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Comparison of different methods for simulating the effect of specular ion reflection on microtrenching during dry etching of polysilicon
Available online at www.sciencedirect.com
Microelectronic Engineering 85 (2008) 992–995 www.elsevier.com/locate/mee
Comparison of different methods for simulating the effect of specular ion reflection on microtrenching during dry etching of polysilicon D. Kunder *, E. Ba¨r Fraunhofer Institute of Integrated Systems and Device Technology, Schottkystrasse 10, 91058 Erlangen, Germany Received 5 October 2007; received in revised form 11 January 2008; accepted 15 January 2008 Available online 26 January 2008
Abstract For the simulation of etching processes, a key step is the calculation of the etch rates depending on the specific model and depending on the specific geometry of the feature. In this work, we demonstrate the calculation of etch rates using a Monte Carlo, a flux balancing, and an analytical approach. For a relatively simple model for etching of polysilicon in chlorine-based chemistry, the three approaches are compared and microtrenching is studied which results from the specular reflection of ions and depends on different parameters. The results for the different approaches are in good agreement for the cases studied. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Simulation; Etching processes; Microtrenching; Monte Carlo method
Etching is a key process step determining the geometry of relevant feature patterns such as gate electrodes. Etching models, including treatment of specular reflections of ions, have been presented in the literature for the 2D case, e.g. using a Monte Carlo method [1]. 3D implementations (e.g. [2]) published so far do not consider the effect of specular reflection. The flux balancing approach used in this work has not yet been demonstrated at all for modeling of specular reflection. In this paper we present different 3D simulation approaches that take into account specular ion reflection leading to microtrenching. We focus on comparing the different independent approaches rather than on using a full chemistry model of the etching process.
performed in three phases: First, the structure is discretized by a surface triangulation generated by a combination of inhouse structure generation tools with the meshing tools of the Synopsys TCAD software suite [3]. The structures studied in this work all have a cylindrical mask opening. An example for such a structure after the simulation of etching is shown in Fig. 1. In the second phase, the etch rates for all triangles are calculated. In the third phase, the surface is shifted depending on the etch rates. Phase two and three are repeated until the desired etch depth is obtained. The discretization errors inherent in this procedure are due to the finite size of the triangles used for the surface discretization and due to the finite number of time steps employed for updating the etch rates during the simulation run.
2. Simulation
2.2. Calculation of the etch rates
2.1. General
We use a relatively simple model for polysilicon etching in a chlorine plasma where the local etch rate is determined by the local ion flux. It is assumed that the ion flux above the substrate is constant, i.e. it does not depend on the lateral position. Furthermore, a perfectly directional ion flux is assumed which means that the ion velocity has only a
1. Introduction
The simulation based on the Monte Carlo approach and the simulation based on the flux balancing approach are *
Corresponding author. Tel.: +49 9131 761 223. E-mail address: [email protected] (D. Kunder).
0167-9317/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2008.01.038
D. Kunder, E. Ba¨r / Microelectronic Engineering 85 (2008) 992–995
993
Fig. 1. Etched structure (left: 3D view, right: cross sectional view). The etch rates were calculated by the Monte Carlo approach.
