- Email: [email protected]
Kinetic Monte Carlo simulation of surfactant-mediated Cu thin-film growth
Computational Materials Science 50 (2010) 6–9
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Kinetic Monte Carlo simulation of surfactant-mediated Cu thin-film growth q Xiaoping Zheng a, Peifeng Zhang a,b,* a b
Institute of Electronic Information Science and Technology, Lanzhou City University, Lanzhou 730070, Gansu Province, PR China School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China
a r t i c l e
i n f o
Article history: Received 3 March 2010 Accepted 27 May 2010 Available online 16 August 2010 Keywords: Diffusion theory Thin-film growth Exchange reaction RLA model
a b s t r a c t A novel model consisting of basic micro-processes has been developed based upon the classic diffusion theory. It is the first time that the concept of exchange rate has been introduced, and the process of surfactant-mediated epitaxial thin-film growth has been simulated by the Kinetic Monte Carlo (KMC) technique. It is found that the exchange reaction in the Reaction Limited Aggregation (RLA) model is a combination of basic micro-processes. The majority of exchanges are not complete site exchanges and the exchange rate does not always equal one. Both surfactant atoms and adatoms diffuse from one layer to another. The diffusion occurs mostly between single atoms and the diffusing atoms increase with the substrate temperature or the film thickness. Ó 2010 Published by Elsevier B.V.
1. Introduction The characteristics of film growth with surfactant-mediation are completely opposite to the non-surfactant situation. The investigations of the abnormal effects has become a focus of research in the field of film growth in recent years [1–3]. To date, the studies for the mechanism of thin-film growth with surfactant-mediation are not very satisfactory. Among numerous models, the Reaction Limited Aggregation (RLA) [4–6] model successfully explains the abnormal effects found in experiments. The basic principle of RLA model is different from that of the DLA (Diffusion Limited Aggregation) [7–9] model, in the RLA model the nucleation is mainly affected by exchange reaction but in the DLA model the nucleation is affected by diffusion reaction. The key idea of the RLA model is the exchange reaction and it has three basic presuppositions. Firstly, the adatoms could become a stable nucleation centre after conquering a larger potential and exchanging site with surfactant atoms. Secondly, to become a portion of a stable island, the adatoms that deposited subsequently conquers a larger potential and exchange site with the surfactant atoms. The last assumption is that only exchanged atoms could form an island, and the island locating in surfactant layer is stable. q Project supported by Project No. 10574059 of the National Natural Science Foundation of China, No. 0710RJZA074 of the Natural Science Foundation of Gansu Province, No. 0711B-04 of the Second Scientific Research Project of Bureau of Gansu education and ‘Qing Lan’ Talent Engineering Funds of Lanzhou Jiaotong University. * Corresponding author at: Institute of Electronic Information Science and Technology, Lanzhou City University, Lanzhou 730070, Gansu Province, PR China. Tel.: +86 0931 7616828; fax: +86 0931 7601256. E-mail addresses: [email protected], [email protected] (P. Zhang).
