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PurposeThe purpose of this offered research is to articulate a multifaceted kind of highly unstable initial perturbation and further analyze the performance of the plasma particles for time-fractional order evaluation.Design/methodology/approachFor this purpose, the authors designed specific geometry and further interpreted it into the mathematical model using the concepts of the Vlasov Maxwell system. The suggested algorithm is based on the finite-difference and spectral estimation philosophy. The management of time and memory in generic code for computational purposes is also discussed.FindingsThe main purpose is to analyze the fractional behavior of plasma particles and also the capability of the suggested numerical algorithm. Due to initial perturbations, there are a lot of sudden variations that occurred in the formulated system. Graphical behavior shows that SR parameter produces devastation as compared to others. The variation of fractional parameter between the defend domain demonstrates the hidden pictures of plasma particles. The design scheme is efficient, convergent and has the capability to cover the better physics of the problem.Practical implicationsPlasma material is commonly used in different areas of science. Therefore, in this paper, the authors increase the capability of the mathematical plasma model with specific geometry, and further suitable numerical algorithm is suggested with detailed physical analysis of the outcomes. The authors gave a new direction to study the performance of plasma particles under the influence of LASER light.Originality/valueIn the recent era, science has produced a lot of advancements to study and analyze the physical natural process, which exist everywhere in the real word. On behalf of this current developments, it is now insufficient to study the first-order time evaluation of the plasma particles. One needs to be more precise and should move toward the bottomless state of it, that is, macroscopic and microscopic time-evaluation scales, and it is not wrong to say that there exits a huge gap, to study the time evaluation in this discussed manner. The presented study is entirely an advanced and efficient way to investigate the problem into the new directions. The capability of the proposed algorithm and model with fractional concepts can fascinate the reader to extend to the other dimensions.

In Li et al. (J Sci Comput 62:317–335, 2015), we thoroughly investigated the structure of the kernel space of the discrete one-dimensional (1D) non-polar optical phonon (NPOP)-electron scattering matrix, and proposed a strategy to setup grid points so that the uniqueness of the discrete scattering kernel is preserved. In this paper, we extend the above work to the three dimensional (3D) case, and also investigate the polar optical phonon (POP)-electron case. In numerical discretization, it is important to get a discretization scattering matrix that keeps as many properties of the continuous scattering operator as possible. We prove that for the 3D NPOP-electron scattering, (i) the dimension of the kernel space of the discrete scattering matrix is one and (ii) the equilibrium distribution is constant with respect to the angular coordinates as long as the mesh over the energy interval obeys the rule proposed in Li et al. (2015). For the POP-electron scattering, (i) is also kept under the same condition, but to keep (ii) becomes a challenging task, since generally a simple uniform mesh in the azimuth and polar angular coordinates will not preserve (ii). Based on high degree of symmetry of the Platonic solids and regular pyramids, we propose two conditions and prove they are sufficient to guarantee (ii). Numerical experiments strongly support our theoretical findings.

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.

The “Vlasov-Maxwell system” is a groundbreaking algorithm to model, simulate and further analyze the vigorous performance of the collisionless plasma in the presence of the electromagnetic fields. In this frame of reference, the inquiry of this system with the deep conceptions of the time-fractional derivative is a novel benchmark and also the key intentions of this study. For this purpose, higher dimensional semi-relativistic time-fractional Vlasov-Maxwell system is formulated with the physical significances of the geometry. Furthermore, to fabricate the numerical consequences, we suggest an innovative algorithm which based on spectral and finite-difference approximations. The spatial and temporal variables are handled by using shifted Gegenbauer polynomials and finite-difference approximations respectively. Numerous simulations are carried out to validate the reliability and accuracy of the anticipated method. Error bound convergence and stability of the method is inspected numerically. Furthermore, the established technique can be used conveniently to observe the numerical result of other multi-dimensional time-fraction with variable-order problems of physical nature.

We study the stationary Wigner equation on a bounded, one-dimensional spatial domain with inflow boundary conditions by using the parity decomposition of L. Barletti and P. F. Zweifel [Transport Theory Statist. Phys., 2001, 30(4-6): 507–520]. 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.

This paper proposes a systematic procedure to calibrate the parameters of the drift-diffusion (DD) model for a performance evaluation of InGaAs MOSFETs in the quasi-ballistic regime. The simulation results of a deterministic multi-subband Boltzmann transport equation (BTE) solver serve as the standard. The DD model is calibrated both under low and high electric fields. The electrostatic characteristics, low field mobility model, and high field saturation model are calibrated in proper sequence, and a good agreement among the drive current, carrier distribution, and velocity distribution are achieved between the results of the calibrated DD model and the BTE solver. The proposed calibration procedure can also be employed in devices made of other materials.