component perpendicular to the substrate surface. Regarding the latter assumption, however, for the comparison of the different methods shown later on, an isotropic velocity distribution of the ions is assumed. The reason for this assumption is discussed in Section 3. For the interaction of the ions with the surface, it is assumed that an ion is reflected specularly if its angle of incidence is above a given limit (here called alim). For typical etching conditions, typical values of alim are around 80° [4]. Ions not being reflected lead to etching. It is assumed that each sticking ion contributes the same amount to the etching rate, independent of its angle and energy. 2.2.1. Monte Carlo approach The Monte Carlo method is a probabilistic approach in which many particle trajectories (for the results shown in this work 108) are generated to simulate the behavior of the real system. Since the specular reflection and the etching process are taken to be deterministic, the only variation in the particle trajectories is due to their different starting positions. For the simulation, the particles are launched from above the feature and their trajectories are determined based on traveling in the ballistic regime, i.e. scattering can only occur when interacting with the feature surface. The trajectory of a particle ends if it sticks at the surface thus contributing to etching, or escapes from the simulation region. The simulation time for the Monte Carlo approach is much higher (by a factor of 10–100) than the time needed by the flux balancing approach. However, especially regarding the implementation of more complex models, the Monte Carlo approach offers higher flexibility and a more straightforward implementation of etching mechanisms. 2.2.2. Flux balancing approach For the flux balancing approach, a discretization is used which considers the flux to each triangle arriving from discrete spatial directions. The discretization of the spatial directions is based on spherical coordinates, i.e. using the angles # and u. For both angles, a discrete number of intervals are used. The discretized flux Ui;n#; nu is then introduced which represents the flux for triangle i for the spatial direc-
tion represented by the integer numbers n# and nu (see Fig. 2 for illustration). In this context, Ui;n#; nu is defined as the flux to triangle i per solid angle interval, i.e. the number of particles arriving at triangle i from a specific direction is given by Ui;n#; nu times the area of the triangle times the solid angle interval considered for particle delivery. For the interaction of the flux with the surface the reflection condition is employed in the same way as in the Monte Carlo approach, i.e. the particle is reflected if the angle of incidence is larger than the limiting angle for reflection alim. Flux balancing then leads to a system of equations in the unknowns Ui;n#; nu : Ui;n# ;nu ¼ G cosð#loc ðn# ; nu ÞÞ f ð#glob ðn# ; nu Þ; uglob ðn# ; nu ÞÞ X vij Ui;m# ;mu T ij ðn# ; nu Þ ð1Þ þ j; j6¼i
In (1) the first summand gives the direct flux arriving from the reactor at the triangle under consideration which depends on the angles in the reactor related system (subscript glob) for the direction under consideration which appear as argument of the angular distribution function f. The angular distribution function f also includes the effect of flux shadowing due to the feature geometry. The factor G is used for normalization to yield the 1D etching rates (as specified as input to the simulator) in regions far from any 2D or 3D structure. The cosine term of the first summand represents the flux reduction for non-normal incidence and therefore depends on the local angles (subscript loc). The second summand is the sum of contributions of particles reflected from
Discretization of spatial directions (nϑ ,nϕ)
Triangle i
Fig. 2. Discretization used for the flux balancing approach. Refer to the text for further explanation.
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D. Kunder, E. Ba¨r / Microelectronic Engineering 85 (2008) 992–995
other triangles and arriving at triangle i. vij is the view factor which is unity if there is direct view between triangle i and triangle j and zero otherwise. The matrix elements Tij are calculated for each pair of triangles based on geometrical considerations. In particular, for their calculation the reflection condition with alim as mentioned above is taken into account. (1) has to be written for each triangle i and for each value of the integers n# and nu representing the spatial directions. Therefore, the number of variables (which is equal to the number of equations in the system) is the number of triangles for the surface discretization times the number of discrete # directions times the number of discrete u directions and is in the order of magnitude of 107. To obtain the solution of the system, an iterative solver is used. From the obtained solution the etch rate for each triangle is determined by summing up the flux contributions from directions where the ion is not reflected, i.e. the directions with an angle of incidence smaller than alim. For the flux balancing approach, discretization errors are due to the finite triangle size and due to the finite number of directions used for the discretization of the spatial directions. 2.2.3. Analytical calculation The analytical calculations are based on integration of fluxes for different positions on the feature surface. Depending on the ion angular distribution two different calculations have been carried out. In both cases, the rates are determined for the structure at the very beginning of the etching process, i.e. changes to the structure due to feature evolution are not taken into account. The calculation of the etch rates for a directional ion angular distribution is performed by summing up the particles arriving directly from the reactor and the particles arriving due to reflection from the mask. To this end (as shown in Fig. 3) the feature is discretized by rings. For example, the etch rate for the substrate ring 1 results from particles arriving directly from the reactor and particles reflected by the mask ring 1. More generally and for infinitesimally thin rings i, the rate for ring ri is given as Am;i ri ¼ const 1 þ F ð2Þ As;i
