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2〈1 1 1〉{0 1 1} edge dislocations in bcc iron
Computational Materials Science 38 (2006) 39–44 www.elsevier.com/locate/commatsci
Electronic structures of the h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} edge dislocations in bcc iron Li-qun Chen a
a,*
, Chong-yu Wang
a,b,c
, Tao Yu
a
Central Iron and Steel Research Institute, Institute of Functional Materials, No. 76 Xueyuan Nanlu, Beijing 100081, China b Department of Physics, Tsinghua University, Beijing 100084, China c International Center for Materials Physics, Chinese Academy of Science, 110016, China Received 1 October 2005; received in revised form 12 January 2006; accepted 16 January 2006
Abstract The discrete variational method within the framework of density functional theory is used to study the electronic structures of the h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} edge dislocations in bcc Fe, and the density of states, the charge density, the structural energy and the interatomic energy are obtained. The results show that localized electronic states exist in the cores of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} edge dislocations, but not in the 1/2h1 1 1i{0 1 1} edge dislocation. The features of the h1 0 0i{0 1 0} edge dislocation are similar to those of the h1 0 0i{0 1 1} edge dislocation, but different from those of the 1/2h1 1 1i{0 1 1} edge dislocation. In addition, there is an intrinsic hindrance of the lattice to the dislocation motion, namely, an effect of trapping of lattice on the dislocation. For the 1/2h1 1 1i{0 1 1} edge dislocation, the interaction between the atoms along the slip direction is much stronger than that normal to the slip direction. However, in the h1 0 0i type edge dislocation, the interatomic bonds along and normal to the slip direction have almost the same strength, and the bond is even stronger normal to the slip direction in the h1 0 0i{0 1 1} edge dislocation. The results show that the motion of the 1/2h1 1 1i edge dislocation may be easier than that of the h1 0 0i edge dislocation under a stress field. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.72.LK; 71.15.Mb; 71.55.AK Keywords: Edge dislocation; First-principles calculations; Electronic structure
1. Introduction The dislocation is a kind of very important structure defect in crystalline material, which is closely related to mechanical, chemical, and photoelectric properties. The geometrical feature, the elastic theory and the electron microscopic experiment for dislocation have been well developed. However, the elastic theory on the basis of the continuous medium has some difficulty for the description of the dislocation core because of the two reasons: one is the singularity in the core part, and the other is the elastic theory smoothing out the characteristic of discrete lattice structure differences between materials [1], which must be *
Corresponding author. Fax: +86 010 62182756. E-mail address: [email protected] (L.-q. Chen).
0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.01.005
considered for the dislocation core. Thus, the electronic effect of the dislocation core with about 1.25–1.67 Burger vectors [2] may turn into a dominant factor. There is an experimental evidence showing the importance of electronic factors in the properties of dislocations and their interaction with other defects in metals [2–5]. There have been some investigations at either the atomic level or the electronic level on the dislocation core in structure materials [6–10]. De Hosson [11] investigated the localized states of the h1 0 0i{0 1 0} edge dislocation (ED) in bcc Fe by the self-consistent-field Xa method with a multiple-scattering model. He found that the variations of hydrostatic pressure arising from an ED can result in a rearrangement of the conduction electrons of the Fe atom, and electrons tend to flow away from the compression region toward the expansion region. Using a tight-binding-type
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L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
electronic theory, Masude et al. [12] studied the atomic structures and calculated the energies of the 1/ 2h1 1 1i{0 1 1} and the 1/2h1 1 1i{112} EDs in bcc Fe. They showed that the electronic effect on the dislocation core is important and the Peierls stress of the screw dislocation is approximately 10 times larger than that of the edge dislocation. Wang et al. [13] calculated the electronic structure of the h1 0 0i{0 1 0} ED in bcc Fe by use of the recursion method and it is found that the ED results in the splitting of degenerate states, and the movement of Fermi level. Yan et al. [14] investigated the electronic structure of C doped at different types of interstitial sites in the h1 0 0i{0 1 0} edge dislocation core (EDC) of bcc Fe. The results showed that some charge accumulations in the expansion region were found, and the inhomogeneous charge distribution in the dislocation core was induced. Furthermore, the trapping effect on C appears at the dislocation core center. Using the first-principles real space tight-binding linear-muffin-tin-orbital recursion method, Kontsevoi et al. [15,16] investigated the electronic structure of different types of edge dislocations in B2 intermetallic NiAl and bcc transition metals. They observed an unusual localization of electronic states inside the valence band in the cores of the h1 0 0i{0 1 0} and the h1 0 0i{0 1 1} EDs. The studies on the electronic structure of the EDC in bcc Fe are almost concentrated on the h1 0 0i{0 1 0} ED. However, the Burger vector of the dislocation observed experimentally is generally 1/2h1 1 1i in bcc Fe. The structure and energies of the 1/2h1 1 1i ED in bcc Fe were reported by Masude et al. [12]. In order to understand the property of the dislocation in bcc Fe further, the electronic structure for the 1/2h1 1 1i ED should be investigated. In recent years, density functional theory [17,18] as well as the various subsequently developed first-principles methods have shown great potential and advantages in the way of treating low dimension or defect system [19]. In this work, the discrete variational method (DVM) [20–23] within the framework of density functional theory is used to study the electronic structure of the h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} EDs in bcc Fe. The method and the model are presented in Section 2, the calculated results are discussed in Section 3, and some conclusions are given in the last section. 