5±Fe7 dFe3±Fe14 dFe9±Fe10 dFe11±Fe12 Eb a a
Initial DC
Clean DC
N-doped DC
4.20 2.87 2.87 2.35 3.08 3.02 5.26 )432.41
3.61 2.49 3.22 2.59 3.05 2.94 5.20 )433.50
3.42 3.17 3.20 2.62 2.51 3.16 5.24 )441.84
Eb is the binding energy of the system (in eV).
gave out the increased interatomic distance in the section above the slip plane and the decreased interatomic distance under the slip plane. In the stacking layers A and B, the distance between the Fe1 and Fe2 atoms (denoted by dFe1±Fe2 ) reduces by 14.1% and dFe9±Fe10 reduces by 2.6%, while dFe5±Fe7 expands by 10.2%, dFe3±Fe14 and dFe11±Fe12 change little. Along the direction of dislocation line, dFe3±Fe4 reduces by 13.2% but dFe5±Fe6 expands by 12.2%. The dierence of the binding energies between the initial DC system and the clean DC system are small, only 1.09 eV. It appears that for the study of the edge dislocation system in a-iron, the correction on atomic bond length made by ®rst-principles relaxation calculation may be more essential. Similar experiences can also be found in the studies on Fe grain boundary by Tang et al. [30]. We now turn to the discussions on the N-doped DC system. Comparing with the clean DC system, but dFe9±Fe10 increases by dFe1±Fe2 reduces by 0:19 A 0:22 A, while dFe5±Fe6 change little, only 0:02 A. In sequence of distance, the interatomic distances for the selected Fe±N atomic pairs are: dN±Fe1 is dN±Fe3 is 1:83 A, dN±Fe11 is 2:55 A and 1:80 A, dN±Fe5 is 3:16 A, respectively. It is very interesting that in the interstice, N impurity prefers to stay at a stable position in the dilated region. As a result, Fe3 and Fe4 atoms in the dilated region are pu by N impurity along [0 1 0] shed down for 0:54 A direction, and the distance between them increases The binding energy of Nnotably, about 0:68 A. doped DC system is 8.34 eV lower than that of clean DC system, which imply that the N-doped DC system is more stable. These analyses show
that N impurity could induce a remarkable distortion of the environment atomic sites in edge DC system in a-iron. 3.2. Energy analysis In order to study the interaction of N impurity with the DC system, we have calculated two energetic quantities: the structural energy in realspace Green Function method [31] and the interatomic energy [32,33]. Structural energy El of atom l is expressed as: Z EF El nl EE dE; 1
where nl E is the local density of states of atom l, and EF is the Fermi energy of the system. It was used to de®ne the impurity formation energy in treating defect system [34]. An impurity at its stable atomic site usually displays a low structural energy. Thus structural energy can re¯ect the interaction of the impurity with the environment in host. Interatomic energy Elm describes the interaction strength between atom l and m. It is de®ned as XX Elm Nn anal anbm Halbm ; n
ab
where Nn is the electron occupation number in molecular orbital un , anal h/al jun i, and Halbm is the Hamiltonian matrix element connecting the atomic orbital /bm of atom m to the atomic orbital /al of atom l. Interatomic energy diers from the usual bond order parameter by that Hamiltonian
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Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
matrix and overlap matrix are used, respectively, for the two quantities' de®nitions. Interatomic energy was successfully applied to the studies on the Ni3 Al grain boundary [27±29]. Usually the lower the interatomic energy, the stronger the interaction is. In this work, the structural energy and the interatomic energy are calculated using the data obtained by DV method. The structural energies for some interesting atoms are listed in Table 2. In the N-doped DC system, N impurity has the lowest structural energy about )8.90 eV, which shows that N impurity stays at a site of energy valley. The result re¯ects the trapping eect of DC on N impurity. In addition, the structural energies of Fe1 and Fe3 atoms display noticeable changes compared to those in the clean DC system, which implies the strong interactions of N impurity with Fe1 and Fe3 atoms. The interactions re¯ect the pinning eect of N impurity on the dislocation. In Table 3, the interatomic energies of some typical atomic pairs in clean DC system and Ndoped DC system are presented. We can see that when N impurity enters into the DC system, the bonding strength between its neighboring Fe atoms is weakened. Comparing to the clean DC
Table 2 The structural energies (in eV) of the selected atoms in clean DC system and N-doped DC system N Fe1 Fe3 Fe5 Fe9
Clean DC
N-doped DC
)2.83 )2.76 )2.98 )2.77
)8.90 )2.72 )2.95 )2.97 )2.76
Table 3 The interatomic energies (in eV) of the typical atomic pairs in clean DC system and N-doped DC system Atomic pair
Clean DC
N-doped DC
DEa
Fe1±Fe3 Fe3±Fe4 Fe3±Fe5 Fe3±Fe9 Fe5±Fe6 Fe5±Fe7 N±Fe1 N±Fe3 N±Fe5 N±Fe9
)2.13 )1.83 )0.83 )1.70 )0.33 )1.66
)0.35 0.25 )0.15 )1.47 )0.32 )1.73 )4.50 )3.95 0.17 0.28
1.75 2.08 0.68 0.23 0.01 )0.07
a
DE is the dierence of the interatomic energy of atomic pair in N-doped DC system and clean DC system.