where Am,i and As,i give the areas of the substrate ring i and of the mask ring i, respectively, and F is the direct flux to the substrate. For an isotropic angular distribution, the flux for a location at the substrate is calculated by integration using angles in spherical coordinates. The integration limits depend on the aspect ratio of the structure, the shortest distance of the point to the mask, and alim. For the example shown below, a sidewall angle of 90° was assumed. Part of the integration has to be carried out numerically. For more complex etching models this approach cannot be used anymore. 3. Results First, we compare the three approaches for a test case assuming an isotropic angular distribution of the ions. Though an isotropic distribution of the ions is not realistic, for comparing different methods, assuming such a distribution is advantageous, as the etch rate shows variations for the full range of positions between the feature center and the mask edge, whereas for directional distributions only the vicinity of the mask edge exhibits etch rate variations. Therefore, we have chosen this hypothetical case for comparing the different methods (i.e. Monte Carlo, flux balancing, and analytical calculations) as shown in Fig. 4. It can be seen that there is good agreement between the three fully independent approaches when predicting the etch rate along an etched feature. The differences can be explained by the finite size of the surface triangles and, in case of the flux balancing approach, by discretization of the spatial directions. The escape of the curves for the flux balancing and Monte Carlo method at the very right of the figure is due to the triangle nodes at the mask edge which are kept fixed during the shift of the surface. Next, the effect of specular ion reflection on microtrenching is studied. For the simulation shown in Fig. 1, a perfectly directional distribution of incoming ions has been used. The sidewall angle of the mask was 86°. In the cross section, the microtrenching near the mask edge is clearly visible. The positions where the reflected ions etch the substrate have an etch rate which is at least twice as large as it is for positions at the substrate not reached by
Fig. 3. Discretization used for the analytical approach and a directional ion angular distribution; side view (left) and top view (right).
D. Kunder, E. Ba¨r / Microelectronic Engineering 85 (2008) 992–995
Fig. 4. Shift of the substrate after one etch step for the three different approaches. The angular distribution of ions is isotropic. Furthermore, perfectly steep sidewalls and an aspect ratio of 1 have been assumed.
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reflected ions. However, more realistic etch models here would lead to a lower value for this ratio. In this simple model the only parameters influencing the degree of microtrenching are the aspect ratio of the mask opening and the mask sidewall angle (for a given angular distribution of ions). In Fig. 5, the Monte Carlo approach and the analytical calculations have been used to study this dependence. The ratios shown in Fig. 5 correspond to the values at the very beginning of the etching process. Included are the results of the Monte Carlo approach as well as the results of the analytical calculations. The difference observed between the Monte Carlo approach and the analytical one is due to the finite triangle size which leads to an averaging of the etching rates over a lateral range corresponding to the spatial extension of the triangle. 4. Conclusions Three completely independent approaches taking into account the effect of specular reflection during the dry etching of polysilicon are in good agreement. Parameters influencing the formation of microtrenches have been identified and simulation studies varying theses parameters have been carried out. Further work will deal with extending the Monte Carlo and the flux balancing approach including also the effects of chemistry thus allowing more realistic modeling. However, it is expected that the basic findings, at least qualitatively, remain the same. References
Fig. 5. Simulated influence of the aspect ratio (A) of the mask opening and of the mask sidewall angle on microtrenching characterized as ratio between maximum and minimum etch rate at the very beginning of the etching process. A perfectly directional angular distribution of the ions has been assumed.
[1] A.P. Mahorowala, H.H. Sawin, J. Vac. Sci. Technol. 20 (2002) 1064. [2] E. Ba¨r, J. Lorenz, H. Ryssel, Proc. SISPAD 2004, p. 339. [3] Synopsys TCAD Software, Release 2007.03, Synopsys, Montain View, CA, 2007. [4] R.J. Hoekstra, M.J. Kushner, V. Sukharev, P. Schoenborn, J. Vac. Sci. Technol. B 16 (1998) 2102.
