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Computational methods for semiclassical and quantum transport in semiconductor devices

Published online by Cambridge University Press:  07 November 2008

Christian Ringhofer
Christian Ringhofer
Affiliation:
Department of Mathematics, Arizona State UniversityTempe, AZ 85287-1804, USA E-mail: ringhofer@asu.edu
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Abstract

The progressive miniaturization of semiconductor devices, and the use of bulk materials other than silicon, necessitates the use of a wide variety of models in semiconductor device simulation. These include classical and semiclassical models, such as the Boltzmann equation and the hydrodynamic system, as well as quantum transport models such as the quantum Boltzmann equation and the quantum hydrodynamic system. This paper gives an overview of recently developed numerical methods for these systems. The focus is on Galerkin methods for the semiclassical and quantum kinetic systems and on difference methods for the classical and quantum hydrodynamic systems. The stability and convergence properties of these methods and their relation to the analytical properties of the continuous systems are discussed.

Type
Research Article
Information
Acta Numerica , Volume 6 , January 1997 , pp. 485 - 521
Copyright
Copyright © Cambridge University Press 1997

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References

REFERENCES

Arnold, A. and Ringhofer, C. (1995 a), ‘An operator splitting method for the Wigner–Poisson problem’, SIAM J. Numer. Anal. 32, 1895–1921.Google Scholar
Arnold, A. and Ringhofer, C. (1995 b), ‘Operator splitting methods applied to spectral discretizations of quantum transport equations’, SIAM J. Numer. Anal. 32, 1876–1894.CrossRefGoogle Scholar
Arnold, A., Degond, P., Markowich, P. and Steinrück, H. (1989), ‘The Wigner–Poisson equation in a crystal’, Appl. Math. Lett. 2, 187–191.CrossRefGoogle Scholar
Ashcroft, N. and Mermin, M. (1976), Solid State Physics, Holt-Saunders, New York.Google Scholar
Baccarani, G. and Wordeman, M. (1985), ‘An investigation of steady state velocity overshoot effects in Si and GaAs devices’, Solid State Electr. 28, 407–416.CrossRefGoogle Scholar
Chen, Z., Cockburn, B., Gardner, C. and Jerome, J. (1995), ‘Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode’, J. Comput. Phys. 117, 274–280.CrossRefGoogle Scholar
Cordier, S. (1994 a), ‘Hyperbolicity of Grad's extension of hydrodynamic models for ionospheric plasmas I: the single species case’, Math. Mod. Meth. Appl. Sci. 4, 625–645.CrossRefGoogle Scholar
Cordier, S. (1994 b), ‘Hyperbolicity of Grad's extension of hydrodynamic models for ionospheric plasmas II: the two species case’, Math. Mod. Meth. Appl. Sci. 4, 647–667.CrossRefGoogle Scholar
Engquist, B. and Majda, A. (1977), ‘Absorbing boundary conditions for the numerical simulation of waves’, Math. Comput. 31, 629–651.CrossRefGoogle Scholar
Ferry, D. and Grubin, H. (1995), ‘Modelling of quantum transport in semiconductor devices’, Solid State Phys. 49, 283–448.CrossRefGoogle Scholar
Gardner, C. (1991 a), ‘Numerical simulation of a steady state electron shock wave in a submicrometer semiconductor device’, IEEE Trans. Electr. Dev. 38, 392–398.CrossRefGoogle Scholar
Gardner, C. (1991 b), Shock waves in the hydrodynamic model for semiconductor devices, in IMA Volumes in Mathematics and its Applications, Vol. 59, pp. 123–134.Google Scholar
Gardner, C. (1992), Upwind simulation of a steady state electron shock wave in a semiconductor device, in Viscous Profiles and Numerical Methods for Shock Waves (Shearer, M., ed.), pp. 21–30.Google Scholar
Gardner, C. (1993 a), The classical and the quantum hydrodynamic models, in Proc. Int. Workshop on Computational Electronics, Leeds 1993 (Snowden, J., ed.), pp. 25–36.Google Scholar
Gardner, C. (1993 b), ‘Hydrodynamic and Monte Carlo simulations of an electron shock wave in a 1μm n + − n − n − diode’, IEEE Trans. Electr. Dev. 40, 455–457.CrossRefGoogle Scholar
Gardner, C. (1994), ‘The quantum hydrodynamic model for semiconductor devices’, SIAM J. Appl. Math. 54, 409–427.CrossRefGoogle Scholar
