Physical and Mathematical Fundamentals

  • Bernd Lemaitre
  • Christoph Sohrmann
  • Lutz Muche
  • Joachim Haase


This chapter provides a short overview on the basics of CMOS transistor modeling with respect to deep submicron requirements and mathematical approaches to analyze variations in the design process. Technical terms are going to be defined and explained; physical processes and mathematical theories will be illustrated.


Response Surface Probability Density Function Random Vector Singular Value Decomposition Importance Sampling 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Shichman, H., Hodges, D.A.: Modeling and simulation of insulated-gate field-effect transistor switching circuits. IEEE J. Solid-State Circuits 3(5), 285–289 (1968)CrossRefGoogle Scholar
  2. 2.
    Meyer, J.E.: MOS models and circuit simulation. RCA Review 32, 42–63 (1971)Google Scholar
  3. 3.
    Ward, D.E., Dutton, R.W.: A charge-oriented model for MOS transistor capacitances. IEEE J. Solid-State Circuits 13(5), 703–708 (1978)CrossRefGoogle Scholar
  4. 4.
    Foty, D.P.: MOSFET Modeling with SPICE - Principles and Practice. Prentice Hall, Upper Saddle River, NJ (1997)Google Scholar
  5. 5.
    Sheu, B.J., Scharfetter, D.L., Ko, P.K., Jen, M.C.: BSIM Berkeley short-channel IGFET model for MOS transistors. IEEE J. Solid-State Circuits 22(4), 558–566 (1987)CrossRefGoogle Scholar
  6. 6.
    Synopsys: HSPICE MOSFET Models Manual, version z-2006.03 edn. (2007). Chapter 6Google Scholar
  7. 7.
    Liu, W.: MOSFET Models for SPICE Simulation, Including BSIM3v3 and BSIM4. John Wiley & Sons, New York (2001)Google Scholar
  8. 8.
    Enz, C.C., Krummenacher, F., Vittoz, E.A.: An analytical MOS transistor model valid in all regions of operation and dedicated to low-voltage and low current applications. J. Analog Integr. Circuits Signal Process 8, 83–114 (1995)CrossRefGoogle Scholar
  9. 9.
  10. 10.
  11. 11.
    Miura-Mattausch, M., Feldmann, U., Rahm, A., Bollu, M., Savignac, D.: Unified complete MOSFET model for analysis of digital and analog circuits. IEEE Trans. CAD/ICAS 15(1), 1–7 (1996)Google Scholar
  12. 12.
    Gildenblat, G., Li, X., W.Wu, Wang, H., Jha, A., van Langevelde, R., Smit, G., Scholten, A., Klaassen, D.: PSP: An advanced surface-potential-based MOSFET model for circuit simulation. Electron Devices, IEEE Transactions on 53(9), 1979–1993 (2006)Google Scholar
  13. 13.
  14. 14.
  15. 15.
    Tsividis, Y.: Operation and Modeling of the MOS Transistor, 2nd Edn. McGraw-Hill, New York (1999)Google Scholar
  16. 16.
    Taur, Y., Ning, T.: Fundamentals of modern VLSI devices. Cambridge University Press (1998)Google Scholar
  17. 17.
    Wang, A., Calhoun, B.H., Chandrakasan, A.P.: Sub-threshold Design for Ultra Low-Power Systems. Springer (2006)Google Scholar
  18. 18.
    Moore, G.E.: Cramming more components onto integrated circuits. Electronics 38, 114 ff. (1965)Google Scholar
  19. 19.
    Dennard, R., Gaensslen, F., Rideout, V., Bassous, E., LeBlanc, A.: Design of ion-implanted MOSFET’s with very small physical dimensions. IEEE Journal of Solid-State Circuits 9(5), 256–268 (1974)CrossRefGoogle Scholar
  20. 20.
    Rabaey, J.: Low Power Design Essentials. Springer, Boston, MA (2009). DOI 10. 1007/978-0-387-71713-5Google Scholar
  21. 21.
