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Atomistic simulation of the 60∘ dislocation mobility in silicon crystal
Superlattices and Microstructures 40 (2006) 113–118 www.elsevier.com/locate/superlattices
Atomistic simulation of the 60◦ dislocation mobility in silicon crystal Cheng-xiang Li ∗ , Qing-yuan Meng 1 , Gen Li, Li-jun Yang Department of Astronautical Science and Mechanics, Harbin Institute of Technology, P.O. Box 344, Harbin, 150001, PR China Received 22 February 2006; received in revised form 19 April 2006; accepted 19 May 2006 Available online 11 July 2006
Abstract Dislocation velocity and mobility are studied via molecular dynamics simulation for a 60◦ dislocation dipole in silicon crystal. The atomic interactions are described using the Stillinger–Weber potential and the external stress is applied by means of the Parrinello–Rahman algorithm. It is found that the dislocation begins to move when the applied stress is larger than the Peierls stress, and the calculated Peierls stress decreases as the temperature increases, which is in agreement with the Peierls–Nabarro model. The dislocation velocity at relatively low temperature is insensitive to variation of temperature. In fact, the velocity increases monotonically as the stress increases, and eventually approaches its plateau velocity which is about 2900 m/s. At higher temperature, however, the velocity no longer increases monotonically as the stress increases and the plateau velocity decreases as the temperature increases. In general, the dislocation velocity decreases as the temperature increases, which is consistent with the phonon drag model. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Dislocation dipole; Dislocation velocity; The Peierls stress; Molecular dynamics simulation
1. Introduction The heterostructures of strained-layer semiconductor have received considerable attention for the last three decades due to their promising electronic and optoelectronic properties—especially the SiGe/Si heterostructures, due to the existing silicon technology [1–3]. If the thickness of ∗ Corresponding author.
E-mail addresses: [email protected] (C.-X. Li), [email protected] (Q.-Y. Meng). 1 Tel.: +86 451 86414143. c 2006 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2006.05.004
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the epitaxial film is sufficiently small, the film is cohesively strained to match the substrate. However, misfit dislocations may be introduced at the strained interface when a critical thickness is reached [4]. Since these dislocations lead to a poor device performance, many theoretical and experimental studies have focused on these misfit dislocations, particularly on their sources and equilibrium configurations at atomic level [5–7]. A considerable number of prior investigations on modeling and simulation of the dynamic behavior of dislocations have been conducted [8– 11]. For example, dislocations have been assumed to be continuum line defects whose dynamic characteristics are controlled by a driving force, i.e., the Peach–Koehler force [8], and an assumed dislocation mobility law. Simulations of dislocation motion in metals have also been carried out [10,11], and the results on the velocity of edge and screw dislocations in Al and Ni as a function of applied stress and temperature have been presented. However, atomistic simulation of the dislocation mobility in diamond structure, such as silicon, over a large range of temperature and stress has not appeared yet. In the present work, we establish a model consisting of the dislocation dipole and carry out large scale computational simulations. A 60◦ dislocation motion in silicon for temperature ranging from 100 to 900 K and applied stress ranging from 100 to 3000 MPa is investigated extensively by means of the atomistic simulation method. The dislocation velocity, which is regarded as a function of applied stress and temperature, is described and discussed in detail. 2. Computational model In this simulation, periodic boundary conditions (PBC) are adopted. However, when a single dislocation is introduced in a simulation cell, the overall Burgers vectors are not zero and a step will be formed on the cell boundary after the dislocation glides out of the crystal [12,13], so the periodicity will be violated. In order to eliminate the total Burgers vector and use the periodic boundary conditions in three directions, the dislocations have to be introduced at least in pairs, e.g., in the form of dipoles [14]. The interaction between the two dislocations forming the dipole and the interaction between these two dislocations and their images which are introduced by the PBC are quite small [14]. So these two effects are ignored. Fig. 1 shows the formation procedure for creating a particular 60◦ dislocation dipole. We first ¯ 14[111] and 6[110]. ¯ create a rectangular atomic cell with box vectors at 8[112], That is, the atomic cell sizes in the three directions are 5.32 nm, 13.17 nm and 2.31 nm, respectively. There are 8064 atoms in the cell. Then, as shown in Fig. 1(a), points A, B, C, D and A0 , B0 , C0 , D0 are in the front and back surfaces of the cell, respectively. The atoms bounded by the surfaces ABB0 A0 –BCC0 B0 –DCC0 D0 –ADD0 A0 are removed. 