Atomic-Level Stress Calculation and Two Potentials for Critical

Atomic-Level Stress Calculation and Two Potentials for Critical Conditions of Deposition Process Zheng-Han Hong,† Shun-Fa Hwang,*,† and Te-Hua Fang‡ Graduate School of Engineering Science and Technology, National Yunlin UniVersity of Science and Technology, 123 UniVersity Road, Sec. 3, Douliu, Taiwan 640, P. R. China, and Institute of Mechanical and Electromechanical Engineering, National Formosa UniVersity, 64 Wunhua Road, Huwei, Taiwan 632, P. R. China

CRYSTAL GROWTH & DESIGN 2008 VOL. 8, NO. 4 1191–1199

ReceiVed June 26, 2007; ReVised Manuscript ReceiVed January 9, 2008

ABSTRACT: The Morse two-body potential and the second-moment approximation of the tight-binding (TB-SMA) many-body potential are employed and compared in modeling the sputtering of Cu atoms onto the Cu(100) substrate by Molecular dynamics simulation. The results indicate that when the TB-SMA potential is used under a 5 atom/ps deposition rate, the epitaxy mode of film growth is observed as the incident energy is lower than 3 eV, the film mixing mode clearly occurs from 3 to 5 eV, and the sputtering phenomenon is significant after 10 eV. When the Morse potential is used, the epitaxy mode is observed below 5 eV, the film mixing mode occurs around 5-50 eV, and the sputtering process may be clear only after 50 eV. As for the atomic-level stress, the average normal stress along thickness direction and the average mean biaxial stress are considered. Both the average stresses at the substrate layer are decreased, as the substrate temperature is increased. Increasing the substrate temperature leads to a gradual increase on the standard deviation of the normal stress along thickness direction at the substrate layer. 1. Introduction The ion beam assisted deposition (IBAD) process is one popular process to produce a thin film on a substrate for further applications. To produce a thin film, the IBAD process could be applied in two aspects. One is to enhance the mobility of the deposited atoms and to promote the film growth. The other is to bombard the surface of a solid substrate and to remove its atoms for further deposition. To have these two aspects of applications, it is necessary to adjust the ion incident energy, the ion incident angle, and the ion assisted ratio that is the ratio of ions to neutral atoms. Furthermore, these parameters of IBAD process will affect the quality and morphology of the deposited thin film. To analyze the morphology of the deposited thin film in detail and to understand the growing mechanisms, researchers always adopt molecular dynamics (MD) simulation instead of experiments. Currently, the understanding of IBAD process has been mainly achieved by the use of MD. An important aspect of MD is the selection of potential model, which describes the interaction between atoms. These interatomic potentials could be determined theoretically or empirically, and they can be classified as two-body potentials and many-body potentials. From the viewpoint of simulation, two-body potentials are attractive because of their simple forms and less calculation time. Among them, the Lennard-Jones potential has been extensively used in the studies of the IBAD process. For example, Smith et al.1,2 employed a two-dimensional MD with the classical Lennard-Jones pair potential to investigate the deposition of Ni atoms onto Ni substrate and discussed the film growth process at a low level of incident energy with various conditions of substrate temperature and incident angle. Their results showed that the formation of voids occurred at low incident energy, low substrate temperature, and deposition angles of 45, 60, and 75°. Increasing substrate temperature and incident energy leads * Corresponding author. Tel: 886-5-5342601, ext. 4143. Fax: 886-5-5312062. E-mail: [email protected]. † National Yunlin University of Science and Technology. ‡ National Formosa University.

