- Email: [email protected]
Modeling secondary electron images for linewidth measurement by critical dimension scanning electron microscopy
Microelectronics Reliability 50 (2010) 1407–1412
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Modeling secondary electron images for linewidth measurement by critical dimension scanning electron microscopy Mauro Ciappa a,*, Alexander Koschik a, Maurizio Dapor b, Wolfgang Fichtner a a b
Integrated Systems Laboratory, Swiss Federal Institute of Technology (ETH), ETH-Zentrum, CH-8092 Zurich, Switzerland Center for Materials and Microsystems, FBK-IRST, Trento, Italy
a r t i c l e
i n f o
Article history: Received 6 July 2010 Accepted 19 July 2010 Available online 19 August 2010
a b s t r a c t Modeling of critical dimensions scanning electron microscopy with sub-nanometer uncertainty is required to provide a metrics and to avoid yield loss in the processing of advanced CMOS technologies. In this paper, a new approach is proposed, which includes a new Monte Carlo scheme, a new Monte Carlo code, as well as the coupling with electrostatic fields to take into account self-charging effects. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction With the most advanced CMOS technologies, critical dimensions (CD) measurements with sub-nanometer uncertainty must be performed during the manufacturing process to provide a metrics and to avoid yield loss. As an example, this is the case of the linewidth measurement of photoresist lines (e.g. PMMA) that are largely used in optical and electron beam lithography for device integration down to the sub-16 nm scale. Besides the use of the most advanced equipment, accurate nanometrology requires that the physics of image formation in scanning electron microscopy (SEM) is modeled to extract the relevant quantitative information. Modern CD-SEM mainly operate at very low primary energies (down to 200 eV). This implies the use of quite complicated physical models defined by a variety of parameters that can also become time-dependent due to unwanted effects as electrostatic charging of the sample. Therefore, up to date Monte Carlo (MC) simulation of the generation and transport of secondary electrons (SE) in materials is still the most straightforward approach to the solution. A novel MC simulation code is proposed that is based on the energy straggling principle. Original physical models have been developed for the interactions. Depending on the sample material, generation and transport of SE are calculated according to consistent models for elastic, electron–electron, plasmon, and phonon scattering, as well as for polaron trapping. Special attention is paid to the dielectrics, whose models have been recently revised and accurately recalibrated for very low primary electron (PE) energies [1]. All models are based on well-defined physical measurables to facilitate the generation of complex 3D multilayered structures. The MC scheme has been mutated from a previous work [1] and
* Corresponding author. Tel.: +41 446322436; fax: +41 446321194. E-mail address: [email protected] (M. Ciappa). 0026-2714/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2010.07.120
takes into account quantum–mechanical barrier scattering at the different material interfaces. Thanks to its recursive structure, the scheme easily keeps track of the hundreds of SE that are generated per PE in a typical shower. Section 2 of this paper will introduce the physical models implemented in the developed code and will focus on the structure of the developed code. Finally, Section 3 will present four applications of the novel code to cases from the literature for comparison purposes.
2. Physical models and Monte Carlo scheme 2.1. Physical models In the developed MC code, the interactions of the PE and of the related SE showers are simulated down to electron energies of 0.1 eV. The electron tracking and the physics sections have been completely decoupled. The 3D sample geometry definition, single electron tracking, and quantification of the emitted SE are carried out by the standard MC code PENELOPE [2]. In converse, the scattering physics has been implemented by original, separated library functions for minimum interference of both code parts. Besides the parallelizing capabilities of the code, particular attention has been paid to the implementation of a fully discrete interaction scheme based on the straggling principle, as well as of efficient tracking and ray tracing functions of SE in 3D geometries with arbitrary shape and in the presence of electromagnetic fields. The transport equations of the electron in the materials and in the vacuum are solved under consideration of the local electrostatic field to take into consideration a possible deflection of the primary beam and SE as well as the effects of the induced accumulation/inversion regions in the semiconductor substrate. At present, a static charge distribution in the dielectric has been assumed. The global Poisson equation is solved for the given geometry and boundary conditions
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(e.g. extraction field) by the external tool SENTAURUS DEVICE [3] that takes into account a rigorous approach for semiconductor materials. The MC code provides the charge distribution for the SENTAURUS DEVICE simulator, which returns to the MC code the local electric field for the calculation of the electron trajectories. In the case of a calculation under consideration of the electrostatics, the interface/surface potentials as provided by the internal physical models are switched off and replaced by the local potential delivered by the device simulator, in order to obtain a systematic treatment. The proposed code includes physical models for the main scattering mechanisms governing the transport of electrons in silicon [1], silicon dioxide, and PMMA in an energy domain ranging from the energy of the primary electron beam down to 0.1 eV. Special attention has been paid to the physical models for dielectric materials (PMMA, SiO2) that are original and refer to the results presented in a recent publication [1]. Scattering models for dielectrics take into account elastic scattering, electron–electron interactions, scattering with phonons, and trapping by polarons. The models for the inelastic mean free path and the SE yield as a function the polaron trapping are shown in Figs. 1 and 2, respectively. The proposed code also includes quantum–mechanical modeling of barrier scattering occurring either at the material interfaces, or at the sample surface [1]. This option is disabled, in the case of simulations taking into account the electrostatic field, since in this case, the local potential is directly calculated by an auxiliary device simulator. All these models are used in conjunction with a dedicated Monte Carlo simulation scheme, which is also described in detail in [1] and that accounts for the entire cascade of secondary electrons.
