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Modeling the effects of cohesive energy for single particle on the material removal in chemical mechanical polishing at atomic scale
Applied Surface Science 253 (2007) 9137–9141 www.elsevier.com/locate/apsusc
Modeling the effects of cohesive energy for single particle on the material removal in chemical mechanical polishing at atomic scale Yongguang Wang, Yongwu Zhao *, Wei An, Jun Wang School of Mechanical Engineering, Southern Yangtze University, Wuxi, 214122, PR China Received 13 April 2007; received in revised form 13 May 2007; accepted 17 May 2007 Available online 29 May 2007
Abstract This paper proposes a novel mathematical model for chemical mechanical polishing (CMP) based on interface solid physical and chemical theory in addition to energy equilibrium knowledge. And the effects of oxidation concentration and particle size on the material removal in CMP are investigated. It is shown that the mechanical energy and removal cohesive energy couple with the particle size, and being a cause of the nonlinear size-removal rate relation. Furthermore, it also shows a nonlinear dependence of removal rate on removal cohesive energy. The model predictions are in good qualitative agreement with the published experimental data. The current study provides an important starting point for delineating the micro-removal mechanism in the CMP process at atomic scale. # 2007 Elsevier B.V. All rights reserved. Keywords: Chemical mechanical polishing; Binding energy; Modeling; Atomic scale
1. Introduction In the semiconductor industry chemical mechanical polishing (CMP), developed at IBM [1] during early 1991s, has emerged to be most promising because it can provide the planarity necessary to build multilevel interconnect schemes. With the decrease in line widths and an increase in the number of metal layers, CMP has become a vital processing step for the realization of circuits with current line width of as small as 0.05 mm [2]. Despite its crucial importance in the microelectronic industry, many aspects of the CMP process remain unclear [3,4]. Understanding of the fundamental of CMP material removal mechanisms may offer insights into the control and optimization of the polishing process. Many researchers modeled the CMP material removal rate based on different removal mechanisms, such as indentationsliding mechanism [5–7], adhesion removal mechanism [8], and hydrodynamic slurry [9,10], etc. Despite its seemingly correct cause-and-effect sequence, the validity of these continuum descriptions of the CMP process has been questioned. Firstly,
* Corresponding author. Tel.: +86 510 85140186. E-mail addresses: [email protected] (Y. Wang), [email protected] (Y. Zhao). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2007.05.043
they are incapable of delineating the variable trends of particle size on material removal despite its significant influence observed experimentally [11–13]. In some instances, it could be decreasing [11] or increasing [12], and Zhou et al. [13] found that the material removal rate initially increases with the particle size, then decreases with it. Secondly, the real CMP chemical action time during two successive particles is of an order of 1.0 108 s [14]. Wang and Zhao [15] concluded that the thickness of the oxidized layer is evaluated on the order of 0.01 nm. In addition, the indentation of the abrasive particles into the wafer surface is on the order of 0.1 nanometers or less [16,17] under the conventional CMP conditions. Therefore, under the polishing equilibrium process, no softened wafer surface film is formed and the slurry chemistry, at best, has time only to react with the surface monolayer of the wafer surface. More recently, the abrasive particle size provided by Cabot Company is around 20 nm. With the decrease of the particle size, the material removal mechanism may be more sensibly described by noncontinuum theories. The atomic-scale removal mechanism during CMP process was investigated by many researchers in combination with examinations through AFM [18], XPS [19], TEM [20] and molecular dynamics simulation (MD) [21]. However, the above discrete mechanisms in relation to molecular bonds and binding energies have been established through empirical investigation.
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Science-based model of CMP discrete material removal mechanism has not been developed yet, largely due to the complex interactions between multiple variables during polishing process. Because of the complexity of the polishing system, it is highly desirable to characterize the physical and chemical behaviors of the individual interaction while other components are fixed, or nearly so. The present research will only focus on the aspect of a single slurry particle. In this paper, a mathematical model as a function of binding energy, chemical parameter and mechanical removal energy is presented for CMP. The model is proposed based on interface solid physical and chemical theory as well as energy balance knowledge. It shows a nonlinear dependence of removal rate on binding energy and particle size. The model predictions are compared with the experimental data to verify the current model. 2. Modeling As the knowledge of CMP process and role of consumable improved over years, much present work has investigated the material removal from molecular/atomic-scale perspective [22]. As shown in Fig. 1, a more likely scenario for the process of the material removal has three steps: (1) Chemical actions covert strongly the fresh wafer surface atoms/molecules into reacted molecular species. (2) Mechanical actions deliver the energy that removes the weakly bonded reacted molecular species. (3) The slurry fluid washes the wafer surface, and the fresh surface sites are exposed, which is subsequently etched and removed. 2.1. Cohesive energy of surface atom Silicon wafers and metal layers have been extensively used during the CMP process. The following assumptions are made to develop the material removal rate model, which were also adopted by Xu et al. [23], Qi et al. [24] and Zhao et al. [14]. (1) The cohesive energy can be regarded as the required energy to divide the crystal into isolated atoms by destroying the bonded atoms.
