High Order Finite Difference Schemes for the Transparent Boundary Conditions and Their Applications in the 1D Schrödinger-Poisson Problem

Abstract

The 1D Schrödinger equation closed with the transparent boundary conditions(TBCs) is known as a successful model for describing quantum effects, and is usually considered with a self-consistent Poisson equation in simulating quantum devices. We introduce discrete fourth order transparent boundary conditions(D4TBCs), which have been proven to be essentially non-oscillating when the potential vanishes, and to share the same accuracy order with the finite difference scheme used to discretize the 1D Schrödinger equation. Furthermore, a framework of analytic discretization of TBCs(aDTBCs) is proposed, which does not introduce any discretization error, thus is accurate. With the accurate discretizations, one is able to improve the accuracy of the discretization for the 1D Schrödinger problem to arbitrarily high levels. As numerical tools, two globally fourth order compact finite difference schemes are proposed for the 1D Schrödinger-Poisson problem, involving either of the D4TBCs or the aDTBCs, respectively, and the uniqueness of solutions of both discrete Schrödinger problems are rigorously proved. Numerical experiments, including simulations of a resistor and two nanoscale resonant tunneling diodes, verify the accuracy order of the discretization schemes and show potential of the numerical algorithm introduced for the 1D Schrödinger-Poisson problem in simulating various quantum devices.

Figures (12)

• Figure 1: Norm of numerical wave functions of the Schrödinger problem, where different discrete boundary conditions are compared
• Figure 2: Skeleton of a three dimensional $n^{++}-n^+-n^{++}$ resistor, where the transport of electrons in the x-direction is of concern
• Figure 3: Numerical results about the potential functions $V(x)$(on the left) and density functions $n(x)$(on the right) simulated with two samplings of $V_{ds}$, i.e., $0\ \rm (V)$ and $0.25\ \rm (V)$, respectively. The DSP1(in blue solid line) and the DSP2(in red dash line) are both considered, for both values of $V_{ds}$, separately. Corresponding results(in star) solved with the second order NEGF method are shown as benchmarks
• Figure 4: Tendency of $\log_{10}\left(\boldsymbol{\delta} \boldsymbol{V}_{\Delta x}\right)$ with respect to decreasing $\Delta x$, where the DSP1(a) and the DSP2(b) are considered separately
• Figure 5: I-V characteristic curves of the 1D resistor simulated with decreasing $\Delta x$, where the DSP1(a) and the DSP2(b) are considered, respectively

Theorems & Definitions (9)

• remark 1
• lemma 1
• proof
• theorem 1
• lemma 2
• proof
• theorem 2
• proof
• theorem 3