to less voids and smoother film surfaces. Similarly, Ju et al.3,4 reported that voids and vacancies could be removed at the incident energy of 4 eV and the deposition rate of 200 atom/ ps. Because the Lennard-Jones potential was initially developed to represent noble gases, it could be used to model metal atoms, but not so suitable. Hence, Girifalco et al.5 proposed the Morse potential for metals and it was suitable to represent the cohesive energy of close-packed crystal materials. Yamaguchi et al.6 used the Morse potential to model the deposition of YBa2Cu3O7-x (YBCO) on a three-dimensional highly oriented pyrolytic graphite (HOPG) substrate and discussed the rearrangement behavior of the hot clusters on a low-temperature substrate. It was shown that high temperature nanoscale clusters could be deformed easily even on a room-temperature substrate because of their high internal energy. In recent years, it was found that two-body potentials could not accurately fit the long-range forces acting among atoms. To accurately describe the interaction between atoms, manybody potentials such as embedded-atom method (EAM) potential, the second-moment approximation of the tight-binding (TBSMA) potential,7 and Stillinger-Weber potential8,9 are popular. These potentials are attractive choices for sputtering simulation because they could display good transferability environments. Chu et al.10 used the EAM potential to model the Cu-Cu interaction for a three-dimensional substrate and discussed the morphology of the film deposited under low substrate temperature of 500 K and high incident energy of 10–15 eV. The results indicated that it would be less helpful for layer coverage when the substrate temperature became too high, and larger incident energy could be more effective to produce smoother surface rather than higher substrate temperature. Güvenc et al.11 observed the bombardment of Ni substrate with low-energy argons by using the EAM potential to describe the solid and a Morse-like function to model the interaction between the projectile and the metal atoms. They found that the maximum penetration depth was a function of incident energy and deposition rate. Recently, the TB-SMA many-body potential has also been developed for sputtering and it has been proven to be more accurate than EAM potential.12,13 Su et al.14

10.1021/cg070584h CCC: $40.75  2008 American Chemical Society Published on Web 02/28/2008

1192 Crystal Growth & Design, Vol. 8, No. 4, 2008

investigated the influence of energetic Ar ions from 10 to 40 eV onto a three-dimensional Cu substrate. The TB-SMA potential and the Moliere potential were used to simulate the Cu-Cu and Cu-Ar interactions, respectively. They found that the dependency between the surface roughness and the incident angle becomes apparent at higher incident energy. Marcon et al.15 used the inter-ring torsion potential and Lennard-Jone potential to simulate the orientational effect of the surface of potassium hydrogen phthalate upon the epitaxy growth of tetrathiophene (4T) film from the vapor phase. The results clearly showed the presence of two preferred orientations of the long axes of the 4T molecules. Ju et al.16 used the TBSMA potential to simulate the Cu-Co interaction in the deposition process onto a three-dimensional trench model. An ideal trench-filling morphology was found at higher incident energy and higher substrate temperatures, and voids and vacancies could be removed. Recently, the stress of the IBAD process has concerned in a deposited thin film on a substrate for further applications, such as ultralarge scale of integration (ULSI) used in aluminum or aluminum-alloy interconnection. It has been observed by experiments that the deposition process will produce very large stress in the deposited film. To understand the stress evolution is imperative to the reliability of the deposited film, and this could be provided by MD simulation. Chocyk et al.17 observed the stress evolution in Cu thin films during and after the deposition process at room temperature by using the LennardJones potential. Their results indicate that there is compressivetensile-compressive evolution of stress under Volmer–Weber mode at various deposition rates. The stresses obtained by simulation were very close to the results of experiment as the deposited film was thin. Zhang et al.18 studied the generation mechanisms of virial stresses under various incident energies by using the Tersoff-Brenner potential to model the thin carbon film. The stress at 40 eV incident energy approached a steadystate value when the number of the deposited atoms was over 600. Furthermore, biaxial tensile stresses occurred at the deposited film as the incident energy was low, and a transition from tensile to compressive stress was observed at around 10 eV. This compressive stress started to decrease when the incident energy was over 60 eV. From the above discussion, it will be very interesting to compare the difference between a two-body potential and a many-body potential in the MD simulation for IBAD process. In this study, the Morse two-body potential and the TB-SMA many-body potential are chosen to model the atom interaction in the system of Cu onto Cu(001) substrate. By using these two potentials, the conditions for epitaxy, mixing, and sputtering modes in the IBAD process will be investigated systematically by controlling the deposition parameters including incident energy, substrate temperature, and incident angle. To have an insight on the stress distribution of the thin film after the deposition process, the distribution of BDT stress defined by Basinski, Duesbery, and Tayor19 for the Frank-van der Merwe mode is observed. Furthermore, the BDT stress is calculated under various substrate temperatures. 2. Molecular Dynamics Method The Morse two-body potential and the TB-SMA many-body potential are adopted to model the atomic interactions among the Cu atoms. Similar to the Lennard-Jones potential, the Morse potential contains an attractive part and a repulsive part. However, the Lennard-Jones potential is commonly used to model the interaction of gases, whereas the Morse potential is

