Wigner-Boltzmann Equation Research Articles

Quantum transport for electron quasiparticles interacting with a disordered binary alloy: A Wigner approach

Quantum transport is developed in the Wigner function representation for a Bloch electron quasiparticle interacting with a disordered binary alloy in the presence of a homogeneous electric field of arbitrary time dependence and amplitude. The electron quasiparticle is described by a single-band effective Hamiltonian, and the homogeneous electric field is treated in the vector potential gauge. The methodology for the quantum transport analysis proceeds by first transforming the Liouville equation to one in which the interaction Hamiltonian (the binary alloy Hamiltonian) appears quadratically. The basis states employed in evaluating the requisite matrix elements are the instantaneous eigenstates of the electron quasiparticle Hamiltonian in the presence of the electric field. The Wigner quantum transport equations are derived, and the binary alloy collision term is suitably ensemble averaged over the disordered binary alloy matrix elements. In addition, the general drift and diffusion terms are exactly obtained, resulting in the complete Wigner-Boltzmann equation for the binary alloy system. In approximating the collision term for the two separate cases of parabolic energy dispersion and the long-wavelength limit, it is found that the reduced Wigner-Boltzmann equation includes the manifestation of the intracollisional field effect and other quantum generalities. As a contrast to the actual random alloy treatment, attention is given to the canonical problem of quantum transport for a virtual crystal (VC). As an alternative to the random binary alloy scattering problem, the VC Hamiltonian adopted for this treatment is derived by ensemble averaging the random binary alloy Hamiltonian. The resulting Wigner transport equation for the VC case is descriptive of Bloch dynamics in a graded semiconductor alloy.

Quantum energy-transport and drift-diffusion models for electron transport in graphene: an approach by the wigner function

The present work aims at formulating quantum energy-transport and drift-diffusion equations for charge transport in graphene from a quantum hydrodynamic model proposed in Luca and Romano (Ann. Phys. 406:30–53, 2019), obtained from the Wigner-Boltzmann equation via the moment method. Here, we only sketch the main ideas and show the most relevant formulas. Further developments are under current investigation by the authors. The interested reader can find the omitted details in the references. In analogy with the semiclassical case, we are confident that the energy-transport and drift-diffusion models have mathematical properties which allow an easier numerical treatment.

Quantum corrected hydrodynamic models for charge transport in graphene

A quantum hydrodynamic model for charge transport in graphene is derived from a moment expansion of the Wigner–Boltzmann equation. The needed closure relations are obtained by adding quantum corrections based on the equilibrium Wigner function to the semiclassical model formulated in Camiola and Romano (2014) and Luca and Romano (2018) [2,3] by exploiting the Maximum Entropy Principle. The expression of the equilibrium Wigner function which takes into account the energy band of graphene has been obtained by solving the corresponding Bloch equation.

Introduction to the special issue on Wigner functions

The Wigner function was first derived by E.P. Wigner around 1931 as an exotic outcome of the wave mechanics which, being defined in the phase space, does not favor either coordinate or momentum variables. Pioneered by H.J. Groenewold and J.E.Moyal theWigner formulation of quantum mechanics evolved until the end of last century, to a fully autonomous, independent alternative to Hilbert space mechanics and path integral formulations. As S. MacLane put it (Ladies and) gentlemen: There is lots of room left in Hilbert Space, the same holds true for the phase space, where Wigner and alternative approaches, likeHusimi andBohmian distributions, are easily accommodated. Applied to challenges in computational electronics, the Wigner approach offers the unique ability to describe transient problems for open systems by simultaneously accounting for different levels of approximation for the effects of the degrees of freedom associated with the environment. For example, the Wigner–Boltzmann equation enables the formalism with the efficient Boltzmann model of collisions (local in position/time) with phonons and impurities featuring the classical era of microelectronics. The equation bridges the gap between classical and quantum electron dynamics, ensuring a seamless transition between coherent and scattering dominated modes of transport. This approach

A simple method to obtain the equilibrium solution of Wigner-Boltzmann equation with all higher order quantum corrections

A simple method has been introduced to furnish the equilibrium solution of the Wigner equation for all order of the quantum correction. This process builds up a recursion relation involving the coefficients of the different power of the velocity. The technique greatly relies upon the proper guess work of the trial solution and is different from the Wigner s original work. The solution is in a compact exponential form with a polynomial of velocity in the argument and returns the Wigner s form when expansion of the exponential factor is carried out. The study keeps its importance in studying various close as well as open quantum mechanical system. In addition, this solution may be employed to obtain the non equilibrium one particle wigner distribution in the relaxation-time approximation and under near-equilibrium conditions.

Wigner distribution functions for complex dynamical systems: The emergence of the Wigner-Boltzmann equation

The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.

Physical scales in the Wigner–Boltzmann equation

The Wigner–Boltzmann equation provides the Wigner single particle theory with interactions with bosonic degrees of freedom associated with harmonic oscillators, such as phonons in solids. Quantum evolution is an interplay of two transport modes, corresponding to the common coherent particle-potential processes, or to the decoherence causing scattering due to the oscillators. Which evolution mode will dominate depends on the scales of the involved physical quantities. A dimensionless formulation of the Wigner–Boltzmann equation is obtained, where these scales appear as dimensionless strength parameters. A notion called scaling theorem is derived, linking the strength parameters to the coupling with the oscillators. It is shown that an increase of this coupling is equivalent to a reduction of both the strength of the electric potential, and the coherence length. Secondly, the existence of classes of physically different, but mathematically equivalent setups of the Wigner–Boltzmann evolution is demonstrated.