Gardner, C., Jerome, J. and Rose, D. (1989), ‘Numerical methods for the hydro-dynamic device model’, IEEE Trans. CAD 8, 501–507.CrossRefGoogle Scholar
Goldsman, N., Henrickson, L. and Frey, J. (1991), ‘A physics based analytical-numerical solution to the Boltzmann equation for use in semiconductor device simulation’, Solid State Electr. 34, 389.CrossRefGoogle Scholar
Goldsman, N., Wu, J. and Frey, J. (1990), ‘Efficient calculation of ionization coefficients in silicon from the energy distribution function’, J. Appl. Phys. 68, 1075.CrossRefGoogle Scholar
Grad, H. (1949), ‘On the kinetic theory of rarefied gases’, Comm. Pure Appl. Math. 2, 331–407.CrossRefGoogle Scholar
Grad, H. (1958), ‘Principles of the kinetic theory of gases’, Handbooks Phys. 12, 205–294.Google Scholar
Kersch, A. and Morokoff, W. (1995), Transport Simulation in Microelectronics, Birkhäuser, Basel.CrossRefGoogle Scholar
Lanzkron, P., Gardner, C. and Rose, D. (1991), ‘A parallel block iterative method for the hydrodynamic device model’, IEEE Trans. CAD 10, 1187–1192.Google Scholar
Markowich, P. and Ringhofer, C. (1989), ‘An analysis of the quantum Liouville equation’, ZAMM 69, 121–127.CrossRefGoogle Scholar
Markowich, P., Mauser, N. and Poupaud, F. (1994), ‘A Wigner function approach to semiclassical limits’, J. Math. Phys. 35, 1066–1094.CrossRefGoogle Scholar
Markowich, P., Ringhofer, C. and Schmeiser, C. (1990), Semiconductor Equations, Springer.CrossRefGoogle Scholar
Poupaud, F. (1991), ‘Diffusion approximation of the linear Boltzmann equation: analysis of boundary layers’, Asympt. Anal. 4, 293–317.Google Scholar
Poupaud, F. and Ringhofer, C. (1995), ‘Quantum hydrodynamic models in semiconductor crystals’, Appl. Math. Lett. 8, 55–59.CrossRefGoogle Scholar
Ringhofer, C. (1990), ‘A spectral method for the numerical solution of quantum tunneling phenomena’, SIAM J. Numer. Anal. 27, 32–50.CrossRefGoogle Scholar
Ringhofer, C. (1992), ‘On the convergence of spectral methods for the Wigner-Poisson problem’, Math. Mod. Meth. Appl. Sci. 2, 91–111.CrossRefGoogle Scholar
Ringhofer, C. (1994 a), Galerkin methods for kinetic equations in time variant coordinate systems, in Proc. ‘Mathematical Methods in Semiconductor Simulation’ (Natalini, R., ed.), pp. 32–49.Google Scholar
Ringhofer, C. (1994 b), A series expansion method for the Boltzmann transport equation using variable coordinate systems, in Proc. Int. Wkshp. on Comp. Electr. (Goodnick, S., ed.), Portland, pp. 128–132.Google Scholar
Ringhofer, C. (1997), ‘An adaptive Galerkin procedure for the Boltzmann transport equation’, Math. Mod. Meth. Appl. Sci. To appear.Google Scholar
Ringhofer, C., Ferry, D. and Kluksdahl, N. (1989), ‘Absorbing boundary condition for the simulation of quantum transport phenomena’, Transp. Theory and Stat. Phys. 18, 331–346.CrossRefGoogle Scholar
Schmeiser, C. and Zwirchmayr, A. (1995), Galerkin methods for the semiconductor Boltzmann equation, in Proc. ICIAM 95, Hamburg.Google Scholar
Schmeiser, C. and Zwirchmayr, A. (1997), ‘Convergence of moment methods for the semiconductor Boltzmann equation’, SIAM J. Numer. Anal. To appear.Google Scholar
Selberherr, S. (1981), Analysis of Semiconductor Devices, 2nd edn, Wiley, New York.Google Scholar
Tatarski, V. (1983), ‘The Wigner representation of quantum mechanics’, Soviet. Phys. Uspekhi 26, 311–372.CrossRefGoogle Scholar
Taylor, M. (1981), Pseudodifferential Operators, Princeton University Press, Princeton.CrossRefGoogle Scholar
Ventura, D., Gnudi, A. and Baccarani, G. (1991), One dimensional simulation of a bipolar transistor by means of spherical harmonics expansions of the Boltzmann equation, in Proc. SISDEP 91 Conference (Zürich) (Fichtner, W., ed.), pp. 203–205.Google Scholar
Ventura, D., Gnudi, A., Baccarani, G. and Odeh, F. (1992), ‘Multidimensional spherical harmonics expansions for the Boltzmann equation for transport in semiconductors’, Appl. Math. Lett. 5, 85–90.CrossRefGoogle Scholar
Wigner, E. (1932), ‘On the quantum correction for thermodynamic equilibrium’, Phys. Rev. 40, 749–759.CrossRefGoogle Scholar