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The influence of the Coulomb interaction effect in the electron beam on the developed resist structure for the projection lithography
Microelectronic Engineering 57–58 (2001) 255–261 www.elsevier.com / locate / mee
The influence of the Coulomb interaction effect in the electron beam on the developed resist structure for the projection lithography a, a a b c c c M. Kotera *, M. Sakai , K. Yamada , K. Tamura , Y. Tomo , I. Simizu , A. Yoshida , c c Y. Kojima , M. Yamabe a
Department of Electronic Engineering, Osaka Institute of Technology, Omiya, Asahi-ku, Osaka 535 -8585, Japan b Department of Physics and Electronics, Osaka Prefecture University, Gakuen-cho, Sakai 599 -8531, Japan c Advanced Technology Research Department, Semiconductor Leading Edge Technologies, Inc., Yoshida-cho, Totsuka-ku, Yokohama 244 -0817, Japan Abstract A series of simulations are introduced to express the processes of the electron projection lithography quantitatively. It consists of the following three major steps: (1) electron trajectory simulation in an electron beam projection lithography optical system considering the Coulomb interaction effect among electrons and the lens aberrations, (2) electron energy deposition simulation in a resist on Si substrate, and (3) time evolution three-dimensional resist-development simulation. They are used to predict the error propagation characteristics in the sequence of lithography processes up to the three-dimensional resist structure after development. The influence of a variation in the exposure intensity on the final resist structure is demonstrated as an example. 2001 Elsevier Science B.V. All rights reserved. Keywords: Electron beam projection lithography; Coulomb interaction; Resist structure
1. Introduction Electron beam projection lithography (EPL), is a potential candidate to realize both demands of high-resolution and high-throughput for the next generation lithography tool. In order to tune up the system performance, it is necessary to explore the most preferable parameter at each portion of the lithography sequence. However, the preferable parameters may not be always realized, and it is necessary to estimate how much the influence of the deviation in the parameter propagates through the rest of the lithography processes. This analysis gives the process-margin for the parameter in the lithography process, and it is used for the CD control of the lithography. In the present study, we
* Corresponding author. E-mail address: [email protected] (M. Kotera). 0167-9317 / 01 / $ – see front matter PII: S0167-9317( 01 )00486-5
2001 Elsevier Science B.V. All rights reserved.
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
256
introduce a series of simulations for the EPL, which we built by ourselves, to obtain the error propagation characteristics in the lithography. 2. Simulations The simulations are categorized by the following three major programs: (1) electron trajectory simulation in EPL optical system considering the Coulomb interaction effect among electrons and the lens aberrations, (2) electron energy deposition simulation in a resist on Si substrate, (3) time evolution resist-development simulation. The result of each simulation is used as the input data for the subsequent simulation, and the three-dimensional resist structure is finally obtained. Although there have been several papers and even commercial softwares for those analyses as described in the following sections, it is much preferable to have the programs, which are all developed by ourselves covering the whole processes of the lithography, for a discussion of error propagation characteristics in the processes as a total analysis.
2.1. Electron trajectory simulation in EPL optical system The Coulomb interaction effect between electrons in the given optical system is calculated using the following equation of motion, if electrons are closer than 1 mm from each other [1]: e2 r¨ i 5 ]]] 4p´0 m
r 2r O ]]] ur 2 r u N
i
j
3
j51 ( j ±1 )
i
j
Using the thin lens approximation, the spherical and the chromatic aberrations are taken into account [2]. Based on a typical cell-projection optical system [3], electron trajectories are calculated to expose the pattern with the dimensions and parameters, which are defined in Fig. 1. The accelerating voltage of the beam is 50 kV, and the current density at the wafer surface is 10 A / cm 2 . Fig. 2 shows electron distributions at the wafer. Fig. 2a shows the distribution when the wafer surface is at the optical focal plane, and Fig. 2b shows that when the wafer surface is 150 mm farther from the optical focal plane. It is found that by even refocusing 150 mm, the edge blur of the distribution still remains because of the Coulomb interaction among electrons in the beam. Each one of these points indicates the arrival position of an electron at the wafer surface. The number of electrons showing the figure is 40 000. This figure shows the two-dimensional intensity distribution rather clearly to the eye, but statistically, the distribution is quite noisy. This kind of noisy distribution cannot be used, as it is, for the resist development simulation. It is necessary to smooth the distribution by some means. One of the well-known methods to smooth the distribution is to use the point-spread function (PSF) of the optics. Fig. 3 shows the total PSF due not only to the lens aberrations but also to the Coulomb interaction effect among electrons in the electron beam, which is obtained at the defocus length is 150 mm. It is obtained by simulating a large number of electron trajectories in the system. The PSF obtained is often fitted by some known functions. Actually, this PSF can be approximated by the Gaussian and the Lorentzian function as shown in Fig. 3. However, it is found either function does not show good agreement with the original PSF. For a quantitative discussion of the electron exposure density
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
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Fig. 1. Designed pattern of exposure in 5-mm field at the wafer surface.
