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Molecular-dynamics study of mechanical properties of nanoscale copper with vacancies under static and cyclic loading
Microelectronic Engineering 65 (2003) 239–246 www.elsevier.com / locate / mee
Molecular-dynamics study of mechanical properties of nanoscale copper with vacancies under static and cyclic loading Win-Jin Chang Department of Mechanical Engineering, Kun-Shan University of Technology, 949 Tawan Road, 71016 Yung-Kong, Tainan, Taiwan Received 4 June 2002; received in revised form 18 July 2002; accepted 30 July 2002
Abstract A molecular-dynamics simulation of the mechanical properties of nanoscale copper with vacancies under static and cyclic loading has been carried out. From the tensile test, the stress–strain curve for nanoscale copper was obtained first and then the Young’s modulus of the material was determined. The Young’s modulus decreases with increasing vacancy fraction, and it also decreases with increasing temperature. It can be clearly seen that the ultimate tensile stress and elongation rate are high for nanoscale copper, and result in the high fracture energy required to break the nanoscale copper. From the fatigue test, the amplitude stress–number of cycles curve was obtained. Under the lower applied stress, the amplitude stress of the nanoscale copper increases with decreasing vacancy fraction, however, it decreases with decreasing temperature. Furthermore, the fatigue limit of the nanoscale copper increases with decreasing vacancy fraction and increasing temperature, when the material is under a less-than-critical value of applied stress. 2003 Elsevier Science B.V. All rights reserved. Keywords: Nanoscale copper; Molecular-dynamics simulation; Vacancy fraction; Mechanical properties; Tensile test; Fatigue test
1. Introduction Over the past decade, the study of the mechanical properties of nanostructured materials using atomistic simulation has been of significant interest to researchers due to nano-technological development [1–3]. For example, Miyazaki and Shiozaki [4] calculated the elastic constant and thermal expansion coefficient of Fe. Aya and Nakayama [5] investigated the influence of environmental temperature on the yield stress of polymers. The material stiffness is one of the important properties of a material. Miller and Shenoy [6] studied the bending stiffness properties of nanosized E-mail address: [email protected] (W.-J. Chang). 0167-9317 / 03 / $ – see front matter PII: S0167-9317( 02 )00887-0
2003 Elsevier Science B.V. All rights reserved.
W.-J. Chang / Microelectronic Engineering 65 (2003) 239–246
240
structural Al and Si. Since the development of the electronic industry, copper has been one of the important materials in the field, used in, for example, electrical interconnects [7]. Many studies have look at the material properties of copper. Heino et al. [8] investigated the mechanical properties of copper, including the elastic constant and the behavior of crack propagation at room temperature. Schiotz et al. [9] studied the effects of strain rate and porosity on the mechanical deformation of copper at various temperatures. Recently, Kang and Hwang [10] investigated mechanical deformations of copper nanowire. Previous studies focused on the material behavior under a static loading. Only a limited portion of studies published were concerned with the aspect of cyclic loading. This is because the fatigue test for cyclic loading is a time-consuming task, especially when using an experimental method. Inoue et al. [11] studied the fracture mechanisms of nanoscale pure iron under static and cyclic loading using molecular dynamics simulation. Read [12] experimentally studied the tension–tension fatigue behavior of copper thin films at room temperature. Chen et al. [13] investigated the influence of temperature on the low cycle fatigue behavior of nickel-based superalloys, and found that the fatigue life does not monotonously decrease with increasing test temperature. In this paper, using the molecular dynamics simulation [14], the tensile test and the tension– compression fatigue test for nanoscale copper with vacancies are carried out at various temperatures.