2.2. Simulation model and code Typical structures of interest are dielectric lines on silicon substrates with trapezoidal cross-section, while the critical dimensions under investigation are the bottom and/or top line width, as well as the slope of the rising and falling edges. Linescans are generated by scanning the electron beam with circular cross-section perpendicularly to the dielectric lines. For the sake of simplicity, the emitted secondary electrons are detected over a 2p solid angle and if not specified, no extraction field
Fig. 1. Inelastic mean free path (IMFP) of electrons in PMMA. Solid line represents the present calculation, obtained utilizing the Ashley approach for the extension of the energy loss function out from the optical domain. Dashed line describes the original Ashley results. The differences in the two calculations are due to the different optical energy loss functions utilized [1].
Fig. 2. MC simulation of secondary electron yield of PMMA as a function of the primary energy E0, for a normally incident beam. MC data are compared to experimental data coming from different laboratories [1].
is applied. A typical linescan is carried out in a few hundred discrete steps. Each burst includes 103 up to 104 primary mono-energetic electrons. Although it can be defined arbitrarily, the shape of the beam spot is usually assumed as circular with a typical diameter of 2–3 nm (FWHM). This is also the case of the current density within the beam spot, which is usually assumed as a cylinder-symmetrical Gaussian distribution centered in the beam axis. Due to multiple scattering effects in different material the SE signal does not only depend on the local material properties and topography at the landing spot, but also on the properties in near proximity. In this respect, the fact that the surface topology is composed by insulating materials (e.g. SiO2, PMMA) plays an important role. This can lead to negative or positive surface charging depending on the SE yield. The resulting electric fields can cause changes in the SE image formation as a result of the deflection of the primary beam and of the emitted secondary electrons. The standard Monte Carlo code PENELOPE [2] has been assumed as a basis for our own developments, among the choice of potential other codes. It provides benefits in terms of 3D geometry management, ray tracing capabilities in electromagnetic fields, tracking and scoring features and general simulation bookkeeping that is readily available in the code. The original physics of PENELOPE has been replaced completely by the original physics module described before with the scope to implement an energy straggling simulation scheme working down to electron energies of 0.1 eV. A block diagram of the proposed MC simulation tool is shown in Fig. 3. The main program is the PENMAIN code of PENELOPE, which has been adapted to allow calling the subroutines of our physics library package instead of the PENELOPE intrinsic physics subroutines. Care has been taken to very loosely couple our physics routines to the main program. In essence, the two parts only communicate via subroutine calls to the getStepLength(currentMaterial) and doScatter(currentMaterial) functions of our physics library, which completely contain the necessary computation of the mean free path and the energy and momentum changes due to an interaction event. The latter values are returned to the PENELOPE domain for subsequent tracking operation. A further supplement to PENMAIN is a loop, which changes the beam position step by step. Part of this loop is also writing the SE yield data to the output file. Throughout this study either pencil-like beams or beams with Gaussian profile are used, which can additionally have a cone
M. Ciappa et al. / Microelectronics Reliability 50 (2010) 1407–1412
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Fig. 3. Block structure of the developed MC code. The physics and the MC process are completely encapsulated in library functions and loosely coupled to Penelope, which is mainly used for tracking and scoring purposes. The self-consistency can be achieved by an additional feedback loop that includes the re-calculation and the update of the electric field.