    Kenyon, C., Kornfeld, A., Kuhn, K., Liu, M., Maheshwari, A., Shih, W., Sivakumar, S., Taylor, G., VanDerVoorn, P., Zawadzki, K.: Managing process variation in Intel’s 45nm CMOS technology. Intel Technology Journal 12(2) (2008). URL
  22. 22.
    Asenov, A.: Random dopant induced threshold voltage lowering and fluctuations in sub-0.1 um MOSFET’s: A 3-D “atomistic” simulation study. IEEE Transactions on Electron Devices 45(12), 2505–2513 (1998). DOI 10.1109/16.735728Google Scholar
  23. 23.
    Diaz, C.H., Tao, H.J., Ku, Y.C., Yen, A., Young, K.: An experimentally validated analytical model for gate line-edge roughness (LER) effects on technology scaling. IEEE Electron Device Letters 22(6), 287–289 (2001). DOI 10.1109/55.924844CrossRefGoogle Scholar
  24. 24.
    Asenov, A., Kaya, S., Davies, J.H.: Intrinsic threshold voltage fluctuations in decanano MOSFETs due to local oxide thickness variations. IEEE Transactions on Electron Devices 49(1), 112–119 (2002). DOI 10.1109/16.974757CrossRefGoogle Scholar
  25. 25.
    Kaushik, V.S., O’Sullivan, B.J., Pourtois, G., Van Hoornick, N., Delabie, A., Van Elshocht, S., Deweerd, W., Schram, T., Pantisano, L., Rohr, E., Ragnarsson, L.A., De Gendt, S., Heyns, M.: Estimation of fixed charge densities in hafnium-silicate gate dielectrics. IEEE Transactions on Electron Devices 53(10), 2627–2633 (2006). DOI 10.1109/TED.2006.882412CrossRefGoogle Scholar
  26. 26.
    Lucovsky, G.: Intrinsic limitations on the performance and reliability of high-k gate dielectrics for advanced silicon devices. In: Proc. IEEE Int. Integrated Reliability Workshop Final Report (2005). DOI 10.1109/IRWS.2005.1609592Google Scholar
  27. 27.
    Capodieci, L.: From optical proximity correction to lithography-driven physical design (1996-2006): 10 years of resolution enhancement technology and the roadmap enablers for the next decade. In: Proceedings of SPIE, vol. 6154 (3) (2006)Google Scholar
  28. 28.
    Nag, S., Chatterjee, A., Taylor, K., Ali, I., O’Brien, S., Aur, S., Luttmer, J.D., Chen, I.C.: Comparative evaluation of gap-fill dielectrics in shallow trench isolation for sub-0.25 /spl mu/m technologies. In: Proc. Int. Electron Devices Meeting IEDM ’96, pp. 841–845 (1996). DOI 10.1109/IEDM.1996.554111Google Scholar
  29. 29.
    Tsang, Y.L., Chattopadhyay, S., Uppal, S., Escobedo-Cousin, E., Ramakrishnan, H.K., Olsen, S.H., O’Neill, A.G.: Modeling of the threshold voltage in strained si/si1-x gex/si1-ygey(x-y) cmos architectures. IEEE Transactions on Electron Devices 54(11), 3040–3048 (2007). DOI 10.1109/TED.2007.907190CrossRefGoogle Scholar
  30. 30.
    Al-Bayati, A., Graoui, H., Spear, J., Ito, H., Matsunaga, Y., Ohuchi, K., Adachi, K., Miyashita, K., Nakayama, T., Oowada, M., Toyoshima, Y.: Advanced CMOS device sensitivity to USJ processes and the required accuracy of doping and activation. In: Proc. 14th Int. Conf. Ion Implantation Technology 2002, pp. 185–188 (2002). DOI 10.1109/IIT.2002.1257969Google Scholar
  31. 31.
    Lorenz, J., Bär, E., Clees, T., Jancke, R., Salzig, C., S., S.: Hierarchical simulation of process variations and their impact on circuits and systems: Methodology. IEEE Trans. on Electron Devices, Special Issue Vol. 58(8) (2011), pp. 2218–2226Google Scholar
  32. 32.