2. Analysis In this paper, the Lennard–Jones potential model, which is still widely used, is also adopted for the calculation process. It is s 12 s 6 fsrd 5 4´ ] 2 ] (1) r r
FS D S D G
where f represents the potential of the system, ´ and s denote energy and length scales, respectively, and r is the intermolecular distance. The initial velocities of particles form a Maxwell–Boltzmann distribution corresponding to a given temperature. To keep the system temperature, the following correction is required; ] TD new n i 5 ni ] (2) TA
œ
new i
where n is the velocity of the particle i after correction. T D and TA are the desired temperature and actual temperature of the system, respectively. The initial configuration of the molecular dynamics simulation for the tensile test is shown in Fig. 1a. The loading is applied to atoms in the z-direction. The periodic boundary condition is used in the x- and y-directions and the test material is free to move in the z-direction. The stress calculation in the atomistic simulation smn on m plane and in n-direction is calculated by 1 smn 5 ] Ns
OF i
m i n im n in 1 ]]] 2] Vi 2Vi
O j
≠fsr ijd r mij r nij ]] ]] ≠r ij r ij
G
(3)
where m i is the mass of atom i; Vi is the volume assigned to atom i; Ns is the number of particles
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241
Fig. 1. The initial configuration of nanoscale copper with vacancies for (a) the tensile test and (b) the fatigue test. The label A in (b) is a free motion region and B is a tension–compression region.
contained in the region S and S is defined as the region of atomic interaction; r ij is the distance between atoms i and j; r mij and r nij are two components of the vector from atom i to j. The form expressed in Eq. (3) contains two terms on the right-hand side. The first is a kinetic part and is caused by atomic motions and the second is a potential part and is affected by the interactive forces of atoms. The normal strain in the z-direction ´z is calculated by Lz 2 Lzo ´z 5 ]]] Lzo
(4)
where Lz is the average length in the z-direction and Lzo is its average initial length. The stress–strain curve can be yielded by means of Eqs. (3) and (4); and then the elastic modulus of the nanoscale copper can be obtained from the curve. Generally, the modulus is dependent on the kind of material, the loading condition, the effect of defects effects within, and the temperature of the material. A fatigue test for nanoscale copper with vacancies was carried out at various temperatures, and the initial configuration of the molecular dynamics simulation is shown in Fig. 1b. The two ends of the specimen, labeled A, are free motion region, where velocity rescaling is applied, while the atoms in the middle of the specimen, labeled B, are subjected to the reverse uniaxial tension–compression loading in the z-direction. The loading cycles are always fully reversed so that the maximum compressive stress is the same as the maximum tensile stress. The conventional periodic boundary condition is used in the x- and y-directions. The specimens tested are cycled to failure, where failure is defined as separation of the specimen into two halves.
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3. Result and discussion In this paper, the mechanical properties of nanoscale copper were studied. The mass of an atom, m, is 1.0556 3 10 225 kg [15]. The pairwise interaction between atoms is described by a Lennard–Jones potential. The interatomic energy and force vanish at a separation of r 5 2.5s. The motion equations of the atoms are integrated by the Verlet algorithm with a timestep Dt of 3.64 fs. We consider a perfect nanostructure case, N5960 classical atoms are initially collected on the sites of a face-centered cubic lattice and confined to a cubic simulation cell n x 5 4, n y 5 4 and n z 5 15 at initial spatial conditions. However, some defects exist within the material. It is assumed that the specimens have some random vacancies within materials in the simulation. The random distribution of vacancies is modeled by: (i) first calculating the number of vacancies according to the given vacancy fraction and the total atomic numbers; (ii) numbering the random vacancies in order; (iii) obtaining the occupation positions of the vacancies; (iv) converting the atoms and the vacancies to their actual positions in simulation. To understand the effects of the vacancies on the mechanical properties of nanoscale copper, the tensile and fatigue tests are simulated. The tensile loading with a strain rate of 1 3 10 22 / step is applied to the specimen at every time step and the average stress and strain is obtained. Fig. 2 shows the relationship between the tensile stress and the strain for nanoscale copper with various vacancies at T5300 K. From this figure, it can be seen that tensile stress decreases with increasing vacancy fraction, V, of nanoscale copper and the maximum stress occurs about ´z 5 0.6 for every case. When V50% it can be seen that the ultimate tensile stress is about 37 GPa and the elongation rate is more than 100%; this contrasts with 0.29 GPa and 8%, respectively for bulk copper [16]. The results are clearly different. Similarly, high ultimate tensile stress and elongation rate of nanoscale copper were also obtained in a previous study [8]. The
Fig. 2. Relationship between tensile stress and strain for nanoscale copper with various vacancies at T5300 K.