shape narrowing towards the surface. The detector has been assumed as hemispherical. Input parameters like the scanning beam parameters as well as general job parameters are defined in a text input file, the geometry is defined in a separate text input file, both in PENELOPE format. Those files are generated from a Python script. This Python script includes also the logic to parallelize the simulation task and run multiple jobs in parallel on the available load sharing facility (Condor). Once the simulation environment has been read and set up by the main program, the electrons undergo several steps in the main loop. These are in principle four main computational tasks. 1. LOCATE the particle in the geometry and retrieve the corresponding material, 2. GETMEANFREEPATH in the current material, 3. STEP by a random fraction of mean free path, 4. INTERACTION process, which can be either elastic, inelastic, phonon excitation, polaron. An interaction process can lead to the generation of a secondary electron. In this case the particle is put into a LIFO stack. Particles trajectories end, when they either reach the boundary of the simulation model or their energy has reached the lower limit of E = 0.1 eV, whereupon they are supposed to get absorbed and the last-in electron from the stack will be simulated next. Along the trajectory, a boundary in the geometry might be crossed. Depending on whether or not the material changes, barrier scattering does occur, leading to either reflection or transmission of the particle. The barrier scatter algorithm is implemented according to the principles outlined in [1]. Both classical and quantum mechanical bar-
rier scattering have been implemented. Particles leaving the simulation model in upward direction are counted as backscattered (E > 50 eV) or secondary electrons (E < 50 eV) depending on their energy. For secondary electrons leaving the surface, a simple, yet effective scoring algorithm has been implemented. Once particles emerge from the surface their phase space coordinates are kept in memory. If this very particle stays in vacuum and finally leaves the simulation model, its coordinates at the surface have been recorded. However, if the particle undergoes another barrier scatter event and therefore enters the sample again, its trace is eliminated from the memory. This procedure delivers the detected secondary electron distribution at surface level for complicated geometries and even in the presence of EM fields leading to bent trajectories. In a first phase, specimen charging has been modeled by assuming steady-state charging conditions corresponding to a sample pre-conditioning phase. The related 3D Poisson equation is solved for the given geometry and boundary conditions in the different materials and in the vacuum by the external tool SENTAURUS DEVICE [3]. This approach enables to take accurately into account all charges induced in the different materials (e.g. semiconductors and metals) and to calculate the related electrostatic potentials. In the final stage, the SENTAURUS DEVICE simulator returns to the MC code the local, time-invariant electrostatic field and potential for the calculation of the electron trajectories. In a future configuration of the code, dynamic charge up of the sample can be simulated by an appropriate self-consistent simulation scheme, also including transport and generation/recombination models in the different materials. In order to be computationally efficient in the ray-tracing part of the code, a tensor grid has been used for the electric field mesh. Like this the retrieval of field values for a given position
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in Cartesian co-ordinates can be efficiently implemented using indexed look-up. 2.3. Computing time The proposed MC simulation scheme works according to a rigorous energy straggling principle, i.e. only discrete events are considered, i.e. no continuous slowing down approximation (CSDA) steps are involved. All particles are followed down to 0.1 eV, when they are considered as absorbed. Energy is lost only in discrete interaction processes. Nodes in the computer cluster used for this study offer CPU clock cycles in the order of 2.5–2.9 GHz. With these machines a typical computation time of 2–7 ms per started primary electron is reached. Per incidence point a set of usually 104 primary electrons is started to get sufficiently smooth linescans, hence computation time for one point is typically in the order of 20 up to 70 s, depending on material, geometry, and energy of the primary electron. Simulating a linescan of 150–200 points on a single CPU accordingly requires in the best case a computation time of just less than one hour. The use of massive parallelization with typically 100 CPU nodes can simultaneously reduce the computing time down to less than one minute. In this case, additional time is needed for the more complex post-processing. 3. Simulation results The used physical models have been validated with experimental data of SE (and BSE) yields from flat surfaces of bulk materials. Additionally, the energy spectrum of the emitted electrons has been compared with experimental data available. Both the SE yield and the energy spectrum have been shown to fit well with experimentally obtained data [1]. In this section, linescans obtained for 3D geometries by the proposed simulation code are compared in the following with results from other SEM image simulation codes available in the literature. In addition, the capabilities of the proposed code are shown based on several realistic examples. 3.1. Step structures Step structures have been simulated to investigate the SE yield response of the proposed code. Fig. 4a and b show the situation for a step in silicon, only. The linescan signal in Fig. 4a shows the expected quantitative behavior. On the flat surface far away from the step, the SE yield corresponds to that of normal incidence. Approaching the step from the left side, trajectories of emerging SEs are intercepted by the topography step and an increasing shadowing effect is observed. As expected, the higher the step and/or the steeper the side wall angle, the more shadowing occurs. At the step itself the SE signal shows a discontinuity. The signal minimum is at the bottom edge position where emerging SEs experience maximum geometric shadowing. The signal maximum is at the position where the generation and escape probability of the secondary electrons reach their maximum. For a pencil beam, this condition is reached close to the top edge. For larger side wall angles, an additional intermediate level within the transition is observed. The distribution of the emerging electrons is calculated by the scoring algorithm for SE and is plotted in Fig. 4b in conjunction with the local value of the SE yield. For a flat silicon sample at 0.5 keV the 3r interval of the lateral distribution of emerging SEs is 10 nm. When the beam approaches the step, two effects are observed. First, the angular distribution shows a clear asymmetry because of the SEs, which are absorbed at the edge. Additionally, the
Fig. 4. Linescan from a 50 nm silicon step with 10° side wall angle scanned with a pencil beam at 0.5 keV. (a) Simulated linescan and (b) lateral and angular distributions of the emerging secondary electrons are shown at the edge.
