Comprehensive model of electron energy deposition*

Transcription

1 Comprehensive model of electron energy deposition* Geng Han, Mumit Khan, Yanghua Fang, and Franco Cerrina a) Electrical and Computer Engineering and Center for NanoTechnology, University of Wisconsin Madison, Wisconsin Received 10 June 2002; accepted 14 October 2002 We present our effort in developing a complete model of electron energy transfer from fast electrons kev to the photoresist. Our model is based on the direct Monte Carlo method, instead of using continuous slowing down approximation, model and a stopping power curve. We separate the interaction events into four types: Elastic, ionization, excitation, and plasmon. Our results show that: First, secondary electrons are major mechanism of energy distribution; and second, plasmons are very efficient friction mechanism but do not create molecular changes; and finally, excitations lead to molecular changes American Vacuum Society. DOI: / I. INTRODUCTION In electron-beam lithography, the photoresist is exposed by the chemical changes associated with the electron energy loss. In x-ray and extreme UV lithographies, the excitation of core electrons creates fast photoelectrons that lose energy by scattering processes, similar to the case of electron-beam lithography. 1 In the energy range of high-energy lithography 20 ev, the main source of energy loss is due to interactions with the bound electrons in the medium. The loss of energy to the medium through plasmon excitation is akin to a friction process, removing energy from the beam to the material in the form of heat. Plasmons are relatively insensitive to the detail of the valence band orbitals because their frequency is determined mainly by the average electron density n(e) and average energy E. To understand the electron scattering process, the continuous slowing down approximation CSDA has been widely used in Monte Carlo simulations. 1 3 However, the CSDA model gives us only an expected value of energy deposition, and no information on generations of secondary electrons can be achieved from it. In this article, we present our effort in developing a complete model of electron energy transfer from fast electrons to the photoresist. Section II gives the overview and implementation method of our model. Section III focuses on generations of secondary electrons in the photoresist. Section IV investigates the treatment of plasmons in our model. Section V discusses modeling at the material interfaces. II. MODEL DESCRIPTION To build a complete model of electron energy loss in high-energy lithographies, two issues have to be addressed: The first is the electron scattering process in the materials propagation ; the second is how the energy is transferred from electrons to the photoresist deposition. Previously, a theoretical model of inelastic events has been proposed by Ashley. 4 The electron photoresist interaction model has been proposed by us based on the method of virtual quanta. 5 *No proof corrections received from author prior to publication. a Author to whom correspondence should be addressed; electronic mail: cerrina@nanotech.wisc.edu To model the propagation and the electron scattering process, the direct Monte Carlo DMC method 6 has been used in our simulation. In a nutshell, given an electron of energy E, we compute the cross sections for the global processes elastic, ionization, etc.. From we obtain a mean-free path, and the interaction location is computed using a standard exponential mean free path e x/. Similar approaches are used in hot electron device modeling. 7 Figure 1 a shows a typical trajectory of three 1 kev primary electrons, along with the identification of each type of events, and Fig. 1 b shows one of the primary electrons along with its secondaries. The propagation of an electron can be well described by the CSDA, but this model does not give any information on local chemical processes. In our new DMC model, we separate the scattering events into four types: elastic, ionization, excitation, and plasmon, completely replacing CSDA and stopping power. This model is stochastic, in the sense that the electrons travel in free flight between interactions. Cross section data for elastic, ionization, and excitation events are obtained from the EPDL97 database of the Lawrence Livermore National Laboratory. 8 Plasmon cross section is computed using the dielectric function and experimental data. 9 Figure 2 illustrates the four types of events considered in this DMC simulation. The elastic events only change the scattering angle of the electrons without depositing any energy. If an ionization event happens, this model will help us to first find out the related element and subshell involved in this ionization event. Then, some of the energy of the primary electron is lost through two ways: One part is passed to the outgoing secondary electron, the other part is deposited as bind energy of the relative subshell to the relative atom. Thus, this atom will be left with a hole in the subshell that involved in this ionization event. This hole will contribute to later electron recombination processes, which both Auger and fluorescence processes are included. These processes could lead to chemical changes if the valence band is involved e.g., LVV Auger. The treatment of secondary electrons is computationally challenging. In our approach, each electron trajectory is followed until the energy falls below a threshold, typically 10 ev, where the electron is considered stopped and that energy is deposited in the material. As 2666 J. Vac. Sci. Technol. B 20 6, NovÕDec Õ2002Õ20 6 Õ2666Õ6Õ$ American Vacuum Society 2666