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243
Fig. 3. Relationship between tensile stress and strain for nanoscale copper with V56% at various temperatures.
relationship between tensile stress and strain for nanoscale copper with constant fraction of vacancy, V56%, at various temperatures is shown in Fig. 3. The tensile stress obtained from the simulation is slightly higher in the case of a lower temperature than that obtained in the case of a higher temperature under small strain, while the stress is inconsistent with the temperature when ´z . 0.5. The Young’s moduli can be determined from the tensile tests for the strain ´z ,2% in terms of linear regression. To assess the effect of vacancies on Young’s modulus for nanoscale copper, the relationship between Young’s modulus and vacancy fraction at various temperatures can be obtained after curve fitting as shown in Fig. 4. According to a previous study [16], the elastic response of a solid material containing porosity can be written as the following empirical equation: E 5 E0 ? exps 2 b Pd
(5)
where E is the apparent Young’s modulus, E0 is a reference value, P is the volume fraction of pores, the pore is equivalent to the vacancy of the material during the simulation, and b is a constant with a value of about 4 for nanoscale copper [16]. The values of E0 and b are obtained from the simulation by a least-squares fitting procedure and they are listed in Table 1. The b value in this table is lower than that obtained from the previous study. The discrepancy may be caused by the effect of the size of the material, and different types of defect within the material. From Fig. 4, it can be found that the Young’s modulus of the material decreases with increasing vacancy fraction or increasing temperature. In the case of zero vacancy and T5300 K, the modulus is about 138.8 GPa, compared with 100 GPa for bulk copper obtained by Shen et al. [17], 107 GPa obtained by Read [12] and 124 GPa obtained by Sanders et al. [16]. The value for nanoscale copper is larger than that for bulk copper due to the fact that the nanoscale copper in the simulation is a single crystal and the grain boundaries were not taken into consideration.
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Fig. 4. Young’s modulus as a function of vacancy fraction for nanoscale copper at various temperatures.
The stress amplitude against number of cycles for nanoscale copper with various vacancies at T5300 K is shown in Fig. 5. In the higher stress amplitude region, the stress increases with increasing values of V, while the stress amplitude decreases with increasing values of V when V becomes larger than 6%. In the lower stress amplitude region, the stress decreases with increasing values of V. It implies that the lower the vacancy fraction, the higher the fatigue limit. Generally, increasing vacancies, say V56%, can increase the elongation of the material, while it only slightly decreases the Young’s modulus, which can be seen from Fig. 4. Consequently, the amplitude stress is high in the case of V56% and T5300 K; the other temperature cases can obtain similar results and are not shown. The stress amplitude against number of cycles for nanoscale copper with V56% at various temperatures is shown in Fig. 6. Diffusion of vacancies occurs during cyclic loading in the simulation, and the diffusion phenomena clearly increase with increasing temperature. The stress amplitude decreases with decreasing temperature and the influence of temperature on the stress amplitude is significant under the lower cyclic stress. It can be seen that the nanoscale copper has a higher fatigue limit at higher temperatures when the value of applied stress is less-than-critical.
Table 1 The values of E0 and b in Eq. (5), obtained using a least-squares fitting procedure Temperature (K) E0 (GPa) b
10 150.2 1.76
100 146.1 1.52
300 140.2 1.31
500 132.9 1.25
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Fig. 5. Cyclic stress against number of cycles to failure for nanoscale copper with various vacancies at T5300 K.
Fig. 6. Cyclic stress against number of cycles to failure for nanoscale copper with V56% at various temperatures.
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4. Conclusions The tensile and fatigue behavior of nanoscale copper with vacancies at various temperatures has been studied by means of molecular-dynamics simulation. The stress–strain curve for nanoscale copper was obtained. It can be seen from the curve that the tensile stress decreases with increasing vacancy fraction of the material and the maximum stress occurs at about ´z 5 0.6. The nanoscale copper shows very high ultimate tensile stress and elongation rate. In addition, the Young’s modulus for the nanoscale copper was obtained from the curve, and was found to decrease with increasing temperature. Increasing vacancy fraction was also found to decrease the modulus. From the fatigue test, the cyclic stress–number of cycles curve was obtained. It was observed from the curve that the nanoscale copper has a higher fatigue limit when the vacancy fraction is lower and the temperature is higher, and when the value of applied stress is less-than-critical. Furthermore, the copper material subjected to cyclic loading in microelectromechanical devices and copper interconnects in microelectronics is important, and requires further research.