Hong et al. Table 1. Parameters for Morse Potential

Cu

D(ev)

R (Å1)

r0 (Å)

0.343

1.358

2.626

Table 2. Parameters for TB-SMA Potential A (eV)

ξ (eV)

p

q

r0 (nm)

0.085

1.224

10.960

2.278

0.255

Cu

used to model metals.20 Unlike many-body potentials, the Morse potential just considers the interaction between two atoms without including the simultaneous influence of their neighbor atoms. Because of this simplification, the form of the Morse potential is simple and it is fast to execute a simulation. Therefore, it has been used to simulate the sputtering process and reasonable results were claimed.21 The Morse potential energy U(rij)could be described with three free parameters as U(rij) ) D(e-2R(rij-r0) - 2e-R(rij-r0))

(1)

where D is the dimer energy, r0 is the equilibrium displacement, and R is fitted to the bulk modulus of material. The separation distance between atoms i and j is denoted as rij. For Cu-Cu atoms, the values of these parameters are listed in Table 1. The TB-SMA potential energy U(rij) contains a repulsive pair potential and a cohesive band energy term, and it could be expressed as follows N

U(rij) )

∑ i)1

[ (∑ -

j

( ( ))

rij ξ exp - 2q -1 r0 2

1 2

+

[ ( ))]] rij -1 r0

Aexp -p

(2)

where ξ is an effective hopping integral, r0 is the first-neighbor distance, and N is the number of atoms considered. The parameters ξ, q, p, and A are fitted to the experimental values of cohesive energy. For Cu-Cu atoms, the values of these parameters are listed in Table 2. Sometimes it is very interesting to know the stress distribution of the thin film after the deposition process. One possibility to represent the stress of the thin film is to use the atomic or BDT stress. This stress is calculated from the volume Vi of atom i, and it includes two parts, the kinetic energy of the atom and the interatomic forces. The definition of BDT stress (σ) can be expressed as19 σ)

[

N

∑

]

1 1 m V X Vi + r Xf Vi i i 2 j*i ij ij

(3)

where mi is the mass of the atom i, Vi is its velocity, rij is the distance between atoms i and j, fij is the interatomic force, and X denotes the tensor product of two vectors. The three-dimensional model of the deposition process is presented in Figure 1. The substrate size is composed of 15a × 15a × 10a (a ) lattice constant) from a face-centered-cubic bulk crystal. The Cu(001) substrate perpendicular to the z-axis consists of 9000 Cu atoms. The lowest layer of the substrate is fixed to prevent the substrate from being moved by the incident atoms during deposition. The other layers of the substrate are called the thermal control layers, and the atomic velocities of these layers are rescaled at every ten-time step according to the prescribed substrate temperature. Hence, the kinetic energy of the incident atoms is absorbed by these thermal control layers during the deposition processes. The velocities of the atoms of

Two Potentials for Critical Conditions of Deposition

Crystal Growth & Design, Vol. 8, No. 4, 2008 1193

the Morse and TB-SMA potentials under different mechanisms of IBAD process. 3. Results and Discussions 3.1. Difference on the Effect of Incident Energy. To study the effects of incident energy by using the Morse potential and TB-SMA potential, defect formation and an accumulation ratio are observed during the IBAD process. The accumulation ratio Ra is defined as Ra ) Ns ⁄ Nt