The “Vlasov-Maxwell system” is a groundbreaking differential procedure to visualize, model, simulate and further analyze the vigorous performance of plasma (collisionless) in the presence of the different fields (electromagnetic). In this frame of reference, the analysis of “Vlasov-Maxwell system” with the deep conceptions of (time-fractional) calculus is a novel benchmark and also the key intentions of this study. For this purpose, (1D + 1P) dimensional semi-relativistic time-fractional Vlasov-Maxwell system is formulated with the physical significances of the geometry. Furthermore, to fabricate the numerical consequences, we suggest (and also implement) an innovative algorithm which based on spectral and finite-difference estimations. The spatial and temporal variables are handled by using sifted Gegenbauer polynomials and finite-difference calculations respectively. Numerous simulations are executed to validate the reliability and accuracy of anticipated method. Error bound convergence and stability of the method is inspected numerically. Moreover, the established technique can be used conveniently to observe the numerical result of other multi-dimensional fraction (variable) order problems of physical nature.

In this work, the transient characteristics of nanoscale field-effect transistors (FETs) have been investigated using a deterministic solver based on the time-dependent multi-subband Boltzmann transport equation (BTE). The response to a step signal superimposed on the gate or drain electrode is simulated. The transient process can be understood as a combination of electrostatic and transport relaxation. The extracted transient relaxation time for the drain current, which is unrelated to the direct-current (DC) shift, is important for transient device modeling.

We investigate the discretization of of an electron---optical phonon scattering using a finite volume method. The discretization is conservative in mass and is essentially based on an energy point of view. This results in a discrete scattering system with elegant mathematical features, which are fully clarified. Precisely the discrete scattering matrix is thoroughly studied, including its sparsity pattern and its symmetries, the structure of its eigenvalues and eigenvectors. It makes us reveal the strategy to setup grid points, so that the proper scattering matrix can be obtained to preserve the unique discrete scattering kernel. Numerical results are presented to validate these theoretical findings.

A globally hyperbolic moment system upto arbitrary order for the Wigner equation was derived in [6]. For numerically solving the high order hyperbolic moment system therein, we in this paper develop a preliminary numerical method for this system following the NRxx method recently proposed in [8], to validate the moment system of the Wigner equation. The method developed can keep both mass and momentum conserved, and the variation of the total energy under control though it is not strictly conservative. We systematically study the numerical convergence of the solution to the moment system both in the size of spatial mesh and in the order of the moment expansion, and the convergence of the numerical solution of the moment system to the numerical solution of the Wigner equation using the discrete velocity method. The numerical results indicate that the high order moment system in [6] is a valid model for the Wigner equation, and the proposed numerical method for the moment system is quite promising to carry out the simulation of 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.

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)., 13 pages, 2 figures

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.

By the moment closure of the Boltzmann transport equation, the extended hydrodynamic models for electron transport have been derived in Cai et al. (J Math Phys 53:103503, 2012). With the numerical scheme developed in Li et al. (2012) recently, it has been demonstrated that the derived extended hydrodynamic models can capture the major features of the solution of kinetic equations. As the application of the models and the numerical scheme proposed therein, we in this paper carry out the numerical simulation to investigate the carrier transport in $$n^{+}$$ n + - $$n$$ n - $$n^{+}$$ n + structures by solving the moment system derived from the Boltzmann---Poisson equations. Without any additional empirical parameters than that used in directly solving the kinetic equations, we obtain numerical results by the extended hydrodynamic models with very satisfied agreement with the solution of the kinetic equations, even in case that the length of the channel is as short as 50 nm.

In the first of a series of papers, we will study a discontinuous Galerkin (DG) framework for many electron quantum systems. The salient feature of this framework is the flexibility of using hybrid physics-based local orbitals and accuracy-guaranteed piecewise polynomial basis in representing the Hamiltonian of the many body system. Such a flexibility is made possible by using the discontinuous Galerkin method to approximate the Hamiltonian matrix elements with proper constructions of numerical DG fluxes at the finite element interfaces. In this paper, we will apply the DG method to the density matrix minimization formulation, a popular approach in the density functional theory of many body Schrödinger equations. The density matrix minimization is to find the minima of the total energy, expressed as a functional of the density matrix ρ(r,r′), approximated by the proposed enriched basis, together with two constraints of idempotency and electric neutrality. The idempotency will be handled with the McWeeny’s purification while the neutrality is enforced by imposing the number of electrons with a penalty method. A conjugate gradient method (a Polak-Ribiere variant) is used to solve the minimization problem. Finally, the linear-scaling algorithm and the advantage of using the local orbital enriched finite element basis in the DG approximations are verified by studying examples of one dimensional lattice model systems.