angular distribution evidences some SEs generated and/or emerging from the side wall region. In the case of a silicon step this component is still small compared with the geometric shadowing effect. If the beam impinges on the sidewall, the effect of the angle of incidence can be clearly observed. The angular distribution becomes very asymmetric, since the SEs emerge with a distribution centered around the normal to the sidewall surface. The lateral distribution also shows a strong asymmetry, since no SE is emitted in the direction of the increasing slope. 3.2. Comparison with JMONSEL Step structures have also been used to benchmark the results obtained by the proposed simulation code against published results from the established code JMONSEL [4]. The linescans have been scaled to one for the flat surface level of silicon for quantitative comparison. In Fig. 5a both codes are directly compared for a silicon step structure. The linescan obtained by the proposed code is in good agreement with JMONSEL even if the latter uses different physical models [4]. This also demonstrates the proper treatment of 3D geometry and boundary crossing in the present approach. The observed discrepancies can mainly be explained by the differences in the physical models used for the secondary electron yield in the different materials. In Fig. 5b, adjacent lines with trapezoidal cross-section have been used to assess the performance of the proposed code against JMONSEL in the case of a structure with three SiO2 lines on a silicon
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(a)
JMONSEL Bin. JLN GBS
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Fig. 6. Linescans at the side wall of a Si trenches with different aspect ratio. Beam conditions have been set to 0.8 keV and 2.0 nm (FWHM) beam width. The results are compared to the linescans in [5].
1.5
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substrate. Again the qualitative agreement is very good. No scaling is applied in this case and absolute signals are shown. The main difference observed here is an increased geometry shadowing effect in JMONSEL. This is due to a different angular distribution of the emerging electrons due to different barrier scattering models. Furthermore, the apparent increase of the SE yield from the Si area between the lines is mainly caused by multiple scattering of the SE emerging from Si with the SiO2 walls that exhibit a much higher SE yield.
Fig. 7. Simulation of three adjacent PMMA lines with and without charging effects simulated with a 2p detector.
respect to the outer flat surface. The comparison in Fig. 6 shows that both simulation codes deliver results, which are in excellent agreement.
3.3. Side-wall signal in trench geometries
3.4. Self-charging effects
The SE emission yield from the side walls of trench geometries has been simulated and compared with the data published in [5]. As shown in the latter paper, the signal at the side wall can vary depending on the aspect ratio of the trench. Here, the same geometry as in [5] has been assumed for a quantitative comparison. The depth of the silicon trench is 288 nm with a side wall angle of four degrees. The bottom width has been assumed to be 80 nm and 300 nm, respectively. A 2.0 nm (FHMW) Gaussian profile has been taken for the primary beam at 0.8 keV. For the sake of comparison, the obtained and the literature linescans have been normalized in
Time-invariant electrostatic fields have been assumed in the following, which have been calculated by the commercial tool SENTAURUS DEVICE [0]. Fig. 7 shows a comparison of the simulation of a primary beam impinging onto three charged PMMA lines (50 nm in height) on a silicon substrate. It is assumed that positive charges are trapped in the dielectrics according to a Gaussian distribution along the z-axis (5 1019 cm 3, r = 7 nm). The trajectories of the SE inside the materials and after emerging from the surface are deflected by the local electrostatic field. This impacts especially the low-en-
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ergy tail of the SE and results in a different SE yield as in the neutral materials. In Fig. 7, this effect is less evident due to the fact that a 2p detector has been used in the simulation. 4. Summary and conclusions A novel MC simulator tool for the quantification of CD-SEM images has been proposed. Particular emphasis has been put on scattering models for the low-energy range. The simulator has been compared to other existing codes and its features and capabilities have been assessed in examples. Based on the proposed code an application using a model-based library strategy has been demonstrated. In conclusion, the proposed simulator has been shown to provide results that are in good agreement with published data.
References [1] Dapor M, Ciappa M, Fichtner W. Monte Carlo modeling in the low-energy domain of the secondary electron emission of PMMA for critical dimension scanning electron microscopy. J Micro/Nanolith MEMS MOEMS 2010;9(2): 023001. [2] Salvat F, Fernández-Varea JM, Sempau J. PENELOPE-2008: a code system for Monte Carlo simulation of electron and photon transport; NEA-6416. OECD Publishing; 2009. [3] Synopsys, sentaurus manual B-2009.06, Mountain View; 2009. [4] Villarrubia JS, Ritchie N, Lowney JR. Monte Carlo modeling of secondary electron imaging in three dimensions. Proc SPIE 2007;6518 (JMONSEL). [5] Abe H, Hamaguchi A, Yamazaki Y. Evaluation of CD-SEM measurement uncertainty using secondary electron simulation with charging effect. Proc SPIE 2007;6518:65180L.
Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
Modeling secondary electron images for linewidth measurement by critical dimension scanning electron microscopy Mauro Ciappa a,*, Alexander Koschik a, Maurizio Dapor b, Wolfgang Fichtner a a b
Integrated Systems Laboratory, Swiss Federal Institute of Technology (ETH), ETH-Zentrum, CH-8092 Zurich, Switzerland Center for Materials and Microsystems, FBK-IRST, Trento, Italy
a r t i c l e
i n f o
Article history: Received 6 July 2010 Accepted 19 July 2010 Available online 19 August 2010
a b s t r a c t Modeling of critical dimensions scanning electron microscopy with sub-nanometer uncertainty is required to provide a metrics and to avoid yield loss in the processing of advanced CMOS technologies. In this paper, a new approach is proposed, which includes a new Monte Carlo scheme, a new Monte Carlo code, as well as the coupling with electrostatic fields to take into account self-charging effects. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction With the most advanced CMOS technologies, critical dimensions (CD) measurements with sub-nanometer uncertainty must be performed during the manufacturing process to provide a metrics and to avoid yield loss. As an example, this is the case of the linewidth measurement of photoresist lines (e.g. PMMA) that are largely used in optical and electron beam lithography for device integration down to the sub-16 nm scale. Besides the use of the most advanced equipment, accurate nanometrology requires that the physics of image formation in scanning electron microscopy (SEM) is modeled to extract the relevant quantitative information. Modern CD-SEM mainly operate at very low primary energies (down to 200 eV). This implies the use of quite complicated physical models defined by a variety of parameters that can also become time-dependent due to unwanted effects as electrostatic charging of the sample. Therefore, up to date Monte Carlo (MC) simulation of the generation and transport of secondary electrons (SE) in materials is still the most straightforward approach to the solution. A novel MC simulation code is proposed that is based on the energy straggling principle. Original physical models have been developed for the interactions. Depending on the sample material, generation and transport of SE are calculated according to consistent models for elastic, electron–electron, plasmon, and phonon scattering, as well as for polaron trapping. Special attention is paid to the dielectrics, whose models have been recently revised and accurately recalibrated for very low primary electron (PE) energies [1]. All models are based on well-defined physical measurables to facilitate the generation of complex 3D multilayered structures. The MC scheme has been mutated from a previous work [1] and
* Corresponding author. Tel.: +41 446322436; fax: +41 446321194. E-mail address: [email protected] (M. Ciappa). 0026-2714/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.microrel.2010.07.120
takes into account quantum–mechanical barrier scattering at the different material interfaces. Thanks to its recursive structure, the scheme easily keeps track of the hundreds of SE that are generated per PE in a typical shower. Section 2 of this paper will introduce the physical models implemented in the developed code and will focus on the structure of the developed code. Finally, Section 3 will present four applications of the novel code to cases from the literature for comparison purposes.
2. Physical models and Monte Carlo scheme 2.1. Physical models In the developed MC code, the interactions of the PE and of the related SE showers are simulated down to electron energies of 0.1 eV. The electron tracking and the physics sections have been completely decoupled. The 3D sample geometry definition, single electron tracking, and quantification of the emitted SE are carried out by the standard MC code PENELOPE [2]. In converse, the scattering physics has been implemented by original, separated library functions for minimum interference of both code parts. Besides the parallelizing capabilities of the code, particular attention has been paid to the implementation of a fully discrete interaction scheme based on the straggling principle, as well as of efficient tracking and ray tracing functions of SE in 3D geometries with arbitrary shape and in the presence of electromagnetic fields. The transport equations of the electron in the materials and in the vacuum are solved under consideration of the local electrostatic field to take into consideration a possible deflection of the primary beam and SE as well as the effects of the induced accumulation/inversion regions in the semiconductor substrate. At present, a static charge distribution in the dielectric has been assumed. The global Poisson equation is solved for the given geometry and boundary conditions
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(e.g. extraction field) by the external tool SENTAURUS DEVICE [3] that takes into account a rigorous approach for semiconductor materials. The MC code provides the charge distribution for the SENTAURUS DEVICE simulator, which returns to the MC code the local electric field for the calculation of the electron trajectories. In the case of a calculation under consideration of the electrostatics, the interface/surface potentials as provided by the internal physical models are switched off and replaced by the local potential delivered by the device simulator, in order to obtain a systematic treatment. The proposed code includes physical models for the main scattering mechanisms governing the transport of electrons in silicon [1], silicon dioxide, and PMMA in an energy domain ranging from the energy of the primary electron beam down to 0.1 eV. Special attention has been paid to the physical models for dielectric materials (PMMA, SiO2) that are original and refer to the results presented in a recent publication [1]. Scattering models for dielectrics take into account elastic scattering, electron–electron interactions, scattering with phonons, and trapping by polarons. The models for the inelastic mean free path and the SE yield as a function the polaron trapping are shown in Figs. 1 and 2, respectively. The proposed code also includes quantum–mechanical modeling of barrier scattering occurring either at the material interfaces, or at the sample surface [1]. This option is disabled, in the case of simulations taking into account the electrostatic field, since in this case, the local potential is directly calculated by an auxiliary device simulator. All these models are used in conjunction with a dedicated Monte Carlo simulation scheme, which is also described in detail in [1] and that accounts for the entire cascade of secondary electrons.
2.2. Simulation model and code Typical structures of interest are dielectric lines on silicon substrates with trapezoidal cross-section, while the critical dimensions under investigation are the bottom and/or top line width, as well as the slope of the rising and falling edges. Linescans are generated by scanning the electron beam with circular cross-section perpendicularly to the dielectric lines. For the sake of simplicity, the emitted secondary electrons are detected over a 2p solid angle and if not specified, no extraction field
Fig. 1. Inelastic mean free path (IMFP) of electrons in PMMA. Solid line represents the present calculation, obtained utilizing the Ashley approach for the extension of the energy loss function out from the optical domain. Dashed line describes the original Ashley results. The differences in the two calculations are due to the different optical energy loss functions utilized [1].
Fig. 2. MC simulation of secondary electron yield of PMMA as a function of the primary energy E0, for a normally incident beam. MC data are compared to experimental data coming from different laboratories [1].
is applied. A typical linescan is carried out in a few hundred discrete steps. Each burst includes 103 up to 104 primary mono-energetic electrons. Although it can be defined arbitrarily, the shape of the beam spot is usually assumed as circular with a typical diameter of 2–3 nm (FWHM). This is also the case of the current density within the beam spot, which is usually assumed as a cylinder-symmetrical Gaussian distribution centered in the beam axis. Due to multiple scattering effects in different material the SE signal does not only depend on the local material properties and topography at the landing spot, but also on the properties in near proximity. In this respect, the fact that the surface topology is composed by insulating materials (e.g. SiO2, PMMA) plays an important role. This can lead to negative or positive surface charging depending on the SE yield. The resulting electric fields can cause changes in the SE image formation as a result of the deflection of the primary beam and of the emitted secondary electrons. The standard Monte Carlo code PENELOPE [2] has been assumed as a basis for our own developments, among the choice of potential other codes. It provides benefits in terms of 3D geometry management, ray tracing capabilities in electromagnetic fields, tracking and scoring features and general simulation bookkeeping that is readily available in the code. The original physics of PENELOPE has been replaced completely by the original physics module described before with the scope to implement an energy straggling simulation scheme working down to electron energies of 0.1 eV. A block diagram of the proposed MC simulation tool is shown in Fig. 3. The main program is the PENMAIN code of PENELOPE, which has been adapted to allow calling the subroutines of our physics library package instead of the PENELOPE intrinsic physics subroutines. Care has been taken to very loosely couple our physics routines to the main program. In essence, the two parts only communicate via subroutine calls to the getStepLength(currentMaterial) and doScatter(currentMaterial) functions of our physics library, which completely contain the necessary computation of the mean free path and the energy and momentum changes due to an interaction event. The latter values are returned to the PENELOPE domain for subsequent tracking operation. A further supplement to PENMAIN is a loop, which changes the beam position step by step. Part of this loop is also writing the SE yield data to the output file. Throughout this study either pencil-like beams or beams with Gaussian profile are used, which can additionally have a cone
M. Ciappa et al. / Microelectronics Reliability 50 (2010) 1407–1412
1409
Fig. 3. Block structure of the developed MC code. The physics and the MC process are completely encapsulated in library functions and loosely coupled to Penelope, which is mainly used for tracking and scoring purposes. The self-consistency can be achieved by an additional feedback loop that includes the re-calculation and the update of the electric field.