    Lorenz, J., Bär, E., Clees, T., Jancke, R., Salzig, C., S., S.: Hierarchical simulation of process variations and their impact on circuits and systems: Results. IEEE Trans. on Electron Devices, Special Issue Vol. 58(8) (2011), pp. 2218–2226Google Scholar
  33. 33.
    Jancke, R., Kampen, C., Kilic, O., Lorenz, J.: Hierarchischer ansatz für die monte-carlo-simulation komplexer mixed-signal-schaltungen. In: 11. ITG/GMM-Fachtagung ANALOG. Erfurt (2010)Google Scholar
  34. 34.
    Yamaoka, M., Onodera, H.: A detailed vth-variation analysis for sub-100-nm embedded SRAM design. In: Proc. IEEE Int. SOC Conf, pp. 315–318 (2006). DOI 10.1109/SOCC.2006.283905Google Scholar
  35. 35.
    Pelgrom, M.J.M., Duinmaijer, A.C.J., Welbers, A.P.G.: Matching properties of mos transistors. IEEE Journal of Solid-State Circuits 24(5), 1433–1439 (1989). DOI10.1109/JSSC.1989.572629Google Scholar
  36. 36.
    Petzold, L., Li, S., Cao, Y., Serban, R.: Sensitivity analysis of differential-algebraic equations and partial differential equations. Computers & Chemical Engineering 30(10-12), 1553 – 1559 (2006). DOI 10.1016/j.compchemeng.2006.05.015CrossRefGoogle Scholar
  37. 37.
    Özyurt, D.B., Barton, P.I.: Cheap second order directional derivatives of stiff ODE embedded functionals. SIAM J. Sci. Comput. 26, 1725–1743 (2005). DOI 10.1137/030601582MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Cao, Y., Li, S.T., Petzold, L., Serban, R.: Adjoint sensitivity analysis or differential-algebraic equations: The adjoint DAE system and its numerical solution. Siam Journal on Scientific Computing 24(1), 1076–1089 (2003). DOI 10.1137/ S1064827501380630MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Sakurai, T., Newton, A.R.: Alpha-power law MOSFET model and its applications to CMOS inverter delay and other formulas. IEEE Journal of Solid-State Circuits SC 25(2), 584–594 (1990)CrossRefGoogle Scholar
  40. 40.
    Bowman, K.A., Austin, B.L., Eble, J.C., Tang, X., Meindl, J.D.: A physical alpha-power law mosfet model. IEEE Journal of Solid-State Circuits 34(10), 1410–1414 (1999). DOI 10.1109/4.792617CrossRefGoogle Scholar
  41. 41.
    Rabaey, J.M., Chandrakasan, A., Nikolic, B.: Digital Integrated Circuits: A Design Perspective. Prentice Hall (2003)Google Scholar
  42. 42.
    Stolk, P.A., Widdershoven, F.P., Klaassen, D.B.M.: Modeling statistical dopant fluctuations in MOS transistors. IEEE Transactions on Electron Devices 45(9), 1960–1971 (1998). DOI 10.1109/16.711362CrossRefGoogle Scholar
  43. 43.
    Narendra, S.G.: Effect of MOSFET threshold voltage variation on high-performance circuits. Ph.D. thesis, Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science (2002)Google Scholar
  44. 44.
    Roy, K., Mukhopadhyay, S., Mahmoodi-Meimand, H.: Leakage current mechanisms and leakage reduction techniques in deep-submicrometer CMOS circuits. In: Proceedings of the IEEE, pp. 305–327 (2003)Google Scholar
  45. 45.
    Veendrick, J.M.H.: Nanometer CMOS ICs: From basics to ASICs, 1st Edn. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  46. 46.
    Srivastava, A., Blaauw, D., Sylvester, D.: Statistical Analysis and Optimization for VLSI: Timing and Power. Springer Science+Business Media Inc, Boston, MA (2005). DOI 10.1007/b137645Google Scholar
  47. 47.
    Chan, T.Y., Chen, J., Ko, P.K., Hu, C.: The impact of gate-induced drain leakage current on mosfet scaling. In: Proc. Int. Electron Devices Meeting, vol. 33, pp. 718–721 (1987). DOI 10.1109/IEDM.1987.191531Google Scholar
  48. 48.