Figure 1. Simulation model of the IBAD process.

the thermal control layers are given by the Maxwell–Boltzmann distribution23 of the prescribed substrate temperature. Periodic boundary conditions are enforced in the x and y directions and there is no periodic boundary condition along the z direction. The position of the incident atoms is random in the x and y directions, and the z position of the incident atoms is at 20fold-lattice length above the substrate surface. During the MD simulation, the deposition rate is controlled as 5 atoms/ps, and the time step is chosen as 1 × 10-15 s on considering of stability of the simulation. In addition, 14 incident energies from 0.01, 0.05, 0.1, 0.3, 0.5, 1, 3, 5, 10, 20, 30, 40, 50, and 200 eV are considered for the 300 ps simulation. Also, two substrate temperatures with 300 and 450 K as well as two incident angles of 0 and 30° are discussed. The reason to discuss so many different incident energies is to compare the performance of

(4)

where Nt is the total number of the incident atoms and substrate atoms, and Ns is the number of the accumulated atoms of the substrate, which is the total atom number subtracted to the number of the free atoms. Instead of the sputtering yield that describes the ratio of the number of deposited atoms to the number of incident atoms, the reason to use the accumulation ratio is because it considers both the increase of atom number due to the film growth process and the reduction of atom number due to the sputtering process. The accumulation ratio is calculated at the thirteen incident energies under the deposition rate of 5 atom/ps, the substrate temperature of 300 K, and the incident angle of 0°. During the simulation of the IBAD process under different conditions, two different modes of film growth may occur. One is the epitaxy mode in which the incident atoms attach to the surface of the substrate and do not penetrate the first layer of the substrate. The other is the mode of film mixing in which the incident atoms penetrate into the substrate and mix with the substrate atoms. In addition to the two modes of film growth, the sputtering process in which the substrate atoms are bombed away by the incident atoms may be also observed. When the TB-SMA potential is used, the accumulation ratio as shown in Figure 2 is increased from 0.949 to 0.989 as the incident energy is increased from 0.3 to 5 eV. After that, the accumulation ratio starts to decrease as the incident energy increases. To differentiate the epitaxy mode from the film mixing mode, one could examine the morphology of the grown film. The epitaxy mode of film growth is observed as the incident energy is lower than 3 eV, while the mixing mode of film growth

Figure 2. Accumulation ratio as a function of incident energy at substrate temperature of 300 K, incident angle of 0°, and deposition rate of 5 atom/ps.


Hong et al.

Figure 5. Deposition morphology at incident energy of 200 eV, deposition rate of 5 atom/ps, substrate temperature of 300 K, and incident angle of 0° by using TB-SMA potential under different simulation times: (a) 100, (b) 200, and (c) 300 ps.

Figure 6. Accumulation ratio as a function of incident energies at deposition rates of 5 atom/ps, substrate temperature of 300 K, and incident angle of 0 and 30°.

Figure 3. Average kinetic energy of incident atoms at deposition rate of 5 atom/ps, substrate temperature of 300 K, incident angle of 0°, and different incident energies: (a) 0.3, (b) 3, and (c) 30 eV.

Figure 7. Accumulation ratio as a function of different substrate temperatures at incident energy of 30 eV, deposition rate of 5 atom/ ps, and incident angle of 0 and 45°.

Figure 4. Deposition morphology at incident energy of 200 eV, deposition rate of 5 atom/ps, substrate temperature of 300 K, and incident angle of 0° by using Morse potential under different simulation times: (a) 100, (b) 200, and (c) 300 ps.

clearly occurs at the incident energy from 3 to 5 eV. When the incident energy is larger than 10 eV, the sputtering phenomenon is significant. When the incident energy is lower than 1 eV, the surface of the grown film is not smooth at the end of simulation,

and vacancies and voids occur almost everywhere. These phenomena may be attributed to the low incident energy with which the incident atoms could not move energetically and are absorbed easily by small clusters. If the incident energy is increased to 1 eV, the surface of the grown film is smooth, and vacancies and voids are less observed. Also the film growth mechanism is changed to a layer-by-layer form because of better thermal diffusion or mobility. As the incident energy is further increased to 3 eV, the mode of film mixing is observed, while the accumulation ratio still increases as shown in Figure 2.


