- Email: [email protected]
Texture competition during thin film deposition – effects of grain boundary migration
Computational Materials Science 23 (2002) 190–196 www.elsevier.com/locate/commatsci
Texture competition during thin film deposition – effects of grain boundary migration Hanchen Huang a
a,*
, George H. Gilmer
b
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong b Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, USA Accepted 1 June 2001
Abstract In this paper, we describe an implementation of grain boundary migration in the atomistic simulator of thin film deposition (ADEPT), and apply the simulator to study effects of the grain boundary migration on texture evolution. In the implementation, atoms are classified into two categories: those belong to a single grain and those at grain boundaries. An atom is defined as one at a grain boundary if it has more than half of its neighbors occupied and not all of the neighboring atoms are in the same grain. The grain boundary atom is attempted to re-align with neighboring grains to represent the grain boundary migration; the attempt probability is defined by the grain boundary migration coefficient. Our studies show that grain boundary migration does not always assist formation of texture with a top surface of the lowest energy. At the nucleation stage of thin film deposition, high migration coefficient of grain boundaries may enhance the formation of grain nuclei with top surfaces of higher energy, and therefore effectively may suppress formation of textures with a top surface of the lowest energy. This effect may provide an extra dimension to engineer textures of thin films. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Microstructure evolution of thin films is determined by a complex array of processes. The performance and macroscopic structures of thin films are determined mainly by atomic-level process occurring during deposition. In modern integrated circuits, the interconnects are made of aluminum or copper thin films [1,2]. The lifetime of an interconnect is a sensitive function of microstructure, and it has been found that h1 1 1i texture is *
Corresponding author. E-mail addresses: [email protected] (H. Huang) and [email protected] after summer 2002.
the most effective for suppressing electromigration which controls the lifetime. Similarly, titanium nitride thin films are predominantly h1 1 1i and h1 0 0i textures for semiconductor and mechanical applications, respectively [3]. For more effective applications of thin films, better understanding and control of the deposition processes are required. Some of the microstructure evolution processes can be directly investigated by experiments, while others are not accessible by available experimental techniques. Computer simulations can often complement experimental investigations. In a computer simulation, the processing conditions can be precisely controlled, and the mechanisms occurring during deposition can be analyzed in detail.
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 2 3 4 - 8
H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
Computer simulations have advanced substantially in recent years because of rapid improvements in computer technology and in computational methods. While computers become faster and faster, models of thin film growth have been developed to cover a wider range of length and time scales, from detailed electronic structure calculations to continuum treatments of macroscopic structures. At the atomic level, Dong and Srolovitz [4] studied the texture development at a high deposition rate using the molecular dynamics method. In this model, atomic vibration and diffusion are rigorously taken into account. However, the physical time of one molecular dynamics simulation is only a few nanoseconds, although millions of atoms can be treated; in contrast, most thin film deposition processes take minutes in reality. As a result, the deposition rate must be extremely high in order to simulate growth of a few atomic layers of thin film. One artifact of the high deposition rate is the large fluctuations of deposition and diffusion rate during the growth process. This can lead to extreme surface roughening and enhance columnar growth. Recent development of hypermolecular dynamics methods may extend the physical time for molecular dynamics simulations of the growth process up to microseconds [5]. Still, the simulation time scale is several orders of magnitude shorter than its experimental counterpart in a typical vapor deposition process. Being flexible in time scale, continuum models such as EVOLVE [2] have proven to be effective for studies of thin film processing over the past years. In the continuum models, surfaces of thin film are represented by a series of nodal points, and advancement of the nodal points is tracked as a function of time. These models provide insight into surface morphology and may also represent void formation under the surface. To avoid numerical difficulties when two surface fronts pinch off, the level set method [6] has been adopted in most of these models. One major drawback of continuum models is the lack of microstructure information, such as textures. In an attempt to simulate microstructure evolution up to a time scale of minutes, both lattice Monte Carlo and continuum Monte Carlo
191
methods have been used. The discrete nature of particles is explicitly represented in these Monte Carlo methods. A simulator, based on a twodimensional continuum Monte Carlo model, was developed at the University of Alberta [7]; in this model, two-dimensional disks representing the particles are allowed to move around in a continuum space. Domain structures, somewhat representative of polycrystalline films, are simulated in this approach. The two-dimensional Monte Carlo model was extended to three-dimensions by Baumman and Gilmer [8]. However, the extension is computationally so demanding that it is beyond the capacity of present computers. The lattice Monte Carlo method seems to be the only choice, in order to simulate deposition processes up to a time scale of minutes and to incorporate the microstructure evolution. The computational efficiency is a direct result of atomic motion in a discrete space as defined by the lattice. A group at the University of Virginia developed such a model focusing on incorporation of the details of the atomic collision processes, such as resputtering and heat dissipation during the initial stage of particle – surface interaction [9,10]. Due to the use of a single lattice, only one pre-defined texture is allowed to form during thin film deposition. Competition of textures during the deposition is missing in these simulations. Aiming at simulation of texture competition at the atomic level, we have devised an atomistic simulator (ADEPT) that is based on a multi-lattice Monte Carlo method. This model has gone through three stages: single-lattice model [11,12], dual-lattice model [13], and multi-lattice model [14]. At present, the multi-lattice model is twodimensional, and all possible lattices are mapped onto a single lattice. According to this model [14], diffusion along grain boundaries is approximately represented by diffusion along free surfaces. However, migration perpendicular to the grain boundaries is absent. In this paper, we implement a representation of grain boundary migration in ADEPT and apply it to study effects of the grain boundary migration on texture evolution. In Section 2, the implementation and the simulation method are presented. In Section 3, ADEPT is applied to simulate the texture
192
H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
competition as a function of grain boundary migration coefficient. Finally, a summary of the studies is presented in Section 4.