2. Method and computational model The DVM, which is a first-principles numerical method for solving the local density functional equations [20–23], has been used successfully to study electronic structure of metals [23], alloys [21], and intermetallic compounds [24]. In the work, we employ DVM to calculate the energetics and the electronic structure of the dislocation cores in bcc Fe. In the one-electron-like wave equation, the Hamiltonian is H ¼ r2 =2 þ V C þ V xc ;
ð1Þ
where VC is the electron–nucleus and electron–electron Coulomb potential, and Vxc derived by Von Barth and Hedin [25] is the exchange–correlation potential. The eigenstates (or molecular orbitals) wn(r) are expanded as linear combinations of atomic orbitals /i(r) X cni /i ðrÞ. ð2Þ wn ðrÞ ¼ i
The structural energy [26] can be used to describe the total effects of other atoms in the system on the interest atom, and it is defined as R EF nðEÞE dE Es ¼ R1 ; ð3Þ EF nðEÞ dE 1 where n(E) is the local state density of this atom, and EF is the Fermi energy. In order to study the interaction between atoms, the interatomic energy between atom l and atom m is derived [27,28] XX Elm ¼ N n anal anbm H bmal ; ð4Þ n
ab
where Nn is the occupation number for molecular orbital wn, anal = h/al(r)jwn(r)i, and Hbmal is the Hamiltonian matrix element connecting the atomic orbital b of atom m and the atomic orbital a of atom l. In the computation, the single site basis set with the frozen core mode is used in the atomic orbitals and the funnel potential is added for realizing the bound states. The nonspin-polarized secular equations are solved using the self-consistent charge (SCC) approximation, and the convergence is accepted if the root-mean-square of all atomic orbital populations differs by less than 105 between two successive iterations. The h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and h1 1 1i{0 1 1} EDCs in bcc Fe are taken as the computational models. In order to get the atomic configurations of the dislocation cores for the first-principles calculation, we first adopt the molecular dynamics (MD) method with the Finnis–Sinclair potential [29,30], a well-known semi-empirical N-body potential for transition metals, to simulate the three kinds of EDs. The initial atomic configurations of the dislocation systems for the MD simulation were determined by use of the elastic stress-field theory. After the relaxation, we obtained the atomic configurations with C2V symmetry for the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs, and C2 symmetry for the 1/2h1 1 1i{0 1 1} ED, respectively. The structural model for the h1 0 0i{0 1 0} ED consists of the two non-equivalent (0 0 1) planes (respectively signed as A and B), perpendicular to the direction of the dislocation line, the two nonequivalent (0 1 1) planes (respectively also signed as A and B) for the h0 1 0i{0 1 1} ED, and the six non-equivalent (1 1 2) planes (respectively signed as A, B, C, D, E and F) for the 1/2h1 1 1i{0 1 1} ED. Then, for the first-principles calculation, we constructed the computational models of the three kinds of dislocation cores from the MD results. The computational models of the h1 0 0i{0 1 0} and
L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
h1 0 0i{0 1 1} EDs consist of 209 and 220 Fe atoms, respectively, and contain seven layers with the stacking sequence
(a)
[010] direction
19 5
3
1
4
2
9
6
12
10
7 8 14
16
13
11
15
41
ABABABA along the direction of the dislocation line. In order to fully analyze for each nonequivalent (1 1 2) plane, we constructed the two computational models for the 1/2h1 1 1i{0 1 1} ED, which include 202 and 203 Fe atoms with the stacking sequence ABCDEFA and DEFABCD along the direction of the dislocation line, respectively. The computational models are illustrated in Fig. 1. The sizes of the computational models are determined in terms of the radius of the effective dislocation core region. The essential features of the electronic structure can be well described in the above computational models without loss of significant accuracy.
17
3. Results and discussion
18
3.1. Density of states
[100] direction (b)
4
[011] direction
2
5
3
25 1
6
9
7
8 10 12 26
The density of states (DOS) is obtained by broadening the discrete eigenvalue spectrum with a set of Lorentz functions. The local densities of states (LDOS) for some atoms in the h1 0 0i{0 1 0} ED is shown in Fig. 2. The LDOS for pure bcc Fe is also calculated for comparison. In Fig. 2,
11
40
bcc Fe
Fe 4
Fe 5
Fe 12
Fe 13
Fe 3
Fe 2
Fe 19
Fe 17
Fe 11
Fe 18
Fe 10
30 20
16
13 14 15
17
19 18
21 23 20
24
10 0 40
22
30 20 10 0 40
[01-1] direction
(c)
1 3 5 7 9 11 1416 18 202224 2 4 6 8 1012 1315 1719 2123 26 28 30 32 34 37 39 41 43 45 47 25 27 29 31 33 35 36 38 40 42 44 46 48
DOS (arbitrary unit)
[100] direction
30 20 10 0 40
30 20 10 0 40
30 20 10 0 40
[111] direction Fig. 1. Atomic models of the three types of EDCs. A repeated unit along the direction of dislocation line is drawn for each kind of dislocation. The square, solid circle, up triangle, down triangle, rhombus and left triangle symbols denote the atoms on the stacking planes A, B, C, D, E and F, respectively along the dislocation line direction. The dashed line is the slip plane. (a) The h1 0 0i(010) ED; (b) the h1 0 0i(011) ED; (c) the 1/2h1 1 1i{0 1 1} ED.
30 20 10 0
-10 -8 -6 -4 -2
0
Energy (eV)
2
4 -10 -8 -6 -4 -2
0
2
4
Energy (eV)
Fig. 2. The LDOSs for Fe atoms in the core of the h1 0 0i{0 1 0} ED. The Fermi level is shifted to zero.