7848 atoms remain. Finally, as shown in Fig. 1(b), the linearly distributed displacements are applied to each part of the atoms separated by the half-plane removed. Let the atoms in the ABB0 A0 and DCC0 D0 planes move a distance be to the right and left respectively, and similarly, move a distance bs along the positive and negative Z directions. Finally, let each atom in a few atomic planes adjacent to the ABB0 A0 and DCC0 D0 planes move a distance which varies linearly from the values of be and bs to zero. The Burgers vectors bEe and bEs are half of the edge and screw components of the 60◦ dislocation Burgers E vector b. The time step in this simulation is 1 fs. The system is relaxed for 2000 steps at 0 K. After relaxation, as shown in Fig. 2, we obtain an atomic configuration with two 60◦ dislocations ¯ and (a/2)[101], ¯ (shuffle-set) with two opposite Burgers vectors (a/2)[101] each lying on a (111) glide plane and separated from the other by a distance of 6a[111]. Obviously, the dislocation line ¯ The insets in Fig. 2 show the relaxed atomic arrangement around the dislocation is along [110].
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Fig. 1. (a) Setting up the 60◦ dislocation dipole by removing the atoms bounded by the surfaces ABB0 A0 –BCC0 B0 –DCC0 D0 –ADD0 A0 , where points A, B, C, D and A0 , B0 , C0 , D0 are in the front and back surfaces of the cell, respectively. the initial atomic displacements around the 60◦ dislocation √ (b) Schematic diagram showing √ core, where be = [( 2a/2) × sin(π/3)]/2, bs = [( 2a/2) × cos(π/3)]/2, a is a Si lattice constant.
core in detail. This atomic arrangement is saved as the initial configuration for later use in simulations. 3. Simulation method The molecular dynamics simulation was carried out with the use of a 60◦ dislocation dipole for a constant applied stress and constant temperature. The atomic interactions are described using the Stillinger–Weber potential, and periodic boundary conditions are adopted in the three directions. The system temperature is kept constant by rescaling the atom velocities [15]. We increase the temperature from 0 K to the desired temperature T by 10 K every 2000 steps. Then, a constant shear stress is applied by means of the Parrinello–Rahman method [16]. When the shear stress acts on a dislocation, only the component parallel to the Burgers vector does work. So let E During the simulation, two dislocations the shear stress σE be parallel to the Burgers vector b. are observed moving in opposite directions on their (111) glide planes. After relaxation for 5000 steps, around 5 ps, a steady-state motion of the dislocation is achieved. In order to obtain the dislocation velocity, we record the instantaneous position of the dislocation core, which can be extracted by identifying the core atoms in terms of the measurement of the maximum local energy. In this simulation, the range of temperature is from 100 to 900 K and the stress is applied from 100 to 3000 MPa, which is shown to be sufficient for the simulations. On a single AMD 2800+ MHz processor, the simulation at one temperature (for example 900 K) takes about 33 CPU hours.
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Fig. 2. Atomic configuration with the 60◦ dislocation dipole after relaxation at 0 K; the insets show the relaxed atomic arrangement around the dislocation core in detail.
Fig. 3. The velocity versus temperature under different applied shear stresses.
4. Results and discussion We first address the issue of the temperature dependence of the dislocation velocity under the given stresses. Fig. 3 shows the dislocation velocity versus temperature under different stresses. It can be seen that the velocity decreases as the temperature increases. According to the phonon drag model [8,17], the existing thermal phonons in a crystal lattice scatter from a moving dislocation, thereby damping the dislocation motion. The phonon damping effect is
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Fig. 4. The Peierls stress versus temperature.
