Perfectly Matched Layers versus discrete transparent boundary conditions in quantum device simulations

Abstract

Discrete transparent boundary conditions (DTBC) and the Perfectly Matched Layers (PML) method for the realization of open boundary conditions in quantum device simulations are compared, based on the stationary and time-dependent Schrödinger equation. The comparison includes scattering state, wave packet, and transient scattering state simulations in one and two space dimensions. The Schrödinger equation is discretized by a second-order Crank-Nicolson method in case of DTBC. For the discretization with PML, symmetric second-, fourth, and sixth-order spatial approximations as well as Crank-Nicolson and classical Runge-Kutta timeintegration methods are employed. In two space dimensions, a ring-shaped quantum waveguide device is simulated in the stationary and transient regime. As an application, a simulation of the Aharonov-Bohm effect in this device is performed, showing the excitation of bound states localized in the ring region. The numerical experiments show that the results obtained from PML are comparable to those obtained using DTBC, while keeping the high numerical efficiency and flexibility as well as the ease of implementation of the former method.

Uncertainty Quantification of Charge Transfer through a Nanowire Resonant-Tunneling Diode with an ADHIE-FDTD Method

“…A wave packet is inserted into the computational domain by using a Total-Field Scattered-Field (TFSF) boundary [7] and the computational domain is terminated by Perfectly Matched Layers (PML) [2123].…”

Nonuniform and Higher-order FDTD Methods for the Schrödinger Equation

“…However, for computational efficiency, the real and imaginary parts should be split and only one of both leapfrog schemes should be computed, either ( 19)-( 20) or ( 21)- (22), as was also proposed in [3]. Even though only (18) will be considered in the remainder of this paper, the results are valid for all schemes.…”

“…where L is either L L or L H , for the lower-or higher-order scheme, respectively. This update scheme (18) naturally leads to two independent leapfrog schemes, as the real and imaginary parts at even and odd time steps, respectively, are decoupled from the real and imaginary parts at odd and even time steps, respectively (Fig. 2).…”

“…For notational simplicity, it is easier to write the wave function as the complex scalar equation, as in (18). However, for computational efficiency, the real and imaginary parts should be split and only one of both leapfrog schemes should be computed, either ( 19)-( 20) or ( 21)- (22), as was also proposed in [3].…”

“…The proposed schemes can incorporate a perfectly matched layer (PML) [14] to allow a particle to leave the computational domain. The PML technique for the Schrödinger equation, where a coordinate stretching is performed into the complex plane [15][16][17][18], is applied to (18) both for the lower-order and higher-order accurate scheme. The cells have to be stretched as…”