Microelectronic Engineering 85 (2008) 992–995 www.elsevier.com/locate/mee
Comparison of different methods for simulating the effect of specular ion reflection on microtrenching during dry etching of polysilicon D. Kunder *, E. Ba¨r Fraunhofer Institute of Integrated Systems and Device Technology, Schottkystrasse 10, 91058 Erlangen, Germany Received 5 October 2007; received in revised form 11 January 2008; accepted 15 January 2008 Available online 26 January 2008
Abstract For the simulation of etching processes, a key step is the calculation of the etch rates depending on the specific model and depending on the specific geometry of the feature. In this work, we demonstrate the calculation of etch rates using a Monte Carlo, a flux balancing, and an analytical approach. For a relatively simple model for etching of polysilicon in chlorine-based chemistry, the three approaches are compared and microtrenching is studied which results from the specular reflection of ions and depends on different parameters. The results for the different approaches are in good agreement for the cases studied. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Simulation; Etching processes; Microtrenching; Monte Carlo method
Etching is a key process step determining the geometry of relevant feature patterns such as gate electrodes. Etching models, including treatment of specular reflections of ions, have been presented in the literature for the 2D case, e.g. using a Monte Carlo method [1]. 3D implementations (e.g. [2]) published so far do not consider the effect of specular reflection. The flux balancing approach used in this work has not yet been demonstrated at all for modeling of specular reflection. In this paper we present different 3D simulation approaches that take into account specular ion reflection leading to microtrenching. We focus on comparing the different independent approaches rather than on using a full chemistry model of the etching process.
performed in three phases: First, the structure is discretized by a surface triangulation generated by a combination of inhouse structure generation tools with the meshing tools of the Synopsys TCAD software suite [3]. The structures studied in this work all have a cylindrical mask opening. An example for such a structure after the simulation of etching is shown in Fig. 1. In the second phase, the etch rates for all triangles are calculated. In the third phase, the surface is shifted depending on the etch rates. Phase two and three are repeated until the desired etch depth is obtained. The discretization errors inherent in this procedure are due to the finite size of the triangles used for the surface discretization and due to the finite number of time steps employed for updating the etch rates during the simulation run.
2. Simulation
2.2. Calculation of the etch rates
2.1. General
We use a relatively simple model for polysilicon etching in a chlorine plasma where the local etch rate is determined by the local ion flux. It is assumed that the ion flux above the substrate is constant, i.e. it does not depend on the lateral position. Furthermore, a perfectly directional ion flux is assumed which means that the ion velocity has only a
1. Introduction
The simulation based on the Monte Carlo approach and the simulation based on the flux balancing approach are *
Corresponding author. Tel.: +49 9131 761 223. E-mail address: [email protected] (D. Kunder).
0167-9317/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.mee.2008.01.038
D. Kunder, E. Ba¨r / Microelectronic Engineering 85 (2008) 992–995
993
Fig. 1. Etched structure (left: 3D view, right: cross sectional view). The etch rates were calculated by the Monte Carlo approach.