References [1] L. Ristic (Ed.), Sensor Technology and Devices, Artech Hiuse, Boston, 1994. [2] B. Bhushan (Ed.), Handbook of Micro / Nano Tribology, CRC Press, New York, 1995. [3] U.S. National Report, WTEC Panel Report on Nanostructure Science and Technology R&D Status and Trends in Nanoparticles, Nanostructured Materials, and Nanodevices, 1999. [4] N. Miyazaki, Y. Shiozaki, JSME Int. J. Ser. A 39 (1996) 606. [5] T. Aya, T. Nakayama, JSME Int. J. Ser. A 40 (1997) 343. [6] R.E. Miller, V.B. Shenoy, Nanotechnology 11 (2000) 139. [7] C.K. Hu, B. Luther, F.B. Kaufman, J. Hummel, C. Uzoh, D.J. Pearson, Thin Solid Films 262 (1995) 84. [8] P. Heino, H. Hakkinen, K. Kaski, Phys. Rev. B 58 (1998) 641. [9] J. Schiotz, T. Vegge, F.D. Di Tolla, K.W. Jacobsen, Phys. Rev. B 60 (1999) 11971. [10] J.W. Kang, H.J. Hwang, Nanotechnology 12 (2001) 295. [11] H. Inoue, Y. Akahoshi, S. Harada, Mater. Sci. Res. Int. 1 (1995) 95. [12] D.T. Read, Int. J. Fatigue 20 (1998) 203. [13] L.J. Chen, Z.G. Wang, G. Yao, J.F. Tian, Int. J. Fatigue 21 (1999) 791. [14] J.M. Haile, Molecular Dynamic Simulation, Wiley, New York, 1992. [15] R. Smith, M. Jakas, Atomic and Ion Collisions in Solids and at Surfaces: Theory, Simulation and Application, Cambridge University Press, USA, 1997. [16] P.G. Sanders, J.A. Eastman, J.R. Weertman, Acta Mater. 45 (1997) 4019. [17] T.D. Shen, C.C. Koch, T.Y. Tsui, G.M. Pharr, J. Mater. Res. 10 (1995) 2892.
Molecular-dynamics study of mechanical properties of nanoscale copper with vacancies under static and cyclic loading Win-Jin Chang Department of Mechanical Engineering, Kun-Shan University of Technology, 949 Tawan Road, 71016 Yung-Kong, Tainan, Taiwan Received 4 June 2002; received in revised form 18 July 2002; accepted 30 July 2002
Abstract A molecular-dynamics simulation of the mechanical properties of nanoscale copper with vacancies under static and cyclic loading has been carried out. From the tensile test, the stress–strain curve for nanoscale copper was obtained first and then the Young’s modulus of the material was determined. The Young’s modulus decreases with increasing vacancy fraction, and it also decreases with increasing temperature. It can be clearly seen that the ultimate tensile stress and elongation rate are high for nanoscale copper, and result in the high fracture energy required to break the nanoscale copper. From the fatigue test, the amplitude stress–number of cycles curve was obtained. Under the lower applied stress, the amplitude stress of the nanoscale copper increases with decreasing vacancy fraction, however, it decreases with decreasing temperature. Furthermore, the fatigue limit of the nanoscale copper increases with decreasing vacancy fraction and increasing temperature, when the material is under a less-than-critical value of applied stress. 2003 Elsevier Science B.V. All rights reserved. Keywords: Nanoscale copper; Molecular-dynamics simulation; Vacancy fraction; Mechanical properties; Tensile test; Fatigue test
1. Introduction Over the past decade, the study of the mechanical properties of nanostructured materials using atomistic simulation has been of significant interest to researchers due to nano-technological development [1–3]. For example, Miyazaki and Shiozaki [4] calculated the elastic constant and thermal expansion coefficient of Fe. Aya and Nakayama [5] investigated the influence of environmental temperature on the yield stress of polymers. The material stiffness is one of the important properties of a material. Miller and Shenoy [6] studied the bending stiffness properties of nanosized E-mail address: [email protected] (W.-J. Chang). 0167-9317 / 03 / $ – see front matter PII: S0167-9317( 02 )00887-0
