## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

For the stationary Wigner equation with inflow boundary conditions, its numerical convergence with respect to the velocity mesh size are deteriorated due to the singularity at velocity zero. In this paper, using the fact that the solution of the stationary Wigner equation is subject to an algebraic constraint, we prove that the Wigner equation can be written into a form with a bounded operator $\mathcal{B}[V]$, which is equivalent to the operator $\mathcal{A}[V]=\Theta[V]/v$ in the original Wigner equation under some conditions. Then the discrete operators discretizing $\mathcal{B}[V]$ are proved to be uniformly bounded with respect to the mesh size. Based on the therectical findings, a signularity-free numerical method is proposed. Numerical reuslts are proivded to show our improved numerical scheme performs much better in numerical convergence than the original scheme based on discretizing $\mathcal{A}[V]$.

To read the full-text of this research,

you can request a copy directly from the authors.

... Furthermore, it is found and discussed in [31] that singularity exists around v = 0 and it may bring difficulties to solve the SWIBVP numerically. In [31], the authors propose an integral constraint along with the Wigner equation to remove the singularity at v = 0, and thus a regularized SWIBVP is considered. In this paper, the integral constraint is transformed into a penalty term appending to Φ[θ ] and a modified optimization model is thus generated. ...

... It is found in literatures (e.g. [14,31]) that the SWIBVP has singularity at v = 0, which causes dilemmas when solving the SWIBVP numerically since the Wigner function will tend to infinity when v → 0. Based on the results shown in [31], an additional equation ...

... It is found in literatures (e.g. [14,31]) that the SWIBVP has singularity at v = 0, which causes dilemmas when solving the SWIBVP numerically since the Wigner function will tend to infinity when v → 0. Based on the results shown in [31], an additional equation ...

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.

In this paper, the ground state Wigner function of a many-body system is explored theoretically and numerically. First, an eigenvalue problem for Wigner function is derived based on the energy operator of the system. The validity of finding the ground state through solving this eigenvalue problem is obtained by building a correspondence between its solution and the solution of stationary Schrödinger equation. Then, a numerical method is designed for solving proposed eigenvalue problem in one dimensional case, which can be briefly described by i) a simplified model is derived based on a quantum hydrodynamic model [Z. Cai et al, J. Math. Chem., 2013] to reduce the dimension of the problem, ii) an imaginary time propagation method is designed for solving the model, and numerical techniques such as solution reconstruction are proposed for the feasibility of the method. Results of several numerical experiments verify our method, in which the potential application of the method for large scale system is demonstrated by examples with density functional theory.

We analyse a quantum-mechanical model for the transport of electrons in semiconductors. The model consists of the quantum Liouville (Wigner) equation posed on the bounded Brillouin zone corresponding to the semiconductor crystal lattice, with a self-consistent potential determined by a Poisson equation. A global existence and uniqueness proof for this model is the main result of the paper.

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).

The Wigner--Poisson equation describes the quantum-mechanical motion of electrons in a self-consistent electrostatic field. The equation consists of a transport term and a non-linear pseudodifferential operator. In this paper we analyze an operator splitting method for the linear Wigner equation and the coupled Wigner--Poisson problem. For this semidiscretization in time, consistency and nonlinear stability are established in an L2-framework. We present a numerical example to illustrate the method.

Small semiconductor devices can be separated into regions where the electron transport has classical character, neighboring with regions where the transport requires a quantum description. The classical transport picture is associated with Boltzmann-like particles that evolve in the phase-space defined by the wave vector and real space coordinates. The evolution consists of consecutive processes of drift over Newton trajectories and scattering by phonons. In the quantum regions, a convenient description of the transport is given by the Wigner-function formalism. The latter retains most of the basic classical notions, particularly, the concepts for phase-space and distribution function, which provide the physical averages. In this work we show that the analogy between classical and Wigner transport pictures can be even closer. A particle model is associated with the Wigner-quantum transport. Particles are associated with a sign and thus become positive and negative. The sign is the only property of the particles related to the quantum information. All other aspects of their behavior resemble Boltzmann-like particles. The sign is taken into account in the evaluation of the physical averages. The sign has a physical meaning because positive and negative particles that meet in the phase space annihilate one another. The Wigner and Boltzmann transport pictures are explained in a unified way by the processes drift, scattering, generation, and recombination of positive and negative particles. The model ensures a seamless transition between the classical and quantum regions. A stochastic method is derived and applied to simulation of resonant-tunneling diodes. Our analysis shows that the method is useful if the physical quantities do not vary over several orders of magnitude inside a device.