component perpendicular to the substrate surface. Regarding the latter assumption, however, for the comparison of the different methods shown later on, an isotropic velocity distribution of the ions is assumed. The reason for this assumption is discussed in Section 3. For the interaction of the ions with the surface, it is assumed that an ion is reflected specularly if its angle of incidence is above a given limit (here called alim). For typical etching conditions, typical values of alim are around 80° [4]. Ions not being reflected lead to etching. It is assumed that each sticking ion contributes the same amount to the etching rate, independent of its angle and energy. 2.2.1. Monte Carlo approach The Monte Carlo method is a probabilistic approach in which many particle trajectories (for the results shown in this work 108) are generated to simulate the behavior of the real system. Since the specular reflection and the etching process are taken to be deterministic, the only variation in the particle trajectories is due to their different starting positions. For the simulation, the particles are launched from above the feature and their trajectories are determined based on traveling in the ballistic regime, i.e. scattering can only occur when interacting with the feature surface. The trajectory of a particle ends if it sticks at the surface thus contributing to etching, or escapes from the simulation region. The simulation time for the Monte Carlo approach is much higher (by a factor of 10–100) than the time needed by the flux balancing approach. However, especially regarding the implementation of more complex models, the Monte Carlo approach offers higher flexibility and a more straightforward implementation of etching mechanisms. 2.2.2. Flux balancing approach For the flux balancing approach, a discretization is used which considers the flux to each triangle arriving from discrete spatial directions. The discretization of the spatial directions is based on spherical coordinates, i.e. using the angles # and u. For both angles, a discrete number of intervals are used. The discretized flux Ui;n#; nu is then introduced which represents the flux for triangle i for the spatial direc-
tion represented by the integer numbers n# and nu (see Fig. 2 for illustration). In this context, Ui;n#; nu is defined as the flux to triangle i per solid angle interval, i.e. the number of particles arriving at triangle i from a specific direction is given by Ui;n#; nu times the area of the triangle times the solid angle interval considered for particle delivery. For the interaction of the flux with the surface the reflection condition is employed in the same way as in the Monte Carlo approach, i.e. the particle is reflected if the angle of incidence is larger than the limiting angle for reflection alim. Flux balancing then leads to a system of equations in the unknowns Ui;n#; nu : Ui;n# ;nu ¼ G cosð#loc ðn# ; nu ÞÞ f ð#glob ðn# ; nu Þ; uglob ðn# ; nu ÞÞ X vij Ui;m# ;mu T ij ðn# ; nu Þ ð1Þ þ j; j6¼i
In (1) the first summand gives the direct flux arriving from the reactor at the triangle under consideration which depends on the angles in the reactor related system (subscript glob) for the direction under consideration which appear as argument of the angular distribution function f. The angular distribution function f also includes the effect of flux shadowing due to the feature geometry. The factor G is used for normalization to yield the 1D etching rates (as specified as input to the simulator) in regions far from any 2D or 3D structure. The cosine term of the first summand represents the flux reduction for non-normal incidence and therefore depends on the local angles (subscript loc). The second summand is the sum of contributions of particles reflected from
Discretization of spatial directions (nϑ ,nϕ)
Triangle i
Fig. 2. Discretization used for the flux balancing approach. Refer to the text for further explanation.
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D. Kunder, E. Ba¨r / Microelectronic Engineering 85 (2008) 992–995
other triangles and arriving at triangle i. vij is the view factor which is unity if there is direct view between triangle i and triangle j and zero otherwise. The matrix elements Tij are calculated for each pair of triangles based on geometrical considerations. In particular, for their calculation the reflection condition with alim as mentioned above is taken into account. (1) has to be written for each triangle i and for each value of the integers n# and nu representing the spatial directions. Therefore, the number of variables (which is equal to the number of equations in the system) is the number of triangles for the surface discretization times the number of discrete # directions times the number of discrete u directions and is in the order of magnitude of 107. To obtain the solution of the system, an iterative solver is used. From the obtained solution the etch rate for each triangle is determined by summing up the flux contributions from directions where the ion is not reflected, i.e. the directions with an angle of incidence smaller than alim. For the flux balancing approach, discretization errors are due to the finite triangle size and due to the finite number of directions used for the discretization of the spatial directions. 2.2.3. Analytical calculation The analytical calculations are based on integration of fluxes for different positions on the feature surface. Depending on the ion angular distribution two different calculations have been carried out. In both cases, the rates are determined for the structure at the very beginning of the etching process, i.e. changes to the structure due to feature evolution are not taken into account. The calculation of the etch rates for a directional ion angular distribution is performed by summing up the particles arriving directly from the reactor and the particles arriving due to reflection from the mask. To this end (as shown in Fig. 3) the feature is discretized by rings. For example, the etch rate for the substrate ring 1 results from particles arriving directly from the reactor and particles reflected by the mask ring 1. More generally and for infinitesimally thin rings i, the rate for ring ri is given as Am;i ri ¼ const 1 þ F ð2Þ As;i