2003 Elsevier Science B.V. All rights reserved.
W.-J. Chang / Microelectronic Engineering 65 (2003) 239–246
240
structural Al and Si. Since the development of the electronic industry, copper has been one of the important materials in the field, used in, for example, electrical interconnects [7]. Many studies have look at the material properties of copper. Heino et al. [8] investigated the mechanical properties of copper, including the elastic constant and the behavior of crack propagation at room temperature. Schiotz et al. [9] studied the effects of strain rate and porosity on the mechanical deformation of copper at various temperatures. Recently, Kang and Hwang [10] investigated mechanical deformations of copper nanowire. Previous studies focused on the material behavior under a static loading. Only a limited portion of studies published were concerned with the aspect of cyclic loading. This is because the fatigue test for cyclic loading is a time-consuming task, especially when using an experimental method. Inoue et al. [11] studied the fracture mechanisms of nanoscale pure iron under static and cyclic loading using molecular dynamics simulation. Read [12] experimentally studied the tension–tension fatigue behavior of copper thin films at room temperature. Chen et al. [13] investigated the influence of temperature on the low cycle fatigue behavior of nickel-based superalloys, and found that the fatigue life does not monotonously decrease with increasing test temperature. In this paper, using the molecular dynamics simulation [14], the tensile test and the tension– compression fatigue test for nanoscale copper with vacancies are carried out at various temperatures.
2. Analysis In this paper, the Lennard–Jones potential model, which is still widely used, is also adopted for the calculation process. It is s 12 s 6 fsrd 5 4´ ] 2 ] (1) r r
FS D S D G
where f represents the potential of the system, ´ and s denote energy and length scales, respectively, and r is the intermolecular distance. The initial velocities of particles form a Maxwell–Boltzmann distribution corresponding to a given temperature. To keep the system temperature, the following correction is required; ] TD new n i 5 ni ] (2) TA
œ
new i
where n is the velocity of the particle i after correction. T D and TA are the desired temperature and actual temperature of the system, respectively. The initial configuration of the molecular dynamics simulation for the tensile test is shown in Fig. 1a. The loading is applied to atoms in the z-direction. The periodic boundary condition is used in the x- and y-directions and the test material is free to move in the z-direction. The stress calculation in the atomistic simulation smn on m plane and in n-direction is calculated by 1 smn 5 ] Ns
OF i
m i n im n in 1 ]]] 2] Vi 2Vi
O j
≠fsr ijd r mij r nij ]] ]] ≠r ij r ij
G
(3)
where m i is the mass of atom i; Vi is the volume assigned to atom i; Ns is the number of particles
W.-J. Chang / Microelectronic Engineering 65 (2003) 239–246
241
Fig. 1. The initial configuration of nanoscale copper with vacancies for (a) the tensile test and (b) the fatigue test. The label A in (b) is a free motion region and B is a tension–compression region.
contained in the region S and S is defined as the region of atomic interaction; r ij is the distance between atoms i and j; r mij and r nij are two components of the vector from atom i to j. The form expressed in Eq. (3) contains two terms on the right-hand side. The first is a kinetic part and is caused by atomic motions and the second is a potential part and is affected by the interactive forces of atoms. The normal strain in the z-direction ´z is calculated by Lz 2 Lzo ´z 5 ]]] Lzo
(4)
where Lz is the average length in the z-direction and Lzo is its average initial length. The stress–strain curve can be yielded by means of Eqs. (3) and (4); and then the elastic modulus of the nanoscale copper can be obtained from the curve. Generally, the modulus is dependent on the kind of material, the loading condition, the effect of defects effects within, and the temperature of the material. A fatigue test for nanoscale copper with vacancies was carried out at various temperatures, and the initial configuration of the molecular dynamics simulation is shown in Fig. 1b. The two ends of the specimen, labeled A, are free motion region, where velocity rescaling is applied, while the atoms in the middle of the specimen, labeled B, are subjected to the reverse uniaxial tension–compression loading in the z-direction. The loading cycles are always fully reversed so that the maximum compressive stress is the same as the maximum tensile stress. The conventional periodic boundary condition is used in the x- and y-directions. The specimens tested are cycled to failure, where failure is defined as separation of the specimen into two halves.