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, 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.

The Wigner–Boltzmann model is a partial integro-differential equation which describes the time dependent dynamics of quantum mechanical phenomena including the effects of lattice vibration as a second-order approximation. Recently a Monte Carlo technique exploiting the concept of signed particles has been developed for its ballistic counterpart, in one and two-dimensional space. In this work, we introduce an extension to the Wigner–Boltzmann model in three-dimensional geometries adapted for the treatment of the scattering term. As an application, we study the dynamics of an electron wave packet in proximity of a Coulombic potential in the presence of absorbing boundary conditions. This mimics the presence of a dopant atom buried in a semiconductor substrate. By using this method, one can observe how the lattice temperature eventually affects the dynamics of the wave packet.

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.

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.

A quantum corrected dynamical model for charged particles is derived. The corrections to the Newton motion are obtained within the Liouville formalism and are expressed by an effective nonlinear force. This effective force describes the non-local character of the quantum particles. The well posedness of the quantum-corrected problem is studied from the mathematical point of view. Existence and uniqueness of the solution are proved for a mollified version of the effective force. Numerical tests are performed and the range of validity of the model is investigated.

A few high order numerical calculation schemes for the Wigner transport equation were studied and compared with conventional first-order (FDS) and second-order differencing schemes (SDS). Though the SDS model is generally accepted, there has been no trial to check the accuracy of this model by comparing it with those of higher order differencing schemes. In this report, it is shown that although the SDS model has a non-negligible numerical error, it can be regarded and accepted as a relatively accurate and efficient numerical model compared with the FDS model.

We present an analysis of the quantum Liouville equation under the assumption of a globally bounded potential energy. By using methods of semigroup theory we prove existence and uniqueness results. We also show the existence of the particle density. The last section is concerned with the classical limit. We show that the solutions of the quantum Liouville equation converge to the solution of the classical Liouville equation as the Planck constant h tends to zero.
Unter der Annahme global beschränkter potentieller Energie geben wir eine Analyse der Quanten-Liouville-Gleichung. Indem wir Methoden der Halbgruppentheorie anwenden, beweisen wir Existenz- und Eindeutigkeitsergebnisse. Ferner zeigen wir die Existenz der Partikel-Dichte. Der letzte Anschnitt beschäftigt sich mit dem klassischen Grenzfall. Wir zeigen, daß die Lösungen der Quanten-Liouville-Gleichung gegen die Lösung der klassischen Liouville-Gleichung konvergieren, wenn die Planksche Konstante h gegen Null strebt.

A theoretical criterion for the origin of high-frequency current oscillations in double-barrier quantum well structures (DBQWS’s) is presented. The origin of the current oscillations is traced to the development of a dynamic emitter quantum well (EQW) and the coupling of that EQW to the main quantum well, which is defined by the double-barrier quantum well system. The relationship between the oscillation frequency and the energy-level structure of the system is demonstrated to be ν=ΔE0/h. Insight into DBQWS’s as potential devices for very high-frequency oscillators is facilitated through two simulation studies. First, a self-consistent, time-dependent Wigner-Poisson numerical investigation is used to reveal sustained current oscillations in an isolated DBQWS-based device. Furthermore, these terahertz-frequency oscillations are shown to be intrinsic. Second a multisubband-based procedure for calculating ΔE0, which is the energy separation of the quantum states in the system that are responsible for the instability mechanism, is also presented. Together, these studies establish the fundamental principals and basic design criteria for the future development and implementation of DBQWS-based oscillators. Furthermore, this paper provides physical interpretations of the instability mechanisms and explicit guidance for defining structures that will admit enhanced oscillation characteristics.

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.

We introduce a semi-discretized version of the Wigner equation. The model fulfills the following fundamental requirements: it is well-posed in a natural functional framework; as the mesh size of the discretization goes to 0, the solutions converge (with spectral accuracy) to the solution of the continuous equation; it is consistent with the semi-classical limit. We also discuss an operator splitting implementation of this model.

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.

Differences between the «classical» and the «nuclear» Vlasov equation are shortly discussed and some mathematical ideas which
can be carried over from the classical case are described.
Si discutono brevemente le differenze tra l'equazione di Vlasov «classica» e «nucleare» e si descrivono alcune idee matematiche
che si possono riportare dal caso classico.
Вкратце обсуждаются различия между «классическим» и «ядерным» уравнениями Власова. Описываются математические идеи, которые
могут быть перенесены из классического случая.

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.

We present a study of the Wigner–Poisson problem in a bounded spatial domain with non-homogeneous and time-dependent “inflow” boundary conditions. This system of nonlinearly coupled equations is a mathematical model for quantum transport of charges in a semiconductor with external contacts. We prove well-posedness of the linearized n-dimensional problem as well as existence and uniqueness of a global-in-time, regular solution of the one-dimensional nonlinear problem.

We introduce a semidiscretized version of the Wigner equation---discretization concerning the velocity variable. We show that the corresponding discrete velocity problem is well-posed and permits us to approach the solution of the continuous problem when the mesh size of the discretization vanishes. The approximation shows spectral accuracy because the rate of convergence corresponds to the (Sobolev) regularity of the solution of the continuous problem. We also discuss the behavior of the solution with respect to the Planck constant.

RESUMEN RESUMEN
Tunneling effects in solid-state physics are frequently described via a quantum mechanical formulation based on the Wigner Distribution Function. In this paper a mixed finite-differencespectral-collocation method for the solution of the corresponding equation of motion is developed and analyzed. Numerical examples illustrating the convergence results are presented.

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.

In the present work, we compare the efficiency, accuracy, and robustness of four basic iteration methods for implementing self-consistency in Wigner function-based quantum device simulation. These methods include steady-state Gummel, transient Gummel, steady-state Newton, and transient Newton. In a single mathematical framework and notation, we present the numerical implementation of each of these self-consistency iteration methods. As a test case to compare the iteration methods, we simulate the current-voltage (I-V) curve of a resonant tunneling diode. Standard practice for this task has been to rely solely on either a steady-state or a transient iteration method. We illustrate the dangers of this practice, and show how to take advantage of the complimentary strengths of both steady-state and transient iteration methods where appropriate. Thus, because the steady-state methods are vastly more efficient (i.e., have a much lower computational cost), and are usually equal in accuracy to the transient methods, the former are preferable for wide-ranging initial device investigations such as tracing the I-V curve. Implementation difficulties which we address here may have reduced the use of the steady-state methods in practice. On the other hand, the transient methods are inherently more robust and accurate (i.e., they reliably and correctly reproduce device physics). However, the high computational cost of the transient methods makes them more appropriate for a narrower range of directed investigations where transient effects are inherent or suspected, rather than for full I-V curve traces. Finally, we found the two Gummel methods to be generally preferable to their (theoretically more accurate) Newton counterparts, since the Gummel methods are equally accurate in practice, while having a lower computational cost.

Intrinsic high-frequency oscillations (≊2.5 THz) in current and corresponding quantum-well density have been simulated for the first time for a fixed-bias voltage in the negative differential resistance (NDR) region of the current-voltage (I-V) characteristics of a resonant tunneling diode. Scattering and self-consistency are included. Hysteresis and ‘‘plateaulike’’ behavior of the time-averaged I-V curve are simulated in the NDR region. Intrinsic bistability is manifested by the phenomenon of unstable electron charge buildup and ejection from the quantum well.

Partial Differential Equations

- L C Evans

L.C. Evans. Partial Differential Equations. American Mathematical Society, Providence RI, 2nd edition, 2010.