Time dependent transport in 1D micro- and nanostructures: Solving the Boltzmann and Wigner–Boltzmann equations

For many decades the Boltzmann distribution function has been used to calculate the non-equilibrium properties of mobile particles undergoing the combined action of various scattering mechanisms and externally applied force fields. When the latter give rise to the occurrence of inhomogeneous potential profiles across the region through which the particles are moving, the numerical solution of the Boltzmann equation becomes a highly complicated task. In this work we highlight a particular algorithm that can be used to solve the time dependent Boltzmann equation as well as its quantum mechanical extension, the Wigner–Boltzmann equation. As an illustration, we show the calculated distribution function describing electrons propagating under the action of both a uniform and a pronouncedly non-uniform electric field.

Quantum Transport and Boltzmann Operators

In this paper the transport of quantum particles in time-dependent random media is studied. In the white noise limit, a quantum model for collisions is obtained. At the level of Wigner equation, this limit is described by a linear Wigner-Boltzmann equation.

Incorporation of quantum corrections to semiclassical two-dimensional device modeling with the Wigner–Boltzmann equation

A new method for accounting for quantum effects in semiclassical device simulation is presented. The approach is based on the solution to the Wigner–Boltzmann equation, which is derived from the Schrodinger equation. The Wigner–Boltzmann equation is truncated to order ℏ 2, and then formulated using spherical harmonics. This facilitates analytical evaluation of the collision integral, and allows for reduction of dimensionality. The Wigner–Boltzmann equation is solved self-consistently with the Poisson and hole-current continuity equations for a BJT and a MOSFET. The results show that the carrier concentrations predicted by the Wigner–Boltzmann equation near the base–emitter junction and the MOSFET channel are less than that predicted by the semiclassical Boltzmann model. Calculations show terminal currents to be 2% and 7% lower for the quantum simulations than the semiclassical results for the BJT and MOSFET, respectively.

Semiconductor device simulation: The hydrodynamic model

VLSI (very large scale integrated) chips incorporate hundreds of millions of semiconductor circuits. To predict the performance of VLSI circuits, the current-voltage (I-V) characteristics of the semiconductor devices are required. Semiconductor device simulation codes provide a way of predicting I-V curves as device parameters are varied, without having to fabricate the device first. Thus, many different designs for devices and circuits can be explored efficiently using computer simulations. Promising designs then can be selected for actual fabrication and testing. A fundamental approach to modeling the quantum transport of electrons and holes in semiconductor devices is the Wigner-Boltzmann equation, the quantum generalization of the Boltzmann equation. Simulating these kinetic equations is computationally expensive. Thus, a hydrodynamic approximation to the kinetic equations, where the density, velocity and temperature of a charge carrier are functions only of three spatial dimensions plus time, offers enormous computational speedups in simulating devices. The article discusses the use of classical and quantum hydrodynamic models in semiconductor device simulation.

PARTICLE MODELS FOR DEVICE SIMULATION

A theoretical analysis of the Monte Carlo (MC) method for both semiclassical and quantum device simulation is presented. A link between physically-based MC methods for semiclassical transport calculations and the numerical MC method for solving integrals and integral equations is established. The integral representations of the transient and the stationary Boltzmann equations are presented as well as the respective conjugate equations. The structure of the terms of the Neumann series and their evaluation by MC integration is discussed. Using this formal approach the standard MC algorithms and a variety of new algorithms is derived, such as the backward and the weighted algorithms, and algorithms for linear small-signal analysis. Applying this theoretical framework to the Wigner-Boltzmann equation enables the development of particle models for quantum transport problems.

The Chapman-Enskog Expansion and the Quantum Hydrodynamic Model for Semiconductor Devices

A “smooth” quantum hydrodynamic (QHD) model for semiconductor devices is derived by a Chapman-Enskog expansion of the Wigner-Boltzmann equation which can handle in a mathematically rigorous way the discontinuities in the classical potential energy which occur at heterojunction barriers in quantum semiconductor devices. A dispersive quantum contribution to the heat flux term in the QHD model is introduced.

The Quantum Hydrodynamic Model for Semiconductor Devices

The classical hydrodynamic equations can be extended to include quantum effects by incorporating the first quantum corrections. These quantum corrections are $O( {\hbar ^2 } )$. The full three-dimensional quantum hydrodynamic (QHD) model is derived for the first time by a moment expansion of the Wigner–Boltzmann equation. The QHD conservation laws have the same form as the classical hydrodynamic equations, but the energy density and stress tensor have additional quantum terms. These quantum terms allow particles to tunnel through potential barriers and to build up in potential wells.The three-dimensional QHD transport equations are mathematically classified as having two Schrodinger modes, two hyperbolic modes, and one parabolic mode. The one-dimensional steady-state QHD equations are discretized in conservation form using the second upwind method.Simulations of a resonant tunneling diode are presented that show charge buildup in the quantum well and negative differential resistance (NDR) in the current-v...