    Bouhdada, A., Bakkali, S., Touhami, A.: Modelling of gate-induced drain leakage in relation to technological parameters and temperature. Microelectronics and Reliability 37(4), 649–652 (1997). DOI 10.1016/S0026-2714(96)00062-5CrossRefGoogle Scholar
  49. 49.
    Mulaik, S.A.: Foundations of factor analysis, 2nd Edn. Chapman & Hall/CRC statistics in the social and behavioral sciences series. CRC Press, Boca Raton, FL (2010)MATHGoogle Scholar
  50. 50.
    Johnson, R.A., Wichern, D.W.: Applied multivariate statistical analysis, 6th Edn. Pearson Prentice Hall, Upper Saddle River N.J. (2007)Google Scholar
  51. 51.
    Weisstein, E.W.: Regularized gamma function. From MathWorld – A Wolfram Web Resource
  52. 52.
    Rencher, A.C.: Methods of multivariate analysis (2002). DOI 10.1002/0471271357Google Scholar
  53. 53.
    Johnson, N., Kotz, S.: Distribution in Statistics I. Continuous univarite distributions. Wiley (1970)Google Scholar
  54. 54.
    Karian, Z.A., Dudewicz, E.J.: Fitting statistical distributions: The Generalized Lambda Distribution and Generalized Bootstrap methods. CRC Press, Boca Raton (2000)MATHCrossRefGoogle Scholar
  55. 55.
    Shlens, J.: Tutorial on Principal Component Analysis. Tech. Rep. Version 2, Systems Neurobiology Laboratory, Salk Insitute for Biological Studies and Institute for Nonlinear Science, University of California, San Diego (2005). CiteSeerX Scholar
  56. 56.
    Jolliffe, I.T.: Principal Component Analysis, 2nd Edn. Springer (2002)Google Scholar
  57. 57.
    Jackson, J.E.: A User’s Guide to Principal Components. Wiley Series in Probability and Statistics. Wiley (2003)Google Scholar
  58. 58.
    Skillicorn, D.B.: Understanding complex datasets: Data mining with matrix decompositions. Chapman & Hall/CRC data mining and knowledge discovery series. Chapman & Hall/CRC Press, Boca Raton (2007)MATHCrossRefGoogle Scholar
  59. 59.
    Kalman, D.: A singulary valuable decomposition : The SVD of a matrix. The College Mathematical Journal 27(1), 1–23 (1996). URL
  60. 60.
    Shamsi, D., Boufounos, P., Koushanfar, F.: Noninvasive leakage power tomography of integrated circuits by compressive sensing. In: ISLPED ’08: Proceedings of the 2003 international symposium on Low power electronics and design, pp. 341–346. ACM, NY, USA, Bangalore (2008)Google Scholar
  61. 61.
    Box, G.E.P., Draper, N.R.: Empirical model-building and response surfaces. Wiley series in probability and mathematical statistics. Wiley, New York (1987)MATHGoogle Scholar
  62. 62.
    Poggio, T.: On optimal nonlinear associative recall. Biol. Cybernetics 19, 201–209 (1975)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Lauridsen, S., Vitali, R., van Keulen, F., Haftka, R.T., Madsen, J.: Response surface approximation using gradient information. In: World Congress of Structural and Multidisciplinary Optimization WCSMO-4. Dalian, China (2001). CiteSeerX URL
  64. 64.
    Park, S., H.J., K., Cho, J.I.: Recent advances in linear models and related areas. In: Optimal Central Composite Designs for Fitting Second Order Response Surface Linear Regression Models, pp. 323–339. Physica-Verlag HD (2008). DOI 10.1007/978-3-7908-2064-5_17Google Scholar
  65. 65.
    Cheng, L., Xiong, J., He, L.: Non-Gaussian statistical timing analysis using second-order polynomial fitting. Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on 28(1), 130–140 (2009). DOI 10.1109/TCAD.2008.2009143CrossRefGoogle Scholar
  66. 66.