distribution, the original numerical PSF is much preferable to smooth the distribution. For example, Fig. 4a–d show the exposure intensities, which are obtained by a convolution between the designed pattern and the PSFs. The PSFs used to obtain the distributions of Fig. 4a–c are the Gaussian, the Lorentzian, which are approximated as shown in Fig. 3, and the numerical PSF, respectively. It is observed that both figures approximated by the analytical expressions do not agree with the one, which is obtained by using the numerical PSF. On the other hand, the distribution shown in Fig. 4d is
Fig. 2. Two-dimensional electron arrival position distribution at the wafer surface made by 40 000 electrons. (a) Shows the distribution when the wafer surface is at the optical focal plane, and (b) shows the distribution when the wafer surface is 150 mm farther from the optical focal plane.
258
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
Fig. 3. Total PSF due not only to the lens aberrations but also to the Coulomb interaction effect in the electron beam. The PSF is fitted by Gaussian and Lorentzian function in this figure.
obtained by smoothing directly the two-dimensional intensity distribution of Fig. 3 mathematically by using the Spline function [4]. In the optical system used here, since the exposure field size at the wafer is small, as 535 mm, the PSF almost does not show any dependence on the position of exposure. On the other hand, if the area of exposure is large, like SCALPEL [5] and PREVAIL [6], the PSF is a function of the position in the pattern. It should be understood that the single PSF could not be used. It may not be realistic to use the position dependent PSF with the exposure pattern. It turns out that the two-dimensional Spline smoothing may be the most flexible and even a quite accurate method to express the smoothed exposure intensity distribution out of the calculated noisy distribution for the analysis of the resist development in various exposure systems.
2.2. Energy deposition distribution in a resist Monte Carlo simulation of electron trajectories is performed in a resist on Si substrate. The electron energy deposition is obtained in the resist for a point electron beam incidence at the surface. Almost all simulations presented recently treat fast secondary electron trajectories in addition to the incident electron trajectories in the resist [7–9]. In contrast, trajectories of not only incident fast electrons, but also slow secondary electrons with e.g. several eV are calculated in the present simulation by using the dielectric function [10]. The exposure dose for the resist is 100 mC / cm 2 . The resist assumed here is PMMA at 300 nm thick. The energy deposited by each electron trajectory is accumulated for a large number of incident electrons, and a statistical energy deposition distribution is obtained for a point electron beam. By convoluting a given exposure intensity distribution at the wafer surface with the energy deposition distribution in the resist, the three-dimensional accumulated energy distribution is obtained.
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
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Fig. 4. Exposure intensity distribution, which are obtained by a convolution between the designed pattern and the PSFs. (a–c) These results are obtained by using the Gaussian, the Lorentzian, and the numerical PSF, respectively. (d) Shows results obtained by using the two-dimensional Spline function.
2.3. Resist development simulation The time evolution resist development phenomenon is simulated. Here, the three-dimensional cell-removal model [11] is applied. The solubility-rate as a function of the energy deposited is determined by Neureither’s equation [12]. Development time for each cell is determined by the structure of the resist surface [11]. The time evolution resist development is demonstrated in Fig. 5 as the cropped cross sectional view by using the PSF, which is extracted from the electron distribution obtained at 150 mm defocusing. The height of the resist is three times enlarged.
