The preceding section reviewed the transport models commonly used in device TCAD and nanoelectronics research. In light of this the main advantages of using Wigner-based simulations for quantum transport are i) the classical analogies that arise from the phase-space formalism, ii) the quite straight-forward inclusion of scattering effects allowing decoherence to be investigated and iii) the ability to simulate time-resolved quantum transport with reasonable computational effort. These three points are elaborated upon in the following, whereafter some problems well-suited to be solved in this formalism are presented.
The Wigner formulation of quantum mechanics retains many classical concepts and notions, which makes it a convenient approach to describe the transport phenomena characterizing the evolution of electrons in nanostructures. The Wigner formalism expresses quantum mechanics using functions and phase space variables, as opposed to wave functions and operators, as used in the Schrödinger equation. This phase space formulation offers a more intuitive interpretation of quantum phenomena.
The phase space representation of the Wigner formalism provides a clear analogy to classical notions. The most striking example of this is the (initially phenomenological) augmentation of the Wigner equation with the semi-classical scattering models used in the Boltzmann transport equation; a formal derivation introducing the scattering terms to the Wigner equation has now been shown [41]. Furthermore, the phase space formulation is advantageous to specify and recover classical distributions at boundaries [42].
Some caution, however, is justified when trying to apply concepts from classical physics to interpret the Wigner picture. A common mistake is interpreting the Wigner function as a true probability density function, which it is not. Unlike a distribution function, the Wigner function may attain negative values, which are a manifestation of the uncertainty relation in the phase space [43, 44]. Nonetheless, the Wigner function retains the necessary properties of a true distribution function, which allows the calculation of physical averages using the same expressions as used in the case of the Boltzmann formulation. Therefore, the Wigner function is sometimes called a quasi-distribution function. Alternative phase space formulations, like the Husimi function [45], recover a positive definite function by smoothing the Wigner function over a wavelength with a Gaussian kernel, but fail in other respects [45].
The straight-forward inclusion of scattering mechanisms in the Wigner formalism enables the description of decoherence processes which are of fundamental interest, when investigating the evolution of quantum states. A hierarchy of Wigner transport models with scattering can be derived: These begin with the simple relaxation time approximation, the Wigner-Boltzmann equation [41, 46], which accounts for scattering by phonons and impurities at the classical transport level, and end with the quite complicated Levinson and Barker-Ferry equations, which account for the quantum character of the interaction with the sources of decoherence [47]. Of central interest is the Wigner-Boltzmann equation which, as the name suggests, unifies the two theories and ensures a seamless transition between purely coherent and classical transport [48] – the Wigner function gradually turns into the Boltzmann distribution function. This transition occurs either, when phonon scattering is significant or the potential varies very smoothly [20, 49]. The Wigner formalism bridges the gap between purely quantum (ballistic) and classical (diffusive) transport in a seamless fashion.
Currently, the Wigner-Boltzmann equation presents the only computationally attractive quantum model to consider scattering effects in multi-dimensional simulations. Nanoscale devices, e.g. silicon nanowires [50], which exploit coherent quantum phenomena, should be able to operate at room temperature to be viable as a commercial technology. Since phonon scattering increases with temperature, the inclusion of scattering in simulation is of significant importance to simulate the decoherence effects taking place.
The transient evolution of quantum states is of considerable interest in nanoelectronic devices as “Many basic concepts remain to be clarified in the area of time-dependent current flow as well as current fluctuations” [9]. Nano-circuits, which are formed by simple nanostructures, exhibit an electrical behaviour which cannot be explained by classical theory. Time-resolved simulations of quantum transport will help to resolve the questions which surround the frequency response of quantum capacitors and resistances. The Wigner formalism easily allows the time-dependent behaviour, like oscillations and switching times, to be investigated. This is an aspect which cannot currently be investigated by NEGF simulations due to the excessive computational costs.
Wigner-based simulations require the consideration of seven dimensions, whereas NEGF simulations require eight. The higher dimensionality of the NEGF simulations makes it inherently more computationally demanding, but allows the description of temporal correlations. This theoretical advantage has not been exploited up to now due to the exorbitant computational demands, although the first promising efforts are emerging [29, 30]. Transient simulations, neglecting temporal correlations, can be naturally treated in the Wigner picture. Admittedly, this is also possible with the Schrödinger equation/density matrix, however, the inclusion of scattering effects is problematic.
Problems with (some of) the following properties are deemed well-suited to be investigated using the Wigner formalism [42, 51]: