Paul Ellinghaus

Implications of the Coherence Length on the Discrete Wigner Potential

The solution of the Wigner equation, using the Monte Carlo method [1] along with the signed-particle technique [2], requires a finite coherence length to be chosen. We investigate how the choice of the coherence length influences computational aspects of the calculation of the Wigner potential, like momentum resolution. Additionally, the physical interpretation attributed to a chosen coherence length is discussed.

The Wigner Monte Carlo method for accurate semiconductor device simulation

The Wigner equation can conveniently describe quantum transport problems in terms of particles evolving in the phase space. Improvements in the particle generation scheme of the Wigner Monte Carlo method are shown, which increase the accuracy of simulations as validated by comparison to exact solutions of the Schrödinger equation. Simulations with a time-varying potential are demonstrated and issues which arise in devices with an externally applied voltage between the contacts are treated, thereby further advancing the Wigner Monte Carlo method for the simulation of semiconductor devices.

Optimized Particle Regeneration Scheme for the Wigner Monte Carlo Method

The signed-particle Monte Carlo method for solving the Wigner equation has made multi-dimensional solutions numerically feasible. The latter is attributable to the concept of annihilation of independent indistinguishable particles, which counteracts the exponential growth in the number of particles due to generation. After the annihilation step, the particles regenerated within each cell of the phase-space should replicate the same information as before the annihilation, albeit with a lesser number of particles. Since the semi-discrete Wigner equation allows only discrete momentum values, this information can be retained with regeneration, however, the position of the regenerated particles in the cell must be chosen wisely. A simple uniform distribution over the spatial domain represented by the cell introduces a ‘numerical diffusion’ which artificially propagates particles simply through the process of regeneration. An optimized regeneration scheme is proposed, which counteracts this effect of ‘numerical diffusion’ in an efficient manner.

Deterministic Solution of the Discrete Wigner Equation

The Wigner formalism provides a convenient formulation of quantum mechanics in the phase space. Deterministic solutions of the Wigner equation are especially needed for problems where phase space quantities vary over several orders of magnitude and thus can not be resolved by the existing stochastic approaches. However, finite difference schemes have been problematic due to the discretization of the diffusion term in this differential equation. A new approach, which uses an integral formulation of the Wigner equation that avoids the problematic differentiation, is shown here. The results of the deterministic method are compared and validated with solutions of the Schrodinger equation. Furthermore, certain numerical aspects pertaining to the demanded parallel implementation are discussed.

Stochastic analysis of surface roughness models in quantum wires

Abstract We present a signed particle computational approach for the Wigner transport model and use it to analyze the electron state dynamics in quantum wires focusing on the effect of surface roughness. Usually surface roughness is considered as a scattering model, accounted for by the Fermi Golden Rule, which relies on approximations like statistical averaging and in the case of quantum wires incorporates quantum corrections based on the mode space approach. We provide a novel computational approach to enable physical analysis of these assumptions in terms of phase space and particles. Utilized is the signed particles model of Wigner evolution, which, besides providing a full quantum description of the electron dynamics, enables intuitive insights into the processes of tunneling, which govern the physical evolution. It is shown that the basic assumptions of the quantum-corrected scattering model correspond to the quantum behavior of the electron system. Of particular importance is the distribution of the density: Due to the quantum confinement, electrons are kept away from the walls, which is in contrast to the classical scattering model. Further quantum effects are retardation of the electron dynamics and quantum reflection. Far from equilibrium the assumption of homogeneous conditions along the wire breaks even in the case of ideal wire walls.

Parallelization of the Two-Dimensional Wigner Monte Carlo Method

A parallelization approach for two-dimensional Wigner Monte Carlo quantum simulations using the signed particle method is introduced. The approach is based on a domain decomposition technique, effectually reducing the memory requirements of each parallel computational unit. We depict design and implementation specifics for a message passing interface-based implementation, used in the Wigner Ensemble Monte Carlo simulator, part of the free open source ViennaWD simulation package. Benchmark and simulation results are presented for a time-dependent, two-dimensional problem using five randomly placed point charges. Although additional communication is required, our method offers excellent parallel efficiency for large-scale high-performance computing platforms. Our approach significantly increases the feasibility of computationally highly intricate two-dimensional Wigner Monte Carlo investigations of quantum electron transport in nanostructures.

Efficient calculation of the two-dimensional Wigner potential

The solution of the two-dimensional (2D) Wigner equation has become numerically feasible in recent times, using the Monte Carlo method fortified with the notion of signed particles. The calculation of the Wigner potential (WP) in these 2D simulations consumes a considerable part of the computation time. A reduction of the latter is therefore very desirable, in particular, if self-consistent solutions are pursued, where the WP must be recalculated many times. An algorithm is introduced here - named box discrete Fourier transform (BDFT) - that reduces the computational effort roughly by a factor of five.

Memory-efficient particle annihilation algorithm for Wigner Monte Carlo simulations

The Wigner Monte Carlo solver, using the signed-particle method, is based on the generation and annihilation of numerical particles. The memory demands of the annihilation algorithm can become exorbitant, if a high spatial resolution is used, because the entire discretized phase space is represented in memory. Two alternative algorithms, which greatly reduce the memory requirements, are presented here.

Improved drive-current into nanoscaled channels using electrostatic lenses

The contact regions in nanoscaled transistors play an increasingly important role in the overall performance of the devices. An electrostatic lens in the source contact region to focus a beam of electron wave packets into a nanoscaled channel is investigated here, using a Wigner Ensemble Monte Carlo simulator. An improvement in the drive-current is achieved by reducing reflections from the surrounding oxide. The associated modifications to the momentum distributions are readily shown by using the phase space description of the Wigner formalism.

Optimization of the Deterministic Solution of the Discrete Wigner Equation

The development of novel nanoelectronic devices requires methods capable to simulate quantum-mechanical effects in the carrier transport processes. We present a deterministic method based on an integral formulation of the Wigner equation, which considers the evolution of an initial condition as the superposition of the propagation of particular fundamental contributions.