L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
3.2. Energy analysis The structural energy can be used to describe the effects of other atoms in the system on the interest atom. An atom with a more negative structural energy implies that the atom is more stable at the site. According to Eq. (3), the structural energies of some atomic sites in the three types of dislocation cores are calculated. In order to understand
40
PDOS (arbitrary unit)
the LDOSs of some atoms (for example, Fe5 and Fe13 atoms) at EF are decreased as compared with that in pure bcc Fe, indicating that the electronic density in the system is redistributed, and giving rise to some localized electronic states. For example, a well-defined localized peak can be found at about 4.6 eV on the LDOS for Fe5 atom in the h1 0 0i{0 1 0} ED. A well-defined localized peak is also found at about 1.9 eV on the LDOS for Fe13 atom. For the LDOS of Fe13 atom, another interesting phenomena is the remarkable splitting between bonding and antibonding states near EF. The localized electronic states are absent from the LDOS of pure bcc Fe and all other Fe atoms beyond the central atoms (Fe5 and Fe13 atoms are regarded as the ‘‘central atoms of the dislocation core’’). The formation of the localized electronic states may be associated with the ‘‘broken bonds’’, or a change of coordination number in comparison with the pure bcc Fe. The LDOSs for some atoms in the h1 0 0i{0 1 1} ED are similar to those in the h1 0 0i{0 1 0} ED. The localized electronic states are found from the LDOSs of Fe7 and Fe19 atoms (Fe7 and Fe19 atoms are also regarded as the ‘‘central atoms of the dislocation core’’). The splitting between bonding and antibonding states near EF is also observed on the LDOS for Fe19 atom. We also calculate the LDOSs for some atoms in the 1/2h1 1 1i{0 1 1} ED, which are different from those of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs. A well-defined localized peak is not found on the LDOS for any atom in the 1/2h1 1 1i{0 1 1} EDC. This indicates that the localized state is closely related with the type of the dislocation. In comparison with the atomic core configurations for the three types of dislocations, it can be seen that the configurations of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDCs have C2V symmetry, which has the mirror symmetry with respect to the reflection in the extra plane. However, the configuration of the 1/2h1 1 1i{0 1 1} EDC is C2 symmetry, with no mirror symmetry. This implies that the appearance of the localized electronic states may be related to the local symmetry of the atoms in the core. In addition, the d-electron of the central atom of the dislocation core plays an important role in the appearance of the localized electronic states. In Fig. 3, the partial density of states (PDOS) of the d-electron for the central atoms in the h1 0 0i{0 1 0} and the h1 0 0i{0 1 1} EDCs is shown with the PDOS for Fe in pure bcc iron. It is obvious that d-electrons give the main contribution to the localized electronic states. They are the main factor leading to the strong localization.
Fe 5
(a)
Fe 13
30
20
10
0
-10 -8
-6
-4
-2
0
2
4
-10 -8
Energy (eV)
40
PDOS (arbitrary unit)
42
-6
-4
-2
0
2
4
Energy (eV)
Fe 7
(b)
Fe 19
30
20
10
0
-10 -8
-6
-4
-2
0
Energy (eV)
2
4
-10 -8
-6
-4
-2
0
2
4
Energy (eV)
Fig. 3. The PDOS for d states on the central atoms in the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs compared with the d-PDOS in pure bcc Fe. The solid line denotes the dislocation core; the dashed line denotes the pure bcc Fe. The Fermi level is shifted to zero. (a) The h1 0 0i{0 1 0} ED; (b) the h1 0 0i{0 1 1} ED.
the variation of the structural energy of the atomic sites in the dislocation core more clearly, a plot of the absolute value of the structural energy of the atomic sites in the compression region and expansion region vs the order of the atoms is drawn in Fig. 4. The structural energy of Fe atomic site in pure bcc iron is also calculated for comparison, and its value is 3.15 eV. From Fig. 4, the structural energies of some atomic sites above the slip plane (illustrated in Fig. 1) in three types of dislocation cores are lower than that in pure bcc iron, which indicates that those atoms may be more stable than that in pure bcc iron. Fig. 4(a) is similar to Fig. 4(b): there are three barriers in the compression region, and there is an energy well in the expansion region; the fluctuation of the structural energy is large. However, Fig. 4(c) is obviously different from Fig. 4(a) and (b): in the compression region, the fluctuation of the structural energy is smaller than that in the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs; in the expansion region, the change of the structural energy is in oscillation. From Fig. 4, it can be seen that Fe3, Fe5, and Fe7 atoms in the h1 0 0i{0 1 0}, Fe4, Fe7 and Fe9 atoms in the h1 0 0i{0 1 1}, and Fe4–Fe21 atoms in the 1/2h1 1 1i{0 1 1} EDs must surmount the barrier during the dislocation moving. This reflects an intrinsic hindrance of the lattice to the dislocation motion, namely, an effect of trapping of lattice on dislocation. However, the resistance in different types of dislocation is different.
L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
3.0
11
8
6
4
15
14
12
3.0
16
2.8 2.6
13
0
1
2
3 4 5 Order of atom
Es (eV)
2 13
3.0
5
3
1
3.2
7
6
2
Es (eV)
4
4 5 6 78
9
3.0 2.8 2.6 0
4
10-23 12-24
1.5 1.0
8-13 4-10
6-12
3-7
4-6
12-17
5-8 7-10
1-5
13-19 8-12
25-3 1-3
1-10
15-21
10-26 2-4
3-5
5-7
4-13(6-13)
8-15
2-11
7-9
9-16
7-14
3-12
24
6-8
0.0
10
16
15
37
12
(c)
17
40 39
41
18 20 19 21 2223 24 46
42 44 45
43
48 47
38
8 12 Order of atom
8-21
2
3
4
5
6
7
Order of atom
14 13
36 35
2.0
1
11 12
5-16 7-19 10-15
1-13
0.5
21
6 8 Order of atom
10 27 29 31 33 25 26 28 30 32 34
3.2
12
3-14
2.5
19
2.6
3 1 2
10 11
20
17 18
3.4
8
22 23 16
0
8
(b)
14 15
2.8
7
9
4
3.4
6
Interatomic energy (eV)
Es (eV)
2 10
<100>{010}ED <100>{011}ED 1/2<111>{011}ED
7
3
3.2
3.5
(a)
5
3.4
43
16
20
24
Fig. 4. The alteration of the structural energies (absolute value) of the atomic sites in the dislocation core. The solid circles show the energies of the selected atomic sites in the compression region, and the open ones are relevant to those in the expansion region. The dashed line indicates the structural energy of Fe atom in pure bcc iron by our calculation. The numbers correspond to those in Fig. 1. (a) The h1 0 0i{0 1 0} ED; (b) the h1 0 0i{0 1 1} ED; (c) the 1/2h1 1 1i{0 1 1} ED.