Figure 8. Number of deposition atom at the substrate layers at incident energies of 200 eV, deposition rates of 5 atom/ps, substrate temperature of 300 and 450 K, and incident angle of 0°.

Figure 9. Morphology of the epitaxy mode of the Cu onto Cu substrate at incident energy of 1 eV, deposition rate of 5 atom/ps, incident angle of 0°, and substrate temperatures of 300 K after 500 ps simulation.

Moreover, the grown film is smooth and vacancies and voids seldom occur. This layer-by-layer film growth mechanism was also verified by experimental methods24 with the incident energy of 3 eV. A s imilar situation occurs as the incident energy is increased to 5 eV. If the incident energy is larger than 5 eV, the accumulation ratio starts to decrease as shown in Figure 2. The decreasing of the accumulation ratio is due to that some of the incident atoms and the substrate atoms are desorbed. This may also represent that the process of sputtering in which the substrate atoms are removed by the incident atoms is triggered. If the incident energy is further increased, the substrate atoms will be removed more, and the phenomenon of sputtering process is clearer, because more incident atoms have the energy to penetrate into the substrate and to break up the bond between the atoms of the substrate. When the Morse potential is used, the accumulation ratio increases as the incident energy is increased from 0.3 to 50 eV and slightly decreases after 50 eV, as shown in Figure 2. From the morphology of the grown film, it indicates that the epitaxy mode occurs if the incident energy is lower than 5 eV, while


the film mixing mode occurs from 5 eV to about 50 eV. When the incident energy is 0.3 eV, the accumulation ratio obtained by using the Morse potential is larger than that obtained by TB-SMA potential. To further explain this phenomenon, the average kinetic energy per incident atom is shown in Figure 3a for the two potentials as the incident energy is 0.3 eV. The average kinetic energy for the Morse potential decreases instantly from 0.3 to 0.1 eV once the incident atom is released. This may represent that the incident atom is absorbed by the substrate at once. Because the incident atom has low energy and mobility, voids and vacancies may be easily produced. On the other hand, the average kinetic energy obtained by the TBSMA potential has a lot of variation, and its value is even higher than the initial value. This may be attributed to some of the incident atoms escaping away before they arrive at the substrate because of many-body interaction and they have a lot of energy. This may also explain why the accumulation ratio of the TBSMA potential is lower than that of the Morse potential. A similar phenomenon is observed for the incident energy of 0.5 eV. However, both potentials have a very close accumulation ratio as the incident energy is 0.1 eV. This may be due to the occurring of clusters before they attach to the substrate by the Morse potential, and its accumulation ratio is reduced. When the incident energy is 3 eV, the average kinetic energy per incident atom obtained by both potentials is shown in Figure 3b. The result obtained by the Morse potential has a significant drop from 3 to 0.7 eV at the beginning of simulation and then slowly decreases to near zero. This may indicate that the incident atoms are directly absorbed by the substrate. On the other hand, the average kinetic energy obtained by the TB-SMA potential does not have this drop at the beginning, but later its value significantly decreases to a stable value. This may represent the incident atoms attaching to the substrate but not so fast. Because both potentials show the attachment of the incident atom to the substrate, they have very close accumulation ratios and sputtering yields in this case. Similar results are found for the incident energy of 1 and 5 eV. Unlike the results obtained from the TB-SMA potential, the accumulation ratio from the Morse potential still slightly increases when the incident energy is lager than 5 eV. Its accumulation ratio only slightly decreases after 50 eV. This result implies that the process of sputtering is not clearly observed by using the Morse potential even under high incident energy. The average kinetic energy by the TBSMA potential is clearly higher than that by the Morse potential as shown in Figure 3c, and it may explain the clear sputtering process by using the TB-SMA potential. Experimental observation of this sputtering process was made at the energy of 80 eV for Cu atoms incident upon Cu substrate.25 The deposition process at the incident energy of 200 eV by the Morse potential is shown in Figure 4a-c. In these figures, it is demonstrated that some incident atoms are continuously sucked into the bottom of the substrate as the increasing of the simulation time, which could be explained by the two-body nature and short-range effect of the Morse potential. Hence, the incident atoms could continuously attach to the substrate, and only few substrate atoms are bombed away by these incident atoms with high energy such that its accumulation ratio only slightly decreases from the maximum value. On the other hand, the deposition process at the incident energy of 200 eV by the TB-SMA potential is shown in Figure 5a-c. The incident atoms do not continuously penetrate into the bottom of the substrate and mainly remain in the surface layers because of the character of the many-body interaction provided by the TB-SMA potential. Therefore, the atoms in the surface layers of the substrate