In this paper, we extend the method in Cai et al. (J Math Phys 53:103503, 2012) to derive a class of quantum hydrodynamic models for the density-functional theory (DFT). The most popular implement of DFT is the Kohn–Sham equation, which transforms a many-particle interacting system into a fictitious non-interacting one-particle system. The Kohn–Sham equation is a non-linear Schrodinger equation, and the corresponding Wigner equation can be derived as an alternative implementation of DFT. We derive quantum hydrodynamic models of the Wigner equation by moment closure following Cai et al. (J Math Phys 53:103503, 2012). The derived quantum hydrodynamic models are globally hyperbolic thus locally wellposed. The contribution of the Kohn–Sham potential is turned into a nonlinear source term of the hyperbolic moment system. This work provides a new possible way to solve DFT problems.

In this study, we simulate double-gate MOSFET using a 2-D direct Boltzmann transport equation solver. Simulation results are interpreted by quasi-ballistic theory. It is found that the relation between average carrier velocity at virtual source and back-scattering coefficient needs to be modified due to the oversimplified approximations of the original model. A 1-D potential profile model also needs to be extended to better determine the kT-layer length. The key expression for back-scattering coefficient is still valid, but a field-dependent mean free path is needed to be taken into account.

We investigate the scattering mechanism in ultrashort double gate In 0.53 Ga 0.47 As nMOSFETs by deterministically solving Boltzmann transport equation (BTE). The intra-valley acoustic phonon scattering, optical phonon scattering, intervalley optical scattering, polar optical scattering, and surface roughness (SR) scattering are considered. The impacts of scattering on the performance of device under high/low biases are compared. Results show that the ballistic ratio (Iscat/Iball) decreases from 96.8% to 94.5% when the drain bias increases from 0.05V to 0.6V, which is mainly caused by the inter-valley scatterings.

A nano-scale double gate In0.53Ga0.47As nMOSFET device structure is simulated by deterministically solving the time dependent Boltzmann Transport Equation (BTE). The results show that the contribution of the L valleys cannot be ignored even if the energy gap between r and L valleys are very large. Moreover, the quasi-ballistic transport is observed despite the existence of scattering.

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 $\mathcal{A}[V]$., Comment: 15 pages, 4 figures

We propose a deterministic solver for the time-dependent multi-subband Boltzmann transport equation (MSBTE) for the two dimensional (2D) electron gas in double gate metal oxide semiconductor field effect transistors (MOSFETs) with flared out source/drain contacts. A realistic model with six-valleys of the conduction band of silicon and both intra-valley and inter-valley phonon-electron scattering is solved. We propose a second order finite volume method based on the positive and flux conservative (PFC) method to discretize the Boltzmann transport equations (BTEs). The transport part of the BTEs is split into two problems. One is a 1D transport problem in the position space, and the other is a 2D transport problem in the wavevector space. In order to reduce the splitting error, the 2D transport problem in the wavevector space is solved directly by using the PFC method instead of splitting into two 1D problems. The solver is applied to a nanoscale double gate MOSFET and the current-voltage characteristic is investigated. Comparison of the numerical results with ballistic solutions show that the scattering influence is not ignorable even when the size of a nanoscale semiconductor device goes to the scale of the electron mean free path.

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 proposed 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 local 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 accuracy, conservation, convergence and a reduction of the cost using adaptive meshes.

This paper presents the study of coupling efficiencies between two-dimensional (2-D) waveguides and microring resonators with a newly developed high-order discontinuous Galerkin time domain (DGTD) method for Maxwell's equations. The DGTD method is based on a unified formulation for the physical media and the artificial media in the uniaxial perfectly matched layer (UPML) regions used to truncate the computational domain. The DGTD method employs finite element type meshes and uses piecewise high-order polynomials for spatial discretization of the Maxwell's equations and Runge-Kutta methods for time integration. After demonstrating the high-order convergence of the DGTD method, the effect of separation gap between the waveguides and one and two microrings on the coupling efficiency and transmittance for pulse propagations is studied.