By analyzing the density of states and the structural energy, we can find that the features of the h1 0 0i{0 1 0} ED are similar to those of the h1 0 0i{0 1 1} ED, but not to those of the 1/2h1 1 1i{0 1 1} ED. It shows that the property of the edge dislocation may be closely related with the intrinsic parameter—Burgers vector. In order to investigate the effect of different types of EDs in bcc Fe on bonding behavior more clearly, the interatomic energies defined by Eq. (4) were calculated for the three types of EDs, and shown in Fig. 5. Our computational result of the interatomic energy indicates that the non-bond characteristic will exist when the distance between the adjacent atoms is larger than 7.5 au. The distances between the atoms normal to the slip direction in the 1/2h1 1 1i{0 1 1} EDC all are almost larger than 7.5 au. This implies that the interaction between the atoms normal to the slip direction in the 1/2h1 1 1i{0 1 1} EDC is very weak, so it is not drawn in Fig. 5. From Fig. 5, it can be seen that the interaction between the atoms along the slip direction is the strongest in the 1/2h1 1 1i{0 1 1} EDC, the weakest in the h1 0 0i{0 1 0}
Fig. 5. The interatomic energy (absolute value) of the selected atom-pairs in the three types of EDCs. The numbers correspond to those in Fig. 1. The solid lines indicate the interatomic energies of atom pairs along the slip direction, and the dashed lines indicate the interatomic energies of atom-pairs crossing the slip plane.
except Fe4–Fe6 and the middle in the h1 0 0i{0 1 1} EDC. However, the interaction between the atoms crossing the slip plane is the weakest in the 1/2h1 1 1i{0 1 1} EDC, the strongest in the h1 0 0i{0 1 1} and the middle in the h1 0 0i{0 1 0} EDC. It shows that the motion of the 1/2h1 1 1i{0 1 1} ED is easier than that of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs under the stress field, which is in agreement with the experimental result [31]. 3.3. Charge density From the charge density of the dislocation core, one can obtain a direct understanding of the interatomic bonding characteristics. The charge-density difference is obtained by subtracting the charge density of free atoms at the superposition from the charge density of the defect system. For each type of ED, the distribution of charge on the nonequivalent planes (A–F) is similar. So we only analyze the charge density on the plane A for the three types of EDCs. Fig. 6 represents the charge-density difference on the plane A in the h1 0 0i{0 1 0} EDC. It is obvious that the charge distribution is inhomogeneous in the dislocation cores. This may be because the introduction of dislocations leads to the charge redistribution, which may give rise to a significant local reconstruction of the electronic states in the dislocation core. From Fig. 6, we can observe that the strength of the bonds between the atoms (Fe4–Fe6) along slip direction is slightly stronger than those between the atoms (Fe4– Fe13, Fe6–Fe13) crossing the slip plane. For the h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} EDCs, the charge distributions are also inhomogeneous in the core regions. However, the characteristics of bonding between atoms are different from those in the h1 0 0i{0 1 0} ED. For the h1 0 0i{0 1 1} EDC, the bond strength between the atoms
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L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
21
22
[010]
A plane
4
2
11
6
13
20 19
an effect of trapping of lattice on dislocation. The hindrance is related with the kind of dislocation. The strength of the bonding along the slip direction is stronger in the 1/ 2h1 1 1i ED than in the h1 0 0i ED, but the strength of the bonding normal to the slip direction is weaker. The results show that the motion of the 1/2h1 1 1i ED may be easier than that of the h1 0 0i ED under a stress field. Acknowledgements The authors are grateful to Dr. Hongli Dang for beneficial discussion. This work was supported by ‘‘973’’ Project from the Ministry of Science and Technology of China (Grant No. G2000067102) and National Natural Science Foundation of China (Grant No. 90306016).
[100] Fig. 6. The charge-density difference on the plane A in the h1 0 0i{0 1 0} ˚ 3. The solid lines and dashed EDC. The contour spacings are 0.002 eA lines indicate the gain and the loss of electrons, respectively. The numbers marked in the plots correspond to Fe atoms in Fig. 1.