shape narrowing towards the surface. The detector has been assumed as hemispherical. Input parameters like the scanning beam parameters as well as general job parameters are defined in a text input file, the geometry is defined in a separate text input file, both in PENELOPE format. Those files are generated from a Python script. This Python script includes also the logic to parallelize the simulation task and run multiple jobs in parallel on the available load sharing facility (Condor). Once the simulation environment has been read and set up by the main program, the electrons undergo several steps in the main loop. These are in principle four main computational tasks. 1. LOCATE the particle in the geometry and retrieve the corresponding material, 2. GETMEANFREEPATH in the current material, 3. STEP by a random fraction of mean free path, 4. INTERACTION process, which can be either elastic, inelastic, phonon excitation, polaron. An interaction process can lead to the generation of a secondary electron. In this case the particle is put into a LIFO stack. Particles trajectories end, when they either reach the boundary of the simulation model or their energy has reached the lower limit of E = 0.1 eV, whereupon they are supposed to get absorbed and the last-in electron from the stack will be simulated next. Along the trajectory, a boundary in the geometry might be crossed. Depending on whether or not the material changes, barrier scattering does occur, leading to either reflection or transmission of the particle. The barrier scatter algorithm is implemented according to the principles outlined in [1]. Both classical and quantum mechanical bar-
rier scattering have been implemented. Particles leaving the simulation model in upward direction are counted as backscattered (E > 50 eV) or secondary electrons (E < 50 eV) depending on their energy. For secondary electrons leaving the surface, a simple, yet effective scoring algorithm has been implemented. Once particles emerge from the surface their phase space coordinates are kept in memory. If this very particle stays in vacuum and finally leaves the simulation model, its coordinates at the surface have been recorded. However, if the particle undergoes another barrier scatter event and therefore enters the sample again, its trace is eliminated from the memory. This procedure delivers the detected secondary electron distribution at surface level for complicated geometries and even in the presence of EM fields leading to bent trajectories. In a first phase, specimen charging has been modeled by assuming steady-state charging conditions corresponding to a sample pre-conditioning phase. The related 3D Poisson equation is solved for the given geometry and boundary conditions in the different materials and in the vacuum by the external tool SENTAURUS DEVICE [3]. This approach enables to take accurately into account all charges induced in the different materials (e.g. semiconductors and metals) and to calculate the related electrostatic potentials. In the final stage, the SENTAURUS DEVICE simulator returns to the MC code the local, time-invariant electrostatic field and potential for the calculation of the electron trajectories. In a future configuration of the code, dynamic charge up of the sample can be simulated by an appropriate self-consistent simulation scheme, also including transport and generation/recombination models in the different materials. In order to be computationally efficient in the ray-tracing part of the code, a tensor grid has been used for the electric field mesh. Like this the retrieval of field values for a given position
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in Cartesian co-ordinates can be efficiently implemented using indexed look-up. 2.3. Computing time The proposed MC simulation scheme works according to a rigorous energy straggling principle, i.e. only discrete events are considered, i.e. no continuous slowing down approximation (CSDA) steps are involved. All particles are followed down to 0.1 eV, when they are considered as absorbed. Energy is lost only in discrete interaction processes. Nodes in the computer cluster used for this study offer CPU clock cycles in the order of 2.5–2.9 GHz. With these machines a typical computation time of 2–7 ms per started primary electron is reached. Per incidence point a set of usually 104 primary electrons is started to get sufficiently smooth linescans, hence computation time for one point is typically in the order of 20 up to 70 s, depending on material, geometry, and energy of the primary electron. Simulating a linescan of 150–200 points on a single CPU accordingly requires in the best case a computation time of just less than one hour. The use of massive parallelization with typically 100 CPU nodes can simultaneously reduce the computing time down to less than one minute. In this case, additional time is needed for the more complex post-processing. 3. Simulation results The used physical models have been validated with experimental data of SE (and BSE) yields from flat surfaces of bulk materials. Additionally, the energy spectrum of the emitted electrons has been compared with experimental data available. Both the SE yield and the energy spectrum have been shown to fit well with experimentally obtained data [1]. In this section, linescans obtained for 3D geometries by the proposed simulation code are compared in the following with results from other SEM image simulation codes available in the literature. In addition, the capabilities of the proposed code are shown based on several realistic examples. 3.1. Step structures Step structures have been simulated to investigate the SE yield response of the proposed code. Fig. 4a and b show the situation for a step in silicon, only. The linescan signal in Fig. 4a shows the expected quantitative behavior. On the flat surface far away from the step, the SE yield corresponds to that of normal incidence. Approaching the step from the left side, trajectories of emerging SEs are intercepted by the topography step and an increasing shadowing effect is observed. As expected, the higher the step and/or the steeper the side wall angle, the more shadowing occurs. At the step itself the SE signal shows a discontinuity. The signal minimum is at the bottom edge position where emerging SEs experience maximum geometric shadowing. The signal maximum is at the position where the generation and escape probability of the secondary electrons reach their maximum. For a pencil beam, this condition is reached close to the top edge. For larger side wall angles, an additional intermediate level within the transition is observed. The distribution of the emerging electrons is calculated by the scoring algorithm for SE and is plotted in Fig. 4b in conjunction with the local value of the SE yield. For a flat silicon sample at 0.5 keV the 3r interval of the lateral distribution of emerging SEs is 10 nm. When the beam approaches the step, two effects are observed. First, the angular distribution shows a clear asymmetry because of the SEs, which are absorbed at the edge. Additionally, the
Fig. 4. Linescan from a 50 nm silicon step with 10° side wall angle scanned with a pencil beam at 0.5 keV. (a) Simulated linescan and (b) lateral and angular distributions of the emerging secondary electrons are shown at the edge.