Fig. 5. The velocity versus applied shear stress at different temperatures.
approximately proportional to the temperature, so the temperature resistance to the dislocation motion becomes quite significant as the temperature increases. In order to move the dislocation, the Peierls force in silicon crystal must be overcome. So, if the applied stress is lower than the Peierls stress, the dislocation is not able to glide. But the Peierls stress is different at different temperatures. As shown in Fig. 4, The dislocation begins to glide when about 930 MPa is exerted at 100 K and about 100 MPa at 900 K. It can be found that the Peierls stress calculated in this simulation decreases as the temperature increases, which is consistent with the Peierls–Nabarro model [8]. Fig. 5 shows the dislocation velocity as a function of the stress at different given temperatures. When the temperature is lower than about 500 K, the velocity increases monotonically as the stress increases. It is shown that the velocity curves are quite similar and, therefore, the velocity is dependent mainly on the stress rather than the temperature variation. At relatively high temperature (T > 500 K), the velocity increases gradually as the stress increases, but no longer monotonically.
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As shown in Fig. 5, when the stress is lower than about 1500 MPa, the velocity versus stress increases quite rapidly. However, the velocity varies very slowly when the stress is larger than 1500 MPa and eventually approaches its plateau velocity. The plateau velocity shown in Fig. 5 is about 2900 m/s at 100 K, and only about 1000 m/s at 900 K. The behavior is obviously attributable to the effect of thermal phonon interactions. It is known that the phonon damping effect is closely related to the temperature; furthermore, the influence of the temperature becomes more and more effective as the temperature increases. 5. Conclusions On the basis of the Stillinger–Weber potential and the simulation technique of molecular dynamics, a 60◦ dislocation is created in silicon and the results for dislocation motion as a function of applied stress and temperature are presented. It is found that when the applied stress is larger than the Peierls stress, the dislocation begins to move, and the Peierls stress decreases as the temperature increases. At relatively low temperature (about T < 500 K), the dislocation velocity increases monotonically as the stress increases, but the velocity is not very sensitive to the temperature variation. At relatively high temperature (about T > 500 K), however, the velocity is quite sensitive to temperature, and it increases gradually as the stress increases, but no longer monotonically. When the applied stress is below about 1500 MPa, the velocity increases rapidly as the stress increases, especially for lower temperature. However, when the stress is above 1500 MPa, the velocity varies very slowly and irregularly, especially for higher temperature. The dislocation motion eventually approaches its plateau velocity as the stress increases. The plateau velocity decreases as the temperature increases especially when the temperature is above 500 K. In general, the dislocation velocity decreases as the temperature increases, which is obviously due to the thermal phonon interaction effects. Acknowledgement The authors gratefully acknowledge many helpful discussions with HeChong (Northeast Agricultural University, China). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
F.K. LeGoues, B.S. Meyerson, J.F. Morar, Phys. Rev. Lett. 66 (1991) 2903. H. Chen, L.W. Guo, Q. Cui, Q. Huang, J.M. Zhou, J. Appl. Phys. 79 (1996) 1167. K.K. Linder, F.C. Zhang, J.S. Rieh, P. Bhattacharya, Appl. Phys. Lett. 70 (1997) 3224. J.W. Matthews, A.E. Blakeslee, J. Cryst. Growth 27 (1974) 118. J. Zou, D.J.H. Cockayne, Appl. Phys. Lett. 69 (1996) 1083. W.C. Liu, S.Q. Shi, H. Huang, C.H. Woo, Comput. Mater. Sci. 23 (2002) 155. T. Yamamoto, A. Sakai et al., Appl. Surf. Sci. 224 (2004) 108–112. J.P Hirth, J. Lothe, Theory of Dislocations, second edn., Wiley, New York, 1982. A.N. Gulluoglu, C.S. Hartley, Model. Simul. Mater. Sci. Eng. 1 (1992) 383. D.L. Olmsted, L.G. Hector et al., Model. Simul. Mater. Sci. Eng. 13 (2005) 371. M. Li, W.Y. Chu, K.W. Gao, L.J. Qiao, J. Phys.: Condens. Matter 15 (2003) 3391. A.T. Blumenau, R. Jones, T. Frauenheim, J. Phys.: Condens. Matter 15 (2003) 2951. A. Marzegalli, F. Montalenti, M. Bollani et al., Microelectron. Eng. 76 (2004) 290. J. Chang, W. Cai et al., Comput. Mater. Sci. 23 (2002) 112. H. Andersen, J. Chem. Phys. 72 (1980) 2384. M. Parrinello, A. Rahman, Phys. Rev. Lett. 45 (1980) 1196. J.D. Eshelby, Proc. Phys. Soc. B 69 (1956) 1013.
