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We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition in (Barletti and weifel, Trans. Theory Stat. Phys., 507--520, 2001). The decomposition reduces the half-range, two-point boundary value problem into two decoupled initial value problems of the even part and the odd part. Without using a cutoff approximation around zero velocity, we prove that the initial value problem for the even part is well-posed. For the odd part, we prove the uniqueness of the solution in the odd $L^2$-space by analyzing the moment system. An example is provided to show that how to use the analysis to obtain the solution of the stationary Wigner equation with inflow boundary conditions.

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... The wellposedness of the stationary Wigner inflow boundary value problem (SWIBVP), i.e., (1)-(3), is still an open problem, where many efforts have been made to solve it [14][15][16][17]. In [16], the wellposedness of the SWIBVP with periodic potential has been proved. ...

... In [17], the authors have discussed the wellposedness in general cases by using the paritydecomposition and the moment method, of which the wellposedness is obtained partially. Besides Frensley's method [13], plenty of works have been done in solving the Wigner equation by using the finite difference method [18][19][20][21], spectral methods [22][23][24][25], moment methods [26,27], the Monte-Carlo method [28,29], etc. Due to lack of a unified theory of wellposedness of the SWIBVP, numerical results solved with different numerical methods are controversial. ...

... , then the solution of the discrete velocity SWIBVP, i.e., (17) and (18), is also symmetric about x = L 2 , i.e., ...

The stationary Wigner inflow boundary value problem (SWIBVP) is modeled as an optimization problem by using the idea of shooting method in this paper. To remove the singularity at \(v=0\), we consider a regularized SWIBVP, where a regularization constraint is considered along with the original SWIBVP, and a modified optimization problem is established for it. A shooting algorithm is proposed to solve the two optimization problems, involving the limited-memory BFGS (L-BFGS) algorithm as the optimization solver. Numerical results show that solving the optimization problems with respect to the SWIBVP with the shooting algorithm is as effective as solving the SWIBVP with Frensley’s numerical method (Frensley in Phys Rev B 36:1570–1580, 1987). Furthermore, the modified optimization problem gets rid of the singularity at \(v=0\), and preserves symmetry of the Wigner function, which implies the optimization modeling with respect to the regularized SWIBVP is successful.

... long-standing open problem even in a single spatial dimension [1,14,15]. ...

... 1. Setting k = 0 in Eq. (8) gives the following integrability constraint In this degenerate case, the left hand side in Eq. (8) vanishes and we do not get a differential equation. This special case is called an algebraic constraint in [1,15] and is needed to avoid poles on the right hand side of Eq. (17). The integrability constraint has also a physical interpretation in the particle picture: The total potential inscattering rate at k = 0 must vanish in the steady state. ...

We discuss boundary value problems for the characteristic stationary von Neumann equation (stationary sigma equation) and the stationary Wigner equation in a single spatial dimension. The two equations are related by a Fourier transform in the non-spatial coordinate. In general, a solution to the characteristic equation does not produce a corresponding Wigner solution as the Fourier transform will not exist. Solution of the stationary Wigner equation on a shifted k-grid gives unphysical results. Results showing a negative differential resistance in IV-curves of resonant tunneling diodes using Frensley’s method are a numerical artefact from using upwinding on a coarse grid. We introduce the integro-differential sigma equation which avoids distributional parts at k=0 in the Wigner transform. The Wigner equation for k=0 represents an algebraic constraint needed to avoid poles in the solution at k=0. We impose the inverse Fourier transform of the integrability constraint in the integro-differential sigma equation. After a cutoff, we find that this gives fully homogeneous boundary conditions in the non-spatial coordinate which is overdetermined. Employing an absorbing potential layer double homogeneous boundary conditions are naturally fulfilled. Simulation results for resonant tunneling diodes from solving the constrained sigma equation in the least squares sense with an absorbing potential reproduce results from the quantum transmitting boundary with high accuracy. We discuss the zero bias case where also good agreement is found. In conclusion, we argue that properly formulated open boundary conditions have to be imposed on non-spatial boundaries in the sigma equation both in the stationary and the transient case. When solving the Wigner equation, an absorbing potential layer has to be employed.