Fig. 2. Micro-area DS on wafer.
(2) Only a monolayer of reacted molecular species is formed on the wafer surface. (3) Since the polishing pad surface is much rougher than the wafer surface, the wafer is regarded as a smooth surface. Various parameters considered in the present analysis and the underlying assumptions are described in detail as the model develops. The number of the atoms at a given micro-cylinder, n, may be calculated using quantitative equivalent equation, as illustrated in Fig. 2. nV 0 ¼ pR2 d
(1)
where V0 is the effective occupied volume by an atom in the crystal micro-cylinder, R is its micro-area radius, as shown in Fig. 2, and d is the thickness of the wafer. Eq. (1) can be rewritten as n¼
pR2 d V0
(2)
And the volume V0 is, assuming the atom to be of a spherical shape is given by 4 V 0 ¼ pr03 3
(3)
in which r0 is the effective occupied radius of the atom. The contribution of the interface atom to the interface area is assumed to Sa ¼ pr02
(4)
Then the number of the interface atom is N ¼
pR2 Sa
(5)
Substituting Eqs. (3) and (4) into Eq. (5) yields N¼
Fig. 1. Schematic diagrams of molecular-scale removal mechanism.
R2 ð3V 0 =4pÞ2=3
(6)
It is acknowledged that the surface atoms are less stable. We can regard the bond as the interaction between different atoms. For simplicity, the interactions between the nearest atoms are only taken into consideration in this paper [25]. Therefore, each bond is formed by a pair of nearest atoms, and belongs to two atoms. As shown in Fig. 3, it is postulated that the number of bonds of an inside atom is h. For the surface relaxation, less than half
Y. Wang et al. / Applied Surface Science 253 (2007) 9137–9141
where b is ratio of the unreacted surface sites in A0 that undergo the chemical reaction during the two successive actions of the slurry particles. Eq. (13) is the basic relation accounting for chemical effects dependence of the removal cohesive energy. And the removal energy Er is a function of the particle size, and the chemical reactions parameters (b, j). In addition, the removal cohesive energy Er also depends on the wafer properties (r0, Eb).
Fig. 3. The inside atom and surface atom of the wafer.
of the volume of each surface atom, exists in the lattice, which suggests more than half of the bonds of the surface atoms are dangling bonds. Following the approach of Qi et al. [24], we can regard the number of the bonds of a surface atom with inside atoms as h/4. Therefore, the cohesive energy of the micro-cylinder E is given by 1 1 E¼ hN þ hðn NÞ Eb (7) 2 4 where Eb is bond energy of the each atom of the micro-cylinder. Substituting Eqs. (2) and (6) into Eq. (7) and rewriting it yields the cohesive energy of n atoms. 1 3V 0 E ¼ nhEb 1 (8) 2 4pdð3V 0 =4pÞ2=3 Then, the cohesive energy of per atom is E 1 3V 0 Ep ¼ ¼ hEb 1 n 2 4pdð3V 0 =4pÞ2=3
A0 Sa
(9)
(10)
The area A0 of the wafer/particle contact is given by [26] A0 ¼ pDðD dp Þ
(11)
where D is the average diameter of the abrasive particles, and dp is the depth of penetration of particle into the polishing pad. Substituting Eq. (11) into Eq. (10) yields Es ¼ Ep
pDðD dp Þ Sa
2.2. Material removal rate Three important elements in the CMP energy system are the mechanical removal energy Em, surface atoms cohesive energy and defect energy Eq including the lattice distortion energy, the dislocation energy, the grain boundary energy and the surface energy. The mechanical energy is delivered to the surface atoms by a single particle. And the kinetic energy of the surface atom is raised by an increase in value of Em. Therefore, the increase of the kinetic energy may cause the atoms leaving away from the surface at the velocity v. Ek is defined to describe the doing work, contributing to the surface atoms leaving away from the wafer/particle interface at the velocity v. Then, Ek = Er Eq. On the basis of energy balance, the kinetic energy is expressed as 1 2 mv ¼ Em Ek ¼ Em ðEr Eq Þ 2
The interface atoms existing in the area A0 of particle in contact with the wafer surface are initially removed by the particle. The removal cohesive energy Es in the area A0 is written as Es ¼ Ep
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(12)
The removal cohesive energy at atomic scale plays a significant role in removal rate and interacts strongly with chemical effects. Chemical reactions covert strongly bonded surface atoms or molecules to weakly bonded molecular species. Furthermore, according to our assumption, only the reacted molecule is removed. Defining the chemical reactions parameter j gives the removal cohesive energy DðD dp Þ 1 3V 0 Er ¼ hEb bj 1 ; 0
(14)
In Eq. (14), the kinetic energy is a positive constant, mv2 =2 > 0. It implies that, the mechanical energy Em is higher than removal cohesive energy Er, Em Er. Value of b as a function of oxidation concentration is determined by Zhao et al. in ref. [14]. Zhao et al. [14] also proposed that the ratio of the removal g is a function of mechanical removal energy Em and the removal cohesive energy Er, which can be expressed in the below Em g¼ f Er For simplicity, it is assumed that the value of g is proportional to the mechanical removal energy and is inverse proportional to the removal cohesive energy Er. To qualitatively analyze the relationship between the material removal rate and the particle size, the polishing rate MRR can be obtained by combing Eqs. (13) and (14) with Zhao et al.’s [14] Eq. (32), while at constant other parameters, such as the properties of polishing pad and the wafer. MRR / 1 1=2hEb bjDðD
dp Þ=Em r02 ½1
3V 0 =4pdð3V 0 =4pÞ2=3 þ 1=b (15)
which is plotted in Fig. 4 at different values of j as Es = Em. It is illustrated that, the increase of Em results in the increase in removal rate. An important extension of the model is to
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Fig. 5. Schematic diagrams of the relationship between removal rate and particle size.