0927-0256/$ - see front matter Ó 2010 Published by Elsevier B.V. doi:10.1016/j.commatsci.2010.05.054
Based on the first two assumptions, exchanges must satisfy the following requirements: (i) At the beginning, exchange must occur between nuclei (at least two atoms) and it is impossible that exchange can occur between single atoms. (ii) Exchange is purely the site exchange, i.e., the completeexchange. (iii) The number of adatoms is equal to one of the surfactant atoms during exchange processes. The thin-film growth is a result of atom diffusing and congregating, the micro-mechanism of exchange reaction in the RLA model has not hitherto been reported. In this paper, on the basis of diffusion theory we simulated the mechanism of thin-film growth with surfactant-mediation, and focused our investigation on the essence of exchange reaction. 2. Description of simulator The growth of thin film is due to atoms diffusing and congregating. The micro-processes in most of the models consist of the processes: (1) atoms deposit on the substrate, (2) a single atom diffuses on the surface of substrate, (3) the diffusing atom encounters another and then both of them nucleate, (4) the diffusing atom is captured by existent island, (5) atoms locating at the edge of island detach the island at a certain probability, (6) the atom locating at island edge retains bonding with an island and removes along an island edge, (7) after diffusing, the atom deposited directly on island falls on the substrate again, (8) adatoms nucleate on an island,
X. Zheng, P. Zhang / Computational Materials Science 50 (2010) 6–9
(9) the cluster which is composed of two or more atoms diffuses together, and so on. Most of the listed micro-processes are propounded for specific models and a lot of processes are not included, for example the exchange reaction in RLA model. Therefore, most of the models based on the above processes cannot satisfactorily explain the abnormal effects. Taking this into account, we generalize the basic microprocesses of film growth into three basic events: (i) The incident atom deposits on the growing surface and is attached, namely the absorption events. (ii) All kinds of atoms diffuse on the growing surface, namely the diffusion events. (iii) The adatoms detach from the growing surface, namely the evaporation events. In fact, this process could be incorporated into a special kind of diffusion event. Based upon the diffusion theory and the basic micro-process, we developed a novel three-dimensional model in our study. After generalization, the new model comprises not only all the processes listed above, but also others, such as corner diffusion, layer-tolayer diffusion, and so on. More importantly the new model might include some unexpected diffusion processes, for example the exchange reaction, and other processes which are combinations of the above three processes. The difference between the new model and the existing ones is that in the new model there are no compulsory assumptions and extra conditions on the occurrence of basic micro-processes. The occurrence probability of a certain event is determined completely by Monte Carlo algorithm, the target site of the next jump is all vacant sites in the effective sphere, including the stable site, the metastable sites and the unstable sites. The absorption rate of atoms was given by the number of incident atoms per unit time and consisted of: (i) An incident atom deposits at a lattice site on the growing surface. (ii) If the site has three atoms as nearest neighbors (stable site), the incident atom sticks and occupies the stable site. Otherwise, it will diffuse to one of the nearest or the next nearest sites. If the site is not stable, the incident atom will continue to diffuse until it finds a stable site or evaporates from the growing surface. The whole process is regarded as one event. The related effects caused by the atom diffusion were considered in the model. As illustrated in Fig. 1, since atom A has only two nearest neighbors, it will diffuse to a stable site. As a result,
the site of atom B is no longer a stable site, it will diffuse to another stable site, and so will atom C. Though different from the normal diffusion, it is regarded as one event rather than three events. The diffusion rate of a single adatom was defined as the probability of a diffusion jump per unit of time and is given by the Arrhenius-type expression
r ¼ m0 expðDE=kB TÞ;
ð1Þ
where m0 is the frequency of atom vibrations, assigned a value of m0 = (2kBT)/h, where kB is the Boltzmann constant, T is the substrate temperature and DE is the activation energy. The evaporation rate was calculated in the same way as the diffusion rate, but the final target site is that without any neighbors. We implemented the simulation onto face-centered cubic (1 0 0) surface, the substrate is 50 50, and we adopted the periodical border condition. The surface of substrate atom A is surfactant S and the incident atom is atom A. Morse potential was used to model the interaction between two atoms, where we considered not only the interaction between A and A, but also the interaction between A and S and between S and S,
VðijÞ ¼ V 0 exp 2a r ij =r 0 1 2 exp aðr ij =r 0 1Þ ;
ð2Þ
where a is a constant depending on the interaction range of atoms, V0 is the interactional potential between the nearest neighbor atoms, in the simulation, we referred to the interaction potentials [4] and took VAA = 0.87 ev, VAS = 0.46 ev and VSS = 0.13 ev, respectively. rij is the distance between atoms i and j, r0 is the distance between the nearest neighbors; rij/r0 is determined by the geometric relation of face-centered cubic structure. It has been demonstrated that Morse potential could reasonably represent the interaction between two atoms and easily calculate the potential change during the atom diffusion. The calculation of the activation barrier was improved further on the basis of our previous methods [10–12]. In order to find the maximum activation barrier Emax between two atoms, we took 99 points Pi (i = 1, 2, . . ., 99) on the line connecting the initial lattice site to the next jump target. We took another two points P0 and P100, the distances from P0 to the initial site and from P100 to the next jump target are 0.01a (where a is the lattice constant). The respective potentials are Ei (i = 0–100), Eini (potential in the initial site) and Efin (potential in the next jump site). The total potential for each point of interest can be obtained by summing up the contribution from all atoms within a sphere of radius of 1.5a. Emax is the maximum of Ei (i = 0–99), and Efin. The activation energy DE is determined by taking the difference between Emax and Eini. In this paper, we introduce the concept of exchange rate in order to study the micro-mechanism of exchange reaction and defined it as
s ¼ n2 =n1 ;
ð3Þ
where n1 is the number of surfactant atoms diffusing up from surfactant layer into thin film surface, n2 is the numbers of adatoms diffusing down from the film surface into surfactant vacant sites. It is obvious that the exchange rate always equals to one according to the RLA model.