... The well-posedness of the stationary Wigner transport equation with inflow boundary conditions have attracted the attention of many mathematicians, but it is still an open problem and is only partially solved in [1,2,7,8]. One big issue is that if L 2 (R) is a suitable solution space for (1.10). ...

In this work, the abstract Cauchy problem for an initial value system with singular integral is considered. The system is of closed form of two evolution equations for a real-valued function and a function-valued function. By proposing an appropriate Banach space, we prove the existence and uniqueness of classical solutions to the evolution system under assumptions on the boundedness and smoothness of data. Furthermore, we connect by an isomorphism the solution of the evolution system and a class of integral-differential equations with singular convolution kernels and extend our results to the corresponding problem. It is revealed that our findings also improve the understanding of the open problem on the well-posedness of the stationary Wigner equation with inflow boundary conditions.

The Wigner equation is a remarkable tool to model complex problems of quantum physics in phase space. The main objective of this paper is to propose a new hybrid algorithm for the time-dependent Wigner equation. This scheme is based on sinc-Galerkin and finite difference approximations and is moderately simple but highly efficient. Error estimation, stability, and convergence are also investigated concretely. Numerical experiments validate the theoretical results and present the reliability and efficiency of the proposed algorithm to simulate quantum effects.

The Wigner equation describing stationary quantum transport has a singularity at the point \(k=0\). Deterministic solution methods usually deal with the singularity by just avoiding that point in the mesh (e.g., Frensley’s method). Results from such methods are known to depend strongly on the discretization and meshing parameters.

A long time description of electrostatic Schrödinger-Poisson states, satisfying i∂tψ = − 1 2 ∆xψ + C |x| * x |ψ| 2 ψ , is provided in terms of a non-Markovian Wigner formalism through the choice of the simplest charge-preserving scale group, ψε(t, x) = ψ(ε −1 t, x), in which the position variable x ∈ R 3 remains unscaled while time t ∈ R + is sent to infinity as ε → 0. Typically, the introduction of this group of scale transformations leads to high frequency, time oscillatory states that may not converge in such a good topology as to deal with the nonlinear term. To overtake this drawback, the sequence of wavefunctions is Wignerized via the action of the so-called " τ-convoluted space-time Wigner transform ". The main goal of this extended Wigner operator consists in producing an attenuating effect on the temporal oscillations as time grows up, which in turn allows to overcome the eventual lack of time compactness. In a certain sense, it transforms high frequency asymptotics into a low oscillating limit that allows us to rigorously find the stationary Wigner-Poisson equation at the long time, as well as to recover some of the macroscopic features of the solutions to the original Schrödinger-Poisson problem. This methodology is also shown to apply successfully to Schrödinger-Poisson states subject to parabolic confinement.

Making use of the Whittaker-Shannon interpolation formula with shifted
sampling points, we propose in this paper a well-posed semi-discretization of
the stationary Wigner equation with inflow BCs. The convergence of the
solutions of the discrete problem to the continuous problem is then analysed,
providing certain regularity of the solution of the continuous problem.

Based on the well-posedness of the stationary Wigner equation with inflow
boundary conditions given in (A. Arnold, H et al. J. Math. Phys., 41, 2000), we
prove without any additional prerequisite conditions that the solution of the
Wigner equation with symmetric potential and inflow boundary conditions will be
symmetric. This improve the result in (D. Taj et al. Europhys. Lett., 74, 2006)
which depends on the convergence of solution formulated in the Neumann series.
By numerical studies, we present the convergence of the numerical solution to
the symmetric profile for three different numerical schemes. This implies that
the upwind schemes can also yield a symmetric numerical solution, on the
contrary to the argument given in (D. Taj et al. Europhys. Lett., 74, 2006).