Fig. 4. MRR vs. b.
remove the surface reacted atoms. That is to say, the removal rate may not be capable to be increased as the particle size is further increased. Value of Em as a function of particle size D is shown below
discuse the chemical effects (b, j) and particle size D on removal rate.
Em ¼ f ðDÞ
3. Experimental verification
Eq. (13) suggests that removal cohesive energy Er can be converted to the following expression
The qualitative concepts can be used to understand the role of chemistry and particle size in CMP systems. The present study indicates variable trends on particle size. To evaluate the proposed model, we will compare it next to groups of experimental material removal rate measurements, gathered from literature under different experimental conditions. 3.1. Effects of oxidation concentration It is reported that the ratio of b is proportional to oxidation concentration of the slurry [14]. As shown in Fig. 4, material removal rate MRR is proportional to b at lower value of j. This means an increase in the oxidation concentration allowing for an increase in the MRR rapidly. With the increase in parameters j, MRR increases rapidly at low oxidation concentration, and approaches a constant at high oxidation concentration. An interpretation of this phenomenon is that the number of reacted atoms (in Fig. 1) is raised by an increase in the oxidation concentration and appears to enhance the MRR. However, the removal cohesive energy also increases with value of b, as given in Eq. (13), which generates opposite effects to this of value b on the removal rate. The relation MRR with b at high value of j shows similar trend with the results of the experimental data [27,28]. 3.2. Effects of particle size It is assumed that Em is proportional to abrasive particle size. Therefore, Eq. (15) also appears to reveal some insights into the effects of particle size. The increase of particle size could significantly enhance the micro-contact area A0 between particle and wafer. Eq. (10) states that increasing A0 leads to a higher cohesive energy. And this suggests that more mechanical energy should be delivered to the particle to
Er ¼ gðA0 Þ: Consider the particle diameter. Its effects on the removal rate are reflected in two terms. On one hand, an increase in particle size results in the increase in the mechanical energy Em, contributing to a proportional increase in the removal rate. On the other hand, Eq. (13) shows it enhances the removal cohesive energy as well in the micro-area A0, reducing the removal rate. As the increment of Em is much higher than that of Er with the increase in D, a higher D will tend to increase the removal rate during the CMP process as shown in Eq. (16). Experimental results lend some support to this relation of particle diameter and the removal rate. Fig. 5 shows schematic diagrams of the relationship between removal rate and particle size. Three different schematic drawings for particle size ranges are illustrated in Fig. 5. For typical CMP process, the values of the particle size published and quoted range from 20 nm to 2 mm. It may be plausible to assume the particle size D ranges from D1 to D3 (i.e. D1 < D2 < D3). And Tamboli et al. [12] reported the Ta CMP removal rate increases with particle size, as show in Fig. 5. D f ðDÞ > DgðA0 Þ (16) Em Er However, referring to Eq. (17), the increase of D will lead to reducing the MRR. It is observed that the model predictions from Eqs. (15) and (17) compare to the experimental trend in Fig. 5 [11]. D f ðDÞ < DgðA0 Þ (17) Em Er Another set of experimental measurements, studying the effect of particle size on the SiO2 CMP was carried out by Zhou et al. [13]. As shown in Fig. 5, the MRR initially increases with
Y. Wang et al. / Applied Surface Science 253 (2007) 9137–9141
the particle size, and then decreases with it, which follows the present model. Eqs. (15) and (18) show the MRR increases in the similar trend as the polishing rate in Fig. 5. 8 < D f ðDÞ > DgðA0 Þ D1 < D < D2 D f ðDÞ < DgðA0 Þ D2 < D < D3 : Em Er
(18)
Since both of the mechanical energy and removal cohesive energy couple with the particle size, the predicted removal rate indicates variable trends on the particle size, which are in consistent with the published experiment data [11–13].