Atomistic simulation of the 60◦ dislocation mobility in silicon crystal Cheng-xiang Li ∗ , Qing-yuan Meng 1 , Gen Li, Li-jun Yang Department of Astronautical Science and Mechanics, Harbin Institute of Technology, P.O. Box 344, Harbin, 150001, PR China Received 22 February 2006; received in revised form 19 April 2006; accepted 19 May 2006 Available online 11 July 2006
Abstract Dislocation velocity and mobility are studied via molecular dynamics simulation for a 60◦ dislocation dipole in silicon crystal. The atomic interactions are described using the Stillinger–Weber potential and the external stress is applied by means of the Parrinello–Rahman algorithm. It is found that the dislocation begins to move when the applied stress is larger than the Peierls stress, and the calculated Peierls stress decreases as the temperature increases, which is in agreement with the Peierls–Nabarro model. The dislocation velocity at relatively low temperature is insensitive to variation of temperature. In fact, the velocity increases monotonically as the stress increases, and eventually approaches its plateau velocity which is about 2900 m/s. At higher temperature, however, the velocity no longer increases monotonically as the stress increases and the plateau velocity decreases as the temperature increases. In general, the dislocation velocity decreases as the temperature increases, which is consistent with the phonon drag model. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Dislocation dipole; Dislocation velocity; The Peierls stress; Molecular dynamics simulation
1. Introduction The heterostructures of strained-layer semiconductor have received considerable attention for the last three decades due to their promising electronic and optoelectronic properties—especially the SiGe/Si heterostructures, due to the existing silicon technology [1–3]. If the thickness of ∗ Corresponding author.
E-mail addresses: [email protected] (C.-X. Li), [email protected] (Q.-Y. Meng). 1 Tel.: +86 451 86414143. c 2006 Elsevier Ltd. All rights reserved. 0749-6036/$ - see front matter doi:10.1016/j.spmi.2006.05.004
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the epitaxial film is sufficiently small, the film is cohesively strained to match the substrate. However, misfit dislocations may be introduced at the strained interface when a critical thickness is reached [4]. Since these dislocations lead to a poor device performance, many theoretical and experimental studies have focused on these misfit dislocations, particularly on their sources and equilibrium configurations at atomic level [5–7]. A considerable number of prior investigations on modeling and simulation of the dynamic behavior of dislocations have been conducted [8– 11]. For example, dislocations have been assumed to be continuum line defects whose dynamic characteristics are controlled by a driving force, i.e., the Peach–Koehler force [8], and an assumed dislocation mobility law. Simulations of dislocation motion in metals have also been carried out [10,11], and the results on the velocity of edge and screw dislocations in Al and Ni as a function of applied stress and temperature have been presented. However, atomistic simulation of the dislocation mobility in diamond structure, such as silicon, over a large range of temperature and stress has not appeared yet. In the present work, we establish a model consisting of the dislocation dipole and carry out large scale computational simulations. A 60◦ dislocation motion in silicon for temperature ranging from 100 to 900 K and applied stress ranging from 100 to 3000 MPa is investigated extensively by means of the atomistic simulation method. The dislocation velocity, which is regarded as a function of applied stress and temperature, is described and discussed in detail. 2. Computational model In this simulation, periodic boundary conditions (PBC) are adopted. However, when a single dislocation is introduced in a simulation cell, the overall Burgers vectors are not zero and a step will be formed on the cell boundary after the dislocation glides out of the crystal [12,13], so the periodicity will be violated. In order to eliminate the total Burgers vector and use the periodic boundary conditions in three directions, the dislocations have to be introduced at least in pairs, e.g., in the form of dipoles [14]. The interaction between the two dislocations forming the dipole and the interaction between these two dislocations and their images which are introduced by the PBC are quite small [14]. So these two effects are ignored. Fig. 1 shows the formation procedure for creating a particular 60◦ dislocation dipole. We first ¯ 14[111] and 6[110]. ¯ create a rectangular atomic cell with box vectors at 8[112], That is, the atomic cell sizes in the three directions are 5.32 nm, 13.17 nm and 2.31 nm, respectively. There are 8064 atoms in the cell. Then, as shown in Fig. 1(a), points A, B, C, D and A0 , B0 , C0 , D0 are in the front and back surfaces of the cell, respectively. The atoms bounded by the surfaces ABB0 A0 –BCC0 B0 –DCC0 D0 –ADD0 A0 are removed. 7848 atoms remain. Finally, as shown in Fig. 1(b), the linearly distributed displacements are applied to each part of the atoms separated by the half-plane removed. Let the atoms in the ABB0 A0 and DCC0 D0 planes move a distance be to the right and left respectively, and similarly, move a distance bs along the positive and negative Z directions. Finally, let each atom in a few atomic planes adjacent to the ABB0 A0 and DCC0 D0 planes move a distance which varies linearly from the values of be and bs to zero. The Burgers vectors bEe and bEs are half of the edge and screw components of the 60◦ dislocation Burgers E vector b. The time step in this simulation is 1 fs. The system is relaxed for 2000 steps at 0 K. After relaxation, as shown in Fig. 2, we obtain an atomic configuration with two 60◦ dislocations ¯ and (a/2)[101], ¯ (shuffle-set) with two opposite Burgers vectors (a/2)[101] each lying on a (111) glide plane and separated from the other by a distance of 6a[111]. Obviously, the dislocation line ¯ The insets in Fig. 2 show the relaxed atomic arrangement around the dislocation is along [110].