2. Model implementation As in our recent publication [14], a twodimensional simple cubic lattice is used to demonstrate the concept and the application of the multi-lattice Monte Carlo model. In addition, we consider only the first nearest neighbors for simplicity; as a result, each lattice site has four nearest neighboring sites. An extension to three-dimensions is straightforward, although more complex programming is involved. In this section, we first describe representation of grain boundary migration, and then present energetics used in the simulation; all presentations are based on the two-dimensional Monte Carlo model. In the simulations of polycrystalline thin films, atoms can be classified into two categories: those belonging to a single grain and those at grain boundaries. An atom is considered to be in a single grain when neighbored by atoms of the same grain, and it is considered to be at a grain boundary when neighbored by at least one atom from other grains and at least half of its nearest neighbours are occupied. Deposition and diffusion of the atoms in a single grain are simulated in the same way as described in [14]. The atoms at grain boundaries are treated differently. In addition to diffusion along grain boundaries, these atoms also migrate perpendicular to the grain boundaries; the latter process represents the grain boundary migration. The migration of grain boundary atoms is represented by re-alignment of the atoms with neighboring grains. The attempt frequency (m) of atomic re-alignment is determined by the grain boundary migration coefficient (D) according to m¼
2D ; a2
ð1Þ
where a is the nearest neighbor distance of the simple cubic lattice, and is taken to be 0.3 nm. At each attempt of grain boundary migration, one atom at grain boundary is randomly chosen for re-
alignment. Since all grain boundaries are assumed to have the same migration coefficient, all atoms at grain boundaries have equal probability to be chosen. For each chosen atom, its neighboring atoms are searched and listed if (i) they are nearest neighbors of the chosen atom or (ii) the chosen atom is one of their nearest neighbors. These two conditions are not identical because an atom may not be nearest neighbor of its own nearest neighbor; this happens only at grain boundaries. The chosen atom is temporarily assigned to belong to one of the grains that the listed atoms belong to, and the configuration energy of the entire system is calculated as Ei ; for the two-dimensional model, one angle is sufficient to represent orientation of each grain. After calculating the energy (Ei ) for each orientation of the listed atoms, a final orientation of the chosen atom is assigned. The probability of assigning the orientation to be hk is X Ej =kT Pk ¼ eEk =kT e ; ð2Þ j
where j runs over all the listed atoms, and kT is the conventional Boltzmann factor. Since only the listed atoms are affected by the re-alignment, the j does not have to run over those atoms that are not listed. It is worthwhile mentioning that the orientation of a neighboring atom maybe the same as the orientation of the migrating atom. In such a case, the grain boundary migration leads to no change in atomic states, although a grain boundary migration has occurred. The controlling parameter of the grain boundary migration, the migration coefficient, is chosen to be 104 e0:5 eV=kT cm2 =s as a reference value. The prefactor is taken to be the same as those of adatom diffusion coefficients, and the activation energy is taken to be the same as that of twocoordinated adatoms. The attempt frequency of the re-alignment is related to the migration coefficient according to Eq. (1). The update of time is carried out in the same fashion as in [11]. The grain boundary migration, or re-alignment of the grain boundary atoms, is counted as an additional category of diffusion in the simulation. To study the effects of grain boundary migration, this coefficient is multiplied by a numerical factor, ranging from 104 to 101 . Details of other energetics have been
H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
presented in [14], and only a summary of them is presented here. The potential energy of each atom is assumed to be a linear function of its coordination, and it is )1.0 eV with coordination being 1. Thin film atoms from other grains contribute to the coordination but do not change the potential energy. When a deposited atom is neighbored by one or more substrate atoms, it is always taken to have two neighbors from the substrate; however, an increase of 0.5 eV in potential energy is added to the atom to account for the interface energy. The potential energy of the substrate atoms is calculated by directly counting the number of nearest neighbors, including substrate atoms and deposited atoms. The substrate atoms are assigned random orientation at the start, and assumed to be immobile and diffusion coefficient of each thin film atom is assumed to be 104 eQ=kT cm2 =s; Q is 0.2, 0.5, 0.7 eV, and infinity when coordination is 1, 2, 3, and 4, respectively. The proposed energy calculation and the diffusion energetics are to mimic deposition of a thin film on an amorphous substrate of the same chemical composition.