4. Conclusion Based on atomic-scale removal mechanism, interface solid physical and chemical theory and energy balance knowledge, a mathematical model as a function of binding energy, chemical energy parameters and mechanical removal energy is presented in this paper. The model presented here describes how the CMP material removal depends on the oxidation concentration and the particle size. It predicts that increasing the oxidation can enhance the removal rate, up to some asymptotic constant value. The effects of oxidation concentration and particle size on material removal are reflected in two terms in the model. On one hand, the chemical reaction decreases the molecular binding energy and increases the number of reacted molecules, contributing to an increase in removal rate. On the other hand, it enhances the removal cohesive energy, which leads to a lower removal rate. The effect of particle size on removal rate is also reflected in two opposite effects, complicating the outcome and being a cause of the non-linear size-removal rate relationships. In this regard, large particle size increases the mechanical removal energy to a higher removal rate. On the other hand, it contributes to an increase in cohesive energy, which allows for a decrease in material removal. Furthermore, the removal cohesive energy can be calculated from Eq. (13), which is a function of the particle size, wafer properties and the chemical reactions parameters. The model predictions are presented in graphical form and show good qualitative agreement with the published experimental date. The results and analysis may provide some insights into the removal mechanism from molecular/ atomic perspective.
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Acknowledgments Financial support of this research work was provided in part by the Natural Science Foundation of Jiangsu Province in China (Grant No. BK2004020), Tribology Science Foundation of the State Key Laboratory of Tribology in Tsinghua University in China (Grant No. SKLT04-06) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (Grant No.2004527). References [1] M. Fury, Solid State Technol. 40 (1997) 81. [2] P.O. Hahn, Microelectron. Eng. 56 (2001) 3. [3] C.J. Evans, E. Paul, D. Dornfeld, D.A. Lucca, G. Byme, M. Tricard, F. Klocke, O. Dambon, B.A. Mullany, Ann. CIRP 152 (2003) 611. [4] Y.W. Zhao, J.J. Liu, Tribology 24 (2004) 283 (in Chinese). [5] T.F. Zeng, S. Thomas, IEE Trans. Semicond. Manufact. 16 (2005) 655. [6] J.X. Su, Study on material removal mechanism of wafer chemical mechanical polishing in IC manufacturing, Dalian Univ. Technol. (2007) (in Chinese). [7] Y.R. Jeng, P.Y. Huang, J. Tribol. 127 (2005) 190. [8] M. Bastaninejad, G. Ahmadi, J. Electrochem. Soc. 152 (2005) G720. [9] H. Hocheng, H.Y. Tsai, Y.T. Su, J. Electrochem. Soc. 148 (2001) G581. [10] J.F. Lin, J.D. Chern, Y.H. Chang, K.L. Ping, T.S. Ming, J. Tribol. 126 (2004) 185. [11] M. Bielman, U. Mahajan, R.K. Singh, Electrochem. Soild State Lett. 2 (1999) 401. [12] D. Tamboli, G. Banerjee, M. Waddell, Electrochem. Soild State Lett. 7 (2004) F62. [13] C. Zhou, L. Shan, J.R. Hight, D. Steven, S.H. Ng, J.P. Andrew, Tribol. Trans. 45 (2002) 232. [14] Y.W. Zhao, L. Chang, S.H. Kim, Wear 254 (2003) 332. [15] Y.G. Wang, Y.W. Zhao, Chin. J. Semicond. 28 (2007) 130 (in Chinese). [16] J. Luo, D. Dornfield, IEE Trans. Semicond. Manufact. 16 (2003) 45. [17] Y.G. Wang, Y.W. Zhao, J. Gu, J. Mater. Process Technol. 183 (2007) 374. [18] A. Miyoshi, H. Nakagawa, K. Matsukawa, Microsyst. Technol. 11 (2005) 1102. [19] D. Gadf, A. Schnegg, R. Schmolke, M. Subren, H.A. Gerber, P. Wagner, Electrochem. Soc. Pro. 96 (2000) 186. [20] J. Xu, J.B. Luo, L.L. Wang, Tribol. Int. 40 (2007) 285. [21] A. Rajendran, Y. Takahashi, M. Koyama, A. Miyamoto, Appl. Surf. Sci. 244 (2005) 34. [22] L. Chang, J. Tribol. 129 (2007) 436. [23] J. Xu, J.B. Luo, L.L. Wang, G.S. Pan, S.Z. Wen, Nanotechnology 16 (2005) 859. [24] W.H. Qi, M.P. Wang, G.Y. Xu, Chem. Phys. Lett. 372 (2003) 632. [25] X.D. Xie, D. Chen, Solid Band Theory, Shang hai: FuDan University Press, 1998,, p. 102 (in Chinese). [26] Y.W. Zhao, L. Chang, Wear 252 (2002) 220. [27] J.S. David, D.L. Hetherington, J.L. Cecchi, J. Electrochem. Soc. 1 (1999) 376. [28] E. Paul, J. Electrochem. Soc. 6 (2001) G355.