B A C
7
B
Fig. 1. The sketch plot of related-effect.
3. Results and discussion Fig. 2 is a snapshot of film morphology at the initial stage with surfactant-mediated at a temperature of 300 K and deposition rate of 0.32 lay./s, where the layers from 1 to 3 are substrate atoms and the layers from 4 to 5 are surfactant atoms, white and yellow spheres represent respectively the adatoms and surfactant atoms.
8
X. Zheng, P. Zhang / Computational Materials Science 50 (2010) 6–9
300
Atom numbers
250
Ascending numbers Descending numbers 150
100
50
Fig. 2. Film morphology at the initial stages at the deposition rate of 0.32 lay./s and the substrate temperature of 300 K.
0 200
300
400
500
600
700
Substrate temperature£¨K£© Fig. 3. The dependence of atoms crossing layers on substrate temperatures.
0.38 0.36 0.34
Exchange rate£¨%£©
We see from the Fig. 2 that some of the surfactant atoms diffuse into the surface and simultaneously some adatoms diffuse into the surfactant during the film growth, and the majority of diffusion is layer-to-layer diffusion of single atoms. A trace file was created during the simulation, which records the initial and the next jump sites of every jump. By analyzing the trace file, we found that both surfactant and adatoms diffuse by means of layer-to-layer under various interactions of atom kinetic movement and interactions between atoms. On the one hand some surfactant atoms diffuse up into the surface and leave vacant sites, whilst the existence of the vacant sites causes the surfactant atoms to diffuse constantly in the surfactant layers so that the vacant sites move continually. On the other hand adatoms diffuse on the film surface and might arrive at one of the nearest neighbors of a vacant site (in the deposited surface). Because of the lower energy of the vacant site, the adatom diffuses more easily downwards and holds the vacant site in the surfactant and then forms site to be filled. At the moment, if the surfactant atom, which diffused into the surface, diffuses to other nearest neighbors of the filled site (in the surface, but not the site where the adatom located before filling), and the exchange is called quasi-exchange. If the vacant site just diffuses into the site where the adatom located before filling and the exchange is called complete-exchange. It was found that most of the layer-to-layer diffusions occur between single atoms during the simulation process. Although there might exist a few quasi-exchange and few complete-exchange of position, these exchanges are a combination which is constructed by some basic micro-processes. The possibility, in which the adatom nucleates first and then quasi-exchange or complete-exchange of nuclei occurs, is comparatively small. The dependence of the number of diffusing atoms on the substrate temperature is presented in Fig. 3. In this figure, both the surfactant atoms which diffuse upwards into surface and the adatoms which diffuse downwards into surfactant layer increase sharply with the temperature increasing. But the tendency becomes obviously slower while the temperature is higher than the valve (approximately 320 K), and which might be the maximum of the saddle energy of layer-to-layer diffusion. Because the atom cannot be activated effectively under the low temperature and most of atoms possess the larger barrier while diffusing, the atom can only diffuse in the same layer. However, the movement becomes more and more acute with increasing temperature so that many atoms can climb over the saddle and cross through the interface when the temperature arrives at the valve. The dependence of exchange rate on substrate temperature for the deposition rate 0.32 lay./s and 32 lay./s is shown in Fig. 4. It is observed that exchange rate increases rapidly with temperature increasing under the lower temperature, and arrives at the maximum of 320 K, afterwards it decreases and then becomes stable. It is well known that the deposition rate decides diffusion time.