193
Using the proposed representation of the grain boundary migration and the energetics, we investigate effects of grain boundary migration on texture competition in the following section.
3. Results and discussions Staring with a flat substrate, we deposit thin films using various grain boundary migration coefficients. To focus on how the grain boundary migration affects dominance of the texture with a top surface of the lowest energy, we use a 1:1 collimation to filter the incident atoms. Source atoms are injected toward the substrate at an incident angle a, defined relative to the vertical axis, with a relative probability sinð2aÞ tanð45° aÞ for a 6 45°. No source atoms with a > 45° are allowed to reach the substrate under the 1:1 collimation. At first, simulations are performed at a deposition rate of 5 lm/min, keeping the substrate temperature at 300 K; 50 atomic layers are deposited in each simulation. The simulation results with the
Fig. 1. The thin films of 50 atomic layers deposited at 300 K under 5 lm/min. From top (a) to bottom (f), the grain boundary migration coefficient is 101 , 100 , 101 , 102 , 103 , and 104 times that of the reference value, respectively. The solid circles in four different gray scales represent atoms in the lattices of four different orientations. The four types of lattices are generated by rotating the reference lattice count clockwise 0°, 22.5°, 45°, and 67.5°, respectively. The darkness of the circles increases with the rotation angle.
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H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
grain boundary migration coefficient scaled by 101 , 100 , 101 , 102 , 103 , and 104 are shown in Fig. 1(a)–(f), respectively. The texture with a surface of the lowest energy is the h1 0i texture, shown as the lightest color in the figure. There is an indication that the h1 0i texture is the most dominant when the grain boundary migration coefficient is scaled by 102 . For either a larger or a smaller grain boundary migration coefficient, the dominance of the h1 0i texture is reduced. In order to confirm this observation, we repeat the simulations using a lower deposition rate, 1 lm/min. The simulation results, as shown in Fig. 2(a)–(f), indicate that (i) an optimal migration coefficient for the dominance of the h1 0i texture exists, and (ii) the optimal migration coefficient corresponds to the same scaling factor, 102 . To test the temperature dependence of this phenomenon, we repeat the simulations of Fig. 1 using a higher substrate temperature, 350 K. The simulation results, as shown in Fig. 3(a)–(f), confirm what was observed in Figs. 1 and 2. Due to the qualitative nature of the two-dimensional model, no emphasis should be placed on the absolute value of the optimal migration coefficient.
However, existence of the optimal migration coefficient for the dominance of the h1 0i texture deserves further discussion. It is easier to understand that the h1 0i texture is more dominant when the migration coefficient is increased, since a larger migration coefficient leads to thin films closer to the thermodynamic equilibrium. This is indeed the case as the scaling factor of the migration coefficient goes from 104 to 102 . When the migration coefficient is further increased, this reasoning clearly does not apply anymore. To explain this anomaly, we compare surface energies of two clusters, as shown in Fig. 4. The cluster (a) has two {1 0} surfaces on the top, resembling grains with rotation angles of 22.5° and 67.5°; this is the case when the horizontal surface is not one of the {1 0} surfaces and is faceted. The cluster (b) has one horizontal {1 0} surface and two vertical {1 0} surfaces on the top, resembling grains of h1 0i texture. For simplicity, we assume the interfacial area (meaning length in twodimensions) with the substrate is the same, that is L, for both clusters. Further, we assume the number of atoms in the two clusters is the same;
Fig. 2. Thin films deposited at the same conditions as in Fig. 1, except that the deposition rate is smaller, 1 lm/min.