Modeling the effects of cohesive energy for single particle on the material removal in chemical mechanical polishing at atomic scale Yongguang Wang, Yongwu Zhao *, Wei An, Jun Wang School of Mechanical Engineering, Southern Yangtze University, Wuxi, 214122, PR China Received 13 April 2007; received in revised form 13 May 2007; accepted 17 May 2007 Available online 29 May 2007
Abstract This paper proposes a novel mathematical model for chemical mechanical polishing (CMP) based on interface solid physical and chemical theory in addition to energy equilibrium knowledge. And the effects of oxidation concentration and particle size on the material removal in CMP are investigated. It is shown that the mechanical energy and removal cohesive energy couple with the particle size, and being a cause of the nonlinear size-removal rate relation. Furthermore, it also shows a nonlinear dependence of removal rate on removal cohesive energy. The model predictions are in good qualitative agreement with the published experimental data. The current study provides an important starting point for delineating the micro-removal mechanism in the CMP process at atomic scale. # 2007 Elsevier B.V. All rights reserved. Keywords: Chemical mechanical polishing; Binding energy; Modeling; Atomic scale
1. Introduction In the semiconductor industry chemical mechanical polishing (CMP), developed at IBM [1] during early 1991s, has emerged to be most promising because it can provide the planarity necessary to build multilevel interconnect schemes. With the decrease in line widths and an increase in the number of metal layers, CMP has become a vital processing step for the realization of circuits with current line width of as small as 0.05 mm [2]. Despite its crucial importance in the microelectronic industry, many aspects of the CMP process remain unclear [3,4]. Understanding of the fundamental of CMP material removal mechanisms may offer insights into the control and optimization of the polishing process. Many researchers modeled the CMP material removal rate based on different removal mechanisms, such as indentationsliding mechanism [5–7], adhesion removal mechanism [8], and hydrodynamic slurry [9,10], etc. Despite its seemingly correct cause-and-effect sequence, the validity of these continuum descriptions of the CMP process has been questioned. Firstly,
* Corresponding author. Tel.: +86 510 85140186. E-mail addresses: [email protected] (Y. Wang), [email protected] (Y. Zhao). 0169-4332/$ – see front matter # 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2007.05.043
they are incapable of delineating the variable trends of particle size on material removal despite its significant influence observed experimentally [11–13]. In some instances, it could be decreasing [11] or increasing [12], and Zhou et al. [13] found that the material removal rate initially increases with the particle size, then decreases with it. Secondly, the real CMP chemical action time during two successive particles is of an order of 1.0 108 s [14]. Wang and Zhao [15] concluded that the thickness of the oxidized layer is evaluated on the order of 0.01 nm. In addition, the indentation of the abrasive particles into the wafer surface is on the order of 0.1 nanometers or less [16,17] under the conventional CMP conditions. Therefore, under the polishing equilibrium process, no softened wafer surface film is formed and the slurry chemistry, at best, has time only to react with the surface monolayer of the wafer surface. More recently, the abrasive particle size provided by Cabot Company is around 20 nm. With the decrease of the particle size, the material removal mechanism may be more sensibly described by noncontinuum theories. The atomic-scale removal mechanism during CMP process was investigated by many researchers in combination with examinations through AFM [18], XPS [19], TEM [20] and molecular dynamics simulation (MD) [21]. However, the above discrete mechanisms in relation to molecular bonds and binding energies have been established through empirical investigation.
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Y. Wang et al. / Applied Surface Science 253 (2007) 9137–9141
Science-based model of CMP discrete material removal mechanism has not been developed yet, largely due to the complex interactions between multiple variables during polishing process. Because of the complexity of the polishing system, it is highly desirable to characterize the physical and chemical behaviors of the individual interaction while other components are fixed, or nearly so. The present research will only focus on the aspect of a single slurry particle. In this paper, a mathematical model as a function of binding energy, chemical parameter and mechanical removal energy is presented for CMP. The model is proposed based on interface solid physical and chemical theory as well as energy balance knowledge. It shows a nonlinear dependence of removal rate on binding energy and particle size. The model predictions are compared with the experimental data to verify the current model. 2. Modeling As the knowledge of CMP process and role of consumable improved over years, much present work has investigated the material removal from molecular/atomic-scale perspective [22]. As shown in Fig. 1, a more likely scenario for the process of the material removal has three steps: (1) Chemical actions covert strongly the fresh wafer surface atoms/molecules into reacted molecular species. (2) Mechanical actions deliver the energy that removes the weakly bonded reacted molecular species. (3) The slurry fluid washes the wafer surface, and the fresh surface sites are exposed, which is subsequently etched and removed. 2.1. Cohesive energy of surface atom Silicon wafers and metal layers have been extensively used during the CMP process. The following assumptions are made to develop the material removal rate model, which were also adopted by Xu et al. [23], Qi et al. [24] and Zhao et al. [14]. (1) The cohesive energy can be regarded as the required energy to divide the crystal into isolated atoms by destroying the bonded atoms.