200
0.32 0.30 0.28 0.26 Deposition rate 0.32 lay./s Deposition rate 32 lay./s
0.24 0.22 0.20 0.18 0.16 0.14 200
300
400
500
600
700
Substrate temperature(K) Fig. 4. The dependence of The exchange rate on substrate temperature, the deposition rates are 0.32, 32 lay./s.
For the same temperature, although the atom number of layerto-layer diffusion is different under different deposition rate, the diffusion time for the adatoms and surfactant atoms decreases with deposition rate increasing so that the atoms of layer-to-layer diffusion decreases, but the exchange rate remains constant. The exchange rate vs. the film average thickness with deposition rate 0.32 lay./s for temperature 300 K and 400 K is shown in Fig. 5. When there are fewer adatoms, the exchange rate is lower, increasing continuously as the number of adatoms increases. Although the vacant sites formed by surfactant atoms diffusing up can effectively capture adatoms locating in its nearest neighbors, there does not always exist adatoms in the nearest neighbors of the vacant site. The vacant can therefore be empty temporarily or move within surfactant layer. With the number of adatoms increasing, as long as there exists a vacant site, the possibility of adatoms existing in the nearest neighbors and capturing adatoms increases so that exchange rate becomes high. At temperatures of 300 K and 400 K, the exchange rate is bigger than 0.8 when the average thickness of the film reaches one layer. We also found that both of the exchange rates are almost equal. It is indicated that the
X. Zheng, P. Zhang / Computational Materials Science 50 (2010) 6–9
0.9
Exchange rate£¨%£©
0.8
0.7
0.6
0.32 lay./s 300k 0.32 lay./s 400k
0.5
0.4
0.3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
9
exchanges and a few complete-exchanges of position, but these exchanges are a combination of some basic microprocesses. (ii) Both surfactant atoms and adatoms diffuse from one layer to another. The majority of diffusion is layer-to-layer diffusion of single atom and the atoms of layer-to-layer diffusion increase with the temperature increasing. (iii) The layer-to-layer diffusion is not simple site exchange, and there are more surfactant atoms which diffuse into surface than adatoms diffusing into the surfactant, namely the exchange rate is less than one. The exchange rate increases rapidly with the temperature increasing and reaches the maximum at the valve, it then begins to decrease and tends to be stable. (iv) The exchange rate increases rapidly with adatoms increasing and is more than 0.8 under the temperature 300–400 K when the average thickness of the film reaches one layer.
Average depth (lay.) Fig. 5. The dependence of the exchange rate on average thickness, the deposition rates is 0.32, and the temperature is 300 K and 400 K.