angular distribution evidences some SEs generated and/or emerging from the side wall region. In the case of a silicon step this component is still small compared with the geometric shadowing effect. If the beam impinges on the sidewall, the effect of the angle of incidence can be clearly observed. The angular distribution becomes very asymmetric, since the SEs emerge with a distribution centered around the normal to the sidewall surface. The lateral distribution also shows a strong asymmetry, since no SE is emitted in the direction of the increasing slope. 3.2. Comparison with JMONSEL Step structures have also been used to benchmark the results obtained by the proposed simulation code against published results from the established code JMONSEL [4]. The linescans have been scaled to one for the flat surface level of silicon for quantitative comparison. In Fig. 5a both codes are directly compared for a silicon step structure. The linescan obtained by the proposed code is in good agreement with JMONSEL even if the latter uses different physical models [4]. This also demonstrates the proper treatment of 3D geometry and boundary crossing in the present approach. The observed discrepancies can mainly be explained by the differences in the physical models used for the secondary electron yield in the different materials. In Fig. 5b, adjacent lines with trapezoidal cross-section have been used to assess the performance of the proposed code against JMONSEL in the case of a structure with three SiO2 lines on a silicon
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substrate. Again the qualitative agreement is very good. No scaling is applied in this case and absolute signals are shown. The main difference observed here is an increased geometry shadowing effect in JMONSEL. This is due to a different angular distribution of the emerging electrons due to different barrier scattering models. Furthermore, the apparent increase of the SE yield from the Si area between the lines is mainly caused by multiple scattering of the SE emerging from Si with the SiO2 walls that exhibit a much higher SE yield.
Fig. 7. Simulation of three adjacent PMMA lines with and without charging effects simulated with a 2p detector.
respect to the outer flat surface. The comparison in Fig. 6 shows that both simulation codes deliver results, which are in excellent agreement.
3.3. Side-wall signal in trench geometries
3.4. Self-charging effects
The SE emission yield from the side walls of trench geometries has been simulated and compared with the data published in [5]. As shown in the latter paper, the signal at the side wall can vary depending on the aspect ratio of the trench. Here, the same geometry as in [5] has been assumed for a quantitative comparison. The depth of the silicon trench is 288 nm with a side wall angle of four degrees. The bottom width has been assumed to be 80 nm and 300 nm, respectively. A 2.0 nm (FHMW) Gaussian profile has been taken for the primary beam at 0.8 keV. For the sake of comparison, the obtained and the literature linescans have been normalized in
Time-invariant electrostatic fields have been assumed in the following, which have been calculated by the commercial tool SENTAURUS DEVICE [0]. Fig. 7 shows a comparison of the simulation of a primary beam impinging onto three charged PMMA lines (50 nm in height) on a silicon substrate. It is assumed that positive charges are trapped in the dielectrics according to a Gaussian distribution along the z-axis (5 1019 cm 3, r = 7 nm). The trajectories of the SE inside the materials and after emerging from the surface are deflected by the local electrostatic field. This impacts especially the low-en-
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ergy tail of the SE and results in a different SE yield as in the neutral materials. In Fig. 7, this effect is less evident due to the fact that a 2p detector has been used in the simulation. 4. Summary and conclusions A novel MC simulator tool for the quantification of CD-SEM images has been proposed. Particular emphasis has been put on scattering models for the low-energy range. The simulator has been compared to other existing codes and its features and capabilities have been assessed in examples. Based on the proposed code an application using a model-based library strategy has been demonstrated. In conclusion, the proposed simulator has been shown to provide results that are in good agreement with published data.
References [1] Dapor M, Ciappa M, Fichtner W. Monte Carlo modeling in the low-energy domain of the secondary electron emission of PMMA for critical dimension scanning electron microscopy. J Micro/Nanolith MEMS MOEMS 2010;9(2): 023001. [2] Salvat F, Fernández-Varea JM, Sempau J. PENELOPE-2008: a code system for Monte Carlo simulation of electron and photon transport; NEA-6416. OECD Publishing; 2009. [3] Synopsys, sentaurus manual B-2009.06, Mountain View; 2009. [4] Villarrubia JS, Ritchie N, Lowney JR. Monte Carlo modeling of secondary electron imaging in three dimensions. Proc SPIE 2007;6518 (JMONSEL). [5] Abe H, Hamaguchi A, Yamazaki Y. Evaluation of CD-SEM measurement uncertainty using secondary electron simulation with charging effect. Proc SPIE 2007;6518:65180L.