where Am,i and As,i give the areas of the substrate ring i and of the mask ring i, respectively, and F is the direct flux to the substrate. For an isotropic angular distribution, the flux for a location at the substrate is calculated by integration using angles in spherical coordinates. The integration limits depend on the aspect ratio of the structure, the shortest distance of the point to the mask, and alim. For the example shown below, a sidewall angle of 90° was assumed. Part of the integration has to be carried out numerically. For more complex etching models this approach cannot be used anymore. 3. Results First, we compare the three approaches for a test case assuming an isotropic angular distribution of the ions. Though an isotropic distribution of the ions is not realistic, for comparing different methods, assuming such a distribution is advantageous, as the etch rate shows variations for the full range of positions between the feature center and the mask edge, whereas for directional distributions only the vicinity of the mask edge exhibits etch rate variations. Therefore, we have chosen this hypothetical case for comparing the different methods (i.e. Monte Carlo, flux balancing, and analytical calculations) as shown in Fig. 4. It can be seen that there is good agreement between the three fully independent approaches when predicting the etch rate along an etched feature. The differences can be explained by the finite size of the surface triangles and, in case of the flux balancing approach, by discretization of the spatial directions. The escape of the curves for the flux balancing and Monte Carlo method at the very right of the figure is due to the triangle nodes at the mask edge which are kept fixed during the shift of the surface. Next, the effect of specular ion reflection on microtrenching is studied. For the simulation shown in Fig. 1, a perfectly directional distribution of incoming ions has been used. The sidewall angle of the mask was 86°. In the cross section, the microtrenching near the mask edge is clearly visible. The positions where the reflected ions etch the substrate have an etch rate which is at least twice as large as it is for positions at the substrate not reached by
Fig. 3. Discretization used for the analytical approach and a directional ion angular distribution; side view (left) and top view (right).
D. Kunder, E. Ba¨r / Microelectronic Engineering 85 (2008) 992–995
Fig. 4. Shift of the substrate after one etch step for the three different approaches. The angular distribution of ions is isotropic. Furthermore, perfectly steep sidewalls and an aspect ratio of 1 have been assumed.
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reflected ions. However, more realistic etch models here would lead to a lower value for this ratio. In this simple model the only parameters influencing the degree of microtrenching are the aspect ratio of the mask opening and the mask sidewall angle (for a given angular distribution of ions). In Fig. 5, the Monte Carlo approach and the analytical calculations have been used to study this dependence. The ratios shown in Fig. 5 correspond to the values at the very beginning of the etching process. Included are the results of the Monte Carlo approach as well as the results of the analytical calculations. The difference observed between the Monte Carlo approach and the analytical one is due to the finite triangle size which leads to an averaging of the etching rates over a lateral range corresponding to the spatial extension of the triangle. 4. Conclusions Three completely independent approaches taking into account the effect of specular reflection during the dry etching of polysilicon are in good agreement. Parameters influencing the formation of microtrenches have been identified and simulation studies varying theses parameters have been carried out. Further work will deal with extending the Monte Carlo and the flux balancing approach including also the effects of chemistry thus allowing more realistic modeling. However, it is expected that the basic findings, at least qualitatively, remain the same. References
Fig. 5. Simulated influence of the aspect ratio (A) of the mask opening and of the mask sidewall angle on microtrenching characterized as ratio between maximum and minimum etch rate at the very beginning of the etching process. A perfectly directional angular distribution of the ions has been assumed.
[1] A.P. Mahorowala, H.H. Sawin, J. Vac. Sci. Technol. 20 (2002) 1064. [2] E. Ba¨r, J. Lorenz, H. Ryssel, Proc. SISPAD 2004, p. 339. [3] Synopsys TCAD Software, Release 2007.03, Synopsys, Montain View, CA, 2007. [4] R.J. Hoekstra, M.J. Kushner, V. Sukharev, P. Schoenborn, J. Vac. Sci. Technol. B 16 (1998) 2102.