3. Influence of the Coulomb interaction effect on the resist pattern By using the series of simulations of the EPL mentioned above, a quantitative discussion can be made on the influence of the Coulomb interaction effect on the resist pattern. Because of an edge blur, the cross section shows a curvature, and interference between patterns can be observed. If we quantify the relation between data for the exposure intensity and the resist height after development, the following results are extracted as an example: Although it is found that the exposure intensity at the center of the side edge of the large pattern
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
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Fig. 5. Cropped cross sectional views of the PMMA resist structure for various development time.
opposite to the small pattern is almost 55% of the maximum intensity, which is found at the center of the large pattern in Fig. 4c, after the development of the resist, as shown in Fig. 5, when the bottom of the large pattern reach the Si surface, the pattern edge reaches only 30% of the total thickness of the resist. When the electron exposure intensity is smoothed by using the numerical PSF, it is found that the length needed for the intensity to rise from 20 to 80% of its maximum at the center of the right edge of the pattern is 161 nm, as the defocus length is 150 mm. If the wafer surface is at the optical focal plane, the length is 350 nm. Because of the inter-proximity effect between the small and the large patterns, the length is 167 nm for 150 mm defocusing, and it is 392 nm at the optical focal plane, respectively. After the resist is developed and the center of the large pattern reaches the Si surface, the length needed for the developed resist thickness varies from 80 to 20% of its original thickness for both right and left edges of the large pattern is almost 150 nm, if the defocus length is 150 mm. Those lengths are 200 and 300 nm for right and left of the edge of the large pattern, if the wafer surface is at the optical focal plane. These kinds of discussions can be made in detail using the present simulation.
4. Conclusions In the present study we developed: (1) electron trajectory simulation in EPL optical system, (2) electron energy deposition simulation in a resist on Si substrate, and (3) time evolution resist development simulation. Two-dimensional Spline smoothing is found to be quite flexible and even accurate to express the practical EPL exposure intensity distribution to give an applicable data form in the subsequent simulations. It is now possible for us to predict the error propagation in the series of lithography processes up to three-dimensional resist structure after development. It is expected to extract the information on the error-budget and the CD control of the lithography system using these simulations. Various conditions will be treated using the present simulations, and the validity will be checked in future.
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
261
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
G.H. Jansen, Advances in Electronics and Electron Physics, Academic Press, New York, 1990, Suppl. 21. E. Munro, Nucl. Instrum. Method Phys. Res. A 258 (1987) 443. M. Kotera, K. Yamaguchi, T. Okagawa, K. Matsuoka, Y. Kojima, M. Yamabe, J. Vac. Sci. Technol. B 17 (1999) 2921. K. Ichida, F. Yoshimoto, Kyouiku Shuppan (1979) (in Japanese). L.R. Harriott, J. Vac. Sci. Technol. B 15 (1997) 2130. H.C. Pfeiffer, W. Stickel, Microelectron. Eng. 27 (1995) 6658. K. Murata, D.F. Kyser, C.H. Ting, J. Appl. Phys. 52 (1981) 4396. C.R.K. Marrian, F.K. Perkins, D. Park, E.A. Dobisz, M.C. Peckerar, K.-W. Rhee, R. Bass, J. Vac. Sci. Technol. B 14 (1996) 3864. D.-H. Kim, Y.-M. Ham, C. Lee, K. Chun, Microelectron. Eng. 41–42 (1998) 179. M. Kotera, R. Ijichi, T. Fujiwara, H. Suga, D.B. Wittry, Jpn. J. Appl. Phys. 29 (1990) 2277. Y. Hirai, S. Tomida, K. Ikeda, M. Sasago, M. Endo, S. Hayama, N. Nomura, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 10 (1991) 802. A.R. Neureuther, D.F. Kyser, C.H. Ting, IEEE Trans. Electron. Devices ED–26 (1979) 686.