H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
195
Fig. 3. Thin films deposited at the same conditions as in Fig. 1, except that the substrate temperature is higher, 350 K.
Fig. 4. Simple shapes of grain nuclei. L and h are horizontal and vertical dimensions, respectively. The shaded bottom represents interface with an amorphous substrate.
this is equivalent to assuming the same cross-section area in two-dimensions. The latter assumption requires that the height of cluster (a) is twice as that of cluster (b), as shown in Fig. 4. In our model, the substrate is taken to be amorphous. As a result, the interfacial energies of the two clusters are the same, and the thermodynamic preference of the two clusters is determined by their relative surface energies. Because all surfaces are of {1 0}
Fig. 5. Comparison of surface areas (that is lengths in twodimensions) of the triangular cluster and rectangular cluster in Fig. 4.
type and have the same surface energy per unit area, the preference is determined by the relative surface areas of the two clusters. As shown in Fig. 5, the triangular cluster of (a) type has a smaller surface area, when the height h is less than one quarter of the interfacial length L. To minimize the total energy, the nuclei of the triangular shape will therefore be preferred thermodynamically.
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H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
Certainly, this preference is not always the case. In particular, when h is larger than L=4, it is not possible for the two assumed clusters to have the same number of atoms; hence, the above reasoning does not apply anymore. Although the analysis in Figs. 4 and 5 is simple, it does show the possibility for the h1 0i texture to lose dominance at the thermodynamic equilibrium. When the assumptions in Figs. 4 and 5 do not apply, the h1 0i texture is thermodynamically preferred, according to traditional understanding [11]. These two mechanisms lead to reduction and increase of the h1 0i texture dominance, respectively. Consequently, they are responsible for the existence of the optimal grain boundary migration coefficient. If confirmed, the existence of an optimal grain boundary migration coefficient implies that thin film texture can be engineered by modifying grain boundary conditions. For example, segregation of impurity atoms at grain boundaries [15] may block the open path for grain boundary diffusion, and may also slow down the grain boundary migration.
4. Summary In this paper, we have presented an implementation of grain boundary migration in the atomistic simulator (ADEPT), and applied the simulator to study the effects of grain boundary migration on texture competition. Our studies show that an optimal grain boundary migration coefficient exists for the dominance of the h1 0i texture, the one with a top surface of the lowest energy. Further increase or decrease of the migration coefficient suppresses the dominance of this texture. This effect may provide an additional means of engineering thin film textures.
Acknowledgements The work described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (PolyU 5146/99E), and partially by grants from the Hong Kong PolyU (A-PB32 and G-YB91). Support from the NSF/ DARPA VIP program through the University of Illinois is also acknowledged.
References [1] S.R. Wilson, C.J. Tracy, J.L. Freeman, Handbook of Multilevel Metallization for Integrated Circuits, Noyes Publications, NJ, USA, 1993. [2] T.S. Cale, D.F. Richards, D. Yang, J. Comput. Aided Mater. Des. 6 (1999) 283. [3] B.W. Kar, I. Petrov, D.G. Cahill, J.E. Greene, Appl. Phys. Lett. 70 (1997) 1703. [4] L. Dong, D.J. Srolovitz, J. Appl. Phys. 84 (1998) 5261. [5] A. Voter, Phys. Rev. Lett. 78 (1997) 3908. [6] J. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Sciences, Cambridge University Press, London, 1996. [7] T. Smy, R. Tait, M. Breit, IEEE Trans. Comput. Aided Des. 10 (1991) 130. [8] F. Baumann, G.H. Gilmer, IEDM Technical Digest, 1995, p. 89. [9] Y. Yang, R. Johnson, H. Wadley, Acta Mater. 45 (1997) 1445. [10] X. Zhou, R. Johnson, H. Wadley, Acta Mater. 45 (1997) 1513. [11] H. Huang, G.H. Gilmer, T. Diaz de la Rubia, J. Appl. Phys. 84 (1998) 3636. [12] G.H. Gilmer, H. Huang, T. Diaz de la Rubia, C. Roland, Comput. Mater. Sci. 12 (1998) 354. [13] G.H. Gilmer, H. Huang, T. Diaz de la Rubia, J.D. Torre, F. Baumann, Thin Solid Films 365 (2000) 189. [14] H. Huang, G.H. Gilmer, J. Comput. Aided Mater. Des. 6 (1999) 117. [15] H. Huang, T. Diaz de la Rubia, M. Fluss, in: W. Filter, et al. (Eds.), Materials Reliability in Microelectronics VI, MRS, Pittsburgh, 1997, p. 177.