Hong et al.

Figure 10. Stress of each atom at incident energy of 1 eV, deposition rate of 5 atom/ps, incident angle of 0°, and different substrate temperatures: (a) 300, (b) 450, (c) 600, and (d) 750 K.

have more chance to be bombed away by the incident atoms with high energy and mobility. 3.2. Difference on the Effects of Incident Angle and Substrate Temperature. The effects of the incident angle on the accumulation ratio are shown in Figure 6 by using the Morse potential and the TB-SMA potential. In this figure, the incident angles are 0 and 30°, the incident energies considered are 0.3, 3, 30, and 150 eV, the substrate temperature interested is 300 K, and the deposition rate is 5 atom/ps. It is very interesting to note that as the incident energy is 3 eV, the accumulation ratio is about the same by using the two potentials and the two incident angles. As described in the previous section, it is in the mode of film mixing at this incident energy, and one could say that at this condition the two potentials do not cause clear difference on the effect of incident angle. When the incident energy is 0.3 eV, the Morse potential predicts that the incident angle of 30° has smaller accumulation ratio than the incident angle of 0°. Because it is in the mode of film growth, inclined incident angle may reduce the sputtering yielding. Similar situation occurs as the TB-SMA potential is used. When the Morse potential is used and the incident angle is 0°, the accumulation ratio for 30 eV incident energy is larger than that for 3 eV incident energy, and it indicates that it is still in the mode of film mixing. However, as the incident angle is increased to 30°, the accumulation ratio for 30 eV incident energy is slightly decreased as compared to that for 3 eV incident energy. In this case the sputtering process may be triggered but not clear. If the TB-SMA potential is used, the accumulation ratios at both incident angles for 30 eV incident energy are clearly less than those for 3 eV incident energy, and the sputtering process is clearly indicated. Also, the inclined incident angle could assist the occurring of the sputtering process. When the incident energy is 150 eV, both potentials predict that the inclined incident angle may have high possibility to assist the escape of the substrate atoms. From these discussions, both potentials show similar effect of the incident angle.

Because the two potentials have big difference under high incident energy, the incident energy of 30 eV is chosen to discuss the effect of substrate temperature. The accumulation ratio in terms of substrate temperature under different potentials and different incident angles is illustrated in Figure 7. When the incident angle is 0°, the accumulation ratio predicted by both potentials is decreased as the increasing of substrate temperature. However, when the incident angle is 45°, the accumulation ratio obtained by both potentials just slightly decreases as the substrate temperature is increased from 300 to 750 K. Furthermore, the accumulation ratio of 45° incident angle is far lower than that of 0° incident angle, and the Morse potential could result in higher accumulation ratio than the TBSMA potential. Hence, even though the two potentials have different accumulation ratio, they predict similar coupling effects between the incident angle and the substrate temperature. When the incident angle is small, the substrate temperature has a significant effect on the sputtering process. Instead, the substrate temperature has little effect on the sputtering process as the incident angle is high. It seems that the effect of the incident angle is larger than that of the substrate temperature. To get a deep insight, the number of the atoms penetrated into the substrate is distributed along the thickness direction in Figure 8 for the incident energy of 200 eV with two different substrate temperatures. In this figure, the substrate layer is numbered from the bottom to the top, and the twentieth layer is the surface layer of the substrate. As shown in this figure, the penetrated atoms estimated by the TB-SMA potential mainly remain around the surface layers, whereas those predicted by the Morse potential are distributed deeply into the substrate. It is interesting to note that when the Morse potential is used, there are a lot of penetrated atoms distributed between the 11th layer and the 18th layer, and there are some penetrated atoms presented at the second layer. Because some atoms penetrate deeply into the substrate, the accumulation ratio predicted by the Morse potential remains high, and the sputtering process is