The influence of the Coulomb interaction effect in the electron beam on the developed resist structure for the projection lithography a, a a b c c c M. Kotera *, M. Sakai , K. Yamada , K. Tamura , Y. Tomo , I. Simizu , A. Yoshida , c c Y. Kojima , M. Yamabe a
Department of Electronic Engineering, Osaka Institute of Technology, Omiya, Asahi-ku, Osaka 535 -8585, Japan b Department of Physics and Electronics, Osaka Prefecture University, Gakuen-cho, Sakai 599 -8531, Japan c Advanced Technology Research Department, Semiconductor Leading Edge Technologies, Inc., Yoshida-cho, Totsuka-ku, Yokohama 244 -0817, Japan Abstract A series of simulations are introduced to express the processes of the electron projection lithography quantitatively. It consists of the following three major steps: (1) electron trajectory simulation in an electron beam projection lithography optical system considering the Coulomb interaction effect among electrons and the lens aberrations, (2) electron energy deposition simulation in a resist on Si substrate, and (3) time evolution three-dimensional resist-development simulation. They are used to predict the error propagation characteristics in the sequence of lithography processes up to the three-dimensional resist structure after development. The influence of a variation in the exposure intensity on the final resist structure is demonstrated as an example. 2001 Elsevier Science B.V. All rights reserved. Keywords: Electron beam projection lithography; Coulomb interaction; Resist structure
1. Introduction Electron beam projection lithography (EPL), is a potential candidate to realize both demands of high-resolution and high-throughput for the next generation lithography tool. In order to tune up the system performance, it is necessary to explore the most preferable parameter at each portion of the lithography sequence. However, the preferable parameters may not be always realized, and it is necessary to estimate how much the influence of the deviation in the parameter propagates through the rest of the lithography processes. This analysis gives the process-margin for the parameter in the lithography process, and it is used for the CD control of the lithography. In the present study, we
* Corresponding author. E-mail address: [email protected] (M. Kotera). 0167-9317 / 01 / $ – see front matter PII: S0167-9317( 01 )00486-5
2001 Elsevier Science B.V. All rights reserved.
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
256
introduce a series of simulations for the EPL, which we built by ourselves, to obtain the error propagation characteristics in the lithography. 2. Simulations The simulations are categorized by the following three major programs: (1) electron trajectory simulation in EPL optical system considering the Coulomb interaction effect among electrons and the lens aberrations, (2) electron energy deposition simulation in a resist on Si substrate, (3) time evolution resist-development simulation. The result of each simulation is used as the input data for the subsequent simulation, and the three-dimensional resist structure is finally obtained. Although there have been several papers and even commercial softwares for those analyses as described in the following sections, it is much preferable to have the programs, which are all developed by ourselves covering the whole processes of the lithography, for a discussion of error propagation characteristics in the processes as a total analysis.
2.1. Electron trajectory simulation in EPL optical system The Coulomb interaction effect between electrons in the given optical system is calculated using the following equation of motion, if electrons are closer than 1 mm from each other [1]: e2 r¨ i 5 ]]] 4p´0 m
r 2r O ]]] ur 2 r u N
i
j
3
j51 ( j ±1 )
i
j
Using the thin lens approximation, the spherical and the chromatic aberrations are taken into account [2]. Based on a typical cell-projection optical system [3], electron trajectories are calculated to expose the pattern with the dimensions and parameters, which are defined in Fig. 1. The accelerating voltage of the beam is 50 kV, and the current density at the wafer surface is 10 A / cm 2 . Fig. 2 shows electron distributions at the wafer. Fig. 2a shows the distribution when the wafer surface is at the optical focal plane, and Fig. 2b shows that when the wafer surface is 150 mm farther from the optical focal plane. It is found that by even refocusing 150 mm, the edge blur of the distribution still remains because of the Coulomb interaction among electrons in the beam. Each one of these points indicates the arrival position of an electron at the wafer surface. The number of electrons showing the figure is 40 000. This figure shows the two-dimensional intensity distribution rather clearly to the eye, but statistically, the distribution is quite noisy. This kind of noisy distribution cannot be used, as it is, for the resist development simulation. It is necessary to smooth the distribution by some means. One of the well-known methods to smooth the distribution is to use the point-spread function (PSF) of the optics. Fig. 3 shows the total PSF due not only to the lens aberrations but also to the Coulomb interaction effect among electrons in the electron beam, which is obtained at the defocus length is 150 mm. It is obtained by simulating a large number of electron trajectories in the system. The PSF obtained is often fitted by some known functions. Actually, this PSF can be approximated by the Gaussian and the Lorentzian function as shown in Fig. 3. However, it is found either function does not show good agreement with the original PSF. For a quantitative discussion of the electron exposure density
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
257
Fig. 1. Designed pattern of exposure in 5-mm field at the wafer surface.