Texture competition during thin film deposition – effects of grain boundary migration Hanchen Huang a
a,*
, George H. Gilmer
b
Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong b Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974, USA Accepted 1 June 2001
Abstract In this paper, we describe an implementation of grain boundary migration in the atomistic simulator of thin film deposition (ADEPT), and apply the simulator to study effects of the grain boundary migration on texture evolution. In the implementation, atoms are classified into two categories: those belong to a single grain and those at grain boundaries. An atom is defined as one at a grain boundary if it has more than half of its neighbors occupied and not all of the neighboring atoms are in the same grain. The grain boundary atom is attempted to re-align with neighboring grains to represent the grain boundary migration; the attempt probability is defined by the grain boundary migration coefficient. Our studies show that grain boundary migration does not always assist formation of texture with a top surface of the lowest energy. At the nucleation stage of thin film deposition, high migration coefficient of grain boundaries may enhance the formation of grain nuclei with top surfaces of higher energy, and therefore effectively may suppress formation of textures with a top surface of the lowest energy. This effect may provide an extra dimension to engineer textures of thin films. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Microstructure evolution of thin films is determined by a complex array of processes. The performance and macroscopic structures of thin films are determined mainly by atomic-level process occurring during deposition. In modern integrated circuits, the interconnects are made of aluminum or copper thin films [1,2]. The lifetime of an interconnect is a sensitive function of microstructure, and it has been found that h1 1 1i texture is *
Corresponding author. E-mail addresses: [email protected] (H. Huang) and [email protected] after summer 2002.
the most effective for suppressing electromigration which controls the lifetime. Similarly, titanium nitride thin films are predominantly h1 1 1i and h1 0 0i textures for semiconductor and mechanical applications, respectively [3]. For more effective applications of thin films, better understanding and control of the deposition processes are required. Some of the microstructure evolution processes can be directly investigated by experiments, while others are not accessible by available experimental techniques. Computer simulations can often complement experimental investigations. In a computer simulation, the processing conditions can be precisely controlled, and the mechanisms occurring during deposition can be analyzed in detail.
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 1 ) 0 0 2 3 4 - 8
H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
Computer simulations have advanced substantially in recent years because of rapid improvements in computer technology and in computational methods. While computers become faster and faster, models of thin film growth have been developed to cover a wider range of length and time scales, from detailed electronic structure calculations to continuum treatments of macroscopic structures. At the atomic level, Dong and Srolovitz [4] studied the texture development at a high deposition rate using the molecular dynamics method. In this model, atomic vibration and diffusion are rigorously taken into account. However, the physical time of one molecular dynamics simulation is only a few nanoseconds, although millions of atoms can be treated; in contrast, most thin film deposition processes take minutes in reality. As a result, the deposition rate must be extremely high in order to simulate growth of a few atomic layers of thin film. One artifact of the high deposition rate is the large fluctuations of deposition and diffusion rate during the growth process. This can lead to extreme surface roughening and enhance columnar growth. Recent development of hypermolecular dynamics methods may extend the physical time for molecular dynamics simulations of the growth process up to microseconds [5]. Still, the simulation time scale is several orders of magnitude shorter than its experimental counterpart in a typical vapor deposition process. Being flexible in time scale, continuum models such as EVOLVE [2] have proven to be effective for studies of thin film processing over the past years. In the continuum models, surfaces of thin film are represented by a series of nodal points, and advancement of the nodal points is tracked as a function of time. These models provide insight into surface morphology and may also represent void formation under the surface. To avoid numerical difficulties when two surface fronts pinch off, the level set method [6] has been adopted in most of these models. One major drawback of continuum models is the lack of microstructure information, such as textures. In an attempt to simulate microstructure evolution up to a time scale of minutes, both lattice Monte Carlo and continuum Monte Carlo
191
methods have been used. The discrete nature of particles is explicitly represented in these Monte Carlo methods. A simulator, based on a twodimensional continuum Monte Carlo model, was developed at the University of Alberta [7]; in this model, two-dimensional disks representing the particles are allowed to move around in a continuum space. Domain structures, somewhat representative of polycrystalline films, are simulated in this approach. The two-dimensional Monte Carlo model was extended to three-dimensions by Baumman and Gilmer [8]. However, the extension is computationally so demanding that it is beyond the capacity of present computers. The lattice Monte Carlo method seems to be the only choice, in order to simulate deposition processes up to a time scale of minutes and to incorporate the microstructure evolution. The computational efficiency is a direct result of atomic motion in a discrete space as defined by the lattice. A group at the University of Virginia developed such a model focusing on incorporation of the details of the atomic collision processes, such as resputtering and heat dissipation during the initial stage of particle – surface interaction [9,10]. Due to the use of a single lattice, only one pre-defined texture is allowed to form during thin film deposition. Competition of textures during the deposition is missing in these simulations. Aiming at simulation of texture competition at the atomic level, we have devised an atomistic simulator (ADEPT) that is based on a multi-lattice Monte Carlo method. This model has gone through three stages: single-lattice model [11,12], dual-lattice model [13], and multi-lattice model [14]. At present, the multi-lattice model is twodimensional, and all possible lattices are mapped onto a single lattice. According to this model [14], diffusion along grain boundaries is approximately represented by diffusion along free surfaces. However, migration perpendicular to the grain boundaries is absent. In this paper, we implement a representation of grain boundary migration in ADEPT and apply it to study effects of the grain boundary migration on texture evolution. In Section 2, the implementation and the simulation method are presented. In Section 3, ADEPT is applied to simulate the texture
192
H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
competition as a function of grain boundary migration coefficient. Finally, a summary of the studies is presented in Section 4.