242
W.-J. Chang / Microelectronic Engineering 65 (2003) 239–246
3. Result and discussion In this paper, the mechanical properties of nanoscale copper were studied. The mass of an atom, m, is 1.0556 3 10 225 kg [15]. The pairwise interaction between atoms is described by a Lennard–Jones potential. The interatomic energy and force vanish at a separation of r 5 2.5s. The motion equations of the atoms are integrated by the Verlet algorithm with a timestep Dt of 3.64 fs. We consider a perfect nanostructure case, N5960 classical atoms are initially collected on the sites of a face-centered cubic lattice and confined to a cubic simulation cell n x 5 4, n y 5 4 and n z 5 15 at initial spatial conditions. However, some defects exist within the material. It is assumed that the specimens have some random vacancies within materials in the simulation. The random distribution of vacancies is modeled by: (i) first calculating the number of vacancies according to the given vacancy fraction and the total atomic numbers; (ii) numbering the random vacancies in order; (iii) obtaining the occupation positions of the vacancies; (iv) converting the atoms and the vacancies to their actual positions in simulation. To understand the effects of the vacancies on the mechanical properties of nanoscale copper, the tensile and fatigue tests are simulated. The tensile loading with a strain rate of 1 3 10 22 / step is applied to the specimen at every time step and the average stress and strain is obtained. Fig. 2 shows the relationship between the tensile stress and the strain for nanoscale copper with various vacancies at T5300 K. From this figure, it can be seen that tensile stress decreases with increasing vacancy fraction, V, of nanoscale copper and the maximum stress occurs about ´z 5 0.6 for every case. When V50% it can be seen that the ultimate tensile stress is about 37 GPa and the elongation rate is more than 100%; this contrasts with 0.29 GPa and 8%, respectively for bulk copper [16]. The results are clearly different. Similarly, high ultimate tensile stress and elongation rate of nanoscale copper were also obtained in a previous study [8]. The
Fig. 2. Relationship between tensile stress and strain for nanoscale copper with various vacancies at T5300 K.
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243
Fig. 3. Relationship between tensile stress and strain for nanoscale copper with V56% at various temperatures.
relationship between tensile stress and strain for nanoscale copper with constant fraction of vacancy, V56%, at various temperatures is shown in Fig. 3. The tensile stress obtained from the simulation is slightly higher in the case of a lower temperature than that obtained in the case of a higher temperature under small strain, while the stress is inconsistent with the temperature when ´z . 0.5. The Young’s moduli can be determined from the tensile tests for the strain ´z ,2% in terms of linear regression. To assess the effect of vacancies on Young’s modulus for nanoscale copper, the relationship between Young’s modulus and vacancy fraction at various temperatures can be obtained after curve fitting as shown in Fig. 4. According to a previous study [16], the elastic response of a solid material containing porosity can be written as the following empirical equation: E 5 E0 ? exps 2 b Pd
(5)
where E is the apparent Young’s modulus, E0 is a reference value, P is the volume fraction of pores, the pore is equivalent to the vacancy of the material during the simulation, and b is a constant with a value of about 4 for nanoscale copper [16]. The values of E0 and b are obtained from the simulation by a least-squares fitting procedure and they are listed in Table 1. The b value in this table is lower than that obtained from the previous study. The discrepancy may be caused by the effect of the size of the material, and different types of defect within the material. From Fig. 4, it can be found that the Young’s modulus of the material decreases with increasing vacancy fraction or increasing temperature. In the case of zero vacancy and T5300 K, the modulus is about 138.8 GPa, compared with 100 GPa for bulk copper obtained by Shen et al. [17], 107 GPa obtained by Read [12] and 124 GPa obtained by Sanders et al. [16]. The value for nanoscale copper is larger than that for bulk copper due to the fact that the nanoscale copper in the simulation is a single crystal and the grain boundaries were not taken into consideration.
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Fig. 4. Young’s modulus as a function of vacancy fraction for nanoscale copper at various temperatures.