Figure 11. Layer stress at incident energy of 1 eV, deposition rate of 5 atom/ps, incident angle of 0°, and different substrate temperatures: (a) average normal stress and (b) average mean biaxial stress.

seldom observed. On the other hand, the penetrated atoms estimated by the TB-SMA potential mainly remain around the surface layers, and it has high chance to bomb away the atoms of the surface layers. Hence, the accumulation ratio is low and the sputtering process is evident. From these discussions, one could say that the results may be in doubt as the Morse potential

is used at high incident energy or at the sputtering process. From Figure 8, it is also found that for both potentials the distribution of the number of the penetrated atoms does not have a big difference as the substrate temperature is increased from 300 to 450 K. This may explain why the accumulation ratios at these two substrate temperatures are close, as shown in Figure 7.


Hong et al.

Figure 12. Standard deviation of atomic stresses of each layer at incident energy of 1 eV, deposition rate of 5 atom/ps, incident angle of 0°, and different substrate temperatures.

3.3. Stress Evolution under the Effect of Substrate Temperature. As shown in Figure 9, the epitaxy mode could be observed at the incident energy of 1 eV, the deposition rate of 5 atom/ps, the incident angle of 0°, and the substrate temperatures of 300 K after simulation times of 500 ps by using the TB-SMA potential. It clearly indicates that the surface of the grown film is smooth. Moreover, the film growth mechanism is changed from a Frank-van der Merwe mode to a layer-bylayer form because of better thermal diffusion or mobility.25 To have an insight on the stress distribution of the deposited thin film, the BDT stress of each atom, σzz, is calculated after the simulation and represented by different colors in Figure 10a-d under the substrate temperature of 300, 450, 600 and 750 K, respectively. Because the sign convention for force is positive for repulsion and negative for attraction, a positive stress indicates compression and a negative stress is tensile. From these figures, it is evident that the atoms with tensile stress are mainly distributed at the surface layers, and the atoms at the bottom layers have compressive stresses. To quantify the BDT stress at each layer, an average stress σmnavg is defined for each layer as N

avg σmn )

∑

1 σi N i)1 mn

(5)

where σmni is the BDT stress of the ith atom of the layer, m and n are indices of the stress tensor, and N is the number of atoms of the layer. Since the shear stresses are one order less than the normal stresses in the deposition case17 the average normal stress σzzavg and the average mean biaxial stress σxxavg + σyyavg/2 are considered. The average mean biaxial stress is chosen because it is related to the residual stress of the layer. These two stresses for all layers are illustrated in panels a and b in Figure 11 for different substrate temperatures. Again, the layers are numbered from the bottom to the top, and layer numbers 21-30 are the deposited layers. Because layer number 1 is fixed and layer

