Local Discontinuous Galerkin Methods for the 2D Simulation of Quantum Transport Phenomena on Quantum Directional Coupler

In this paper, we present local discontinuous Galerkin methods (LDG) to simulate an important application of the 2D stationary Schrodinger equation called quantum transport phenomena on a typical quantum directional coupler, which frequency change mainly reflects in y -direction. We present the minimal dissipation LDG (MD-LDG) method with polynomial basis functions for the 2D stationary Schrodinger equation which can describe quantum transport phenomena. We also give the MD-LDG method with polynomial basis functions in x -direction and exponential basis functions in y -direction for the 2D stationary Schrodinger equation to reduce the computational cost. The numerical results are shown to demonstrate the accuracy and capability of these methods.

[1]  Lan Wang,et al.  Spectral-like resolution compact ADI finite difference method for the multi-dimensional Schrödinger equations , 2012, Math. Comput. Model..

[2]  Ling Yuan,et al.  Discontinuous Galerkin method based on non-polynomial approximation spaces , 2006, J. Comput. Phys..

[3]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[4]  Luming Zhang,et al.  A compact finite difference scheme for the nonlinear Schrödinger equation with wave operator , 2012, Appl. Math. Comput..

[5]  Pingwen Zhang,et al.  CONSERVATIVE LOCAL DISCONTINUOUS GALERKIN METHODS FOR TIME DEPENDENT SCHR˜ ODINGER EQUATION , 2005 .

[6]  Hao Wu,et al.  High Order Scheme for Schrödinger Equation with Discontinuous Potential I: Immersed Interface Method , 2011 .

[7]  Peter A. Markowich,et al.  A Wigner-Measure Analysis of the Dufort-Frankel Scheme for the Schrödinger Equation , 2002, SIAM J. Numer. Anal..

[8]  Eric Polizzi,et al.  Subband decomposition approach for the simulation of quantum electron transport in nanostructures , 2005 .

[9]  Harry Yserentant,et al.  A spectral method for Schrödinger equations with smooth confinement potentials , 2012, Numerische Mathematik.

[10]  Kai Fan,et al.  A generalized discontinuous Galerkin (GDG) method for Schrödinger equations with nonsmooth solutions , 2008, J. Comput. Phys..

[11]  Bo Dong,et al.  An Analysis of the Minimal Dissipation Local Discontinuous Galerkin Method for Convection–Diffusion Problems , 2007, J. Sci. Comput..

[12]  Shi Jin,et al.  Numerical Study of Time-Splitting Spectral Discretizations of Nonlinear Schrödinger Equations in the Semiclassical Regimes , 2003, SIAM J. Sci. Comput..

[13]  Chi-Wang Shu,et al.  The WKB Local Discontinuous Galerkin Method for the Simulation of Schrödinger Equation in a Resonant Tunneling Diode , 2009, J. Sci. Comput..

[14]  Sihong Shao,et al.  Adaptive Conservative Cell Average Spectral Element Methods for Transient Wigner Equation in Quantum Transport , 2011 .

[15]  Chi-Wang Shu,et al.  Local discontinuous Galerkin methods for nonlinear Schrödinger equations , 2005 .

[16]  Wei Cai,et al.  Boundary treatments in non-equilibrium Green's function (NEGF) methods for quantum transport in nano-MOSFETs , 2008, J. Comput. Phys..

[17]  Christophe Besse,et al.  A Review of Transparent and Artificial Boundary Conditions Techniques for Linear and Nonlinear Schrödinger Equations , 2008 .

[18]  Tiao Lu,et al.  A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger-Poisson equations with discontinuous potentials , 2008 .

[19]  Hao Wu,et al.  A Generalized Stationary Algorithm for Resonant Tunneling: Multi-Mode Approximation and High Dimension , 2011 .

[20]  P. Markowich,et al.  On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime , 2002 .

[21]  Peter A. Markowich,et al.  Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit , 1999, Numerische Mathematik.