2 2667 Han et al.: Comprehensive model of electron energy deposition 2667 FIG kev electron-beam simulation setup: 100 nm PMMA as photoresist, Au and Cr as two 50 nm thin film layers, with a Si substrate underneath. the incident electron to a nearby residual atom. We notice that the method predicts which atoms are excited and through what processes. No new electron is generated in this case, and the lost energy is considered deposited in that material. And finally, if a plasmon event is picked, both direction and energy of the incident electron are changed. The energy lost to plasmon is considered to be eventually deposited in the material as heat. 10 After the scattering event is decided, the corresponding electron mean-free path is calculated, also based on the cross section data, and the electron position is updated. FIG. 1. Electron scattering process in the photoresist. a The trajectories of three primary electrons. b An example of secondary electrons. the primary electron generates secondaries, the starting coordinates are put in a queue. When the primary stops, the first secondary is traced. Any further secondaries are added to the queue and the process continues until all energy has been expended. In this way, our implementation considers the contributions from all the possible electrons of every generation. When an excitation event is picked, the energy is lost from III. GENERATION OF SECONDARY ELECTRONS In order to show the effect of secondary electrons, a simulation of 25 kev electrons passing through a three layers structure was performed. Figure 3 shows the setup of the simulation. Figure 4 a shows the generations of electrons by ionization events in the resist. We use 0 to label the primary electrons, 1 to label the first secondaries, 2 to label the second secondaries, etc. Figure 4 b shows the relative elements and subshells involved in the ionization events. Figure 5 gives us the energy deposited in the resist versus the lateral range of the generations of electrons. The simulation results have good agreement with published data. 11 Our data show that the secondary electrons have the largest contribution to the distribution of electron energy deposition in the photoresist. FIG. 2. Four kinds of electron scattering events considered in the simulation: Elastic, ionization, excitation, and plasmons. JVST B-Microelectronics and Nanometer Structures

3 2668 Han et al.: Comprehensive model of electron energy deposition 2668 FIG. 4. Secondary electrons generated by ionization events in the photoresist, simulation setup as shown in Fig. 3. a Generations of secondaries in the resist. b Elements and relative subshells involved in ionization events. FIG. 6. Loss function and the differential cross section of plasmon events of PMMA. a Calculated loss function at 0, E 0 1 kev. b Calculated differential cross section of plasmon events. FIG. 5. Energy deposited in the photoresist proximity effects. Simulation setup as shown in Fig. 3. IV. PLASMONS The inelastic scattering, according to the energy of the primary electron, can be with the outer-shell conduction or valence electrons and inner-shell core-level electrons. For inner-shell scattering, as it has been considered in the ionization or excitation events, the electron atom scattering theory can be used and is included in the atomic cross section tables. For outer-shell scattering, the collective effect of plasmon excitation dominates. These plasmon excitations are longitudinal waves, and thus cannot couple to electronic or molecular transitions in three-dimensional 3D systems and result in nonspecific energy deposition. However, due to the large cross section of this type of event, they are the largest contributor to the electron energy loss. We developed our plasmon model based on the standard dielectric response of the material to an incident electron, which gives good results for describing the collective exci- J. Vac. Sci. Technol. B, Vol. 20, No. 6, NovÕDec 2002

4 2669 Han et al.: Comprehensive model of electron energy deposition 2669 FIG. 7. Energy distribution of plasmon events in the resist. Simulation setup as shown in Fig. 3. FIG. 8. Calculated total cross section of plasmon events comparing to ionization and excitation events for PMMA. tations. In this case, the dielectric function can be written as p q 2 avg 2 p i exp, 1 q 2 p 2 3 F 2 q 2 /5 2 q 4 /4m 2, p 4 ne2 m, 3 where p is the plasma frequency, avg is the expected value of all the possible oscillators in the material, is the standard deviation of the oscillators, q is the dispersions term coming from the momentum transfer of the plasmon events, n is electron density, and F is the Fermi velocity obtained from the valence band electron density. Using this model, the loss function and the differential cross section of plasmon excitation for poly methylmethacrylate PMMA are calculated, as shown in Fig. 6. This result has good agreement with the experimental data. 9 Figure 7 shows the energy distribution of the plasmon events in the simulation with the layout as shown in Fig. 3. As we expected, the plasmon events are centered at about 23 ev, with a wide energy distribution. The total cross section of plasmon excitation can be calculated by integrating over all the possible energies. Figure 8 shows the total cross section of plasmon events, compared to the ionization and excitation events. As can be seen, when the electron energy is greater than 100 ev, plasmon events will be the major contributor to the electron energy loss process. To further illustrate this point, we present a simulation of 100 kev electron beam on 200 nm PMMA. Figure 9 shows the energy loss peak of electrons after transmission through the PMMA sample. The plasmon peak can be clearly seen in the energy loss process. Similar results can be found in Ritsko s experimental work. 9 2 V. INTERFACE SIMULATION There are two issues need to be discussed about the simulation at the material interfaces: 1 calculations of the electron step length when the electron mean free path crosses the material interfaces; and 2 electron refractions at the interface. This may be important at low energies. A. Electron step length at the interface Let us consider a system with three layers of thickness (z 1,z 2,z 3 ). An electron is located in material No. 1 will have a mean-free path with z* P 1 (e z/ 1;r) and r 0,1 being a predicted event. If an event z* z 1, then the scattering occurs within material No. 1. If z* z 1, then the event will occur in material Nos. 2 or 3: We compute the new location using a smooth function e z 1 / 1 e (z z 1 )/ 2 suit- FIG. 9. Energy loss peak of 100 kev electron beam on a 200 nm PMMA sample. JVST B-Microelectronics and Nanometer Structures