exchange rate being invariable in spite of the atoms of layer-tolayer diffusion increasing when the temperature rising. 4. Conclusion A three-dimensional Kinetic Monte Carlo (KCM) technique has been developed for simulating surfactant-mediated epitaxial growth of Cu thin film. The following results are obtained: (i) Most of the layer-to-layer diffusion is single atoms diffusion during the film growth. There might exist a few quasi-
Reference [1] J.M. Pomeroy, J. Jacobsen, C.C. Hill, B.H. Cooper, J.P. Sethna, Phys. Rev. B 66 (2002) 235412. [2] N. Combe, P. Jensen, Phys. Rev. Lett. 85 (2000) 110–113. [3] G. Henkelman, H. Jonsson, Phys. Rev. Lett. 90 (2003) 116101. [4] E.G. Wang, Phys. Adv. 23 (2003) 1–61. [5] W.G. Zhu, Francesco Buatier de Mongeot, U. Valbusa, E.G. Wang, Phys. Rev. Lett. 92 (2004) 106102. [6] J. Wu, E.G. Wang, Kvarga, B.G. Liu, S.T. Pantelides, Z.Y. Zhang, Phys. Rev. Lett. 89 (2002) 146103. [7] Z.Y. Zhang, M.G. Lagally, Science 276 (1997) 377–383. [8] M.G. Lagally, Z.Y. Zhang, Nature 417 (2002) 907–910. [9] L.G. Wang, P. Clancy, Surf. Sci. 473 (2001) 25–38. [10] P.F. Zhang, X.P. Zheng, D.Y. He, Vacuum 72 (2004) 405–410. [11] P.F. Zhang, X.P. Zheng, S.P. Wu, D.Y. He, Comput. Mater. Sci. 30 (2004) 331– 336. [12] P.F. Zhang, X.P. Zheng, D.Y. He, J. Korean Phys. Soc. 46 (2005) 92–95.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Kinetic Monte Carlo simulation of surfactant-mediated Cu thin-film growth q Xiaoping Zheng a, Peifeng Zhang a,b,* a b
Institute of Electronic Information Science and Technology, Lanzhou City University, Lanzhou 730070, Gansu Province, PR China School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, PR China
a r t i c l e
i n f o
Article history: Received 3 March 2010 Accepted 27 May 2010 Available online 16 August 2010 Keywords: Diffusion theory Thin-film growth Exchange reaction RLA model
a b s t r a c t A novel model consisting of basic micro-processes has been developed based upon the classic diffusion theory. It is the first time that the concept of exchange rate has been introduced, and the process of surfactant-mediated epitaxial thin-film growth has been simulated by the Kinetic Monte Carlo (KMC) technique. It is found that the exchange reaction in the Reaction Limited Aggregation (RLA) model is a combination of basic micro-processes. The majority of exchanges are not complete site exchanges and the exchange rate does not always equal one. Both surfactant atoms and adatoms diffuse from one layer to another. The diffusion occurs mostly between single atoms and the diffusing atoms increase with the substrate temperature or the film thickness. Ó 2010 Published by Elsevier B.V.
1. Introduction The characteristics of film growth with surfactant-mediation are completely opposite to the non-surfactant situation. The investigations of the abnormal effects has become a focus of research in the field of film growth in recent years [1–3]. To date, the studies for the mechanism of thin-film growth with surfactant-mediation are not very satisfactory. Among numerous models, the Reaction Limited Aggregation (RLA) [4–6] model successfully explains the abnormal effects found in experiments. The basic principle of RLA model is different from that of the DLA (Diffusion Limited Aggregation) [7–9] model, in the RLA model the nucleation is mainly affected by exchange reaction but in the DLA model the nucleation is affected by diffusion reaction. The key idea of the RLA model is the exchange reaction and it has three basic presuppositions. Firstly, the adatoms could become a stable nucleation centre after conquering a larger potential and exchanging site with surfactant atoms. Secondly, to become a portion of a stable island, the adatoms that deposited subsequently conquers a larger potential and exchange site with the surfactant atoms. The last assumption is that only exchanged atoms could form an island, and the island locating in surfactant layer is stable. q Project supported by Project No. 10574059 of the National Natural Science Foundation of China, No. 0710RJZA074 of the Natural Science Foundation of Gansu Province, No. 0711B-04 of the Second Scientific Research Project of Bureau of Gansu education and ‘Qing Lan’ Talent Engineering Funds of Lanzhou Jiaotong University. * Corresponding author at: Institute of Electronic Information Science and Technology, Lanzhou City University, Lanzhou 730070, Gansu Province, PR China. Tel.: +86 0931 7616828; fax: +86 0931 7601256. E-mail addresses: [email protected], [email protected] (P. Zhang).