The stress amplitude against number of cycles for nanoscale copper with various vacancies at T5300 K is shown in Fig. 5. In the higher stress amplitude region, the stress increases with increasing values of V, while the stress amplitude decreases with increasing values of V when V becomes larger than 6%. In the lower stress amplitude region, the stress decreases with increasing values of V. It implies that the lower the vacancy fraction, the higher the fatigue limit. Generally, increasing vacancies, say V56%, can increase the elongation of the material, while it only slightly decreases the Young’s modulus, which can be seen from Fig. 4. Consequently, the amplitude stress is high in the case of V56% and T5300 K; the other temperature cases can obtain similar results and are not shown. The stress amplitude against number of cycles for nanoscale copper with V56% at various temperatures is shown in Fig. 6. Diffusion of vacancies occurs during cyclic loading in the simulation, and the diffusion phenomena clearly increase with increasing temperature. The stress amplitude decreases with decreasing temperature and the influence of temperature on the stress amplitude is significant under the lower cyclic stress. It can be seen that the nanoscale copper has a higher fatigue limit at higher temperatures when the value of applied stress is less-than-critical.
Table 1 The values of E0 and b in Eq. (5), obtained using a least-squares fitting procedure Temperature (K) E0 (GPa) b
10 150.2 1.76
100 146.1 1.52
300 140.2 1.31
500 132.9 1.25
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245
Fig. 5. Cyclic stress against number of cycles to failure for nanoscale copper with various vacancies at T5300 K.
Fig. 6. Cyclic stress against number of cycles to failure for nanoscale copper with V56% at various temperatures.
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4. Conclusions The tensile and fatigue behavior of nanoscale copper with vacancies at various temperatures has been studied by means of molecular-dynamics simulation. The stress–strain curve for nanoscale copper was obtained. It can be seen from the curve that the tensile stress decreases with increasing vacancy fraction of the material and the maximum stress occurs at about ´z 5 0.6. The nanoscale copper shows very high ultimate tensile stress and elongation rate. In addition, the Young’s modulus for the nanoscale copper was obtained from the curve, and was found to decrease with increasing temperature. Increasing vacancy fraction was also found to decrease the modulus. From the fatigue test, the cyclic stress–number of cycles curve was obtained. It was observed from the curve that the nanoscale copper has a higher fatigue limit when the vacancy fraction is lower and the temperature is higher, and when the value of applied stress is less-than-critical. Furthermore, the copper material subjected to cyclic loading in microelectromechanical devices and copper interconnects in microelectronics is important, and requires further research.
References [1] L. Ristic (Ed.), Sensor Technology and Devices, Artech Hiuse, Boston, 1994. [2] B. Bhushan (Ed.), Handbook of Micro / Nano Tribology, CRC Press, New York, 1995. [3] U.S. National Report, WTEC Panel Report on Nanostructure Science and Technology R&D Status and Trends in Nanoparticles, Nanostructured Materials, and Nanodevices, 1999. [4] N. Miyazaki, Y. Shiozaki, JSME Int. J. Ser. A 39 (1996) 606. [5] T. Aya, T. Nakayama, JSME Int. J. Ser. A 40 (1997) 343. [6] R.E. Miller, V.B. Shenoy, Nanotechnology 11 (2000) 139. [7] C.K. Hu, B. Luther, F.B. Kaufman, J. Hummel, C. Uzoh, D.J. Pearson, Thin Solid Films 262 (1995) 84. [8] P. Heino, H. Hakkinen, K. Kaski, Phys. Rev. B 58 (1998) 641. [9] J. Schiotz, T. Vegge, F.D. Di Tolla, K.W. Jacobsen, Phys. Rev. B 60 (1999) 11971. [10] J.W. Kang, H.J. Hwang, Nanotechnology 12 (2001) 295. [11] H. Inoue, Y. Akahoshi, S. Harada, Mater. Sci. Res. Int. 1 (1995) 95. [12] D.T. Read, Int. J. Fatigue 20 (1998) 203. [13] L.J. Chen, Z.G. Wang, G. Yao, J.F. Tian, Int. J. Fatigue 21 (1999) 791. [14] J.M. Haile, Molecular Dynamic Simulation, Wiley, New York, 1992. [15] R. Smith, M. Jakas, Atomic and Ion Collisions in Solids and at Surfaces: Theory, Simulation and Application, Cambridge University Press, USA, 1997. [16] P.G. Sanders, J.A. Eastman, J.R. Weertman, Acta Mater. 45 (1997) 4019. [17] T.D. Shen, C.C. Koch, T.Y. Tsui, G.M. Pharr, J. Mater. Res. 10 (1995) 2892.