2. Model implementation As in our recent publication [14], a twodimensional simple cubic lattice is used to demonstrate the concept and the application of the multi-lattice Monte Carlo model. In addition, we consider only the first nearest neighbors for simplicity; as a result, each lattice site has four nearest neighboring sites. An extension to three-dimensions is straightforward, although more complex programming is involved. In this section, we first describe representation of grain boundary migration, and then present energetics used in the simulation; all presentations are based on the two-dimensional Monte Carlo model. In the simulations of polycrystalline thin films, atoms can be classified into two categories: those belonging to a single grain and those at grain boundaries. An atom is considered to be in a single grain when neighbored by atoms of the same grain, and it is considered to be at a grain boundary when neighbored by at least one atom from other grains and at least half of its nearest neighbours are occupied. Deposition and diffusion of the atoms in a single grain are simulated in the same way as described in [14]. The atoms at grain boundaries are treated differently. In addition to diffusion along grain boundaries, these atoms also migrate perpendicular to the grain boundaries; the latter process represents the grain boundary migration. The migration of grain boundary atoms is represented by re-alignment of the atoms with neighboring grains. The attempt frequency (m) of atomic re-alignment is determined by the grain boundary migration coefficient (D) according to m¼
2D ; a2
ð1Þ
where a is the nearest neighbor distance of the simple cubic lattice, and is taken to be 0.3 nm. At each attempt of grain boundary migration, one atom at grain boundary is randomly chosen for re-
alignment. Since all grain boundaries are assumed to have the same migration coefficient, all atoms at grain boundaries have equal probability to be chosen. For each chosen atom, its neighboring atoms are searched and listed if (i) they are nearest neighbors of the chosen atom or (ii) the chosen atom is one of their nearest neighbors. These two conditions are not identical because an atom may not be nearest neighbor of its own nearest neighbor; this happens only at grain boundaries. The chosen atom is temporarily assigned to belong to one of the grains that the listed atoms belong to, and the configuration energy of the entire system is calculated as Ei ; for the two-dimensional model, one angle is sufficient to represent orientation of each grain. After calculating the energy (Ei ) for each orientation of the listed atoms, a final orientation of the chosen atom is assigned. The probability of assigning the orientation to be hk is X Ej =kT Pk ¼ eEk =kT e ; ð2Þ j
where j runs over all the listed atoms, and kT is the conventional Boltzmann factor. Since only the listed atoms are affected by the re-alignment, the j does not have to run over those atoms that are not listed. It is worthwhile mentioning that the orientation of a neighboring atom maybe the same as the orientation of the migrating atom. In such a case, the grain boundary migration leads to no change in atomic states, although a grain boundary migration has occurred. The controlling parameter of the grain boundary migration, the migration coefficient, is chosen to be 104 e0:5 eV=kT cm2 =s as a reference value. The prefactor is taken to be the same as those of adatom diffusion coefficients, and the activation energy is taken to be the same as that of twocoordinated adatoms. The attempt frequency of the re-alignment is related to the migration coefficient according to Eq. (1). The update of time is carried out in the same fashion as in [11]. The grain boundary migration, or re-alignment of the grain boundary atoms, is counted as an additional category of diffusion in the simulation. To study the effects of grain boundary migration, this coefficient is multiplied by a numerical factor, ranging from 104 to 101 . Details of other energetics have been
H. Huang, G.H. Gilmer / Computational Materials Science 23 (2002) 190–196
presented in [14], and only a summary of them is presented here. The potential energy of each atom is assumed to be a linear function of its coordination, and it is )1.0 eV with coordination being 1. Thin film atoms from other grains contribute to the coordination but do not change the potential energy. When a deposited atom is neighbored by one or more substrate atoms, it is always taken to have two neighbors from the substrate; however, an increase of 0.5 eV in potential energy is added to the atom to account for the interface energy. The potential energy of the substrate atoms is calculated by directly counting the number of nearest neighbors, including substrate atoms and deposited atoms. The substrate atoms are assigned random orientation at the start, and assumed to be immobile and diffusion coefficient of each thin film atom is assumed to be 104 eQ=kT cm2 =s; Q is 0.2, 0.5, 0.7 eV, and infinity when coordination is 1, 2, 3, and 4, respectively. The proposed energy calculation and the diffusion energetics are to mimic deposition of a thin film on an amorphous substrate of the same chemical composition.
