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Approximations based on Faber polynomials for the numerical solution of Maxwell’s equations in the time domain are investigated. The exponential convergence properties of the Faber polynomials allow to construct highly accurate time propagation schemes with time steps exceeding the Courant–Friedrichs–Lewy stability limit by several orders of magnitude. Additional material models, such as Drude and Lorentz models, and, moreover, perfectly matched layers to truncate the computational domain are included using additional differential equations. A novel approach to approximate the spectrum of the system matrix is introduced to enable the Faber polynomial expansions for arbitrary device models. [ABSTRACT FROM AUTHOR]

The numerical analysis of transient quantum effects in heterostructure devices with conventional numerical methods tends to pose problems. To overcome these limitations, a novel numerical scheme for the transient non-equilibrium solution of the quantum Liouville equation utilizing a finite volume discretization technique is proposed. Additionally, the solution with regard to the stationary regime, which can serve as a reference solution, is inherently included within the discretization scheme for the transient regime. Resulting in a highly oscillating interference pattern of the statistical density matrix as well in the stationary as in the transient regime, the reflecting nature of the conventional boundary conditions can be an additional source of error. Avoiding these non-physical reflections, the concept of a complex absorbing potential used for the Schrödinger equation is utilized to redefine the drift operator in order to render open boundary conditions for quantum transport equations. Furthermore, the method allows the application of the commonly used concept of inflow boundary conditions. [ABSTRACT FROM AUTHOR]

A novel fully explicit vectorial time-domain algorithm for the analysis of photonic devices capable of solving Maxwell’s equations with arbitrarily large time steps is presented. Furthermore, the time-domain propagation algorithm ensures energy conservation and stability. For this purpose, a unitary time-domain propagation scheme is defined by a suitable polynomial expansion of the propagator. [ABSTRACT FROM AUTHOR]

A new numerical approach is presented for the determination of the statistical density matrix as a solution of the Liouville-von Neumann equation in center-mass coordinates. The numerical discretization is performed by utilizing a finite volume method, which leads to a discretized drift and diffusion operator. The solution is based on the eigenvector basis of the discretized diffusion operator with its corresponding eigenvalues and on the introduction of the self-energy concept. More specifically, the self-energy concept is essential to describe open-boundary problems adequately. Furthermore, this approach allows the definition of inflow and outflow conditions. The method presented is investigated with regard to the conventional Wigner transport equation and the quantum transmitting boundary method, when investigating coherent effects. [ABSTRACT FROM PUBLISHER]

The Wigner function formalism leads to several difficulties in order to investigate quantum effects in heterostructure devices. The corresponding spatial variation of the effective mass distribution is neglected in most methods. Furthermore, conventional methods for the numerical analysis of the Wigner transport equation disregard the quantum effects at the boundaries caused by the nonlocal kinetic operator. A method is proposed, which considers the spatially varying effective mass distribution for the stationary equilibrium case in the ballistic regime and considers an adequate inclusion of the nonlocal kinetic operator. Due to the interaction between the nonlocal kinetic operator and the quantum statistical distribution function at the boundaries, the boundary conditions inherently include the nonlocal character of quantum mechanics. For this purpose a slowly varying envelope function is introduced, which considers the modification of the Wigner function for the spatially varying effective mass distribution on the basis of the Wigner function for the case of a spatially constant effective mass distribution. The envelope function approach is presented and validated. [ABSTRACT FROM PUBLISHER]

The application of the Wigner–Veyl transform for the analysis of quantum transport effects leads to several difficulties and to criticism. One of the difficulties is the numerical discretization of the Wigner transport equations, after the Wigner–Veyl transform is applied onto the statistical density matrix as a part of the quantum Liouville equation. The conventional discretization methods are based on a finite-difference approximation of the resulting Wigner transport equation applying upwind difference schemes or hybrid difference schemes. The results obtained do not match with the statistical density matrix based on the solution of the Schrödinger equation. The contribution addresses the difficulties when the conventional discretization approaches are applied. The difficulties arising from this approach are discussed, and the prerequisites for an improved approach are defined. On the basis of this discussion and the conclusions obtained, a new approach is proposed. This approach is based on an evaluation of the characteristics of the Wigner transport equation in the eigenvalue domain. The new approach is discussed and is applied onto a standard resonant tunneling diode structure. The numerical results are compared with the results from the application of the conventional techniques. [ABSTRACT FROM AUTHOR]