This paper presents a newly developed high-order discontinuous Galerkin time-domain (DGTD) method for solving Maxwell's equations in linear dispersive media with UPML boundary treatment. A unified formulation is derived for linear dispersive media of Debye type and the artificial material in the UPML regions with the help of auxiliary differential equations. The DGTD employs finite-element-type meshes, and uses piecewise high-order polynomials for spatial discretization and Runge-Kutta method for time integrations. Arbitrary high-order accuracy can be obtained for scattering of various objects in dispersive media. After validating the numerical convergence of the DGTD method together with the second-order Yee's scheme, we apply this new method to the ground-penetrating radar for the detection of buried objects in a lossy half space.

In this paper, we will present a unified formulation of discontinuous Galerkin method (DGM) for Maxwell's equations in linear dispersive and lossy materials of Debye type and in the artificial perfectly matched layer (PML) regions. An auxiliary differential equation (ADE) method is used to handle the frequency-dependent constitutive relations with the help of auxiliary polarization currents in the computational and PML regions. The numerical flux for the dispersive lossy Maxwell's equations with the auxiliary polarization current variables is derived. Various numerical results are provided to validate the proposed formulation.

The transient response to the drain voltage of a nano scale Ultra-Thin Body and BOX (UTBB) nMOSFET are investigated by deterministically solving the time dependent Boltzmann Transport Equation (BTE). The relaxation process of subbands profile, electron density and current is investigated. The relaxation time of the device is about 0.1ps at typical biases, which is several times larger than the electron transit time.

A globally hyperbolic high-order moment method of the Boltzmann transport equation (BTE) is proposed in [1], [2], and here it is extended for the BTE with the electron-phonon scattering term to simulate a silicon nano-wire (SNW). Convergence with respect to the order of the moment system and the characteristics of SNW including the I-V curve are studied.

Quasi-ballistic asymmetric DG-MOSFET has been simulated using a multi-subband Boltzmann transport equation solver and important parameters regarding to back-scattering at the top of barrier are carefully studied in this work. It is observed that the simulated results are in good agreement with established theory and phonon scattering still plays an important role in limiting the performance of MOSFET even when gate length is scaled down to sub-10nm.

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 first proposed by Grad [“On the kinetic theory of rarefied gases,” Commun. Pure Appl. Math. 2(4), 331–407 (1949)10.1002/cpa.3160020403]. The Grad's moment method was originally developed for the Boltzmann equation. Recently, a regularization method for the Grad's moment system of the Boltzmann equation was proposed by Cai et al. [Commun. Pure Appl. Math. “Globally hyperbolic regularization of Grad's moment system” (in press)] to achieve the global 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 applies. The moment expansion of the nonlocal Wigner potential term in the Wigner equation turns out to be a linear source term, which can only...

In this paper, the ballistic and quasi-ballistic transport characteristics of Sub-30 nm double gate MOSFETs are simulated by deterministic solver of the Time Dependent Muti-subbands Boltzmann Transport Equation [1]. The scattering effect on the Sub-30 nm double gate MOSFETs is investigated not only through analyzing ON current, but also the source to channel barrier height, average electron velocity and electron density at virtual source (VS: top of the barrier).

As MOSFETs scaling down to nano-scale, short channel effect(SCE) become a critical issue. Multiple channel MOSFET structure such as FINFET [1,2] has well gate controllability on channel charge, and will be used in nano-scale CMOS technology. In this work the performance of 20nm bulk FINFET is investigated by Using 3D full band Monte Carlo Method with Effective Potential Quantum Correction. Gate and drain bias affect on the carrier density, velocity and energy distribution are introduced. The transit time and SSEC C gs and C gd as a function of V ds are showed. Results show about 0.1 psec intrinsic transit time at on state in this 20nm gate length device.

The lattice scattering is carefully involved in a direct solution of the BTE and Poisson-Schrodinger equation method. Simulating results of a 9nm DG MOSFET shows the lattice scattering effects on the barrier height and the positions of barrier peak are small, but the effects on the carrier drift velocity are strongly. Not only intra-valley scatterings but also the inter-valley scatterings affect the electron energy, drift velocity and density distribution in channel region strongly. Thus the scattering effect must be considered when discussion carrier energy related device characteristics such as reliability, heat generation.

We present a self-consistent multi subband deter- ministic solver of the Boltzmann transport equation of the two dimensional (2D) electron gas. The Sch¨ odinger equation at each slice in the confinement direction and the two dimensional Poisson equation are self-consistently solved with the Boltzmann transport equation. The energy quantization and the scattering of the 2D electron gas are included. We apply this solver to an ultra-thin body double gate MOSFET and show the influence of the 2Dk scattering to the electron transport.