(Fe3–Fe7, Fe7–Fe10) along slip direction as well as that between the atoms (Fe3–Fe14, Fe7–Fe19, Fe10–Fe21) crossing the slip plane is the same. For the 1/ 2h1 1 1i{0 1 1} EDC, the strength of the bonds between the atoms (Fe9–Fe16) along the slip direction is much stronger than those between the atoms (Fe9–Fe37, Fe16– Fe37) crossing the slip plane. It shows that the motion of the 1/2h1 1 1i{0 1 1} ED is easier than that of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs under the stress field. 4. Conclusion We have performed first-principles calculation for the electronic structures of three types of edge dislocations. The results present that the introduction of dislocations in bcc Fe leads to a local reconstruction of the electronic states. The reconstruction results in the formation of localized electronic states inside the valence band. The possible conditions for the appearance of localized states are (1) a decrease of the coordination number around the central atom of the dislocation core; (2) a specific local symmetry of the atomic arrangement in the region of the dislocation core; and (3) the contribution of d-electron to the localized electronic states. By the analysis of the density of states, the charge density, the structural energy, and the interatomic energy, it can be found that the features of the h1 0 0i{0 1 0} ED are similar to those of h1 0 0i{0 1 1} ED, but not those of the 1/2h1 1 1i{0 1 1} ED. This shows that the property of the edge dislocation may be closely related with the intrinsic parameter—Burgers vector. Secondly, there is an intrinsic hindrance of the lattice to the dislocation motion, namely,
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Electronic structures of the h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} edge dislocations in bcc iron Li-qun Chen a
a,*
, Chong-yu Wang
a,b,c
, Tao Yu
a
Central Iron and Steel Research Institute, Institute of Functional Materials, No. 76 Xueyuan Nanlu, Beijing 100081, China b Department of Physics, Tsinghua University, Beijing 100084, China c International Center for Materials Physics, Chinese Academy of Science, 110016, China Received 1 October 2005; received in revised form 12 January 2006; accepted 16 January 2006
Abstract The discrete variational method within the framework of density functional theory is used to study the electronic structures of the h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} edge dislocations in bcc Fe, and the density of states, the charge density, the structural energy and the interatomic energy are obtained. The results show that localized electronic states exist in the cores of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} edge dislocations, but not in the 1/2h1 1 1i{0 1 1} edge dislocation. The features of the h1 0 0i{0 1 0} edge dislocation are similar to those of the h1 0 0i{0 1 1} edge dislocation, but different from those of the 1/2h1 1 1i{0 1 1} edge dislocation. In addition, there is an intrinsic hindrance of the lattice to the dislocation motion, namely, an effect of trapping of lattice on the dislocation. For the 1/2h1 1 1i{0 1 1} edge dislocation, the interaction between the atoms along the slip direction is much stronger than that normal to the slip direction. However, in the h1 0 0i type edge dislocation, the interatomic bonds along and normal to the slip direction have almost the same strength, and the bond is even stronger normal to the slip direction in the h1 0 0i{0 1 1} edge dislocation. The results show that the motion of the 1/2h1 1 1i edge dislocation may be easier than that of the h1 0 0i edge dislocation under a stress field. Ó 2006 Elsevier B.V. All rights reserved. PACS: 61.72.LK; 71.15.Mb; 71.55.AK Keywords: Edge dislocation; First-principles calculations; Electronic structure
1. Introduction The dislocation is a kind of very important structure defect in crystalline material, which is closely related to mechanical, chemical, and photoelectric properties. The geometrical feature, the elastic theory and the electron microscopic experiment for dislocation have been well developed. However, the elastic theory on the basis of the continuous medium has some difficulty for the description of the dislocation core because of the two reasons: one is the singularity in the core part, and the other is the elastic theory smoothing out the characteristic of discrete lattice structure differences between materials [1], which must be *
Corresponding author. Fax: +86 010 62182756. E-mail address: [email protected] (L.-q. Chen).
0927-0256/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2006.01.005
considered for the dislocation core. Thus, the electronic effect of the dislocation core with about 1.25–1.67 Burger vectors [2] may turn into a dominant factor. There is an experimental evidence showing the importance of electronic factors in the properties of dislocations and their interaction with other defects in metals [2–5]. There have been some investigations at either the atomic level or the electronic level on the dislocation core in structure materials [6–10]. De Hosson [11] investigated the localized states of the h1 0 0i{0 1 0} edge dislocation (ED) in bcc Fe by the self-consistent-field Xa method with a multiple-scattering model. He found that the variations of hydrostatic pressure arising from an ED can result in a rearrangement of the conduction electrons of the Fe atom, and electrons tend to flow away from the compression region toward the expansion region. Using a tight-binding-type
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L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
electronic theory, Masude et al. [12] studied the atomic structures and calculated the energies of the 1/ 2h1 1 1i{0 1 1} and the 1/2h1 1 1i{112} EDs in bcc Fe. They showed that the electronic effect on the dislocation core is important and the Peierls stress of the screw dislocation is approximately 10 times larger than that of the edge dislocation. Wang et al. [13] calculated the electronic structure of the h1 0 0i{0 1 0} ED in bcc Fe by use of the recursion method and it is found that the ED results in the splitting of degenerate states, and the movement of Fermi level. Yan et al. [14] investigated the electronic structure of C doped at different types of interstitial sites in the h1 0 0i{0 1 0} edge dislocation core (EDC) of bcc Fe. The results showed that some charge accumulations in the expansion region were found, and the inhomogeneous charge distribution in the dislocation core was induced. Furthermore, the trapping effect on C appears at the dislocation core center. Using the first-principles real space tight-binding linear-muffin-tin-orbital recursion method, Kontsevoi et al. [15,16] investigated the electronic structure