A new adaptive cell average spectral element method (SEM) is proposed to solve the time-dependent Wigner equation for transport in quantum devices. The proposed cell average SEM allows adaptive non-uniform meshes in phase spaces to reduce the high-dimensional computational cost of Wigner functions while preserving exactly the mass conservation for the numerical solutions. The key feature of the pro-posed method is an analytical relation between the cell averages of the Wigner function in the k-space (local electron density for finite range velocity) and the point values of the distribution, resulting in fast transforms between the local electron density and lo-cal fluxes of the discretized Wigner equation via the fast sine and cosine transforms. Numerical results with the proposed method are provided to demonstrate its high ac-curacy, conservation, convergence and a reduction of the cost using adaptive meshes.

In this paper, we derive the quantum hydrodynamics models based on the moment
closure of the Wigner equation. The moment expansion adopted is of the Grad
type firstly proposed in \cite{Grad}. The Grad's moment method was originally
developed for the Boltzmann equation. In \cite{Fan_new}, a regularization
method for the Grad's moment system of the Boltzmann equation was proposed to
achieve the globally hyperbolicity so that the local well-posedness of the
moment system is attained. With the moment expansion of the Wigner function,
the drift term in the Wigner equation has exactly the same moment
representation as in the Boltzmann equation, thus the regularization in
\cite{Fan_new} applies. The moment expansion of the nonlocal Wigner potential
term in the Wigner equation is turned to be a linear source term, which can
only induce very mild growth of the solution. As the result, the local
well-posedness of the regularized moment system for the Wigner equation remains
as for the Boltzmann equation.

This is the first part of what will be a two-part review of distribution functions in physics. Here we deal with fundamentals and the second part will deal with applications. We discuss in detail the properties of the distribution function defined earlier by one of us (EPW) and we derive some new results. Next, we treat various other distribution functions. Among the latter we emphasize the so-called P distribution, as well as the generalized P distribution, because of their importance in quantum optics.

In this paper, the accuracy of the Frensley inflow boundary condition of the Wigner equation is analyzed in computing the I–V characteristics of a resonant tunneling diode (RTD). It is found that the Frensley inflow boundary condition for incoming electrons holds only exactly infinite away from the active device region and its accuracy depends on the length of contacts included in the simulation. For this study, the non-equilibrium Green’s function (NEGF) with a Dirichlet to Neumann mapping boundary condition is used for comparison. The I–V characteristics of the RTD are found to agree between self-consistent NEGF and Wigner methods at low bias potentials with sufficiently large GaAs contact lengths. Finally, the relation between the negative differential conductance (NDC) of the RTD and the sizes of contact and buffer in the RTD is investigated using both methods.

We present results of ultrascaled double-gate MOSFET operation and performance obtained from a new self-consistent particle-based quantum Monte Carlo (MC) approach. The simulation of quantum transport along the source-drain direction is based on the Wigner transport equation and the mode-space approximation of multi subband description. An improved method for correctly reproducing the Wigner function in the phase space by means of pseudo-particles is proposed. Our approach includes scattering effects for a two-dimensional (2-D) electron gas via standard MC algorithm. Detailed comparisons with both ballistic nonequilibrium Green's function and semiclassical multi subband Monte Carlo approaches show the ability of this Wigner transport model to incorporate correctly quantum effect into particle ensemble Monte Carlo simulation together with accurate description of scattering. This study of 6-nm-long MOSFET emphasizes the prevalent contribution of source-drain tunneling in subthreshold regime and the significant effect of quantum reflections in on-state. The influence of scattering in both the source access region and the gated part of the channel appears to be of prime importance for the correct evaluation of the on-state current, even for such small device in which the fraction of ballistic electrons is high

The paper is devoted to review, from a mathematical point of view, some fundamental aspects of the Wigner formulation of quantum mechanics. Starting from the axioms of quantum mechanics and of quantum statistics, we justify the introduction of the Wigner transform and eventually deduce the Wigner equation.