0927-0256/$ - see front matter Ó 2010 Published by Elsevier B.V. doi:10.1016/j.commatsci.2010.05.054
Based on the first two assumptions, exchanges must satisfy the following requirements: (i) At the beginning, exchange must occur between nuclei (at least two atoms) and it is impossible that exchange can occur between single atoms. (ii) Exchange is purely the site exchange, i.e., the completeexchange. (iii) The number of adatoms is equal to one of the surfactant atoms during exchange processes. The thin-film growth is a result of atom diffusing and congregating, the micro-mechanism of exchange reaction in the RLA model has not hitherto been reported. In this paper, on the basis of diffusion theory we simulated the mechanism of thin-film growth with surfactant-mediation, and focused our investigation on the essence of exchange reaction. 2. Description of simulator The growth of thin film is due to atoms diffusing and congregating. The micro-processes in most of the models consist of the processes: (1) atoms deposit on the substrate, (2) a single atom diffuses on the surface of substrate, (3) the diffusing atom encounters another and then both of them nucleate, (4) the diffusing atom is captured by existent island, (5) atoms locating at the edge of island detach the island at a certain probability, (6) the atom locating at island edge retains bonding with an island and removes along an island edge, (7) after diffusing, the atom deposited directly on island falls on the substrate again, (8) adatoms nucleate on an island,
X. Zheng, P. Zhang / Computational Materials Science 50 (2010) 6–9
(9) the cluster which is composed of two or more atoms diffuses together, and so on. Most of the listed micro-processes are propounded for specific models and a lot of processes are not included, for example the exchange reaction in RLA model. Therefore, most of the models based on the above processes cannot satisfactorily explain the abnormal effects. Taking this into account, we generalize the basic microprocesses of film growth into three basic events: (i) The incident atom deposits on the growing surface and is attached, namely the absorption events. (ii) All kinds of atoms diffuse on the growing surface, namely the diffusion events. (iii) The adatoms detach from the growing surface, namely the evaporation events. In fact, this process could be incorporated into a special kind of diffusion event. Based upon the diffusion theory and the basic micro-process, we developed a novel three-dimensional model in our study. After generalization, the new model comprises not only all the processes listed above, but also others, such as corner diffusion, layer-tolayer diffusion, and so on. More importantly the new model might include some unexpected diffusion processes, for example the exchange reaction, and other processes which are combinations of the above three processes. The difference between the new model and the existing ones is that in the new model there are no compulsory assumptions and extra conditions on the occurrence of basic micro-processes. The occurrence probability of a certain event is determined completely by Monte Carlo algorithm, the target site of the next jump is all vacant sites in the effective sphere, including the stable site, the metastable sites and the unstable sites. The absorption rate of atoms was given by the number of incident atoms per unit time and consisted of: (i) An incident atom deposits at a lattice site on the growing surface. (ii) If the site has three atoms as nearest neighbors (stable site), the incident atom sticks and occupies the stable site. Otherwise, it will diffuse to one of the nearest or the next nearest sites. If the site is not stable, the incident atom will continue to diffuse until it finds a stable site or evaporates from the growing surface. The whole process is regarded as one event. The related effects caused by the atom diffusion were considered in the model. As illustrated in Fig. 1, since atom A has only two nearest neighbors, it will diffuse to a stable site. As a result,
the site of atom B is no longer a stable site, it will diffuse to another stable site, and so will atom C. Though different from the normal diffusion, it is regarded as one event rather than three events. The diffusion rate of a single adatom was defined as the probability of a diffusion jump per unit of time and is given by the Arrhenius-type expression
r ¼ m0 expðDE=kB TÞ;
ð1Þ
where m0 is the frequency of atom vibrations, assigned a value of m0 = (2kBT)/h, where kB is the Boltzmann constant, T is the substrate temperature and DE is the activation energy. The evaporation rate was calculated in the same way as the diffusion rate, but the final target site is that without any neighbors. We implemented the simulation onto face-centered cubic (1 0 0) surface, the substrate is 50 50, and we adopted the periodical border condition. The surface of substrate atom A is surfactant S and the incident atom is atom A. Morse potential was used to model the interaction between two atoms, where we considered not only the interaction between A and A, but also the interaction between A and S and between S and S,
VðijÞ ¼ V 0 exp 2a r ij =r 0 1 2 exp aðr ij =r 0 1Þ ;
ð2Þ