Table 4 The electron occupation number n n0 in valence orbital of the selected atoms in N-doped DC system (clean DC system)a Clean cluster n0 N
Fe1
Fe3
Fe9
a
2s 2p Q
N-doped cluster n
Dn
1.68 3.87 5.55
DQ
+0.55
3d 4s 4p Q
6.34 0.76 0.87 7.97
6.37 0.62 0.72 7.71
+0.03 )0.14 )0.15
3d 4s 4p Q
6.35 0.78 0.91 8.04
6.34 0.69 0.82 7.85
)0.01 )0.09 )0.09
3d 4s 4p Q
6.34 0.75 0.80 7.89
6.36 0.72 0.80 7.88
+0.02 )0.03 )0.00
)0.26
)0.19
)0.01
Q is the summation of the electron occupation number in valence orbitals. For Fe atom, Dn n Q N-doped DC Q clean DC. For N impurity, DQ Q N-doped DC Q free atom.
n0 , and DQ
Y. Niu et al. / Computational Materials Science 22 (2001) 144±150
149
(a)
13
[ 010] direction
11
1
(b)
13
12
2 N
9
N 3
4
14
15
10
[100] direction
[001] direction
Fig. 2. The charge density dierences for two planes: (a) the center stacking plane A along the [0 0 1] direction containing N, Fe1 and Fe2 atoms; (b) a (1 0 0) plane containing N, Fe3 and Fe4 atoms. The circled numbers correspond to those in Fig. 1. The contour 3 . Solid lines and dashed lines indicate the gain and the loss of electrons, respectively. spacings are 0:001 e=A
system, the interatomic energies between the Fe atoms which are adjacent to N impurity such as Fe1 ±Fe3 and Fe3 ±Fe5 in the N-doped DC system decrease by about 17.8% and 18.1%, respectively. In the direction of the dislocation line, the sign of interatomic energy between Fe3 and Fe4 changes from minus to plus, which implies the strongly weakened interaction between Fe3 and Fe4 atoms because of their increased interatomic distance induced by the N impurity. In addition, N displays strong interactions with its adjacent Fe atoms. The interatomic energies of atomic pairs of N±Fe1 and N±Fe3 are )4.50 and )3.95 eV, respectively. The above results indicate that N impurity could attract the adjacent Fe atoms in DC, and conversely DC could exert the trapping eect on N impurity. 3.3. Charge distribution Naturally, the charge transfers among the impurity and the host Fe atoms play an important
role in the formation of atom bond. Table 4 lists the electron occupation number in the valence orbitals of N impurity and its neighboring Fe atoms obtained by Mulliken population [35]. It can be seen that N impurity obtains 0.55 electrons from the adjacent Fe atoms and assigns them mainly to its 2p valence orbital. The adjacent Fe1 , Fe3 and Fe11 atoms lose 0.26, 0.19 and 0.01 electrons, respectively, and parts of the electrons from the 4s and 4p orbitals are transferred to N impurity. One can also obtain directly an insight into the bonding character by charge density distribution. Figs. 2(a) and (b) illustrate the charge density dierences on the center stacking plane A which contains N, Fe1 and Fe2 atoms, and on a (0 1 0) plane which contains N, Fe3 and Fe4 atoms, respectively. The charge density dierences are obtained by subtracting the charge density of the clean DC system and that of a free N atom from that of N-doped DC system. Thus the impurity-
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induced charge redistribution in the DC system can be seen clearly. We can see that a remarkable number of electrons accumulate near the N impurity, and there are strong charge correlation curves connecting the N impurity with its adjacent Fe atoms from Fe1 to Fe4. The results indicate that the covalent-like bonds are formed between the N impurity and Fe atoms from Fe1 to Fe4. The feature of the charge density distribution re¯ects again that the N impurity has a strong interaction with its adjacent Fe atoms, which agrees well with the analyses on interatomic energy results. These results imply the nature of quantummechanical eect of Cottrell atmosphere in DC system. 4. Conclusion We have studied the electronic eect of N impurity in an h1 0 0i edge DC system in a-iron by use of ®rst-principles methods. We found that the N impurity prefers to stay at a stable position in the dilated region in the edge DC system. The calculated energetic parameters indicate that the N impurity has a strong interaction with its adjacent Fe atoms. There are charge transfers from the host Fe atom to the N impurity. The charge density distribution shows that the N impurity forms covalent-like bond with its adjacent Fe atoms. These results suggest the formation of N impurity±Fe edge dislocation complex, which implies the trapping eect of DC on N impurity. Acknowledgements The authors are grateful to Dr. Hongbo Liu for his helpful work in the MD simulation of the dislocation. This work was supported by the National Natural Science Foundation of China (59971041) and 973 National Pandeng Project (G2000067102).
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