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A Generalized Stationary Algorithm for Resonant Tunneling: Multi-Mode Approximation and High Dimension

Published online by Cambridge University Press:  20 August 2015

Hao Wu*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, 10084, China Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France
*
Corresponding author.Email:[email protected]
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Abstract

The multi-mode approximation is presented to compute the interior wave function of Schrödinger equation. This idea is necessary to handle the multi-barrier and high dimensional resonant tunneling problems where multiple eigenvalues are considered. The accuracy and efficiency of this algorithm is demonstrated via several numerical examples.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2011

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References

[1]Ben Abdallah, N., On a multidimensional Schrädinger-Poisson scattering model for semiconductors, J. Math. Phys., 41(2000), no. 7, 42414261.Google Scholar
[2]Ben Abdallah, N., Degond, P. and Markowich, P.A., On a one-dimensional Schrädinger-Poisson scattering model, Z. angew. Math. Phys., 48(1997), 135155.Google Scholar
[3]Ben Abdallah, N., Negulescu, C., Mouis, M. and Polizzi, E., Simulation Schemes in 2D Nanoscale MOSFETs: A WKB Based Method, J. Comput. Elect., 3(2004), 397400.Google Scholar
[4]Abdallah, N.Ben and Pinaud, O., Multiscale simulation of transport in an open quantum system: Resonances and WKB interpolation, J. Comput. Phys., 213(2006), no. 1, 288310.Google Scholar
[5]Bao, W.Z., Jin, S. and Markowich, P.A., On time-splitting spectral approximations for the Schrädinger equation in the semiclassical regime, J. Comput. Phys., 175(2002), 487524.Google Scholar
[6]Bao, W.Z., Jin, S. and Markowich, P.A., Numerical studies of time-splitting spectral discretizations of nonlinear Schrädinger equations in the semiclassical regime, SIAM J. Sci. Comput., 25(2003), no. 1, 2764.Google Scholar
[7]Datta, S., Electronic Transport in Mesoscopic Systems, Cambridge University Press, 1995.Google Scholar
[8]Datta, S., Quantum Transport: Atom to Transistor, Cambridge University Press, 2005.Google Scholar
[9]Faraj, A. and Ben Abdallah, N., An improved transient algorithm for resonant tunneling, preprint.Google Scholar
[10]Ferry, D.K. and Goodnick, S.M., Transport in Nanostructures, Cambridge University Press, 1997.Google Scholar
[11]Gummel, H.K., Self-consistant iterative scheme for one-dimensional steady state transistor calculation, IEEE Trans. Electron Devices, 11(1964), 455465.CrossRefGoogle Scholar
[12]Jiang, H.Y., Cai, W. and Tsu, R., Accuracy of the Frensley inflow boundary condition for Wigner equations in simulating resonant tunneling diodes, J. Comput. Phys., 230(2011), 20312044.Google Scholar
[13]Jiang, H.Y., Shao, S.H., Cai, W. and Zhang, P.W., Boundary treatments in non-equilibrium Green’s function(NEGF) methods for quantum transport in nano-MOSFETs, J. Comput. Phys., 227(2008), no. 13, 65536573.Google Scholar
[14]Jin, S., Wu, H. and Yang, X., Gaussian beam methods for the Schrädinger equation in the semi-classical regime: Lagrangian and Eulerian formulations, Commun. Math. Sci., 6(2008), no. 4, 9951020.Google Scholar
[15]Jin, S., Wu, H. and Yang, X., A numerical study of the Gaussian beam method for Schrädinger-Poisson equations, J. Comput. Math., 28(2010), no. 2, 261272.Google Scholar
[16]Jona-Lasinio, G., Presilla, C. and Sjöstrand, J., On Schrädinger equations with concentrated nonlinearities, Ann. Phys., 240(1995), 121.Google Scholar
[17]Lent, C. and Kirkner, D., The quantum transmitting boundary method, J. Appl. Phys., 67(1990), 63536359.Google Scholar
[18]Markowich, P.A., Pietra, P. and Pohl, C., Numerical approximation of quadratic obaservables of Schrädinger-type equations in the semiclassical limit, Numerische Mathematik, 81(1999), no. 4, 595630.Google Scholar
[19]Markowich, P.A., Pietra, P., Pohl, C. and Stimming, H.P., A Wigner-measure analysis of the Dufort-Frankel scheme for the Schrädinger equation, SIAM J. Numer. Anal., 40(2002), no. 4, 12811310.CrossRefGoogle Scholar
[20]Markowich, P.A., Ringhofer, C. and Schmeiser, C., Semiconductor Equations, Springer Verlag Wien, 1990.Google Scholar
[21]Mizuta, H. and Tanou, T., The Physics and Applications of Resonant Tunnelling Diodes, Cambridge University Press, 1995.Google Scholar
[22]Pathria, D., Morris, J.LL., Pseudo-spectral solution of nonlinear Schrädinger equations, J. Comput. Phys., 87(1990), no.1, 108125.Google Scholar
[23]Pinaud, O., Transient simulations of a resonant tunneling diode, J. Appl. Phys., 92(2002), no. 4, 19871994.CrossRefGoogle Scholar
[24]Polizzi, E. and Ben Abdallah, N., Subband decomposition approach for the simulation of quantum electron transport in nanostructures, J. Comput. Phys., 202(2005), no. 1, 150180.Google Scholar
[25]Presilla, C. and Sjöstrand, J., Transport properties in resonant tunneling heterostructures, J. Math. Phys., 37(1996), no. 10, 48164844.Google Scholar
[26]Shao, S.H., Cai, W. and Tang, H.Z., Accurate calculation of Green’s function of the Schrädinger equation in a block layered potential, J. Comput. Phys., 219(2006), no. 2, 733748.Google Scholar
[27]Weisbuch, C. and Vinter, B., Quantum Semiconductor Structures: Fundamentals and Applications, Academic Press, 1991.Google Scholar