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Fig. 1. (a) Setting up the 60◦ dislocation dipole by removing the atoms bounded by the surfaces ABB0 A0 –BCC0 B0 –DCC0 D0 –ADD0 A0 , where points A, B, C, D and A0 , B0 , C0 , D0 are in the front and back surfaces of the cell, respectively. the initial atomic displacements around the 60◦ dislocation √ (b) Schematic diagram showing √ core, where be = [( 2a/2) × sin(π/3)]/2, bs = [( 2a/2) × cos(π/3)]/2, a is a Si lattice constant.
core in detail. This atomic arrangement is saved as the initial configuration for later use in simulations. 3. Simulation method The molecular dynamics simulation was carried out with the use of a 60◦ dislocation dipole for a constant applied stress and constant temperature. The atomic interactions are described using the Stillinger–Weber potential, and periodic boundary conditions are adopted in the three directions. The system temperature is kept constant by rescaling the atom velocities [15]. We increase the temperature from 0 K to the desired temperature T by 10 K every 2000 steps. Then, a constant shear stress is applied by means of the Parrinello–Rahman method [16]. When the shear stress acts on a dislocation, only the component parallel to the Burgers vector does work. So let E During the simulation, two dislocations the shear stress σE be parallel to the Burgers vector b. are observed moving in opposite directions on their (111) glide planes. After relaxation for 5000 steps, around 5 ps, a steady-state motion of the dislocation is achieved. In order to obtain the dislocation velocity, we record the instantaneous position of the dislocation core, which can be extracted by identifying the core atoms in terms of the measurement of the maximum local energy. In this simulation, the range of temperature is from 100 to 900 K and the stress is applied from 100 to 3000 MPa, which is shown to be sufficient for the simulations. On a single AMD 2800+ MHz processor, the simulation at one temperature (for example 900 K) takes about 33 CPU hours.
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Fig. 2. Atomic configuration with the 60◦ dislocation dipole after relaxation at 0 K; the insets show the relaxed atomic arrangement around the dislocation core in detail.
Fig. 3. The velocity versus temperature under different applied shear stresses.
4. Results and discussion We first address the issue of the temperature dependence of the dislocation velocity under the given stresses. Fig. 3 shows the dislocation velocity versus temperature under different stresses. It can be seen that the velocity decreases as the temperature increases. According to the phonon drag model [8,17], the existing thermal phonons in a crystal lattice scatter from a moving dislocation, thereby damping the dislocation motion. The phonon damping effect is
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Fig. 4. The Peierls stress versus temperature.