distribution, the original numerical PSF is much preferable to smooth the distribution. For example, Fig. 4a–d show the exposure intensities, which are obtained by a convolution between the designed pattern and the PSFs. The PSFs used to obtain the distributions of Fig. 4a–c are the Gaussian, the Lorentzian, which are approximated as shown in Fig. 3, and the numerical PSF, respectively. It is observed that both figures approximated by the analytical expressions do not agree with the one, which is obtained by using the numerical PSF. On the other hand, the distribution shown in Fig. 4d is
Fig. 2. Two-dimensional electron arrival position distribution at the wafer surface made by 40 000 electrons. (a) Shows the distribution when the wafer surface is at the optical focal plane, and (b) shows the distribution when the wafer surface is 150 mm farther from the optical focal plane.
258
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
Fig. 3. Total PSF due not only to the lens aberrations but also to the Coulomb interaction effect in the electron beam. The PSF is fitted by Gaussian and Lorentzian function in this figure.
obtained by smoothing directly the two-dimensional intensity distribution of Fig. 3 mathematically by using the Spline function [4]. In the optical system used here, since the exposure field size at the wafer is small, as 535 mm, the PSF almost does not show any dependence on the position of exposure. On the other hand, if the area of exposure is large, like SCALPEL [5] and PREVAIL [6], the PSF is a function of the position in the pattern. It should be understood that the single PSF could not be used. It may not be realistic to use the position dependent PSF with the exposure pattern. It turns out that the two-dimensional Spline smoothing may be the most flexible and even a quite accurate method to express the smoothed exposure intensity distribution out of the calculated noisy distribution for the analysis of the resist development in various exposure systems.
2.2. Energy deposition distribution in a resist Monte Carlo simulation of electron trajectories is performed in a resist on Si substrate. The electron energy deposition is obtained in the resist for a point electron beam incidence at the surface. Almost all simulations presented recently treat fast secondary electron trajectories in addition to the incident electron trajectories in the resist [7–9]. In contrast, trajectories of not only incident fast electrons, but also slow secondary electrons with e.g. several eV are calculated in the present simulation by using the dielectric function [10]. The exposure dose for the resist is 100 mC / cm 2 . The resist assumed here is PMMA at 300 nm thick. The energy deposited by each electron trajectory is accumulated for a large number of incident electrons, and a statistical energy deposition distribution is obtained for a point electron beam. By convoluting a given exposure intensity distribution at the wafer surface with the energy deposition distribution in the resist, the three-dimensional accumulated energy distribution is obtained.
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
259
Fig. 4. Exposure intensity distribution, which are obtained by a convolution between the designed pattern and the PSFs. (a–c) These results are obtained by using the Gaussian, the Lorentzian, and the numerical PSF, respectively. (d) Shows results obtained by using the two-dimensional Spline function.
2.3. Resist development simulation The time evolution resist development phenomenon is simulated. Here, the three-dimensional cell-removal model [11] is applied. The solubility-rate as a function of the energy deposited is determined by Neureither’s equation [12]. Development time for each cell is determined by the structure of the resist surface [11]. The time evolution resist development is demonstrated in Fig. 5 as the cropped cross sectional view by using the PSF, which is extracted from the electron distribution obtained at 150 mm defocusing. The height of the resist is three times enlarged.