of different types of edge dislocations in B2 intermetallic NiAl and bcc transition metals. They observed an unusual localization of electronic states inside the valence band in the cores of the h1 0 0i{0 1 0} and the h1 0 0i{0 1 1} EDs. The studies on the electronic structure of the EDC in bcc Fe are almost concentrated on the h1 0 0i{0 1 0} ED. However, the Burger vector of the dislocation observed experimentally is generally 1/2h1 1 1i in bcc Fe. The structure and energies of the 1/2h1 1 1i ED in bcc Fe were reported by Masude et al. [12]. In order to understand the property of the dislocation in bcc Fe further, the electronic structure for the 1/2h1 1 1i ED should be investigated. In recent years, density functional theory [17,18] as well as the various subsequently developed first-principles methods have shown great potential and advantages in the way of treating low dimension or defect system [19]. In this work, the discrete variational method (DVM) [20–23] within the framework of density functional theory is used to study the electronic structure of the h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} EDs in bcc Fe. The method and the model are presented in Section 2, the calculated results are discussed in Section 3, and some conclusions are given in the last section. 2. Method and computational model The DVM, which is a first-principles numerical method for solving the local density functional equations [20–23], has been used successfully to study electronic structure of metals [23], alloys [21], and intermetallic compounds [24]. In the work, we employ DVM to calculate the energetics and the electronic structure of the dislocation cores in bcc Fe. In the one-electron-like wave equation, the Hamiltonian is H ¼ r2 =2 þ V C þ V xc ;
ð1Þ
where VC is the electron–nucleus and electron–electron Coulomb potential, and Vxc derived by Von Barth and Hedin [25] is the exchange–correlation potential. The eigenstates (or molecular orbitals) wn(r) are expanded as linear combinations of atomic orbitals /i(r) X cni /i ðrÞ. ð2Þ wn ðrÞ ¼ i
The structural energy [26] can be used to describe the total effects of other atoms in the system on the interest atom, and it is defined as R EF nðEÞE dE Es ¼ R1 ; ð3Þ EF nðEÞ dE 1 where n(E) is the local state density of this atom, and EF is the Fermi energy. In order to study the interaction between atoms, the interatomic energy between atom l and atom m is derived [27,28] XX Elm ¼ N n anal anbm H bmal ; ð4Þ n
ab
where Nn is the occupation number for molecular orbital wn, anal = h/al(r)jwn(r)i, and Hbmal is the Hamiltonian matrix element connecting the atomic orbital b of atom m and the atomic orbital a of atom l. In the computation, the single site basis set with the frozen core mode is used in the atomic orbitals and the funnel potential is added for realizing the bound states. The nonspin-polarized secular equations are solved using the self-consistent charge (SCC) approximation, and the convergence is accepted if the root-mean-square of all atomic orbital populations differs by less than 105 between two successive iterations. The h1 0 0i{0 1 0}, h1 0 0i{0 1 1} and h1 1 1i{0 1 1} EDCs in bcc Fe are taken as the computational models. In order to get the atomic configurations of the dislocation cores for the first-principles calculation, we first adopt the molecular dynamics (MD) method with the Finnis–Sinclair potential [29,30], a well-known semi-empirical N-body potential for transition metals, to simulate the three kinds of EDs. The initial atomic configurations of the dislocation systems for the MD simulation were determined by use of the elastic stress-field theory. After the relaxation, we obtained the atomic configurations with C2V symmetry for the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs, and C2 symmetry for the 1/2h1 1 1i{0 1 1} ED, respectively. The structural model for the h1 0 0i{0 1 0} ED consists of the two non-equivalent (0 0 1) planes (respectively signed as A and B), perpendicular to the direction of the dislocation line, the two nonequivalent (0 1 1) planes (respectively also signed as A and B) for the h0 1 0i{0 1 1} ED, and the six non-equivalent (1 1 2) planes (respectively signed as A, B, C, D, E and F) for the 1/2h1 1 1i{0 1 1} ED. Then, for the first-principles calculation, we constructed the computational models of the three kinds of dislocation cores from the MD results. The computational models of the h1 0 0i{0 1 0} and
L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
h1 0 0i{0 1 1} EDs consist of 209 and 220 Fe atoms, respectively, and contain seven layers with the stacking sequence
(a)
[010] direction
19 5
3
1
4
2
9
6
12
10
7 8 14
16
13
11
15
41
ABABABA along the direction of the dislocation line. In order to fully analyze for each nonequivalent (1 1 2) plane, we constructed the two computational models for the 1/2h1 1 1i{0 1 1} ED, which include 202 and 203 Fe atoms with the stacking sequence ABCDEFA and DEFABCD along the direction of the dislocation line, respectively. The computational models are illustrated in Fig. 1. The sizes of the computational models are determined in terms of the radius of the effective dislocation core region. The essential features of the electronic structure can be well described in the above computational models without loss of significant accuracy.
17
3. Results and discussion
18
3.1. Density of states
[100] direction (b)
4
[011] direction
2
5
3
25 1
6
9
7
8 10 12 26
The density of states (DOS) is obtained by broadening the discrete eigenvalue spectrum with a set of Lorentz functions. The local densities of states (LDOS) for some atoms in the h1 0 0i{0 1 0} ED is shown in Fig. 2. The LDOS for pure bcc Fe is also calculated for comparison. In Fig. 2,
11
40
bcc Fe
Fe 4
Fe 5
Fe 12
Fe 13
Fe 3
Fe 2
Fe 19
Fe 17
Fe 11
Fe 18
Fe 10
30 20
16
13 14 15
17
19 18
21 23 20
24
10 0 40
22
30 20 10 0 40
[01-1] direction
(c)
1 3 5 7 9 11 1416 18 202224 2 4 6 8 1012 1315 1719 2123 26 28 30 32 34 37 39 41 43 45 47 25 27 29 31 33 35 36 38 40 42 44 46 48
DOS (arbitrary unit)
[100] direction
30 20 10 0 40
30 20 10 0 40
30 20 10 0 40
[111] direction Fig. 1. Atomic models of the three types of EDCs. A repeated unit along the direction of dislocation line is drawn for each kind of dislocation. The square, solid circle, up triangle, down triangle, rhombus and left triangle symbols denote the atoms on the stacking planes A, B, C, D, E and F, respectively along the dislocation line direction. The dashed line is the slip plane. (a) The h1 0 0i(010) ED; (b) the h1 0 0i(011) ED; (c) the 1/2h1 1 1i{0 1 1} ED.
30 20 10 0
-10 -8 -6 -4 -2
0
Energy (eV)
2
4 -10 -8 -6 -4 -2
0
2
4
Energy (eV)
Fig. 2. The LDOSs for Fe atoms in the core of the h1 0 0i{0 1 0} ED. The Fermi level is shifted to zero.
L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
3.2. Energy analysis The structural energy can be used to describe the effects of other atoms in the system on the interest atom. An atom with a more negative structural energy implies that the atom is more stable at the site. According to Eq. (3), the structural energies of some atomic sites in the three types of dislocation cores are calculated. In order to understand
40
PDOS (arbitrary unit)
the LDOSs of some atoms (for example, Fe5 and Fe13 atoms) at EF are decreased as compared with that in pure bcc Fe, indicating that the electronic density in the system is redistributed, and giving rise to some localized electronic states. For example, a well-defined localized peak can be found at about 4.6 eV on the LDOS for Fe5 atom in the h1 0 0i{0 1 0} ED. A well-defined localized peak is also found at about 1.9 eV on the LDOS for Fe13 atom. For the LDOS of Fe13 atom, another interesting phenomena is the remarkable splitting between bonding and antibonding states near EF. The localized electronic states are absent from the LDOS of pure bcc Fe and all other Fe atoms beyond the central atoms (Fe5 and Fe13 atoms are regarded as the ‘‘central atoms of the dislocation core’’). The formation of the localized electronic states may be associated with the ‘‘broken bonds’’, or a change of coordination number in comparison with the pure bcc Fe. The LDOSs for some atoms in the h1 0 0i{0 1 1} ED are similar to those in the h1 0 0i{0 1 0} ED. The localized electronic states are found from the LDOSs of Fe7 and Fe19 atoms (Fe7 and Fe19 atoms are also regarded as the ‘‘central atoms of the dislocation core’’). The splitting between bonding and antibonding states near EF is also observed on the LDOS for Fe19 atom. We also calculate the LDOSs for some atoms in the 1/2h1 1 1i{0 1 1} ED, which are different from those of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs. A well-defined localized peak is not found on the LDOS for any atom in the 1/2h1 1 1i{0 1 1} EDC. This indicates that the localized state is closely related with the type of the dislocation. In comparison with the atomic core configurations for the three types of dislocations, it can be seen that the configurations of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDCs have C2V symmetry, which has the mirror symmetry with respect to the reflection in the extra plane. However, the configuration of the 1/2h1 1 1i{0 1 1} EDC is C2 symmetry, with no mirror symmetry. This implies that the appearance of the localized electronic states may be related to the local symmetry of the atoms in the core. In addition, the d-electron of the central atom of the dislocation core plays an important role in the appearance of the localized electronic states. In Fig. 3, the partial density of states (PDOS) of the d-electron for the central atoms in the h1 0 0i{0 1 0} and the h1 0 0i{0 1 1} EDCs is shown with the PDOS for Fe in pure bcc iron. It is obvious that d-electrons give the main contribution to the localized electronic states. They are the main factor leading to the strong localization.
Fe 5
(a)
Fe 13
30
20
10
0
-10 -8
-6
-4
-2
0
2
4
-10 -8
Energy (eV)
40
PDOS (arbitrary unit)
42
-6
-4
-2
0
2
4
Energy (eV)
Fe 7
(b)
Fe 19
30
20
10
0
-10 -8
-6
-4
-2
0
Energy (eV)
2
4
-10 -8
-6
-4
-2
0
2
4
Energy (eV)
Fig. 3. The PDOS for d states on the central atoms in the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs compared with the d-PDOS in pure bcc Fe. The solid line denotes the dislocation core; the dashed line denotes the pure bcc Fe. The Fermi level is shifted to zero. (a) The h1 0 0i{0 1 0} ED; (b) the h1 0 0i{0 1 1} ED.
the variation of the structural energy of the atomic sites in the dislocation core more clearly, a plot of the absolute value of the structural energy of the atomic sites in the compression region and expansion region vs the order of the atoms is drawn in Fig. 4. The structural energy of Fe atomic site in pure bcc iron is also calculated for comparison, and its value is 3.15 eV. From Fig. 4, the structural energies of some atomic sites above the slip plane (illustrated in Fig. 1) in three types of dislocation cores are lower than that in pure bcc iron, which indicates that those atoms may be more stable than that in pure bcc iron. Fig. 4(a) is similar to Fig. 4(b): there are three barriers in the compression region, and there is an energy well in the expansion region; the fluctuation of the structural energy is large. However, Fig. 4(c) is obviously different from Fig. 4(a) and (b): in the compression region, the fluctuation of the structural energy is smaller than that in the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs; in the expansion region, the change of the structural energy is in oscillation. From Fig. 4, it can be seen that Fe3, Fe5, and Fe7 atoms in the h1 0 0i{0 1 0}, Fe4, Fe7 and Fe9 atoms in the h1 0 0i{0 1 1}, and Fe4–Fe21 atoms in the 1/2h1 1 1i{0 1 1} EDs must surmount the barrier during the dislocation moving. This reflects an intrinsic hindrance of the lattice to the dislocation motion, namely, an effect of trapping of lattice on dislocation. However, the resistance in different types of dislocation is different.
L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
3.0
11
8
6
4
15
14
12
3.0
16
2.8 2.6
13
0
1
2
3 4 5 Order of atom
Es (eV)
2 13
3.0
5
3
1
3.2
7
6
2
Es (eV)
4
4 5 6 78
9
3.0 2.8 2.6 0
4
10-23 12-24
1.5 1.0
8-13 4-10
6-12
3-7
4-6
12-17
5-8 7-10
1-5
13-19 8-12
25-3 1-3
1-10
15-21
10-26 2-4
3-5
5-7
4-13(6-13)
8-15
2-11
7-9
9-16
7-14
3-12
24
6-8
0.0
10
16
15
37
12
(c)
17
40 39
41
18 20 19 21 2223 24 46
42 44 45
43
48 47
38
8 12 Order of atom
8-21
2
3
4
5
6
7
Order of atom
14 13
36 35
2.0
1
11 12
5-16 7-19 10-15
1-13
0.5
21
6 8 Order of atom
10 27 29 31 33 25 26 28 30 32 34
3.2
12
3-14
2.5
19
2.6
3 1 2
10 11
20
17 18
3.4
8
22 23 16
0
8
(b)
14 15
2.8
7
9
4
3.4
6
Interatomic energy (eV)
Es (eV)
2 10
<100>{010}ED <100>{011}ED 1/2<111>{011}ED
7
3
3.2
3.5
(a)
5
3.4
43
16
20
24
Fig. 4. The alteration of the structural energies (absolute value) of the atomic sites in the dislocation core. The solid circles show the energies of the selected atomic sites in the compression region, and the open ones are relevant to those in the expansion region. The dashed line indicates the structural energy of Fe atom in pure bcc iron by our calculation. The numbers correspond to those in Fig. 1. (a) The h1 0 0i{0 1 0} ED; (b) the h1 0 0i{0 1 1} ED; (c) the 1/2h1 1 1i{0 1 1} ED.
By analyzing the density of states and the structural energy, we can find that the features of the h1 0 0i{0 1 0} ED are similar to those of the h1 0 0i{0 1 1} ED, but not to those of the 1/2h1 1 1i{0 1 1} ED. It shows that the property of the edge dislocation may be closely related with the intrinsic parameter—Burgers vector. In order to investigate the effect of different types of EDs in bcc Fe on bonding behavior more clearly, the interatomic energies defined by Eq. (4) were calculated for the three types of EDs, and shown in Fig. 5. Our computational result of the interatomic energy indicates that the non-bond characteristic will exist when the distance between the adjacent atoms is larger than 7.5 au. The distances between the atoms normal to the slip direction in the 1/2h1 1 1i{0 1 1} EDC all are almost larger than 7.5 au. This implies that the interaction between the atoms normal to the slip direction in the 1/2h1 1 1i{0 1 1} EDC is very weak, so it is not drawn in Fig. 5. From Fig. 5, it can be seen that the interaction between the atoms along the slip direction is the strongest in the 1/2h1 1 1i{0 1 1} EDC, the weakest in the h1 0 0i{0 1 0}
Fig. 5. The interatomic energy (absolute value) of the selected atom-pairs in the three types of EDCs. The numbers correspond to those in Fig. 1. The solid lines indicate the interatomic energies of atom pairs along the slip direction, and the dashed lines indicate the interatomic energies of atom-pairs crossing the slip plane.
except Fe4–Fe6 and the middle in the h1 0 0i{0 1 1} EDC. However, the interaction between the atoms crossing the slip plane is the weakest in the 1/2h1 1 1i{0 1 1} EDC, the strongest in the h1 0 0i{0 1 1} and the middle in the h1 0 0i{0 1 0} EDC. It shows that the motion of the 1/2h1 1 1i{0 1 1} ED is easier than that of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs under the stress field, which is in agreement with the experimental result [31]. 3.3. Charge density From the charge density of the dislocation core, one can obtain a direct understanding of the interatomic bonding characteristics. The charge-density difference is obtained by subtracting the charge density of free atoms at the superposition from the charge density of the defect system. For each type of ED, the distribution of charge on the nonequivalent planes (A–F) is similar. So we only analyze the charge density on the plane A for the three types of EDCs. Fig. 6 represents the charge-density difference on the plane A in the h1 0 0i{0 1 0} EDC. It is obvious that the charge distribution is inhomogeneous in the dislocation cores. This may be because the introduction of dislocations leads to the charge redistribution, which may give rise to a significant local reconstruction of the electronic states in the dislocation core. From Fig. 6, we can observe that the strength of the bonds between the atoms (Fe4–Fe6) along slip direction is slightly stronger than those between the atoms (Fe4– Fe13, Fe6–Fe13) crossing the slip plane. For the h1 0 0i{0 1 1} and 1/2h1 1 1i{0 1 1} EDCs, the charge distributions are also inhomogeneous in the core regions. However, the characteristics of bonding between atoms are different from those in the h1 0 0i{0 1 0} ED. For the h1 0 0i{0 1 1} EDC, the bond strength between the atoms
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L.-q. Chen et al. / Computational Materials Science 38 (2006) 39–44
21
22
[010]
A plane
4
2
11
6
13
20 19
an effect of trapping of lattice on dislocation. The hindrance is related with the kind of dislocation. The strength of the bonding along the slip direction is stronger in the 1/ 2h1 1 1i ED than in the h1 0 0i ED, but the strength of the bonding normal to the slip direction is weaker. The results show that the motion of the 1/2h1 1 1i ED may be easier than that of the h1 0 0i ED under a stress field. Acknowledgements The authors are grateful to Dr. Hongli Dang for beneficial discussion. This work was supported by ‘‘973’’ Project from the Ministry of Science and Technology of China (Grant No. G2000067102) and National Natural Science Foundation of China (Grant No. 90306016).
[100] Fig. 6. The charge-density difference on the plane A in the h1 0 0i{0 1 0} ˚ 3. The solid lines and dashed EDC. The contour spacings are 0.002 eA lines indicate the gain and the loss of electrons, respectively. The numbers marked in the plots correspond to Fe atoms in Fig. 1.
(Fe3–Fe7, Fe7–Fe10) along slip direction as well as that between the atoms (Fe3–Fe14, Fe7–Fe19, Fe10–Fe21) crossing the slip plane is the same. For the 1/ 2h1 1 1i{0 1 1} EDC, the strength of the bonds between the atoms (Fe9–Fe16) along the slip direction is much stronger than those between the atoms (Fe9–Fe37, Fe16– Fe37) crossing the slip plane. It shows that the motion of the 1/2h1 1 1i{0 1 1} ED is easier than that of the h1 0 0i{0 1 0} and h1 0 0i{0 1 1} EDs under the stress field. 4. Conclusion We have performed first-principles calculation for the electronic structures of three types of edge dislocations. The results present that the introduction of dislocations in bcc Fe leads to a local reconstruction of the electronic states. The reconstruction results in the formation of localized electronic states inside the valence band. The possible conditions for the appearance of localized states are (1) a decrease of the coordination number around the central atom of the dislocation core; (2) a specific local symmetry of the atomic arrangement in the region of the dislocation core; and (3) the contribution of d-electron to the localized electronic states. By the analysis of the density of states, the charge density, the structural energy, and the interatomic energy, it can be found that the features of the h1 0 0i{0 1 0} ED are similar to those of h1 0 0i{0 1 1} ED, but not those of the 1/2h1 1 1i{0 1 1} ED. This shows that the property of the edge dislocation may be closely related with the intrinsic parameter—Burgers vector. Secondly, there is an intrinsic hindrance of the lattice to the dislocation motion, namely,
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