The theoretical accomplishments of the Wigner formalism are accompanied by challenging and sometimes peculiar numerical aspects, when solving the associated transport equation. Several numerical methods have been explored over the years to solve the WTE. An overview of the development of the most important deterministic and stochastic solution approaches is given in the following. All the solvers mentioned below are restricted to a single spatial dimension, due to computational constraints.
The first deterministic solvers for the WTE applied the finite difference (FD) scheme [62, 77, 80, 83, 84] and used a relaxation time approximation for the collision operator. The numerical solution of the WTE allowed the study of physically relevant boundary conditions and demonstrated the feasibility of applying the Wigner formalism to study quantum structures, like the resonant tunnelling diode (RTD). Various refinements and additions to the FD-based solvers were made over the years [85]. However, two disadvantages of applying the finite difference scheme have become clear: The discretization of the WTE yields a dense matrix, which is numerically expensive to invert. Furthermore, the solution is sensitive to the chosen discretization of the diffusion term, due to the highly oscillatory nature of the Wigner function in regions with rapid changes in the electrostatic potential [64]. As a result, finite difference schemes remain limited in their application to single-dimensional structures of few tens of nanometres, with moderate potential variations in the active regions. Nonetheless, the high precision offered by deterministic methods remains very desirable, which motivates the continued pursuit of novel deterministic approaches.
The need to find a more efficient discretization of the non-local Wigner kernel, motivated the first alternative approach to the FD method using the spectral collocation method [86], which was later augmented with an operator-splitting technique [87]. After some years of relative inactivity, fresh efforts have started on the deterministic solvers: The spectral element method has recently been introduced [88], which intrinsically has the property of mass conservation. A weighted essentially non-oscillatory (WENO) finite difference solver [89], has tackled the numerical difficulties of the advection term with an adaptive grid in the k-space. A solution based on the integral formulation has also been demonstrated [90].
Phase space formulations always suffer under the curse of dimensionality, which makes a deterministic solution challenging in higher dimensions due to the fine discretization required to accommodate the highly oscillatory nature of the Wigner function. A good finite-dimensional characterization of the solution is required to reduce the size of the linear systems. Recent advancements in deterministic solvers for the BTE using wavelets may also become useful to solve the Wigner equation [91]. The application of a spherical harmonics expansion [92, 93] may also prove promising, but has only been attempted for steady-state solutions [94].
A deterministic solver for multi-dimensional problems still remains out of reach, which motivates the use of stochastic (Monte Carlo) approaches.
Stochastic methods offer an alternative to deterministic methods and their application to solve the Wigner equation has been inspired by the great success of the Monte Carlo approaches to the very similar Boltzmann transport equation [95, 96]. Many classical concepts have been revised and adapted to develop numerical models for computing the quantum quasi-distribution function. Nonetheless, the basis of the method remains the association of trajectories to a single or an ensemble of particles.
Wigner trajectories have been defined with the help of a quantum force [97]. They give insight in quantum phenomena like tunnelling processes, but can be created or destroyed making the important consequences of the Liouville theorem invalid for this particle model. Another particle model introduces the concept of Wigner paths [98]. Here, the action of the Wigner potential operator is interpreted as scattering, which links pieces of classical trajectories to Wigner paths.
Two more recent particle models – the affinity and signed-particle method – exhibit improved numerical efficiency and higher functionality. They unify classical and quantum regions within a single transport picture and allow the consideration of fully three-dimensional wavevector spaces in multi-dimensional devices. The affinity model represents the Wigner function as a sum of Dirac excitations in the phase-space, each weighted by an amplitude, called affinity [99]. The affinities are updated by the Wigner potential during the particle evolution and contain all the information on the quantum state of the system. The affinities can assume positive or negative values, which act as weighting factors in the reconstruction of the Wigner function and consequently in the computation of all physical averages [100]. This approach was also adopted in [64, 101] where the potential is decomposed in a ’slow’ and a ’fast’ varying part, representing the classical electric field (first derivative of potential) and higher-order quantum effects of the potential, respectively; only the fast-varying quantum part of the potential is used to update particle affinities.
The signed-particle method is based on the alternative interpretation of the Wigner potential as a generator of signed particles. The signed-particle method makes use of integer affinities (+1 and -1), which is very advantageous from a computational point of view. In all other aspects the evolution of the particle is field-less and classical. Two particles with opposite sign, which meet in the phase space, may annihilate each other, since they have an equivalent probabilistic future but make an opposite contribution in the process of averaging. Due to the ergodicity of such systems, a single particle Monte Carlo algorithm has been developed [63] and more recently the method has been generalized to also treat transient transport [49].
Indeed, currently the signed-particle method is the only computationally tractable method to solve the WTE in multiple dimensions. The signed-particle method forms the basis of this thesis and will be extensively discussed in Chapters 3 to 5.