In this paper, an improved inflow boundary condition is proposed for Wigner equations in simulating a resonant tunneling diode (RTD), which takes into consideration the band structure of the device. The original Frensley inflow boundary condition prescribes the Wigner distribution function at the device boundary to be the semi-classical Fermi-Dirac distribution for free electrons in the device contacts without considering the effect of the quantum interaction inside the quantum device. The proposed device adaptive inflow boundary condition includes this effect by assigning the Wigner distribution to the value obtained from the Wigner transform of wave functions inside the device at zero external bias voltage, thus including the dominant effect on the electron distribution in the contacts due to the device internal band energy profile. Numerical results on computing the electron density inside the RTD under various incident waves and non-zero bias conditions show much improvement by the new boundary condition over the traditional Frensley inflow boundary condition.

The stationary Wigner equation is studied on a bounded, one-dimensional, spatial domain with inflow boundary conditions assumed. By means of a parity decomposition in the velocity variable, the half-range, two-point boundary problem is reduced to an initial-value problem which is studied in a suitable cutoff approximation around zero velocity. In the final section existence and uniqueness of a regular solution is proved.

This second edition of [Zbl 0902.35002] differs from the first one mainly by a new chapter 12 on nonlinear wave equations, by new Sections 4.1.2 on Turing instabilities, 4.3.2 on Radon transforms, 8.2.5 on local minimizers and 8.6.2 on Nether's Theorem as well as by new exercises.par Chapter 12 concerns the initial value problem for semilinear wave equations $u_tt-Delta u= f(u)$ or for mildly quasilinear wave equations $u_tt-Delta u= f(Du,u_t, u)$. Results on local existence, critical power nonlinearities and blow-up are presented.

The advent of semiconductor structures whose characteristic dimensions are smaller than the mean free path of carriers has led to the development of novel devices, and advances in theoretical understanding of mesoscopic systems or nanostructures. This book has been thoroughly revised and provides a much-needed update on the very latest experimental research into mesoscopic devices and develops a detailed theoretical framework for understanding their behavior. Beginning with the key observable phenomena in nanostructures, the authors describe quantum confined systems, transmission in nanostructures, quantum dots, and single electron phenomena. Separate chapters are devoted to interference in diffusive transport, temperature decay of fluctuations, and non-equilibrium transport and nanodevices. Throughout the book, the authors interweave experimental results with the appropriate theoretical formalism. The book will be of great interest to graduate students taking courses in mesoscopic physics or nanoelectronics, and researchers working on semiconductor nanostructures.

This paper is concerned with the one-dimensional stationary linear Wigner equation, a kinetic formulation of quantum mechanics. Specifically, we analyze the well-posedness of the boundary value problem on a slab of the phase space with given inflow data for a discrete-velocity model. We find that the problem is uniquely solvable if zero is not a discrete velocity. Otherwise one obtains a differential-algebraic equation of index 2 and, hence, the inflow data make the system overdetermined. © 2000 American Institute of Physics.

The probability of a configuration is given in classical theory by the Boltzmann formula exp[−VhT] where V is the potential energy of this configuration. For high temperatures this of course also holds in quantum theory. For lower temperatures, however, a correction term has to be introduced, which can be developed into a power series of h. The formula is developed for this correction by means of a probability function and the result discussed.

Source-to-drain tunneling in deca-nanometer double-gate MOSFETs is studied using a Monte Carlo solver for the Wigner transport equation. This approach allows the effect of scattering to be included. The subband structure is calculated by means of post-processing results from the device simulator Minimos-NT, and the contribution of the lowest subband is determined by the quantum transport simulation. By separating the potential profile into a smooth classical component and a rapidly varying quantum component the numerical stability of the Monte Carlo method is improved. The results clearly show an increasing tunneling component of the drain current with decreasing gate length. For longer gate lengths the semi-classical result is approached.