193
Using the proposed representation of the grain boundary migration and the energetics, we investigate effects of grain boundary migration on texture competition in the following section.
3. Results and discussions Staring with a flat substrate, we deposit thin films using various grain boundary migration coefficients. To focus on how the grain boundary migration affects dominance of the texture with a top surface of the lowest energy, we use a 1:1 collimation to filter the incident atoms. Source atoms are injected toward the substrate at an incident angle a, defined relative to the vertical axis, with a relative probability sinð2aÞ tanð45° aÞ for a 6 45°. No source atoms with a > 45° are allowed to reach the substrate under the 1:1 collimation. At first, simulations are performed at a deposition rate of 5 lm/min, keeping the substrate temperature at 300 K; 50 atomic layers are deposited in each simulation. The simulation results with the
Fig. 1. The thin films of 50 atomic layers deposited at 300 K under 5 lm/min. From top (a) to bottom (f), the grain boundary migration coefficient is 101 , 100 , 101 , 102 , 103 , and 104 times that of the reference value, respectively. The solid circles in four different gray scales represent atoms in the lattices of four different orientations. The four types of lattices are generated by rotating the reference lattice count clockwise 0°, 22.5°, 45°, and 67.5°, respectively. The darkness of the circles increases with the rotation angle.
194
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grain boundary migration coefficient scaled by 101 , 100 , 101 , 102 , 103 , and 104 are shown in Fig. 1(a)–(f), respectively. The texture with a surface of the lowest energy is the h1 0i texture, shown as the lightest color in the figure. There is an indication that the h1 0i texture is the most dominant when the grain boundary migration coefficient is scaled by 102 . For either a larger or a smaller grain boundary migration coefficient, the dominance of the h1 0i texture is reduced. In order to confirm this observation, we repeat the simulations using a lower deposition rate, 1 lm/min. The simulation results, as shown in Fig. 2(a)–(f), indicate that (i) an optimal migration coefficient for the dominance of the h1 0i texture exists, and (ii) the optimal migration coefficient corresponds to the same scaling factor, 102 . To test the temperature dependence of this phenomenon, we repeat the simulations of Fig. 1 using a higher substrate temperature, 350 K. The simulation results, as shown in Fig. 3(a)–(f), confirm what was observed in Figs. 1 and 2. Due to the qualitative nature of the two-dimensional model, no emphasis should be placed on the absolute value of the optimal migration coefficient.
However, existence of the optimal migration coefficient for the dominance of the h1 0i texture deserves further discussion. It is easier to understand that the h1 0i texture is more dominant when the migration coefficient is increased, since a larger migration coefficient leads to thin films closer to the thermodynamic equilibrium. This is indeed the case as the scaling factor of the migration coefficient goes from 104 to 102 . When the migration coefficient is further increased, this reasoning clearly does not apply anymore. To explain this anomaly, we compare surface energies of two clusters, as shown in Fig. 4. The cluster (a) has two {1 0} surfaces on the top, resembling grains with rotation angles of 22.5° and 67.5°; this is the case when the horizontal surface is not one of the {1 0} surfaces and is faceted. The cluster (b) has one horizontal {1 0} surface and two vertical {1 0} surfaces on the top, resembling grains of h1 0i texture. For simplicity, we assume the interfacial area (meaning length in twodimensions) with the substrate is the same, that is L, for both clusters. Further, we assume the number of atoms in the two clusters is the same;
Fig. 2. Thin films deposited at the same conditions as in Fig. 1, except that the deposition rate is smaller, 1 lm/min.