where a is a constant depending on the interaction range of atoms, V0 is the interactional potential between the nearest neighbor atoms, in the simulation, we referred to the interaction potentials [4] and took VAA = 0.87 ev, VAS = 0.46 ev and VSS = 0.13 ev, respectively. rij is the distance between atoms i and j, r0 is the distance between the nearest neighbors; rij/r0 is determined by the geometric relation of face-centered cubic structure. It has been demonstrated that Morse potential could reasonably represent the interaction between two atoms and easily calculate the potential change during the atom diffusion. The calculation of the activation barrier was improved further on the basis of our previous methods [10–12]. In order to find the maximum activation barrier Emax between two atoms, we took 99 points Pi (i = 1, 2, . . ., 99) on the line connecting the initial lattice site to the next jump target. We took another two points P0 and P100, the distances from P0 to the initial site and from P100 to the next jump target are 0.01a (where a is the lattice constant). The respective potentials are Ei (i = 0–100), Eini (potential in the initial site) and Efin (potential in the next jump site). The total potential for each point of interest can be obtained by summing up the contribution from all atoms within a sphere of radius of 1.5a. Emax is the maximum of Ei (i = 0–99), and Efin. The activation energy DE is determined by taking the difference between Emax and Eini. In this paper, we introduce the concept of exchange rate in order to study the micro-mechanism of exchange reaction and defined it as
s ¼ n2 =n1 ;
ð3Þ
where n1 is the number of surfactant atoms diffusing up from surfactant layer into thin film surface, n2 is the numbers of adatoms diffusing down from the film surface into surfactant vacant sites. It is obvious that the exchange rate always equals to one according to the RLA model.
B A C
7
B
Fig. 1. The sketch plot of related-effect.
3. Results and discussion Fig. 2 is a snapshot of film morphology at the initial stage with surfactant-mediated at a temperature of 300 K and deposition rate of 0.32 lay./s, where the layers from 1 to 3 are substrate atoms and the layers from 4 to 5 are surfactant atoms, white and yellow spheres represent respectively the adatoms and surfactant atoms.
8
X. Zheng, P. Zhang / Computational Materials Science 50 (2010) 6–9
300
Atom numbers
250
Ascending numbers Descending numbers 150
100
50
Fig. 2. Film morphology at the initial stages at the deposition rate of 0.32 lay./s and the substrate temperature of 300 K.
0 200
300
400
500
600
700
Substrate temperature£¨K£© Fig. 3. The dependence of atoms crossing layers on substrate temperatures.
0.38 0.36 0.34
Exchange rate£¨%£©
We see from the Fig. 2 that some of the surfactant atoms diffuse into the surface and simultaneously some adatoms diffuse into the surfactant during the film growth, and the majority of diffusion is layer-to-layer diffusion of single atoms. A trace file was created during the simulation, which records the initial and the next jump sites of every jump. By analyzing the trace file, we found that both surfactant and adatoms diffuse by means of layer-to-layer under various interactions of atom kinetic movement and interactions between atoms. On the one hand some surfactant atoms diffuse up into the surface and leave vacant sites, whilst the existence of the vacant sites causes the surfactant atoms to diffuse constantly in the surfactant layers so that the vacant sites move continually. On the other hand adatoms diffuse on the film surface and might arrive at one of the nearest neighbors of a vacant site (in the deposited surface). Because of the lower energy of the vacant site, the adatom diffuses more easily downwards and holds the vacant site in the surfactant and then forms site to be filled. At the moment, if the surfactant atom, which diffused into the surface, diffuses to other nearest neighbors of the filled site (in the surface, but not the site where the adatom located before filling), and the exchange is called quasi-exchange. If the vacant site just diffuses into the site where the adatom located before filling and the exchange is called complete-exchange. It was found that most of the layer-to-layer diffusions occur between single atoms during the simulation process. Although there might exist a few quasi-exchange and few complete-exchange of position, these exchanges are a combination which is constructed by some basic micro-processes. The possibility, in which the adatom nucleates first and then quasi-exchange or complete-exchange of nuclei occurs, is comparatively small. The dependence of the number of diffusing atoms on the substrate temperature is presented in Fig. 3. In this figure, both the surfactant atoms which diffuse upwards into surface and the adatoms which diffuse downwards into surfactant layer increase sharply with the temperature increasing. But the tendency becomes obviously slower while the temperature is higher than the valve (approximately 320 K), and which might be the maximum of the saddle energy of layer-to-layer diffusion. Because the atom cannot be activated effectively under the low temperature and most of atoms possess the larger barrier while diffusing, the atom can only diffuse in the same layer. However, the movement becomes more and more acute with increasing temperature so that many atoms can climb over the saddle and cross through the interface when the temperature arrives at the valve. The dependence of exchange rate on substrate temperature for the deposition rate 0.32 lay./s and 32 lay./s is shown in Fig. 4. It is observed that exchange rate increases rapidly with temperature increasing under the lower temperature, and arrives at the maximum of 320 K, afterwards it decreases and then becomes stable. It is well known that the deposition rate decides diffusion time.