Fig. 5. The velocity versus applied shear stress at different temperatures.
approximately proportional to the temperature, so the temperature resistance to the dislocation motion becomes quite significant as the temperature increases. In order to move the dislocation, the Peierls force in silicon crystal must be overcome. So, if the applied stress is lower than the Peierls stress, the dislocation is not able to glide. But the Peierls stress is different at different temperatures. As shown in Fig. 4, The dislocation begins to glide when about 930 MPa is exerted at 100 K and about 100 MPa at 900 K. It can be found that the Peierls stress calculated in this simulation decreases as the temperature increases, which is consistent with the Peierls–Nabarro model [8]. Fig. 5 shows the dislocation velocity as a function of the stress at different given temperatures. When the temperature is lower than about 500 K, the velocity increases monotonically as the stress increases. It is shown that the velocity curves are quite similar and, therefore, the velocity is dependent mainly on the stress rather than the temperature variation. At relatively high temperature (T > 500 K), the velocity increases gradually as the stress increases, but no longer monotonically.
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As shown in Fig. 5, when the stress is lower than about 1500 MPa, the velocity versus stress increases quite rapidly. However, the velocity varies very slowly when the stress is larger than 1500 MPa and eventually approaches its plateau velocity. The plateau velocity shown in Fig. 5 is about 2900 m/s at 100 K, and only about 1000 m/s at 900 K. The behavior is obviously attributable to the effect of thermal phonon interactions. It is known that the phonon damping effect is closely related to the temperature; furthermore, the influence of the temperature becomes more and more effective as the temperature increases. 5. Conclusions On the basis of the Stillinger–Weber potential and the simulation technique of molecular dynamics, a 60◦ dislocation is created in silicon and the results for dislocation motion as a function of applied stress and temperature are presented. It is found that when the applied stress is larger than the Peierls stress, the dislocation begins to move, and the Peierls stress decreases as the temperature increases. At relatively low temperature (about T < 500 K), the dislocation velocity increases monotonically as the stress increases, but the velocity is not very sensitive to the temperature variation. At relatively high temperature (about T > 500 K), however, the velocity is quite sensitive to temperature, and it increases gradually as the stress increases, but no longer monotonically. When the applied stress is below about 1500 MPa, the velocity increases rapidly as the stress increases, especially for lower temperature. However, when the stress is above 1500 MPa, the velocity varies very slowly and irregularly, especially for higher temperature. The dislocation motion eventually approaches its plateau velocity as the stress increases. The plateau velocity decreases as the temperature increases especially when the temperature is above 500 K. In general, the dislocation velocity decreases as the temperature increases, which is obviously due to the thermal phonon interaction effects. Acknowledgement The authors gratefully acknowledge many helpful discussions with HeChong (Northeast Agricultural University, China). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
F.K. LeGoues, B.S. Meyerson, J.F. Morar, Phys. Rev. Lett. 66 (1991) 2903. H. Chen, L.W. Guo, Q. Cui, Q. Huang, J.M. Zhou, J. Appl. Phys. 79 (1996) 1167. K.K. Linder, F.C. Zhang, J.S. Rieh, P. Bhattacharya, Appl. Phys. Lett. 70 (1997) 3224. J.W. Matthews, A.E. Blakeslee, J. Cryst. Growth 27 (1974) 118. J. Zou, D.J.H. Cockayne, Appl. Phys. Lett. 69 (1996) 1083. W.C. Liu, S.Q. Shi, H. Huang, C.H. Woo, Comput. Mater. Sci. 23 (2002) 155. T. Yamamoto, A. Sakai et al., Appl. Surf. Sci. 224 (2004) 108–112. J.P Hirth, J. Lothe, Theory of Dislocations, second edn., Wiley, New York, 1982. A.N. Gulluoglu, C.S. Hartley, Model. Simul. Mater. Sci. Eng. 1 (1992) 383. D.L. Olmsted, L.G. Hector et al., Model. Simul. Mater. Sci. Eng. 13 (2005) 371. M. Li, W.Y. Chu, K.W. Gao, L.J. Qiao, J. Phys.: Condens. Matter 15 (2003) 3391. A.T. Blumenau, R. Jones, T. Frauenheim, J. Phys.: Condens. Matter 15 (2003) 2951. A. Marzegalli, F. Montalenti, M. Bollani et al., Microelectron. Eng. 76 (2004) 290. J. Chang, W. Cai et al., Comput. Mater. Sci. 23 (2002) 112. H. Andersen, J. Chem. Phys. 72 (1980) 2384. M. Parrinello, A. Rahman, Phys. Rev. Lett. 45 (1980) 1196. J.D. Eshelby, Proc. Phys. Soc. B 69 (1956) 1013.