3. Influence of the Coulomb interaction effect on the resist pattern By using the series of simulations of the EPL mentioned above, a quantitative discussion can be made on the influence of the Coulomb interaction effect on the resist pattern. Because of an edge blur, the cross section shows a curvature, and interference between patterns can be observed. If we quantify the relation between data for the exposure intensity and the resist height after development, the following results are extracted as an example: Although it is found that the exposure intensity at the center of the side edge of the large pattern
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
260
Fig. 5. Cropped cross sectional views of the PMMA resist structure for various development time.
opposite to the small pattern is almost 55% of the maximum intensity, which is found at the center of the large pattern in Fig. 4c, after the development of the resist, as shown in Fig. 5, when the bottom of the large pattern reach the Si surface, the pattern edge reaches only 30% of the total thickness of the resist. When the electron exposure intensity is smoothed by using the numerical PSF, it is found that the length needed for the intensity to rise from 20 to 80% of its maximum at the center of the right edge of the pattern is 161 nm, as the defocus length is 150 mm. If the wafer surface is at the optical focal plane, the length is 350 nm. Because of the inter-proximity effect between the small and the large patterns, the length is 167 nm for 150 mm defocusing, and it is 392 nm at the optical focal plane, respectively. After the resist is developed and the center of the large pattern reaches the Si surface, the length needed for the developed resist thickness varies from 80 to 20% of its original thickness for both right and left edges of the large pattern is almost 150 nm, if the defocus length is 150 mm. Those lengths are 200 and 300 nm for right and left of the edge of the large pattern, if the wafer surface is at the optical focal plane. These kinds of discussions can be made in detail using the present simulation.
4. Conclusions In the present study we developed: (1) electron trajectory simulation in EPL optical system, (2) electron energy deposition simulation in a resist on Si substrate, and (3) time evolution resist development simulation. Two-dimensional Spline smoothing is found to be quite flexible and even accurate to express the practical EPL exposure intensity distribution to give an applicable data form in the subsequent simulations. It is now possible for us to predict the error propagation in the series of lithography processes up to three-dimensional resist structure after development. It is expected to extract the information on the error-budget and the CD control of the lithography system using these simulations. Various conditions will be treated using the present simulations, and the validity will be checked in future.
M. Kotera et al. / Microelectronic Engineering 57 – 58 (2001) 255 – 261
261
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
G.H. Jansen, Advances in Electronics and Electron Physics, Academic Press, New York, 1990, Suppl. 21. E. Munro, Nucl. Instrum. Method Phys. Res. A 258 (1987) 443. M. Kotera, K. Yamaguchi, T. Okagawa, K. Matsuoka, Y. Kojima, M. Yamabe, J. Vac. Sci. Technol. B 17 (1999) 2921. K. Ichida, F. Yoshimoto, Kyouiku Shuppan (1979) (in Japanese). L.R. Harriott, J. Vac. Sci. Technol. B 15 (1997) 2130. H.C. Pfeiffer, W. Stickel, Microelectron. Eng. 27 (1995) 6658. K. Murata, D.F. Kyser, C.H. Ting, J. Appl. Phys. 52 (1981) 4396. C.R.K. Marrian, F.K. Perkins, D. Park, E.A. Dobisz, M.C. Peckerar, K.-W. Rhee, R. Bass, J. Vac. Sci. Technol. B 14 (1996) 3864. D.-H. Kim, Y.-M. Ham, C. Lee, K. Chun, Microelectron. Eng. 41–42 (1998) 179. M. Kotera, R. Ijichi, T. Fujiwara, H. Suga, D.B. Wittry, Jpn. J. Appl. Phys. 29 (1990) 2277. Y. Hirai, S. Tomida, K. Ikeda, M. Sasago, M. Endo, S. Hayama, N. Nomura, IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 10 (1991) 802. A.R. Neureuther, D.F. Kyser, C.H. Ting, IEEE Trans. Electron. Devices ED–26 (1979) 686.