5 2670 Han et al.: Comprehensive model of electron energy deposition 2670 case, we follow the treatment by Williams, 13 where the angle between the electron and the outward pointing surface normal can be calculated as sin 1 m* E V 0 me 1/2 sin, where V 0 is the inner potential, is the angle between the electron and the inward pointing surface normal of the material, E is the measured kinetic energy of the outgoing electron, m* is the effective mass of the electron, and m is the free electron mass. Figure 10 b shows the angle difference between the incident and outgoing electrons at the PMMA Si interface. In the calculation, we have treated m* m. As can be seen, at smaller incident angles and high electron energies, the electron refraction is very small. However, as incident angle becomes larger and electron energy lower than 30 ev, the refraction can be large. 5 FIG. 10. Interface modeling. a Electron step length calculation at material interfaces. b Electron refraction angle calculation at PMMA Si interface angles in degrees. ably normalized. Again, if z* is in material No. 2, the event is accepted; if not the process is repeated. This can be summarized in the formula ln r s 1 s s 2 du 1 2, 4 sn n where, 1, 2,..., n, are the mean-free paths in the n different materials, r is a uniform random number, s 1,s 1,...,s n,is the distance the electron travels in correspond materials, and s is the total step length electron travels. Figure 10 a shows the distribution function P(z) for different lengths of free flight. B. Electron refraction One of the important surface effects during low-energy electron scattering in the material is the electron refraction as it crosses the potential barrier at the surface. This particular effect has received some theoretical consideration. 12 In our VI. SUMMARY AND CONCLUSIONS We have developed a comprehensive model of the physics of electron energy loss, and implemented a simulation code using the DMC method. This code has the ability to simulate both electron and high-energy photon lithographies. It is also capable of handling any resist composition, any thin film/ substrate interface, including multilayer system, and arbitrary geometrical layout. In contrast with other work in this area, we keep track of generations of secondary electrons, and consider plasmon events in the energy loss process. This simulation will also be applied to LER studies. DMC is a preprocessor for computation of the atomic and molecular excitations. It can also be used as a standard electron propagation code. The models as presented here are still relatively crude and need refinement. In particular, one must use a more complete dielectric function to describe the plasmon process. Future work will focus on the definition of a more exhaustive dielectric function, and the calculation of cross sections of molecular bonds. ACKNOWLEDGMENTS The authors acknowledge extensive discussions with Dr. L. Ocola Argonne National Lab on the physics of the process and on relation of the previous code, LESIS, to our new model. Discussions with Dr. D. Joy University of Tennessee helped us with the definition of the role of plasmons. This work is based in part by a grant from the Semiconductor Research Corporation, No MJ-985. The Center for NanoTechnology, University of Wisconsin Madison, is supported in part by DARPA/ONR Grant No. MDA , MDA L. Ocola and F. Cerrina, J. Vac. Sci. Technol. B 11, D. F. Kyser and K. Murata, IBM J. Res. Dev. 18, M. Kotera, K. Murata, and K. Nagami, J. Appl. Phys. 52, J. C. Ashley, J. Appl. Phys. 63, G. Han and F. Cerrina, J. Vac. Sci. Technol. B 18, J. Vac. Sci. Technol. B, Vol. 20, No. 6, NovÕDec 2002

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