Fig. 2. Micro-area DS on wafer.
(2) Only a monolayer of reacted molecular species is formed on the wafer surface. (3) Since the polishing pad surface is much rougher than the wafer surface, the wafer is regarded as a smooth surface. Various parameters considered in the present analysis and the underlying assumptions are described in detail as the model develops. The number of the atoms at a given micro-cylinder, n, may be calculated using quantitative equivalent equation, as illustrated in Fig. 2. nV 0 ¼ pR2 d
(1)
where V0 is the effective occupied volume by an atom in the crystal micro-cylinder, R is its micro-area radius, as shown in Fig. 2, and d is the thickness of the wafer. Eq. (1) can be rewritten as n¼
pR2 d V0
(2)
And the volume V0 is, assuming the atom to be of a spherical shape is given by 4 V 0 ¼ pr03 3
(3)
in which r0 is the effective occupied radius of the atom. The contribution of the interface atom to the interface area is assumed to Sa ¼ pr02
(4)
Then the number of the interface atom is N ¼
pR2 Sa
(5)
Substituting Eqs. (3) and (4) into Eq. (5) yields N¼
Fig. 1. Schematic diagrams of molecular-scale removal mechanism.
R2 ð3V 0 =4pÞ2=3
(6)
It is acknowledged that the surface atoms are less stable. We can regard the bond as the interaction between different atoms. For simplicity, the interactions between the nearest atoms are only taken into consideration in this paper [25]. Therefore, each bond is formed by a pair of nearest atoms, and belongs to two atoms. As shown in Fig. 3, it is postulated that the number of bonds of an inside atom is h. For the surface relaxation, less than half
Y. Wang et al. / Applied Surface Science 253 (2007) 9137–9141
where b is ratio of the unreacted surface sites in A0 that undergo the chemical reaction during the two successive actions of the slurry particles. Eq. (13) is the basic relation accounting for chemical effects dependence of the removal cohesive energy. And the removal energy Er is a function of the particle size, and the chemical reactions parameters (b, j). In addition, the removal cohesive energy Er also depends on the wafer properties (r0, Eb).
Fig. 3. The inside atom and surface atom of the wafer.
of the volume of each surface atom, exists in the lattice, which suggests more than half of the bonds of the surface atoms are dangling bonds. Following the approach of Qi et al. [24], we can regard the number of the bonds of a surface atom with inside atoms as h/4. Therefore, the cohesive energy of the micro-cylinder E is given by 1 1 E¼ hN þ hðn NÞ Eb (7) 2 4 where Eb is bond energy of the each atom of the micro-cylinder. Substituting Eqs. (2) and (6) into Eq. (7) and rewriting it yields the cohesive energy of n atoms. 1 3V 0 E ¼ nhEb 1 (8) 2 4pdð3V 0 =4pÞ2=3 Then, the cohesive energy of per atom is E 1 3V 0 Ep ¼ ¼ hEb 1 n 2 4pdð3V 0 =4pÞ2=3
A0 Sa
(9)
(10)
The area A0 of the wafer/particle contact is given by [26] A0 ¼ pDðD dp Þ
(11)
where D is the average diameter of the abrasive particles, and dp is the depth of penetration of particle into the polishing pad. Substituting Eq. (11) into Eq. (10) yields Es ¼ Ep
pDðD dp Þ Sa
2.2. Material removal rate Three important elements in the CMP energy system are the mechanical removal energy Em, surface atoms cohesive energy and defect energy Eq including the lattice distortion energy, the dislocation energy, the grain boundary energy and the surface energy. The mechanical energy is delivered to the surface atoms by a single particle. And the kinetic energy of the surface atom is raised by an increase in value of Em. Therefore, the increase of the kinetic energy may cause the atoms leaving away from the surface at the velocity v. Ek is defined to describe the doing work, contributing to the surface atoms leaving away from the wafer/particle interface at the velocity v. Then, Ek = Er Eq. On the basis of energy balance, the kinetic energy is expressed as 1 2 mv ¼ Em Ek ¼ Em ðEr Eq Þ 2
The interface atoms existing in the area A0 of particle in contact with the wafer surface are initially removed by the particle. The removal cohesive energy Es in the area A0 is written as Es ¼ Ep
9139
(12)
The removal cohesive energy at atomic scale plays a significant role in removal rate and interacts strongly with chemical effects. Chemical reactions covert strongly bonded surface atoms or molecules to weakly bonded molecular species. Furthermore, according to our assumption, only the reacted molecule is removed. Defining the chemical reactions parameter j gives the removal cohesive energy DðD dp Þ 1 3V 0 Er ¼ hEb bj 1 ; 0
(14)
In Eq. (14), the kinetic energy is a positive constant, mv2 =2 > 0. It implies that, the mechanical energy Em is higher than removal cohesive energy Er, Em Er. Value of b as a function of oxidation concentration is determined by Zhao et al. in ref. [14]. Zhao et al. [14] also proposed that the ratio of the removal g is a function of mechanical removal energy Em and the removal cohesive energy Er, which can be expressed in the below Em g¼ f Er For simplicity, it is assumed that the value of g is proportional to the mechanical removal energy and is inverse proportional to the removal cohesive energy Er. To qualitatively analyze the relationship between the material removal rate and the particle size, the polishing rate MRR can be obtained by combing Eqs. (13) and (14) with Zhao et al.’s [14] Eq. (32), while at constant other parameters, such as the properties of polishing pad and the wafer. MRR / 1 1=2hEb bjDðD
dp Þ=Em r02 ½1
3V 0 =4pdð3V 0 =4pÞ2=3 þ 1=b (15)
which is plotted in Fig. 4 at different values of j as Es = Em. It is illustrated that, the increase of Em results in the increase in removal rate. An important extension of the model is to
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Fig. 5. Schematic diagrams of the relationship between removal rate and particle size.