200
0.32 0.30 0.28 0.26 Deposition rate 0.32 lay./s Deposition rate 32 lay./s
0.24 0.22 0.20 0.18 0.16 0.14 200
300
400
500
600
700
Substrate temperature(K) Fig. 4. The dependence of The exchange rate on substrate temperature, the deposition rates are 0.32, 32 lay./s.
For the same temperature, although the atom number of layerto-layer diffusion is different under different deposition rate, the diffusion time for the adatoms and surfactant atoms decreases with deposition rate increasing so that the atoms of layer-to-layer diffusion decreases, but the exchange rate remains constant. The exchange rate vs. the film average thickness with deposition rate 0.32 lay./s for temperature 300 K and 400 K is shown in Fig. 5. When there are fewer adatoms, the exchange rate is lower, increasing continuously as the number of adatoms increases. Although the vacant sites formed by surfactant atoms diffusing up can effectively capture adatoms locating in its nearest neighbors, there does not always exist adatoms in the nearest neighbors of the vacant site. The vacant can therefore be empty temporarily or move within surfactant layer. With the number of adatoms increasing, as long as there exists a vacant site, the possibility of adatoms existing in the nearest neighbors and capturing adatoms increases so that exchange rate becomes high. At temperatures of 300 K and 400 K, the exchange rate is bigger than 0.8 when the average thickness of the film reaches one layer. We also found that both of the exchange rates are almost equal. It is indicated that the
X. Zheng, P. Zhang / Computational Materials Science 50 (2010) 6–9
0.9
Exchange rate£¨%£©
0.8
0.7
0.6
0.32 lay./s 300k 0.32 lay./s 400k
0.5
0.4
0.3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
9
exchanges and a few complete-exchanges of position, but these exchanges are a combination of some basic microprocesses. (ii) Both surfactant atoms and adatoms diffuse from one layer to another. The majority of diffusion is layer-to-layer diffusion of single atom and the atoms of layer-to-layer diffusion increase with the temperature increasing. (iii) The layer-to-layer diffusion is not simple site exchange, and there are more surfactant atoms which diffuse into surface than adatoms diffusing into the surfactant, namely the exchange rate is less than one. The exchange rate increases rapidly with the temperature increasing and reaches the maximum at the valve, it then begins to decrease and tends to be stable. (iv) The exchange rate increases rapidly with adatoms increasing and is more than 0.8 under the temperature 300–400 K when the average thickness of the film reaches one layer.
Average depth (lay.) Fig. 5. The dependence of the exchange rate on average thickness, the deposition rates is 0.32, and the temperature is 300 K and 400 K.
exchange rate being invariable in spite of the atoms of layer-tolayer diffusion increasing when the temperature rising. 4. Conclusion A three-dimensional Kinetic Monte Carlo (KCM) technique has been developed for simulating surfactant-mediated epitaxial growth of Cu thin film. The following results are obtained: (i) Most of the layer-to-layer diffusion is single atoms diffusion during the film growth. There might exist a few quasi-
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