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Fig. 3. Thin films deposited at the same conditions as in Fig. 1, except that the substrate temperature is higher, 350 K.
Fig. 4. Simple shapes of grain nuclei. L and h are horizontal and vertical dimensions, respectively. The shaded bottom represents interface with an amorphous substrate.
this is equivalent to assuming the same cross-section area in two-dimensions. The latter assumption requires that the height of cluster (a) is twice as that of cluster (b), as shown in Fig. 4. In our model, the substrate is taken to be amorphous. As a result, the interfacial energies of the two clusters are the same, and the thermodynamic preference of the two clusters is determined by their relative surface energies. Because all surfaces are of {1 0}
Fig. 5. Comparison of surface areas (that is lengths in twodimensions) of the triangular cluster and rectangular cluster in Fig. 4.
type and have the same surface energy per unit area, the preference is determined by the relative surface areas of the two clusters. As shown in Fig. 5, the triangular cluster of (a) type has a smaller surface area, when the height h is less than one quarter of the interfacial length L. To minimize the total energy, the nuclei of the triangular shape will therefore be preferred thermodynamically.
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Certainly, this preference is not always the case. In particular, when h is larger than L=4, it is not possible for the two assumed clusters to have the same number of atoms; hence, the above reasoning does not apply anymore. Although the analysis in Figs. 4 and 5 is simple, it does show the possibility for the h1 0i texture to lose dominance at the thermodynamic equilibrium. When the assumptions in Figs. 4 and 5 do not apply, the h1 0i texture is thermodynamically preferred, according to traditional understanding [11]. These two mechanisms lead to reduction and increase of the h1 0i texture dominance, respectively. Consequently, they are responsible for the existence of the optimal grain boundary migration coefficient. If confirmed, the existence of an optimal grain boundary migration coefficient implies that thin film texture can be engineered by modifying grain boundary conditions. For example, segregation of impurity atoms at grain boundaries [15] may block the open path for grain boundary diffusion, and may also slow down the grain boundary migration.
4. Summary In this paper, we have presented an implementation of grain boundary migration in the atomistic simulator (ADEPT), and applied the simulator to study the effects of grain boundary migration on texture competition. Our studies show that an optimal grain boundary migration coefficient exists for the dominance of the h1 0i texture, the one with a top surface of the lowest energy. Further increase or decrease of the migration coefficient suppresses the dominance of this texture. This effect may provide an additional means of engineering thin film textures.
Acknowledgements The work described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region (PolyU 5146/99E), and partially by grants from the Hong Kong PolyU (A-PB32 and G-YB91). Support from the NSF/ DARPA VIP program through the University of Illinois is also acknowledged.
References [1] S.R. Wilson, C.J. Tracy, J.L. Freeman, Handbook of Multilevel Metallization for Integrated Circuits, Noyes Publications, NJ, USA, 1993. [2] T.S. Cale, D.F. Richards, D. Yang, J. Comput. Aided Mater. Des. 6 (1999) 283. [3] B.W. Kar, I. Petrov, D.G. Cahill, J.E. Greene, Appl. Phys. Lett. 70 (1997) 1703. [4] L. Dong, D.J. Srolovitz, J. Appl. Phys. 84 (1998) 5261. [5] A. Voter, Phys. Rev. Lett. 78 (1997) 3908. [6] J. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Sciences, Cambridge University Press, London, 1996. [7] T. Smy, R. Tait, M. Breit, IEEE Trans. Comput. Aided Des. 10 (1991) 130. [8] F. Baumann, G.H. Gilmer, IEDM Technical Digest, 1995, p. 89. [9] Y. Yang, R. Johnson, H. Wadley, Acta Mater. 45 (1997) 1445. [10] X. Zhou, R. Johnson, H. Wadley, Acta Mater. 45 (1997) 1513. [11] H. Huang, G.H. Gilmer, T. Diaz de la Rubia, J. Appl. Phys. 84 (1998) 3636. [12] G.H. Gilmer, H. Huang, T. Diaz de la Rubia, C. Roland, Comput. Mater. Sci. 12 (1998) 354. [13] G.H. Gilmer, H. Huang, T. Diaz de la Rubia, J.D. Torre, F. Baumann, Thin Solid Films 365 (2000) 189. [14] H. Huang, G.H. Gilmer, J. Comput. Aided Mater. Des. 6 (1999) 117. [15] H. Huang, T. Diaz de la Rubia, M. Fluss, in: W. Filter, et al. (Eds.), Materials Reliability in Microelectronics VI, MRS, Pittsburgh, 1997, p. 177.