Fig. 4. MRR vs. b.
remove the surface reacted atoms. That is to say, the removal rate may not be capable to be increased as the particle size is further increased. Value of Em as a function of particle size D is shown below
discuse the chemical effects (b, j) and particle size D on removal rate.
Em ¼ f ðDÞ
3. Experimental verification
Eq. (13) suggests that removal cohesive energy Er can be converted to the following expression
The qualitative concepts can be used to understand the role of chemistry and particle size in CMP systems. The present study indicates variable trends on particle size. To evaluate the proposed model, we will compare it next to groups of experimental material removal rate measurements, gathered from literature under different experimental conditions. 3.1. Effects of oxidation concentration It is reported that the ratio of b is proportional to oxidation concentration of the slurry [14]. As shown in Fig. 4, material removal rate MRR is proportional to b at lower value of j. This means an increase in the oxidation concentration allowing for an increase in the MRR rapidly. With the increase in parameters j, MRR increases rapidly at low oxidation concentration, and approaches a constant at high oxidation concentration. An interpretation of this phenomenon is that the number of reacted atoms (in Fig. 1) is raised by an increase in the oxidation concentration and appears to enhance the MRR. However, the removal cohesive energy also increases with value of b, as given in Eq. (13), which generates opposite effects to this of value b on the removal rate. The relation MRR with b at high value of j shows similar trend with the results of the experimental data [27,28]. 3.2. Effects of particle size It is assumed that Em is proportional to abrasive particle size. Therefore, Eq. (15) also appears to reveal some insights into the effects of particle size. The increase of particle size could significantly enhance the micro-contact area A0 between particle and wafer. Eq. (10) states that increasing A0 leads to a higher cohesive energy. And this suggests that more mechanical energy should be delivered to the particle to
Er ¼ gðA0 Þ: Consider the particle diameter. Its effects on the removal rate are reflected in two terms. On one hand, an increase in particle size results in the increase in the mechanical energy Em, contributing to a proportional increase in the removal rate. On the other hand, Eq. (13) shows it enhances the removal cohesive energy as well in the micro-area A0, reducing the removal rate. As the increment of Em is much higher than that of Er with the increase in D, a higher D will tend to increase the removal rate during the CMP process as shown in Eq. (16). Experimental results lend some support to this relation of particle diameter and the removal rate. Fig. 5 shows schematic diagrams of the relationship between removal rate and particle size. Three different schematic drawings for particle size ranges are illustrated in Fig. 5. For typical CMP process, the values of the particle size published and quoted range from 20 nm to 2 mm. It may be plausible to assume the particle size D ranges from D1 to D3 (i.e. D1 < D2 < D3). And Tamboli et al. [12] reported the Ta CMP removal rate increases with particle size, as show in Fig. 5. D f ðDÞ > DgðA0 Þ (16) Em Er However, referring to Eq. (17), the increase of D will lead to reducing the MRR. It is observed that the model predictions from Eqs. (15) and (17) compare to the experimental trend in Fig. 5 [11]. D f ðDÞ < DgðA0 Þ (17) Em Er Another set of experimental measurements, studying the effect of particle size on the SiO2 CMP was carried out by Zhou et al. [13]. As shown in Fig. 5, the MRR initially increases with
Y. Wang et al. / Applied Surface Science 253 (2007) 9137–9141
the particle size, and then decreases with it, which follows the present model. Eqs. (15) and (18) show the MRR increases in the similar trend as the polishing rate in Fig. 5. 8 < D f ðDÞ > DgðA0 Þ D1 < D < D2 D f ðDÞ < DgðA0 Þ D2 < D < D3 : Em Er
(18)
Since both of the mechanical energy and removal cohesive energy couple with the particle size, the predicted removal rate indicates variable trends on the particle size, which are in consistent with the published experiment data [11–13].