number 29 or 30 does not have enough stable atoms, their average stresses are not included in the figure. From these figures, the substrate layers numbered 2-20 have compressive stresses, because the substrate suffers from the deposition of incident atoms and the substrate atoms are also compressed due to periodic boundary conditions along the x and y directions. Also, both the average stresses of the substrate layers have oscillation and it may result from the internal interaction between consecutive layers. On the other hand, there is no clear oscillation on both average stresses of the deposited layers. From Figure 11a, the average normal stresses σzzavg of the deposited layers numbered 21-26 are still compressive as the substrate layers, whereas the stresses are tensile after layer number 27. These tensile stresses may come from the unstable and loose packing of atoms at these layers. From Figure 11b, the average mean biaxial stress remains positive in both the substrate layer and the deposited film layer. It looks like this stress has a minimum value around the transition layers between the substrate and the film, and after these transition layers, the stress may increase. As for the effect of the substrate temperature, increasing the substrate temperature leads to a gradual decrease in both average stresses of the substrate layer. However, there is no clear effect on both average stresses of the deposited layer as the substrate temperature is increased. Note that as the substrate temperature is increased, the oscillation of the averaged mean biaxial stress around the surface layer of the substrate may disappear as shown in Figure 11 b. This result may come from higher intermixing of the incident atom and the substrate atom occurring in these layers because of higher substrate temperature. From panels a and b in Figure 11, it is interesting to note that the σzzavg stresses are higher than the average mean biaxial stresses. Because there are some atoms under tensile stress and some atoms under compressive stress in each layer, it is interesting



to know the standard deviation of the BDT stress. A standard deviation, Smn, for stress σmn is defined for each layer as 2 Smn )

∑

n i)1

i avg 2 (σmn - σmn )

N-1

(6)

The standard deviation of σzz in different layers is shown in Figure 12. It clearly shows that the standard deviation of this stress becomes higher as the layer number is increased. That is to say, the layer of the deposited film has a lot of variation on the stress, while the layer of the substrate has narrow distribution of the stress values. As the substrate temperature is increased, the standard deviation in the substrate layer is slightly increased. However, the standard deviation in the deposited film has no clear increase when the substrate temperature is changed. A similar situation occurs on the standard deviation of the mean biaxial stress. 4. Conclusion This paper investigates in detail the film growth process as well as the sputtering process and compares the difference of a two-body potential and a many-body potential in the MD simulation for IBAD process. In the study, the Morse two-body potential and TB-SMA many-body potential are chosen to model the atom interaction in the system of Cu onto Cu(001) substrate. By using these two potentials, we systematically investigated the conditions for epitaxy, mixing, and sputtering modes in the IBAD process by controlling the deposition parameters including incident energy, substrate temperature, and incident angle. When the deposition rate is 5 atom/ps and the TB-SMA potential is used, the epitaxy mode of film growth is observed as the incident energy is lower than 3 eV. The mixing mode of film growth clearly occurs at the incident energy from 3 to 5 eV. When the incident energy is larger than 10 eV, the sputtering phenomenon is significant. As the Morse potential is used, the epitaxy mode is observed below 5 eV, film mixing mode occurs around 5-50 eV, and the sputtering process may be clear only after 50 eV. Also, when the incident atoms are deposited onto the substrate by using the Morse potential, these atoms are absorbed by the substrate at once. However, the incident atoms do not directly attach to the substrate by using the TB-SMA potential. Hence, the average kinetic energy by the TB-SMA potential is clearly higher than that by the Morse potential. Furthermore, the accumulation ratio of 45° incident angle is far lower than that of 0° incident angle, and the Morse potential has higher accumulation ratio than the TB-SMA potential. On the other hand, the substrate temperature has little effect on the sputtering process as the incident angle is high. It also seems that the effect

of the incident angle is larger than that of the substrate temperature. As for the atomic-level stress, the results indicate that both the average normal stress along thickness direction and the average mean biaxial stress of the substrate layers are compressive due to the substrate suffered from the deposition of incident atoms. Both the average stresses of the substrate layers have oscillation and there is no clear oscillation of the average stresses at the deposited layers. Both the average stresses at the substrate layer are decreased, as the substrate temperature is increased. Increasing the substrate temperature leads to a gradual increase on the standard deviation of the normal stress at the substrate layer.

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CG070584H

Atomic-Level Stress Calculation and Two Potentials for Critical

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