The development of a more accurate numerical scheme for simulating double‐barrier semiconductor structures has highlighted sensitivities of the computational results to numerical parameters for the different approximation schemes. In numerically evaluating the time evolution of the Wigner function, a second‐order differencing scheme (SDS) was used instead of a simple up/down wind differencing scheme (UDS). In our investigations of the numerical aspects of these schemes, we have found: (a) the proximity of the ‘‘computational box’’ boundaries to the double‐barrier region affects the peak‐to‐valley ratio of the I‐V curve and the value of the bias at peak current; (b) the peak‐to‐valley ratio is larger for the SDS than it is for the UDS; (c) the current at the resonant bias for SDS is larger than that calculated using UDS; (d) the rise in the current in the nonresonant regions for both SDS and UDS is dependent on how the bias is applied; and (e) the presence of an accumulation of electrons in the first heterojunction of the first barrier provides a closer correspondence between simulation and experimentally observed I‐V.

A more efficient and accurate discretization of the Wigner-Poisson model for double barrier resonant tunneling diodes is presented. This new implementation uses nonuniform grids and higher order numerical methods to improve the accuracy of the solutions at a significantly lower computational cost. Using the new implementation, devices with short and long contact regions are analyzed as well as the effect of a correlation length parameter that defines the degree of nonlocality effects. The results show that devices with longer contact regions reduce numerical inconsistencies present when modeling shorter devices, and that longer correlation lengths generally improve the correspondence of the numerical solutions with those typically expected from experimental measurement. These new numerical simulation tools will enable researchers to successfully apply the Wigner-Poisson model to describe electron transport in nanoscale semiconductor tunneling devices. More specifically, the computationally more efficient numerical algorithms presented will be shown to allow for the quantum-based studies of resonant tunneling devices useful as sources and detectors at very high frequencies (e.g., THz regime). These types of devices are very important for use in sensors and sensing systems where very long wavelength characterization capabilities are important (e.g., interrogation of chemical and biological systems) as well as an array of other electronics applications.

A model of an open quantum system is presented in which irreversibility is introduced via boundary conditions on the single-particle Wigner distribution function. The Wigner function is calculated in a discrete approximation by solution of the Liouville equation in steady state, and the transient response is obtained by numerical integration of the Liouville equation. This model is applied to the quantum-well resonant-tunneling diode. The calculations reproduce the negative-resistance characteristic of the device, and indicate that the tunneling current approaches steady state within a few hundred femtoseconds of a sudden change in applied voltage.

The semiclassical (local) approximation of mass discontinuity in the Wigner theory, when applied to resonant-tunneling diodes (RTD) with wide or high barriers, is shown to lead to unphysical results such as negative valley currents and negative peak-to-valley ratios (PVR). To deal with such problems, we propose to include the full nonlocal effect of mass discontinuity and restore the positive definiteness of the diagonal part of the density matrix. We have studied several examples and established the following points: (i) in the low narrow-barrier case, we obtain substantially lower PVR than that with the local approximation; (ii) in the wide- and/or high-barrier case, we obtain positive valley currents and hence meaningful PVR, indicating the applicability of the present theory to a wide range of RTD; specifically those which are actually in use; (iii) the peak currents are reduced substantially compared to that in the local approximation. In addition, effects of the mass discontinuity on ac characteristics of resonant-tunneling diodes are also presented for the ac admittance and the second-harmonic coefficients.

The authors report progress in quantum-mechanical simulation based
on the Wigner function model. An exact nonlocal formulation in the
Wigner representation due to a spatially varying effective mass and its
discretization for numerical calculation are discussed. To verify the
validity of such a formulation, the current-voltage characteristics of
resonant tunneling diodes are simulated to compare with the conventional
Wigner function model. The authors also point out the importance of
self-consistent calculation in the electrostatic potential for precise
device simulation. The emphasize that the Wigner function model is
superior to the alternative method based on the transmission probability
method even for the static simulation of quantum transport