4. Conclusion Based on atomic-scale removal mechanism, interface solid physical and chemical theory and energy balance knowledge, a mathematical model as a function of binding energy, chemical energy parameters and mechanical removal energy is presented in this paper. The model presented here describes how the CMP material removal depends on the oxidation concentration and the particle size. It predicts that increasing the oxidation can enhance the removal rate, up to some asymptotic constant value. The effects of oxidation concentration and particle size on material removal are reflected in two terms in the model. On one hand, the chemical reaction decreases the molecular binding energy and increases the number of reacted molecules, contributing to an increase in removal rate. On the other hand, it enhances the removal cohesive energy, which leads to a lower removal rate. The effect of particle size on removal rate is also reflected in two opposite effects, complicating the outcome and being a cause of the non-linear size-removal rate relationships. In this regard, large particle size increases the mechanical removal energy to a higher removal rate. On the other hand, it contributes to an increase in cohesive energy, which allows for a decrease in material removal. Furthermore, the removal cohesive energy can be calculated from Eq. (13), which is a function of the particle size, wafer properties and the chemical reactions parameters. The model predictions are presented in graphical form and show good qualitative agreement with the published experimental date. The results and analysis may provide some insights into the removal mechanism from molecular/ atomic perspective.
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Acknowledgments Financial support of this research work was provided in part by the Natural Science Foundation of Jiangsu Province in China (Grant No. BK2004020), Tribology Science Foundation of the State Key Laboratory of Tribology in Tsinghua University in China (Grant No. SKLT04-06) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry (Grant No.2004527). References [1] M. Fury, Solid State Technol. 40 (1997) 81. [2] P.O. Hahn, Microelectron. Eng. 56 (2001) 3. [3] C.J. Evans, E. Paul, D. Dornfeld, D.A. Lucca, G. Byme, M. Tricard, F. Klocke, O. Dambon, B.A. Mullany, Ann. CIRP 152 (2003) 611. [4] Y.W. Zhao, J.J. Liu, Tribology 24 (2004) 283 (in Chinese). [5] T.F. Zeng, S. Thomas, IEE Trans. Semicond. Manufact. 16 (2005) 655. [6] J.X. Su, Study on material removal mechanism of wafer chemical mechanical polishing in IC manufacturing, Dalian Univ. Technol. (2007) (in Chinese). [7] Y.R. Jeng, P.Y. Huang, J. Tribol. 127 (2005) 190. [8] M. Bastaninejad, G. Ahmadi, J. Electrochem. Soc. 152 (2005) G720. [9] H. Hocheng, H.Y. Tsai, Y.T. Su, J. Electrochem. Soc. 148 (2001) G581. [10] J.F. Lin, J.D. Chern, Y.H. Chang, K.L. Ping, T.S. Ming, J. Tribol. 126 (2004) 185. [11] M. Bielman, U. Mahajan, R.K. Singh, Electrochem. Soild State Lett. 2 (1999) 401. [12] D. Tamboli, G. Banerjee, M. Waddell, Electrochem. Soild State Lett. 7 (2004) F62. [13] C. Zhou, L. Shan, J.R. Hight, D. Steven, S.H. Ng, J.P. Andrew, Tribol. Trans. 45 (2002) 232. [14] Y.W. Zhao, L. Chang, S.H. Kim, Wear 254 (2003) 332. [15] Y.G. Wang, Y.W. Zhao, Chin. J. Semicond. 28 (2007) 130 (in Chinese). [16] J. Luo, D. Dornfield, IEE Trans. Semicond. Manufact. 16 (2003) 45. [17] Y.G. Wang, Y.W. Zhao, J. Gu, J. Mater. Process Technol. 183 (2007) 374. [18] A. Miyoshi, H. Nakagawa, K. Matsukawa, Microsyst. Technol. 11 (2005) 1102. [19] D. Gadf, A. Schnegg, R. Schmolke, M. Subren, H.A. Gerber, P. Wagner, Electrochem. Soc. Pro. 96 (2000) 186. [20] J. Xu, J.B. Luo, L.L. Wang, Tribol. Int. 40 (2007) 285. [21] A. Rajendran, Y. Takahashi, M. Koyama, A. Miyamoto, Appl. Surf. Sci. 244 (2005) 34. [22] L. Chang, J. Tribol. 129 (2007) 436. [23] J. Xu, J.B. Luo, L.L. Wang, G.S. Pan, S.Z. Wen, Nanotechnology 16 (2005) 859. [24] W.H. Qi, M.P. Wang, G.Y. Xu, Chem. Phys. Lett. 372 (2003) 632. [25] X.D. Xie, D. Chen, Solid Band Theory, Shang hai: FuDan University Press, 1998,, p. 102 (in Chinese). [26] Y.W. Zhao, L. Chang, Wear 252 (2002) 220. [27] J.S. David, D.L. Hetherington, J.L. Cecchi, J. Electrochem. Soc. 1 (1999) 376. [28] E. Paul, J. Electrochem. Soc. 6 (2001) G355.