An efficient 3-D spectral-element method for Schrödinger equation in nanodevice simulation

A three-dimensional (3-D) spectral-element method (SEM) based on Gauss-Lobatto-Legendre (GLL) polynomials is proposed to solve the Schro/spl uml/dinger equation in nanodevice simulation. Galerkin's method is employed to obtain the system equation. The high-order basis functions employed are orthogonal and the numerical quadrature points are the same as the GLL integration points, leading to a diagonal mass matrix and a more sparse stiffness matrix. Thus, the proposed method leads to a regular eigenvalue problem, rather than a generalized eigenvalue problem, greatly reducing the computer-memory requirement and central-processing-unit (CPU) time in comparison with the conventional finite-element method (FEM). Furthermore, the SEM is implemented for high geometrical orders, where curved structures can be modeled up to the accuracy comparable to the interpolation accuracy afforded by the basis functions. Numerical examples verify a spectral accuracy with the interpolation orders, and confirm that higher geometrical orders are essential for curved structures to achieve overall spectral accuracy. Examples of quantum dots in various structures, including a waveguide, are analyzed with mixed boundary conditions. Numerical results show that the SEM is an efficient alternative to conventional FEM and to the finite-difference method (FDM) for nanodevice simulation.

[1]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[2]  Qing Huo Liu,et al.  Nonuniform fast cosine transform and the Chebyshev PSTD algorithm , 1999, IEEE Antennas and Propagation Society International Symposium. 1999 Digest. Held in conjunction with: USNC/URSI National Radio Science Meeting (Cat. No.99CH37010).

[3]  J. M. Rorison,et al.  Quantum Dot Heterostructures , 2000 .

[4]  Jean-Pierre Leburton,et al.  Stark Effect and Single-Electron Charging in Silicon Nanocrystal Quantum Dots , 2001, Physical Models for Quantum Dots.

[5]  D. Komatitsch,et al.  Wave propagation near a fluid-solid interface : A spectral-element approach , 2000 .

[6]  D. Vasileska,et al.  3D simulation of GaAs/AlGaAs quantum dot point contact structures , 1998 .

[7]  Juang,et al.  Stark shift and field-induced tunneling in AlxGa1-xAs/GaAs quantum-well structures. , 1990, Physical review. B, Condensed matter.

[8]  Chuang,et al.  Exact calculations of quasibound states of an isolated quantum well with uniform electric field: Quantum-well Stark resonance. , 1986, Physical review. B, Condensed matter.

[9]  Yia-Chung Chang,et al.  Dynamic behavior of electron tunneling and dark current in quantum well systems under an electric field , 1999 .

[10]  H. Massoud,et al.  Spectral Element Method for the Schrödinger-Poisson System , 2004, 2004 Abstracts 10th International Workshop on Computational Electronics.

[11]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[12]  E. Austin,et al.  Electronic structure of an isolated GaAs-GaAlAs quantum well in a strong electric field. , 1985, Physical review. B, Condensed matter.

[13]  Atta,et al.  Nanoscale device modeling: the Green’s function method , 2000 .

[14]  Chemla,et al.  Numerical calculation of the optical absorption in semiconductor quantum structures. , 1996, Physical review. B, Condensed matter.

[15]  W. Jaskólski,et al.  Resonant tunnelling lifetimes in multi-barrier structures - a complex coordinate approach , 1996 .

[16]  Qing Huo Liu,et al.  Nonuniform Fast Cosine Transform and Chebyshev Pstd Algorithms , 2000 .

[17]  Alain Nogaret,et al.  Left and right tunnelling times of electrons from quantum wells in double-barrier heterostructures investigated by the stabilization method , 1994 .

[18]  Bengt Fornberg,et al.  A practical guide to pseudospectral methods: Introduction , 1996 .

[19]  Chuang,et al.  Exciton Green's-function approach to optical absorption in a quantum well with an applied electric field. , 1991, Physical review. B, Condensed matter.

[20]  Qing Huo Liu,et al.  The 2.5-D multidomain pseudospectral time-domain algorithm , 2003 .

[21]  Q.H. Liu,et al.  A pseudospectral frequency-domain (PSFD) method for computational electromagnetics , 2002, IEEE Antennas and Wireless Propagation Letters.

[22]  D. Gottlieb,et al.  Numerical analysis of spectral methods , 1977 .

[23]  Wood,et al.  Electric field dependence of optical absorption near the band gap of quantum-well structures. , 1985, Physical review. B, Condensed matter.

[24]  P. Matagne,et al.  Three-dimensional self-consistent simulations of symmetric and asymmetric laterally coupled vertical quantum dots , 2004 .

[25]  B. K. Panda,et al.  Electric field effect on the diffusion modified AlGaAs/GaAs single quantum well , 1996 .

[26]  Martha L. Zambrano,et al.  Stark-resonance densities of states, eigenfunctions, and lifetimes for electrons in GaAs/(Al, Ga)As quantum wells under strong electric fields: An optical-potential wave-packet propagation method , 2002 .

[27]  B. K. Panda,et al.  Application of Fourier series methods for studying tunnelling of electrons out of quantum wells in an electric field , 1999 .

[28]  B. K. Panda,et al.  Analytic methods for field induced tunneling in quantum wells with arbitrary potential profiles , 2001 .

[29]  Qing Huo Liu,et al.  The spectral grid method: a novel fast Schrodinger-equation solver for semiconductor nanodevice simulation , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[30]  Qing Huo Liu,et al.  The PSTD algorithm: A time-domain method requiring only two cells per wavelength , 1997 .

[31]  L. Esaki,et al.  Variational calculations on a quantum well in an electric field , 1983 .

[32]  M G Pala,et al.  A three-dimensional solver of the Schrödinger equation in momentum space for the detailed simulation of nanostructures , 2002 .

[33]  Qing Huo Liu,et al.  Multidomain pseudospectral time-domain simulations of scattering by objects buried in lossy media , 2002, IEEE Trans. Geosci. Remote. Sens..

[34]  Branislav M. Notaros,et al.  Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling , 2003 .

[35]  M. Koshiba,et al.  Finite-element analysis of quantum wells of arbitrary semiconductors with arbitrary potential profiles , 1989 .

[36]  Ieee Circuits,et al.  IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems information for authors , 2018, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[37]  Aaron Thean,et al.  Three-Dimensional Self-Consistent Simulation of the Charging Time Response in Silicon Nanocrystal Flash Memories , 2002, Physical Models for Quantum Dots.

[38]  Jan S. Hesthaven,et al.  Spectral Simulations of Electromagnetic Wave Scattering , 1997 .

[39]  P. Harrison.  Quantum wells, wires, and dots : theoretical and computational physics , 2016 .

[40]  William H. Press,et al.  Numerical recipes , 1990 .