    Sohrmann, C., Muche, L., Haase, J.: Accurate approximation to the probability of critical performance. In: 2. GMM/GI/ITG-Fachtagung Zuverlässigkeit und Entwurf, pp. 93–97 (2008)Google Scholar
  67. 67.
    Zhang, M., Olbrich, M., Seider, D., Frerichs, M., Kinzelbach, H., Barke, E.: CMCal: An accurate analytical approach for the analysis of process variations with non-Gaussian parameters and nonlinear functions. In: Design, Automation & Test in Europe Conference & Exhibition, 2007. DATE ’07, pp. 1–6 (2007). DOI 10.1109/DATE.2007.364598Google Scholar
  68. 68.
    Saucier, R.: Computer generation of statistical distributions. Tech. rep., Army Research Laboratory (2000). URL
  69. 69.
    Cheng, R.C.: Boostrap methods in computer simulation experiments. In: Proceedings of the 1995 Winter Simulation Conference, pp. 171–177 (1995)Google Scholar
  70. 70.
    Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer Publishing Company, Incorporated (2008)MATHGoogle Scholar
  71. 71.
    Rao, R., Srivastava, A., Blaauw, D., Sylvester, D.: Statistical estimation of leakage current considering inter- and intra-die process variation. In: Proceedings of the 2003 International Symposium on Low Power Electronics and Design ISLPED ’03, pp. 84–89 (2003), DOI 10.1109/LPE.2003.1231840Google Scholar
  72. 72.
    Denny, M.: Introduction to importance sampling in rare-event simulations. EUROPEAN JOURNAL OF PHYSICS 22(4), 403–411 (2001)CrossRefGoogle Scholar
  73. 73.
    Robert, C.P., Casella, G.: Monte Carlo statistical methods, 2nd Edn. Springer texts in statistics. Springer, New York, NY (2004). ISBN 978-0-387-21239-5MATHGoogle Scholar
  74. 74.
    Hesterberg, T.: Advances in importance sampling. Statistics Department, Stanford University (1998)Google Scholar
  75. 75.
    Hein, A.: Parameter- und Quantilschätzung in der Extremwerttheorie. Uni Kaiserslautern (2001)Google Scholar
  76. 76.
    Reiss, R., M., T.: Statistical Analysis of Extreme Values. Birkhäuser (2007)Google Scholar
  77. 77.
    de Haan, L., Ferreira, A.: Extreme value theory. An Introduction. Springer series in operations research and financial engineering. Springer (2000)Google Scholar
  78. 78.
    Li, X., Le, J., Pileggi, L.T.: Projection-based statistical analysis of full-chip leakage power with non-log-normal distributions. In: Proc. DAC 2006, pp. 103–108 (2006)Google Scholar
  79. 79.
    GSA & IET International Semiconductor Forum, London UK, 18-19 May 2010 “Better Analog Modeling and Integration with iPDKs”Google Scholar
  80. 80.
    Hu, C.: Future CMOS scaling and reliability. Proceedings of the IEEE, 81(5) (1993)Google Scholar
  81. 81.
    Wong, B.P., Mittal, A., Cao, Y., Starr, G.: NANO-CMOS Circuit and physical design, John Wiley & Sons, New York (2005)Google Scholar
  82. 82.
    Robertson, J.: High dielectric constant gate oxides for metal oxide Si transistors. Rep. Prog. Phys. 69, 327–396 (2006) Institute physics publishingGoogle Scholar
  83. 83.
    Ge precursors for strained Si and compound semiconductors, semiconductor international, (2006)Google Scholar
  84. 84.
    Risch, L.: Pushing CMOS beyond the roadmap, Proceedings of ESSCIRC, p. 63 (2005)Google Scholar
  85. 85.
    Subramanian, V.: Multiple gate field-effect transistors for future CMOS technologies. IETE Technical review 27, 446–454 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Bernd Lemaitre
    • 1
  • Christoph Sohrmann
    • 2
  • Lutz Muche
    • 2
  • Joachim Haase
    • 2
  1. 1.MunEDA GmbHMunichGermany
  2. 2.Design Automation Division EASFraunhofer Institute for Integrated Circuits IISDresdenGermany

Personalised recommendations